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Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

A98-30806 AIAA-98-2208

Numerical Simulation of Sound Generation by Flow past a Cylinder

San-Yih Lin, Bae-Yih Hwung, Sheng-Yu Wen,Institute of Aeronautics and Astronautics

National Cheng Kung University70101, Tainan, Taiwan, R. O. C.

and Sheng-Chang ShihFar East College, Hsin-Shin, Tainan, Taiwan, R.O.C.

Abstract

The aeroacoustic sound generation by flowpast a two-dimensional circular cylinder is inves-tigated numerically. The compressible Navier-Stokes equations are solved by a proposed finitevolume method. The numerical method is basedon a third-order upwind finite-volume scheme inspatial derivatives. The Roe's approximation Rie-mann solver is used for the numerical flux andan explicit second-order three-step Runge-Kuttascheme is used for the time marching. The flowvariables, such as density, momentum, and pres-sure, around the near and middle field are ob-tained by the proposed numerical solver. Thenthe pressure field and the associated sound wavesare computed by the Kirchhoff's method. Inthis paper, a series flow parameters are consid-ered, such as Mach number from 0.2 to 0.6 andReynolds number from 900 to 90,000. A polargrid system is used in the calculations. The gridand time resolution studies are performed to seeconvergence of the computational solutions. Toaccurately apply the Kirchhoff's method, the in-tegral surface used in the method should includethe nonlinear region. In this study, several in-tegral surfaces for the Kirchhoff's method aretested to check accuracy and convergence. Thenumerical results indicate that the decay rate ofthe strength of the pressure fluctuation is pro-portional to l/R, where R is the radius from thecenter of the circular cylinder.

1. Introduction

The flow around a cylinder has already beenstudied intensively. Most of them considered the

^Supported by NSC grant 86-2212-E006-105.Copyright c 1998 by San-Yih Lin. Published bythe Confederation of European Aerospace Soci-eties, with permission.

incompressible flows. A few papers have dealt .with the compressible flows. 1~3 The vortex streetbehind a cylinder is the main flow structure in theincompressible flow. But, at higher Mach num-bers, there are shocks sitting in the near wake ofthe cylinder. The flow may become quasisteady.In this paper, we numerically investigate the flowstructures with a series of flow parameters, suchas Mach number from 0.2 to 0.6 and Reynoldsnumber from 900 to 90,000. The two-dimensionalcompressible Navier-Stokes equations are solvedby a finite volume method, MOC scheme.4'5 Thisscheme, in strong conservation form, is based ona third-order upwind finite-volume scheme forspatial derivatives and an explicit, second-order,three-step Runge-Kutta scheme for time march-ing.

In order to accurately solve the acoustic field,a very fine grid system is required. Howeverfor numerical study, this grid system wastes toomuch computational source and this is not aneconomic way. To prevent the waste of CPUtimes, the Kirchhoff's method6 can overcome thedifficulties. It is an integral formulation on a ap-propriate control surface. Assume all the nonlin-ear effect is included in the control surface andonly linear effect is present outside the surface.The far-field solutions can be accurately solvedby means of the integral formulation on the con-trol surface. Therefore a finer grid system is usedaround the near and middle fields to obtain theaccurate flow variables. The far acoustic field isthen obtained from the Kirchhoff's formulation.

2. Computational Methods

The near and mid-field flow fields of the sub-sonic flow past a circular cylinder are obtained bya compressible Navier-Stokes solver. The solveris based on a third-order upwind finite-volumescheme in space and an explicit second-order three-step Runge-Kutta scheme in time. Far field noise

46


is then obtained from the Kirchhoff 's method.

2.1 Navier-Stokes Solver. MOC scheme

The flows of two dimensional, compressibleand viscous fluid can be described in conservationform by the Navier-Stokes equations:

V-( / ,<?) = 0 (1)where

w =

with

Txy =

Tyy = fj.-

= UTXX + VTxy

UTXy + VTyy

4Uj,)/3

yU

here p, p, u, v, and e are the pressure, density,x-directional and y-directional velocity compo-nents, and the total energy per unit mass, re-spectively, fj, is the dynamic viscosity determinedby Sutherlands law. The Reynolds number andPrandtl number are denoted as Re and Pr, re-spectively, "a" is the speed of sound. The pres-sure p is given by the equation of state for a per-fect gas:

12'

(2)

where 7 (= 1.4 for air) is the ratio of specificheats.

