BEM for compressible fluid dynamics L. gkerget & N. Samec Faculty of Mechanical Engineering, Institute of Power, Process and Environmental Engineering, University of Maribor, Slovenia Abstract The fully developed boundary element method (BEM) numerical model of compressible fluid dynamics is presented. In particular, the singular boundary domain integral approach, which has been established for the viscous incompressible flow problem, is modified and extended to capture the compressible fluid state. As test cases a natural convection of compressible fluid in a closed cavity and an L-shape cavity are studied. 1 Introduction This paper deals with a BEM numerical scheme developed for simulation of motion of compressible viscous fluid. The method is based on the approximate solution of the set of Navier-Stokes equations in the velocity-vorticity formulation. Particular attention is given to proper transformation of the governing differential equations into corresponding integral representations, which satisfy the continuity equation exactly, e.g. velocity and mass density field functions. The pressure field is computed by using the Poisson equation for pressure for known velocity, vorticity and mass density functions. 2 Conservation laws The analytical description of the continuous homogenous medium motion is based on conservation of mass, momentum and energy with associated rheological models and equation of state. The present development will be focused on laminar flow of compressible viscous isotropic fluid in solution domain $2 bounded by boundary T. The field functions of interest are velocity
124 Boutrdary Elcmatr~~ XXV vector field vi(rj,t), pressure field p(rj,t), mass density field p(rj,t) and temperature field T(rjt) such that the mass, momentum and energy equations are satisfied, written in the Cartesian tensor notation xi, where p and c,, are fluid mass density and isobaric specific heat per unit mass, t is time, gi is gravitational acceleration vector, qi denotes the components of the total stress tensor, g; stands for heat flux vector. Using the Stokes mass time derivative in general form --- ~t at the following alternative form of the conservation laws can be stated where c is the isobaric specific heat per unit volume, c -- cp P. For an incompressible fluid, the rate of change of mass density following the motion is zero, that is and the mass conservation equation takes simple form V4=0, expressing the solenoidality constraint for the velocity vector. The set of field eqns (4)-(6) has to be closed and solved in conjunction with appropriate rheological models of the fluid and boundary and initial conditions
of the flow problem. Boundary conditions in general depend on the dependent variables applied, i.e. primitive or velocity-vorticity variables formulation. For a compressible fluid, the Cauchy total stress au can be decomposed into a pressure contribution plus an extra deviatoric stress tensor field function 0.. = -p6.. +r.. 1 'I II ' (9) where fiij is the Kronecker delta function. 3 Rheological models In general, real fluid in motion sustains shear stresses. The most general relationship between the extra stress tensor qj and the strain rate tensor E, is given by the Reiner-Rivlin model r0 = a6,i + P&ii + y2ik&kj, (10) where the coefficients a, p and y are functions of three scalar invariants of strain rate tensor Eij. For a simple viscous shear compressible fluid in motion one can consider the relations a = -2qD / 3 and P = 2q, such that the following constitutive model can be stated n where quantity D = div6 = Cii, and r] is dynamic viscosity. For a most heat transfer problems of practical importance Fouier model of heat diffusion is accurate enough where k is heat conductivity. 4 Summary of governing equations Combining constitutive models for stress tensor and heat diffusion flux, eqns (1 1) and (12) in conservation eqns (5) and (6) the following system of nonlinear equations is developed
126 Bnurzdary Elcmatzt~ XXV Using an extended form of the operator divz, i.e. the momentum eqn (14) can be written in the form appropriate for development of the velocity-vorticity formulation DU 4 p- = -grad p -I- pi - rot(r]g)+ -grad(q~)+ 2grad6 gradv Dt 3 (17) +2 grad 17 X c3-2213 grad 7. 5 Velocity-vorticity formulation The divergence and the curl of a vector field function are fundamental differential operators in vector analysis. Applied to the velocity vector field they give local rate of expansion D and local vorficity vectoor representing a solenoidal vector by definition, the fluid motion computation procedure is partitioned into its kinetics and kinematics [l]. The vorticity transport in fluid domain is governed by non-linear parabolic difhsive-convective equation obtained as a curl ot the momentum eqn (17), i.e. written in general vector form D ac5 1 - -=-+(~V)CZ=V,AW+(U~~V)~~-WD+-VXP, (20) ~t at PO or in Cartesian notation form The pseudo body force vector Fincludes the effects of variable material properties F=~"(~-Z)-~~VXG-GXV~+~VU.V~-~DV~, (22) the following tensor notation form is also valid
