On the Frequency Scaling of the Forced Flow Above a Low Aspect Ratio Wing

Transcription

1 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition January 2012, Nashville, Tennessee AIAA On the Frequency Scaling of the Forced Flow Above a Low Aspect Ratio Wing Stefan Vey and Christian N. Nayeri and Christian O. Paschereit Institut für Strömungsmechanik und Technische Akustik, TU Berlin, Germany. David Greenblatt Technion - Israel Institute of Technology, Haifa, Israel. The flowfield dynamics of a low aspect ratio wing with leading edge flow control were investigated at post stall angles of attack. Load measurements and frequency scans were used to determine the optimum forcing frequency. A separation classification scheme was applied to the separated flowfield. Reynolds stress terms were computed from a large number of PIV snapshots and their spatial distribution was compared to standard cases from the literature. The turbulent kinetic energy was decomposed into a coherent and an incoherent component by using a POD-based method. The natural shedding frequency of the wing was determined through a temporal stability analysis of the wake velocity profiles. Nomenclature a real wavenumber c chord length [m] b wing span [m] t thickness [mm] AR aspect ratio α angle of attack [ ] ω complex frequency F + dimensionless forcing frequency F = F + sin(α) C L lift coefficient f forcing frequency [Hz] f ion ionization frequency [khz] P el normalized electric power [W/m] peak-to-peak voltage [kv] V p2p U freestream velocity [m/s] u, v, w instantaneous velocities [m/s] ũ, ṽ, w coherent velocities [m/s] u, v, w stochastic velocities [m/s] u, v, w mean velocities [m/s] u 2, v 2, w 2 Reynolds normal stresses [m 2 /s 2 ] u v Reynolds shear stress [m 2 /s 2 ] k turbulent kinetic energy [m 2 /s 2 ] l characteristic length [m] x, y, z spatial directions [m] Superscript stochastic periodic and stochastic I. Introduction The aerodynamics of low aspect ratio wings at low Reynolds numbers have only recently been studied in more detail. This is because of an increasing interest in small, autonomous flying vehicles known as micro air vehicles (MAVs). 1 Within the operation envelope of MAVs are phases of low speed flight, such as take off and landing and loitering. In these flight phases an increased lift coefficient is required. Active flow control can significantly increase the lift coefficient and shift the stall angle of attack to higher angles. Another application of flow control on MAVs is roll control and gust alleviation. These topics are also investigated within this project. PhD student, Institut für Strömungsmechanik und Technische Akustik, TU Berlin, Germany. Student Member AIAA. Research Assistant, Institut für Strömungsmechanik und Technische Akustik, TU Berlin, Germany. Senior Member AIAA. University Professor, Institut für Strömungsmechanik und Technische Akustik, TU Berlin, Germany. Senior Member AIAA. Senior Lecturer, Technion - Israel Institute of Technology, Haifa, Israel. Senior Member AIAA. 1 of 10 Copyright 2012 by Stefan Vey. Published by the, Inc., with permission.

2 II. Experimental Setup The flat plate wing (AR = 2.7, c = 0.15m, t = 3mm) was mounted vertically on a support arm in the center of the 1.4m 2.0m test section (see figure 1). The vertical mounting allowed using the wind tunnels turn table to adjust the angle of attack. A load cell measuring the normal force on the wing was incorporated into the support arm. A dielectric barrier discharge (DBD) actuator was attached to the wings blunt leading edge. The actuator was operated in pulsed mode with a duty cycle of dc = 20% at a peak-to-peak voltage of V p2p = 8kV and an ionization frequency of f ion = 4.75kHz. The power consumption of the actuator was measured to be approximately P el = 9W/m. 2 By further reducing the duty cycle the power consumption can be reduced. Further details on DBD actuators and their application in flow control can be found in. 3 7 The flowfield was captured by stereoscopic particle image velocimetry in a plane perpendicular to the wings upper surface at a spanwise station b/4 from the wings symmetry-plane as indicated in figure 1. The light sheet was generated by a high energy Nd:Yag laser with 170mJ/pulse. For every configuration 942 snapshotsweretakenand averagedto obtain the mean velocityfield. The Reynolds number wasre = 18000, corresponding to a freestream velocity of U = 2.09m/s. The freestream turbulence level was about %. Figure 1. Rectangular planform wing mounted vertically on support arm in the wind tunnel. III.A. Optimum Forcing Parameters III. Results Frequency scans were performed at a number of post stall angles of attack. The optimum reduced frequency F + (equation 1) was found to be approximately 0.24, whereas it would be expected to be in flow control applications. 8 F + = f c U (1) By scaling the frequency f with the transverse height of the wing (i. e.: c sin(α)) the frequency scan data for different angles of attack (figure 2) was found to be self similar for F < F opt, with F according to equation 2. 4 F = F + sin(α) (2) Thus the forcing frequency scales with the width of the wings wake. This holds true for frequencies smaller than the optimum forcing frequency. A similar effect was observed by Sigurdson, 9 who studied the drag 2 of 10

