[M�todo Muskingum]   [Ondas Cinem�ticas]   [Ondas Difusivas]   [M�todo Muskingum-Cunge]   [Ondas Din�micas]   [Quest�es]   [Problemas]   [Refer�ncias]     

CAP�TULO 9: 
PROPAGA��O  
DE CHEIA EM CANAIS 

"When we started our research at Cornell University in 1964, we intended to show that the St. Venant equations
without simplification were required for the overland flow problem,
although we came to quite different conclusions."

"Quando come�amos nossa pesquisa na Universidade de Cornell em 1964, pretendemos mostrar que as equa��es de St. Venant sem simplifica��o foram necess�rias para o problema do fluxo terrestre, embora tenhamos chegado a conclus�es bem diferentes ".
David A. Woolhiser (1996)


This chapter is divided into five sections. Section 9.1 describes the Muskingum method, the most widely used method of hydrologic stream channel routing. Sections 9.2 and 9.3 discuss simplified hydraulic routing techniques: kinematic and diffusion waves, respectively. Section 9.4 describes the Muskingum-Cunge method. Section 9.5 introduces the subject of dynamic wave routing, the most complete hydraulic routing technique.

Este cap�tulo est� dividido em cinco se��es. A Se��o 9.1 descreve o m�todo Muskingum, o m�todo mais amplamente usado de roteamento de canais de correntes hidrol�gicas. As se��es 9.2 e 9.3 discutem t�cnicas simplificadas de roteamento hidr�ulico: ondas cinem�ticas e de difus�o, respectivamente. A Se��o 9.4 descreve o m�todo Muskingum-Cunge. A Se��o 9.5 apresenta o assunto do roteamento din�mico de ondas, a mais completa t�cnica de roteamento hidr�ulico.


9.1  M�TODO MUSKINGUM

[Ondas Cinem�ticas]   [Ondas Difusivas]   [M�todo Muskingum-Cunge]   [Ondas Din�micas]   [Quest�es]   [Problemas]   [Refer�ncias]      [Top]  

Stream Channel Routing

Roteamento de canal de fluxo

Stream channel routing uses mathematical relations to calculate outflow from a stream channel once inflow, lateral contributions, and channel characteristics are known.

O roteamento de canal de fluxo usa rela��es matem�ticas para calcular a sa�da de um canal de fluxo assim que a entrada, as contribui��es laterais e as caracter�sticas do canal s�o conhecidas.

Stream channel routing usually implies open channel flow conditions, although there are exceptions, such as storm sewer flow, for which mixed open channel-closed conduit flow conditions may prevail. In this chapter, stream channel routing refers to unsteady flow calculations in streams and rivers. Channel reach refers to a specific length of stream channel possessing certain translation and storage properties. The hydrograph at the upstream end of the reach is the inflow hydrograph; the hydrograph at the downstream end is the outflow hydrograph. Lateral contributions consist of point tributary inflows and/or distributed inflows; e.g., interflow and groundwater flow.

O roteamento de canal de fluxo geralmente implica condi��es de fluxo de canal aberto, embora existam exce��es, como o fluxo de esgoto pluvial, para o qual possam prevalecer condi��es de fluxo misto de canal aberto e canal fechado. Neste cap�tulo, o roteamento de canal de fluxo se refere a c�lculos de fluxo inst�veis em c�rregos e rios. O alcance do canal refere-se a um comprimento espec�fico do canal de fluxo que possui certas propriedades de convers�o e armazenamento. O hidrograma na extremidade a montante do alcance � o hidrograma de entrada; o hidrograma na extremidade a jusante � o hidrograma de sa�da. As contribui��es laterais consistem em entradas tribut�rias pontuais e / ou entradas distribu�das; por exemplo, interfluxo e fluxo de �guas subterr�neas.

The terms stream channel routing and flood routing are often used interchangeably. This is attributed to the fact that most stream channel routing applications are in flood flow analysis, flood control design, or flood forecasting (Fig. 9-1).

Os termos roteamento de canal de fluxo e roteamento de inunda��o s�o frequentemente usados %G​​%@de forma intercambi�vel. Isso � atribu�do ao fato de que a maioria das aplica��es de roteamento de canal de fluxo � na an�lise de fluxo de inunda��o, projeto de controle de inunda��o ou previs�o de inunda��o (Fig. 9-1).

Flood stage in a tropical river

Figure 9-1  Flood stage in a tropical river.

Two general approaches to stream channel routing are recognized: (1) hydrologic and (2) hydraulic. As in the case of reservoir routing (Chapter 8), hydrologic stream channel routing is based on the storage concept. Conversely, hydraulic channel routing is based on the principles of mass and momentum conservation. Hydraulic routing techniques are of three types: (1) kinematic wave, (2) diffusion wave, and (3) dynamic wave. The dynamic wave is the most complete model of unsteady open channel flow. Kinematic and diffusion waves are convenient and practical approximations to the dynamic wave.

S�o reconhecidas duas abordagens gerais para o roteamento de canais de corrente: (1) hidrol�gico e (2) hidr�ulico. Como no caso do roteamento de reservat�rio (Cap�tulo 8), o roteamento de canal de corrente hidrol�gica � baseado no conceito de armazenamento. Por outro lado, o roteamento de canais hidr�ulicos � baseado nos princ�pios de conserva��o de massa e momento. As t�cnicas de roteamento hidr�ulico s�o de tr�s tipos: (1) onda cinem�tica, (2) onda de difus�o e (3) onda din�mica. A onda din�mica � o modelo mais completo de fluxo de canal aberto inst�vel. As ondas cinem�ticas e de difus�o s�o aproxima��es pr�ticas e convenientes da onda din�mica.

An alternate approach to hydrologic and hydraulic routing has emerged in recent years. This approach is similar in nature to the hydrologic routing methods yet contains sufficient physical information to compare favorably with the more complex hydraulic routing techniques. This hybrid approach is the basis of the Muskingum-Cunge method of flood routing.

Uma abordagem alternativa ao roteamento hidrol�gico e hidr�ulico surgiu nos �ltimos anos. Essa abordagem � de natureza semelhante aos m�todos de roteamento hidrol�gico, mas cont�m informa��es f�sicas suficientes para comparar favoravelmente com as t�cnicas de roteamento hidr�ulico mais complexas. Essa abordagem h�brida � a base do m�todo Muskingum-Cunge de roteamento de inunda��es.

At the outset of the study of stream channel routing, it is necessary to introduce a few basic modeling concepts. A typical hydrologic model consists of: (1) input, (2) system, and (3) output (Fig. 9-2). In surface water hydrology, the system is usually a catchment, a reservoir, or a stream channel. In the case of a catchment, the input is a storm hyetograph. For reservoirs and stream channels, the input is an inflow hydrograph. For all three cases, catchments, reservoirs, and channels, the output is an outflow hydrograph.

No in�cio do estudo do roteamento de canal de fluxo, � necess�rio introduzir alguns conceitos b�sicos de modelagem. Um modelo hidrol�gico t�pico consiste em: (1) entrada, (2) sistema e (3) sa�da (Fig. 9-2). Na hidrologia das �guas superficiais, o sistema geralmente � uma bacia hidrogr�fica, um reservat�rio ou um canal de corrente. No caso de uma bacia hidrogr�fica, a entrada � um het�grafo de tempestade. Para reservat�rios e canais de corrente, a entrada � um hidrograma de entrada. Nos tr�s casos, bacias hidrogr�ficas, reservat�rios e canais, a sa�da � um hidrograma de vaz�o.

Input, system, and output in a typical hydrologic model

Figure 9-2  Input, system, and output in a typical hydrologic model.

In general, modeling problems are classified into three types: (1) prediction, (2) calibration, and (3) inversion. In the prediction problem, input and system are known and described by properties or parameters, and the task is to calculate the output based on the knowledge of system and input. For instance, with known inflow hydrograph, lateral contributions, and channel reach parameters, the outflow hydrograph from a stream channel can be computed using routing techniques (Example 9-1).

Em geral, os problemas de modelagem s�o classificados em tr�s tipos: (1) previs�o, (2) calibra��o e (3) invers�o. No problema de previs�o, entrada e sistema s�o conhecidos e descritos por propriedades ou par�metros, e a tarefa � calcular a sa�da com base no conhecimento do sistema e da entrada. Por exemplo, com o hidr�grafo de entrada conhecido, as contribui��es laterais e os par�metros de alcance do canal, o hidrograma de sa�da de um canal de fluxo pode ser calculado usando t�cnicas de roteamento (Exemplo 9-1).

In the calibration problem, input and output are known, and the objective is to determine the properties or parameters describing the system. In the case of a stream channel, with known upstream inflow, lateral contributions, and outflow hydrograph, the routing parameters are calculated by a calibration procedure (Example 9-2).

No problema de calibra��o, entrada e sa�da s�o conhecidas, e o objetivo � determinar as propriedades ou par�metros que descrevem o sistema. No caso de um canal de corrente, com fluxo a montante conhecido, contribui��es laterais e hidrograma de vaz�o, os par�metros de roteamento s�o calculados por um procedimento de calibra��o (Exemplo 9-2).

The inversion problem is the third type of modeling problem. In this case, system and output are known, and the task is to calculate the inflow or inflows. This is accomplished by reversing the routing process in a technique known as inverse channel routing. For instance, with known upstream inflow, outflow, and channel reach parameters, the lateral contributions can be calculated by inverse routing.

O problema de invers�o � o terceiro tipo de problema de modelagem. Nesse caso, o sistema e a sa�da s�o conhecidos, e a tarefa � calcular as entradas ou entradas. Isso � feito revertendo o processo de roteamento em uma t�cnica conhecida como roteamento de canal inverso. Por exemplo, com par�metros conhecidos de entrada, sa�da e alcance de canal a montante, as contribui��es laterais podem ser calculadas por roteamento inverso.

The prediction problem is the more common type of modeling application; however, a calibration is usually required in advance of the prediction. Model verification is the process of testing the model with actual data to establish its predictive accuracy. To calibrate and verify a model, it is usually necessary to assemble two different data sets. The first set is used in model calibration and the second set is used in model verification. A close agreement between calculated and measured data is an indication that the model has been verified. A detailed discussion of these subjects is given in Chapter 13.

O problema de previs�o � o tipo mais comum de aplicativo de modelagem; no entanto, geralmente � necess�ria uma calibra��o antes da previs�o. A verifica��o do modelo � o processo de testar o modelo com dados reais para estabelecer sua precis�o preditiva. Para calibrar e verificar um modelo, geralmente � necess�rio montar dois conjuntos de dados diferentes. O primeiro conjunto � usado na calibra��o do modelo e o segundo conjunto � usado na verifica��o do modelo. Um acordo pr�ximo entre os dados calculados e medidos � uma indica��o de que o modelo foi verificado. Uma discuss�o detalhada desses assuntos � apresentada no cap�tulo 13.

Muskingum Method M�todo Muskingum

The Muskingum method of flood routing was developed in the 1930s in connection with the design of flood protection schemes in the Muskingum River Basin, Ohio (Fig. 9-3) [11]. It is the most widely used method of hydrologic stream channel routing, with numerous applications in the United States and throughout the world.

O m�todo Muskingum de roteamento de inunda��es foi desenvolvido na d�cada de 1930 em conex�o com o projeto de esquemas de prote��o contra inunda��es na Bacia do Rio Muskingum, Ohio (Fig. 9-3) [11]. � o m�todo mais usado de roteamento de canais de corrente hidrol�gica, com in�meras aplica��es nos Estados Unidos e em todo o mundo.

Flood stage in a tropical river
Tim Kiser

Figure 9-3  The Muskingum river near Marietta, Ohio.

The Muskingum method is based on the differential equation of storage, Eq. 8-4, reproduced here:

O m�todo Muskingum � baseado na equa��o diferencial de armazenamento, Eq. 8-4, reproduzido aqui:

             dS
I - O = _____
             dt
(8-4)

In an ideal channel, storage is a function of inflow and outflow. This is in constrast with an ideal reservoir, in which storage is solely a function of outflow (see Eqs. 8-5 to 8-7). In the Muskingum method, storage is a linear function of inflow and outflow:

Em um canal ideal, o armazenamento � uma fun��o da entrada e sa�da. Isso est� em contraste com um reservat�rio ideal, no qual o armazenamento � apenas uma fun��o da vaz�o (consulte as Eqs. 8-5 a 8-7). No m�todo Muskingum, o armazenamento � uma fun��o linear de entrada e sa�da:

S = K [ X I + ( 1 - X ) O ] (9-1)

in which S = storage volume; I = inflow; O = outflow; K = a time constant or storage coefficient; and X = a dimensionless weighting factor. With inflow and outflow in cubic meters per second, and K in hours, storage volume is in (cubic meters per second)-hour. Alternatively, K could be expressed in seconds, in which case storage volume is in cubic meters.

em que S = volume de armazenamento; I = entrada; O = vaz�o; K = uma constante de tempo ou coeficiente de armazenamento; e X = um fator de pondera��o adimensional. Com entrada e sa�da em metros c�bicos por segundo e K em horas, o volume de armazenamento fica em (metros c�bicos por segundo)-hora. Como alternativa, K pode ser expresso em segundos, caso em que o volume de armazenamento � em metros c�bicos.

Equation 9-1 was developed in 1938 and has been widely used since then [11]. It is esentially a generalization of the linear reservoir concept (Eq. 8-7). In fact, for X = 0, Eq. 9-1 reduces to Eq. 8-7. In other words, linear reservoir routing is a special case of Muskingum channel routing for which X = 0.

A equa��o 9-1 foi desenvolvida em 1938 e tem sido amplamente utilizada desde ent�o [11]. � essencialmente uma generaliza��o do conceito de reservat�rio linear (Eq. 8-7). De fato, para X = 0, a Eq. 9-1 reduz para a Eq. 8-7. Em outras palavras, o roteamento linear de reservat�rio � um caso especial do roteamento de canal de Muskingum para o qual X = 0.

To derive the Muskingum routing equation, Eq. 8-4 is discretized on the x-t plane (Fig. 8-2), to yield Eq. 8-13, repeated here:

Para derivar a equa��o de roteamento de Muskingum, Eq. 8-4 � discretizado no plano x-t (Fig. 8-2), para produzir a Eq. 8-13, repetido aqui:

    I1 + I2          O1 + O2            S2 - S1
__________  -  ___________  =  ___________
       2                    2                     Δt
(8-13)

Equation 9-1 is expressed at time levels 1 and 2:

A equa��o 9-1 � expressa nos n�veis de tempo 1 e 2:

S1 = K [ X I1 + ( 1 - X ) O1 ] (9-2)

S2 = K [ X I2 + ( 1 - X ) O2 ] (9-3)

Substituting Eqs. 9-2 to 9-3 into Eq. 8-13 and solving for O2 yields Eq. 8-15, repeated here:

Substituindo Eqs. 9-2 a 9-3 na Eq. 8-13 e a solu��o para O2 produz a Eq. 8-15, repetido aqui:

O2 = C0 I2 + C1 I1 + C2 O1 (8-15)

in which C0, C1 and C2 are routing coefficients defined in terms of Δt, K, and X as follows:

em que C0, C1 e C2 s�o coeficientes de roteamento definidos em termos de ~t, K e X da seguinte maneira:

                ( Δt / K ) - 2X
C0 = _______________________
            2(1 - X) + ( Δt / K )
(9-4)

                ( Δt / K ) + 2X
C1 = _______________________
            2(1 - X) + ( Δt / K )
(9-5)

            2(1 - X) - ( Δt / K )
C2 = _______________________
            2(1 - X) + ( Δt / K )
(9-6)

Since C0 + C1 + C2 = 1, the routing coefficients may be interpreted as weighting coefficients. For X = 0, Eqs. 9-4, 9-5, and 9-6 reduce to Eqs. 8-16, 8-17, and 8-18, respectively.

Como C0 + C1 + C2 = 1, os coeficientes de roteamento podem ser interpretados como coeficientes de pondera��o. Para X = 0, Eqs. 9-4, 9-5 e 9-6 s�o reduzidos �s Eqs. 8-16, 8-17 e 8-18, respectivamente.

Given an inflow hydrograph, an initial flow condition, a chosen time interval Δt, and routing parameters X and K, the routing coefficients can be calculated with Eqs. 9-4 to 9-6, and the outflow hydrograph with Eq. 8-15. The routing parameters K and K are related to flow and channel characteristics, K being interpreted as the travel time of the flood wave from upstream end to downstream end of the channel reach. Therefore, K accounts for the translation (or concentration) portion of the routing (Fig. 9-3).

Dado um hidrograma de entrada, uma condi��o de fluxo inicial, um intervalo de tempo escolhido ~t e par�metros de roteamento X e K, os coeficientes de roteamento podem ser calculados com as Eqs. 9-4 a 9-6, e o hidrograma de vaz�o com a Eq. 8-15. Os par�metros de roteamento K e K est�o relacionados �s caracter�sticas de fluxo e canal, sendo K interpretado como o tempo de viagem da onda de inunda��o da extremidade a montante at� a extremidade a jusante do alcance do canal. Portanto, K � respons�vel pela parte da convers�o (ou concentra��o) do roteiro (Fig. 9-3).

The parameter X accounts for the storage portion of the routing. For a given flood event, there is a value of X for which the storage in the calculated outflow hydrograph matches that of the measured outflow hydrograph. The effect of storage is to reduce the peak flow and spread the hydrograph in time (Fig. 9-4). Therefore, it is often used interchangeably with the terms diffusion and peak attenuation.

O par�metro X � respons�vel pela parte de armazenamento do roteamento. Para um dado evento de inunda��o, existe um valor de X para o qual o armazenamento no hidrograma de vaz�o calculado corresponde ao do hidr�grafo de vaz�o medido. O efeito do armazenamento � reduzir o pico de fluxo e espalhar o hidrograma com o tempo (Fig. 9-4). Portanto, � frequentemente usado de forma intercambi�vel com os termos difus�o e atenua��o de pico.

Translation and storage processes in stream channel routing

Figure 9-4  Translation and storage processes in stream channel routing.

The routing parameter K is a function of channel reach length and flood wave speed; conversely, the parameter X is a function of the flow and channel characteristics that cause runoff diffusion. In the Muskingum method, X is interpreted as a weighting factor and restricted in the range 0.0 ≤ X ≤ 0.5. Values of X greater than 0.5 produce hydrograph amplification (i.e., negative diffusion), which does not correspond with reality (under the Froude numbers applicable to flood flows). With K = Δt and X = 0.5, flow conditions are such that the outflow hydrograph retains the same shape as the inflow hydrograph, but it is translated downstream a time equal to K. For X = 0, Muskingum routing reduces to linear reservoir routing (Section 8.2).

O par�metro de roteamento K � uma fun��o do comprimento de alcance do canal e velocidade da onda de inunda��o; por outro lado, o par�metro X � uma fun��o das caracter�sticas de fluxo e canal que causam a difus�o do escoamento. No m�todo Muskingum, X � interpretado como um fator de pondera��o e restrito na faixa de 0,0 ~ X ~ 0,5. Valores de X maiores que 0,5 produzem amplifica��o por hidrografia (ou seja, difus�o negativa), que n�o corresponde � realidade (sob os n�meros de Froude aplic�veis %G​​%@aos fluxos de inunda��o). Com K = ~t e X = 0,5, as condi��es de fluxo s�o tais que o hidrograma de vaz�o mant�m a mesma forma que o hidrograma de entrada, mas � traduzido a jusante um tempo igual a K. Para X = 0, o roteamento de Muskingum reduz o roteamento linear de reservat�rio ( Se��o 8.2).

In the Muskingum method, the parameters K and X are determined by calibration using streamflow records. Simultaneous inflow-outflow discharge measurements for a given channel reach are coupled with a trial-and-error procedure, leading to the determination of K and X (see Example 9-2). The procedure is time-consuming and lacks predictive capability. Values of K and X determined in this way are valid only for the given reach and flood event used in the calibration. Extrapolation to other reaches or to other flood events (of different magnitude) within the same reach is usually unwarranted.

No m�todo Muskingum, os par�metros K e X s�o determinados por calibra��o usando registros de fluxo. As medi��es simult�neas de descarga de entrada e sa�da para um determinado alcance do canal s�o acopladas a um procedimento de tentativa e erro, levando � determina��o de K e X (consulte o Exemplo 9-2). O procedimento � demorado e carece de capacidade preditiva. Os valores de K e X determinados desta maneira s�o v�lidos apenas para o evento de alcance e inunda��o fornecido na calibra��o. Extrapola��o para outros alcances ou para outros eventos de inunda��o (de magnitude diferente) dentro do mesmo alcance geralmente � injustificada.

When sufficient data are available, a calibration can be performed for several flood events, each of different magnitude, to cover a wide range of flood levels. In this way, the variation of K and X as a function of flood level can be ascertained. In practice, K is more sensitive to flood level than X. A sketch of the variation of K with stage and discharge is shown in Fig. 9-5.

Quando dados suficientes est�o dispon�veis, uma calibra��o pode ser realizada para v�rios eventos de inunda��o, cada um com magnitude diferente, para cobrir uma ampla gama de n�veis de inunda��o. Desta forma, a varia��o de K e X em fun��o do n�vel de inunda��o pode ser verificada. Na pr�tica, K � mais sens�vel ao n�vel de inunda��o do que X. Um esbo�o da varia��o de K com est�gio e descarga � mostrado na Fig. 9-5.

