Vortex pairing in an axisymmetric jet using two

Experiments in Fluids 25 (1998) 305—315 ( Springer-Verlag 1998

Vortex pairing in an axisymmetric jet using two-frequency acoustic forcing at low to moderate strouhal numbers S. K. Cho, J. Y. Yoo, H. Choi

305 Abstract Hot-wire measurement and multi-smoke wire flow visualization method are employed to study vortex pairing in the jet column mode under two-frequency forcing with controlled initial phase differences. For the range of 0.3\St \0.6, vortex pairing can be easily controlled by means D of the fundamental and its subharmonic forcing with varying initial phase differences. As stable vortex pairing dominates, the variation of the subharmonic component with the initial phase difference changes from a sine shape to a cusp-like shape. The harmonics of the subharmonic also show similar trends. The detuning induces the amplitude and phase modulations of the u-signal in the time trace and the sideband growth in the spectra. The u-signal reflects the subharmonic variation with the initial phase difference in its envelope. For 0.6\St \0.9, non-pairing advection of vortices due D to improper phase difference is sometimes observed under single-frequency forcing. In this case, vortex pairing can be made to occur by the addition of a subharmonic with very small amplitude. As the initial level of this subharmonic is increased, the onset position of vortex pairing moves upstream. In this range, the initial phase difference is not an effective parameter in controlling vortex pairing.

List of symbols a f a s D f r r 0.1 St D St he

fundamental amplitude at jet exit (u@ /U D ) f e x/0 subharmonic amplitude at jet exit (u@ /U D ) s e x/0 jet diameter fundamental frequency radial coordinate radial position where U/U \0.1 c Strouhal number based on jet diameter ( fD/U ) e Strouhal number based on initial momentum thickness ( f h /U ) e e

Received: 22 May 1997 /Accepted: 16 October 1997 S. K. Cho, J. Y. Yoo, H. Choi Dept. of Mechanical Engrg., College of Engrg., Seoul National University Seoul 151-742, Korea Correspondence to: J. Y. Yoo The authors wish to express their deepest gratitude for the financial support of Turbo and Power Machinery Research Center throughout this work.

U u v x h / d / de

streamwise mean velocity streamwise velocity fluctuation radial velocity fluctuation streamwise coordinate momentum thickness (:r0.1 U/U (1[U/U ) dr) c c 0 phase difference between fundamental and subharmonic phase difference at jet exit (/ D ) d x/0

Superscripts * @

complex conjugate root mean square

Subscripts c e f 2f 3f/2 r s t

jet center jet exit fundamental fourth harmonic of subharmonic third harmonic of subharmonic incoherent turbulent fluctuation subharmonic total fluctuation

1 Introduction Initial disturbances near the jet exit have significant effects on the development of large scale vortical structures in the near field of an axisymmetric jet. For the last few decades a number of studies have been performed on dynamics and control of the jet flow field by means of initial disturbances (Ho and Huerre 1984; Thomas 1991; Mankbadi 1992). The flow structure in an axisymmetric jet can be classified into two modes depending on the length scales, the exit boundary-layer thickness (say, the initial momentum thickness h ) and the jet diameter D. One is e the shear layer mode where the vortical structure is formed very close to the jet exit and develops in a similar process as that of a two-dimensional free shear layer. For this mode, the Strouhal number based on the momentum thickness at the jet exit, St , is the important dimensionless parameter. It is he known that the initially laminar shear layer is most likely to roll up at the high frequency of St \0.012 (the most amplified he frequency, Zaman and Hussain 1980), even with the axisymmetric disturbances of extremely low level. The other is the jet column mode which is dominant at the end of the potential core where the mixing layer of the axisymmetric jet thickens and the momentum thickness h becomes comparable to the jet

