AIAA-2000-2564
Large Eddy Simulation and Turbulence Control S. Scott Collis∗, Yong Chang†, Steven Kellogg‡, and R.D. Prabhu§ Mechanical Engineering and Materials Science Rice University, Houston, TX 77005-1892 (713) 348-3617,
[email protected] Abstract This paper reviews LES methods, based on the dynamic subgrid-scale model, that greatly improve the efficiency of turbulence control simulations in the context of drag reduction for plane turbulent channel flow. We begin by performing simulations of opposition control at Reynolds numbers in the range Reτ = 100 to 590 which demonstrate a decrease in effectiveness of this control strategy with increased Reynolds number. We then review techniques for optimal control of turbulent flows and discuss our implementation using an instantaneous control framework where the flow sensitivity is computed from the adjoint LES equations. Detailed optimal control results are presented that demonstrate excellent agreement with available DNS at a fraction of the computational expense. Given the added efficiency of LES, a receding horizon control framework is also explored that results in increased drag reduction along with improved control distributions. Results are also presented for a hybrid LES/DNS method where optimization is performed using LES but the flow is advanced using DNS. This approach demonstrates that even coarse grid LES can serve as a viable reduced order model for DNS. In our ongoing efforts to seek greater model reductions for predictive control, we also explore the influence of control on the basis functions obtained from proper orthogonal decomposition. 1
Introduction
Flow control offers the potential for modifying turbulent flows to achieve a variety of objectives including skin-friction drag reduction, noise suppression, ∗ Assistant
Professor, Member AIAA Candidate ‡ Masters Student § Postdoctoral Research Associate c Copyright 2000 by Rice University. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. † Doctoral
and heat transfer modification. Previously, both passive and active turbulence control schemes have been studied and Moin et al.25 have reviewed the subject recently. The present article focuses on drag reduction primarily using optimal control, which provides a framework to systematically derive the most efficient means of control to achieve a desired effect. The control distribution that results from optimal control theory can be used to guide the development of heuristic control approaches as well as identify the merits of various actuator/sensor combinations. However, optimal control theory typically requires a high-fidelity means of predicting the flow field response which makes the method computationally intensive. In the context of turbulence control, optimal control theory has been applied to reduce the drag in a low Reynolds number, turbulent channel flow using Direct Numerical Simulation (DNS) to predict the flow response4, 22, 26 and recent results indicate that, for certain objective functionals, relaminarization is possible.5 Unfortunately, the computational expense of DNS makes the method intractable for the high Reynolds number complex-flows typical of engineering applications. The current work partially addresses this limitation by modeling the flow field using Large Eddy Simulation (LES) with a dynamic subgrid-scale model. LES greatly reduces the computational expense, compared to DNS, allowing solutions at higher Reynolds numbers and for more complex flow systems. In the parlance of control, LES is a reduced order model for the Navier–Stokes equations that can either supplement or entirely replace DNS. For example, in our earlier work8, 11 we demonstrate that LES accurately predicts both the control and the flow response for both opposition control and optimal control of turbulent channel flow. In this case, LES replaces DNS within a control simulation resulting in efficient simulations of controlled turbulent flows. While the results are self-consistent in that the model and state are the same, since LES is only an approximation to DNS, the reliability of the predicted control
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distributions must be validated. Therefore, we have explored hybrid LES/DNS methods12 where LES is used to predict the control (using optimal control) but the flow response is obtained from DNS. This demonstrates that even coarse grid LES is a viable reduced order model for DNS in the context of nearwall turbulence control. In the current paper, we review these results and present extensions including opposition control at Reτ = 590 and optimal control at Reτ = 180 representing the highest Reynolds numbers yet reported for turbulence control simulations. Given the added efficiency of the LES based methods discussed here, we also present results using a receding horizon instantaneous control framework. In this approach, one solves optimization problems over a time horizon Tp but only uses a fraction of the predicted control Ta , where Ta < Tp . Previously with DNS based optimization, the idea of throwing away a portion of the control was not practical. With the reduced expense of LES-based optimization this idea can be explored to determine the advantage of considering the longer term impact of control on the state of the flow. Although LES serves as an effective reduced order model for DNS, it is still relatively expensive so that further model reductions are desirable. One option for additional model reduction is to use a model based on a truncated basis arising from Proper Orthogonal Decomposition (POD) which is a well-known technique for deriving an empirical basis optimal in the energy sense for any given dataset. We explore the viability of using a reduced order model based on a truncated POD basis from a no-control flow to represent controlled flows. The paper begins in §2 with a brief discussion of LES using the dynamic subgrid scale model. In §3 LES is used to study opposition control while §4 presents results for optimal control for both LES and hybrid LES/DNS. Receding horizon control is explored in §5 and POD based reduced order models are discussed in §6 with conclusions given in §7. 2
Large Eddy Simulation
Consider incompressible, fully-developed turbulent flow in a planar channel with coordinates of x1 in the streamwise direction, x2 in the wall-normal direction, and x3 in the spanwise direction. The flow in the streamwise and spanwise directions is assumed to be periodic. Large Eddy Simulation is performed on the turbulent flow by removing the small scale turbulent structures through a low-pass filter leading to the filtered, nondimensional, incompressible Navier–
Stokes equations ui,t + (ui uj ),j −
1 ui,jj + p,i + f i + τij,j = 0 (1a) Re ui,i = 0 (1b)
where a bar denotes the grid filter operator, f i = δi1 Px is the body force required to enforce the mean pressure gradient for turbulent channel flow, and τij is the subgrid-scale (SGS) stress which can be modeled using the Smagorinsky model35 1 2 τij − δij τkk = −2C∆ |S|S ij 3
(2)
where ∆ is the grid-filter width, S ij is the strain rate tensor defined as S ij = 12 (ui,j + uj,i ), and |S| = (2S ij S ij )1/2 . The dynamic procedure,14, 23 which has been successfully used to study a variety of complex inhomogeneous flows,1, 3, 14, 15, 28, 29 is used to calculate the dimensionless model coefficient C, C=
hLij Mij i hMkl Mkl i
(3)
b 2 |S| bS b + bi u bj and Mij = −2∆ where Lij = ud i uj − u ij 2 d 2∆ |S|S ij . The h i denotes the average over planes parallel to the walls and the hat denotes the testb which satisfies filtering operator with the width ∆, b = 2∆. b > ∆. We use ∆ ∆ The LES equations for turbulent channel flow are solved using a hybrid Fourier-spectral and finite difference method1, 7, 26 which is designed to run efficiently on workstation class computers and shared memory parallel computers. Fourier transforms are used to compute spatial derivatives in the homogeneous directions and a conservative second-order finite difference scheme is used to compute spatial derivatives in the wall-normal direction. The computational grid is staggered in the wall-normal direction. The flow is advanced in time using an implicit Crank-Nicholson method for wall-normal derivative terms, an explicit third-order Runge-Kutta method for terms involving derivatives in homogeneous directions, and a fractional step algorithm is used for divergence free condition. For most of the computations reported here, the computational domain is (4πδ, 2δ, 4πδ/3) in the x1 , x2 , and x3 directions respectively, where δ is the channel half-height. We choose δ as the reference length, and uτ = (τw /ρ)1/2 as reference velocity, where τw is the average stress on the walls. Thus the Reynolds number is Reτ = uτ δ/ν and the reference (convective) time scale is δ/uτ . Results are frequently presented in wall units, with t+ = tu2τ /ν, y + = yuτ /ν and u+ = u/uτ .
