Ultrasound as a probe of turbulence

Physica D 37 (1989) 508-514 North-Holland, Amsterdam ULTRASOUND AS A PROBE OF TURBULENCE Fernando LUND and Cristifin ROJAS Departamento de Fisicag Facultad de C~encias Fisicas y Matemrticas, Universidad de Chile, Casilla 487-3, Santiago, Chile Formulae relating the two-point, space time, vorticity correlation tensor to the intensity of ultrasound scattered by a bounded region of nonvanishing vorticity are given both in two and in three dimensions. The incident wave is supposed to be of low intensity and of frequency high in comparison with typical frequencies of the target flow, which is assumed to be of low Math number; viscosity is assumed not to affect sound propagation. This result suggests a non-intrusive, direct, way of measuring vorticity correlations. The derivation emphasizes the central role played by vorticity as the basic scattering mechanism of the incident sound. Well-developed turbulence is one of the most commonplace natural phenomena that, in spite of the very considerable advances made in its study, still defies understanding in terms of basic physics [1]. One of the obstacles that stand in the way of such understanding appears to be that precise quantitative experiments allowing the test of various theories are, by comparison with other branches of physics, relatively scarce. In particular, turbulence, taken in the broad sense of any disordered flow, is inextricably linked with vorticity as without the latter, at least for a Navier-Stokes incompressible fluid, there is no disorder. And vorticity is a quantity particularly hard to measure using the traditional methods of hot wire anemometry and laser Doppler velocimetry [2]. In this paper we derive a linear relztion between two-point dynamic vorticity correlations and the intensity of scattered ultrasound, both in two and in three dimensions, under assumptions that are spelled out below. The experimental realization of the conditions under which such assumptions are valid would then provide a direct, non-intrusive way of measunng vorticity properties just as electromagnetic scattering measures, say, density correlations, and neutron scattering measures spin correlations [3]. The basic intuition we wish to emphasize is that any fluid velocity, V, may be decomposed as v = vL + where VL, a longitudinal curl-free vector, is associated with sound, and V x, a transverse divergenceless vector, is associated with vorticity [4]. In a linear approximation the equations for both components decouple, with sound propagating according to the linear wave equation, and vorticity staying put. When the nonlinear terms in the full fluid equations are included, these two modes interact, and there is a relation between them that is not unlike that of field and seurce [5] (as in electromagnetic field and charged particle) in that nonstationary vorticity generates sound and sound convects vorticity [5, 6]. The set-up we envisage is shown in fig. 1. An ultrasonic plane wave, with velocity Finc = V o c o s ( k o x - rot ) is incident on a bounded body of vordcity ~ (the target), ~hose velocity, in the absence of the sound wave, is u (hence - - V A u), with IV01 << lul << c - ~,o/ko. Moreover, the typical time scales associated with u, say ~2-~, are long compared to the sound time scale: ~'o>~ ~2. We can handle both two- and three-dimensional situations. The target flow is assumed to be bounded so that linear sound waves exist away from it, and 0167-~.789/89/$03.50 0 Elsevier Science Publishers B.V. !North-Holland Physics Publishing Division) F. Lund and C. Rojas / Ultrasound OS(IIprobe of turbulence Fig. 1. A plane ultrasonic wave is incident on a target, which scattered. The geometry may be two- or three-dimensional. about the vorticity vibration will wave sam@g We assume Navier-Stokes 509 of a bounded region of nonvanishing zyxwvutsrqponmlkjihgfedcb vo r ticity, and zyxwvutsrqponmlkji is thereby vorticity or, ound wave will induce on a time scale 1 in turn generate soun the target flow at distance scales that, in the absence of the so e uation with a typical time scale S2 arger t ore 510 F. Lund and C Rojas/ Ultrasoundas a probe of turbulence Call p s = P - p and g = V - u . They oscillate with frequency vo and we expect p ~ = p i , c + ( s m a l l corrections), where Pi~ is the pressure of the incident wave, and similarly for vs. We shall neglect terms quadratic in vs . The viscous terms are of the order o f (5) t ) = - ~ (~v p ) - v ' v + ~ v ~(v- v), ,v- ~v'v with ~ = ~ / p the kinematic viscosity and we shall compare them with V . ( ( V . V ) V ) . We have ( V p / o ) VoVJC 2 and V " V - p - l ( a p / O t + ( V ' V ) O ) - p o V J C . Also V2Vs - voEvs/c2 and V2u - uv2/c 2, so that the ,fight-hand side of (5) is - t~vaovs/'c3, while V . ( ( V . V ) V - v2V2/c 2, so viscous terms are negligible when vo << V2c/vs~ which allows v0 to go well into the MHz range for, say, water. Physically speaking, eq. (4) describes the evolution of the compressible (i.e. longitudinal) part of the flow and we take this to happen on a time scale too short for viscosity to influence it. The term ( O p / ~ t ) 2 / p 2-- 1)0202/C2 Call also be neglected, as well as the term (Vp" V P ) / p 2 - (vovJc2p)V(Ps + P) - ~'oVs(~2u + VoVs)/c2. Finally p - t V 2p_ polV 2p_ p~p-1 V 2p but psp-lV 2P ~ VsVgU2/c3, and 102p OoOt ~ O(V • ) -~ 7o v0 + v . ( ( v . v ) v ) = - - - v l 2P. 00 Since the flow is adiabatic, p~ = (~p/Op)p~ and we have 1 02ps ¢2 Or2 V 2Ps = OoV " ( ( V " V ) V ) - Po~ (v)°2(, -~o " V P + - ~ "-~ - "~ Ps . (6) The last term on the right-hand side of (6) is of order poV20OsU2/c3, and can be omitted. We have then a wave equation for Ps with a source term. This is of course no real advance yet because Ps appears also in the source term. However, we can turn it into a scattering integral equation: ps=pinc+G,s, (7) where the second term on the fight is a convolution of c = c ( = - =', t - t') = (4,~1=- = ' ! ) - ' " ~ ( t - t ' - ~ - ' l x - ='!) the Green's function for the wave equation wRh vanishing boundary conditions at infinity, with the source ~=pov . ( ( v . v ) v ) -(~~ ,Vo vp). (8) Now, the source s, up to terms linear in the compressible velocity v~, is a ~= po(V . ( ( . . v ) , ) + v . ( ( . . v ) ~ ) + v .((~. v ) , ) ) - -~7( , . vp), (9) so that the pressure Ps is given by Ps =Pinc + Pvort + Pscat, (10) F. Lund and C Rojas / Ultrasound as a probe of turbulence 511 where (1i) p.o. = 0 o r . ( v - ( ( , . v ) = ) ) is the sound [5, 6] (vortex sound) radiated by the target flow in the absence of any incoming sound wave. It has frequency -- $2 and, if we tune our detectors to the much higher frequency :'0, will not be heard. The last term is the scattered pressure ( Pseat=O°G* V ' [ V ( U ' V s ) - V s A ° ~ ] 10 ) Oo a t ( U ' v P ) ' (12) where we have used that V ^ Vs= 0 to the desired order of accuracy. Eq. (11) is an integral equation for Ps- Since the scattered pressure will be small in comparison to the incident pressure, we use the first Born approximation: solve (12) by substituting in it the values for vs and p corresponding to the incident wave. Actually one ought to replace the sum Pinc + Pvort but Pvort wiU contribute to Ps in the frequencies - $2 which will not be heard by a detector tuned to %. We then have P~at = PsX + Ps2, where (~3) psa = p0c.(v .(,~ ^ ~,.o)), 1 ~a psz = poG * ( V 2 ( u ' % c ) - ~o ("'vp,.c)) (14) The first term, Pst, comes from the direct interaction of the sound wave with vorticity. The second, Ps2, is an interaction with the velocity field generated by the vorticity. At large distances from the target flow G may be replaced by its asymptotic form, using which one has that, for the convolutions in (13), V = ( - P / c ) a / a t where ~ is the direction of observation (fig. 1). Using this, the fact that one has an incident plane sound wave m~d integrating by parts in (13) it is possible to show that (both in two and in three dimensions) Ps2= (1-cosO) 1+ 70 SO -cos0 ) Pscat= 1 - c o s 0 Psx- (16) This relation involves only the polar angle 8 and not the azimuthal angle. Taldng the Fourier transform in time of (13) we have, in three dimensions [8], Pscat(x, v) ~ v e ivlxl/c -cose ) Po 1 - cos 8 ~r2i cgxg 512 E Lund and C Rojas / Ultrasound as a probe of turbulence where o3,(q,v) = (2~) 1 4f d3xdtel~"t-'')~,(x, t) is the Fourier transform (in space arId time) of the vorticity and q=c k is the momentum transfer. Eq. (17) provides a linear relation between scattered pressure and vorticity in Fourier space. A measurement of the former would give a direct measurement of the latter. The analogous relation in two dimensions is [7] 1/2 . /Sseat(r' V) ~ 1 --COSO "~ PoOoe'("r/c+3~r/4)~(q'v-- vO)" (18) This relation, which is of course not a special case of eq. (17), appears particularly relevant in view of the recent advances in the experimental realization of two-dimensional geometries using soap films [9]. In a study of disordered motion, one is interested more in vorticity correlations than in vorticity itself. Information about correlations may be obtained by considering the scattered acoustic intensity: If I(r) is the energy per unit area arriving at a detector placed at r, I ( r ) =.r with I(x, yields d~.,I ( r , g), p)=4crlp(x)12/COo~., where COS2 ~ I ( x , v) = Jo (1 - cose) 2 where A k = (P A ~3)k, J0 = ,~ , ~- is the time the measurement process lasts. Substitution of (17) ,n.