Numerical study of wave propagation in a nonuniform

Numerical study of wave propagation in a nonuniform compressible flow Alex Povitsky Citation: Physics of Fluids 14, 2657 (2002); doi: 10.1063/1.1490137 View online: http://dx.doi.org/10.1063/1.1490137 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/14/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Linear stability and acoustic characteristics of compressible, viscous, subsonic coaxial jet flow Phys. Fluids 25, 084102 (2013); 10.1063/1.4816368 Propagation of hydrodynamic interactions between particles in a compressible fluid Phys. Fluids 25, 046101 (2013); 10.1063/1.4802038 A numerical study of the generation and propagation of thermoacoustic waves in water Phys. Fluids 16, 3786 (2004); 10.1063/1.1791191 Oscillatory limited compressible fluid flow induced by the radial motion of a thick-walled piezoelectric tube J. Acoust. Soc. Am. 114, 1314 (2003); 10.1121/1.1603769 Vortex breakdown in compressible flows in pipes Phys. Fluids 15, 2208 (2003); 10.1063/1.1586272

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PHYSICS OF FLUIDS

VOLUME 14, NUMBER 8

AUGUST 2002

Numerical study of wave propagation in a nonuniform compressible flow Alex Povitskya) Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Quebec, Canada H3G 1M8

共Received 26 February 2001; accepted 8 May 2002; published 20 June 2002兲 The propagation of acoustic waves originating from cylindrical and spherical pulses in a nonuniform mean flow, and in the presence of a reflecting wall, is investigated by solving linearized Euler equations in terms of disturbances using high-order compact approximation of spatial derivatives. The two-dimensional and three-dimensional stagnation flows and a flow around a cylinder are taken as prototypes of real-world flows with strong gradients of mean pressure and velocity. The intensity and directivity of acoustic wave patterns appear to be quite different from the benchmark solutions obtained in a static environment for the same geometry. The physical reasons for amplification of sound are discussed in terms of redistribution of acoustic energy and its potential and kinetic components. The role of acoustic energy exchange with the mean flow is investigated. Compressibility of the background mean flow is taken into account and its effect on amplification of acoustic pressure is discussed. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1490137兴

I. INTRODUCTION

flight, fountain flow occurs when the downwash from the rotors strikes the wings and is redirected laterally toward the centerline of the aircraft fuselage. The recirculating fountain flow phenomenon is a principal noise source mechanism and a barrier issue for use of tiltrotor in civil aviation.4 The above-listed phenomena lead to propagation of acoustic waves in nonuniform flow upstream of a bluff body; therefore, stagnation flow and a potential flow around a circular cylinder could be taken as typical representatives of background steady flows. Also, these idealized mean flows mimic real-world flows in areas of strong sound refraction such as a leading edge of a turbine blade, a wall cavity, a wing-fuselage intersection, and an impingement area of a jet. While the Doppler effect 共downstream amplification of acoustic pressure and shift in wave frequency兲 is well documented,5,6 the upstream amplification of sound received much smaller attention. Howe7 and Taylor8 presented studies about downstream Doppler sound amplification in the presence of rigid sphere in the flow. Upstream amplification of sound occurs only in flows around noncompact rigid bodies, where the flow deceleration area is larger than the wavelength. While downstream convective amplification of sound holds for uniform flows, the upstream amplification takes place only in nonuniform flows. The latter phenomenon was called upstream Doppler amplification in Ref. 8, although the governing physical mechanisms are different from those for Doppler effect. The current study investigates the qualitative reasons for upstream amplification of sound in terms of deformation of wave profile and transformation of acoustic energy. Sound propagation in nonuniform flows is often regarded as refraction, i.e., the alternation of the direction of propagation of acoustic energy. For a traveling plane wave in the static ambient conditions, potential and kinematic components of acoustic energy are locally equal 共Ref. 9, p. 256兲.

This study investigates numerically the influence of strong mean flow gradients in the presence of a stagnation point on the directivity and strength of sound waves propagating in such a flow. The practical motivation comes from the problem of airframe noise. The problem of airframe noise is important in the landing phase of flight where engines do not operate at full thrust and the trailing-edge flaps and undercarriage gear are deployed. The noise is generated by the flap cove and the trailing edge of the main airfoil. Sound waves reach the flap leading edge, reflect by its surface, and amplify by interaction with the nonuniform flow upstream of the flap. The flap side-edge vortex forms close to the leading edge of the extended flap in the cove region between the flap and the main element. This vortex is fed from the spanwise flow from the lower boundary layers.1 A major source of noise is that from the interaction of flap side-edge vortices with large brackets which support the flap during its deflection. The noise radiation by the slot 共gap兲 ahead of the flap leading edge becomes the major source of aircraft noise in landing stage of the flight. The undercarriage components, i.e., wheels, axles, and struts generate unsteady separated flow behind them and wake interactions with downstream members lead to broadband noise.2 The separated flow from wheels ahead impinges in downstream wheels in the multi-wheel undercarriage while the vortices, shedding from struts, impinge in deployed flaps. The high noise level in helicopter descent flight operations are caused by an impulsive noise-generating mechanism known as blade-vortex interaction.3 In the tiltrotor a兲

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© 2002 American Institute of Physics


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The current study shows that the local ratio of kinetic-topotential components of acoustic energy appears to be far from unity while sound propagates in a nonuniform mean flow. Therefore, the kinetic-to-potential redistribution within acoustic energy in the current direction appears to be as important as the angular redistribution of acoustic energy. The potential energy is closely related to the averaged-in-time root mean square of acoustic pressure, which is usually measured by experimentalists as an indicator of noise intensity. Nonharmonic sources of sound are characteristic for broadband noise; therefore, a momentary acoustic pulse is taken as a typical representative. Our recent study10 has shown that the generation of sound waves originated from entropy sources, which are not immediate sources of sound, is caused by baroclinic generation of disturbance vorticity as a result of interaction of the mean flow pressure gradient and the wave-associated disturbance density gradient. In turn, the generated vortices produce sound waves originated from local disturbance pressure gradients which are caused by centrifugal forces in vortices and/or by interactions of vortices with rigid surfaces. Transfer of disturbance energy by means of interaction of background nonuniform velocity field and disturbance vorticity is also included in the acoustic energy equation.11 This source term can not be neglected by order-of-magnitude analysis and its value is estimated numerically. It will be shown in the current study, that the major effect of the nonuniformity of the mean flow on the wave propagation 共as opposed to sound generation兲 is the angular redistribution of acoustic energy and kinetic-to-potential transform in acoustic energy whereas the newly generated acoustic energy appears to be minor. The effect of compressibility of background flow and its components 共variable density, speed of sound, and spatial density gradients兲 on the amplification of acoustic energy will be obtained from simulations. In this paper, the propagation of acoustic waves in nonuniform flows is studied using direct numerical simulation of wave propagation since other methods are restricted to special cases. For instance, geometrical acoustics 共also known as ray acoustics兲 has been applied to sound refraction in the case of underwater propagation with variable speed of sound and in the case of propagation in the atmosphere with the presence of wind gradients. The geometric acoustics assumes that the acoustic pressure amplitude varies only slightly over distances comparable to a wavelength and that the radii of curvature of the wavefront are substantially larger than a wavelength.12 Atassi and Grzedzinsky13 considered propagation of unsteady disturbances in flows around bodies with a stagnation point. For incompressible potential mean flows, the aeroacoustic problem is formulated in terms of an integral equation of the Fredholm type for an acoustic velocity potential. The considered streamlined body and the corresponding potential mean flow result from superposition of a uniform flow and a source. Yet, additional assumptions and simplifications that depend on the type of potential background flow are needed to perform integration. Suzuki and Lele14 derived Green’s functions for wave propagation in an isothermal

