A wave propagation method for compressible multicomponent problems on quadrilateral grids

A wave propagation method for compressible multicomponent problems on quadrilateral grids K.-M. Shyue Department of Mathematics, National Taiwan University, Taipei 16, Taiwan Abstract We describe a simple interface-capturing approach for the numerical computation of compressible multicomponent flow with a stiffened gas equation of state on body-fitted quadrilateral grids in complex two-dimensional geometries. The algorithm uses a generalized curvilinear coordinate formulation of a fluid-mixture type model system that is composed of the Euler equations of gas dynamics for the basic conserved variables and an additional set of effective equations for the problem-dependent material quantities. In this approach, as in its Cartesian coordinates counterpart devised by the author (K.-M. Shyue, An efficient shock-capturing algorithm for compressible multicomponent problems, J. Comput. Phys. 142 (1998) 28-242), the latter equations are introduced in the algorithm primarily for an easy computation of the pressure from the equation of state, and are derived so as to ensure a consistent modeling of the energy equation near the interfaces where two or more fluid components are present in a grid cell, and also the fulfillment of the mass equation in the other single component regions. A standard finite-volume method based on wave propagation viewpoint is employed to Email address: shyue@math.ntu.edu.tw (K.-M. Shyue) Preprint submitted to Journal of Computational Physics March 26, 29

solve the proposed multicomponent flow model with the dimensional-splitting technique incorporated for an efficient implementation of the method to multidimensional problems. Several calculations are presented with an HLLC approximate Riemann solver that show accurate results obtained using the method without introducing any spurious oscillations in the pressure near the interfaces. This includes solutions for numerical simulations of underwater explosions with circular obstacles and water wave breaking over a reef. Key words: Compressible multicomponent flow, Body-fitted quadrilateral grids, Wave propagation method, Stiffened gas equation of state, HLLC approximate Riemann solver 2 MSC: 65M6, 65M5, 65Y15, 35L6, 58J45 1. Introduction This paper is devoted to the development of a simple interface-capturing approach for numerical resolution of compressible multicomponent flow problems in complex two-dimensional geometries. For simplicity, we assume that the constitutive law for the fluid components of interest satisfies the stiffened gas equation of state for approximating materials including compressible liquids and solids (cf. [15, 27]). The algorithm uses a generalized curvilinear coordinate formulation of a mathematical model that is composed of the Euler equations of gas dynamics for the basic conserved variables and an additional set of effective equations for the problem-dependent material quantities. In this approach, as in its Cartesian coordinates counterpart proposed by the author (cf. [31, 32]), the latter equations are derived so as to ensure a consistent modeling of the energy equation near the interfaces where two or more 2

fluid components are present in a grid cell, and also the fulfillment of the mass equation in the other single component regions. This will be discussed further in Section 2. There are quite a few numerical approaches available in the literature for compressible multicomponent flows in complex geometries. Some typical ones are the unstructured meshes methods [2, 44], the overlapping grids method [5], and the Cartesian grid embedded boundary method [37]. See [5] for a concise survey of the up-to-date multicomponent methods. In this work, we want to generalize a state-of-the-art shock-capturing method that was devised originally for single-component flows on boundaryfitted curvilinear grids to the case of a multicomponent flow. It is well known that the principal problem in the usual extension is the occurrence of spurious pressure oscillations when two or more fluid components are presence in a grid cell (cf. [5] and references therein). Here by following essentially the same approach as discussed before for a fluid-mixture type algorithm in Cartesian grids, we show that by choosing the correct set of model equations, see Section 2, accurate results can be obtained using a standard method, such as the dimensional-splitting version of the high-resolution wave propagation method developed by LeVeque and coworkers [1, 11, 22], for a single-component flow. It is an efficient and yet accurate scheme without any spurious oscillations in the pressure near an interface as illustrated by numerical results presented in this paper. It should be noted that the methodology we have given here is by no means limited to the current case with the stiffened gas equation of state. Extension of the method to a hybrid barotropic and non-barotropic flow [35] 3

and a barotropic two-fluid flow [34] can be done in a straightforward manner. Without going into the details for that, our goal is to establish the basic solution strategy and validate its use via some sample numerical experimentations; this is a necessary step for our further development of the method for problems with moving geometries as well as of the volume tracking method for the improved resolution of the interfaces on quadrilateral grids (cf. [3, 36]). The format of this paper is as follows. In Section 2, we describe our mathematical model for a simplified homogeneous multicomponent problems in generalized curvilinear coordinates. In Section 3.1, we review briefly the dimensional-splitting version of the wave propagation method, and give discussion of the HLLC approximate Riemann solver for the solution of the normal Riemann problem at the cell edge in Section 3.2. Numerical results of some sample examples are presented in Section 4. 2. Mathematical models This work is concerned with a simplified compressible multicomponent flow problem, where a linearized version of the Mie-Grüneisen (i.e., the linearly density-dependent stiffened gas) equation of state of the form p(ρ, e) = (γ 1) ρe + (ρ ρ ) B (1) is assumed for the fundamental characterization of the thermodynamic behavior of the fluid component of interests (cf. [19, 25, 26, 27]). Here p, ρ, e, γ, ρ, and B are in turn the pressure, density, specific internal energy, adiabatic constant (γ > 1), reference density, and reference speed of sound. In addition, we assume a homogeneous equilibrium flow that there are no jumps 4

in the pressure and velocity (the normal component of it) across interfaces that separate two regions of different fluid components. If we ignore the physical effects such as the viscosity, surface tension, and heat conduction in the problem formulation, and consider a two-dimensional flow as an example, the basic conservation laws for the fluid mixtures of mass, momentum in the x 1 -direction, momentum in the x 2 -direction, and energy can be written as ρ t + (ρu 1 ) + (ρu 2 ) =, x 1 x 2 t (ρu 1) + ( ρu 2 x 1 + p ) + (ρu 1 u 2 ) =, 1 x 2 t (ρu 2) + (ρu 1 u 2 ) + ( ρu 2 x 1 x 2 + p ) =, 2 E t + (Eu 1 + pu 1 ) + (Eu 2 + pu 2 ) =, x 1 x 2 respectively, where u i is the particle velocity in the x i -direction for i = 1, 2, and E = ρe + ρ 2 i=1 u2 i /2 is the total energy. Clearly, (2) takes the same form as the standard Euler equations of gas dynamics for a single component flow [12]. 2.1. Governing equations in generalized coordinates We want to use a boundary-fitted quadrilateral grid for numerical discretization of our model multicomponent flow problem in a general domain with complex stationary geometries. (2) In this case, following the standard procedure as described in [4, 18, 43], for instance, we first introduce a coordinate mapping from the Cartesian (x 1, x 2 ) in a physical domain Ω to the generalized curvilinear (ξ 1, ξ 2 ) in a logical domain ˆΩ via the relations dx 1 = a 1 dξ 1 + a 2 dξ 2, dx 2 = b 1 dξ 1 + b 2 dξ 2, (3) 5

