Experimental time resolved flow features of separation over...

Experimental time resolved flow features of separation over an elliptic leading edge

Daniel R. Morse1 and James A. Liburdy2

Mechanical Engineering Oregon State University, Corvallis, OR 97331, USA

This study investigates the flow over a flat airfoil of low aspect ratio using experimental, time-resolved particle image velocimetry methods. The flow conditions are characterized by low Reynolds flow at two relatively high angles of attack, 15° and 20°. A high pass Gaussian filter was employed to help identify structures within the flow field. Also, a velocity gradient tensor method, λ2, was used to visualize vortical structures within the flow. The Proper Orthogonal Decomposition was applied to analyze primary features within the flow. A few, high energy modes are found which recreate the process of vortex shedding for both angles of attack.

Nomenclature ak(t) = amplitude time series function for mode k d(x,t) = function representing all velocity data points over all time steps D = NxM matrix of velocity at N locations and M times fk(x) = spatial eigenfunction of velocity for mode k G = 2x2 matrix of 2D velocity vector gradient α = angle of attack λ2 = vortex detection parameter, second eigenvalue of G λk = total variance of amplitude time series for mode k Σλk = cumulative modal variance up through mode k Σλtot = cumulative modal variance of all modes

I. Introduction There is a growing interest in the aerodynamic flow characteristics of low Reynolds number flight with the

development of unmanned aircraft and micro-air vehicles. Generally these types of vehicles have small characteristic lengths and low velocities. In addition, understanding the flight phenomena of many insects and animals requires a greater knowledge of low Reynolds number conditions. Of great interest is the separation process at low Reynolds numbers for unconventional airfoil shapes. It is known that separation from airfoils can be a complex process that is not defined by a single separation point in space and time. The nature of the flow instability leading to separation tends to include large scale flow structures with fairly low frequency events that can have high amplitudes. Understanding this process is important for flight control and overall stability.

There is a growing body of knowledge on the lift and drag characteristics of low Reynolds number airfoils. For example Shyy et al. [1999] examined both rigid and flexible airfoils, Selig and Guglielmo [1997] showed that high lift conditions could be obtained using a concave pressure recovery and aft loading. Gopalarathnam et al. [2003] applied boundary layer trips coupled with transition ramps in the geometric design. They showed low Reynolds number performance characteristics resulting in higher lift to drag ratios when operating near the maximum lift coefficient condition. Within the lower lift regime the performance enhancement is not apparent. Pelletier and Mueller [2000], used small aspect ratio wings with and without camber. The cambered design was shown to obtain higher lifting conditions.

The unsteady flow phenomena and corresponding lift in a static stall condition has been studied by Sicot et al. [2006] using local pressure measurements. They identify the three separation conditions based on the fraction of

1 Ph. D. candidate in Mechanical Engineering, 204 Rogers Hall, Corvallis OR, 97331 2 Professor of Mechanical Engineering, 204 Rogers Hall, Corvallis OR, 97331

1 American Institute of Aeronautics and Astronautics

time a particular location on the surface experiences detachment. These were identified by Simpson et al. [1981] as incipient detachment, transitory detachment and detachment each with an increasing fraction of time of detached flow (1% and 50% backflow occurrence, and time averaged wall stress equal to zero, respectively). Hoarau et al. [2003] studied unsteady separation on a wing and show a link between the separation and von Karman instability conditions. This is consistent with flow over a cylinder with an oscillating separation point, Nishimura and Taniike [2001]. Sicot et al. [2006] present PIV data showing the instantaneous vorticity field with a strong Kelvin-Helmholtz instability originating near the leading edge of a wing during stall.

The desire to better understand the detailed flow phenomena and their evolution in transient conditions leads to a need to identify data processing techniques that reveal qualitative and quantitative measures of flow events. In the leading edge separation resulting in a Kelvin-Helmholtz instability large scale vertical structures occur. Different vortex detection schemes have emerged whose success may be linked to the particular basic flow to which it is applied. One such method proposed by Jeong and Hussain [1995] was the association of a local pressure minimum with the velocity vector field using the eigenvalue of the gradient of the Navier Stokes equations. Several identification methods were discussed by Adrian et. al. [2000]. Another method was the characterization of a vortex by the swirl over a small area proposed by Graftieaux et. al. [2001]. Weiland and Vlachos [2007] used the Proper Orthogonal Decompostion (POD) of Time Resolved Particle Image Velocimetry (TRPIV) data to elucidate the high energy modes associated with an airfoil with leading edge blowing in the wake of a cylinder.