306

diameter D. For this mode, St is the important dimensionD less parameter, where the characteristic length scale is the jet diameter and the characteristic frequency is the passage frequency of large-scale vortical structures which is relatively low. Moreover, this mode can be formed immediately downstream of the jet exit bypassing the shear layer mode when high-amplitude forcings of a low frequency are imposed (collective interaction, Ho and Huang 1982). Crow and Champagne (1971) performed experiments on an artificially forced axisymmetric jet in the range of St O0.6. D They showed that the preferred mode was St \0.3 for the case D of external single-frequency forcing over that range. The preferred mode was determined on the basis of maximum amplification of the initial fundamental, at which strong vortices are formed in the jet flow field. Pairing of the neighboring vortices in the jet column mode was found to occur in the range of St [0.6, and most stably at St \0.85 D D (Zaman and Hussain 1980). Broze and Hussain (1994) investigated the effect of St and a on vortex pairing and classified D f the states into stable pairing, stable double pairing, nearly periodic modulations of pairing, fundamental only and so on, utilizing the concept of chaotic and periodic attractors. They showed that, for single-frequency forcing over the range of St O0.5 and a \0.2, there did not occur any stable vortex D f paring. In the mean time, Ho and Huang (1982) showed that the subharmonic is not just a by-product of the vortex pairing but its rapid growth is a prerequisite to vortex pairing. After the fundamental grows exponentially and reaches a critical amplitude, the subharmonic grows rapidly from nonlinear interaction with the fundamental until its energy reaches a maximum at the location of pairing, which is commonly referred to as subharmonic resonance. Pressure perturbation induced by the saturated subharmonic propagates upstream, acts as the initial subharmonic disturbance and evolves downstream of the jet exit. The subharmonic growth is extensively associated with nonlinear interaction between the fundamental and its subharmonic. In particular, the amplitudes and the phase differences between these waves play important roles in subharmonic growth and vortex pairing. Arbey and Ffowcs Williams (1984) showed from an experimental study that the subharmonic component under two-frequency forcing can be either amplified or suppressed depending on the phase difference between the fundamental and its subharmonic. Based on this finding, Monkewitz (1988) theoretically investigated the conditions for vortex pairing and shredding in the mixing layer. Husain and Hussain (1995) showed three kinds of vortex interactions in the shear layer mode, namely, enhanced pairing, non-pairing advection of vortices and shredding, depending on the initial amplitude ratio and the initial phase difference between the fundamental and its subharmonic. Paschereit et al. (1995) also investigated the effect of the Strouhal number, forcing level and phase difference on vortex pairing and energy transfer among the waves and mean velocity components. In those studies, the disturbance was kept low in amplitude and high in frequency so that the distortion of the induced flow was not significant. In general, this leads to qualitative agreements between experiment and theory. However, the life-span of this high-frequency disturbance is very short and limited to the region very close to the jet exit. Mankbadi (1992) predicted that the streamwise

life-span of the wave was inversely proportional to the Strouhal number. In order to control the jet flow farther downstream, it is necessary to apply a forcing of low frequency and high amplitude. In addition, at this band of low frequency, vortex pairing by two-frequency forcing with various phase differences would be valuable in practical applications. For a jet issued with a turbulent exit boundary layer, Raman and Rice (1991) showed that the subharmonic component can be enhanced or suppressed under the fundamental and its subharmonic forcing of relatively low frequencies. The objective of the present study is to further investigate, by means of hot wire measurements and multi-smoke wire flow visualization, the effect of the fundamental and its subharmonic forcing with controlled phase difference on vortex pairing in the range of low to moderate St based on the D fundamental frequency. Also, the effect of the detuned forcing on the modulation of the subharmonic is investigated. The jet issued with a laminar exit boundary layer is forced over the range of 0.3\St \0.6, where it is known that stable vortex D pairing does not occur with single-frequency forcing, and over the range of 0.6\St \0.9, where vortex pairing can easily D occur under single-frequency forcing (Broze and Hussian 1994).

2 Experimental apparatus Experimental apparatus consisted of an air jet facility, acoustic forcing system, data acquisition system and flow visualization system, as shown in Fig. 1. The flow from the blower passed through a silencer box, a diffuser and a settling chamber equipped inside with three screens and a honeycomb straightener before exiting through the nozzle. An ASME flow nozzle with a contraction ratio of 76 : 1 and an exit diameter of 32.1 mm was selected to achieve very thin boundary layer thickness at the exit. At U \8.7 m/s, non-uniformity of the e flow at the exit was less than 0.8% and the turbulence level at the exit was 0.3%, the largest portion of which was at very low frequency (\20 Hz). The exit velocity profile measured at x\0.1 mm is shown in Fig. 2 as a top-hat velocity profile, surrounded by a thin shear layer with a shape factor of 2.6 which agrees well with the Blasius profile. Maximum value of turbulence level in this shear layer was 0.5%. Therefore,

Fig. 1. Schematic layout of the experimental apparatus

The fluctuation velocity forced at the jet exit was of the following form:

u/U \J2(a sin(2nft[/ )]a sin(nft[/ )) e f f s s

Fig. 2. Velocity profile at the nozzle exit (L: U \8.7 m/s, D/h \178; e e K: U \14.4 m/s, D/h \241; —: Blasius profile) e e