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Figure 1: Mean and Root-Mean-Square velocity profiles at Reτ = 180 with different resolutions, the channel domain is (4π, 2, 4π/3). : LES results with : LES results with (32×65×32); (48×65×48); ◦ : filtered DNS.14, 18 (a) Mean velocity profile. (b) (c) (d) Turbulence intensities (u0i u0i )1/2 .
Quantitative comparisons of LES and DNS statistics for Reτ = 180 are shown in Figures 1 and 2. In Figure 1, LES statistics with different resolutions are compared with the results of a filtered DNS.14, 18, 27 Our LES results match the DNS results nearly perfectly with a resolution of (48 × 65 × 48) and adequately with only (32× 65 × 32) grid points compared to (128 × 129 × 128) grid points in the DNS calculations — by a factor of 14 and 32 reduction respectively. In Figure 1(a), our mean-flow profile in wall units is in excellent agreement, only slightly overpredicting the wall shear stress at the lower resolution of (32 × 65 × 32). Similarly, rms velocities from the LES are in good agreement with the filtered DNS, as shown in Figure 1. Figure 2 presents additional statistics for LES at (48 × 65 × 48). Figure 2(a) shows the spanwise velocity correlations which demonstrate that the turbulence is uncorrelated over the width of the computational domain. The spanwise energy spectra in Figure 2(b) show that the grid and test filter cutoffs are both in the inertial range for this resolution. The total stress is linear in Figure 2(d) indicating that second order statistics are well converged. The rms velocity profiles for LES in global coordinates are compared in Figure 2(c) with unfiltered DNS and good agreement is achieved with the LES slightly lower than the DNS as expected. Similar results are obtained for resolution of (32 × 65 × 32). Exploiting the capability of LES to model higher Reynolds number flows than DNS, we also compare
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Figure 2: LES statistics with resolution of (48 × 65 × 48). (a) Spanwise two-point correlations at y + = 12. : R11 ; :R22 ; :R33 . (b) Spanwise one-dimensional energy spectra at y + = 12. : E11 ; :E22 ; :E33 . (c) Profiles of rms velocity fluctuations. : urms ; :vrms ; :wrms . ◦ : DNS.18 (d) Profiles of Reynolds : Reynolds stress, −u0 v 0 ; stress and total stress: : total shear stress, −u0 v 0 + (1/Re)∂u/∂y. statistics for Reτ = 590, the highest Reynolds number DNS currently available.27 Here the computational domain is (2πδ, 2δ, πδ), the same as DNS and the resolution is (72 × 129 × 72), 57 times smaller than the DNS resolution of (384 × 257 × 384). Figure 3 compares the mean velocity and rms velocities of the LES in wall coordinates with the DNS27 which are seen to be in good agreement. The mean velocity in the log layer is lower than DNS, which indicates that the drag on the wall is slightly overpredicted. Discrepancies between LES and DNS shown in this figure are mainly due to the fairly low resolution of the LES. Figure 1 shows that similar discrepancies occured at Reτ = 180 at the low resolution of (32 × 65 × 32) and that LES agrees much better with DNS at slightly higher resolution. Therefore we believe that at Reτ = 590, better LES results could be achieved with increased resolution. Nevertheless, the LES results presented here are still in the error range of typical LES and therefore judged to be acceptable. Comparing the results at Reτ = 590 with the results at Reτ = 180, we can see that at Reτ = 590, the log layer extends over a wider range of y + , whereas for Reτ = 180, the log layer is very short. The peaks of the streamwise velocity intensity urms for both Reτ = 180 and Reτ = 590 are at y + ≈ 14, which is in excellent agreement with previous results.20, 34, 36 In summary, our no-control LES results are in ex-
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Drag Reduction 30% 28% 21%
Table 1: Drag reductions and power savings for opposition control at different Reynolds numbers. The sensing plane locations are ys+ = 14.3 for Reτ = 590, ys+ = 14.0 for Reτ = 180, ys+ = 14.9 for Reτ = 100. 1.1
0.5
no control turbulent (normalized)
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Figure 3: Mean and Root-Mean-Square velocity profiles at Reτ = 590 with resolution of (72 × 129 × 72), the channel domain is (2π, 2, π). : LES results; : DNS results with resolution of (384 × 257 × 384).27 (a) Mean velocity profile. (b) (c) (d) Turbu1/2 lence intensities hu02 . i i
0.9
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t+ cellent quantitative agreement with available DNS data at Reτ = 180 and Reτ = 590, and the LES results are obtained at a fraction of the computational Figure 4: Drag histories for opposition control at expense of DNS. different Reynolds numbers. The time is in viscous units. Drag is normalized by the average drag for 3 Opposition Control Simulations uncontrolled turbulent flow. Heuristic control schemes are feedback control laws based on physical arguments. An active cancellation scheme, generally called opposition control, was used for Reτ = 100, (32 × 65 × 32) for Reτ = 180, and by Choi et al.9 to reduce the drag in a fully-developed (72 × 129 × 72) for Reτ = 590. The opposition controlled drag histories for these turbulent channel flow, and about 20% drag reducthree Reynolds numbers are compared in Figure 4. tion was achieved when the control was chosen to The drag reduction values are compared in Table 1. oppose the vertical motion on a sensing plane lo+ In these runs, increased Reynolds numbers tend to cated at ys = 10. The idea is simply to choose reduce the effectiveness of opposition control. Here the control to oppose the vertical motion near the = 100 is subcritical, and Reτ = 180 and Reτ = Re τ wall therefore lifting the high shear region away from 590 are supercritical. As Reynolds numbers increases, the wall so that drag is reduced. More recent DNS 16, 19 the flow becomes more unstable and chaotic, thus it is demonstrated about 25% drag reduction studies reasonable that the drag reduction achieved by oppowhen the control was set to oppose the vertical mo+ sition control decreases for higher Reynolds number. tion at ys = 15. Both studies show that drag actually From Figure 4 we also see that at Reτ = 590, the increases when the control is set to counter motions transient phase of the control history occurs more too far from the wall, say at ys+ = 25.9, 19 All DNS investigations of opposition control are quickly than at low Reynolds numbers even when prefor flow at Reτ = 180. We have previously obtained sented in viscous units. This suggests that for high similar results using LES.8 As an efficient tool for tur- Reynolds number flow, actuators need to have faster bulent flow simulation at higher Reynolds numbers, response times, and that real time active control is we use LES to extend opposition control simulation more difficult. The “virtual” wall8, 16, 19 concept is evident in Figto Reτ = 590, as well as Reτ = 100 and Reτ = 180. The computational conditions are identical to the no- ure 5, which shows the average v 2 profile in wall cocontrol simulation with resolutions of (32 × 49 × 32) ordinate, for Reτ = 590 and Reτ = 180. For both
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savings ratio of around 1.4%. Note that this ratio of power input to savings increases with increasing Reynolds number. This observation confirms that high Reynolds number flows are more difficult to control. Based on this investigation of opposition control, we conclude that LES is a viable tool for simulating turbulence control based on wall transpiration.
