THE RADIATED NOISE FROM ISOTROPIC TURBULENCE WITH APPLICATIONS TO THE THEORY OF JET NOISE

Journal of Sound and Vibration (1996) 190(3), 463–476

THE RADIATED NOISE FROM ISOTROPIC TURBULENCE WITH APPLICATIONS TO THE THEORY OF JET NOISE† G. M. L‡ Department of Aeronautics and Astronautics, University of Southampton, Southampton SO17 1BJ, England (Received 1 November 1995) Lighthill [1], in his Theory of Aerodynamic Noise, considered the noise from a pseudo-finite yet unbounded domain of compressible unsteady flow. The first application of this theory was given by Proudman [2] for the case of isotropic turbulence at low Mach numbers and high Reynolds numbers. More recently, Lilley [3] and Sarkar and Hussaini [4], using Direct Numerical Simulation (DNS), have reconsidered this problem, and evaluated for the first time the fourth order space-retarded time covariance which is central to Lighthill’s theory for the determination of the acoustic radiated sound power. In this paper the previous work is extended to include the effects of a hot fluid in motion immersed in an external medium at rest. On the introduction of a simple hypothesis these results for the noise radiated from isotropic turbulence are used to predict the noise power radiated from a gaseous hot turbulent jet. The results are found to be qualitatively in agreement with far field experimental data on hot jets at subsonic and supersonic speeds, provided the jets are fully expanded and are devoid of shock waves. The theory has its origins in the 1950s following the publication of Lighthill’s theory of aerodynamic noise, when Professor E. J. Richards, the author and their colleagues were striving to predict the noise from jet engines and establish methods for their noise reduction, without loss in performance. 71996 Academic Press Limited

PREFACE

The noise disturbance from gaseous jets has created one of the greatest nuisances to mankind, and it was realized soon after the Second World War by many aircraft engineers and scientists, including the late Professor E. J. Richards, that the introduction of the jet engine as the power plant for civil air transport would be dependent on limiting the noise from its jet exhaust. In 1949 little published work existed on the noise from turbulent jets and experimental work was commenced to establish a database, from which could be exposed the major characteristics of the noise and its relation to the flow properties of the jet. Soon a few research groups were set up in the United Kingdom, including Professor Richards’ new laboratory at the University of Southampton, to explore the characteristics of jet noise and means for its reduction. † This paper is a contribution to the 80th birthday celebrations for Professor E. J. Richards and to commemorate his sad passing away in this same year. ‡ Present address: Center for Turbulence Research, Bldg 500 Stanford University, Stanford, California 94305-2030, U.S.A.

463 0022–460X/96/080463+14 $12.00/0

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. . 

Our understanding of aerodynamic noise, however, has its foundations in the work of Lighthill on ‘‘sound generated aerodynamically’’, which was the first major advance in acoustics since the pioneering work of Lord Rayleigh in the last century. The combination of Lighthill’s theory of aerodynamic noise, and the experimental database which was slowly gathering momentum, was quickly exploited by various jet propulsion engine designers in reducing the noise from civil transport aircraft powered by jet engines at takeoff and landing to levels marginally acceptable to communities living in the neighbourhoods of airports. The rapid growth of fast economical civil transport aircraft worldwide was thereby achieved. The author was an old friend and colleague of Elfyn Richards for 45 years. We first met during World War Two at the National Physical Laboratory, Teddington, when I was an Assistant Engineer at Vickers Armstrongs Weybridge. Later we became close colleagues in our work for the Aeronautical Research Council, after I joined the College of Aeronautics and Elfyn Richards became the first Professor of Aeronautics and Astronautics at the University of Southampton. We had many discussions on aircraft noise research in the 1950s and, arising from these, a number of initiatives were undertaken, including proposals for the setting up of the Noise Research Committee of the ARC, the Institute of Sound and Vibration Research, the British Acoustical Society and the Noise Advisory Council. We were also joint consultants for Rolls Royce on turbo-fan noise research. In 1963 I joined Elfyn Richards at Southampton to become the second holder of the chair of Aeronautics and Astronautics, following his resignation to become the first Director of the ISVR and Professor of Acoustics.

