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# Stability and accuracy of periodic flow solutions obtained by a POD-penalty method

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We develop a new penalty method to derive low-dimensional Galerkin models for fluid flows with time-dependent boundary conditions. We then outline a procedure based on bifurcation analysis in selecting the proper values of the penalty parameter(s) that yield asymptotically stable periodic solutions of the highest possible accuracy. We illustrate this new approach by studying flow past a circular cylinder using direct numerical simulation (DNS) data, and a wave-structure interaction problem using particle image velocimetry (PIV) data.
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Physica D 202 (2005) 218–237
Stability and accuracy of periodic ﬂow solutions obtained
by a POD-penalty method
Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA
Received 14 May 2004; accepted 11 February 2005
Communicated by I. Mezic
Abstract
We develop a new penalty method to derive low-dimensional Galerkin models for ﬂuid ﬂows with time-dependent boundary
conditions. We then outline a procedure based on bifurcation analysis in selecting the proper values of the penalty parameter(s)
that yield asymptotically stable periodic solutions of the highest possible accuracy. We illustrate this new approach by studying
ﬂow past a circular cylinder using direct numerical simulation (DNS) data, and a wave-structure interaction problem using
particle image velocimetry (PIV) data.
MSC: 37E99; 65P99
Keywords: Penalty methods; Low-dimensional; Dynamical systems; Galerkin projections
1. Introduction
Low-dimensional systems for unsteady ﬂuid ﬂows, based on the proper orthogonal decomposition (POD), have
had mixed success in predicting the correct dynamics even at exactly the same set of parameters for which the POD
modes were obtained. Speciﬁcally, an erroneous state may be obtained after long-time integration even if the correct
periodic state is set to initialize the simulation—the numerical solution eventually drifts to a new erroneous state.
This has also been observed in other systems, for example in the Kuramoto–Sivashinsky equations [1]. Empirical
ﬁxes based on artiﬁcial dissipation, e.g. [2], can only correct the dynamics in short-term integration, and more
rigorous procedures need to be followed to guarantee asymptotic stability, e.g. see [3].
Corresponding author. Tel.: +1 401 863 1217; fax: +1 401 863 3369.
0167-2789/$– see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2005.02.006 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 219 A distinction, however, should be made between autonomous low-dimensional dynamical ﬂow systems and non-autonomous ones. We have observed, for example in many POD studies with diverse ﬂow systems [4,5], that autonomous systems are more susceptible to this “drifting” while non-autonomous systems may reach asymptot- ically stable and accurate states without incorporating any special treatment. An example of a non-autonomous system is oscillatory ﬂow past a circular cylinder. We have observed through accurate numerical integration of the corresponding POD system (for millions of time steps) that asymptotic stable and accurate states can be reached, at least for an external frequency close to the natural frequency of the system, i.e. the vortex shedding frequency in this case. The implementation of complicated boundary conditions in Galerkin systems has historically been a matter of some controversy [6]; see also [7–9]. An in-depth study of boundary conditions for Galerkin POD systems was performed in [10]. Herein we introduce a penalty method, similar in spirit with the “tau” method in spec- tral methods [11], but more ﬂexible in many aspects as we will see in this study. In particular, we will study two different systems based on data obtained from direct numerical simulations (DNS) and experimental results using particle image velocimetry (PIV) [12]. A new aspect of the current work is the use of the penalty param- eter(s) as bifurcation parameter(s) in order to perform stability analysis using a standard package, e.g. AUTO [13]. Penalty methods have been used in the past successfully in implementing boundary conditions for different types of numerical discretizations. For example, ﬁnite difference schemes on complex geometries have been de- veloped in [14] using a penalty method to impose Dirichlet boundary conditions. Also, a penalty method was developed in [15] to enforce boundary conditions for shock-free compressible Navier–Stokes simulations. A similar penalty method was used in imposing boundary vorticity constraints in [16]. In general, the penalty ap- proach enforces the boundary conditions but also accounts for the governing equation at the boundary in a con- tinuous manner, thus relaxing some of the numerical stiffness associated with very steep gradients at Dirichlet boundaries. From the fundamental point of