On the mode development in the developing region of a plane jet

PHYSICS OF FLUIDS VOLUME 11, NUMBER 7 JULY 1999 On the mode development in the developing region of a plane jet Jiann-Min Huang a) Aeronautical Research Laboratory, Chung Shan Institute of Science and Technology, Taichung, Taiwan 400, Republic of China Fei-Bin Hsiao b) Institute of Aeronautics and Astronautics, National Cheng-Kung University, Tainan, Taiwan 701, Republic of China Received 6 March 1998; accepted 4 March 1999 The development of the mode arrangement in the developing region of an acoustically excited plane jet is extensively studied using hot-wire measurements. Two single-wire probes are located within the two shear layers of the jet to detect the flow structure patterns simultaneously. The jet is excited at its fundamental frequency in either varicose mode (m 0) or sinuous mode (m 1) condition. Experimental results show that different excitation mode conditions lead to different spreading rates and velocity fluctuation distributions in the developing flow field. The frequency spectral distributions for the m 1 excitation exhibit double peaks of f a and f b, rather than a single subharmonic f 0 /2 for the m 0 excitation. These three components constitute a relationship of ( f a f b )/2 f 0 /2. The mode switching phenomenon is found to be prominent under the m 0 case, while less pronounced for the m 1 case. The mode development is mainly governed by the evolution of the primary instabilities in the jet shear layers. 1999 American Institute of Physics. S1070-6631 99 03906-9 NOMENCLATURE A 0 amplitude of a disturbance E( f ) energy content of streamwise velocity fluctuations at specific frequency f 0 fundamental frequency f a, f b the double peaks induced at m 1 condition f frequency offset for the double-peak components with the subharmonic wave G 12 ( f ) cross-spectral distribution between points 1 and 2 H height of the plane jet at the exit m 0 varicose mode m 1 sinuous mode Re Reynolds number ( U 0 H/v) R 12 ( ) cross-correlation function between points 1 and 2 St 0 initial Strouhal number ( f 0 0 /U 0 ) t time scale U streamwise mean velocity U 0 mean velocity at the nozzle exit U c mean velocity at the jet center line u streamwise RMS velocity fluctuation u c streamwise velocity fluctuation at the jet centerline a Associate Researcher. b Author to whom correspondence should be addressed. Professor Fei-Bin Hsiao, Institute of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan 701, Republic of China. Tel: 886-6-275-7575 ext. 63667; Fax: 866-6-238-9940; electronic mail: fbhsiao@mail.ncku.edu.tw u p peak streamwise velocity fluctuation along the Y axis u ( f ) amplitude of streamwise velocity fluctuation at specific frequency u ( f )/U 0 integrated amplitude of the instability at specific frequency X, Y streamwise and transverse coordinates Y a transverse position where U au c, a 0.99, 0.9, etc. v kinetic viscosity 0 initial instability wavelength ( U 0 /2f 0 ) momentum thickness 0 initial boundary layer momentum thickness 12 ( f ) phase spectral distribution between points 1 and 2 relative phase difference of the excitation disturbances between the two jet shear layers relative phase difference between f 0 and f 0 /2 time delay in the cross correlation function wave number I. INTRODUCTION Research into the concept of coherent structures in turbulent shear flow has been carried out for more than a decade. 1 3 This concept is of technological importance for the control of aerodynamic noise and the enhancement of flow mixing. It has been proven that the free shear flow having inflection points is always inviscidly unstable due to Rayleigh instability, and the initial unstable disturbances grow exponentially with the downstream direction. Through 1070-6631/99/11(7)/1847/11/$15.00 1847 1999 American Institute of Physics

1848 Phys. Fluids, Vol. 11, No. 7, July 1999 J.-M. Huang and F.-B. Hsiao nonlinear interactions, these amplifying instabilities can roll up into vortices, and then merge with each other to form larger coherent structures. Thus, a convenient approach to the study of the free shear flow is from the stability point of view. The evolution of the coherent structures can then be considered as a composition of interacting instability waves that propagate and amplify in the streamwise direction. Providing the ratio between the diameter or height of the jet and the initial momentum thickness is large, the initial region of the shear layers may be considered to be parallel and two dimensional. The initially most unstable mode is usually a symmetric or axisymmetrical one due to the geometrical symmetry of the plane jet or round jet. However, with further downstream distance, the antisymmetrical mode or helical mode becomes more prominent. In fact, both symmetrical or axisymmetrical and antisymmetrical or helical modes of the instability waves could have existed in the initial jet column. With the theoretical analysis, Mattingly and Chang 4 investigated the spatial growth of the instability waves in an axisymmetrical jet column. They found that the dominant disturbances in the initial region of the jet were axisymmetrical. With further downstream, where the thickness of the shear layer attained 55% of the radius of the potential core, the helical mode dominated the flow field. Similar results were also identified by others. 