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1 Computers & Fluids 48 (2011) Contents lists available at SieneDiret Computers & Fluids journal homepage: A high-order spetral differene method for unstrutured dynami grids M.L. Yu, Z.J. Wang, H. Hu Department of Aerospae Engineering, Iowa State University, Ames, IA 50011, United States artile info abstrat Artile history: Reeived 8 May 2010 Reeived in revised form 22 February 2011 Aepted 31 Marh 2011 Available online 12 April 2011 Keywords: High-order Unstrutured dynami grids Spetral differene Navier Stokes Bio-inspired flow A high-order spetral differene (SD) method has been further extended to solve the three dimensional ompressible Navier Stokes (N S) equations on deformable dynami meshes. In the SD method, the solution is approximated with piee-wise ontinuous polynomials. The elements are oupled with ommon Riemann fluxes at element interfaes. The extension to deformable elements neessitates a time-dependent geometri transformation. The Geometri Conservation Law (GCL), whih is introdued in the time-dependent transformation from the physial domain to the omputational domain, has been disussed and implemented for both expliit and impliit time marhing methods. Auray studies are performed with a vortex propagation problem, demonstrating that the spetral differene method an preserve high-order auray on deformable meshes. Further appliations of the method to several moving boundary problems inluding bio-inspired flow problems are shown in the paper to demonstrate the apability of the developed method. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introdution Computational fluid dynamis (CFD) has attrated a surge of researh ativities during the last three deades, and it has beome a routine tool in the aerodynami design of airraft, wind turbines, entrifugal pumps, et. For general engineering appliations, nearly all prodution flow solvers are based on at most seond-order numerial methods. Although they proved very useful, the seond-order methods may not be aurate enough for problems requiring high auray, suh as vortex-dominated flows, and aousti noise preditions. Therefore, there has been a growing interest in the development of high-order methods for unstrutured grids in reent years. The reasons for this are obvious. High-order methods enjoy remarkably high auray with low numerial dissipations, and unstrutured grids an provide flexibility in handling omplex geometries. A review of the high-order methods for the Euler and Navier Stokes equations an be found in [31]. The spetral differene (SD) method [12] is a reently developed high-order method to solve ompressible flow problems on simplex meshes. Its preursor is the onservative staggered-grid Chebyshev multi-domain method [11]. The general formulation of the SD method was first desribed in [12] and applied for omputational eletromagneti problems. It is then extended to 2D Euler [33] and Navier Stokes equations [14,34]. After that, Sun et al. [22,23] implemented the SD method for 3D N S Corresponding author. addresses: mlyu@iastate.edu (M.L. Yu), zjw@iastate.edu (Z.J. Wang), huhui@iastate.edu (H. Hu). equations on unstrutured hexahedral meshes. Later, a weak instability in the original SD method was found independently by Vanden Adeele et al. [26] and Huynh [8]. Huynh [8] further found that the use of Legendre Gauss quadrature points as flux points results in a stable SD method. This was later proved by Jameson [9] for the one dimensional linear advetion equation. The present study is based on Sun et al. [22,23] and further extends the method to 3D deformable meshes. The basi idea to ahieve high-order auray in the SD method is to use a high degree polynomial to approximate the exat solution in a standard element (a loal ell). However, unlike the disontinuous Galerkin (DG) [3] method and spetral volume (SV) method [32], the SD method is in the differential form, whih is effiient and simple to implement. As all the omputations are performed on the fixed standard element in the omputational domain, it is reasonable to expet that the SD method an preserve high-order features for moving boundary problems in the physial domain. Sine a time-dependent urvilinear transformation from the physial domain to the standard element is needed in the SD method, the Geometri Conservation Law (GCL), first disussed in [25], should be stritly enfored in order to eliminate the grid motion indued errors. For high-order methods, an approah to guarantee GCL for the finite differene method has been proposed in [27].Itis straightforward to extend this approah to the present SD method. In addition, a GCL ompliant high-order time integration method is developed for the impliit sheme with a similar method used in [13]. Note that there is an alternative way to deal with moving boundary problems, whih is alled the arbitrary Lagrangian Eulerian (ALE) method [4]. In that approah, a mapping from a fixed referene onfiguration to the physial domain is needed. In /$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi: /j.ompfluid

