Experiments in Fluids

, Volume 37, Issue 1, pp 22–28

Forcing a planar jet flow using MEMS


    • Department of Mechanical EngineeringMIT
  • Elizabeth Bradley
    • University of Colorado
  • Jean Hertzberg
    • University of Colorado
  • Yung-Cheng Lee
    • University of Colorado

DOI: 10.1007/s00348-004-0780-8

Cite this article as:
Peacock, T., Bradley, E., Hertzberg, J. et al. Exp Fluids (2004) 37: 22. doi:10.1007/s00348-004-0780-8


We present the results of an experimental study in which a planar laminar jet of air was forced by an array of micro-electromechanical systems (MEMS) micro-actuators. In the absence of forcing, the velocity profile of the experimental jet matched the classic analytic solution. Driving actuators on either side of the jet in-phase or anti-phase, respectively, excited the symmetric or anti-symmetric mode of instability of the jet. Asymmetric forcing, using MEMS actuators on only one side of the jet, was also investigated.

List of symbols

x, y, z

Streamwise, cross-stream, and spanwise coordinates


Streamwise velocity


Exit slit width


Centerline streamwise velocity


Jet half-width


Kinematic viscosity


Reynolds number=U0b/v


Momentum flux


Fluctuating velocity component in x direction


Forcing frequency of MEMS actuators


Dimensionless frequency of velocity fluctuations=fb/U0

1 Introduction

The prediction and control of fluid flows is a major research activity. The ability to do so, particularly in aerospace and combustion applications, can generate huge financial and environmental savings. Many experimental studies of flow control have been performed using conventionally machined actuators. For examples of this, see Wiltse and Glezer (1993), Jacobson and Reynolds (1998), and Wiltse and Glezer (1998). Recently, however, there has been much interest in using MEMS-based (micro-electromechanical systems) technology for flow control, as it offers several advantages over conventional machining, such as batch processing of devices on the micron scale. Reviews of the use of MEMS devices for flow control are given by Ho and Tai (1998) and Löfdahl and Gad-el-Hak (1999).

Here, we present the results of an experimental study in which MEMS micro-actuators were used to force a low Reynolds number planar jet of air, which is realized by considering a cross-section of flow out of a long thin slit. The intent of this study is to undertake a basic, qualitative study of the effect of some different types of forcing upon this system. This experiment is part of a larger research plan, whose goal is to obtain closed-loop control of coherent features of fluid flows, using a combination of experimental, numerical, and analytic techniques. We chose to work with an idealized flow in the interests of reducing the simulation burden and simplifying the process of closing the loop between simulation and experiment, i.e., assimilating data directly into the numerical code. Axisymmetric jet flow, in contrast, possesses a more intricate family of instabilities (Cohen and Wygnanski 1987), and high Reynolds number flows would provide a much more significant challenge for numerical simulation.

The 2D jet is a canonical problem in fluid dynamics that has been studied for many different purposes, and it remains a test bed problem for numerical, theoretical, and experimental techniques. To the best of our knowledge, however, this is the first modern experimental study of instability in the low Reynolds number planar jet. Since the early work of Sato (1960) on the planar jet, there has been significant progress in understanding forced laminar and turbulent shear layers in different geometries (Ho and Huerre 1984). Much of the theoretical advancement has focused on the notions of absolute and convective instability, and the ability to consider a spatially varying base flow (Huerre and Monkewitz 1990). Experimental work on the planar jet has almost exclusively focused on high Reynolds number flows (Huang and Hsiao 1999), which are relevant for engineering applications and have the mathematical benefit of permitting the assumption of a parallel base flow.

The velocity profile of a planar jet was first obtained by Schlichting (1933) and Bickley (1937), using a boundary layer approximation, and is commonly referred to as the Bickley jet. This profile gives an accurate description of two-dimensional jet flow, provided one is not too close to the source of the jet, and was corroborated experimentally by Andrade (1939). Sato and Sakao (1964) noted that, in experimental studies, the velocity profile at the exit slit is likely to be parabolic, but evolves into that of the Bickley jet further downstream.