where a 1, a 2, b 1, and b 2 are the metric terms of the mapping, see Fig. 1 for an illustration of the spatial domains of concerned. Now under the mapping (3), the Euler Eqs. (2) can be transformed into the new coordinate system as ρ t + 1 (ρu 1 ) + 1 (ρu 2 ) =, J ξ 1 J ξ 2 t (ρu 1) + 1 (ρu 1 U 1 + pb 2 ) + 1 (ρu 1 U 2 pb 1 ) =, J ξ 1 J ξ 2 t (ρu 2) + 1 (ρu 2 U 1 pa 2 ) + 1 (ρu 2 U 2 + pa 1 ) =, J ξ 1 J ξ 2 E t + 1 (EU 1 + pu 1 ) + 1 (EU 2 + pu 2 ) =, J ξ 1 J ξ 2 that are essential in the devise of numerical methods on boundary-fitted grids (cf. [9]). Here J = det (x 1, x 2 )/ (ξ 1, ξ 2 ) = a 1 b 2 a 2 b 1 is the Jacobian of the mapping, and U 1, U 2 defined by U 1 = u 1 b 2 u 2 a 2, U 2 = u 1 b 1 + u 2 a 1, are the contravariant velocities in the ξ 1 - and ξ 2 -direction in a respective manner. Note that during the initialization step for computations, the metric terms a i, b i for i = 1, 2, and the Jacobian J would be determined and remained fixed at all time when a curvilinear grid is constructed by a chosen numerical grid generators (cf. [11, 4]). Having obtained the equations of motion for ρ, ρu 1, ρu 2, and E in a curvilinear coordinate, our goal next is to derive a computational model that may prevent spurious oscillations in pressure near the interfaces, when solving the problem numerically with standard interface-capturing methods. To do so, by following the same approach as discussed in [31, 33, 34, 35], we start with an interface only problem where both the pressure and each component of the particle velocities are constant in the domain, while the other variables 6 (4)

x2.5 physical domain Ω 1 1.5 1 1 x 1 mapping ξ 1 = ξ 1 (x 1, x 2 ) ξ 2 = ξ 2 (x 1, x 2 ) ξ2.5 1 1.5 logical domain ˆΩ 2 1.5.5 ξ 1 Figure 1: An example of a general non-rectangular domain Ω in two dimensions on the left that is mapped to a logical domain ˆΩ on the right via the mapping (3). such as the density and the material quantities are having jumps across some interfaces. In this instance, from (4), it is easy to obtain two basic transport equations for the motion of ρ and ρe as ρ t + U 1 J t (ρe) + U 1 J ρ + U 2 ξ 1 J ρ =, ξ 2 (ρe) + U 2 ξ 1 J ξ 2 (ρe) =. By inserting the stiffened gas equation of state (1) into the latter one, we find an alternative description of the energy equation ( p t γ 1 ρ ρ ) γ 1 B + U ( 1 p J ξ 1 γ 1 ρ ρ ) γ 1 B + ( U 2 p J ξ 2 γ 1 ρ ρ ) γ 1 B = that is in relation to not only the pressure, but also the density and the material quantities: γ, ρ, and B. In our algorithm, to maintain the pressure in equilibrium as it should be for our model interface only problem, we split (5) into the following two equations for the fluid mixture of 1/(γ 1) and (ρ ρ )B/(γ 1) as ( ) 1 + U ( ) 1 1 + U ( ) 2 1 = (6) t γ 1 J ξ 1 γ 1 J ξ 2 γ 1 7 (5)

and ( ) ρ ρ t γ 1 B + U 1 J ξ 1 ( ) ρ ρ γ 1 B + U 2 J ξ 2 ( ) ρ ρ γ 1 B =, (7) respectively. It is important to note that in order to have the correct pressure equilibrium in (5) near the interfaces, these are the two key equations that should be satisfied and approximated consistently. On the other hand, as a practical matter, it is obvious that, in addition to (6) and (7), we need to impose one more additional condition so as to have enough equations for the determination of all the three material quantities in (1). In our approach, this is done by simply breaking (7) into the following two parts: ( ) ρb + U 1 t γ 1 J ( ) ρ B + U 1 t γ 1 J ξ 1 ξ 1 ( ) ρb + U 2 γ 1 J ( ) ρ B + U 2 γ 1 J ξ 2 ξ 2 ( ) ρb =, (8) γ 1 ( ) ρ B =. (9) γ 1 Thus, we arrive at a primitive form of the transport equations (6), (8), and (9) for the variables 1/(γ 1), ρb/(γ 1), and ρ B/(γ 1) that is sufficient to have all the material quantities determined at all times. Up to this point, our discussion stresses only on an approach that is capable of keeping the pressure in equilibrium for a model interface-only problem. Since in practice we are interested in shock wave problems as well, we should take the equations, i.e., (6), (8), and (9), in a form such that all the three material quantities remain unchanged across both shocks and rarefaction waves. In this regard, it is easy to see that with 1/(γ 1) and ρ B/(γ 1) governed by (6) and (9), respectively, there is no problem to do so (cf. [1, 31]). For ρb/(γ 1), however, due to the dependence of the 8

density term, it turns out that, in a time when such a situation occurs, for consistent with the mass conservation law of the fluid mixture in (4), the primitive form of (8) should be modified by ( ) ρb + 1 ( ) ρb t γ 1 J ξ 1 γ 1 U 1 + 1 J ξ 2 ( ) ρb γ 1 U 2 =, (1) so that the mass-conserving property of the solution in the single component region can be acquired also (cf. [32, 33, 34, 35]). In summary, combining the Euler equations (4) and the set of effective equations: (6), (9), and (1), yields a so-called γ-based model system as ρ t + 1 (ρu 1 ) + 1 (ρu 2 ) = J ξ 1 J ξ 2 t (ρu 1) + 1 (ρu 1 U 1 + pb 2 ) + 1 (ρu 1 U 2 pb 1 ) = J ξ 1 J ξ 2 t (ρu 2) + 1 (ρu 2 U 1 pa 2 ) + 1 (ρu 2 U 2 + pa 1 ) = J ξ 1 J ξ 2 E t + 1 (EU 1 + pu 1 ) + 1 (EU 2 + pu 2 ) = J ξ 1 J ξ (11) ( ) 2 ρb + 1 ( ) ρb t γ 1 J ξ 1 γ 1 U 1 + 1 ( ) ρb J ξ 2 γ 1 U 2 = ( ) 1 + U ( ) 1 1 + U ( ) 2 1 = t γ 1 J ξ 1 γ 1 J ξ 2 γ 1 ( ) ρ B + U ( ) 1 ρ B + U ( ) 2 ρ B = t γ 1 J ξ 1 γ 1 J ξ 2 γ 1 that is fundamental in our method on quadrilateral grids in two space dimensions, see Section 3. With a system expressed in this way, there is no problem to compute all the state variables of interests, including the pressure from the equation of state [ 2 i=1 p = E (ρu i) 2 + 2ρ ( ) ( ) ] / ( ) ρb ρ B 1. (12) γ 1 γ 1 γ 1 9