In this study the flow characteristics over a fixed surface, flat, low aspect ratio thin wing is investigated. Of interest is the dynamic separation process for a range of angle of attacks, for a given chord Reynolds number, particularly the time dependent nature of the vortex development, convection and interactions. Using discrete vortex detection schemes coupled with a high pass filtering and Proper Orthogonal Decompostion analysis, the time dependent characteristics of this instability layer is elucidated.

II. Experimental method The PIV data were collected in a throughpass wind tunnel with test region cross section measuring

approximately 30cm x 30cm. The seeded flow was illuminated using laser light generated by a New Wave Research Pegasus Laser at wavelength 532 nm. A laser light sheet with a width of 1 mm was generated by a Dantec 9080x0651 light sheet optical module and used to illuminate a vertical plane in the center span of the airfoil. A full diagram of the experimental setup is shown in Figure 1. It should be noted that the imaging field of view occupied only the first quarter of the airfoil chord.

Seed for the flow was generated using a Laskin nozzle operating at 10 psi to atomize the working liquid, Canola oil. Particle size distribution was centered at 6 µm. The particle response time was calculated from the Stokes flow drag model described by Hinze [1959]. A two time constant estimation yields an upper frequency limit of 5 kHz.

The airfoil was a flat plate of span and chord dimensions of 10.25 cm by 20.5 cm, respectively, giving an aspect ratio of 0.5. The leading and trailing edges were shaped as an ellipsoid with length to thickness ratio of 5:1; the airfoil side edges were rounded. The thickness of the airfoil was approximately 2% of the chord length, equaling 4 mm. The airfoil was supported from the downstream edge by a sting balance.

Free stream velocity in the wind tunnel was set at 1.75 m/s with a corresponding Reynolds number, based on chord length, of 23,500. Tests were performed over a range of angles of attack from 0 to 20 degrees. Major, large vortical structures at separation were observed for the 15 and 20 degree cases only.

Images were obtained using a high-speed IDT camera with 1280x1040 resolution and field of view approximately 40 mm x 35 mm. This camera was equipped with an image intensifier. Laser pulses and image capture were coordinated using Dantec’s FlowManager TRPIV system. Time delay between laser pulses ranged between 20 and 40 μs and was based on a characteristic time scale of the flow, the magnification of the field, the pixel spacing of the camera’s CCD and a desired particle movement between frames of approximately 8 pixels. The frequency of velocity sampling was set to 500 Hz (time between velocity field capture was 2 ms).

The original images were processed to remove glare from the surface of the airfoil. This background removal consisted of pixel averaging the first 100 images with subsequent removal of the average values from each image. This process did not degrade the overall seed particle detection. A velocity field of 79x63 grid points was generated from the processed images using a one step adaptive cross correlation of 32x32 pixel sub-regions with 50% overlap. The region for the adaptive correlation step was 64x64 pixels. A 3x3 median filter was applied to the resulting velocity data to remove large, single vectors assumed to erroneous. This represented less than 1% of the total velocity field vectors.

III. Data Analysis


There exist several methods of filtering data to assist in the identification of vortical structures, some of which are discussed by Adrian et. al. [2000]. One such method is a Gaussian high pass filter, labeled a “Large Eddy Simulation” (LES) filter. To clarify, this did not involve the common Large Eddy Simulation style turbulence modeling typically performed in many Computational Fluid Dynamics (CFD) algorithms. Gaussian low/high pass filters are common in the field of image processing and usually contain a specified range of influence which determines the size of features to be removed or enhanced and a spatial weighting parameter determined by the rate of exponential decay from the centered filter. The full computational details are not discussed here. For this study, the Gaussian function provided a low-pass, smoothed velocity field. This low-pass field was subtracted from the original field to produce a high-pass, LES filtered field containing the fluctuations of a smaller size than the Gaussian filter diameter. Many filter ranges were evaluated but the final range value used was determined by associating it with the size of the major vortical structures which were formed at the leading edge. This was 19 grid points or approximately 10 mm diameter with a standard deviation of the Gaussian kernel equal to 2.