The planar jet is unstable when the Reynolds number, based on the half-width and peak velocity of the jet, exceeds a critical value (Andrade 1939). Small-scale velocity disturbances are amplified as they progress downstream, generating coherent structures that dominate the flow field. Linear stability analysis, assuming a parallel flow, results in the well-known Orr-Sommerfeld equation. For a summary of the linear stability analysis and its results, see Drazin and Reid (1981). Solving the Orr-Sommerfeld equation, subject to the appropriate boundary conditions, shows that the planar jet is unstable to two different modes, one symmetric about the jet centerline and the other anti-symmetric (Howard 1958). The perturbation velocity of the symmetric mode has a maximum on the jet centerline, and oscillations on either side are in phase. In contrast, the perturbation velocity is zero on the jet centerline for the anti-symmetric mode, with oscillations on either side of the jet centerline in anti-phase. As the Reynolds number is increased, the first mode predicted to become unstable is the anti-symmetric mode, at Re≈4; the symmetric mode is predicted to remain stable until Re≈80. For this reason, experimental studies have assumed that the growth rate of the anti-symmetric mode exceeds that of the symmetric mode for all Reynolds numbers. This assumption was later corroborated numerically by Silcock (Drazin and Reid 1981). Linear stability analysis that accounts for a non-uniform base flow was performed by Garg (1981). The spreading of the base flow was found to significantly increase the predicted critical Reynolds number for the anti-symmetric mode. An intriguing aspect of the theory is that different quantities grow at different rates, with kinetic energy typically being the quantity considered most relevant for experiments. Furthermore, a growth rate can no longer be simply assigned to a mode. Rather, for the quantity under consideration, one must determine a growth factor between two specific points in the jet.

Instabilities can be excited using acoustic forcing, an effect first reported for the planar jet by Brown (1935), who was interested in the phenomenon of sensitive flames (LeConte 1858). It was Sato (1960), however, who first observed the unstable modes, using a hotwire anemometer. A loudspeaker was placed near the base of the jet and the growth rate and propagation velocity of disturbances were measured for different frequencies of excitation. For Reynolds numbers between 500 and 1500, the instabilities of the initially laminar flow were excited acoustically, and good agreement was found with numerical results obtained using the Orr-Sommerfeld equation. At Reynolds numbers around 50, however, Sato and Sakao (1964) were unable to excite the symmetric mode of the planar jet using acoustic forcing, or make a close comparison with theoretical predictions of the growth rates (Tatsumi and Kakutani 1958).

Using MEMS micro-actuators, we have been able to selectively excite both the anti-symmetric and symmetric instabilities of the planar jet at low Reynolds numbers. This was possible because the micro-actuators allowed us to provide localized disturbances, with an arbitrary phase relationship, on either side of the experimental jet. This is in contrast to acoustic techniques, for which the entire flow field is subject to forcing of uniform phase. Furthermore, we have been able to use actuators to generate asymmetric disturbances by providing forcing on only one side of the exit nozzle. We proceed, in Sect. 2, with a description of the experimental jet facility and the MEMS devices. In Sect. 3, we present results for the steady flow, demonstrating that a Bickley jet is realized experimentally. Then, in Sect. 4, we describe the effects of the micro-actuators upon the flow.

2 Experimental arrangement

A schematic of the experiment is presented in Fig. 1. Filtered air entered the base of a plenum, whose internal cross-section was 400 mm by 15 mm, and was passed through three flow treatment screens constructed using 24 mesh with 0.0075-in. diameter wire. Upon reaching the top, the air entered a matched cubic contraction of area ratio 6 to 1, shown as the shaded area in Fig. 1, to further reduce turbulence and produce a more uniform flow. Flow emerged from an exit slit 400 mm long and 2.5±0.01 mm wide, shown in both Figs. 1 and 2, which we define to lie in the yz plane. The system was highly sensitive to mechanical vibrations and motion of the ambient air. To minimize these effects, the entire experiment stood on vibration control mounts and was semi-enclosed by a 1-m3 plexiglass box.
Fig. 1

Schematic of the experimental apparatus. This side view shows the plenum and the contraction

Fig. 2

Schematic showing MEMS actuators positioned along either side of the exit slit. The actuators, which were mounted on ceramic substrates, are shown positioned flush with the edge of the exit slit. The flaps deflect downward from the shown position. All dimensions are in mm