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Figure 5: Profile of v 2 in wall coordinate at different Reynolds numbers for opposition control. : Reτ = 590; : Reτ = 180. Reynolds numbers, the locations of the virtual walls in viscous units are almost identical because the sensing plane are located near ys+ ≈ 14 in both cases. Far away from the wall, the wall-normal average velocity for Reτ = 590 is larger, which means that the wall-normal momentum transport is larger, the flow is more difficult to control, and larger amplitude control velocities are required. We can see that the wallnormal velocities on the virtual wall are nearly zero, but the value for Reτ = 590 is slightly larger than the value for Reτ = 180, which may partially explain why a smaller drag reduction is achieved for Reτ = 590. In flow control for drag reduction, we hope to achieve a large ratio of power savings to the power input by the control. The power input by the control is defined as 1 Pφ = T
tZ 0 +T Z
φ t0
Γw
φ2 + p dΓdt, 2
(4)
Optimal Control Based On LES Instantaneous Control
We consider optimal control of fully developed turbulent flow in a planar channel that is subject to wallnormal transpiration ui = Φni on the walls with the initial condition ui = g i at t = 0, where gi is an instantaneous velocity field from an uncontrolled LES calculation. One possible formulation of optimal control is to determine the control Φ such that the drag is minimized while also minimizing the magnitude of the control. However, prior numerical experiments using DNS show that this is not the most effective way to reduce the drag.4 Since the turbulence is responsible for increasing the momentum transport from the center of the channel to the near-wall region and thus increasing the drag, it is reasonable for the cost functional to target the turbulence directly. This appears to be advantageous since the kinetic energy responds to changes in the flow more quickly than does the drag. In other words, turbulent kinetic energy is the cause and drag is the effect. In addition, temporary increases in the turbulent kinetic energy are allowable, as long as a reduction is achieved by the end of the optimization interval.5 This leads to a cost functional that minimizes the turbulent kinetic energy only at the end of each optimization period, tZ 0 +T Z
J t (Φ) = t0 Γw
` 2 Φ dt dΓ + 2
Z Ω
u0i u0i 2
dΩ (6) t=t0 +T
where φ = −n2 u2 is the control on the wall Γw . The The first term regularizes the control and is intepower savings due to drag reduction is given by grated over the wall boundaries, Γw , and over a fixed tZ 0 +T Z time interval [t0 , t0 + T ]. The penalty coefficient ` is 1 Px(no−control) − Px u1 dxdt, (5) used to represent the cost of the control. The secPdrag = T ond term minimizes the TKE at the end of each time t0 Ω window. Note that for LES, the cost functional only (no−control) is the streamwise pressure gradi- acts on the resolved scales of motion, the implications where Px ent for the no-control flow and Px is the streamwise of which will be discussed below. In optimal control, the objective is to find the control Φ that minimizes pressure gradient for the controlled flow. The power saving ratios are also presented in Ta- (6). This is achieved using the methods of calcuble 1 for different Reynolds numbers. We see that op- lus of variations and leads to adjoint equations in a position control at Reτ = 590 achieves a good power form similar to the LES equations, though with terms 5 American Institute of Aeronautics and Astronautics
t=0
t=Tp
... ... ...
...
0
Ta
2Ta
3Ta
...
t
Figure 6: Instantaneous control approach (Ta = Tp ) and receding horizon control (Ta < Tp ).
arising from the dynamic model. The first derivation and solution of the adjoint LES equations for the dynamic subgrid-scale model were presented by Collis and Chang.8, 11 The gradient of the cost functional with respect to the control can be obtained through the solution of the adjoint equations. Then the control is updated using an iterative method based on the Polak-Ribiere conjugate gradient algorithm. The details of the adjoint LES equations, the numerical methods for the adjoint equations, and the implementation of the conjugate gradient algorithm can be found in the authors’ previous papers.7, 8, 11, 12 The selection of a suitable optimization time interval, T , is very important. Previous optimal control simulation based on DNS5 shows that for the channel flow of Reτ = 100, the turbulent flow can be relaminarized in 2000 to 3000 viscous units. However, over such a long time period, both the state solution and the adjoint solution are sensitive to the numerical errors due to the non-linearity of the state equations and the exponential growth of the adjoint variables. Thus, the optimization becomes more difficult and may even fail to converge for sufficiently large T . Although the optimal control objective is to minimize the cost functional over a time interval [0, Ttotal ], in order to overcome the computational difficulty mentioned above, this large control interval is divided into many smaller time windows with size T . The cost functional is then minimized on each smaller window [(n − 1)T, nT ] in succession, where n = 1, · · · , Ttotal /T . This procedure is referred to as instantaneous control. Although instantaneous control will not lead to the same control as optimizing over [0, Ttotal ], it does greatly reduce the computational overhead and experience from DNS shows that
viable control strategies are produced.5 A generalization of instantaneous control, called receding horizon control, is illustrated in Figure 6. In this figure, the dashed arrows represent the optimization in each prediction window of length Tp . The solid arrows indicate the flow advancement once the “best” control is obtained and it is possible that the flow only advances some portion Ta (Ta < Tp ) of the prediction period Tp . After advancement, the optimization process is repeated again in another prediction window. In this section the advance window is the same as the prediction window, T = Ta = Tp , yielding the basic instantaneous control approach first implemented by Bewley et al.5 in the context of DNS for optimal control of turbulent channel flow. In §5 we explore the use of receding horizon optimal control, in which Ta < Tp . 4.2
Terminal TKE Control
For the simulation of turbulent channel flow with Reτ = 100, the dimensionless computational domain is (4π, 2, 4π/3) with a baseline mesh resolution of (32 × 49 × 32) and a time step of ∆t = 0.003. This domain size is the same as the DNS by Bewley et al.5 where they use a resolution of (42 × 65 × 42). The penalty coefficient ` in the cost functional (6) is set to be ` = 1 here. Figure 7 shows the evolution of drag and turbulent kinetic energy (TKE) at Reτ = 100 using optimal control with the terminal TKE objective functional described in §4.1. The LES-based optimal control achieves relaminarization with T + ≥ 25 and results are in excellent agreement with previous DNS5 at all values of T + . For small time windows, such as T + = 1.5 or T + = 10, the drag and TKE saturate after t+ ≈ 500. In general, the larger the time window is, the more drag reduction can be achieved. For large time windows (T + ≥ 25), the flow is relaminarized after t+ ≈ 2000. TKE and drag decrease more rapidly as the size of the optimization window increases. Similar to DNS results,5 the drag and TKE decrease quickly in the LES during the early part of the control. But after t+ ≈ 500 for T + > 25, the LES does not reduce TKE as quickly as DNS. One possible explanation for this is that for large t+ with T + > 25 the TKE eventually decays exponentially as would be expected in a process dominated by viscous dissipation. Since the smallest scales where dissipation would first act are modeled in LES, this could explain the delayed decay in TKE compared to DNS. At Reτ = 100 the flow is subcritical which means the flow is stable to small disturbances and that turbulent flow at this Reynolds number is relatively easy
