Evolution of vortex structures in an electromagnetically excited separated flow
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- Cierpka, C., Weier, T. & Gerbeth, G. Exp Fluids (2008) 45: 943. doi:10.1007/s00348-008-0512-6
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Time periodic wall parallel Lorentz forces have been used to excite the separated flow on the suction side of an inclined flat plate. Experiments for a Reynolds number of 104 and an angle of attack of α = 13° are reported. The controlled flow is characterised by a small number of relatively large scale vortices, which are related to the control mechanism. The influence of the main parameters, i.e. the excitation frequency, amplitude and wave form on the suction side flow structures was investigated by analysing time resolved particle image velocimetry (TR-PIV) measurements using continuous wavelet analysis for vortex detection and characterisation. Statistical analysis of the coherent structures of the flow was performed on a large amount of data samples.
Such large load factors result into two important consequences. On one hand the force density distribution is independent of the flow field. On the other hand, a large load factor means a small efficiency of momentum generation since the ratio of mechanical (∼ju∞B0) to electrical (∼jE0) power is the reciprocal of ϕ (Shatrov and Gerbeth 2007). The low energetic efficiency hindered up to now the technical use of the Lorentz force actuator for practical applications. However, for fundamental research in the lab frame it has several appealing features: momentum is directly generated in the fluid without associated mass flux, the frequency response of the actuation is practically unlimited and the wave form of the excitation is freely adjustable. The physics of momentum generation is fully understood and can be computed from first principles. Plasma actuators deliver a body force as well, but their physical mechanism is still under active discussion (see, e.g., Boeuf et al. 2007; Jukes et al. 2006; Likhanskii et al. 2007).
Essential features of separation control by periodic Lorentz forces have been investigated on a NACA 0015 at chord length Reynolds numbers 5.2 × 104 < Re < 1.5 × 105 by Weier and Gerbeth (2004). Characteristic excitation frequencies, effective momentum coefficients, and resulting lift gain compare well to that found with conventional methods for periodic addition of momentum. Regarding periodic excitation with different wave forms, Weier and Gerbeth (2004) have shown that the lift gain scales under certain conditions with the peak momentum coefficient. This momentum coefficient is calculated with the peak value of the applied current instead of the usual rms value. Hence, excitations using pulsed wave forms seem to be very attractive. Keeping the rms value constant, the peak momentum input can be increased by shortening the duty cycle. Because of the non-varying rms momentum coefficient, the energy expenditure remains constant as well.
Motivated by the prospect of efficiency improvement, the main focus of the present paper is on the effect of pulsed wave forms with various duty cycles. Force measurements in deep stall on a NACA 0015 demonstrate the effect of pulse width on the integral forces. The main part of the paper presents time resolved PIV measurements of the flow around a generic profile, i.e., an inclined flat plate. The driving phenomenon in both flow configurations is supposed to be the modification of vortex interaction caused by the excitation. Coherent structure eduction using the continuous wavelet transform (CWT) may therefore be a helpful tool to extract characteristic flow features.
A short introduction to the experimental setup and the applied CWT-algorithm is given in Sect. 2. In the following, the different effects of frequency and wave form of the excitation are discussed. A brief conclusion on the results and an outlook for further investigations can be found in Sect. 4.
2 Experimental setup and postprocessing
2.1 Experimental setup
The body of the plate was made of polyvinyl chloride (PVC). It has a circular leading edge, a span width of 140 mm, a chord length of 130 mm, and a thickness of 10 mm. Shape and material were chosen to provide durability in the used electrolyte solution (0.25 M NaOH) and ease of manufacturing. A high power amplifier FM 1295 from FM Elektronik Berlin has been used to feed the electrodes. The power supply is driven by a frequency generator 33220A from Agilent. This allows for easy adjustment of frequency, wave form and amplitude of the Lorentz force.
The time resolved PIV measurements were performed in the small electrolyte channel at Forschungszentrum Dresden-Rossendorf. This free surface channel is driven by a centrifugal pump and has been operated with a mean velocity of u∞ = 8 cm/s resulting in a chord length Reynolds number of Re = 1.04 × 104 for the plate. The plate was inclined with α = 13° resulting in strong leading edge separation which also occurs for the NACA 0015 at α = 20°. A settling chamber equipped with a filter pad, two honeycombs and a set of four screens results in a relatively small turbulence level in the test section (≈1%). The latter is 1 m long and has a 0.2 × 0.2 m2 cross section. For further details we refer to Weier (2005). To reduce end effects, the plate has been mounted between rectangular plates with rounded edges made from perspex extending from the bottom of the test section to the free surface in vertical and from 3 cm in front of the leading edge to 3 cm behind the trailing edge in horizontal direction. Only small variations of the flow in spanwise direction have been found by PIV measurements and flow visualisations.
