The acoustic waves popagation in a cylindical waveguide with the lamina flow Alexande Petov 1, and Valentina Rumyantseva, 1 MISIS National Univesity of Science and Technology Bauman Moscow State Technical Univesity Abstact. The poblem of modeling of acoustic wave popagation in inhomogeneous flow is consideed. Thee is an appoximate analytical solution of the hydodynamics equations in the pesence of annula acoustic oscillations souce in the case of lamina flow. Special attention is to paid to the popagation of acoustic waves modes. The amplitudes and phases dependences of the individual modes on the Mach numbe in the linea appoximation wee established. 1 Intoduction The task of the gas-ai flows measuement is elevant fo many aeas of science and technology, fo example, such as medicine spiomety), occupational safety, ventilation contol, etc. A numbe of equiements to the sensos of ai flow speeds, ae imposed, such as: accuacy, eliability, inetia-fee, lage dynamic ange, elatively low cost. The ability to meet these equiements depends on the measuement pinciple. In ou opinion, the most pomising method fo measuing of the velocity of gas-ai flow is the acoustic phase method poposed by Pofesso Shkundin Russia) [1]. The method consists of mounting two piezoelectic ing tansduces main pats of the senso) into the cylindical waveguide-ai duc s wall. One of them is an acoustic wave tansmitte and the othe is a eceive. Peiodically they tun thei oles. An extenal flow ceates a flow within the senso as well. The phase of the acoustic signal depends on the flow ate in the waveguide-ai duct. The eceive and tansmitte change oles at each measuement step. Flow velocity values ae detemined by the phase diffeence between the signals popagating along and against the flow. Samples of such sensos aleady exist and successfully opeate as pat of mine anemometes. A theoetical desciption of this method was developed. Howeve, all models of aeoacoustic inteaction use the appoximation of unifom flow. This appoximation is coect in the case of lage Reynolds numbes, when the aveage velocity plot is simila to ectangula. In this pape, we conside the desciption of the aeoacoustic inteaction with a lamina flow, i.e. a flow with a paabolic velocity plot. To impove the existing devices, as well e-mail: petovipmech@gmail.com e-mail: vala@bmstu.u The Authos, published by EDP Sciences. This is an open access aticle distibuted unde the tems of the Ceative Commons Attibution License 4.0 http://ceativecommons.og/licenses/by/4.0/).
as to develop new ones: spiometes, anemometes, flowmetes, it is necessay to theoetically descibe the physical pocesses undelying the measuement pinciple, i.e. the model of aeoacoustic inteaction in the waveguide-ai duct. Figue 1. Cilindical waveguad Poblem statement Conside an acoustic waveguide-ai duct with a souce of acoustic vibations having the fom of a ing mounted in the wall of the waveguide. We obtain the acoustic field of the wave popagating fom the souce. The ing oscillates in the adial diection accoding to the hamonic law with a fequency of ω. Since we will not conside eflections fom the ends of the waveguide, let us assume that in this appoximation the waveguide of unlimited length is consideed. The walls of the waveguide ae consideed infinitely igid. We intoduce a cylindical coodinate system, as shown in Fig. 1. The poblem has cylindical symmety, so vectos have only two components: =, z). Let a lamina flow with a velocity pofile constant along the z axis be established in the waveguide. Then, accoding to [], its field will be detemined by the expession 1). ) u) = u 1 1) whee u -the aveage in coss-section velocity. To descibe the wave field, we use the acoustic appoximation. We believe that the paametes of the medium ae the sum of constant and small oscillatoy component. v z = u) + v z,v = v,ρ = ρ 0 + ρ, p = p 0 + p, whee v = v, v z ) = v, t), ρ = ρr, t), p = p, t) - speed, density and pessue of the medium, espectively. The bounday condition fo the velocity on the inne wall of the waveguide will be: R v =R = { V0 e iωt, z h/ 0, z > h/ ) Hee V 0 is the suface of the ing vibation speed amplitude, ω is the angula fequency of oscillation, i is the imaginay unit. We solve the basic system of hydodynamics [3, 5 9]. Fo the ai in the acoustic appoximation the viscosity is small. Theefoe, the equation of motion fo us will be the Eule
