Experiments in Fluids

, Volume 45, Issue 5, pp 943–953

Evolution of vortex structures in an electromagnetically excited separated flow

Authors

    • Forschungszentrum Dresden-Rossendorf
  • Tom Weier
    • Forschungszentrum Dresden-Rossendorf
  • Gunter Gerbeth
    • Forschungszentrum Dresden-Rossendorf
Research Article

DOI: 10.1007/s00348-008-0512-6

Cite this article as:
Cierpka, C., Weier, T. & Gerbeth, G. Exp Fluids (2008) 45: 943. doi:10.1007/s00348-008-0512-6

Abstract

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.

1 Introduction

Flow separation and its control is a persistent topic of fluid dynamic research. A comprehensive review can be found in Gad-el Hak (2000). Shape modifications and vortex or turbulence generators belong to the passive flow control methods. Active flow control methods, such as blowing and suction, are characterized by an additional input of energy. This energy input can be applied steadily or varying with time. While steady blowing is a tool investigated for more than eight decades, separation control by periodic addition of momentum has been a subject of intense research only since the early 1990s. Its most striking feature is that a control goal, e.g., a specific lift increase, can typically be attained by orders of magnitude smaller momentum input compared to steady actuation (Greenblatt and Wygnanski 2000). Widely accepted as the most significant parameters are the time averaged momentum input and the excitation frequency. The influence of the excitation wave form was investigated, among others, by Bouras et al. (2000), Margalit et al. (2005) and Cierpka et al. (2007). Although a very pronounced effect of the excitation wave form on the lift gain has been reported (Weier and Gerbeth 2004), the subject has received comparably little attention up to now. Seifert et al. (2004) reviewed the state of the art of “active flow control”, a term now commonly used for periodic excitation. In traditional aerodynamics the addition of momentum is almost exclusively accompanied by mass-flux. For electrically conducting fluids, like seawater or ionized air, momentum can be directly generated by electromagnetic Lorentz forces. The Lorentz force density F appears as a body force term on the right hand side of the Navier–Stokes-equation for incompressible flow:
$$\frac{\partial {{\mathbf{u}}}}{\partial t} + ({{\mathbf{u}}} \cdot \nabla){{\mathbf{u}}} = - \frac{\nabla p}{\rho} + \nu \nabla^2 {{\mathbf{u}}} +\frac{{\mathbf{F}}}{\rho}, $$
(1)
where u denotes the velocity, p the pressure, and t the time, respectively. ρ is the density and ν the kinematic viscosity of the fluid. The Lorentz force density itself is the vector product of a current density j and a magnetic induction B
$${{\mathbf{F}}}={{\mathbf{j}}} \times {{\mathbf{B}}}. $$
(2)
For moving conductors Ohm’s law
$${{\mathbf{j}}}=\sigma_e ({{\mathbf{E}}} + {{\mathbf{u}}} \times {{\mathbf{B}}}) $$
(3)
describes the current density. E denotes the electric field, and σe the electrical conductivity of the fluid. In the case of liquid metal magnetohydrodynamics (MHD) σe is typically very high (∼106–107 S/m), resulting in a strong coupling of the Lorentz force density and the flow. For seawater and other electrolytes, σe is small (∼10 S/m). Therefore, the induced currents are very low for moderate applied magnetic fields (B0 ∼ 1 T) and the induced Lorentz forces due to these currents are negligible. An additional electric field has to be applied in order to generate forces strong enough to act on the flow. The ratio of the applied electric field E0 to the electric field induced by the free stream velocity u in the presence of the applied magnetic field B0 is commonly termed load factor (Sutton and Sherman 1965)
$$\phi=\frac{E_0}{u_{\infty}B_0}. $$
(4)
For seawater flow control with moderate magnetic fields, it follows from above: ϕ ≫ 1.

