A Polynomial Adaptive LCP Scheme for Viscous Compressible Flows

Transcription

1 A Polynomial Adaptive LCP Sceme for Viscous Compressible Flows J.S. Cagnone, B.C. Vermeire, and S.. Nadaraja Department of Mecanical Engineering, McGill University, Montreal, Canada, H3A 2S6 ABSTRACT Tis paper presents a polynomial-adaptive liftingcollocation-penalty (LCP) formulation for te compressible Navier-Stokes equations. Te problem of non-conforming polynomial approximations is dealt wit an interface element approac. Empasis is put on te treatment of diffusive fluxes, and p-adaptive viscous compressible flow simulations are performed. 1 INTRODUCTION In recent years, a number of very efficient elementbased ig-order unstructured scemes ave been proposed. Tese formulations, collectively referred to as nodal discontinuous metods, strive to improve over te more establised discontinous Galerkin (DG) finite-element formulation. Ways to acieve iger efficiency include employing nodal instead of modal bases, casting te sceme in differential form, and favoring collocated solution/flux point placements. Te lifting collocation penalty (LCP) of Wang et al. [9, 6] combines tese attributes in a single unifying formulation, free of reconstruction step, integral evaluation and mass-matrix inversion. It is applicable to unstructured triangular meses, wit arbitrary-ig order of accuracy. However, tese advantages also come at te expense of flexibility in te application of formulation. Namely, a one-to-one connectivity of te edge flux points is mandatory to adequately evaluate te inter-element flux excange. A direct implication of tis requirement is tat using independent polynomial degree in eac element is no longer straigtforward. In reference [5], we explored te use of specially designed interface elements to tackle spatially-varying polynomial approximations. Tese interface elements ave a different number of nodes on eac face, and are employed to recover conforming flux interpolation at interfaces were te polynomial degrees differ. In tis paper, te applicability of te interface element approac to diffusion problems is studied. Section 2 covers te formulation of te numerical sceme, and section 3 presents results for te Navier-Stokes equations. 2 NUMERICAL FORMULATION 2.1 Inviscid discretization We consider Ω, an approximation to te pysical domain Ω, subdivided by a triangulation T = {} consisting of sape-regular elements. Using standard finite-element notation [2], we introduce te broken polynomial spaces V and Σ associated wit te triangulation T, and denote by P() and Σ() teir restriction to element. In te following, we denote ( ) and ( ) te traces taken from te interior and exterior of element wit boundary. To discuss te inviscid discretization, we start by considering te Euler equations in conservation form u(x, t) t F(u) = 0 in Ω, (1) were u are te state variables and F = (f(u),g(u)) is te inviscid flux vector. We seek an approximate solution to (1) consisting of piecewise polynomial functions by invoking te variational formulation: find u V suc tat for all T v u t dx v ˆF ds v F(u ) dx = 0 v P(), (2) were n denotes te outward normal unit vector to. Here, ˆF denotes a consistent and conservative numer-