Alex Povitsky

transversely sheared boundary layer and showed good accordance with direct numerical simulation for the high frequency limit of waves, but not for the low frequency case. In this study, sound waves originated from an initial noncompact acoustic pulse propagate so that the size of computational domain is equal to a few wavelengths 共Fig. 3兲. Therefore, the low-frequency and high-frequency limits are not applied to the case. Direct simulation of aeroacoustics implies consideration of a flowfield as a sum of the mean velocity and a disturbance field. Solution for unknown disturbance variables in time and space domain is found by explicit integration in time and use of appropriate spatial discretization. The computational methodology used in the current study is based on the above approach. In the current study, linearized Euler equations are explicitly solved using the fourth-order Runge–Kutta scheme for time marching and the fourth-order compact 共Pade´兲 approximation of spatial derivatives.15 The approach of numerical simulation is capable of handling complex geometry and/or mean flows for which analytical methods and asymptotics are poorly suited. In order to use an analytical solution for compressible mean flows considered, the approach exploited by Eversman and Okunbor16 is adopted in the current study for modeling of sound propagation in the stagnation flow. The background velocity flowfield is assumed to be potential, i.e., inviscid and incompressible. The mean flow density and the local speed of sound are based on the isentropic equation of state and the given velocity field with specified conditions on Mach number, density, and speed of sound in a reference point. This is a first-order compressibility correction in terms of a Mach number expansion about an incompressible flow. The flow around the 2D cylinder is subsonic if the M ⬁ ⭐0.405 共Ref. 17, p. 637兲 and becomes transonic for higher Mach number. It has been shown by Salas18 that the flow streamlines around a cylinder are qualitatively similar to those for incompressible flow if M ⬁ ⭐0.4. For higher Mach numbers, strong recirculating zones appear at the rear part of the cylinder and the shock waves take place at the midsection of the cylinder. Lighthill has shown the influence of O(M 2 ) and higher-order terms in the expansion in powers of the Mach number about the incompressible potential flow for the flow around a two-dimensional 共2D兲 cylinder with high subsonic M ⬁ ⫽0.4 共Ref. 19, p. 13兲. For the flow around cylinder 共Sec. V兲, the O(M 2 ) corrections to the potential flow are taken into account. This study can be expanded to cases where a compressible mean flow is computed by a computational fluid dynamics method and the acoustic source distribution is extracted from appropriate turbulence modeling. The goal of this study is to gain computational insight into physical mechanisms of angular redistribution of acoustic energy and acoustic pressure. In all cases considered, the obtained acoustic pressure is compared with that for the wave propagation under static ambient conditions and for the same geometry of surrounding rigid boundaries. The paper is setup as follows. In Sec. II, the developed mathematical model, numerical methods, and the validation of computer code are described. In Sec. III, the wave propa-



Numerical study of wave propagation

gation originated from a cylindrical 2D source in 2D compressible stagnation-type flows is studied and numerical results about amplification of acoustic pressure, alternation of wave pattern, and acoustic energy redistribution are presented. In Sec. IV, the wave propagation from a spherical source is considered in three-dimensional 共3D兲 stagnation flows. The symmetric 3D stagnation flow with a single stagnation point and the 3D stagnation flow with stagnation line flow are taken as typical representatives of three-dimensional stagnation mean flows. In Sec. V the inviscid compressible flow around a circular cylinder is taken for the Mach number of the mean flow up to M ⬁ ⫽0.4. Amplification of sound for upstream wave propagation in such a flow is studied in this section. Here, the results are compared with available unifrequency analytical solution8 and the bounds of its applicability to the current case are discussed. II. MATHEMATICAL MODEL, NUMERICAL METHOD, AND CODE VALIDATION

The adapted mathematical model and numerical algorithms used follow those in our study10 being expanded to the isentropic background flows with variable density, 3D flows, and the polar coordinate system. Introducing a disturbance, instantaneous velocities and density are considered as sums of the known steady compressible mean flow and the unsteady disturbance, u⫽U⫹u ⬘ ,

v ⫽V⫹ v ⬘ ,

p⫽ P⫹p ⬘ ,

␳ ⫽R⫹ ␳ ⬘ ,

共1兲

where U, V, P, and R are velocities, pressure, and density in the steady flow and u ⬘ , v ⬘ , p ⬘ , and ␳ ⬘ are corresponding disturbance variables. Substituting above sums to the Euler equations, the dynamic equations for unsteady 共disturbance兲 components of mass and momentum fluxes are obtained qt ⫹fx ⫹gy ⫽0, where

冉冉冉

共2兲

冊

␳⬘ R⫹ ␳ 兲共 U⫹u ⬘ 兲 , 共 ⬘ q⫽ 共 R⫹ ␳ ⬘ 兲共 V⫹ v ⬘ 兲

冊冊

共 R⫹ ␳ ⬘ 兲共 U⫹u ⬘ 兲 2 f⫽ 共 R⫹ ␳ ⬘ 兲共 U⫹u ⬘ 兲 ⫹ 共 P⫹p ⬘ 兲 , 共 R⫹ ␳ ⬘ 兲共 U⫹u ⬘ 兲共 V⫹ v ⬘ 兲

共3兲

共 R⫹ ␳ ⬘ 兲共 V⫹ v ⬘ 兲 g⫽ 共 R⫹ ␳ ⬘ 兲共 U⫹u ⬘ 兲共 V⫹ v ⬘ 兲 . 共 R⫹ ␳ ⬘ 兲共 V⫹ v ⬘ 兲 2 ⫹ 共 P⫹p ⬘ 兲

The fluxes f and g can be considered as sums of fluxes F and G containing only steady variables and the disturbance fluxes f⬘ and g⬘ which include the remaining components of fluxes f and g. The energy equation is taken in the form of the isentropic equation of state

冉冊

␳ p ⫽ p0 ␳0

␥

.

共4兲

For simplicity of presentation, the primes are omitted in the governing systems 共5兲, 共11兲, 共26兲, 共27兲, and 共31兲 so vari-

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ables u, v , p, ␳ denote disturbance variables there. Above equations are linearized and solved with respect to unknown disturbance variables in the space-time domain

冉

冊

⳵␳ ⳵u ⳵v ⳵␳ ⫺U ⫽⫺R ⫹ ⳵t ⳵x ⳵y ⳵x ⫺V

冉

冊

⳵␳ ⳵R ⳵R ⳵U ⳵V ⫺u ⫺v ⫺␳ ⫹ , ⳵y ⳵x ⳵y ⳵x ⳵y

⳵u ⳵u ⳵U 1 ⳵p ⫽⫺ ⫺U ⫺ u ⳵t R ⳵x ⳵x ⳵x ⫺V

冉

冊

⳵u ⳵U ⳵U ⳵U ␳ ⫺ ⫹V , v⫺ U ⳵y ⳵y ⳵x ⳵y R

⳵v ⳵v ⳵v 1 ⳵p ⫽⫺ ⫺V ⫺U ⳵t R ⳵y ⳵y ⳵x ⫺

冉

共5兲

冊

⳵V ⳵V ⳵V ⳵V ␳ u⫺ ⫹V , v⫺ U ⳵x ⳵y ⳵x ⳵y R

where p⫽c 2 ␳ , and c is the local speed of sound. The local velocity components U and V are defined by corresponding potential flow solution 共see the next sections兲. The local density R and the local speed of sound c in the background flow are given by

