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/ PREDICTION OF SECONDARY VORTEX FLOWFIELDS GENERATED BY AN INTERJ!ACTING MULTIPLE FR'EE-JET CONFIGIJIRATION by A. J. Baker,, J., A. Orzechowski, and' G. E. Stungis presented as Tecfinical Paper AIAA 83-fl289 at ttne AIAA 21st Aerospace Sciences ~!{eet~ing, Reno, NV, Jarnuary 1993, SSubmitted:, May, 1983 Revisedt March, 198'4 Synoptic: August, 1994 Department of Engineering Science and Mechanics University of Tennessee 1Knoxvillle, TN 3799b-203'0'

PRE'DICTION. OF SECONDARY VORTEX FLOWFI'ELDS GENERATED BY AN INTERACTING MULTIPLE FREE-3ET CONFIGURATION A. 3. Baker•, J. A. Orz.ecFiiowski* *, and G. E. Stcmgis+ ABSTRACT A penalty finite element numerical algorithm, f'or soiutiion: of the three- dimensional parabolic Navier-Stokes equations for subsonic turbulent flows, is examined for prediction, of- secondary vortex flowf ields induced by a multiple free-jet configuratiion. The combined action of decay of the initial high speed jets, turbulence level, induced entrainment from the: farfield,, and geometric discreteness for a four-jet configurafiion is predicted to produce a persistent system of eight counter-rotating vortex pairs in the plane normal to the jet axis. The magnitude of the inducedi transverse vortex vellociity components can approximate tu% of the jet initial velociAy. The results of a range.oi numerical predictions are interpreted and compared with available experimental data. L INTRODUiCTION! An important problem class in steady subsonic aerodynamics iss characterized' by the merging of' viscous andl (perhaps) turbulent unidirectiional flows fol'liowing abrupt termination o1 a surface off separation. The classic exampie is the confluence of upper and lower surface boundary layers at an airfoil trailing, edge. A second illustration, is a jet issuing into a quiescent *'IBM Professor of' Engineering Science and Mechanics, University of Tennessee, Knoxville, TN, Associate ffellow AIAA. *'* Principal Progra!mmer, Computational Mechanics Consultants, Inc., Knoxville, TN +Research Consultant, Brown & Willliamson Tobacco Corp., Louisvillle, KY. 1.

chamber or merging with a coflowing stream with distinct initial momenta. Each problem def~inutibn corresponds to abrupt, infusion of an (axial) momentum defect, the relief of which can result in generation of substantial perpendicular velocity components usually termed "entmainrnent " This is state& rnathematically by the continuity equatibn, which requires that an incompressible (or small Mach numberY velbcity (vector field) be always divergence-f ree. Two- dimensional or axisymmetr'ic definit'ibns for laminar flow and certain turbulent flows are amenable to exact andl/or similarity analyses, e.g., the inf'inite slot jet (SchlGchtingl')„ triple deck theory for the trailing edge problem (Melhick2) and/or an, isolated jet downfield of the potential core (Schetz3). These analysis techniques do not usually generalize to three-dimensiional! geometries, or to most two-dimensional turbulent nearfield flows of broader interest in aerodynamics. The characterization of'such flows generates the need to1 develop1 a suitable discrete: approximate (numerical) solution to an appropriately simplified form off the governGng, Navier-Stokes equation system. The abiding character of an aerodynamic flbw in this class is the predominance o2 a preferred (axial) flbw direction. An elementary extension of the convent'iional boundary layer order-af- magniRude analysis confirms that diffusion processes parallell to this preferred - direction are of the order Re Ys smaller than: all convection and transverse plane diffusion processes, where Re is the characteristic Reynolds number. The deletion of these terms yields the so-called thin-layer Navier-Stokes (TLNS) approximation, which corresponds to ai singular perturbation definition since the deletion of' the highest (second) order derivatives in, the axial direction removes ~I the ability (requirement) to specify a, downstream outflbw boundary condition. Provided a suitable approximation procedure exists to enforce the remaining, fully el'liiptic pressure coupling, the thin-layer equation system can be further