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1.1
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Figure 7: Drag and TKE histories for LES based terminal TKE control for different optimization winFigure 8: Comparison of LES results and available dows T : (32 × 49 × 32), Reτ = 100. DNS results5 for optimal control at Reτ = 180. Here for LES, the iteration number is the maximum iterto control since for sufficiently small fluctuations, the ation number for the conjugate gradient method to flow will converge to laminar state with no control. find the best control. The next step is to extend the LES-based optimal control simulation to a supercritical Reynolds number, Reτ = 180. At this Reynolds number, DNS requires a very high resolution of (170 × 129 × 170)5 to resolve the small scales in the turbulent flow and adjoint velocities. Based on results in §2 for flow without control and in §3 for opposition control we present results for LES at (48 × 65 × 48) and (32 × 65 × 32), confident that LES can give good results at such low resolutions. The optimal terminal TKE control is implemented first using a time window of T + = 36 with time step of ∆t = 0.002 and 3 conjugate gradient iterations for the optimization in each time window. Results are shown in Figure 8. We can see that even with this relatively large time window, optimal control at Reτ = 180 fails to relaminarize the flow. The drag and TKE saturate after t+ ≈ 600. Nevertheless, 40% drag reduction can be achieved, a considerable im-
provement over the drag reduction obtained using opposition control (see §3). From the TKE history in Figure 8, we see that a burst-like phenomena happens at t+ ≈ 800, greatly increasing the TKE and leading to a statistically stable state with approximately 40% drag reduction. In Figure 8, we also compare our LES results with available DNS results.5 Unfortunately, DNS is very expensive for optimal control at this Reynolds number, and the available DNS5 only reaches t+ = 480. In terms of both the drag reduction and the TKE reduction, our LES results with 3 gradient iterations are not as good as DNS results. To identify the cause of this difference, a new simulation with the time window of T + = 40 was performed using 6 conjugate gradient iterations that yields results in much better agreement with the DNS. However, the flow still does
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1.1 1 0.9 0.8 Drag
not converge to the laminar state as at Reτ = 100. Nevertheless, about 50% drag reduction is achieved. Similar to opposition control, the higher the Reynolds number, the more difficult it is to control the flow. At Reτ = 180 with T + = 40 we are unable to relaminarize the flow as was done for T + ≥ 25 at Reτ = 100. Simulations at larger T + are required to determine whether relaminarization is possible at supercritical Reτ .
LES(32x49x32) filtered DNS Hybrid(LES32x49x32) DNS(42x65x42)
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tZ 0 +T Z
Jt (Φ) = t0 Γw
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LES is based on the SGS model which may or may not represent the dynamic behavior of turbulence very well, especially for near wall turbulence control. In wall bounded turbulence control, it is essential that the dynamic behavior of near wall coherent structures is captured. In order to explore the viability of the controls predicted by LES, we have implemented a novel hybrid LES/DNS algorithm in which the optimization is performed based on LES and, once the optimal control is found, the flow is advanced using DNS. Our experience indicates that the most time consuming portion of optimal control simulations is the optimization procedure. Thus, the hybrid method not only validates the viability of the controls predicted by LES, but also utilizes LES as a low cost reduced order model for DNS. In the hybrid LES/DNS algorithm, the initial condition for every control time window is at DNS resolution. We first interpolate the initial condition to the LES mesh and then perform the optimization using LES. Once the optimal control is found, it is interpolated back to the DNS mesh, and flow is advanced using DNS with this control distribution to the start of next time window. This procedure is repeated for every time window. Figure 9 compares the results for Reτ = 100 with T + = 30 for LES, DNS, and hybrid LES/DNS with LES resolution of (32 × 49 × 32) and DNS resolution of (42 × 65 × 42). In interpreting these results, it is important to distinguish between the different forms of the objective functions used in each simulation. Although the general form of the objective functional based on terminal observation of TKE is the same for all cases, for the LES and the hybrid LES/DNS, both the velocity field and the controls are on the grid filtered mesh as indicated in the definition of the cost functional (6). In the DNS, however, the full control Φ and velocities ui are used in the objective functional
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2500
3000
Figure 9: Drag and TKE comparison for LES, DNS, filtered DNS and Hybrid DNS/LES at T + = 30, Reτ = 100. As seen in Figure 9, all methods converge to the laminar state with T + = 30, although the LES takes more time to relaminarize. The quickest convergence to the laminar state is obtained with DNS, which is not surprising since the DNS possesses the full stage of the turbulence. Interestingly, the hybrid LES/DNS where the resolution used during the optimization is the same as the full LES relaminarizes at nearly the same rate as the full DNS. It appears that accurate state prediction is more critical than having the exact DNS control distribution. To explain this further, results from a filtered DNS run are also included in the figure where the velocity field is filtered to the LES resolution in x and z in the cost functional (6), and the control is projected to LES resolution. The filtered DNS results in Figure 9 shows that filtering delays the relaminarization, making the filtered DNS nearly identical to the hybrid LES/DNS. This again is strong evidence that the accuracy of the state is
8 American Institute of Aeronautics and Astronautics
1.1
advancing the flow using DNS becomes larger and the TKE reduction rates are smaller than TKE reduction rates for pure DNS. In summary, LES with the dynamic subgrid-scale model not only yields optimal control results that are consistent with DNS, but the controls computed based on LES optimization are also valid for DNS as shown by the hybrid LES/DNS method. This indicates that LES can be reliably used as a reduced order model for optimal control of turbulence.
