numerical simulation of the noise generated by a low mach number, low reynolds number jet

Fluid Dynamics Research35 (2004) 425–447

Numerical simulation of the noise generated by a low Machnumber, low Reynolds number jet

Bendiks Jan Boersma

J.M. Burgerscentre, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands

Received 6 March 2003; received in revised form 29 September 2004; accepted 1 October 2004Communicated by T. Mullin

Abstract

In this paper we study the sound field produced by a turbulent round jet with a Mach number of 0.6 based onthe centerline velocity and the ambient speed of soundc∞. The turbulent flow field is found by solving the fullycompressible Navier–Stokes equations with help of high-order compact finite difference schemes. It is shown thatthe simulated flow field is in good agreement with experiments. The corresponding sound field has been obtainedwith help of the Lighthill equation using two different formulations for the Lighthill stress tensorTij . In the firstformulation ofTij the fluctuating density is taken into account. In the second formulation the density is assumed tobe constant. As an additional check we have also performed an acoustic calculation using a formulation in which ahomogeneous wave equation is solved. The boundary conditions for this homogeneous wave equation are obtainedfrom the numerical simulation of the Navier–Stokes equation. The results obtained with both formulations of theLighthill stress tensor are nearly identical. This implies that an incompressible formulation of the conservationslaws could be used to predict jet noise at low Mach numbers.© 2004 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved.

Keywords:DNS; Turbulence; Aeroacoustics

1. Introduction

Aeroacousticalsound, with respect to its characteristics and generation, forms in many cases a technicalproblem. Its solution is needed in view of various demands for instance in relation to user comfort or

E-mail address:[email protected](B.J. Boersma).

0169-5983/$30.00 © 2004 Published by The Japan Society of Fluid Mechanics and Elsevier B.V.All rights reserved.doi:10.1016/j.fluiddyn.2004.10.003

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426 B.J. Boersma / Fluid Dynamics Research 35 (2004) 425–447

environmental regulations. Everybody is familiar with the sound produced by a jet engine of a commercialairliner. Strict environmental measures around airports have put strong limitations on this type of sound.Although significant sound reduction of these jet engines has been obtained over the last few decades,it is nevertheless required to reduce the sound of jet engines even more in view of strong growth in airtraffic foreseen in the future. Other examples of aeroacoustical sound production are: the noise producedby high speed trains, wind noise around buildings, the sound comfort in cars but also ventilator noise invarious household appliances. In this study we will focus on sound produced by turbulent jets becausethis flow is one of the benchmark flows for which a reasonable amount of experimental data is available.

Here we only want to mention a few of the experimental studies on jet noise, for a more detailedoverview we refer to (Goldstein (1976); Ribner (1996)). Lush (1971)reports acoustic pressure spectra ofa Mach 0.3, 0.6 and 0.9 turbulent jet.Mollo-Christensen (1967)also reports spectra for Mach 0.6, 0.8 andMach 0.9 turbulent jets and gives detailed information about the directivity of the sound. Unfortunatelyno information is given on the structure of the flow field.Stromberg et al. (1980)reports acoustic data ina Mach 0.9 turbulent jet, and they also give some information on the mean velocity profiles. Informationon the jet flow field has been presented in various studies (see for instance,Bradshaw et al., 1964;Panchapakesan and Lumley, 1993; Hussein et al., 1994; Boersma et al., 1998). As far as we know thereis no detailed experimental study in which both the flow field and the acoustic field are presented.

Traditional jet mixing-noise predictions are based on a statistical description of the jet turbulence;space-time correlations of the Lighthill quadruples (Lighthill, 1952, 1954) are specified in the jet. Furtherrefinements are, however, needed to obtain a reasonable prediction of the directivity and spectral distribu-tion of the radiated noise (Goldstein, 1976), and significant improvements are achieved by modeling theeffects of source-convection (Lighthill, 1954) and mean-flow refraction (Mani, 1976). Reynolds-averagedmean flow calculations can be used to provide the source-strength distribution and the length and time-scale estimates needed in the source models (Bechara et al., 1994; Khavaran and Krejsa, 1998). However,the empirical input needed in such an approach places limits on the range of applicability of this method.It is, therefore, desirable to develop methods which obtain the unsteady flow data with less empiricalinput.

Recently, as a result of increasing computer power, it has become possible to calculate the acousticfield of a jet using direct numerical simulation (DNS), (Colonius et al., 1997; Mitchell et al., 1999). Directnumerical simulations of high Mach number turbulent jets have been performed byFreund (1999). Inthese simulations the sound is calculated direct. In flows with very low Mach numbers direct methodsare computationally very inefficient and unreliable, due to the very small amplitude of the acoustic waveresulting in a low signal-to-noise ratio for the acoustic field. In the low Mach number regime it is likelythat acoustic analogons like the Lighthill equation, in combination with incompressible flow solvers, willgive more reliable results because they are less contaminated by numerical errors. In the past acousticanalogons were mainly used to provide qualitatively information (integral quantities). In the present paperwe will use the Lighthill equation quantitatively to predict the sound produced by a turbulent jet with aMach number of 0.6. The source term in the Lighthill equation will be calculated directly from the flowfield which has been obtained from a DNS. The Mach number of 0.6 should be high enough to performa direct noise calculation with sufficient accuracy and also low enough for a prediction of the sound fieldusing the Lighthill equation. This gives us the opportunity to carry out a detailed comparison betweenboth methods.

