NASA/CR--2002-212013 AIAA-2003-0365

Near Field Trailing Edge

Noise Computation

Ching Y. Loh

Taitech, Inc., Cleveland, Ohio

Tone

December 2002
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NASA/CR--2002-212013 AIAA-2003-0365

Near Field Trailing Edge

Noise Computation

Tone

Ching Y. Loh

Taitech, Inc., Cleveland, Ohio

Prepared for the

41st Aerospace Sciences Meeting and Exhibit

sponsored by the American Institute of Aeronautics and Astronautics

Reno, Nevada, January 6-9, 2003

Prepared under Contract NAS3-97186

National Aeronautics and

Space Administration

Glenn Research Center

December 2002
Acknowledgments

This work received support from the Supersonic Propulsion Technology Project Office at NASA Glenn Research

Center. The author is grateful to Dr. Richard A. Blech and Dr. James Scott for carefully reviewing the paper.

This report contains preliminary

findings, subject to revision as

analysis proceeds.

This report is a preprint of a paper intended for presentation at a conference. Because

of changes that may be made before formal publication, this preprint is made

available with the understanding that it will not be cited or reproduced without the

permission of the author.

The Aerospace Propulsion and Power Program at

NASA Glenn Research Center sponsored this work.

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

Available from

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Available electronically at _sa. _ov
NEAR FIELD TRAILING EDGE TONE NOISE COMPUTATION

Ching Y. Loh*

Taitech, Inc.

Cleveland, Ohio 44135, USA

e-mail: fsloh@turbot.grc.nasa.gov

Abstract

Blunt trailing edges in a flow often generate tone noise due to wall-jet shear layer and

vortex shedding. In this paper, the spacetime conservation element and solution element (CE/SE) method is employed to numerically study the near-field noise of blunt trailing

edges. Two typical cases, namely, flow past a

circular cylinder (aeolian noise problem ) and

flow past a flat plate of finite thickness are considered. The computed frequencies and the estimated sound pressure level (SPL) compare

well with experimental data. For the aeolian

noise problem, comparisons with the results of

other numerical approaches are also presented.

1 Introduction

In today's aviation industry, everyday, thousands of

commercial airliners take off and land at the airports

around the world, accompanied by various uncomfortable noises, polluting the environment of the nearby areas. The noises due to airframe components, e.g. airfoil

slat, landing gear and opeings, etc. are one of the major

noise sources. They are generated by the interaction of

the vortex street in the turbulent wake. A blunt trailing

edge often generates certain tone noise in the flow. Numerical investigations of airframe noise based on direct

numerical simulation (DNS) need very fine grids and are

hence expensive. Large eddy simulations (LES) allow

a coarser grid since only large eddies are resolved and

the smaller ones are modeled by a subgrid scale (SGS)

model. Furthermore, some researchers use Reynolds

averaged Navier-Stokes (RANS) methods to save more

storage and CPU time [ 1-5].

In the present paper, we use a DNS type approach to

investigate two benchmark noise problems, namely, the

noise generated by the flow past a circular cylinder (aeolian noise) and the noise generated by flow past a flat

plate of finite thickness. Both may be categorized as

*Taitech,Inc., MemberAIAA

blunt trailing edge noise. By 'DNS type', we mean that

the grid may not be fine enough for the Kolmogorov scale

of turbulence but is fine enough for the large turbulence

structure (or large eddies) associated with the tone noise

frequencies. This approach seems somewhat similar to

LES but there is no explicit filtering or SGS model. More

details are given later.

The space-time conservation element and solution element method (CE/SE) [6-8] is adopted for the computation. As demonstrated in our previous papers, the

CE/SE scheme is well suited for aeroacoustics computation [9,10]. Because of the CE/SE non-reflecting boundary conditions (NRBC), which are based on flux balance

[ 11], a smaller near field computational domain can be

used in the present numerical simulation and helps to

save both memory and CPU time.