5 8 Plaschko 9 studied the inviscid spatial growth of the spiral modes for slowly diverging jets by using the linear stability analysis, suggesting that the evolution of the modes depended on the Strouhal number of the instability waves. When the Strouhal number was higher, the axisymmetrical mode could be amplified more strongly than the helical mode; while for a small Strouhal number situation, the amplification of the helical mode became stronger. From experimental investigations, Drubka 10 suggested that both axisymmetrical and helical modes could originate in the initial region of the axisymmetrical jet. They found that the frequency of the most amplified wave associated with the helical mode was about 20% higher than that of the axisymmetrical mode. With the aid of short-time spectral analysis, Corke et al. 11 also showed that these two fundamental modes did not exist at the same time or space. This suggested that each is a basin of attraction, which will suppress the existence of the other. The nondeterministic switching between these modes was observed to be the result of the response of the jet to stochastic input of axisymmetric or nonaxisymmetric disturbances. For the plane jet study, Sato 12 performed a complete experimental investigation on the plane jet transition. Both varicose symmetric and sinuous antisymmetric modes of instabilities were observed in the initial region of the shear layers. The occurrence of each mode was strongly dependent upon the jet exit velocity profiles. Varicose mode instability was found to be associated with the undeveloped flat exit mean velocity profiles, while more well-developed parabolic profiles led to the dominance of sinuous mode fluctuations. Hussain and Thompson 13 studied the near field of a plane air jet by introducing the sinusoidal acoustic perturbations. They found that the induced symmetric mode remains symmetric as it travels downstream, while the plane jet acts as a nondispersive waveguide. By applying the conditional phase technique to the time traces of the streamwise velocity in a plane jet, Hsiao 14 found that the dominance of the instability modes depended not only on the jet exit velocity (U 0 ), but also on the transverse positions. Hsiao and Ho 15 further solved the quasiparallel Rayleigh equation for the jet and compared it with the experimental results, discovering that before the first merging location, the varicose mode was dominant over the sinuous mode. After the first merging location, the sinuous mode grew faster than the varicose mode and became dominant in the shear layers. Apart from the symmetrical excitation, the jet flow dynamics can also be controlled by higher order modes excitation. Wat 16 studied the velocity and acoustic field of an axisymmetrical jet under axisymmetrical and helical excitation at its fundamental frequency. He suggested that both axisymmetrical and helical structures had similar influences on the jet column, and could exhibit similar effects on the acoustic field. Strange 17 investigated the structural response of a subsonic turbulent jet to the acoustic excitation in the form of circumferential modes with the azimuthal wave number m 0, 1, and 2. He found that the m 0 and m 1 mode had comparable axial growth rates. However, when the excitation was applied in a combination of the first (m 1) and the second (m 2) modes, substantial changes in the mean flow were evident over the first twelve diameters of the jet. More recently, Cohen and Wygnanski 8 observed that the mean flow of the jet would lose its axial symmetry by artificially introducing two different azimuthal modes at the same frequencies on the jet column. By forcing helical modes at a frequency near the most amplified, Corke and Kusek 18 found the enhanced growth of near-subharmonic modes, as well as numerous sum and difference modes. Their amplitude development could be well represented by the nonlinear amplitude equation. Thus, the employment of the active control on the jet flow field by using dual-mode acoustic excitation has become an attractive subject for manipulating and enhancing the entrainment and mixing effects of the shear layers. Although the variations of the instability modes play an important role in the structural dynamics of the jet flow, the mechanism for the mode switching phenomenon is still not well identified. The main objective of this paper is to investigate the characteristics of the instability mode developments in a plane jet. The local acoustic disturbances with a relative phase difference between the two sides of the plane jet exit are applied to manipulate the initial conditions of the jet column. Their effects on the instability interactions and the mode switching behaviors within the jet shear layers are extensively studied. The resulting mean flow properties are also compared to determine the enhancement of the shear flow entrainment. II. EXPERIMENTAL SETUP AND TEST CONDITIONS The experiments were performed in a plane jet system. The air supply facility consists of a noise reduction chamber together with an open-circuit tunnel. The wind tunnel includes a diffuser, a settling chamber, and a contraction