Sketch of travel time as a function of discharge and stage

Figure 9-5  Sketch of travel time as a function of discharge and stage.

 Example 9-1.

An inflow hydrograph to a channel reach is shown in Col. 2 of Table 9-1. Assume baseflow is 352 m3/s. Using the Muskingum method, route this hydrograph through a channel reach with K = 2 d and X = 0.1 to calculate an outflow hydrograph.

Um hidrograma de entrada para o alcance do canal � mostrado na Col. 2 da Tabela 9-1. Suponha que o fluxo base seja de 352 m3 / s. Usando o m�todo Muskingum, direcione esse hidrograma atrav�s de um alcance de canal com K = 2 de X = 0,1 para calcular um hidrograma de vaz�o.


First, it is necessary to select a time interval Δt. In this case, it is convenient to choose Δt = 1 d. As with reservoir routing, the ratio of time-to-peak to time interval (tpt) should be greater than or equal to 5. In addition, the chosen time interval should be such that the routing coefficients remain positive. With Δt = 1 d, K = 2 d, and X = 0.1, the routing coefficients (Eqs. 9-4 to 9-6) are: C0 = 0.1304; C1 = 0.3044; and C2 = 0.5652. It is verified that C0 + C1 + C2 = 1. The routing calculations are shown in Table 9-1.

Primeiro, � necess�rio selecionar um intervalo de tempo ~t. Nesse caso, � conveniente escolher ~t = 1 d. Como no roteamento do reservat�rio, a raz�o entre o tempo e o pico do intervalo de tempo (tp / ~t) deve ser maior ou igual a 5. Al�m disso, o intervalo de tempo escolhido deve ser tal que os coeficientes de roteamento permane�am positivos. Com ~t = 1 d, K = 2 d e X = 0,1, os coeficientes de roteamento (Eqs. 9-4 a 9-6) s�o: C0 = 0,1304; C1 = 0,3044; e C2 = 0,5652. Verifica-se que C0 + C1 + C2 = 1. Os c�lculos de roteamento s�o mostrados na Tabela 9-1.

  • Column 1 shows the time in days.

    A coluna 1 mostra a hora em dias.

  • Column 2 shows the inflow hydrograph ordinates in cubic meters per second.

    A coluna 2 mostra as ordenadas do hidrograma de entrada em metros c�bicos por segundo.

  • Columns 3-5 show the partial flows.

    As colunas 3-5 mostram os fluxos parciais.

  • Following Eq. 8-15, Cols. 3-5 are summed to obtain Col. 6, the outflow hydrograph ordinates in cubic meters per second.

    Ap�s a Eq. 8-15, Cols. 3 a 5 s�o somados para obter a Col. 6, o hidrograma de sa�da ordena em metros c�bicos por segundo.

To explain the procedure briefly, the outflow at the start (day 0) is assumed to be equal to the inflow at the start: 352 m3/s. The inflow at day 1 multiplied by C0 is entered in Col. 3, day 1: 76.6 m3/s. The inflow at day 0 multiplied by C1 is entered in Col. 4, day 1: 107.1 m3/s. The outflow at day 0 multiplied by C2 is entered in Col. 5, day 1: 199 m3/s. Columns 3-5 of day 1 are summed to obtain Col. 6 of day 1: 76.6 + 107.1 + 199.0 = 382.7 m3/s. The calculations proceed in a recursive manner until all outflows in Col. 6 have been obtained. Inflow and outflow hydrographs are plotted in Fig. 9-6. The outflow peak is 6352.6 m3/s, which shows that the inflow peak, 6951 m3/s, has attenuated to about 91 % of its initial value. The peak outflow occurs at day 9, 2 d after the peak inflow, which occurs at day 7. The time elapsed between the occurrence of peak inflow and peak outflow is generally equal to K, the travel time.

Para explicar brevemente o procedimento, presume-se que a vaz�o no in�cio (dia 0) seja igual � entrada no in�cio: 352 m3 / s. A entrada no dia 1 multiplicada por C0 � inserida na Col. 3, dia 1: 76,6 m3 / s. A entrada no dia 0 multiplicada por C1 � inserida na Col. 4, dia 1: 107,1 m3 / s. A vaz�o no dia 0 multiplicada por C2 � inserida na Col. 5, dia 1: 199 m3 / s. As colunas 3-5 do dia 1 s�o somadas para obter a Col. 6 do dia 1: 76,6 + 107,1 + 199,0 = 382,7 m3 / s. Os c�lculos prosseguem de maneira recursiva at� que todas as sa�das na Col. 6 tenham sido obtidas. Os hidrogramas de entrada e sa�da s�o plotados na Fig. 9-6. O pico de vaz�o � de 6352,6 m3 / s, o que mostra que o pico de entrada, 6951 m3 / s, atenuou cerca de 91% do seu valor inicial. O pico de vaz�o ocorre no dia 9, 2 d ap�s o pico de entrada, que ocorre no dia 7. O tempo decorrido entre a ocorr�ncia do pico de entrada e o pico de sa�da � geralmente igual a K, o tempo de viagem.

Table 9-1  Channel Routing by the Muskingum Method:  Example 9-1.
(1) (2) (3) (4) (5) (6)
Time
(d)
Inflow
(m3/s)
Partial Flows (m3/s) Outflow
(m3/s)
C0I2 C1I1 C2I1
0 352.0 ___ ___ ___ 352.0
1 587.0 76.6 107.1 199.0 382.7
2 1353.0 176.5 178.6 216.3 571.4
3 2725.0 355.4 411.8 323.0 1090.2
4 4408.5 575.0 829.4 616.2 2020.6
5 5987.0 780.9 1341.7 1142.1 3264.7
6 6704.0 874.4 1822.1 1845.3 4541.8
7 6951.0 906.7 2040.3 2567.1 5514.1
8 6839.0 892.0 2115.5 3116.7 6124.2
9 6207.0 809.6 2081.5 3461.5 6352.6
10 5346.0 697.3 1889.1 3590.6 6177.0
11 4560.0 594.8 1627.0 3491.4 5713.2
12 3861.5 503.7 1387.8 3229.2 5120.7
13 3007.0 392.2 1175.2 2894.3 4461.7
14 2357.5 307.5 915.2 2521.8 3744.5
15 1779.0 232.0 717.5 2116.5 3066.0
16 1405.0 183.3 541.4 1733.0 2457.7
17 1123.0 146.5 427.6 1389.1 1963.2
18 952.5 124.2 341.8 1109.6 1575.6
19 730.0 95.2 289.9 890.6 1275.7
20 605.0 78.9 222.2 721.0 1022.1
21 514.0 67.1 184.1 577.7 828.9
22 422.0 55.1 156.4 468.5 680.0
23 352.0 45.9 128.4 384.4 558.7
24 352.0 45.9 107.1 315.8 468.8
25 352.0 45.9 107.1 265.0 418.0

calculator image 

ONLINE CALCULATION. Using ONLINE ROUTING04, the answer is essentially the same as that of Col. 6, Table 9-1.

Stream channel routing by Muskingum method:  Example 9-1

Figure 9-6  Stream channel routing by Muskingum method:  Example 9-1.

Unlike reservoir routing, stream channel-routing calculations exhibit a definite (time) lag between inflow and outflow. Furthermore, in the general case (X ≠ 0), maximum outflow does not occur at the time when inflow and outflow coincide.

Diferentemente do roteamento de reservat�rio, os c�lculos de roteamento de canal de fluxo exibem um atraso (tempo) definido entre a entrada e a sa�da. Al�m disso, no caso geral (X ~ 0), a vaz�o m�xima n�o ocorre no momento em que a entrada e a sa�da coincidem.

Example 9-1 has illustrated the predictive stage of the Muskingum method, in which the routing parameters are known in advance of the routing. If the parameters are not known, it is first necessary to perform a calibration. The trial-and-error procedure to calibrate the routing parameters is illustrated by Example 9-2.

O Exemplo 9-1 ilustrou o est�gio preditivo do m�todo Muskingum, no qual os par�metros de roteamento s�o conhecidos antes do roteamento. Se os par�metros n�o forem conhecidos, � necess�rio primeiro executar uma calibra��o. O procedimento de tentativa e erro para calibrar os par�metros de roteamento � ilustrado pelo Exemplo 9-2.

 Example 9-2.

Use the outflow hydrograph calculated in the previous example together with the given inflow hydrograph to calibrate the Muskingum method, that is, to find the routing parameters K and X.

Use o hidrograma de vaz�o calculado no exemplo anterior, juntamente com o hidrograma de entrada fornecido, para calibrar o m�todo Muskingum, ou seja, para encontrar os par�metros de roteamento K e X.


The procedure is summarized in Table 9-2.

O procedimento est� resumido na Tabela 9-2.

  • Column 1 shows the time in days.

    A coluna 1 mostra a hora em dias.

  • Column 2 shows the inflow hydrograph in cubic meters per second.

    A coluna 2 mostra o hidrograma de entrada em metros c�bicos por segundo

  • Column 3 shows the outflow hydrograph in cubic meters per second.

    A coluna 3 mostra o hidrograma de vaz�o em metros c�bicos por segundo.

  • Column 4 shows the channel storage in (cubic meters per second)-days.

    A coluna 4 mostra o armazenamento do canal em (metros c�bicos por segundo) dias.

  • Channel storage at the start is assumed to be 0, and this value is entered in Col. 4, day 0.

    O armazenamento do canal no in�cio � assumido como 0 e esse valor � inserido na Col. 4, dia 0.

  • Channel storage is calculated by solving Eq. 8-13 for S2:

    O armazenamento do canal � calculado resolvendo-se a Eq. 8-13 para S2:

    S2 = S1 + ( Δt / 2 ) ( I1 + I2 - O1 - O2 ) (9-7)

  • Several values of X are tried, within the range 0.0 to 0.5, for example, 0.1, 0.2 and 0.3.

    S�o tentados v�rios valores de X, dentro do intervalo de 0,0 a 0,5, por exemplo, 0,1, 0,2 e 0,3.

  • For each trial value of X, the weighted flows [ XI + ( 1 - X ) O ] are calculated, as shown in Cols. 5-7.

    Para cada valor experimental de X, os fluxos ponderados [XI + (1 - X) O] s�o calculados, conforme mostrado em Cols. 5-7.

  • Each of the weighted flows is plotted against channel storage (Col. 4), as shown in Fig. 9-7.

    Cada um dos fluxos ponderados � plotado contra o armazenamento de canal (Col. 4), conforme mostrado na Fig. 9-7.

  • The value of X for which the storage versus weighted flow data plots closest to a line is taken as the correct value of X. In this case, Fig. 9-7 (a): X = 0.1 is chosen.

    O valor de X para o qual os dados de armazenamento versus fluxo ponderado mais pr�ximos de uma linha � considerado o valor correto de X. Nesse caso, a Fig. 9-7 (a): X = 0,1 � escolhida.

  • Following Eq. 9-1, the value of K is obtained from Fig. 9-7 (a) by calculating the slope of the storage vs weighted outflow curve.

    Ap�s a Eq. 9-1, o valor de K � obtido da Fig. 9-7 (a) calculando a inclina��o da curva de armazenamento versus a vaz�o ponderada.

  • In this case, the value of K = [2000 (m3/s)-d]/(1000 m3/s) = 2 d.

    Nesse caso, o valor de K = [2000 (m3 / s) -d] / (1000 m3 / s) = 2 d.

  • Thus, it is shown that K = 2 days and X = 0.1 are the Muskingum routing parameters for the given inflow and outflow hydrographs.

    Assim, � mostrado que K = 2 dias e X = 0,1 s�o os par�metros de roteamento de Muskingum para os hidrogramas de entrada e sa�da dados.


Table 9-2  Calibration of Muskingum Routing Parameters:  Example 9-2.
(1) (2) (3) (4) (5) (6) (7)
Time
(d)
Inflow
(m3/s)
Outflow
(m3/s)
Storage
(m3/s)-d
Weighted Flow (m3/s)
X = 0.1 X = 0.2 X = 0.3
0 352.0 352.0 0 ___ ___ ___
1 587.0 382.7 102.2 403.0 423.5 443.9
2 1353.0 571.4 595.2 649.6 727.7 805.9
3 2725.0 1090.2 1803.4 1253.7 1417.2 1580.6
4 4408.5 2020.6 3814.7 2259.4 2498.2 2737.0
5 5987.0 3264.7 6369.8 3536.9 3809.2 4081.4
6 6704.0 4541.8 8812.1 4758.0 4974.2 5190.5
7 6951.0 5514.1 10611.6 5657.8 5801.5 5945.2
8 6839.0 6124.2 11687.5 6195.7 6267.2 6338.6
9 6207.0 6352.6 11972.1 6338.0 6323.5 6308.9
10 5346.0 6177.0 11483.8 6093.9 6010.8 5927.7
11 4560.0 5713.2 10491.7 5597.9 5482.6 5367.2
12 3861.5 5120.7 9285.5 4994.8 4868.9 4742.9
13 3007.0 4461.7 7928.5 4316.2 4170.8 4025.3
14 2357.5 3744.5 6507.7 3605.8 3467.1 3328.4
15 1779.0 3066.0 5170.7 2937.3 2808.6 2679.9
16 1405.0 2457.7 4000.8 2352.4 2247.2 2141.9
17 1123.0 1963.2 3054.4 1879.2 1795.2 1711.1
18 952.5 1575.6 2322.7 1513.4 1451.1 1388.7
19 730.0 1275.7 1738.2 1221.1 1166.6 1112.0
20 605.0 1022.1 1256.8 980.4 938.7 897.0
21 514.0 828.9 890.8 797.4 765.9 734.4
22 422.0 680.0 604.4 654.2 628.4 602.6
23 352.0 558.7 372.0 537.9 517.3 496.6
24 352.0 468.8 210.3 457.1 445.4 433.8
25 352.0 418.0 118.9 411.4 404.8 398.2

Calibration of Muskingum routing parameters:

Figure 9-7  Calibration of Muskingum routing parameters: Example 9-2.

The estimation of routing parameters is crucial to the application of the Muskingum method. The parameters are not constant, tending to vary with flow rate. If the routing parameters can be related to flow and channel characteristics, the need for trial-and-error calibration would be eliminated. Parameter K could be related to reach length and flood wave velocity, whereas X could be related to the diffusivity characteristics of flow and channel. These propositions are the basis of the Muskingum-Cunge method (Section 9.4).

A estimativa dos par�metros de roteamento � crucial para a aplica��o do m�todo Muskingum. Os par�metros n�o s�o constantes, tendendo a variar com a vaz�o. Se os par�metros de roteamento puderem estar relacionados �s caracter�sticas de fluxo e canal, a necessidade de calibra��o por tentativa e erro ser� eliminada. O par�metro K pode estar relacionado ao comprimento e velocidade da onda de inunda��o, enquanto X pode estar relacionado �s caracter�sticas de difusividade do fluxo e canal. Essas proposi��es s�o a base do m�todo Muskingum-Cunge (Se��o 9.4).


9.2  ONDAS CINEM�TICAS

[Ondas Difusivas]   [M�todo Muskingum-Cunge]   [Ondas Din�micas]   [Quest�es]   [Problemas]   [Refer�ncias]      [Top]   [M�todo Muskingum]  

Three types of unsteady open-channel flow waves are commonly used in engineering hydrology: (1) kinematic, (2) diffusion, and (3) dynamic waves. Kinematic waves are the simplest type of wave and dynamic waves are the most complex; diffusion waves lie somewhere in between kinematic and dynamic waves. Kinematic waves are discussed in this section and diffusion waves are discussed in Section 9.3. An introduction to dynamic waves is given in Section 9.5.

Tr�s tipos de ondas de fluxo de canal aberto inst�veis %G​​%@s�o comumente usados %G​​%@na hidrologia de engenharia: (1) cinem�tico, (2) difus�o e (3) ondas din�micas. As ondas cinem�ticas s�o o tipo mais simples de onda e as ondas din�micas s�o as mais complexas; ondas de difus�o est�o em algum lugar entre ondas cinem�ticas e din�micas. As ondas cinem�ticas s�o discutidas nesta se��o e as ondas de difus�o s�o discutidas na Se��o 9.3. Uma introdu��o �s ondas din�micas � apresentada na Se��o 9.5.

Kinematic Wave Equation

Equa��o de Ondas Cinem�ticas

The derivation of the kinematic wave equation is based on the principle of mass conservation within a control volume. This principle states that the difference between outflow and inflow within one time interval is balanced by a corresponding change in volume. In terms of finite intervals (i.e., finite differences) it is:

A deriva��o da equa��o da onda cinem�tica � baseada no princ�pio de conserva��o de massa dentro de um volume de controle. Este princ�pio afirma que a diferen�a entre sa�da e entrada dentro de um intervalo de tempo � equilibrada por uma mudan�a correspondente no volume. Em termos de intervalos finitos (ou seja, diferen�as finitas), �:

( Q2 - Q1 ) Δt  +  ( A2 - A1 ) Δx = 0 (9-8)

in which Q = flow; A = flow area; Δt = time interval; and Δx = space interval. In differential form, Eq. 9-8 can be written as:

em que Q = fluxo; A = �rea de fluxo; ~t = intervalo de tempo; e ~x = intervalo de espa�o. De forma diferencial, a Eq. 9-8 pode ser escrito como:

 ∂Q        ∂A
____  +  ____  =  0
 ∂x          ∂t
(9-9)

which is the equation of conservation of mass, or equation of continuity.

que � a equa��o de conserva��o de massa ou equa��o de continuidade.

The equation of conservation of momentum (Eq. 4-22) contains local inertia, convective inertia, pressure gradient (due to flow depth gradient), friction (friction slope), gravity (bed slope), and a momentum source term (Section 4.2). In deriving the kinematic wave equation, a statement of uniform flow is used in lieu of conservation of momentum. Since uniform flow is strictly a balance of friction and gravity, it follows that local and convective inertia, pressure gradient, and momentum source terms are excluded from the formulation of kinematic waves. In other words, a kinematic wave is a simplified wave that does not include these terms or processes. As shown later in this section, this simplification imposes limits to the applicability of kinematic waves.

A equa��o de conserva��o do momento (Eq. 4-22) cont�m in�rcia local, in�rcia convectiva, gradiente de press�o (devido ao gradiente de profundidade do fluxo), fric��o (declive de atrito), gravidade (declive de leito) e um termo de fonte de momento (Se��o 4.2 ) Na deriva��o da equa��o das ondas cinem�ticas, uma declara��o de fluxo uniforme � usada no lugar da conserva��o do momento. Como o fluxo uniforme � estritamente um equil�brio de atrito e gravidade, segue-se que os termos de in�rcia local e convectiva, gradiente de press�o e fonte de momento s�o exclu�dos da formula��o de ondas cinem�ticas. Em outras palavras, uma onda cinem�tica � uma onda simplificada que n�o inclui esses termos ou processos. Como mostrado mais adiante nesta se��o, essa simplifica��o imp�e limites � aplicabilidade das ondas cinem�ticas.

Uniform flow in open channels is described by the Manning or Chezy formulas (Section 2.4). The Manning equation is:

O fluxo uniforme em canais abertos � descrito pelas f�rmulas de Manning ou Chezy (Se��o 2.4). A equa��o de Manning �:

        1
Q = ___ A R 2/3 Sf 1/2
        n
(9-10)

in which R is the hydraulic radius in meters, Sf is the friction slope in meters per meter, and n is the Manning friction coefficient. A pictorial on n is given by Barnes (1967).

em que R � o raio hidr�ulico em metros, Sf � a inclina��o de atrito em metros por metro en � o coeficiente de atrito de Manning. Uma imagem em n � dada por Barnes (1967).

The Chezy equation is:

A equa��o de Chezy �:

Q = C A R 1/2 Sf 1/2
(9-11)

in which C = Chezy coefficient. Notice that in unsteady flow, friction slope is used in Eqs. 9-10 and 9-11 in lieu of channel slope.

em que C = coeficiente de Chezy. Observe que no fluxo inst�vel, a inclina��o de atrito � usada nas Eqs. 9-10 e 9-11 em vez da inclina��o do canal.

The hydraulic radius is R = A/P, in which P is the wetted perimeter. Substituting this into Eq. 9-10, leads to:

O raio hidr�ulico � R = A / P, no qual P � o per�metro �mido. Substituindo isso na Eq. 9-10, leva a:

        1       Sf 1/2
Q = ___ _________ A5/3
        n      P 2/3
(9-12)

Assume for the sake of simplicity that n, Sf, and P are constant. This may be the case of a wide channel in which P can be assumed to be essentially independent of A. Equation 9-12 can then be written as:

Suponha, por uma quest�o de simplicidade, que n, Sf e P s�o constantes. Pode ser o caso de um canal amplo no qual P pode ser assumido como essencialmente independente de A. A equa��o 9-12 pode ser escrita como:

Q = α Aβ (9-13)

in which α and β are parameters of the discharge-area rating (see rating curve, Section 2.4), defined as follows:

em que ~ e ~ s�o par�metros da classifica��o da �rea de descarga (ver curva de classifica��o, se��o 2.4), definidos da seguinte forma:

          1       Sf 1/2
α  =  ___ _________
          n      P 2/3
(9-14)

         5
β  =  ___
         3
(9-15)

In Eq. 9-13, differentiating Q with respect to A leads to:

Na Eq. 9-13, diferenciar Q em rela��o a A leva a:

  dQ             Q
_____  =  β ____  =  β V
  dA             A
(9-16)

in which V is the mean flow velocity.

em que V � a velocidade m�dia do fluxo.