The Proper Orthogonal Decomposition (POD) is used to illuminate the major periodic processes occurring in this leading edge separation. The basic premise behind the POD of the velocity field is that a function d(x,t) can be approximated as the sum of the products of functions ak(t) and fk(x) where k is the mode number.

( ) ( ) (1

,M

k kk

d x t a t f x=

≈ ∑ )

k

(1)

If we take the functions f to be orthonormal basis functions, then the coefficient function, sometimes called the amplitude time series, can be expressed as:

( ) ( , ) ( )kX

a t d x t f t dx= ∫ (2) An NxM design matrix, D is constructed from the experimental data where N is the number of time steps and M

is the number of modes. The components of D are d(x,t) which are the experimental data points. A singular value decomposition (SVD) of the design matrix, D, is performed to determine the basis functions, or eigenfunctions, and amplitude time series’. This decomposition is performed for both velocity components within the same set.

Combining the resulting eigenfunctions from this analysis, velocity fields representative of each mode were formed. In addition, the cumulative effect of the higher energy modes when multiplied by the amplitude time series was used to reconstruct the velocity field from a low number of modes. Note that mode 0 represents the time averaged flow field.

The contribution of each eigenfunction to the total variance of the data is expressed by the sum of the squared amplitude time series components for a given mode, k. This characteristic allows the calculation of the energy contribution of each mode. Figure 2 shows the energy accumulation when summed cumulatively over individual modes.

One proposed method of empirically quantifying a rotating structure is to evaluate the local velocity gradient tensor, determined from the PIV field. This will be limited to the two dimensional case as set by the PIV data. This requires forming a 2x2 matrix of the velocity components u and v and their derivatives with respect to x and y as:

u ux y

Gv vx y

∂ ∂⎡ ⎤⎢ ⎥∂ ∂⎢ ⎥=∂ ∂⎢ ⎥⎢ ⎥∂ ∂⎣ ⎦

(3)

The method proposed by Jeong and Hussain [1995] associates a local pressure minimum with the second eigenvalue

of the velocity gradient tensor. For 2D application, if the quantity: 2

4u v u vx y y

⎛ ⎞x

∂ ∂ ∂+ −⎜ ⎟

∂∂ ∂ ∂⎝ ⎠ ∂

is greater than zero then λ1

and λ2 are two real roots, otherwise they form a complex conjugate pair. Eigenvalues for regions of strong rotation will have a large imaginary component. For purposes of this paper, the positive imaginary component corresponding to the eigenvalue, λ2 is used.

IV. Results

The two angle of attack cases investigated revealed identical primary shedding frequencies at the leading edge of approximately 28 Hz. This was found by performing autocorrelations as well as Fourier transforms on the u and v component time series as well as a vortex detection criterion. Further explanation is provided by Morse and Liburdy [2007]. The Strouhal number based on freestream velocity and chord length for the leading edge vortex shedding was 3.1. In some literature the Strouhal number for an airfoil is based on the approach thickness, the


height of the frontal projected area. This modified Strouhal number is 0.81 and 1.06 for 15° and 20° angles of attack, respectively.

As an example, Figure 3 illustrates three sequential vector fields separated by 2 ms, each with a gray mask applied to the region under the wing which was not illuminated by the laser sheet. Visible in the vector field is a large vortical structure approximately 25 mm from the leading edge. This structure appears to move in the streamwise direction and grow in size. Identifying similar structures was done using the velocity gradient tensor eigenvalue technique, λ2. Figure 4 overlays contours of λ2 with the velocity vector fields for the same three time steps as shown in Figure 3. This identification technique shows a concentration near the center of the observed feature in the vector field. It is of interest to note the decrease in λ2 for the third time step. This coincides with a qualitative observation of the increase in size of the feature. This indicates that this vortex, generated in the high shear Kelvin-Helmholtz instability near the leading edge of the airfoil, becomes weaker as it convects downstream and begins to break up or dissipate. The convection of this structure is more apparent when the contours of λ2 are shown to outline the flow structure location at each time step. The ‘wake’ of the large structure contains a series of smaller, weaker structures which change over time.