To give an outline of the paper: in Section 2 we will give the governing equations for the flow and theacoustic field. In Section 3 the numerical techniques to solve the flow and acoustic field are discussed. In

B.J. Boersma / Fluid Dynamics Research 35 (2004) 425–447 427

Section 4 the results for the flow and acoustic field are given. Finally some conclusions are presented inSection 5.

2. Governing equation

In this section we will present the governing equations for the flow and describe how these equationsare non-dimensionalized.

2.1. Flow equations

The flow is described by the well-known compressible Navier–Stokes equations, see for instance(Batchelor, 1967). The equation for conservation of mass reads.

��

�t+ ��ui

�xi= 0, (1)

where� is the fluids density andui the fluids velocity component in theith coordinate direction. Theequation which describes the conservation of momentum reads

��ui�t

+ ��ujui�xj

= − �p

�xi+ �

�xi�ij . (2)

In whichp is the pressure and�ij is the viscous stress tensor given by

�ij = 2�Sij = �

(�ui�xj

+ �uj�xi

− 2

3�ij

�uk�xk

). (3)

The dynamic viscosity� is a weak function of the temperature in the gas,1 but for the moment we willneglect this and assume that� is constant. Sometimes a bulk viscosity�b is added to the main diagonalterms of the stress tensor. The bulk viscosity acts as a crude model for molecular relaxation and non-perfect gas effects (Vincentu and Kruger, 1965). For the flow we consider here this effect will be verysmall and is thus neglected.

For the energy equation in a compressible flow various formulations are possible. Here we choose fora formulation in terms of the total energy, i.e., the sum of internal energy (expressed in its form for anideal gas�CvT ) and kinetic energy (potential energy being negligible)

E = �CvT + 1

2�uiui. (4)

In whichCv is the specific heat at constant volume andT is the temperature. The transport equation forthe total energyE reads

�E

�t+ �

�xiui[E + p] = �

�xi�

(�T

�xi

)+ �

�xiuiSij . (5)

1 In the literature the following relation is frequently used:�=�0(T /T0)0.76. In whichT is the temperature and the subscript

0 denotes the reference state.


In which� is the thermal diffusion coefficient, which is again a weak function of the temperature, whichwill be neglected. The formulation of the energy equation given above has the advantage that no sourceterms appear in the right-hand side which would be the case in formulations that use the temperatureinstead of the energy. The temperatureT , the pressurepand the density� are related to each other by theequation of state for an ideal gas

p = �RT . (6)

In whichR is the gas constant.

2.2. Acoustic field

The acoustic field in a low Mach number flow can be described by different acoustic equations. Afamous equation has been derived byLighthill (1952). This equation describes the evolution of the smallacoustic density fluctuation�′ = � − �∞ in a quiescent flow and reads

�2�′

�t2− c2 �2�′

�x2i

= �2

�xi�xjTij , (7)

wherec is the speed of sound andTij the Lighthill stress tensor which is given by the following relation:

Tij = �uiuj + 2�Sij + �ij (p′ − c2�′). (8)

HereSij denotes the strain rate in the fluid (see Eq. (3)), and�ij the Kronecker delta function. Eqs. (7) and(8) are an exact restatement of the continuity (1) and Navier–Stokes equations (2). For turbulent flows, atsufficient high Reynolds number, the contribution of the nonlinear term�uiuj will be much larger thanthe contribution of the dissipation�Sij and the entropy term�ij (p

′ − c2�′). So for a turbulent flow it canbe assumed thatTij is approximately equal to�uiuj . The fluctuation density distribution�′ is then givenby

�2�′

�t2− c2 �2�′

�x2i

= �2

�xi�xj�uiuj . (9)

The right-hand side of this equation is only significant in regions where there is a spatial non-uniformfluid motion. Outside this region�uiuj will be small and Eq. (9) reduces to an ordinary wave equation,which is valid for the propagation of sound in the linear approximation.

In the simplified form of the Lighthill equation (9) the fluid density� is occurring in the source term. Inincompressible flows� is assumed to be constant and the source term�uiuj can be rewritten as�0uiuj .Thus Eq. (9) becomes

�2�′

�t2− c2 �2�′

�x2i

= �0�2

�xi�xjuiuj . (10)

The validity of this assumption is one of the points we want to address in this study. We will do this bypresenting results for the acoustic field with both formulations for the Lighthill stress, i.e.,Tij = �uiuj(Eq. (9)) andTij = �0uiuj (Eq. (10)). The outcome of the Lighthill equation will be validated by a directmethod in which a homogeneous wave equation for�′ is solved outside the DNS domain subject toboundary conditions for�′ provided by the DNS.