we could assume that the boundary layer at the jet exit was laminar. In addition, D/h was greater than 100 at the exit, e indicating that the boundary layer was indeed very thin. The most amplified frequencies at the jet exit in the shear layer mode based on St \0.012 (Zaman and Hussain 1980) he would be 579 Hz at U \8.7 m/s and 1297 Hz at U \14.4 m/s, e e respectively, which are much higher than the present forcing frequencies (94, 135 and 270 Hz), indicating that the jet column mode of the present study was dissociated from the shear layer mode. A constant-temperature hot-wire anemometer (Dantec Streamline) was used to obtain velocity data with an I-type probe (Dantec 55P11). In order to minimize the interference of the probe supporter, it was aligned at an angle of 30° from the streamwise direction. The output signals of the hot-wire were converted through a 16-Bit A/D converter (DT 2838), stored and linearized afterward to velocity data with a fourth-order polynomial in an IBM PC486. The mean velocity and fluctuation level were obtained by a long time average calculation. The FFT technique was employed to perform the energy spectral analysis. The probe movement was controlled threedimensionally by the PC. Acoustical forcing was introduced by a 4 in diameter speaker mounted on the pipe wall preceding the diffuser. The pipe wall facing the speaker had a number of small holes (less than 2 mm in diameter) to minimize the disturbance of the flow. The signal transmitted to the speaker was produced by superimposing two sine waves with a discrepancy of one octave in frequency and a variable relative phase difference in an arbitrary wave form generator (AWG Tektronix AWG2020) and amplified by an audio amplifier (Inkel AD972). For the detuned forcing, a detuned subharmonic sine wave was generated in a function generator and added to the fundamental sine wave by an adder circuit. This added signal was also transmitted to the audio amplifier. In order to obtain high amplitude forcing, experiments were carried out near resonance frequencies of the air jet facility. We can obtain the desired St at these fixed frequencies by adjusting the jet exit D velocity over the range of Re({U D/l)\1.3]104—3.1]104. e

As an initial condition at the jet exit, amplitudes (a and f a ) and the initial phase difference (/ \/ [2/ ) were also s de f s obtained through FFT calculation of the instantaneous streamwise velocity component. In the present experiment, a was s always kept lower than a . f Uncertainty estimates of hot-wire data were made in accordance with Yavuzkurt (1984), which are 1.6% for both the streamwise mean velocity U and the streamwise velocity fluctuation u along the centerline. Uncertainties in the FFT calculation were almost negligible. The multi-smoke wire technique (Makita and Hasegawa 1993) was employed to visualize the time sequence of vortex pairing downstream of the jet exit. In the test section, a number of thin nichrome wires 80 lm in diameter were placed perpendicularly to the jet axis at intervals of 0.5D up to x/D\3 and at intervals of 1D therefrom. The same voltage (about 24 V) from a power supply was applied to all wires. The lengths of two wires adjacent to the jet exit were adjusted to be shorter in order to acquire more electric power than the other wires in the downstream region. Thus, it was possible that all streaklines over the test section were visible at one time. The heating-up of the wires was triggered in advance and the stroboscope was synchronized with the subharmonic component of the superimposed signal in the AWG. All the photos were taken at several phases of the subharmonic at about 70° increments with the camera lens open. The exit velocity was set at 6 m/s for visualization.

3 Results and discussion Actually, it is difficult to precisely identify the St limits in D terms of vortex pairing, since they strongly depend on the initial fundamental amplitude a and there is no method for f clearly defining the stable vortex pairing and fundamental-only regions. It is noted that there exists a transient regime between the two regions, which consists of various states of vortex pairing, that is, nearly periodic modulations of pairing, aperiodic modulations of pairing and intermittent pairing (Broze and Hussain 1994). Moreover, in this transient regime, these states randomly intermingle with each other temporally. Thus, the division of the two regions in the present paper was made mainly on the basis of the previous study made by Broze and Hussain (1994), who presented a phase diagram which classified the states of vortex pairing in terms of St and D a utilizing the concept of the periodic and chaotic attractors. f In their results, stable vortex pairing occurred near St \0.6 D only if a is higher than 10%. Furthermore, at St \0.5 and up f D to a \20%, there was no stable vortex pairing. This forcing f level is considered in general to be very high. Therefore, it was determined that the division of the two regions could roughly be made at St \0.6 for moderate forcing level. Thus, we are to D consider two forcing cases, i.e., those for 0.3\St \0.6 and D 0.6\St \0.9, since stable vortex pairing can be achieved D readily by single-frequency forcing for 0.6\St \0.9, while D

307

only the fundamental exists under single-frequency forcing for 0.3\St \0.6. D

3.1 Fundamental-only region under single-frequency forcing (0.3\StD\0.6) 3.1.1 Vortex pairing under the fundamental and its subharmonic forcing 308

In this region it is known that stable vortex pairing can not be achieved by single-frequency forcing. However, vortex pairing can still be achieved by means of the fundamental and its subharmonic forcing with proper phase differences (Raman and Rice 1991). The u-signal from the hot-wire measured at x/D\3 is shown in Fig. 3a for four different forcing cases of St \0.5. When the subharmonics at / \23° and 272° are D de added to the fundamental at the jet exit, frequency-halvings of the u-signal from the hot-wire due to vortex pairing are observed downstream although their states look a little

