SECOND-ORDER FDS SCHEME ON ADAPTIVE GRID FOR 1-D TRANSIENT FLOWS WITH CONTROL STRUCTURES

 

AKHILESH KUMAR JHA, JUICHIRO AKIYAMA and MASARU URA

 

Dept. of Civil Engineering, Kyushu Institute of Technology

Sensui-cho 1-1, Tobata-ku, Kitakyushu-shi 804-8550, Japan

Tel: +81 93 884-3117, e-mail: amarnath@tobata.isc.kyutech.ac.jp

 

 

ABSTRACT

The second-order accurate flux difference splitting scheme based on Lax-Wendroff numerical flux is implemented on a self-adjusting grid for solving one-dimensional transient free surface flows. The finite-difference grid adjusts itself by averaging the local characteristic velocities with respect to the signal amplitude. This grid adjusting procedure, developed by Harten and Hyman, further enhances shock-resolution of the second-order scheme. The Roe's approximate Jacobian is used for conservation and consistency while theoretically sound treatment for satisfying entropy inequality condition ensures physically realistic solutions. Improvement in resolution of discontinuities by the self-adjusting grid is examined through numerical examples. The numerical results are verified against analytical and experimental results. The model's capability to simulate presence and operation of is investigated through some exacting problems.

 

Keywords: unsteady flow, mathematical modeling, flux-difference splitting, shock-resolution, varying grid, hydraulic structures

 

INTRODUCTION

Mathematical modeling is widely recognized as an efficient tool for solving important flow problems. Recently, advances made in the field of gas dynamics in obtaining high resolution of discontinuous flows has shifted the focus of research from classical schemes to more sophisticated high-resolution, shock-capturing schemes for solving flow problems with strong discontinuities. The application of flux-splitting technique ( e.g. Fennema and Chaudhry (1986), Jha et al.(1996)), flux-difference-splitting technique (e.g. Glaister (1988), Jha et al. (1995)) and higher-order TVD and ENO schemes (Yang et al. (1993)) to one-dimensional transient free surface flows have been reported with varying success.

Rapidly varying flow or flow with control structures are likely to have some form of flow discontinuities. The computation of discontinuous flows by shock-fitting technique involves problem of detecting and tracking a bore. On the other hand, the shock-capturing technique, unless extended to higher-order accuracy, might smear a shock when applied to a fixed-grid finite-difference scheme. An alternative to these techniques is to track a shock and make its location coincide with a mesh point and then use a finite-difference scheme capable of perfectly resolving a stationary shock. Harten and Hyman (1983) devised a self-adjusting grid that, when used with appropriate finite-difference schemes, yields perfect shock-resolution by ensuring that a shock always lies on a mesh point. Jha (1995) applied this concept to shallow water equation and reported high shock-resolution with Roe's (1981) first-order scheme.

In this paper, a second-order flux-difference-splitting scheme is developed using Lax-Wendroff numerical flux and Roe's (1981) approximate jacobian on the adaptive grid developed by Harten and Hyman (1983). Numerical examples examining improvements in shock-resolution due to the self-adjusting grid and model's capability to simulate operation of hydraulic structures are also presented.

 

GOVERNING EQUATIONS

The governing equations for one-dimensional transient free surface flows in a prismatic channel of arbitrary cross section are

 

; ; (1a,b,c)

 

S is the column vector of source terms. A = cross-sectional area of flow; u = velocity; g = acceleration due to gravity; S_o = bed slope and S_f = friction slope; F_h = hydrostatic pressure force. The flux vector E is related to vector U through its Jacobian J as

 

: (4a,b)

 

where W(h) is the channel width at distance h from channel bottom. The governing equations are known to be hyperbolic which implies that J has a complete set of independent and real eigenvectors expressed as

 

; (5a,b)

 

where c = celerity. The eigenvalues of J are given by

 

(6)

 

For conservative evaluation of E by Eq.(4a), Roe (1981) constructed an approximate Jacobian that uses following average values of velocity and celerity

 

; (7a,b,c)

 

SECOND-ORDER SCHEME ON FIXED GRID

The second-order accurate flux-difference-splitting scheme for one-dimensional transient free surface flows can be written as

 

; g = Dt/Dx (8)

 

where i and t = space and time indices, respectively; = time increment and = finite difference grid size in space (Fig.1).