Phys. Fluids, Vol. 11, No. 7, July 1999 Mode development in the developing region of a plane jet 1849 FIG. 1. a Schematic representation of the jet flow field. b, c Visualized jet vortex arrays under different mode conditions, U 0 3.15 m/s. FIG. 2. Excitation level effects on the a peak fluctuation evolution with downstream direction, and b peak fluctuation measured at X 1.8 0 (m 0). nozzle. The jet exit velocity can be changed with a3hp variable speed centrifugal blower. The noise reduction chamber is designed to reduce the passage noise generated from the blower. The jet nozzle is a fifth-order polynomial profile with a height of 15 mm and a width of 300 mm at the exit. The area contraction ratio is 20. One honeycomb and eight fine screens are installed throughout the tunnel for further managing the flow quality. Throughout the experiments, the jet is externally excited on the both sides of the trailing edge of the nozzle. Twenty earphones used as loudspeakers are placed on both sides of the nozzle exit. They are evenly distributed and are glued on two aluminum beams, which are mounted near the lips of the nozzle. A phase controller is connected to the function generator to produce specific phase difference of the acoustic disturbances on both sides of the shear layers. The Cartesian coordinate system is used in the experiment. As shown in Fig. 1 a, the origin is located on the middle height of the nozzle. The flow field is divided into the shear layer region from Y 0.99 to Y 0.1 ) and the potential core region where the streamwise velocity distributions remain flat. The movement of the individual probes is facilitated through two twodimensional traversing mechanisms. Measurements of the flow field are made by using DISA 55M10 constant-temperature anemometers. Two single hotwire probes with 5 m Pt wire as the sensing elements are used simultaneously for the mode analysis. Its frequency response can reach 40 khz. The data acquisition system consists of a 12-bit A/D converter with sample-and-hold capability. A personal computer is used for measurement control and subsequent data processing. The initial boundary layers at the nozzle exit of the natural jet are first examined to ensure that they are laminar for various jet exit velocities. The fundamental stability frequency ( f 0 ) of the shear layers are found to vary with the jet exit velocity by the relationship of f 0 U 0 3/2. Throughout the experiments, the operating jet velocity is fixed at 10 m/s. The corresponding Reynolds number Re U 0 H/v is 1.03 10 4. Its turbulence intensity at the nozzle exit center is below 0.3%. A most effective method on promoting the coherent structures in the jet shear layers is by applying acoustic disturbances at its fundamental frequency. In order to get more organized flow structures appreciated for signal detection and data analysis, the jet is then acoustically excited at its fundamental frequency of f e f 0 484 Hz. The relative phase differences can be changed to produced varicose mode (m 0) or sinuous mode (m 1) condition. The initial Strouhal number based on the initial momentum thickness is St 0 f 0 0 /U 0 0.0092, but when based on the nozzle exit width of the plane jet, it is St H f 0 H/U 0 0.726. The ratio of the nozzle height and the fundamental instability wavelength, H/ 0, is 1.45. Although the disturbance level can affect the evolution, the frequency effect is observed to be the most effective

1850 Phys. Fluids, Vol. 11, No. 7, July 1999 J.-M. Huang and F.-B. Hsiao FIG. 3. Comparison for the downstream variations of a mean velocities, and b streamwise velocity fluctuations along the jet centerline under different initial conditions. FIG. 4. Comparison for the downstream developments of a peak velocity fluctuations, and b momentum thickness for the jet under different initial conditions. parameter in controlling the shear layer properties. Figure 2 a shows the excitation level effect on the peak velocity fluctuation u p as the jet is initially excited at its fundamental frequency (m 0). It is clear that the fluctuation energy of the flow is greatly increased due to the excitation. However, the increment of u p tends to become saturated as the excitation level increased up to a threshold value. As one specifically considers the position for the occurrence of the first local maximum at X 1.8 0, at which the velocity fluctuations are mainly due to the growth of fundamental instability, the peak fluctuation intensities have become saturated when the excitation level exceeds 0.06 Pa see Fig. 2 b. This indicates that even higher acoustic disturbances can no longer enhance the evolution of the coherent structures significantly. Thus, the excitation level of 0.099 Pa is selected in the present study at which the saturation condition has attained. III. RESULTS AND DISCUSSION There are two types of instability modes in a plane jet. One is the varicose