BEM for turbulentfluidflow. L. Skerget & M. Hribersek Faculty of Mechanical Engineering, University ofmaribor, Slovenia.

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1 BEM for turbulentfluidflow L. Skerget & M. Hribersek Faculty of Mechanical Engineering, University ofmaribor, Slovenia. Abstract Boundary-Domain Integral Method is applied for the solution of incompressible turbulentfluidflow.the standard two-equation near-wall k - c turbulence model is applied. Integral representations are derived for flow kinematics and flow kinetic transport equations. A special iterative algorithm is applied for solving the k and e nonlinear transport equations and the resulting set of averaged Navier-Stokes equations. Effect of buoyancy force on production of turbulence is accounted for. The numerical results for side-heated quadratic enclosure are presented for a Prandtl number equal to 0.71 with the Rayleigh number values 10 and 2 * Introduction The paper deals with the numerical simulation of a turbulentfluidflowusing the Boundary-Domain Integral Method (BDIM). A time-averaged form of the Navier-Stokes equations formulated for the velocity-vorticity variables is employed through the Reynolds decomposition of the instantaneous value of each field function. Turbulent stress tensor is interpreted in the Boussinesq manner. Two-equation near wall A: - e or turbulent kinetic energy - energy dissipation rate model is used. Particular attention is given to formulation of appropriate integral representations for all field functions. For the kinematics the parabolic diffusion fundamental solution is used, while the integral statements of all kinetic transport equations are based on elliptic diffusion-convection fundamental solution. The finite difference approximation is considered for the time derivatives of field functions.

2 2 Governing Equations 2.1 Primitive Variables Formulation Advances in Fluid Mechanics III With the assumption of incompressibility within Boussinesq approximation the Reynolds Averaged Navier-Stokes (RANS) equations set, governing the transport of time mean flow quantities in a turbulent incompressible fluid motion confined in a bounded domain fi of R*(d = 2,3), can be written as +/f, (2) Dt +, (3) for ij = 1,2,3. Vectors Vi and a% represent mean velocityfieldand spatial position, t is time, P = p-poqjtj is the mean modified pressure, p and % are the mean pressurefieldand the gravity acceleration vector, T stands for the mean temperature, /» = Fgi is vector of mean body buoyancy force with p =&>. fi(t - TO) is the normalized density temperature variation function" / is the heat source, */ and a are molecular kinematic viscosity and thermal diffusivity. The mass density p and the specific isobaric heat Cp are assumed to be constant parameters. Sij is the symmetrical part of the velocity gradient, and are the Reynolds stress or turbulent momentum flux tensor and the turbulent heat flux vector, respectively, where v\ and T' are the fluctuating velocity and temperature, and the overbar denotes Reynolds average. The Reynolds stress tensor ty and turbulent heat flux vector QJ require modeling. The Boussinesq assumption consists in taking tij and % as where Iij is the identity tensor, k is the turbulent kinetic energy, defined as &=-v^, ^ (7) while vt and at are turbulent or eddy kinematic viscosity and turbulent diffusivity, which are unknown functions of position and flow quantities and have to be modeled.

3 Advances in Fluid Mechanics III 157 If assumptions (5) and (6) are used the transport equations (2) and (3) can be written in terms of an effective or total viscosity v<> = v + i/t and diffusivity o<, = o4- (% = a+ ^-, -<** ">- +* < > Dt ~ d where the volumetric part of (*_, has been included in the pressure term 2.2 Velocity- Vorticity Formulation P* = P+lp,k. (10) o With the mean vorticity vector w% representing the curl of the mean velocity field [1] UJbj UJUj the fluid motion computation scheme is partitioned into its kinematics, given by the elliptic mean velocity vector equation - + and kinetics given by vorticity transport equation obtained as a curl of the momentum eq.(8): Dui dujvj. d ( dwi\ df dfij -=TT = -^ -- H ^ I "e^ + ^U&g&a--^ A- ' Dt dxj dxj \ dxj) dxj dxj which is in plane motion case simplified to F dfj where the additional term is equated to tensor quantity /# = Vz/e x 5, which reduces in plane flow case to the following vector quantity - _ fdi/e fdv^, dvy^ _ dve_dv^ _&/, 9^ dvy % \ /'" I &/ %, 3a/ g%/ gz ' ^ ^9%/ ^ 9z ^ 9z 9i/ '/ ' (15) To accelerate the convergency and the stability of coupled velocityvorticity iterative scheme the false-transient approach is applied to eq.(12), rendering the following parabolic diffusion velocity equation a where a is a relaxation parameter. It is obvious that the continuity or compatibility eq.(12) is exactly satisfied only at the steady state (t -4 oo), when the false transient derivative term vanishes.

