Page 1

Influence of Temperature Gradients on the Sound

radiated from Flames

R. Piscoya and M. Ochmann

Technische Fachhochschule Berlin, Univ. of Applied Sciences, Luxemburger Str. 10, 13353

Berlin, Germany

piscoya@tfh-berlin.de

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The far field pressure of a turbulent flame can be determined using the standard boundary element method

(BEM) if the sound pressure or its derivative is known at a closed surface (control surface) surrounding the

flame, as long as the medium outside the control surface is homogeneous. If temperature gradients are present,

the homogeneous Helmholtz equation is no more valid. In that case, the wave equation can be rewritten in form

of an inhomogeneous Helmholtz equation with a source term that depends also on the unknown pressure. Using

the “Dual Reciprocity BEM” the integral form of this wave equation can be solved involving only surface

integrals, so that the sound field can still be computed from field values at the control surface. The cases under

study consider a volume of hot gas with a temperature distribution that is prescribed or obtained from a CFD

simulation. The influence of the temperature gradients on the sound field can be evaluated by comparison of

characteristic quantities like sound power and radiation patterns, with and without temperature gradient.

When the medium outside S is homogeneous (Fig. 1a), the

sound field can be obtained by applying the standard BEM.

Thus, the sound pressure at all points outside S is given by

[4]

?

?

???

1

Introduction

∫

S

+

∂

∂

=

n

y dSyxgyvj

yn

y

)

x

(

g

ypxpxC)(),()(

),

?

(

)()()(

0

????

ωρ

(1)

The prediction of the sound radiated from a turbulent flame

is a very difficult task and cannot be handled efficiently

with one method alone because of the disparity in time and

length scales of the sound production and the sound

propagation.

In previous works [1-3], a hybrid approach combining a

Large Eddy Simulation (LES) with a Boundary Element

Method (BEM) was implemented and used to compute the

sound radiation of open turbulent flames. A key condition

for the validity of this approach is to place the Kirchhoff

surface in a homogeneous medium.

In many cases, it may be difficult to ensure that the

Kirchhoff surface lies in a homogeneous medium.

However, a large source region would demand a large

computational domain leading to high computational effort.

For these cases, the presence of an inhomogeneous medium

has to be taken into account.

In the present work, the propagation of sound waves in an

inhomogeneous medium is studied using an extension of

the BEM, namely the Dual Reciprocity BEM (DRBEM).

with the normal vector pointing to the outside,

yx

e

4

yx

y

?

x jk

?

?

?

G

??

−

=

−−

π

),(

0

and C

. (2)

∈

∈=

V x

Sx

V

x

0

5 . 0

outside

?

1

)(

?

?

When the medium outside S is not homogeneous, Eq. (1) is

no more valid. But a similar expression can be deduced if

the inhomogenities are written in form of a source

distribution q, as it will be shown next.

Here, it is assumed that the inhomogeneous region occupies

only a volume Ω and the rest of the fluid is homogeneous

(see Fig. 1b). In this region, the density ρ and sound speed

c may vary locally and differ from the ambient values

ρ , .

00 c

To find the sound field outside S, the exterior space is

divided in two regions and S is subdivided in two surfaces

S0 and S1 as shown in Fig. 2. Region I is given by the

volume Ω and limited by the surfaces S2 and S1. Region II

is the homogeneous zone outside S0∪S2.

2

Sound radiation of the flame

For the calculation of the radiated sound of a turbulent

flame, it is assumed that at least one acoustic quantity, for

example pressure p or particle velocity vn is known at a

closed surface S that encloses the flame (see Fig. 1).

Fig. 2 Surfaces and domains for the sound field

determination.

Two differential equations have to be solved

(

(

+∇

)

)

IIRegion 0

I Region

2

0

2

2

0

2

=

=+∇

II

I

pk

qpk

(3)

Fig.1 Models for an open flame; a) in a homogeneous

medium and b) with a non homogeneous zone.

with the boundary conditions

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10

0

2

0

2

on

1

on

11

ρ

on

SSv

n

p

∂

S

n

p

∂

n

p

Spp

nS

II

III

III

∪=

∂

∂

=

∂

∂

=

ρ

ρ

, (4)

where vnS is the known particle velocity.

The integral equation for region II is given by Eq. (1) with

the integration surface being S0∪S2. The integral equation

for region I is given by [5]

. )

y

(),()(

)(),()(

)(

),

y

?

(

n

)()()(

21

∫

Ω

∫

∪

−

+

∂

∂

−=

dVyxgyq

y dSyxgyvj

yxg

ypxpxC

SS

nII

???

????

?

??

???

