Jounlal o f Sound attd Vibration (1978) 59(3), 313-333 COHERENT STRUCTURES SUBSONIC IN THE HOT MIXING FREE ZONE OF A JETJ" C. DAtIAN, G. ELIAS, J. MAULARD AND M. PERULLI Office National d'Etudes et de Recherches Agrospatiales, 92320 Chatillon, France (Receired 3 October 1977, and ht revised form 28 February 1978) In order to characterize the acoustical consequences of the existence of regular structures which have been emphasized in previous work, we proceeded with analysis of the spectral relations measured between a radiometric signal and microphonic signals picked up in both the near and far fields. In the near field, the pressure field is organized and possesses phase surfaces whose velocity passes gradually from the convection velocity inside the jet to the sound velocity in free space. At the same time, one observes the initiation of a directlvity effect. In the far field, this directivity effect is confirmed, the radiation of the coherent structures reaching its maximum in the angular sector between 10~ and 50 ~ A signal processing method has therefore been developed, to effect the decomposition of the far pressure field into a "coherent" field and a field related to the small scale turbulence, and which makes it possible to give amplitude values to the former. It then appears that, at the maximum emission angle and at the dominant frequency, the contribution of the coherent structure represents 50% of the total acoustic pressure radiated. In order to assess the validity of this type of decomposition, another signal treatment, based on conditional sampling rather than on second order analysis, has been implemented, in connection with other diagnostics, such as hot wires and laser Doppler anemometry. In particular, the existence of large scale structures has been confirmed by means of a conditional schlieren visualization. Moreover, conditional sampling led us to extract typical shapes in different signals and to reveal causality relations between signals exhibiting very weak correlations (radiometer and microphone at 0 = 90 ~ in the far field, hot wire in the mixing layer and microphone in the far field). i. INTRODUCTION The complexity of jet noise studies involves many iterative improvements involving the picture of the phenomenon under study, the measurement tools t.led to c h a r a c t e r i z . .:: sources, and the signal treatmeut procedures. Concerning the last l,oint, it has been recently recognized in turbulence research that second order signal analysis is probably insufficient to describe turbulence properties. Nevertheless, up to now, this kind of signal processing remains preferred in noise studies, as far as acoustic intensity is the relevant quantity to be accounted for, which is linked to the cross-correlations of fluctuations in the source region by means of the Lighthili formulation. We will show here, however, that second order analysis suggests an approach for building up a physical model of the turbulent field (at least for some mechanisms involved in the jet mixing) and of the associated noise field. Keeping in mind this new point of view, we will describe and discuss the first results pro,'!ded by conditional sampling in noise studies, and the future possibilities of this method in noise generation mechanism characterization. t A summarized version of this work was presented at the symposium on turbulence, Technische Universittit Berlin, FRG, I-5 August 1977. 313 314 c. DAHANET AL. 2. THE GLOBAL APPROACH 2.1. THEORETICAL BASIS In the Lighthill approach for the computation of the acoustic field of free jets emphasis is placed on the fact that, independently of the physical mechanisms involved to build up and sustain the turbulent field, the knowledge of a limited number of parameters (governing the Lighthili source term) is sufficient to provide a good estimation of the radiated noise. The experimental difficulties arise when one actually tries to measure the Lighthill source term over the whole jet volume, so that an intermediate step was soon introduced between turbulence and sound: that is, a model able to reasonably represent the quantities responsible for noise emission. Such models--and in the first place the small eddies model (statistical homogeneous and isotropie turbulence)--are quite convenient for providing indications of what needs to be measured (as far as noise is concerned), and, in the same way, for designing a test facility appropriate for jet noise studies. 