The Navier-Stokes solver (Modified Osher-Chakravarthy scheme, MOC scheme4'5) solves thetwo-dimensional, unsteady, Navier-Stokes equa-tions. The scheme, in strong conservation form,

is based on a third-order upwind finite-volumescheme for the convection terms and an explicitsecond-order Runge-Kutta scheme for the timemarching. For the description of the scheme andits applications, please see the papers.4'5 Here,we only describe the treatment of the boundaryconditions for the cylinder flow computations.On the surface of the circular cylinder, the no-slip boundary conditions are used. The density iscomputed by the extrapolation for the adjacentpoints, then the pressure is solved by the normalmomentum equation. The far-field flow variablesare solved by the characteristic boundary condi-tions.

2.2 Kirchhoff's Method

For the numerical researches of the aeroa-coustic sound generation by flow past a two di-mensional circular cylinder, fine mesh system isnecessary to precisely capture the acoustic per-turbation. However such grid system will resultin waste of CPU time and large memory stor-age. These cause computational distress. Kirch-hoff's formulation is a convenient approach forthe flowfield evaluation by means of informationon a closed control surface in the mid field re-gion. All the nonlinear effects are assumed tobe contained within the control surface. In theexterior of the surface, linear wave patterns arepresent. Therefore we only require fine meshin the mid-field domain and utilize the numer-ical solutions on the control surface, which arethe input of the Kirchhoff's formula, to calcu-late the far field value. Take advantage of theGreen function, Kirchhoff's formula can be ob-tained. Pierce7 derived the formula for a fixedsurface. The Kirchhoff's formulation for an arbi-trarily moving piecewise smooth deformable sur-face is performed by Farssat and Myers.8

Assume that all the nonlinear effects andacoustic sources are present merely within a con-trol surface 5 and three dimensional waves prop-agate outward from the surface S. The pressureat the far field region can be modeled from theKirchhoff's formulation. The pressure field out-side the control surface 5 satisfies the wave equa-tion

where and are free stream sound speed

47


and velocity, respectively. For fixed control sur-face S and utilizing the Prandtl-Glauert trans-formation

XQ = x, y0 = (3y, z0 = j3z, r0 = r0, (4)

pressure outside the surface S is described as

1 dp.dr0- -£ ——dt dn0

/ ^ 1 f{ 1 dpp(x, y,z,t} = — \ —— ——4?r Js r0 dnQ

~n——) ~\~ ~~o T;——\Ta&0an0 rg <9n0

where

r0 = {(x - xs)2 + {32[(y - ysf + (z -T= [rp - Moo (X-X8)} /Coo/32

(6)

MOO is the free stream Mach number. T repre-sents the time delay for wave to propagate fromsource (xs, ys, za] to observation position (x, y, z}.The subscript 0 expresses the transformed value,T expresses that all the values are calculated attime ts = t—r. n is the outward normal directionof 5.

The strip theory is used in spanwise direc-tion and the two-dimensional numerical solutionis accepted for every strip. The space encircledby the control surface is expected to be largeenough to contain all the nonlinear effect, butthe numerical solutions are not accurate enoughon this large surface if without the match of finegrids. Therefore the appropriate selection of thesurface is essential. Since the contribution fromthe tip surface can be neglected6, the surface in-tegration of Equation (5) excludes tip surface ef-fects here. The aspect ratio is assumed to be 20,and 800 strips are used in the spanwise direction.

2.3 Sound Pressure Level

The SPL is defined by

SPL(dB] = 20logw( Prms •

Pref '(7)

where pref = 2 x 10~°. The root-mean-squarevalue of pressure. prms is obtained from t = 100to 124.

In this paper, SPL is computed in the com-putation domain to see the range of the acoustic

sources. Then SPL is computed at several pointsat far field to see the strength of noise and toanalyze the type of acoustic sources for the flowover a circular cylinder.

2.4 Numerical Test

Flow through a NACA0012 airfoil with M^ =0.5, Re = 5000 and angle of attack a = 0° istested. The grid system of 120 x 40 is used.Figure 1 is the pressure coefficient on the airfoil.The results is compared well with Swanson andTurkel's numerical results.9 Then flow througha NACA0012 airfoil with M^ = 0.602, Re -3 x 106 and angle of attack a = —0.14° is consid-ered. The grid system of 120 x 50 is used. Undersuch circumstances, flow is subsonic and turbu-lent, the Baldwin-Lomax turbulence model10 isused in this tested. Figure 2 is the pressure coef-ficient on the airfoil. Compared with Harris's ex-perimental results11, it also has satisfactory con-clusion.