1. INTRODUCTION

Our understanding of the theory of jet noise has its foundations in Lighthill’s theory of aerodynamic noise [1, 5–8]. Lighthill’s theory is based on an acoustic analogy whereby the exact Navier–Stokes equations for fluid flow are rearranged to form an inhomogeneous wave equation for the fluctuating fluid density. The forcing function on its right-hand side represents a distribution of acoustic sources in the ambient flow at rest, replacing the complete unsteady flow. In Lighthill’s theory, 1 2Tij /1xi 1xj is the strength of the acoustic sources per unit volume, and Tij is Lighthill’s instantaneous applied acoustic stress tensor. In this acoustic analogy the equivalent acoustic sources may move, but not the fluid. The central property of Lighthill’s theory, when applied to the noise radiation from turbulent flows, is the evaluation of the fourth order two-point space-retarded time covariance of 1 2Tij /1t 2 and its distribution throughout the given flow field. The acoustic power radiated to the far field is found by integration of this space-retarded time covariance over the entire flow volume. In an incompressible flow this integral is exactly zero, since although it is positive for small spatial separation distances this is cancelled by negative values at greater distances. Soon after Lighthill’s theory was published, Proudman [2] applied it to the noise radiated from decaying isotropic turbulence at low Mach numbers and high Reynolds numbers. Proudman confirmed that the radiated noise power per unit volume of turbulence, ps , in this case of near-incompressible flow, was proportional to the eighth power of the turbulent velocity according to the formula 5 ps=ara u 8/ca L,

(1)

where u=zu 2 and u 2=2K/3. K is the kinetic energy of the turbulence per unit volume, L is the integral scale of the turbulence, and ra and ca are, respectively, the

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ambient density and speed of sound. a, which we call the Proudman constant, is a numerical constant dependent on the model of turbulence used. In the early 1950s it was realized that Proudman’s model of the noise from isotropic turbulence was unsuitable for predicting the noise from the inhomogeneous turbulent mixing region of a jet. However, until recently no reliable method was available to predict the properties of the space-retarded time covariance and its integration over all space for a compressible turbulent flow. In order to predict the radiated acoustic power it was therefore necessary to introduce models of the turbulence which had to be validated against ‘‘good’’ reliable and repeatable experimental data. The only previous attempt at this had been made by the present author in 1958, in a paper of restricted publication [9], using an approximate solution for the turbulent pressure fluctuations in isotropic turbulence in the presence of a uniform mean shear and applying this to model the noise radiated from the turbulent mixing region of a jet. However, this method simply established that until an exact evaluation of the space-retarded time covariance was available approximate methods had little or no advantages over the results obtained from dimensional analysis. These were shown to be qualitatively correct, but quantitative answers were then only found by the introduction of numerical constants, the values of which had to be validated against experiments, as discussed above. The turbulence models in use by the turbulence community are, in general, not applicable for the prediction of radiated noise, since they model only the average or mean characteristics and not the instantaneous properties of the turbulent flow. The recent advances in the use of supercomputers and their application to the understanding of the structure and dynamic properties of turbulent motion, and in particular of turbulent shear flows, from direct numerical simulation (DNS), have enabled, for the first time, the characteristics of the space-retarded time covariance to be uncovered. The new field of numerical simulation in problems involving aerodynamic noise has been called computational aeroacoustics (CAA). The present paper discusses the recent re-evaluation by Lilley [3] of Proudman’s constant for weak compressible isotropic turbulence based on the (DNS) results of Sarkar and Hussaini [4]. A hypothesis is then introduced whereby Proudman’s theory is applied to model the space-retarded time covariance in a turbulent heated jet. The results of the model for the total acoustic power are compared with experimentald data over a wide range of jet/ambient temperature ratios and exit Mach numbers.

2. DIRECT EVALUATION OF THE TOTAL ACOUSTIC POWER

The fluid flow conservation equations of mass, momentum and total energy for a Newtonian compressible gas are, respectively, 1r/1t+9 · rv=0,

(2)

1rv/1t+9 · rvv=−9p+9 · t,

(3)

1rhs /1t+9 · rvhs−1p/1t=9 · (q+t),

(4)

where r, p, v, t and q are, respectively, the fluid density, pressure, velocity vector, shear stress tensor and the heat flux vector. hs=h+v 2/2, where h is the local enthalpy, which in a perfect gas is given by CP T. CP is the specific heat at constant pressure and T is the absolute temperature. The equation of state of the gas is p={(g−1)/g}rh,

(5)

. . 