冉冉

R⫽ 1⫺

冊冊

␥ ⫺1 2 共 U ⫹V 2 ⫺M 20 兲 2

␥ ⫺1 2 c⫽ 1⫺ 共 U ⫹V 2 ⫺M 20 兲 2

1/共 ␥ ⫺1 兲

,

1/2

共6兲

,

where M 0 is the Mach number in the reference point where the speed of sound and the density of mean flow are set equal to unity. This way of calculating the background velocity field was used by the authors of Ref. 16. Although the obtained background flowfield does not satisfy Eq. 共3兲, the nonphysical source terms do not appear in above linearized equations since the sum of flux derivatives Fx ⫹Gy is assumed equal to zero in Eq. 共5兲. Numerical algorithm and the code validation

The solution is advanced in time in five sub-stages per time step using a low-storage explicit-in-time fourth-order Runge–Kutta 共RK兲 scheme proposed by Williamson20 and implemented by Wilson et al.21 The spatial derivatives of unknown disturbance variables are approximated using the compact fourth-order Pade´ scheme.22 At all boundaries, except a rigid plate, characteristic inflow or outflow boundary conditions are applied to disturbance variables. Assuming one-directional background flow perpendicular to the boundary with constant speed U and constant density, the Riemann invariants are w 1 ⫽0.5共 u⫹ p 兲 , w 2 ⫽0.5共 u⫺ p 兲 ,

共7兲


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FIG. 1. Centerline acoustic pressure for the case 2-A. Computational domain 关 ⫺1,1兴 ⫻ 关 0,1兴 is covered with the following numerical grids 共1兲 80⫻80; 共2兲 120 ⫻120; 共3兲 160⫻160; and 共4兲 200⫻200.

where velocity component u is perpendicular to the boundary. For the right boundary (x⫽x max), the invariant w 1 belongs to the incoming characteristic while the invariant w 2 corresponds to the outcoming characteristic,

⳵w1 ⳵w1 ⫹U ⫽0, ⳵t ⳵x w 2 ⫽w 2 共 ⬁ 兲 ⫽0.

共8兲

The first of the above equations is solved numerically by Runge–Kutta temporal integration 共see above兲. Then, the system 共7兲 is solved with respect to unknown u and p at the boundary. At a rigid surface, reflection boundary conditions are used for pressure disturbance and for the parallel to the surface component of velocity disturbance, whereas the normal to the rigid surface component of velocity is equal to zero. Discretization of spatial derivatives in the direction perpendicular to a boundary are computed by one-sided finite differences at all boundaries.23 At outer boundaries, perfect matching layer 共PML兲24 with thickness of 10 grid nodes is implemented to suppress disturbances originated from boundary conditions. The proposed numerical algorithm has been validated for 2D cases with static ambient conditions and incompressible stagnation background flow in our previous study.10 For the former case, the solution approaches the known analytical solution25 if the 160⫻160 numerical grid were used. For the latter case, the grid refinement study has shown that the solutions on the 160⫻160 and 200⫻200 grids are identical. In the current study, the code is verified for the background compressible 2D stagnation flow 共see Sec. III and Fig. 1兲, for the 3D stagnation flow 共see Sec. IV and Fig. 7兲, and for the 2D flow around a cylinder 共see Sec. V and Fig. 11兲. The considered cases seem harder for accurate numeri-

cal treatment than those in the study10 because of nonconstant coefficients associated with variable density and speed of sound and additional terms originated from nonzero spatial density gradients 关see Eq. 共5兲兴. The third spatial direction 关see Eqs. 共26兲 and 共27兲兴 and the polar coordinate system 关Eq. 共31兲兴 might be sources of complications. Also, the profiles of acoustic pressure are steeper for 3D spherical pulses than for 2D cylindrical pulses 共see discussion in Sec. IV and Fig. 10兲. Nevertheless, the grid refinement study for abovementioned cases shows good convergence either to an analytical solution where available or to a numerical solution. The results of the grid refinement study for the 2D sound propagation in the 2D stagnation compressible flow are presented in Fig. 1共b兲 for 关 ⫺1,1兴 ⫻ 关 0,1兴 computational domain covered with 80⫻80, 120⫻120, 160⫻160, and 200⫻200 numerical grids. The visible wiggles exist only on the coarsest grid. The acoustic pressure computed on the above series of grids approaches the solution on the finest grid elsewhere including peaks of the transmitted and reflected waves. Solutions on the 200⫻200 and 160⫻160 grid practically coincide; therefore, the grid 160⫻160 has been adopted for simulations in the next section.

III. SOUND PROPAGATION IN A 2D STAGNATIONTYPE MEAN FLOW

The common type of mean flow is that the flux across a given surface is equal to zero, either because the surface is a surface of symmetry or because the surface is the boundary of a rigid body.26 If two straight zero-flux boundaries intersect at an angle ␲ /n, the stream-function field for incompressible flow is given by

␺ ⫽Ar n sin共 n ␪ 兲 ,

共9兲




where (r, ␪ ) are polar coordinates. Note that ␺ ⫽0, if ␪ ⫽0 or ␪ ⫽ ␲ /n. The spatial derivatives of the mean flow variables for arbitrary n are presented in the Appendix. The 2D stagnation flow U⫽⫺x,

共10兲

V⫽y

is a union of two flows with n⫽2 where the dividing streamline is the symmetry axis of the flow. These velocity components of the stagnation mean flow are substituted to the generic linearized Euler equations 共5兲,

冉

冊

⳵␳ ⳵u ⳵v ⳵␳ ⳵␳ ⳵R ⳵R ⫽⫺R ⫹ ⫺v , ⫹x ⫺y ⫺u ⳵t ⳵x ⳵y ⳵x ⳵y ⳵x ⳵y ⳵u ⳵u ⳵u ␳ 1 ⳵p ⫽⫺ ⫹x ⫹u⫺y ⫺x , ⳵t R ⳵x ⳵x ⳵y R

共11兲

⳵v ⳵p ⳵v ⳵v ␳ ⫽⫺ ⫺y ⫹x ⫺ v ⫺y . ⳵t ⳵y ⳵y ⳵x R The velocity components 共10兲 are substituted in Eqs. 共6兲 to compute the density, speed of sound, and spatial derivatives of density,

冉冉

R⫽ 1⫺ c⫽ 1⫺

冊冊

␥ ⫺1 2 2 共 x ⫹y ⫺M 20 兲 2

␥ ⫺1 2 2 共 x ⫹y ⫺M 20 兲 2

冉冉

1/共 ␥ ⫺1 兲

,

1/2

,

⳵R ␥ ⫺1 2 2 ⫽⫺ 1⫺ 共 x ⫹y ⫺M 20 兲 ⳵x 2 ⳵R ␥ ⫺1 2 2 ⫽⫺ 1⫺ 共 x ⫹y ⫺M 20 兲 ⳵y 2

冊冊

共12兲

共 2⫺ ␥ 兲 / 共 ␥ ⫺1 兲

x, 共 2⫺ ␥ 兲 / 共 ␥ ⫺1 兲

y,

where M 0 ⫽0.5. The initial 2D cylindrical acoustic pulse is given by

冋

p⫽ ␳ ⫽ ⑀ exp ⫺d 2

册

共 x⫺x c 兲 2 ⫹ 共 y⫺y c 兲 2 , a

共13兲

where ⑀ ⫽0.01, a⫽ln(2)/9, d⫽60 is the normalization coefficient, and (x c ,y c ) is the pulse center coordinates. A. Acoustic pressure