The 2-D unstructured-grid Navier-Stokes iN-S)

CE/SE scheme used here is briefly discussed in Section

2. Section 3 illustrates the initial and boundary conditions for these blunt trailing edge noise problems, in particular, the CE/SE NRBC. The numerical results are

presented and compared to available experimental

findings [12,15] in Section 4. Conclusions are drawn

in Section 5.

2 The Numerical Scheme

The space-time conservation element and solution element (CE/SE) method is a low order (second order accurate in time and space) scheme with low dissipation. It is

chosen in the present numerical investigation. A detailed

description of the method can be found in [6- 8]. In this

Section, only a brief sketch of the scheme is given.

2.1 Conservation Form of the 2-D Unsteady

Compressible Navier-Stokes Equations

In general, the CE/SE method systematically solves a

set of integral equations derived directly from the physical conservation laws and naturally captures shocks and

other discontinuities in the flow. Both conservative variables and their derivatives are computed simultaneously

as unknowns.

NASA/CR 2002-212013 1

American Institute of Aeronautics and Astronautics
Consider adimensionless conservation frmoftheunsteady2-DNavier-Stokes equationsofaperfectgas.Let

p, u, v, p, and "7be the density, streamwise velocity component, transverse velocity component, static pressure,

and constant specific heat ratio, respectively. The 2-D

Navier-Stokes equations then can be written in the following vector form:

Ut + F. + Gy = O, (1)

where x, y, and t are the streamwise and transverse coordinates and time, respectively. The conservative flow

variable vector U and the flux vectors in the streamwise

and radial directions, F and G, are given by:

U= U2 F= F2 G= G2

0-3 ' F3 ' G3 '

U4 F4 G4

with

Ul = p, U2 = pu, U3 = pv,

u4 = p/(9`- 1)+ p(_2+ v2)/2.

The flux vectors are further split into inviscid and viscous

fluxe s:

P = Fi - Fv, G = Gi - Gv,

where the inviscid fluxes are the same as in the Euler

equations:

F_I= Y2,

Fi2 = (7 --1)[/-4+ [(3--7)U @ - (7 - I)U2] /2U1,

Fi3 = U2U3/Ul,

F_ = 9`u2u_/ui - (9`- 1)u2 [v_+ u_]/2u_,

Gil = U3, Gi2 = U2U3/U1,

Gi3 = (9` -- 1)0-4 + [(3 -- 9`)U32- (9' - 1)U22]/2171,

a_ = 9`v_v_/u1- (9`- 1)u_ [v_+ u_]/2U_l,

and the viscous fluxes are:

Fvl = O, Fv2 = #(2_tx - 2V,V),

Fv3= _(vx+ u_),

r,_ = ,[2_x + (_ + _x)v- _(V.V)_+

9` 0 (U4 U2-'_-V 2

Pr Oy U1 2 )]'

Gvl = O, Gv2 = #(Vx + Uy),

av3 = _(2v_- _V.V),

a_ = ,[2vv_ + % + vx)_ - _(V.V)v+

C_

F' E'

I ',\["

It !

[ I ' II I

IF l!.C lIE I

/"

time level n -_

hexagoncylinder ADBECFA- A'D'B'E'C'F'A'

wilhils3 CEEOADB-O'AD'B', OBEC-O'B'E'C',

andOCFA- O'C'F'A'

B_

B

Figure 1: CE/SE unstructured grid in space-time E3

space.

9` 2.(v_ _._+v_)],

Pr Oy U1 2

where u, v, u., Uy, v., Vy are respectively the x- and yflow velocity components and their derivatives, which

can be written in terms of the conservative variables

U1, U2, 0-3 and U4, with Pr (= 0.72) being the Prandtl

number, p the viscosity, the velocity divergence

V.V = u_ + %.