466

where g is the ratio of the specific heats. For a perfect gas, CP and g are constants. The corresponding speed of sound is c=z(g−1)h . The disturbance created by unsteady turbulent flow which results in alternate compressions and expansions of a fluid element as it is convected by the flow is given by the time rate of change, following the fluid, in the volume of this fluid elment, dV, per unit volume of fluid, s dV:0

1 DdV D ln r =− =9 · v, dV Dt Dt

(6)

and although the dilation, u=9 · v, is zero in an incompressible flow it is always finite in compressible flows, and similarly with respect to 9 · rv. In order to ensure the finiteness of the latter in calculations concerning aerodynamic noise, Lighthill [1] derived the inhomogeneous wave equation for the density fluctuations by eliminating 9 · rv between the equations of conservation of mass and momentum. Here we follow Lighthill’s approach and derive the inhomogeneous wave equation for the fluctuating pressure in the form derived by Lilley [10] 1 2p (g−1) 1 2rv 2 1 rv(hs−ha ) 2 −9 · 2 2 2−9 p=9 · 9 · (rvv−t)− ca 1t ha 2ca 1t 2 1t +

(g−1) 1 9 · (q+v · t)0A(x, t), 2 ca 1t

(7)

2 where 1(p−rca )/1t has been replaced by its equivalent terms from the energy equation. The unbounded solution is

(p−pa )(x, t)=

1 4p

g

V

[A(y, t)]

d3y , =x−y=

(8)

where the [. . .] denotes that the function is evaluated at the retarded time, t=t−=x−y=/ca . The far field approximation is found by noting that

$

%0 1

0 1

1fi (y, t) 1fi (x −yi ) 1fi = + i , 1yi t =x−y= ca 1t y 1yi

(9)

and since the terms leading to the integral of a complete divergence reduce to a surface integral, which is zero in an unbounded flow, we find (p−pa )(x, t)1

1 2 4pca

g

0

xi xj 1 2 (g−1) 2 rv dij 3 2 rvi vj−tij− x 1t 2 V

+(g−1)

0

1 1

xj 1 2 rvj (hs−ha )(qj+vk tkj ) d3y. cax 2 1t 2

(10)

We find the integrand in equation (10) is identical with the component of Lighthill’s stress 2 r)dij taken in the direction joining the source at y to the far tensor, Tij=rvi vj−tij+(p−ca 2 )/1t has been replaced by its equivalent field observer at x, when, as stated above, 1(p−rca terms from the energy equation. Apart from the noise generated by the diffusive terms, q and t, which at high Reynolds numbers are shown to be very small and can be neglected, the major sources of sound in a turbulent flow involve the fluctuations of the momentum flux, rvv, and the fluctuations of the total enthalpy flux, rv(hs−ha ). The fluctuations of the kinetic energy, rv 2/2, make a small contribution to the radiated noise. (In an inviscid incompressible flow the time gradient of the integral of the kinetic

     

467

energy would be zero. The integral is, however, finite in a weak compressible viscous flow.) Lighthill’s integral can be evaluated in a frame of reference moving with a mean convection speed, VC=ca MC , where MC is known as the acoustic convection Mach number, to distinguish it from the true Mach number, which is evaluated relative to the local speed of sound in place of the ambient speed of sound, ca . The moving co-ordinates are defined by h=y−ca MC t,

(11)

where the source emits as it crosses the fixed point y at time t=t. The intensity, I(x), of the radiated sound in the far field is I(x, t)=(p−pa )2/ra ca,

(12)

and the autocorrelation, I(x, t×), and spectral density, I(x, v), for a stationary turbulent flow are, respectively, I(x, t×)=

I(x, v)=

1 2p

g

1 5 16p x 2ra ca 2

g g d3y

V

I(x, t×) exp(ivt×) dt×=

14 P (y, r, t) d3r, 1t 4 xx, xx pv 4 5 2x 2ra ca

g

Pxx, xx (y, k, v) d3y,

(13)

(14)

V

where r is the spatial separation in fixed co-ordinates, t is the retarded time difference, defined by t=t×+x · r/xca , t× is the far field autocorrelation time separation, Pxx, xx is the source (y)–observer ( x) aligned space-retarded time covariance of Tij , and Pxx, xx (y, k, v)=