Note that the initialization of the state variables in (11) for fluid-mixture cells can be made in a standard way as described in [31, 32] for numerical simulation. For the ease of the latter reference, it is useful to write (11) into a more compact expression by with q = f 1 = f 2 = q t + 1 J ξ 1 f 1 (q) + 1 J [ ρ, ρu 1, ρu 2, E, ξ 2 f 2 (q) + B 1 (q) 1 J ρb γ 1, 1 γ 1, ] T ρ B, γ 1 q ξ 1 + B 2 (q) 1 J [ ρu 1, ρu 1 U 1 + pb 2, ρu 2 U 1 pa 2, EU 1 + pu 1, [ ρu 2, ρu 1 U 2 pb 1, ρu 2 U 2 + pa 1, EU 2 + pu 2, B 1 = diag [,,,,, U 1, U 1 ], B 2 = diag [,,,,, U 2, U 2 ]. q ξ 2 = (13) ] T ρb γ 1 U 1,,, ] T, ρb γ 1 U 2,, (14) It is interesting to note that if the grid mapping quantities: J = 1, a 1 = 1, a 2 =, b 1 =, and b 2 = 1 are substituted in (13), we find the equations in Cartesian coordinates as with q t + f 1 = f 2 = x 1 f1 (q) + f2 (q) + x B 1 (q) q + 2 x B 2 (q) q =, (15) 1 x 2 [ ρu 1, ρu 2 1 + p, ρu 2 u 1, Eu 1 + pu 1, [ ρu 2, ρu 1 u 2, ρu 2 2 + p, Eu 2 + pu 2, B 1 = diag [,,,,, u 1, u 1 ], B 2 = diag [,,,,, u 2, u 2 ]. ] T ρb γ 1 u 1,,, ] T, ρb γ 1 u 2,, (16) 1

2.2. Characteristic structure of equations To examine the hyperbolicity of this multicomponent model (13), we assume the proper smoothness of the solutions, and inquire the characteristic structure of the quasi-linear form of the equations q t + A 1(q) 1 J q ξ 1 + A 2 (q) 1 J q ξ 2 =. (17) Here the matrices A 1 and A 2 are expressed by b 2 a 2 Kb 2 u 1 U 1 u 1 Mb 2 + U 1 u 1 a 2 u 2 Γb 2 Γb 2 Γb 2 pγb 2 Γb 2 Ka 2 u 2 U 1 u 2 b 2 + u 1 Γa 2 u 2 Ma 2 + U 1 Γa 2 Γa 2 pγa 2 Γa 2 A 1 = (K H)U 1 Hb 2 u 1 U 1 Γ Ha 2 u 2 U 1 Γ γu 1 ΓU 1 pγu 1 ΓU 1, ϕu 1 ϕb 2 ϕa 2 U 1 U 1 U 1 b 1 a 1 Kb 1 u 1 U 2 u 1 Mb 1 + U 2 u 1 a 1 + u 2 Γb 1 Γb 1 Γb 1 pγb 1 Γb 1 Ka 1 u 2 U 2 u 2 b 1 u 1 Γa 1 u 2 Ma 1 + U 2 Γa 1 Γa 1 pγa 1 Γa 1 A 2 = (K H)U 2 Hb 1 u 1 U 2 Γ Ha 1 u 2 U 2 Γ γu 2 ΓU 2 pγu 2 ΓU 2, ϕu 2 ϕb 1 ϕa 1 U 2 U 2 U 2 where we have Γ = γ 1, K = Γ 2 i=1 u2 i /2, H = (E + p)/ρ, M = 1 Γ, and ϕ = B/Γ. 11

For convenience, we define a scaled version of the metric elements and of the contravariant velocity by â i = a i S i, ˆbi = b i S i, Û i = U i S i, where S i = a 2 i + b2 i for i = 1, 2. With these notations, the eigenvalues and the corresponding eigenvectors of the matrices are: for matrix A 1, Λ 1 = diag (λ 1,1, λ 1,2,..., λ 1,7 ) ) = diag (Û1 c, Û 1, Û 1 + c, Û 1,..., Û 1 S 1, R 1 = [r 1,1, r 1,2,..., r 1,7 ] 1 1 1 u 1 ˆb 2 c u 1 u 1 + ˆb 2 c â 2 u 2 + â 2 c u 2 u 2 â 2 c ˆb2 = H Û1c K/Γ H + Û1c ˆV1 1 p 1, ϕ ϕ 1 1 1 12

and for matrix A 2, Λ 2 = diag (λ 2,1, λ 2,2,..., λ 2,7 ) ) = diag (Û2 c, Û 2, Û 2 + c, Û 2,..., Û 2 S 2, R 2 = [r 2,1, r 2,2,..., r 2,7 ] 1 1 1 u 1 + ˆb 1 c u 1 u 1 ˆb 1 c â 1 u 2 â 2 c u 2 u 2 + â 1 c ˆb1 = H Û2c K/Γ H + Û2c ˆV2 1 p 1 ; ϕ ϕ 1 1 1 A i r i,k = λ i,k r i,k, i = 1, 2, and k = 1, 2,..., 7. Here c = γ(p + p )/ρ is the speed of sound of the fluid, where p = ρ B/γ is the pressure at a reference state, and ˆV 1 = â 2 u 1 +ˆb 2 u 2, ˆV 2 = â 1 u 1 +ˆb 1 u 2 are the scaled velocity components in the transverse direction to ξ 1 - and ξ 2 -axis, respectively. Note that in this work we assume that the thermodynamic description of the materials of interest is limited by the stability requirement that the speed of sound c belongs to a set of real numbers, and so the eigenvalues that appear in the above are all real. Moreover, because there exists a complete set of linearly independent right eigenvectors for each of the matrix appearing in (17), we conclude that our multicomponent model is hyperbolic. Regarding discontinuous solutions of the system, such as shock waves or contact discontinuities, it is not difficult to show that (11) has the usual form of the Rankine-Hugoniot jump conditions across the waves. 13