equation fo the ideal gas. The equation of continuity: ρ t + z ρv z) + ρv ) + ρv The equations of motion longitudinal and adial pojections) ae + 1 v z t v t v z + v z z + v v z = p ρz v + v z z + v v = p ρ = 0 3) whee = + z Howeve, the inhomogeneity of the flow does not make the velocity field potential viscosity is taken into account when calculating the shape of the velocity plot), so the solution will be expessed in the fom of the acoustic potential Φ and the cuent function Ψ. The vibation speed dependence on these functions has the fom: 3 Poblem solution 3.1 System of equations obtaining v z = Φ z + 1 Ψ, v = Φ + 1 Ψ z We obtain a system of diffeential equations with espect to potentials and acoustic pessue. To do this we substitute the expession of the medium paametes into the main system and apply linea appoximation fo acoustic vibational) components. We assume that sound popagation is adiabatic, supplementing the system with p = c ρ, whee c = p ρ, c is the adiabatic speed of sound. In this case, we intoduce the auxiliay function F, and also epesent the density as a elative dimensionless value: ρ = ρ ρ 0. Substituting expessions 6) into the system 3-5) : t + u z ) 1 Ψ 1 Ψ + 1 Ψ z 4) 5) 6) ) 4u Φ = 0 7) R t Φ+uΦ z + c ρ + F = 0 8) t + u ) ρ + Φ = 0 9) z F z = u Φ 1 ) Ψ + z t + u ) 1 Ψ z = 0 10) We will look fo a solution in the fom of: Let us intoduce the opeatos: Φ, z, t) = φ)e i αct+βz R, Ψ, z, t) = ψ)e i αct+βz R 11) ρ, z, t) =Ω)e i αct+βz R, F, z, t) = f )e i αct+βz R A = + 1, 3 B = 1 1)
Substituting the fom of the solution 11) into equations 7-10) and afte a seies of tansfomations, we obtain equations fo the adial dependences of the acoustic potential and cuent function: 4β α + 8β M 1 ) + 16M 3 ) φ)+ 13) +iβ B β ) ψ) 4iαM ψ) = 0; i α + βm 1 )) B β ) ψ) 4 A β ) φ) = 0. We will look fo a solution in the linea appoximation of the Mach numbe, in the fom of φ = φ 0 + Mφ 1, ψ = ψ 0 + Mψ 1, β = β 0 + Mβ 1, 14) Substituting the expession 14) into equation 13) we goup the components of the equations by powes of M. We neglect the second ode of smallness and divide the system into two: zeo appoximation and the fist. Get the equation sepaately fo ψ 0 and φ 0 : ) A + α β 0 φ0 = 0 15) iα B β 0) ψ0 = 4 α φ 0 16) Then we obtain the equations fo the fist appoximation. The equation fo detemining φ 1 : A β 0 + α ) φ 1 = 4α 1 ) β 0 + β 1 β 0 4α 1 ) φ 0 + iα β 0 β 0 ψ 0; 17) and the equation to detemine ψ 1 : ) B β 0 ψ1 = 8i 1 ) β 0 + 16i 3 1 ) φ 0 + 18) β 0 +4i αφ 1 + β 1 β 0 + 4α 1 ) ψ 0. β 0 We got equations fo detemining all the components. We solve them. 3. Zeo appoximation solution We have to stat the solution fo the zeo appoximation with Eq. 15). The bounday conditions ) must be taken into account. This poblem is a case of acoustic waves popagation in a cylindical waveguide in the pesence of a hamonic oscillations cicula souce in the absence of a flow. The solution of this poblem is pesented in [4] if the flow ate is set to zeo. It is the sum of the hamonic components of z - nomal modes popagating in two diections fom the souce. In ou notation we may wite ϕ ±, z) = A n J 0 µ n ) e isnz, whee A n = irv 0 sin ) h s n R 19) J 0 µ n ) s n n=0 hee the uppe sign applies when z > 0, lowe at z < 0, β 0 = s n = α µ n, α = ωr c, J 0 is the Bessel function of the fist kind, of zeo ode, µ n,n = 0, 1,...) ae zeos of the fist kind, fist ode Bessel functions J 1 ). 4