A TSI 1210 hotwire probe, in conjunction with a TSI 1010 hotwire anemometer, was used to measure the velocity profile and velocity fluctuations in the jet. The probe was mounted on translation stages and could be moved in the streamwise (x) direction and cross-stream (y) directions with an accuracy of 0.025 mm. The hotwire was operated by a constant temperature method, and its output digitized using a 12-bit A/D converter. First, the centerline velocity was measured along the length of the exit slit, and was found to be uniform to within 2%, supporting our assumption of a 2D flow near the exit slit. Thereafter, hotwire measurements of both mean and fluctuating velocity profiles were made in the xy plane across the center of the exit slit. The peak velocity could be varied in the range from 0.1 to 2.0 m/s, giving Reynolds numbers of the order of 10 through 170, based on the centerline velocity at the exit slit and the half-width of the slit. Over this range, the turbulent intensity of the jet, in the absence of any forcing, was less than 0.5%. Large-scale images of the flow were obtained by seeding the air supply with fine oil droplets and illuminating the jet with an argon-ion sheet laser beam.

External forcing of the jet was provided by arrays of MEMS devices positioned on either side of the exit slit, as shown in the schematic in Fig. 2. The individual actuators, of length 0.98 mm and width 0.28 mm, were fabricated using the commercial MUMPS surface micro-machining process (for details of the MUMPS process, see http://www.memsrus.com); a SEM photograph of one of the actuators is presented in Fig  3. Each was constructed from two layers of polysilicon of thickness 1.5 µm and 2.0 µm, and operated using the bi-layer effect (Löfdahl and Gad-el-Hak 1999): passing a current through the two layers of polysilicon caused them to be differentially heated and, therefore, expand by different amounts, which caused the flap to deflect downward. A more detailed description of the micro-actuators is given in Ma et al. (2001; 2003).
Fig. 3

SEM image of the one of the MEMS micro-actuators, taken from above

The MEMS devices were mounted on a ceramic substrate using a “flip-chip” solder assembly technique (Lee and Basavanhally 1994). In this technique, an electrode pattern is etched onto a ceramic substrate, and the pattern is wetted with Pb/Sn solder. The MEMS devices were placed atop the solder pads, where surface tension served to align the actuators on the substrate. The gap between the surface of the ceramic and the MEMS flap is set by the quantity of solder used; in this case, the gap was of the order of 50 µm. Custom-built micro-positioning devices were then used to position the ceramic substrate along the edge of the exit slit. The substrates could be arranged so that the actuators were either mounted flush with exit slit, or intruding into the flow by a known distance. In the experiments described here, arrays of ten micro-actuators, with a combined length of 1 cm, were positioned on either side of the exit slit. As the actuators only covered the central part of the nozzle, 3D effects were inevitable, and evident on flow visualization, even when the actuators were stationary.

The response characteristics of the MEMS actuators are presented in Fig. 4. These were obtained by reflecting a HeNe laser beam off the surface of a vibrating flap and tracking the displacement of the reflected beam. The resonant frequency of an actuator was around 25 Hz, as shown by the results in Fig. 4a, which were obtained for a driving voltage of 5 V; above this value, the amplitude of oscillation decreased monotonically. The results presented in Fig. 4b show that, when the driving voltage was increased from 5 V to 10 V, at a frequency of 25 Hz, the oscillation amplitude increased linearly from around 10 µm to over 25 µm. The driving voltage could not be increased above 10 V, as the MEMS devices then suffered permanent damage. An estimate of the axial velocity disturbance that the actuators apply to a flow can be obtained by multiplying the amplitude of deflection by the angular frequency of oscillation. From the measurements presented in Fig. 4, this was approximately 7.5 mm/s, which compares favorably with the amplitude of disturbances provided by acoustic forcing (Sato and Sakao 1964). The amplitude of disturbances generated by the MEMS actuators could not be directly measured using the hotwire probe because the driving voltage for the actuators interfered with the hotwire signal when the two were placed in close proximity. The Reynolds number associated with the flaps, based on their displacement and the jet flow velocity, was of the order 1, so we expect them to provide an essentially laminar disturbance to the flow.
Fig. 4

The response characteristics of the MEMS actuators. a Deflection vs. frequency for a driving voltage of 5 V. b Deflection vs. voltage for a driving frequency of 25 Hz