2.3. Remark on volume-fraction model It should be mentioned that, by following the basic procedure described in our previous work (cf. [31, 34, 35]), it is an easy matter to derive a socalled volume-fraction model for the current compressible multicomponent flow problem. Without going into the detail for that, in a two-fluid flow case, for example, the equations written in the compact form of (13) are with [ ] T ρb q = ρ, ρu 1, ρu 2, E, γ 1, α, f 1 = f 2 = [ ρu 1, ρu 1 U 1 + pb 2, ρu 2 U 1 pa 2, EU 1 + pu 1, [ ρu 2, ρu 1 U 2 pb 1, ρu 2 U 2 + pa 1, EU 2 + pu 2, B 1 = diag [,,,,, U 1 ], B 2 = diag [,,,,, U 2 ], ] T ρb γ 1 U 1,, ] T, ρb γ 1 U 2, (18) where α [, 1] denotes the volume fraction of the fluid component of interest. With that, a direct computation of the pressure from the equation of state is done by [ 2 i=1 p = E (ρu i) 2 + 2ρ ( ) ρb γ 1 2 i=1 α i ρ i B i γ i 1 ] / 2 i=1 α i γ i 1. (19) Here we have set α 1 = α and α 2 = 1 α for the volume fractions occupied by the fluid-component 1 and 2, in a respective manner, and the mixtures of γ and ρ B = γp are computed by / 2 α i γ = 1 + 1 and ρ B = γ i 1 i=1 2 i=1 / 2 ρ i B i α i α i γ i 1 γ i=1 i 1. As in the γ-based model case, it is not difficult to show that this volumefraction model is hyperbolic, and has a similar mathematical structure of the solutions as discussed in Section 2.2. 14

2.4. Include source terms To end this section, we note that if ξ 1 is the axisymmetric direction, an axisymmetric version of the current multicomponent model can be written as q t + 1 J ξ 1 f 1 (q) + 1 J ξ 2 f 2 (q) + B 1 (q) 1 J q ξ 1 + B 2 (q) 1 J q ξ 2 = ψ(q), (2) where ψ is the source term derived directly from the geometric simplification, ψ = 1 x 1 [ρu 1, ρu 2 1, ρu 1 u 2, (E + p)u 1, ] T ρb γ 1 u 1,,. (21) In addition to that, if gravity is the only body force in the problem formulation, we may include the following source term: ψ = [,, ρg, ρgu 2,,, ] T, (22) in the model (2) as well. Here g denotes the gravitational constant, and is set to be 9.81m/s 2 in the computations done in Section 4.6. As to the other source terms such as the one arises from the surface tension force at the interface, we may use a continuum surface force model of Brackbill et al. [7] for that, see the work done by Perigaud and Saurel [28] and the references therein for more details. Since it is beyond the scope of this paper, we will not discuss this further. 3. Numerical approximation To find approximate solutions of our multicomponent flow model (11) in a generalized curvilinear coordinate for complex two-dimensional geometries, we use a standard high-resolution wave propagation method developed 15

by LeVeque [2, 21] with the dimensional-splitting technique incorporated in the method for multidimensional problems. This method is a variant of the fluctuation-and-signal scheme of Roe [29, 3] in that we solve onedimensional Riemann problems in the direction normal to each cell interface, and use the resulting waves (i.e., discontinuities moving at constant speeds) to update the solutions in neighboring grid cells. To achieve second-order accurate on smooth solutions, and sharp and monotone profiles on discontinuous solutions, we introduce slopes and limiters to the method as in many other high-resolution schemes for hyperbolic conservation laws [13, 22]. 3.1. Wave propagation methods For simplicity, we assume a uniform rectangular grid with a fixed mesh spacing ξ 1 in the ξ 1 -direction and ξ 2 in the ξ 2 -direction that discretizes a computational domain as illustrated in Fig. 2, for instance. The method we consider is based on a finite-volume formulation in which the approximate value Q n ij of the cell average of the solution over the (i, j)th grid cell at time t n can be written as Q n ij 1 q(ξ 1, ξ 2, t n ) dξ 1 dξ 2, ξ 1 ξ 2 C ij where C ij denotes the rectangular region occupied by the grid cell (i, j). Note that in the current setup the numerical solution on the rectangular grid cell C ij in the computational domain gives distinctively the result on the mapped quadrilateral grid cell Ĉij in the physical domain for all the grid cell (i, j), see Fig. 2. The time step from the current time t n to the next t n+1 is denoted by t. 16

physical grid computational grid x 2 j + 1 j i 1 Ĉ ij i mapping x 1 = x 1 (ξ 1, ξ 2 ) x 2 = x 2 (ξ 1, ξ 2 ) j + 1 j ξ 2 ξ 2 ξ 1 C ij x 1 ξ 1 n i 1/2,j i 1 i Figure 2: A sample grid system in our two-dimensional numerical method on a quadrilateral grid. The numerical solution on the rectangular grid cell C ij in the computational domain gives distinctively the result on the mapped quadrilateral grid cell Ĉij in the physical domain for all the grid cell (i, j). In a simple dimensional-splitting approach, the equations to be solved, e.g., (13), are split into a sequence of one-dimensional problems as ξ 1 -sweeps: q t + 1 f 1 (q) + B 1 (q) 1 q =, (23) J ξ 1 J ξ 1 ξ 2 -sweeps: q t + 1 f 2 (q) + B 2 (q) 1 q =, (24) J ξ 2 J ξ 2 and so in each time step a dimensional-splitting (or called Godunov-splitting) version of the first-order wave-propagation method in two dimensions can be written as Q ij = Qn ij 1 t [ (A + J ij ξ 1 Q ) n + ( ] A i 1/2,j 1 Q) n, (25) i+1/2,j 1 Q n+1 ij = Q ij 1 t [ (A + J ij ξ 2 Q ) + ( ] A i,j 1/2 2 Q). (26) i,j+1/2 2 17

Here in the ξ 1 -sweeps we start with cell average Q n ij at time t n and solve (23) along each row of cells C ij with j fixed, updating Q n ij to Q ij by the use of (25) with the fluctuations and m w (A + 1 Q)n i 1/2,j = (A 1 Q) n i+1/2,j = m=1 m w m=1 ( λ + 1,m W 1,m ) n i 1/2,j ( λ 1,m W 1,m ) n i+1/2,j, where (λ 1,m ) n ι 1/2,j and (W 1,m) n ι 1/2,j are in turn the wave speed and the jump of the wave for the mth family of the solutions obtained from solving the onedimensional Riemann problems in the direction normal to the cell interface between C ι 1,j and C ιj with a properly chosen solver, see Section 3.2, for ι = i, i + 1. We have the usual definition for the notations λ = min (λ, ) and λ + = max (λ, ). Then in the ξ 2 -sweeps we can use the Q ij values as data for solving (24) along each column of cells C ij with i fixed, which gives us the solution of the next time step Q n+1 ij and m w (A + 2 Q) i,j 1/2 = (A 2 Q) i,j+1/2 = m=1 m w m=1 from (26) with the fluctuations ( λ + 2,m W 2,m ) i,j 1/2 ( λ 2,m W 2,m ) i,j+1/2. It is clear that this method belongs to a class of upwind schemes, and is stable when the typical CFL (Courant-Friedrichs-Lewy) condition: ν = t max m (λ 1,m, λ 2,m ) min ( ξ 1, ξ 2 ) 1, (27) is satisfied (cf. [22]). Moreover, it is not difficult to show that the method is quasi-conservative in the sense that when applying the method to (13) not 18