3. Results and Discussions

Numerical results are now given for the twodimensional circular cylinder at various freestreamReynolds and Mach numbers. First, flow proper-ties are compared with previous data and grid in-dependence is performed. Then the aeroacousticsound generation by flow past a two-dimensionalcircular cylinder is investigated.

3.1 Flow Properties

First Reynolds number is fixed to be 106.flow past a circular cylinder with mach numberof 0.9, 1.0, 1.2, and 1.5 are simulated. Accord-ing Murthy and Rose's experimental results2, theflow is quasisteady. Figure 3 shows the pressureand vorticity contours with mach number of 1.2.The symmetric and steady solution is obtained.Figure 4 shows the drag coefficient. The com-puted data are little higher than the experimen-tal data.12 In such a high Reynolds number, oneexpects that the flow should be turbulent.

Then Reynolds number is fixed to be 900and Mach number varies from 0.1 to 0.6. Table1 shows the Strouhal numbers. One can see thatthe Strouhal numbers increases at lower Machnumbers and then decreases as Mach number in-creases from 0.4 to 0.6. Finally, Mach numberis fixed to be 0.2 and Reynolds number varies

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from 900 to 90000. Table 2 shows the Strouhalnumbers.

3.2 Grid Independence

Four grids are chosen, 96 x 96,128 x 64,128 x96,128 x 128, to investigate the grid indepen-dence. Figure 5 shows the SPL contours at Reynoldsnumber of 900 and Mach number of 0.2 with fourgrids. The results obtained by the 128 x 96 and128 x 128 grids are close to each other. Thedistributions of CL and CD obtained from thesetwo grids also show that there converge together.Now the grid system is fixed to be 128 x 96,the effect on the choice of the Kirchhoff 's inte-gral surfaces are tested. Four integral surfaceswhich are the circles with radius ID, I.5D,2D,and 5D (D is the diameter of the circular cylin-der) are chosen. Figure 6 shows the pressure dis-tributions at the point (17.5,17.5) x/2Z? obtainedby the Kirchhoff's method with the four integralsurfaces under the above four grids. Figure 7shows the power spectra of the pressure at theabove point obtained by the Kirchhoff's methodwith the integral circle with radius ID under theabove four grids. From the above two figures,the results obtained from the two integral circleswith radius 1.5D and 21? are close together. Onecan conclude that the almost of the nonlinear ef-fects and acoustic sources include in the circlewith radius 1.5D or 2D.

3.3 Acoustic Radiation

Aeolian tones, sound generation by flow overa cylinder, are relevant to automobile antennaand airframe noise.12 The purpose of this study isto demonstrate computation of sound generationby viscous.

First, consider uniform flow at Reynolds num-ber 900 over a two-dimensional cylinder. TheMach number varies from 0.1 to 0.6. The nearand middle flow fields are obtained by the Navier-Stokes solver with the 128 x 96 grid system. Fig-ure 8 shows the SPL contours with Mach num-ber from 0.1 to 0.6. One can see that there arehigher SPL values around the rear region of thecircular for all cases. For Mach number form 0.1to 0.4, the main acoustic sources include in thecircle with radius 2D. But for the Mach number0.6. SPL value is very high in the region withradius between ID and 6D. One can conclude

that the Kirchhoff's integral surface chosen to be2D is good enough for Mach number from 0.1 to0.4. but is not good for the Mach number 0.6.Figures 9 and 10 show the pressure fluctuationat the points, (35D,0) and \/2(17.5,17.5)£> re-spectively, obtained by the Kirchhoff's methodunder the above Mach numbers. Figures 11 and12 show the power spectra of the pressure fluctu-ation at the above points under the above Machnumbers. One can find that the main frequencyof the power spectrum of each case are exactlytwice to its Strouhal number. For Mach num-ber form 0.1 to 0.4, the results obtained by theKirchhoff's integral surface with radius 1.5D and2D are close together. But for the Mach number0.6, the results are not quite consistent with eachother. This is consistent to our observation fromthe SPL contours. Now fixed Re = 900 and Machnumber = 0.2, Figure 13 shows the strength ofthe pressure fluctuation varied with the distanceto the center of the cylinder. The results indicatethat the decay rate of the strength of the pressurefluctuation is proportional to l/R, where R is theradius from the center of the circular cylinder.Moreover, assume that the pressure fluctuationbe the function of the radius as following