Multiplying Eqs. 9-9 and 9-16 and applying the chain rule, the kinematic wave equation is obtained:

Multiplicando Eqs. 9-9 e 9-16 e aplicando a regra da cadeia, a equa��o da onda cinem�tica � obtida:

  ∂Q           dQ      ∂Q
_____  +  (_____) ______  =  0
  ∂t             dA       ∂x
(9-17)

or, alternatively

ou alternativamente

  ∂Q                    ∂Q
_____  +  (β V) ______  =  0
  ∂t                      ∂x
(9-18)

Equation 9-17 (or 9-18) describes the movement of waves which are kinematic in nature . These are referred to as kinematic waves, i.e., waves for which inertia and pressure (flow depth) gradient have been neglected [10]. Equation 9-17 is a first-order partial differential equation. Therefore, kinematic waves travel with wave celerity dQ/dA (or βV) and do not attenuate. Wave attenuation can only be described by a second-order partial differential equation.

A Equa��o 9-17 (ou 9-18) descreve o movimento das ondas de natureza cinem�tica. Estes s�o chamados de ondas cinem�ticas, isto �, ondas para as quais a in�rcia e o gradiente de press�o (profundidade do fluxo) foram negligenciados [10]. A equa��o 9-17 � uma equa��o diferencial parcial de primeira ordem. Portanto, as ondas cinem�ticas viajam com a celeridade das ondas dQ / dA (ou ~V) e n�o atenuam. A atenua��o das ondas s� pode ser descrita por uma equa��o diferencial parcial de segunda ordem.

The absence of wave attenuation can be further explained by resorting to a mathematical argument. Since dQ/dA is the celerity of the unsteady (i.e. , wavelike) Q, it can be replaced by dx/dt. Therefore, in Eq. 9-17:

A aus�ncia de atenua��o das ondas pode ser explicada ainda mais recorrendo a um argumento matem�tico. Como dQ / dA � a celeridade do Q inst�vel (isto �, ondulado) Q, ele pode ser substitu�do por dx / dt. Portanto, na Eq. 9-17:

  ∂Q           dx       ∂Q
_____  +  (_____) ______  =  0
  ∂t             dt        ∂x
(9-19)

which is equal to the total derivative dQ/dt. Since the right side of Eq. 9-19 is zero, it follows that Q remains constant in time for waves traveling with celerity dQ/dA.

que � igual ao derivado total dQ / dt. Desde o lado direito da Eq. 9-19 � zero, segue-se que Q permanece constante no tempo para as ondas que viajam com celeridade dQ / dA.

Discretization of Kinematic Wave Equation

Discretiza��o da Equa��o de Ondas Cinem�ticas

Equation 9-18 (or 9-17) is a nonlinear first-order partial differential equation describing the change of discharge Q in time and space. It is nonlinear because the wave celerity βV (or dQ/dA) varies with discharge. The nonlinearity, however, is usually mild, and therefore, Eq. 9-18 can also be solved in a linear mode by considering the wave celerity to be constant.

A equa��o 9-18 (ou 9-17) � uma equa��o diferencial parcial de primeira ordem n�o linear que descreve a altera��o da descarga Q no tempo e no espa�o. N�o � linear porque a celeridade das ondas ~V (ou dQ / dA) varia com a descarga. A n�o linearidade, no entanto, � geralmente leve e, portanto, a Eq. 9-18 tamb�m pode ser resolvido em um modo linear, considerando a celeridade das ondas constante.

The solution of Eq. 9-18 can be obtained by analytical or numerical means. The simplest kinematic wave solution is a linear numerical solution. For this purpose, it is necessary to select a numerical scheme with which to discretize Eq. 9-18 on the x-t plane (Fig. 9-8). A review of basic concepts of numerical analysis is necessary before discussing numerical schemes.

A solu��o da Eq. 9-18 pode ser obtido por meios anal�ticos ou num�ricos. A solu��o mais simples de ondas cinem�ticas � uma solu��o num�rica linear. Para esse fim, � necess�rio selecionar um esquema num�rico com o qual discretizar a Eq. 9-18 no plano x-t (Fig. 9-8). Uma revis�o dos conceitos b�sicos da an�lise num�rica � necess�ria antes de discutir esquemas num�ricos.

Space-time discretization of kinematic wave equation

Figure 9-8  Space-time discretization of kinematic wave equation.

Order of Accuracy of Numerical Schemes. The order of accuracy of a numerical scheme measures the ability of the scheme to reproduce (i.e., recreate) the terms of the differential equation. In general, the higher the order of accuracy of a scheme, the better it is able to reproduce the terms of the differential equation. Forward and backward finite differences have first-order accuracy, i.e., discretization errors of first order. Central differences have second-order accuracy, with discretization errors of second order.

Ordem de precis�o dos esquemas num�ricos. A ordem de precis�o de um esquema num�rico mede a capacidade do esquema de reproduzir (isto �, recriar) os termos da equa��o diferencial. Em geral, quanto maior a ordem de precis�o de um esquema, melhor � capaz de reproduzir os termos da equa��o diferencial. As diferen�as finitas para frente e para tr�s t�m precis�o de primeira ordem, ou seja, erros de discretiza��o de primeira ordem. As diferen�as centrais t�m precis�o de segunda ordem, com erros de discretiza��o de segunda ordem.

When solving Eq. 9-18 by numerical methods, first-order schemes create numerical diffusion and numerical dispersion, while second-order schemes create only numerical dispersion. A third-order scheme creates neither numerical diffusion nor dispersion. Numerical diffusion and/or dispersion are caused by the finite grid size and are not necessarily related to the physical problem.

Ao resolver a Eq. 9-18 por m�todos num�ricos, os esquemas de primeira ordem criam difus�o e dispers�o num�rica, enquanto os esquemas de segunda ordem criam apenas dispers�o num�rica. Um esquema de terceira ordem n�o cria difus�o nem dispers�o num�rica. A difus�o e / ou dispers�o num�rica s�o causadas pelo tamanho finito da grade e n�o est�o necessariamente relacionadas ao problema f�sico.

Second-order-accurate Numerical Scheme. The discretization of Eq. 9-18 following a linear second-order-accurate scheme, i.e., using central differences in space and time, leads to (Fig. 9-8):

Esquema num�rico preciso de segunda ordem. A discretiza��o da Eq. 9-18, seguindo um esquema linear preciso de segunda ordem, ou seja, usando diferen�as centrais no espa�o e no tempo, leva a (Fig. 9-8):


   M                    N
______  +  βV ______  =  0
   Δt                  Δx
(9-20)


            Q j+1n+1 + Q j n+1           Q j+1n + Q j n
M  =  ___________________  -  _________________
                        2                                2
(9-20a)

            Q j+1n + Q j+1 n+1           Q j n + Q j n+1
N  =  ___________________  -  _________________
                        2                                2
(9-20b)

in which βV  has been held constant (linear mode), leading to:

em que ~V foi mantido constante (modo linear), levando a:

Q j+1 n+1 = C0 Q j n+1 + C1 Q j n + C2 Q j+1 n (9-21)

in which

           C - 1
C0 = _______
          1 + C
(9-22)

C1 = 1 (9-23)

          1 - C
C2 = _______
          1 + C
(9-24)

and C is the Courant number, defined as follows:

e C � o n�mero do Courant, definido da seguinte forma:

               Δt
C = βV  _____
               Δx
(9-25)

Note that Courant number is the ratio of physical wave celerity βV to grid celerity Δxt. The Courant number is a fundamental concept in the numerical solution of hyperbolic partial differential equations.

Observe que o n�mero de Courant � a raz�o entre a celeridade da onda f�sica ~V e a celeridade da grade ~x / ~t. O n�mero de Courant � um conceito fundamental na solu��o num�rica de equa��es diferenciais parciais hiperb�licas.

 Example 9-3.

Use Eq. 9-21 with the routing coefficients of Eqs. 9-22 to 9-24 (linear kinematic wave numerical solution using central differences in space and time) to route the following triangular flood wave. Consider the following three cases: (1) V = 1.2 m/s and Δx = 7200 m; (2) V = 1.2 m/s and Δx = 4800 m; and (3) V = 0.8 mls and Δx = 4800 m. Use β = 5/3, and Δt = 1 h.

Use a Eq. 9-21 com os coeficientes de roteamento das Eqs. 9-22 a 9-24 (solu��o num�rica de ondas cinem�ticas lineares usando diferen�as centrais no espa�o e no tempo) para direcionar a seguinte onda de inunda��o triangular. Considere os tr�s casos a seguir: (1) V = 1,2 m / se ~x = 7200 m; (2) V = 1,2 m / se ~x = 4800 m; e (3) V = 0,8 ml e ~x = 4800 m. Use ~ = 5/3 e ~t = 1 h.

Time (h) 0 1 2 3 4 5 6 7 8 9 10
Inflow (m3/s 0 30 60 90 120 150 120 90 60 30 0


  1. Using Eq. 9-25: C = 1. Using Eqs. 9-22 to 9-24: C0 = 0; C1 = 1; C2 = 0. The routing by Eq. 9-21 shown in Table 9-3 depicts the pure translation of the hydrograph a time equal to Δt. In other words, for βV = Δxt (i.e., C = 1), the central difference scheme is of third order, and the numerical solution is exactly equal to the analytical solution.

    Usando a Eq. 9-25: C = 1. Usando Eqs. 9-22 a 9-24: C0 = 0; C1 = 1; C2 = 0. O roteamento pela Eq. 9-21 mostrado na Tabela 9-3 representa a tradu��o pura do hidrograma um tempo igual a ~t. Em outras palavras, para ~V = ~x / ~t (isto �, C = 1), o esquema de diferen�a central � de terceira ordem e a solu��o num�rica � exatamente igual � solu��o anal�tica.

  2. Using Eq. 9-25: C = 1.5. Using Eqs. 9-22 to 9-24: C0 = 0.2; C1 = 1.0; C2 = -0.2. The routing by Eq. 9-21 shown in Table 9-4 depicts the translation of the hydrograph a time approximately equal to Δt, but it also shows a small amount of numerical dispersion because βV is not equal to Δxt. The dispersion, including the notorious negative outflows at the trailing end of the hydrograph, are caused by errors associated with the scheme's second-order accuracy.

    Usando a Eq. 9-25: C = 1,5. Usando Eqs. 9-22 a 9-24: C0 = 0,2; C1 = 1,0; C2 = -0,2. O roteamento pela Eq. 9-21, mostrada na Tabela 9-4, representa a tradu��o do hidrograma um tempo aproximadamente igual a ~t, mas tamb�m mostra uma pequena quantidade de dispers�o num�rica porque ~V n�o � igual a ~x / ~t. A dispers�o, incluindo as not�rias sa�das negativas no final do hidrograma, � causada por erros associados � precis�o de segunda ordem do esquema.

  3. Using Eq. 9-25, C = 1. Therefore, the solution is the same as in the first case, exhibiting pure hydrograph translation.

    Usando a Eq. 9-25, C = 1. Portanto, a solu��o � a mesma do primeiro caso, exibindo tradu��o pura em hidrografia.

Table 9-3  Kinematic Wave Routing:  Pure Translation, Example 9-3, Part 1.
(1) (2) (3) (4) (5) (6)
Time
(h)
Inflow
(m3/s)
Partial Flows (m3/s) Outflow
(m3/s)
C0 I2 C1 I1 C2 I1
0 0 ___ ___ ___ 0
1 30 0 0 0 0
2 60 0 30 0 30
3 90 0 60 0 60
4 120 0 90 0 90
5 150 0 120 0 120
6 120 0 150 0 150
7 90 0 120 0 120
8 60 0 90 0 90
9 30 0 60 0 60
10 0 0 30 0 30
11 0 0 0 0 0

Table 9-4  Kinematic Wave Routing:  Translation and Dispersion,
Example 9-3, Part 1.

(1) (2) (3) (4) (5) (6)
Time
(h)
Inflow
(m3/s)
Partial Flows (m3/s) Outflow
(m3/s)
C0 I2 C1 I1 C2 I1
0 0 ___ ___ ___ 0
1 30 6 0 0 6
2 60 12 30 -1.20 40.80
3 90 18 60 -8.16 69.84
4 120 24 90 -13.97 100.03
5 150 30 120 -20.91 129.99
6 120 24 150 -26.00 148.00
7 90 18 120 -29.60 108.40
8 60 12 90 -21.68 80.32
9 30 6 60 -16.06 49.94
10 0 0 30 -9.99 20.01
11 0 0 0 -4.00 -4.00
12 0 0 0 0.80 0.80
13 0 0 0 -0.16 -0.16


The three cases of Example 9-3 illustrate the properties of kinematic waves. The second-order-accurate scheme has no numerical diffusion. In addition, for Courant number C = 1, i.e., the wave celerity βV equal to the grid celerity Δxt, the scheme has no numerical dispersion, with the hydrograph being translated downstream without change in shape. In other words, the numerical solution by Eqs. 9-21 to 9-25 is exact only for Courant number C = 1. For other values of C, the numerical solution exhibits perceptible amounts of numerical dispersion.

Os tr�s casos do Exemplo 9-3 ilustram as propriedades das ondas cinem�ticas. O esquema preciso de segunda ordem n�o tem difus�o num�rica. Al�m disso, para o n�mero de Courant C = 1, isto �, a velocidade da onda ~V igual � velocidade da grade ~x / ~t, o esquema n�o tem dispers�o num�rica, com o hidrograma sendo traduzido a jusante sem altera��o de forma. Em outras palavras, a solu��o num�rica pelas Eqs. 9-21 a 9-25 � exato apenas para o n�mero de Courant C = 1. Para outros valores de C, a solu��o num�rica exibe quantidades percept�veis de dispers�o num�rica.

First-order-accurate Numerical Scheme. The numerical solution of Eq. 9-18 can also be attempted using a first-order-accurate scheme, i.e., one featuring forward or backward finite differences. The discretization of Eq. 9-18 in a linear mode, using backward differences in both space and time yields (Fig. 9-8):

Esquema num�rico preciso de primeira ordem. A solu��o num�rica da Eq. 9-18 tamb�m pode ser tentado usando um esquema preciso de primeira ordem, isto �, um apresentando diferen�as finitas para frente ou para tr�s. A discretiza��o da Eq. 9-18 em um modo linear, usando diferen�as inversas nos rendimentos de espa�o e tempo (Fig. 9-8):

  Q j+1n+1  -  Q j+1 n                 Q j+1n+1  -  Q j n+1
____________________  +  βV  ___________________  =  0
             Δt                                        Δx
(9-26)

from which

Q j+1n+1 = C0 Q j n+1 + C2 Q j+1n (9-27)

in which

                C
C0  =  _________
             1 + C
(9-28)

                1
C2  =  _________
             1 + C
(9-29)

and C = Courant number, defined by Eq. 9-25.

e C = n�mero de Courant, definido pela Eq. 9-25.

 Example 9-4.

Use Eq. 9-27 with the coefficients calculated by Eq. 9-28 and 9-29 to route the same inflow hydrograph as in the previous example. Use V = 1.2 m/s; Δx = 7200 m; β = 5/3; and Δt = 1 h.

Use a Eq. 9-27 com os coeficientes calculados pela Eq. 9-28 e 9-29 para encaminhar o mesmo hidrograma de entrada que no exemplo anterior. Use V = 1,2 m / s; ~x = 7200 m; p = 5/3; e ~t = 1 h.


Using Eq. 9-25, C = 1. Therefore, C0 = 0.5, and C2 = 0.5. The routing using Eq. 9-27 is shown in Table 9-5. It is observed that off-centering the derivatives by using backward differences has caused a significant amount of numerical diffusion, with peak outflow of 120.93 m3/s as compared to peak inflow of 150 m3/s. The conclusion is that different schemes for solving Eq. 9-18 lead to different answers, depending on the time and space intervals, Courant number, order of accuracy of the scheme, and associated numerical diffusion and/or dispersion.

Usando a Eq. 9-25, C = 1. Portanto, C0 = 0,5 e C2 = 0,5. O roteamento usando a Eq. 9-27 � mostrado na Tabela 9-5. Observa-se que a descentraliza��o dos derivados pelo uso de diferen�as inversas causou uma quantidade significativa de difus�o num�rica, com vaz�o de pico de 120,93 m3 / s em compara��o ao pico de entrada de 150 m3 / s. A conclus�o � que diferentes esquemas para resolver a Eq. 9-18 levam a respostas diferentes, dependendo dos intervalos de tempo e espa�o, n�mero de Courant, ordem de precis�o do esquema e difus�o e / ou dispers�o num�rica associada.

Table 9-5  Kinematic Wave Routing:  Translation and Diffusion,
Example 9-4.

(1) (2) (3) (4) (5) (6)
Time
(h)
Inflow
(m3/s)
Partial Flows (m3/s) Outflow
(m3/s)
C0 I2 C1 I1 C2 I1
0 0 ___ ___ ___ 0
1 30 15 ___ 0 15.00
2 60 30 ___ 7.50 37.50
3 90 45 ___ 18.75 63.75
4 120 60 ___ 31.87 91.87
5 150 75 ___ 45.93 120.93
6 120 60 ___ 60.46 120.46
7 90 45 ___ 60.23 105.23
8 60 30 ___ 52.62 82.62
9 30 15 ___ 41.31 56.31
10 0 0 ___ 28.15 28.15
11 0 0 ___ 14.08 14.08
12 0 0 ___ 7.04 7.04
13 0 0 ___ 3.52 3.52
14 0 0 ___ 1.76 1.76
15 0 0 ___ 0.88 0.88


Convex Method. The convex method of stream channel routing belongs to the family of linear kinematic wave methods. Through the 1970s, it was part of the SCS TR-20 model for hydrologic simulation (Chapter 13). The routing equation for the convex method is obtained by discretizing Eq. 9-18 in a linear mode using a forward-in-time, backward-in-space finite difference scheme, to yield (Fig. 9-8):

M�todo convexo. O m�todo convexo de roteamento de canal de fluxo pertence � fam�lia de m�todos de ondas cinem�ticas lineares. Nos anos 70, fazia parte do modelo SCS TR-20 para simula��o hidrol�gica (cap�tulo 13). A equa��o de roteamento para o m�todo convexo � obtida discretizando a Eq. 9-18 em um modo linear, usando um esquema de diferen�as finitas para a frente no tempo e para tr�s no espa�o, para produzir (Fig. 9-8):

  Q j+1n+1 - Q j+1n                 Q j+1n - Q j n
__________________  +  βV  ________________  =  0
              Δt                                    Δx
(9-30)

from which

Q j+1n+1 = C1 Q j n + C2 Q j+1n (9-31)

in which

C1  =  C (9-32)

C2  =  1 - C (9-33)

and C = Courant number (Eq. 9-25), restricted to C ≤ 1 for numerical stability reasons. In the convex method, C is regarded as an empirical routing coefficient. Example 9-5 illustrates the application of the convex method.

e C = n�mero de Courant (Eq. 9-25), restrito a C ~ 1 por raz�es de estabilidade num�rica. No m�todo convexo, C � considerado como um coeficiente de roteamento emp�rico. O Exemplo 9-5 ilustra a aplica��o do m�todo convexo.

The convex method is relatively simple, but the solution is dependent on the routing parameter C. The latter could be interpreted as a Courant number and related to kinematic wave celerity and grid size, as in Eq. 9-25. However, for values of C other than 1, the amount of diffusion introduced in the numerical problem is unrelated to the true diffusion, if any, of the physical problem. Therefore, the convex method, as well as all kinematic wave methods featuring uncontrolled amounts of numerical diffusion, are regarded as a somewhat crude approach to stream channel routing.

O m�todo convexo � relativamente simples, mas a solu��o depende do par�metro de roteamento C. Este �ltimo pode ser interpretado como um n�mero de Courant e relacionado � celeridade das ondas cinem�ticas e ao tamanho da grade, como na Eq. 9-25. No entanto, para valores de C diferentes de 1, a quantidade de difus�o introduzida no problema num�rico n�o est� relacionada � difus�o real, se houver, do problema f�sico. Portanto, o m�todo convexo, bem como todos os m�todos de ondas cinem�ticas que apresentam quantidades descontroladas de difus�o num�rica, s�o considerados uma abordagem um tanto grosseira para o roteamento de canais de fluxo.

 Example 9-5.

Use Eq. 9-31 (the convex method) to route the same inflow hydrograph as in Example 9-3. Assume C = 2/3.

Use a Eq. 9-31 (o m�todo convexo) para direcionar o mesmo hidrograma de entrada que no Exemplo 9-3. Suponha que C = 2/3.