ωρ

(5)

Eq. (5) has an additional term with respect to Eq. (1), which

corresponds to a volume integral over the source

distribution q. To avoid the computation of the volume

integral, the Dual Reciprocity method is applied to replace

the volume integral with a series of surface integrals. This

substitution is accomplished by expanding the source

distribution in a set of functions fj

∑

j

=

jjfq

α

(6)

which are associated to another set of functions ψj through

the inhomogeneous Helmholtz equation

(

k

+∇

)

ψ

jj

f

=

22

. (7)

After applying the BE procedure to Eq. (7) and inserting the

results together with Eq. (6) into Eq. (5), the final

expression for the sound pressure in region I is

∂

−=

∪

SS

n

21

∑

j

∫

∪

∫

∂

∂

−

∂

∂

+

+

+

∂

SS

j

n

jjIj

nIIII

dSyxg

y

n

yxg

yxxC

dSyxgyvj

yxg

ypxpxC

21

),(

)(

),(

)()()

ψ

(

),()(

),(

)()()(

,

??

?

??

???

???

??

??

ψ

ψα

ρω

(8)

with

Ω

∪∈

Ω∈

S

=

outside0

5 . 0

1

)(

21

Sx

x

xCI

?

?

?

.

Assuming that the source term q is known at some points of

the volume, the coefficients αj can be determined.

Considering N points at the surface S1∪S2 and L points

inside the volume Ω, and truncating the series (6) at

M=N+L terms, M coefficients αj can be computed by

solving the matrix equation:

bF

1

−

=α

. (9)

Here α and b are vectors with M components and F is a

M×M matrix.

The discretization of Eq. (8) leads to the following matrix

equation

α

ψ

∂

n

ψψ

I

ρω

j

∂

−+=++

GHCGvHppC

S

SS

In

S

III

. (10)

If the discretization is made at the same N points at the

surface and L points inside the volume, and the boundary

conditions of Eq. (4) are taken into account, the pressure at

the discretization points can be obtained.

3 Temperature gradient

In this work, the inhomogeneous region will be considered

to have a local temperature distribution which is constant in

time.

Since the sound speed and the density depend on the

temperature, the wave equation becomes [6]

(

2

∇⋅∇

pc

c

)

0

1

22

=+

II

pk

(11)

By inserting the relation for perfect gases c2=γ RT in Eq.

(11), with γ and R constant, we get

0

22

=

∇⋅

T

∇

++∇

pT

pkp

I

II

. (12)

The wave number k=ω/c depends also on the temperature.

Adding and subtracting the term

terms we obtain:

and rearranging the

I pk2

0

()

T

pT

pkkqqpkp

I

III

∇⋅∇

−−==+∇

22

0

2

0

2

,

(13)

Eq. (13) shows that the source term q contains the

derivatives of the unknown variable pI. In this case, pI has

to be expanded in a series of functions dj in a similar way as

was performed for q, so that its derivatives can be defined

in terms of the derivatives of the known functions dj:

∑

j

∑

j

∇=∇→=

jjIjjI

xdxpxdxp)()( )()(

????

ββ

. (14)

4 Numerical example

The procedure described in section 3 for an open flame can

be very well applied to study the sound radiation of a semi

closed flame. It is assumed that the flame is placed inside a

cylindrical combustion chamber that has one open end. The

sound waves coming out through the opening are

characterized by the velocity at the opening. On the other

hand, the sound waves in the chamber may induce

vibrations of the chamber walls that radiate sound to the

outside and contribute to the total emitted sound.

Fig. 3 Velocity distribution at the cylindrical surface.

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In our example, the chamber walls are assumed to be rigid,

so that no vibrations arise. Thus, the normal velocity at the

walls is zero. At the open end, a radial velocity distribution

is considered (see Fig. 3). Outside the combustion chamber,

next to the open end, a temperature distribution

3

j

2

j

0

4

0

2

0

1

?

))cos(

r

1 (

2

1

,

−

1

+

j

jj

j

jj

rd

rk

kk

r

rf

+=

−=

+=

ψ

(16)

()

−

+

=

xx

zyA

T x,y,zT

m

0

22

- exp)(

µ

(15)

where

the surface or interior point y?.

jj

yxr

?−=

is the distance from the field point x? to

in a region of length LT is prescribed. In Eq. (15), A and x0

are constants, µ=ln(Tm/ Ta) and Tm and Ta are the maximum

and ambient temperature respectively.

The effect of the temperature distribution was studied by

varying Tm and LT. Three different values of LT were

considered: 0.7R, 1.4R and 2R. The maximum temperatures

investigated were 50°C, 100°C, 200°C, 300°C, 400°C and

500°C. In Fig. 4, the temperature distribution is shown for

two values of Tm.

