A two-fluid solver for hydraulic applications

Transcription

1 A two-fluid solver for hydraulic applications L. Qian, D. M. Causon, D. M. Ingram&C. G. Mingham Department of Computing and Mathematics, I%e Manchester Metropolitan University, Manchester Ml 5 GD, United Kingdom Abstract A two-fluid solver for hydraulic applications including the prediction of violent overtopping of waves at seawalls has been developed. The scheme is based on the solution of the incompressible Navier-Stokes equations for a variable density fluid system using the artificial compressibility method. The computational domain encompasses both water and air regions and the interface between the two fluids is treated as a contact discontinuity in the density field which is captured automatically as part of the solution. A time+accurate solution has been achieved by using an implicit dual-time iteration technique. Several classical test cases includhg the low amplitude sloshing tank and the broken dam problems, as well as a collapsing water column hitting a downstream obstacle have been calculated using the present approach and the results compare very well with other theoretical and experimental results. 1 Introduction Traditional approaches to flow problems with free surfaces separating a liquid and a gas such as water and air are surface-fitting methods and surfacetracking methods. Surface-fitting methods solve the flow in the liquid region only and the free surface is treated as a moving boundary of the computational domain. This method is very efficient for simple free surface problems but its validlty is limited by the skewness of the resulting grid, thus precluding it from being applied to, for example, wave breaking problems. The surface-tracking method, however, solves both fluid regions on a fixed grid system and the free surface is identified by a marker function such as the

2 volume fraction function in the widely used VOF method. The shape of the free surface can then be reconstructed from the distribution of the marker function. This method can define very sharp interfaces and is robust. However, the tracking and the reconstruction of the free surfaces remain complicated and difficult, especially in three dimensions. More recently, another approach, referred to as the surface-capturing method, has been developed by Kelecy and Pletcher[l], which views a free surface as a contact discontinuity in the density field. Similar to the shock-capturing method in the compressible flow, in this approach, the interface is automatically captured as part of the solution along with other flow variables such as pressure and velocity by the enforcement of a conservation law, thereby eliminating the need for complex surface tracking and reconstruction procedures. The robustness and simplicity of the method represents a new development in the field and is the basis of the present study in developing numerical methods for hydraulic applications. More specifically, the mathematical model of the tw~fluid (water and air) system can be formulated as a set of partial differential equations which govern the motion of an inviscid, incompressible, variable density fluid. These equations consist of mass conservation (density) equation, momentum equation and an incompressibility y constraint (continuity equation) that are solved simultaneously using the finite volume method. The formulation is based on the artificial compressibility method in which the pressure, density and velocity fields are directly coupled to produce a hyperbolic system of equations. To achieve a time-accurate solution for unsteady flow problems, an implicit dual-time iteration technique has been used, in which the solution at each real time step is obtained by solving a steady-state problem in the pseudotime domain. To evaluate the inviscid fluxes, Roe s flux function is adopted locally at each cell edge, assuming a 1-D Riemann problem in the direction normal to the cell edge. To achieve a second-order accurate solution in space, a piece-wise linear model for the cell variables is constructed in conjunction with a slope limiter to prevent over-shooting or under-shooting of the interpolated variable values before the two Riemann states at each cell edge are computed. For the pseudotime iteration, however, a first order upwind scheme can be used to calculate the inviscid fluxes and the resultant linear equations are solved using an approximate LU factorization scheme[2]. At every real time step, once the flow variables including density have been calculated, the position of the interface can be defined as the contour with the average density value of water and air. In the following sections, the mathematical formulation of the method is reviewed, followed by a detailed description of the numerical implementation. The results for three typical test cases including the classical low amplitute sloshing tank problem, the dam-breaking problem and a collapsing water column hitting a downstream obstacle are presented using this approach. Finally, some conclusions are drawn from the study and a brief dkicussion of future work is given.

3 2 Numerical method 2.1 Governing equations and boundary conditions The integral form of the 2D incompressible Euler equations for a fluid system with variable density field can be written m wqdo+hf nds=./lbdo (1) where fl is the domain of interest, S is the boundary surrounding 0, n is the unit normal to S in the outward direction, Q is the vector of conserved variables, F is the vector of flux function through S and B is the source term for body forces. By using the artificial compressibility method and assuming the only body force is the gravity, Q, F and B are given as Q = k, PU,PV,P/P]T, 1?= j~n. + grnv and B,= [0,0, pg, O], where j I = [F, W2 + P, PJV, U]T, 9 = [PV,puv, pv + P,v], u and v are the velocity components, p is the density, p is the pressure, B is the coefficient of artificial compressibility and g is the gravitational acceleration. Introducing a fictitious time derivative of pressure into the continuity equation produces a system of hyperbolic equations which can then be solved by any of the recently developed upwind finite volume techniques, such as the characteristics-based Godunov-type schemes. Clearly, from the above formulation, any meaningful solution can only be achieved when a divergenc~free velocity field is recoverd, i.e. dp/8t -) O. For a steady-state calculation, this should not be a problem. For unsteady flow problems, however, a divergence-free velocity should be attained at every time step, which can be done by using the dual-time stepping technique and subiterating the equations in the pseudotime domain to achieve a steady-state solution for each physical time step. For the current problem, the velocity boundary conditions consist of the following two parts: at the body surface, we have a no-penetration condition for inviscid flows U.n =() (2) and at infinity, we have freestream conditions U=um (3) The boundary conditions for pressure and density can also be specified accordingly. 2.2 Numerical solution In the present study, a cell centred finite volume approach has been adopted to discretize the governing equations on Cartesian grids. For each control