3 reduction due to forcing on a cylinder aligned coaxially with the flow. When he normalized the drag reduction data and scaled the forcing frequency with a length corresponding to the free-streamline height he found self similarity for frequencies smaller than the optimum frequency. The optimum reduced frequency was found to be in the range of 0.16 to 0.4. For higher frequencies a similar behaviour to the one seen in figure 2 was observed. The linear response for low frequencies can be explained by an increasing number of vortices being shed over the wing per time frame. For low frequencies these vortices do not interact. This results in a linear lift increase for low frequencies. When the frequency is increased beyond the optimum frequency the shedding vortices start to interact. It is assumed that vortex shedding then takes place within the shear layer emanating from the leading edge. This might explain why the data no longer scales with c sin(α). In this case the shear layer thickness might be the appropriate scaling length. max Figure 2. Frequency scan data for three post stall angles of attack at Re = Leading edge actuator running at 20% duty cycle. III.B. Flowfield Classification & Characterization A separated flowfield can be classified depending on the geometric arrangement of the shear layers. The classificationschemeof Leder 10 is usedtodescribe the flowfieldstructureabovethe lowaspectratiowing for the baseline and the control case. The two characteristic flowfield configurations are schematically depicted in figure 3, reproduced from, 10 p. 44. Class a (figure 3(a)) represents a single-sided shear layer spreading. This configuration is typical for airfoils at prestall angles of attack, backward-facing steps, fences, and bodies with splitter-plates in the wake region. A class b separation comprises flowfields with two-sided shear-layer spreading such as found on airfoils at post-stall angles of attack, cylinders, and flat plates perpendicular to the incoming flow. In this class the two separating shear layers can interact downstream of the body. A third class of separated flows can be introduced for axisymmetric configurations which are characterized by helical vortex structures as found behind disks and spheres. For generic flow configurations a class b separation has a pronounced shedding frequency and the Reynolds stress terms in the wake are an order of magnitude larger than in the case of a class a separation. The term generic configurations refers to backward facing steps and fences (class a) as well as cylinders and inclined flat plates (class b). The low aspect ratio wing investigated here involves active flow control and as such does not represent a generic configuation. The similarities and differences between the generic configurations and the low aspect ratio wing with flow control are discussed in the following. Before the baseline and control flowfields above the low aspect ratio wing are characterized according to the above separation classes some definitions are necessary. The Reynolds normal and shear stresses need to be regarded. These were calculated from a set of 942 PIV snapshots. The Reynolds normal stress in the freestream direction is defined as: u 2 = 1 N (u i u) 2 (3) N i=1 Where u is the mean velocity and u i is the instantaneous velocity-component in the freestream direction. The -superscript denotes that the variable includes periodic as well as stochastic components. Later the 3 of 10