By considering (x,y,t) as coordinates of a threedimensional Euclidean space, E3, and using Gauss' divergence theorem, it follows that Eq. (1) is equivalent to

the following integral conservation law:

._ H m • dS = = (2) 0, m 1,2,3,4,

(v)

where S(V) denotes the surface around a volume V in

E3 andHm = (F,_,G_,U,_).

2.2 Unstructured Grid for CE/SE

The CE/SE scheme is constructed to take advantage of

an unstructured triangle grid. The unstructured geometry used with the CE/SE scheme is illustrated in Fig. 1.

Here, AABC is a typical triangle cell and D, E, F are

the triangle centers of the neighboring cells. The flow

variables at the previous time step are stored at the triangle cell centers. Three quadrilateral cylinders (conservation elements) are formed by the edges that connect the

NASA/C_2002-212013 2

American Institute of Aeronautics and Astronautics
verticesandthecenterofthetriangleanditsthreeneighbors.Inthespace-time E3 space, Eq. (2) is applied to

the hexagon cylinder ADBECF - 'el DBE' C'F' of

volume V that consists of these 3 quadrilateral cylinder

CEs, see Fig 1.

In the CE/SE scheme, the above flux conservation relation, Eq. (2), in space-time is the only mechanism that

transfers information between node points. A conservation element CE (here, quadrilateral cylinders) is the finite volume to which Eq. (2) is applied. Discontinuities

are allowed to occur in a conservation element. A solution element SE associated with a grid node (e.g., D,

E, or F in Fig. 1) is here a set of interface planes in Ea

that passes through this node (e.g. D_A D', DHB D',

EI_B E', EUC E', etc.). Each surface S(CE) is made

up of segments belonging to two neighboring CE's.

Within a given solution element SE(j, n), where j, n are

the node index and time step, respectively, the flow variables are not only considered continuous but are also approximated by the linear Taylor series expansions. The

surface flux can be calculated accurately and easily by

first evaluating the flux vectors at the geometrical center of the surface through the above Taylor series expansions.

At time level n, the solution variables U, Ux, and

Uv are given at the three nodes D,E,F in Fig. 1 and

U, Ux and Uy at O' at the new time level n + 1 are

to be computed. In principle, each of the 3 CEs provides 4 scalar equations when Eq. (2) is applied to the element. Hence, the 12 scalar equations needed for the 12

scalar unknowns at O' are available. All the unknowns

are computed based on these relations. No extrapolations

(interpolations) across a stencil of cells are needed.

More details about the unstructured CE/SE method

can be found in [8].

3 The Trailing Edge Noise Problems

It is well known that when the trailing edge of a body

is sharp, the turbulent wake generates broadband noise,

while a blunt trailing edge produces certain tone noises

due to the presence of wall-jets and the consequent large

scale turblulence. Typical blunt trailing edges include,

e.g., the flat plate of finite thickness and the circular

cylinder. They are investigated separately in this section.

3.1 The Aeolian Noise Problem

Aeolian noises, or the sounds generated by flow past a

circular cylinder, are closely related to airframe noise

and noise in automobiles, e.g. noise induced by autoantenna. As it is important, the problem is chosen as one

of the benchmark problems in CAA Workshop II [1].

Consider a Mach number M = 0.2 uniform flow past

a 2-D circular cylinder of diameter D = 1.9cm. Fig. 2

shows a sketch of the computational domain and the triangulated grid. As shown in the figure, the grid is obtained by dividing a rectangular cell into 4 triangles. By

so doing, the advantage is that the cell size, which is important in aeroacoustics computation, is well-controlled.

The diameter D of the circular cylinder, the ambient

speed of sound ao and the ambient flow density are chosen as the scales of length, velocity and density respectively.

top NRBC

bottom NRBC

Figure 2: Geometry of the computational domain and

unstructured grid,-2.1D _< x _< 5.5D,-4D _< y _<

4D; 7 exponentially growing rectangular cells (as sponge

zones) at the top and bottom are not shown.