4 158 Advances in Fluid Mechanics III 2.3 'Turbulence Model In order to solve dynamical equations the turbulent viscosity must be specified. Different turbulence models are obtained depending on the way in which vt is computed. In this paper, we focus on the two equation nearwall k - e turbulence model, in which the turbulent viscosity is given by relation [5], [6], [7], y,=cy/y, (17) e where e is the rate of turbulent energy dissipation / (18) OXjOXj Both turbulence quantitiesfcand e are determined from the individual transport equations. The equations also have to be modeled, which means that several assumptions have to be considered in order to close the mathematical description, as these equations involve moments of fluctuating velocity of order higher than two. In the near wall k e model, the differential transport equations for the individual turbulent quantities to be solved are» «(»)+P-«-B. (19) Dt dxj \ dxjj 57 - where the production term P is given by P = ^s^ (21) and i/efc = v 4- and i/ee = i/ +. The values of the model constants are k e as follows: Cy = 0.09, at = 0.95, <r* = 1.0, ^ = 1.3, <7«i = 1.4, C& = 1.8. (22) The near wall and low-reynolds number functions /,/,/ci,/e2, > and E should assure the correct near-wall asymptotic behavior of turbulence quantities. These functions, proposed by several investigators, are based on different characteristic velocity scales. The dimensionless variables such as v+ and y+ based on friction velocity v* have been successfully applied as characteristic parameters in the near wall region of steady turbulent flows. In unsteady turbulent wall bounded flow, where the near-wall region is characterized by a rapid phase change in flow field quantities, the local turbulence velocity scale such as turbulent Reynolds number, defined as or & = - (23) ve

5 Advances in Fluid Mechanics III 159 is believed to be a better velocity scale. The following near-wall empirical correlations are used [7] /el = 1.0 (25) where the near-wall damping function /%,, accounting for the wall damping effects, is defined as The correction functions D and E are chosen as E = Q, (28) 3 Boundary Conditions In order to solve the complete dynamical nonlinear equations system some physically justified boundary conditions must be specified [2]. Boundary conditions, assigned to elliptic diffusion kinematics equation (12) and to parabolic diffusion-convective kinetics equations (13), (9) (19) and (20), are in general of thefirst,second, mixed and outflow-convective type, imposed on the part of the boundary FI, F2, Fa and F^ respectively. The most physically based boundary conditions for the kinematics arise when velocity Vi is specified over the whole boundary. In this case, normal fluxes of velocity components ^ are the unknown boundary values in the set of kinematics equations, assuming known vorticity distribution in the domain. More difficulties arise when the velocity vector is not known a priori over a part of the boundary, i.e. at the outflow region. In such cases a reasonable choice is to assume zero velocity normal flux values or some kind of convective type boundary conditions over the specific part of the outflow boundary. The boundary conditions applied at the specific parts of the surface are as follows, e.g. specified Dirichlet conditions for the variables (vj,fc,c) at the inflow boundaries Vi Vi, k k, and e = e, on FI, (1)

6 16Q Advances in Fluid Mechanics III zero Dirichlet values for the functions (v*,a;,e) at the solid wall boundaries (no slip conditions) _ Vi = k = e = 0 on I\, (2) and the outflow boundary conditions for the field quantities (%*,fc,c) as zero Neumann normalfluxvalues dk de /g\ = = = 0 on TS, (3) on on on or in the form of outflow-convective boundary conditions du du /,\ + v =Q on F4, (4) ot on where thefieldfunction u can in general be equated to quantities (vi, T, k, e) and v is the mean outflow velocity. The most critical computational part of the kinematics is the determination of the new boundary vorticity values for the vorticity kinetics. The important advantages of the boundary integral formulations as a solution technique is that the integral formulation of the kinematics itself provides the new boundary vorticity values through the application of the vorticity definition as a derived variable or compatibility constraint condition between the velocity and vorticity field on the boundary of the solution domain, e.g. eijk-~-="i on T, (5) OXj which can be explicitly expressed for the plane motion in x - y plane by the following relation dv* dv dvy combining the normal and tangential velocity component fluxes. Thus the only physically correct boundary condition associated with the parabolic diffusion-convective vorticity equation (13) are the specified vorticity values over the whole boundary, e.g. while the vorticity normal flux values (jj = uji on F, (7) dui dui /*\ a nj = -^- on r, («j * are the only boundary unknown quantities in the vorticity kinetics. The mathematical description of the transport phenomena in a nonisothermal fluid motion is completed by providing appropriate boundary