In a previous work [7], these functions were tested using a

“spherical flame” with different source distributions. For

some specific type of sources, the problem has an analytical

solution. The numerical solutions showed very good

agreement with the analytical ones.

The results of the calculations are presented in figures 6 –

9. In first place, we analyze the effect of the temperature

distribution on the sound power. The sound power increases

at low frequencies and reaches some approximately

constant value at high frequencies. This constant value

decreases uniformly with increasing Tm. This effect could

be explained considering that more energy is reflected back

into the hot region if the temperature is higher.

Fig. 4 Temperature distribution.

The variation of the sound speed and the density with the

temperature was obtained by using the relations:

00336 . 1

−

)( 77819. 360 , )( 05. 20

°=°=

KTKTc

ρ

.

Fig. 6 Dependence of the sound power with the maximum

temperature.

For the numerical computation of the sound field, the

combustion chamber was modeled with a cylinder of length

0.5 m and a radius of 0.22 m. The cylinder had 768

elements. The inhomogeneous region was limited by a

paraboloid of revolution. For LT=0.7R the surface had 224

elements, for LT=4R, the surface had 800 elements. The

number of interior points for the approximation of the

source term was 200 for the shorter region and 500 for the

larger one. The surface models and interior points are

shown in Fig. 5.

The length of the hot region appears to have less influence

on the sound power than the temperature itself. In Fig. 7,

the sound power for Tm=773°K and three different values of

LT is shown. The differences can be seen principally at high

frequencies and they are small.

Fig. 5 Surface models and interior points (LT=2R).

The approximation functions fj and associated functions ψj

that were used are given by

Fig. 7 Dependence of the sound power with the length of

the hot region.

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References

The influence of the temperature on the radiation patterns is

illustrated in Fig. 8. The polar plots show the sound

pressure level normalized to the maximum value at a

spherical surface with a radius of 100 m. It can be observed

that the higher the temperature, the broader the radiation

patterns become. This effect is caused by the refraction of

the sound waves in the hot region due to the variable sound

speed.

[1] H. Brick , R. Piscoya, M. Ochmann, P. Költzsch,

"Prediction of the Sound Radiated from Open Flames

by Coupling a Large Eddy Simulation and a Kirchhoff-

Method ", Proc. Forum Acusticum, Budapest (2005).

[2] R. Piscoya, H. Brick , M. Ochmann, P. Költzsch,

"Application of equivalent sources to the determination

of the sound radiation from flames ", Proc. 13th

International Congress on Sound and Vibration, Viena

(2006).

[3] R. Piscoya, H. Brick, M. Ochmann, P. Költzsch,

"Equivalent Source Method and Boundary Element

Method for calculating combustion noise", Acta

Acustica united with Acustica, accepted for publication,

(2008).

[4] M. Ochmann, "Analytical und Numerical Methods in

Acoustics" in Mechel. F.P.: Formulas of Acoustics,

930-1026, Springer, (2002).

Fig. 8 Dependence of the radiation pattern with the

maximum temperature.

[5] L. C. Wrobel, The boundary element method - Vol. 1:

Applications in thermo-fluids and acoustics, Wiley,

(2002).

The radiation patterns appear to be more sensitive to the

length of the hot region than the sound power. The larger

the hot region, the broader the radiation pattern becomes.

This effect is shown in Fig. 9. The polar plots are calculated

for two frequencies for Tm=773°K.

[6] S. W. Rienstra, A. Hirschberg, An Introduction to

Acoustics, Eindhoven University of Technology,

(2004).

[7] R. Piscoya, M. Ochmann, " Sound propagation in a

region of hot gas using the DRBEM ", Proceedings

14th International Congress on Sound and Vibration

(ICSV14), Cairns, Australia (2007).

[8] Combustion

http://www.combustion-noise.de

Noise Initiative, URL:

Fig. 9 Dependence of the radiation pattern with the length

of the hot region.

5 Conclusion

The sound propagation in an inhomogeneous medium can

be treated by using the Dual Reciprocity BEM if the

differential equation can be written as an inhomogeneous

Helmholtz equation with source terms appearing at the right

hand side. This approach was applied to study the

propagation of sound waves coming from a combustion

chamber in the presence of a hot region with temperature

gradient at the chamber exit. The sound waves are refracted

away from the axis producing broader radiation patterns

and partially reflected leading to a decrease of the radiated

power.

Acknowledgments

This work is integrated in the research unit “Combustion

Noise Initiative”, supported by the German Research

Foundation (DFG) [8].

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