The routing coefficients are C1 = C = 2/3; and C2 = 1 - C = 1/3. The routing is shown in Table 9-6. The convex method leads to a significant amount of diffusion, with peak outflow of 135.06 m3/s as compared to peak inflow of 150 m3/s. The calculated diffusion amount is a function of C, with practical values of C being restricted in the range 0.5 to 0.9. For C = 1, the hydrograph is translated with no diffusion or dispersion, as in the first and third parts of Example 9-3. Values of C > 1 render the calculation unstable (large negative values of discharge) and are, therefore, not recommended. It should be noted that the instability of the convex method for C > 1 has a parallel in the instability of the Muskingum method for X > 0.5.

Os coeficientes de roteamento s�o C1 = C = 2/3; e C2 = 1 - C = 1/3. O roteamento � mostrado na Tabela 9-6. O m�todo convexo leva a uma quantidade significativa de difus�o, com vaz�o de pico de 135,06 m3 / s em compara��o com pico de entrada de 150 m3 / s. A quantidade de difus�o calculada � uma fun��o de C, com valores pr�ticos de C sendo restritos no intervalo de 0,5 a 0,9. Para C = 1, o hidrograma � traduzido sem difus�o ou dispers�o, como na primeira e terceira partes do Exemplo 9-3. Os valores de C> 1 tornam o c�lculo inst�vel (grandes valores negativos de descarga) e, portanto, n�o s�o recomendados. Deve-se notar que a instabilidade do m�todo convexo para C> 1 tem um paralelo na instabilidade do m�todo Muskingum para X> 0,5.

Table 9-6  Kinematic Wave Routing:  Convex Method, Example 9-5.
(1) (2) (3) (4) (5) (6)
Time
(h)
Inflow
(m3/s)
Partial Flows (m3/s) Outflow
(m3/s)
C0 I2 C1 I1 C2 I1
0 0 ___ ___ ___ 0
1 30 ___ 0 0 0.00
2 60 ___ 20 0 20.00
3 90 ___ 40 6.67 46.67
4 120 ___ 60 15.56 75.56
5 150 ___ 80 25.19 105.19
6 120 ___ 100 35.06 135.06
7 90 ___ 80 45.02 125.02
8 60 ___ 60 41.67 101.67
9 30 ___ 40 33.89 73.89
10 0 ___ 20 24.63 44.63
11 0 ___ 0 14.88 14.88
12 0 ___ 0 4.96 4.96
13 0 ___ 0 1.65 1.65
14 0 0 0.55 0.55


Kinematic Wave Celerity

Celeridade das Ondas Cinem�ticas

The kinematic wave celerity is dQ/dA, or βV. A value of β = 5/3 was derived for the case of a hydraulically wide channel governed by Manning friction. The kinematic wave celerity is also known as the Kleitz-Seddon, or Seddon, law [8, 19]. In 1900, Seddon [19] published a paper in which he studied the nature of unsteady flow movement in rivers and concluded that the celerity of long disturbances was equal to:

A celeridade das ondas cinem�ticas � dQ / dA ou ~V. Um valor de ~ = 5/3 foi derivado para o caso de um canal hidraulicamente amplo governado pelo atrito de Manning. A celeridade das ondas cinem�ticas tamb�m � conhecida como lei de Kleitz-Seddon, ou Seddon, [8, 19]. Em 1900, Seddon [19] publicou um artigo no qual estudou a natureza do movimento de fluxo inst�vel nos rios e concluiu que a celeridade de longas perturba��es era igual a:

        1     dQ
c = ____ _____
        T     dy
(9-34)

in which dQ/dy = slope of the discharge-stage rating (Q versus y), and T = stage, or water surface elevation. The quantity c is the kinematic wave celerity. Since dA = T dy, the kinematic wave celerity is equal to dQ/dA (see Eq. 9-17) [10].

em que dQ / dy = inclina��o da classifica��o do est�gio de descarga (Q versus y) e T = est�gio ou eleva��o da superf�cie da �gua. A quantidade c � a celeridade das ondas cinem�ticas. Como dA = T dy, a celeridade das ondas cinem�ticas � igual a dQ / dA (ver Eq. 9-17) [10].

From Eq. 9-34 it is concluded that the kinematic wave celerity is a function of the slope of the discharge-stage rating. This slope is likely to vary with stage; therefore, the kinematic wave celerity is not constant but varies with stage and flow level. If c = βV is a function of Q, then Eq. 9-18 is a nonlinear equation requiring an iterative solution. Nonlinear kinematic wave solutions account for the variation of kinematic wave celerity with stage and flow level. The simpler linear solutions, as in Examples 9-3 and 9-4, assume a constant value of kinematic wave celerity βV. Notice that there is a striking similarity between the linear kinematic wave solutions and the Muskingum method. This subject is further examined in Section 9.4.

Da Eq. 9-34, conclui-se que a celeridade das ondas cinem�ticas � uma fun��o da inclina��o da classifica��o do est�gio de descarga. � prov�vel que esta inclina��o varie com o est�gio; portanto, a celeridade das ondas cinem�ticas n�o � constante, mas varia com o est�gio e o n�vel do fluxo. Se c = ~V � uma fun��o de Q, ent�o a Eq. 9-18 � uma equa��o n�o linear que requer uma solu��o iterativa. As solu��es de ondas cinem�ticas n�o lineares s�o respons�veis %G​​%@pela varia��o da celeridade das ondas cinem�ticas com o est�gio e o n�vel do fluxo. As solu��es lineares mais simples, como nos Exemplos 9-3 e 9-4, assumem um valor constante da celeridade da onda cinem�tica ~V. Observe que h� uma semelhan�a impressionante entre as solu��es lineares de ondas cinem�ticas e o m�todo Muskingum. Este assunto � examinado em mais detalhes na Se��o 9.4.

Theoretical β values other than 5/3 can be obtained for other friction formulations and cross-sectional shapes. For turbulent flow governed by Manning friction, β has an upper limit of 5/3, and it is usually greater than 1. For laminar flow in wide channels, β = 3; for mixed or transitional flow-between laminar and turbulent Manning, it is in the range 5/3 < β < 3. For flow in a hydraulically wide channel described by the Chezy formula, β = 3/2 (Section 4.2). The calculation of β as a function of frictional type and cross-sectional shape is illustrated by the following example.

Valores ~ te�ricos diferentes de 5/3 podem ser obtidos para outras formula��es de fric��o e formas de se��o transversal. Para fluxo turbulento governado pelo atrito de Manning, ~ tem um limite superior de 5/3 e geralmente � maior que 1. Para fluxo laminar em canais largos, ~ = 3; para fluxo misto ou transit�rio entre Manning laminar e turbulento, est� na faixa de 5/3 <~ <3. Para fluxo em um canal hidraulicamente amplo descrito pela f�rmula de Chezy, ~ = 3/2 (Se��o 4.2). O c�lculo de ~ em fun��o do tipo de atrito e da forma da se��o transversal � ilustrado pelo exemplo a seguir.

 Example 9-6.

Calculate the value of β for a triangular channel with Manning friction.

Calcule o valor de ~ para um canal triangular com atrito de Manning.


Equation 9-10 is the Manning equation. Substituting R = A/P leads to Eq. 9-12. Since P is a function of A, Eq. 9-12 can be written as follows:

A equa��o 9-10 � a equa��o de Manning. Substituir R = A / P leva � Eq. 9-12. Como P � uma fun��o de A, Eq. 9-12 pode ser escrito da seguinte maneira:

               A 5/3
Q = K1  ________
               P 2/3
(9-35)

in which K1 is a constant containing n and Sf . The latter have been assumed to be independent of either A or P. For the triangular-shaped channel of Fig. 9-9, the top width is proportional to the flow depth, say T = Kd, in which T is the top width, d is the flow depth, and K is a proportionality constant.

em que K1 � uma constante contendo n e Sf. Presume-se que este �ltimo seja independente de A ou P. Para o canal de forma triangular da Fig. 9-9, a largura superior � proporcional � profundidade do fluxo, digamos T = Kd, na qual T � a largura superior, d � a profundidade do fluxo e K � uma constante de proporcionalidade.

Properties of a triangular channel cross section.

Figure 9-9  Properties of a triangular channel cross section.

The flow area is:

A �rea de fluxo �:

              d 2
A = K  ______
               2
(9-36)

and the wetted perimeter is:

e o per�metro �mido �:

                            K 2
P  =  2 d  ( 1  +  _____ ) 1/2
                             4
(9-37)

Eliminating d from Eqs. 9-36 and 9-37:

Eliminando d das Eqs. 9-36 e 9-37:

             2 (21/2) A1/2                 K 2
P  =  _________________ ( 1  +  _____ ) 1/2
                   K 1/2                       4
(9-38)

from which

P 2/3 = K2 A1/3
(9-39)

in which K2 is a constant containing K. Substituting Eq. 9-39 into Eq. 9-35 leads to:

em que K2 � uma constante contendo K. Substituindo a Eq. 9-39 na Eq. 9-35 leva a:

Q = K3 A 4/3 (9-40)

in which K3 is a constant containing K1 and K2. From Eq. 9-40:

em que K3 � uma constante contendo K1 e K2. Da Eq. 9-40:

  dQ                  Q
_____  =  (4/3) _____
  dA                  A
(9-41)

and the value of β for a triangular channel with Manning friction is β = 4/3.

e o valor de ~ para um canal triangular com atrito de Manning � ~ = 4/3.


Kinematic Waves with Lateral Inflow

Ondas Cinem�ticas com Aflu�ncia Lateral

Practical applications of stream channel routing often require the specification of lateral inflows. The latter could be either concentrated, as in the case of tributary inflow at a point along the channel reach, or distributed along the channel, as with groundwater exfiltration (for effluent streams) or infiltration (for influent streams). As with Eq. 9-9, a mass balance leads to:

Aplica��es pr�ticas de roteamento de canal de fluxo geralmente exigem a especifica��o de entradas laterais. Este �ltimo poderia estar concentrado, como no caso de afluentes tribut�rios em um ponto ao longo do alcance do canal, ou distribu�do ao longo do canal, como na exfiltra��o das �guas subterr�neas (para fluxos de efluentes) ou infiltra��o (para fluxos de influentes). Como na Eq. 9-9, um balan�o de massa leva a:

 ∂Q        ∂A
____  +  ____  =  qL
 ∂x          ∂t
(9-42)

which, unlike Eq. 9-9, includes the source term qL, the lateral flow per unit channel length. For Q given in cubic meters per second and x in meters, qL is given in cubic meters per second per meter [L2 T -1].

ao contr�rio da Eq. 9-9, inclui o termo fonte qL, o fluxo lateral por unidade de comprimento de canal. Para Q dado em metros c�bicos por segundo ex em metros, qL � dado em metros c�bicos por segundo por metro [L2 T -1].

Multiplying Eq. 9-42 by ∂Q / ∂A (or βV), as with Eq. 9-17 (or Eq. 9-18), leads to:

Multiplicando a Eq. 9-42 por ~Q / ~A (ou ~V), como na Eq. 9-17 (ou Eq. 9-18), leva a:

  ∂Q                    ∂Q
_____  +  (β V) ______  =  (β V) qL
  ∂t                      ∂x
(9-43)

which is the kinematic wave equation with lateral inflow (or outflow). For qL positive, there is lateral inflow (e.g., tributary flow); for qL negative, there is lateral outflow (e.g., channel transmission losses).

que � a equa��o da onda cinem�tica com entrada ou sa�da lateral. Para qL positivo, h� influxo lateral (por exemplo, fluxo tribut�rio); para qL negativo, h� vaz�o lateral (por exemplo, perdas de transmiss�o de canal).

Applicability of Kinematic Waves Aplicabilidade de ondas cinem�ticas

The kinematic wave celerity is a fundamental streamflow property. Flood waves which approximate kinematic waves travel with the kinematic wave celerity (c = βV) and are subject to very little or no attenuation.

A celeridade das ondas cinem�ticas � uma propriedade fundamental do fluxo. As ondas de inunda��o que se aproximam das ondas cinem�ticas viajam com a celeridade das ondas cinem�ticas (c = ~V) e est�o sujeitas a muito pouca ou nenhuma atenua��o.

In practice, flood waves are kinematic if they are of long duration (Fig. 9-10) or travel on a channel of steep slope. Criteria for the applicability of kinematic waves to overland flow [20] (Section 4.2) and stream channel flow [14] have been developed. The stream channel criterion states that in order for a wave to be kinematic, it should satisfy the following dimensionless inequality:

Na pr�tica, as ondas de inunda��o s�o cinem�ticas se forem de longa dura��o (Fig. 9-10) ou se deslocarem em um canal de declive acentuado. Crit�rios para a aplicabilidade das ondas cinem�ticas ao fluxo terrestre [20] (Se��o 4.2) e fluxo do canal de corrente [14] foram desenvolvidos. O crit�rio do canal de fluxo afirma que, para que uma onda seja cinem�tica, ela deve satisfazer a seguinte desigualdade sem dimens�o:

  tr So Vo
__________  ≥  M
       do
(9-44)

in which tr is the time-of-rise of the inflow hydrograph, So is the bottom slope, Vo is the average velocity, and do is the average flow depth. For 95% accuracy in one period of translation, a value of M = 85 is indicated [14].

em que tr � o tempo de subida do hidrograma de entrada, tamb�m � a inclina��o inferior, Vo � a velocidade m�dia e faz � a profundidade m�dia do fluxo. Para uma precis�o de 95% em um per�odo de tradu��o, � indicado um valor de M = 85 [14].

Properties of triangular channel cross section.

Figure 9-10  Flood stage in a large tropical river.

 Example 9-7.

Use the kinematic wave criterion (Eq. 9-44) to determine whether a flood wave with the following characteristics is a kinematic wave: time-of-rise tr = 12 h; bottom slope So = 0.001; average velocity Vo = 2 m/s; and average flow depth do = 2 m.

Use o crit�rio de ondas cinem�ticas (Eq. 9-44) para determinar se uma onda de inunda��o com as seguintes caracter�sticas � uma onda cinem�tica: tempo de subida tr = 12 h; inclina��o inferior So = 0,001; velocidade m�dia Vo = 2 m / s; e profundidade m�dia do fluxo = 2 m.


For the given channel and flow characteristics, the left side of Eq. 9-44 is equal to 43.2, which is less than 85. For values greater than 85, the wave would be kinematic-therefore, subject to negligible diffusion. Since the value is 43.2, this wave is not kinematic and is likely to experience a significant amount of diffusion. If this wave is routed as a kinematic wave with zero diffusion and dispersion, as in Example 9-3 (Part 1), the peak outflow would be much larger than in reality. If this wave is routed as a kinematic wave with diffusion or dispersion, as in Examples 9-3 (Part 2) and 9-4, it is likely that the amount of numerical diffusion and/or dispersion would be different from the actual amount of physical diffusion. It should be noted that had the bottom slope been So = 0.01, the left side of Eq. 9-44 would be 432, satisfying the kinematic wave criterion. Therefore, it is concluded that the steeper the channel slope, the more kinematic the flow is.

Para as caracter�sticas de canal e fluxo fornecidas, o lado esquerdo da Eq. 9-44 � igual a 43,2, que � menor que 85. Para valores maiores que 85, a onda seria cinem�tica, portanto, sujeita a difus�o desprez�vel. Como o valor � 43,2, essa onda n�o � cinem�tica e provavelmente experimentar� uma quantidade significativa de difus�o. Se essa onda for roteada como uma onda cinem�tica com difus�o e dispers�o zero, como no Exemplo 9-3 (Parte 1), o pico de vaz�o seria muito maior do que na realidade. Se essa onda for roteada como uma onda cinem�tica com difus�o ou dispers�o, como nos Exemplos 9-3 (Parte 2) e 9-4, � prov�vel que a quantidade de difus�o e / ou dispers�o num�rica seja diferente da quantidade real de difus�o f�sica. Deve-se notar que, se a inclina��o inferior fosse So = 0,01, o lado esquerdo da Eq. 9-44 seria 432, satisfazendo o crit�rio de ondas cinem�ticas. Portanto, conclui-se que quanto mais �ngreme a inclina��o do canal, mais cinem�tico � o fluxo.

calculator image 

ONLINE CALCULATION. Using ONLINE KINEMATIC WAVE APPLICABILITY, the answer is M = 43.2. The wave is not kinematic.


9.3  ONDAS DIFUSIVAS

[M�todo Muskingum-Cunge]   [Ondas Din�micas]   [Quest�es]   [Problemas]   [Refer�ncias]      [Top]   [M�todo Muskingum]   [Ondas Cinem�ticas]  

In Section 9.1, the Muskingum method was used to calculate unsteady flows in a hydrologic sense. In Section 9.2, the principle of mass conservation was coupled with a uniform flow formula to derive the kinematic wave equation. Solutions to this equation have been widely used in hydrologic practice, particularly for overland flow and other routing applications involving steep slopes or slow-rising hydrographs.

Na Se��o 9.1, o m�todo Muskingum foi usado para calcular fluxos inst�veis %G​​%@no sentido hidrol�gico. Na Se��o 9.2, o princ�pio de conserva��o de massa foi acoplado a uma f�rmula de fluxo uniforme para derivar a equa��o da onda cinem�tica. As solu��es para essa equa��o t�m sido amplamente utilizadas na pr�tica hidrol�gica, particularmente para fluxo terrestre e outras aplica��es de roteamento que envolvem declives acentuados ou hidrogramas de crescimento lento.

The Muskingum method and linear kinematic wave solutions show striking similarities. Both methods have the same type of routing equation. The Muskingum method, however, can calculate hydrograph diffusion, whereas the kinematic wave can do so only by the introduction of numerical diffusion. The latter is dependent on the grid size and type of numerical scheme.

O m�todo Muskingum e as solu��es lineares de ondas cinem�ticas mostram semelhan�as impressionantes. Ambos os m�todos t�m o mesmo tipo de equa��o de roteamento. O m�todo Muskingum, no entanto, pode calcular a difus�o hidrogr�fica, enquanto a onda cinem�tica pode faz�-lo apenas com a introdu��o da difus�o num�rica. Este �ltimo depende do tamanho da grade e do tipo de esquema num�rico.

Kinematic wave theory can be enhanced by allowing a small amount of physical diffusion in its formulation [10]. In this way, an improved type of kinematic wave can be formulated, a kinematic-with-diffusion wave, for short, a diffusion wave. A definite advantage of the diffusion wave is that it includes the diffusion which is present in most natural unsteady open channel flows.

A teoria das ondas cinem�ticas pode ser aprimorada permitindo uma pequena quantidade de difus�o f�sica em sua formula��o [10]. Deste modo, pode ser formulado um tipo melhorado de onda cinem�tica, uma onda cinem�tica com difus�o, para abreviar, uma onda de difus�o. Uma vantagem definitiva da onda de difus�o � que ela inclui a difus�o que est� presente na maioria dos fluxos de canal aberto inst�veis e naturais.

Diffusion Wave Equation

Equa��o da onda de difus�o

In Section 9.2, the kinematic wave equation was derived by using a statement of steady uniform flow (i.e., friction slope is equal to bottom slope) in lieu of momentum conservation. In deriving the diffusion wave, a statement of steady nonuniform flow (i.e., friction slope is equal to water surface slope) is used instead (Fig. 9-11). This leads to:

Na Se��o 9.2, a equa��o da onda cinem�tica foi derivada usando uma declara��o de fluxo uniforme constante (isto �, a inclina��o da fric��o � igual � inclina��o inferior) em vez da conserva��o do momento. Na deriva��o da onda de difus�o, uma declara��o de fluxo n�o uniforme constante (isto �, a inclina��o da fric��o � igual � inclina��o da superf�cie da �gua) � usada (Fig. 9-11). Isto leva a:

        1                            dy
Q = ___ A R 2/3 ( So  -  ____ ) 1/2
        n                            dx
(9-45)

in which the term [So - (dy/dx)] is the water surface slope. The difference between kinematic and diffusion waves is in the term dy/dx. From a physical standpoint, the term dy/dx accounts for the natural diffusion processes present in unsteady open channel flow phenomena.

em que o termo [So - (dy / dx)] � a inclina��o da superf�cie da �gua. A diferen�a entre ondas cinem�ticas e de difus�o est� no termo dy / dx. Do ponto de vista f�sico, o termo dy / dx � respons�vel pelos processos naturais de difus�o presentes em fen�menos de fluxo de canal aberto inst�veis.

Diffusion wave assumption

Figure 9-11  Diffusion wave assumption.

To derive the diffusion wave equation, Eq. 9-45 is expressed in a slightly different form:

Para derivar a equa��o da onda de difus�o, Eq. 9-45 � expresso de uma forma ligeiramente diferente:

                          dy
m Q 2  =  So  -  ____
                          dx
(9-46)

in which m is the reciprocal of the square of the channel conveyance K, defined as:

em que m � o inverso do quadrado do transporte de canal K, definido como:

        1                       
K = ___ A R 2/3
        n                        
(9-47)

With dA = T dy, in which T = top width, Eq. 9-46 changes to:

Com dA = T dy, em que T = largura superior, Eq. 9-46 altera��es para:

   1       dA
_____ ______  +  m Q 2  -  So  =  0
   T       dx
(9-48)

Equations 9-9 and 9-48 constitute a set of two partial differential equations describing diffusion waves. These equations can be combined into one equation with Q as dependent variable. However, it is first necessary to linearize the equations around reference flow values. For simplicity, a constant top width is assumed (i.e., a wide channel assumption).