The PIV setup used a Spectra Physics continuous wave Ar+-Laser type 2020–5 as light source. Image recording has been done with a Photron Fastcam 1024PCI 100K operated at 60 Hz frame rate. The light sheet with a thickness of Δz ≈ 1 mm was formed by two cylindrical lenses. It illuminated the x–y-plane at mid span of the plate from the channel bottom. The electronic camera shutter was used to set the single image exposure time to 2 ms. For each configuration, 6,400 single images of 1,024 × 512 pixel2 have been recorded synchronised to the excitation signal. Polyamide particles (Vestosint) of 25 μm mean diameter provided the necessary seeding. x and y velocity components have been calculated from the images using PIVview-2C 2.4 from PivTec. Each image was correlated with its successor using multigrid interrogation with a final window size of 16 × 16 pixel2 and 50% overlap. Image deformation and sub pixel shifting has been applied as well.
Obviously, the separated flow under investigation is inherently instationary and the mechanisms involved are quite complex. The averaged velocity fields can give a first impression of the flow but will hide important features. In former studies, e.g., Cierpka et al. (2007), phase averaging was applied to overcome these limitations to a certain extent. However, the phase averaging is based on the excitation frequency which is a reasonable choice for sine wave excitation. Using it for other wave forms, there is the “risk of obtaining results that are more representative of the conditional event than of the coherent structure” (Adrian 1996). Applying arbitrary excitation wave forms, peaks of comparable height may occur in the power spectra for several frequencies. These are connected to the Fourier coefficients of the excitation signal. The choice of a proper frequency for phase averaging becomes difficult since each constriction to a certain frequency will hide features on other time scales. Nevertheless, it is a necessary precondition for data interpretation to condense the vast amount of information provided by the TR-PIV measurements. To extract the coherent structures from the velocity fields, several methods have been proposed and are discussed, e.g., by Bonnet et al. (1998). The present paper applies basically a vortex detection method proposed by Schram et al. (2004) to characterize the flow field. Their method is based on the continuous wavelet transform of the enstrophy field. For a complete treatment of the approach, we refer to Schram (2002). In the following, only a short introduction is given in order to facilitate the interpretation of the results.
2.3 Continuous wavelet transform
An extensive review of wavelet transforms and their application to fluid dynamics can be found in Farge (1992). The wavelet analysis is selective in space and scale and is therefore more applicable for the investigation of coherent structures than Fourier analysis. Schram et al (2004) proposed a special kind of CWT. A 2D Marr wavelet, also known as Mexican hat wavelet, is applied to the time resolved vorticity fields. That way, different vortex characteristics such as vortex size and circulation, trajectories, and mean convection velocities can be determined. Since the flow considered in the present study is dominated by vortex structures, this information can be used to determine the action of different excitation modes in more detail.
The vorticity field is in principle a well suited basis to detect vortices, since vortices are regions of high vorticity. It has the additional advantage of being Galilean invariant. However, its major drawback in respect to vortex detection is the appearance of high vorticity not only in regions of swirling motion, i.e. vortices, but in regions of pure shear, i.e. shear layers, as well. In the flow field under investigation, both phenomena are present. It is impossible to differentiate between them by applying a threshold based on vorticity alone. In order to distinguish between vortices and shear layers the λ2-criterion as proposed by Jeong and Hussain (1995) was used. Since vortices exist only in regions of negative λ2, the vorticity field has been set to zero for λ2 > 0. The CWT algorithm was then applied to this modified vorticity field.
There should however be a limit on the gain in lift with decreasing pulse width. One would expect that owing to inertia the flow is not able to react to very short pulses. Obviously, this expected limit is not yet reached in the measurements shown in Fig. 6. F+ = 0.75 seems to work best with a pulsed rectangular wave form of a duty cycle of only 6%.
Using a pulsed signal for excitation, one forcing cycle consists of consecutive times with downstream and upstream forcing interrupted by pauses. During the first half-cycle, vortices form in the phase of no forcing (“off time”). They are then accelerated and pushed in streamwise direction by the pulse generating a downstream Lorentz force. Natural vortex shedding is reestablished in the following off time. In the second half-cycle, a compact structure of high vorticity is produced by the pulse generating now an upstream Lorentz force. This structure remains in the vicinity of the actuator during upstream forcing and leaves this place in the next off time. Shortly after, it merges with the naturally shed vortices from the previous off time.
For F+ = 1 the convective velocity of the structure accelerated by the downstream Lorentz force is higher than that of the vortex generated by the upstream Lorentz force. Consequently, the vortices merge approximately at mid chord and generate one persistent vortex. It can be tracked until it leaves the field of view. Summed up, three vortex structures of different convective velocity and different vorticity are produced in one period close to the actuator. Due to dissipation and merging, only one large structure per period can be observed at the trailing edge.
Phase averaged vorticity contours for a higher excitation frequency F+ = 4 are shown in the right column of Fig. 9. The main difference to F+ = 1 is that the forcing is too fast to allow for the formation of more than one vortex per period. A vortex grows at the leading edge in the off time of the first half-cycle. During upstream forcing there is still an acceleration, however, its duration is too short to markedly push the vortex downstream. In the following off time and the subsequent upstream forcing, a second vortical structure grows and is finally shed. This structure merges soon with the one released in the first half-cycle. This is the reason why, at the end, only one relatively compact but small structure is produced per cycle. This structure dissipates comparatively quickly further downstream. No further interaction between the vortices can be observed.