Fig. 3a, b. Comparison of four forcing cases at St \0.5. a Time D domain; b frequency domain

different from each other depending on / . In contrast, vortex de pairing is completely suppressed in the forcing case of / \120°. Therefore, it is noted that vortex pairing depends de sensitively on the initial phase difference / (this is further de investigated in the following section). The u-spectra are shown in Fig. 3b. The higher peaks of the subharmonic than those of the fundamental at / \23° and 272° indicate the appearance de of stable vortex pairing. As expected, at / \120°, the peak of de the subharmonic is still lower than that of the fundamental. Also, note that the harmonics of the subharmonic grow under all two-frequency forcings through nonlinear interactions between the fundamental and its subharmonic while only the harmonics of the fundamental grow under single-frequency forcing. This can be explained by the relationship of sum and difference wave interaction (Cohen and Wygnanski 1987). The sequence of the vortex pairing process subject to two-frequency forcing is visualized by the multi-smoke wire technique, as is illustrated in Fig. 4a—e. To capture stable vortex pairing, photos were taken at a \6.3%, a \3.5% and f s / \266°. At x/D\0.5, vortex starts to roll up and then de interacts with the neighboring downstream vortex which rolled up earlier. At x/D\3—4, the two neighboring vortices merge and eventually break down farther downstream. This is a typical vortex pairing process and is part of leap-frog motion (Van Dyke 1982). Under single-frequency forcing at a \7.1% (which is f approximately equal to the overall forcing level of the preceding vortex pairing shown in Fig. 4a—e), however, vortices nearly equally spaced are advected downstream without vortex pairing as shown in Fig. 4f. The location of vortex roll-up is identical with that of the two-frequency forcing case. In this case, convection velocity of the vortex in the middle of the shear layer could be simply deduced from u \fj #0/7 :St U \0.5U based on the result that the distance j between D e e the centers of the vortices is about 1D. The streamwise evolution of each disturbance component under single-frequency forcing along the centerline is shown in Fig. 5a. The subharmonic component does not rapidly grow downstream. In contrast, under two-frequency forcing at / \23°, the subharmonic rapidly grows from subharmonic de resonance as shown in Fig. 5b, eventually saturates and becomes equal in magnitude to the total streamwise fluctuation velocity u@ at x/D\3.5—4.5. In this region, it seems that vortex t pairing is completed. At / \120°, however, the subharmonic de component is damped near the jet exit as shown in Fig. 5c, which results in non-pairing advection of vortices similar to the case shown in Fig. 4f. The streamwise evolutions of the fundamental and total streamwise velocity fluctuations are also similar to those shown in Fig. 5a. In Husain and Hussain’s work (1995), non-pairing advection of vortices occurred when both of the following conditions were satisfied; (1) phase difference / \90° along the jet-lip line during the interaction d of the vortices, (2) forcing of a low initial amplitude ratio of the subharmonic to the fundamental. Although the initial amplitude ratio of a /a \0.14 seems to meet the second condition, it s f is not certain whether the first condition is satisfied due to the randomness of the phase difference. The evolutions of the phase difference between the fundamental and its subharmonic for stable vortex pairing are shown in Fig. 6. The phase difference can be obtained from the

309

Fig. 4a–f. Flow visualization using multi-smoke wires at St \0.5 and U \6 m/s. a–e Time sequence (Dt:0.0021 s) of vortex pairing at D e a \6.3%, a \3.5% and / \266°; f no vortex pairing at a \7.1% and a \0% f s de f s

following equation (Hajj et al. 1993):

A

real(A( f,[f/2)) / \tan~1 d imag(A( f,[f/2))

B

where A( f,[f/2)\E(X( f )X*( f/2)X*( f/2)) (auto-bispectrum). In this equation, X( f ) and X( f/2) are the complex Fourier transforms of the streamwise velocity fluctuations at the frequencies of the fundamental and its subharmonic and the notation E (2) denotes a statistical average. The regions of constant phase difference, that is, non-dispersive regions, correspond to subharmonic resonance where the fundamental and subharmonic waves are advected downstream at the same speed (about 0.5U ) as that in Fig. 4f and the subharmonic e component rapidly grows from direct energy exchanges with the fundamental and mean velocity. In the case of / \272°, de the region of constant phase difference appears farther upstream than in the case of / \23°. It is consistent that the de frequency-halving at / \272° is more developed at x/D\3 as de shown in Fig. 3a. However, the local phase differences for two / ’s are nearly equal to about 230° when resonance occurs. de At / \120°, the phase differences were also obtained (not de shown) in the same manner but the data were random in time and their auto-bicoherences (Hajj et al. 1993) were lower than 0.3, indicating that the two waves were not phase-locked. The total streamwise velocity fluctuations u@ along the t centerline are shown in Fig. 7a. The highest peak appears when / \272° and is farther upstream than the other three forcing de cases. The streamwise variation of the mean velocity along the centerline shown in Fig. 7b generally reveals a reverse trend to the total streamwise velocity fluctuations in the range of x/D:0.5—4.5: as the total streamwise velocity fluctuation

increases, the mean velocity decreases, and vice versa. This can be explained by the following equation, which is simply obtained through the Reynolds averaging of the decomposed streamwise momentum equation (Hussain and Reynolds 1970) in the cylindrical coordinate by assuming that the pressure term, viscous term, and radial component of the forcing velocities along the centerline are negligible