view, we pose the following question: Is there a range in the penalty parameter τfor which the periodic solutions of the ﬂow system are asymptotically stable, and are there any particular values of τfor which the solution is most accurate. For full numerical discretizations of any type, the accuracy of the solution scales as the inverse of the penalty parameter in well-resolved simulations. However, efﬁciency considerations require that a ﬁnite value for τmust be used. As τ→∞we have a strong imposition of the boundary condition while for small values of τwe have a weak imposition of the boundary conditions. The question then is how do low-dimensional discrete systems differ from their “full-blown” counterparts in this respect? More importantly, should we impose the boundary conditions for low-dimensional systems in a strong form or in a weak form. Intuitively, we expect to have stability above a threshold value in τbut that does not imply good accuracy in predicting the ﬂow dynamics. The above are some of the issues we address in the current work. The outline of the paper is as follows: in the next section, we describe the data sets based on which the POD-penalty system is derived. Then, the POD-penalty formulation is given in some detail. Subsequently, the results of the bifurcation study using the penalty parameter are presented for the two cases corresponding to DNS and experimental data. Finally, a summary and a brief discussion is included in the last section. 2. Data gathering We construct POD modes based on data obtained both from DNS as well as from experiments. We report here on two prototype cases we have studied with the penalty method. 220 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 Fig. 1. Computational domain for DNS. 2.1. Direct numerical simulation We consider ﬂow past a circular cylinder for which both two- and three-dimensional POD models have been constructedin[17]and[18],respectively.Inparticular,fortheconceptsdevelopedhere,weconsidertwo-dimensional time-dependentinﬂowpastacircular cylinder at Reynolds number Re =100and500. The inﬂow velocity is uniform but oscillatory in time, and is given by U=(1.0+Asin(ωt),0).(1) The amplitude of the forcing term A, is kept the same at A=0.1 for both Reynolds numbers. The forcing frequency is chosen so that we have a lock-in (resonant) state; this is done by choosing the frequency to be close to the Strouhal number which is 0.1667 for Re =100 and 0.22 for Re =500. The computational domain is shown in Fig. 1. A time-dependent boundary condition is imposed at the inﬂow boundary 1; periodicity is imposed on 3and 4while on 2the zero Neumann condition on velocity is im- posed and the pressure is set to be zero. On the cylinder surface 5the no-slip boundary condition is prescribed. Converged solutions were obtained using the spectral/hp element method [19]. The domain is discretized into 412 triangular elements while seventh-order Jacobi polynomial basis are used to obtain resolution independent solutions. 2.2. Particle imaging velocimetry (PIV) experiment Here we consider wave interaction with a vertical, surface-piercing cylinder, see Fig. 2. This ﬂow gives rise to complex forms of wake structure due to the orbital particle trajectories of the incident wave, and the sweeping of previously generated vortices past the cylinder due to the oscillatory nature of the wave. The cylinder is ﬁxed and the ﬂow motion is sustained by the wave action. We will employ the experimental results obtained using particle image velocimetry (PIV) by Yang and Rockwell [12]. For this ﬂow there are two important non-dimensional parameters that need to be speciﬁed: First, the Keulegan–Carpenter number deﬁned by KC =2πAo D, in which Aois either the displacement amplitude of the cylinder motion or the amplitude of the oscillatory ﬂow and Dis the cylinder diameter. Secondly the Stokes number β=fD2 ν, in which fis the frequency of the motion and νis the kinematic viscosity. S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 221 Fig. 2. Experimental set up [12]. Quantitative images were obtained using a technique of high-image-density particle velocimetry (PIV). There are 13 phased-averaged snapshots available for a time period T=0.89. These images will be used to extract the POD modes. Details of the experiments and the imaging approach are described in detail in [12]. A brief summary is given next. The vertical, rigidly suspended cylinder, which is shown in the schematic of Fig. 2, was maintained stationary during all experiments. It had a diameter of D=12.7mm and a length of L=876mm. The submerged length of the cylinder was 700mm. The value of the Keulegan–Carpenter number were KC =2πAo/D =13.9atthe depth of the laser sheet, which is indicated in Fig. 2 as 51mm beneath the quiescent free-surface. The amplitude Aocorresponds to the radius of the particle orbit of the wave, which was also determined at the depth of 51mm. Furthermore, the value of the Stokes number was β=fD2=164 for this experiment. The corresponding value of Reynolds number is Re =KC ×β=2280. 