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t(~' )= a'-2L _ 2 . az. 0 u iY = 0' (15) 2x~ r A first integral of' equation 13' confirms tham ~ exhibits the character of' a scalar velocity potential. Therefore, the boundary condition statement for 4~ on the boundary a!li;' of Tt2 is, ~ (~ ) = a~ + b ~I •~iLI= 0 (1~6). where a= 0 yield5 a non-through flow constraint' while b= 0 defines a porous boundary. Since ~ is arbitrary to within al constant, 0 = 0 may be define& anywhere on a boundary segment where b= 0. Hence,: for equations 15-16 ~h -> 0 in any norrn as p~ approaches a divergence free field. The implementation of ~, h as, the penalty measure is embedded within the iteration procedure associatedl with the PNS: algorithm solution statement. For the remaining initial-valiue dependent variables (u i, kh, Eh), as wel4 as the boundary-value variables (pP; f h)„ and letting qh denote the generalizedi dependent variable set„ the weighed residuals semi-discrete approximation error extremizatiion statement: is12 JR2kt { N (x d} t(q) = { 0 } ('17) Equations 13 and 17, for u h kh and c h each produce a coupled ordinary differential equation, system of the form, IA(~1) dl{ ~i} r { g(qi)) = { 0~ }' li N 10

which is employed to complete the Taylor series on, the interval xji l- xj = e x1, {Fl } E {QI}jil & x1,°LV,. (19) _ { p} dxl j+14 to define the non-linear algebraic statement on the discrete approximation {Ql(x1)} (where I is the discrete index denoting,members of qh) For the Poisson variebles p~ and ~h, equatiorn 17 produces { FI} _{ 0} directly. The soliution of the combined non-linear algebraic equation system is obtained using a modified Newton iteration alg,oritfirn of the form 13QQIJ,I ' I p 1{'~~QI }'~+ 1=-{ Fl'~} P I In equation 20,, p is the iteration index, and the p+ist discrete approximate iterate is obtained in terms of'the solution as {QI},p+l ' {'Ql}J+ 1 + {aQ{}j+l (21) For the PNS' algorithm, implementation1'3, the Newton algorithm Jacobian [' J( {FI })), is replaced by two sparse matrices. The initial-value dependent variables u h lrh, andi E h, i.e.,, {Ql }; i< I< S, are solved as multiple right side substitutions to equation 20 using the u i Jacobian [ J I 1) , where [Jl'ljl = a{' F1}' R7_qff N The f ield variables p~ and (P h, Le., { Qd }, 6 < 1< 7, are thereafter solved as ~ rnultiple right side substitutions using the p Jacobian ( 3661 . The six P 0411 components ofuj~uj, are subsequent'lycalculated using an adgebraicassembly~ 11 4

procedure eqwivalent to solving equation 20 using [ J88] with multiple right side substitutions. The PNS ~ algorithm timing uti'izes Ahe sequence (31. 1] ,[ J66j , ,and [398], , with update of the non-linear Jacobian at' each iteration. Hence, the pth level of the Poisson field 4 P' is calculated using the pth approximation to the velocity field' uP. This predictioct is'added algebraically to the previous p-1s4lutions to construct the functional form of, the penalty term in equation, 13 as I _ 61 2 ~ {Nk) E f: h (23) l i'_ 1 R !f a. R2-£ Hence, each iterate f ~, modifies the previous p-1 contributions to the.penalty term, as the correction requiredl to move the next iterate puprl towards satisfaction of the divergence-free requirement. In the limit as p becomes large, Vh- p+I approaches zero in some measure, hence II ~~+li 0 due to: the definition statement, equations 15-16. In practice, following a few extra iterations to homogenize the inital-condition discrete approximationerror, the penalty f irnite element PNS algorithm converges to VQI Irnax < 3. x 10-4 in - typically 4-5 iterations, for S z L in equation 23. The intrinsic measure for ~, h P is the energy semi-norm,, a~h' aO h 11~pl~~F = ~ R z ~~ ~ (24) and typically 114 p 11E < OC10-S) at convergence. This measure can be interpreted in terms of, kinetic energy, since o~A, P b p(ul- u~) is a direct measure of discrete approximation error. For comparison, the kinetic energy of' the mean: flow is «100), and the kinetic energy o,f the turbulent kinetic energy field is 12

