Application of the relaxat ion met hod to model hydraulic jumps

Transcription

1 Application of the relaxat ion met hod to model hydraulic jumps P. J. Montgomery Mathematics and Computer Science Program, University of Northern British Columbia, Prince George, Canada. Abstract A finite difference method, known as the relaxation method, is generalized from a class of relaxation schemes with the ultimate aim of numerically modelling hydraulic jumps at a phase interface. This method has been applied previously to model gravity currents arising from the instantaneous release of a dense volume of fluid. The relaxation scheme is an iterative, second order accurate, timemarching method which is able to capture shocks and interfaces without front tracking or calculation of the eigenvalues of the Jacobian matrix for the flux vector. In this paper, the relaxation scheme will be described, with specific attention paid to the new generalizations included to account for boundary conditions, spatially dependent flux terms, and simple forcing terms. Numerical results will be compared with simple theory using the inviscid Burgers equation, permitting the simplicity of the scheme to be portrayed through this example. The more general case of the shallowwater equations for a single layer in one spatial dimension will then be modelled numerically for an initial release boundary value problem to show that the method is widely applicable to problems involving two fluids of large density differences such as air and water where the systems are sufficiently decoupled. 1 Introduction Gravity currents resulting from the initial release of a dense volume of fluid

2 within a region of less dense fluid have been modelled extensively by using the shallow-water equations [l]. One physical limit of this model, for large density differences between the fluids, recovers the two phase dam-break problem for water flowing beneath a layer of air. The study of gravity current front prediction thus is connected with the problem of finding a phase interface. During a study of the model equations for gravity currents [2], a firstorder hyperbolic system of equations was discussed in the form where R = (-m, m) represents the real number line, a subset of which is defined by R+ = [O, m). In equation (l), the real vectors are defined as U -- u(x,t) E Rn, f = f(u,x) E R", and b = b(u,x) E R" for a positive integer n 2 1. Throughout this article, vectors will appear in boldface notation. Numerical systems such as equation (1) are used to model gravity currents which exhibit the properties of shallow layer flow: accelerations in the horizontal direction are much greater than those in the vertical direction. An excellent review of the properties and various modelling equations for gravity currents has recently been completed by Simpson (l]. Previous work by Montgomery and Moodie [3] has revealed a connection between the Froude number for a hydraulic jump and the Rankine-Hugoniot jump conditions at a discontinuity in the solution. These results have spurred the author to investigate a more general use for the relaxation method in multiphase flow. To solve systems such as (1) with various initial and boundary values, a numerical method presented by Jin and Xin [4] was modified [2]. The original class or relaxation schemes is applicable to initial value problems for hyperbolic systems of conservation laws in several space dimensions. These schemes are finite-difference, iterative, and have been shown to be total variation diminishing (TVD) for scalar hyperbolic conservation laws. For nonlinear hyperbolic systems, the second-order relaxation schemes are stable, conservative, and capture discontinuous solutions in a nonoscillatory manner corresponding to the correct shock speed for initial value problems. The original method is limited to flux vectors of the specific form f = f(u) and b = 0, while the generalized relaxation method encompasses the system (1) with both initial and boundary values as appropriate to the specific problem under study. This paper contains a brief description of the relaxation method with its modifications. Complete details may be found elsewhere [2], although it is hoped that there is enough information provided herein to allow researchers working in the field to be able to use the method without further references. This exposition is found in Section 2. Section 3 includes a comparison of the relaxation method with the exact solution for a scalar initial value

3 problem: Burgers' Equation with a nonhomogeneous forcing term. Section 4 presents a brief survey of some numerical results for an initial boundary value problem for the single layer shallow-water equations in one spatial dimension, or the river flow problem. Some concluding comments follow in Section 5. 2 The generalized relaxation method In this section, the relaxation method proposed by Jin and Xin [4] is briefly described in its generalized form [2] to include problems for which the system has spatial dependence in the flux vector, boundary values and nonzero forcing terms. Although the scheme may be expressed for systems of more than one spatial variable, it is described for systems of conservation laws in one spatial dimension so that the ideas may be expressed in as simple a notation as possible. Associated with the system (1) is a larger system, called the m,odi,fied relaxation system. This system consists of 2n equations derived from the,n equations in (l), and it can be expressed as d d -W + -F(w) = B(w, b, f), (X, t) E R X R', dt dx where W, F, and B are all vector-valued functions in R2n. f in (2) is the same vector-valued function that appears in equation (1). These new vectors are defined as and B = for real scalars cr and E. Using the definition (3), the system (2) may be restated as two separate systems of n equations each by and The modified relaxation system interpreted in the form (4), (5) is now interpreted as a linear system, with the special property that in the limit as E -+ 0, equation (5) yields the solution f = v, which subtitutes back into (4)

