COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Numerical Simulation of 2D Flood Waves Using TVD Scheme
Wenli Wei 1*, W. Y. He 2
1
2
Institute of Water Conservancy and Hydraulic Engineering, Xi’an University of Technology, Xi’an, 710048 China
Science school, Xi’an University of Technology, Xi’an, 710048 China
E-mail: [email protected]
Abstract This paper is concerned with a High- Resolution mathematical model for 2D shallow water equations; and
the two-dimensional (2D) shallow water equations were split into two systems of equations by using the Strang type
operator splitting method; and were solved with the one-dimensional upwind total variation diminishing (TVD)
scheme and finite volume method (FVM). And this model is used to predict the flood evolution process caused by the
instantaneous partial dam-break, and the simulating results are analyzed qualitatively. The analysis indicates that this
model is fairly effective for simulating dam-break flood waves.
Key words: Shallow water equations; Difference splitting; FVM method; TVD scheme; Dam break waves
INTRODUCTION
In recent studies, the type of shock-capturing scheme called total variation-diminishing (TVD) scheme, was proposed
by Harten [1], and was applied widely in gas dynamics. The TVD scheme has second-order accuracy, is
oscillation-free across discontinuities, and does not require additional artificial viscosity. Therefore it was introduced
to solve hydrodynamics for free-surface flows, in particular, for recent complex dam-break flows. Delis and Skeels
made a comparison with several different TVD schemes (i.e., symmetric, upwind, TVD-MacCormack, and MUSCL
scheme) to predict one-dimensional (1D) dam-break flows [2]. Based on the above research results, a high-resolution
algorithm is proposed for the solution of the 2D shallow water equations by adopting the numerical flux for the TVD
scheme, the Strang type operator splitting method and the finite volume method (FVM).
GOVERNING FLOW EQUATIONS
The conservation laws of the governing flow equations for the physical domain, assuming that the flow is
homogeneous, incompressible, 2D and viscous with hydrostatic pressure distribution and absence of Coriolis and
wind forces, are
∂E ∂F ( E ) ∂G ( E )
+
+
= Q (E)
∂t
∂x
∂y
(1)
In which the variables E, F, G, and Q are defined in matrix forms as follows:
E = (h,qx , qy )
T
(2a)
⎛ q x2 1 2 q xqy ⎞⎟
F ( E ) = ⎜⎜qx , + gh ,
⎟
⎜⎝ h
2
h ⎠⎟
T
(2b)
⎛ qxqy qy2 1 2 ⎞⎟
G ( E ) = ⎜⎜⎜q y ,
, + gh ⎟⎟
⎜⎝
h h
2
⎠⎟
T
(2c)
T
Q ( E ) = ⎢⎡ 0, gh ( s0 x − s fx ), gh (s 0y − s fy )⎥⎤
⎣
⎦
(2d)
⎯ 241 ⎯
Where qx and qy=components of discharge per unit width along x- and y-directions, respectively; sox and soy=bottom
slopes along x- and y-directions, respectively; and sfx and sfy=friction slopes along x-and y-directions, respectively.
The friction slopes sfx and sfy are determined by the Manning formula
S fx =
n 2qx qx2 + qy2
h 4/3
S fy =
;
n 2qy qx2 + qy2
(3a, b)
h 4/3
Where n=Mannings’ flow friction coefficient.
By utilizing a local characteristic approach, the equivalent nonconservative form of equation (1) can be expressed as
∂E
∂E
∂E
+A
+B
= Q (E )
∂t
∂x
∂y
(4)
The Jacobian of the fluxes are,
⎡
⎢ 0
⎢
⎢
∂F
q2
A=
= ⎢⎢c 2 − x2
h
∂E
⎢
⎢ q xqy
⎢− 2
⎢⎣ h
1
qx
h
qy
2
h
⎡
⎢
⎢ 0
⎢
∂G ⎢ q x q y
; B=
= ⎢− 2
∂E
⎢ h
⎢
2
⎢ 2 qy
c
−
⎢
h2
⎣⎢
⎤
0⎥
⎥
⎥
0 ⎥⎥
⎥
qx ⎥
⎥
h ⎥⎦
0
qy
h
0
⎤
⎥
1 ⎥
⎥
qx ⎥
⎥
h ⎥
⎥
qy ⎥
2 ⎥
h ⎦⎥
(5a, b)
Where c is celerity, c2=gh.