LU 1 L 2 L(u l ) c:[ r r (u2]u2]u2 )[ f s r Lx U Lx U Lr c c where overbar denotes the Reynolds average. It implies that the streamwise gradient of the mean velocity along the centerline has an opposite sign to that of the total velocity fluctuation component (the forcing velocity components plus incoherent turbulent velocity components). In particular, for the most stable vortex pairing (/ \272°), the mean velocity keeps on de decreasing along the centerline, indicating that the incoherent turbulent velocity components grow and take over as soon as the subharmonic component saturates. This mechanism can be explained by energy exchanges among the mean velocity, the forcing velocity components and the incoherent turbulent velocity components. Detailed investigations on this energy exchange were made by Liu (1988), Hsiao and Huang (1994) and Paschereit et al. (1995). The mixing rate, which can be represented by the two-dimensional momentum thickness h, is also related to the growth of disturbances and vortex pairing. It increases rapidly at the location of vortex roll-up and pairing as shown in Fig. 7c: we note from Fig. 4 that under two-frequency forcing, vortex roll-up is completed at x/D\1, where the fundamental starts increasing as shown in Fig. 5b, and vortex pairing is completed at x/D\4, where the subharmonic

310

Fig. 6. Evolutions of the phase difference between the fundamental and the subharmonic along the centerline for St \0.5 (K: a \7%, D f a \1%, / \23°; e: a \7%, a \1%, / \272°). Arrows denote s de f s de the regions of constant phase differences

Fig. 5a–c. Evolutions of the fundamental, subharmonic and total streamwise fluctuation velocities along the centerline for St \0.5(L: D u@ /U ; K: u@ /U ; n: u@ /U ). a Fundamental-only (a \7%, a \0%); s e f e t e f s b stable vortex pairing (a \7%, a \1%, / \23°); c suppressed f s de vortex pairing (a \7%, a \1%, / \120°) f s de

saturates. Then, the ratio of the momentum thicknesses between the two locations is about 2 (Ho and Huang 1982). This is due to the existence of two ring vortices concentrically aligned on the plane normal to the streamwise direction.

3.1.2 The effects of the initial phase difference The dependence of the subharmonic component u@ on the s initial phase difference / is examined at several streamwise de

locations. Initial conditions at the jet exit under two-frequency forcing with a \7% and a \1% for various / are shown in f s de Fig. 8a, where / is varied in increments of about n/8, and de u@ and u@ at the jet exit are kept nearly constant over the entire f s range of / . In the downstream region, however, u@ shows s de significant variations with / . At x/D\2, the variation of u@ is s de similar to a sine shape as shown in Fig. 8b and u@ is lower than s u@ over the entire range of / . The variation of u@ at x/D\3 s f de is shown in Fig. 8c, over which there exists a region where u@ exceeds u@ and vortex pairing occurs. Thus, it is noted from s f these figures that the development of vortex pairing can be precisely controlled by varying the initial phase difference. It is further noticed that the fundamental is slightly lower in the region where the subharmonic exceeds the fundamental as shown in Fig. 8c. This may be attributed to the energy exchange from the fundamental to its subharmonic, that is, some deviation from parametric resonance (Hajj et al. 1993) between the fundamental and its subharmonic. As the amplitude of the subharmonic at the jet exit is increased, the variation of u@ does not show a sine shape any s longer but changes into a cusp-like shape as shown in Fig. 9a. The initial phase difference at which the subharmonic component is minimum is changed from 120° to 0° due to the change in the location of nonlinear interaction. A similar cusp-like variation was shown in Husain and Hussain’s measurements (1995) along the lip line in the shear layer mode and in Raman and Rice’s measurements (1991) along the centerline in the jet column mode, and the theory (Monkewitz 1988) with the assumption of parallel non-diverging flow predicted that the subharmonic was suppressed at a critical value of the phase difference and enhanced over a wide range of the phase difference. It is noteworthy that a cusp-like behavior of the subharmonic with respect to the initial phase difference can also be observed in the jet column mode, even though the forcing with high amplitude and low frequency induced a significant distortion of the mean flow that affects nonlinear interaction among the waves. In this forcing

311

Fig. 7a–c. Effects of single- and two-frequency forcings on flow characteristics at St \0.5. a Total streamwise velocity fluctuation; D b centerline mean velocity; c momentum thickness (L: a \7%, f a \0%; K: a \7%, a \1%, / \23°; n: a \7%, a \1%, s f s de f s / \120°; e: a \7%, a \1%, / \272°; — — — : no forcing) de f s de

condition, an instability mode immediately enters the region of nonlinear interaction among the waves. As St is lowered to St \0.4, higher amplitudes of the D D fundamental and its subharmonic at the jet exit are required for stable vortex pairing. As shown in Fig. 9b, the variation of the subharmonic component is also a sine shape and its amplitude is in general lower than that of the fundamental, although two-frequency forcing with higher amplitudes than