3. POD-penalty systems In order to employ time-dependent boundary conditions in low-dimensional models, we formulate a new method to construct Galerkin systems. In particular, we incorporate the boundary conditions directly into the Navier–Stokes equations as constraints, enforced via suitable penalty parameters. In the next section we will demonstrate how to select the penalty parameters through bifurcation analysis in order to achieve asymptotically stable and accurate periodic solutions. 222 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 Herewe employ the hierarchical POD modes as a trial basis to represent the velocity ﬁeld. In addition, we employ a Galerkin projection of the Navier–Stokes equations onto these modes to derive dynamical systems to simulate the ﬂow. Let us decompose the total ﬂow ﬁeld Vas V(x,t)=U0(x)+u(x,t), where U0is the time-averaged ﬁeld. We express uas the linear combination of the POD modes as written in the summation conventions: u(x, y, t)=φu j(x, y)aj(t),v(x, y, t)=φv j(x, y)aj(t), where aj(t) are the unknown coefﬁcients and φ=(φu v) deﬁnes the vector of the POD modal basis. In the following we derive separately the low-dimensional system for the DNS data and the experimental data. 3.1. DNS: POD-penalty system The Galerkin projection of the Navier–Stokes equations with penalty terms included onto the jth POD mode is φj·V ∂t +(V·∇)V+∇p1 Re 2V+τ1ϒ(x)(VU)dx=0,(2) where τ1is the penalty parameter and Uis the imposed velocity at the inﬂow boundary 1(see Fig. 1). The function ϒ(x) is deﬁned as ϒ(x)=1,ifxon 1 0,otherwise.(3) We note here that on boundary 1we do not impose any boundary conditions as it is now treated as part of the interior domain. The treatment of the pressure term is of particular importance, so we analyze the corresponding Galerkin projection by using the Gauss’s theorem to obtain φj·∇pdx=− ∇·φjpdx+∂ φj·npds. (4) Obtaining the POD modes from DNS of an incompressible ﬂow ﬁeld leads to divergence-free eigenmodes, and thus the pressure term inside the domain is eliminated (ﬁrst term in the above equation). On the side bound- aries 3and 4we assume periodicity and hence the pressure boundary terms cancel each other. On the outﬂow boundary 2the pressure is set to zero in the corresponding DNS. The inﬂow boundary 1should not be in- cluded in the computation of the second term of Eq. (4) since we have already included it in the Navier–Stokes equations. Therefore, there is no contribution from the pressure on this boundary in the integration by parts pro- cedure. Finally, on the cylinder boundary the test function is zero and thus there is no pressure contribution there either. In summary, the Galerkin projection leads to the dynamical system: daj dt=fj(a)Gj(a) (5) S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 223 with a=[a1,a 2,...,a M], where Mis the number of POD modes. The term fj(a) includes the convective and viscous terms and has the form: fj(a)=− φj·((φi·∇)φk)dxaiak 1 Re φj·∇ 2φidx+ φj·((φi·∇)U0)dx+ φj·((U0·∇)φi)dxai φj·((U0·∇)U0)dx1 Re φj·∇ 2U0dx. Also, Gj(a) is the boundary penalty term, which is written as follows (in summation convention): Gj(a)=τ1ai1 φi(y|1)·φj(y|1)dy1 (UU0(y|1)) ·φj(y|1)dy, where φm(y|n) means the function of yobtained by evaluating φmon n. Since U(t) is time-dependent, we obtain a non-autonomous system. More details on the derivation of the above formulation for the Navier–Stokes equation are presented in [20]. 3.2. Experiment: POD-penalty system The POD-penalty system for the experimental data is derived similarly. By referring to Fig. 3, we now employ twopenalty terms τ1andτ2for the boundaries 1and 2, respectively.The Galerkin projection of the Navier–Stokes equations with penalty terms onto the jth POD mode is now φj·V ∂t +(V·∇)V+∇p1 Re 2V+τ1ϒ1(x)(VU1 )+τ2ϒ2(x)(VU2 )dx=0,(6) where the projection vector φjis deﬁned as previously, and U1 ,U2 are the velocity vectors at the boundaries 1 and 2, respectively (see Fig. 3). The function ϒi(x) is deﬁned as ϒi(x)=1,ifxon i 0,otherwise.(7) Unlike the earlier DNS study where the ﬂow is two-dimensional, in this case the true ﬂow is three-dimensional but only a two-dimensional slice is visualized via PIV. Correspondingly, imposing the divergence-free condition on the two-dimensional POD modes is not appropriate. To this end, we will employ the divergent POD modes and let the penalty terms “counteract” the divergent contributions (ﬁrst term of Eq. (4)); how accurate is this procedure will be tested by the results presented in the next section. Intuitively, it can be justiﬁed as the penalty term controls effectively the boundary mass ﬂuxes (on 1and 2), and thus by adjusting the value of τ1and τ2, respectively, we can counteract any mass sources or sinks due to the pressure contributions in the domain interior. The pressure contributions from the boundaries vanish due to periodicity and Dirichlet boundary conditions, similarly to DNS