2.3. Scaling of equations

All the variables in the equations above are made non-dimensional with help of the ambient speed ofsoundc∞ as reference velocity scale, the ambient density�∞ as reference density,�∞c2∞ as referencepressure andc2∞/Cp as reference temperature. This scaling will give a Reynolds number based on thespeed of sound. Such a Reynolds number is not very useful, because it is not related to the fluid motions.Therefore we will use, throughout this paper, the following definitions for the Reynolds and Mach number

Re = �U0D

�, Ma = U0

c∞, (11)

whereU0 is the jet orifice velocity andD= 2R0 the jet diameter. The total energy in dimensionless formis equal to

E = �T

�+ 1

2�uiui, (12)

where� = Cp/Cv is the specific heat ratio, which is usually equal to 1.4 which is also the value we willuse in this study. Using the fact thatR = Cp − Cv the equation of state becomes in a non-dimensionalform

p = �� − 1

�T . (13)

The Prandtl number which appears in the non-dimensional energy equation

Pr = Cp�

�, (14)

was taken equal to unity.

3. Numerical method

In the previous section we have presented the governing equation for the compressible flow, i.e., Eqs.(1)–(6). In this section we will describe how these equations are solved numerically.

A natural choice for the computation of a round jet would be to use a cylindrical coordinate system.In previous computational studies such a coordinate system has been used (Freund, 2001; Boersmaet al., 1998). However, in such a coordinate system a numerical problem arises when dealing with theartificial singularity at the centerline(r = 0) of the coordinate system. In the literature various methodsare discussed to circumvent these problems for a detailed overview we refer toMohensi and Colonius(2000). These methods are not able to retain a high order of numerical accuracy at the axis(r = 0) of thesystem. In physical space this axis will represent the jet centerline. An accurate simulation of the flow atthe centerline of the jet is necessary because we expect that in this area most of the sound will be producedand any numerical inaccuracies can lead to artificial sound sources. In view of the problems mentionedabove we have decided to use a Cartesian coordinate system for the complete flow domain.


The computational grid in the physical domain is non-uniform. Mapping functionsXi = �i(xi), withXi = i�X, are used to map differential equation on a uniform grid in the computational domain, i.e.,

�f

�x= �f

�X

�X

�x. (15)

The mapping functionXi = �i(xi) is chosen in such a way that�X/�x can be integrated analytically toobtain the physical distribution of the gridpointsxi . The derivative�f/�X has been calculated with a 6thorder compact finite difference scheme (Lele, 1992):

�f

�X

∣∣∣∣i

= f ′i ,

f ′i−1 + 3f ′

i + f ′i+1 = 7

3

fi+1 − fi−1

�X+ 1

12

fi+2 − fi−2

�X+ O(�X)6. (16)

At the boundaries of the computational domain the accuracy of the compact scheme was reduced to thirdorder (Lele, 1992). If we would have used a cylindrical system we would also have to reduce the order atthe jet centerline to third order. Which on its turn would give an unreliable prediction of the source termsin Eqs. (9) and or (10). The discretization given by Eq. (16) results in a tridiagonal system which can beinverted efficiently with the Thomas algorithm.

All the spatial derivatives in the continuity, momentum and energy equation are discretized with the6th order approximation given above. The time integration has been performed with a standard 4th orderRunge–Kutta method. The time step was fixed and the corresponding CFL number(ui�t/�xi) wasapproximately 0.5. It is well known that high-order compact finite differences can generate short wavenumerical instabilities. In this study these instabilities are removed by filtering the flow field with a sixthorder compact filter (Lele, 1992). The filter was applied once every acoustic time scaleR0/c∞ (onceevery 200 timesteps).

Due to the large memory and CPU requirements, the algorithm outlined above had to be implementedon parallel computing platforms. For the parallelization we have used two different data distributions.One distribution in which the computational grid in thex direction is distributed over the processors anda distribution in which thez direction is distributed over the processors, i.e., the number of grid pointsin thex andz direction has to be a multiple of the number of processors. The distribution inx is usedto calculate all the derivatives in they andz direction, and the distribution in thez direction is used tocalculate thexderivatives. The MPI routine MPI_ALLTOALL is used to switch from thexdistribution tothezdistribution and vice versa. The transfer of the data has to be performed during every Runge–Kuttasub step and takes approximately 10–15% of the total CPU time (depending on the number of CPUs).

3.1. Boundary conditions

A correct formulation of the boundary conditions is very important in computational aero acoustics(CAA). Small non-physical fluctuations introduced at boundaries will have almost no influence on thevelocity field but can have a very large influence on the acoustic pressure fluctuations which in low Machnumber flows are many orders of magnitude smaller than the pressure fluctuations generated by the flow.