4 (a) class a: one-sided shear layer, α < α stall (b) class b: two-sided shear layer, α > α stall Figure 3. Classification scheme for separated flows on airfoils. Reproduced from, 10 p. 44. velocity, Reynolds stresses, and kinetic energy terms will be decomposed into a mean, a periodic, and a stochastic component. Half the sum of the three Reynolds normal stresses yields the turbulent kinetic energy of the flow. ) k = 1/2 (u 2 +v 2 +w 2 (4) The Reynolds stresses and the kinetic energy terms are normalized with the squared freestream velocity U 2. According to this classification scheme the baseline flowfield above the low aspect ratio wing (figure 4) can be identified as a class b separation (figure 3(b)). The two shear layers can be discerned by the vorticity distribution (red and blue contours) overlayed over the lic-plot. The shear-layers originate from the leading and the trailing edge respectively. In the time mean flowfield two recirculating flow regions are formed. As it is typical for poststall angles of attack the flow is completely separated from the wing. Note the free stagnation point downstream of the trailing edge (marked with a white circle). The separation region forms a virtual body shape that reaches down to the stagnation point. Note the anisotropy in the distribution of the Reynolds normal stresses u 2 /U 2 and v 2 /U 2 in figure 6. The Reynolds stress component in freestream direction u 2 /U 2 is distributed along the developing shear layers (compare figures 6(a) and 4). It reaches a maximum value of 9 in the trailing edge shear layer downstream of the wing. A distinctively different distribution ( ) is observed for the transversal Reynolds stress component (figure 6(b)). The maximum value v 2 /U 2 = is reached at the downstream end of the separation region in the viscinity of the max stagnation point. The location of the stagnation point is marked by the blue spot in figure 6(b) (compare to lic visualization in figure 4). Experimental results reported in Leder 10 showed that the maximum values of the Reynolds normal stresses in the separation region of a cylinder are in the range of 0.25 for u 2 /U 2 and 0.35 for v 2 /U 2. The control flowfield features a one-sided shear-layer that originates from the leading edge (figure 5). It is therefore identified as a class a separation. In the time mean flowfield a separation region with a recirculation zone is formed on the suction side of the wing. The flow reattaches at approximately = 0.8. Due to the reattachment there is only a small velocity difference at the trailing edge between the pressure side flow and the suction side flow. Therefore the vorticity in this region is negligible. In contrast to the baseline case the Reynolds normal stresses are larger and the two components have a similar spatial distribution. Maximum values are reached within the separation ( region. As in the baseline ( case ) the transversal component is larger than the streamwise component ( u 2 /U )max 2 = 0.2 and v 2 /U 2 = 0.33 respectively). While the max streamwise Reynolds normal stress is restricted to the separation region the transversal component reaches into the wake behind the wing. The transversal component has two maxima in the viscinity of the wings upper surface, one at 0.4 and the second in the viscinity of the reattachment point at 0.7. For the generic configurations introduced above the Reynolds stresses of a class b separation are an order of magnitude higher than the Reynolds stress terms of a class a separation. 10 This is not the case for the low aspect ratio wing flowfield configurations. Comparing figures 6 and 7 an increase by a factor of almost two in the Reynolds normal stresses is observed when the actuator is turned on. 4 of 10