2 ical flux approximating F(u) n on. In our implementation, te Roe flux [8] is retained for tat purpose. Integrating by parts equation (2) leads to te strong variational statement v u t dx v F(u ) dx v ( ˆF F(u ) n) ds = 0. (3) In tis alternative formulation, te differential operator is sifted from te test function to te fluxes. Moreover, te boundary integrand now consists of te difference between te desired Riemann flux and te internal normal flux. In te LCP sceme, te influence of tis normal flux difference is extended from te boundary to te volume of eac element by considering a correction field δ (x) V defined by te following lifting operator vδ dx = v ( ˆF F(u ) n) ds v P(). () Substituting () into (3) allows to factor and eliminate te test fuction v, giving te differential formulation u t F(u ) δ = 0. (5) In te practical implementation of te sceme of order p 1, te unknowns are stored at N p = (p 2)(p 1)/2 solution points. Te numerical fluxes are ten calculated at N fp = 3(p1) flux points, collocated wit te edge solution points. Te interpolation witin te element is performed using N p twodimensional Lagrange basis functions denoted ϕ i (x). Likewise, te numerical fluxes are interpolated along all tree edges using N fp one-dimensional basis functions denoted φ k (s). Using tis nodal format, te semidiscrete sceme can be expressed from te vectors of nodal values as d dt u D x f D y g L[ ˆF F(u ) n] = 0, (6) were D x and D y are N p N p differentiation matrices and L = M 1 N is te N p N fp algebraic lift operator. Here, M and N denote te mass and edge-mass matricies given by M i j = ϕ i (x)ϕ j (x) dx, N ik = ϕ i (x)φ k (s) ds. 2.2 Viscous discretization edge In order to develop a discretization suitable for diffusion problems, we introduce an auxiliary gradient variable q = u and cast te Navier-Stokes equations as a first-order system q u = 0, Euler F v (u,q ) = 0. (7a) (7b) Applying te ideas presented in te previous section, we consider (7a) in strong variational form: find u V and q Σ suc tat for all T, τ Σ() and v P() τ q dx τ u dx (8a) τ (û u )n ds = 0 τ Σ(), Euler v F v (u,q ) dx v ( ( ˆF v F v u,q ) ) (8b) n ds = 0 v P(), were û and ˆF v are numerical fluxes approximating u and F v n on. Te next step consists of defining operators to lift te jump in fluxes at te boundary. We introduce te necessary correction fields R Σ and δ v V τ R dx = τ (û u )n ds, (9a) vδ v dx = v ( ( ˆF v F v u,q ) ) n ds. (9b) Te LCP discretization of te gradient and viscous term is tus q = u R, Euler F v (u, u R ) δ v = 0. (10a) (10b) Since te correction fields R and δ v are completely analogous to δ, te algebraic operators used to lift te inviscid fluxes can be utilized in (10) as well. Te formulation is complete once te numerical fluxes û and ˆF v are specified in terms of u and q. Here, te second metod of Bassi and Rebay (BR2) is employed [3, ]. To make te sceme compact, tey suggest splitting te gradient correction field into edge-wise contributions R = r, were τ r dx = τ (û u )nds τ Σ(). (11) edge Te BR2 numerical fluxes are ten given by û = 1 ( 2 u u ), ˆF v = 1 ( ( 2 Fv u, u ) ( r Fv u, u )) r n. Te key aspect of te BR2 formulation is tat ˆF v is evaluated from partially corrected edge gradients. Using te fully corrected gradients q and q is referred to as te BR1 metod, wic leads to sub-optimal convergence rates and an extended stencil connecting nonneigbor elements.

3 2.3 Non-conforming polynomial approximations Te LCP formulation presented in te preceding section is only directly applicable to conforming polynomial approximations were te degree of eac element is te same trougout te domain. In order to andle non-conforming polynomial approximations, an interface element approac is used. Tese specially designed elements ave a variable number of nodes per face, and are inserted at junctions between zones were te interpolation degree differs. Teir construction guarantees C 0 -continuity of te fluxes along eac edge, tus ensuring local conservation. Tis approac also retains te efficient collocated nodal formulation of te original sceme, and is free of additional interpolation or projection operations. Construction of tese interface elements is possible since te LCP formulation is not strictly dependent on te element type, or coice of nodal support. Consequently, we are free to consider elements wit supplementary edge nodes, as long as a proper interpolation basis can be defined. To transition from degree p to p 1, we consider te nodal set wit N p points, and add a new node along one, or possibly two faces of te element. Tis leads to six possible configurations, as illustrated by Fig. 1 sowing te linear-to-quadratic interface elements. Next, te interpolation polynomial space is defined in terms of monomial expansion in (ξ, η), te usual local coordinates of te canonical reference element. Starting from te order-complete basis ψ(ξ,η) = {1,ξ,η,ξ 2,ξη,η 2,...,η p } T, additional components needed by te supplementary face-nodes are included. Tese components, cosen to avoid a singular Vandermonde matrix, are determined as follows ξ p1 for face 1 ξ n η m, n m = p 1, (n,m) > 0 for face 2 η p1 for face 3 Te Lagrange interpolation functions are ten obtained by te relation provided by te Vandermonde matrix V V T ϕ = ψ, V i j = ψ j (ξ i,η i ). A Gram-Scmidt ortonormalization of te monomial expansion can be employed to improve te conditioning of te Vandermonde matrix. Finally, wit te nodes and polynomial basis properly defined, te differentiation and lift matrices are computed, leading to a seamless integration of te interface element approac in te original framework (a) < 1 ξ η ξ 2 > (c) < 1 ξ η η 2 > 2 3 (e) < 1 ξ η ξ 2 η 2 > (b) < 1 ξ η ξη > 2 3 (d) < 1 ξ η ξ 2 ξη > (f) < 1 ξ η ξη η 2 > Figure 1: Interface elements for te P1-P2 interface 3 NUMERICAL RESULTS 3.1 Laminar NACA0012 airfoil We evaluate our p-adaptive sceme on by considering laminar flow around a NACA0012 airfoil. Te free stream conditions cosen are M = 0.5, α = 0 o and Re = An interesting aspect of tis flow is te presence of a small recirculation bubble aft of te airfoil caused by laminar separation. We begin by studying te impact of te discretization s order on te flow solution. Figure 2 presents te computed Mac contours using te P1, P and p-adaptive scemes. It is seen tat te P1 solution in Fig. 2a is unsatisfactory. Specifically, te boundary layer and wake ticknesses are severely under-predicted, and te solution displays a general lack of smootness. Tis situation is corrected wen employing te fift-order sceme (P) wic recovers smoot Mac contours and sows an increased resolution of te viscous effects. It is also verified tat te solution obtained wit