where a = DV l Dt represents acceleration vector. For the two dimensional plane notation the vorticity vector i7i has just one component perpendicular to the plane of the flow, and it can be treated as a scalar field function. The stretching-twisting term is identically zero, reducing the vector vorticity eqn (20) to a scalar one for the vorticity w where pseudo body force term is - am a17 avi aq F;. = p(gi -a,)-r7e..-+e..m-+2---2d-, 317 (25) rj lj axj axj ax, axi or by putting together last three terms represented also as In eqns above, the material properties are considered as a sum of constant and variable part, i.e. 17 =% +ff and p=po+p. (27) Applying the curl operator to the vorticity deffinition PX~~~=TX(V~V)=V(V.~)-A~~, (28) and by using the continuity eqn (13), the following elliptic Poisson eqn is obtained AU+VX~~~-VD=O, (29) or in tensor notation form a *V, am, m -0 +eij, a.,' axj axi The eqn (30) represents the kinematics of a compressible fluid motion expressing the compatibility and restriction conditions between velocity and solenoidal vorticity vector field functions at a given point in space and time. TO accelerate the convergence of the coupled velocity-vorticity iterative scheme the false transient approach is applied. Thus, in the solution scheme the eqn (30) is rewritten as parabolic diffusion eqn for velocity vector with a as a relaxation parameter. It is obvious that the governing velocity eqn (30) is exactly satisfied only at the steady state (t+-) when the false time derivative or false accumulation term vanishes.
6 Pressure equation Let us rewrite momentum cqn (17) for pressure gradient gradp vp=fp =P(X-ri)-~x(rlui)+~~(rl~)+2~~~~0+2~~xoi-2~~ (32) 3 whcrc in vector function fr, inertia, gravitational, diffusion and non-linear malcrial effects are incorporated. To derive pressure equation dependent on known field function values the divergence of eqn (32) should bc considered A~-V.&, =O. (33) Considering the normal component of the eqn (32) thc Neuman pressure boundary conditions ar specified. 7 Integral representations The unique advantage of BEM originates from the application of Green fundamental solutions as particular weighting functions. Sincc they only consider the linear transport phenomena, an appropriate selection of a linear differential operator is of main importance in establishing stable and accurate singular integral representations of thc original differential conservation equations. 7.1 Kinematics Consider an integral representation of false transient velocity eqn (31), which can be recognized as a non-homogenous parabolic PDE of the form the following corresponding boundary-domain integral eqn can be obtained by applying weighted residual statement, e.g. written in a time incremental form with a timc step At = tf - tf-, where U* is the parabolic diffusion fundamental solution
Equating pseudo body force term b with rotational and compressible part of fluid motion and assuming constant variation of all field functions within individual time increment, one can derive the following integral statement c(<@({,r,)+ J(vu* = ~(vu* xii)xi#+ JGXVU*~Q r I- R R t~ where U* =a ju*dt DVU*~~ + J"F-lu;-,d~, t~-l The kinematics of planar fluid motion is given by R (37) Eqn (37) is equivalent to continuity equation also recognized as compatibility and restriction conditions between velocity and mass density field functions, and vorticity definition expressing the kinematics of general compressible fluid motion in the integral form. Velocity boundary conditions are incorporated in the boundary integrals, while the first two domain integrals express the influence of the local vorticity and expansion rate on the velocity field. The last domain integral is taking into account the influence of initial velocity conditions of false transient phenomena. In eqn (37) normal and tangential derivative of fundamental solution au * /an and au * /at are employed. To compute boundary values of field fuctions normal or tangential form of vector eqn (37) [2] is demanded. The boundary vorticity values are expressed in integral form within the domain integral. One has to use tangential component of eqn (37) to determine the boundary vorticity values ~(<~(~)xc(<.t,)+n(~)xj(vu*.ri)im= r 7.2 Kinetics Considering the vorticity kinetics in an integral representation one has to consider the parabolic diffusive-convective character of the vorticity eqn. Since