At the inflow of the computational box we specify all the velocity components, density and energy.This can only be done in the case that the convection velocity in the equations is larger than the speed of


SoundPhysical domainSpunge layer

Fig. 1. A sketch of the geometry of the jet. The dashed line denotes the border between the physical part of the domain and thenonphysical part, i.e., the part whereA(x, y, z) andU(x) are non zero.

sound, i.e., the flow is supersonic and acoustic perturbations cannot travel upstream. This is accomplishedby adding a small (nonphysical) supersonic inflow zone to the domain.2 Such a layer has also been addedat the outflow boundary. At the lateral boundaries simple characteristic boundary conditions are used. Toavoid reflections, sponge layers are added on all sides of the domain. In these layers the solution of thegoverning equations is driven to a specified state by adding a source/sink term−A(x, y, z) · (�−�target)

to the equations. In this relation� is a flow variable�target its required value in the sponge layer andA(x, y, z) is a smooth function which is zero in the physical part of the domain and has a small positivevalue in the sponge layer. The value ofA depends on the size of the computational domain and also onthe time step of the numerical scheme (Fig. 1).

3.2. Wave equation

The Lighthill equation contains a second-order time derivative and a standard Runge–Kutta timeintegration method can not be used. So we rewrite the wave equation as a coupled set of first-orderpartial differential equations:

�q

�t− c2 �2�′

�x2i

= �2

�xixj�uiuj ,

��′

�t= q. (17)

For which we can use again standard Runge–Kutta methods. The spatial derivatives in the equation abovehave been calculated in a similar manner as for the momentum equations. The second derivative is first

2 The advective term is rewritten as�(U + ux)uj /�x − uxujdU/dx in which the advection velocityU(x) is a functionwhich is larger thanc∞ in a small zone close to the inflow and outflow and zero in elsewhere.


mapped on a uniform grid by applying the chain rule

�2f

�x2 = �2f

�X2

(�X

�x

)2

− �f

�X

(�X

�x

)3(�2X

�x2

)−2

. (18)

The second derivative is then evaluated with help of a 6th order compact finite difference scheme for thesecond derivative (Lele, 1992):

�2f

�X2

∣∣∣∣i

= f ′′i ,

2f ′′i−1 + 11f ′′

i + 2f ′′i+1 = 12

fi+1 − 2fi + fi−1

�X2 + 3

4

fi+2 − fi−2

�X2 + O(�X)6. (19)

On all sides of the computational domain for the wave equation we use a radiation boundary conditions(Goldstein, 1976):

��

�t= c

��

�xini, (20)

whereni is the unit normal on the boundary of the computational domain. It has been found that additionaldamping layers, as have been used in the solution of the Navier–Stokes equations, are not necessary forthe acoustic field.

The boundary value method which has been used to validate the Lighthill equation solves a simplifiedversion of Eq. (17):

�q

�t− c2 �2�′

�x2i

= 0,

��′

�t= q. (21)

The acoustic information is fed into the problem by the boundary conditions,

�′wave= �′

Navier–Stokes.

This method can only be used in region where the fluid velocity is negligible (Goldstein, 1976). Thisimplies that the bounding surface cannot be completely closed and some acoustic information cannot becalculated (see Figs. 11, 12).

To avoid short wave numerical instabilities in the solution of the Lighthill equation we had to filter thewave field once every few hundred timesteps. To be consistent with the filtering of the Navier–Stokesequations we did the filtering once every 200 timesteps. It turned out that filtering of the wave fieldobtained with the boundary value method was not necessary.

4. Results

In this section we will present results obtained from the direct numerical simulations. Different simu-lations have been performed (seeTable 1). In all cases the velocity profile at the jet inflow plane is given


Table 1The computational grids which have been used

Navier–Stokes Wave domain Re Ma Method

Nx × Ny × Nz 160× 144× 144 288× 256× 256 5000 0.6 Lighthill�uiujNx × Ny × Nz 160× 144× 144 288× 256× 256 5000 0.6 Lighthill�uiujNx × Ny × Nz 160× 144× 144 288× 256× 256 5000 0.6 Direct

Nx,Ny andNz denote the number of gridpoints inx, y andzdirection respectively.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Uc/

Ma c

r/D

Fig. 2. The velocity profile at the jet inflow, e.g. Eq. (22).

by the following equation (Michalke, 1984):

U(r) = Ma

(1

2− 1

2tanh[�(r − R0)]

), (22)

whereMa is the Mach number based on the centerline velocity,� is the momentum thickness andR0 theradius. In all our simulations we have usedRe = 5000, � = 20 andR0 = D/2 = 1

4. The correspondingvelocity profile is shown inFig. 2. On this profile small white noise random perturbations with a maximumamplitude of 0.001Ma are superimposed to trigger the transition of the jet to a turbulent state. It hasbeen found that the development of the jet is not very sensitive to the amplitude of the random numbers.The Reynolds number of 5000 is far above the Reynolds number at which the transition in a jet occurs.The Reynolds number at which transition occurs is not unique, it depends strongly on the shape of theinitial velocity profile and the conditions in the far field, but has in general values in the range 10..500.Our Reynolds number is at least one order higher than the transitional Reynolds number.


Fig. 3. The computational grid, consisting of 160× 144× 144 points in thex, y andzdirection respectively.