To study the effect of nonuniform background flow on propagation of acoustic wave, the propagation of sound waves in the 2D stagnation flow, in the static ambient conditions, and in the backward stagnation flow are considered in this section. The cases are denoted as 2-A, 2-B, and 2-C, respectively, where the number ‘‘2’’ denotes the 2D background flow. The initial position of acoustic pulse 共0.25, 0兲 is the same in all three cases. In the case 2-A, the propagation of sound is governed by the system 共11兲. In the case 2-B, the propagation of sound satisfies the wave equation and corresponding linearized Euler equations were presented in Ref. 10. The backward 2D stagnation flow 共case 2-C兲 is considered to study the influence of the mean flow direction on the wave patterns. In the case 2-C, the mean velocities are given by

U⫽x,

V⫽⫺y.

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共14兲

In the case 2-C, the mean flow streamlines and the mean pressure distribution are the same as for the case 2-A. The system of governing equations is similar to Eq. 共11兲. For the cases 2-A and 2-C, the isobars of acoustic pressure are presented in Figs. 2共a兲 and 2共b兲. In the case 2-A, the approximate centerline coordinates of the maximums of acoustic pressure for the transmitted and reflected waves are 0.56 and 0.26, respectively. In the case 2-A, the x coordinate of the wave front at the neighborhood of the centerline is smaller and the y coordinate of the wavefront near the wall is larger than those in the case 2-C. The explanation is straightforward and is based on the fact that the speed of sound in a steady frame is equal to c⫹U, where c is the local speed of sound and U is the local velocity of the flowfield. The density of isobars is larger at the neighborhood of the centerline and smaller near the wall in the case 2-A in comparison with those in the case 2-C. In Fig. 3共a兲, the acoustic pressure profiles at the centerline are presented for both cases at t⫽0.5. The maximum absolute value of acoustic pressure is approximately 80% larger for the transmitted wave and two times larger for the reflected wave than those in the case 2-B 关compare curves 1 and 6 in Fig. 3共a兲兴. Recall that the speed of wave propagation is c⫹U. While the wave propagates upstream the decelerating subsonic flow, its front moves slower than its back. Therefore, the wave front becomes narrower and higher at the neighborhood of the centerline, in comparison with that in the case 2-B. The direction of propagation of the maximum acoustic pressure has been considerably changed from being perpendicular to the wall in the case 2-A to being parallel to it in the case 2-B 关compare Fig. 2共b兲 with Fig. 2共a兲兴. Again, the maximum acoustic pressure corresponds to the direction where the sound propagates upstream of the mean flow. The difference between acoustic pressure profiles corresponding to the incompressible model of the background flow (R⫽c⫽1) and compressible isentropic flow is small in comparison with the difference between the wave propagation in the static ambient conditions and in the presence of stagnation flow 关compare curves 1, 2, and 6 in Fig. 3共a兲兴. In terms of the maximum acoustic pressure of transmitted wave, the difference between these two models of the background flow is about 6%. At this point, the Mach number of the background flow is equal to 0.56. To study the effect of various components of compressibility of the background flow on the amplification of acoustic pressure in the case 2-B, Eq. 共11兲 was modified to treat the background flow as incompressible 共curve 2 in Fig. 3兲, as a constant-density flow with variable speed of sound by Eq. 共12兲共curve 3兲, as a constant-speed-of-sound flow with variable density 共curve 4兲, and as a flow where density and speed of sound obey Eq. 共12兲 but ⳵ R/ ⳵ x⫽ ⳵ R/ ⳵ y⫽0 共curve 5兲. For the reflected wave, the curves 2 and 4 are close to each other and show slight delay in wave propagation in comparison with the isentropic compressible model 共curve 1兲. Therefore, the variable speed of sound mostly causes the difference in the wave propagation in the immediate neigh-


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FIG. 2. Acoustic pulse propagation in the 2D stagnation flow: 共a兲 forward stagnation flow 共case 2-A兲, 共b兲 backward stagnation flow 共case 2-C兲, 共c兲 case 2-A, vorticity generation source 共15 equally spaced levels between ⫺0.011 and 0.011兲, and 共d兲 case 2-A: acoustic energy source 共10 equally spaced levels between 0.2e⫺08 and 0.65e⫺07兲. In 共a兲 and 共b兲 acoustic pressure isobars are taken with 15 equally spaced intervals between ⫺0.889e⫺03 and 0.185e⫺02.

borhood of stagnation point. The effect of variation of density on the wave propagation is negligible 共curves 1 and 3 almost coincide兲, however, there is some effect of spatial derivatives of density 共compare curves 1 and 5兲. For the transmitted wave, taking ⳵ R/ ⳵ x⫽ ⳵ R/ ⳵ y⫽0 leads to the 10% reduction of the maximum acoustic pressure in comparison with the isentropic compressible background flow. Therefore, the spatial derivatives ⳵ R/ ⳵ x and ⳵ R/ ⳵ y in Eq. 共11兲 represent the major contribution of compressibility of the background flow on the level of acoustic pressure.

B. Distribution of acoustic energy

To get physical insight into amplification of sound, distribution of acoustic energy in nonuniform flows is discussed here. Transport equation of acoustic energy density for irrotational and isentropic mean flow is given by Ref. 12, p. 422,

⳵E ⫹ⵜI⫽0, ⳵t

共15兲

where




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FIG. 3. Acoustic pressure along the centerline. Assumptions about the background steady flow: 1-isentropic; 2-incompressible; 3-constant density; 4-constant speed of sound; 5-spatial derivatives of density are equal to zero; and 6-static ambient conditions.

p ⬘2 1 ␳ ⬘ u⬘ •U 2 E⫽ 2 ⫹ Ru⬘ ⫹ 2Rc 2 c2

共16兲

is the acoustic energy density and

冉

I⫽ 共 p ⬘ ⫹Ru⬘ •U兲 u⬘ ⫹

p ⬘2U Rc 2

冊

共17兲

is the acoustic energy flux. The acoustic energy is a sum of potential energy P⫽p ⬘ 2 /2Rc 2 , and kinetic energy K ⫽Ru⬘ 2 /2⫹ ␳ ⬘ u⬘ •U/c 2 共see also Ref. 12, p. 39兲. To make a fair comparison, an acoustic power output of the source 共13兲 should be the same for all cases considered. Thus, the initial profile of acoustic density is normalized by the ratio of total initial acoustic energies for the static ambient conditions and in the presence of background flow, p ⬘2 ␳ ⬘2c 2 dx dy 兰 dx dy 兰 2Rc 2 2R ⫽ , E 0⫽ 兰 p ⬘ 2 dx dy 兰 ␳ ⬘ 2 dx dy

共18兲

where ␳ ⬘ is computed by 共13兲. The initial acoustic kinetic energy is equal to zero because u⬘ ⫽0 at t⫽0. The initial pulse is computed by