7 Advances in Fluid Mechanics III 161 conditions for the energy equation (9) e.g. prescribed temperature values, normal heat flux values q or heat transfer to the surrounding ambient at the temperature T«with a heat transfer coefficient a, 4 Integral Representations General considerations T = T on TI, dt q - = -- on A on Fg, I - -5<r-'»»> It is of main importance in the context to formulate appropriate stable and accurate boundary-domain integral representation to divide the governing equation into a homogenous linear and non-homogenous nonlinear part [2], e.g. C[u] + b = 0 in 3fJ, (1) where C[ ] is a linear differential operator, e.g. parabolic diffusion, modified Helmholtz, elliptic or parabolic diffusion-convective etc., u(xj,t) is an arbitrary field function, and non-homogenous part b stands for a pseudo body force or source term. 4.1 Integral representations for Kinematics The velocity equation (16) can be given as a nonhomogenous parabolic diffusion equation of the form a Q d*" Q <?H Qj. + b = 0, ' \ (2) OXjOXj Ot / with the following corresponding boundary-domain integral equation written in a time incremental form for a time step A = tp tp-i' + ft* bu*dtdtt+ f UF-iu*p_i(Kl, (3) Q *^~* n where u* is the parabolic diffusion fundamental solution [1]. Assuming a linear variation of field quantities within the individual time increment by the use of interpolation polynomials {#} = {#jp-i, #} tp t,. t tf~i /,\

8 Advances in Fluid Mechanics III the time integrals in eq. (3) may be evaluated analitically. The integral statement eq. (3) can now be rewritten as r + f ^hi^, ^ + 1 /^[/^n + 1 /\f _i^_i<m J on a J a J r n n + I UF-\U*F-i<Kl. (5) Q The boundary-domain integral statement for the false transient kinematics equation (16) can be derived by equating thefieldfunction u with the velocity vector V{ and the pseudo-body force term 6 with the rotational flow part, resulting in the following statement, e.g. for the plane motion I UF-injUp_idT - eij I UF-g^-dft F-iu*F_idtt. (6) 4.2 Integral representations for Kinetics Due to the diffusion-convective character of the kinetics transport equations the convection dominated flows, when the hyperbolicity of the equation prevails its parabolicity, are subjected to numerical instabilities. To suppress such instabilities known to all domain type numerical methods, first or higher order upwinding schemes have to be considered. Although this increase the stability of the numerical scheme, it at the same time introduces the artificial diffusivity resulting in a non physical numerical solution. Partitioning the velocity field and the diffusivity into a constant and perturbed part, such as Vj = Vj 4- Vj and a a -f a, the transport equations for the specific flow field functions can be generalized as a non-homogenous parabolic diffusion-convective equation for an arbitrary scalar field u. By using afinitedifference approximation of thefieldfunction time derivative for a time increment Af Af

9 Advances in Fluid Mechanics III 163 the equation can be written as with the following integral representation c( )ti(0 +a/" 12^-dT = a / VdT - f uvjuju^dt + ^ 6u*d«(9) where %* is the elliptic diffusion-convective fundamental solution [2]. The pseudo body force term b includes convective flux for the perturbed velocity field, nonlinear diffusion, source term and initial conditions, rendering the following integral representation + a fu^dt = af^u^dt- fuvjuju^dt r r r f du ^_ f^dudu*.^ f. * _ + / a u*dt- I a^ ^ dq- I uvj7iju*dt J on J dxj oxj J r n r v^dti + j Ivfdto + ^ y tif-it**dn. (10) ^ The integral representations for the time averaged field functions vorticity, temperature, turbulent kinetic energy and dissipation rate of the turbulent kinetic energy can be readily formulated using integral statement for the general scalar function u, (10), and corresponding dynamical equations, e.g. dn J On J r r OUJ. ^, f d(j OU* _,_ f ~ * rr. p u ot / P a$6 / (jjvjuju al r Q ^ ^ r r ^ du* f + f p&u*jri J ^ dxj ** J * ** J * 9xj n r n /( du* 1 f * finju*dt j j I fj dq, * /ti» -h-r /\ f / / cjf_iw*dfi, (11) V azj zat 7 -f a r

10 1 54 Advances in Fluid Mechanics III n a u^dt- fa^^-d^- [TvjnjU*dT dn J dxj dxj J a r vj^dtt + f u*dtt + -Jr /2>-iti*dn, (12) * dxj J Cpp AtJ r o r n + /(P - c - >)u*dfi + ^ / *F-i«*dn, (13). (14) 5 Numerical Solution Searching for an approximate numerical solution the corresponding integral equations have to be formulated in a discretized manner. The proposed discrete model is based in the limit version of a subdomain technique, such that each subdomain or macro element consists of a quadrilateral internal cell bounded by four boundary elements. It should be pointed out that the boundary integrals of the corresponding linear differential operator are approximated by a 3-node quadratic discontinues boundary elements, while the boundary and domain integrals representing the pseudo-body force term have to be approximated by continuous polynomials, i.e. 3-node quadratic continuous boundary elements and 9-node quadratic continuous internal cell are applied. 5.1 Solution Procedure In order to obtain a solution of the fluid dynamics problem, a sequential computational algorithm was developed [6]. The main steps in this algorithm are: S 1 Start with some initial values for: v^a^t, k, e.