The eigenvalues of matrices A and B are,
e A = (qx / h + c, qx / h, qx / h − c ) ; eB = (qy / h + c, qy / h, qy / h − c )
T
T
(6a, b)
The governing equations are known to be hyperbolic, which implies that A and B have complete sets of independent
and real eigenvectors. Therefore, the Jacobians can be written in diagonalized form as
A = eeAe −1;
B = feA f −1
(7a, b)
where e, f and e-1, f-1 are matrices and inverse matrices of eigenvectors of A and B, respectively, and are given by
⎛
⎞⎟
⎜⎜ 1
⎟⎟
0
1
⎜⎜
⎟⎟
⎜⎜q
⎟
qx
x
− c ⎟⎟⎟ ;
e = ⎜⎜ + c 0
⎟⎟
⎜⎜ h
h
⎜⎜ qy
qy ⎟⎟⎟
⎜
⎟
c
⎜⎝ h
h ⎠⎟
⎛
⎞⎟
⎜⎜ 1
⎟⎟
1
0
⎜⎜
⎟⎟
⎜⎜ q
⎟
qx
x
c ⎟⎟⎟ ;
f = ⎜⎜
⎟⎟
⎜⎜ h
h
⎟⎟
⎜⎜qy
q
⎜ + c y − c 0⎟⎟⎟
⎜⎝ h
⎠
h
e −1
f −1
⎛ ⎛q x
⎞
⎞
⎜⎜⎜− ⎜⎜⎝ − c ⎠⎟⎟ 1 0⎟⎟⎟
⎟⎟
⎜⎜ h
⎟⎟
⎜
q
1
= ⎜⎜ −2 y
0 2⎟⎟⎟
h
⎟⎟
2c ⎜⎜
⎜⎜
⎟
q
⎜⎜ ⎛⎜ x + c ⎞⎟ −1 0⎟⎟⎟
⎜⎝ ⎜⎝ h
⎠⎟
⎠⎟
⎛ ⎛q y
⎞
⎞
⎜⎜⎜− ⎝⎜⎜ − c ⎠⎟⎟ 0 1 ⎟⎟⎟
⎟⎟
⎜⎜ h
⎟⎟
1 ⎜⎜ ⎛qy
⎞⎟
= ⎜⎜ ⎜⎜ + c ⎟ 0 −1⎟⎟⎟
⎠
⎟⎟
2c ⎜⎜ ⎝ h
⎟⎟
⎜⎜
q
2 0 ⎟⎟⎟
⎜⎜ −2 x
⎜⎝
h
⎠⎟
(8a-d)
NUMERICAL ALGORITHMS
By using the Strang type operator-splitting method [3] that has been enlarged to include source terms into the splitting. Eq.
(1) can be solved with the one-dimensional upwind and symmetric TVD schemes as
Ein, j+2 = Lx (Δt )Ly (Δt )La (Δt )La (Δt )Ly (Δt )Lx (Δt )Ein, j
with
⎯ 242 ⎯
(9)
Lx Ein, j = Ei*, j = Ein, j −
Δt n
Fi +1/ 2 − Fi−n 1/ 2 ),
(
Δx
1
La Ei**, j = Ein, j+1 = [Ei**, j + E i, j − Δt ⋅ Q(E i, j )],
2
Ly Ei*, j = Ei**, j = Ei*, j −
Δt *
G j +1/ 2 − G j*−1/ 2 )
(
Δy
E i, j = Ei**, j − Δt ⋅ Q(Ei**, j )
(10a, b)
(10c, d)
where Δt is the time step determined from the stability requirement, and Δx and Δy are the mesh spacing in the x
and y directions, respectively. The functions Fi +1/ 2 andG j +1/ 2 are the numerical fluxes in the x and y directions
evaluated at cell faces (i+1/2,j) and (i, j+1/2), respectively (see Fig. 1). In the above, the ordinary differential operator La
adopts the Euler predictor-corrector method for second-order time accuracy. The numerical flux F i +1/ 2 is defined as
Fi +1/ 2 =
1⎡
F (Ei, j ) + F (Ei +1, j ) + Ri +1/ 2Φi +1/ 2 ⎤⎥⎦
2 ⎢⎣
(11)
where Ri +1/ 2 is a matrix whose column vectors are the right eigenvectors of the flux Jacobian ∂F / ∂E evaluated
with some symmetric average of Ei , j and Ei +1, j . Although a simple arithmetic average can be used for the symmetric
average, Roe’s gas dynamic average [5] was used, since it has the ability to satisfy correctly the jump condition over a
discontinuity. The last Ri +1/ 2Φi +1/ 2 represent the anti-diffusive flux contribution that corrects the excessive dissipation of
fist-order numerical flux in a nonlinear way. This term makes the resulting numerical scheme higher-order accurate and
TVD. The numerical flux G j +1/ 2 is similarly defined.