Turbulent
1 0.9
Drag
0.8 0.7 0.6 0.5 0.4 0.3 0
Laminar 500
1000
1500 t+
2000
2500
3000
hybrid(24x49x24), T +=51 DNS(42x65x42), T+=51 + hybrid(24x49x24), T =102 + DNS(42x65x42), T =100
100 TKE
(by Bewley, et al. 1999)
-1 10
0
500
1000
1500 t+
2000
2500
3000
Figure 10: Drag and TKE comparison for DNS, and Hybrid DNS/LES at T + = 51 and T + = 102 with a coarse resolution for LES optimization, Reτ = 100. The DNS result at T + = 100 comes from Bewley et al.5 more important than the accuracy of the control. In fact, for the first 500 viscous time units, the drag and TKE evolution for all methods, including full LES, are nearly identical. This suggests that the optimization initially targets the large scale structures in the turbulence and it is only after the large scales (near the wall) are significantly reduced that differences due to the filtered scales and model error become noticeable. Since for LES both the state and control represent the large scale structures in the flow, for t+ ≤ 500, the TKE is initially reduced more rapidly for LES than for DNS and hybrid LES/DNS. We have also performed a hybrid LES/DNS with control time windows of T + = 51 and T + = 102 using very coarse LES resolution (24× 49 × 24). Results are shown in Figure 10. The flow is still relaminarized and these simulations are three times faster than the full DNS run. We can see that with this coarse resolution for LES optimization, the control error for
5
Receding Horizon Optimal Control
Optimal Control is a subset of Model-Based Predictive Control (MBPC), a scheme that finds good control distributions by testing potential controls on a mathematical model of the system before applying them. In industrial applications, errors in the mathematical model of the system cause the accuracy of predicted state and corresponding controls to diminish if frequent measurements of the state are not made. In the channel flow problem, however, the system and the model are typically the same thing — numerical solutions to the state equation. Consequently, past experiments with optimal control have not re-sampled the state; calculated control distributions have been used in their entirety. Although it seems wasteful to accrue the computational expense of calculating a control distribution and then throwing a fraction of it away, the increase in speed provided by our LES formulation allows for some flexibility to experiment with a more traditional type of MBPC. Furthermore, it is evident from plots of control effort, e.g. Figure 12, that the greatest control magnitude is applied at the beginning of each time window for some cost functionals, suggesting that truncating the end of the window and starting a new one may make controls more dense, more regular, and more effective. In the instantaneous control framework, calculating a control distribution for a longer time window than necessary may be referred to as Receding Horizon (RH) optimal control. The RH approach is illustrated by Figure 6. First, the flow is predicted with no control. Then the adjoint is marched backward in time and the extracted gradient information is used to update the control. The, adjoint, control, and state are updated successively until the criteria for an optimal control is met, and then the flow is advanced. The time window for which controls are predicted, TP , may be larger than the window for which the flow is advanced, TA . In the receding horizon framework TP > TA , whereas traditional instantaneous control, hereafter referred to as Full Horizon (FH) control, uses T = TA = TP .
9 American Institute of Aeronautics and Astronautics
Case FH FH RH FH FH RH FH FH RH FH FH RH
TA+ 6.3 25.5 6.3 12.6 25.5 12.6 18.9 25.5 18.9 12.6 25.5 12.6
TP+ 6.3 25.5 25.5 12.6 25.5 25.5 18.9 25.5 25.5 12.6 25.5 25.5
TA
/TP 1.00 1.00 0.25 1.00 1.00 0.50 1.00 1.00 0.75 1.00 1.00 0.50
Drag 0.788 0.775 0.699 0.793 0.775 0.695 0.766 0.775 0.731 0.630 0.430 0.431
2
Φ 0.0147 0.0311 0.0378 0.0313 0.0311 0.0333 0.0308 0.0311 0.0347 1.57E-2 1.72E-6 3.83E-6
RMS Drag 1.3
+ TA= 6.3 + T = 6.3 P
1.2
Normalized RMS Drag
J Jd Jd Jd Jd Jd Jd Jd Jd Jd Jt Jt Jt
TA+= 6.3 TP+= 25.5 D average = 0.6993
D average = 0.7881 1.1
TA+= 25.5 TP+= 25.5 D average = 0.7753
1
0.9
0.8
0.7
0.6 15
20
25
30
35
40
t+/100
45
50
55
60
Table 2: Comparison of average drag and control efFigure 11: In traditional optimal control with J d , fort for FH and RH cases for the distributed cost increasing the time window from T + = T + = 6.3 to A P functional, J d , and the terminal cost functional, J t . T + = T + = 25.5 caused drag reduction to improve A P from 17% to 22%. RH with TA+ = 6.3 and TP+ = 25.5 Results are presented here for the (4π, 2, 4π/3) tur- had a drag reduction of 26%. bulent channel with Reτ = 100. The RH framework is applied with two cost functionals: the terminal observation cost functional J t described by equation (6) that targets TKE at the end of the predictive win- The success of RH is less sensational with J t : using + + dow and the distributed observation cost functional TP = 25.5 and TA = 12.6, the control is still able J d that targets the average TKE over the predictive to relaminarize the flow, but at higher control cost. The average drag appears higher because the RH case window, requires slightly longer to relaminarize the flow and t0Z+TZ t0Z+TZ A A part of the transient is included in the average. 0 0 ` 2 u iu i dΩ dt. (8) J d (Φ) = Φ dΓ dt+ 2 2 The distribution of control effort over a few optit0 Γw t0 Ω mization windows is shown in Figure 12 for the FH To determine the impact of the relative size of the cases with TA+ = TP+ = 25.5 and 6.3 and the RH the advancement and predictive windows, ratios of case with TA+ = 6.3 and TP+ = 25.5. For FH conTA /TP of 1.0, 0.75, 0.50, and 0.25 are considered. trol effort diminishes toward the end of the window. For all RH cases, TP+ = 25.5. For RH, however, the diminishing control effort at As shown in Table 2, increasing the optimization the end of the predictive windows is truncated to window TA+ = TP+ for FH cases from 6.3 to 12.6 to begin a new time window. Consequently, the con18.9 to 25.5 generally improves drag reduction and re- trol is not required to cycle on and off, and the more quires additional control input. Although not shown uniform control distributions maintain more uniform here, the correlation between optimization window TKE and drag profiles which in turn require fewer size and TKE reduction is even stronger. For the high amplitude spikes of control effort to keep the RH cases that use J d , it can be seen that as TA is flow in check. We can summarize these results by reduced from TP , additional increases in drag reduc- drawing on Tom Bewley’s chess analogy:5 Just as tion are achieved. In general, control effort is larger in FH, longer time windows are more effective, the for RH cases than for comparable FH cases. Figure chess player who thinks furthest ahead in the game 11 shows drag histories for the first three rows of Ta- is most successful. Just as in RH, smaller TA /TP ble 2; the drag reduction of the RH scheme over the ratios are more effective, the chess player who reconsiders his strategy most often is most successful. comparable FH schemes is clear. For FH using J t , Table 2 shows that T + = 25.5 Finally, the success of RH with J d and the unimpresis able to relaminarize the flow, whereas T + = 12.6 sive performance of RH with J t can be reconciled, as is not. Once the flow is relaminarized, control ef- well: just as RH works better with some cost funcfort is minimal, so the two FH cases show four or- tionals than others, some chess strategies work better ders of magnitude spread in control effort required. together than others. 10 American Institute of Aeronautics and Astronautics