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 (∼juB0) 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 actuator design follows the idea of Gailitis and Lielausis (1961) and Rice (1961). They proposed the arrangement of flush mounted electrodes and permanent magnets shown in Fig. 1 to generate a wall parallel Lorentz force. Magnets and electrodes are of the same width a = 5 mm. That way, the attainable integral force density is maximized (Grienberg 1961). Apart from end effects, both electric as well as magnetic fields have only components in wall normal (y) and spanwise (z) direction. From the vector product in Eq. 2 follows that the Lorentz force possesses a streamwise (x) component Fx mainly. For the experiments described here, the electrode/magnet-array sketched in Fig. 2 has been mounted close to the leading edge at the suction side of an inclined flat plate as sketched in Fig. 1. The magnetic induction at the surface of the magnetic poles can be calculated from the geometry of the magnets and their magnetization. Due to the short length of the actuator, the end effects cause a force density distribution showing non-uniformities in all directions as shown in Fig. 2. For the z-direction they are in the range of 0 ≤ y ≤ a (Weier et al. 2001). Averaged over z, the mean force density decreases exponentially with increasing wall distance
$$F_x=\frac{\pi}{8} j_0 M_0 \, \hbox{exp}\left({-\frac{\pi} {a} y}\right). $$
(5)
M0 denotes the magnetization of the permanent magnets at the surface and j0 the applied current density, respectively. The difference between the calculated exact Lorentz force and the spanwise averaged expression from Eq. 5 is relatively small. Even for the quadratic surface geometry of the electrodes and magnets it amounts to just 4%.
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Fig. 1

Electrode/magnet-array generating a mainly wall-parallel Lorentz force at the leading edge in streamwise direction

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Fig. 2

Close up to the short electrode/magnet/electrode-array for time periodic forcing at the leading edge of the inclined plate. Isosurfaces are of the Lorentz force density, ribbons showing selected electric field lines

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.

Since an alternating current was used, the generated Lorentz force oscillates in streamwise direction with the frequency fe of the applied current. The normalized excitation frequency
$$F^+ =\frac{f_{e} c}{u_{\infty}} $$
(6)
is defined with the chord length c and the free stream velocity u. For time periodic actuation, the applied Lorentz force strength is characterized by the ratio of the rms momentum added by the Lorentz force to the free stream momentum. The effective momentum coefficient is defined with the rms value of the applied electric current density
$$c^{\prime}_{\mu}=\frac{1}{2} \cdot \frac{ab B_0}{\rho u^2_{\infty} c} \cdot \sqrt{\frac{1}{T}\int\limits_0^T j(t)^2\,\mathrm{d} t}$$
(7)
where a denotes the width and b (= a) the length of an electrode, j(t) the time dependent current density, T the period of oscillation, B0 the magnetic induction at the surface of the permanent magnets and ρ the fluid density. Using a Hall-probe, a mean magnetic induction of B0 = 0.35 T was determined.

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 xy-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.

2.2 Postprocessing

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.

For some general aspects of the wavelet analysis the interested reader is referred to the short introduction of the wavelet analysis for data processing by Schram et al. (2004). Vortex detection is based on the absolute value of the vorticity field
$$\omega_{z} = \frac{\partial v}{\partial x} - \frac{\partial u} {\partial y}. $$
(8)
This is different from Schram’s approach, who used the enstrophy field. In Eq. 8u and v denote the velocity components in x and y-direction, respectively. The vorticity field ωz was computed using a least square differentiation scheme (Raffel et al. 1998). Prior to the vorticity computation, three consecutive vector fields have been averaged. To further remove noise an additional median filtering with a kernel size of 3 × 3 was applied to the final vorticity field.

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.