y
i,j+1/2
i-1/2,j
i+1/2,j
Δy
i,j
i,j-1/2
Δx
x
Figure1: A finite control volume element
BOUNDARY CONDITIONS
The inclusion of the boundary is very important in the successful application of any numerical technique. Hyperbolic
equations are particularly sensitive because errors introduced at the boundaries are propagated and reflected
throughout the grid. This, in many cases, may result in instability. In the present application, two types of boundaries
are encountered: solid boundaries and water boundaries. For the solid boundaries, the governing equations do not
include the turbulent viscosity, but the bottom friction, free-slip conditions may be considered, and the normal
discharge to the wall is set to zero in order to represent no flux through the solid boundaries. The water boundaries, in
particular, need to be treated. The local value of the Froude number, or whether the flow is subcritical or supercritical,
is the basis for determining the number of boundary conditions. For the 2D subcritical flow, two external conditions
are specified at the inflow boundary, and one is specified at the outflow boundary. For the supercritical flow, three
boundary conditions at the inflow boundary and none at the outflow boundary have to be specified [4].
2D FRICTIONLESS PARTIAL DAM-BREAK
This problem models a partial dam-break or the rapid opening of a sluice gate with a symmetrical breach. The
discontinuous initial conditions impose severe difficulties, and most of the presently used non-Godunov-type
numerical schemes create enormous errors or fail under such conditions. Here, the computational domain is defined
by a channel 200m long, 200m wide, with a horizontal friction bottom. The symmetric breach is 120m wide and the
thickness of dam is 5m (Fig. 2(a)). At downstream in front of the breach there are two rectangular cylinder barriers
that occupy an area of 15×15m2, the distance among which is shown in Fig. 2(a). Initially, the upstream and
downstream water depths are set at 10 m and 5m, respectively. The dam wall is then breached; and instantaneously
water discharges from the higher to the lower level as a downstream-directed bore while a depression wave
⎯ 243 ⎯
propagates upstream. The water body is discretized by a uniform 5×5m2 mesh that exactly fits the domain. The
time-step Δt is 0.1s. Fig. 2(a) shows computational velocity vectors; and Fig. 2(b) shows the 3D view of the water
surface elevation 8s after dam failure when the waves have not yet reached all the boundaries. At both ends of the
breach, the water depths are smaller than that at the center of the breach. Flow separates from the truncated dam
walls just downstream of the breach and forms counter rotating eddies. One can clearly see that the structure, shape,
and location of vortices, as well as the reflection (bore front of the barrier), and the discontinuities change as time
increases.
200
180
160
y(m)
140
120
100
80
60
40
20
20 40 60 80 100 120 140 160 180 200
x(m)
(a)
(b)
Figure 2: (a) Computational velocity vectors, 8s after Dam-Break; (b) 3D View of Water Surface,8s after Dam-Break
CONCLUSION
In the present paper we have developed the High- Resolution mathematical model for 2D shallow water equations for the
computation of dam-break flows, which combines the second-order TVD scheme, the finite volume method (FVM), the
local characteristic approach and the operator splitting technique. Numerical simulations for the flood evolution process
caused by the instantaneous partial dam-break indicate that this model is fairly effective for simulating dam-break flood
waves. As for applying the approach to solve dam-break problems with natural river geometries, numerous difficulties
need to be overcome under the frame of the finite-difference method. The next step is to adopt a body-fitted coordinate
transformation system to deal with irregular boundaries, and so the High- Resolution mathematical model will be further
modified and extended to multi-dimensional hyperbolic conservation laws in a generalized coordinate system. The results
will be given in a future work.
REFERENCES
1.
Harten A. High resolution schemes for hyperbolic conservation laws. J. Comp. Phys., 1983; 49: 357-393.
2.
Delis AI, Skeels CP. TVD schemes for open channel flow. Int J. Numer. Methods in Fluids, 1998; 26:
791-809.
3.
Strang G. On the construction and comparison of difference schemes. SIAM J. Numer., Anal., 1968; 5: 506.
4.
Garcia R, Kahawita RA. Numerical solution of the St. Venant equations with MacComack finite-difference
scheme. Int. J. Numer. Methods in Fluids, 1986; 6: 259-274.
5.
Roe PL. Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys., 1981; 43:
357.
⎯ 244 ⎯