0(10 -2'): Hence, the discrete approximate energy error in the velocity field (measure) deviation from exact divergence-freeness at convergence of the Newton iteration is nominally insignif icant. IW: DISCUSSION AND RESULTS The penalty finite element solution algorithm for the PNS equation system, as operational in the CMC:3DPN'S computer programl3-1S, is well documented for a variety of subsonic two- and three-dimensional aerodynamics problem definitions. Within the inviscid interaction, framework, results are published for laminar and turbulent flows in a three-dimensional juncture region geometry and in, a square cross-section ductk. Data on convergence history and satisfaction of the continuity requirement.are presented, as well as detailed, quantitative comparisons between experimental'. data, and PNS' prediction of distributions of the Reynolds stress tensor, equations 6 and 10, for a flow witti a secondary vortex structure. Similar comparisons are reported7 for a turbulent two- dimensiional airfoil~ wake geometry, including, excellent quantitative agreement with experimental data for mean veloci'ty profiles and Reynolds stress distributions over the entire strong interaction region. The robustness of the penalty algorithm for the subsonic PNS interaction problem, class is thus well substantiated, as well as the pertinence of the algebraic Reynolds stress clbsure model implementation using publishe& correlation coefficients. The specific multiple free-jet problem dass of interest is graphed in Figure 1. Four jets, symmetricalPy disposed about the circumference of a circle of radius r= R, exhaust into the half space xli/R > 0 with an initial veliocity (dustribution) strictly parallel to t'he xl-axis. Dependent upon design parameters, this four jet system can, undergo a rapid loss of' initial momentum, with the 13'

consequence that a substantial transverse plane velocity f ield must become induced. The close proximity of the initial jets yields dominating interference effects, such that' the resuDtant three-dimensional velbcity field exhibits (according to the PNS sollution) a systematic secondary vortex distribution. Figure 1 also graphs in perspective the PNS algorithm prediction of a typical induced vortex flowfieldl evolution, in a symmetric quadrant of the transverse plane on 0:25 < xl/>;t' < 1.5.. The engineering, design requirement for the multi-jet configuration is to induce a rapid mixing, ofl fluid, initially contained within the interior region r < R, into the exterior region r<Pt. Experimental data conf'ums tham t'his multi-jieR configuration operating at design conditions efficiently accomplishes this requirement6 A video recording, of the device in operation, and with the multi'- jet system shut-off,, was shouvn during the oral presentation16 at the AIAA 21st Aerospace Sciences Meeting. Using smoke tracer photography,, these visual comparisons provided strong qualitative evidence of the robustness of the mixing, with, the jets in operation. Conversely, the retained' integrity of the smoke tracer column verifies the persistence of a unidirectional fliow with the jiets inoperable. Figure 2a is a still frame from the video verifying this persistence of the smoke tracer column, with the multi-jet system off. Figure 2b is aa corresponding, stilli framewit'~h the jjetsystem, operating. Eventhough the inlet flow conditions for both tests are strictly unidirectional (to the right, parallel to the experiment' axis), the annihilation of the column of smoke appears almost complete with, the multi-jet system operating. If' the mixing process was diffusion dominated, a well defined spreading strvcture would be evident. That it is not indicates that a convection process must dominate, i.e., the multi-jet 14

