A multiple grid approach for open channel flows with strong shocks

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Abstract

Explicit finite difference schemes are being widely used for modeling open channel flows accompanied with shocks. A characteristic feature of explicit schemes is the small time step, which is limited by the CFL stability condition. To overcome this limitation, in this work, we demonstrate the advantages of using a multiple grid formulation coupled to MacCormack scheme. This formulation is tested by simulating a stationary hydraulic jump and the numerical results compared with the experimental findings. Focus in this work is equally distributed between the accuracy of the solution and on the computational savings.

Introduction

At the outset the motivation in performing this work is outlined. The equations that describe the flow in open channels are a set of non-linear partial differential equations, the analytical solutions of which are available only for ideal cases. Hence they are often solved using various numerical schemes. A review of the ongoing numerical work in the field of computational hydraulics indicates that the finite difference techniques are being widely used [1], [2], [3], [4] for solving this set of equations. The results indicate that the finite difference schemes can satisfactorily simulate a wide variety of free surface flows with the added advantage being their ease in coding. The family of finite difference schemes can be classified to either as implicit schemes or explicit schemes based on the choice of time-step. For flows accompanied with shocks, there is a wide consensus among the researchers for using second- and higher-order explicit schemes. This decision is largely based on the accuracy of the solution when compared to the one obtained using first-order schemes. A characteristic feature of the solution obtained using higher-order (⩾2) schemes is the presence of oscillations in the vicinity of the shock front. These oscillations (also addressed to in literature as `wiggles', `over shoots/under shoots') are purely a numerical phenomenon and are absent in actual physical flow. They need to be smoothed or suppressed, at the instant of their generation as they have the potential to amplify with time and space. Since there is no natural damping mechanism present in the governing, the solution is filtered by adding some viscosity. To this end, the procedure originally suggested by Jameson et al. [5] has found wide application.

The choice of the time-step while implementing all the explicit schemes is limited by the CFL stability criteria. For flow problems, where the simulation is carried out for long time periods, a small time-step warrants additional computational resources. This category of simulations includes all the flows for which the steady-state solution is sought. A general practice to arrive at steady-state (also known as stationary state) values is by using a false transient approach. In this approach, time is used as an iterative parameter. Starting from the initial time level for which all the variables are known, the solution is marched with time, until the desired time period is reached. For arriving at stationary profiles, the desired time period is set very large. As the time-step resulting from the CFL criteria is very small, a researcher has to wait a substantial time before the code converges to steady state. This problem is more manifest in the routine life of design engineers, who while trying to arrive at an optimum cross-section of the channel needs to run and rerun their codes for various flow scenarios coupled with different channel properties.

Readers familiar with the state of art numerical algorithms in computational hydraulics will agree that any algorithm which can accelerate the convergence of solution to steady state is of immense use. Characteristic features of this algorithm should include: (i) it can be readily implemented into the existing codes; (ii) it should accelerate the convergence of solution and (iii) the solution obtained by using this technique should be consistent to the one obtained by using the unmodified code. It is in this direction that the multiple grid methodology looks promising. As we discuss in the course of this document, a multiple grid formulation can satisfy all the three criteria. To show that (iii) holds good, which is perhaps the most important for the design community, we have tested this formulation for the most critical flow, hydraulic jump. A hydraulic jump in a open channel is formed when ever flow changes from supercritical to sub-critical. It is a widely used phenomenon for mixing chemicals in water, for dissipating the excess energy of water and for raising the water levels in channels. Given its wide importance, proper simulation involves a satisfactory prediction of the location and height of jump.

The intent of this paper is to show the robustness of the adapted multiple grid strategy for simulating stationary hydraulic jump profiles. The obtained numerical results are compared with the experimental data reported in the literature. To maintain uniformity in the paper while making it self contained, we have dealt more in detail on the implementation issues of the multigrid approach, while only briefly touching on the numerical scheme, which is well documented in the cited literature. The effect of various coarsening strategies (e.g., V-cycle and W-cycle) on the computed shock front are indicated. Focus in this work is equally distributed between the accuracy of the solution and the associated computational savings. The performance of this approach on rectangular and trapezoidal channels is studied. Since the authors are not aware of any comparable work in computational hydraulics, it is hoped that this work will stimulate the interest and application of multiple grid methods for modeling free surface flows.

Section snippets

Governing equations

Based on the principles of conservation of mass and momentum, the basic flow equations that describe the one-dimensional flow in open channels can be written as [3]At+Qx=0,Qt+xQ2A+gIp=gA(S0−Sf),where A is the cross-sectional area of flow, Q is the flow discharge, g is the acceleration due to gravity, Ip is the hydrostatic pressure term, S0 is the bottom slope and Sf is the friction slope of the channel. The pressure term is given asIp=∫0h(h−η)b(η)dη,in which h is the flow depth, η is

Numerical scheme

The MacCormack finite difference scheme, which has found wide application [2], [3], [6] has been used in this investigation for the numerical solution of the flow equations, , . It is second-order accurate in space and time. Starting from the initial time level, the solution at the new time level is computed in a two-tier process utilizing predictor and corrector steps. These can be written as:
Predictor step:ft=fip−finΔt,fx=fi+1n−finΔx.Corrector step:ft=fic−fipΔt,fx=fip−fi−1pΔx.Here f

Stability condition

The choice of time-step is based on the CFL stability criteria. For the set of , , this condition can be written as [3]Δt=CnΔxmax(|u|+gh),where Cn is the Courant number (⩽1), and Δx is the uniform grid spacing. With selected grid spacing and known flow conditions, the time-step is computed using Eq. (11).

Multiple-grid preliminaries

Before we start the discussion we briefly discuss the implementation of standard finite difference schemes. In these formulations, starting from the initial time level (t=0), where the flow variables are known, the time-step (Δt) is first computed by using Eq. (11). At this new time level (t=t+Δt), Eq. (7) is used for computing the values of the flow variables at all the interior nodes. Applying appropriate boundary conditions at the end nodes completes the solution at this time level. The flow

Application

The multiple grid approach coupled with the above-mentioned MacCormack scheme has been applied for simulating mixed flow conditions (Fig. 3). A flow which is supercritical (Fr>1) at the upstream end and sub-critical (Fr<1) at the downstream, as in hydraulic jump, is a typical example for mixed flow conditions. The present numerical model has been run to the experimental data of Gharangik and Chaudhry [6]. Their test flume is of rectangular section 14-m long and 0.46-m wide. The experiments were

Conclusions

In this work the equations that govern the flow in open channels are numerically solved using a finite differences scheme coupled to a multiple grid algorithm. The focus was laid on simulating flows accompanied with hydraulic jump. The effect of various coarsening strategies, both on the accuracy of the solution and on computational savings has been studied. The tests conducted over a wide range of flows and channel cross-sections indicate the reliability of the present formulation. The

References (10)

  • M Rahman et al.

    Simulation of dam break flow with grid adaptation

    Adv. Water Res.

    (1998)
  • Z Zhmed et al.

    Finite difference scheme for longitudinal dispersion in open channels

    J. Hydraul. Res.

    (1999)
  • M.H Chaudhry

    Open-Channel Flow

    (1993)
  • M.B Abbott

    Computational Hydraulics: Elements of Theory of Free Surface Flows

    (1979)
  • A. Jameson, W. Schmidt, E. Turkel, Numerical solutions of the Euler equations by finite volume methods using...
There are more references available in the full text version of this article.

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