mode in which the traveling waves in the shear layers perform a succession of symmetrical swellings and contractions with respect to the jet centerline. The other is sinuous mode, where the instability waves exhibit a rhythmic undulation like the shape of a sausage. The latter is antisymmetrical with respect to the jet centerline. The induction of the two modes are observed to be effectively enhanced by artificially acoustic excitation. The resulting vortex patterns under different mode conditions can then be easily observed from the flow visualization, which is shown in Figs. 1 b and 1 c. For the varicose mode case, the arrangement of the arrays of the rolled-up vortices are found to be symmetrical with respect to the jet center line. However, the vortex arrays become staggered in the two shear layers for the sinuous mode case. In the present experiments, the jet is excited in the varicose mode and the sinuous mode conditions, respectively, in order to investigate the effects of the mode development on the shear flow dynamics. A. On the mean and fluctuation flow field Figures 3 a and 3 b show the variations of the streamwise mean velocity (U c ) and the fluctuation velocity along the jet centerline (u c ) at different initial conditions. The acceleration phenomenon of U c can be observed for the m 0 case after the first vortex merging region i.e., 4 0 X 7 0 ), but not for the m 1 case see Fig. 3 a. The increase of U c is associated with the decrease of the growth rate of u c. It is speculated that the fluctuation energy tends

Phys. Fluids, Vol. 11, No. 7, July 1999 Mode development in the developing region of a plane jet 1851 FIG. 5. a, b Downstream variations of the energy spectra measured along inner shear layer under m 0 and 1 conditions; natural jet. to be transferred back to the mean flow by the negative production mechanism. 19 Thus, the growth of u c for the m 1 case can obtain a higher value before X 5.5 0, but it saturates earlier and becomes lower than that for the m 0 case in the region of X 0. Figures 4 a and 4 b present the downstream development of the peak streamwise velocity fluctuations (u p ) and the momentum thickness for the jet under different initial conditions. In the rolled-up region for the fundamental vortex formation, both m 0 and m 1 cases can obtain a similar amplification rate of the fluctuation energy and shear layer spreading. However, u p and are found to perform distinctive behaviors for the two mode conditions after the region for the induction of nonlinear vortex merging processes i.e., X 2.5 0 ). More significant discrepancies occur within the region of 2.5 0 X 4 0, where the first vortex merging process dominates the flow field referred to Hsiao and Huang 20. This indicates that the nonlinear resonance of instability evolution and the resulting vortex structure dynamics are effectively changed by the mode arrangements in both sides of the shear layers of the jet column. This will be studied in detail in the next section. B. On the instability evolution Figures 5 a and 5 b show the energy spectra measured along the inner edge of the shear layers (Y 0.99 ) as the jet is excited at m 0 and m 1, respectively. For the m 0 case, the prescribed dominant fundamental and subharmonic instabilities of f 0, f 0 /2, and f 0 /4, exhibit clear peaks in the flow field. However, it is interesting to find that the peak of f 0 /2 disappears in the m 1 case Fig. 4 b. Now the energy spectra perform a double-peak at the frequencies of f a and f b near that of f 0 /2, and holds the relationship of ( f a f b )/2 f 0 /2. Similar phenomenon also observed in the boundary layer transition by Kachanov and Levchenko. 21 They found that the parametric resonance was the major mechanism which lead to the instability evolution with a narrow spectral subharmonic f 0 /2, but with a rather broadened spectra of low-frequency fluctuations. Such a parametric subharmonic resonance is due to the relative phase resonance between the waves of f 0 and f 0 /2. For an amplifying subharmonic wave f 0 /2 drifting its phase linearly with time, say (t) 2 ft, relative to the fundamental wave f 0, the wave pattern u (t) can be represented as a disturbance of frequency f a f 0 /2 f without relative phase shift. That is, u t A 0 exp I X 2 f 0 /2 t t A 0 exp i X 2 f 0 /2 f t, where A 0 and are, respectively, the amplitude and the wave number of the disturbances. Its evolution in the spectral distribution is qualitatively demonstrated in Fig. 6. Ac-