 

Fig.1 Fixed and Varying Grid

 

The source term has been dropped from the present consideration as it can be treated separately once general formulation for the homogeneous part has been worked out. All variables are computed at known time level t, if not indicated otherwise. F_{i +1/2} and F_{i -1/2} are called numerical fluxes. The Lax-Wendroff numerical flux using Roe's approximate can be expressed as

 

(9)

 

where k= wave number, a= wave strength and f is flux limiter. a is defined as

 

(10)

 

The flux limiter is designed to prevent oscillations that might appear due to second order of accuracy. It is a non-linear function of

 

(11)

 

There are several types of non-linear functions of r (Sweby, 1984) available in the literature. We use Van Albada limiter in this study which is expressed as

 

(12)

 

SELF-ADJUSTING GRID

The details of self-adjusting grid is referred to Harten and Hymen(1983) and Jha(1995) and only important equations are presented herein. The self-adjusting grid and the underlying fixed grid are shown in Fig.1. The end-points of the variable grid are computed as

 

(13)

 

where The other terms in Eq.(13) are computed as

 

; (14a,b)

 

; (15a,b)

 

(16)

 

The computed interval end-points are finally modified by the following tests

 

(17)

 

where

At each step of computation, new interval end points are computed and the grid is automatically adjusted according to evolving solutions.

 

SECOND-ORDER SCHEME ON SELF-ADJUSTING GRID

A conservative scheme on a variable mesh can be written as (Harten and Hyman(1983))

 

(18a,b)

 

and the Lax-Wendroff numerical flux on the self-adjusting grid is expressed as

 

(19)

 

; (20a,b)

 

NUMERICAL SIMULATIONS AND RESULTS

The examples are suitably designed to examine effect of varying grid on shock-resolution and capability of the model to respond to sudden operations of hydraulic structures. All computations have been done with the Courant number 0.9.

Eq.18 yields integral solution of the problem which may be significantly different from the point-wise values at the finite-difference nodes in the case of rarefaction waves. Consequently, plotting of these integral values as point-wise values may indicate the existence of several constant states, which is entirely a problem of plotting algorithm. The problem can be avoided to some extent by suitable averaging procedure ( Harten and Hyman 1983). However, we have ignored this aspect as it only affects to the appearance of the plotted profile.

 

Sudden Opening of Sluice Gate

The first example considers sudden opening of a sluice gate in a channel (the so called dam-break problem). The rectangular channel is 20 m long, horizontal and frictionless. The sluice gate is placed at 10 m from either end of the channel. The gate retains 10 m deep still water to its left side while the remaining half of the channel has 0.05 m deep stationary water. The discontinuity in depth at the initiation of the computation simulates the hydraulic condition resulting from sudden opening of the sluice gate. Both upstream and downstream boundaries are kept closed. The computations are done with fixed-grid size of 0.10 m.

 

 

The computed depth and velocity profiles soon after 0.2 seconds are shown in Figs.2 and 3, respectively. The analytical solutions are also plotted therein. A noticeable improvement in the shock-resolution is seen as a result of using varying-grid. The depth and velocity computed by both fixed-grid and varying-grid schemes agree perfectly with the analytical solution.

 

Sudden Closure of Sluice Gate

This example considers the flow resulting from sudden closure of a sluice gate in the channel of previous example. Initially the sluice gate is fully open and the channel has 0.064 m deep water flowing at 1.82 m/s (Froude no.=2.30) throughout it's length. At time 0.05 seconds the sluice gate is instantaneously closed. Upstream of the sluice gate, a reflected bore is formed that travels upstream, leaving still water behind. Downstream of sluice gate, a negative wave is formed whose profile stretches in the downward direction. At supercritical inflow upstream, both depth and velocity is specified while no boundary condition is required at the supercritical outflow downstream.