One method proposed to help elucidate structures in vector fields is Large Eddy Simulation (LES) filtering. The LES filtered fields are shown in Figure 5, again for the same three time frames as previously shown. This resultant vector field more strongly illustrates the shear layer from the leading edge as well as the vortical feature seen in the previous figures. The third time step in Figure 5 also shows the increase in size of the vertical structure as is observed in the unfiltered case. Application of this specific filter, however, illustrates that this structure in the third time step does not have strong circulation, i.e. the shape of the LES filtered structure seems less coherent. There exist larger structures in the flow that are eliminated with the high pass, LES technique. This indicates that a different method may be more appropriate in the detection of these vortices, such as the POD of the velocity field, discussed below.

Performing the λ2 analysis on the LES filtered fields shows a more concentrated distribution of λ2 for the first two time steps as seen in Figure 6. This would indicate a possibly more complex structure within each large scale feature. This same analysis indicates a complete breakup of the original vortex in the final time step. The contours of λ2 are broken up into many smaller regions instead of a with high concentration in the center.

The POD analysis was performed on the data sets to identify major contributions to the periodic nature of the flow. The u and v components of velocity were combined into a single set for analysis. Figure 7 illustrates the first three eigenfunctions obtained from the POD of the velocity field time sequence for the 20° angle of attack case. The first mode, which accounts for approximately 30% of the variance of the velocity, is shown in Figure 7a. The primary feature in this first mode is a large up-wash in the region between x equal 30 and 40 mm. This upwash might be a consequence of a vortex pair, namely the upstream structure shown and one structure further downstream. In addition, the λ2 technique detects a small rotating flow feature upstream between x equal 20 and 25. The second mode, shown in Figure 7b, accounts for slightly less than 20% of the total variance. A single counterclockwise rotating feature is seen centered at approximately x = 25 mm which dominates this eigenfunction. A lesser, clockwise feature is seen downstream and somewhat above the vortex shown in mode 1. These two rotating features bracket a strong upwash similar to that found in the first mode. While this mode contains a lower energy content than the first mode, it has a stronger rotating feature as defined by the λ2 method. The third mode shown in Figure 7c contains less than 10% of the energy. There exist several counter rotating structures, all weaker than the vortex found in the second mode.

By combining the mean flow with these first three modes the velocity field can be constructed within some level of accuracy, accounting for about 55% of the energy within the flow as shown in Figure 7d. To reconstruct the flow the first three modes are multiplied by their respective time series values at time step 1 and then summed together with the mean flow. A strong shear region is seen at the leading edge followed downstream by a large single vortex. Comparing this reconstructed mode with the original vector field in Figure 3, it is seen that the modal vortex detected using λ2 is weaker than the original flow field value. Also, the center location of the reconstructed vortex is slightly upstream of the original. The time averaged flow, referred to here as mode 0, is provided for clarity in Figure 7e. The flow is fully separated from the leading edge across the whole field of view.

The amplitude time series’ of the first four modes are shown in Figure 8 and illustrate the cyclic nature of the flow. The first two modes appear to be similar, yet out of phase with each other. Plotting the first mode time series against the second mode (normalized by 2√λk where λk is the mean variance for mode k, not related to the vortex detection scheme λ2) produces a phase pattern shown in Figure 9a, indicating a π/2 out of phase relation between modes 1 and 2. These two modes were combined to show a single moving vortex as seen in Figure 10. There appears to be no such obvious phase relationships between modes 2-3, 3-4 and 4-5 as shown in Figure 9b, c and d.