4 volume, Eq. (1)can be written as ~Qivi = _ l% / aci F nds + Bw = R(Qi) (4) where Qi are the average quantities at cell i stored at the cell centre, and 8Ci and Vi denote the boundary of the cell and area of cell i, respectively. The surface integration on the right hand side of Eq. (4) is evaluated by summing the flux vectors over each edge of a cell and the discrete form of the integral is! F.nds = ~ Fi,jAlj (5) ac; j=k(i) where k(i) is a list of the neighboring cells to cell i, Fi,j is the numerical flux through the interface of cells i and j, and Alj is the length of edge j. In present study, k(i) is a list of four cells. In order to evaluate the inviscid numerical fluxes F~j, Roe s flux function is adopted locally at each cell edge, assuming a ID Riemann problem in the direction normal to the cell edge, as follows: F:j = ;[F (Q:J + F1(Q;j) \Al(Q:j) - Q;j)], IA[ = RIAIL, (6) where Q~j and Q;j are the reconstructed right and left states at the cell face between cell i and cell j, A is the flux Jacobian evaluated by Roe s average state, which is the average of Q+ and Q. The quantities R, L and A are the right and left eigenvectors of A and the eigenvalues of A[1][2]. To achieve second-order accuracy, a piecewise linear model for the cell variables must first be reconstructed from the solution before the two Riemann states at each cell edge are computed. For a given cell with centre point A for example, this requires the construction of the cell variables in the form Q(x, y) = QA + AQ~ r (7) where r is the vector from the cell centre A to any point (z, y) within the cell, QA is the cell centre value of the cell, and AQA is the gradient of cell A evaluated using the neighboring cell centre values. To prevent spurious over- or under-shoots, a slope limiter function such as the van Leer limiter should be applied before calculating the gradient. By discretising Eq. (4) in time, the first-order Euler implicit difference scheme for example can be used: (QV) +l - (QV) At R(Q +l), (8)

5 t Iidnmlic /)7/owf(tliotI,\/CI}IC[cycImeHt 331 where V is the cell area. To achieve a time-accurate solution for each time step for unsteady flow problems, Eq. (8) must be further modified to obtain a divergence free velocity field. This is accomplished by introducing a pseudotime derivative into the system of equation, as (QV) +l~+i (QV)n+l,m + ~~~(QV)n+ m+i- (Qv)n AT At = _~(Q~+l>~+l), (9) where T is the pseudotime and It. = diag[l, 1, 1, O]. The right-hand side (R.HS) of Eq. (9) can be linearized using Newton s method at the m + 1 pseudotime level to yield [l v + 19R(Qn+11m) m i3q ](Q~+l,m+l _ fy+l,m) = _pta (Qn+ m- Qn)v At +~(Qn+lIm)l> where Im = diag[l/at + l/at, l/ar + l/at, l/ar + I/At, l/at]. When A(Q +l)~ = Qn+l,m+l _ Q +l m is iterated to zero, the density and momentum equations are satisfied and the divergence of the velocity at time level n + 1 is zero. The system of the equation can be written in matrix form as (D+ L + U)AQ = RHS, (lo) where D is a block diagonal matrix, L is a block lower triangular matrix, and U is a block upper triangular matrix. Each of the elements in D, L and U is a 4 x 4 matrix. An approximate LU factorization(alu) scheme as proposed by Pan and Lomax[2] can be adopted to form the inverse of Eq. (10) in the form (D+ L) D-l(D + U)AQ8 = RHS, (11) Within each time step of the implicit integration the subiteration is terminated when the L2 norm of the iteration process is less than a specified limit. N L2 = [~(Qs+l Qs)2]/N, (12) i 1/2 3 Results Three test cases including the classical low amplitude sloshing tank problem, the collapse of a water column and the collapse of a water column with