1 16p 4

g

exp(ik · r) d3r

g

exp(ivt)Pxx, xx (y, r, t) dt

(15)

is the four-dimensional wavenumber–frequency spectrum function corresponding to the aligned space-retarded time covariance of Tij . The frequency of the sound, v, is the same as in the turbulence, and the wavenumber vector of the sound, k=−vx/xca , equals the wavenumber vector in the turbulence. The total acoustic power per unit volume of turbulence is found by integrating the intensity per unit volume at the given source position, y, over a large spherical surface, so that for isotropic turbulence ps=

1 5 4pra ca

g

14 P (y, r, t) d3r. 1t 4 xx, xx

(16)

When the acoustic sources are in uniform motion with the eddy convection speed, VC , and the space-retarded time covariance of Tij is measured in the moving frame, the spectral density of the sound intensity per unit volume is given from the Lighthill–Ffowcs Williams eddy convection theory [11] in the form 5 i(x, v)=(pv 4/2x 2ra ca )Pxx, xx (y, k, vT ),

(17)

where the radiated sound to the far field at frequency v arises from turbulence in the moving frame with frequency vT , which is the Do¨ppler shifted frequency, with vT=v(1−MC · x/x). The wavenumber in the turbulence, k=−vx/xca , is unaffected by the eddy motion. When the direction to the far field is near the Mach wave direction, where normal to the Mach wave (MC · x/x=1), detailed analysis shows that the relation

. . 

468

between the frequencies in the turbulence and that of the radiated sound becomes vT=v(=1−MC · x/x=2+ST2 MT2 )1/2 ,

(18)

where ST and MT=vT /ca are, respectively, the characteristic Strouhal number and the Mach number of the turbulence. The reference Strouhal number of the turbulence, which we assume to be a constant throughout a given turbulent flow, and is of order unity, is defined by, ST=VL/vT , where L is the local integral scale of the turbulence, and V is the reference frequency in the turbulence. The reference velocity is given as vT=z2K/3, where K is the local kinetic energy of the turbulence. In isotropic turbulence, vT is equal to zu 2, where u is the velocity component in any direction. The corresponding result for the intensity per unit volume, found by integrating equation (17) over all frequencies, is i(x)=

1 2 2 2 −5/2 2 5 (=1−MC · x/x = +ST vT /ca ) 16p x 2ra ca 2

g

14 P (y, d, t) d3d, 1t 4 xx, xx

(19)

where d is the separation distance in the moving frame, and t is the corresponding retarded time difference, showing the preferential direction for sound radiation in the downstream direction of the convecting eddies, with a sharp peak in the direction normal to the Mach angle when the eddy convection Mach number is supersonic. We will assume the turbulence has a uniform density, r0 , and ratio of specific heats, g0 , compared with the ambient medium values of ra and ga . The mean pressure in the turbulent flow is assumed to be equal to that of the external medium. We found above there were three dominant source terms in Lighthill’s aligned stress tensor, Txx , and if we further assume they are statistically independent, we find their separate contributions to the radiated sound power are, in the case of stationary isotropic turbulence at rest, ps(1)=

ps(2)=

ps(3)=

1 r02 u 8ST4 5 L 4p ra ca

g

1 4 (ux )2A (ux )2B−u 22 3 d r, 1t 4 u 22

1 r02 u 8ST4 (g0−1)2 5 4p ra ca L 4

0 1g

1 r02 u 6ST4 g0−1 3 4p ra ca L ga−1

2

g

1 4 qA2 qB2 −q 22 3 d r, u 22 1t 4

1 4 (ux )A (h')A (ux )B (h')B−ux h'2 3 d r, 2 1t 4 u 2ha

(20)

(21)

(22)

where h' is the fluctuation of the enthalpy and . . . denotes a mean value. Suffixes A and B denote the two source positions, distance r apart, forming the respective space-retarded time covariance. Let us consider the evaluation of the aligned velocity squared space-retarded time covariance that appears in ps(1) in equation (20): (1) 2 2 2 2 Pxx, xx (r)=(uA uB−u  ).