only the conservation laws but also the transport equations are approximated in a consistent manner by the method. To extend this splitting method to a high-resolution version (i.e., secondorder accurate on smooth solutions, and sharp and monotone profiles on discontinuous solutions), it is a common practice to modify (25) and (26), in a respective manner, as and Q ij := Q ij 1 J ij Q n+1 ij := Q n+1 ij 1 J ij [ t ( ) n F1 ξ 1 [ t ( ) F2 ξ 2 i+1/2,j i,j+1/2 ( ) n F1 ( ) F2 where the add in correction terms such as ( F 1 ) n i 1/2,j example, may take the form and ( F 1 ) n i 1/2,j = 1 2 ( F 2 ) i,j 1/2 = 1 2 m w m=1 m w m=1 [ λ 1,m ( 1 t ) λ 1,m J ξ 1 [ ( λ 2,m 1 t ) λ 2,m J ξ 2 i 1/2,j ] i,j 1/2 ], and ( F 2 ) i,j 1/2, for W 1,m ] n i 1/2,j ] W 2,m, i,j 1/2 respectively, see [22] for more expositions. Note that J is an averaged value of the grid Jacobian (say, the arithmetic average of J for grid cells to the left and right of the cell edge), and W ι,m is a limited value of W ι,m obtained by comparing W ι,m with the corresponding W ι,m from the neighboring Riemann problem to the left (if λ ι,m > ) or to the right (if λ ι,m < ) for ι = 1, 2. It is a known fact that, except for some simple problems, there will be generally splitting error of the method just described. But from numerical experiences it turns out that the splitting error is often no worse than the 19

errors introduced by the numerical methods in each sweep, and hence as a practical matter it is typically not necessary to use a more accurate splitting approach such as the Strang splitting [38] to reduce the splitting error, see [22] for some discussion of why one might not want to use a higher order splitting method. In addition, we have also observed good results for many practical problems obtained using the present method as compared to the fully discrete wave propagation method (cf. [2, 21]). For these reasons, we will use the dimensional-splitting method as we have just described for all the multidimensional tests done in Section 4. 3.2. Riemann problem and HLLC approximate solver To determine the waves, speeds, and fluctuations in the aforementioned wave propagation methods, we need to solve the one-dimensional Riemann problems normal to each cell interfaces. If we consider the case at edge between cells C i 1,j and C ij as illustrated in Fig. 2, for example, it amounts to solving a Cauchy problem in the direction n i 1/2,j that consists of (23) as for the equations and the piecewise constant data Q n i 1,j if ξ 1 < (ξ 1 ) i 1/2, q(ξ 1, t n ) = (28) Q n ij if ξ 1 > (ξ 1 ) i 1/2 as for the initial condition at time t n. To do so, here we take a popular approach (cf. [5, 22]) in that the normal and tangential components of velocity are determined first from Q n i 1,j and Q n i,j, relabeling these as the x 1 and x 2 components, and then the Riemann problem in the x 1 direction is solved with this modified data. We therefore recombine the velocity components of the resulting Riemann problem solution to obtain the proper 2

updates in the physical space. This is conceptually easier to implement in the current case because a Riemann solver has already been written for an analogous model system in Cartesian coordinates (cf. [32, 34, 35]). Note that in the applications concerned here we have chosen the data in (28) well enough so that the solution of the Riemann problem would consist of genuinely nonlinear waves such as shock and rarefaction, and linearly degenerate wave such as contact discontinuity; there is no vacuum region occurring in the solution. Without loss of generality, we consider the γ-based model (13) as an example. In this case, with n i 1/2,j = (ˆb 2, â 2 ) i 1/2,j given a priori, our first step of the Riemann solver is to transform the data Q n i 1,j and Q n i,j into the new data Q l and Q r via Q l = R i 1/2,j Q n i 1,j, Qr = R i 1/2,j Q n i,j, where R i 1/2,j is a rotation matrix defined by 1 (ˆb R i 1/2,j = 2 ) i 1/2,j (â 2 ) i 1/2,j (â 2 ) i 1/2,j (ˆb 2 ) i 1/2,j I with I in it as being a 4 4 identity matrix. It is clear that this rotation matrix rotates the velocity components of Q into components normal and tangential to the cell edge, and leaves the remaining components unchanged. Having obtained the Riemann data Q l and Q r, our next step is to solve q t + f1 (q) + x B 1 (q) q = (29) 1 x 1 21

in the x 1 direction with f 1 and B 1 defined by (16). For that, among various methods proposed in the literature (cf. [41]), here we are interested in a simple and yet accurate variant of the approximate solver devised by Harten, Lax, and van Leer [16] for hyperbolic systems of conservation laws in that rather than introducing two discontinuities propagating at constant speeds λ l and λ r to the left and right, λ l λ r, separating three constant states in the space-time domain, an additional middle wave of speed λ m is included in the solution structure for modeling the speed of contact discontinuity, yielding a so-called 3-wave HLL (or called HLLC) solver (cf. [6, 41, 42] and references therein). Note that if we assume further that λ l and λ r are known a priori by some simple estimates based on the local information of the wave speeds, say, by taking for instance, with v ι = λ l = min (v l c l, v r c r ), λ r = max (v l + c l, v r + c r ), (2) (1) Q ι / Q ι and c ι = ( γ(p + p )/ρ) ι for ι = l or r, then it is easy to find the constant state in the middle region of the original 2-wave HLL solver as and set λ m = Q m = λ r Q r λ l Ql f 1 ( Q r ) + f 1 ( Q l ) λ r λ l, (3) (2) (1) Q m / Q m, where Q (i) m is the ith component of the vector Q m, see [41] for the other possible choices on the wave speeds. With that, it is not difficult to find the constant states Q ml and Q mr in the regions m l and m r to the left and right of the middle wave, respectively, 22