3 The mean velocity distribution

Mean velocity distributions were measured using the hotwire anemometer at several different downstream locations, in the absence of any MEMS forcing. A representative set of results is presented in Fig. 5a, for which the peak velocity at the exit slit was 0.9 m/s. The velocity on the centerline of the jet decreased with downstream position, x/d, where d=2.5 mm is the exit slit width, and the width of the jet increased. Solid curves drawn through the data points are fitted velocity profiles of the form obtained by Bickley (1937):
$$u = U_{0} \operatorname{sech} ^{2} {\left( {\frac{{ay}}{b}} \right)}. $$
Fig. 5a, b

The unforced jet profile. a The streamwise velocity measurements at three different downstream locations, plotted as a function of cross-stream position. The solid lines are fitted Bickley profiles. b The non-dimensionalized profiles; the solid line is the theoretical profile of Bickley

Here, U0 is the centerline velocity, a=0.88136 is a constant, and b is the half-width of the jet, i.e., the distance from the centerline to where the velocity has decreased to U0/2 . Note that at x/d=2.5, the profile is nearly that of the analytic solution. The local Reynolds number of the flow, defined as Re=U0b/v, where v is the kinematic viscosity, increased from 64 to 75 between x/d=2.5 and 10.0.

The self-similarity of the experimental jet is demonstrated in Fig. 5b, in which the velocities presented in Fig. 5a have been non-dimensionalized by the maximum speed, U0, at each downstream location, and the cross-stream position, y, has, likewise, been non-dimensionalized by the half-width of the jet, b . The individual profiles collapse on top of one another, demonstrating geometrical similarity. Furthermore, they agree well with the theoretical profile of Bickley, which is the solid curve drawn through the data.

The spatial variation of the peak velocity and jet half-width is presented in Fig. 6. The analysis of Bickley (1937) predicted that \(U_{0} ^{{ - 3}} \) and \(b^{{3/2}} \) are both proportional to the downstream position, x: \( U_{0} = {\left( {3M^{2} /32\nu x} \right)}^{{1/3}} \) and \( b = a{\left( {48\nu ^{2} x^{2} /M} \right)}^{{1/3}} \), where
$$M = {\int_{ - \infty }^\infty {u^{2} {\text{d}}y} } $$
Fig. 6

The streamwise variation of the centerline velocity and half-width with x. The virtual origin of the jet is at x=–0.009±0.001 m

is related to the momentum flux of the jet. The experimental results presented in Fig. 6 are in good agreement with this prediction. The linear fits to these results, which were obtained using a least-squares method, intersect the abscissa near x=–0.009 m. This point of intersection is the virtual origin of the jet: the location of a point source of momentum from which the jet can be considered to originate.

4 Instability and forcing

The aim of the experiments described in this paper is to demonstrate that MEMS actuators make it possible to excite both the anti-symmetric and symmetric modes of the planar jet. Marginal stability curves for the symmetric and anti-symmetric modes, determined from the linear stability analysis assuming parallel flow, are presented in Fig. 7. The curves were obtained by numerically solving the Orr-Sommerfeld equation using a Newton-Raphson relaxation scheme, and are in agreement with those presented in Drazin and Reid (1981). In Fig. 7, the ordinate is the dimensionless frequency, ω, and the abscissa is the Reynolds number; the centerline velocity and jet half-width have been used as the characteristic velocity and length scales. It can be seen that the planar jet is unstable to anti-symmetric disturbances when Re≈4. At this Reynolds number, the linear stability analysis is no longer valid, however, as the flow is not parallel. Experimentally, the critical Reynolds number is found to be of the order of 30 (Andrade 1939; Sato and Sakao 1964) and this is corroborated by more recent stability calculations (Garg 1981), which account for slow spreading of the jet. The critical Reynolds number for the symmetric mode is predicted to be Re≈80 (Drazin and Reid 1981).
Fig. 7

The marginal stability curves for the anti-symmetric and symmetric modes

The Reynolds number for our experiments, based on the jet half-width and centerline velocity, increased with downstream position. We chose a characteristic Reynolds number of 70 for the experimental flow, based on the nozzle conditions. At this Reynolds number, according to linear stability theory, the anti-symmetric mode is unstable and the symmetric mode is weakly stable. In this case, it should be possible to observe both modes, depending on the type of forcing used, without having to vary the flow conditions. We note that, in addition to generating perturbations, the MEMS flaps had an effect on the mean flow profile. When activated, they reduced the centerline velocity and increased the half-width of the jet. This effect was small, however, and the streamwise velocity at a given location never differed by more than 5% of the unforced value, and was typically around 1% less.