case. Regarding the representation of the time-dependent velocity boundary condition at 1and 2, we have found that it is accurate to use a Fourier series with 16 Fourier modes to represent the time-periodic forcing at those boundaries. A systematic investigation of this has been presented in [5]. The Galerkin projection of the two-dimensional governing equations leads to the dynamical system: daj dt=fj(a)G j(a) (8) 224 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 Fig. 3. Computational and PIV domains for KC =13.9. with a=[a1,a 2,...,a M], where Mis the number of POD modes. The term fj(a) has the same form as in Eq. (6). The boundary penalty term G j(a) is modiﬁed as follows: G j(a)=τ1ai1 φi(y|1)·φj(y|1)dy1 (U1 U0(y|1)) ·φj(y|1)dy +τ2ai2 φi(y|2)·φj(y|2)dy2 (U2 U0(y|2)) ·φj(y|2)dy 3.3. Transformation to an autonomous system In the next section we will show how to track periodic branches of the dynamical systems described by Eqs. (5) and (8). However, in order to effectively use the AUTO dynamical system package [13] to track the periodic branch, these non-autonomous systems need to be transformed to autonomous systems. To this end, we introduce the nonlinear oscillators dp dt=p+ˆ βq p(p2+q2),dq dt=qˆ βp q(p2+q2). This particular system has an asymptotically stable solution given by p(t)=sin(ˆ βt) and q(t)=cos(ˆ βt). We then incorporate the nonlinear oscillator to the POD-penalty system in order to obtain an equivalent autonomous system. S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 225 3.3.1. DNS: POD-penalty autonomous system WerecallthatinEq.(6)theonlytime-dependenttermisU,whichisgivenbyEq.(1).Theequivalentautonomous system is daj dt=fj(a)Gj(a,p),dp dt=p+ωq p(p2+q2),dq dt=qωp q(p2+q2), where a=[a1,a 2,...,a M], fj(a) is given by Eq. (6) and Gj(a,p) is now deﬁned as Gj(a,p)=τ1ai1 φi(y|1)·φj(y|1)dy1 ((1.0+Ap, 0) U0(y|1)) ·φj(y|1)dy. Therefore, we have replaced the term sin(ωt)inEq.(1) with p(t). 3.3.2. Experiment: POD-penalty autonomous system The transformation of the non-autonomous system to an equivalent autonomous one for the case of POD-penalty system derived from experimental data is somewhat more complicated. For this POD-penalty system, we have the representation of the velocity vectors at the boundaries 1and 2in the form of Fourier series as Ui (yj,t)=A(i, j) 0+ N n=1 A(i,j ) ncosnπt T+B(i,j ) nsinnπt T,(9) where Tis the period, Nthe number of Fourier modes (for this case N=16), yja grid boundary point, and i=1 or 2 for velocity vectors at the boundaries 1and 2, respectively. We then can transform Eq. (8) into an equivalent autonomous system as follows: daj dt=fj(a)G j(a,p,q),dpn dt=pn+ Tqnpn(p2 n+q2 n),dqn dt=qn Tpnqn(p2 n+q2 n), where with a=[a1,a 2,...,a M], fj(a) is the same as Eq. (6),p=[p1,p 2,...,p N], q=[q1,q 2,...,q N] and n=1...N. Note that here ˆ β=nπ/ T . Correspondingly, G j(a,p,q) is deﬁned as G j(a,p,q)=τ1ai1 φi(y|1)·φj(y|1)dy τ1ny k=1A(1,k) 0+ N n=1 A(1,k) nqn+B(1,k) npnU0(yk|1)·φj(yk|1)wk +τ2ai2 φi(y|2)·φj(y|2)dy τ2ny k=1A(2,k) 0+ N n=1 A(2,k) nqn+B(2,k) npnU0(yk|2)·φj(yk|2)wk. Here nyis the number of grid points on 1and 2, and wkis the weight for the trapezoid integration. 226 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 4. Results 4.1. DNS: Galerkin POD-penalty system The Galerkin POD-penalty systems for Reynolds number Re =100 and 500 are derived by employing 100 snapshots per period for both cases. We ﬁrst present representative results of the stability of these solutions and subsequently we investigate their accuracy. 4.1.1. Stability of periodic solutions Here, we study stability of the solutions of the Galerkin POD-penalty model through bifurcation analysis. We choose the bifurcation parameter to be the penalty constant τ1. In order to use the AUTO bifurcation tracking package, for this case, the asymptotically stable periodic solution must be provided—this of course is not known a priori. To this end, we assume a constant (typically large) value of τ1and obtain the corresponding solution of the non-autonomous system. However, it is not certain that this solution will have the same period as the forcing period orevenbeingperiodic [21–23]. Toovercomethis,we will studythe stability of theparticular solution for thatspeciﬁc value of τ1using Poincar´ e maps, following the work of [24–26]. Speciﬁcally, we have obtained the stability of the periodic solution using Poincar´ e maps, which we used to ﬁnd the return times of the periodic solution, following procedures outlined in [24,25]. We then employed the algorithm in [26] in order to ﬁnd the Floquet multipliers of the periodic solution. Let us examine a speciﬁc case to illustrate this approach. We consider the low Reynolds number Re =100 case with the number of POD modes M=6 and integrate Eq. (5) for a few values of the penalty parameter, say in the range of τ1[2000,3000]. We found, through the Poincar´ e map, that in this range an asymptotically stable periodic solution does indeed exist. Hence, we can choose any value of τ1[2000,3000] to