As equa��es 9-9 e 9-48 constituem um conjunto de duas equa��es diferenciais parciais que descrevem ondas de difus�o. Essas equa��es podem ser combinadas em uma equa��o com Q como vari�vel dependente. No entanto, � necess�rio primeiro linearizar as equa��es em torno dos valores de fluxo de refer�ncia. Por uma quest�o de simplicidade, � assumida uma largura superior constante (isto �, uma suposi��o de canal amplo).

The linearization of Eqs. 9-9 and 9-48 is accomplished by small perturbation theory [4]. This procedure, while heuristic, has seemed to work well in a number of applications. The variables Q, A, and m can be expressed in terms of the sum of a reference value (with subscript o) and a small perturbation to the reference value (with superscript '): Q = Qo + Q' ; A = Ao + A' ; m = mo + m'. Substituting these into Eqs. 9-9 and 9-48, neglecting squared perturbations, and subtracting the reference flow leads to:

A lineariza��o das Eqs. 9-9 e 9-48 � realizado pela teoria das pequenas perturba��es [4]. Esse procedimento, embora heur�stico, pareceu funcionar bem em v�rias aplica��es. As vari�veis Q, A e m podem ser expressas em termos da soma de um valor de refer�ncia (com o subscrito o) e de uma pequena perturba��o no valor de refer�ncia (com sobrescrito '): Q = Qo + Q'; A = Ao + A '; m = mo + m '. Substituindo-os nas Eqs. 9-9 e 9-48, negligenciar perturba��es ao quadrado e subtrair o fluxo de refer�ncia leva a:

 ∂Q'        ∂A'
____  +  ____  =  0
 ∂x          ∂t
(9-49)

and

   1       ∂A'
_____ ______  +  Qo2 m'  +  2 mo Qo Q'  =  0
   T       ∂x
(9-50)

Differentiating Eq. 9-49 with respect to x and Eq. 9-50 with respect to t gives:

Diferenciando a Eq. 9-49 em rela��o a x e Eq. 9-50 em rela��o a t d�:

  ∂2Q'         ∂2A'
______  +  _______  =  0
  ∂x2          ∂xt
(9-51)

   1         ∂2A'                  ∂m'                        ∂Q'
_____ _________  +  Qo2 _____  +  2 mo Qo ______  =  0
   T       ∂xt                    ∂t                          ∂t
(9-52)

Using the chain rule and Eq. 9-49 yields:

Usando a regra da cadeia e a Eq. 9-49 rendimentos:

  ∂m'          ∂m'       ∂A'                 ∂m'       ∂Q'
_____  =  _______ _______  =    -  _______ _______
   ∂t            ∂A'        ∂t                   ∂A'        ∂x
(9-53)

Combining Eq. 9-52 with Eq. 9-53:

Combinando a Eq. 9-52 com a Eq. 9-53:

   1        ∂2A'                  ∂m'     ∂Q'                         ∂Q'
_____ ________  -  Qo2 ______ ______  +  2 mo Qo ______  =  0
   T       ∂xt                  ∂A'      ∂x                           ∂t
(9-54)

Combining Eqs. 9-51 and 9-54 and rearranging terms, yields:

Combinando Eqs. 9-51 e 9-54 e termos de reorganiza��o, produz:

  ∂Q'           Qo       ∂m'     ∂Q'                  1             ∂Q'2
______  -  _______ ______ ______  =  _____________ _______
   ∂t           2 mo      ∂A'      ∂x            2 T mo Qo      ∂x2
(9-55)

Since by definition: mQ 2 = Sf, it follows that

Como, por defini��o: mQ 2 = Sf, segue-se que

  ∂Q'          ∂Q                 Qo
_____  =  _______  =  -  _______
  ∂m'          ∂m               2 mo
(9-56)

and also

                   So
mo Qo  =  ______
                   Qo
(9-57)

Substituting Eqs. 9-56 and 9-57 into Eq. 9-55, using the chain rule, and dropping the superscripts for simplicity, the following equation is obtained:

Substituindo Eqs. 9-56 e 9-57 na Eq. 9-55, usando a regra da cadeia e eliminando os sobrescritos para simplificar, � obtida a seguinte equa��o:

  ∂Q              ∂Q        ∂Q                 Qo            ∂2Q
______  +  ( ______ ) ______  =  ( __________ ) _______
   ∂t               ∂A         ∂x               2 T So         ∂x2
(9-58)

The left side of Eq. 9-58 is recognized as the kinematic wave equation, with ∂Q/∂A as the kinematic wave celerity. The right side is a second-order (partial differential) term that accounts for the physical diffusion effect. The coefficient of the second-order term has the units of diffusivity [L2T -1], being referred to as the hydraulic diffusivity, or channel diffusivity.

O lado esquerdo da Eq. 9-58 � reconhecida como a equa��o da onda cinem�tica, com ~Q / ~A como a celeridade da onda cinem�tica. O lado direito � um termo de segunda ordem (diferencial parcial) que explica o efeito da difus�o f�sica. O coeficiente do termo de segunda ordem possui as unidades de difusividade [L2T -1], sendo denominadas difusividade hidr�ulica ou difusividade do canal.

The hydraulic diffusivity is a characteristic of the flow and channel, defined as:

A difusividade hidr�ulica � uma caracter�stica do fluxo e canal, definida como:

               Qo               qo
νh  =  _________  =  _______
            2 T So          2 So
(9-59)

in which qo = Qo/T is the reference flow per unit of channel width. From Eq. 9-59, it is concluded that the hydraulic diffusivity is small for steep bottom slopes (e.g., those of small mountain streams), and large for mild bottom slopes (e.g., tidal rivers).

em que qo = Qo / T � o fluxo de refer�ncia por unidade de largura do canal. Da Eq. 9-59, conclui-se que a difusividade hidr�ulica � pequena para encostas �ngremes do fundo (por exemplo, aquelas de pequenos riachos de montanhas) e grande para encostas suaves do fundo (por exemplo, rios das mar�s).

Equation 9-58 describes the movement of flood waves in a better way than Eq. 9-17 or 9-18. It falls short from describing the full momentum effects, but it does physically account for peak flow attenuation.

A Equa��o 9-58 descreve o movimento das ondas de inunda��o de uma maneira melhor que a Eq. 9-17 ou 9-18. � insuficiente para descrever os efeitos do momento, mas explica fisicamente a atenua��o do pico de fluxo.

Equation 9-58 is a second-order parabolic partial differential equation. It can be solved analytically, leading to Hayami's diffusion analogy solution for flood waves [7], or numerically with the aid of a numerical scheme for parabolic equations such as the Crank-Nicolson scheme [3]. An alternate approach is to match the hydraulic diffusivity with the numerical diffusion coefficient of the Muskingum scheme. This approach is the basis of the Muskingum-Cunge method [4, 12] (Section 9.4).

A equa��o 9-58 � uma equa��o diferencial parcial parab�lica de segunda ordem. Pode ser resolvido analiticamente, levando � solu��o de analogia de difus�o de Hayami para ondas de inunda��o [7], ou numericamente com a ajuda de um esquema num�rico para equa��es parab�licas, como o esquema de Crank-Nicolson [3]. Uma abordagem alternativa � combinar a difusividade hidr�ulica com o coeficiente de difus�o num�rico do esquema de Muskingum. Essa abordagem � a base do m�todo Muskingum-Cunge [4, 12] (Se��o 9.4).

Applicability of Diffusion Waves

Aplicabilidade de ondas de difus�o

Most flood waves have a small amount of physical diffusion; therefore, they are better approximated by the diffusion wave rather than by the kinematic wave. For this reason, diffusion waves apply to a much wider range of practical problems than kinematic waves. Where the diffusion wave fails, only the dynamic wave can properly describe the translation and diffusion of flood waves. The dynamic wave, however, is very strongly diffusive, especially for flows well in the subcritical regime [14]. In practice, most flood flows are only mildly diffusive, and therefore, are subject to modeling with the diffusion wave.

A maioria das ondas de inunda��o tem uma pequena quantidade de difus�o f�sica; portanto, eles s�o melhor aproximados pela onda de difus�o do que pela onda cinem�tica. Por esse motivo, as ondas de difus�o se aplicam a uma gama muito maior de problemas pr�ticos do que as ondas cinem�ticas. Onde a onda de difus�o falha, apenas a onda din�mica pode descrever adequadamente a tradu��o e difus�o das ondas de inunda��o. A onda din�mica, no entanto, � muito fortemente difusiva, especialmente para fluxos bem no regime subcr�tico [14]. Na pr�tica, a maioria dos fluxos de inunda��o � levemente difusiva e, portanto, est� sujeita a modelagem com a onda de difus�o.

To determine if a wave is a diffusion wave, it should satisfy the following dimensionless inequality [14]:

Para determinar se uma onda � uma onda de difus�o, ela deve satisfazer a seguinte desigualdade sem dimens�o [14]:

             g
tr So ( ____ )1/2  ≥  N
            do
(9-60)

in which tr is the time-of-rise of the inflow hydrograph, So is the bottom slope, do is the average flow depth, and g is the gravitational acceleration. The greater the left side of this inequality, the more likely it is that the wave is a diffusion wave. In practice, a value of N = 15 is recommended for general use.

em que tr � o tempo de ascens�o do hidrograma de entrada, o mesmo ocorre com a inclina��o inferior, o que � a profundidade m�dia do fluxo eg � a acelera��o gravitacional. Quanto maior o lado esquerdo dessa desigualdade, maior a probabilidade de a onda ser uma onda de difus�o. Na pr�tica, um valor de N = 15 � recomendado para uso geral.

 Example 9-8.

Use the criterion of Eq. 9-60 to determine whether the flood wave of Example 9-7 can be considered a diffusion wave.

Use o crit�rio da Eq. 9-60 para determinar se a onda de inunda��o do Exemplo 9-7 pode ser considerada uma onda de difus�o.


For tr = 12 h, So = 0.001, and do = 2 m, the left side of Eq. 9-60 is 95.7, which is greater than 15. In the previous example, this wave was shown not to satisfy the kinematic wave criterion. This example shows, however, that this wave is a diffusion wave. Had Eq. 9-60 not been satisfied, the flood wave would have been properly a dynamic wave, subject only to dynamic wave routing. Dynamic wave routing takes into account the complete momentum quation, including the inertia terms (local and convective) that were neglected in the formulation of kinematic and diffusion waves. Section 9.5 contains a brief introduction to dynamic waves.

Para tr = 12 h, So = 0,001 e do = 2 m, no lado esquerdo da Eq. 9-60 � 95,7, que � maior que 15. No exemplo anterior, essa onda mostrou n�o satisfazer o crit�rio de onda cinem�tica. Este exemplo mostra, no entanto, que esta onda � uma onda de difus�o. Tinha Eq. 9-60 n�o foi satisfeita, a onda de inunda��o teria sido adequadamente uma onda din�mica, sujeita apenas ao roteamento din�mico de ondas. O roteamento din�mico de ondas leva em considera��o a completa din�mica do momento, incluindo os termos de in�rcia (local e convectivo) que foram negligenciados na formula��o de ondas cinem�ticas e de difus�o. A Se��o 9.5 cont�m uma breve introdu��o �s ondas din�micas.

calculator image 

ONLINE CALCULATION. Using ONLINE KINEMATIC WAVE APPLICABILITY, the answer is N = 95.657. The wave is a diffusion wave.


9.4  M�TODO MUSKINGUM-CUNGE

[Ondas Din�micas]   [Quest�es]   [Problemas]   [Refer�ncias]      [Top]   [M�todo Muskingum]   [Ondas Cinem�ticas]   [Ondas Difusivas]  

The Muskingum method can calculate runoff diffusion, ostensibly by varying the parameter X. A numerical solution of the linear kinematic wave equation using a third-order accurate scheme (C = 1) leads to pure flood hydrograph translation (see Example 9-3, Part 1). Other numerical solutions to the linear kinematic wave equation invariably produce a certain amount of numerical diffusion and/or dispersion (See Example 9-3, Part 2). The Muskingum and linear kinematic wave routing equations are strikingly similar. Furthermore, unlike the kinematic wave equation, the diffusion wave equation does have the capability to describe physical diffusion.

O m�todo Muskingum pode calcular a difus�o do escoamento, ostensivamente variando o par�metro X. Uma solu��o num�rica da equa��o linear de ondas cinem�ticas usando um esquema preciso de terceira ordem (C = 1) leva � tradu��o pura do hidrograma de inunda��o (consulte o Exemplo 9-3, Parte 9). 1) Outras solu��es num�ricas para a equa��o linear das ondas cinem�ticas produzem invariavelmente uma certa quantidade de difus�o e / ou dispers�o num�rica (Veja o Exemplo 9-3, Parte 2). As equa��es de roteamento de ondas cinem�ticas e de Muskingum s�o surpreendentemente semelhantes. Al�m disso, diferentemente da equa��o da onda cinem�tica, a equa��o da onda de difus�o tem a capacidade de descrever a difus�o f�sica.

From these propositions, Cunge [4] concluded that the Muskingum method is a linear kinematic wave solution and that the flood wave attenuation shown by the calculation is due to the numerical diffusion of the scheme itself. To prove this assertion, the kinematic wave equation (Eq. 9-18) is discretized on the x-t plane (Fig. 9-12) in a way that parallels the Muskingum method, centering the spatial derivative and off-centering the temporal derivative by means of a weighting factor X:

A partir dessas proposi��es, Cunge [4] concluiu que o m�todo Muskingum � uma solu��o linear de ondas cinem�ticas e que a atenua��o das ondas de inunda��o mostrada pelo c�lculo se deve � difus�o num�rica do pr�prio esquema. Para provar essa afirma��o, a equa��o da onda cinem�tica (Eq. 9-18) � discretizada no plano xt (Fig. 9-12) de uma maneira que se assemelha ao m�todo de Muskingum, centralizando a derivada espacial e descentralizando a derivada temporal por meios de um fator de pondera��o X:

  X (Q j n+1 - Q j n )  +  (1 - X) (Q j+1n+1 - Q j+1 n )                 
________________________________________________  + 
                                        Δt                             

      (Q j+1 n - Q j n )  +  (Q j+1n+1 - Q j n+1 )                 
c  _______________________________________  =  0
                                   2 Δx                             
(9-61)

in which c = βV  is the kinematic wave celerity.

em que c = ~V � a celeridade da onda cinem�tica.

Space-time discretization of kinematic wave equation paralleling Muskingum method

Figure 9-12  Space-time discretization of kinematic wave equation paralleling Muskingum method.

Solving Eq. 9-61 for the unknown discharge leads to the following routing equation:

Resolvendo a Eq. 9-61 para a descarga desconhecida leva � seguinte equa��o de roteamento:

Q j+1 n+1 = C0 Q j n+1 + C1 Q j n + C2 Q j+1 n (9-62)

The routing coefficients are:

Os coeficientes de roteamento s�o:

                c ( Δt / Δx )  -  2X
C0  =  ________________________
            2(1 - X)  +  c ( Δt / Δx )
(9-63)

                c ( Δt / Δx )  +  2X
C1  =  ________________________
            2(1 - X)  +  c ( Δt / Δx )
(9-64)

            2(1 - X)  -  c ( Δt / Δx )
C2  =  ________________________
            2(1 - X)  +  c ( Δt / Δx )
(9-65)

By defining the travel time

           Δx
K  =  ______
            c
(9-66)

it is seen that the two sets of Eqs. 9-63 to 9-65 and Eqs. 9-4 to 9-6 are the same.

� visto que os dois conjuntos de Eqs. 9-63 a 9-65 e Eqs. 9-4 a 9-6 s�o os mesmos.

Equation 9-66 confirms that K is in fact the flood wave travel time, i.e., the time it takes a given discharge to travel the reach length Δx with the kinematic wave celerity c. In a linear mode, c is constant and equal to a reference value; in a nonlinear mode, it varies with discharge.

A Equa��o 9-66 confirma que K � de fato o tempo de viagem das ondas de inunda��o, isto �, o tempo que uma determinada descarga leva para percorrer o comprimento de alcance ~x com a celeridade da onda cinem�tica c. Em um modo linear, c � constante e igual a um valor de refer�ncia; no modo n�o linear, varia com a descarga.

It can be seen that for X = 0.5, Eqs. 9-63 to 9-65 reduce to the routing coefficients of the linear second-order-accurate kinematic wave solution, Eqs. 9-22 to 9-24. For X = 0.5 and C = 1 (C = cΔtx = βV Δtx, the Courant number, Eq. 9-25) the routing equation is third-order accurate, i.e., the numerical solution is equal to the analytical solution of the kinematic wave equation. For X = 0.5 and C ≠ 1, it is second-order accurate, exhibiting only numerical dispersion (recall that in the context of convection-diffusion, numerical dispersion is the error of the second-order scheme). For X< 0.5 and C ≠ 1, it is first-order accurate, exhibiting both numerical diffusion and dispersion. For X < 0.5 and C = 1, it is first: order accurate, exhibiting only numerical diffusion (effectively, the numerical dispersion vanishes for C = 1). These relations are summarized in Table 9-7.

Pode ser visto que para X = 0,5, Eqs. 9-63 a 9-65 reduzem os coeficientes de roteamento da solu��o linear de ondas cinem�ticas precisas de segunda ordem, Eqs. 9-22 a 9-24. Para X = 0,5 e C = 1 (C = c~t / ~x = ~V ~t / ~x, o n�mero de Courant, Eq. 9-25), a equa��o de roteamento � de terceira ordem precisa, ou seja, a solu��o num�rica � igual � solu��o anal�tica da equa��o da onda cinem�tica. Para X = 0,5 e C ~ 1, � preciso de segunda ordem, exibindo apenas dispers�o num�rica (lembre-se de que no contexto de difus�o por convec��o, dispers�o num�rica � o erro do esquema de segunda ordem). Para X <0,5 e C ~ 1, � preciso de primeira ordem, exibindo difus�o e dispers�o num�rica. Para X <0,5 e C = 1, � o primeiro: ordem precisa, exibindo apenas difus�o num�rica (efetivamente, a dispers�o num�rica desaparece para C = 1). Essas rela��es est�o resumidas na Tabela 9-7.

Table 9-7  Numerical Properties of Muskingum-Cunge Method.
Parameter
X
Parameter
C
Order of
Accuracy
Numerical
Diffusion
Numerical
Dispersion
0.5 1 Third No No
0.5 ≠ 1 Second No Yes
< 0.5 ≠ 1 First Yes Yes
< 0.5 1 First Yes No

In practice, the numerical diffusion can be used to simulate the physical diffusion of the actual flood wave. By expanding the discrete function Q (jΔx, n Δt) in Taylor series about grid point (jΔx, n Δt), the numerical diffusion coefficient of the Muskingum scheme is derived (see Appendix B):

Na pr�tica, a difus�o num�rica pode ser usada para simular a difus�o f�sica da onda de inunda��o real. Expandindo a fun��o discreta Q (j~x, n ~t) na s�rie de Taylor sobre o ponto de grade (j~x, n ~t), � obtido o coeficiente de difus�o num�rico do esquema de Muskingum (consulte o Ap�ndice B):

                      1
νn  =  c Δx ( ____  -  X )
                      2
(9-67)

in which νn is the numerical diffusion coefficient of the Muskingum scheme. This equation reveals the following:

em que ~n � o coeficiente de difus�o num�rico do esquema de Muskingum. Esta equa��o revela o seguinte:

  • For X = 0.5 there is no numerical diffusion, although there is numerical dispersion for C ≠ 1;

    Para X = 0,5, n�o h� difus�o num�rica, embora haja dispers�o num�rica para C ~ 1;

  • For X > 0.5, the numerical diffusion coefficient is negative, i.e., numerical amplification, which explains the behavior of the Muskingum method for this range of X values;

    Para X> 0,5, o coeficiente de difus�o num�rico � negativo, isto �, amplifica��o num�rica, o que explica o comportamento do m�todo Muskingum para esta faixa de valores X;

  • For Δx = 0, the numerical diffusion coefficient is zero, clearly the trivial case.

    Para ~x = 0, o coeficiente de difus�o num�rico � zero, claramente o caso trivial.

A predictive equation for X can be obtained by matching the hydraulic diffusivity νh (Eq. 9-59) with the numerical diffusion coefficient of the Muskingum scheme νn (Eq. 9-67). This leads to the following expression for X:

Uma equa��o preditiva para X pode ser obtida combinando a difusividade hidr�ulica ~h (Eq. 9-59) com o coeficiente de difus�o num�rico do esquema de Muskingum ~n (Eq. 9-67). Isso leva � seguinte express�o para X:

          1                   qo   
X  =  ___ ( 1  -  __________ )
          2              So c Δx
(9-68)

With X calculated by Eq. 9-68, the Muskingum method is referred to as Muskingum-Cunge method [12]. Using Eq. 9-68, the routing parameter X can be calculated as a function of the following numerical and physical properties: ]

Com X calculado pela Eq. 9-68, o m�todo Muskingum � referido como m�todo Muskingum-Cunge [12]. Usando a Eq. 9-68, o par�metro de roteamento X pode ser calculado em fun��o das seguintes propriedades num�ricas e f�sicas:

  1. Reach length Δx,

    Comprimento de alcance ~x,

  2. Reference discharge per unit width qo,

    Descarga de refer�ncia por unidade de largura qo,

  3. Kinematic wave celerity c, and

    Celeridade das ondas cinem�ticas c, e

  4. Bottom slope So.

    Inclina��o inferior

It should be noted that Eq. 9-68 was derived by matching physical and numerical diffusion (i.e., second-order processes), and does not account for dispersion (a third-order process). Therefore, in order to simulate wave diffusion properly with the Muskingum-Cunge method, it is necessary to optimize numerical diffusion (with Eq. 9-68) while minimizing numerical dispersion by keeping the value of C as close to 1 as practicable.