1852 Phys. Fluids, Vol. 11, No. 7, July 1999 J.-M. Huang and F.-B. Hsiao FIG. 6. Qualitative demonstration for the induction of the double-peak behavior in the spectrum Ref. 21. cording to the parametric resonance by the linear phase jittering of f 0 /2 relative to that f 0, the spectrum would initially exhibit two peaks at the frequencies of f 0 and f a f 0 /2 f see Fig. 6 a. During the amplification process of the subharmonic wave in the shear layers, the nonlinear interaction between the waves of f 0 and f 0 /2 f then leads to the formation of another component at f b f 0 ( f 0 /2 f ) f 0 /2 f, as shown in Figs. 6 b and 6 c. In the developing region of the jet, the flow dynamics is also governed by the amplifying fundamental and its subharmonic instabilities. These evolving waves have been proven to induce effectively nonlinear interaction in the shear layers. Thus, it is considered that the antisymmetrical vortex arrangement in the two shear layers of the jet column would cause the evolution of the induced subharmonic wave to exhibit linear FIG. 8. a, b Energy spectral distributions measured at (3 0,Y 0.99 ) as the jet is excited by the combination of ( f 0 f 0 /2) at a fixed relative phase difference under different mode conditions. FIG. 7. Energy spectral distributions measured at (3 0,Y 0.99 ) as the jet is excited at different phase differences between the two shear layers. phase jittering with that of the fundamental wave and then lead to an occurrence of nonlinear parametric resonance between f 0 and f 0 /2. As a result, the shear flow shows a double-peak behavior at f a and f b, instead of a single peak at f 0 /2, in the frequency spectral distributions. For more detailed investigations into the relative mode shifts of the initial wave structures on the downstream shear flow dynamics, the jet is excited at various phase differences between the two shear layers. The resulting energy spectra measured in the inner shear layers at X 3 0 are plotted in Fig. 7. When disturbances with a relative phase difference of approximately 75 are produced, only a pronounced single peak of subharmonic f 0 /2 is obtained. This reveals that the phase variation of f 0 /2 is well locked with that of the fundamental f 0. If the relative phase differences are beyond 90, a double-peak starts growing and becomes more prominent with the increase of. Approaching the sinuous mode condition ( 180 ), the frequency peak at f 0 /2 disappears, and the double-peak f a and f b replaces f 0 /2. According to the parametric resonance mechanism proposed by Kachanov and Levchenko, 21 a definitely linear phase jittering between the primary instabilities of f 0 and f 0 /2 should occur in the jet shear layers under the sinuous mode condition.

Phys. Fluids, Vol. 11, No. 7, July 1999 Mode development in the developing region of a plane jet 1853 mode condition has more potential for producing the nonlinear parametric resonance between the primary instabilities of f 0 and f 0 /2. From the global point of view, the overall amplitudes of the instability waves at a specific streamwise distance are obtained by integrating the eigenfunctions from the jet center line through the shear layer region, and are normalized by the local momentum thickness. That is, u f /U 0 1/ 0 Y0.1 u f /U0 dy. FIG. 9. a, b Downstream variations of the integrated amplitude of primary instabilities under m 0 and m 1 conditions. However, when the jet is excited at the varicose mode condition, no phase jittering occurs between f 0 and f 0 /2 such that only a single subharmonic peak at f 0 /2 obtained in the spectra. In order to further identify the role of the parametric resonance of f 0 and f 0 /2 on the induction of the double-peak phenomenon in the jet flow filed, the jet is excited simultaneously by the combination of two waves ( f 0 f 0 /2) with phase difference between them. Now, the relative phase difference between f 0 and f 0 /2 is artificially locked by excitation. Both the m 0 and m 1 cases are found to produce an apparent frequency peak at f 0 /2 only in the energy spectra, which are shown in Figs. 8 a and 8 b. This further confirms that the linear phase jittering between the fundamental and its subharmonic instabilities would effectively change the characteristics of the subharmonic in the frequency spectral distributions. Note that even the relative phase difference between f 0 and f 0 /2 is initially locked, substantial energy contents at frequencies of f a and f b can still be obtained in the nonlinear region downstream of the shear layers under the m 1 case although they are in a broadband pattern and less pronounced than that of peak f 0 /2, as shown in Fig. 8 b. Thus, the antisymmetrical arrangement of the two vortex arrays in a jet column i.e., the sinuous The downstream evolution of the primary instabilities for m 0 and m 1 cases are shown in Figs. 9 a and 9 b, respectively. Within the linear region near the nozzle exit, the dominant fundamental instability ( f 0 ) can obtain a similar growth rate at both m 0 and m 1 conditions. Also, the double-peak of f a and f b due to parametric resonance of f 0 and f 0 /2 in m 1 case in association with the single peak f 0 /2 in m 0 case are all found to exhibit similar evolution patterns, except that the lower frequency f a tends to obtain larger energy content than that of the higher one f b. Thus, the occurrence and the characteristics of the double-peak are indeed governed by the first subharmonic instability f 0 /2. Figures 10 a and 10 b are the isoamplitude contours of the primary instabilities in the jet under m 0 and m 1 conditions, respectively. Note that the component of f a still contains significant amplitude when approaching the potential core region. It will be seen latter that the mode behavior of the f a component is