2 the mapping, a time-dependent GCL is introdued for the referene domain [17 19]. It is quite similar to the oordinate transformation approah aforementioned in the SD or the finite differene methods in [24,27]. It an be shown that the final form of the time-dependent GCL is exatly the same for both approahes. The remainder of the paper is organized as follows. In Setion 2, the SD method is briefly reviewed inluding both the spae disretization proedure and time integration approah. The GCL of the transformation from the physial domain to the omputational one is then disussed in detail. After that, the implementation of GCL into the numerial shemes is desribed for different time marhing methods. An algebrai grid deformation method together with the orresponding blending strategy is given in Setion 2 as well. Then several numerial test ases are presented in Setion 3. For a single flapping airfoil, the numerial results are obtained with both a rigid moving grid and a deformable grid. The omparisons of these results with experimental data are also presented. Moreover, some superior features of high-order methods over the lower ones are also illustrated in Setion 3. Setion 4 briefly onludes the paper. M.L. Yu et al. / Computers & Fluids 48 (2011) x n x g x f x y; z; tþ J g; f; sþ ¼ y n y g y f y s z n z g z f z s 5 : ð2:4þ For a non-singular transformation, its inverse transformation must also exist, and the transformation matrix is 2 3 n x n y n z n t J g; f; sþ y; z; tþ ¼ g x g y g z g t f x f y f z f t 5 : ð2:5þ It should be noted that all the information onerning grid veloity ~v g ¼ðx s ; y s ; z s Þ is ontained in n t, g t and f t, whih an be written as 8 >< n t ¼ ~v g rn g t ¼ ~v g rg : ð2:6þ >: f t ¼ ~v g rf 2. Numerial method 2.1. Governing equations We onsider the unsteady ompressible Navier Stokes (N S) equations in onservation form in the physial @z ¼ 0; ð2:1þ where Q is the vetor of onservative variables, and F, G, H are the total fluxes inluding both the invisid and visous flux vetors. After introduing a time-dependent oordinate transformation (Fig. 1a) from the physial domain (t,x,y,z) to the omputational domain (s,n,g,f), Eq. (2.1) an be rewritten Q G H ¼ 0; ð2:2þ where 8 eq ¼jJjQ >< ef ¼jJjðQn t þ Fn x þ Gn y þ Hn z Þ : ð2:3þ eg ¼jJjðQg t þ Fg x þ Gg y þ Hg z Þ >: eh ¼jJjðQf t þ Ff x þ Gf y þ Hf z Þ Herein, s = t, and (n,g,f) 2 [ 1,1] 3, are the loal oordinates in the omputational domain. In the transformation shown above, the Jaobian matrix J takes the following form: 2.2. Spae disretization A brief review of the SD method is given here for ompleteness. A more detailed desription of this numerial method is available in [22]. In the SD method, two sets of points are given, namely the solution and flux points, as shown in Fig. 1b. Conservative variables are defined at the solution points, and then interpolated to flux points to obtain loal fluxes. In this study the flux points are seleted to be the Legendre Gauss points plus both end points 1 and 1. The fluxes are omputed at the flux points using Lagrange interpolation polynomials. It should be pointed out that this solution polynomial is only ontinuous within a standard element, but disontinuous at the ell interfaes. Therefore, for the invisid flux, a Riemann solver is neessary to ompute a ommon flux on the interfae. For a moving boundary problem, sine the eigenvalues of the Euler equations are different from those for a fixed boundary problem by the grid veloity, the design of the Riemann solver should onsider the grid veloity. Taking the Rusanov flux [22] as an example, the reonstruted fluxes in three diretions an be written as 8 e F i ¼ 1½f F i 2 L þ f F i R ðjv n v gn jþþðq R Q L ÞjJjjrnjsignð~n rnþš >< e G i ¼ 1½f G i 2 L þ G fi R ðjv n v gn jþþðq R Q L ÞjJjjrgjsignð~n rgþš >: f H i ¼ 1½f H i 2 L þ H fi R ðjv n v gn jþþðq R Q L ÞjJjjrfjsignð~n rfþš; ð2:7þ Fig. 1. (a) Transformation from a moving physial domain to a fixed omputational domain. (b) Distribution of solution points (as denoted by irles) and flux points (as denoted by squares) in a standard quadrilateral element for a third-order SD sheme.

