A Non-Linear Sub-Grid Embedded Finite Element Basis ... - CiteSeerX

A Non-Linear Sub-Grid Embedded Finite Element Basis for Accurate Monotone Steady CFD Solutions - Part II : Benchmark Navier-Stokes Solutions Subrata Roy Computational Mechanics Corporation 601 Concord St., Suite 116, Knoxville, TN 37919-3382 A.J. Baker Mechanical & Aerospace Engineering and Engineering Science University of Tennessee, Knoxville, TN 37996-2030

Abstract A non-linear sub-grid embedded (SGM) nite element basis is established for generating monotone solutions via a CFD weak statement algorithm. The theory con rms that only the NavierStokes dissipative ux vector term is appropriate for implementation of SGM, which thereafter employs element-level static condensation for eciency and nodal-rank homogeniety. Numerical results for select benchmark compressible and incompressible steady-state Navier-Stokes problem de nitions are presented con rming theoretical prediction for attainment of monotone solutions devoid of excess numerical diusion on minimal degree-of-freedom meshes.

1

A c,cj d dete [Dk ]e [DS ]e e E f (
) fj fvj F , Fij fFQg g,gi fGg h,he,hij [JAC ] k L; R

L

m [M ] fNk g fNS g p; P Pr q jjqjjE Q

NOMENCLATURE

scalar constant continuum SGM parameter (set) dimension of the problem transformation matrix determinant Lagrange diusion matrix SGM diusion matrix nite element volume speci c total energy function of (
) kinematic ux vector viscous ux vector SGM (correlation) function Newton residual vector SGM element embedding function nodally distributed SGM vector nite element length measure jacobian matrix Lagrange polynomial degree element left/right of node j partial dierential equation operator momentum ux assembled mass (interpolation) matrix Lagrange basis function of degree k SGM basis function of degree S static pressure Prandtl number continuum state variable energy semi-norm of q discrete state variable 2

r,rj ,rij distributed SGM parameter (set) R statically reduced matrix fRg solution residual vector 1 basis Lagrange element in 1-D simply returns the k = 1 form, [17]. Therefore, the SGM theory augments the diusion matrix via an embedding function g(x; c), hence the name \SubGrid eMbedded." The de nite integral form of the SGM basis function set, denoted fNS g, for [D]e in (15) is T R Z @ fNS g @ fNS gT Z @ f N k g @ fNk g (22)

e @x @x dx  e g(x; c) @x @x dx

where the statically condensed, reduced Rank form is denoted as jR. The embedded polynomial g(x; c) contains one arbitrary parameter c for each additional Lagrange degree k  2. The form of the 1-D SGM element basis set fNS g for k = 2 = S is expressed, analogous to the k = 1 Lagrange basis, (11), as 8 9 > > > < 1? > = fNS g = >> ; (23) > :  > ; where  = P1 a   is a polynomial function of an expansion coecient set, a , dependent on i=1 i i

i

embedding degree k, and the k = 1 local coordinate  , and  is a function of c. A detailed computation of the SGM 1-D basis is provided in [17]. If the embedded polynomial in (22) is selected as quadratic, then g2 = f1; c; 1gfN2g where fN2g is the Lagrange quadratic basis. Then, for S = 2 = k, the 3  3 element matrix form for [D]e prior to condensation is 3 2 (7 ? 2c) ?(16c + 24) 77 66 (18c + 17) 7 6 (24) [Dk=2]e = 151h 666 (7 ? 2c) (18c + 17) ?(16c + 24) 777 7 e6 4 ?(16c + 24) ?(16c + 24) (32c + 48) 5

Static condensation of this matrix yields

3 2 6 (2c + 1) ?(2c + 1) 77 7 [Dk=2 ]Re ) [DS=2 ]e = 31h 664 e ?(2c + 1) (2c + 1) 5

(25)

The discussion in [17] (Appendix B) con rms c  1 is the2requirement. 3 For c  1, [DS=2 ]e in (25) 6 1 ?1 77 7. This degeneracy is termed is identical to the linear Lagrange matrix, [Dk=1 ]e = h1e 664 ?1 1 5 9

consistent, which occurs only for d = 1. Speci cally, for d > 1 dimensions, static condensation of a k > 1 diusion matrix does not yield the k = 1 form. The d-dimensional form in the dissipative ux vector diusion matrix [D]e in (11), formed via the SGM element (denoted by subscript S ) is T R @ f N k g @ fNk g [DS ]e  e g(x; c) @ x @ x d T Z Sg = e @ f@NxS g @ fN @ x d and many selections are possible. For d  1, g2(x; c) is written as