siimplifiedl to the "parabolic Navier-Stokes" (P;VS)i equations. A discrete approximate solution technique for the PNS definition utilizes the first order axial convection term to construct an iniitial-valve solution procedure. The PNS equation system remains an elliptic problem, definition in the plane transverse t+oo the axial coordinate direction, and definifiion, of a multiple-sweep viscous- inviscid interaction algorithm " permits an enforcerment of the subsonic flow three-dimensional elliptic pressure couplling, cf., Baker and Orzechowski4. As developed in this paper, a penalty finite element numerical sollution algorithm is well suited to construction of' discrete approximation solutions to a class of' subsonic, three-dimensional multiple free-jet problem definitions. The PNS ordering confirms that the continuity equation governs to first order the development of the transverse plane velocity distribution, while the two transverse plane momentum equations describe first order modifications to the otherwise uniform static pressure distribution. The sole known initial condition is typically the jet velocity (ratio, to the freestream), and'the farfield transverse plane boundary conditions must admit self -generation of a transverse mass flux distribution (corresponding to development of the associated entrainment field),. The key elements of the established penalty finite element algorithm applied to the free-jet problem class include, 1) establishment of a quasi-linear pressure. Pousson, equation, with complementary and particular solution fields that readily admit associated farfiel& boundary condition speci f iicat ions, and! 2) a specific form for the continuity constraint that provides an efficient and accurate procedure for gilobal communication of the local momentum defect relaxatiion.. For an initally turbulent jet, a parabolized' form of the two-equation turbuJent. kinetic energy-isotropic dissipation functiion, (4ty c) different ial equation system is employed, coupled with an appropriately ordered algebraic Reynolds stress 3'

c+onstit~ut~iveequatibn. This penalty finit'eelerment ail'goriitihm, is developed and app'lied to prediction ofal substantial secondary vortex fibwfield assaciartea with a eloseliy coupled multiple firee-jet problem definition. ll. PROBLEM' STATEMENT The three.-dimensional PNS equation system describing thesteady, subsonicc turbulent flow of an isothermal variable-density fluid to the principal scale of ordering4 is a j ~ J- _ 9 X ~ Ltw !Y= a.a,Xl~ 1 ~u !; l~ + d ~ t Lr k) = 3x k+~zR ~ k`°iC = 0 L(k) = a,x! [p u lk] + 2 XR I P uI k+'TP 2u! + pu~- +p£= 0. X a ~ L(c) = aX,! [PulJ + axI Cp utE + QCE €~Jt ax' 1 au 1 2 E 2 +CE Pulu~, k ax~ + e p~ = 0 4 a~ ~l+pu~ - e ax~ ak~ k- ~~ £ uj~ - v~ i ~.»I, / a x li (4)

The variables appearing in equat'ions' 1-S have their usual interpretation: in fluid mechaniks, and the superscript bar' denotes time averaging5. The tensor index summation convention is implied, with xl, aligned with the principal flow direction and' 1< j,c 3 an&2 c(k,,R) < 3. The turbulence kinetic energy k is the trace of' the Reynolds stress tensor,, c is the isotropic dissipation function, and Re is the characteristic Reynolds number. An algebraic Reynolds stress constitutive equation is employed to close equations 1-5. The PNS: approximation yields the significant order off components of u_u.' proportional to Re-Y1, and' the Reynolds stress modeli 1l simplif ied to, this order yields4 T- I> k3 a U l ulu,l, = Clk - C2C4-~ ax E k3 a'u1 2 "iui = C3k - C2Cw~ aX e 2 3 a,; C3k - C2C4 ~ a x3 uluZ = ulu3, _ u7u3 = 2 3u1 ,2, +(a~x1,, k3 au l a u' 1 - C2C4 -~ x2 ax3 (6) The various coefficients in equations 4-5' are modell constants with suggested' "standard"' valiues6 C = 1. C- 1= 1.3' C I = 1!.44 and! C2 = 1.92. The stress k ~ E ~ E E 5

rnodell coefficients Ca, l< a< 4, equation 6 are defined in terms of' two: empirical constants in the form 22(C401 - 1) - 60 C02 - S). C 1 - 33(C 01 - 2C 02) C2 = 33(C 0,1 - 2C02, Y C 44~C ~1 - 22C41 CL2 - t28iC4~2 - 4' = 1165(C~ i - 2C 02)2 C?., + 110 (7) where C'~1 = 2.8 and' Ct2'= 0.45' are suggested "standard" values, Hanjalic and Launder6. The resultant values for the a are { 0.94, 0:067, 0.56, 0.068 Summing the first three terms in equation 6 and dividing by two then yields 1'' _ 1 1t3 a!u li a7u 1 uiuj S i9 _ 2 (Cl + 2C3)k - C2C4 C Z _"~ a:x1 f . . . C (k ~ 2 aul aw l ..-,i ~ l~ 1.033 - O.OOS {£ ax , aX Jr s' ~. (9) The second term in equation 8 is uniformly non-negative and couplles transverse plane mean flow, strain rates into the Reynolds normad, stresses. Numerical experience 4'7 indicates the magnitude of' this term is at most a few percent off k; hence, equation 8 approximates an identity for the parabolic flow problem class. 4(3C 2 1') 11I(C ~ - 2C~ 2J - 22(C~1__1) - 1'2(3C 2 - 1). 6