The computational grid (seeFig. 3) is strongly non uniform in theyandzdirection. The grid stretchingin thex direction is moderate.3 The acoustic domain is always larger than the Navier–Stokes domain.For the acoustic domain we use the same mapping functions as we have used for the Navier–Stokesdomain, i.e., every point of the Navier–Stokes domain corresponds with a point of the acoustic domain.In this way information from the Navier–Stokes domain can be transferred to the acoustic domain withoutinterpolation errors.

The computations have been started with as initial conditions a flow field at rest. After 180 acoustictime scales(R0/c∞) the flow field is developed and a start is made with the acoustic calculation. Thecalculation is continued over 80 acoustic timescales. During which 80 data fields, equally separated intime, are stored for further statistical analysis.

In Fig. 4 we show a contour plot of the instantaneous density� the instantaneous axial velocitycomponentux and the instantaneous vorticity magnitude multiplied by the axial coordinate. (We havemultiplied the vorticity withx to make the vorticity structure also visible for large values of the axialcoordinatex). We observe that in the first couple of diameters the flow is laminar, which is clearlydemonstrated by the plot of the axial velocity. Further downstream a helical instability starts to developwhich is clearly visible in the density field. The jet breaks up aroundx=15R0 and then becomes graduallyfully turbulent.

3 The mapping function,�Y/�y and�X/�x are given by the following relations (�Y/�y and�Z/�z are identical):

�Yj =[1 + 5

2

(t −

2

)]�yi, t = j

Ny,

�Xi = 22[0.9t2 − 0.6t + 0.7]�xi, t = i

Nx.


Fig. 4. The instantaneous density distribution in the jet (top), the instantaneous axial velocityux , and the magnitude of theinstantaneous vorticity multiplied with the axial coordinate,x|| (bottom).


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 50

u x,ρ

ux

x/D

uxρ ux

0

1

2

3

4

5

6

7

8

9

10

0 10 20 30 40 50

U0/

Uc

x/D

uxPL

HCG

Fig. 5. Left: the centerline velocityux and flux �ux as a function of the downstream coordinatex/R0. Right: the in-verse of the centerline velocity, together with the decay rates reported from the experiments ofPanchapakesan and Lumley(1993)andHussein et al. (1994).

4.1. Velocity statistics

It is a well known fact that the mean centerline velocity profile in a turbulent jet should decay linearlywith the distance to the virtual origin of the jet, see for instance (Tennekes and Lumley, 1972; Boersmaet al., 1998). The location of the virtual origin of the jet and to a lesser extend the decay constant, dependson the jet inflow conditions (Boersma et al., 1998). In Fig. 5 (left) the axial velocityux and the axialflux �ux along the jet centerline have been plotted as a function of the downstream coordinate. There isalmost no difference between the profiles forux and�ux indicating that the compressibility effects onthe turbulent statistics are very small.

In Fig. 5(right) we have plotted the inverse of the centerline velocity, i.e.,u−1x . In this figure we have

also included the curve fits obtained from the experiments ofPanchapakesan and Lumley (1993)andHussein et al. (1994). These experiments are performed at very low fluid velocities(Ma ≈ 0.01–0.02),but at higher Reynolds numbers. InFig. 5 (left) we have showed that compressibility effects in theDNS are very small. Therefore, we do not expect large differences between the experimental data atMa ≈ 0.01–0.02 and the DNS data with a Mach number of 0.6. Furthermore, a jet is a free shear flow(Tennekes and Lumley, 1972) and its dynamics are strongly dominated by the large flow scales. Theselarge scales mainly depend on the geometry and it is not expected that the value of the Reynolds number,which is a measure for the micro structure, will play an important role. This is confirmed by the rathergood agreement between the decay rate of the DNS and the experiments ofPanchapakesan and Lumley(1993)andHussein et al. (1994).

In Fig. 6 we show the mean axial velocity profiles at various downstream locations. The first twoprofiles are for downstream locations where the centerline velocity is still equal to the orifice velocity, seeFig. 5, and the velocity profiles are almost identical to the inflow profiles. The other three profiles havebeen taken at positions where the centerline velocity has dropped considerably and the shape of the profilesis completely different. For large values ofr/R0 a very small negative velocity is observed. This is causedby the boundary conditions atr= rmax which will not give the correct entrainment rate for the jet and as aresult of this a very small negative fluid motion close to the boundaries of the computational domain will


-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 2 4 6 8 10 12

Ux(

r/R

0)

x = 1.9 R0x = 12.5 R0x = 22.4 R0x = 32.7 R0

x = 46 R0

r/R0

Fig. 6. The mean axial velocity as a function of the radius for various different downstream locations.

