Vortex Models for the Control of Stall

Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003

WeM12-5 Vortex Models for the Control of Stall A.C. Smith [email protected]

J. Baillieul [email protected]

Aerospace/Mechanical Engineering Boston University Boston, MA 02215

Abstract— A model-based control theory for boundary flows requires a model for the relevant fluid dynamics that accurately captures the physics involved, including interactions between the actuators and the system, while also being simple enough to run in real time. In the present work, we develop a simple model for simulating stalled flow past a flat plate. Additionally, we show that analysis of results provided by the model motivates a control scheme using active vortex generator jets.

I. I NTRODUCTION There are many reasons for interest in controlling separation of the boundary layer over an airfoil. Effective control of separation can improve lift, reduce drag, and lessen turbulent velocity fluctuations, leading to improvements in performance and safety. Previous experimental work has shown that the use of pulsed-jet actuators at the leading edge of an airfoil can exert significant control over the separation of flow at high angles of attack [8]. One promising avenue for creating a model of these separation dynamics is the use of point vortex methods. In these methods, vortices are treated as particles that influence each other in an otherwise irrotational flow. This paper presents an attempt to develop such a model and apply it to the problem of flow separation control. II. T HE MODEL A. Vortices Incompressible flow in two dimensions is governed by the continuity equation and the viscous Navier-Stokes equation: ∇ · U = 0, (1) 1 DU = − ∇P + ν∇2 U. (2) Dt ρ Operating with ∇× on eq. 2 and defining ωk = ∇×U gives the vorticity transport equation: Dω = ν∇2 ω, (3) Dt where the capital D indicates the material derivative. At high Reynolds numbers, momentum effects dominate viscosity effects, and hence the diffusion of vorticity can be neglected in favor of evaluating the convection of vorticity. Under these conditions, the vorticity can be discretized into a set of vortex particles that have convective effects on each other.

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The simplest implementation is to treat the vortex particles as point vortices, with a velocity profile of Γ 1 , vr = 0 (4) 2π r However, the existence of a pole at r=0 means that the differential equations of motion for point vortices will become stiff when the vortices are in close proximity, making point vortices problematic for numerical applications. Because of this, it is desirable to treat vortices in close proximity as blob-like distributions of vorticity, with velocity vθ =

r Γ , vr = 0 (5) 2π r2 + a2 where a is the “core radius” of the blob. Because vorticity blobs spread the vorticity over an area in the flowfield, it is impossible to formulate a potential function for them. However, it is possible to formulate a stream function for a vorticity blob: vθ =

ψ(r, θ) =

Γ r ln . 2π r2 + a2

(6)

In the present application, we treat the vortices as vorticity blobs, but approximate their interactions over large distances by treating them as point vortices. B. Flat Plate Our flat plate is divided into N equal-length line segments, or panels. A set of N −1 node points is placed at the dividing points between the panels, and one additional node point is placed at each end of the plate. Vortices are placed at the node points to represent the vorticity distributed along the surface of the plate. Since there are N + 1 node vortices, determining their strengths at any point in time requires N + 1 conditions. N of these conditions can be derived by enforcing a nonpenetration condition along the plate–requiring that the net mass flux through each of the n panels be zero. The final condition is that vorticity in the flowfield must be conserved. If the plate starts from rest in a zero-vorticity flowfield, then the total strength of the node vortices at any point in time will be the negative of the total free vorticity in the field. Because the flow near the surface of the plate will be parallel to the plate, the plate’s surface vorticity will only

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Fig. 1. The model used to implement a two-dimensional jet in our simulation. The source above the plate provides a mass influx, while the vortices keep the jet collimated.

single vortex is reduced from O(n) to O(log n), and thus the work required to evaluate the velocities of all vortices is O(n log n). The “Fast Multipole Method” of Greengard and Rokhlin [4][5] reduces computation time further by replacing a number of cluster-vortex interactions with a single clustercluster interaction. This is accomplished by shifting the center of a multipole expansion from a “source cluster” to a “target cluster”. The work required to compute the velocities of n vortices is reduced to O(n). Running on a 1.0 GHz Pentium III workstation, our implementation of the Fast Multipole Method requires approximately 50ms to compute the velocities of 2500 vortices. III. R ESULTS A. No jet

be convected away from the plate at its edges. The transfer of the plate’s surface vorticity into the free stream is handled by allowing the vorticity at the two end nodes to be convected away from the plate as free vortices at each time-step. These free vortices then persist through subsequent simulation timesteps. C. Jet Model We implement a jet by placing a point source just above the upper surface of the plate. A point source has stream function m θ (7) ψ(r, θ) = 2π and flow velocity m1 , vθ = 0, (8) 2π r where 2πm is the rate of fluid flow from the source. In order to keep the flow from the jet collimated, a pair of counterrotating vortices is released next to the jet at each time set. Over time, the successive vortex pairs behave like a pair of vortex sheets along the boundaries of the jet, as in fig. 1. vr =