(23)

Now this fourth order isotropic tensor can be shown to be a function of the longitudinal and lateral velocity squared covariances which are functions of r only. When the turbulence follows Gaussian statistics, as assumed by Proudman [2], we find according to Millionshtchikov’s hypothesis as given by Batchelor [12] that the velocity squared covariances can be replaced by the sum of the squares of the corresponding second order covariances involving f(r) and g(r), where the second order longitudinal and lateral

     

469

covariances are, respectively, up (x)up (x+r)=u 2f(r)

un (x)un (x+r)=u 2g(r).

and

(24, 25)

Lighthill [13] has shown more generally that the fourth order longitudinal velocity covariance can be replaced by the square of the corresponding second order covariance giving [up (x)2up (x+r)2]−u 22=[up (x)up (x+r)]2[(u 4/u 2 )−1], 2

(26)

and a similar relation holds for the fourth order lateral covariance by replacing the suffix p by the suffix n. The relationship between the respective fourth and second order covariances holds for the given retarded time difference time, t. The velocity flatness factor, 2 T1=u 4/u 2 has the value 3 in Gaussian statistics, and is found by Townsend [14] to be nearly 3 in decaying isotropic turbulence. A similar result was obtained in the (DNS) results of Sarkar and Hussaini [4] and Dubois [15]. Noting that the points A and B are separated by the vector distance r, which is in a direction at angle u to the propagation direction x, we find from equation (23), together with equations (24), (25) and (26), that (1) 2 2 2 2 2 Pxx, xx (r, t)=(T1−1)u  (f(r, t) cos u+g(r, t) sin u) .

(27)

In weakly compressible flown, the turbulent Mach number is very small, and in this case we may assume that the modulus of the wavenumber k in the turbulence is also small. Hence the contribution to the acoustic power spectral density is ps(1) (v)=

ra v 4u 22(T1−1) 5 15pca

g

a

r 2 dr

0

g

a

(3f 2+4fg+8g 2 ) cos vt dt.

(28)

0

The relation between f(r, t) and g(r, t) in isotropic turbulence must satisfy the equation of continuity, and as shown by Batchelor [12], g(r, t)=f(r, t)+(r/2) 1f(r, t)/1r. a

(29)

a

It follows that since f0 (3rf 2+2r 3f 1f/1r) dr=f0 [1(r 3f 2 )/1r dr=0,

g

a

g 01 a

r 2(3f 2+4fg+8g 2 ) dr=2

0

r4

0

1f 1r

2

dr,

(30)

and then equation (28) becomes ps(1) (v)=

ra v 4u 22(T1−1) 2 5 ca 15p

g

a

cos vt dt

0

g 01 a

r4

0

1f 1r

2

dr.

(31)

The integrals in equation (31) can only be evaluated when the distribution f(r, t) is known. Lilley [3] used the (DNS) databases obtained by Sarkar and Hussaini [4] and Dubois [15], and the (DNS) and (LES) databases obtained by Witkowska [16] to obtain the spatial and temporal covariances. Thus, upon using the data derived from these databases the value of the Proudman constant, aP(1) , in 5 ps(1)=aP(1) r0u 8/ra ca L

(32)

aP(1)=1·80(T1−1)ST4 .

(33)

becomes

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. . 

When the flatness factor, T1=3, as discussed above, and the reference Strouhal number ST=1, we find the Proudman constant aP(1)=3·6. The available (DNS) databases gave values of ST between 1 and 1·25. The temporal covariance was checked between the (DNS) space–time covariance results of Dubois [15] and the far field acoustic spectra obtained by Sarkar and Hussaini [4]. These results are for low Reynolds numbers and low Mach numbers, and there are doubts as to their applicability to higher Reynolds numbers and Mach numbers. The low Reynolds numbers of the (DNS) data precludes the existence of an inertial subrange and there is less than a decade of separation between wavenumbers in the energy containing and Taylor microscale ranges of eddies. The peak frequency of the radiated noise is at a frequency slightly higher than that of the energy containing eddies. This suggests that the dominant eddies responsible for the generation of sound are slightly smaller than those in the energy containing range. This is consistent with the deductions of Lighthill and Proudman. At high Reynolds numbers all simplified models of turbulence, along with dimensional analysis, suggest it is the eddies of scales close to the energy containing range which are responsible for the bulk of the sound generation. In a recent paper, Zhou and Rubinstein [17] consider the noise radiated from the turbulent inertial subrange and find that the temporal correlations derived by Lilley [3] are consistent with the sweeping hypothesis of Kraichnan [18], and Praskovsky et al. [19], involving a non-local property of the energy containing eddies. Zhou et al. deduce that the noise power generated at high Reynolds numbers should have a spectral decay of v−4/3 . The current low Reynolds number database, as shown in Figure 1, suggests the decay law is of order v−2 over a wide range of frequencies before falling exponentially in the dissipation range. In addition, Zhou et al. [20] have examined a large databank of high Reynolds number atmospheric and wind tunnel turbulence data at around the peak and higher wavenumbers to derive values for the Proudman constant using the formulas derived by Lilley [3] and discussed above. Although this data is largely for anisotropic turbulence, it is regarded as a useful guide to the Reynolds number dependence of the integral properties of isotropic turbulence which govern noise generation and its acoustic power. The calculated value of the Proudman constant