Note-se que a Eq. 9-68 foi obtido por correspond�ncia de difus�o f�sica e num�rica (isto �, processos de segunda ordem) e n�o � respons�vel pela dispers�o (um processo de terceira ordem). Portanto, para simular a difus�o de ondas corretamente com o m�todo Muskingum-Cunge, � necess�rio otimizar a difus�o num�rica (com as Eq. 9-68) enquanto minimiza a dispers�o num�rica, mantendo o valor de C o mais pr�ximo poss�vel de 1.

A unique feature of the Muskingum-Cunge method is the grid independence of the calculated outflow hydrograph, which sets it apart from other linear kinematic wave solutions featuring uncontrolled numerical diffusion and dispersion (e.g., the convex method). If numerical dispersion is minimized, the calculated outflow at the downstream end of a channel reach will be essentially the same, regardless of how many subreaches are used in the computation. This is because X is a function of Δx, and the routing coefficients C0, C1, and C2 vary with reach length.

Uma caracter�stica exclusiva do m�todo Muskingum-Cunge � a independ�ncia da grade do hidrograma de vaz�o calculado, que o diferencia de outras solu��es de ondas cinem�ticas lineares que apresentam difus�o e dispers�o num�rica n�o controlada (por exemplo, o m�todo convexo). Se a dispers�o num�rica for minimizada, a vaz�o calculada na extremidade a jusante de um alcance de canal ser� essencialmente a mesma, independentemente de quantas subcategorias forem usadas na computa��o. Isso ocorre porque X � uma fun��o de ~x e os coeficientes de roteamento C0, C1 e C2 variam com o comprimento do alcance.

An improved version of the Muskingum-Cunge method is due to Ponce and Yevjevich [15]. The C value is the Courant number, i.e. , the ratio of wave celerity c to grid celerity Δx/ Δt:

Uma vers�o melhorada do m�todo Muskingum-Cunge � devida a Ponce e Yevjevich [15]. O valor C � o n�mero de Courant, ou seja, a raz�o entre a celeridade da onda c e a celeridade da grade ~x / ~t:

              Δt
C  =  c ______
              Δx
(9-69)

The grid diffusivity is defined as the numerical diffusivity for the case of X = 0. From Eq. 9-67, the grid diffusivity is:

               Δx
νg  =  c  _____
                2
(9-70)

The cell Reynolds number [18] is defined as the ratio of hydraulic diffusivity (Eq. 9-59) to grid diffusivity (Eq. 9-70). This leads to:

O n�mero de Reynolds da c�lula [18] � definido como a raz�o entre a difusividade hidr�ulica (Eq. 9-59) e a difusividade da grade (Eq. 9-70). Isto leva a:

               qo   
D  =  __________
           So c Δx
(9-71)

in which D = cell Reynolds number. Therefore:

em que D = n�mero de Reynolds da c�lula. Portanto:
          1                  
X  =  ___ ( 1  -  D )
          2             
(9-72)

Equations 9-71 and 9-72 imply that for very small values of Δx, D may be greater than 1, leading to negative values of X. In fact, for the characteristic reach length

As equa��es 9-71 e 9-72 implicam que, para valores muito pequenos de ~x, D pode ser maior que 1, levando a valores negativos de X. De fato, para o comprimento de alcance caracter�stico

                qo   
Δxc  =  ________
               So c
(9-73)

the cell Reynolds number is D = 1, and X = 0. Therefore, in the Muskingum-Cunge method, reach lengths shorter than the characteristic reach length result in negative values of X. This should be contrasted with the classical Muskingum method (Section 9.1), in which X is restricted in the range 0.0 ≤ X ≤ 0.5. In the classical Muskingum, X is interpreted as a weighting factor. As shown by Eqs. 9-71 and 9-72, nonnegative values of X are associated with long reaches, typical of the manual computation used in the development and early application of the Muskingum method.

o n�mero de Reynolds da c�lula � D = 1 e X = 0. Portanto, no m�todo Muskingum-Cunge, comprimentos menores que o comprimento caracter�stico resultam em valores negativos de X. Isso deve ser contrastado com o m�todo cl�ssico de Muskingum (Se��o 9.1 ), em que X � restrito no intervalo 0,0 ~ X ~ 0,5. No Muskingum cl�ssico, X � interpretado como um fator de pondera��o. Como mostrado pelas Eqs. 9-71 e 9-72, valores n�o negativos de X est�o associados a alcances longos, t�picos da computa��o manual usada no desenvolvimento e na aplica��o precoce do m�todo Muskingum.

In the Muskingum-Cunge method, however, X is interpreted in a moment-matching sense [2] or diffusion-matching factor. Therefore, negative values of X are entirely possible. This feature allows the use of shorter reaches than would otherwise be possible if X were restricted to nonnegative values.

No m�todo Muskingum-Cunge, no entanto, X � interpretado em um sentido de correspond�ncia de momentos [2] ou em um fator de difus�o. Portanto, valores negativos de X s�o inteiramente poss�veis. Esse recurso permite o uso de alcances mais curtos do que seria poss�vel se X fosse restrito a valores n�o negativos.

The substitution of Eqs. 9-69 and 9-72 into Eqs. 9-63 to 9-65 leads to routing coefficients expressed in terms of Courant and cell Reynolds numbers:

A substitui��o de Eqs. 9-69 e 9-72 nas Eqs. 9-63 a 9-65 levam a coeficientes de roteamento expressos em termos de n�meros de Courant e de c�lula de Reynolds:

             -1 + C + D
C0  =  ______________
              1 + C + D
(9-74)

              1 + C - D
C1  =  ______________
              1 + C + D
(9-75)

              1 - C + D
C2  =  ______________
              1 + C + D
(9-76)

The calculation of routing parameters C and D, Eqs. 9-69 and 9-71, can be performed in several ways. The wave celerity can be calculated with either Eq. 9-16 or Eq. 9-34. With Eq. 9-16, c = βV; with Eq. 9-34, c = (1/T) dQ/dy. Theoretically, these two equations are the same. For practical applications, if a stage-discharge rating and cross-sectional geometry are available (i.e., stage-discharge-top width tables), Eq. 9-34 is preferred over Eq. 9-16 because it accounts directly for cross-sectional shape. In the absence of a stage-discharge rating and cross-sectional data, Eq. 9-16 can be used to estimate flood wave celerity.

O c�lculo dos par�metros de roteamento C e D, Eqs. 9-69 e 9-71, podem ser executadas de v�rias maneiras. A celeridade das ondas pode ser calculada com a Eq. 9-16 ou Eq. 9-34. Com Eq. 9-16, c = pV; com a Eq. 9-34, c = (1 / T) dQ / dy. Teoricamente, essas duas equa��es s�o iguais. Para aplica��es pr�ticas, se houver uma classifica��o de descarga do est�gio e uma geometria de se��o transversal (ou seja, tabelas de largura do est�gio de descarga-superior), Eq. 9-34 � prefer�vel � Eq. 9-16, porque responde diretamente pelo formato da se��o transversal. Na aus�ncia de uma classifica��o de descarga do est�gio e dados transversais, a Eq. 9-16 pode ser usado para estimar a celeridade das ondas de inunda��o.

With the aid of Eqs. 9-69 and 9-71, the routing parameters may be based on flow characteristics. The calculations can proceed in a linear or nonlinear mode. In the linear mode, the routing parameters are based on reference flow values and kept constant throughout the computation in time. The choice of reference flow has a bearing on the calculated results [2, 15], although the overall effect is likely to be small. For practical applications, either an average or peak flow value can be used as reference flow. The peak flow value has the advantage that it can be readily ascertained, although a better approximation may be obtained by using an average value [15]. The linear mode of computation is referred to as the constant-parameter Muskingum-Cunge method to distinguish it from the variable-parameter Muskingum-Cunge method, in which the routing parameters are allowed to vary with the flow. The constant parameter method resembles the Muskingum method, with the difference that the routing parameters are based on measurable flow and channel characteristics instead of historical streamflow data.

Com a ajuda de Eqs. 9-69 e 9-71, os par�metros de roteamento podem ser baseados nas caracter�sticas de fluxo. Os c�lculos podem prosseguir em um modo linear ou n�o linear. No modo linear, os par�metros de roteamento s�o baseados em valores de fluxo de refer�ncia e mantidos constantes durante todo o c�lculo no tempo. A escolha do fluxo de refer�ncia influencia os resultados calculados [2, 15], embora seja prov�vel que o efeito geral seja pequeno. Para aplica��es pr�ticas, um valor de fluxo m�dio ou de pico pode ser usado como fluxo de refer�ncia. O valor do pico de fluxo tem a vantagem de poder ser prontamente determinado, embora uma melhor aproxima��o possa ser obtida usando um valor m�dio [15]. O modo linear de computa��o � referido como o m�todo Muskingum-Cunge de par�metro constante para distingui-lo do m�todo Muskingum-Cunge de par�metro vari�vel, no qual os par�metros de roteamento podem variar com o fluxo. O m�todo de par�metro constante se assemelha ao m�todo Muskingum, com a diferen�a de que os par�metros de roteamento s�o baseados em caracter�sticas mensur�veis %G​​%@de fluxo e canal em vez de dados hist�ricos de fluxo.

 Example 9-9.

Use the constant-parameter Muskingum-Cunge method to route a flood wave with the following flood and channel characteristics: peak flow Qp = 1000 m3/s; baseflow Qb, = 0 m3/s; channel bottom slope So = 0.000868; flow area at peak discharge Ap = 400 m2; top width at peak discharge Tp = 100 m; rating exponent β = 1.6; reach length Δx = 14.4 km; time interval Δt = 1 h.

Use o m�todo Muskingum-Cunge de par�metro constante para rotear uma onda de inunda��o com as seguintes caracter�sticas de inunda��o e canal: fluxo de pico Qp = 1000 m3 / s; fluxo base Qb, = 0 m3 / s; inclina��o inferior do canal So = 0,000868; �rea de vaz�o no pico de descarga Ap = 400 m2; largura superior na descarga m�xima Tp = 100 m; expoente de classifica��o ~ = 1,6; alcance comprimento ~x = 14,4 km; intervalo de tempo ~t = 1 h.

Time (h) 0 1 2 3 4 5 6 7 8 9 10
Flow (m3/s 0 200 400 600 800 1000 800 600 400 200 0


The mean velocity (based on the peak discharge) is V = Qp/Ap = 2.5 m/s. The wave celerity is c = βV = 4 m/s. The flow per unit width (based on the peak discharge) is qo = Qp/Tp = 10 m2/s. The Courant number (Eq. 9-69) is C = 1. The cell Reynolds number (Eq. 9-71) is D = 0.2. The routing coefficients (Eqs. 9-74 to 9-76) are C0 = 0.091; C1 = 0.818; and C2 = 0.091. It is confirmed that the sum of routing coefficients is equal to 1. The routing calculations are shown in Table 9-8.

A velocidade m�dia (com base no pico de descarga) � V = Qp / Ap = 2,5 m / s. A celeridade das ondas � c = ~V = 4 m / s. O fluxo por largura da unidade (com base no pico de descarga) � qo = Qp / Tp = 10 m2 / s. O n�mero de Courant (Eq. 9-69) � C = 1. O n�mero de Reynolds da c�lula (Eq. 9-71) � D = 0,2. Os coeficientes de roteamento (Eqs. 9-74 a 9-76) s�o C0 = 0,091; C1 = 0,818; e C2 = 0,091. Confirma-se que a soma dos coeficientes de roteamento � igual a 1. Os c�lculos de roteamento s�o mostrados na Tabela 9-8.

Table 9-8  Channel Routing by Muskingum-Cunge Method, Example 9-9.
(1) (2) (3) (4) (5) (6)
Time
(h)
Inflow
(m3/s)
Partial Flows (m3/s) Outflow
(m3/s)
C0 I2 C1 I1 C2 I1
0 0 ___ ___ ___ 0.0
1 200 18.2 0.0 0.0 18.20
2 400 36.4 163.6 1.66 201.66
3 600 54.6 327.2 18.35 400.15
4 800 72.8 490.8 36.41 600.01
5 1000 91.0 654.4 54.60 800.00
6 800 72.8 818.0 72.80 963.60
7 600 54.6 654.4 87.69 796.69
8 400 36.4 490.8 72.50 599.70
9 200 18.2 327.2 54.57 399.97
10 0 0.0 163.6 36.40 200.00
11 0 0.0 0.0 18.20 18.20
12 0 0.0 0.0 1.66 1.66
13 0 0.0 00.0 0.16 0.16

calculator image 

ONLINE CALCULATION. Using ONLINE ROUTING05, the answer is essentially the same as that of Col. 6, Table 9-8.


Resolution Requirements

Requisitos de resolu��o

When using the Muskingum-Cunge method, care should be taken to ensure that the values of Δx and Δt are sufficiently small to approximate closely the actual shape of the hydrograph. For smoothly rising hydrographs, a minimum value of tpt = 5 is recommended. This requirement usually results in the hydrograph time base being resolved into at least 15 to 25 discrete points, considered adequate for Muskingum routing.

Ao usar o m�todo Muskingum-Cunge, deve-se tomar cuidado para garantir que os valores de ~x e ~t sejam suficientemente pequenos para aproximar de perto a forma real do hidrograma. Para hidrogramas em subida suave, � recomendado um valor m�nimo de tp / ~t = 5. Esse requisito geralmente resulta na resolu��o do tempo do hidrograma em pelo menos 15 a 25 pontos discretos, considerados adequados para o roteamento de Muskingum.

Unlike temporal resolution, there is no definite criteria for spatial resolution. A criterion borne out by experience is based on the fact that Courant and cell Reynolds numbers are inversely related to reach length Δx. Therefore, to keep Δx sufficiently small, Courant and cell Reynolds numbers should be kept sufficiently large. This leads to the practical criterion [16]:

Diferentemente da resolu��o temporal, n�o h� crit�rios definidos para resolu��o espacial. Um crit�rio confirmado pela experi�ncia baseia-se no fato de que os n�meros de Courant e de c�lula de Reynolds est�o inversamente relacionados ao comprimento ~x. Portanto, para manter ~x suficientemente pequeno, os n�meros de Courant e de c�lula de Reynolds devem ser mantidos suficientemente grandes. Isso leva ao crit�rio pr�tico [16]:

C + D ≥ 1 (9-77)

which can be written as follows:  -1 + C + D ≥ 0. This confirms the necessity of avoiding negative values of C0 in Muskingum-Cunge routing (see Eq. 9-74). Experience has shown that negative values of either C1 or C2 do not adversely affect the method's overall accuracy [16].

que pode ser escrito da seguinte forma: -1 + C + D ~ 0. Isso confirma a necessidade de evitar valores negativos de C0 no roteamento Muskingum-Cunge (consulte as Eq. 9-74). A experi�ncia mostrou que valores negativos de C1 ou C2 n�o afetam adversamente a precis�o geral do m�todo [16].

Notwithstanding Eq. 9-77, the Muskingum-Cunge method works best when the numerical dispersion is minimized, that is, when C ≅ 1. Values of C substantially less than 1 are likely to cause the notorious dips, or negative outflows, in portions of the calculated hydrograph. This computational anomaly is attributed to excessive numerical dispersion and should be avoided.

N�o obstante a Eq. 9-77, o m�todo Muskingum-Cunge funciona melhor quando a dispers�o num�rica � minimizada, ou seja, quando C %G≅%@ 1. Valores de C substancialmente menores que 1 provavelmente causam quedas not�rias, ou sa�das negativas, em partes do c�lculo calculado. hidrograma. Essa anomalia computacional � atribu�da � dispers�o num�rica excessiva e deve ser evitada.

Nonlinear Muskingum-Cunge Method

M�todo n�o linear de Muskingum-Cunge

The kinematic wave equation, Eq. 9-18, is nonlinear because the kinematic wave celerity varies with discharge. The nonlinearity is mild, among other things because the wave celerity variation is usually restricted within a narrow range. However, in certain cases it may be necessary to account for this nonlinearity. This can be done in two ways: (1) during the discretization, by allowing the wave celerity to vary, resulting in a nonlinear numerical scheme to be solved by iterative means; and (2) after the discretization, by varying the routing parameters, as in the variable-parameter Muskingum-Cunge method [15]. The latter approach is particularly useful if the overall nonlinear effect is small, which is often the case.

A equa��o da onda cinem�tica, Eq. 9-18, n�o � linear porque a celeridade das ondas cinem�ticas varia com a descarga. A n�o linearidade � leve, entre outras coisas, porque a varia��o da celeridade das ondas � geralmente restrita dentro de uma faixa estreita. No entanto, em certos casos, pode ser necess�rio considerar essa n�o linearidade. Isso pode ser feito de duas maneiras: (1) durante a discretiza��o, permitindo que a celeridade das ondas varie, resultando em um esquema num�rico n�o linear a ser resolvido por meios iterativos; e (2) ap�s a discretiza��o, variando os par�metros de roteamento, como no m�todo de par�metro vari�vel Muskingum-Cunge [15]. A �ltima abordagem � particularmente �til se o efeito n�o linear geral for pequeno, o que geralmente ocorre.

In the variable parameter method, the routing parameters are allowed to vary with the flow. The values of C and D are based on local qo and c values instead of peak flow or other reference value as in the constant-parameter method. To vary the routing parameters, the most expedient way is to obtain an average value of qo and c for each computational cell. This can be achieved with a direct three-point average of the values at the known grid points (see Fig. 9-11), or by an iterative four-point average, which includes the unknown grid point. To improve the convergence of the iterative four-point procedure, the three-point average can be used as the first guess of the iteration. Once qo and c have been determined for each computational cell, the Courant and cell Reynolds numbers are calculated by Eqs. 9-69 and 9-71. The value of bottom slope So remains unchanged within each computational cell.

No m�todo de par�metro vari�vel, os par�metros de roteamento podem variar com o fluxo. Os valores de C e D s�o baseados nos valores locais de qo e c em vez do pico de fluxo ou outro valor de refer�ncia, como no m�todo de par�metro constante. Para variar os par�metros de roteamento, a maneira mais conveniente � obter um valor m�dio de qo e c para cada c�lula computacional. Isso pode ser alcan�ado com uma m�dia direta de tr�s pontos dos valores nos pontos de grade conhecidos (veja a Fig. 9-11) ou com uma m�dia de quatro pontos iterativa, que inclui o ponto de grade desconhecido. Para melhorar a converg�ncia do procedimento iterativo de quatro pontos, a m�dia de tr�s pontos pode ser usada como o primeiro palpite da itera��o. Depois de determinados qo e c para cada c�lula computacional, os n�meros de Courant e de Reynolds da c�lula s�o calculados pelas Eqs. 9-69 e 9-71. O valor da inclina��o inferior So permanece inalterado dentro de cada c�lula computacional.

The variable parameter Muskingum-Cunge method represents a small yet sometimes perceptible improvement over the constant parameter method. The differences are likely to be more marked for very long reaches and/or wide variations in flow levels. Flood hydrographs calculated with variable parameters show a certain amount of distortion, either wave steepening in the case of flows contained inbank or wave attenuation (flattening) in the case of typical overbank flows. This is a physical manifestation of the nonlinear effect, i.e., different flow levels traveling with different celerities. On the other hand, flood hydrographs calculated using constant parameters do not show wave distortion.

O par�metro vari�vel m�todo Muskingum-Cunge representa uma pequena, mas �s vezes percept�vel, melhoria em rela��o ao m�todo de par�metro constante. As diferen�as provavelmente ser�o mais acentuadas para alcances muito longos e / ou grandes varia��es nos n�veis de fluxo. Os hidrogramas de inunda��o calculados com par�metros vari�veis %G​​%@mostram uma certa distor��o, seja a inclina��o das ondas no caso de fluxos contidos no banco ou atenua��o das ondas (achatamento) no caso dos fluxos t�picos do excesso de margens. Esta � uma manifesta��o f�sica do efeito n�o linear, isto �, diferentes n�veis de fluxo que viajam com diferentes celeridades. Por outro lado, os hidrogramas de inunda��o calculados usando par�metros constantes n�o mostram distor��o das ondas.

Assessment of Muskingum-Cunge Method

Avalia��o do m�todo Muskingum-Cunge

The Muskingum-Cunge method is a physically based alternative to the Muskingum method. Unlike the Muskingum method where the parameters are calibrated using streamflow data, in the Muskingum-Cunge method the parameters are calculated based on flow and channel characteristics. This makes possible channel routing without the need for time-consuming and cumbersome parameter calibration. More importantly, it makes possible extensive channel routing in ungaged streams with a reasonable expectation of accuracy. With the variable-parameter feature, nonlinear properties of flood waves (which could otherwise only be obtained by more elaborate numerical procedures) can be described within the context of the Muskingum formulation.