␳⫽

1

冑E 0

冋

⑀ exp ⫺d 2

册

共 x⫺x c 兲 2 ⫹ 共 y⫺y c 兲 2 , a

p⫽ ␳ c 2 . 共19兲

The normalization coefficient 1/冑E 0 is equal to 1.027 72 while the center of pulse is located in 共0.25, 0兲 and M 0 ⫽0.5. The normalization coefficient computed by values of R and c taken at the center of pulse 关i.e., without performing

integration in 共18兲兴 is equal to 1.028, i.e., quite close to its exact value. The 2.8% difference in terms of acoustic density or pressure leads to noticeable 2.82 ⫽7.84% difference in terms of acoustic energy. The local density R⬇1.11 in the center of the pulse and, therefore, the straightforward multiplication of initial disturbance density 共13兲 by R leads to unequal initial acoustic energy in cases 2-A and 2-B. To discuss the redistribution of total acoustic energy and its components, the integrals I E⫽

冕

I P⫽

冕

L

0

E 共 l 兲 dl

共20兲

P共 l 兲 dl

共21兲

and L

0

are plotted as functions of linear coordinate L along beams 0°, 45°, and 75° at the time moment t⫽0.5 关see Figs. 4共a兲– 4共c兲兴. Obviously, the integrals reach their maximum when the coordinate y exceeds the front coordinate of the transmitted wave. The values of these maximums for integrals 共20兲 differ only within 5%, whereas the integral of potential energy 共21兲 is 70% larger in the case 2-A than that in the case 2-B. Thus, the amplification of sound along the centerline occurs mainly by increase of the potential part in total acoustic energy. For the bisector direction 共45°兲 and for a near-wall direction 共75°兲 the part of potential energy in acoustic energy in the case 2-A approaches that in the case 2-B 共60.5% and 70% for 45° and 39% and 47.6% for 75°, respectively兲.


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Alex Povitsky

comparison with the case 2-B occurs mainly due to the angular redistribution of acoustic energy. The direction-dependent behavior of acoustic energy is explained by the relative value of the last term in Eq. 共16兲. If sound propagates upstream of the mean flow, the vectors in the inner product u⬘ •U are collinear, therefore, the variation of U in a nonuniform mean flow leads to more prominent reduction of the kinetic energy in comparison with other directions. The kinetic energy transforms into potential energy and leads to an increase of potential part in the acoustic energy. If disturbance vorticity is nonzero, the energy equation becomes as follows:11,27

⳵E ⫹ⵜI⫽Ru⬘ • 共 U⫻ ␻⬘ 兲 . ⳵t

共22兲

To study the generation of disturbance vorticity by means of interaction of the wave with a 2D nonuniform irrotational mean flow, we present here the simplified form of the vorticity transport equation10 D␻⬘ ⫽ⵜ ␳ ⬘ ⫻ⵜ P. Dt

FIG. 4. Integral of acoustic energy and its potential part in various spatial directions: 共a兲 0° 共centerline兲, 共b兲 45°, and 共c兲 75°. Curves 共1兲 and 共2兲 correspond to cases 2-A and 2-C. Solid lines denote integral of acoustic energy 共20兲, dashed lines denote integral of potential part of acoustic energy 共21兲.

While the level of acoustic energy is approximately the same in the bisector direction for the cases 2-A and 2-B 关see Fig. 4共b兲兴, for the 75° direction the acoustic energy is 20% smaller in the case 2-A than that in the case 2-B. Therefore, the weakening of sound near the wall in the case 2-A in

共23兲

The presence of the pressure gradient in the mean flow 共i.e., the nonuniformity of the flow兲 leads to the nonzero righthand side of the above equation. Otherwise, the vorticity ␻ ⬘ remains equal to zero as in an irrotational mean flow. In turn, the nonzero ␻ ⬘ leads to the nonzero source term in the energy equation 共22兲. The vorticity is generated where the acoustic wave passes, i.e., ⵜ ␳ ⬘ ⫽0. Since the disturbance velocity u⬘ is a multiplier in the inner product on the righthand side of Eq. 共22兲, the exchange of energy between the steady mean flow and the unsteady disturbance occurs only in the presence of the wave where u⬘ ⫽0. The generated vorticity flows with the local speed of background flow and eventually the vorticity is left behind the wave, that moves with the sonic speed. The instantaneous patterns of the vorticity generation source 关the right-hand side of Eq. 共23兲兴 and the acoustic energy source 关the right-hand side of Eq. 共22兲兴 at l⫽0.5 are shown in Figs. 2共c兲 and 2共d兲. Areas of maximum acoustic energy generation coincide with instantaneous positions of the reflected and transmitted waves. Roughly, the maximum value of the energy source term is located in the bisector direction 共45°兲 at t⫽0.5. In Fig. 5, the acoustic energy and its source are shown as functions of the distance from the origin along the bisector line at t ⫽0.5. The maximum value of this source of acoustic energy is more than one order of magnitude smaller than the level of the acoustic energy itself. Therefore, the gain of acoustic energy is small in comparison with the angular redistribution of acoustic energy and kinetic-to-potential transforms considered above. C. Sound propagation in the corner flow

To study the influence of the background flow on the acoustic wave propagation while the acoustic pulse is located off the centerline and two reflecting surfaces are present, the 90° corner geometry is considered 共Fig. 6兲. The rigid plains




W⫽⫺z,

U⫽0.5x,

2665

共24兲

V⫽0.5y.

The flowfield in stagnation flow with the stagnation line 共case 3-B兲 is W⫽⫺z,

U⫽x,

共25兲

V⫽0.

The developed code is readily available for 3D parallel computations as well.28 In the case 3-A, the propagation of disturbance is described by the following system of equations:

冉

冊

冉

⳵␳ ⳵u ⳵v ⳵w ⳵␳ ⳵␳ ⫽⫺R ⫹ ⫹0.5 ⫺x ⫺y ⫺ ⳵t ⳵x ⳵y ⳵z ⳵x ⳵y ⫹z FIG. 5. Acoustic energy and its source along the bisector ␣ ⫽45° at t ⫽0.5, where 共1兲 is the acoustic energy 共solid line兲 and 共2兲 is the source of energy 共dashed line兲. L(45°) is the distance from the origin along the bisector.

coincide with the Cartesian axes. The potential flow in this geometry is described by a single stagnation flow with n ⫽2 flowing to the right. Initial position of acoustic pulse is taken 共0.25, 0.25兲, where the local Mach number of the mean flow is approximately equal to 0.35. Computational results in terms of acoustic pressure are compared with those obtained in the static ambient conditions for the same geometry. In the latter case, the maximum acoustic pressure is in the bisector direction and the cylindrical shape of wave pattern is symmetric with respect to the bisector 关Fig. 6共b兲兴. In presence of the mean flow, the waves flatten and become almost perpendicular to the y axis 关see Fig. 6共a兲兴. The maximum acoustic pressure increases about 40% in comparison with that in the static ambient conditions. IV. ACOUSTIC PULSE IN 3D STAGNATION FLOWS

In this section we consider a spherical pulse propagating in a 3D stagnation flow. The 3D stagnation flow with a single stagnation point at the origin is denoted as case 3-A and its flowfield is given by

⳵␳ ⳵R ⳵R ⳵R ⫺u ⫺v ⫺w , ⳵z ⳵x ⳵y ⳵z

冉冉冉

冊

冊冊

⳵u ⳵u ⳵u ␳ ⳵u 1 ⳵p ⫽⫺ ⫹0.5 ⫺x ⫺u⫺y ⫺x , ⫹z ⳵t R ⳵x ⳵x ⳵y R ⳵z 共26兲 ⳵v ⳵v ⳵v ␳ ⳵v 1 ⳵p ⫽⫺ ⫹0.5 ⫺y ⫺x ⫺ v ⫺y , ⫹z ⳵t R ⳵y ⳵y ⳵x R ⳵z