5 Figure 4. Line integral convolution (lic) of mean baseline velocity field. Flow from left to right, Re = 18000, α = 28. Yellow line represents the wing. The white circle denotes a stagnation point within the flowfield. Vorticity overlayed to emphasize the shear layers (red shading: negative, blue shading: positive). III.C. Identification of Coherent Structures Forcing the flow at the leading edge triggers the formation of spanwise vortices which are then shed into the wake. These vortices form coherent structures which carry a significant share of the turbulent kinetic energy in the flow. The Reynolds normal stresses discussed above consist of a coherent (ũ, ṽ, w) and an incoherent or stochastic velocity component (u, v, w ). In order to distinguish between coherent ( k) and incoherent (k ) kinetic energy content a proper orthogonal decomposition (POD) was performed on the PIV snapshots. The resulting POD spectra for the baseline and the control cases are shown in figure 8. In the control case (figure 8(b)) a substantial amount of energy is contained within the first two POD modes ( 10% and 12%). These two modes represent the shedding vortex structure that is triggered by the leading edge actuation. Using the phase reconstruction method described by Oberleithner et. al. 11 the coherent velocity components can be reconstructed. The decomposed energy components of the control flowfield are shown in figure 10. The coherent kinetic energy k is concentrated in a spatial region close to the wings upper surface with the maximum value located in the viscinity of the reattachment point at 0.7 (figure 10(a)). Moving away from the wing in the downstream direction k rapidly decays. The incoherent kinetic energy is concentrated further away from the wing within the separation region. There ) is a higher 2 level of incoherent kinetic energy downstream of the wing in the wake region. With ( k/u = 0.14 max the ( maximum coherent kinetic energy is only marginally smaller than the incoherent kinetic energy with k /U ) 2 = 0.2. max In the baseline case the first two POD modes contain a smaller amount of the total kinetic energy compared to the control case (figure 8(a)). The coherent kinetic energy is spatially concentrated around the stagnation point in the flowfield (figure 9(a)). Similarities to the spatial distribution of the transversal Reynolds normal stress are obvious (compare figures 9(a) and 6(b)). The two separated shear layers interact in this region. The distributions of the incoherent kinetic energy and the Reynolds normal stress in flow direction coincide (compare figures 9(a) and 6(a)). Note that the coherent kinetic energy is almost an order of magnitude smaller than the incoherent kinetic energy. The low levels of coherent kinetic energy suggest that vortex shedding does not occur at a clearly defined 5 of 10

6 Figure 5. Line integral convolution (lic) of mean control velocity field. Flow from left to right, Re = 18000, α = 28. Yellow line represents the wing. Vorticity overlayed to emphasize the shear layers (red shading: negative, blue shading: positive). frequency. Previous attempts to measure the natural shedding frequency with a hotwire probe did also not yield a clear shedding frequency. Presumably this is due to the three dimensional characteristics of the separation on the low aspect ratio wing. III.D. Stability Analysis of Wake Flow The question how the optimum forcing frequency is related to the natural shedding frequency can only be answered by performing a stability analysis on the baseflow. More specifically the growth rates of spatial or temporal modes can be determined by performing a spatial or temporal stability analysis respectively. Therefore the next step was to perform a temporal stability analysis of the wake velocity profiles. The goal was to determine the most amplified modes of the wake profiles of the flowfields. A Chebychev collocation method was used to set up the eigenvalue problem for the temporal stability analysis. The numerical implementation described in Oertel 12 was modified and applied to the wake flowfield. The velocity profiles were nondimensionalized by the freestream velocity and the transverse height c sin(α) of the wing. A least squares fitting function consisting of a superposition of four hyperbolic tangent functions was then fitted to the experimental data. A range of wavenumbers was given as an input and the eigenvalue problem was then solved for the complex frequency ω using a standard eigenvalue solver of the numerical Python library NumPy. The unstable eigenvalues correspond to a positive imaginary part of ω. The temporal stability analysis was performed for the wake downstream of the wings trailing edge. The cumulative temporal growth rates of certain dimensionless frequencies F for the baseline (left) and control (right) case over the streamwise position are shown in figure 11. The diagram can be interpreted as follows: The largest amplitudes in the flowfield at a certain downstream position are obtained when the flow is forced at the frequency where the local cumulative growth rate is largest. For the baseline case the maximum amplitudes are obtained at the natural frequency of F = 0.2. This is marked by the upper dashed line in figure 11(a). This is a typical dimensionless shedding frequency, or Strouhal number, for bluff body flows. The second, lower dashed line, marks the optimum forcing frequency F opt =. Note that at this frequency there is a reduced cumulative growth rate, visible through the 6 of 10

7 u 2 /U v 2 /U (a) Reynolds normal stress u 2 /U 2. (b) Reynolds normal stress v 2 /U 2. Figure 6. Normalized Reynolds normal stresses. Baseline case. Blue spot marks stagnation point in flowfield. Note anisotropic spatial distribution of the two Reynolds normal stress terms. Re = 50000, α = u 2 /U v 2 /U (a) Reynolds normal stress u 2 /U 2. (b) Reynolds normal stress v 2 /U 2. Figure 7. Normalized Reynolds normal stresses. Control case at F opt =, Re = 18000, α = 28. Note similar spatial distribution of the two Reynolds normal stress terms increased transversal component in the downstream wake. relative energy content mode number (a) Baseline case (no actuation). relative energy content mode number (b) Control case (actuation at F opt = ). Figure 8. Energy content over POD mode number for (a) baseline and (b) control case. 7 of 10