D. Numerical Methods Direct evaluation of vortex motions for a system of n vortices requires O(n2 ) work. Our flows typically contain between 103 and 104 vortices, making such direct calculation prohibitively expensive. One method for reducing the computation burden of calculating vortex interactions is to employ a scheme adapted from Barnes and Hut [1] in which the square flow domain is partitioned into a tree hierarchy of nested square boxes. Some threshold number of vortices s is specified beforehand. Any box containing more than s vortices is partitioned in to four smaller boxes, and the process is repeated for each child box until all childless boxes have s or fewer vortices. Then, the influence of a distant cluster on a vortex can be approximated by using a multipole expansion. Using this technique, the work required to calculate the velocity of a

Figure 2 shows the development of flow past a flat plate at an angle of attack of 15 degrees, starting from rest. Initially, a large cluster of counterclockwise vorticity is shed from the trailing edge to satisfy the Kutta condition. Meanwhile, the clockwise vorticity shed from the leading edge forms a cluster that remains near the plate (image a). This cluster grows as more clockwise vorticity is shed from the leading edge, while a continuous stream of counterclockwise vorticity is shed from the trailing edge (b). Lift increases as the total circulation near the plate grows. At some point, the continued growth of the clockwise cluster above the plate causes the flow from the trailing edge to be swept back over the top of the plate. The counterclockwise vorticity being shed from the trailing edge then forms a cluster above the plate (c). Both clusters continue to grow as lift decreases. Eventually, the clockwise cluster will begin to move up and away from the plate (d). The clockwise vorticity being shed from the plate’s leading edge will no longer be fed into the large cluster; instead, it will form a new cluster near the plate. The large clusters will drift away together as the new cluster near the plate begins to grow, thereby causing lift to increase (e). This cycle repeats, causing lift to oscillate as pairs of vortex clusters grow and shed (fig. 3). The development of the flow can be understood by examining the development of streamlines, which are shown schematically in fig. 4. Of particular importance is the streamline that includes the plate surface. The evolution of the flow during each shedding cycle can be divided into three periods (marked with capital letters in the figure), with a critical event (marked with numbers) marking the transition from one period to the next. During each of the three periods, the flow will evolve but the topology of the plate-surface streamline will remain unchanged. This topology will only shift at the three critical events. Two of these critical events (1,2) are the separation of flow stagnation points from the surface of the plate. The third (3) is the passing of the plate-surface streamline through the rear stagnation point (which is separated from the plate

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AoA = 15 degrees; VR = 1.0; Jet at 20% chord

(a)

3 Jet off

Lift

2

1

0 3 Jet on

(b) Lift

2

1

0 300

320

340

360

380

400

Time

Fig. 3. Lift over time for a flat plate at a 15 degree angle of attack, with and without a steady jet.

(c)

(d)

(e)

Fig. 2. Development of flow past a flat plate in stall starting from rest. Dark areas represent clockwise vorticity; light areas represent counterclockwise vorticity. The superimposed arrows show the flow direction. The bulk flow is moving left to right.

at that point). The qualitative dynamics will only change when the flow topology changes at the critical events. Thus. understanding the nature of these critical events is the key to understanding the development of a stalled flow. During the first phase (A), the circulation bubble above the plate grows, causing lift to increase until the stagnation point at the trailing edge of the plate separates from the surface, creating a counterclockwise circulation bubble at the trailing edge (1). Lift then decreases in the second phase (B), as both the clockwise and counterclockwise vorticity clusters grow. The growth of the clockwise cluster ceases when the stagnation point on the plate’s upper surface separates (2), causing the vorticity being shed from the plate’s leading edge to form a new cluster (C). However, the counterclockwise cluster continues to grow and lift continues to decrease until the streamline leaving the trailing edge of the plate passes through the rear stagnation point (3). After this point, the vorticity being shed from the plate’s trailing edge is no longer feeding into the counterclockwise cluster. Lift begins to increase as the two clusters drift away from the plate. A new cluster of clockwise vorticity grows above the plate, setting the stage for another cycle (A). Figure 5 shows a portion of the lift curve from the simulation, and several snapshots of the flow streamlines from the simulation. The images show the three critical points at which the flow topology changes, as illustrated in fig. 4. Note especially the changes in the lift curve at the first and third critical points. The effect of these changes in flow topology on the lift can be understood by examining the path of the plate-surface streamline. Lift is generated when the flow near the plate