Figure 1. The acoustic energy spectrum in isotropic turbulence. ——, (v/vm )4 [1+2(v/vm )2 ]−3; Q, DNS, Sarkar and Hussaini [4]; – – –, (w/wm )−4/3; · · · ·, (w/wm )−7/2 .

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obtained by Zhou et al. [20] is within the range found by Lilley [3], based on the databases described abve, suggesting that there is only a weak dependence on Reynolds number. The contribution ps(2) can be combined with ps(1) , and their combined contribution is similar to that when enthalpy fluctuations are absent. In the evaluation of ps(3) we need the value of the fourth order covariance (ux h')A (ux h')B . If we assume Gaussian statistics and impose Millionshtchikov’s hypothesis, and note that in isotropic turbulence ux h' is zero, then (ux h')A (ux h')B−ux h'2=(ux )A (ux )B (h')A (h')B .

(34)

On the assumption that the correlation function for the enthalpy fluctuations is equal to f(r, t), then the acoustic power spectral density arising from ps(3) is similar to that arising from ps(1) and ps(2) . We find that ps(3)=

4z2 r02 u 6ST4 (h')2 (g0−1)2 . 3 2 p ra ca L ha (ga−1)2

(35)

Our final values for the two terms in the contributions to the acoustic power output are ps=aP

r02 u 8 r2 u6 +aH 0 3 , 5 ra ca L ra ca L

(36)

where aP =

0

1

4z2 3(g −1)2 (T1−1)ST4 1+ 0 p 4

and

aH =

0 1

4z2 g0−1 2 4 (h')2 S . 2 ga−1 T ha p (37, 38)

We find that the term involving the enthalpy fluctuations generates acoustic power proportional to u 6 and hence dominates over the u 8 contribution. These results show that, typically, the dipole contribution equals the quadrupole contribution when MT=0·28. Our conclusion is that our analysis of Lighthill’s Tij space-retarded time covariance for a volume of isotropic turbulence and its incorporation into Lighthill’s integral for the acoustic power radiated to the far field, per unit volume of turbulence, may have application to the estimation of the radiated acoustic power from a wider variety of turbulent flows. In such estimations, the critical parameters will be the local values of the turbulence kinetic energy, its integral scale(s), and the peak frequency of the radiated noise, where the latter is captured by introduction of the reference Strouhal number, ST . It appears that ST will not vary greatly throughout a given turbulent flow, and will always be of order unity. This procedure, at least in low Mach number unbounded stationary turbulent flows where the equivalent acoustic sources will be compact, (i.e., where the scale of the turbulence is small compared with the wavelength of the radiated sound, with the compactness ratio L/l=ST MT /2pW1), requires an efficient flow solver to provide the average properties of the turbulent flow throughout the entire flow field and in a frame moving at the local convection speed. Since the turbulent Mach number, MT , is generally less than unity even in very high speed flows, our method is likely to have application over a wide range of flow Mach numbers. Once the flow properties are known for the flow at a give Mach number the radiated acoustic power can be obtained from application of the Lighthill–Ffowcs Williams [11] convection theory as discussed above.