apply AUTO in order to study stability of periodic solutions more systematically. We also found that the period of the asymptotic state is T=5.9988 while the corresponding period from the full DNS is identical to this value. As we will show in the next section, agreement in the period does not imply agreement in the ﬂow ﬁeld dynamics between the DNS and the low-dimensional system. Repeating this procedure with higher truncations at M=10 and 20 at the same Reynolds number, we found similarly asymptotically stable periodic states, which can be used as starting points for the AUTO bifurcation analysis. A similar study was performed for the POD-penalty system at Reynolds number Re =500 for M=6, 10 and 20 POD modes. With the penalty parameter τ1=3000, the system posses an asymptotically stable solution with period T=4.5454, which is in agreement with the results from the full DNS. Other large values of the penalty parameter yield similar results. After obtaining the asymptotically stable periodic solution, we use it as a starting point for AUTO, and track the stability of the periodic solution by decreasing τ1until loss of stability is detected. This produces the lowest value of the penalty parameter that guarantees stability. Speciﬁcally, for all the systems examined here loss of stability shows bifurcation into a torus. The results of this analysis for both Reynolds numbers are listed in Table 1, where in the third column the minimum values of τ1for stability are presented. Table 1 Values of penalty parameter for stability and accuracy requirements Re Modes Lowest penalty parameter for stability Best penalty parameter for accuracy Re =100 6 5.24127 5200 10 5.23467 6000 20 3.20842 6500 Re =500 6 1.39258 9000 10 1.26819 2.5×105 20 1.27168 3.0×106 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 227 Fig. 4. Simulation using M=10 POD modes for Re =100, τ1=5×105. DNS: ; POD-penalty simulation: solid line. 4.1.2. Accuracy of periodic solutions We now turn our attention to the accuracy of the ﬂow dynamics predicted by the low-dimensional system at different values of the τ1parameter. In Fig. 4 we plot the phase portraits predicted by the POD-penalty system for τ1=500,000 against the DNS corresponding results for the system with truncation at M=10. At this value of τ1an asymptotically stable state is obtained with the correct time period but as can be observed in this plot the accuracy in the ﬂow dynamics is poor. In order to improve this accuracy we ﬁrst deﬁne a relative error for each penalty parameter by Eτ1=M i=1(a iQ i)2 M i=1(Q i)2,(10) where a iand Q iare the maximum of the predicted POD modal coefﬁcients corresponding to the low-dimensional system and DNS, respectively. In Fig. 5 we plot the results of the bifurcation analysis for Re =100 and M=10 for the ﬁrst four POD modes. We see that for the higher values of the penalty parameter τ1the accuracy of a icompared to Q iis worse than for 228 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 Fig. 5. Bifurcation diagram using M=10 POD modes for Re =100. a icorresponds to solid line and denotes the maximum of the POD-penalty coefﬁcients. Q i, denoted by dash line, is the corresponding maximum coefﬁcient of the DNS predictions. the lower values of τ1. Similar results are also observed for the case of Re =100 with M=6 and 20, and also at Re =500 with M=6. However, this trend is not universal. For example, for the truncation M=20 at Re =500 (see results in Fig. 6), the agreement in the ﬂow dynamics is good for large values of τ1but there is a lower bound of the penalty parameter τ1below which this agreement is lost. Similar results were obtained at Re =500 with M=10. Therefore, it is the combination of the penalty parameter and truncation parameter for certain complexity in the ﬂow dynamics (here governed by the Reynolds number) that determines the quality of the prediction in the POD-penalty system. InFig.7,therelativeerror, deﬁned in Eq. (10), is plottedagainstthepenaltyparameterinordertodeterminethebest penalty parameter for the case Re =100 with M=10. The best penalty parameter is found to be approximately 6000. A qualitatively different result is provided in Fig. 8, where the relative error for the case Re =500 with Fig. 6. Bifurcation diagram using M=20 POD modes for Re =500. The legend is as in Fig. 5. S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 229 Fig. 7. Left: relative error for the POD-penalty model with respect to DNS with M=10 POD modes for Re =100, and a close-up on the right. M=20 is plotted against the penalty parameter. Here we ﬁnd that the error decreases monotonically with the penalty parameter but above some value, much greater than the stability bound, the accuracy saturates. A summary of our studies to determine the best values of the penalty parameter τ1for the best possible accuracy is presented in the last column of Table 1. Following the results of the bifurcation analysis, we then performed integration of the POD-penalty system using