2.2. THE TEST FACIL1TY The large number Of parameters on which jet noise depends invites one to study and in a detailed way define the jet within its whole volume, and leads to the definition of a test facility with the following main characteristics: (a) the capability for generation of a hot jet (up to 1l OO~K); (b) provision of the possibility to probe all the jet elements both for average characteristics and fluctuations (velocity, temperature); (c) access to a powerful, real time signal processing unit, allowing the data compression necessary for monitoring the test. 2.3. MEASUREMENT PROBES AND SIGNAL PROCESSING Besides the use of microphones to explore the near and far pressure fields, and of aerothermal probes to plot velocity and temperature profiles, we focused our attention on thermal fluctuations inside the jet, which can provide a picture of the spatio-temporal structure of turbulence. Many measurement devices have been implemented in recent years, to characterize the fluctuations responsible for the Lighthill source term (hot wires in cold subsonic jets, laser strioscopy, infrared emission and absorption, etc. [I ]). Dealing with hot jets, we chose to probe the jet volume with infrared radiometers, operating in a crossed beam configuration [2]. We could in this way obtain knowledge ofthe local properties of turbulence, and measure correlation lengths, eddy lifetimes and convection speeds. We here have to re-emphasize-and this is a part of the iterative process previously mentioned--that these interpretations of the crossed beam results are based on the implicit assumption of small scale fluctuations (which is postulated in the model). 2.4. DISCUSSIONOF THE RESULTS A typical set of results is presented in Figures 1-3. Such information, introduced into an appropriate computer program, enables one to make a prediction of the acoustic far field directivity (Figure 4), making the usual additional assumption of Gaussian space-time correlation. Despite the apparent self-consistency of this approach, many experimental features led us to question the pertinency of a small scale fluctuations model. 2.4.1. Inside the j e t The shape of the correlation function obtained when the axial separation of the radiometer beams is zero is not symmetrical with respect to z = 0; this indicates that it would be mis- COIIERENT STRUCTURES IN A tlOT JET I I , i I I 0 I i ~ v , I I I I l 5 315 I J i i i [ I I I I I [ I0 15 X/D Figure I. Axial evolution of the convection speed. +, V] = 590 m/s, Tj = 1 | 0 0 ~ K ; o , V] = 420 m/s, Tj = 900 ~ K. R/D=0"45; D = 80 ram. P- I I I o o + I I I 5 I0 15 xlP Figure 2. Axial evolution of the axial length scale, x, .2"0: Vj = 420 m/s, Tj = 900 ~ K ; o , -.~'o: Vj = 600 m/s, Tj = 1100 ~ K ; a , Uc x to: Vj = 420 m/s, Tj = 900 ~ K ; +, U, x to: Vj = 780 m/s, Tj = 1000' K. D = 80 ram. i I I I I l i I I I i I i I I I I I 16 fl I0~ 0 0 ~ ~ 2 I 2 4 6 i I 8 i I I0 I IZ I 14 X/D Figure 3. Axial evolution of the integral time scale, x, Vj = 420 m/s, Tj = 900= K ; @, Vj = 600 m/s, Tj = I 1 O0 ~ K. D = 80 mm. I I ] I I I x- - ~ , ~ x " ~ x I,~ dB \X\,, X"X"-X~ ~ ~ ~ 0 I 30 I 60 I 90 I 120 I 150 ~X I 180 0 (degrees) Figure 4. Ar far field d i r e c t i v i l y . - - , Measurement; - - x - - , computation. C. DAIIAN ETAL. 316 leading 9 to consider this function as the auto-correlation function of the fluctuations at the crossing of the two beams. Moreover, the Gaussian assumption used for modelling the cross-correlation functions is obviously not adequate. These functions exhibit large oscillations in space and time ranges much greater than those derived from integral scales, as shown in Figure 5 (the time oscillations can also be detected by other probes in a jet: e.g., hot ~(r) 0.5 /1 "r Figure 5. Cross-correlation of infrared signals at zero axial separation. Figure 6. Data processing laboratory. wires). It appears that at least a part of the turbulent field is not correctly described by such models, so that a more comprehensive analysis in the frequency domain is desirable. This narrowband analysis requires an extensive use of Fourier transforms and, accordingly, a real time Fast Fourier Transform calculator became a major tool in our data processing laboratory (see Figure 6) allowing us to measure the pertinent quantities in several frequency bands exhibiting different behaviour. 2.4.2. Acoustic for fiehl Turning now to the computation of the acoustic far field, we admitted still the assumption that the turbulence can be described by a Gaussian model, but with characteristic scales varying with frequency. This kind of"intermediate" assumption, that we class for the sake of simplicity in the global approach, led us to (i) a correct prediction of the acoustic spectrum at the maximum emission angle and (ii) a reasonable estimation of the ratio between the maximum and the minimum of the acoustic intensity around the jet. ~ ~ 0 c~ E -? .-D. i< c. DAHANETAL. 