3 86 M.L. Yu et al. / Computers & Fluids 48 (2011) where supersript i indiates the invisid flux, subsript n indiates the normal diretion of the interfae. It should be noted that Qn t, Qg t and Qf t are inluded in the invisid fluxes. The reonstrution of the visous flux an be found in [22] Geometri Conservation Law (GCL) The GCL for the metris of the transformation from the physial domain to the omputational one an be expressed as @ ðjjjg x xþ¼0 @ ðjjjg y @ ðjjjg z ðjjjf zþ¼0 ðjjjg t tþ¼0: It is obvious that the first three formula of the GCL only depend on the auray of the spae disretization, while the last one is related to the time evolution of the moving grid. Sine the spatial metris are omputed exatly, the first three equations are automatially satisfied. If the mesh undergoes rigid-body motion without deformation, jjj is independent of time. Due to the disretization error, the time-dependent GCL may not be stritly satisfied if one does not pay attention to how the mesh veloity is omputed. However, for a dynami mesh, spurious flows an be indued if the GCL is not stritly enfored. Therefore, GCL is a ritial element for dynami meshes. In the present study, the GCL error in the numerial simulation is aneled by adding a soure term to the N S equations in the omputational domain. In [17 19], the enforement of GCL is ahieved by using the same time integration form for the Jaobian as the onservative variables. An extra equation for the Jaobian needs to be solved iteratively. However, the present approah alulates the Jaobian diretly, and then eliminates the errors generated by the disagreements between Jaobian and the orresponding grid veloity through a soure term. Herein, treatments of the GCL are introdued separately for expliit and impliit shemes due to their different harateristis Expliit sheme The semi-disrete form of the N S equation in the omputational domain e ¼ Rð e Q n G H : ð2:9þ The equation is solved with a multi-stage strong-stability-preserving (SSP) Runge Kutta sheme. The following equation is obvious by the hain Q ð2:10þ Substitute the last formula of Eqs. (2.8) into Eq. (2.10), we Q ðjjjg ðjjjf tþ ð2:11þ Thus Eq. (2.9) is hanged to the following ¼ F G H e! @n ðjjjg t ðjjjf tþ ( ¼ F G H e! ) þ where ð2:12þ soure ðjjjg ðjjjf tþ : ð2:13þ h Note e F e H ontains a term as ðjjjn tþþ ðjjjg t tþš. It is lear that GCL is satisfied stritly as this term will be aneled by the soure term when Q is a onstant (i.e. the free stream flow). The benefits of this method are that the soure term is easy to ompute and implement for the original solver for stationary grids and the alulation an be avoided, whih might generate additional errors and inrease the omputational ost Impliit sheme At eah ell, using the bakward Euler sheme for the time derivative, fq nþ1 f Q n Dt h R ðq e nþ1 Þ R ðq e i n Þ ¼ R ðq e n Þ; ð2:14þ further performing the Taylor expansion and keeping the first-order term, we obtain R ðq e nþ1 Þ R ðq e Q f DQ f þ Q g DQ g nb ; ð2:15þ nb where DQ f ¼ Q f nþ1 Q f n, nb indiates all the neighboring ells ontributing to the residual of ell. Combining (2.14) and (2.15), we obtain f Q! D f Q g Q nb D g Q nb ¼ R ð e Q n Þ: ð2:16þ However, it is expensive in memory to store the full LHS impliit Jaobian matries. Therefore, a preonditioned LU-SGS sheme is adopted in the development of the impliit sheme. Herein, we just introdue a preonditioning matrix as D ¼ f Q! ; ð2:17þ and the iterative sheme beomes! DDQ f ðkþ1þ ¼ Q f D f ðkþ1þ Q ¼ R ðq e n Þþ Q g DQ g nb ; nb ð2:18þ where supersript (k + 1) is an iterative index, and indiates the most reently updated solutions. It should be noted that D f Q ðkþ1þ an be written as DQ f ðkþ1þ ¼ Q f ðkþ1þ Q f n ¼ Q f ðkþ1þ Q f ðkþ þ Q f ðkþ Q f n ; with Q f ðkþ ¼ Q f : ð2:19þ Sine we do not want to store the =@ Q g nb, (2.18) is further manipulated as follows: R ðq e n Þþ Q g D g Q nb ¼ R Q e n ;f Q e n nb g nb þ nb Q g D g Q nb nb R ð e Q n ;f e Q nb g nb Þ R ðq e ;f Q e nb g Q f D f Q ¼ R ðq Q f D f Q or R ðq Q f D f ðkþ Q! ð2:20þ

4 M.L. Yu et al. / Computers & Fluids 48 (2011) In (2.20), note that both approximations an be obtained using the first-order Taylor series expansion. Combining (2.18) (2.20), we obtain D Q f ðkþ1þ Q f ðkþ ¼ ¼ R! Q f fq ðkþ1þ Q f ðkþ Q e Df Q ; ð2:21þ Dt Sine matrix D merely serves as a preonditioner, the auray of the iteration will be determined by the right-hand side (RHS) of the Eq. (2.21). Note e jjjðqn t þ Fn x þ Gn y þ Hn z ¼ jjjðfn x þ Gn y þ Hn z ; ð2:22þ Fig. 2. (a) Pressure oeffiient distribution and grid deformation; (b) omparison between numerial and analytial solutions of pressure oeffiient along y =0att = 0.1. The solid line denotes the analytial result, and the dash dot line with triangles indiates the numerial result. Fig. 3. The onvergene of the vortex propagation problem using the deformable grid with and without GCL orretion, as well as for the stationary grid. Figure (a) and (b) displays results from the third-order and fourth-order SD methods respetively. In both ases, four mesh sizes are used and error representations in both 2-norm (as denoted by L 2 ) and infinity-norm (as denoted by L 1 ) are given. Fig. 4. The onvergene of the free stream preservation test using the deformable grid with and without GCL orretion. Results from the fourth-order SD method with (a) expliit SSP-RKS and (b) impliit BDF2 time integration shemes are displayed. In both ases, four time steps are used and error representations in both 2-norm (as denoted by L 2 ) and infinity-norm (as denoted by L 1 ) are given.