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2

Ux(

η )

DNS

PL

η =r/(x-x0)

Fig. 7. The axial velocity profiles scaled with the centerline velocity, plotted as a function of the self similarity coordinate� = r/[x − x0]. The symbols denote the results of the experiment ofPanchapakesan and Lumley (1993).

develop. InFig. 7we show again the mean velocity profiles. However, this time the profiles are normalizedwith the centerline value and plotted as a function of the self-similarity coordinate� = r/[x − x0] wherex0 is the virtual origin of the jet. The virtual origin of the jet has been found by fitting the centerlinevelocity of Fig. 5a withB/[x − x0] over the rangex = 20..50R0 resulting in a decay constantB of 6.2(Panchapakesan and Lumley, 1993found 6.06) and a virtual originx0 of 9.2D (for more details about the


0.040.060.08

0.10.120.140.160.18

0.20.220.240.26

0 0.05 0.1 0.15 0.2

u′(η

)/U

c

η r/[x-x0]

DNSPL

0.040.060.08

0.10.120.140.160.18

0.20.22

0 0.05 0.1 0.15 0.2

v′(η

)/U

c

η r/[x-x0]

DNSPL

0.00150.00160.00170.00180.00190.002

0.00210.00220.00230.0024

0 0.05 0.1 0.15 0.2

ρ′ /

ρ 0DNS

η r/[x-x0]

Fig. 8. Top and Middle: the axial and azimuthal rms velocity profiles as a function of the self similarity coordinate�. Bottom: therms profiles of the density as a function of the self similarity coordinate�. The lines are obtained from the DNS and the symbolsfrom the paper ofPanchapakesan and Lumley (1993). (No density fluctuations are reported byPanchapakesan and Lumley(1993).

self similarity scaling of jet data we refer toBoersma et al. (1998)andLubbers et al. (1999)). In Fig. 7we have also included the experimental data given byPanchapakesan and Lumley (1993). The agreementbetween the DNS and experiment is satisfactory.

In Fig. 8we show the root mean square profiles of the axial and azimuthal velocity components in theself similar region of the jet. In these figures we have also included the experimental data reported byPanchapakesan and Lumley (1993). The agreement between the axial rms profiles from the simulationsand experiments is reasonable good. It seems that the DNS under predicts the axial rms somewhat, thisis most likely caused by the difference in Reynolds number between the DNS and the experiment. Theagreement between simulation and experiment for the azimuthal velocity components is very good. InFig. 8we also show the rms profile of the density fluctuations, normalized with the ambient density, asa function of the self-similarity coordinate. This figure shows clearly that the effect of compressibility inthe flow is very small(�′/�∞�0.25%). However, if we convert the density rms in acoustic quantities wefind very large levels(SPLmax = 144dB).

To summarize this section we can conclude that the simulation of the flowfield of the jet is in goodagreement with experimental data. This gives us a good starting point to study the acoustic field generatedby this jet.

4.2. Acoustic field

In Section 2.2 we have presented the governing equations for the acoustic field, i.e., Eq. (9) or (10). Inprinciple a closed form solution of this equation is possible (using Green’s functions) (see for instance(Goldstein, 1976; Mitchell et al., 1999) . In this study we decided to perform a full numerical simulation


Fig. 9. The dilatation(∇u) in the jet. The black box denotes the boundary of the physical part of the domain, i.e., inside the boxA(x, y, z) = 0 outside the boxA(x, y, z) = 0.

of the Lighthill equation. Because such a numerical technique can also be used to study more complicatedflow geometries than the present one.

The numerical method used to solve Eqs. (9) and (10) has been outlined in Section 3.2.From similarity argumentsGoldstein (1976)derives the following relation for the acoustic intensityI

I (r) ∝ K�0�

−4u′4l3

162c5((1 − Mc cos�)5r2). (23)

In whichK is a dimensionless constant,u′ andl′ are turbulent velocity and length scales,� a correlationtime Mc the convective Mach number, andr and� denote the position of the observer measured withrespect to the acoustic source which is assumed to be very close to the jet nozzle (� = 0 correspondswith the jet centerline). The convective Mach numberMc in Eq. (23) is approximately 0.62 times the jetcenterline velocity. IfMc is smaller than unity the maximum acoustic intensity according to relation (23)is always at the jet centerline(� = 0). This observation is not confirmed by experiments (Lush, 1971;Mollo-Christensen et al., 1964) and is due to the refraction of sound wave by the jet fluid. IfMc is largerthan unity there will be an angle� for which the denominator of Eq. (23) 1−Mc cos� ≈ 0, and the soundfield will have a distinct direction. In the present study we investigate low speed flows and the convectiveMach number will be approximately 0.6× 0.62= 0.37 and according to (23) most of the sound will thusbe emitted in the forward direction.