A similar analysis was done for the 15° angle of attack case. As shown before in Figure 2, the contribution to energy from each mode is similar to the 20° case with the exception that the lower modes contribute slightly less energy than the 20° case. This may be a consequence of a slightly stronger shear layer resulting in more concentrated energy at the lower modes for the higher angle of attack case. The eigenfunctions for modes 1,2 and 3 are shown in Figure 11a, b and c, respectively. Mode 1 has an upwash region similar to the 20° case. It is bracketed by a pair of counter-rotating features. Mode 2 also exhibits a counter-rotating pair similar to the 20° case. Mode 3 differs in that it contains several small, intense vortical structures. In all three modes the 15° case displays more cohesive structures compared to the 20° case. The time averaged flow, referred to here as mode 0, is also provided for clarity in Figure 11e. Similar to the 20° case, the flow is fully separated from the leading edge across the whole field of view.

Analysis of the amplitude time series’ in Figures 12 and 13a reveal a similar π/2 out-of-phase relationship between modes 1 and 2, with mode 2 preceding mode 1 by π/2 (or mode 2 follows mode 1 by 3π/2). These combine with the mean flow to form the convected vortex seen in Figure 14. Phase plots of modes 2-3, 3-4 and 4-5 show interesting patterns unlike the 20° case. Figure 13b shows modes 2-3 avoiding the first quadrant, where both amplitudes are positive. Inspection of the mode 2 and 3 fields reveals the downwash between the rotating features of mode 2 is close to the upwash between the first two vortical structures of mode 3. Simultaneous large positive values for the amplitudes of modes 2 and 3 would result in these two modes tending to cancel each other out in those upwash/downwash regions located between x of 25 and 30. Figure 13c displays a less cohesive π/2 relationship between modes 3 and 4 compared to modes 1 and 2.

One approach to determining how the amplitude time series’ are related is to perform a cross-correlation of the time series’ from two separate modes. Figure 15 shows the cross correlation for amplitude time series’ of adjacent modes, i.e. modes 1 and 2, modes 2 and 3, etc. The primary frequency associated with the cyclic feature found in cross correlation of modes 1-2, and 2-3 is the same as the vortex shedding frequency of approximately 28 Hz discussed previously by Morse and Liburdy [2007].

In order to identify cross correlations between non-adjacent modes Figure 16 maps the maximum absolute value of the correlation ratio between all modes. Autocorrelations, correlating one mode’s time series with itself, by definition has a maximum of 1 and is not included in this map. In addition, the upper left half of the plot is not plotted because of symmetry. Figure 16a maps the maximum correlation for the 20° case. Modes 1 and 2 show a strong relationship. Modes 2-3 and 1-3 are also notable. An interesting feature is the relatively high correlation between modes 10 and 11. Closer inspection shows a higher correlation at very short time lags (2 to 4 ms) between those two modes as shown in Figure 17a. The vector plots and λ2 contours shown in Figures 17b and c for modes 10 and 11, respectively, reveals the existence of much smaller spatial scales which is consistent with the smaller time scales observed in the correlation.

Figure 16b maps the maximum correlation for the 15° case. There exist strong correlations between modes 1-2 and 3-4. The primary difference from the 20° case is the much lower correlation for the lower modes. The right side of the color map is much darker when compared with Figure 16a, showing a decreased correlation between modes. That is not observed here as the higher angle of attack shows a higher cross-correlation on average in Figure 16a. But this follows with the modal energy plot from Figure 2 which shows a higher energy contained in the low numbered modes for the lower angle of attack.

In many applications the POD can be used as a filtering technique to remove lower energy fluctuations in the flow field. A reduced order model of the flow reconstructed for the two cases is shown in Figures 18 and 19. These results display an increase in the local maximum of the λ2 value at the center of the vortex structure as the number of modes increases. In addition, the location of the local maximum appears to vary slightly as the number of modes increases. The large scale structure is apparent at these lower mode reconstructions when compared to the original flow, also shown in Figures 18 and 19.

V. Conclusion

The flow over a flat airfoil was analyzed using TRPIV methods for low Reynolds number flow at two angles of attack, 15° and 20°. The velocity gradient tensor eigenvalue method, λ2, was used to help elucidate regions of swirling flow. A high pass Gaussian filter, sometimes referred to as an LES filter, was employed to assist in vortex identification. Some of the major vortical features of the flow were observed to be too large for this technique to be effective. The POD was performed on the velocity fields to determine high energy flow features. The 20° case displayed a higher energy concentration in the primary modes when compared with the 15° case. In addition, the amplitude time series’ for the 20° case show higher maximum correlation of the lower order modes, resulting in more well defined vertical structures. For both cases a primary modal pair, modes 1 and 2, exists with their amplitude time series’ π/2 out of phase with each other. They can be combined to recreate the primary vortex


shedding which occurs at the leading edge. Reconstruction of the flow using a reduced number of modes was shown to recreate the primary vortices with some variation in location as the number of included modes increased.