3.1.1 Initial Conditions Initially, the flow of the

entire domain is set at the ambient conditions, i.e.,

po=l.0, uo=0, vo=0, po=1/%

All spatial derivatives are set to zero.

3.1.2 Non-Reflecting Boundary Condition

(NRBC) for CE/SE In the CE/SE scheme, NRBCs

are constructed so as to allow fluxes from the interior

domain to a boundary CE smoothly exit to the exterior

of the domain. There are various variants of the NRBCs

[9-11 ], the following are the ones employed in this paper.

For a grid node (j, R) lying at the outer border of the

domain, where j is the grid node index number and n the

time step, the Type I NRBC requires that

= = o,

while U_ is kept fixed at the initially given steady boundary value. At the outflow boundary, where there are

NASA/C_2002-212013 3

American Institute of Aeronautics and Astronautics
substantial gradientsintheradialdirection, theTypeII

NRBCrequiresthat

(vx)} = o,

while U} _and (Us)} _are now defined by simple extrapolation from the nearest interior node jr, i.e.,

Ujn = U j,n--1 (Uy);_ = _-1 .

As will be observed later, these NRBCs are robust

enough to allow a near field computation without disturbing or distorting the flow and acoustic fields.

3.1.3 Boundary Conditions At the inflow

boundary, the flow variables are specified as the M = 0.2

uniform flow:

pi=l.0, ui=0.2, vi=0, pi=1/7,

and their spatial derivatives are conveniently set to zero.

Note that these inlet conditions also play a role as absorbing boundary conditions in the CE/SE scheme. At

the surface of the circular cylinder, no-slip wall condition

is applied. At the domain top and bottom Type I NRBC

of the CE/SE method is used. The CE/SE outflow NRBC

(Type II) is applied at the outflow boundary. Due to the

inconsistency of the boundary conditions at the top, bottom and the inflow one, unpleasant non-physical gradients appear at the upper and lower left comers, although

they do not propagate to the interior. To remove them, 7

rectangular cells with exponentially fast growth in size

in the y direction are added to the top and bottom as

"sponge zones". Each of these rectangular cells is then

divided into 4 triangle cells as in the interior.

3.2 The Trailing Edge Tone Noise of a Flate

Plate of Finite Thickness

In the case of a flat plate of finite thickness, due to flow

separation, a wall-jet is formed downstream of the blunt

trailing edge of the flat plate, tone noise is thus generated

from the consequent Karman vortex street.

Consider a Mach number M = 0.3 uniform flow past

a 2-D flat plate of thickness D = 1" (25 4cm). The

flat plate extends into the computational domain with

a length of 3D. Fig. 3 shows a sketch of the computational domain. Similar to the previous problem, The

plate thickness D , the ambient speed of sound an and

the ambient flow density are used to scale length, velocity and density respectively. The domain shown in Fig. 3

is exactly the computational domain, no sponge zone is

used for this problem. As in the previous problem, the

'unstructured' grid is obtained based on cutting a structured rectangular cell into 4 triangles.

top NRBC 10D

..............M=0.3

outflow

NRBC

28D

bottom NRBC

Figure 3: The computational domain for the flat plate

noise problem; totally 77,600 triangular cells, no sponge

zones.

3.2.1 Initial Conditions Initially, the flow of the

entire domain is set at the ambient conditions, i.e.,

po = 1.0, uo = 0, vo = 0, po = 1/7,

All spatial derivatives are set to zero.

3.2.2 Boundary Conditions At the inflow

boundary, the flow variables are specified as the M = 0.3

uniform flow:

p_=l.0, u_=0.3, vi=0, Pi=1/7,

and their spatial derivatives are conveniently set to zero.

The no-slip wall boundary condition is applied to the flat

plate. The top, bottom and outflow NRBC are the same

as the previous problem, but no sponge zones are used.

4 Numerical Results

In this section, numerical results for both the aeolian

and flat plate trailing edge noises are presented and compared to their respective experimantal data and results by

other numerical approaches when they are available.