4 46 Computational method^ irl Mulriphase F lo~ to yield the original system (1). In this way, it is theorized that for small enough values of E, solutions of (1) can be obtained as limits of solutions to the linear, and much simpler to deal with, system (2). It should be noted that there are two parameters, E and a, introduced in the notation (3) and present in equation (5). The relaxation parameter E is a small positive constant, chosen as where At is the width of the discretization for the time variable t. In general, a value of E = 10-l0 or 10-l1 was found to be sufficient to satisfy (6) while being small enough so that the approximate solutions to (2) were indistinguishable from those of (1). The parameter a is a dissipation constant whose choice depends on the magnitude of the eigenvalues of the Jacobian matrix fj(u, X), where the derivative is taken with respect to U alone. The eigenvalues provide a lower bound for a, which is bounded above for stability through the Courant- Friedrichs-Lewy condition [5]. Such a range for a may be expressed as In the inequality (7), X = maxi=l,,,., IXi(u, X) l is the supremum of the eigenvalues Xi of f'(u, X). The effects of choosing a are such that it is desirable to choose a as small as practicable while satisfying the lower bound in (7), and then to fix the grid widths At and Ax so that the upper bound holds. Conceptually, a must be large enough such that the characteristic curve X - cut creates a wide enough cone in (X, t) space to encompass the characteristic curves X - Xit from (l), while being small enough so that shocks are permitted to remain for enough time steps to be observable. If a is too large, then the resolution of discontinuities is poor, and the system becomes first order [4]. Once a and E are fixed, a finite-difference numerical method may be chosen to calculate solutions to the system (2). The method is fully described elsewhere [2] and is a second-order TVD Runge-Kutta splitting scheme, which employs Van Leer's slope limiter [5] to remove oscillations near any shocks. The simplicity of the scheme is its iterative procedure, which is by the following procedure starting from a known initial guess un and vn, and resulting in the next iteration.

5 The specific spatial discretization for the operator D is chosen as a secondorder scheme with a slope limiter to remove oscillations. Figure 1: Graph of lower layer height at a given time for varying cu The procedure consisting of equations (8) through (17) is explicit in each step excepting equations (8) and (12). Here, the values of U* and U** must be found implicitly through inverting the values of b(u*, X) and

6 48 Con~pirtutior~ul Methods in Mdtiphase Flow, b(u**, X) respectively. Although this implicit step may occasion to cause instability in the overall iterative procedure, this may be overcome by using the previous values of U as appropriate. Such a limitation was not found to be limiting in the types of equations considered. Discontinuities in the system are resolved through a finite difference across any jumps or shocks. The position may then be passively traced at each time step without any need for more general and time-consuming front-tracking methods [6]. Typically, it is sufficient to set a tolerance level for one of the system variables (e.g. U) to be above this level, and record the position of this point as the iteration progresses. Two pictures which show the typical resolution for the relaxation for the shallow water equations follow. The first, shown in Figure 1 on the previous page, is for changing values of cr in the calculation. These tend to have a sniall effect in the final solution, although change the computation t>ime accordingly so that larger values of cr necessitate smaller time grid spacing. The second diagram, Figure 2 below, shows the effect of changing the value of E. As may be observed from Figure 2, such an effect is more drastic on the resolution of the discontinuity. In practice, a few calculations with different choices of E permit the setting of the parameter for subsequent siniulations. Figure 2: Graph of lower layer height at a given time for varying E.