system must self-create a substantiall multi-dimensional velocity field from a unidirectional inlet flow conditiion.. A measure of the magnitude of' this generated velocity f'ieldl can be estimated in Figure 3„ which is a close-up view of the strong, interaction region! immediately downstream of the plane xl/rt = 0, see Figure 1; and 2b). The initial smoke column is uniformly distributed' on r < R at, xl/R = 0 for all experiments. With the multi-jet system operating, this column is eroded tola conical focus by xl/R :r 1', whereupon stringers of smoke appear to, be propelled outwards and'downstream aibng planar trajectories. Assuming the flow is steady, these streaklines must identify corresponding streamtubes, hence the corresponding tangent angle is a measure of induced flow angle. Defining p as the angle between the xli axis and! a streakline trajectory, Figure 3 yields -39° < 0< 27° as the range of' flow angularity. WhiFe this does not quantify the nature of the flowfield induced by the multi-jet system, it does give strong evidence that it must be substantially three-dimensional. The requirement of the PN'S analysis is to quantitatively predict the inducedl flowfield and to characterize important design parameters. The characteristic dimension of the multi-jet problem is R = 4mm. The jet flows, and the co-parallel interior and.' exterior (r > R) inlet unidirectional flows, are initiated by, applying a reduced pressure everywhere on the half-space x1JR > -1. The resultant induced initial condition velocity field is, ~ uo(xt +xl/D = 0) = 5o(x~,x11LD = 0)a + Oj + 01~ (25) 15

The design initial' velbcity for eacK jet is ua' = 12 rn/s, while on the interior regibn~ (r < R), uo x 0:25 m/s, andl im the exterior regiom (r > R), u o= 0.1 rnLs. Therefbre, the jet' system velocity ratio is X> S0. The initiali cross-section of each jet is non-circular,, as formed by the region interior to the approximate bisectiion ol a nominal 1 mm diameter hole by the circle r= R, see Figure li. This yields a hydraul'iic diameter dh z I mm4 Since each jet flow channel! thus contains two interior, corners approximating right angles, and each is approximately 20 mm in length and has quite rougta walls, and the jet Reynolds number is Redh z 0:6 x 104, the initial jets probably possess a turbulent structure. Hence, the requirement exists to estimate ko(x,,xl = 0) > 0 and o(xI,xl = 0) > 0, for the PNS analysis,, whilk everywhere exterior to the initial' jets ko Z 0 : eo. This completes identification of' the initial-conditions that must be specified for a PNS analysis. The other requirement is to select an appropriate region of R2 for application of boundary conditions. Computations could be executed on the entire transverse plane, but symmetry in the initial conditions andl geometry permits use of a quarter-plane, see Figure 1. The transverse plane region used for most PNS computations is the domain 0 < x./R < 2.0, as graphed in Figure 4 with boundary O-A.-B-C. A non-uniforrn discretization U'Re was defined to permit "adequate" definition: of the initial jet region using: an M = 19 x 19 mesh,, the nodal lbcat!ions of' which are shown as dots. For later comparison with tlhe smoke streakline datay an, inert' species mass fraction was initialized at' all nodes of the interior region, denoted as "smoke column"' inN. Figure 4. The corresponding PN'S algorithm boundary conditions are summarize in Table 1. 16

Tablle 1 PN5' Algorithm Boumda;ry Conditions for Multi-Jet Symmetric Quarter-Plane Domain, Figure 4 Boumdary Boundary Condifion Scgment Vanishing Neumann Diridhlet Comments O- A u I, a 3, k,, E, pp, m u 2= 0 Non-porous O-& ul,u2,k,e,pp,~ u3=0 Non-porous A- C u! i, u 2, u 31 k, ~ ~= 0' = pp Porous B - C ul,u2,u3,k,4 ~ ~ =0=pP Porous While the basic penalty finite element al'goriRhm is well' verified,, benchmark tests of specific pertinence to the free jet problem are required as relates to initial-condition specification for k andi e and flow symmetries. The simple two-dimensional slot flow geometry,, Figure 5, serves the requirement. Figure 6 summarizes initial cond'utions, and the penalty algorithm prediction for two representative downstream distributions of ur 1, u 2, k and~ E, for an assumed symmetric slot jet flow at velocity ratio a= 50. The initial conditions are uo = 10 mi/s on 0< x2/Hf < 1 and w,o = 0:2 rn/s on 1< x2/'H f< 2.5, where. Hf = 5: cm is the slot half-width. Vanishing normal derivative boundary conditions are appllied'. for all variables at x2/H f = 0 and x2/H,f = 2.5,, except ~= 0 at x2/Hf, = 2.5, and an M = 50 element uniforrm, discretization is employed. No other specificat'ions are required for a liaminar flow siinulation,, but both ko and o distributions are required for a turbullent flow simulation. In the absence off' definitive detailed experimental data, one procedure to self-ini'tialize levels is based on an order of magnitude assumption for a"turbulent eddy viscosity"' vt = C4 k2/'e, see equation 6. For "parabolic" aerodynarnic flows, the extremum level of ko (non-dimensio,nad), may be assumed' of' the ordler 17