Z

(26) (27)

g2 = fGgTe fN2g

(28)

For the d = 2 vector c = (cx; cy ), the element array is fGgTe = f1; 1; 1; 1; cx; cy ; cx; cy ; cxcy g and fN2g is the corresponding Lagrange quadratic basis set. The array fGge may be represented by the matrix outer product

fGgT ) f1; cx; 1g f1; cy ; 1g

(29)

hence pictured as the tensor product A B of A = f1; cx; 1g and B = f1; cy ; 1g, i.e.,

cx

1

=

1

1

cy

1

1 cy

1

cx c xc y

cx

1 cy

1

The d = 3 extension is obvious. Element constructions for d = 2 are illustrated in Figures 2e-f. Figure 2e graphs a p-hierarchical 2-D element with two quadratic sides (p = 2), one cubic side (p = 3) and a linear side. The k = 2 (S = 2) condensed 2-D SGM element, graphed in Figure 2f, is very distinct from both the p-element and the Lagrange k = 2 element.

10

The diusive ux vector matrix [D]e in (11), formed with the Lagrange bilinear (k = 1) basis in 2-D for a generic cartesian element of unit span is 2 3 66 4 ?1 ?2 ?1 77 66 77 6 77 Z ? 1 4 ? 1 ? 2 det 77 (30) [D1]e = e rfN1g  rfN1gT d = 6 e 6666 66 ?2 ?1 4 ?1 777 64 7 ?1 ?2 ?1 4 5 where dete is the transformation matrix determinant, equal to one-fourth the plane area of e. The S = 2 extremized form for (26) for 2-D is constructed as 2 3 66 d11 d12 d13 d14 77 66 77 6 7 d d d d e6 66 21 22 23 24 777 [DS ]e = det (31) 6 666 d31 d32 d33 d34 777 64 7 d41 d42 d43 d44 5 where each matrix element dij in (31) is a distinct polynomial function of cx and cy , [18]. For example, assuming cx = c = cy , the rst term in (31) becomes  3 1 2 + 1838c + 549 656 c + 2032 c (32) d11 = 15 (148c22c++164 c + 73) Simplifying further to cx = cy = c = 1, the S = 2 SGM diusion matrix form is 2 3 66 29 ?11 ?7 ?11 77 66 77 6 77 ? 11 29 ? 11 ? 7 det 6 77 (33) [DS ]e = 66e 666 66 ?7 ?11 29 ?11 777 64 75 ?11 ?7 ?11 29 The distinctions between (30) and (33) are apparent. For general applications, a nodally distributed (subscript j ) SGM parameter is preferable to an element parameter. Therefore, de ning rj = (2cj +1)=3, the monotonicity constraint form becomes 2cj + 1  r  juj jhj = juj jhj Re (34) j 3 F F where Reynolds number Re replaces  and F > 0 is a real number. In 1-D, F is precisely determined from the eigenvalue analysis, hence F = 2 for the linear convection-diusion (Peclet) problem, F = 6 for the linear stationary wave de nition, while F = 3 for non-linear convection-diusion (viscous 11

Burgers) equation, [17]. However, for multidimensional problems, determining a suitable functional form for F involves de nition of a correlation function Fij = f (Re, dete ), the form of which must be validated. For velocity eld uj = (u1j ; u2j ; u3j ), principal coordinate mesh measures h1j ; h2j and h3j , and principal coordinate diusion parameter set j ) (1j ; 2j ; 3j ), the condition for a d-dimensional monotone solution is expressed for scalar components of rj = (r1j ; r2j ; r3j ) in (29) and this correlation function Fij is 11=d 0 j u AV ij jhij e (35) rij  F  ; where Fij = @ d?1 A ; 1  i  d; and 1  j  Nnode ij hij ij ij Hence, only the scalar 0 < A < 2 remains unde ned, and must be estimated. Ve (= 2d dete) is the volume (area) of the d-dimensional element, and the form for Fij was determined using computational data from the 2-D and 3-D linear Peclet problem solutions [17]. Sub-Grid eMbedding solution process : The steps forming the SGM element GWSh solution process include: 1. Use the monotonicity constraint (34)-(35) to estimate rj = f (uj ; hj ;Re). Z NS gT @ fNS gT d , where g (c) = fGgT fN g. 2. Form [D]e = e g2(c) @ f@x 2 2 @xj j 3. Rank reduce [D]e to [DS ]e via static condensation. 4. Form the weak statement WSh (17)-(21) for the discretized domain h Se e . 5. Solve (19). 6. Return to step 1 and repeat process until solution is converged.