In the futue we will conside sepaately n - th mode. Then it is clea that φ 0 ) = A n )J 0 µ n ). We obtain an expession fo the adial component of the cuent function in the zeo appoximation of ψ 0 ). We substitute this expession 19) fo φ 0 ) and β 0 fo n th mode in the ight pat of Eq. 16). Then we get 1 ) β 0 ψ 0 = 4iα A n ) J 0 µ n ) 0) We obtain the solution of the Eq. 0) with zeo bounday conditions on the waveguide wall, whee the souce was taken into account in the calculation 19): ψ 0 ) = 4iα A n) J α 0 µ n ) + 8iα A n) µ n J α 4 1 µ n ) + 1) + A n ) 4i J 0 µ n ) α J 1 iβ 0 ) J 1 iβ 0 ). 3.3 Solution in the fist appoximation We find the equations fo the fist appoximation by collecting fist-ode summands of M. We get the equation fo φ 1 ) by substituting solutions 19) and 1) into the Eq. 17). The solution of this equation is: φ 1 ) = A n ) J 1 µ n ) 8β 0/α 4αβ 0 /3 4αβ 0 + β 0 3 ) + µ 0 3µ 3 n 3µ n + A n ) αβ 0 J 0 µ n ) 3µ n 4iαA nj 0 µ n ) ) J 0 iβ 0 ) α J 1 i α µ n Upon obtaining this solution, it was shown that zeo bounday conditions ae satisfied fo this function when: β 1 = 4α 3. ) Conside the elationship between the acoustic potential phase and the aveage aiflow velocity. It is defined by expession ). It can be shown that the phase diffeence of the acoustic signals popagating along and against the flow at a distance of l fom the souce will be equal: φ lam = β 1 lm = 8 ω lu 3) 3 c In [1], we obtain a simila expession as Eq. ) fo the phase diffeence depending on the velocity of the homogeneous flow with a ectangula plot). Fom it in a linea appoximation of M it can be obtained: φ tub = ω lu 4) c Compaison of expessions 3) and 4) allows to estimate influence of inhomogeneity of a steam on indications of the device measuing speed of a flow by an acoustic phase method. It can be seen that the popotionality facto between the measued velocity and the phase diffeence in the lamina flow will be 3/4, if we talk about the aveage coss-section velocity and 3/, if about the maximum. 5
4 Conclusion We consideed the poblem of acoustic wave popagation in a cylindical waveguide in an inhomogeneous flow with a paabolic velocity plot. The solution was obtained fo the acoustic potential and cuent function. The amplitude and phase of acoustic waves wee expessed in a linea Mach-numbe epesentation. This appoximation holds because the lamina flow flow with paabolic plot) is set at low flow ates. Expession ) makes it possible to detemine the influence of the shape of the velocity plot on the phase of the acoustic signal, which is an infomative paamete in the phase acoustic method fo measuing flow velocities. This solution can be used to descibe the acoustic pinciple of measuing ai flow and velocity, as well as to calibate the acoustic spiomete by flow. The wok was suppoted by the Russian Science Foundation, use poject No 14-19-01633. Refeences [1] S.Z. Shkundin, V. B. Lashin Metology. 7 39-43 1990) in Russian) [] D. D. Landau, E. M. Lifshits Hydodynamics Nauka, Moscow, 1988) 733 in Russian) [3] L. G. Loytzyansky Mechanics of liquid and gas Dofa, Moscow, 003) 840 in Russian) [4] A.D. Lapin Akustiko-aehodinamicheskie issledovaniya: sbonik. pod ed. Rimskogo- Kosakova.[ Acoustic-aeodynamic studies: a collection. ed. by Rimsky-Kosakov] Moscow, 1975 ) 57-60 in Russian) [5] N.I. Sidnyaev, N.M. Godeeva, Jounal of Applied and Industial Mathematics 9, Issue 1, 110-118 015) [6] I.K. Machevskii, V.V. Puzikova, Jounal of Machiney Manufactue and Reliability 46, Issue, 114-119 017) [7] A.A. Pozhalostin, D.A. Gonchaov, Russian Aeonautics 58, Issue 1 37-41 015) [8] E.V. Muashkin, Y. N. Radayev J. Phys.: Conf. Se. 937 01031 017) [9] A.Yu. Vaaksin High Tempeatue 53, No. 3 43-444. 015) 6