4.1 Aeolian Noise Problem

In order to have a sizable sampling time series for FFT

(fast Fourier transform) analysis, with non-dimensional

time step size At = .005, over half million time steps

are run. Fig. 4 - Fig. 6 demonstrate the isobars or numerical Schlierens at time step 410,000 and 545,000. The

unsteadiness and the Karman vortex street in the wake

are clearly displayed. A point located at (0D, 3._D)

(Fig. 4) is selected for recording the time series. At this

location, there is no direct inluence from the Karman vortex street. Fig. 7 illustrates the non-dimensional pressure

history at this point. Fig. 8 depicts the result of FFT, i.e.,

the sound pressure level (SPL) vs. frequency at this selected location. The SPL is based on the relative level to

20 #Pc.

4.1.1 Tone noise frequency At Reynolds

number Re= 90,000 the highest peak of SPL in the

plot corresponds to a computed frequency of 617 Hz

NASA/CR 2002-212013 4

American Institute of Aeronautics and Astronautics
I •

7.7D

15.4R

_!_iii_iiii_iiiiiiiiiiiiii!ii!ii!_i!!i!i!iiii_ii_iii_iii_ii_!_i_i_i!_i!i_ii_

IM = 0.2_ isobars at time step 410,0001 Figure 6: Instantaneous isobars at time step 545,000.

Figure 4: Instantaneous isobars at time step 410,000.

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

_i_iiiii! num_i_a_iis_ihi_iiei_ei"iiiiiiiiiiiiiiiiiiii!_!_!_!_!_!_!_!_!_!_iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii_!!!!!!!!!!i!i!i!i!i!i!i!i!i!i!i

!ii_!iiii_atiiitimeiiistepiiii4i_i_i!i_i_i_iiiiiiiiiiiiiiiiiiii!iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii!!! _!i!_iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii

Figure 5: Instantaneous numerical Schlierens at time step

410,000.

(with a 3db 'bandwidth' or binwidth of 88 Hz), which

is quite close to the experimental one of 643 Hz [1] at

this M = 0.2 Mach number, despite the crude resolution

in frequency. A numerical experiment with a "very

high" Reynolds number is also tested. The computed

frequency is 652 Hz, slightly higher but still within a 4%

range of the experimental 643 Hz. The phenomenon that

the frequency ( or Strouhal number) around this range of

Reynolds number does not significantly change with Re

is justified by experimental observation (see Schlichting

[13]). and also agrees with Townsend's observation

[14] that the large scale turbulence only weakly depends

on the Reynolds number. The term "very high" used

here is a synonym of turning off the viscous terms in

the scheme. As a result of the numerical damping in

the scheme, viscous effects still exist. According to the

chart in Schlichting's book (p.32, [13] ), we speculate

that the effective Re is probably in the order of O(104 )

to O(105).

It is interesting to compare the computed frequency

with the results by other numerical approaches from the

CAA Workshop II [ 12].

Table 1 lists the computed frequencies by several other

researchers [2 - 5]. Generally, they use either compressible (Comp.) or incompressible (Incomp.) flow solver

with RANS (Reynolds averaged Navier-Stokes), LES, or

DNS and then an acoustic solver is applied. It is observed that, compared to the experimental result of 643

Hz, there is an error varying from 10 - 60%, while for

Re from 90,000 to O(105), the current method yields a

error within only 4%, and no acoustic solver is employed

at all.

NASA/C_2002-212013 5

American Institute of Aeronautics and Astronautics
Table1:Comparisonof computed frequencies

Experiment

643 Hz.

Flow

Brentner

et al.

815-1031

Comp.

Kumarasamy

et al.

710

Incomp.

Pope

753

Incomp.

Spyropoulos CE/SE

and Holmes

550 617

652

Incomp. Comp.