冊

⳵w ⳵w ⳵w ␳ ⳵w 1 ⳵p ⫹w⫺z ⫹z ⫽⫺ ⫹0.5 ⫺y ⫺x , ⳵t R ⳵z ⳵y ⳵x R ⳵z where the computations of R and c are performed by expressions similar to 共12兲 and M 0 ⫽0.5. In the case 3-B, the governing system of equations is given by

冉

冊

⳵␳ ⳵u ⳵v ⳵w ⳵␳ ⳵␳ ⫺ ⫽⫺R ⫹ ⫺x ⫹z ⳵t ⳵x ⳵y ⳵z ⳵x ⳵z ⫺u

⳵R ⳵R ⫺w , ⳵x ⳵z

⳵u ⳵u ␳ ⳵u 1 ⳵p ⫽⫺ ⫺x ⫺u ⫺x ⫹z , ⳵t R ⳵x ⳵x R ⳵z ⳵v ⳵v ⳵v 1 ⳵p ⫽⫺ ⫺x ⫹z , ⳵t R ⳵y ⳵x ⳵z

共27兲

⳵w ⳵w ␳ ⳵w 1 ⳵p ⫽⫺ ⫺x ⫹w⫺z ⫹z . ⳵t R ⳵z ⳵x R ⳵z

FIG. 6. Acoustic pulse propagation in a 90° corner at t⫽0.5: 共a兲 in presence of the inviscid flow and 共b兲 in the static ambient conditions.


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Alex Povitsky

FIG. 7. 3D acoustic pulse propagation in the static ambient conditions in presence of rigid wall. Computational domain: 关 ⫺1,1兴 ⫻ 关 ⫺1,1兴 ⫻ 关 0,1兴共1兲 803 ; 共2兲 1203 ; and 共3兲 analytical solution and numerical solution on the 1603 grid.

The initial 3D spherical acoustic pulse is given by an expression similar to 共13兲. To validate the code, the computations are performed on a set of numerical grids where the wave spreads under static ambient conditions in the presence of the rigid reflecting plate 共case 3-C兲. The analytic solution for acoustic pressure in case of the infinite domain and the static ambient conditions is given by p ⬘⫽

⑀ 兵共 r⫺t 兲 exp关 ⫺a 共 r⫺t 兲 2 兴 2r ⫹ 共 r⫹t 兲 exp关 ⫺a 共 r⫹t 兲 2 兴其 ,

共28兲

where r is the location of the pulse center at t⫽0. Solution for the semi-infinite domain 共i.e., case 3-C兲 is obtained by use of the image pulse located at ⫺r: p⫽p 共 r 兲 ⫹p 共 ⫺r 兲 .

共29兲 3

3

cation of sound at the neighborhood of the centerline. In case 3-B, the wave pattern is direction dependent: the maximum elongation of the wave pattern along the rigid plane occurs in the section x – z, whereas the minimum elongation is in the section y – z. In the latter case, the elongation is closed to that for the case 3-C 关compare Fig. 8共a兲 to Fig. 8共d兲兴. Yet, the centerline position of the pulse is the same in cases 3-A and 3-B 关see Figs. 8共b兲, 8共c兲, and 8共d兲兴. In Fig. 9, the centerline acoustic pressure for cases 3-A and 3-B is presented together with its counterpart for the case 3-C at t⫽0.5. In spite of the different wave pattern in cases 3-A and 3-B, the centerline profiles of acoustic pressure coincide for these cases. In this special situation, both the initial conditions of pulse and the background flow velocity are the same along the streamline, that coincide with the z axis, for two different flows 共24兲 and 共25兲. The maximum of absolute magnitude of acoustic pressure in presence of stagnation flow is approximately 2 times larger for the transmitted wave and 2.5 times larger for the reflected wave than those for the static ambient conditions 共see Fig. 9兲. The amplification of sound originated from spherical pulse in the 3D stagnation flows is larger than that originated from cylindrical pulse and propagated in the 2D stagnation flow. Recall that in all cases considered, the background flow velocity and pressure are the same along the centerline. To get a qualitative explanation of such a difference, the analytic solutions for acoustic pressure at the centerline in the static ambient conditions 共the cases 2-B and 3-C兲 are presented in Fig. 10共a兲. The spatial derivative of acoustic pressure along the centerline is shown in Fig. 10共b兲. By inspection of Fig. 10共a兲 and by comparison of derivatives in Fig. 10共b兲, one can conclude that the back and front wave fronts are sharper in the 3D case than those in the 2D case. The front of the wave propagating upstream moves slower than the back of it, that leads to the amplification of sound 共see the preceding section兲. The sharper the wave front is, the more prominent is the phenomenon. Therefore, the acoustic wave originated from the 3D pulse in a 3D stagnation flow amplifies larger than that in the 2D case. The difference in the maximum of acoustic pressure between the compressible and incompressible 关R⫽c⫽1 in Eqs. 共26兲 and 共27兲兴 models of stagnation flow is within 5% 共compare curves 1 and 2 in Fig. 9兲.

3

Results of computations on 80 , 120 , and 160 numerical grids are compared to the above analytic solution. Profiles of the centerline acoustic pressure on the above set of grids at t⫽0.5 are presented in Fig. 7. The numerical solution on the 1603 numerical grid practically coincides with the analytic solution 共29兲. The results of the grid refinement study for systems 共26兲 and 共27兲 on the 803 , 1203 , 1603 , and 2003 are similar to those obtained for the case 2-A 共see Sec. III兲. Therefore, the 1603 numerical grid is used for numerical simulations in this section. To compare acoustic fields in cases 3-A, 3-B, and 3-C, the isolines of acoustic pressure at t⫽0.5 are shown in Fig. 8. In cases 3-A and 3-C 关Figs. 8共a兲 and 8共b兲兴, the acoustic field is uniform with respect to the direction in the x – y plane. The presence of the stagnation flow leads to amplifi-

V. AEROACOUSTICS OF THE FLOW OVER A CIRCULAR CYLINDER

The propagation of sound waves originated from a single acoustic pulse in the 2D, inviscid isentropic mean flow over an infinite circular cylinder is considered here. The circle of unit diameter (R cyl⫽0.5) is the inner boundary and a circle of 5R cyl is the outer boundary. The mean incompressible velocity is computed using the potential flow model for nonlifting flow over a circular cylinder17

冉

U⫽U ⬁ cos ␪ 1⫺

2 R cyl

r2

冊

,

冉

V⫽⫺U ⬁ sin ␪ 1⫹

2 R cyl

r2

冊

. 共30兲




2667

FIG. 8. Isobars of acoustic pressure for 3D pulse at t⫽0.5: 共a兲 static ambient conditions, 共b兲 stagnation mean flow 共24兲, 共c兲, 共d兲 stagnation flow 共25兲, where 共c兲 x – z section and 共d兲 y – z section.

