Transactions on Modelling and Simulation vol 35, © 2003 WIT Press, www.witpress.com, ISSN 1743-355X 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 Transactions on Modelling and Simulation vol 35, © 2003 WIT Press, www.witpress.com, ISSN 1743-355X 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 cpP . -- 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 Transactions on Modelling and Simulation vol 35, © 2003 WIT Press, www.witpress.com, ISSN 1743-355X 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 aucan be decomposed into a pressure contribution plus an extra deviatoric stress tensor field function 0.. = -p6.. +r.. 11 '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 Transactions on Modelling and Simulation vol 35, © 2003 WIT Press, www.witpress.com, ISSN 1743-355X 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 Transactions on Modelling and Simulation vol 35, © 2003 WIT Press, www.witpress.com, ISSN 1743-355X 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 +eij, -0 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. Transactions on Modelling and Simulation vol 35, © 2003 WIT Press, www.witpress.com, ISSN 1743-355X 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~~~1, (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 =O. A~-V.&, (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 Transactions on Modelling and Simulation vol 35, © 2003 WIT Press, www.witpress.com, ISSN 1743-355X 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 (37) DVU*~~+ J"F-lu;-,d~, R R t~ where U* =a ju*dt t~-l The kinematics of planar fluid motion is given by 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.