Engineering Approaches for Active Flow Control Simulation

S. Palaniswamy, U. Goldberg, and S. Chakravarthy

Metacomp Technologies, Inc., Agoura Hills, CA 91301

Current practice in flow control simulation often entails unsteady calculation of jet actuators, resolving their geometries in detail. This approach requires: (a) fine grids to resolve flow inside and outside actuators; (b) many time-steps to resolve high-frequency actuator flow-field; (c) many more time-steps to resolve multiple time-scales if the main flow is also unsteady. This method is very time-consuming and renders the current approach impractical from an engineering standpoint. The paper addresses these issues and suggests strategies that enable active flow control simulation within engineering design cycles.

Nomenclature Ko = doublet’s pulsation amplitude e = total energy per unit volume p = pressure r = radial distance from origin S = doublet’s strength T = time u, v = mean velocity components x0, y0 = doublet’s origin x, y = Cartesian coordinates α = angle-of-attack γ = ratio of specific heats ϕ = doublet’s orientation ρ = density ω = actuator’s pulsation frequency

I. Introduction Active flow control has been increasingly utilized by aeronautical engineers to augment efficiency through

modification of flow behavior. Examples are maintaining laminar flow over wings to reduce drag, projectile maneuvering with pulsating jets and flow separation control to increase wing L/D at high angles-of-attack. In the past few years pulsating synthetic jets have emerged as a promising approach to active flow control due to their high efficiency/cost ratio. Work by Smith et al1, Amitay et al2 and Honohan et al3 has provided much experimental insight into aerodynamic flow control using synthetic jet actuators. More recently CFD tools have been used to predict active flow control effects. Parekh et al4 demonstrated the ability of unsteady CFD to predict experimentally observed suppression of flow separation from airfoils at high angles-of-attack using synthetic jets. However, time-accurate CFD analysis of turbulent flow, inherently 3D, is very time consuming, especially when the capture of flow details requires LES or hybrid RANS/LES methodology. This prevents traditional CFD approaches from fitting into the often tight schedules necessary in engineering analysis and design cycles. In order to enable CFD to be an engineering tool for flow control analysis and design, simplifications to the prevailing approach must be introduced. The work presented here describes two viable simplifications; one is still based on unsteady flow whereas the other introduces a steady flow approach: The first simplification consists of the following attributes:

1. Jet actuators are replaced with momentum source terms in time-dependent doublet form, retaining original actuator frequency

1 American Institute of Aeronautics and Astronautics

3rd AIAA Flow Control Conference5 - 8 June 2006, San Francisco, California

AIAA 2006-3681

Copyright © 2006 by Metacomp Technologies, Inc. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

2. Doublets are placed in locations of original actuators 3. Mesh complexity required for original actuators is eliminated 4. Amplitude of doublet pulsation must be calibrated based on experimental data

The second simplification consists of the following attributes: 1. Introduce steady-state momentum source terms representing unsteady doublets averaged over a cycle 2. This eliminates the need to resolve high-frequency actuator action 3. Only low frequency main flow unsteadiness (if it exists) needs to be resolved 4. This renders the method a true engineering tool

These two approaches are presented in the paper and illustrated through several examples.

II. Methodology Current practice in flow control simulation is based on unsteady calculation of detailed jet actuator geometries.

This approach requires: 1. Fine grids to resolve flow inside and outside actuators 2. Many time-steps to resolve high-frequency actuator flow-field 3. Many more time-steps to resolve multiple time-scales if main flow is also unsteady

This method is very time-consuming and renders the current approach impractical from an engineering standpoint.

The paper addresses these issues and suggests strategies that enable active flow control simulation within engineering design cycles. The proposed approaches are shown schematically in Fig. 1:

Figure 1. Simplified approaches to jet actuator flow control modeling. (a) The first simplification consists of the following attributes:

1. Jet actuator is replaced with momentum source terms in time-dependent doublet form, retaining original actuator frequency

2. Doublets are placed in locations of original actuators 3. Mesh complexity required for original actuators is thus eliminated 4. Amplitude of doublet pulsation must be calibrated based on experimental data

Fig. 2 shows the relationship between original actuator orientation and that of the doublet axis.