Normalized RMS Control
RMS Control
0.00 45.0
TA+= 25.5 TP+= 25.5
TA+= 6.3 TP+= 25.5
+ TA= 6.3 + TP= 6.3
45.2
45.4
t+/100
45.6
45.8
46.0
Figure 12: A cartoon of RMS control distribution over several time windows for TA+ = TP+ = 6.3 and TA+ = TP+ = 25.5 optimal control cases compared with the control distribution for the RH case with TA+ = 6.3 and TP+ = 25.5. Controls for the RH case are more regular and require a maximum amplitude equivalent to that of the short TA = TP case, but are more effective than those of the long TA = TP case. 6
Reduced order models and POD
Reduced order models hold the promise of capturing the essential physics of a given problem within a few degrees of freedom thus leading to more effective and less intensive predictive control strategies. The hybrid LES/DNS results presented in §4.3 demonstrate that LES can serve as an effective reduced order model for computationally intensive DNS simulations of turbulent channel flows. The success of this approach is partially due to the dynamic subgrid-scale model which has the ability to adjust the model coefficient according to the resolved scales of motion. This is critical for turbulence control simulations where the physics of the controlled flow may change dramatically over the course of the control, i.e. fully developed turbulence may be relaminarized. An essential aspect of a reduce order model for turbulence control requires that a model accurately represent both the uncontrolled and controlled dynamics. Although LES is an effective reduced order model, it is still relatively expensive so that further model reductions are desired. One option for further model reductions is to form a model based on a truncated basis arising from Proper Orthogonal Decomposition (POD). POD is a well-known technique for deriving an empirical basis optimal in the energy sense for any given dataset17 and the success of POD in representing the dynamics of near-wall flows has prompted the development of POD based reduced-order models
(see e.g. ref. 2,10,30,32,37) that could be used in the context of turbulence control.6, 24 A reduced-order model constructed from a subset of POD basis functions for one particular flow, often taken to be the uncontrolled flow, would certainly not be optimal and may not even successfully represent the dynamics of the controlled flow under some circumstances.6, 13, 21 This in turn calls for a modification to the POD basis that constitutes a reducedorder model, which if not done could reduce the effectiveness of the model. Hence it becomes imperative to first study the influence of typical controls on the very basis that we might employ to model our system. In order to ascertain the impact of control on the POD bases for turbulent channel flow, three different flows are examined: a flow with no control, a flow employing opposition control achieving 25% drag reduction and a flow with optimal control achieving up to 40% drag reduction. All the flows are at Reτ = 180. The channel dimensions in terms of wall units are + + L+ x = 2260, Ly = 360, and Lz = 753, where x, y, and z are the streamwise, wall normal, and spanwise directions, respectively. The streamwise and spanwise directions are homogeneous. Details of the LES simulations and POD analysis are described in Prabhu et al..31 For the purpose of this paper it suffices to state that POD allows a decomposition of the three different flow-fields uno (x, tn ) = uopp (x, tn ) = uopt (x, tn ) =
∞ X
no ano k (tn )φk (x)
(9)
opp aopp k (tn )φk (x)
(10)
k=1 ∞ X k=1 ∞ X
opt aopt k (tn )φk (x)
(11)
k=1
where u denotes the flow-field, the superscripts no, opp and opt stand for no-control, opposition-control and optimal-control respectively. Here, x is the vector of spatial coordinates, tn is the nth time instant, k is the kth POD eigenmode, ak (t) is the temporal eigenfunction, and φ(x) is the spatial eigenfunction. Since the streamwise and spanwise directions are homogeneous, the POD eigenfunctions are periodic along these directions and have the general form X
Nz /2−1
φ(x) =
X
Nx /2−1
ψ(nx , nz ; y)
nz =−Nz /2 nx =0
nx x nz z + exp 2πi Lx Lz
11 American Institute of Aeronautics and Astronautics
(12)
where nx and nz are Fourier series indices and Nx , Nz are the number of modes in the x and z directions. The eigenfunctions ψ are found by solving the integral equation
0.8
κ(nx , nz ; y, y 0 ) · ψ(nx , nz ; y)dy = λψ(nx , nz ; y)
0.6
κij (nx , nz ; y, y 0 ) = hˆ ui (nx , nz ; y)ˆ u†j (nx , nz ; y 0 )i . (14) where u ˆi is the ith component of the fluctuating velocity in Fourier space in x and z and hi stands for the ensemble average. In this expression, the complex conjugate is denoted by †. The kernel κ is Hermitian, non-negative and square integrable such that there exists a complete set of vector eigenfunctions ψ(nx , nz ) requiring an eigenfunction calculation for each Fourier mode pair (nx , nz ). In the discrete case, each eigenfunction calculation gives rise to set of 3Ny POD eigenmodes. Thus, a given POD eigenmode k comprises the triplet (nx , q, nz ) where q has been introduced to track the number of the eigenmode within each mode pair (nx , nz ). The eigenmodes are arranged in the order of their decreasing contribution to the energetics. Usually for low Reynolds number flows, the first few POD modes tend to capture a substantial portion of the total energy, which makes POD an optimal basis in the energy sense compared to any other decomposition such as Fourier, Chebyshev or the like. For turbulent channel flow, the most energetic modes are the ones which are streamwise independent i.e. modes with k = (0, q, nz ). These are called the roll modes.33 On the other hand, modes with streamwise dependence are called propagating modes since they resemble oblique waves propagating along the streamwise direction. Figure 13 shows a comparison of a typical roll mode with k = (0, 1, 3) for all the three cases. The eigenfunctions are complex in general, but for the roll modes the streamwise and wall-normal components are purely real and the spanwise component is purely imaginary. From Figure 13 one may note that control introduces a shear layer in the streamwise component (ψx ), a virtual wall in the wall-normal component (ψy ) and there are only slight changes in the near-wall features of the spanwise component (ψz ). For opposition-control, the virtual wall forms with slip along the spanwise and streamwise directions. In contrast, optimal control forms a virtual wall with slip only along the spanwise direction. The
x
Real(ψ )
0.4
(13) where λ is the eigenvalue that is equal to twice the kinetic energy in the particular POD mode. The cospectrum tensor κ has components
0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0
20
40
60
80
y
+
100
120
140
160
180
100
120
140
160
180
100
120
140
160
180
0.15
0.1
0.05 y
−δ
Real (ψ )
δ
0
−0.05
−0.1
−0.15
−0.2 0
20
40
60
80
+
y 0.1
0.05
0
Imag(ψz)
Z
1
−0.05
−0.1
−0.15
−0.2
−0.25 0
20
40
60
80
y+
Figure 13: Effect of control on the spatial eigenstructure for the roll mode (0, 1, 3): (a) Real(ψx ) component, (b) Real(ψy ) component, and (c) Imag(ψz ) no control; opposition concomponent. trol; optimal control.
