STABILIZED FINITE ELEMENT METHOD WITH AN ALE STRATEGY TO SOLVE MOVING BOUNDARIES PROBLEMS

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1 STABILIZED FINITE ELEMENT METHOD WITH AN ALE STRATEGY TO SOLVE MOVING BOUNDARIES PROBLEMS M. Viale and N. Nigro Universidad Nacional de Rosario Pelegrini 5, () Rosario, Argentina Abstract. Fluid flow problems with moving boundaries are one of the challenge problems to be solved in this decade. Our target is oriented to develop a finite element code to simulate the fluid flow inside the cylinder of an internal combustion engine in order to understand how this flow field is related with the combustion, the heat release and the intake and exhaust charging process. In this paper we only present some preliminary results using the ALE technique coupled with a stabilized finite element method applied to compressible fluid flow problems with moving boundaries. Keywords. 1. Introduction Fluid mechanics Computational fluid dynamics, moving boundaries, in-cylinder flows Fluid flow problems with moving boundaries are one of the challenge problems to be solved in this decade. They are inherently unsteady problems where the domain is changing according to a user-defined law commonly associated with a prefixed boundary motion or in a coupled way with the flow itself. The former case is simpler than the later, commonly named fluid-structure interaction, because of many reasons. One of them is associated with the grid generation. While in the former the mesh may be produced in advance with no enough care about the evolution of the flow field in the later the boundaries move in an unpredictable way and the grid generation should be solved at flow field computation time. Other important subject is related to the different time scales that may characterize the behavior of fluid and solid materials. Flow with moving boundaries can be encountered in many practical situations. For example flow motion around a ship with a changing free surface has attracted a great deal of research interest in recent years. In this case the free surface is an unknown moving boundary and the ship may be treated as a rigid body. The landing of an aircraft with the air flowing around the moving composed flaps is another example of great interest. Other field with a rapid development is that associated with the simulation of the complex haemodynamics in order to solve the blood flow in moving arteries. Recently, with the enormous development of computer capabilities, the simulation of in-cylinder flow inside an internal combustion engine is feasible with the target in lower emissions and higher performances. In this case the combustion chamber is a variable volume vessel with moving boundaries represented by the piston and the intake and exhaust valves. This project is oriented to simulate the in-cylinder flow in order to understand how this flow field is related with the combustion, the heat release and the intake and exhaust charging process. The numerical solution of this problem was originally performed via a finite volume method using an ALE (Arbitrary Lagrangian Eulerian) strategy. The list of references is huge and only some of the most cited references are included here (I. Demirdzic (199),J. Trepanier and Camarero (1991),J. Trepanier and Camarero (1993),C. Farhat and Stern (1997)). One of the main difficulties of the finite volume formulation is the satisfaction of the geometric conservation law (GCL). Several different proposals were published in order to bypass this difficulty. On the other hand the finite element formulation has the ability of being completely consistent with the ALE strategy satisfying the GCL with no special care (I. Lomtev (1999)). Therefore a compressible Navier-Stokes code based on finite elements with an ALE strategy is adopted (J. Donea (198),T. Nomura (199),T. Sheu (1999),I. Lomtev (1999),W. Tworzydlo (199)). An SUPG formulation for the spatial discretization (Brooks and Hughes (198); Hughes and Tezduyar (1984)) and a finite difference scheme for the time evolution is chosen. This work plan is divided in three steps, the first considers the development of an scalar transport element with an ALE strategy, the second adds the inviscid flow equations in a moving domain formulation and the third step considers the original target, the compressible Navier-Stokes solved via a finite element and ALE code. In the first stage the model originally presented by Kershaw and coworkers (Kershaw and Milovich (1998)) based on the usage of a transformation to a reference grid selected as the initial mesh was used. Even though this strategy is valid it presents several disadvantages specially when vector fields having tensor diffusion terms are involved. In the scalar case in an isotropic medium this inconvenience is not relevant because the diffusion coefficient is transformed in a diffusion tensor. In Euler equations this problem is not present because there is no diffusion term. However in the Navier-Stokes equations the stress tensor playing the role of diffusion introduces Christoffel symbols when the transformation to the reference grid is performed. This fact increases the computational cost in a drastic way. To avoid this situation it is possible to use a standard ALE strategy