symmetrical across the jet column. Hence, its amplitude distributions in the two shear layers of the jet tend to constructively interact with each other when close to the center line of the potential core region. C. On the mode development In order to investigate the mode behaviors between the two shear layers across the jet, two hot-wire probes are simultaneously used to detect the flow structures. The measuring points are symmetrical with respect to the jet center line e.g., points 1 and 2 in Fig. 1 a. Figures 11 14 show the simultaneous time traces of the streamwise velocity fluctuations, u 1 (t) and u 2 (t), as the probes are arranged close to the inner shear layers say, Y 0.8 to Y 0.8 ) at different downstream locations for m 0 and m 1 cases. Their corresponding cross-correlation functions, R 12 ( ), associated with the cross spectra, G 12 ( f ), and their relative phase spectra, 12 ( f ), are also plotted in the figures. Near the initial region of X 2 0, where the jet flow is dominated by the fundamental instability, the time traces in the two shear layers clearly exhibit an in phase or out of phase pattern for m 0 or m 1 case see Fig. 11. This indicates that the initial mode developments can be effectively controlled by the artificial acoustic excitation. Also in Fig. 11, the crosscorrelation function, R 12 ( ), obtains the same period as that of f 0 with the positive maximum or negative minimum near the zero time delay ( 0). It reveals a global indication for identifying the varicose mode or sinuous mode characteristics of the flow structures. Moreover, the crossspectrum, G 12 ( f ), performs dominant peaks at the prescribed

1854 Phys. Fluids, Vol. 11, No. 7, July 1999 J.-M. Huang and F.-B. Hsiao FIG. 10. a, b Isoamplitude contours of the primary instabilities under m 0 and m 1 conditions. primary instabilities. Its corresponding phase spectrum, 12 ( f ), also provides the information for the relative phase difference of the individual instabilities across the jet column. The value of 12 (f 0 ) for the fundamental instability at X 2 0 is observed to be indeed near 0 or 180 for the m 0 or m 1 case. An important feature of the mode switching phenomenon is found to occur in the nonlinear region for the induction of the subharmonic instabilities. For the m 0 case, the traveling waves of f 0 in the two shear layers are symmetrical i.e., in phase with each other. However, it is interesting to find that the subsequent subharmonic f 0 /2 performs in an antisymmetrical i.e., out of phase pattern throughout the jet column region. The characteristics of the subharmonic evolution is considered to be the major mechanism for the mode switching phenomenon which has been pointed out in some literature. 4,6,8 Near the nozzle exit region e.g., X 2 0 ), the jet is dominated by the amplifying fundamental instability ( f 0 ) which is in varicose mode behavior. The two arrays of the rolled-up vortices then exhibit a symmetrical pattern Fig. 11 a. Further downstream, the first subharmonic instability ( f 0 /2) becomes the locally most amplified instability in the shear layers. Because f 0 /2 performs a sinuous mode behavior under the m 0 case, the mode switching phenomenon in the jet flow can then proceed. As the energy content of f 0 /2 rises higher than that of f 0, such that the shear flow becomes dominated by the vortex merging processes e.g., X 3 0, the resulting flow structures now present an antisymmetrical pattern across the jet refer to the features of u 1 (t), u 2 (t), and R 12 ( ) in Fig. 13 a. The flow structures can retain an antisymmetrical pattern in the region for occurring the second vortex merging process, because then f 0 /4 also tends to perform sinuous mode behavior. For the m 1 case, the initial region of the jet e.g., X 3 0 shown in Fig. 10 b exhibits an antisymmetrical pattern of vortex arrays because the dominant fundamental instability is now in sinuous mode behavior. However, the induced double-peak f a and f b, which are due to the parametric resonance between the waves of f 0 and f 0 /2, are found to show opposite mode features. The lower frequency component f a is of varicose mode, but the higher frequency component is of sinuous mode. No apparent mode switching phenomenon is observed in the m 1 case. The resulting flow structures during the first vortex merging process i.e., near X 3 0 or 4 0 ) are dominated by the combination of the components f a and f b, so that the global mode pattern cannot be well identified see Figs. 13 b and 14 b. Figures 15 and 16 depict the downstream variations of the relative phase difference for the individual instabilities