2 American Institute of Aeronautics and Astronautics

Figure 2. Relation between actuator and doublet orientations. The doublet gives rise to momentum fluxes given (in 2D) by:

( ) ( )[ ]

( )tcosKK

,sinyycosxxrKS

,ryy

Stv,

rxx

Stu

0

003

00

ωω=

ϕ−+ϕ−−=

−=

∂∂−

=∂∂

&

&

Figure 3. Basic jet actuator nomenclature (φ=doublet orientation). Referring to Fig. 3, the following relationships exist between ω, K and the commonly used parameters F+ and Cµ:

3 American Institute of Aeronautics and Astronautics

2jj

uu

cbC

,u

cF

ρ

ρ=

ω=

∞∞µ

∞

+

where

( ) ( )[ ] ( )( ) .)2cos(yyxx)2sin(yyxx21

rKu 00

20

204j

ϕ−−−ϕ−−−=

II(a). Results The following example demonstrates the performance of the unsteady doublet in modeling a jet actuator

reattaching a massively separated flow over the AVIA3 airfoil. Here U∞=18.5 m/s and α=17.5o. As seen in Fig. 4 the baseline flow is massively separated; it becomes fully attached once the doublet is activated with ϕ=30o, ω=4576 s-1 and K0=10-10. The doublet’s orientation and frequency correspond to those of the original jet actuator, whereas its amplitude is iterated upon to match experimental pressure data, shown in Fig. 5, corresponding to an active jet actuator.

Figure 4. AVIA airfoil (L) baseline, (R) doublet activated.

Figure 5. Effect of doublet strength and orientation on Cp profile.

4 American Institute of Aeronautics and Astronautics

The pressure side leading edge data mismatch, observed in Fig. 5, is due to the doublet’s shadow effect, discussed later.

Unlike the onset of reattachment, which occurs 15 cycles after source activation, it takes on the order of 100 cycles for the flow to completely re-separate after source deactivation. This hysteresis was observed experimentally (D. Parekh, private communication).

Fig. 6 compares Cp distributions predicted using three turbulence models5,6 among those available in CFD++7, the solver used in this effort. It is evident that the variation among the models is relatively small.

Figure 6. Cp distributions using three turbulence models.

The next case involves an oscillating airfoil to mimic a helicopter rotor. Fig. 7 depicts schematically the change in effective angle-of-attack due to increased downwash at the aft portion of the rotor relative to the front part. Fig. 8 shows the equivalent effect modeled by an oscillating airfoil. The pivot is chosen at the jet actuator location (represented by the doublet) so that it remains stationary during the oscillations.

Figure 8. Equivalent airfoil pitching.

Figure 7. Rotor blade fore-aft pitching (W – downwash).

5 American Institute of Aeronautics and Astronautics

A realistic free-stream speed of U∞=180 m/s was chosen. The angle-of-attack was set to α=12º and the

oscillation frequency to ω=5Hz with an amplitude of 0.17c. Fig. 9 is a snapshot without flow control whereas Fig. 10 is with active control. The massively separated baseline flow reattaches (except for a few isolated separation bubbles) once the doublet is activated. This reduces drag while increasing lift, as seen in Fig. 11.

Figure 9. Oscillating airfoil without flow control.

Figure 10. Same airfoil with flow control.

Figure 11. Increased L/D of oscillating airfoil due to flow control. (b) The second simplification consists of the following attributes:

1. Introduce steady-state momentum source terms representing unsteady doublet averaged over a cycle 2. This eliminates the need to resolve high-frequency actuator action 3. Only low frequency main flow unsteadiness needs to be resolved (e.g. helicopter rotor) 4. It renders the method a true engineering tool

Derivation of the momentum source terms is as follows. From the steady-state 2D Euler equations:

( ).vu21

pe,

pvuvv

v)pe(

g,

uvpu

uu)pe(

f

)1(,0gf

22

2

2

yx

+ρ

+−γ

=

+ρρρ+

=

ρ+ρ

ρ+

=

=+

Assuming ρ and p remain constant within a cycle:

( )

( )

( ) .

2

)(

,2

)()(

+

+++

≅

+

+++

≅

y

yy

y

yyy

y

xx

x

x

xxx

x

vvvuuv

vvvuuvvpe

g

vuuvuuu

vvuuuupe

f

ρρ

ρρ

ρρρρ

6 American Institute of Aeronautics and Astronautics

From 2D doublet:

( ) ( ) ( ) ( ) ϕ−+ϕ−=ηη−−=η−−= sinyycosxx,yyrKv,xx

rKu 000404

Thus, for example,

( ) ( )[ ]

( ) ( ) ( )[ ] .cosxxxxr4xx

rKuu

,cosxxxxr4

rKu

02

020

2

4x

02

024x

ϕ−+η−−ηη−

−=

ϕ−+η−−η=

Averaging over one oscillation cycle:

t.t̂,2

ˆ21 2

02

0

22 ωπ

π

=== ∫K

tdKK

The integrals over a cycle of all odd powers of sine and cosine vanish, e.g., ∫ =π

ψψ2

0

3 .0sin d

Thus, only momentum components remain since they involve even powers of sine and cosine. For example,

[ ]

[ ] .cos)0(22)0(28)0(8

2

,cos)0(2)0(24)0(8

2

−+−−−−=+

−+−−−−=

ϕηηη

ϕηηη

xxxxr

yyrK

xvuxuv

xxxxr

xxrK

xuu

Finally, the averaged Eq. (1) becomes:

.