472

. .  3. THE MIXING REGION NOISE

The physical process of noise generation in a mixing region is assumed to be similar to that in isotropic turbulence, as discussed above. However, the turbulence is now anisotropic and inhomogeneous and is dependent on the mean rate of strain. Its Reynolds stress tensor contains both shear and normal stress components. Nevertheless, with respect to the principal axes of stress only the direct stresses act. The sum of these enables us to find the local values of the average kinetic energy of the turbulence. The turbulence intensity is assumed to be proportional to the velocity difference across the shear layer. In the fully developed mixing region of a jet, the turbulent intensity depends on the velocity difference between the centreline velocity of the jet, which decays with downstream distance, and the external velocity. The integral scale of the turbulence is assumed to be proportional to the local width of the mixing region based on the vorticity thickness. The mean flow growth is governed by entrainment and the mean shear. The intense turbulence is found to exist near the centre of the mixing region. The turbulence is intermittent, but a useful model is to assume that the turbulence is approximately uniform over the mean vorticity thickness of the jet and zero outside. The average convection speed, VC , of the main energy containing eddies in a turbulent mixing region over a wide range of different gases, velocities and temperatures can be obtained from the work of Papamoshou and Roshko [21]. For the mixing region of a jet near the nozzle exit, VC is about 0·58Vj . With these properties we may assume that the turbulence is quasi-isotropic, having a mean convection speed, VC . The noise radiated to the far field of a mixing region is estimated, based on the hypothesis that the fourth order space-retarded time covariance has properties in shear flow turbulence similar to those in isotropic turbulence, apart from changes in the scales of length and velocity. The local reference turbulent velocity is assumed on the basis of the local kinetic energy, and a local reference integral length scale, corresponding to the scale of the energy containing eddies, is defined at each section of the mixing region or jet. The spectrum of turbulence is assumed to be similar to that of isotropic turbulence, but the frequency of the peak energy, fp , proportional to the mean velocity gradient. The turbulent Strouhal number, ST , in the case of the mixing region is assumed to be of order 1·7.

4. EFFECTS OF TEMPERATURE DIFFERENCE AND MACH NUMBER

We showed in section 2 that at low Mach numbers Lighthill’s Theory of Aerodynamic Noise is based on an acoustic analogy whereby the flow is replaced by a distribution of acoustic sources, the instantaneous applied stress tensor of which is 2 Tij=rvi vj+(p−rca )dij if diffusive effects are neglected. In the analogy, the acoustic sources are embedded in a medium at rest having the same constant properties as the ambient medium, but whereas the medium is at rest the acoustic sources may move. At low Mach numbers it is legitimate to neglect density fluctuations in the flow interior, but not outside, where the sound waves are formed and propagate to the far field. When the flow temperature differs from ambient the average value of r varies throughout the flow, except in the special case of isotropic turbulence. In a mixing region an approximation may be introduced whereby the density is assumed to be constant across the mixing region with a value based on the density at the position at which the local mean velocity is equal to the mean convection speed. The mean flow is assumed to be self-preserving and the mean density, temperature, stagnation enthalpy and velocity profiles are calculated throughout the flow by using a simple eddy viscosity model and

     

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based on the boundary conditions of the flow. In this model the equations of momentum and total enthalpy are similar, and hence the mean velocity is a linear function of the mean total enthalpy. The reference density, compatible with the convection speed, is then determined at each downstream station. The effects of turbulence convection can be applied as discussed in section 2. 5. THE NOISE RADIATED FROM A CIRCULAR JET

The total acoustic power radiated from a circular heated jet can be evaluated from the analysis in section 2 for isotropic turbulence with the modifications discussed in section 3 and section 4 to allow for the effects of anisotropy, mean density variation and convection. The theory can be used for jet noise prediction for hot and cold jets, both under static and flight conditions provided the convection Mach number is subsonic. The present theory does not address the acoustic power from supersonic jets when shock waves are present and the ‘‘mixing region’’ noise is augmented by shock-cell noise and ‘‘screech’’. The contributions to the acoustic power are integrated over the complete volume of the flow. However, a large number of flow parameters must be specified. These include the jet exit Mach number and temperature, the flight Mach number and the corresponding convection Mach number. Also required is the corresponding mean density and enthalpy ratios, the length of the potential core, the growth of the jet in the initial mixing region and far downstream, the mean turbulent intensity and its law of decay, and the ratio of the integral turbulence scale to the local jet width. All these parameters are functions of the jet exit Mach number and the ratio of the jet to flight Mach numbers. For the hot jet, we require the mean square of the enthalpy fluctuations. In this treatment the directivity of the noise is not discussed, although the angular dependence of the noise arising from convection is included when integrating the sound intensity over a large spherical surface to determine the total acoustic power. It is found, however, that the axial distribution of the effective acoustic sources as determined by the model is supported by the experimental results obtained by using the far field polar correlation technique. These results have been discussed by Lilley [22]. A comparison of the present results with experimental data is shown in Figures 2 and 3. The results show the correct trends for the heated jet at low Mach numbers and the changes in the acoustic power in the upper end of the subsonic jet Mach number range. For the static cold jet tested by Lush [23], the measured total acoustic power was 135 dB (re 10−13 watts) compared with 137 dB (re 10−13 watts) as found from the present results. 6. FLOW–ACOUSTIC INTERACTION