RESULTS AND DISCUSSION Computational results verifying theory and summarizing performance for select Navier-Stokes problem statements are presented. The problem domains lie on > < 1:75 ? 0:75 cos[2(x ? 0:5)] ; 0:0  x  0:5 A(x) = > (36) > : 1:25 ? 0:25 cos[2(x ? 0:5)] ; 0:5  x  1:0 The non-dimensional initial condition for state variable q is the sonic throat, shock-free isentropic solution for (36). The reference state is in ow, hence the boundary conditions are (0; t) = 1:0 = in , E (0; t) = 1:0 = Ein, and p(1; t) = pout = 0:909. The corresponding inlet Mach number is Main = 0:2395. The veri cation problem de nition is the steady state following an impulsive change from isentropic ow by decreasing the exit pressure pout from 0.909 to 0.864. The resultant expansion wave propagates upstream to the throat, hence triggers formation of a (normal) shock which moves downstream. The analytical solution shock is located at x = 0:65 and the shock Mach number is Mas = 1:37. First, the nozzle domain is uniformly discretized into 50 elements. The nondissipative GWS algorithm solution diverges for this problem. The TWS dissipative algorithm for ) q = 0:1 ? 0:2 converges to a steady state with shock location and sharpness, hence nonmonotone character, as a function of , Figure 3b. A close look near the shock region reveals that the TWS = 0:2 solution is substantially diused, while the = 0:1 solution is non-monotone (oscillatory). The SGM solution Mach number distribution for F = 9; FE = 10; Fm = 12:0 at Re = 106 is also plotted in Figure 3b, is monotone and follows the analytical solution more closely that either TWS solution. Figure 3c veri es the monotonicity of the SGM solution during time evolution to steady state. Finally, the SGM solution on a non-uniform mesh is monotone and accurately captures the steady state shock on two nodes at Ma = 1.369 and 0.78, Figure 3d. The analytical solution is plotted thereon for comparison, documenting that this non-uniform mesh SGM solution as essentially the Lagrange interpolant of the exact solution. The iteratively-determined SGM parameter set, is determined via (35) and the distributed rsgmq, is plotted in Figure 3e. For Re = 106 , the parameters range between 103 ? 104, and a sharp shock point operation is con rmed present. 13

Shear-Driven Cavity Flow

The laminar ow of an incompressible uid in a square cavity with top lid translating at constant velocity in its own plane is a standard benchmark problem. Despite the boundary condition singularities at the two corners, on moderately high values of the Reynolds number Re, published comparative results of accepted accuracy are available, c.f., [27]-[30], on moderate to very dense meshes. For example, [30] used a 51  51 uniform mesh for a Least Squares Finite Element Method (LSFEM) solution, while [27] employed up to a 257  257 uniform mesh via a multigrid approach for 100  Re  10; 000. For Re  100, all results agree well indicating that coarser grids usually employed by practitioners are adequate. As Re increases, however, coarse mesh inadequacy becomes fully apparent. This is particularly evident in the solutions reported in [28]. Nevertheless, the fourth-order accurate spline method [29] solution remains satisfactory computed on a 17  17 mesh at Re = 1000, but "......... the corresponding computer time becomes large." The driven cavity problem, via the vorticity-streamfunction form (5a)-(5b), was studied using GWSh and SGMh algorithms for 100  Re  3200 on coarse (17  17) to moderately dense (65  65) meshes, [18]. The solution comparisons are documented herein for Re = 3200 computed on nonuniform meshings. Figure 4 compares two GWSh and SGMh vorticity solution distributions in perspective with corresponding 17  17 meshes overlaid. The GWSh vorticity solution, Figure 4a, is totally polluted by a mesh-scale dispersive error oscillation near the moving lid. The companion SGMh vorticity solution, obtained with optimal nodally distributed parameter rsgmj  1 on the identical non-uniform 17  17 mesh, Figure 4b, veri es attainment of a monotone solution exhibiting very sharp corner (singularity) extrema. This solution resolution is comparable to that obtained via the GWSh algorithm on a double-density (33  33) highly non-uniform mesh, Figure 4c. While neither were optimized, the Newton algorithm computation time for the GWSh solution was nearly six times that required for the SGMh solution, Figure 4b and the memory requirement was four times larger. The accuracy comparisons to benchmark data for the 33  33 GWSh solution is essentially identical to the 17  17 SGMh solution summarized in Figure 5. The normal arrowheads on each secondary vortex region graph locate the attachment points documented in [27, Figure 3] on a 129  129 uniform mesh; the non-uniform 17  17 SGM solution agreement is excellent. The SGM solution u- and v-velocity pro les through the geometric center, Figure 5, also agree within  1% 14