5 88 M.L. Yu et al. / Computers & Fluids 48 (2011) It should be noted that in the above equation the disrete form of DjJj /Dt is exatly the same as D f Q =Dt. This onsisteny an help minimize the errors indued by disretization shemes. For example, the seond-order bakward differene sheme (BDF2) for the two derivatives an be written as below, D f Q Dt ¼ 3f Q 4Q f n þ Q f n 1 ; 2Dt DjJj Dt ¼ 3jJj 4jJj n þjjj n 1 2Dt ð2:24þ 2.4. General grid deformation strategies Fig. 5. Pithing angle evolution during the hold-pith-up-hold-pith down proess. whih is ontained in RðQ e Þ. Thus, the GCL is introdued in the RHS as follows.! Q f ðq f ðkþ1þ Q f ðkþ Þ ¼ R ðq e Þ Df Q þ Q DjJj Dt ðjjj n ðjjj g ðjjj f t Þ ð2:23þ In order to solve problems with moving grids, it is neessary to design a grid moving algorithm. As the first step, the boundary motion of the physial domain is speified aording to the physial problem. Then traditionally two methods an be used to manipulate the rest of the mesh nodes. The first one is to use the algebrai proedure to smooth the whole field [5,17 19,30]. Another approah is to solve differential equations (usually ellipti, like equations of linear elastiity) with the speified boundary onditions [21,30]. For the sake of omputational effiieny, an algebrai methodology is performed in the present study, whih has been widely used by other researhers [17 19]. The first implementation of the algebrai method is to make the whole physial domain perform a rigid-body motion. Obviously, this approah annot handle relative motions among several Fig. 6. (a) Overview of the deformable grid; (b) lose-up view of the deformable grid near the moving boundary; () overview of the rigidly moving grid.

6 M.L. Yu et al. / Computers & Fluids 48 (2011) omponents. Another implementation is to use blending funtions to reonstrut the whole physial domain. In the present study, a fifth-order polynomial blending funtion proposed in [19], r 5 ðsþ ¼10s 3 15s 4 þ 6s 5 ; s 2½0; 1Š ð2:25þ is adopted. It is obvious that r 0 5 ð0þ ¼0; r0 5ð1Þ ¼0, whih an generate a smooth variation at both end points during the mesh reonstrution. Herein, s is a normalized ar length, whih reflets the distane between the present node and the moving boundaries. Speifially, s = 0 means that the present node will move with the moving boundary, while s = 1 means that the present node will not move. Therefore, for any motion (transition, rotation), the hange of the position vetor ~ P is D ~ P present ¼ð1 r 5 ÞD ~ P rigid : ð2:26þ After these manipulations, a new set of mesh nodes an be alulated based on D ~ P. In the present study, for the deformable grid approah, in order to maintain the grid quality near the wall boundaries, rigid motions are enfored in the viinity of the wall boundaries. The outer boundaries far from the wall are speified as stationary referene. Between the rigidly displaed grid and the stationary grid, the blending funtion (2.25) is used to interpolate and smooth the grid motion. It should be mentioned that the same smoothing method an also be used in problems with two or more objets with relative motions. In the present study, a tandem airfoil problem is investigated using this approah, as will be disussed in the next setion. In that ase, the hange of the position vetor ~ P an be written as D ~ P present ¼ D ~ P rigid1 ; if s 1 ¼ 0 D ~ P present ¼ D ~ P rigid2 ; if s 2 ¼ 0 D ~ P present ¼ sn 2 ½1 r s n 5 ðs 1 ÞŠD ~ P rigid1 þ sn 1 ½1 r 1 þsn s n 5 ðs 2 ÞŠD ~ P rigid2 ; otherwise 2 1 þsn 2 ð2:27þ and it is made sure that there is no region with both s 1 = 0 and s 2 =0. Fig. 7. Comparison between numerial and experimental results for Re = 10,000, k = 0.2, a =20 when tu 1 /C = (a) Experimental results (ourtesy of OL [15]). From left to right: flow visualization with dye; u veloity ontour (PIV); vortiity ontour in the spanwise diretion (PIV). (b) Numerial results with deformable grids. Left: u veloity ontour (u/u 1 ); right: vortiity ontour in the spanwise diretion. () Numerial results with rigidly moving grids. Left: u veloity ontour; right: vortiity ontour in the spanwise diretion.