In Lighthill’s acoustic analogy the acoustic sources are embedded in a medium at rest. All acoustic sources irrespective of their location in the flow emit sound waves which then cross the source region without interference to the far field. If the acoustic stress tensor has been evaluated including all unsteady effects in the flow arising from the turbulent flow, its sound generation and propagation, then in principle the sound radiation field would correctly represent all interference effects between the sound and its interaction with the turbulence as it crosses the flow. In general, Tij includes only the effects of the turbulent flow, and some interference exists between the turbulent flow and sound waves in the flow, which we refer to as flow–acoustic interaction. An easily observable influence of flow–acoustic interaction occurs at high frequencies where the sound waves are refracted by the flow, resulting in a near zone of silence in the high frequencies close to the jet

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. . 

Figure 2. The overall noise power for a cold jet. Nozzle exit area 0·000507 m2, Hstj /Ha=1, ga=gj=1·4. R, Lush [23]; W, Olsen et al. [25]; ——, with refraction; – – –, refraction neglected; — – —, Vja .

boundary, as shown in Figure 4. The present results, shown in Figures 2 and 3, include the elementary effects of refraction. An alternative theory of flow–acoustic interaction, which embraces the effects of refraction, has been discussed by Goldstein [24]. 7. CONCLUSIONS

The sources of sound in isotropic turbulence have been found for hot and cold flows at low turbulent Mach numbers by using Lighthill’s acoustic analogy. The present results have used Lighthill’s relationship relating the fourth order covariance of the acoustic stress tensor to the square of corresponding second order covariances. The results have been compared with recent (LES) and (DNS) results. The corresponding acoustic power per unit

Figure 3. The overall noise power for a hot jet. Nozzle exit area 1·0 m2, ga=1·4, gj=1·3. — – —, V J8 ; – – – V J6 , Tstj /Ta values: (subsonic, Hoch et al. [26] ×, 1·2 +, 1·4; · · · ·, 2·0 (theory); (supersonic, Tanna [27] R, 2·0; Q, 6·25; ——, 6·25 (theory).

     

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Figure 4. The directivity of jet noise. Mj=0·88. ——, Lighthill–Ffowcs Williams convection theory with high frequency refraction added; Q, Lassiter and Hubbard [28]; W, Westley and Lilley [29]; R, Fitzpatrick and Lee [30].

volume radiated to the far field has been derived. The effects of turbulence convection at a speed VC have been included by using the Lighthill–Ffowcs Williams convection theory. The effects of variable density and temperature are included and it has been shown that the low Mach number heated jet generates an acoustic power proportional to MT6 , rather than the proportionality of MT8 . These results for isotropic turbulence are used as an approximate model for the noise radiated from turbulent shear flows, and the present results for the acoustic power radiated from a circular jet are in fair agreement with measurements over a wide range of jet speeds and temperature. ACKNOWLEDGMENTS

This work was completed during my stay at the Center for Turbulence Research at Stanford University and NASA Ames Research Center; I am grateful to both these Institutions for their hospitality. An earlier version of this work was reported at the Second International Symposium on Transport Noise, St Petersburg, Russia, in October 1994. REFERENCES 1. M. J. L 1952 Proceedings of the Royal Society of London A211, 564–587. On sound generated aerodynamically: 1. General theory. 2. I. P 1952 Proceedings of the Royal Society of London A214, 119–132. The generation of noise by isotropic turbulence. 3. G. M. L 1994 Theoretical and Computational Fluid Dynamics 6, 281–301. The radiated noise from isotropic turbulence. 4. S. S and M. Y. H 1993 ICASE Report 93-74. Computation of the sound generated by isotropic turbulence. 5. M. J. L 1954 Proceedings of the Royal Society of London A222, 1–32. On sound generated aerodynamically: 2. Turbulence as a source of sound. 6. M. J. L 1962 Proceedings of the Royal Society of London A267, 147–182. Sound generated aerodynamically: the Bakerian Lecture. 7. M. J. L 1963 American Institute of Aeronautics and Astronautics Journal 1, 1507–1517. Jet noise: the Wright Brothers Lecture. 8. M. J. L 1978 Waves in Fluids. Cambridge: Cambridge University Press.

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