Phys. Fluids, Vol. 11, No. 7, July 1999 Mode development in the developing region of a plane jet 1855 FIG. 11. a, b Flow structure behaviors across the jet column for u 1 (t), u 2 (t), R 12 ( ), G 12 ( f ), 12 ( f ) under m 0 and m 1 conditions. The two measuring probes are located at (2 0, Y 0.8 ). FIG. 12. a, b Flow structure behaviors across the jet column for u 1 (t), u 2 (t), R 12 ( ), G 12 ( f ), 12 ( f ) under m 0 and m 1 conditions. The two measuring probes are located at (2.5 0, Y 0.8 ). across the jet column under m 0 and m 1 cases, respectively. Four measuring pairs at different transverse positions, namely (Y 0.99, Y 0.99 ), (Y 0.8, Y 0.8 ), (Y 0.5, Y 0.5 ), and (Y 0.2, Y 0.2 ), are selected here. For the m 0 case in Fig. 15, the fundamental instability ( f 0 ) clearly shows in-phase behavior before it becomes decaying (X 2 0 ). It tends to develop toward out-of-phase pattern beyond X 2 0, especially near the outer shear layer regions. With the aid of the conditional phase technique, Hsiao and Ho 15 also found that the lower speed side of the shear layers could obtain higher probability density of sinuous mode than that of the higher speed side of the shear layers. Furthermore, the wave of f 0 /2 always performs out of phase behavior throughout the jet flow field, which is the major mechanism leading to the occurrence of the mode switching phenomenon. Note that the wave of f 0 /4 is observed to approach an in-phase pattern as the measuring points are along inner shear layers of (Y 0.99, Y 0.99 ), as shown in Fig. 15 a. This is because the jet potential core region becomes thinner, and the two measuring points are closer to each other. The interaction of the flow structures near the inner portion of the two shear layers now becomes more significant as they are approaching the region near the end of the potential core. However, the mode pattern of f 0 /4 still appears to approach out of phase as the separation distance of the measuring points is increased see Figs. 15 b 15 d. As for the m 1 case shown in Fig. 16, the components of f 0 and f b behave in an out-of-phase pattern throughout the jet flow field, while f a and f a /2 always exhibit an in-phase pattern. The resulting mode patterns of the flow structures are then dependent on the evolution and dominance of these primary instabilities. In view of global flow structural dynamics, the downstream variations of the cross-correlation function of zero time delay, R 12 (0), for m 0 and m 1 are plotted in Figs. 17 a and 17 b, respectively. For the m 0 case in Fig. 17 a, the value of R 12 (0) depends on the local dominance of the evolving instabilities. Its positive maximum which represents varicose mode behavior occurs near X 2 0 due to higher energy of f 0 here. However, the amplification of f 0 /2 over that of f 0 causes R 12 (0) dropping to a local negative minimum which represents sinuous mode behavior within the region where the first vortex merging process evolves. Its location is observed to move more upstream, closer to the outer shear layers due to earlier saturation of f 0 /2, which has been verified by Hsiao and Huang. 22 Because the effect of turbulent diffusion becomes more important, the characteristics of the global mode pattern can no longer be well identified beyond the region of X 4 0. Note that the value of R 12 (0) tends to decrease or even becomes negative when close to the nozzle exit (X 0 ). The negative behavior also spreads to the region over X 4 0 in the outer shear layers of (Y 0.2, Y 0.2 ). This is mainly due to the feedback effect of the subharmonic f 0 /2, such that the initial energy content of

1856 Phys. Fluids, Vol. 11, No. 7, July 1999 J.-M. Huang and F.-B. Hsiao FIG. 13. a, b Flow structure behaviors across the jet column for u 1 (t), u 2 (t), R 12 ( ), G 12 ( f ), 12 ( f ) under m 0 and m 1 conditions. The two measuring probes are located at (3 0, Y 0.8 ). FIG. 14. a, b Flow structure behaviors across the jet column for u 1 (t), u 2 (t), R 12 ( ), G 12 ( f ), 12 ( f ) under m 0 and m 1 conditions. The two measuring probes are located at (4 0, Y 0.8 ). the amplifying f 0 is comparable or still lower than that of f 0 /2. On the other hand, for the m 1 case in Fig. 17 b, the value of R 12 (0) shows a negative minimum in the initial vortex formation region of X 2 0. Further downstream, because of the competition of the components f a in varicose mode and f b in sinuous mode in the nonlinear region of the jet column, the distributions of R 12 (0) begin to approach zero beyond the region of X 3 0. As previously described, the amplifying instabilities tend to saturate earlier in the outer edge than that in the inner edge of the shear layers. The distributions of R 12 (0) then exhibit earlier saturation to a zero condition when the measuring points are closer to the outer edge of the shear layers. It is noted that R 12 (0) continues increasing beyond X 4 0 in the inner edge of the shear layers of (Y 0.99, Y 0.99 ) for both m 0 and m 1 cases. This is because the measuring points of 1 and 2 are close to each other as approaching the end of the potential core. In this region, the interaction of the flow structures in the inner portion of the two shear layers is prominent and results in a higher value of the correlation function between them. IV. CONCLUDING REMARKS The mode development of the coherent structures in the developing region of a plane jet is investigated experimentally by means of hot-wire measurements. The relative mode FIG. 15. a d Downstream variations of the relative phase differences for the individual instabilities across the jet column under the m 0 condition., f 0 ;, f 0 /2;, f 0 /4.