and the results are [ Q mι = h (1) mι, λ mh (1) mι, Q(3) where we have set ι h (1) mι (1) / Q ι, h (4) mι + λ ( m λm h (1) mι h(2) h (i) mι = λ (i) ι Q ι λ ι λ m f (i) 1 ( Q ι ) for ι = l or r. It should be mentioned that the states mι), h (5) mι, Q(6) ι, Q(7) ι Q (i) ml and ] T, (31) Q (i) mr satisfy the basic consistency condition of the integral form of the conservation laws, ( ) ( ) λm λ l Q (i) ml λ r λ + λr λ m Q (i) (i) mr = Q m, i = 1, 2,..., 5. l λ r λ l The solution of this Riemann problem in the x 1 direction is then composed of the three moving discontinuities: λ 1 = λ l, λ 2 = λ m, λ 3 = λ r and the jumps across each of them by W 1 = Q ml Q l, W2 = Q mr Q ml, W3 = Q r Q mr. To get the proper Riemann problem solution in the ξ 1 direction that was considered originally in the beginning of this subsection, we should set the speeds by (λ 1,m ) n i 1/2,j = (S 2) i 1/2,j λ m, m = 1, 2, 3, where (S 2 ) i 1/2,j = ( a 2 2 + b2 2 ) i 1/2,j is a scale factor, and the waves by (W 1,m ) n i 1/2,j = R 1 i 1/2,j W m, m = 1, 2, 3. 23

Here R 1 i 1/2,j is the inverse of the rotation matrix given above. Having obtained the speeds and waves, we can then compute the fluctuations at the cell edge as 3 (A ± 1 Q)n i 1/2,j = ( ) λ ± n 1,m W 1,m, i 1/2,j m=1 and this would be used in the wave propagation method for the solution updates. 4. Numerical Results We now present some sample numerical results obtained using our numerical algorithm described in the previous section for compressible multicomponent problems in general two-dimensional geometries. Without stated otherwise, we have carried out all the tests using the Courant number ν =.9 defined by (27), and the minmod limiter in the high-resolution version of the finite-volume wave propagation method. The material-dependent parameters in the stiffened gas equation of state are set by (γ, ρ, B) = (1.4, 1.2kg/m 3, ) and (4.4, 1 3 kg/m 3, 2.64 1 6 (m/s) 2 ) for the gas- and liquid-phase (i.e., for the air and water), respectively. We note that in this section we have only present numerical solutions obtained using the volume-fraction model described in Section 2.3 to the method. Since we find little difference between the results as compared to the ones obtained using the γ-based model to the method for simulations, we omit the presentation of the results for that here. 24

4.1. Interface only problem We begin by considering an interface only problem that the solution consists of a circular water column evolving in air with uniform equilibrium pressure p = 1 5 Pa and constant particle velocity (u 1, u 2) = (1 3, 1 3 )m/s throughout the domain. Initially, inside the column of radius r =.2m, the fluid is water with the data (ρ, α) r r = ( 1 3 kg/m 3, 1 ), while outside the column the fluid is air with the data (ρ, α) r>r = ( 1.2kg/m 3, ). Here r = (x 1 x c 1) 2 + (x 2 x c 2) 2 is the distance from a point (x 1, x 2 ) in a quarter annulus to the center of the water column (x c 1, xc 2 ) = (.8,.8)m. Note that despite the simplicity of the solution structure, this problem is one of the popular tests for the numerical validation of a compressible multicomponent flow solver (cf. [31, 44]). form To discretize the quarter-annulus region, we use polar coordinates of the x 1 = ξ 1 cos (ξ 2 ), x 2 = ξ 1 sin (ξ 2 ) for.5m ξ 1 2.5m and ξ 2 π/2, see Chapter 23 of [22] for an illustration. Numerical results of the density and pressure obtained using our algorithm with non-reflecting boundary conditions on all sides and a 1 1 grid are shown in Fig. 3, where the 3D surface plots and the cross-section plots along ξ 2 = π/4 are presented at time t = 52µs. From the displayed profiles, it is easy to observe good agreement of the numerical 25

PSfrag replacements ρ(kg/m 3 ) p(mpa) Density Pressure r(m) air t = t = 52µs PSfrag replacements water Density Pressure 1.2.11 1 ρ(kg/m 3 ).8.6.4.2 air water water p(mpa).1 t = t = 52µs.5 1 1.5 r(m).999.5 1 1.5 r(m) Figure 3: Numerical results for an interface only problem in a quarter annulus. Top row: Surface plots of the density and pressure. Bottom row: Cross-sectional plots of the density and pressure along line ξ 2 = π/4. The solid line in the cross-sectional plot is the exact solution, the dotted points are the numerical results, and the dashed line is the initial condition at time t =. solutions as compared with the exact results. Notice that the computed pressure remains in the correct equilibrium state p (to be more accurate, the difference of these two is only on the order of machine epsilon), without any spurious oscillations near the bubble interface. Moreover, the bubble retains its circular shape and appears to be very well located also. 26

4.2. Radially symmetric flow We are next concerned with a radially symmetric flow where the computed solutions in two space dimensions can be compared to the one-dimensional results. In this test, we take a quarter of a circular domain of radius r = 1m, and use the following two-phase (gas-liquid) flow data for experiments in which, in the gas phase, the state variables are (ρ, p, α) = ( 125 kg/m 3, 1 9 Pa, 1 ) and (ρ, p, α) = ( 1.2 kg/m 3, 1 5 Pa, 1 ) if r < r 1 and r 2 < r r, respectively, while in the liquid phase they are (ρ, p, α) = ( 1 3 kg/m 3, 1 5 Pa, ) if r 1 < r r 2. Here we have r 1 =.2m, r 2 =.7m, and r = x 2 1 + x2 2. We note that initially all the fluid components are in a resting state with zero total velocity, but due to the pressure difference between the fluids at r = r 1, breaking of the inner circular membrane occurs instantaneously, yielding an outward-going shock wave in liquid, an inward-going rarefaction wave in gas, and a material interface lying in between that separates the gas and liquid. As times go along, the inward-going wave would be reflected from the geometric center that generates a new outward-going wave and induces the subsequent interaction of waves. In the meantime, the outward-going shock wave would be collided with the outer gas-liquid interface at r = r 2 that results in a wave pattern consisting of a transmitted shock wave, an interface, and a reflected rarefaction wave. At a later time, the transmitted shock wave would be reflected from the boundary of the domain. 27

To solve this problem numerically, we are interested in a mapped grid approach proposed by Calhoun et al. [11] in that a grid point (ξ 1, ξ 2 ) in the computational domain [, 1] [, 1] would be mapped to a grid point (x 1, x 2 ) in the current quarter-circular domain, see Fig. 4 for an example of such a quadrilateral grid, and also the web-site: http://www.amath.washington.edu/ rjl/pubs/circles/index.html for the detailed numerical implementation. Figure 5 shows the contours for the density, radial velocity (defined as ū = u 2 1 + u 2 2), and pressure, at three stopping times t = 15, 36, and 52µs, where the test has been carried out by using solid wall boundary conditions on all sides with a 1 1 grid in the computational domain. From the figure, we observe good resolution of the solution structure (i.e., both the shock and interface remain circular and appear to be very well located) after the breaking of the membrane and also the interaction of the shock, the outer interface, and the boundary. The scatter plots presented in Fig. 6 provide the quantitative comparison of our two-dimensional results to the true solution obtained from solving the one-dimensional model with appropriate source terms for the radial symmetry using the high-resolution method with 1 mesh points in a unit length domain (cf. [32]). It is clear that our results agree quite well with the true solutions at all the selected times, and are free of spurious oscillations in the pressure near the inner and outer interfaces before and after the various wave interactions. 28