the parameters in Table 1. The corresponding results for two typical cases are shown in Fig. 9 for Re =100 and Fig. 10 for Re =500. For the latter case we observed the following: for the simulation of the POD-penalty system with M=10, the system can predict correctly up to the ﬁfth mode while the prediction of higher modes is erroneous. When we increase the number of modes in the system to M=20, the accuracy of prediction of the dynamics is much better, i.e. good accuracy is now obtained up to 15th mode, see Fig. 11. We recall that these two cases correspond to a lower bound in the penalty parameter for best accuracy as determined by the bifurcation analysis. This ﬁnding could possibly suggest that for cases with a lower bound for the most effective penalty parameter a higher truncation is required to achieve better accuracy; it is not clear if this result will be true in other ﬂow problems, however. We also note that integrating the POD-penalty Galerkin system Fig. 8. Relative error for the POD-penalty model with respect to DNS with M=20 POD modes for Re =500, and a close up on the right. 230 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 Fig. 9. Simulation using M=10 POD modes for Re =100, τ1=6000. DNS: ; POD-penalty simulation: solid line. requires approximately 10 periods to reach an asymptotically stable state even if the “exact” DNS conditions are used in the initialization process. In summary, the lesson learned from the DNS study is that above a certain threshold in the value of penalty parameter, stability of the periodic solution is obtained. However, the best accuracy may be obtained for speciﬁc values of the penalty parameter that seem to depend strongly on the ﬂow dynamics and the truncation in the POD-penalty low-dimensional system. 4.2. Experiment: Galerkin POD-penalty system In this section, we will present the results for the POD-penalty for the experimental data. We have used two truncations in the number of modes here, M=6 and 12. The number of Fourier modes that represents the periodic time-dependent boundary condition is set to N=16. For this POD-penalty model, there are two penalty parameters τ1andτ2that need to be speciﬁed. However,wewill adopt here a procedure where we trackthese penalty parameters by ﬁxing one of them at a point that the asymptotically stable periodic solution for the POD-penalty system is obtained. As in the previous study with DNS data, this asymptotically stable periodic solution might not be accurate S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 231 Fig. 10. Simulation using M=20 POD modes for Re =500, τ1=4×106. DNS: ; POD-penalty simulation: solid line. but it will be used as a starting point for AUTO to track the minimum value of either τ1and τ2for asymptotic stability of the periodic solutions. 4.2.1. Stability of periodic solutions From preliminary numerical experiments for both M=6 and 12 we have determined that at τ1=1000 and τ2=1000 the corresponding POD-penalty systems possess an asymptotically stable solution; see Fig. 12 for these speciﬁc parameters. As in the previous case with the DNS data, the periodic solution is then studied through the Poincar´ e map to examine its stability. We have found that the periodic solution for these speciﬁc penalty parameters is indeed asymptotically stable with period of T=0.89, which is in agreement with the data from PIV. Havingdeterminedthe starting pointfor AUTO,thetracking of stabilityof the periodicsolution is then performed by ﬁxing τ1=1000 and decreasing τ2until loss of stability is detected. We then switch the role of τ1and τ2and perform an analogous analysis. This study produces the lowest values of the penalty parameter for stability, and the corresponding results are presented in Table 2. We have found that for both values of M=6 and 12 the periodic solution loses its stability when one of the Floquet multipliers crosses the unit circle at 1. This was also observed 232 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 Fig. 11. Higher modes from the simulation using M=20 POD modes for Re =500, τ1=4×106. DNS: ; POD-penalty simulation: solid line. for M=6 with τ2=1000 ﬁxed while varying τ1. However, in the case of the truncation with M=12 the periodic solution loses its stability when one of the Floquet multipliers crosses the unit circle at 1. 