Solver LES DNS

Acoustic

Solver

Re

RANS

FW-H

lk-90k

RANS

Helmholtz

90k

DNS

A/VS

20k

Curie none

90k 90k

very

high

pressure history at ( 1 75D, 325D)

0 7320

0 7080

00150 0 0250 0 0350 0 O450 0 0550

Time in sec

Figure 7: History of pressure fluctuation at the FFT point

(recorded every 20 time steps).

SPL at (0,3 25D) in dB

140

2000 4000 6000 8000

F_quencyin Hz

Figure 8: Sound pressure Level for Re=90,000 at (0,

3.25D), the highest peak corresponds to 617 Hz and 138

dB.

NASA/C_2002-212013 6

American Institute of Aeronautics and Astronautics
4.1.2 Sound pressure level of the tone noise

Since the computation is two-dimensional, and the computational domain is in the near field (within 8 diameters), there is no direct comparison of the sound pressure

level (SPL) to the experimental data which is of 3-D nature and measured at a location 35 diameters away from

the circular cylinder center [12]. However, after some

approximation and simple analysis from the first principle, an estimated SPL for 3-D flow SPL at ro = 35D in

the direction of 90° can be obtained.

Based on the 2-D computation as shown in Fig. 8, at a

point of(0D, 3.25D) (see the 'FFT point' in Fig. 4) which

is in the 90 ° direction with x axis, the SPL is 138.5 dB.

If the noise source is regarded to be a simple one and locates approximately at the circular cylinder centerline,

for 2-D flow, the SPL attenuation is inversely proportional to r2, where r is the distance from the source to

the measuring point. Therefore, at the point Q (0, 35D)

in the 90° direction the SPL is estimated to be:

SPL2d = 138- 201og(35D/3.25D) = l17.9dB.

This result is obtained under the 2-D assumption that the

circular cylinder is infinitely long.

According to the experiment [3, 12], a cyliner length

of 10D is used. Q is assumed at the mid-plane normal

to the cylinder centerline ( the z direction), the cylinder

rod ranges from z = -5D to z = 5D. For truly 3-D

flow, let p(z) be the r.m.s, pressure contribution from a

line segment dz at z (along the cylinder centerline) to Q.

Then

- + Apo, po= p(o).

Generally, let L be the half length of the cylinder, the

square of the total sound pressure at Q is

P2L = Lpg _ _ ropo_[_o+O.5_/_(2_o)], [1+ (Z/ro)2]2

where C_o = tan-l(L/ro). Let L = 5D and L =_

respectively and the ratio

R -- P_D _ 0.1794612

Therefore, the sound pressure level at Q is estimated as:

SPL3d = SPL,2d + lOlogR = llO.4dB

Table 2 lists the numerical SPLs at these two locations at

an angle of 90 ° and comparison to the experimental data.

Recall that the basic computation is a two dimensional

one, while the experimental data is for a circular cylinder

of finite length in 3-D space, the agreement is excellent.

Table 2: SPL of aeolian tone noise at 90 ° angle

r Experiment CE/SE

(distance) [ 12] numerical

3.25D 138.5 dB

35D 111 dB 110.4 dB

(estimated)

presure history

0 050 0 080 0 110

Figure 10: Pressure history at the FFT point (recorded at

every 2 time steps).

4.2 Trailing Edge Noise Tone of a flat Plate

With non-dimensional time step size At = .005,390,000

time steps are run. Fig. 9 demonstrates the instantaneous

isobars at the last time step. The unsymmetric unsteadiness and the Karman vortex street in the wake are evident. Since there is no sponge zone, the 'matching layer'

between the steady Type I NRBC at the top and bottom

boundaries are observed. Also, at the upper and lower

left comers, non-physical gradients due to inconsistency

of inflow boundary condition and the Type I NRBCs are

present. However, they have practical no influence on the

interior flow. A point located at (15D, 3D) is selected

for recording the time series. At this location, there is no

direct inluence from the Karman vortex street. Fig. 10

illustrates the non-dimensional pressure history at this

point. Fig. 11 depicts the PSD vs. frequency at this

selected location and the experimental results by Heinemann et al [15] for similar cases. The Reynolds number

is in the magnitude of 104 to 105 ("very high"). The

highest peak of PSD in the plot corresponds to a computed frequency of 791 Hz (with a 3db binwidth 33 Hz).