Phys. Fluids, Vol. 11, No. 7, July 1999 Mode development in the developing region of a plane jet 1857 FIG. 16. a, b Downstream variations of the relative phase differences for the individual instabilities across the jet column under the m 1 condition., f 0 ;, f a ;, f b ;, f a /4. arrangements of the rolled-up vortex arrays in the two shear layers of a jet are found to be effectively controlled by artificial acoustic excitation. Different mode excitation would change the spreading rate, the velocity fluctuation evolution and the frequency spectral distributions in the nonlinear region of the jet, where the shear flow is dominated by the vortex merging processes. For the sinuous mode excitation case (m 1), the spectral peak at f 0 /2 disappears and the frequency spectrum exhibits a double-peak of f a and f b around f 0 /2. These three components constitute a relationship of ( f a f b )/2 f 0 /2. The double-peak phenomenon is considered to be due to the parametric subharmonic resonance which is caused by the relative linear phase jittering between the primary instabilities of f 0 and f 0 /2. The mode behavior of the individual instabilities can be obtained from the phase information of the cross spectrum. The induced primary instabilities are observed to sustain specific mode features of their own in the jet column. The resulting mode patterns of the flow structures are then dependent of the amplification and dominance of these primary instabilities. The mode switching phenomenon is found to be prominent under the varicose mode excitation case (m 0). It is mainly governed by the evolution of the subharmonic instability f 0 /2, which performs a sinuous mod behavior throughout the jet column. As for the m 1 case, the double-peaks of f a and f b appear to exhibit opposite mode patterns in the shear layers. Therefore, no apparent mode switching phenomenon is observed here because of the mode competition between the components of f a and f b. ACKNOWLEDGMENT This work is supported by the National Science Council, Republic of China under contract No. NSC 86-2212-E-006-038. FIG. 17. a, b Downstream variations of R 12 ( ) under m 0 and m 1 conditions. 1 S. C. Crow and F. B. Champagne, J. Fluid Mech. 48, 547 1971. 2 F. K. Brown and A. Roshko, J. Fluid Mech. 64, 775 1974. 3 C. D. Winant and F. K. Browand, J. Fluid Mech. 63, 237 1974. 4 G. E. Mattingly and C. C. Chang, J. Fluid Mech. 65, 541 1974. 5 G. K. Batchelor and A. E. Gill, J. Fluid Mech. 14, 529 1962. 6 P. J. Morris, J. Fluid Mech. 77, 511 1976. 7 H. V. Fuchs and U. Michael, AIAA J. 16, 871 1978. 8 J. Cohen and I. Wygnanski, J. Fluid Mech. 176, 221 1987. 9 P. Plaschko, J. Fluid Mech. 92, 209 1979. 10 R. E. Drubka, Ph.D. dissertation, Illinois Institute of Technology, Chicago, 1981. 11 T. C. Corke, F. Shalib, and H. M. Nagib, J. Fluid Mech. 223, 253 1991. 12 H. Sato, J. Fluid Mech. 7, 53 1960. 13 A. K. M. F. Hussain and C. A. Thompson, J. Fluid Mech. 100, 397 1980. 14 F. B. Hsiao, Ph.D. dissertation, University of Southern California, Los Angeles, 1985. 15 F. B. Hsiao and C. M. Ho, Bull. Am. Phys. Soc. 29, 1519 1984. 16 J. K. Wat, Ph.D. dissertation, University of Southern California, Los Angeles, 1988. 17 P. J. R. Strange, Ph.D. dissertation, University of Leeds, England, 1981. 18 T. C. Corke and S. M. Kusek, J. Fluid Mech. 249, 307 1993. 19 F. K. Browand and C. M. Ho, J. de Mecan. Theor. Appl. 8, 99 1983. 20 F. B. Hsiao and J. M. Haung, J. Fluids Eng. 116, 714 1994. 21 Y. S. Kachanov and Y. Y. Levchenko, J. Fluid Mech. 138, 209 1984. 22 F. B. Hsiao and J. M. Huang, Exp. Fluids 9, 2 1990.