4.2.2. Accuracy of periodic solutions In order to ﬁnd the most effective values of the penalty parameters that produces the most accurate asymptotically stable periodic solution compared to the data from PIV, we also track the periodic branch by increasing τ1or Table 2 Values of penalty parameters for stability limit and accuracy requirement for the POD-penalty system corresponding to experiment with Re =2280 Modes Lowest penalty parameter for stability Best penalty parameters for accuracy τ1=1000 τ2=1000 τ1=1000 τ2=1000 M=6 201.237 27.935 2×105,7.5×104 M=12 142.652 601.109 2×1077.5×107 Corresponding asymptotically stable periodic solution has a relative error greater than 5%. S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 233 Fig. 12. Simulation using M=12 POD modes for PIV data, τ1=1000, τ2=1000. PIV: ; POD-penalty simulation: solid line. τ2while ﬁxing the other one at 1000. Then, we compute the relative error in order to ﬁnd the best value of the penalty parameter for accuracy. In Fig. 13, the relative error is presented for the case of M=12 with ﬁxed τ2=1000. A summary of best values of the penalty parameter for all cases is presented in the last column of Table 2. Using the values from the table we then integrate in time the POD-penalty system to reach the asymptotic stable periodic states. The results of such simulations are presented in Figs. 14 and 15 as phase portraits. There is very good agreement with the corresponding experimental data with the higher truncation giving higher accuracy, as expected. 5. Summary We have developed a Galerkin POD-penalty method to construct low-dimensional dynamical systems for un- steady ﬂuid ﬂows with time-dependent boundary conditions. Penalized boundaries are incorporated directly in the Galerkin statement of the Navier–Stokes equations, and thus information about the pressure ﬁeld on such bound- aries is not required. The resulting dynamical system is non-autonomous, so we couple it to an equivalent nonlinear oscillator in order to study its stability using standard bifurcation analysis. 234 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 Fig. 13. Relative error for the POD-penalty model from PIV data. Here, M=12 with ﬁxed τ2, and a close up on the right. Fig. 14. Simulation using M=6 POD modes for PIV data, τ1=7.5×104,τ2=1000. PIV: ; POD-penalty simulation: solid line. S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 235 Fig. 15. Simulation using M=12 POD modes for PIV data, τ1=8×107,τ2=1000. PIV: ; POD-penalty simulation: solid line. We study the stability and accuracy of periodic solutions using the penalty parameter(s) as bifurcation param- eter(s). We consider two prototype ﬂows based on results from direct numerical simulations (ﬂow past a circular cylinder), and from experiments (wave–structure interaction). The results from both studies are qualitatively similar. We ﬁnd that there is a threshold value of the penalty parameter above which asymptotic stability of the periodic solution is guaranteed. This is an expected result, similar to what is known for numerical discretizations of Navier Stokes equations. The surprising, however, ﬁnding is that the accuracy of the solution predicted by the Galerkin POD-penalty system does not improve as the penalty parameter increases, as it is the case for full numerical dis- cretizations. Instead, there is a speciﬁc range within which the solution is accurate. In particular, depending on the number of modes (i.e. truncation) and the ﬂow complexity (i.e. Reynolds number) the best solution may correspond to a speciﬁc value of the penalty parameter or a range well above the threshold value for stability. In numerical discretizations that employ the penalty approach to impose Dirichlet or other type of boundary conditions,asthe penalty parameter approachesa very large number (e.g. inverseofmachineprecision)theboundary conditions are imposed exactly, i.e. in a strong form. Correspondingly, the error in the solution scales inversely proportional to the penalty parameter. Our ﬁndings here suggest that for low-dimensional systems, imposing the boundary conditions in a strong form may lead to an erroneous solution. Similar trends have been observed in spectral penalty methods for simulations of high Reynolds number turbulence at relatively low resolution [27]. This 236 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 can have great consequences in constructing effective low-dimensional dynamical systems as well as in formulating proper boundary conditions in large-eddy simulations. However, generalization of this conclusion to other ﬂow problems has to be tested very carefully. Acknowledgments The ﬁrst author gratefully acknowledges the Development and Promotion of Science and Technology Talents (DPST) project from Thailand for providing his scholarship during his graduate studies at Brown University. The authors would like to thank Dr. Y. Yang and Prof. D. Rockwell for providing PIV data. This work was supported by ONR and NSF, and computations were performed at the facilities of NCSA (University of Illinois at Urbana- Champaign) and at DoDs NAVO MSRC. References [1] N. Aubry, W.Y. Lian, E.S. Titi, Preserving symmetries in the proper orthogonal decomposition, SIAM J. Sci. Comput. 