7 90 M.L. Yu et al. / Computers & Fluids 48 (2011) Fig. 8. Comparison between numerial and experimental results for Re = 10000, k = 0.2, a =40 when tu 1 /C = (a) Experimental results (ourtesy of OL [15]). From left to right: flow visualization with dye; u veloity ontour (PIV); vortiity ontour in the spanwise diretion (PIV). (b) Numerial results with deformable grids. Left: u veloity ontour (u/u 1 ); right: vortiity ontour in the spanwise diretion. () Numerial results with rigidly moving grids. Left: u veloity ontour; right: vortiity ontour in the spanwise diretion. Fig. 9. (a) Drag oeffiient history and (b) lift oeffiient history for Re = 10,000, k = 0.2, alulated using both the rigidly moving grid (as denoted by the solid line) and the deformable grid (as denoted by the dash dot line with triangles).

8 M.L. Yu et al. / Computers & Fluids 48 (2011) Another point is that if s in the blending funtion (2.25) is set to be 0 at any grid point, then a rigidly moving grid approah is ahieved. In this ase, the whole domain will have the same motion. Generally speaking, a rigid grid is only suitable for simple motions of one objet. For the ase of multiple objets with relative motions, it will generate overset ells. From a numerial perspetive, the Jaobian of the transformation from the physial domain to the omputational domain will be the same all the time, and theoretially this will introdue less error when performing simulations, as Jaobian needs to be alulated only one. On the other hand, a deformable grid is desirable in more general ases. But extra efforts are needed to alulate the hanging Jaobian as the grid evolves. It is lear that for systems with omplex relative motions, the algebrai algorithm for the grid motion an be hard to design. However, for many ases this method enjoys its remarkable simpliity and effiieny. Several examples will be shown in the next setion. 3. Numerial results 3.1. Auray study using an isentropi vortex propagating problem In order to verify that the SD method an preserve its highorder auray for deformable meshes, a 2D Euler vortex propagation ase is performed in the present study. SSP third-order RungeKutta (SSP-RK3) time integration is used for this study. The definition of the isentropi vortex and its evolution proess an be desribed as [7] uðrþ ¼ U0 1 max r2 b re1 2 b 2 ; qðrþ ¼ 1 1 1=ð 1Þ 2 ð 1ÞU02 max e1 r2 b 2 ; pðrþ ¼ 1 1 =ð 1Þ 2 ð 1ÞU02 max e1 r2 b 2 ; and qðx; y; tþ 0 qðrþ uðx; y; tþ B vðx; y; tþ A ¼ U 0 B V 0 A þ uðrþ sin h B uðrþ os h A ; pðx; y; tþ 0 pðrþ where u(r), q(r), p(r) are the veloity, density and pressure distribution of the vortex respetively; U 0 and V 0 are the advetion veloities of the main stream in the x- and y -diretions; r ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx U 0 tþ 2 þðy V 0 tþ 2, is the radial distane from the vortex enter; b is a onstant. Fig. 10. Grids used for the simulations of the sinusoidally pithing airfoil. (a) Overview of the deformable grid; (b) lose-up view of the deformable grid near the moving boundary; () overview of the rigidly moving grid; (d) airfoil surfae grid for the 3D simulations.

9 92 M.L. Yu et al. / Computers & Fluids 48 (2011) Fig. 11. (a) Convergene history of the energy error for the steady solution of the flow over a stationary NACA0012 airfoil with impliit (LU-SGS) time integration; (b) pressure oeffiient ontours for the onverged steady flow. Fig. 12. Vortiity field for Re = 12600, k = 11.5, S t = (a) Phase-averaged experimental results (ourtesy of Bohl and Koohesfahani [2]). (b) Instantaneous numerial results with deformable grid. () Instantaneous numerial results with rigidly moving grid. Fig. 13. Averaged flow fields for Re = 12600, k = 11.5, S t = (a) Vortiity field, experimental results, (ourtesy of Bohl and Koohesfahani [2]); (b) vortiity field, numerial results; () u veloity field, experimental results, (ourtesy of Bohl and Koohesfahani [2]); (d) u veloity field, numerial results.