318 Nevertheless, the angle of maximum emission was greatly under-estimated. One of the usual assumptions to explain this kind of discrepancy is that such a computation does not take into account refraction phenomena, which are indeed included in principle in the Lighthill formulation, but not easily tractable with an analog approach. To quantify the refraction effects, we developed two-dimensional, and then three-dimensional refraction computer programs (Figure 7). At this point, it is interesting to notice that the introduction of refraction is a departure from the internal logic of the analog approach, in the sense that it represents an attempt to describe some physical phenomenon arising in the jet region itself. The existence of large scales revealed by the spatio-temporal structure of the temperature fluctuation field pushed our research in the same direction, inciting us to formulate the point of view that an approach able to isolate some source mechanisms (turbulence organization), and to link them directly to the noise field, would be helpful for understanding the acoustical behaviour o f jets. 3. A MECHANISM: COHERENT STRUCTURES 3.1. A THEORETICALBASIS Our aim to identify some mechanisms governing the mixing region process fits with earlier work tending to associate the large scale phenomenon with wavelike perturbations predicted by the theory of hydrodynamic instability [3]. In order to test this hypothesis for a i | 19: 0"5 XID =2 0.I /~X/D =3 I 0,I I 0,5 St Figure 8. Axial evolution of growth rates in a hot jet. UE= 400 m/s, T,~= 900~K. subsonic hot jet (TE = 900 ~ K, UE = 420 m/s) we performed a numerical exploration of the master equation of this theory (in its linear form). At low azimuthal numbers, the predicted ranges of amplified frequencies and phase speeds are compatible with the measurement data on the characteristics of the large scale motions (Figures 8 and 9). (For more detailed comparisons between theory and experiments, see reference [4].) 3.2. CHARACTERIZATIONOF COHERENTWAVEblOTIONS 3.2.1. Radial lengths o.f coherency Analyzing the coherency function of radiometer signals issued from two parallel beams at a given axial distance, we observed that the radial coherent length is of the same order of 319 COHERENT STRUCTURES IN A tlOT JET II I . XID \ ~ x l D :',.5 0~ I I o.I 0.5 St Figure 9. Axial evolution of phase speeds in a hot jet. UE = 400 m/s, TE = 900 ~ K. I | ,~ I s \ I \ / (kHz) 0.4\\ Brood- I~ndq "~-I / \\ 9 I I ' - 11 "\,~C>F"4 ~ -o I I \\ I X I ! 5 xm : z.7 \ \ \ x~ "~.~. ! l 5 f (kHz) X/D Figure 10. Radial lengths of coherency for infrared signals. Hot jet, U~ = 400 m/s, TE = 900 ~ K. I I I 0.5 0.2 I 0-2 I 0"4 St I 0-6 Figure 11. Coherency between radiometers at X[D = 2 and X[D = 5. magnitude as the jet diameter, for frequencies pertaining to the range predicted by theory (Figure 10). 3.2.2. Axial lengths of coherency It is now useful to point out that, because the radial scale is large, the crossed beam technique, fully justified in a fine scale structure model, is no longer the best way to proceed if we 320 c. DAIIANET AL. try to throw light on the coherent phenomena arising in the mixing zone of the jet. For instance, the coherency function between two parallel beams axially separated by three nozzle diameters is sensitive to the existence of large axial scales (which are more weighted for parallel than for crossed beam configurations) in a frequency range which is unambiguously defined (Figure I I). 3.2.3. Azhmtthal structure of the near pressure field A more precise picture of the spatial organization of the wavelike perturbations is reached through the Fourier series decomposition of azimuthal correlations between two microphones, one being fixed and the other rotating around the jet. Still in the same frequency range, one can see that the major part of the pressure power spectral density is concentrated in the axisymmetric mode (Figure 12). I • l i 1' I I I i , I x I0 % x x I I I I I I I 0"2 0.4 0,6 0"8 I 1.2 1.4 1.6 St Figure 12. Power spectral density per mode, rotating microphones. X I D = 3, riD = 2. 3.2.4. Further measurements and signal processhzg The hope introduced by the conception of a unique wavelike field extending over the whole space, from the inside of the jet to the acoustic far field, led us to use extensively correlations between a radiometric signal picked up in the mixing zone and pressure signals in the near and far fields. 3.2.5. Near field organization Following the spatial evolution of a frequency component from the mixing zone towards the far pressure field, one obtains interesting information from the cross-spectra between a radiometer focused in the jet and several microphones in the near field. The map of equal phase lines, for instance, reveals a well organized structure of wave fronts, and allows us to determine axial wave numbers and phase speeds associated with this coherent pattern (Figure 13). As well, the lines of equal cross-spectra amplitudes for a given frequency emphasizes the existence of two regions (Figure 14): (a) a downstream region, with a very marked direction ofradiation, which makes a small angle with the jet axis; 321 COttERENT STRUCTURES IN A llOT JET Microphones 0 0 0 - ) \\\ 3r < ) X=3D Figure 13. Phase o f cross-spectra (radiometer-microphones). St = 0'2. / f \ o///~'-o x, I, -22 ~ ~.-16 - 1 4 ^ -12 ^ -2 - - . - " ' ~ . ~, Figure 14. Equi-values o f cross-spectra amplitudes, between radiometer a n d m i c r o p h o n e s . St = 0'4. (b) an upstream region, characterized by a small coherence between the radiometer signal in the jet and the pressure signals in the near field. 