Numerical Solution of Boussinesq Equations to Simulate Dam-Break Flows Pranab K. Mohapatra 1 and M. Hanif Chaudhry, F.ASCE 2
Abstract: To investigate the effect of nonhydrostatic pressure distribution, dam-break flows are simulated by numerically solving the one-dimensional Boussinesq equations by using a fourth-order explicit finite-difference scheme. The computed water surface profiles for different depth ratios have undulations near the bore front for depth ratios greater than 0.4. The results obtained by using the Saint Venant equations and the Boussinesq equations are compared to determine the contribution of individual Boussinesq terms in the simulation of dam-break flow. It is found that, for typical engineering applications, the Saint Venant equations give sufficiently accurate results for the maximum flow depth and the time to reach this value at a location downstream of the dam. 001: 10. 1061/(ASCE)0733-9429(2004) 130:2(156) CE Database subject headings: Dam failure; Boussinesq equations; Finite difference method; Pressure distribution; Water flow; Simulation.
Introduction The analysis of dam-break flow (DBF) is important for emergency planning and preparedness, as it poses high risk to life and property. Two important parameters obtained from the DBF studies are the maximum flow depth at a downstream location and time to reach this value. Traditionally, DBF is studied by numerical solution of the Saint Venant equations, which are derived assuming hydrostatic pressure distribution in the vertical direction. However, pressure distribution is nonhydrostatic immediately after the failure of the dam (Pohle 1952; Kosorin 1983; Strelkoff 1986). Basco (1983, 1989) presented the limitations of the Saint Venant equations for dam-break flow calculations. Mohapatra et al. (1999) studied dam-break flows by using two-dimensional Euler equations in a vertical plane to show the effects of nonhydrostatic pressure distribution. Boussinesq theory can be applied to finite amplitude, quasilong waves propagating in shallow water. Palaniappan (1981) used the Boussinesq equations to study the flow in the tidal region of rivers with curved boundaries and sloping bottoms. Gharangik and Chaudhry (1991) simulated a hydraulic jump by the numerical solution of Boussinesq equation by a finite-difference formulation. However, they neglected the first Boussinesq term containing the mixed time-space derivative. Raman and Chaudhry IResearch Associate, Civil and Env. Engrg., Univ. of South Carolina, Columbia, SC 29208. 2Mr. and Mrs. Irwin B. Kahn Professor and Dept. Chair, Civil and Environmental Engineering, Univ. of South Carolina. E-mail:
[email protected] Note. Discussion open until July 1, 2004. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this technical note was submitted for review and possible publication on November 4, 2002; approved on July 11, 2003. This technical note is part of the Journal of Hydraulic Engineering, Vol. 130, No.2, February I, 2004. ©ASCE, ISSN 0733-9429/200412156-159/$18.00.
(1996) solved the same problem with an adaptive grid technique. Carmo et al. (1993) applied the MacCormack finite-difference scheme to solve the two-dimensional Serre equations. They studied the propagation of a solitary wave, dam-break flow, and a solitary wave overpassing an island. Frazao (2002) used the Boussinesq equations to study the secondary, free-surface undulations due to opening of a sluice gate, which cannot be reproduced by numerical models based on hydrostatic pressure distribution. Walkley (1999) presented an extensive list of references on the Boussinesq approach. In most of the references cited above, steady Boussinesq equations are used to study rapidly varied flows in open channels and the first Boussinesq term is excluded. On the use of Boussinesq equations for studying DBF, a detailed quantitative approach is missing and no definite conclusions are drawn, e.g., the capabilities to simulate long amplitude waves, the contributions of individual Boussinesq terms, the effects of bed slope, and roughness characteristics on the evolution of the free surface. In this paper, Boussinesq equations are solved numerically to simulate DBF with nonhydrostatic pressure distribution. The governing equations are first presented. The numerical solution procedure, and the initial and boundary conditions are then discussed. This is followed by a presentation of the results and the conclusion of the investigations.
Governing Equations One-dimensional Boussinesq equations may be written as (Chaudhry 1993):
Con ffn uity Equaffon
156/ JOURNAL OF HYDRAULIC ENGINEERING © ASCE / FEBRUARY 2004
ah
auh
at
ax
-+-=0
(1)
Momentum Equation 2 ) auh a gh at+ ax u-h+T+ B l+ B 2+ B 3 =gh(So-Sj)
(?
Predictor Part (2)
The predicted variables are obtained from the known variables by using the forward finite differencing for both the time and space derivatives.
In the above equations, x=longitudinal direction; u=depthaveraged velocity in the x-direction; t=time; h=flow depth; g=acceleration due to gravity; So=bed slope in the x direction; and Sj=friction slope in the x direction. The Boussinesq terms, B 1 , B 2, and B 3 in Eq. (2) account for the vertical acceleration:
(5)
p_~[
k
u i - hfJ (Uh)i
~ ~t
+6
k
~x {(T)i+2
_
k
k
S(T)i+l +7(T)J
I
3
__ h B 13
(~)
(6)
axat' where
2 3 2 3 T=u 2h+ gh _ uh a u + h (au)2 2 3 ax 2 3 ax
(3)
It may be noted here that the Saint Venant equations are a special case of the Boussinesq equations when the Boussinesq terms are dropped. The first Boussinesq term, B 1 , contains mixed space and time derivatives. The friction slope, S j' is calculated from the Manning equation
(4)
where n = Manning roughness coefficient and the channel is assumed to be wide and rectangular. An important assumption in the derivation of the above equations is that the velocity in the vertical direction varies linearly from zero at the bed to the maximum value at the surface. It is important to note that the governing equations do not account for the effective stresses arising due to laminar viscous stresses, turbulence stresses and stresses due to depth averaging. In addition, the flow may be two-dimensional and there may be transport of sediments. These phenomena are not taken into consideration in the governing equations.