4.1 Anti-symmetric forcing

To excite the anti-symmetric mode, the actuators on either side of the exit slit were driven in anti-phase, providing velocity disturbances of equal and opposite amplitude on either side of the jet centerline. It was found that the anti-symmetric mode could be readily excited with the actuators mounted flush with the edges of the exit slit, so that they did not intrude into the flow.

The RMS fluctuation profile at x/d=5 is presented in Fig. 8, for a forcing frequency of 40 Hz. This corresponds to a non-dimensional frequency ω=0.44 at x/d=5. At this frequency, the amplitude of velocity forcing was estimated from frequency response measurements of the MEMS actuators to be 7 mm/s (that is, approximately 1% of the mean centerline flow). We note that, at this downstream location, the measured velocity fluctuations were approximately 3% of the centerline velocity, due to a combination of the spatial growth of disturbances and a decrease in U0. The results are characteristic of the anti-symmetric mode: there is a minimum on the jet centerline, where the fluctuation amplitude is predicted to be zero, and a maximum on either side. The oscillations on either side of the jet were in anti-phase. The solid curve drawn through the experimental data is a numerical RMS fluctuation profile, obtained by solving the Orr-Sommerfeld equation for the appropriate Reynolds number at x/d=5 (Re=64). Although the amplitude of the numerical solution is arbitrary, as it is an eigenfunction of the Orr-Sommerfeld equation, the geometrical form is not. The experimental results do not extend far enough to determine the existence of a second maximum on either side. However, in contrast to the numerical results, there does not appear to be any minima away from the centerline in the experimental results. This disagreement between experiment and theory was also observed by Sato (1960) for high Reynolds number jet flow.
Fig. 8

The anti-symmetric velocity fluctuation profile at x/d=5. There is a minimum on the jet centerline and a maximum either side. The solid line is the corresponding numerical solution

The flow could be excited over a wide range of frequencies. Fourier spectra of time series taken from the jet for forcing frequencies of 5 Hz (ω=0.055) and 50 Hz (ω=0.55) are shown in Fig. 9. The spectra are dominated by peaks at the driving frequency and their respective harmonics, demonstrating that the flow responds selectively to the imposed forcing. For a given forcing frequency, the response of the system depends on two factors: the response of the MEMS flaps to a particular driving frequency and voltage, and the growth rate of the unstable mode. It is worth noting that there is a secondary peak in the Fourier spectra at a frequency of 16.8 Hz. We believe this is due to mechanical vibrations in the experiment, as it persisted even when the Reynolds numbers was changed significantly. Coincidentally, however, it corresponds closely to the frequency of the most unstable mode at Re≈70.
Fig. 9a, b

Fourier spectra plotted on a logarithmic scale, showing the response of the jet to anti-symmetric forcing. The time series were taken at x/d=5 and y=1 mm. a 5 Hz forcing (ω=0.055). b 50 Hz forcing (ω=0.55)

4.2 Symmetric forcing

The symmetric mode of instability has previously been observed for a high-speed flow (Sato 1960), for which the velocity profile was flat near the centerline, but has never been reported for the low Reynolds number planar jet. In our experiment, this mode was excited by driving actuators on either side of the exit slit in-phase, providing an equivalent disturbance on either side of the jet centerline.

The RMS fluctuation profiles at x/d=2.5 (Re=61) and x/d=5 (Re=64) are presented in Fig. 10. In this case, it was found that the mode could not be suitably excited with the actuators mounted flush with the exit slit. Instead, the flaps were arranged so as to intrude into the flow (0.28 mm) at their full extent. This provided a more significant perturbation at the regions of strongest shear within the jet, which are most susceptible to disturbances. This was necessary because the symmetric mode is weakly stable at the Reynolds numbers investigated, and therefore inclined to decay as it progresses downstream. The driving frequency was 58 Hz; this corresponded to ω=0.64 at x/d=5. Near the exit slit, at x/d=0.25 (ω=0.46), there is still evidence of the initial forcing, as there is a velocity maximum on either side of the jet centerline. However, no longer are fluctuations absent from the jet centerline. Rather, the amplitude of oscillation on the centerline is comparable to those maxima on either side. Upon reaching five gap-widths downstream, the RMS fluctuation profile evolved into that of the symmetric mode, with a velocity maximum on the centerline. The solid curve drawn through the results is the corresponding numerical solution, which agrees very well with the experimental results. No secondary maxima were observed in the experiment, and the velocity remained essentially zero beyond y=±2 mm. Significantly, the amplitude of fluctuations on the centerline decreased between x/d=2.5 and x/d=5, indicating that the mode was indeed stable, in agreement with the predictions of linear stability theory.
Fig. 10