The computed results soon after 3.0 seconds are shown in Fig.4 along with the analytical solution. The bore upstream, as well as the negative wave downstream, are very well simulated by models on fixed as well as varying-grid. The effect of varying-grid in terms of enhanced shock-resolution is less obvious in this case than in the previous example.

 

 

 

Hydraulic Jump in a Channel

The next problem considers hydraulic jump in a channel. This example is taken from experiments conducted by Gharangik(1988) in a 13.9m long and 0.45 m wide straight, horizontal, rectangular channel with the Manning's values between 0.008 and 0.011. The constant discharge was 0.053 m³/s. The upstream flow depth was 0.064m (velocity =1.82 m/s,Foude no. 2.3) and the conjugate depth was 0.17m (velocity=0.69m/s, Froude no.=0.53). The grid size for this problem is 0.05 m. At the upstream end, both depth and velocity is specified as required for supercritical inflow. The downstream end is provided with an overflow weir that is designed to maintain required depth for the formation of hydraulic jump. The weir across the channel width has crest at 0.048.

The steady state results are compared with experimental data in Fig.5. The location of jump at about 1.8 m agrees well, obtained with the Manning's n equal to 0.009, with the experimental data and so does the jump height. It may be noted, however, that the shallow water equation yields sharply resolved discontinuity against diffused experimental jump profile. There is no noticeable difference between results by fixed and varying-grid formulations. It may also me noted that the downstream weir is very well simulated by the models.

 

CONCLUSIONS

The second-order accurate flux difference splitting scheme based on Lax-Wendroff numerical flux is implemented on a self-adjusting grid for solving one-dimensional transient free surface flows. The model is applied to several specially designed problems and the results are compared with the model on fixed-grid and also with analytical solutions. It is concluded that the models yield very good results for all the problems considered so far. The models can also accurately compute the operation of such hydraulic structures as weir and sluice gates, placed at or within the boundaries. It is also concluded that, unlike the first-order accurate models (Jha (1995)), the improvement in shock-resolution due to varying-grid is not significant. This may be due to the fact that the shock-resolution by the second-order accurate scheme on fixed-grid itself is very good. However, in case of very coarse spatial mesh, some advantage may be gained by using varying-grid.

 

REFERENCES

Fennema, R.J. and M.H. Chaudhry (1986), Explicit Numerical Schemes for Unsteady Free Surface Flows with Shocks, Water Resources Res., AGU, 22(13), 1923-1930.

Glaister, P. (1988), Approximate Riemann Solution of the Shallow Water Equations, J. of Hydr. Res., IAHR, 26(3), 293-306.

Harten, A. and J.M. Hyman (1983), Self-Adjusting Grid Method for One-Dimensional Hyperbolic Conservation Laws, J. Computational Phys., 50, 235-269.

Jha, A.K., J. Akiyama and M. Ura (1996), A Fully Conservative Beam and Warming Scheme for Transient Open Channel Flows, J. Hydr. Res., IAHR, 34(5), 605-621.

Jha, A.K., J. Akiyama and M. Ura (1995), First and Second-Order Flux Difference Splitting Schemes for Dam-Break Problem, J. Hydr. Engrg., ASCE,121(12), 877-884.

Jha, A. K. (1995), Flux Difference Splitting on Self-Adjusting Grid for 1-D Transient Free Surface Flows, J. Hydroscience and hydraulic Engrg., 13(2), 43-54.

Roe, P.L. (1981), Approximate Riemann Solvers, Parameter Vectors and Difference Schemes, J. Computational Phys., 43, 357-372.

Yang, J.Y., C.A. Hsu and S.H. Chang (1993), Computation of Free Surface Flows, PartOne-dimensional dam-break flows, J. Hydr. Res., IAHR, 31(1),19-34.

Sweby, P. K. (1984), High Resolution Schemes using Flux Limiters for Hyperbolic Conservation Laws., SIAM J. Numer. Anal., 21, 995-1101.

Gharangik, A. M. (1988), Numerical simulation of hydraulic jump, M.Sc. thesis, Washington State University, Wash. U.S.A.