Experimentally, from Heinemann's chart in Fig. 11, for

Mach number M = 0.3, the corresponding Strouhal

number S = 0.195. Assuming D = 2.54cm and speed

of sound equal to 343m/s, it is equivalent to 790 Hz,

NASA/CR 2002-212013 7

American Institute of Aeronautics and Astronautics
FFT data pt.

ID

28D

Figure 9: Instantaneous isobars at time step 390,000, for flow past the flat plate.

0.3

0.2

0.!

0.0

0.3 0.6 M= 0,9

I I i i i i ;, i 0 2 3 l, 5 6 8x_O s

CE/SE result

Experimental

data for flat

plate vortex

street noise:

790 Hz.

(Heinemann et

al. )

791 Hz

Figure 11 : PSD and experimental data.

10000

Freq Hz

Table 3: Flate plate tone noise frequency

Experiment CE/SE

(Heinemann eta/) numerical

790 Hz. 791 Hz

almost identical to the computed value, as shown in Table 3. No SPL data is collected since there is no experimental data in [15] to compare with.

5 Concluding Remarks

In this paper, a DNS type numerical approach is

adopted to simulate the trailing edge tone noises arising

from large scale turbulence. Good results in frequency

and SPL are obtained and compare favorably to the experimental ones and the results by other numerical approaches.

In the DNS type approach, we attempt to simulate only

the large scale turbulence, which is believed to be the

tone noise generating source. The smaller scale turbulences (higher frequencies and wave numbers) are practically ignored without a subgrid scale model, since we are

interested mostly in the aeroacoustic data. The CE/SE

is a scheme of conservation laws. When the divergence

theorem (2) is applied to find the cell average, it also

plays a role as a filter ( such as the one in LES approach).

High frequency and high wave number disturbances are

filtered out and aliasing errors are unlikeky to occur.

The novel NRBC based on flux balance is simple and

genuinely multi-dimensional. It allows the use of a small

computational domain in the noise computation.

NASA/C_2002-212013 8

American Institute of Aeronautics and Astronautics
References

[1] C.K.W.Tam, J.C.Hardin, Eds., "Second Computational Aeroacoustics (CAA) Workshop on Benchmark problems", NASA CP-3352, June,1997.

[2] K.S.Brentner, J.S.Cox, C.L.Rumsey and

B.A.Younis, 'Computation of Sound generated

by Flow over a Circular Cylinder: an Acoustic

Analogy Approach', in "Second Computational

Aeroacoustics (CAA) Workshop on Benchmark problems", NASA CP-3352, June,1997.

C.K.W.Tam and J.C.Hardin, Eds.

[3] S.Kumarasamy, R.A.Korpus J.B.Barlow, ' Computation of Noise due to the Flow over a Circular Cylinder', in "Second Computational Aeroacoustics (CAA) Workshop on Benchmark problems", NASA CP-3352, June,1997. C.K.W.Tam

and J.C.Hardin, Eds.

[4] D.S.Pope, 'A Viscous/Acoustic Splitting Technique

for Aeolian Tone prediction', in "Second Computational Aeroacoustics (CAA) Workshop on Benchmark problems", NASA CP-3352, June,1997.

C.K.W.Tam, J.C.Hardin, Eds.

[5] E.T.Spyropoulos and B.S.Holmes, 'Large-Eddy of

a High Reynolds Number Flow Around a Cylinder Including Aeroacoustic prediction' in "Second Computational Aeroacoustics (CAA) Workshop on Benchmark problems", NASA CP-3352,

June, 1997. C.K.W.Tam and J.C.Hardin, Eds.