PSfrag replacements Density Radial velocity Pressure gas liquid t = t = 15µs PSfrag replacements t = 36µs t = 52µs 1 Computational grid 1 Physical grid.8.8 ξ2.6.4.2.5 1 ξ 1 x2.6.4.2.5 1 x 1 Figure 4: Rectangular computational grid (left) and the mapped quadrilateral grid (right) in a quarter-circular domain. ξ 1 ξ 2 x 1 x 2 Computational grid Physical grid Density Radial velocity Pressure t = 52µs gas liquid t = 36µs t = t = 15µs Figure 5: Contours of the density, radial velocity, and pressure for a two-phase (gas-liquid) radially symmetric problem in a quarter of a circular domain. Numerical solutions plotted in a clockwise manner are at times t =, 15, 36, and 52µs with a 1 1 grid. The dashed line shown in the graph is the approximate location of the gas-liquid interface. 29

PSfrag replacements r(m) PSfrag replacements t = 15µs ρ(kg/m3 ) gas liquid r(m) u (km/s) t = 15µs p(gpa) PSfrag replacements t = 36µs 3 liquid ρ(kg/m ) gas liquid.2 u (km/s) gas t = 52µs 1.2 1.8.6.4.2 t =p(gpa) 15µs.4.6 t = 52µs t = 36µs gas 1.2 1.8.6.4.2.8 1.2 1.8.6.4.2.2.4.6.8.6.6.6.4.4.4.2.2.2.2.4.6.8.2.4.6.8.2.4.6.8.2.4.6.8.2.4.6.8 t = 52µs ρ(kg/m3 ) u (km/s) p(gpa) t = 36µs.8.8.8.6.6.6.4.4.4.2.2.2.2.4.6 r(m).8.2.4.6 r(m).8 r(m) Figure 6: Scatter plots of the results for the run shown in Fig. 5. The solid line is the true solution obtained from solving the one-dimensional model with appropriate source terms for the radial symmetry using the high-resolution method and 1 mesh points. The dotted points are the two-dimensional results. The dashed line the approximate location of the gas-liquid interface. 3

4.3. Moving cylindrical vessel To test our algorithm further on a circular geometry, we consider a moving cylindrical vessel problem studied by Banks et al. [5]. Here, the circular domain we use is of radius r =.8, and is centered at the origin x c 1 = x c 2 =. Inside the circle, the initial condition is a planar material interface located at x 1 = that separates air on the left with the state variables (ρ, u 1, u 2, p, γ, B, α) = (1, 1,, 1, 1.4,, 1), and helium on the right with the state variables (ρ, u 1, u 2, p, γ, B, α) = (.138, 1,, 1, 1.67,, ). Note that in this setup we have imposed a uniform flow velocity (u 1, u 2 ) = ( 1, ) throughout the domain, and so we are in the frame of the vessel moving with speed one in the x 1 -direction. To find approximate solution of this problem, we use a similar mapped grid as illustrated in Fig. 4, but is now defined in the whole circular region that can be constructed from this quarter-circular grid via a mirror-reflection approach to both x 1 = and x 2 = axes. Figure 7 gives a sample results of a run with a 8 8 grid, where the pseudo-color images of the density and pressure are shown at four different times t =.25,.5,.75, and 1.. From the figure, it is easy to see that due to the impulsive motion of the vessel a rightward-going shock wave and a leftward-going rarefaction wave emerge from the left- and right-side boundary, respectively. Subsequently, these two waves would be interacting with the material interface that leads to collision of various transmitted and reflected waves. We note that when we 31

compare our results with those ones appeared in the literature (cf. [5, 44]), as far as the global wave structures are concerned, we observe good qualitative agreement of the solutions. The cross section of the density and pressure for the same run along the circular boundary is drawn in Fig. 8 giving some quantitative information of our computed solutions. Clearly, our algorithm works in a satisfactory manner without introducing any spurious oscillations in the pressure near the air-helium interface. 4.4. Shock-bubble interaction in a nozzle As an example to show how our algorithm works on shock waves in a more general two-dimensional geometry, we are interested in a shock-bubble interaction problem in a nozzle. Here the shape of the nozzle is taken to be the witch of Agnesi x b 2 (x 1 ) = 8a3 x 2 1 + 4a 2 on the bottom for a =.2m and 2m x 1 3m, and the flat curve x t 2 = 1m on the top. For this problem, we use the initial condition that is composed of a planarly rightward-going Mach 1.422 shock wave located at x 1 = 1.8m in liquid traveling from left to right, and a stationary gas bubble of radius r =.2m and center (x c 1, xc 2 ) = ( 1,.5)m in the front of the shock wave. Inside the gas bubble, we have the data (ρ, u 1, u 2, p, α) = ( 1.2 kg/m 3,,, 1 5 Pa, 1 ), while outside the gas bubble where the fluid is liquid, we have the preshock state (ρ, u 1, u 2, p, α) = ( 1 3 kg/m 3,,, 1 5 Pa, ), 32

PSfrag replacements Density Pressure t =.25 t =.5 t =.75 PSfrag replacements t = 1 Density Pressure t =.25 t =.5 t =.75 PSfrag replacements t = 1 Density Pressure t =.25 t =.5 t =.75 PSfrag replacements t = 1 Density Pressure t =.25 t =.5 t =.75 t = 1 Figure 7: Pseudo-color images of the density (left) and pressure (right) for an impulsively driven cylinder containing an air-helium material interface. Solutions from top to bottom are at times t =.25,.5,.75, and 1.. 33