Acknowledgments This work was funded in part by a grant from the Air Force Office of Scientific Research (AFOSR grant #FA-9550-05-1-0041).

References Adrian, R.J., Christiansen, K.T., Liu, Z-C., 2000, “Analysis and interpretation of instantaneous velocity fields”, Experiments in Fluids, Vol. 29, pp. 275-290. Gopalarathnam, A., Broughton, B.A., McGrananhan, B.D. and Selig, M.S., 2003, “Design of Low Reynolds Number Airfoils with Trips”, J. of Aircraft, Vol. 40, No. 4, pp 768-775. Graftieaux, L., Michard, M. and Grosjean, N., 2001. “Combining PIV, POD and vortex identification algorithms for the study of unsteady turbulent swirling flows”, Meas. Sci. Technology., Vol. 12, pp. 1422-1429.

Hinze, J. O. “Turbulence”, McGraw-Hill, New York, 1959.

Hoarau, Y., Braza, M., Ventikos, Y., Faghani, D. and Tzabiras, G., 2003, “Organized Modes and the Three dimensional Transition to Turbulence in the Incompressible flow Around a NACA0012 Wing”, J. Fluid Mechanics, Vol. 496, pp 63-72.

Jeong, J. and Hussain, F., 1995, “On the Identification of a Vortex”, J. Fluid Mechanics, Vol. 285, pp 69-94. Morse, D. and Liburdy, J., 2007, “Dynamic Characteristics of Flow Separation from a Low Reynolds Number Airfoil”, ASME Fluids Engineering Division Summer Meeting (FEDSM2007-37083), Jul. 30-Aug. 2, 2007, San Diego, CA

Mueller, T.J., Pohlen, L.J., Conigliaro, P.E. and Hansen, B.J., 1983, “The Influence of Free Stream Disturbances on Low Reynolds Number Airfoils”, Experiments in Fluids, Vol. 1, No. 1, pp 3-14. Nishimura, H. and Taniike, Y., 2001, “Aerodynamic Characteristics of Fluctuating Forces on a Circular Cylinder”, J. Wind Eng, Ind, Aerodynamics, Vol. 89, pp 713-723. Pelletier, A. and Mueller, T.J., 2000, “Low Reynolds Number Aerodynamics of Low-Aspect Ratio, Thin/Flat/Cambered-Plate Wings”, Journal of Aircraft, Vol. 37, No. 5, pp825-832. Selig, M.S., Donovan, J.F. and Fraser, D.B., 1989, Airfoils at Low Speeds, Stokely, Virginia Beach, VA. Selig, M.S. and Guglielmo, J.J., 1997, “High Lift Low Reynolds Number Airfoil Design”, J. of Aircraft, Vol. 34, No. 1, pp 72 - 79. Shyy, W., Klevebring, F., Nilsson, M., Sloan, J., Carroll, B., and Fuentes, C., 1999, “Rigid and Flexible Low Reynolds Number Airfoils”, J. of Aircraft, Vol. 36, No. 3, pp 523-529. Sicot, C., Auburn, S., Loyer, S. and Devinant, P., 2006, “Unsteady characteristics of the Static Stall of an Airfoil Subjected to Freestream Turbulence Level up to 16%”, Experiments in Fluids, Vol. 41, pp 641-648. Simpson, R.L., Chew, Y.T. and Shivaprasad, B.G., 1981, “The structure of a separating Turbulent Boundary Layer: Part I, Mean Flow and Reynolds Stresses”, and “Part II, Higher Order Turbulence Results”, Journal of Fluid Mechanics, Vol. 113, pp 23-73.