10 M.L. Yu et al. / Computers & Fluids 48 (2011) The isentropi vortex was originally entered at(0,0), with the initial ondition given by (U 0,V 0 ) = (0.5,0), U 0 max = 0.5U 0, b = 0.2. The physial domain of this problem is set to be [ 2,2] [ 2,2] with one ell in the z -diretion. The grid deformation strategy follows [13], whih analytially defines the grid motion as ~xðtþ ¼~xðtÞþd~xðtÞ with dxðtþ ¼A x L x dt=t max sinðf n tþ sinðf x xþ sinðf y yþ dyðtþ ¼A y L y dt=t max sinðf n tþ sinðf x xþ sinðf y yþ for 2D problems. Herein, A x,y is the amplitude in x and y diretions; L x,y and t max depit the referene length and time; dt is the time step, and f n ¼ n t p=t max ; f x ¼ n x p=l x ; f y ¼ n y p=l y : The motion ontrol parameters of the deformable grid are set as L x = L y =4,t max = 0.1, and n x = n y =2,n t = 1, and A x = A y = 0.2. Sine at t = 0.1 the grid has the largest deformation, the errors are analyzed at that instane. In order to ensure that the time integration errors have no effets on the auray analyses, a fixed time step is hosen as Dt = Pressure oeffiient (defined as C p ¼ ðp p 1 Þ=ð0:5qU 2 1ÞÞ distribution of the vortex is displayed in Fig. 2aatt = 0.1. From Fig. 2b, it is obvious that the analytial result agrees well with the numerial one. Results of the grid refinement study are displayed in Fig. 3, whih demonstrate the auray of the SD method for the deformable domain. The errors are measured with both L 2 and L 1 norms, and an optimal onvergene has been ahieved in all ases. It is also found that the shemes with and without GCL for the isentropi vortex propagation tests almost obtain the same error values and auray. However, for the free stream preservation test, it is obvious from Fig. 4 that for both expliit (SSP-RK3) and impliit (BDF2) shemes, if the GCL is not enfored, the error level an reahup to nine-orders larger than mahine zero. But with a GCL ompliant sheme, mahine zero an be ahieved. In this test, the fourth-order sheme is used on the grid with ells, and the errors are omputed at t = 0.1 as well Bio-inspired flow simulations Reently, there is a growing interest in the study of bio-inspired flows in the fluid dynamis ommunity. One of the major objetives is to investigate the wake strutures after flapping airfoils Fig. 14. (a) Thrust oeffiient history and (b) lift oeffiient history for Re = 12600, k = 11.5, S t = 0.19, alulated using both the rigidly moving grid (as denoted by the solid line with squares) and the deformable grid (as denoted by the dash dot line with triangles). Fig. 15. Instantaneous spanwise vortiity field for Re = 12,600, k = 11.5, S t = (a) 2D simulation with the deformable grid; (b) 2D simulation with the rigidly moving grid; () 3D simulation with the rigidly moving grid; (d) iso-surfae of Q olored by the spanwise vortiity from the 3D simulation results.

11 94 M.L. Yu et al. / Computers & Fluids 48 (2011) or wings [1,2,6,10,15,16,20,28,29,35]. The reason is that based on the evolution of these wake strutures, the thrust and lift generation mehanism in agile flight an be learly revealed. As mentioned before, suh flows are unsteady vortex-dominated flows. In order to resolve the subtle vortex strutures, a high-order method is neessary, as first- and seond-order flow solvers may dissipate the unsteady vorties quikly. Moreover, these problems all involve moving boundaries. Therefore, several numerial simulations of the flapping-related motions are arried out to examine the performane of the high-order SD method for deformable meshes. Unless otherwise noted, the default numerial sheme used in the simulations is the third-order SD sheme. For the two dimensional simulations, the impliit BDF2 time integration is used; and for the three dimensional simulations, the expliit SSPRK3 time integration is employed. For all the simulations presented in this setion, the free stream Mah number is hosen as Flat plate pith-up proess A series of anonial unsteady experimental studies on the flat plate pith-up problem was onduted in [15,16]. This problem is also studied using the high-order SD method. The aim of the study Fig. 16. (a) Thrust oeffiient onvergene history and (b) lift oeffiient onvergene history for Re = 12,600, k = 11.5, St = 0.33, for 2D simulations using the rigidly moving grid (as denoted by the solid line with squares) and the deformable grid (as denoted by the dash dot line with triangles) and 3D simulations using rigidly moving grid (as denoted by the dash line with diamonds). () and (d) are the orresponding lose-up views of (a) and (b). Fig. 17. (a) Overview of the deformable grid; (b) lose-up view of the deformable grid between the two moving boundaries.