3.2.6. Esthnation of coherent power spectral density hz the acousticfieM [4] Let us recall briefly the underlying assumptions allowing us to interpret the mutual coherency properties of pressure signals in the far field and infrared signals in the jet. Hyp I There exists in the infrared signal R a part Rc which is linked to the coherent wave structure, and a part Ba linked to the small scale fluctuations: R = Rc + R~. The same discrimination is also valid for the pressure field: p = Pc + P~. ltyp H The random processes Ra and Pc are linearly independent, as also are the processes Rc and po, so that the cross-spectrum lo. R can be written in the following form: ]o, R = ]Pc" Rc 4- [p~ R," 322 c. DAHAN AL. ET Hyp H ! The cross spectrum Ip,.R, is dominant since the incoherent radiation comes from uncorrelated sources distributed over the whole volume of the jet. Hyp IV Pc and Rc are linearly related, so that the cross-power spectrum can be written down in the form of a product of power spectra IIp.,I = (spo),,' (s,.),,=. ~ IIp,.,,I Considering then two radiometers focused at the points xi and xj, far enough from one another (Ixj - xtl > 3/)) and a microphone at the point.~(R, 0), one can deduce from the set of hypotheses the following relationships: I•.p =(So, Ira (Sz<,(x,)] ''=, B"'P(T;x';R'O)=(s.sp)"" ~,sp } \ S~(x,)] pI~,p(f;xj;I{IO)=(Spcll 2[SRe(Xj)II[2,~ . [s~,(xj)~ 112 [s~,Cxj)~ 112 so that /~,. ,(f; x, ; R, O) (1) ~,s,(xj)} ~,s~--GS,)) Sp,(R, O) flgo(f; x,; R, O)fl.p(f; xj; R, O) S.(R, O) = f l . . ( f ; x,; x j) (2) One can immediately write two necessary conditions for the validity of our signal splitting: (a) flRp(f;xt, R,O)/flRo(f;xj;R,O) must be independent of R and 0; the independence with respect to the angular position of the microphone is proved in Figure .15; I I I I I I I 0 5"/= 0.i 0,4 0 X X X I ,0 I 20 X X X I 30 S t : 0 . 1 .= X f 40 I I 50 ~;o I 70 0 (degrees) Figure 15'. Angular dependence of the fl, lB5 ratio. I RO 323 COHERENT STRUCTURES IN A HOT JET 0-5 I I \ • I I I o \ 0.4 \ \ 0.3 \ X\k \ 0.2 x\ \A 0.1 I I0 I 20 I 30 0 Figure 16. Dependence of the I 40 I 50 I 60 70 (degrees) fl, flJ/Pu ratio on the positions x,, xj. St = 0.3. x, fllfl4lfll,; A, fl,fls/P,5; O, P,~dB,8. I ~ I . i I I I I 0.2 0.4 0.6 0.8 St Figure 17. Total (i) and cohcrent (ii) power spectral density perccived by a radiometer. X/D = 2, riD = 0.5. (b) flap(xi)flRp(xj)/fl~a(xl,x~) must be independent of the positions xt,x I used to make the measurement, which is also verified (Figure 16). A coherent field can thus be extracted both (i) from the radiometer signal (Figure 17), and (ii) from the microphone signals, indicating a directivity diagram strongly peaked along the jet axis, as one should expect from the axisymmetric character of the perturbations in the jet region (Figure 18). 3.3. DISCUSSION OF TttE RESULTS It is easy to demonstrate that, conversely, if the two conditions (a) and (b) of section 3.2.6 were strictly fulfilled, the decomposition of radiometric and microphone signals would be legitimate, unique and unambiguous. But the physical meanings of p, and Rc may some- 324 C. DAIIAN E T AL. iimes become a matter of definition, especially in the cases where the link between Pc and R, is not linear (travel times randomly distributed, for instance), or when the frequency to frequency correspondence in the Rc and Pc signals is far from being satisfied (non-linearities or Doppler effects, for instance). In such cases, the method, rejecting in the non-coherent part of the decomposition some features connected to the coherent wave packets, provides an under-estimation of the power spectral density actually associated with the so-called coherent motions. i 90 ~ • I I I ~ I 80 -13 70 - I0 30 50 ZO 8 (degrees) Figure i 8. Directivitydiagram for the coherent pressure field. S t = 0-2. x, Total fieldi t~, coherent i~eld. One can see that these limitations are mainly due to the second order analysis which, ~mphasizing the behaviour of individual frequency components in the Fourier space, does not lead to a simple physical interpretation when one tries to associaie them witil the usual picture of vortices. To avoid the fuzziness of this correspondence, 6ne is thus invited to deal with wave packets, in the physical space, and this can be achieved as soon as a detection criterion, relevant to the basic phenomenon to be depicted, is specified. Thus, the large scale structure radiating properties involve the development of a new class of signal detection and processing called conditional sampling. Let us recall that pioneering work Of this kind was developed to picture the intermittency of boundary layers (turbulent/non-turbulent decision [5]), and to search for underlying phenomena in the mixing zones ofjets [6]. 