Fig. 8a–c. Variations of the fundamental and subharmonic velocity components with the initial phase difference at three different streamwise locations at St \0.5, a \7% and a \1% (L: u@ /U ; K: D f s s e u@ /U ). a x/D\0; b x/D\2; c x/D\3 f e

those for the case shown in Fig. 8c is applied. For the same St , D when the level of the fundamental and its subharmonic at the jet exit was increased, the variation of the subharmonic with the initial phase difference was also changed to a cusp-like shape (not shown). The results for St \0.3 are shown in D Fig. 9c. Although there are some jitters in the variation due to a limitation on the present forcing system at high exit

312

Fig. 10a, b. Variations of the third and fourth harmonic velocity components of the subharmonic with the initial phase difference at St \0.5 (L: u@ /U ; K: u@ /U ). a a \7%, a \1% x/D\3; D 3f/2 e 2f e f s b a \7%, a \5%, x/D\3 f s

Fig. 9a–c. Variations of the fundamental and subharmonic velocity components with the initial phase difference under different forcing conditions (L: u@ /U ; K: u@ /U ). a St \0.5, a \7%, a \5%, x/D\3; s e f e D f s b St \0.4, a \13%, a \2%, x/D\3; c St \0.3, a \15%, a \6%, D f s D f s x/D\4.5

velocity and large uncertainties in arrival time of the vortices at x/D\4.5, it still nearly shows a sine shape. This sine shape of the subharmonic variation is a manifestation of the transitional stage to the cusp-like shape and possibly results from a relatively low frequency forcing. The lower the forcing frequency is, the longer its wavelength is. The streamwise length, where the subharmonic is interacting with

the fundamental and mean velocity components, growing and saturating before breaking into incoherent turbulent velocity components, is approximately as long as the potential core, but not long enough for the subharmonic of low frequency to grow completely over the entire range of / . Thus, the variation can de be approximately described as a sine shape. However, if either the initial level of the fundmental and its subharmonic or St D is high enough and the subharmonic wave can be advected farther downstream, then the subharmonic rapidly grows over the entire range of / except at a critical value and its variation de can no longer be described as a sine shape but a cusp-like shape. The harmonics of the subharmonic also depend on the initial phase difference similarly, as shown in Fig. 10. The subharmonic seems to play important roles in the amplification of its harmonics. The variations of the third (3f/2) and fourth (2 f ) harmonics of the subharmonic with the initial phase difference also change from a sine shape to a cusp-like shape when the subharmonic at the jet exit is increased. Moreover, the third harmonic of the subharmonic grows higher, significantly depending more on the initial phase difference than the fourth harmonic does. The growth of the 3f/2 component is due to a second nonlinear interaction

mechanism in which the fundamental and its subharmonic interact with each other to form a new component ( f]f/2). The fourth harmonic (2f ) seems to be generated mostly from the nonlinear interaction of 3f/2]f/2, rather than f]f (Hsiao and Huang 1990), based on the results that its variation with the initial phase difference is similar to that of the subharmonic.

3.1.3 The fundamental and the detuned subharmonic forcing The detuned forcing, in which the secondary forcing frequency is slightly different from the subharmonic frequency, causes the initial phase difference / to linearly change in time. Thus, de the effect of the initial phase difference on vortex pairing can be roughly understood by considering the effect of detuning on the u-signal in the time domain. The experiments were performed at St \0.5 and a \7% for two values of D f a D \1% and 5%, which are the same forcing conditions f/2~ f as shown earlier in Figs. 8c and 9a, except for the detuning of the subharmonic. The detuning frequency D f was 2.5 Hz (Df/f\1.9%). The modulation frequency f under forcing m at the frequencies of f and f/2[Df can be determined

Fig. 11a,b. Effects of detuning on the u-signal at x/D\3 and St \0.5 D for two values of a D \1% and 5% (D f/f\1.9%). a Time domain; f/2~ f b frequency domain

as follows:

C NA

BD

1 ~1 f 1 1 f \ [ m 2 f f/2[Df f/2 2fDf :2Df when fA2Df \ f[2D f When forcing at the frequencies of f\135 Hz and f/2[Df (the carrier frequency, 65 Hz) is introduced, the u-signals at x/D\3 (Fig. 11a) show amplitude modulations with a period of 0.2 s ( f \5 Hz). As a D increases from 1% to 5%, their m f/2~ f envelope shapes change from a sine shape to a cusp-like shape, which are similar to the subharmonic variations with / in de Figs. 8c and 9a, respectively. This proposes an easy way to find out the dependence of the subharmonic on the initial phase difference at an arbitrarily given forcing condition without the complicated FFT calculation. That is, we can roughly estimate the dependence of the subharmonic on / by simply observing de the envelope of the u-signal from the hot-wire output on an oscilloscope. Note that when resonance occurs, the local phase difference / between the fundamental and the subharmonic is d fixed to a nearly constant value regardless of / (for example, de about 230° in Fig. 6). Also, the amplitude of the saturated fundamental is constant to some extent over the range of / . de Therefore, the u-signal, which consists mostly of the fundamental and the modulated subharmonic, similarly reflects the subharmonic variation with / in its envelope (e.g. a sine de shape or a cusp-like shape). The modulations, with envelope shapes similar to the present ones, are also shown in Broze and Hussain’s experiment (1996) under single-frequency forcing at St \0.68. These modulations, denominated nearly periodic D modulations of pairing (NPMP), are due to the inability of the subharmonic to phase-lock. The initial phase difference between the fundamental and the subharmonic induced by pairing feedback does not remain the same from one pairing to the next. In their results, when the initial level of the fundamental increased from 3.4% to 3.7%, the envelope shapes also changed from a nearly sine shape to a cusp-like shape. The increase of the initial fundamental level, leading to the increase of the initial subharmonic level induced by pairing feedback, is equivalent to the increase of the initial subharmonic level in the present two-frequency forcing condition. The u-spectra are shown in Fig. 11b. In addition to the components at forcing frequencies, a lot of sidebands are generated and grown, particularly at f/2]D f, from nonlinear interaction with the fundamental f and the carrier frequency component f/2[Df. As a D increases, these sidebands grow f/2~ f much more, which results in significant modulation of the u-signal in amplitude and phase, and a cusp-like envelope of the u-signal in the time domain (Fig. 11a). Husain and Hussain (1995) showed that the growth of the sidebands was closely related to amplitude and phase modulations of the carrier frequency and that such a modulation produced only the odd sideband frequencies (e.g. f/2](1]2n)Df, n\0,^1,^2, 2 ).

3.2 Pairing region under single-frequency forcing (0.6\StD\0.9) In this region the spacing between neighboring vortices is relatively small, so the vortices easily interact and pair with

313

314

Fig. 13. Variations of the fundamental and subharmonic velocity components with the initial phase difference at St \0.85, a \3.3%, D f a \1.1% and x/D\1.5 (L: u@ /U ; K: u@ /U ) s s e f e

Fig. 12a, b. Distributions of the total streamwise fluctuation velocities along the centerline at St \0.81. a Various fundamental forcing levels D (e: a \1.5%; ]: a \2.2%; n: a \2.5%; L: a \3.3%; d: a \3.3%, f f f f f a \0.26%, £: a \3.9%; — — — : no forcing); b various subharmonic s f forcing levels at a constant fundamental forcing level (a \3.3%) (n: f a \0.012%; K: a \0.036%; L: a \0.117%: £: a \0.26%; — — — : no s s s s forcing)

each other. In Fig. 12a, the streamwise evolutions of u@ at t St \0.81 show the occurrence of vortex pairing under several D single-frequency forcings. In the case of a \2.2% and 3.3%, f however, vortex pairing is delayed or intermittently occurs downstream. Similar phenomena, which are very sensitive to a and are quite intermittent, can be observed at other St ’s. f D This results from improper phase difference between the fundamental and the subharmonic induced from pairing feedback. The rapid changes in the vortex pairing state also appeared for 0.6\St \0.9 in Broze and Hussain (1994). D As shown in Fig. 12b, at a \3.3%, vortex pairing readily f occurs when the initial subharmonics of very low amplitudes are added, which are one or two orders of magnitude lower than that of the initial fundamental, such that the overall forcing levels are maintained within the very limited range of amplitudes of the above single-frequency forcing case. Moreover, as its subharmonic forcing level increases, the location of maximum u@ monotonously moves upstream, because St is t D high enough for / to have little effect on the subharmonic de growth. In fact, in the region where the variation of the subharmonic component shows a sine shape such as in Fig. 8c,

vortex pairing is often delayed farther downstream when under higher-amplitude subharmonic forcing with the improper phase difference than when under lower-amplitude subharmonic forcing with the proper phase difference. In this case, the location of maximum u@ does not move upstream monott onously although the initial subharmonic forcing level increases. The dependence of the fundamental and the subharmonic on the initial phase difference at St \0.85 and x/D\1.5 D is shown in Fig. 13. Note that the initial phase difference has little effect on vortex pairing. The subharmonic component has already grown larger than the fundamental and vortex pairing occurs over the entire range of / (Raman and Rice 1991). de