130 Boutrdary Elcmatr~~ XXV only the linear parabolic diffusion differential operator is employed in this work, the vorticity eqn can be formulated as a non-homogenous parabolic diffusion eqn as follows with the following integral representation wher U* is again the parabolic diffusion Green function. The domain integral in eqn (41) of the non-homogenous non-linear contribution b, represented as includes the convection and effects of variable material properties. Assuming again constant variation of all field functions within the individual time increment, the final integral statement reads as 1 1 au* -dq 770 r. '70 a axj --~pwvjnj(l*~+-jpm Applying similar procedure to heat transport equation, one derives the following integral representation
7.3 Pressure equation Pressure eqn (33) is recognized as an elliptic Poisson equation, thus the following can be stated with the corresponding singular integral representation where U* is the Laplace fundamental solution. By equating body forces with the expression the following final integral statement can be obtained For known Neuman boundary conditions, given by eqn (34), scalar pressure field function can be computed by solving eqn (33) or (48) for known velocity and vorticity fields and material properties for a given instant of time. The solution to eqn (34) is written in an explicit manner of boundary and domain integrals. 8 Computational scheme If one is to solve singular boundary-domain integral representations to get values of field functions one has to transform the derived integral eqn into its discrete algebraic forms. The key to this is partitioning of computational external boundary into boundary elements and interior domain into domain cells [3]. Use of Green fundamental solution results in boundary discretization of linear transport phenomena part, while internal cells consider non-linear transport phenomena part. In the present work, quadratic interpolation functions are used for boundary elements and internal cells. 9 Numerical solution 9.1 Natural convection in closed cavity As a first numerical example natural convection in a closed cavity (shown in Fig. 1 with corresponding boundary conditions) is examined for ~a=10~. The numerical simulation results (velocity components v, and V, at x=0.5 and y=0.5) for the full compressible form of Navier-Stokes eqns are compared with those based on the Bussinesq approximation of Navier-Stokes eqn in Fig. 2 and Fig. 3
132 Boutrdary Elcmatr~~ XXV Figure l: Closed cavity with corresponding boundary conditions coordinates of the velocity components comparisons. and Figure 2: Velocity component v, of compressible (com) and non-compressible (ncom) fluid flow at x=0.5 in closed cavity. The comparison of overall Nusselt number values obtained by De Vahl Davis et al. [4], [S] and BEM for Bussinesq approximation (B) and full compressible flow (C) case is shown in Table 1 for I?a=1o4,
0 0.2 0.4 0.6 0.8 1 X Figure 3: Velocity component V, of compressible (corn) and non-compressible (ncom) fluid flow at y=0.5 in closed cavity. Tablc I : Comparison of Nusselt number values. Dav~s (B) BEM (B) BEM (C) fi 2.243 2.245 2.632 I Figure4: L-shaped cavity with corresponding boundary conditions and coordinates of the velocity components comparisons.
9.2 Natural convection in an L-shaped cavity Natural convection in an L- shaped cavity (shown in Fig. 4) has been applied as the second example case to analyze the differences between velocity field of compressible and non-compressible fluid flow. Velocity components at x=0.75 and y=0.75 have been compared in Figs 5 and 6. -10-5 0 5 10 15 Vx Figure 5: Velocity component v, of compressible (com) and non-compressible (ncom) fluid flow at x=0.75 in L-shaped cavity. Figure 6: Velocity components vy of compressible (corn) and non-compressible (ncom) fluid flow at y=0.75 in L-shaped cavity.
10 Conclusions BEM numerical model of the compressible fluid dynamics has been developed successfully. In particular, singular boundary domain integral approach, which has been established for viscous incompressible flow problem, is modified and extended to capture compressible fluid. As test cases, natural convection of compressible fluid in closed cavity and L-shaped cavity are studied at ~a=10~. In addition, developed model represents good basis for BEM model currently under development for multi-component compressible reacting flows. References [l] Skerget, L., HriberSek, M., Kuhn, G. Computational fluid dynamic by boundary-domain integral method. Int. J. Numer. Meth. Engnng., 46, 1291-1311. [2] $kerget, L. AlujeviE, A., Brebbia, CA., Kuhn, G. Matural and forced convection simulation using the velocity-vorticity approach. Topics in Boundary Element Research, 1989. [3] Wrobel, L.C. The boundary element method. Vol l., Applications in thermo-fluids and acoustics, Wiely, 2002. [4] Davis, G.D.V. Natural convection in a square cavity: A bench mark numerical solution. Int. Jou. for Num. Meth. in Fluids, 3, 1983,249-264. [5] Davis, G.D.V. Natural convection in a square cavity: A comparison exercise. Int. Jou. for Num. Meth. in Fluids, 3, 1983,227-248.