(1)
Won Kim
Senior Researcher, Korea Institute of Construction Technology
2311, Daehwa-dong, Ilsan-gu, Koyang, Kyonggi-do, 411-712, South Korea
Tel : 82-31-910-0265, Fax : 82-31-910-0251, wonkim@kict.re.kr
Kun-Yeun Han
Professor, School of Engineering, Kyungpook National University
1370, Sankeuk-dong, Puk-gu, Taegu, 702-701, South Korea
Tel : 82-53-950-5612, Fax : 82-53-950-6564, kshanj@kyungpook.ac.kr
Abstract: Open channel flow resulting from Dam-break is transcritical flow. Transcitical flow is a term intended to denote the existence of both supercritical and subcritical flows within a computational domain. The present study proposes a new scheme, the implicit ENO scheme, as a high-resolution scheme for dam-break flow. The implicit ENO scheme is based on the ENO scheme, a new class of uniformly high-order-accurate essentially nonoscillatory schemes, and an implicit scheme, which has the advantage of unconditional stability. The implicit ENO scheme developed in this study can accurately calculate a transcritical flow with no oscillations for both weak and very strong shock waves resulting from dam-break. Throughout all the tests conducted in this study, including idealized and real situations, the implicit ENO scheme demonstrated good accuracy as a high-resolution scheme and stability as an implicit scheme.
Keywords: transcritical flow, ENO, implicit, dam-break flow
An unsteady open channel flow, resulting from the sudden destruction of a dam, can be modeled as a one-dimensional flow. A dam-break flow is discontinous and consists of both supercritical and subcritical flows. A transcritical flow is a term intended to denote the existence of both supercritical and subcritical flows within a single computational domain, i.e., it is equivalent to the term transonic flow in gas dynamics. The major problems that need to be addressed while modeling transcritical flows include handling the differing features of signal propagation in subcritical and supercritical flow regions and maintaining conservation.
Jha et al. (1996) developed the Beam-Warming-Roe scheme that uses Roe’s approximate Jacobian to make the Beam and Warming scheme fully conservative and satisfy the entropy inequality condition. Meselhe et al.(1997) proposed a new scheme, the MESH scheme, which is second order accurate in time and space and include a two-step, predictor-corrector, implicit procedure to advance the solution in time. Jin and Fread(1997) used a characteristics-based, upwind, explicit scheme for the conservation form of complete Saint-Venant equations for nonprismatic channels. Vazquez-Cendon(1999) and Garcia-Navarro et al. (1999) dealt with the numerical solution of transcritical flows in channels with irregular geometry.
The ENO(Essentially Nonoscillatory) scheme is a new class of uniformly high-order-accurate essentially nonoscillatory schemes and was developed by Harten and Osher(1987). In contrast to the earlier second-order TVD schemes, which drop to first-order accuracy at local extrema and maintain second-order accuracy in smooth regions, ENO schemes are uniformly high-order accurate throughout, even at critical points.
One of the characteristics that render explicit schemes attractive is their conceptual simplicity. The implicit scheme is more complicated than the explicit scheme, yet it has the advantage of unconditional stability, allowing for bigger time steps. One of the most important features needed in the practical simulation of a transcritical flow is robustness, i.e. the scheme must be flexible enough to accommodate a wide range of applications, nonexact initial conditions, and severe situations. Therefore implicit schemes are preferred in practice over explicit ones, at least in a one-dimensional flow simulation.
This paper offers a new scheme for analyzing transcritical flows. The performance of an implicit method based on the ENO theory is investigated when applied to 1-D shallow water equations for the simulation of dam-break flows. The implicit ENO model developed in the present study includes the source terms in the Saint Venant equations, and does not require any artificial viscosity terms for capturing discontinuities.
The Saint-Venant equations for a 1-D unsteady flow in nonprismatic channels can be formulated in vector form as follows;
(1)
where:
;
;
(2)
where A = wetted cross-sectional area; Q(x,
t) = discharge; g = gravitational
acceleration; and
= hydrostatic pressure force term.
The term
accounts for the pressure forces
exerted by the walls in nonprismatic channels. Finally,
is the bottom slope, and
is the friction slope. The vector
F is related to the flow variables U through the Jacobian J
of F
with respect to U. The matrix J can now be split into two components,
positive and negative, by the testing sign of the eigenvalues. And G
is the Jacobian of the source term vector (
).
The general form of the finite-difference approximation for the Beam and Warming scheme can be written as (Beam and Warming, 1976; Jha et al., 1996)
(3)
where I
= (2
2) unit matrix;
; and n and n+1 denote grid locations in time. The
expressions in the parentheses before
are operators on
.
Yang et al.(1993) used a normalized Jacobian to accommodate the ENO schemes as follows ;
(4)
Here
are split normalized Jacobians
and are defined by
(5)
where T
= the similarity transformation matrix;
= the inverse of matrix T; and
= the normalized eigenvalues defined
by
(6)
Here sgn denotes the sign function.
In equation (4),
is called the modified flux vector
and consists of the original flux vector F and additional terms of higher-order
accuracy which usually include some nonlinear control terms to avoid oscillatory
solutions.
for a uniformly second-order scheme can be expressed as follows;
(7)
The components of the vector E are defined as
(8)
In equation
(8),
are the components of the following
column vector
(9)
where
and
are defined respectively by
(10)
(11)
The minmod function m is defined as
(12)
and the
function is defined by
(13)
When
=0, this produces a second-order TVD scheme. When
=0.5, this produce a uniform second-order nonoscillatory scheme.