The wavelet transform of a signal \((\Uppsi_{l {\bf x}\prime} | f)\) is given by the convolution of the signal f(x) and the wavelet family \(\Uppsi_{l {\bf x}\prime}({\bf x})\)
$$ \left\langle \Uppsi_{l {{\mathbf{x}}\prime}} | f \right\rangle = \int\limits_{R^{n}} f({{\mathbf{x}}}) \Uppsi_{l {{\mathbf{x}}}\prime}^{*}({{\mathbf{x}}}) \hbox{d}^{n} {{\mathbf{x}}} $$
(9)
where Ψ* denotes the complex conjugate of Ψ. x′ describes a translation parameter corresponding to the position and l is the dilation parameter or the scale of the wavelet. As mentioned above, the two dimensional Marr wavelet is a common choice for the detection of coherent structures. It is the second derivative of a Gaussian and reads in polar coordinates
$$\Uppsi_{l}\left({r}\right) =\frac{1}{l} \left({2-\frac{r^{2}} {l^{2}}}\right) \, \hbox{exp} \, \left({-\frac{r^{2}}{2l^{2}}}\right). $$
(10)
The choice of this mother wavelet is consistent with the assumption of a Gaussian distribution of vorticity in the coherent structures. One solution of the Navier–Stokes equations is the so called Lamb–Oseen vortex featuring a Gaussian vorticity distribution over its radius as shown in Fig. 3. This vortex model is defined as follows
$${u_{\phi}}\left(r\right) =\frac{\Upgamma}{2 \pi r} \; \left[{1-\hbox{exp}{\left(\frac{r^{2}}{2\sigma^{2}}\right)}}\right] $$
(11)
$$\omega\left(r\right) = \frac{\Upgamma}{2 \pi \sigma^{2}} \, \hbox{exp}{\left(\frac{-r^{2}} {2\sigma^{2}}\right)}. $$
(12)
uϕ denotes the tangential velocity, Γ the circulation and σ the size of the vortex, respectively. The size of the vortex core Dc is usually defined by the maximum of the tangential velocity and can be determined to be Dc = 3.17 σ. The wavelet coefficient for a Lamb–Oseen vortex is defined to be
$$ \begin{aligned} \, \left\langle \Uppsi_l | \omega \right\rangle =& 2\pi \int\limits^{\infty}_{0} \underbrace{ \frac{\Upgamma}{2 \pi \sigma^{2}} \, \hbox{exp}{\left(\frac{-r^{2}} {2\sigma^{2}}\right)}}_{\omega}\\ & \cdots \underbrace{\frac{1}{l} \left({2-{\frac{r^{2}} {l^{2}}}}\right) \, \hbox{exp} \left({-\frac{r^{2}}{2l^{2}}}\right) r}_{\Uppsi_l} \hbox{d}r\\ \end{aligned} $$
(13)
which gives for the derivative of the scale
$$\frac{\partial\left\langle \Uppsi_l | \omega \right\rangle}{\partial l} = -\frac{2\Upgamma l^{2}\left({l^{2}-3\sigma^{2}}\right)} {\left({l^{2}+\sigma^{2}}\right)^{3}} $$
(14)
and therefore a maximum for \(l / \sigma = \sqrt{3}.\) Using this relation, it is possible to associate a certain vortex core diameter to the maximum wavelet coefficient. In practice, the vorticity field is analysed by a set of wavelets with different scales. In the upper part of Fig. 4 Mexican hat wavelets of three different sizes are plotted together with the corresponding vorticity of the Lamb–Oseen vortex. Ψ2 has optimal size, resulting in the maximum wavelet coefficient for r = 0. The wavelet coefficients for all three scales are marked in the bottom part of Fig. 4. For both scales \(l \neq \sqrt{3}\sigma,\) the coefficient is smaller than that computed for the optimal size. The range of wavelet scales is discretized in 70 steps. In order to adapt the wavelet scales to the actual flow situation, the algorithm calculates size limits of possible structures in advance for every single vorticity field. A kind of “equivalent diameter” is determined for connected regions of negative λ2. The scales for the test wavelets are then preadjusted using this equivalent diameter. This approach allows to cover the proper scale range with best resolution. Additionally, the method decreases the computational time and enhances the accuracy of the algorithm.
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Fig. 3

Vorticity ω and absolute tangential velocity uΦ for a Lamb–Oseen vortex with Γ = 1 and σ = 0.1

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Fig. 4

Vorticity for the Lamb–Oseen vortex and wavelets Ψl for \(l_{1} = \frac{\sqrt{3}}{2}\sigma, l_{2} = \sqrt{3}\sigma\) and \(l_{3} = \frac{3\sqrt{3}}{2}\sigma\) (top) and corresponding wavelet coefficients \(\left\langle \Uppsi | \omega \right\rangle\) (bottom)