O m�todo Muskingum-Cunge � uma alternativa baseada fisicamente ao m�todo Muskingum. Diferente do m�todo Muskingum, em que os par�metros s�o calibrados usando dados de fluxo, no m�todo Muskingum-Cunge os par�metros s�o calculados com base nas caracter�sticas de fluxo e canal. Isso possibilita o roteamento de canais sem a necessidade de calibra��o de par�metros demorada e complicada. Mais importante, torna poss�vel o roteamento extensivo de canais em fluxos n�o calibrados com uma expectativa razo�vel de precis�o. Com o recurso de par�metro vari�vel, as propriedades n�o lineares das ondas de inunda��o (que de outra forma s� poderiam ser obtidas por procedimentos num�ricos mais elaborados) podem ser descritas no contexto da formula��o de Muskingum.

Like the Muskingum method, the Muskingum-Cunge method is limited to diffusion waves. Furthermore, the Muskingum-Cunge method is based on a single-valued rating and does not take into account strong flow non-uniformity or unsteady flows exhibiting substantial loops in discharge-stage rating (i.e., dynamic waves). Thus, the Muskingum-Cunge method is suited for channel routing in natural streams without significant backwater effects and for unsteady flows that classify under the diffusion wave criterion (Eq. 9-60).

Como o m�todo Muskingum, o m�todo Muskingum-Cunge � limitado a ondas de difus�o. Al�m disso, o m�todo Muskingum-Cunge � baseado em uma classifica��o de valor �nico e n�o leva em conta a n�o uniformidade do fluxo forte ou fluxos inst�veis %G​​%@que exibem loops substanciais na classifica��o do est�gio de descarga (ou seja, ondas din�micas). Assim, o m�todo Muskingum-Cunge � adequado para roteamento de canais em riachos naturais sem efeitos significativos de remanso e para fluxos inst�veis %G​​%@que se classificam sob o crit�rio de ondas de difus�o (Eq. 9-60).

An important difference between the Muskingum and Muskingum-Cunge methods should be noted. The Muskingum method is based on the storage concept (Chapter 4) and, therefore, it is lumped, with the parameters K and X being reach averages. The Muskingum-Cunge method, however, is distributed in nature, with the parameters C and D being based on values evaluated at channel cross sections. Therefore, for the Muskingum-Cunge method to improve on the Muskingum method, it is necessary that the routing parameters evaluated at channel cross sections be representative of the channel reach under consideration.

Uma diferen�a importante entre os m�todos Muskingum e Muskingum-Cunge deve ser observada. O m�todo Muskingum � baseado no conceito de armazenamento (cap�tulo 4) e, portanto, � agrupado, com os par�metros K e X sendo alcan�ados em m�dias. O m�todo Muskingum-Cunge, no entanto, � distribu�do na natureza, com os par�metros C e D baseados em valores avaliados nas se��es transversais do canal. Portanto, para que o m�todo Muskingum-Cunge melhore o m�todo Muskingum, � necess�rio que os par�metros de roteamento avaliados nas se��es transversais do canal sejam representativos do alcance do canal em considera��o.

Historically, the Muskingum method has been calibrated using streamflow data. On the contrary, the Muskingum-Cunge method relies on physical characteristics such as rating curves, cross-sectional data and channel slope. The different data requirements reflect the different theoretical bases of the methods, i.e., lumped storage concept in the Muskingum method, and distributed kinematic/diffusion wave theory in the Muskingum-Cunge method.

Historicamente, o m�todo Muskingum foi calibrado usando dados de fluxo. Pelo contr�rio, o m�todo Muskingum-Cunge se baseia em caracter�sticas f�sicas como curvas de classifica��o, dados de se��o transversal e inclina��o do canal. Os diferentes requisitos de dados refletem as diferentes bases te�ricas dos m�todos, isto �, o conceito de armazenamento concentrado no m�todo Muskingum e a teoria de ondas cinem�ticas / de difus�o distribu�das no m�todo Muskingum-Cunge.


9.5  INTRODU��O A ONDAS DIN�MICAS

[Quest�es]   [Problemas]   [Refer�ncias]      [Top]   [M�todo Muskingum]   [Ondas Cinem�ticas]   [Ondas Difusivas]   [M�todo Muskingum-Cunge]  

In Section 9.2, kinematic waves were formulated by simplifying the momentum conservation principle to a statement of steady uniform flow. In Section 9.3, diffusion waves were formulated by simplifying the momentum principle to a statement of steady nonuniform flow. These two waves, in particular the diffusion wave, have been extensively used in stream channel routing applications. The Muskingum and Muskingum-Cunge methods are examples of calculations using the concept of diffusion wave.

Na Se��o 9.2, as ondas cinem�ticas foram formuladas simplificando o princ�pio de conserva��o do momento para uma declara��o de fluxo uniforme e constante. Na Se��o 9.3, as ondas de difus�o foram formuladas simplificando o princ�pio do momento para uma declara��o de fluxo n�o uniforme constante. Essas duas ondas, em particular a onda de difus�o, foram amplamente utilizadas em aplica��es de roteamento de canal de fluxo. Os m�todos Muskingum e Muskingum-Cunge s�o exemplos de c�lculos usando o conceito de onda de difus�o.

A third type of open channel flow wave, the dynamic wave, is formulated by taking into account the complete momentum principle, including its inertial components. As such, the dynamic wave contains more physical information than either kinematic or diffusion waves. Dynamic wave solutions, however, are more complicated than either kinematic or diffusion wave solutions.

Um terceiro tipo de onda de fluxo de canal aberto, a onda din�mica, � formulado levando em considera��o o princ�pio do momento completo, incluindo seus componentes inerciais. Como tal, a onda din�mica cont�m mais informa��es f�sicas do que as ondas cinem�ticas ou de difus�o. As solu��es de ondas din�micas, no entanto, s�o mais complicadas do que as solu��es de ondas cinem�ticas ou de difus�o.

In a dynamic wave solution, the equations of mass and momentum conservation are solved by a numerical procedure, either the method of finite differences, the method of characteristics, or the finite element method. In the method of finite differences, the partial differential equations are discretized following a chosen numerical scheme [9]. The method of characteristics is based on the conversion of the set of partial differential equations into a related set of ordinary differential equations, and the solution along a characteristic grid, i.e. a grid that follows characteristic directions (Fig. 4-16). The method of finite elements solves a set of integral equations over a chosen grid of finite elements.

Em uma solu��o de onda din�mica, as equa��es de conserva��o de massa e momento s�o resolvidas por um procedimento num�rico, seja o m�todo das diferen�as finitas, o m�todo das caracter�sticas ou o m�todo dos elementos finitos. No m�todo das diferen�as finitas, as equa��es diferenciais parciais s�o discretizadas seguindo um esquema num�rico escolhido [9]. O m�todo das caracter�sticas baseia-se na convers�o do conjunto de equa��es diferenciais parciais em um conjunto relacionado de equa��es diferenciais ordin�rias e na solu��o ao longo de uma grade caracter�stica, isto �, uma grade que segue as dire��es das caracter�sticas (Fig. 4-16). O m�todo dos elementos finitos resolve um conjunto de equa��es integrais sobre uma grade escolhida de elementos finitos.

In the past four decades, the method of finite differences has come to be regarded as the most expedient way of obtaining a dynamic wave solution for practical applications [6, 9]. Among several numerical schemes that have been used in connection with the dynamic wave, the Preissmann scheme is perhaps the most popular. This is a four-point scheme, centered in the temporal derivatives and slightly off-centered in the spatial derivatives. The off-centering in the spatial derivatives introduces a small amount of numerical diffusion necessary to control the numerical stability of the nonlinear scheme. This produces a workable yet sufficiently accurate scheme.

Nas �ltimas quatro d�cadas, o m�todo das diferen�as finitas passou a ser considerado a maneira mais conveniente de obter uma solu��o din�mica de ondas para aplica��es pr�ticas [6, 9]. Entre os v�rios esquemas num�ricos utilizados em conex�o com a onda din�mica, o esquema de Preissmann � talvez o mais popular. Este � um esquema de quatro pontos, centrado nas derivadas temporais e ligeiramente descentralizado nas derivadas espaciais. A descentraliza��o nas derivadas espaciais introduz uma pequena quantidade de difus�o num�rica necess�ria para controlar a estabilidade num�rica do esquema n�o linear. Isso produz um esquema vi�vel, mas suficientemente preciso.

The stream channel is divided into several reaches for computational purposes (Fig. 9-12). The application of the Preissmann scheme to the governing equations for the various reaches results in a matrix solution requiring a double sweep algorithm, i.e., one that accounts only for the nonzero entries of the coefficient matrix, which are located within a narrow band surrounding the main diagonal. This technique leads to a considerable savings in storage and execution time. With the appropriate upstream and downstream boundary conditions (Fig. 9-13), the solution of the set of hyperbolic equations marches in time until a specified number of time intervals is completed.

O canal de fluxo � dividido em v�rios alcances para fins computacionais (Fig. 9-12). A aplica��o do esquema de Preissmann �s equa��es governantes para os v�rios alcances resulta em uma solu��o de matriz que requer um algoritmo de varredura dupla, isto �, que � respons�vel apenas pelas entradas diferentes de zero da matriz do coeficiente, localizadas dentro de uma faixa estreita ao redor da principal diagonal. Essa t�cnica leva a uma economia consider�vel no tempo de armazenamento e execu��o. Com as condi��es de contorno a montante e a jusante apropriadas (Fig. 9-13), a solu��o do conjunto de equa��es hiperb�licas marcha no tempo at� que um n�mero especificado de intervalos de tempo seja conclu�do.

Reach subdivision for dynamic wave routing

Figure 9-13  Reach subdivision for dynamic wave routing.

In practice, a dynamic wave solution represents an order-of-magnitude increase in complexity and associated data requirements when compared to either kinematic or diffusion wave solutions. Its use is recommended in situations where neither kinematic nor diffusion wave solutions are likely to represent adequately the physical phenomena. In particular, dynamic wave solutions are applicable to flow over very flat slopes, flow into large reservoirs, strong backwater conditions and flow reversals. In general, the dynamic wave is recommended for cases warranting a precise determination of the unsteady variation of river stages.

Na pr�tica, uma solu��o din�mica de ondas representa um aumento de ordem de magnitude na complexidade e nos requisitos de dados associados quando comparados a solu��es de ondas cinem�ticas ou de difus�o. Seu uso � recomendado em situa��es em que nem as solu��es de ondas cinem�ticas nem de difus�o provavelmente representam adequadamente os fen�menos f�sicos. Em particular, as solu��es de ondas din�micas s�o aplic�veis %G​​%@ao escoamento sobre encostas muito planas, escoamento para grandes reservat�rios, condi��es de mar�s fortes e revers�es de escoamento. Em geral, a onda din�mica � recomendada para casos que garantam uma determina��o precisa da varia��o inst�vel dos est�gios do rio.

Relevance of Dynamic Waves to Engineering Hydrology

Relev�ncia das ondas din�micas para a engenharia hidrol�gica

Dynamic wave solutions are often referred to as hydraulic river routing. As such, they have the capability to calculate unsteady discharges and stages when presented with the appropriate geometric channel data and initial and boundary conditions. Their relevance to engineering hydrology is examined here by comparing them to kinematic and diffusion wave solutions.

As solu��es de ondas din�micas s�o frequentemente chamadas de roteamento hidr�ulico de rios. Como tal, eles t�m a capacidade de calcular descargas e est�gios inst�veis %G​​%@quando apresentados com os dados apropriados do canal geom�trico e as condi��es iniciais e de contorno. Sua relev�ncia para a engenharia hidrol�gica � examinada aqui, comparando-as com solu��es cinem�ticas e de ondas de difus�o.

Kinematic waves calculate unsteady discharges; the corresponding stages are subsequently obtained from the appropriate rating curves. Usually, equilibrium (steady, uniform) rating curves are used for this purpose. Diffusion waves may or may not use equilibrium rating curves to calculate stages. Some methods, e.g., Muskingum-Cunge, use equilibrium ratings, but more elaborate diffusion wave solutions may not.

As ondas cinem�ticas calculam descargas inst�veis; os est�gios correspondentes s�o subsequentemente obtidos a partir das curvas de classifica��o apropriadas. Geralmente, curvas de classifica��o de equil�brio (constante, uniforme) s�o usadas para esse fim. Ondas de difus�o podem ou n�o usar curvas de classifica��o de equil�brio para calcular est�gios. Alguns m�todos, por exemplo, Muskingum-Cunge, usam classifica��es de equil�brio, mas solu��es de ondas de difus�o mais elaboradas podem n�o.

Dynamic waves rely on the physics of the phenomena as built into the governing equations to generate their own unsteady rating. A looped rating curve is produced at every cross section, as shown in Fig. 9-14. For any given stage, the discharge is higher in the rising limb of the hydrograph and lower in the receding limb. This loop is due to hydrodynamic reasons and should not be confused with other loops, which may be due to erosion, sedimentation, or changes in bed configuration (Chapter 15).

As ondas din�micas dependem da f�sica dos fen�menos incorporada nas equa��es governantes para gerar sua pr�pria classifica��o inst�vel. Uma curva de classifica��o em loop � produzida em cada se��o transversal, como mostrado na Fig. 9-14. Para qualquer est�gio, a descarga � maior no membro ascendente do hidrograma e menor no membro recuado. Esse loop � devido a raz�es hidrodin�micas e n�o deve ser confundido com outros loops, que podem ser causados %G​​%@por eros�o, sedimenta��o ou altera��es na configura��o do leito (Cap�tulo 15).

Sketch of the looped rating of dynamic waves

Figure 9-14  Sketch of the looped rating of dynamic waves.

The width of the loop is a measure of the flow unsteadiness, with wider loops corresponding to highly unsteady flow, i.e., dynamic wave flow. If the loop is narrow, it implies that the flow is mildly unsteady, perhaps a diffusion wave. If the loop is practically nonexistent, the flow can be approximated as kinematic flow. In fact, the basic assumption of kinematic flow is that momentum can be simulated as steady uniform flow, i.e., that the rating curve is single-valued.

A largura do la�o � uma medida da instabilidade do fluxo, com la�os mais largos correspondendo ao fluxo altamente inst�vel, isto �, fluxo din�mico da onda. Se o loop for estreito, isso implica que o fluxo � levemente inst�vel, talvez uma onda de difus�o. Se o loop for praticamente inexistente, o fluxo pode ser aproximado como fluxo cinem�tico. De fato, a suposi��o b�sica do fluxo cinem�tico � que o momento pode ser simulado como fluxo uniforme constante, ou seja, que a curva de classifica��o � de valor �nico.

The preceding observations lead to the conclusion that the relevance of dynamic waves in engineering hydrology is directly related to the flow unsteadiness and the associated loop in the rating curve. For highly unsteady flows such as dam-break flood waves, it may well be the only way to properly account for the looped rating. For other less unsteady flows, kinematic and diffusion waves are a viable alternative, provided their applicability can be clearly demonstrated (Eqs. 9-44 and 9-60).

As observa��es anteriores levam � conclus�o de que a relev�ncia das ondas din�micas na hidrologia da engenharia est� diretamente relacionada � instabilidade do fluxo e ao loop associado na curva de classifica��o. Para fluxos altamente inst�veis, como ondas de inunda��o de barragens, pode ser a �nica maneira de contabilizar adequadamente a classifica��o em loop. Para outros fluxos menos inst�veis, as ondas cinem�ticas e de difus�o s�o uma alternativa vi�vel, desde que sua aplicabilidade possa ser claramente demonstrada (Eqs. 9-44 e 9-60).

Diffusion Wave Solution with Dynamic Component

Solu��o de ondas de difus�o com componente din�mico

A simplified approach to dynamic wave routing is that of the diffusion wave with dynamic component [2]. In this approach, the complete governing equations, including inertia terms, are linearized in a similar way as with diffusion waves. This leads to a diffusion equation similar to Eq. 9-58, but with a modified hydraulic diffusivity. The equation is [5]:

Uma abordagem simplificada para o roteamento din�mico de ondas � a da onda de difus�o com componente din�mico [2]. Nesta abordagem, as equa��es completas de governo, incluindo termos de in�rcia, s�o linearizadas de maneira semelhante � das ondas de difus�o. Isso leva a uma equa��o de difus�o semelhante � Eq. 9-58, mas com difusividade hidr�ulica modificada. A equa��o � [5]:

  ∂Q              ∂Q        ∂Q                  Qo                                         ∂2Q
______  +  ( ______ ) ______  =  { ( ________ ) [ 1 - (β - 1)2 Fo2 ] } _______
   ∂t               ∂A         ∂x               2 T So                                       ∂x2
(9-78)

in which the hydraulic diffusivity (i.e., the coefficient of the second-order term) is also a function of the rating curve parameter β and the Froude number, defined as:

em que a difusividade hidr�ulica (isto �, o coeficiente do termo de segunda ordem) tamb�m � uma fun��o do par�metro da curva de classifica��o ~ e do n�mero de Froude, definido como:

                Vo   
Fo  =  ___________
            (g do)1/2
(9-79)

with g = gravitational acceleration, and do = reference flow depth.

com g = acelera��o gravitacional e do = profundidade do fluxo de refer�ncia.

Equation 9-78 can be expressed in terms of the Vedernikov number [17]:

A equa��o 9-78 pode ser expressa em termos do n�mero de Vedernikov [17]:


V  =  (β - 1) Fo

(9-80)

With Eq. 9-80, Eq. 9-78 reduces to:

Com Eq. 9-80, Eq. 9-78 reduz para:

  ∂Q              ∂Q        ∂Q                 Qo                             ∂2Q
______  +  ( ______ ) ______  =  [ ( ________ ) ( 1 - V 2 ) ] _______
   ∂t               ∂A         ∂x               2 T So                          ∂x2
(9-81)

Equations 9-78 and 9-81 provide an enhanced predictive capability for the simulation of diffusion waves including a dynamic component. For instance, for β = 1.5 (i.e., Chezy friction in wide channels) and Fo = 2, the Vedernikov number V = 1 (Eq. 9-80), and the hydraulic diffusivity vanishes, which is in agreement with physical reality [10, 13]. On the other hand, the hydraulic diffusivity of the diffusion wave (Eq. 9-58) is independent of the Vedernikov number. Therefore, Eq. 9-81 is a better model than Eq. 9-58, especially for Froude numbers in the supercritical regime. Most natural flows, however, are in the range well below critical, with Eq. 9-58 remaining a practical model of unsteady open channel flow phenomena [7].

As equa��es 9-78 e 9-81 fornecem uma capacidade preditiva aprimorada para a simula��o de ondas de difus�o, incluindo um componente din�mico. Por exemplo, para ~ = 1,5 (ou seja, atrito Chezy em canais amplos) e Fo = 2, o n�mero Vedernikov V = 1 (Eq. 9-80) e a difusividade hidr�ulica desaparecem, o que est� de acordo com a realidade f�sica [10 13]. Por outro lado, a difusividade hidr�ulica da onda de difus�o (Eq. 9-58) � independente do n�mero de Vedernikov. Portanto, a Eq. 9-81 � um modelo melhor que a Eq. 9-58, especialmente para n�meros de Froude no regime supercr�tico. A maioria dos fluxos naturais, no entanto, est� na faixa bem abaixo do cr�tico, com a Eq. 9-58, permanecendo um modelo pr�tico de fen�menos de fluxo de canal aberto inst�veis %G​​%@[7].


QUEST�ES

[Problemas]   [Refer�ncias]      [Top]   [M�todo Muskingum]   [Ondas Cinem�ticas]   [Ondas Difusivas]   [M�todo Muskingum-Cunge]  

  1. What is routing? What types of waves are used in describing unsteady open channel flow processes?

    O que � roteamento? Que tipos de ondas s�o usados %G​​%@na descri��o de processos de fluxo de canal aberto inst�veis?

  2. What is model calibration? What is model verification?

    O que � calibra��o do modelo? O que � verifica��o de modelo?

  3. In the Muskingum method, what does the parameter K represent? What does the parameter X represent?

    No m�todo Muskingum, o que o par�metro K representa? O que o par�metro X representa?

  4. How does channel routing differ from reservoir routing? What differences are to be noted in the routed hydrographs?

    Como o roteamento de canal difere do roteamento de reservat�rio? Que diferen�as devem ser observadas nos hidrogramas roteados?

  5. What is the kinematic wave celerity? What is the practical range of turbulent flow values of β, the rating constant used in the kinematic wave celerity?

    Qual � a celeridade das ondas cinem�ticas? Qual � a faixa pr�tica dos valores de fluxo turbulento de ~, a constante de classifica��o usada na celeridade das ondas cinem�ticas?

  6. What is the order of accuracy of a numerical scheme? What is the difference between numerical diffusion and numerical dispersion in connection with kinematic wave solutions?

    Qual � a ordem de precis�o de um esquema num�rico? Qual � a diferen�a entre difus�o num�rica e dispers�o num�rica em conex�o com solu��es de ondas cinem�ticas?