4. CONDITIONAL SAMPLING The general idea is that, in the signals under consideration, a limited number of samples exhibit properties pertaining to the phenomenon that we try to isolate. These samples can be detected either in the signal itself or with the use of another signal (called the synchronization signal). Let us give some examples. (a) In the case of jet noise, a microphone signal in the far field could be studied only in the time slots where it cannot be assimilated into a second order homogeneous and stationary process. (b) Such samples could as well be obtained by having their selection triggered by a probe located in the mixing region, and in this case one can proceed to a joint statistical analysis of samples isolated both in the triggering signal and in the microphone signal. COtlERENT STRUCTURESIN A tIOT JET 325 (c) In a more general way, the same event--merged in noise, and appearing in different signals (picked up in the source region and in the far field)--can be isolated by the use of an external synchronization condition. In compressor noise studies, for instance, the tracking signal can come from a revolution Counter. There is, of course, a wide range of triggering criteria, and the proper: choice depends on the a priori information one has about the physical picture of the event to be selected. The first task is thus to establish the relevance ot'the sampling condition. 4.1. COttERENTSTRUCTURE VISUALIZATIONIN MIXINGREGIONOF JETS The current interpretation of the frequency content of the band pertaining to the coherent motions is that of big vortices (i.e., wave packets), well organized (m = 0 mode), occurring at times randomly distributed according to a rather narrowband process. This picture is sustained by arbitrary traces of an infrared signal (Figure 19), in which a characteristic shape is likely associated with these big vortices. This signature is well identified when the level of the fluctuations becomes high, and possesses a positive slope. The first idea is then to use this detection criterion (high level, positive slope) to trigger a schlieren device (Figure 20), which T =3000K ~r T=9OOOK i"~ i~ ) v Ill III ~ I lll IIII _ ~ - - ' ~ " ~ . Mixing zone ~(X/D:4) I I = 'JIA._/2 IIII 1101 I III I III I III I Iill I V I II Shope of the inffored signal Figure 19. Characteristic shape of an infrared signal. i Figure 20. Stroboscopic schlieren system diagram, l, Concave mirror; 2, schlieren system and flash unit; 3, film; 4, infrared radiometer; 5, microphone; 6, threshold detector and synchronization; 7, multiplier; 8, delay; 9, flash and film advance control. 326 c. DAHAN E T AL. is known to fail when no conditional triggering is applied. A second idea is that a single picture is insufficient to emphasize the phenomenon in a hot jet, and thus to take advantage of the different statistical properties of the usable signal and of the noise in which it is merged. This can be done by superimposing several individual pictures on the same film (Figure 21). One can see the results of these operations, confirming the adequacy of the detection criterion, inside the jet, with regard to the proposed interpretation. Figure 21. Synchronized schlieren picture of the jet. The near pressure field exploration with microphones allowed us to build a map of equiphase lines, interpreted as wave fronts accompanying the big vortices along their paths inside the mixing region (existence of a phase line At~ = 0). Thus the signal p(t) of a microphone located on this line, or the product p(t). R(t) (R = infrared signal) should be equivalent to R(t) alone, as far as triggering conditions for visualization purposes are needed. The corresponding images for the same level criterion are actually the same (for more details concerning visualizations, see reference [5]). 4.2. DETECTION AND STATISTICAL ANALYSIS 4.2.1. Detection Among the set of triggering conditions, amplitude detection--which has proven to be adequate for visualization--is convenient to be considered in tile first place, when one takes into account that our data processing laboratory is adapted to deal with analog signals. To sample the processes we try to analyze, we have to fix a time duration T for each sample of signal of interest, together with the triggering level S. To go further in a joint statistical analysis, a constant delay is also introduced to compensate for the propagation times of the information between the various probes. The results we obtain after the signal processing are then functions of the three following quantities: the triggering threshold S on the synchronization signal; the width of the gate T; the propagation time "L 4.2.2. Statistical tools At this stage, the description of the joint probability distribution of two sampled processes proceeds through the computation of its statistical moments. Up to now, we have limited ourselves to the first and second moments. Let Xto .r~(t) be a selected sample of the first signal and YEO.r3(t) be the corresponding sample of the second one. One is led to define 327 ensemble averages (over the samples selected) <X>, < Y>; mean square values <X2>, < y2> ; and a coherency coefficient COI'IERENT STRUCTURES IN A HOT JET -- 112 C(S'T'~) ~ T T f (X2> dt f (Y2> d' 0 0 The parameters S, T, "~must be adjusted to reach the highest value of C, which is a convenient indicator since it allows a direct comparison to be made with the standard coherency coefficient. (a) Ires I ~ (b) mire rap, 2 (7 T ' 0.2 p, (e) 0"72~~(TI (f) PT;p' T Figure 22. Results of conditional sampling. Radiometer at XID = 2; r/D = 0"5. Microphone at RID = 75; 0 = 30~ Triggering condition on the infrared signal. Amplitude >4 x standard deviation. (a) Mean signature of the infrared signal; (b) mean signature of the microphone signal; (c) mean square value of the infrared signal; (d) mean square value of the microphone signal; (e) mean product of infrared and microphone signals; (f) conditional coherency coefficient. Let us give three examples: one in which classical second order analysis allows strong conclusions to be drawn, and two in which it leads to a rather poor resolution. 