PSfrag replacements ρ 3 2 1 p 4 2 1.5 t 1 θ(π) 2 1.5 t 1 θ(π) 2 Figure 8: Cross-sectional plots of the results for the moving vessel run shown in Fig. 7 along the circular boundary. and the postshock state (ρ, u 1, u 2, p, α) = ( 1.23 1 3 kg/m 3, 432.69 m/s,, 1 9 Pa, ). In carrying out the computation, we consider a body-fitted quadrilateral grid with the mapping function ( ) ξ x 1 = ξ 1, x 2 = x b t 2 (ξ 1 ) 2 ξ 2 ξ2 t (32) ξb 2 for 2m ξ 1 3m, ξ 2 1m, ξ2 b =, and ξt 2 = 1m, see Fig. 9 for an illustration with a 1 2 mesh cells. The boundary conditions are the supersonic inflow on the left-hand side, the non-reflecting on the right-hand side, and the solid wall on the remaining sides. In Fig. 1, we show the schlieren-type images of the computed density and pressure at six different times t =.3,.5,.7, 1.2, 1.6, and 2.1ms that is obtained using a 1 2 grid. From the figure, it is easy to see that after the passage of the shock to the gas bubble, the upstream wall begins to spall across the bubble, yielding an refracted air shock traveling within it until its first reflection on the downstream bubble wall. Noticing that this upstream bubble wall would involute 34

1 Sfrag replacements ξ 1 x2 1.5 ξ 2 1 1 2 3 2 x 1 x2.5.5 1 1.5 2 1 1 2 x 1 Figure 9: A sample quadrilateral grid for the shock-bubble interaction in a nozzle (left), and for the underwater explosion with circular obstacles (right). eventually to form a jet which subsequently crosses the bubble and sends an intense blast wave out into the surrounding liquid. In the meantime, the incident shock wave along the bottom curved boundary would be diffracted into a simple Mach reflection. To gain some quantitative understanding of our solutions, we plot the cross section of the density and pressure in Fig. 11 along the ξ 2 =.5m line, observing a variety of interesting wave patterns in the solutions. 4.5. Underwater explosion with circular obstacles Our next example for problems in more general geometries is a underwater explosion flow with circular obstacles, see [24] for a similar computation but with a square obstacle. In this test, the physical domain is a rectangular region of size ([ 2, 2] [ 1.5, 1])m 2 in which inside the domain there are two circular obstacles, denoted by b 1 and b 2, with the centers (x b 1 1, xb 1 2 ) = (.6,.8)m and (x b 2 1, x b 2 2 ) = (.6,.4)m, respectively, and of the same radius r b1 = r b2 =.2m. The initial condition we consider is composed of a horizontal air-water interface at x 2 35 = and a circular gas bubble in

PSfrag replacements Density Pressure air PSfrag replacements water Density Pressure air PSfrag replacements water Density Pressure air PSfrag replacements water Density Pressure air PSfrag replacements water Density Pressure air PSfrag replacements water Density Pressure air water Figure 1: Numerical results for a planar shock wave in liquid over a circular gas bubble in a nozzle. Schlieren-type images for the density (left) and pressure (right) are plotted in a close neighborhood of the gas bubble, where the solutions from top to bottom are at times t =.3,.5,.7, 1.2, 1.6, and 2.1ms. Here a 1 2 grid was used in the computation. 36

PSfrag replacements ρ(kg/m 3 ) 1.5 1.5 2 1 t(ms) 2 x 1 (m) 2 p(gpa) 2 1 2 1 t(ms) 2 x 1 (m) 2 Figure 11: Cross-sectional plots of the density and pressure for the shock-bubble interaction run shown in Fig. 1 along the ξ 2 =.5m line. water that lies below the interface. Here all the fluid components are at rest initially. When x 2, the fluid is air with the state variables (ρ, p, α) = ( 1.2 kg/m 3, 1 5 Pa, 1 ), and when x 2 < and r < r, the fluid is gas with the state variables (ρ, p, α) = ( 125 kg/m 3, 1 9 Pa, 1 ). Now, in the remaining region where the obstacles do not belong to, the fluid is water with the states (ρ, p, α) = ( 1 3 kg/m 3, 1 5 Pa, ). As before, the radial distance r is defined by (x 1 x c 1 )2 + (x 2 x c 2 )2, and we have r =.12m and (x c 1, x c 2) = (,.9)m in the current case. We solve this problem using a mapped quadrilateral grid that have the circular obstacles included in the domain as illustrated in Fig. 9, see [11] for the detailed construction. Figure 12 gives the numerical results of the density and pressure at six different times t =.24,.4,.8, 1.2, 2., and 3.ms, 37

obtained using a 8 5 grid with the non-reflecting boundary on the top, and the solid wall boundary on the remaining sides. From the figure, it is easy to observe that in the early stage of the computation the flow field is essentially radial symmetry. Soon after the outward-going shock wave is reaching at the left obstacle and then the right obstacle, a reflected shock wave results from each of the shock-obstacle collisions. As time goes on, these reflected shock waves would affect the structure of the gas bubble, and that induces numerous other wave-wave and wave-obstacle interactions at the later time. We note that when the outward-going shock is approaching at the air-water interface, we have a typical heavy-to-light shock-contact interaction, and the resulting wave pattern after the interaction would consist of a transmitted shock wave, an accelerated air-water interface, and a reflected rarefaction wave. In Fig. 13, the cross section of the results are drawn along line ξ 1 = at the selected times. For practical applications, it is important to know how the effect of the impinging waves on the obstacles. As a measure of that, we compute the surface pressure force exerted on the boundary of the obstacle by integrating the following line integral numerically: F i (t) = p(t) dl b i for the obstacle b i, i = 1, 2. Figure 14 gives the results for that until time t = 3ms, where a grid sequence, 2 i (2, 125) for i =, 1, 2, is used in the test to show the convergence behavior of the solution. From the graph, the existence of positive pressure force is clearly seen in some time intervals which means the negative value of the pressure around the obstacles. This is, however, permitted in the current model with the stiffened gas equation as 38

long as the pressure stays within the region of the thermodynamic stability of the flow. 4.6. Water wave breaking problem Finally, we consider the simulation of a gravity-included low speed water wave breaking problem studied by Golay and Helluy [14]. In this test, initially, we have a rightward-going stable solitary water wave in air with a crest height.6m over the still water level traveling on a flat bottom from the left toward a slope-like reef on the right of the form x b 2 (x 1 ) = 1 + 1 15 (x 1 5.225) for 5m x 1 22m and 1m x 2 2m, see Fig. 15 for an illustration of the basic structure of the physical states in filled contours. Note that in practice this solitary water wave can be constructed by the Tanaka method [39], and conveniently this can be done by making use of the computer program posted at the web-site http://helluy.univ-tln.fr/soliton.htm (cf. [17]). It should be mentioned, for this problem, the constitutive laws for the water and air are assumed to satisfy the stiffened gas equation of state as before, while the algorithm is based on the balance law (2) together with (22) as the model equations for the computations. Here a standard time-splitting version of the method is considered to deal with the source terms (cf. [23]), and the domain is discretized similar to the nozzle flow case before by the mapping function as ξ 2 ξ 1 5.225 x 1 = ξ 1, x 2 = x b 2 (ξ 1) (ξ2 t ξ 2) / ( ξ2 t 2) ξb ξ1 > 5.225 39