  7. What is a linear model in the context of kinematic wave routing? What is a nonlinear model?

    O que � um modelo linear no contexto do roteamento de ondas cinem�ticas? O que � um modelo n�o linear?

  8. Why are the results of convex routing dependent on the grid size?

    Por que os resultados do roteamento convexo dependem do tamanho da grade?

  9. What is a diffusion wave? How does it differ from a kinematic wave?

    O que � uma onda de difus�o? Como isso difere de uma onda cinem�tica?

  10. What is hydraulic diffusivity? Why is it important in flood routing?

    O que � difusividade hidr�ulica? Por que isso � importante no roteamento de inunda��es?

  11. What values of parameters X and C optimize numerical diffusion and minimize numerical dispersion in the Muskingum-Cunge method?

    Quais valores dos par�metros X e C otimizam a difus�o num�rica e minimizam a dispers�o num�rica no m�todo Muskingum-Cunge?

  12. Why are negative values of X entirely possible in Muskingum-Cunge routing? Why are values of X in excess of 0.5 unfeasible?

    Por que os valores negativos de X s�o inteiramente poss�veis no roteamento Muskingum-Cunge? Por que valores de X acima de 0,5 s�o invi�veis?

  13. What is the Courant number? What is the cell Reynolds number?

    Qual � o n�mero do Courant? Qual � o n�mero de Reynolds da c�lula?

  14. Describe the difference between linear and nonlinear solutions to channel routing problems.

    Descreva a diferen�a entre solu��es lineares e n�o lineares para problemas de roteamento de canal.

  15. What is a dynamic wave? How does it differ from the diffusion and kinematic waves?

    O que � uma onda din�mica? Como ele difere da difus�o e das ondas cinem�ticas?

  16. How does the method of finite differences differ from the method of characteristics? What is a double sweep algorithm?

    Como o m�todo das diferen�as finitas difere do m�todo das caracter�sticas? O que � um algoritmo de varredura dupla?

  17. Discuss the influence of the loop in the rating in determining whether an open channel flow wave is dynamic in nature.

    Discuta a influ�ncia do loop na classifica��o para determinar se uma onda de fluxo de canal aberto � de natureza din�mica.

  18. What is the Vederkinov number? How does it differ from the Froude number?

    Qual � o n�mero de Vederkinov? Qual � a diferen�a do n�mero Froude?

  19. What is the effect of the inclusion of a dynamic component in diffusion wave modeling?

    Qual � o efeito da inclus�o de um componente din�mico na modelagem de ondas de difus�o?


PROBLEMAS

[Refer�ncias]      [Top]   [M�todo Muskingum]   [Ondas Cinem�ticas]   [Ondas Difusivas]   [M�todo Muskingum-Cunge]   [Ondas Din�micas]   [Quest�es]  

  1. Given the following inflow hydrograph to a certain stream channel reach, calculate the outflow by the Muskingum method. Check your results with ONLINE ROUTING 04.

    Dado o seguinte hidr�grafo de entrada para um determinado alcance do canal de fluxo, calcule a vaz�o pelo m�todo Muskingum. Verifique seus resultados com o ROTEAMENTO ONLINE 04.

    Time (h) 0 1 2 3 4 5 6 7 8 9 10 11 12
    Flow (m3/s) 10 20 40 80 120 150 120 60 50 40 30 20 10

    Assume K = 1 h, X = 0.2, and Δt = 1 h.

    Assuma K = 1 h, X = 0,2 e ~t = 1 h.

  2. Given the following inflow hydrograph to a certain stream channel reach, calculate the outflow by the Muskingum method. Check your results with ONLINE ROUTING 04.

    Dado o seguinte hidr�grafo de entrada para um determinado alcance do canal de fluxo, calcule a vaz�o pelo m�todo Muskingum. Verifique seus resultados com o ROTEAMENTO ONLINE 04.

    Time (h) 0 3 6 9 12 15 18 21 24 27 30 33 36
    Flow (m3/s) 100 120 150 200 250 275 250 210 180 150 120 110 100

    Assume K = 2.4 h, X = 0.1, and Δt = 3 h.

    Suponha K = 2,4 h, X = 0,1 e ~t = 3 h.

  3. Given the following inflow and outflow hydrographs for a certain stream channel reach, calculate the Muskingum parameters K and X.

    Dadas as seguintes hidrografias de entrada e sa�da para um determinado alcance do canal de fluxo, calcule os par�metros K e X de Muskingum.

    Time (h) 0 1 2 3 4 5
    Inflow (ft3/s ) 2520 3870 4560 6795 8975 9320
    Outflow (ft3/s ) 2520 2643 3598 4500 6367 8295

    Time (h) 6 7 8 9 10 11
    Inflow (ft3/s ) 7780 6520 5340 4105 3210 2520
    Outflow (ft3/s ) 8900 7971 6808 5628 4439 3482

    Time (h) 12 13 14 15 16 17
    Inflow (ft3/s ) 2520 2520 2520 2520 2520 2520
    Outflow (ft3/s ) 2782 2592 2540 2525 2521 2520

  4. Develop a spreadsheet to solve the Muskingum method of stream channel routing, given the following data: (1) an inflow hydrograph of arbitrary shape, (2) baseflow, (3) storage constant K, (4) weighting factor X, and (5) time interval Δt. Test your spreadsheet with Example 9-1 in the text. Check your results with ONLINE ROUTING 04.

    Desenvolva uma planilha para resolver o m�todo Muskingum de roteamento de canais de fluxo, com os seguintes dados: (1) um hidrograma de influxo de forma arbitr�ria, (2) fluxo de base, (3) constante de armazenamento K, (4) fator de pondera��o X e (5) ) intervalo de tempo ~t. Teste sua planilha com o Exemplo 9-1 no texto. Verifique seus resultados com o ROTEAMENTO ONLINE 04.

  5. Given the following inflow hydrograph, use the spreadsheet developed in Problem 9-4 to calculate the outflow hydrograph by the Muskingum method.

    Dado o hidrograma de entrada a seguir, use a planilha desenvolvida no Problema 9-4 para calcular o hidrograma de sa�da pelo m�todo Muskingum.

    Time (h) 0.00 0.25 0.50 0.75 1.0 1.25 1.50 1.75 2.0 2.25 2.50 2.75
    Flow (m3/s) 0 1 2 4 8 10 8 6 4 2 1 0

    Assume K = 0.4 h, X = 0.15, and Δt = 0.25 h. Check your results with ONLINE ROUTING 04.

    Suponha K = 0,4 h, X = 0,15 e ~t = 0,25 h. Verifique seus resultados com o ROTEAMENTO ONLINE 04.

  6. Develop a spreadsheet to estimate the parameters of the Muskingum method, given a matching set of inflow and outflow hydrographs for a certain channel reach. A suggested algorithm is to search for the value of X that minimizes the root mean square (RMS) of the differences between predicted and measured storage. For this purpose, several values of X (between the range 0.0 to 0.5) are tried. For each trial value, a regression line is fitted to the (measured) storage (calculated using Eq. 9-7) versus weighted flow data, with weighted flow in the abscissas and measured storage in the ordinates. The differences between measured storage and predicted storage, i.e., storage predicted by the regression, are calculated. The RMS is evaluated by the following formula:

    Desenvolva uma planilha para estimar os par�metros do m�todo Muskingum, considerando um conjunto correspondente de hidrogramas de entrada e sa�da para um determinado alcance do canal. Um algoritmo sugerido � procurar o valor de X que minimiza o quadrado m�dio da raiz (RMS) das diferen�as entre armazenamento previsto e medido. Para esse fim, v�rios valores de X (entre o intervalo de 0,0 a 0,5) s�o tentados. Para cada valor de teste, uma linha de regress�o � ajustada ao armazenamento (medido) (calculado usando as Eq. 9-7) versus dados de fluxo ponderado, com fluxo ponderado nas abscissas e armazenamento medido nas ordenadas. As diferen�as entre armazenamento medido e armazenamento previsto, isto �, armazenamento previsto pela regress�o, s�o calculadas. O RMS � avaliado pela seguinte f�rmula:

                          1
    RMS  =  [ ( ______ )  Σ (S - S' )2 ] 1/2
                        n - 1     n
    (9-82)

    in which S = measured storage, S' = predicted storage, and n = number of values. The X corresponding to the minimum RMS value is the estimated X. The Muskingum parameter K is the slope of the regression line corresponding to the chosen X value. Use Example 9-2 in the text to test your program.

    em que S = armazenamento medido, S '= armazenamento previsto en = n�mero de valores. O X correspondente ao valor m�nimo de RMS � o X estimado. O par�metro Muskingum K � a inclina��o da linha de regress�o correspondente ao valor X escolhido. Use o Exemplo 9-2 no texto para testar seu programa.

  7. Use the data of Problem 9-3 to test further the spreadsheet developed in Problem 9-6.

    Use os dados do Problema 9-3 para testar ainda mais a planilha desenvolvida no Problema 9-6.

  8. Route the following flood wave using a linear forward-in-time/backward-in-space numerical scheme of the kinematic wave equation (similar to the convex method).

    Encaminhe a seguinte onda de inunda��o usando um esquema num�rico linear para a frente no tempo / para tr�s no espa�o da equa��o da onda cinem�tica (semelhante ao m�todo convexo).

    Time (min) 0 10 20 30 40 50 60 70 80 90 100
    Flow (m3/s) 0 1 2 4 8 10 8 4 2 1 0

    Assume baseflow 0 m3/s, V = 1m/s, β = 1.5, Δx = 1200 m, and Δt = 10 min.

    Assuma o fluxo de base 0 m3 / s, V = 1m / s, ~ = 1,5, ~x = 1200 me ~t = 10 min.

  9. Derive the routing coefficients for a linear forward-in-space/ backward-in-time numerical scheme of the kinematic wave equation.

    Derive os coeficientes de roteamento para um esquema num�rico linear de avan�o no espa�o / retrocesso no tempo da equa��o de onda cinem�tica.

  10. Use the routing coefficients derived in Problem 9-9 to route the inflow hydrograph of Problem 9-8. Assume a reach length Δx = 800 m.

    Use os coeficientes de roteamento derivados no Problema 9-9 para rotear o hidrograma de entrada do Problema 9-8. Suponha um comprimento de alcance ~x = 800 m.

  11. Calculate the β value for a triangular channel with Chezy friction.

    Calcule o valor ~ para um canal triangular com atrito Chezy.

  12. A large river of nearly constant width B = 900 m is seen to be rising at the rate of 10 cm/h. At the observation point, a stage measurement indicates that the current value of discharge is 2200 m3/s. What is a rough estimate of the discharge at a point 5 km upstream?

    Um rio grande de largura quase constante B = 900 m � visto subindo � taxa de 10 cm / h. No ponto de observa��o, uma medi��o de est�gio indica que o valor atual da descarga � 2200 m3 / s. Qual � uma estimativa aproximada da descarga em um ponto a 5 km a montante?

  13. Solve Problem 9-12 if the tributary contribution between the two points is estimated to be constant and equal to 225 m3/s.

    Resolva o Problema 9-12 se estima-se que a contribui��o tribut�ria entre os dois pontos seja constante e igual a 225 m3 / s.

  14. Determine if a flood wave with the following characteristics is a kinematic wave: time-of-rise tr = 6 h, bottom slope So = 0.015, average flow velocity Vo = 1.5 m/s, and average flow depth do = 3 m.

    Determine se uma onda de inunda��o com as seguintes caracter�sticas � uma onda cinem�tica: tempo de subida tr = 6 h, declive inferior So = 0,015, velocidade m�dia do fluxo Vo = 1,5 m / se profundidade m�dia do fluxo = 3 m.

  15. Determine if a flood wave with the following characteristics is a diffusion wave: time-of-rise tr = 6 h, bottom slope So = 0.005, and average flow depth do = 3 m.

    Determine se uma onda de inunda��o com as seguintes caracter�sticas � uma onda de difus�o: tempo de subida tr = 6 h, inclina��o inferior So = 0,005 e profundidade m�dia do fluxo = 3 m.

  16. Program ONLINE_ROUTING_05 solves the Muskingum-Cunge method of flood routing, with routing parameters based on peak flow. Test this program using Example 9-9 in the text.

    O programa ONLINE_ROUTING_05 resolve o m�todo de roteamento de inunda��o Muskingum-Cunge, com par�metros de roteamento baseados no pico de fluxo. Teste este programa usando o Exemplo 9-9 no texto.

  17. Run ONLINE_ROUTING_05 using the following data: peak discharge = 500 m3/s, time-to-peak = 5 h, time base = 15 h, channel bed slope = 0.0008, flow area corresponding to the peak discharge = 200 m2, channel top width corresponding to the peak discharge = 50 m, rating exponent β = 1.65, reach length = 15 km, time interval Δt = 1 h. Report peak outflow and time-to-peak.

    Execute ONLINE_ROUTING_05 usando os seguintes dados: descarga de pico = 500 m3 / s, tempo a pico = 5 h, base de tempo = 15 h, inclina��o do leito do canal = 0,0008, �rea de fluxo correspondente � descarga de pico = 200 m2, largura superior do canal correspondendo ao pico de descarga = 50 m, expoente nominal ~ = 1,65, alcance = 15 km, intervalo de tempo ~t = 1 h. Relatar vaz�o de pico e tempo a pico.

  18. Given the following inflow hydrograph to a stream channel reach, use ONLINE_ROUTING_05 to calculate the outflow hydrograph.

    Dado o seguinte hidrograma de entrada para um alcance de canal de fluxo, use ONLINE_ROUTING_05 para calcular o hidrograma de sa�da.

    Time (h) 0 1 2 3 4 5 6 7 8 9 10 11 12 13
    Flow (m3/s) 10 20 40 80 160 320 400 320 240 160 80 40 20 10

    Assume channel bed slope = 0.001, flow area corresponding to the peak discharge = 800 m2, channel top width corresponding to the peak discharge = 35 m, rating exponent β = 1.6, reach length = 3 km, and time interval Δt = 1 h.

    Assuma a inclina��o do leito do canal = 0,001, a �rea de fluxo correspondente � descarga de pico = 800 m2, a largura superior do canal correspondente � descarga de pico = 35 m, o expoente de classifica��o ~ = 1,6, alcance de alcance = 3 km e intervalo de tempo ~t = 1 h.

  19. Given the following inflow hydrograph to a channel reach, calculate the outflow hydrographs for: (a) a reach length of 4 km, and (b) for a reach length of 5 km.

    Dado o seguinte hidrograma de entrada para um alcance de canal, calcule os hidrogramas de vaz�o para: (a) um comprimento de alcance de 4 km e (b) para um comprimento de alcance de 5 km.

    Time (h) 0 1 2 3 4 5 6 7 8 9 10 11 12
    Flow (m3/s ) 5 8 12 20 28 33 29 22 19 13 8 6 5

    Assume channel bed slope = 0.0015, flow area corresponding to the peak discharge = 42 m2, channel top width corresponding to the peak discharge = 18 m, and rating exponent β = 1.5.

    Suponha que a inclina��o do leito do canal seja = 0,0015, a �rea de vaz�o correspondente � descarga de pico = 42 m2, a largura superior do canal correspondendo � descarga de pico = 18 m e o expoente nominal ~ = 1,5.

  20. Calculate the hydraulic diffusivity for the following flow conditions: channel bed slope = 0.002, mean flow depth 4 m, mean flow velocity 2 m/s, and rating exponent β = 1.6. Compare the two cases: (a) without inertia, and (b) with inertia.

    Calcule a difusividade hidr�ulica para as seguintes condi��es de fluxo: inclina��o do leito do canal = 0,002, profundidade m�dia do fluxo 4 m, velocidade m�dia do fluxo 2 m / se expoente nominal ~ = 1,6. Compare os dois casos: (a) sem in�rcia e (b) com in�rcia.


REFER�NCIAS

   [Top]   [M�todo Muskingum]   [Ondas Cinem�ticas]   [Ondas Difusivas]   [M�todo Muskingum-Cunge]   [Ondas Din�micas]   [Quest�es]   [Problemas]  

  1. Abbott, M. A. (1975). "Method of Characteristics," in Unsteady Flow in Open Channels, Vol. 1, K. Mahmood and V. Yevjevich, editors, Fort Collins, Colorado: Water Resources Publications.

  2. Agricultural Research Service, U.S. Department of Agriculture. (1973). "Linear Theory of Hydrologic Systems," Technical Bulletin No. 1468, (J. C. 1. Dooge, author), Washington, D.C.

  3. Crandall. S. H. (1956). Engineering Analysis. Engineering Society Monographs. New York: McGraw-Hill.

  4. Cunge. J. A. (1969). "On the Subject of a Flood Propagation Computation Method (Muskingum Method)," Journal of Hydraulic Research. Vol. 7, No.2, pp. 205-230.

  5. Dooge, J. C. 1., W. B. Strupczewski, W.B. and J.J. Napiorkowski. (1982). "Hydrodynamic Derivation of Storage Parameters of the Muskingum Model," Journal of Hydrology. Vol. 54, pp. 371-387.

  6. Fread, D. L. (1985). "Channel Routing," in Hydrological Forecasting. M. G. Anderson and T. P. Burt, editors. New York: John Wiley.

  7. Hayami, S. (1951). "On the Propagation of Flood Waves," Bulletin of the Disaster Prevention Research Institute. Kyoto University, Kyoto, Japan, No.1, December.

  8. Kleitz, M. (1877). "Note sur la Theorie du Mouvement non Permanent des Liquides et sur application a la Propagation del Crues des Rivieres, (Note on the Theory of Unsteady Flow of Liquids and on Application to Flood Propagation in Rivers)," Annales des Ponts et Chaussees, Ser. 5, Yol. 16, 2e semestre, pp. 133-196.

  9. Liggett, J. A., and I. A. Cunge. (1975). "Numerical Methods of Solution of the Unsteady Flow Equations," in Unsteady Flow in Open Channels. Yol. 1, K. Mahmood and V. Yevjevich, editors. Fort Collins, Colorado: Water Resources Publications.

  10. Lighthill, M. J., and G. B. Whitham. (1955). "On Kinematic Waves. I. Flood Movement in Long Rivers," Proceedings of the Royal Society of London. Vol. A229, May, pp. 281-316.

  11. McCarthy, G. T. (1938). "The Unit Hydrograph and Flood Routing," unpublished manuscript, presented at a Conference of the North Atlantic Division, U.S. Army Corps of Engineers, June 24.

  12. Natural Environment Research Council. (1975). Flood Studies Report. Vol. 3: Flood Routing. London. England.

  13. Ponce, V. M., and D. B. Simons. (1977). "Shallow Wave Propagation in Open Channel Flow," Journal of the Hydraulics Division, ASCE, Vol. 103, No. HYI2, December, pp. 1461-1476.

  14. Ponce, V. M., R. M. Li, and D. B. Simons. (1978). "Applicability of Kinematic and Diffusion Models." Journal of the Hydraulics Division, ASCE, Vol. 104, No. HY3, March, pp. 353-360.

  15. Ponce, V. M., and V. Yevjevich. (1978). "Muskingum-Cunge Method with Variable Parameters," Journal of the Hydraulics Division, ASCE, Vol. 104, No. HY12, December, pp. 1663-1667.

  16. Ponce, V. M., and F. D. Theurer. (1982). "Accuracy Criteria in Diffusion Routing," Journal of the Hydraulics Division, ASCE, Vol. 108, No. HY6, June, pp. 747-757.

  17. Ponce, V. M. (1991). "New perspective on the Vedernikov number," Water Resources Research, Vol. 27, No. 7, July, pp. 1777-1779.

  18. Roache. P. (1972). Computational Fluid Dynamics. Hermosa Publishers, New Mexico: Albuquerque.

  19. Seddon, J. A. (1900). "River Hydraulics," Transactions, ASCE, Vol. 43, pp. 179-229.

  20. Woolhiser. M. H., and J. A. Liggett. (1967). "Unsteady One-Dimensional Flow Over a Plane: The Rising Hydrograph." Water Resources Research. Vol. 3, No. 3, pp. 753-771.

SUGGESTED READINGS

  • Agricultural Research Service, U.S. Department of Agriculture. (1973). "Linear Theory of Hydrologic Systems." Technical Bulletin No. 1468, (J. C. I. Dooge, author), Washington, D.C.

  • Cunge, J. A. (1969). "On the Subject of a Flood Propagation Computation Method (Muskingum Method)." Journal of Hydraulic Research, Vol. 7. No.2. 1969, pp. 205-230.

  • Fread, D. L. (1985). "Channel Routing," in Hydrological Forecasting, M. G. Anderson and T. P. Burt, editors, New York: John Wiley.

  • Fread, D. L. (1993). "Flow Routing," Chapter 10 in Handbook of Hydrology, D. R. Maidment, editor, New York: McGraw-Hill.

  • Lighthill, M.J., and Whitham. G. B. (1955). "On Kinematic Waves. I. Flood Movement in Long Rivers," Proceedings of the Royal Society of London, Vol. A229, May, pp. 281-316.

  • Natural Environment Research Council. (1975). Flood Studies Report. Vol. 5: Flood Routing, London, England, 1975.


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