To start we show inFig. 9 a contour plot of the dilatation in the jet. The dilatation, i.e.,�ui/�xi isclosely related to the acoustic pressurep′ via (Colonius et al., 1997):

�p′

�t= −�∞ c2∞

�ui�xi

. (24)

Fig. 9shows that most of the sound is emitted from a region close to the jet centerline at axial locations10–20R0 downstream of the jet orifice. We also observe some high frequency sound which seems to begenerated at the jet orifice. This sound is due to the small random perturbations superimposed on the jet


15 21.371614 18.528613 15.685612 12.842611 9.9996410 7.156669 4.313688 1.470717 -1.372276 -4.215255 -7.058234 -9.901213 -12.74422 -15.58721 -18.4301

10 20

-10

-5

0

5

10

1515 1.6492514 1.4470213 1.2447912 1.0425611 0.84032910 0.6380999 0.4358698 0.2336387 0.0314086 -0.1708225 -0.3730534 -0.5752833 -0.7775132 -0.9797431 -1.18197

X/R0

30 40

10 20X/R0

30 40

Z/R

0

-10

-5

0

5

10

15

Z/R

0

Fig. 10. The double divergence of the Lighthill stressTij = �uiuj (top) and the double divergence of�uiuj − �0uiuj (bottom).

inflow profile (see Section 4). Next we show inFig. 10an instantaneous plot of the double divergenceof the Lighthill stress tensorTij = �uiuj , i.e., the acoustic source term in Eq. (9). In this figure also

the difference between the two formulations of the Lighthill tensor, i.e.,�2(�uiuj − �0uiuj )/�xi�xj isshown. As can be seen fromFig. 10the error made by the simplification of the right-hand side of Eq.(9), i.e., Eq. (10) is less than 5% based on the maximum contour levels shown inFig. 10. FromFig. 10it is again clear that most of the sound is produced in the region 10R0<x <20R0. This is the regionin which the axial velocity drops very fast (seeFig. 5). We also observe a small region with non-zerovalue of�2Tij /�xi�xj at the jet inflow plane. This is caused by the white noise perturbation which issuperimposed on the mean velocity profile, Eq. (22). The high frequency waves at the jet nozzle shownin Fig. 9are due to this source term.

Next we will present some results for the far field quantities. InFig. 11we show a contour plot ofq = ��′/�t (see Eq. (17)) obtained from the solution of the Lighthill equation and from the boundary


Fig. 11. A contour plot ofq = ��′/�t . The left figure is the result obtained from the solution of the Lighthill equation and theright figure the result from the direct method.

Fig. 12. A contour plot ofp′. The left figure is the result obtained from the solution of the Lighthill equation and the right figurefrom the result from the direct method.

value method. (Due to the open bounding surface no sound field can be predicted in the gray area). InFig. 11we observe distinct spherical waves which all seem to emit from the region where the potentialcore of the jet collapses(x ≈ 10−20R0). The agreement between the two methods seems to be reasonablein the area where both methods are valid.

In Fig. 12the contours ofp′ obtained from the Lighthill equation and the direct method are shown.The direct method (right) shows distinct sound waves emitting form the region where the main acousticsources are located (seeFig. 10). The result of the Lighthill equation looks completely different, especiallyin the region close to the jet nozzle. This is caused by our assumption that the dissipation and entropyterms (2�Sij and�ij (p

′ − c2�′)) in Eq. (8) can be neglected. These assumptions will be valid in thefar-field of the jet but not in the near field. We will come back to this issue in Section 4.3.

In Fig. 13we have plotted the acoustic pressure fluctuationp′, obtained from the direct method, ata distance of 80R0 for different angles�, � = 0 corresponds to the jet axis. In this figure we have alsoincluded the experimental result given byLush (1971)rescaled to the same distance. It should be noted


70

75

80

85

90

30 40 50 60 70 80 90 100 110

dB

angle

Lush (1971)

DNS

Fig. 13. The acoustic pressure fluctuationsp′ = �′�∞c2∞ as a function of the angle with the jet axis. The line has been obtainedfrom the DNS and the symbols fromLush (1971).

that the DNS data is not very reliable for small angles(<30..40◦) due to the open bounding surfaces.With this in mind the agreement between simulation and experiment is reasonable good.

4.3. Time series and spectra

In the previous section we have presented some contour plots of�′obtained from different computationalmethods.The agreement between the direct method and the Lighthill equation did not look very promising.Which could be an indication that the Lighthill equation cannot be used in the case where there is a nonnegligible, viscous dissipation, fluid velocity or heat production. In this section we will compare timeseries and spectra of the acoustic date. Unless otherwise stated the results have been obtained with�uiujas source term in the Lighthill equation.

To start we show inFig. 14 time series ofp′ = �′�∞c2∞ at the positions(x, y) = (36R0,48R0),(48R0,62R0), (64R0,62R0). The acoustic calculation is started att = 0 and it takes some time beforethe acoustic disturbances reaches the measuring points, that is why there is no signal fort <20R0/c.There is a clear vertical shift in the mean levels of�′ (see alsoFig. 12).