11 Advances in Fluid Mechanics HI 165 S 2 Time step loop (nloop = number of time steps): S 3 RANS loop: 1. Solve Navier-Stokes equations for non-isothermal flow: (a) flow kinematics loop (nloop=6): -solve for %;,^, compute wp, (b) vorticity transport loop (nloop=3): solve for w, ^, (c) heat transport loop (nloop 3): solve for T,f, (d) Check convergence of the Navier-Stokes loop: if CL > (w go to l(a). 2. Compute z/*. 3. Solve k e turbulence model equations (nloop=3): (a) solve k transport equation (b) Compute the new */*. (c) Check convergence of the k loop: ifc*>efcgoto3(a). (d) solve e transport equation (e) Compute the new i/f (f) Check convergence of the e loop: if Q > ( go to 3(d). 4. Compute at. 5. Check convergence of the RANS loop: If Cv < go to S 4., otherwise goto 1. S 4 Set new time step values: Wo,T^,&o,eo, go to S 3. Convergence criterion for the different iteration loops (general unknown function u = a;, &, e, v) is as follows: V^c /i-fl _i \2 r <r c - ^-^ ^ ^ m) ^u < Cu, ^u - J^e (<+!.)2 ' ^ ^ where i is the inner (loop) iteration counter. The values for e% were set as ^ = 5*1Q-G e& = 5*10-\ (16) ^ = 5*10"*, fy=5*10-\ (17) The parameter nloop shows how many local iterations are used to obtain a solution to a certain nonlinear loop.

12 6 Test example Advances in Fluid Mechanics III Buoyant flow in a two-dimensional side-heated square enclosure [8], is simulated for Prandt number value 0.71 and high Rayleigh number values 1 * 10 and 2 * 10 using a nonuniform numerical grid with aspect ratio 8 : 1 and 28 x 28 subdomains. The calculation was performedfirstfor the Ra number value 1 * 10, where the steady state solution exists, and then continued for the Ra number value 2 * 10, where the periodic flow situation was expected. The time averaged flow and temperaturefieldswill be shown during presentation. 7 Conclusions In this work a numerical procedure, based on boundary element method, for the simulation of two-dimensional time-dependent turbulent flows has been described. Two-equation low-reynolds number k - e model has been considered and corresponding integral representations have been discussed. The suitability of the presented technique to compute buoyant flows has been illustrated by the calculation of the flow in a side-heated cavity forfluidwith Pr number 0.71 at Ra number values 1 * 10 and 2 * 10. References [1] L. Skerget and Z. Rek: Boundary-domain integral method using a velocity-vorticity formulation. Engineering Analysis with Boundary Elements, 15, pp ,1995. [2] Skerget, L., Hribersek, M., Kuhn, G.: Computational Fluid Dynamics by Boundary Domain Integral Method; Int. J. Num. Meth. Eng., 46, pp ,1999. [3] M. Hribersek and L. Skerget: Iterative Methods in Solving Navier-Stokes Equations by the Boundary Element Method, Int.J.Num.Meth.Eng., 39, pp , [4] Hribersek, M., Skerget, L.: Fast Boundary-Domain Integral Algorithm for Computation of Incompressible Fluid Flow Problems; Int. J. Num. Meth. Fluids., 31, pp , [5] Z. Rek, L. Skerget and A. Alujevic: Boundary-Domain Integral Method for Turbulent Fluid Flow. Turbulence, heat and mass transfer. Begell House, New York, pp , [6] Codina, R., Soto, O.: Finite Element Implementation of Two-Equation and Algebraic Stress Turbulence models for Steady Incompressible Flows. Int. J. Num. Meth. Fluids, 30, pp , [7] Fan, S., Lakshiminarayana, B.: Low-Reynolds-Number k - e Model for Unsteady Turbulent Boundary-Layer Flows. AIAA Journal, 31, 10, pp , [8] Nobile, E.: Simulation of Time-Dependent Flow in Cavities with the Additive-Correction Multigrid Method, Part I: Mathematical Formulations. Numerical Heat Transfer, Part B, 30, pp , 1996.