with those generated by the 57 times denser mesh solution of [27, Figure 2a, 2b] as documented by the circles in Figure 5. Pressure prediction is a post-processing operation with a vorticity-streamfunction algorithm solution [20, 24] via solving a Poisson equation with source (the non-linear product of velocity derivatives). Pressure distributions computed from the 172 and 332 GWSh solutions, and the 172 SGMh solution, are compared in Figure 6. The SGMh solution clearly exhibits the \best" local extrema prediction. However, an adverse eect of the coarse centroidal region mesh is also apparent, compare Figure 6b-c. The non-linearly computed nodal distributions of the SGMh theory parameter sets RSGMx and RSGMy are graphed in perspective in Figure 7a-b. To assist in visualizing these data, the local u; v velocity scales are employed as the elevation (z) reference. These distributions clearly indicate that the highest level RSGMx (RSGMy ) is computed where the scalar magnitude of u (v) is the highest, and these data range (1,9) and (1,3) for this non-uniform meshing.

Close-coupled step wall difuser

This 2-D benchmark geometry has been widely studied, speci cally for the wide, close-coupled geometry of [31] reporting laser-Doppler anemometer measurements of velocity distribution and separation region attachment intercepts. These data, for laminar, transitional and turbulent ow measurements in air for 70 < Re < 8000, con rm that the ow downstream of the step is essentially two-dimensional only for Re < 400, and becomes fully turbulent and returns to essential twodimensionality for Re > 6500. Available fully 3-D computational data for this geometry [24] predict that the three-dimensionality for 400  Re  800 is mainly con ned to the lateral sidewall regions, while the symmetry center plane ow eld exhibits an essential two-dimensionality (as the normal velocity component vanishes). In both two and three dimensions, dispersion error control plays a critical role for maintaining CFD solution process stability for Re > 400. Prior to these data, [32] conjectured that the \abrupt change" in the ow structure from two to three dimensional ow was due to a TaylorGortler vortex instability that caused spanwise-periodic counter-rotating vortex pairs aligned with the duct axis. The fully 3-D CFD solutions for 100  Re  800, [24], refute this conjecture, i.e., the ow eld transition to 3-D proceeds smoothly for Re > 400. In either 2-D or 3-D, as the Reynolds number 15

increases, the solution process becomes very sensitive to mesh resolution, hence stability, even using a Re continuation procedure, i.e., using the lower Re solution as the initial condition. The resultant instability in the GWSh algorithm ow eld prediction in 3-D on an inadequate mesh was observed as periodic generation/annihilation of the secondary vortex structure (x4 to x5) along the top wall, [24]. Therefore, a critical validation assessment in two dimensions is to predict the laminar ow solution at Re=800, which according to experiment and the 3-D computational solution, exhibits a steady-state. There has existed some controversy regarding the CFD existence of a two-dimensional steady-state; recently, a quadratic basis GWSh algorithm on a dense mesh, [34], con rmed that the steady solution was stable. The GWSh , TWSh and SGMh algorithms were implemented for the continuity constraint algorithm, (6)-(7b), for two dimensional isothermal ow. The boundary conditions are fully-developed (parabolic) laminar inlet U pro le with the V inlet velocity zero and no-slip on top and bottom walls. At out ow, the Dirichlet boundary conditions for pressure and  in (6)-(7b) were xed at zero and the velocity boundary condition was homogeneous Neumann. Figure 8 summarizes GWSh algorithm computed velocity vector distributions for 100  Re  800, as obtained on a nominal density 7  18+48  35 non-uniform mesh. The arrows perependicular to duct walls indicate the primary and secondary recirculation reattachment coordinates reported in [31]. The solution for the primary reattachment coordinate for 100  Re  400, Figure 8a-c, agrees essentially exactly with the experimental data, and the steady solution predicts the insipient secondary region at Re = 400. However, agreement with data degrades sharply with increasing Reynolds number for Re > 400, as the intrinsic GWSh algorithm dispersive error mechanism yields an unsteady secondary separation bubble which never achieves a steady-state, Figure 8d-e. Figure 9 summarizes data for a range of unsuccessful CFD attempts to obtain a two-dimensional steady-state and accurate solution for 100  Re  800. The application of SGMh in promoting monotonicity, hence controlling the GWSh dispersion error mechanism, constitutes the critical Re = 800 validation. The velocity vector elds computed using the GWSh , dissipative TWSh, and a statically-condensed (but not SGMh ) quadratic basis GWSh algorithm (labelled p- WSh), and the SGMh algorithm are compared in Figure 10a-e. The arrows labelled x1=S, x4=S, and x5=S indicate the measured primary and secondary reattachment coordinates [31]. The GWSh and p-WSh algorithms did not produce steady-state solutions on this 16