4 Conclusions An experimental study has been performed on vortex pairing at low to moderate St under two-frequency forcing with D controlled initial phase differences through hot-wire measurements and a multi-smoke wire flow visualization method. For the range of 0.3\St \0.6, vortex pairing can be D controlled easily by means of two-frequency forcing with varying initial phase differences. A much larger mixing rate can be achieved by two-frequency forcing with the proper phase difference than by single-frequency forcing. As St D decreases, vortex pairing is limited to a narrow region of the initial phase difference between the two disturbances, and higher forcing levels of the fundamental and its subharmonic at the jet exit are required for more stable pairing. As the amplitude of the subharmonic at the jet exit increases for fixed St and the fundamental amplitude, the variation of the D subharmonic component with the initial phase difference changes from a sine shape to a cusp-like shape. Vortex pairing can be controlled more precisely for the former case. The harmonics of the subharmonic, which are formed through nonlinear interaction, also show variations similar to that of the subharmonic itself. The detuning of the subharmonic forcing frequency results in modulations of the u-signal in both

amplitude and phase in the time domain, as well as in sideband growth from nonlinear interaction in the frequency domain. As the initial forcing level of the carrier frequency increases, the u-signal reflects the variation of subharmonic with the initial phase difference in its envelope which changes from a sine shape to a cusp-like shape. This makes it possible to find out the dependence of the subharmonic growth on the initial phase difference by simply observing the envelope of the u-signal on an oscilloscope. For 0.6\St \0.9, non-pairing advection of vortices due to D improper phase difference is sometimes observed in several forcing levels when only single-frequency forcing is applied. However, when its subharmonic is added, vortex pairing occurs readily. As the initial level of this subharmonic is increased, the position of vortex pairing monotonously moves upstream. This is thought to be due to the fact that the variation of the initial phase difference has little effect on vortex pairing in the pairing region under single-frequency forcing. In this region, the initial phase difference is not an effective parameter in controlling vortex pairing.

References Arbey H; Ffowcs Williams JE (1984) Active cancellation of pure tones in an excited jet. J Fluid Mech 149: 445 —454 Broze G; Hussain F (1994) Nonlinear dynamics of forced transitional jets: periodic and chaotic attractors. J Fluid Mech 263: 93—132 Broze G; Hussain F (1996) Transitions to chaos in a forced jet: intermittency, tangent bifurcations and hysteresis. J Fluid Mech 311: 37—71 Cohen J; Wygnanski I (1987) The evolution of instabilities in the axisymmetric jet. Part 2. The flow resulting from the interaction between two waves. J Fluid Mech 176: 221—235 Crow SC; Champagne FH (1971) Orderly structure in jet turbulence. J Fluid Mech 48: 547—591 Hajj MR; Miksad RW; Pewers EJ (1993) Fundamental-subharmonic interaction: effect of phase relation. J Fluid Mech 256: 403—426 Ho CM; Huang LS (1982) Subharmonics and vortex merging in mixing layers. J Fluid Mech 119: 443 —479 Ho CM; Huerre P (1984) Perturbed free shear layers. Ann Rev Fluid Mech 16: 365—424 Hsiao F; Huang J (1990) Near-field flow structures and sideband instabilities of an initially laminar plane jet. Exp Fluids 9: 2—12 Hsiao F; Huang J (1994) On the dynamics of flow structure development in an excited plane jet. J Fluids Eng 116: 715—720 Husain HS; Hussain F (1995) Experiments on subharmonic resonance in a shear layer. J Fluid Mech 304: 343—372 Hussain AKMF; Reynolds WC (1970) The mechanics of an organized wave in turbulent shear flow. J Fluid Mech 41: 241—258 Liu JTC (1988) Contribution to the understanding of large-scale coherent structures in developing free turbulent shear flows. Adv Appl Mech 26: 183 —309 Makita H; Hasegawa T (1993) Acoustic control of vortical structure in a plane jet. In: Eddy structure identification in free shear flows, eds JP Bonnet; MN Glauser, editor, pp. 77—88, Dordrecht: Kluwer Academic Publishers Mankbadi RR (1992) Dynamics and control of coherent structure in turbulent jets. Appl Mech Rev 45: 219—247 Monkewitz PA (1988) Subharmonic resonance, pairing and shredding in the mixing layer. J Fluid Mech 188: 223 —252 Paschereit CO; Wygnanski I; Fiedler HE (1995) Experimental investigation of subharmonic resonance in an axisymmetric jet. J Fluid Mech 283: 365—407

Raman G; Rice EJ (1991) Axisymmetric jet forced by fundamental and subharmonic tones. AIAA J 29: 1114—1122 Thomas FO (1991) Structure of mixing layers and jets. Appl Mech Rev 44: 119—153 Van Dyke M (1982) An album of fluid motion, p 46, Stanford: The Parabolic Press Yavuzkurt S (1984) A guide to uncertainty analysis of hot-wire data. J Fluids Eng 106: 181—186 Zaman KBMQ; Hussain AKMF (1980) Vortex pairing in a circular jet under controlled excitation. Part 1. General jet response. J Fluid Mech 101: 449—491

315