(7)
All terms on the right-hand side of Eq. (7) are at the known time level, k, and the derivatives are approximated by the central finite difference.
Corrector Part The corrected variables are obtained from the predicted variables by using the forward finite differencing for the time derivatives and backward finite differencing for the space derivatives: 1 ~t he=hfJ+ -6 [ -(uh)P1 - 2 +S(uh)P1 - 1 -7(uh)P] 1 I ~x 1
(S)
1 [ 1 ~t u eI =-6 -{-(T)P_ +S(T)P_ -7(T)P} he (uh)fJ+ I ~x 1 2 1 1 1 I
+Mghf(So-Sj)]
(9)
Tin Eq. (9) is the same as that given in Eq. (7); however, it uses the variables obtained in the predicted part.
Intermediate
Numerical Solution Eqs. (1) and (2) are nonlinear partial differential equations and generalized analytical solutions of these equations are not available. Therefore, a numerical approach is used here for their solution. The Boussinesq equations have third-order terms [Eq. (2)] and it is necessary to use third- or higher-order accurate numerical methods to solve these equations (Abbott 1979). In the present work, the method developed by Gotlieb and Turkel (1976) (Chaudhry 1993) is extended to solve the Boussinesq equations for studying the DBF waves. A finite-difference method is used to solve the governing equations [Eqs. (1) and (2)] on a nonstaggered grid. The flow variables (uk+l,h k +1 ) at an unknown time, t+~t, are computed explicitly from the variables (uk,h k) at the known time level, t. First, an intermediate flow field (u,h) is computed by a predictorcorrector procedure, wherein B 2 and B 3 are considered and B 1 is neglected as this contains mixed derivatives. Then, the intermediate flow field is recomputed to obtain the final solution by taking B 1 into account.
Intermediate flow variables (u,h) are evaluated by taking the average of the variables at the known time level, k, and the corrector part:
_ h~+ hf h = 2- 1
(10)
(11)
Final The intermediate flow variables are evaluated excluding the first Boussinesq term, B 1. Therefore, the intermediate velocity is modified by (12)
It results in
JOURNAL OF HYDRAULIC ENGINEERING © ASCE 1 FEBRUARY 2004/157
1.2
-
(13)
.....
Wrth Boussinesq terms
.. - -- Wrthout Boussinesq terms
0.9
g R -2l 0.6
B 1 in Eq. (13) has mixed derivatives [see Eq. (3)] and is evaluated
by using the variables at the known time level and at the intermediate level:
~
"0.3
0 60
(a)
70
80
90
100
110
120
130
140
Distance (m)
Initial and Boundary Conditions 1.2
-Wrth Boussinesq terms
The initial condition is given: At t=O.O,
_. -. _. Wrthout Boussinesq terms
0.9
h;=h u
for
i~idam
(15)
h;=h d
for i>idam
(16)
g R -8 0.6
u;=O.O
for all i
(17)
£ 0.3
The finite-difference approximations presented above cannot be applied at the boundaries. At each computational time level, the values of h and u are specified for three nodes at the boundary [see Eqs. (6) and (9)]: At t>O.O, (18)
o-l--~-~--~---.---~-~-~-~
60
70
80
90
(b)
100
110
120
130
140
Distance (m)
Fig, 1, Surface profiles for t max = 15 s after the dam-break: (a) r =0.2 and (b) r=0.5
(19) Ul=U2=U3=0.0
(20)
Uilast= unast-l = Uilast-2 = 0.0
(21)
In the above equations, idam and ilast represent the computational nodes corresponding to the dam and the outflow boundary. The above initial and boundary conditions are based on the assumptions used in the Stoker solution (Stoker 1957). It is assumed that the disturbance has not reached the boundaries. However, the actual conditions may also be used. For example, a nonzero value of U; or an inflow hydrograph at the upstream end may be specified. The time step, ~ t, is computed by using the stability criterion governed by the Courant condition (Chaudhry 1993) and the variables at the end of each time step are smoothened by utilizing the artificial viscosity procedure (Jameson et al. 1981).
independence. These input parameters are used in subsequent studies to investigate the effect of Boussinesq terms. Flood Wave Propagation The surface profiles at time t max = 15 s after the dam break are presented for two different depth ratios (r = 0.2 and 0.5), in Fig. 1. Other physical and numerical parameters are the same as used earlier. Numerical results are obtained by solving the equations, with and without the Boussinesq terms. All four zones are distinct and similar results are obtained by both sets of equations. The numerical experiments with different depth ratios show that the surface profiles obtained with and without the Boussinesq terms are about the same for depth ratios, r