While the mean flow is directed from left to right, the acoustic waves, originated from the pulse, propagate upstream of the direction of mean flow. The equations of propagation of small disturbances are obtained by linearization about a known inviscid isentropic flow in polar (r, ␪ ) coordinates

冉

冊

⳵␳ ⳵u u 1 ⳵v ⳵␳ 1 ⳵␳ ⫺U ⫺ V ⫽⫺R ⫹ ⫹ ⳵t ⳵r r r ⳵␪ ⳵r r ⳵␪ ⫺u

⫺

冉

⳵u ⳵U 1 V ⫹ v ⫺2V v r ⳵␪ ⳵␪

冉

⫺ U

冊

冉

冉

␾ ⫽U ⬁ 共 ␾ 0 ⫹M ⬁2 ␾ 1 兲 ,

冊

⳵U 1 ⳵U 1 2 ␳ ⫹ V , ⫺ V ⳵r r ⳵␪ r R

冊

冊

where u and v are radial and angular components of the disturbance velocity, p⫽c ␳ 2 is the disturbance pressure, and ␳ is the disturbance density. The compressible velocity potential is computed by

⳵R 1 ⳵R ⫺ v , ⳵r r ⳵␪

⳵u ⳵u ⳵U 1 ⳵p ⫽⫺ ⫺U ⫺ u ⳵t R ⳵r ⳵r ⳵r

⳵v ⳵v ⳵V 1 1 ⳵p ⫺U ⫺ ⫽⫺ u ⳵t r R ⳵␪ ⳵r ⳵r ⳵v ⳵V 1 V ⫹ ⫺ v ⫹U v ⫹Vu r ⳵␪ ⳵␪ ⳵V 1 1 ⳵V ␳ ⫺ U ⫹ UV⫹ V , ⳵r r r ⳵␪ R

共31兲

共32兲

where ␾ 0 is incompressible potential, ␾ 1 is O(M ⬁2 ) correction to incompressible velocity potential given by Janzen and presented in Ref. 19. The density and speed of sound in this mean flow are computed by expressions similar to Eqs. 共6兲. The reference Mach number M 0 ⫽M ⬁ ⫽U ⬁ /c in Eqs. 共6兲.


2668


Alex Povitsky

FIG. 9. Axial profile of acoustic pressure: 共1兲 stagnation compressible isentropic mean flows 共24兲 and 共25兲共cases 3-A and 3-B兲, 共2兲 stagnation incompressible mean flow, and 共3兲 static ambient conditions 共case 3-C兲.

The initial acoustic pulse is given by 共13兲 and is located in some distance from the rigid cylinder. The initial pulse located at the centerline is considered first. To provide the grid refinement study, the numerical solution of above equations is performed on 90⫻90, 180 ⫻180, 360⫻360, and 540⫻540 uniform numerical grids in (r, ␪ ) polar coordinates that cover the computational domain 关 R cyl,5R cyl兴 ⫻ 关 0,360° 兴 . The computed acoustic pressure centerline profiles at time moment t⫽1.5 are presented in Fig. 11. Since the acoustic pressure profiles coincide on the two latter grids, the 360⫻360 grid is used in this section. The upstream propagation of acoustic pulse in the background mean flow with U ⬁ ⭐0.4c and location of initial acoustic pulse at the distance 0.5R cyl from the cylinder is denoted as the case CYL-A. The case CYL-B has the same background flow as the case CYL-A, however, the initial pulse is located at the distance 0.25R cyl . The case CYL-C denotes the sound propagation in the static ambient conditions (M ⫽0) in the same computational domain as in the cases CYL-A and CYL-B. In Fig. 12, isobars of acoustic pressure at t⫽0.5 are presented for the case CYL-A, where U ⬁ ⫽0.4c, and for the case CYL-C. In Fig. 13 the centerline pressure distribution is presented for these cases at time moments t⫽0.5, 1.0, 1.5, and 2. In comparison with the case CYL-C, the acoustic pressure is amplified at the centerline in the case CYL-A in the same time moments. To obtain the time-averaged sound directivity, the root mean square of acoustic pressure is calculated by p rms⫽

冑冕

T

0

p ⬘ 2 dt/T,

共33兲

where p ⬘ is the acoustic pressure (p ⬘ ⫽c 2 ␳ ⬘ ) and T is the time period of summation. This time period is chosen constant and equal for all numerical experiments presented be-

low. For temporal numerical integration, the integral in 共33兲 is computed as sum of values of squares of acoustic pressure at all time steps over the period, T. The time period, T, covers the pass of transmitted and reflected waves. The p rms represents the time-averaged potential part of acoustic energy passing through a fixed point in space. In Fig. 14, the p rms(r, ␪ ) is shown as a function of angle ␣ from the centerline where the radial coordinate r⫽5R cyl . For the cases CYL-A 关Fig. 14共a兲兴 and CYL-B 关Fig. 14共b兲兴, U ⬁ ⫽0.1c, 0.2c, 0.3c, and 0.4c. The case CYL-C 关curve M ⫽0 in Figs. 14共a兲 and 14共b兲兴 is presented as a benchmark for comparison. In Fig. 14 the presented results are normalized in such a way that the p rms at the centerline in the case CYL-C is equal to unity. In Fig. 14 the p rms obtained using the incompressible background flow model 关R⫽c⫽1 in Eq. 共31兲兴 are compared with those obtained by the isentropic flow model. In the latter case, the p rms at the centerline ( ␣ ⫽0) is 3% larger than that in the former case. This corresponds to comparison of background flow models for previously considered stagnation flows 共Figs. 3 and 9兲. The p rms increases with the Mach number M ⬁ and reaches 33% while U ⬁ ⫽0.4c 关Fig. 14共a兲兴. For ␣ ⫽48°, the local p rms in the case CYL-A remains 30% larger than that in the case CYL-C in the same direction. Therefore, the sound amplifies farther apart from the centerline. For the case CYL-C, the acoustic pressure at the centerline reduces about 1% since the location of acoustic pulse becomes 0.25R cyl . When the pulse is located closer to the cylinder, the sound scattering becomes stronger. For the case CYL-B, the centerline amplification of sound reaches 38%, i.e., the amplification of sound is larger when the initial pulse is located more close to the wall. The reason is that the reflected wave propagates in the downstream direction before it hits the wall. In the case CYL-A, it takes more 共in




2669

FIG. 11. Acoustic pulse propagation in compressible flow over cylinder. Numerical grids in (r, ␪ ) coordinates: 共1兲 90⫻90; 共2兲 180⫻180; 共3兲 360 ⫻360; and 共4兲 540⫻540.

In Fig. 15共a兲, the temporal integrals of acoustic energy E( ␣ )⫽ 兰 T0 E dt and its potential part P( ␣ )⫽ 兰 T0 P dt are presented for r⫽5R cyl and 0°⬍ ␣ ⬍48°. The angular redistribution of acoustic energy and potential-to-kinetic transform

FIG. 10. Acoustic pulse propagation and reflection in static ambient conditions 共analytic solution兲: 共a兲 centerline acoustic pressure, 共b兲 the spatial derivative of acoustic pressure along the centerline. The direction of the x-coordinate coincides with the centerline, graphs 共2D兲 and 共3D兲 denote 2D cylindrical pulse 共case 2-B兲 and 3D spherical pulse 共case 3-C兲, respectively.

temporal and spatial scales兲 for the wave to propagate downstream before its reflection from the cylinder than that in the case CYL-C. On the other hand, the stronger scattering of sound by the cylinder in the case CYL-B leads to relative weakening of sound. The weakening effect of scattering is more noticeable for smaller Mach numbers. To compare the strength of sound for these different locations of initial pulse, the p rms curves for the case CYL-C 共dashed–dotted lines兲 and for the case CYL-A 共solid lines兲 are plotted in Fig. 14共b兲. The intersection points of curves of p rms for the cases CYL-A and CYL-C corresponding to the same Mach number are denoted as ␣ cr . If ␣ ⬍ ␣ cr , the p rms is larger for the case CYL-C than that for the case CYL-A. The ␣ cr are shifted towards the larger angles ␣ with increase of M 关see Fig. 14共b兲兴. Therefore, the angular span of the sector where the sound amplifies stronger with the shift of the initial pulse towards the cylinder increases with the Mach number of the flow M ⬁ .