12 M.L. Yu et al. / Computers & Fluids 48 (2011) is to investigate the aerodynami responses of maneuvering flights, suh as perhing. The main features of these problems an be generalized as high-frequeny and high-amplitude pithing proesses, whih an be used to verify the effiieny of the SD method for deformable meshes. In order to ompare the numerial results with the experimental ones, the funtions and parameters used in the present study are defined to be onsistent with the experiment. The maximum pithing angle a m is set to be 40, and a is omputed aording to GðTÞ aðtþ ¼a m MaxðGðTÞÞ ; with a smoothing funtion defined in [6] as GðTÞ ¼ln oshðaðt T 1ÞÞ oshðaðt T 4 ÞÞ ; oshðaðt T 2 ÞÞ oshðaðt T 3 ÞÞ where a is a funtion shape parameter, whih is set to be 11.0, T 1 = DT s, T 2 = T 1 + DT pu, T 3 = T 2 + DT h and T 4 = T 3 + DT pd as shown in Fig. 5. Herein, T is a non-dimensional time with respet toc/u 1, where C stands for the hord length. The start-up interval DT s is set to be 1.0, the redued pith rate K =(Ca m /DT pu,d )/2U 1 is speified as 0.2, and the hold interval DT h is set to be The Reynolds number based on the plate hord length is 10,000. The non-dimensional time step used for the simulations is DtU 1 /C = Fig. 6 shows the details of the deformable grid and the rigidly displaed grid. The grid has ells, and the minimum ell size normalized by the plate hord length in the transverse diretion is The numerial results for two instanes during the pith-up proess, namely tu 1 / C = (orresponding pith angle 20 ) and tu 1 /C = (orresponding pith angle 40 ), are ompared with the experimental results. From Figs. 7 and 8, it is obvious that the omputed instantaneous vortiity and veloity fields agree well with the experimental data. The orresponding fore histories for both deformable and rigidly moving grids are displayed in Fig. 9. Note that the results with different grid deformation algorithms are nearly idential Flow over a sinusoidally pithing airfoil An experimental investigation of the flow over a NACA-0012 airfoil performing a pithing motion with small amplitude and high redued frequeny has been onduted in [2]. The aim of the study is to find the ritial point at whih the von Karman vortex street turns into a reverse von Karman street and to study the parameter dependenies of the thrust generation during the pithing motion. Following this experimental study, a numerial researh is ompleted with the same parameter setting. And some ases are verified both with rigidly moving and deformable grid strategies. In the present study, the airfoil performs a pithing motion expressed as aðtþ ¼a m þ a 0 sinðxt þ /Þ; x ¼ 2pf Fig. 18. Instantaneous vortiity fields of a tandem airfoil onfiguration. (a) and () display the vortiity fields alulated at the phase of the fore plate up and hind plate down position using the third-order and seond-order auray shemes respetively; (b) and (d) display the vortiity fields alulated at the phase of the fore plate down and hind plate up position using the third-order and seond-order auray shemes respetively.

13 96 M.L. Yu et al. / Computers & Fluids 48 (2011) where a m is the mean angle of attak, a 0 is the amplitude of the pithing angle, / is the initial phase. Also, the redued frequeny k and the Strouhal number S t are defined respetively as k ¼ xc 2U 1 ; S t ¼ fa U 1 ; where C is the hord length of the airfoil, A is the pithing amplitude. The Reynolds number based on the airfoil hord length for all the simulations in this setion is 12,600. The non-dimensional time step used for the two dimensional simulations is D tu 1 / C =110 4 ; while that for the three dimensional simulations is DtU 1 /C = For the rigidly moving grid approah, the omputational grid moves with the body and is updated using ( x present x ¼ðx former x Þ osðdaþ ðy former y Þ sinðdaþ ; y present y ¼ðx former x Þ sinðdaþþðy former y Þ osðdaþ where (x,y ) is the pithing enter, and Da = a 0 (sin(x(t + dt)+ / 0 ) sin(xt + / 0 )). The deformable grid and the rigidly moving grid at maximum displaements for the St = 0.33 ase are displayed in Fig. 10. There the grid with ells for the two dimensional simulations and that with ells for the three dimensional simulations are shown. The minimum ell size normalized by the airfoil hord length in the transverse diretion is and that in the spanwise diretion for the three dimensional simulations is A grid refinement study has been performed in [35] to determine this grid setup. The initial onditions for all simulations on the dynami grids in the present setion are set as the steady solutions of the flow fields under the same Reynolds number (Re = 12600) and inlet Mah number (Ma = 0.1). The effets of initial onditions on the bio-inspired flow simulations are disussed in [35], and it is found that the present initial onditions an best imitate the general experimental setups. The onvergene history of the steady flow over the stationary NACA 0012 airfoil and the pressure oeffiient (defined as C p ¼ðp p 1 Þ=ð0:5qU 2 1ÞÞ ontour are shown in Fig. 11. The phase-averaged vortiity field from the experiment [2] and the orresponding instantaneous vortiity fields from the numerial simulations with different grid deformation algorithms are displayed in Fig. 12. In addition, the experimental and numerial results for the time-averaged vortiity and veloity fields are shown in Fig. 13. The numerial results are found to agree well with the experimental results. Thrust and lift oeffiient histories for both deformable and rigidly moving grids are plotted in Fig. 14. Aording to [2], the mean thrust oeffiient for the ase Re = 12600, k = 