A second criterion is applied to validate that the detected structure resembles indeed a vortex. The absolute value of the wavelet coefficient \(\left\langle \Uppsi_l | \omega \right\rangle \) depends on the size of a vortex and its strength. From Eq. 12 the circulation Γ at the vortex centre reads
$$\Upgamma = 2\pi\sigma^{2} \omega_0. $$
(15)
Using the relation \(l / \sigma = \sqrt{3}\) and Γ0 in Eq. 13, the maximum wavelet coefficient depends only on the wavelet scale l and the vorticity ω0 in the centre of the vortex,
$$\left\langle \Uppsi_l | \omega \right\rangle_{th}= \frac{3}{4} \pi \omega_0 \, l. $$
(16)
The ratio β between the detected wavelet coefficient and the theoretical one can be calculated with ω = ω0 as
$$\beta =\frac{\left\langle \Uppsi_l | \omega \right\rangle} {\left\langle \Uppsi_l | \omega \right\rangle_{th}}. $$
(17)
In the following, the β-criterion was set to have at least an 85% agreement of the Lamb–Oseen vortex model and the detected structure.
In summary, a vector field is read, then the vorticity is calculated and thresholded using the λ2-criterion. The range of wavelet scales is adjusted using characteristic sizes of connected regions with λ2 < 0. Wavelets with the selected scales are then calculated. The vorticity field and the wavelets are Fourier transformed using a 2D FFT. Multiplication and back transformation yields the fields of wavelet coefficients for each scale. The maxima among the scales are determined and the reliability, i.e. β-criterion is applied. If a structure is found to comply with the criterion, it is counted as a vortex and certain flow variables are extracted. In Fig. 5 a snapshot of the vorticity field of the unforced separated flow is shown. The circles mark the core size of the detected structures.
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Fig. 5

Snapshot of the unforced separated flow. Filled contours show vorticity and the circles the size of the detected structures

3 Results

The main focus of the present study is to investigate the vortex interactions in the case of pulsed excitation. To motivate these flow field studies, it seems useful to begin with a discussion of some integral force measurements on a stalled NACA 0015 under pulsed excitation. Figure 6 shows the time averaged lift coefficient for a NACA 0015 at post stall (α = 20°, Re = 105) using periodic excitation with pulsed wave forms. A discernible increase in lift compared to the baseline case is observed for excitation with any wave form. In the upper diagram, the effect of the excitation frequency is given. The forcing shows significant differences in lift for different excitation frequencies. At the same time, the lift increase depends on the pulse width. For these measurements, the effective momentum coefficient was kept constant at cμ = 0.15 %. The scaling of the lift gain with the peak momentum coefficient \({\hat{c}}_{\mu}\) found by Weier and Gerbeth (2004) can be recognized in the lower part of the figure as well. \({\hat{c}}_{\mu}\) is defined in analogy to cμ, but instead of the rms value of the applied current density, the peak current density is used. Again, it is clearly visible that the scaling also depends on F+.
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Fig. 6

Lift coefficient at a NACA 0015 at α = 20°, Re = 105 and cμ = 0.15% over excitation frequency for sinusoidal and rectangular wave forms of excitation. Waveforms are given in the insert. DC indicates the length of the duty cycle. Grey symbols mark F+ = 0.75, filled symbolsF+ = 1 and open onesF+ = 1.25

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%.

To obtain some information on the flow structures associated with pulsed excitation, PIV measurements on an inclined flat plate (α = 13°, Re = 1.04 × 104) were performed. Under these conditions, the plate is in deep stall. In Fig. 7, the time averaged length of the separation region LS versus excitation frequency is shown for sinusoidal excitation as well as for excitation with pulses of a one third duty cycle (DC = 1/3, see the insert in Fig. 6 for a graphical representation of the pulsed wave forms). LS was determined by adding up regions of negative streamwise velocity on a line 2.5 mm above and parallel to the plate. For cμ = 2.6 % the flow is almost fully attached for both wave forms up to F+ = 3. Under sinusoidal excitation, LS increases from LS/c = 0.02 at F+ = 4 to LS/c = 0.72 at F+ = 6. An excitation with frequencies F+ ≥ 6 is no longer able to reattach the flow. Under pulsed excitation with DC = 1/3 complete separation is re-established already at lower frequencies.
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Fig. 7

Normalized length of the time averaged separation bubble for excitation with one third duty cycle pulses and sinusoidal wave form. cμ = 2.6%

In Fig. 8 the streamwise component of the velocity is shown for a one third duty cycle and different frequencies. The contour lines represent regions with negative velocity, i.e. regions of back flow. The unforced flow exhibits a large separation region above the whole plate. Forcing with F+ = 0.5–3 leads to a very small or even non-existing separation region. With further increasing excitation frequency the effect on the flow decreases. The separation region re-establishes. As expected from Fig. 7, high frequencies are less effective in terms of separation suppression.
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Fig. 8