This shift is due to the absence ofp′ −c2�′ in the formulation of the Lighthill stress used in Eqs. (9) and(10). By neglecting this term in the Lighthill equation we ignore effects related to a non-constant speedof soundc, convection and refraction effects and deviations from an isentropic behavior. We have chosenfor this approximation of the Lighthill term because in a typical incompressible acoustic calculation thetermp′ − c2�′ is not present. For an ideal gas, as used in this study, the speed of soundc is proportionalto the square root of the temperatureT . FromFig. 4 it can be seen that the density in the source regionis somewhat lower than the reference state, i.e., the temperature is somewhat above the reference state(this is of course due to the viscous heat production). This implies thatc in the source region (seeFig. 10) is slightly higher than in the other parts of the flow and acoustic waves will have a somewhat


-0.0004

-0.00035

-0.0003

-0.00025

-0.0002

-0.00015

-1e-04

-5e-05

0

5e-05

0.0001

10 20 30 40 50 60 70 80

Lighthilldirect

-0.00035

-0.0003

-0.00025

-0.0002

-0.00015

-1e-04

-5e-05

0

5e-05

10 20 30 40 50 60 70 80

Lighthilldirect

-0.00035

-0.0003

-0.00025

-0.0002

-0.00015

-1e-04

-5e-05

0

5e-05

10 20 30 40 50 60 70 80

ρ′ -

ρ0

ρ′ -

ρ0

ρ′ -

ρ0

Lighthilldirect

time (R0/c)

time (R0/c)

time (R0/c)

Fig. 14. Time series ofp′ = �′�∞c2∞ obtained from the Lighthill equation and the direct method at positions(x, y) = (36,48), (48,62), (64,62).


1e-15

1e-14

1e-13

1e-12

1e-11

1e-10

0.1 1

St

Lighthilldirect

1e-16

1e-15

1e-14

1e-13

1e-12

1e-11

1e-10

0.1 1

St

Lighthilldirect

1e-15

1e-14

1e-13

1e-12

1e-11

1e-10

0.1 1

St

Lighthilldirect

M-C

Fig. 15. The power spectra of the time series shown inFig. 14. The dots in the lower figure are the experimental results ofMollo-Christensen.


-0.0006

-0.0005

-0.0004

-0.0003

-0.0002

-0.0001

0

1e-04

0.0002

0.0003

0.0004

10 15 20 25 30 35

ρ′-ρ

0

time (R0/c)

III

Fig. 16. Time series of the density at(x, y) = (36R0,48R0) with as source term in the Lighthill equation�uiuj and�0uiuj(Eqs. (9) and (10)).

larger speed. In the formulation of the Lighthill equation (9) this effect is absent. Furthermore, in thesource region there will be some convection of sound waves by the jet flow. This effect is also not present inour formulation of the Lighthill equation. We have taken the source term in Eq. (9) from a compressibleflow field in which these effects are taken into account. In the compressible flowfield there will be a“balance” between�uiuj andp′ − c2�′. Neglectingp′ − c2p′ will thus lead to an “imbalance” in theright-hand side of the wave equation, which causes the very small vertical shift in theFig. 14. However,as shown inFig. 14, the frequency and amplitude of the acoustic density fluctuations is qualitatively thesame for both methods. Which could indicate that the sound obtained with both methods is not verydifferent. InFig. 15we show the ensemble averaged power spectra obtained from the time series shownin Fig. 14(40< t <48,48< t <56,56< t <64,64< t <72).

Fig. 15shows clearly that the acoustic field obtained with the direct method and with the Lighthillequation is very similar. Which is a clear indication that our way of using the Lighthill equation is avalid way of calculating jet noise. InFig. 15we have also included the far field spectra ofp′ reported byMollo-Christensen et al. (1964). Where only the voltage output of the microphone have been reportedso nothing can be said about the amplitude of the sound. The peak frequency of the experiment ofMollo-Christensen et al. (1964)and our simulation is in good agreement(St = fD/U = 0.2).

Finally in Fig. 16we show the time series of�′ at (x, y) = (36R0,48R0) obtained from the solutionof Eqs. (9) and (10). There is no noticeable difference between both signals presented inFig. 16whichshows that�0uiuj is a valid source term in the Lighthill equation.

5. Conclusion

In this paper we have used DNS to obtain the flowfield of a round turbulent jet. It has been shownthat the flow field of the jet is in good agreement with experimental data. The corresponding sound field


produced by this jet has been obtained with help of the Lighthill equation and with a boundary valuemethod in which a homogeneous wave equation is solved, using boundary value information obtainedfrom the numerical solution of the Navier–Stokes equations. Two different formulations for the Lighthillstress tensorTij , have been used, one in which the density fluctuations have been taken into account,i.e., Tij = �uiuj and one in which the density fluctuations have been ignored, i.e.,Tij = �0uiuj . Allmethods give similar results. The dominant acoustic frequency and the directivity patterns are in goodagreement with the experimental results reported byLush (1971)and byMollo-Christensen (1967). Thesimplification of the Lighthill tensor�uiuj → �0uiuj gives no noticeable difference in the acoustic field.This implies that we can use incompressible codes in whichuiuj is available to predict jet noise.

Acknowledgements

The author gratefully acknowledges the Royal Dutch Academy of Science and Arts (KNAW) for theirfinancial support. The large amount of computing time on the SGI ORIGIN-3800 was made available tothe author by the Dutch supercomputing foundation (NCF).

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numerical simulation of the noise generated by a low mach number, low reynolds number jet

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