2 where the solution is computed over a moving grid in a lagrangian way but with an arbitrary velocity. In general this velocity is different from the fluid one in order to keep the mesh in a good shape. For the solution of a scalar field a model presented by Sheu et.al. (T. Sheu (1999)) was adopted and numerical results over a test case is presented. This development was extended to the inviscid flow model (Euler equations) and some basic tests were solved to demonstrate mainly the geometric conservation law. In future works it is planned to apply this development to in-cylinder flow in an internal combustion engine.. The mathematical model Our target is the simulation of compressible viscous flow inside the cylinder of an internal combustion engine. Then, the mathematical model is defined by compressible Navier-Stokes equations in a variable domain. In the following sections we present first the mathematical formulation in a fixed domain and then the ALE strategy to consider the boundary movement..1. Compressible viscous flow in a fixed domain Compressible Navier-Stokes equations are one of the most used models to predict the behaviour of an internal gas flow. This set of partial differential equations may be seen as an advective-diffusive system of equations like + F a i = F d i + S where the first term is the rate of change of the state variable U expressed in conservative form followed by the divergence of the advective and diffusive fluxes and a source term. The conservation variables are defined as: (1) U = [ρ, ρu 1, ρu, ρu 3, ρe] () where ρ is the density, ρe is the total energy composed by the internal energy i plus the kinetic energy, e = i + 1 u i, i = C v T i and [ρu 1, ρu, ρu 3 ] represents the linear momentum vector. The advective fluxes and the diffusive fluxes are defined as: ρu i ρu 1 u i + pδ i1 Fi a = ρu u i + pδ i ρu 3 u i + pδ i3, F τ i1 i d = τ i τ i3 ρhu i q i + τ ik u k (4) being h = e + p/ρ the enthalpy, q = κ T the heat flux according to the Fourier law and τ ij an element of the deviatoric stress tensor. The source term depends in particular on the applications. In general mass source term is not included, the momentum sources are external forces applied to the system and finally, energy sources are the work of these external forces plus some body heat addition or release between the system and the environment. S = [; ρb 1 ; ρb ; ρb 3 ; ρb u + Q] (5).. ALE strategy In this section we only present a brief summary of the main expressions about the original work of Arbitrary Lagrangian Eulerian (ALE) in a finite element context for the inviscid case written by Donea and coworkers. For more details about this formulation we refer to the original work (J. Donea (198)). It is well known that there are two viewpoints mostly used in the description of the flow motion equations, one is called the Lagrangian approach where the mesh moves with the fluid and the other is the Eulerian approach in which the mesh is fixed and the fluid moves around it. The arbitrary Lagrangian-Eulerian description is a generalization of these two approaches where the computational grid moves with an arbitrary velocity w in the laboratory system. If w = we recover the Eulerian approach and if w = u we recover the Lagrangian one considering u as the fluid flow velocity. A very simple physical scenario is given by figure 1.