4 (a) (b) (c) Figure 2: Laminar NACA0012: Mac number contours. (a) P1 solution. (b) P solution. (c) Adapted solution using levels of refinement (P1-P). P1 P2 P P P3 P3 (a) (b) Figure 3: Laminar NACA0012: adapted polynomial pattern. (a) General view. (b) oom around te airfoil. Te coloring sceme corresponds to P1 (blue) troug P (red), wit te interface elements sown in intermediate sading. te four level p-adaptive solution (P1-P) is essentially identical to te uniformly refined one. Tis agreement confirms te effectiveness of te adaptation procedure and te appropriate treatment of te interfaces were te order of te discretization varies. Indeed, Fig. 3 sows tat polynomial refinement occurs mainly around te stagnation point, boundary layer and wake regions, wic are te crucial features of tis particular flow. A quantitative comparison of te errors in te pressure (c dp ) and viscous drag coefficients (c dv ) is sown in Fig.. A fift-order fine grid solution is employed as reference to estimate te error level of eac computation. We notice tat te uniformly and adaptively refined solutions ave te same error levels, but te adaptation procedure offers a two-fold reduction in total number of degrees of freedom. Turning to table 1, it is seen tat computing te full P solution necessitates 76 seconds, against 176 for te adapted solution, corresponding to a speedup factor of about 2.7. Table 1 also details te location of te laminar separation point for eac calculation. Due to te relatively coarse grid employed, te P1 sceme fails to capture te recirculation region completely. Increasing te sceme s order gives progressively better results, wit te P computation in close agreement wit te reference solution. Once more, it is noticed tat te full-refinement and p- adaptive strategies give essentially identical results at eac refinement level. Finally, te convergence of te Newton-rylov solver is examined in Fig. 5. Macinezero convergence is acieved in four to six iterations for bot uniformly and adaptively refined computations. Te fact tat bot strategies display similar con-

5 Discretization DOF c dp c dv Separation point Wall clock time (sec.) P none 13 P % 6 P % 222 P % 76 P1-P % 0 P1-P % 79 P1-P % 176 Fine grid % Table 1: Summary of te laminar NACA0012 simulations. vergence rates confirms te stability of te interface element approac Uniform p refinement Adaptive p refinement Uniform p refinement Adaptive p refinement Density residual c dp error Newton iterations Degrees of freedom x 10 Figure 5: Laminar NACA0012: convergence of te Newton-rylov solver for four levels of uniform and adaptive p-refinement Uniform p refinement Adaptive p refinement 3.2 Unsteady vortex sedding c dv error Degrees of freedom x 10 Figure : Laminar NACA0012: Pressure (c d p ) and viscous (c dv ) drag error as a function of number of degrees of freedom. Tis second example demonstrates te application of our p-adaptive strategy to unsteady viscous flows. Te problem considered is unsteady vortex sedding occurring beind a cylindrical body of unit diameter. Te flow conditions are given as M = 0.1 and Re = 150. Te simulations are performed on a computational mes consisting of 2810 elements, wit stretced triangles in te boundary layer region and moderately refined wake region beind te cylinder. Time integration is carried-out wit te second-order BDF2 sceme. For all simulations, te time increment is fixed to t = 0.5, corresponding to a Courant- Friedrics-Lewy (CFL) number of about 100. Tis time step was verified to be sufficiently small to let te spatial error dominate For p-adaptive calculations, te polynomial distribution at a new timestep is iteratively determined in a similar fasion as Alauzet et