12 American Institute of Aeronautics and Astronautics
0.4
amplitude of the shear layer is also stronger for the optimal control case. From the spatial eigenstructure φk (x), the temporal eigenfunctions are computed by projecting the flow-fields in the ensemble onto the spatial basis, ak (tn ) = φk (x), u∗ (x, tn ) , (15)
0.3 0.2
0.1
where the symbol (, ) stands for the L2 inner product computed over the entire spatial domain. The temporal eigenfunctions ak (t) for different POD modes k are uncorrelated and obey the relationship: (16) hal (tn )ak (tn )i = δlk λk
0 −0.1 −0.2 −0.3 −0.4 0
50
100
150
+
200
250
300
350
y
where l is a different POD mode. Once the φk (x) and ak (t) are computed, the statistics of the flow field can Figure 14: Comparing the model reconstructions of Reynolds stress using 300 modes. no-control be recovered using the following relationship: opposition control basis; POD babasis; ∞ X sis for optimal control; actual value. j i hui (x, t)uj (x, t)i = λk φk (x)φk (x) (17) k=1
or using equation (16) hui (x, t)uj (x, t)i =
∞ X
hak (tn )ak (tn )iφik (x)φjk (x) .
k=1
(18) This procedure allows us to study the effect of building a model of the optimal-control flow using the nocontrol or the opposition-control basis. The modelstatistics maybe computed using the following procedure: 1) Choose the spatial basis set, φmodel (x). In k opp no our case φmodel (x) could be either φ (x) or φ k k k (x). model 2) Compute the temporal eigenfunctions ak (t) by cross-projecting the flow field (in our case the flow field with optimal-control) onto the basis chosen in step 1. 3) Calculate the ensemble average for each of the temporal modes. According to (16), this yields the amount of energy extracted by the corresponding POD mode from the flow field. 4) Sum up the individual contributions as in (18) to arrive at the model statistics. An examination of the Reynolds stresses for the optimal-control flow-field as recovered by models based on the no-control and opposition-control bases highlights some of the important differences. Figure 14 shows the differences as seen by a superposition of 300 POD modes as calculated by the procedure just described. The thick line denotes the true Reynolds stress profile as calculated from the full dataset. The solid line denotes the Reynolds stress profile reconstructed using 300 POD modes from the original optimal control basis. The dashed line shows a similar reconstruction using the opposition-control
basis and the dotted line is the Reynolds stress profile employing the no-control basis. Control introduces an extra shear layer near the wall. This shear layer is bounded from above by a virtual-wall at a distance of roughly 15 wall units. The no-control basis fails to capture these features and substantially underpredicts the magnitude of the Reynolds stresses. As pointed out by Sirovich,33 the roll modes contribute most to the near wall stress and from Figure 14 it is clear that the no-control basis fails to capture the wall shear layer associated with the controlled roll modes. The opposition-control basis does exhibit a shear layer as well as the virtual wall, yet the amplitude of the shear layer and the location of the virtual wall are mis-predicted. Thus, a naive reduced order model based on a no-control POD basis may be unable to successfully predict the dynamics of a controlled flow. Prabhu et al.31 discuss the implications of this in a comparative study of POD models and emphasize the need to properly augment a given basis set so as to capture the important features of the controlled flow. 7
Conclusions
Given the recent successes in turbulence control based on Direct Numerical Simulation (DNS) and recognizing that the drag due to turbulence in a boundary layer is largely caused by the relatively large-scale coherent structures in the near-wall region, we have explored the use of Large Eddy Simulation (LES) as an approximate flow model for implementing and evaluating flow control strategies. After successfully eval-
13 American Institute of Aeronautics and Astronautics
uating our LES results with available DNS results for uncontrolled channel flow, our findings demonstrate that LES is a viable tool for obtaining quantitatively accurate results for drag reduction based on both opposition control and optimal control in turbulent channel flow using wall transpiration. In particular, our LES results for opposition control at Reτ = 590, which is the highest Reynolds number yet reported for opposition control, indicates that opposition control losses effectiveness as Reynolds number is increased. We successfully implemented optimal control of turbulent flow at Reτ = 100 and Reτ = 180 within an instantaneous control framework. Comparisons of LES results at Reτ = 100 with DNS results shows that agreement is good and that flow relaminarization can be achieved. These results also demonstrate that LES can be implemented much more efficiently than DNS. Our LES results for optimal control at Reτ = 180 show that relaminarization is more difficult to achieve at this supercritical Reynolds number. The drag and TKE saturate above the laminar value, thus yielding about 40% to 50% drag reduction. Additional increases in the optimization interval T , and/or improved optimization algorithms may be required to achieve relaminarization for this flow. Because LES is a reduced order model that does not represent the exact dynamic behavior of the real flow system, the reliability of the optimal control as obtained by LES is evaluated through a novel hybrid LES/DNS approach. With this method, the controls computed based on LES optimization are also found to be valid for DNS. We conclude that LES can be used as a reduced order model for optimal control of turbulence and this conclusions is shown to hold for even low resolution LES. As a generalization of the instantaneous control framework, receding horizon optimal control strategy was studied for different ratios of TA /TP , varying the time intervals for flow advancement TA for a fixed prediction time interval TP+ = 25.5. Our study reveals that the receding horizon approach yields better drag reduction compared to corresponding instantaneous control scheme for regularized TKE control while also providing a more uniform distribution of control effort. However, for terminal control of TKE, the effectiveness of is not as impressive even though the method does achieve results similar to standard instantaneous control although with slightly more control effort in this case. In order to further simplify the complexity of turbulence control simulations, good reduced order models are required. Proper orthogonal decomposition
(POD) is a powerful tool for dimension reduction and hence for reduced order modeling. We examine the assumptions of generality of the POD basis functions in the context of modeling turbulence control. It is found that merely employing a POD basis from a no control simulation in modeling controlled flows may not yield proper model behavior; there is a need to augment such POD bases to better represent the dynamics of the controlled flows. 8
Acknowledgment
The authors are grateful for helpful discussions with Prof. Thomas Bewley of UCSD and Prof. Matthias Heinkenschloss of Rice University. This research has been supported in part by the Texas Advanced Technology Program under Grant No. 003604-017. The computations in this paper were performed on a 16 processor SGI Origin 2000, which was partly funded by the NSF SCREMS grant DMS–9872009 and by the Los Alamos National Laboratory Computer Science Institute (LACSI) through LANL contract number 03891-99-23, as part of the prime contract (W-7405ENG-36) between the Department of Energy and the Regents of the University of California. S. Kellogg greatfully acknowledges the support of a NASA GSRP-Fellowship. References 1. K. Akselvoll and P. Moin. Large Eddy Simulation of Turbulent Confined Coannular Jets and Turbulent Flow Over A Backward Facing Step. PhD thesis, Stanford University, 1995. 2. N. Aubry, P. Holmes, J. Lumley, and E. Stone. The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech., 192:115–173, 1988. 3. E. Balaras, C. Benocci, and U. Piomelli. Finitedifference computations of high Reynolds number flows using the dynamic subgrid-scale model. Theoretical and Computational Fluid Dynamics, 7:207–216, 1995. 4. T. Bewley and P. Moin. Optimal control of turbulent channel flows. In Active Control of Vibration and Noise, ASME-DE, volume 75, 1994. 5. T. R. Bewley, P. Moin, and R. Teman. DNSbased predictive control of turbulence: an optimal target for feedback algorithms. Under preparation for submission to J. Fluid Mech., 1999. 6. P. N. Blossey. Drag Reduction in near wall turbulent flow. PhD thesis, Cornel University, 1999. 7. Y. Chang. Approximate Models for Optimal Control of Turbulent Channel Flow. PhD thesis, Rice University, 2000.