Weiland, C. and Vlachos, P., 2007, “Analysis of the Parallel Blade-Vortex Interaction with Leading Edge Blowing Flow Control using the Proper Orthogonal Decomposition”, ASME Fluids Engineering Division Summer Meeting (FEDSM2007-37275), Jul. 30-Aug. 2, 2007, San Diego, CA


Image field of view

Laser light sheetAirfoil

Dantec iNanosense high speed high resolution camera New Wave

Pegasus laser

Figure 1: Experimental Setup.


Figure 2: Combined modal energy of POD modes for α = 15° and 20°.

k

Tot

λλ

∑∑

Mode number k

Figure 4: Structure identification using the eigenvalue, λ2 of the velocity gradient tensor; Re = 23 500, α = 20°

Figure 3: TRPIV fields separated by 2 ms; Re = 23500, α = 20°


Figure 5: LES filtered velocity fields separated by 2 ms; Re = 23500, α = 20°

Figure 6: Structure identification using the eigenvalue, λ2 of the LES filtered velocity gradient tensor; Re = 23500, α = 20°


Figure 7: Eigenfunctions with λ2 contours for α = 20° (a) mode 1; (b) mode 2; (c) mode 3; (d) a reconstructed velocity field from modes 0 through 3; (e) mode 0, time averaged flow field.

(a) (b)

(c) (d)

(e)


(c)

Figure 9: Normalized amplitude/amplitude plots for the time series’ of (a) modes 1 and 2 (b) modes 2 and 3 (c) modes 3 and 4 and (d) modes 4 and 5. Modes 1 and 2 are the only pair that display a phase relationship, ~90°.

Figure 8: Amplitude time series’ of the first 4 modes for the 20° case. Note the in-phase relationship between modes 1 (green) and 2 (blue).

(a) (b)

Figure 10: Convecting vortex structure reconstructed from the mean flow and modes 1 and 2 for the 20° case. Contours represent the λ2 vortex detection scheme. Time steps are separated by 2 ms.

(d)


(a) (b)

(c) (d)

(e)

Figure 11: Eigenfunctions with λ2 contours for α = 15° (a) mode 1; (b) mode 2; (c) mode 3; (d) a reconstructed velocity field from modes 0 through 3; (e) mode 0, time averaged flow field.


(a) (b)

(c) (d)

Figure 12: Amplitude time series’ of the first 4 modes for the 15° case. Note the in-phase relationship between modes 1 (green) and 2 (blue).

Figure 13: Normalized amplitude/amplitude plots for the time series’ of (a) modes 1 and 2 (b) modes 2 and 3 (c) modes 3 and 4 and (d) modes 4 and 5. More complex modal interactions are observed here for the 15° case.

Figure 14: Convecting vortex structure reconstructed from the mean flow and modes 1 and 2 for the 15° case. Contours represent the λ2 vortex detection scheme. Time steps are separated by 2 ms.



R67

R45

R56

R23

R34

R12

(c)

(b)

(a)

Figure 17: (a) Cross-correlation between modes 10 and 11 for the 20° case. Vector plots with λ2 contours for (a) mode 10 and (b) mode 11

Figure 16: Cross-correlation maximum for each modal combination between modes 1-20. (a) 20° case (b) 15° case.

(b)

(a)

Figure 15: Cross-correlation of amplitude time series’ for modes 1-2, 2-3, etc… of the 20° case.

(a) (b)

Figure 18: Reduced mode reconstruction of instantaneous experimental velocity data with λ2 contours for α = 20° from (a) modes 0 and 1 (b) modes 0 through 2 (c) modes 0 through 10. (d) The original velocity field.

(a) (b)

(d) (c)

16

American Institute of Aeronautics and Astronautics

Figure 19: Reduced mode reconstruction of instantaneous experimental velocity data with λ2 contours for α = 15° from (a) modes 0 and 1 (b) modes 0 through 2 (c) modes 0 through 10. (d) The original velocity field.

(c) (d)

NomenclatureAcknowledgments

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False

/CreateJDFFile false /Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ]>> setdistillerparams> setpagedevice

Date post:	25-Jan-2021
Category:	Documents
Upload:	others
View:	6 times
Download:	0 times

Experimental time resolved flow features of separation over...

Documents