6 an obstacle, have been calculated to validate the numerical scheme. Cartesian uniform grids have been used throughout due to the simplicity of the geometry and the time step At used for advancing the solution is within the range of 5 x 10 6 to 5 x The density ratio between water and air is taken as 1000:1 and viscous effects have been ignored in all calculations. 3.1 Low amplitude sloshing tank The sloshing of a liquid wave with a amplitude under the influence of gravity is a classical test case for free surface flow problems and more recently for evaluating interface tracking methodology[3]. The initial condition of the flow is shown in Fig.(1) for t = 0.0, where the water has an average depth of 0.05m, and its surface is defined by one half of a cosine wave with an amplitude of 0.005m. The computational domain used has a base length and a height of O.lm. For the results presented, the domain is discretised with 50 x 50 cells. Several snapshots of the results for the first period are also shown in Fig.(1), where both the location of the water surface and the velocity vectors are plotted. After a quarter of a period the potential energy of the system has been transferred to kinetic energy and the velocities reach their maxima. After a half period all the kinetic energy has been transferred back into potential energy with the velocity almost back to zero. Fig. (2) shows plots of the position of the interface at the left boundary against time for the first five periods where the theoretical solution for the first mode of the problem with period of sloshing P = s is also shown. As analysed by Tadjibaksh and Keller( 1960), the second mode with half the above period is also important for the problem which causes a slight difference between the predicted and the analytical values. 3.2 Collapse of a water column The initial condition for this problem is shown in Fig(3). A water column with a height of 0.3m and a width of O.15m is initially confined to the left in a container of size 0.6m x 0.6m which is discretised by 100 x 100 cells. The confinement is then suddenly withdrawn at time t = 0.0s. The time evolution of the collapsing column is shown in Fig(3) along with the velocity vectors, which compare well with the images from a video camera (not shown here) taken by Koshizuka(1995) [5]. The non-dimensional height at the left wall and leading-edge positions of the collapsing water column versus the non-dimensional time are shown in Fig.(4) respectively and both are in good agreement with the experimental data presented by Mai-tin and Moyce (1952 )[4].

7 //1 (/) (111//< Il?f;ll llldti[vl.ilm(lgeml nt 333,..A Figure 1: Plots of the wave position and velocity vectors for the first period of the sloshing of an inviscid liquid under the influence of gravity o 04 L-_... J S2 r,me(=l Figure 2: Position of the interface at the left boundary plotted against time

8 r-~ 05,=OOs 1:.:,,,. 6,.06s,=0?6 ::, :,,. 03 }::: ::,,.,,,.!,,,,!!,,,,! ~2::j,!,,!,,\\,,. I Figure 3: Plots of the interface position and velocity vectors for the collapsing water column problem

9 calculated \,.. per. m,. c. \ \... T calculated-.xm?r, m.nt. 0.4 k ~.. > 0.3..>,.., ~.] ~ t.sqrt [g/a). 0.5 /: /,... /: / J Figure 4: The height of the collapsing water column of the leading edge (right) versus time (left) and the position 03 02, O;+ Figure 5: Numerical results of a collapsing water column hitting an obstacle

10 3.3 Collapsing water column hitting an obstacle The initial position and size of the water column are the same as in the previous case, but a small obstacle is placed on the bottom of the tank to the right of the cylinder. The size of the obstacle is 0.024rn x 0.048m. The predicted positions of the interface for times t = 0.2s and 0.4s are shown in Fig(5), in which the corresponding photographs taken by Koshizuka(1995) [5] are also included for comparison. The general agreement between the numerical and experimental results is satisfactory. 4 Conclusions A two-fluid solver has been developed for hydraulic applications by solving a set of conservation equations for an inviscid variable density fluid system. The interface between two fluids can be captured automatically as part of the solution along with other flow variables. The results for several test cases have shown that the method is accurate, robust and simple. Future work will include the incorporation of the Cartesian cut cell method[6,7] to represent complex geometries arising from the real flow problems. References [1] Kelecy F.J. & Pletcher R. H., The development of a free surface capturing approach for multidimensional free surface flows in closed containers, J of Comput. Phys., 139, ,1997. [2] Pan D. & Lomax H., A new approximate LU factorization scheme for the Navier-Stokes equations, AIAA Journal, 26, , [3] Ubbink 0., Numerical prediction of two fluid systems with sharp interface, Ph.D Thesis, Imperial College, 1997, unpublished. [4] Martin,.J.C. & Moyce, W. J., An experimental study of the collapse of liquid column on a rigid horizontal plane. Phdos. Thans. Roy. Sot. London, A244, ,1952. [5] Koshizuka, S. et al, A particle method for incompressible viscous flows with fluid fragmentation, Computational Fluid Dynamics Journal, 113, , [6] Causon D.M. et al, A Cartesian cut cell method for shallow water flows with moving boundaries, Advances in Water Resources, 24, , [7] Qian L., Causon D.M. et al, A Cartesian cut cell method for incompressible viscous flows, Proc. of ECCOMAS CFD 2001, Southend-on Sea, UK, 2001.