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8. Y. Chang and S. S. Collis. Active control of 21. K. Kunisch and S. Volkwein. Control of the burgturbulent channel flows based on large eddy ers equation by a reduced-order approach using simulation. In The Proceedings of the 1999 proper orthogonal decomposition. preprint, 1999. ASME/JSME Joint Fluids Engineering Confer22. C. Lee, J. Kim, and H. Choi. Suboptimal control ence, FEDSM99-6929, 1999. of turbulent channel flow for drag reduction. J. http://www.ruf.rice.edu/∼collis/papers/. Fluid Mech., 358:245–258, 1998. 9. H. Choi, P. Moin, and J. Kim. Active turbulence 23. D. K. Lilly. A proposed modification of the Gercontrol for drag reduction in wall-bounded flows. mano subgrid-scale closure method. Phys. Fluids J. Fluid Mech., 262:75–110, 1994. A, 4(3):633–635, 1992. 10. B. D. Coller, P. Holmes, and J. L. Lumley. Con- 24. J. Lumley and P. Blossey. Control of turbulence. trol of bursting in boundary layer models. Appl. Ann. Rev. Fluid Mech., 30:311–27, 1998. Mech. Rev., 6(2):S139–S143, 1994. 25. P. Moin and T. Bewley. Feedback control of turbulence. Appl. Mech. Rev., 47(6-2):S3–S13, 1994. 11. S. S. Collis and Y. Chang. Computer simulation of active control in complex 26. P. Moin and T. Bewley. Application of control turbulent flows. Modeling and Simulatheory to turbulence. In Twelth Australian Fluid tion Based Engineering, 1:851–856, 1998. Mechanics Conference, pages 109–117, Dec. 10– www.ruf.rice.edu/∼collis/papers/. 15 1995. 12. S. S. Collis and Y. Chang. On the use of LES with 27. R. Moser, J. Kim, and N. Mansour. Direct nua dynamic subgrid scale model for the optimal merical simulation of turbulent channel flow up control of wall bounded turbulence. In D. Knight to reτ = 590. Phys. Fluids, 11(4):943–945, 1999. and L. Sakell, editors, Recent Advances in DNS and LES, pages 99–110. Kluwer Academic Pub- 28. U. Piomelli. High Reynolds number calculations using the dynamic subgrid-scale stress model. lishers, 1999. Phys. Fluids A, 5(6), 1993. 13. J. Delville, L. Cordier, and J. Bennet. Large- 29. U. Piomelli and J. Liu. Large-eddy simulation of scale-structure identification and control in turrotating channel flows using a localized dynamic bulent shear flows. In M. Gad-el-hak, A. Pollard, model. Phys. Fluids, 7(4):839, April 1995. and J. Bonnet, editors, Flow Control: Fundamentals and Practices, pages 199–273. Springer, 1998. 30. B. Podvin and J. Lumley. Reconstructing the flow in the wall region from wall sensors. Phys. 14. M. Germano, U. Piomelli, P. Moin, and W. H. Fluids, 10(5):1182–1190, 1998. Cabot. A dynamic subgrid-scale eddy viscosity 31. R.D.Prabhu, S. Collis, and Y. Chang. The inmodel. Phys. Fluids A, 3(7):1760–1765, 1991. fluence of control on reduced order models of wall-bounded turbulent flows. to be submitted to 15. S. Ghosal, T. S. Lund, P. Moin, and K. AkPhysics of fluids, 2000. selvoll. A dynamic localization model for largeeddy simulation of turbulent flows. J. Fluid 32. D. Rempfer. Investigations of boundary layer Mech., 286:229–255, 1995. transition via galerkin projections on empirical eigenfunctions. Phys. Fluids, 8(1):175, 1996. 16. E. Hammond, T. Bewley, and P. Moin. Observed mechanisms for turbulence attenuation and en- 33. L. Sirovich. Dynamics of Coherent Structures in hancement in opposition-controlled wall-bounded Wall Bounded Turbulence. Computational Meflows. Physics Fluids, 10(9):2421–2423, 1998. chanics Publications, 1997. 17. P. Holmes, J. L. Lumley, and G. Berkooz. Turbu- 34. L. Sirovich, K. Ball, and R. Handler. Propagatlence, Coherent Structures, Dynamical Systems ing structures in wall-bounded turbulent flows. and Symmetry. Cambridge University Press, Theoretical and Computational Fluid Dynamics, 1996. 2:307–317, 1991. 18. J. Kim, P. Moin, and R. Moser. Turbulence 35. J. Smagorinsky. General circulation experiments with the primitive equations. I. The basic experstatistics in fully developed channel flow at low iment. Mon. Weather Rev, 91:99–165, 1963. Reynolds number. J. Fluid Mech., 177:133–166, 1987. 36. K. Sreenivasan. A unified view of the origin and morphology of the turbulent boundary layer 19. P. Koumoutsakos, T. Bewley, E. P. Hammond, structure. In H. Liepmann and R. Narasimha, and P. Moin. Feedback algorithms for turbulence editors, Turbulence Management and Relaminarcontrol–some recent developments. AIAA Paper ization, pages 37–61. Springer-Verlag, New York, 97-2008, 1997. 1988. 20. H. Kreplin and H. Eckelmann. Behavior of the 37. X. Zhou and L.Sirovich. Coherence and chaos three fluctuation velocity components in the wall in a model of turbulent boundary layer. Phys. region of a turbulent channel flow. Physics FluFluids A, 4(12):2855–74, 1992. ids, 22:1233, 1979. 15 American Institute of Aeronautics and Astronautics