The symmetric mode at x/d =2.5 and 5. Fluctuations are a maximum on the jet centerline and decay away on either side. The amplitude of the mode decays as it progresses downstream, as it is stable. The solid line is the numerical solution at the appropriate Reynolds number for x/d =5

4.3 Asymmetric forcing

An important difference between these experiments and those using acoustic forcing is that the MEMS devices provided a localized disturbance at the exit slit, rather than uniformly forcing the entire flow field. It was, therefore, possible to selectively excite either the anti-symmetric or the symmetric mode. However, we need not limit ourselves to these two scenarios; we can also investigate the effect of imposing an arbitrary phase difference between the two actuators. Here, we briefly describe the results of the most straightforward application of these possibilities, that is, only forcing one side of the jet using the MEMS actuators.

The results of asymmetric forcing are presented in Fig. 11 for a driving frequency of 18 Hz (corresponding to ω=0.143 at x/d=2.5 and ω=0.177 at x/d=10.0). The RMS fluctuation profile was initially strongly asymmetric, due to the provision of disturbances on only one side of the exit slit. However, disturbances on the forced side of the jet grew very little as they progressed downstream. In contrast, the fluctuation amplitude on the unforced side of the jet increased significantly until, at x/d= 10, the amplitude of disturbances on either side of the jet centerline became balanced. Again, the solid curve drawn through the results is the corresponding numerical solution. We see that at x/d=10 the anti-symmetric mode had been regained. Within the context of linear stability theory, the symmetric mode, being stable, decayed to the point that it could no longer be observed. The anti-symmetric mode, in contrast, being unstable, grew to dominate the system.
Fig. 11

The effect of asymmetric forcing. Fluctuations are initially maximum on the side of the jet subject to forcing. The profile then becomes increasingly balanced as it progresses downstream

5 Vortex structures

Large-scale flow features were visualized using Canola oil aerosol and a laser sheet, as shown in Fig. 12. In both images, the characteristic Reynolds number for the jet is 25, based on the half-width of the exit slit and the jet velocity at the exit slit. The positioning device for the MEMS actuators can be seen in the lower right corner of each image. In Fig. 12a, we show an image of the unforced jet. The flow is laminar, and there is little entrainment of the surrounding air into the jet, although it is marginally unstable to the anti-symmetric mode. In Fig. 12b, we show an image of the same jet, subject to asymmetric forcing at 6 Hz (ω=0.207) by MEMS actuators. As the perturbations described in the previous sections progress downstream, the instabilities excited by the forcing give rise to large-scale vortices, similar to those observed by Brown (1935). These vortices provide enhanced mixing of the jet with the ambient air.
Fig. 12a, b

Images of the jet at Re=25. a Unforced. b 6 Hz forcing

These observations suggest a wealth of interesting experiments concerning the growth of disturbances as they travel downstream, perhaps the most significant being to develop an understanding of what effects the different forms of forcing have on the nonlinear development of the flow. We are currently performing a wide-ranging, systematic experimental study to address these issues.

6 Conclusions and further work

The aim of the experiments described in this paper is to demonstrate that MEMS actuators make it possible to excite both the anti-symmetric and symmetric modes of the planar jet. This has been accomplished by using a linear array of 1-mm flaps on either side of the jet nozzle. In the absence of any excitation, the mean velocity profile of the jet was found to be in good agreement with theoretical predictions. The mean velocity profile was unstable to anti-symmetric disturbances, and we were able to selectively excite this instability over a range of different frequencies using the MEMS micro-actuators. By providing symmetric disturbances to the base flow, it was also possible to excite the symmetric mode. This mode was subsequently observed to decay as it progressed downstream, as it was stable for the Reynolds number used in this study. A novel feature of the experiment described here was the ability to provide asymmetric forcing to a planar jet. It was found that an initially asymmetric disturbance developed into the anti-symmetric mode of instability.

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