O('10 `'), i.e., the kinetic energy in the turbulence field approximates a few percent (at most) of the mean flow kinetic energy. Since C4, is a constant,, defining v~a_' O(10p); where v is the (laminar) kinematic viscosity and (say). 1< p < 3,: defines a corresponding leveJl of eo Secondly, since 1/e appears throughout the turbulence closure equation system, it is preferable numerically to define a smalll but non-zero farfield level. Numerical experimentation~ with the slot- and free-jet pr©blems confirmed the penalty PNS algorithm will rnaintain levels of co/'Re z O(',10-9) and k° c (X10-4), with vanishing normal derivative boundary conditions, yielding vt/v z 1 in the farf'ield as an approximatiion to a nonrturb;ullent region of' flow. Figure 6 highlights the essential aspects of the PNS algorithm prediction for the two-dimensional slot-jet flow for the specific initial' conditions k~ = 0.05, vo/v = 10 and A = 50: The initial distribution for ii ° is assumed a step function on the nodes of' U Ite; hence, non-zero levels of' ko and eo are correspondingly defined at one node only. The PNS algorithm is marched downstream five steps, using A x l,/H f z 10-3, to smooth somewhat these initial conditions, whereupon thee penalty algorithm for ~h is initialized. The energy norm for 4~ h reaches O00-5) in approximately 110 more integration steps; hence, the entire algorithm iss furnctional at A xl/'H1, -` 10 2. Approximately 150 (progressively larger) integration steps are required to reach xl/'H f= 0.5; where the extremum levelss of k and € are computed, Figure 4c)-d), using, the standard closure model constants. Thereafter, the level of' both variabUes monotonically decreases as the jet spreads liaterally, with the PNS algorithm reaching, the final station xl/H f = 1.0 after 50 additional integraRibn, steps: The PNS integration step size would continue to increase in proceeding, further downstream as the solution field becomes progressively smoother. Changing the initial conditions, to say vtlv - 16A 18

li0?, would not alter siignifiicantly the solution appearance, but woutdl simpl'y compress the history over a shorter axial displacement due to the more vigorous initial mixing levels. Since the multi-jet problem of interest is not of circular initial cross- section, see Figure 4, and the resultant flowfield is not axisymmetmic, Figure ll, a second verificatiion, test must evaluate a three-dimensional problem of a nominally circular cross-section jet interpolated onto the nodes of a non-uniform rectangular discretization. Figure 7 graphs the M! = 13 x 25 symmetric half- plane d'ascretization, withispan 6D x 2D; for a circular jiet, of initial diameter D= 0.05 mL At nodes interior to the circle r= Di/2, the discrete initial specification for jet velocity is wo = 30 m/s, while uo = 3 mi/s everywhere on the exteriior, r> Di/2, yielding a jet initial velocity ratio a= 10. No other initial conditions are required for a laminar flow simulation; for a turbulent flow, the initial condition specifications at nodes on r < D/2 was ko = 0.01 and v©/v = 110, while on the exterior, ko = 10-4 andi v~/'v = 1. The three farfield boundary segments: are porous, while the jjet bisector is a symmetry plane, recall Table 1. Figure 8 graphs the computed transverse plane velbcity distributions at xl/D = 0.25i 0.5' andl 1.0, for the laminar flow simulation; Figure 9 presents comparisoni data for the turbulent flow case. The direction of each arrow is parallel to the local streamline, and't'he length is proportional to tfie magnitude J scaled to the lbcal non-dimensional extremurn val'ue u~'_ ~ u~ax I / The t~. initial jet diameter is also noted. Both PNS solutions exhibit an excellent approximation to nominal radial entrainment, except that by x1/D: = 1'.0,, the laminar flow prediction exhibits lbcal perturbations about the jet boundary.. These are induced by the step interpolation of the iniTiall data, coupled' with, the 19