mesh (these data are \snapshots" in the unsteady solution). The TWSh steady solution comparisons con rm that inaccuarcy in predicting the shallow upper-surface secondary recirculation bubble leads to errors of the order 20% in primary recirculation reattachment location. Conversely, the SGMh algorithm solution primary reattachment coordinate (x1/S) error is about 2%, as the shallow secondary recirculation region is contained in span and with extent within 95% agreement with data, although the x4 intercept is displaced 0.15 x1/S units downstream of the data. The SGMh data for primary recirculation attachment for 400  Re  800 are included in Figure 9, and clearly con rm superior accuracy for the range of solution data plotted. The monotonicity issue distinctions are graphically enforced using perspective plotting of speed isosurfaces. The comparison U (scalar) perspective plots for the GWSh , TWSh and p-WSh algorithms at Re = 800, Figure 11a-d, clearly show the 2
x error mode as \color diamonds" in the blue (-0.15 - 0.016), yellow (1.0 - 1.2) and lemon yellow (0.84 - 1.0) ranges. The linear basis GWSh solution is clearly the most dispersive, while the two TWSh solutions show selective improvement of this error mode. The larger = 0:20 TWSh solution, Figure 11c, does dissipate essentially all local mesh scale error. The non-arti cially dissipative quadratic basis p-WSh solution, Figure 11d, shows an increasing region of yellow level, while evidencing modest non-monotonicity in the orange-red diamonds. In clear distinction, the SGMh algorithm solution, Figure 11e, is genuinely monotone showing no dispersion error (color diamonds). As a consequence, the ranges of red (1.2 - 1.3) and yellow (0.84 - 1.0) fully extend into the solution domain, yielding the indicated agreement with the primary recirculation data. The companion presentation for V solutions, with mesh overlays for emphasis, is presented in Figure 12. Figure 13 summarizes the SGM non-linearly determined algorithm parameter sets rsgmx and rsgmy , computed during the iterative solution process according to (35), as a function of nodal velocity components, and the local directional mesh measures, hx and hy and Re. The near-step nite element mesh and SGMh U and V speed solutions are also shown in perspective. The SGMh theory parameter extrema are logically correlated with velocity and mesh coarseness, Figure 13d-e, and the rsgmj levels for U are three orders larger than for V. The data of Figure 13d could logically lead to a solution adaptive meshing strategy, to restrict the local variation in rsgmj , hence reduce the SGMh intrinsic dissipative levels. 17

CONCLUSIONS The sub-grid embedded (SGM) Lagrange nite element basis construction has been implemented, veri ed and benchmarked for semi-discrete approximate constructions of Galerkin weak statements for steady Navier-Stokes applications. Element-level static condensation of the SGM Lagrange basis S  2 diusion matrix facilitates embedding of arbitrary degree polynomials, guaranteeing the minimal storage requirement associated with a k = 1 Lagrange basis algorithm. The presented results con rm the theory for both compressible and incompressible NavierStokes applications. In comparison to other theories for generating higher-order accurate and/or monotone solutions, the SGM element advantages include guaranteed (non-linear) monotone solution, excellent conditioning of the minimum-band system matrix, and improved stability via retained diagonal dominance. Further, the SGM methodology permits retention of lexicographic ordering for any embedding degree, hence exhibits the eciency of strictly linear basis (or centered FD) algorithms. The SGM nite element Galerkin weak statement algorithm thus exhibits the potential for fundamental impact on CFD methodology, via its intrinsic non-linearity and guarantee of minimum computer memory and cpu requirements for high accuracy monotone solutions on relatively coarse meshes on