In this study, to construct an implicit second-order ENO scheme the space derivative of the flux term F on the right-hand side of (3) is replaced by equations (4) and (7). Finally, a second-order implicit ENO scheme can be constructed as follows;
(14)
The first application was an idealized dam-break problem that has an analytical solution. The channel was 2,000m long with a dam placed in the middle that separated the reservoir and the tailwater channel. The reservoir depth was fixed at 10m, and the tailwater depth was 0.005m. A dam in the wide, horizontal, rectangular, and frictionless channel was instantaneously removed across its entire width at the beginning of the computation. The grid size was 6.25m and the computed results using the implicit ENO scheme at 60s were then compared with the analytical solutions. The analytical solution was obtained using Stoker’s method(Stoker, 1957).
Fig. 1 Dam-break problem, depth ratio=0.0005
Fig. 2 Effect of the Courant number
Fig. 1 shows the computed results when depth ratio was equal to 0.0005. In the front of the wave, the ENO scheme presented more accurate results than the Beam and Warming scheme. Even though the depth ratio was extremely small, the implicit ENO scheme computed very accurate results with no numerical oscillations. In the recently developed implicit MESH scheme (Meselhe et. al., 1997), the maximum depth ratio accepted was 0.1.
Effect of the Courant number on the accuracy of the computed results was examined for the dam-break problem. A reservoir depth of 10m and tailwater depth of 5m were used for the computation. The computed results are shown in Fig. 2. Because the ENO scheme used in this study is implicit, the computed results for a Courant number greater than 1.0 showed a good accuracy, which thereafter more or less deteriorated as the Courant number was increased.
The experiments were conducted in a 122m-long rectangular channel with a
channel width of 1.22m, logitudinal slope of 0.005, and Manning’s n of 0.009. The dam was placed at the
mid-length section giving a reservoir depth of 0.305m just upstream of the dam.
The grid size was 1.0m, and the time step size was 1.0s. The maximum CFL number
wascalculated as 2.7.
Fig.
3 Comparison with WES results (t=2sec)
Fig. 4 Comparison with WES results (t=5sec)
Fig.
5 Comparison with WES results (t=10sec)
Fig. 6 Comparison with WES results (t=30sec)
Fig.3––6 show the longitudinal water-surface profiles at 2s, 5s, 10s, and 30s, respectively, after the dam-break. As shown in the figure, the computed results were in satisfactory agreement with the experimental data. After the dam collapse, the water level changed every second. The implicit ENO scheme was able to accurately calculate this type of rapid change over all periods of the event. Plus, the implicit ENO scheme proved to be numerically very stable with no numerical oscillations.
The present study proposed a new scheme, the implicit ENO scheme, as a high-resolution scheme for transcritical flow. The implicit ENO scheme is based on the ENO scheme, a new class of uniformly high-order-accurate essentially nonoscillatory scheme, and an implicit scheme, which has the advantage of unconditional stability.
The implicit ENO scheme developed in this study is able to accurately calculate a transcritical flow with no oscillations for very strong shock waves. Plus the implicit ENO scheme can accurately calculate a transcritical flow resulting from dam-break. The upwind scheme used in this study accurately reflects the physical propagation of a transcritical flow. A high degree of accuracy is equally maintained throughout the computational domain without numerical oscillation by using a high resolution ENO scheme. And the implicit ENO scheme simulates the hydraulic characteristics of discontinous points accurately with good stability. Throughout all the tests conducted in this study the implicit ENO scheme showed good accuracy as a high-resolution scheme and stability as an implicit scheme.
References
Beam, R.M. and Warming, R.F. (1976). “An implicit finite-difference algorithm for hyperbolic systems in conservation-law form.” Journal of Computational Physics, Vol. 22, pp. 87-110.
Garcia-Navarro,
P., Fras, A., and Villanueva, I. (1999). “Dam-break flow simulation : some
results for one-dimensional models of real cases.” Journal of Hydrology, Vol. 216, pp.
227-247.
Harten, A. and Osher, O. (1987). “Uniformly high-order accurate nonoscillatory schemes I.” SIAM Journal of Numerical Analysis, Vol. 24, No. 2, pp. 279-309.
Jha, A.K.,
Akiyama, J., and Ura, M. (1996). “A fully conservative Beam and Warming scheme
for transient open channel flows.” Journal
of Hydraulic Research, Vol. 34, No. 5, pp. 605-621.
Jin, M. and
Fread, D.L. (1997). “Dynamic flood routing with explicit and implicit
numerical solution schemes.” Journal of
Hydraulic Engineering, ASCE, Vol. 123, No. 3, pp. 166-173.
Meselhe, E.A.,
Sotiropouos, F., and Holly, F.M., Jr. (1997). “Numerical simulation of
transcritical flow in open channel.” Journal
of Hydraulic Engineering, ASCE, Vol. 123, No. 9., pp. 774-783.
Stoker, J.J. (1957). Water waves. Interscience
Publishers Inc., Wiley and Sons, New York, N.Y.
U. S. Corps of Engineers. (1960). “Floods resulting from suddenly breached dams.” Miscellaneous Paper No. 2-374, Report, Conditions of Minimum Resistance, U.S. Army Engineer Waterways Experiment Station, Corps of Engineers, Vicksburg, Mississippi.
Vazquez-Cendon, M.E. (1999). “Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry.” Journal of Computational Physics, Vol. 148, pp. 497-526.
Yang, J.Y., Hsu, C.A., and Chang, S.H. (1993). “Computations of free surface flows.” Journal of Hydraulic Research, Vol. 31, No. 1, pp. 19-34.