1. Coherent structures in a hot jet ( T = 900 ~ K, V = 420 m/s). Here X(t) is the signal of a radiometer focused at X/D = 2, r i d = 0.5, and Y(t) is the signal of a microphone at RID = 75, 0 = 30 ~ In this case, the classical coherency function can reach a value of 0-5 for some frequencies (St = 0.25). Applying the proposed method, one finds that the signatures of the infrared signal and of the corresponding event at the microphone are well isolated, and the conditional coherency coefficient can reach 0.72 (Figure 22). 2. Hot jet; radiometer in the same position asfor example 1, and the microphone at RID = 75, 0 = 90 ~ The classical coherency function is nearly equal to zero, whatever the frequency. The statistical moments used show that, even in this case, the conditional coherency coefficient attains a height of 0.34 (Figure 23). 328 c. D^HAN Er Z L . 3. Cold jet (Uz = 100 in/s),/tot wire hi the mixing zone X/D = 3, r/D = 0"5; microphone ht the for fiehl R/D = 20, 0 = 30:. The classical correlations are usually very weak. In addition to this, this particular experiment was conducted in poor acoustical surroundings, allowing complex reflections (Figure 24). To carry out the analysis, we took advantage o f the fact that .z~.'-,---~ t,~,_^ - ~ I lmSl (a) (b) rap, 0-2 ,o' 0.5~ ~'(T] 0-34 (c) [ ~.... (d) p T r'jp" Figure 23. Results o f c o n d i t i o n a l sampling. R a d i o m c t c r at X/D = 2; r/D = 0-5. M i c r o p h o n e at RID = 75; 0 = 90 ~. Same triggering condition as in Figure 22. (a) Mean signature of the microphone signal; (b) mean square value of the microphone signal; (c) mean product of infrared and microphone signals; (d) conditional coherency coefficient. 0-03 9 Figure 24. Correlation function between a hot wtre signal in the mixing zone and a microphone in lhe far field. 0 = 30'. -- v V L / ,, - . I ms I I (a) (b) CT(c) Figure 25. Conditional sampling on hot wire and microphone signals. (a) Mean signature of the hot wire signal; (b) mean signature of the microphone signal; (c) conditional coherency coefficient. the signal from a hot wire in the potential core o f the jet was rather regular, so that it could provide the condition to sample both the mixing zone hot wire signal and the microphone and thus permit the statistical computations. In this unfavourable case also, the detection process is powerful enough to link definite events occurring in the source region and some wave packets in the far field: the conditional coherency coefficient rises up to 0.3, thus indicating the high signal to noise ratio that can be provided by this method (Figure 25). COHERENTSTRUCTURESIN A t[OT JET 329 4.3. DISCUSSIONOF TIIE RESULTS We have devoted the previous sections to the better signal to noise ratios obtained by using conditional sampling, thus enabling one to obtain visualizations of the axisymmetric big vortices propagating along the mixing layer, accompanied by pressure waves giving a well organized structure of the near field and allowing one to link events occurring during the mixing process inside the jet to pressure perturbations in the far field, in regions where standard second order analysis does not provide a good resolution. --v-v-L I I ms V my, I (a) /.D#o (b) 0.4 ~ " / ~ - - -2.500 ms (c) Figure 26. Conditional sampling on infrared and velocity signals at X/D = 2.5. (a) Mean signature of the infrared signal; (b) mean signature of the velocity signal; (c) comparison between (top) the correlation function of the radiometer and the velocity signals, and (bottom) the correlation function of their mean signatures. Furthermore, consideration of the mean signatures of signals delivered by several probes can help us to answer some questions which arose in the course of our studies. For instance, according to the theoretical basis we recalled previously (section 3.1), coherent fluctuations inside the jet must be linearly related. Taking the signals issued from two probes located in the jet, may we conclude that such a linear dependence exists? We compare signals delivered by a radiometer focused in the potential core (T') (X/D = 2.5, riD = 0) and by a laser Doppler anemometer measuring longitudinal velocity fluctuations (v') at the same point. The mean temperature and velocity signatures are exactly the same (but out ofphase), and the correlation coetficient between these two shapes is _~ I, thus proving the frequency to frequency correspondence, at least as far as mean values are concerned (Figure 26). On the same basis, a comparison between the mean signatures of a radiometer focused in the mixing zone of the jet and a microphone in the far field (Figure 22) led us to conclude that the frequency to frequency relationship existed between what we called the "coherent infrared signal" and the coherent pressure field, a case in which the rules derived in section 3.2 are the most significant. 