FIG. 12. Isobars of acoustic pressure at t⫽1.5: 共a兲 upstream propagation of sound 共case CYL-A兲; and 共b兲 static ambient conditions 共case CYL-C兲.


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FIG. 13. Centerline acoustic pressure: 共a兲 upstream propagation of sound 共case CYL-A兲; and 共b兲 static ambient conditions 共case CYL-C兲. Time moments: 共1兲 t⫽0.5; 共2兲 t⫽1.0; 共3兲 t⫽1.5; and 共4兲 t⫽2.0.

within a given direction ␣ affect the amplification of sound. For instance, the difference in in terms of E( ␣ ) increases from 8.3% to 14.9% when ␣ increases from 0° to 48°. In the case CYL-C, the part of potential energy P( ␣ )/E( ␣ ) * 100% in the case CYL-C remains within the range 49.8%– 49.9%. On the contrary, for the case CYL-A the part of potential energy drops from 80.2% to 67.4% when ␣ increases. Qualitatively, the results are similar to those obtained in Sec. III. The amplification of sound occurs mostly due to kineticto-potential transform where the direction of wave propagation is approximately opposite to the vector of mean flow. Apart from this direction, the angular redistribution of acoustic energy becomes more important. If the initial pulse is located 30° apart from the centerline, the refraction of acoustic energy leads to its decrease at the centerline in comparison with the wave propagation from the same pulse in the static ambient conditions 关see Fig.

Alex Povitsky

FIG. 14. Root mean square 共rms兲 of acoustic pressure for centerline pulse location: 共a兲 R init⫽0.5R cyl where solid lines denote isentropic compressible mean flow 共case CYL-A兲 and dashed lines denote incompressible potential mean flow; and 共b兲 R init⫽0.25R cyl where solid lines denote case CYL-A 共presented for comparison兲 and dashed–dotted lines denote case CYL-B.

15共b兲兴. According to the results of computational modeling, the acoustic pressure at the centerline remains equal to that for the static ambient conditions due to the increase of potential part of acoustic energy. Maximum of acoustic pressure at ␣ ⫽60° is caused by refractive amplification of acoustic energy while the potential part of acoustic energy decreases apart from the centerline. VI. CONCLUSIONS

The behavior of acoustic waves originating from the single pulse and propagated in the non-uniform compressible isentropic background flow is studied by direct numerical simulation. Higher-order compact spatial finite differences




2671

fication of sound and flattening of acoustic waves. The propagation of acoustic wave originated from spherical pulse in 3D stagnation flows was considered and the wave pattern in the neighborhood of the single stagnation point and the stagnation line are discussed. The amplification of sound by a mean flow is more prominent in the 3D case than in the 2D case because of a steeper acoustic wave profile. The acoustic pressure in 3D stagnation flows is more than doubled in comparison with the static ambient conditions. The propagation of acoustic pulse upstream of the flow around the 2D circular cylinder is modeled. The timeaveraged root mean square of acoustic pressure 共rms兲 is presented as a function of angle from the centerline. Refraction of acoustic energy apart from the centerline and increase of the potential part of acoustic energy near the centerline lead to increase of acoustic pressure in comparison with the wave propagation in the static ambient conditions. This amplification holds for the wide angular sector upstream of the solid body. ACKNOWLEDGMENTS

This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS197046 while the author was in residence at ICASE, NASA Langley Research Center, Hampton, VA 23681-2199, by the New Faculty Start Up Funds at Concordia University, and by NSERC Grant No. 249528-02.

FIG. 15. Integral of acoustic energy E( ␣ ) and its potential component P( ␣ ): 共a兲 centerline location of the initial pulse; 共b兲 the initial pulse is located 30° apart from the centerline. Solid curves denote E( ␣ ), dashed curves denote P( ␣ ). Curves 共1兲 correspond to M ⫽0.4 共case CYL-A兲 and curves 共2兲 correspond to the static ambient conditions 共case CYL-C兲.

APPENDIX: MEAN-FLOW VARIABLES FOR A FLOW BETWEEN TWO STRAIGHT STREAMLINES

In Cartesian coordinates, the stream function is given by

␺ ⫽A 共 x 2 ⫹y 2 兲 0.5n sin共 n arctan共 x/y 兲兲 .

共A1兲

The velocity field (U(x,y),V(x,y)) is computed by differentiation of the stream function defined above, and Runge–Kutta temporal integration are used. Results are compared to those obtained for the same geometry in the static ambient conditions. In terms of acoustic energy, the modified sound directivity in the presence of stagnation flow is mainly caused by redistribution of potential and kinetic components of acoustic energy 共while sound propagates upstream兲 and by angular redistribution of acoustic energy 共while directions of the mean flow and sound propagation are far from collinear兲. Taking into account the compressibility of the background flow, the maximum of acoustic pressure moderately increases 共within 6%兲 while local Mach number M ⭐0.6. The pump of acoustic energy from the background flow by means of baroclinically generated vorticity is minor. It is shown that the acoustic pressure is increased 80% while the transmitted wave propagates upstream of the 2D stagnation flow and reaches the point where the Mach number of stagnation flow M ⫽0.56. Alternation of the direction of stagnation flow changes the direction of sound propagation from perpendicular to a wall to parallel to it. Sound propagation in a corner while in the presence of the background inviscid flow leads to ampli-

U 共 x,y 兲 ⫽

⳵␺ , ⳵y

V 共 x,y 兲 ⫽⫺

⳵␺ . ⳵x

共A2兲

Using Mathematica,29 the following expressions for velocities and their derivatives are obtained: U 共 x,y 兲 ⫽Anr n⫺2 共 x cos n ␪ ⫹y sin n ␪ 兲 ,

共A3兲

V 共 x,y 兲 ⫽Anr n⫺2 共 y cos n ␪ ⫺x sin n ␪ 兲 ,

共A4兲

⳵ U 共 x,y 兲 ⫽Anr n⫺4 共共 n⫺1 兲共 y 2 ⫺x 2 兲 cos n ␪ ⳵x ⫹2 共 n⫺1 兲 xy sin n ␪ 兲 ,

共A5兲

⳵ U 共 x,y 兲 ⫽Anr n⫺4 共共 n⫺1 兲共 x 2 ⫺y 2 兲 sin n ␪ ⳵y ⫺2 共 n⫺1 兲 xy cos n ␪ 兲 ,

共A6兲

where r⫽ 冑(x 2 ⫹y 2 , and ␪ ⫽arctan(x/y). For an incompressible and irrotational mean flow,

⳵ U 共 x,y 兲 ⳵ V 共 x,y 兲 ⫽⫺ , ⳵x ⳵y

⳵ U 共 x,y 兲 ⳵ V 共 x,y 兲 ⫽ . ⳵y ⳵x

共A7兲


2672 1


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