)(23

)(23

00

208

20

208

20

−−

−−=+

ηρ

ηρ

yyr

K

xxr

Kgf yx

These are the extra RANS momentum source terms representing the time-averaged doublet.

II(b). Results Figures 12 and 13 compare time-averaged unsteady doublet performance with that of the steady-state (S-S) momentum sources with zero free-stream velocity. The resulting flow-fields are practically identical.

7 American Institute of Aeronautics and Astronautics

Figure 12. Velocity field induced by doublet (L) and by S-S source (R).

Figure 13. Turbulence field induced by doublet (L) and by S-S source (R).

Fig. 14 compares AVIA3 airfoil results under unsteady doublet and steady-state momentum sources. The normally massively separated flow is fully attached by either method.

Figure 14. Flowfield induced by doublet (after 50 cycles, L) and by S-S sources (R). The corresponding pressure distribution comparison is seen in Fig. 15. The time-averaged Cp profile produced by the doublet is close to the one by the steady-state momentum sources.

8 American Institute of Aeronautics and Astronautics

Figure 15. Comparison of Cp profiles with unsteady doublet and steady-state momentum sources.

As a further example, the S-S momentum sources were applied to a realistic airfoil section under the following flow conditions: M∞=0.3 and α=15o. Fig. 16 demonstrates the effect of the source terms in reattaching the massively separated baseline flow-field.

Figure 16. M∞=0.3, α=15o baseline (L) and after S-S source activation (R).

III. Comments Regarding Future Work

A word concerning shadow regions: Fig. 17 shows schematically influence regions induced by two doublets (or their equivalent S-S momentum sources). Besides the near-blade areas where the influence of the sources is sought, there are other areas (e.g. around the fuselage) where such influence is undesirable. It is necessary to modify the analytical treatment of these sources so as to limit their influence where it is most beneficial. Such modifications will be the subject of future work.

9 American Institute of Aeronautics and Astronautics

Figure 17. Influence regions of source potentials.

IV. Concluding Remarks Two flow control simulation methods were proposed and successfully tested: 1. Replacing jet actuators with momentum source terms in time-dependent doublet form, and 2. Introducing steady-state momentum source terms representing unsteady doublet averaged over a cycle.

Advantages of the proposed methods include:

1. Unsteady doublet enables accurate representation of jet actuators without the need to grid up and solve detailed actuator flow-field

2. Steady-state momentum source terms represent cycle-averaged doublet action, eliminating the need to resolve high-frequency unsteadiness

3. With S-S sources only low-frequency main flow unsteadiness requires resolving, leading to relatively large time-steps, hence fast turnaround times

4. Both doublet & S-S sources were validated against available experimental data

References 1. D.R. Smith, M. Amitay, V. Kibens, D. Parekh, and A. Glezer, “Modification of Lifting Body

Aerodynamics using Synthetic Jet Actuators,” AIAA Paper 98-0209, 1998. 2. M. Amitay, V. Kibens, D. Parekh, and A. Glezer, “The Dynamics of Flow Reattachment over a Thick

Airfoil Controlled by Synthetic Jet Actuators,” AIAA Paper 99-1001, 37th AIAA Aerospace Sciences Meeting, Reno, NV, January 1999.

3. A.M. Honohan, M. Amitay, and A. Glezer, “Aerodynamic Control using Synthetic Jets,” AIAA Paper 2000-2401, FLUIDS 2000 Conference, Denver, CO, June 2000.

4. D.E. Parekh, S. Palaniswamy, and U. Goldberg, “Numerical Simulation of Separation Control via Synthetic Jets,” AIAA Paper 2002-3167, 1st Flow Control Conference, St. Louis, MO, June 2002.

5. U. Goldberg, “Turbulence Closure with a Topography-Parameter-Free Single Equation Model,” IJCFD, 17, No. 1, pp. 27-38, 2003.

6. U. Goldberg, and S. Chakravarthy, “A k-ℓ Turbulence Closure for Wall-Bounded Flows,” AIAA Paper 2005-4638, 35th AIAA Fluid Dynamics Conference, Toronto, Canada, June 6-9, 2005.

7. O. Peroomian, S. Chakravarthy, S. Palaniswamy, and U. Goldberg "Convergence Acceleration for Unified-Grid Formulation using Preconditioned Implicit Relaxation," AIAA Paper No. 98-0116, Reno 1998.

10 American Institute of Aeronautics and Astronautics