Mean streamwise velocity for the unforced flow (top left) and excitation with one-third duty cycle pulses for cμ =  2.6% and different F+. Contour lines mark regions of negative streamwise velocity, i.e. regions of backflow

Phase averaged vorticity fields are shown in Fig. 9. The excitation signal has been used as a reference signal for phase averaging the velocity field in 20 bins. The vorticity was then calculated from the phase averaged velocity fields. The contour lines in Fig. 9 indicate regions of negative λ2, i.e. vortices in a phase averaged sense. On the left hand side of Fig. 9, the results for F+ = 1 and on the right hand side for F+ = 4 are depicted. The actual phase of the forcing signal is shown in the small insert in the lower left of each plot.
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Fig. 9

Phase averaged ωz-contours for excitation with one third duty cycle pulses and cμ = 2.6% for F+ = 1 (left) and F+ = 4 (right). Contour lines mark regions of negative λ2

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.

Already Brown and Roshko (1974) have studied vortex merging processes in plane mixing layers by analyzing vortex trajectories extracted from their famous flow visualisations. A very similar approach was followed here. By calculating the probability density function (pdf) of the x-position versus the observation time of a structure, vortex trajectories in a statistical sense are determined. The observation time is normalized by the cycle duration T. Figure 10 shows probability density functions for different excitation frequencies. While the duty cycle is kept at one third as in Fig. 9, the time axis is shifted in such a way that the Lorentz force acts upstream for 0 ≤ t/T ≤ 1/6 and downstream for 1/2 ≤ t/T ≤ 2/3.
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Fig. 10

Probability density function for the occurrence of vortices in the unforced case and for excitation with one third duty cycle pulses of different frequency

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 fu/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.

Since the algorithm fails to detect strongly deformed vortices, it is useful to have a closer look on the single snapshots in order to verify the vortex merging process discussed above. Figure 11 provides a x, t plot clipped from a measurement taken under excitation with one quarter duty cycle pulses and F+ = 1. The upper part of the figure shows detected vortex locations during 40 s of measurement. The total measuring time was limited to 106 s by the available memory in the high speed camera. A zoom of the data corresponding to one excitation period is given in the lower part of Fig. 11. The symbols size and grey value corresponds to the core diameter of the detected structure. The time values marked by dashed lines relates to the forcing cycle as follows: at t1 = 5.2 s downstream forcing ends, at t2 = 5.85 s upstream forcing starts, and t3 = 6.5 s is in the following off time.
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Fig. 11

Vortex trajectories for pulsed excitation with a one quarter duty cycle at F+ = 1 and cμ = 2.6%, timespan t = 0–40 s (top), close up (bottom). Grey scale and size of the symbols correspond to the core diameter DC of the detected structure

Figure 12 shows the vector fields and vorticity contours measured at the three time values marked in Fig. 11. The circles correspond to the detected structures.
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Fig. 12

Snapshots of the vorticity for pulsed excitation with a one quarter duty cycle at F+ = 1 and cμ = 2.6%

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.

At t3, two vortices with different convection velocities are detected. The one at x ≈ 40 mm was shed from the detached shear layer during upstream forcing. It is therefore slower than the one at x ≈ 25 mm. Due to their different velocities they will eventually merge. The influence of the duty cycle on the vortex location can be seen in Fig. 13. Similar to Fig. 10, it shows vortex trajectories in a statistically averaged sense, but for excitation with F+ = 1, cμ = 2.6% and varying duty cycle. Essentially, the same sequence of vortex shedding as discussed above for the one quarter duty cycle can be found for the different pulse durations. However, the shorter the duty cycle the stronger is the vortex interaction, resulting in a larger spreading of the probability density functions. Vortex merging and interaction takes place at a broader range of streamwise coordinates and time instants from a statistical point of view.
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Fig. 13

Probability density function for the occurring of vortices for excitation with F+ = 1 and cμ = 2.6% for pulses with different duty cycle

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.

Acknowledgments

Financial support from Deutsche Forschungsgemeinschaft (DFG) in frame of the Collaborative Research Centre (SFB) 609 is gratefully acknowledged.

Copyright information

© Springer-Verlag 2008