[6] Chang, S. C., "The Method of Space-Time Conservation Element and Solution Element A New

Approach for Solving the Navier-Stokes and Euler

Equations," Journal of Computational Physics. vol.

119, pp. 295-324 (1995).

[7] Chang, S.-C., Wang, X.-Y. and Chow, C.-Y., "The

Space-Time Conservation Element and Solution

Element Metho_A New High Resolution and

Genuinely Multidimensional Paradigm for Solving

Conservation Laws," J. Comp. Phys. vol. 159, pp.

89-136 (1999).

[8] Wang, X.-Y. and Chang S.-C., " A 2-D Nonsplitting Unstructured Triangular Mesh Euler

Solver Based on the Space-Time Conservation Element and Solution Element Method" C.F.D.J. vol.

8, pp309-325 (1999).

[9] Loh, C. Y., Hultgren, L. S. and Chang S.-C.,

"Computing Waves in Compressible Flow Using

the Space-Time Conservation Element Solution Element Method," AIAA J., Vol. 39, pp. 794-801

(2001).

[10] Loh, C. Y., Hultgren, L. S., Chang, S.-C. and Jorgenson, P. C. E., "Vortex Dynamics Simulation in

Aeroacoustics by the Space-Time Conservation Element Solution Element Method," AIAA Paper 99-

0359 (1999).

[11] Loh, C. Y., Hultgren, L. S., and Jorgenson, P. C. E.,

'Near Field Screech Noise Computation for an Underexpanded Supersonic Jet by the CE/SE Method',

AIAA Paper 2001-2252, (2001).

[12] J.C. Hardin, 'Solution Comparisons: Category 4',

in "Second Computational Aeroacoustics (CAA)

Workshop on Benchmark problems", NASA CP3352, June,1997. C.K.W.Tam and J.C.Hardin, Eds.

[13] H.Schlichting, 'Boundary-Layer Theory', McGraw

Hill, 1979.

[14] A.A.Townsend, 'The Structure of turbulent Shear

Flow', 2nd edition, Cambridge University Press,

1976.

[15] H.J.Heinemann, O.Lawaczeck and K.A.Buterfisch,

'Karman vortices and their Frequency Determination in the Wakes of Profiles in the Sub-and Transonic Regime', Sumposium Transonicum II, Goettingen, Germany, Sept. 1975; Springer-Verlag

1976, pp. 75-82.

NASA/CR 2002-212013 9

American Institute of Aeronautics and Astronautics
Form Approved REPORT DOCUMENTATION PAGE

OMB No. 0704-0188

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December 2002 Final Contractor Report

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Near Field Trailing Edge Tone Noise Computation

6. AUTHOR(S)

Ching Y. Loh

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

Taitech, Inc.

21000 Brookpark Road

Cleveland, Ohio 44135

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

WU-708-90-43-00

NAS3-97186

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E-13703

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NASA CR--2002-212013

AIAA-2003-0365

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Prepared for the 41st Aerospace Sciences Meeting and Exhibit sponsored by the American Institute of Aeronautics and

Astronautics, Reno, Nevada, January 6-9, 2003. Project Manager, Richard A. Blech, Turbomachinery and Propulsion

Systems Division, NASA Glenn Research Center, organization code 5880, 216-433-3657.

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This publication is available from the NASA Center for AeroSpace Information, 301-621-0390.

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

Blunt trailing edges in a flow often generate tone noise due to wall-jet shear layer and vortex shedding. In this paper, the

space-time conservation element (CE/SE) method is employed to numerically study the near-field noise of blunt trailing

edges. Two typical cases, namely, flow past a circular cylinder (aeolian noise problem) and flow past a flat plate of finite

thickness are considered. The computed frequencies compare well with experimental data. For the aeolian noise

problem, comparisons with the results of other numerical approaches are also presented.

14. SUBJECT TERMS

Trailing edge tone noise; CE/SE method

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