8 k/u k /U (a) Coherent kinetic energy (bl). (b) Incoherent kinetic energy (bl). Figure 9. Coherent (a) and incoherent (b) kinetic energy for the baseline case k/u k /U 2 (a) Coherent kinetic energy (ctl). (b) Incoherent kinetic energy (ctl). Figure 10. Coherent (a) and incoherent (b) kinetic energy for the control case. 8 of 10

9 deficit in the contours (best viewed at the intersection of the dashed F = line with contourlevel 16 at ). It is emphasized that for the data shown in figure 11(a) the actuator was turned off. When the stability analysis is performed for the forced flowfield there is a notch in the cumulative growth contours for the natural frequency (figure 11(b)). Note that the overall cumulative growth is smaller by almost an order of magnitude in the control case F ωi(,f ) F ωi(,f ) (a) Baseline case (no actuation). (b) Control case (actuation at F opt = ). Figure 11. Cumulative temporal growth rate for (a) baseline and (b) control case. Note the reduction of cumulative growth rate at the natural frequency F nat = 0.2 for the control case (b) and the slight reduction at the optimum forcing frequency F opt = in the baseline case (a). IV. Conclusion The frequency scaling of leading edge actuation of a low aspect ratio wing at post stall angle of attack was investigated. The dimensionless frquency was found to scale with the transverse height of the wing. Lift increase was maximised at a dimensionless frequency of Fopt =. A separation classification scheme was used to characterize the separated flowfield. Differences in the spatial distribution of the Reynolds stress terms were found to be in line with the separation classification scheme. The flow dynamics were investigated by a POD-approach and the mapping of the Reynolds stresses and kinetic energy content of flow. Significant coherent kinetic energy levels were found in the close viscinity of the wings upper surface in the control case. In the baseline case the level of coherent kinetic energy was found to be almost an order of magnitude smaller. A temporal stability analysis was performed on the wake flowfield in order to determine the natural shedding frequency of the baseline case. A natural shedding frequency of Fnat = 0.2 was found for the baseline case. When the stability analysis was performed for the control case a significant reduction of cumulative temporal growth rate was observed at the natural frequency. Acknowledgments This project was funded by the German Research Foundation (DFG). References 1 Mueller, T., Torres, G., and Srull, D., Introduction to the Design of Fixed-Wing Micro Air Vehicles, chap. 2, AIAA Education Series, 2006, pp Vey, S., Greenblatt, D., and Paschereit, C., LOW ASPECT RATIO WING FLOW CONTROL AT MAV REYNOLDS NUMBERS, 51st Israel Annual Conference on Aerospace Sciences, Tel Aviv & Haifa, Vey, S., Nayeri, C., Paschereit, C., and Greenblatt, D., Plasma Flow Control on Low Aspect Ratio Wings at Low Reynolds Numbers, 48th AIAA Aerospace Science Meeting, 4-7 January 2010, Orlando, FL, Vol. AIAA , Vey, S., Greenblatt, D., Nayeri, C., and Paschereit, C., Leading Edge and Wing Tip Flow Control on Low Aspect Ratio Wings, 40th Fluid Dynamics Conference and Exhibit, Chicago, Illinois, June 28- July 1, Vol. AIAA , Greenblatt, D., Kastantin, Y., Nayeri, C., and Paschereit, C., Delta-Wing Flow Control Using Dielectric Barrier Discharge Actuators, AIAA Journal, Vol. 46, No. 6, June 2008, pp , Technical Note. 6 Greenblatt, D., Goeksel, B., Rechenberg, I., Schuele, C., Romann, D., and Paschereit, C., Dielectric Barrier Discharge Flow Control at Very Low Flight Reynolds Numbers, AIAA Journal, Vol. 46, No. 6, 2008, pp of 10

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