6 P1 P2 P3 P Figure 6: Unsteady vortex sedding: Mac contours and polynomial adaptation pattern for te four-level adaptive computation (P1-P). Te snapsots sown correspond instants of maximal (top) and minimal (bottom) lift value. al. [1]. A new refinement pattern is recomputed eac five time steps. Te simulation results are compared in terms of lift and total drag coefficients, as well as non-dimensional sedding frequency expressed by te Stroual number St = L/(TU ), were L is te caracteristic lengt, U te free steam velocity, and T te sedding period. Figure 6 sows te Mac contours and polynomial pattern for te four-level adaptive simulation (P1-P). Te two snapsots presented correspond to instants of maximal and minimal lift value. Most of te P elements are employed in te boundary layer and in te unsteady wake immediately beind te cylinder. Refinement employing P2 and P3 elements also occurs furter downstream, leaving only te farfield region discretized wit P1 elements. Comparing te two time instants also provides visual evidence of te dynamic adaptation occurring conjointly wit te flow evolution. It is verified tat te algoritm successfully follows te location of te primary vortex being sed alternatively above and below te cylinder. To evaluate te impact of te number of adaptation levels, we turn to Fig. 7 sowing te evolution of te lift and total drag coefficients. Te integrated forces are plotted against time units (t t 0 ), were t 0 is a time of a peak value. It is observed tat te resolution of te P1 discretization is insufficient to accurately capture te viscous effects, tus under-predicting te forces exerted on te cylinder. Increasing te number of adaptation levels gradually improves te predictions, eventually leading to close agreement between te tree- and four-levels simulations. Moreover, it is also noticed tat te additional adaptation levels reduce te errors caused by te non-consistent interpolation of te solution at eac time step. Specifically, te drag istory predicted by te P1-P2 simulation presents some unpysical oscillations and oversoots wic disappear completely as te number of refinement steps is increased. To furter examine te accuracy of te different scemes, te output of eac simulation is presented in table 2, along wit te sixt-order finite-difference results of Inoue and Hatakeyama [7]. Te uniform and adaptive refinement strategies bot sow good agreement wit te data reported by tese autors. For bot ap-

7 Table 2: Summary of te unsteady vortex sedding simulations. Discretization DOF c l c d c d St Wall clock time (sec.) P P P P P1-P P1-P P1-P Reference [7] tus acieving a speedup factor of about 2.6. c l P1 P1 P2 P1 P3 P1 P Time (t t ) 0 CONCLUSION Tis paper studied te applicability of te LCP interface elements approac to p-adaptive compressible Navier-Stokes simulations. Tese interface elements ensure te required continuity of te flux interpolation, and tus local conservation. Te efficiency and robustness of te approac was demonstrated for a steady flow over te NACA0012 airfoil, and for an unsteady laminar vortex sedding problem, were a speedup of about 2.7 was reported. c d Time (t t 0 ) Figure 7: Unsteady vortex sedding: lift and drag coefficient istory for p-adaptive simulations. proaces, a converged Stroual number is obtained after te second refinement step, wereas lift and drag require tree levels to settle to teir final values. To close, we note tat te P1-P adaptive strategy predicts similar outputs as te uniformly refined solution, but offers a two-fold reduction in degrees of freedom, REFERENCES [1] F. Alauzet, P. L. George, B. Moammadi, P. Frey, and H. Boroucaki. Transient fixed point-based unstructured mes adaptation. Int. J. Numer. Met. Fluids, 3:729 75, [2] D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini. Unified analysis of discontinuous Galerkin metods for elliptic problems. SIAM J. Numer. Anal., 39(5): , [3] F. Bassi, A. Crivellini, S. Rebay, and M. Savini. Discontinuous Galerkin solution of te Reynoldsaveraged Navier-Stokes and k-ω turbulence model equations. Comput. Fluids, 3:507 50, [] F. Bassi and S. Rebay. A ig order discontinuous Galerkin metod for compressible turbulent flows. In B. Cockburn and G. arniadakis, editors, Discontinuous Galerkin metods. Teory, computation and applications., volume 11 of Lecture Notes in Computational science and engineering. Springer-Verlag, 2000.

8 [5] J. S. Cagnone and S.. Nadaraja. A stable interface element sceme for te p-adaptive lifting collocation penalty formulation. J. Comput. Pys., 231: , [6] H. Gao and. J. Wang. A ig-order lifting collocation penalty formulation for te Navier-Stokes equation on 2D mixed grids. AIAA Paper , [7] O. Inoue and N. Hatakeyama. Sound generation by a two-dimensional circular cylinder in a uniform flow. J. Fluid Mec., 71:285 31, [8] P. L. Roe. Approximate Riemann solvers, parameter vector and difference scemes. J. Comput. Pys., 3: , [9]. J. Wang and H. Gao. A unifying lifting collocation penalty formulation including te discontinuous Galerkin, spectral volume/difference metods for conservation laws on mixed grids. J. Comput. Pys., 228: , also AIAA paper