3 > 8 Figure 1: Different approaches for the description of the flow motion The boat number 1 is attached to the ground by an anchor and therefore its velocity is null and the observer inside this boat has an Eulerian description of the flow. The boat number is free to follow the river stream and its observer has a Lagrangian description of the flow motion. Finally there is a third boat which is powered by an engine with a velocity w non aligned with the river stream, therefore its observer has an ALE description of the flow.!! " $#%&?A@B 9;:=< ')(+*,-(./* (. 134,5.(67 Figure : Definition of different reference frames In order to gain some insight about this general description figure shows the three reference frames and their corresponding coordinates. A particular point P of a portion of fluid with a position a relative to the material domain is plot for both the initial time t = and for some time after. By definition these material coordinates do not change in time. It is also possible to follow the motion of the same point P in the spatial domain by its position x and by an arbitrary motion of the reference domain where its position may be defined by ξ. Initially the material domain (Lagrangian coordinates), the spatial domain (Eulerian coordinates) and the reference domain (mixed coordinates) coincide among them. So, x = a and the position of any point in the reference domain relative to the original configuration is expressed by ξ =. Some time after x a and ξ. In the ALE description a particle is identified by its material coordinates a in the initial configuration but this process of identification is indirect and takes place through the mixed position vector ξ. There is a link between the material coordinates and the mixed coordinates in the form: ξ i = f i (a i, t) (6) ALE description is similar to a mapping between the initial configuration of the continuum in the current configuration of the arbitrary reference frame, whose jacobian determinant is computed by

4 J = ξ i a j (7) Other two important properties of this jacobian are expressed by: dv = J(a, t)dv (8) J = J w..1. Kinematics in the ALE description Following J. Donea (198) we describe a physical property associated with the flow of a continuum media by g(ξ i, t). By (6) we have (9) g(ξ i, t) = g[f i (a i, t)] = g(a i, t) (1) The time derivative of (1) keeping the material coordinates constant is written as: g(a, t) = a g(ξ, t) + ξ g(ξ, t) ξ i ξ i (11) Noting that w i = ξi is the grid velocity and using the following identity (gw) = g w + w g (1) and (9) we arrive to J (gw) = Jg w + Jw g = g J + Jw g (13) Therefore (11) may be rewritten as: g(a, t) J + g(a, t) J Lagrangian Eulerian {}}{{}}{ g(a, t) g(ξ, t) g(ξ, t) ξ i = + a ξ ξ i g(a, t) g(ξ, t) g(ξ, t) ξ i J = J[ + ξ i ] = g(a, t) J (J g) (J g) + J[ g(ξ, t) + g(ξ, t) ξ i ξ i ] = J[ 1 J g J + g + w g] = J[g w + g + w g] (14) (J g) = J[ g + (gw)] (15) Expression (15) appears to be a fundamental relationship which enables us to transform any law expressed in spatial (Eulerian) variables into an equivalent law expressed in mixed variables. The g in (15) is a spatial derivative, i.e., taken with the point fixed in position.

5 ... Conservation laws in the ALE description We rewrite our original set of equations (1) in expanded form as: ρ + (ρu j ) = ρu i + (ρu i u j ) = ρb i p ρe + (ρeu j ) = ρu j b j (pu j ) (16) and using (15) they may be transformed to the following set of equations: (ρj) (ρu ij) (ρej) = J (ρ(w j u j )) = J (ρu i (w j u j )) +J(ρb i p ) = J (ρe(w j u j )) +J(ρu j b j (pu j )) (17) The above set of equations (17) may be rewritten again in a compact form as: + F a i w U = F d i + S (18) similar to (1) but now for the ALE description. In the algebraic manipulations we have dropped the viscous fluxes because they do not change in the ALE description relative to those in fixed grids...3. Weak variational form of the ALE conservation equations We follow the paper of J. Donea (198) but instead of assuming constant density we work with compressible flow formulation. Mass equation After some algebraic manipulations using (9) and (15) and the first equation in (17) we arrive to: ρ = (w u) ρ ρ u = (ρu) + w ρ (19) Even though both expressions are equivalent and useful we have chosen the last one. Multiplying this equation by an arbitrary weighting function ψ and integrating over a control volume we obtain the following weak form of the mass conservation equation: ψ ρ dv = ψ (ρu)dv + ψw ρdv () Energy equation In the same way for the energy conservation equation we have its weak variational form written as: ψ (ρe)dv + ψ ((ρe + p)u)dv ψw (ρe)dv = ψρb u (1)