330 c . DAHAN E T A L . Nevertheless, the aim of the data processing we try to work out is to provide a better estimation of the far pressure field hssociated with the coherent wave packets. Does the information we have obtained with the simple amplitude detection help us in this way? To recast this question in a new form, one is invited to compare the results obtained with those derived on the standard second order analysis basis, when all the information is carried by the correlation functions. The most commonly encountered random processes completely specified by their correlation functions are Gaussian processes, so that we are faced with the following problem: "Are the detection process involved in our signal analysis, and the statistical moments computed, sufficient to ensure that we are not in the situation where the joint probability distribution of the radiometer signal on the one hand, and of the microphone signal on the other hand, is Gaussian, completely specified by their correlation functions?" This class of problems is usually treated by means of the s6-called statistical test technique, a kind of reasoning we take the opportunity to present in the Appendix. Briefly, one can compute, assuming the joint Gaussian distribution, the conditional probability of the microphone signalp(t) when the infrared signal R(t) is known. From this distribution, the standard deviation <p2> _ <p>2 and the conditional coherency coefficient C can be derived and compared to the values provided by our experiments. One can see (in the Appendix) that the measured values are very close to the theoretical one from the Gaussian hypothesis. To go a little bit'deeper in this conclusion, one could say either that the mathematical operations involved are not sufficient, or that the detection process is not powerful enough to provide more information than the usual correlation function. One has to notice that these mathematical operations are to some extent arbitrary, depending on the a priori knowledge of the statistical properties of the process we want to depict. In the same sense, the detection criteria are chosen and adjusted in a heuristic way, insofar as we want to isolate an unknown signal with some random characteristics. Referring to the coherent structure radiation problem, one can imagine that amplitude criteria are not very efficient, so that we are invited to work out sampling procedures taking account of phase relationships, which must exist to build up wave packets. 5. FURTHER DEVELOPMENTS Global approach; identification of some noise generation mechanisms; use of conditional sampling: these are three steps which have marked the evolution of jet noise studies at ONERA. This evolution has proceeded rather through an attempt to visualize in an ever deeper manner the properties of the turbulence responsible for noise production, rather than through a series of breaks with previous concepts. In that sense, the results obtained at each step remain interesting: the global approach for prediction purposes, the coherent structures identification to specify a source mechanism, the statistical analys}s to build a map of real sources inside the jet volume. Concerning this last point, we have pointed out that amplitude criteria, leading to a full success as far as a certain level of visualization is concerned, are not efficient enough to emphasize the part played by wave packets in turbulent processes. It is not very surprising, as the point under study may be summarized as follows: how many variables do we need to represent a phenomenon involving random variations? Must we admit that the turbulent interactions are so strong that no reduction of the number of variables is possible ? The set of presumptions accumulated during the course of jet noise studies prevents us from accepting this extreme opinion, or at least encourages us to operate with conditioning criteria, taking account of simultaneous measurements at several points (large scales) and phase relationships (necessary to build up wave packets in locally weak interactions). This will be done in the near future. C O I I E R E N T S T R U C T U R E S IN A t l O T JET 331 REFERENCES 1. 1973 Proceedings Conference on Noise Alechanisms, Brussels, 19-21 September 1973, AGARD CP 131. 2. J. F. AMPHOUXDE BELLEVAL1974 Publication ONERA no. 159. Relation du champ acoustique d'unjet chaud avec sa turbulence et son 6mission infrarouge. (English translation in 1976 (February) European Space Agency ESA T-T248.) 3. A. MICHALKE1971 Zeitschriftfiir Flugwissenschaften 19, Heft 8/9. Instabilittit eines kompressiblen runden Freistahls unter Berucksichtigung des Einflusses des Strahlgrenzschichtdicke. 