14 (2) (1993) 483–505. [2] W. Cazemier, R.W. Verstappen, A.E. Veldman, Proper orthogonal decomposition and low-dimensional models for driven cavity ﬂows, Phys. Fluids 10 (7) (1998) 1685–1699. [3] S. Sirisup, G.E. Karniadakis, A spectral viscosity method for correcting the long-term behavior of POD models, J. Comp. Phys. 194 (1) (2004) 92–116. [4] X. Ma, G.E. Karniadakis, H. Park, M. Gharib, DPIV-driven ﬂow simulation: a new computational paradigm, Proc. R. Soc. London A 459 (2031) (2003) 547–565. [5] S. Sirisup, G.E. Karniadakis, Y. Yang, D. Rockwell, Wave-structure interaction: simulation driven by quantitative imaging, Proc. R. Soc. London A 460 (2043) (2004) 729–755. [6] X. Zhou, L. Sirovich, Coherence and chaos in a model of turbulent boundary layer, Phys. Fluid A 4 (12) (1992) 2855–2874. [7] G. Berkooz, P. Holmes, J.L. Lumley, N. Aubry, E. Stone, Observations regarding ‘Coherence and chaos in a model of turbulent boundary layer’ by X. Zhou and L. Sirovich [Phys. Fluids A 4 (1992) 2855], Phys. Fluid 6 (4) (1994) 1574–1578. [8] L. 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Sirisup, Issues in low-dimensional modeling of unsteady ﬂows: convergence, asymptotic stability and reconstruction procedures, PhD thesis, Division of Applied Mathematics, Brown University, 2005. [21] T.A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, 1985. [22] T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Springer-Verlag, 1975. S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 237 [23] G.Makay,On some possibleextensionsofMassera’s theorem, ProceedingsoftheSixthColloquium on the QualitativeTheoryofDifferential Equations (In EJ Qualitative Theory of Differential Equations), vol. 16, 2000, pp. 1–8. [24] M. Hannon, On the numerical computation of Poincar´ e maps, Physica D 5 (1982) 412–414. [25] W. Tucker, Computing accurate Poincar´ e maps, Physica D 171 (3) (2002) 127–137. [26] R. Seydel, Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, Springer-Verlag, 1994. [27] P.J. Diamessis, J.A. Domaradzki, J.S. Hesthaven, A spectral multidomain penalty method model for the simulation of high Reynolds number localized incompressible stratiﬁed turbulence, J. Comput. Phys. 202 (1) (2005) 298–322. ... In literature [14][15][16][17], different approaches to control the ROM BCs can be found of which two common approaches are extended and compared in this work: the lifting function method and the penalty method. The aim of the lifting function method [15,17] is to homogenize the BCs of the basis functions contained in the reduced subspace, while the penalty method [14][15][16]18] weakly enforces the BCs in the ROM with a penalty factor. A disadvantage of the penalty method is that it relies on a penalty factor that has to be tuned with a sensitivity analysis or numerical experimentation [18]. ... ... 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Two different stabilization techniques are compared in [25]; the supremizer enrichment of the velocity space in order to meet the inf-sup condition (SUP) and the exploitation of a pressure Poisson equation during the projection stage (PPE). ... ... This requires a huge computing time using standard numerical solvers [1][2][3]. A way to avoid this is the use of reduced-order models, i.e., reduced basis [4][5][6][7][8][9][10] or proper orthogonal decomposition (POD) [11][12][13][14][15][16][17][18][19][20] models. Reduced models are based on the fact that, for dissipative evolution equations, a finite low-dimensional manifold contains the longterm behavior of the system [21][22][23][24][25]. ... ... Thus, the maximum number of grid points is N = 36 × 14 = 504. The integrals in Equations (15)- (18) and the usual L 2 scalar product are performed with the Legendre-Gauss-Lobatto quadrature formulas [57]. 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Two-dimensional unsteady flows in complex geometries that are characterized by simple (low-dimensional) dynamical behavior are considered. Detailed spectral element simulations are performed, and the proper orthogonal decomposition is applied to the resulting data for two examples: the flow in a periodically grooved channel and the wake of an isolated circular cylinder. Low-dimensional dynamical models for these systems are obtained using the empirically derived global eigenfunctions in the spectrally discretized Navier-Stokes equations. The short- and long-term accuracy of the models is studied through simulation, continuation, and bifurcation analysis. Their ability to mimic the full simulations for Reynolds numbers beyond the values used for eigenfunction extraction is evaluated.
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The paper considers the dynamics of coherent structures in the wall region of a turbulent channel flow. The Karhunen-Loeve eigenfunctions and Galerkin procedure are employed to derive the dynamical description. A well-posed Hermitian theory is developed and convergence questions do not arise. No exterior pressure is required by this theory. It is shown that the behavior of the resulting model equations include intermittency, quasi-periodic, and chaotic solutions. Three-dimensional effects are introduced into the dynamics in order to produce a physically more realistic dynamical theory. It is argued that the bursting and ejection events in turbulent boundary layers are explained more satisfactorily within this framework.