7 3 A scalar example The relaxation method may be used to calculate solutions to a simple nonlinear problem, namely the inviscid Burgers' Equation with a forcing t,erm. The initial value problem to be solved is expressed with the aid of the Heaviside function, H (X), as In equation (18), P is a constant parameter, and the Heaviside function is defined by The exact solution of problem (18) can be found through the method of characteristics. There is a propagating discontinuity in the solution initiated by the initial value which is denoted by the position X = s(t). With such notation, the exact solution to the initial value problem (18) can be easily verified as u(x, t) = seche + x o tanh if X 5 s(t), 0, if s(t) < X. (20) The solution (20) is incomplete until the position of the discontinuity is specified precisely. Such a calculation may be completed by using the Rankine- Hugoniot jump conditions, which are straightforward for the homogeneous problem. However, it may be quickly noted that the solution (20) is in fact incorrect in the case that s < X since the constant (0) does not satisfy the initial equation (18). This problem may be rectified by altering the equation (18) to include the undetermined discontinuity by replacing the forcing The position of the discontinuity is found to satisfy the first order ordinary differential equation with initial value s(0) = 0, This result recovers the tradition limit of u(s,t)/2 in the limit as --, 0, and intuitively yields a shock speed which is either slower or faster than the homogeneous case depending on the sign of P. The relaxation method may be applied to this example (18) to produce favourable results to the theoretical solution (20). 4 Application of the relaxation method to the shallowwater equations The one-dimensional shallow-water equations for a fluid with horizontal velocity u(x, t) and layer height h(x, t) overlying a rigid boundary and beneath

8 50 Con~ptational Methods ir~ Multiphase Flow a fluid of much less density may be written as (Whitham [7]) The constant,b represents the angle of the rigid bottom boundary measured from the horizontal, and the constant Cf is a coefficient of friction for a ChBzy-type basal drag law. These equations are also known as the river flow equations (see, for example, [7] page 134). The initial value considered for the river flow problem is that of a finite volume of fluid starting at rest, in a similar geometry to a dam-break problem. The initial variables are expressed as U(%, 0) = 0, h(x, 0) = 1-px -1_<X<l, otherwise. The vertical cross sections of the fluid can be thought of as a representation of an air/water or other phase interface, and the subsequent motion of the discontinuity represents the position of the phase interface. An example of the numerical solution for problem (22),(23) is given below in Figure 3. There, a volume of fluid spreading up a slope can be seen to slow and reverse its flow direction to eventually cause a 'backfill' to a stable configuration. Figure 3: Graph of h at a several times for p = 0.05.

9 5 Concluding remarks Computational Methods in Multiphase Flow 5 1 Wen a fluid dynamic problem involving several phases can be modelled by a first-order hyperbolic system in conservation form (l), then the relaxation method is a straightforward numerical technique which may help to provide insight into the development of time-dependent solutions. The numerical method is used to solve the modified relaxation system (2) as an approximation of the original system of equations, and is an explicit iterative procedure which is well-suited to initial value problems for systems of nonlinear hyperbolic conservation laws. In this paper, some of the modifications to the numerical method are discussed with reference to the river flow problem for an air-water interface. With comparison of the results for the technique for this simplistic situation, it is hoped that the procedure may be applied to more difficult problems in multiphase flow. References [l] Simpson, J.E., Gravity Currents in the Environment and the Laboratory, Second Edition, Cambridge University Press, Cambridge, [2] Montgomery, P. J., Shallow-water Models for Gravity Currents, Ph.D. Thesis, University of Alberta, Edmonton, [3] Montgomery, P.J. & Moodie, T.B., Jump conditions for hyperbolic systems of forced conservation laws with an application to gravity currents, Stud. in Appl. Math, accepted for publication (2000). [4] Jin, S. & Xin, Z., The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure and Appl. Math, 48, pp , [5] Leveque, R. J., Numerical Methods for Conservation Laws, Birkhauser- Verlag, Basel, [6] Davis, S.F., An interface tracking method for hyperbolic systems of conservation laws, Applied Numerical Mathematics, 10, pp , [7] Whitham, G.B., Linear and Nonlinear Waves, Wiley, New York, Acknowledgments The author gratefully recognizes the support given by the University of Northern British Columbia in the form of a Conference Travel Grant. I also wish to thank Professor T. Bryant Moodie at the University of Alberta, for providing a setting in which prior research leading to the ideas contained herein was conducted.