For the baseline case, the same cycle duration as for F+ = 1 has been artificially chosen. A kind of signature is visible in the plot. This means that phenomena with a frequency of f = u∞/c and, indicated by the number of traces, their higher harmonics are present. The traces are roughly parallel due to the same convective velocities of the vortices. It is impossible to clearly identify vortex merging. Compared to the experiments with forcing, where the vortices are mainly generated in the vicinity of the actuator, the region of vortex production in the baseline case is located further downstream in the shear layer.
For F+ = 0.5 the shear layer is pushed downwards during the time with downstream forcing and starts to form a vortex at x ≈ 20–30 mm. At the start of the following off time, the vortex detaches and is convected downstream. Then natural separation sets in at the leading edge and an additional structure forms and detaches. The subsequent upstream forcing generates a vortex structure with strong vorticity. It remains at the actuator until it is released in the next off time and unforced leading edge vortex formation starts again. At the two vortex trail junctions, one close to the leading edge and one at mid chord, the vortices are supposed to merge. However, the algorithm is unable to detect the strongly deformed structures involved in a merging process. The structures are very large and have a long lifetime. From the presence of smaller traces between the more significant vortex trajectories, one can conclude that also higher harmonics are present. This is a sign for recurring natural vortex shedding in the off times. However, looking at the full cycle, vortex shedding is clearly dominated by the forcing.
For actuation with F+ = 1, the picture is much clearer. During downstream forcing, the vortices are accelerated and gain a high convection velocity. Approximately at mid span, they merge with structures of the previous shedding cycle. In the off time, natural vortex shedding recurs. However, the velocity and strength of the shedded structures is lower than that of the one generated by the following upstream forcing. At approximately one quarter of the chord, the weaker structure is almost dissipated and will merge with the following vortex. Also the overall lifetime of the structures is shorter than for F+ = 0.5. Mixing is enhanced and the vortices transfer their energy much faster to the mean flow.
For F+ = 2 a blurred region of vortex occurrence in the vicinity of the actuator is visible for roughly the first half-cycle. This might be taken as an indication for infrequent vortex merging. However, only one vortex is shed per cycle and no direct vortex merging or other interaction has been observed. Because the structures are smaller, their lifetime is reduced with increasing frequency. At F+ = 8 the phase plot looks relatively similar to the one given for the unforced flow, although it appears smoother due to a smaller amount of data per period.
At time t1, the shear layer above the actuator is pushed to the surface of the plate. At the end of the downstream forcing, a small vortex will separate from this region. At x ≈ 40 mm, a vortex which was shed during the last off time is now convected downstream and at x ≈ 120 mm a large vortex can be detected.
When upstream forcing is switched on at t2, the shear layer starts to lift up from the surface and a strong vortical structure is generated close to the actuator. At mid span, the small and fast vortex from the end of the downstream forcing phase and the slower previous vortex have already merged and form now a large vortex structure. This amalgamation appears as a junction in Fig. 11. Just before the merging, two small structures with separated trajectories were detected. Then there is a short gap in both vortex trails after which a single vortex of bigger size appears. This vortex moves then downstream. The gap in the data occurs due to the strong deformation of the structures during their interaction. The shape of the deformed vortices does not resemble that of a Lamb–Oseen vortex and therefore the algorithm is unable to detect the single vortices during amalgamation. Close to the trailing edge a very weak vortex structure is visible.
In an unforced separated flow, vortex merging would occur much further downstream and would probably not influence the performance of the hydrofoil. The major effect of the short pulses is the forced mixing. Especially close to the actuator, the interaction seems to be stronger than for other wave forms (Cierpka et al. 2007). This is mainly due to the vortex shedding during the off times, which are longer for shorter duty cycles. The different convection velocities of the vortices shed during off times and the ones generated by the forcing results in a remarkable momentum transfer in the separated region close to the leading edge. This effect is supposed to be the reason for a better performance of the short pulses in terms of separation suppression.
4 Conclusion and outlook
The wavelet algorithm is able to reduce the vast amount of data measured by the TR-PIV and extract coherent structures. Statistical analysis on this base provides valuable information about the flow characteristics.
For frequencies around F+ ≈ 0.5–1, strong interactions of the generated vortices are the dominant phenomena. Up to mid chord, two merging processes can be observed resulting in strong enhancement of momentum transfer in this region. For increasing excitation frequencies these interactions cease to exist.
With shorter duty cycles the mixing zones are broadened, especially close to the actuator. Separation in a time averaged sense is stronger suppressed by pulses with shorter duty cycles, if all other parameters (Re, α, F+, cμ′) are kept constant. Although the inclined plate is quite a different case, it is supposed that the observed effects are responsible for the lift enhancement using active flow control with shorter pulses at F+∼ 1 for the NACA 0015.
In the future synchronized time resolved PIV and force measurements are scheduled on a NACA 0015 to make the final direct link between the flow structures and the hydrofoil performance.
Financial support from Deutsche Forschungsgemeinschaft (DFG) in frame of the Collaborative Research Centre (SFB) 609 is gratefully acknowledged.