4. C. DAHAN 1976 Pablication ONERA 1976-4. Emission acoustique de structures coh6rentes dans un jet turbulent. 5. L. S. G. KOVASZNAY,V. KIBENSand R. F. BLACKWELDER1970 Jottrnal of Fluid Alechanics 41, 283-325. Large scale motion in the intermittent region of a turbulent boundary layer. 6. J. C. LAU and M. J. FISHER 1975 Journal ofFhdd Alechanics 67, 299-337. The vortex street structure of turbulent jets. 7. L. AVEZARD,C. DAHAN,G. ELIAS, A. LELARGE,J. MAULARDand M. P~RULLI 1977 American Institute of Aeronautics and Astronautics Paper No. 77- i 349. Simultaneous characterization of jet noise sources and acoustic field by a new application of conditional sampling. APPENDIX: STATISTICAL ANALYSIS Our aim, in this appendix, is simply to recall the way in which test procedures are conducted in statistical analysis (simple test), indicating the way in which detection criteria can be classified. The hypothesis we want to test is called the null hypothesis 1to, to be compared with another one, the alternative hypothesis H~. A test rule is fixed, allowing one to choose between Ho and HI. Two "risks" are usually attached to the test procedure: a risk of concluding that Ho is correct, whereas HI is actually the good assumption (error o f first kind quoted, c0; 9a risk o f concluding that tI~ is correct, while Ho is actually true (error of second kind quoted, fl). When one has to extract a signal from noise, applying a detection process, Ho corresponds to the fact that the samples isolated pertain to noise, and Hx to the fact that the usable signal is present in the time slots of signal provided by the sampling condition. The efficiency of the detection i s / ' = I - ft. Let us apply this scheme to the problem of radiation from coherent motions. We wish to go further than the results obtained on a standard second order analysis basis, according to a "wave packet" picture. Thus, we are led to model any probe signal R(t) picked up in the jet by a succession of impulses, each impulse being randomly modulated in width and amplitude (the basic shape being unchanged), the time inte,',al between two impulses being also random. The same model is assumed to be valid also for the microphone signal p(t). This representation is called the hypothesis H1, and we want to compare it to a model where the two processes R(t), p(t) are described by a two dimensional Gaussian distribution, completely specified by the cross-correlation ~ ( 0 (hypothesis tlo of Gaussian noise). To build up a test rule, adapted to the detection criterion previously described (section 4.2), we take advantage of the fact that Gaussian distributions are mathematically tractable [7], leading to rather simple results. Let p and R be described by a Gaussian process of standard deviations ap, aR and of correlation coefficient 5. The conditional mean o f p when R is known is (p) = (crJag) CgR. 332 ET AL. c. DAHAN The conditional mean square o f p is The conditional coherency coefficient is <.> C=<p2>,/2= c# 1+ -1 c6'2 . The conditional staiidard deviation o f p is a9,t 2 = < p : > - <p>2 = a ~ ( l - 52). These equalities allow us to assess the following test rules. I. For samples given by the conditional sampling, c o m p u t e a~,2 and c o m p a r e it to o~(I - 52). 2. F o r several triggering levels R/a~, c o m p u t e the difference q=Cd-C "~R - -~R- I C2J 9 If both a~2 - a~(l - g,2) _ 0 and q _~ 0 conclude Ho; otherwise conclude HI. 0"~2 (arbitrary units) (m volts 2) Unconditional mean square value of the microphone,,~,,~ X l J I I X m ~ I X l I X-- I I I I l 2 3 4 I )_ Triggering level R / o R Figure 27. Conditional standard deviation of the microphone signal, riD = 75; 0 = 50~ These operations were applied when R is an infrared signa! issuing from the point X]D = 2, riD = 0"5 a n d p is a far field microphone signal (RID = 75, 0 = 30 ~ for which the correlation coefficient is cd = 0-23 (Figure 27). One can see that a~,2 - a~(l - 52) = 0 and q = 0 in all cases (Figure 26). Some values o f q are given in the following table: R/aR C q 1"5 3 4 0'42 0'55 0"72 -0"06 -I-0'02 -0'02 To go further, it would be necessary to c o m p u t e the probability o f q when Ho or HI is true: i.e., to specify all the r a n d o m modulations involved in our impulse process. Nevertheless, due to the weak values o f q, one tends to conclude that no reason exists to reject the Ho hypothesis (joint Gaussian probability). But in another way, this means that the error o f the COttERENT STRUCTURES IN A tlOT JET 333 second kind is large: that is, either the mathematical operations involved in the test rules are not sufficient to discrimiflzite the two processes, or the detection process is not efficient enough (P = I - ,6' is weak). The values o f P could be used to classify results, with regard to the p h e n o m e n o n we want to depict, as a classification ciiterion. N o t e : we recall that p is equal to the probability o f g e t t i n g both ap,2 - (I - c6'2)a~-~0and q ~ 0 while the Gaussian assumption is valid.