P:rms,i (Ri + a)

From the four positions, 70, 105. 140, and 175D,we compute three groups of (a, 6) which are (0.63,1.176), (-0.25, 1.167), and (0, 1.169). The val-ues of b are close to a constant 1 which indicatethat the decay rate of the strength of the pres-sure fluctuation is proportional to 1/R. One canexpect that there are dipole sources in the flowover the circular cylinder. But the values of aare not convergent to a constant, which indicatethat the main acoustic source is not precisely de-termined for the flow over the circular cylinderin our numerical computations.

Finally Mach number is fixed to be 0.2 andReynolds number varies from 900 to 90000. Ta-ble 2 shows the Strouhal numbers. The relation-ship between Strouhal and Reynolds numbers iscomplicated. Above Reynolds number 9000, theflow is mixed with laminar and turbulence. Oneshould use turbulent models to solve the flowfields. But the laminar flow solver is still used

49


in our study. Figure 14 shows the power spectraof the pressure fluctuation at the point, (35,0)D,obtained by the Kirchhoff's method with four in-tegral surfaces at Reynolds number 90000. Al-though there are very complicated, the resultsobtained from the integral surfaces with radius1.5D and 2D are consistent with each other. Onecan expect that the main acoustic sources includein the circle with radius 2D.

Conclusions:

The aeroacoustic sound generation by flowpast a two-dimensional circular cylinder is inves-tigated numerically by a proposed Navier-Stokessolver, MOC scheme. The flow variables, suchas density, momentum, and pressure, around thenear and middle field are obtained by the pro-posed numerical solver. Then the pressure fieldand the associated sound waves are computedby the Kirchhoff's method. The grid and timeresolution studies are performed to see conver-gence of the computational solutions. In thisstudy, several integral surfaces for the Kirchhoff'smethod are tested to check accuracy and con-vergence. For most of cases studied except thehigher Mach number, the main acoustic sourcesinclude in the circle with radius 2D. The numer-ical results indicate that the decay rate of thestrength of the pressure fluctuation is propor-tional to 1/R, where R is the radius from thecenter of the circular cylinder.

Acknowledge:

The work is partially supported by the Na-tional Science Council of the Republic of Chinaunder Contracts NSC86-2212-E006-016. Free com-puting time provided by the National Center forHigh-Performance Computing is greatly acknowl-edged.

References:1. A. Roshko, "Experiments on the Flow past

a Circular Cylinder at Very High ReynoldsNumber," Journal of Fluid Mechanics, Vol.10, 1960, pp. 345-356.

2. V. S. Murthy and W. C. Rose, " DetailedMeasurements on a Circular Cylinder in CrossFlow," AIAA Journal, Vol. 16, June 1978,pp. 549-550.

3. O. Rodriguez, "The Circular Cylinder in Sub-sonic and Transonic Flow," AIAA Journal,Vol. 22, December 1984, pp. 1713-1718.

4. S. Y. Lin and Y. S. Chin, "Comparison ofHigher Resolution Euler Schemes for Aeroa-coustic Computations," AIAA Journal, Vol.33. No. 2, Feb. 1995, pp. 237-245.

5. S. Y. Lin and Y. S. Chin, "Numerical Studyof Head-On Blade Vortex Interaction Noise,AIAA Paper 97-0287, Jan. 1997.

6. A. S. Lyrintzis and A. R. George, "Far-FieldNoise of Transonic Blade-Vortex Interaction,"American Helicopter Society Journal, Vol.34. No. 3, 1989, pp. 30-39.

7. A. D. Pierce, Acoustics: An Introduction toIts Physical Principles and Applications, McGra-,Hill, New York, 1981.

8. F. Farassat and M. K. Myers, "Extension ofKirchhoff's Formula to Radiation from Mov-ing Surfaces," Journal of Sound and Vibra-tion, Vol. 123, No. 3, pp. 451-460, 1988.

9. R. C. Swanson and E. Turkel, "A Multi-stage Time-Stepping Scheme for the Navier-Stokes Equations," AIAA Paper 85-0035, June1985.