~2 4C41Xl2A*AtSkt(q'/' PoCV~is the - - ~'0), energy flux of the incident plane wave and 1)4 f d3Rd~ ' . . . . . ~ " "" is the Fourier transform of the two-point vorticity correlation: s,,(R,,2) = f d3x (,ok(R + x,,2/2)o~,(x,-,2/2)), (19) E Lund and C Rojas/Ultrasound as a probe of turbulence 513 in which the bracket ( ) denotes temporal averaging: ¥ The result in two dimensions is [71 .,(,.,,,) o (20) Eqs. (19) and (20) provide a linear relation between dynamic vorticity correlations in Fourier space and the scattered ultrasound intensity, while (17) and (18) relate scattered pressure linearly to vorticity. These relations are valid quite generally, without anything being assumed in the way of isotropy or homogeneity of the target flow. Under such additional assumptions, previous results going back to the work of Kraichnan [101 are recovered [71. It should be of interest to actually carry out the experiment described above; its successful completion would provide a tool to probe vorticity and vorticity correlations in a fluid similar to electromagnetic and neutron scattering in condensed matter physics [3]. Acknowledgements This work was supported by DIB Grant E-2489-8722 and Fondo Naciona! de Ciencia Grant 1285-86. Useful discussions with A. Reisenegger are gratefully acknowledged. References [1] Recent inroads into the vast literature are provided by F. Williams, Phys. Rev. Lctt. 59 (1987) 1%_, C. Meneveau and K.R. Sreenivasan, Phys. Rev. Lett. 49 (1987) 1424; R.M. Kerr, Phys. Rev. Lett. 59 (1987) 783; D. Rouis, Phys. Rev. A 36 (1987) 3322; C. Foias, O.P. Manley and R. Teman, Phys. Fluids 30 (1987) 2007; W. Dannevik, V. Yakhot and S.A. Orszag, Phys. Fluids 30 (1987) 2021 ; R. Kraichnan, Phys. Fluids 30 (1987) 2400; W.T. Ashurst et al., Phys. Fluids 30 (1987) 2343; E. Kit et al., Phys. Fluids 30 (1987) 3323; M.M. Rogers and P. Moin, Phys. Fluids 30 (1987) 2662; R.W. Metcalfe et al., L Fluid Mech. 184 (1987) 207; A. Babiano et al., J. Fluid Mech. 183 (1987) 379. [2] For recent advances in the direct measurement of vorticity and other velocity derivatives see E. Kit et al., Phys. Fluids 30 (1987) 3323; J.L. Balint, P. Vukoslaveevic and J.M. Wallace, in: Advances in Turbulence, G. Compte-Bellot and J. Mathieu. eds. (Springer, Berlin, 1987); J.M. Wallace, Exp. Fluids 4 (1983) 61; R.A. Antonia, D.A. Shah and L.W.B. Browne, Phys. Fluids 30 (1987) 3455. [3] See, for instance, S.W. Lovesey, Condensed Matter Physics, Dynamic Cerrelations (Benjamin, Reading, 1980). [4] The considerations of this paragraph are given in very lucid de~ail by S.C. Crow, S'.~d. Appl. Math. 49 11970) 21. [5] F. Lund and N.J. Zabusky, Phys. Fluids 30 (1987) 2306; F. Lund, in: Instabilities and Nonequilibrium Structares, E. Tirapegui and D. Villarroel, eds. (Re~det, Dordrech~, 1987): in: Quantum Mechanics of Elementary Systems, C. Teitelboim, ed. (Plenum, New York, in press). 514 F. Lund and C Rojas / Ultrasound as a probe of t~.~rbulence [6] M.J. Lighthill, Prcc. Roy. Soc. London A 211 (1952) 564; A 222 (1954) 1; A. Powell, J. Acoust. Soc. Am. 36 (1964) 177; M.S. Howe, J. Fluid Mech. 71 (1975) 625; W. Mohring, J. Huid Mech. 85 (1978) 685; F. Obermeier, Acoustica 42 (1979) 56; T. Kambe and T. Minota, Proc. Roy. SOc. London A 386 (1983) 277; W. Mohring, E.A. Milller and F. Obermeier, Rev. MOd. Phys. 55 (1983) 707. [7] F. Lund and C. Rojas, to be published. [8] For a target flow that is static, the linear relation between scattered pressure and scattered vorticity has been found using a somewhat different method by I. Howe, Sound and Vibration 87 (1983) 567, following ideas of T. Kambe and U. Mya Oo, J. Phys. Soc. Japan 50 (1981) 3507. [9] M. Gharib and P. Derango, Bull. Am. Phys. SOc. 32 (1987) 2031; Y. Couder and C. Basdevant, J. Fluid Mech. 173 (1986) 225. [10] R.H. Kraichnan, J. Acoust. Soc. Am. 25 (1953) 1096; A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics, (MIT Press, Cambridge, MA, 1980), ch. 9, and references therein.