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Fig. 5. Lift over time for a flat plate at stall. The numbered images below the plot show the flowfield at the indicated points on the plot. In image (1), the rear stagnation point (marked by a black dot) has separated from the plate, causing lift to begin to fall. In image (2), a second saddle point has separated from the leading edge of the plate.In image (3), the streamline of the flow coming from the plate’s trailing edge has passed below the rear saddle point, causing the lift to begin to rise. Fig. 4. Streamlines during stall vorticity shedding. The labels show increasing (inc.) and decreasing (dec.) lift. Grey circles show separated stagnation points. Thick dotted lines show the streamline that includes the plate surface. 2410

deflects this streamline downward. During the “A” phase, the growth of the clockwise vorticity cluster causes the total circulation in the neighborhood of the plate to increase, which deflects the streamline leaving the trailing edge further downward, causing lift to increase. After the trailing stagnation point separates from the plate (first critical event), the streamline leaving the plate surface begins to rise in the “B” phase, always remaining above the stagnation point. The elevation of this streamline causes lift to decrease. This process continues through the separation of the forward stagnation point (second critical event) into the “C” phase, when the streamline curves around both vortex clusters. As the clusters leave the plate, the plate-surface streamline continues to elevate, and lift continues to decrease, until the streamline passes through the trailing stagnation point (third critical event). Thereafter, the streamline passes under the stagnation point as the flow re-enters the “A” phase. As the vortex clusters continue to move away from the plate, the streamline leaving the plate’s trailing edge relaxes downward, causing lift to increase. Since the “B” and “C” phases of the flow development are associated with decreases in lift, an effective control strategy would be to shorten one or both of these phases by forcing either the second or third critical event to occur earlier. Since the second critical event–the separation of the forward stagnation point–occurs at the plate surface, it’s an appealing idea to attempt to hasten it through the use of some kind of flow actuator on the upper surface of the plate.

2

1

Lift

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B. Steady jet A steady jet blowing from the upper surface of the plate has exactly the desired effect. Figure 6 shows a portion of the lift-time curve for a flat plate with such a jet, along with corresponding snapshots of the flow streamlines. The primary difference between the flow evolution with the jet and without the jet is that the jet causes the second critical event, the separation of the leading stagnation point from the plate surface (image 2) to occur earlier. This is apparent in fig. 6, in which the “B” period is shortened considerably in comparison to fig. 5. The effect of the early separation of the stagnation point is to arrest the growth of the clockwise vorticity cluster above the plate. As a result, both the clockwise and counterclockwise clusters are smaller, and the pair of clusters separates more quickly from the plate. This, in turn, allows lift to be re-established more quickly. The faster shedding rate and smaller cluster size have the effect of reducing the amplitude and increasing the frequency of the oscillations in lift (fig. 3). The increase in lift caused by the jet varies with the location of the jet on the plate. Simulations show that the optimal location for the jet is about 20% of the plate length from the leading edge (fig. 7).

3

Fig. 6. Lift over time for a flat plate at stall with a steady jet. The jet is located on the upper surface of the plate, 0.2 plate lengths from the leading edge, has a width of 0.01 plate lengths, and the fluid exits at a velocity equal to the freestream velocity. The numbered images correspond to the indicated points on the lift-time plot. The major effect of the jet is to shorten the “B” phase of decreasing lift by causing the forward stagnation point to separate from the plate earlier.

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Steady jet, VR = 1.0

Ultimately, however, a closed-loop control system is likely to provide the greatest improvement in lift and the greatest robustness with regard to changing flow conditions. Ideally, such a system would use sensors on the plate to detect transitions between the various periods of the flow development and activate the jet at appropriate times.

1.4

1.2

Lift

(no jet)

ACKNOWLEDGEMENT The authors gratefully acknowledge the assistance of Prof. Victor Yakhot in sharing his thoughts and insights on this research.

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V. REFERENCES

0.8

0.6

0

0.1 0.2 0.3 Jet location (fraction of chord)

0.4

Fig. 7. Average lift for various locations of the jet along the plate. The width of the jet is 1% of the plate length, and the jet velocity is equal to freestream velocity.

IV. F UTURE W ORK A. Quantitative evaluation While our work to date provides a strong qualitative description of the flow development during stall and a qualitative explanation of the effect of the jet actuator, it is desirable to develop a quantitative method for describing and evaluating stalled flow patterns. Ideally, such a quantitative evaluation of the flow would lead to a representation of the flow as a small number of ordinary differential equations that could be evaluated in real-time and serve as the basis for a model-based control of the system. B. Control It’s clear that the blowing of the jet actuator during the “B” period of the flow evolution (after separation of the rear stagnation point, but before separation of the forward stagnation point) serves to speed the separation of the forward stagnation point. However, it’s unclear whether the jet serves any useful purpose during the “A” or “C” phases of the flow development. Indeed, it seems likely that during the “A” period especially, the jet has little or no effect on the flow at all. Since a physical implementation of this system would require energy to activate the jet, it would be desirable to implement a control scheme that limits the jet actuation to the periods when it’s most useful. Experimental work with pulsed-jet actuators on an airfoil has shown a pulsed jet turning on and off at the wing’s natural shedding frequency can provide a substantial improvement in lift over a steady jet [8]. Presumably the wing’s shedding cycle phase-locks to the jet actuation in such a way that the jet is blowing at the most useful point in the shedding cycle.

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