10. B. S. Baldwin and H. Lomax, "Thin LayerApproximation and Algebraic Model for Sep-arated Turbulent Flows," AIAA Paper 78-257, Jan. 1978.

11. C.D.Harris, "Two-DimensionalAerodynamicCharacteristics of the NAG AGO 12 Airfoil inthe Langley 8-Foot Transonic Pressure Tun-nel," NASA TM 81927, April 1981.

12. C. K. W. Tam, "Proceedings of the SecondComputational Aeroacoustics Workshop onBenchmark Problems." November 1996.

50


Table 1. Strouhal numbers as Reynolds number fixed to be 900:

Mach No.St No.

0.10.232

0.20.244

0.40.244

0.60.208

Table 2. Strouhal numbers as Mach number fixed to be 0.2:

Re No.St No.

9000.244

90000.244

250000.208

900000.280

o presentSwanson

,et

Fig 1. The pressure coefficient on the airfoil withMX, = 0.5, Re = 5000 and angle of attack

1.0

u.

O o.c

Experiment (upper)Experiment (lower)Present (lower)Present (upper)

1.0 0.2 0.4 0.6 0.8

X

1.0

s\.Fig 2. The pressure coefficient on the airfoil with

MX, = 0.602, .Re = 3 x 106 and angle ofattack a = -0.14°.

0 : 2 3

Fig 3. The pressure and vorticity contours with machnumber of 1.2.

'•• M,

Fig 4. The averaged drag coefficient as Reynoldsnumber = 106. The 'real line' is the com-puted results and '•' is Murthy and Rose'sdata.

51


1 0 1 2 3 * 5 6

llfllfITO• l / v 'H if v *h

Fig 5. The SPL contours at Reynolds number of Fig 6. The pressure distributions obtained by the900 and Mach number of 0.2 with four grids ""(a) 96 x 96 (b) 128 x 64 (c) 128 x 96 and fd)128 x 128.

--—-- "j "+^Kirchhoff's method with the four integralsurfaces under the above four grids.

52


Mach=0.1.Re=900

0050

0 040

00»

0020

0010

0060

0050

0.040

0.030

0.020

0.010

•

L

'

:

-

-

ot L . . . . . . . . . . . . . .i 3 3 4 S

-

I

:

;

-

:U., , , , :

Fig 7. The power spectra obtained by the Kirch-hoff's method under the above grids.

Fig 8. The SPL contours with Mach numbers (a)0.1 (2) 0.2 (3) 0.4 and (d) 0.6.

53


."2.0'-5.0

i A: 11 A;, W M il M

| M fill fill |«M V / M V V T / W V J

^ 1' \j !l

Fig 9. The pressure distributions obtained by the Fig 10. The pressure distributions \/2(17.5,17.5)DKirchhoff 's method on the four integral sur- with the above Mach numbers,faces at the point (351?. 0) with the aboveMach numbers.

54


0.050

O.CMO

0.030

0020

00 ID

J

-

:

-

-

-

1 i . . . . . . , , :1 2 3 4 5

O.OSO

0040

0.030

0020

G O T O

-

J

---

. i . .' 2 3 4 S

0.050

0.040

0030

0020

0010

J

-

-

-

.

I i1 2 3 4 S

o.o&o

0.040

0030

0.020

0-0 10

0000

-

-

•1-

-

->, . i ™ , '

0050

004Q

0030

0.020

0010

1 J

- ———— | — . —————— , i . . . . . T . . . - , . . . .

-

-

-

-

L l 1 , . . , . . . . . . . . . . . .1 2 3 4 5

0.050

0.0-10

0.030

0.020

0010

J l ;

, . , . t

^

-

-

I i . , , , '

1 1! 3 4 5

OOSO

U040

0.030

00,0

::

:

.

JJU ,L__ . . , ' •

11. The power spectra at the point (35.0}D withthe above Mach numbers.

Fig 12. The power spectra at the point, \/2(l7.5,17.5)Dwith the above Mach numbers.

55


Fig 13. The strength of the pressure fluctuation var-ied with the distance to the center of thecylinder.

Iff

r-2

1 2 3 4 5

0.005

I

r=5

Fig 14. The power spectra of the pressure fluctua-tion at the point, (35,0)D, obtained by theKirchhoff's method with four integral sur-faces at Reynolds number 90000.

56

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