11.5, S t = 0.19 is around In the present study, the mean thrust oeffiient is alulated to be 0.031, and it is obtained by averaging the data in the ontinuous four yles after twenty-four yles. In addition, an interesting phenomenon disovered in the numerial simulation is that if the pithing amplitude is further inreased, whih means that the Strouhal number is inreased, an asymmetri wake struture appears during the pithing motion. This was first reported in [10] for the plunging motion and has been experimentally studied in [29]. The vortiity fields with both deformable and rigidly moving grid are desribed in Fig. 15. The initial phase / is set to be 180. A three dimensional simulation is then onduted using the same parameters as that of the two dimensional simulations, exept that in the spanwise diretion, periodi boundary onditions are speified. From Fig. 15 and d and Fig. 16, it an be found that results from the 3D simulation are almost the same as those from the 2D simulations. This demonstrates that under the flow onditions speified in the present study, the flow is laminar and 2D simulations an predit the flow features well. The vortex strutures in Fig. 15d are indiated by Q-riterion, whih is desribed by Q ¼ 1 2 ðr ijr ij S ij S ij Þ¼ 1 2 where R ij ¼ 1 2 S ij ¼ 1 j i ; is the angular rotation tensor, and is the rate-of-strain tensor. It also an be disovered from Fig. 16a that the thrust generation proess appears ertain unsteady features aompanying with the asymmetri wake strutures. Again, it an be found from Fig. 16 that the numerial results do not depend on the grid deformation algorithms Flow over Tandem airfoils with inverse initial plunging phases In order to enhane the thrust or lift generation and inrease the propulsive effiieny, the tandem airfoil onfiguration has been studied by some researhers [1,20]. In these problems, the two airfoils have relative motions, whih an be utilized to verify the grid deformation strategy for the SD method. Two flat plates performing plunging motions are studied here. The Reynolds number based on the plate hord length is 10,000. The motions of the two plates are speified as follows. Fore plate : Hind plate : y ¼ h sinðxt þ / 1 Þ y ¼ h sinðxt þ / 2 Þ where h/c = 0.2, the redued frequeny k = 1.5, / 1 =0, and / 2 = 180. The non-dimensional time step used for the simulations is DtU 1 /C = The deformable grid is displayed in Fig. 17. In order to ompare the performanes of high-order methods and their low-order Fig. 19. (a) Thrust oeffiient onvergene history and (b) lift oeffiient onvergene history alulated using both the third-order sheme (as denoted by the dash dot line with triangles) and the seond-order sheme (as denoted by the solid line with squares).

14 M.L. Yu et al. / Computers & Fluids 48 (2011) ounterparts, two sets of grids with almost the same degrees of freedom (DOFs) for the third- and seond-order shemes are used in the simulations. A grid with 46,270 ells (185,080 DOFs) is designed for the seond-order sheme; while another grid with 20,056 ells (180,504 DOFs) is designed for the third-order sheme. The omputed vortiity fields from both third- andseond-order auray shemes are shown in Fig. 18, and remarkable differenes of small vortex strutures near the moving wall boundaries an be observed for different auray approahes. Further, Fig. 19 displays the different aerodynami fore onvergene histories for methods of different auray. The seond-order sheme shows ertain quasi-steady features after several yles, whih is not found from the results of the third-order sheme. This an be explained as follows. Due to the relatively high numerial dissipation, the seond-order sheme an only apture the large vortex strutures as seen from Fig. 18. As a omparison, the third-order sheme an resolve fine vortex strutures near the wall boundaries with the same DOFs. These observations further demonstrate the neessity of high-order methods in vortex-dominated flows. 4. Conlusions A high-order spetral differene method has been extended to solve ompressible Navier Stokes equations on deformable meshes. Sine the present method is based on unstrutured grids, it an handle omplex geometries. Moreover, the differential form of the SD method makes the implementation straightforward even for high-order urved boundaries. Beause a time-dependent transformation from the physial domain to the omputational one has been made in the appliation of the method, the Geometri Conservation Law (GCL) has been arefully onsidered during the proess and implemented for both the expliit and impliit time integration methods. It has been demonstrated that the developed algorithm preserved the high-order auray and works effiiently for several bio-inspired flow problems. Numerial tests learly show that the high-order method with low numerial dissipation an resolve muh more elaborate vortex strutures than the loworder method, and an then help better illuminate the underlying physis of the vortex-dominated flow. Referenes [1] Akhtar I, Mittal R, Lauder GV, Druker E. Hydrodynamis of a biologially inspired tandem flapping foil onfiguration. Theor Comput Fluid Dyn 2007;21: [2] Bohl DG, Koohesfahani MM. MTV measurements of the vertial field in the wake of an airfoil osillating at high redued frequeny. J Fluid Meh 2009;620: [3] Cokburn B, Shu C-W. TVB Runge Kutta loal projetion disontinuous Galerkin finite element method for onservation laws II: general framework. Math Comput 1989;52: [4] Donea J. Arbitrary Lagrangian Eulerian finite element methods. Computational methods for transient analysis (A ). 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