6 Momentum equations Finally the weak variational formulation of the momentum conservation equation is written as: ψ (ρu)dv + ψ (ρu u + pi)dv ψw (ρu)dv = ψρb () where means for tensor product and I represent an identity second order tensor...4. Some remarks We have to remark that this formulation satisfies naturally the geometric conservation law (GCL) as it was demonstrated by I. Lomtev (1999). Another remark is due to the spatial stabilization technique. Equation (18) restricted to inviscid flows may be written in a quasi-linear form as: + F i a w U = + [A i w i I] U = (3) Therefore, the spatial stabilization (upwind) scheme to be used may be computed from the modified advective jacobian instead of the original ones. In the next section we present some details about the numerical method employed with special emphasis on the spatial discretization scheme. 3. The numerical method In order to get a numerical solution of the continuum problem presented in the above section we have to discretize the problem using a particular numerical method. In this work we have used a SUPG technique that is very popular in the context of finite element method and is one of the most referenced in the CFD area. There are a lot of references concerning with this technique that may be included here but for simplicity reasons we forward the readers towards two of the most cited references (Brooks and Hughes (198),Hughes and Tezduyar (1984)). This technique is based on a Petrov-Galerkin weighted residual method which allows to use test functions that can be different from the interpolation ones and not necessarily continuous. This method introduces the numerical dissipation needed to stabilize the system in advection-dominated problems, keeping the consistency with the continuum problem. For each node a there is an interpolation function N a (hat type in 1D, bilinear in D, and multilinear in general) and a test function W a = N a + P a, where P a is called the perturbation function. The standard Galerkin method is recovered when we impose P a. The P a (and, of course, W a ) are not necessarily continuous through the inter-element boundaries Perturbation function Following the procedure employed in the fixed grid Euler equations and adopting for the ALE description that the advective jacobians are transformed according to (3) we define the SUPG perturbation function as: P a = τ[a wi] N a τ = h A wi 1 In this work we have used an extra way for computing the intrinsic time scale matrix τ that is less expensive than (4) and works reasonably good. This procedure is due to G. Le Beau and Tezduyar (1993) and may be written as: (4)

7 τ = τi τ = h λ max 1 λ max = u w + c (5) Rewriting the weak variational formulation presented above (),(1) and () and after using the compact notation with the SUPG method just described we arrive to: W a dv + W a F a dv W a w UdV = W a F d dv + W a SdV (6) 4. Validation tests 4.1. Test 1: Shock-reflection problem This two dimensiona, inviscid, steady problem involves three flow regions separated by an oblique shock and its reflection from a wall. It is a standard benchmark problem (G. Le Beau and Tezduyar (1993), Shakib (1988)). The importance of such a test is to check the ability of our code to resolve flows involving shocks. The computational domain is a rectangular region of dimensions 4.1 in the x direction and 1. in the y direction. The mesh consists of 6 rectangular elements. The boundary conditions are the following: { [ρ, u 1, u, M] = [1, 1,,.9] [ρ, u 1, u, M] = [1.7,.933,.1746,.3781] left, top. (7) While figure (3) shows at left the isobars and at right the density at y = 5 where it may be viewed the three different regions above mentioned. Figure 3: Shock reflection problem. Results 4.. Test : ALE for scalar advection-diffusion problems. This problem proposed by T. Sheu (1999) consists of solving the standard scalar advection-diffusion equation written here as: φ t + yφ x + xφ y y (φ xx + φ yy ) = f(x, y, t) f(x, y, t) = e t y(x 3 y 3 3x y + 4xy x y + 3xy) (8) whose analytic solution is φ(x, y, t) = e t x(1 x)y (9) The domain is defined by x [, 1] y [.5, 1.5] and the bottom side is changing in time. To compute this variation the grid velocity is specified by:

8 w = [, π/ sin(πξ) cos(πt)(η 1)] (3) The following table shows the evolution of the error norm with time. Time L norm Time L norm The results in the above table and in the figure (4) show a very good agreement with those reported by T. Sheu (1999). Figure 4: Scalar advection-diffusion with ALE. Test 4.3. Test 3: Random grid This test proposed by J. Trepanier and Camarero (1991). The test consists in verifying the geometric conservation law (GCL) and for that we move the mesh keeping the boundaries fixed. The initial solution is spatially uniform and should remain steadily frozen. This was verified by the simulation and here we only show some meshes used in the computation, see figure 5. The solution is constant in time and uniform in space at machine tolerance. 5. Conclusions and future trends These are only some preliminary results about solving CFD problems involving variable domain. We have chosen ALE strategy instead of other methodologies like space-time finite element method because of computational cost reasons. We have found our results very promising, specially in the geometric conservation law point of view. We plan to continue our work solving more challenge problems involving inviscid or viscous flow with our final target of simulating the heat and fluid flow inside a cylinder of an internal combustion engine. 6. References Brooks, A. and Hughes, T., 198. Streamline upwind petrov-galerkin formulation for convection dominated flows with particular emphasis on the incompressible navier-stokes equations. Comp. Meth. Applied Mech. and Engineering, volume 3, pp

9 Figure 5: Random grid. Meshes C. Farhat, M. L. and Stern, P., High performance solution of three-dimensional nonlinear aeroelastic problems via parallel partitioned algorithms: methodology and preliminary results. Advances in Enginnering Software, volume 8, pp G. Le Beau, S. Ray, S. A. and Tezduyar, T., Supg finite element computation of compressible flows with the entropy and conservation variables formulations. Comp. Meth. Applied Mech. and Engineering, volume 14, pp Hughes, T. and Tezduyar, T., Finite element methods for first order hyperbolic systems with particular emphasis on the compressible euler equations. Comp. Meth. Applied Mech. and Engineering, volume 45, pp I. Demirdzic, M. P., 199. Finite volulme method for prediction of fluid flow in arbitrarily shaped domains with moving boundaries. Int. Journal for Num. Meth. in Fluids, volume 1, pp I. Lomtev, R. Kirby, G. K., A discontinuous galerkin ale method for compressible viscous flows in moving boundaries. Journal of Comp. Physics, volume 155, pp J. Donea, S. Giuliani, J. H., 198. An arbitrary lagrangian-eulerian finite element method for transient dynamic fluid-structure interactions. Comp. Meth. Applied Mech. and Engineering, volume 33, pp J. Trepanier, M. Reggio, H. Z. and Camarero, R., A finite volume method for solving the euler equations on arbitrary lagrangian-eulerian grids. Computer and Fluids, volume (4), pp J. Trepanier, M. Reggio, M. P. and Camarero, R., Unsteady euler solutions for arbitrarily moving bodies and boundaries. AIAA Journal, volume 31(1), pp Kershaw, Prasad, S. and Milovich, d unstructured mesh ale (arbitrary lagrangian-eulerian) hidrodynamics with the upwind discontinuos finite element method. Comp. Meth. Applied Mech. and Engineering, volume 158, pp Shakib, F., Finite element analysis of the compressible Euler and Navier-Stokes equations. Ph.D. thesis, Department of Mechanical Engineering, Stanford University. T. Nomura, T. H., 199. An arbitrary lagrangian-eulerian finite element method for interaction of fluid and a rigid body. Comp. Meth. Applied Mech. and Engineering, volume 95, pp T. Sheu, H. C., A transient analysis of incompressible fluid flow in vessels with moving boundaries. Int. J. of Num. Meth. for Heat and Fluid Flow, volume 9(8), pp W. Tworzydlo, C. Huang, J. O., 199. Adaptive implicit/explicit finite element methods. Comp. Meth. Applied Mech. and Engineering, volume 97, pp