I{ QI. NUMERICAL SOLUTION ALGORITHM In the primitive derived form, equations 1-3 do not represent a well-posed initial-boundary value statement for the subsonic flow problem class. As a consequence of' the PNS: ordering, simplificatibni both transverse momentum equations are independent of, the corresponding velocity component,, confirming the assertion that the continuity equation governs these effects to first, order.. Subsonic flow PNS algorithms can be distinguished by the manner selected' to establish a well posed numerical statement8+!'2, and each invol!ves augmentation of, equations 1-3 with the higher order (' d _ Re-Y, ) transverse momentum equation sety a~~ La~ k) -' ~ Ip ul,~]I' +' xR Pu~u k" Re ~ z~, '' 0 (9) The use of equation 9 requires inclusion of the appropriate O(d `) terms in the algebraic, Reynolds stress model, equation 6, which are (10) The penalty finite element aligorithm12 i's well suited to the f'ree-jet problem, dass boundary condition definitions. A tmansverse plane Pbisson equation for pressure is establishedl by forming the divergence of equation 3 plus equation 9, yielding I:(p ) = a / L(u k k i. 2- a a a a - _~+ aX~ aX~(p u~ + a xi ~"' iu ~ 2. 1 ~ -o~ - e 2 a Xi 7

For a turbulent flow, the second and third source terms in equation ll are negligibly small compared to the Reynolds stress term, while for a laminar flbw, their inclusion yields predicted (higher order) transverse plane pressure distributions that correctly balance mean flow eonvectioniand diffusion. The solution for the qwasi=linear equation l l is cast in the form p (xi) = pc(xi) + pP(xli,x,) ('12'). The complementary solution -pC satisfiies the homogeneous form of equation 11 with @irichlet farfi+eld boundary conditions determined by the farfield inviscid flbw. For the subsonic free-jet problem, pc reduces to a homogeneous constant. The particular solution p p satisfies the non-homogeneous equation 11 subject to a farfiield homogeneous boundary cond"utiony i6e., pp (x1,xR)= 0. The particular solution pressure field is stored at select axial stafiibns, during a PNS solution, for computation of, a,p%ixl for use in equation 2 during the next, PNS solution sweep. This multi+-pass PNS procedure has been termed "partially paraboliic," and satisfies the cited requirement of a three-dimensional elliptic pressure coupling upon convergence of p(xl) to a stationary f ieldi The second key aspect of' the penalty finite element aligorithm,, applied to the free-jet problem class,, is the functional form selected to enforce the continuity equation. Letting, u~ denote the semi-discrete approximation to u!i, the exact solution to equations 1, 2, and 3+9, a finite element algorithm employs 0 a weighted residuals statement to extremize the associated semi-discrete TJ ~ approximation error12. This algorithm statement for the transverse mormenturn. ~ ~ . 8 ~ ~.

equation must be "penaliaed"' to enforce discrete approximate satisfaction of' the continuity equation. The resulting theoretical statement is, , Lr R) f La (u R)~ +B'~ 2 a { Mk} L(p h) ={ 0} (13). t ! R t where the collurnn matrix {hlktxj)} is the associated finite dimensiional' subspace used'to define the approximation, i.e., u1(X i) z u~(xz,xl) _' U{Llk(x~)}T{ UL(xl)}e Cl4) e In equation 14, the subscript "e"' denotes pertaining, to the finite element domain Re, and the elements of {' UL(xl)} e are the x}-dependent nodal values of uh on the solution domain discretization nh = U'R e x xl. The, functional, form o2 equation 13 is determined as a direct extension of the cllassical analysis for the (linear) Stokes problem, see reference 1'2. Ch.5. i However, direct replacement of' L(p h) with V• p u in equation 13 would be unsatisfactory for the free-jet problem dass, since V hi is only locally supported using the conventional truncated polynomiat basis { Nk},. For a jiet' (or wake) flow, the momentum defect relaxation process is highly intense within the immediate vicinity of the initiation process,, and the intensity decays roughly exponentially into the si'milarity, region. The penalty functional' form must therrforef~acil'utateglobal communication (t'~hroughoutAZ'2, thet'ransverse plane)~ of' local momentum modification processes. This req}uirement is satisfied by definition of the measure of, LGphy by an elliptic boundary value problem on R2. The obvious choice is the harmonic functiion, ~ satisfying the Poisson equation.