Noise generation by turbulent flow in ducts

The aim of this work is to estimate the sound power generated by a turbulent flow in a duct from the knowledge of mean quantities which can be computed with a K-e turbulence model when only the plane wave can propagate, i.e., when the characteristic frequency of the turbulence is low. It does turn out to be possible for the simplest configuration of a duct free of internal obstruction. The result provides a prediction which gives rise to a U scaling law for the variation of the acoustic power. Nevertheless, there is no experimental data to compare our results with for this idealized configuration. That is why we tackled the problem of a duct obstructed by a diaphragm. Using conformal mapping and Matched Asymptotic Expansions, we have determined a two-dimensional Green's function which allow us to assess numerically the acoustic power. This case is more interesting because existing experimental data confirmed the C/ scaling law obtained by calculation for the acoustic power variation. 1Duct free of internal obstruct ion In order to estimate the acoustic power generated by a turbulent flow in a duct, we shall solve the Lighthill inhomogeneous aeroacoustic equation governing the sound field, which reads for the pressure •^ where the turbulence stress tensor is reduced to Tij — poUiiij (inviscid flow, adiabatic motion). 1.1Davies and Ffowcs-Will iams results The first part of this work is based on the paper of Davies and Ffowcs Williams [1] which deals with the problem of estimating the sound field generated by a limited region of turbulence of volume v in an infinitely long, straight, hard walled pipe of square cross-sectional shape and side Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jp4:19945207 C5-948 JOURNAL DE PHYSIQUE IV Equation (1.1) can be solved by means of a Green's function G(2, tlij, T) satisfying the appropriate hard-wall condition on the walls of the duct and the radiation conditions as x j + f m. This function can be expressed in the Fourier domain as a sum of the normal modes of oscillation, these being cosine terms only in a square hard walled pipe. This si~nple form of the Green's function allows theoretical develop~nents which led Davies and Ffowcs-Williams [l] to an expression for the acoustic power due to a limited region v of turbulence. They showed that for low Mach numbers, when the characteristic frequency of the turbulence is so low that only the plane wave Tn = n = 0 can propagate, the acoustic power, or rate at which energy is propagating down the pipe, is proportional to +O0 aZ Pa, = =o where ~ i j k l (ir T) = (Tii ({, t ) Tkl(f S i, t + T)) and the brackets denote an ensemble avera.ging. 1.2Turbulence source modelling Now the difficulty is to model the Lighthill correlation tensor R3j33. We shall neglect the "selfnoise" term uh2 as small compared with the "shear-noise" tern1 Uuh in the Lighthill tensor T33. Assunling that the space-time velocity probability distribution is Gaussian and considering a small convection Mach number M, = U,/co, it is possible to write the acoustic power per unit volume of turbulence as a function of tlle second derivative at the origin of the time velocity correlation where L is the integral longitudinal turbulence scale. Now in the theory of locally isotropic turI~ulence, the longitudinal velocity correlation f ( ~ ) has, for small T << L, a parabolic variation with a radius of curvature known as the Taylor microscale AT , 1 / X $ = f1'(0)/2. Similarly, for time correlation at a fixed point of space one can introduce a Taylor time microscale t~ through l / t+ = -R1'(0)/2. We suggest that the microscales in space and time are related by AT = uo TT, where uo is tlle r.m.s. velocity fluctuation (in one direction), uo = @/3 = 2Ii/3. That is to say we assume that in the convected reference frame, small-time variations are produced 11y the "transport" (not uniform convection) of small space scales by the large e~1dies.A~ can be written AT = d m in terms of energy density I<, kinematic viscosity 7, and energy dissipation rate E per unit volume, using standarc1 results of ho~nogeneous turl~ulence theory. Hence t$,and then R1'(0), can be expressed in terms of IL' and 6. Moreover (ui2) can be written as (u:') = 2K/3. Then (1.4) can be written 8P 4 p o I < U Z L ~ a71 45 7 4 (1.5) It is interesting that all of the quantities occurring on the riglit side of (1.5) can be computed from the widely-used I i -F turbulence model. Codes for models of the I i -F kind calculate U at any position and also directly calculate I<,€ at every point. L can 11e expressecl as an integral of the energy spectrum function E(k) , and if E(k) is chosen to have the von I<QrmQn form, the integral can be evaluated in terms of known function whose para~neters are fixed once I<,€ are known. 1.3Scaling laws and numerical calculations If it is assumed that I< scales with the square of a typical lnean velocity U , then it follows from (1.5) that the acoustic power radiated along the duct 11y a volulne v = D%1)eys the scaling relation in which poU3D2 is a representation, in scaling terms, of the flow power (the nlechanical power required to maintain the given flow), ancl ( U / C ~ ) ~ is the acoustic efficiency factor found for this problem by Davies ancl Ffowcs-Williams [I]. We predict [2] a U7 law because of the extra Reynolds number factor UDlv. This in turn arises fro111 recognition that the source tern1 involves the second derivative of a correlation R1'(0) which is inversely proportional to the scluarecl nlicroscale A$, the lllicroscale being itself proportional to the square root of the overall Reynolds number. Computations were carried out using a ICE code in routine use at Electricit6 de France. The particular case of a duct of 10 cm cross-section side was considered, for mean velocities Uo (averaged over a section) of 10, 20, 40 and 80 m/s, corresponding to Reynolds numbers, UoDIv, equal in air to 6.6 x lo4, 1.32 x lo5, 2.64 x lo5, 5.28 x lo5. The total a.coustic power per unit length, obtained by integrating the results for BPldv across the cluct cross-section, is plotted against the nlean velocity Uo . 2Duct obstructed by a diaphragm 140