The Prediction of Fan Exhaust Noise

Philip J. Morris and Yuan ZhaoDepartment of Aerospace

EngineeringPenn State University

AARC Year End ReviewNovember 18-19, 2004

Outline

Background The basic problem Existing approaches Problems to be overcome

Present Approach General analysis Implementation

Current Status Future Activities

High Bypass Ratio Turbofan Engines

Fan Inlet Noise Radiation

Finite Element Methods – Eversman et al.

Finite Difference Methods – Zhang et al.

Finite Volume Methods – Ozyoruk et al.

Ray Tracing – Kempton, Dougherty•Flow nearly irrotational•Axisymmetric approximation reasonable

Fan Exhaust Noise Radiation

Finite Element Methods – Eversman et al.

Asymptotic Methods – Leib, Goldstein, Dougherty

Finite Difference methods – Zhang et al.

Finite Volume Methods – Ozyoruk et al.

•Mean flow not irrotational•Fan exhaust shear layer generates instabilities in the Linearized Euler Equations

LEE Shear Layer Instabilities

Mathematical entities – result of the use of the LEE rather than the Navier-Stokes Equations

Problem for direct and adjoint solution Suppressed by:

Modification of the LEE Damping Frequency domain solution (Agarwal et

al.)

4th CAA Benchmark Problems

Time domain solution

Time Domain: Realistic Configuration

4th CAA Benchmark Problems

Frequency domain solution

Present Approach

Frequency Domain Solution of LEE Direct and Adjoint Solutions

Issues: Mean flow solution Numerical method

Unstructured grid Grid generation Matrix assembly and solver

Frequency Domain Solutions

Advantages Relatively rapid computation (compared to time

domain) Suppression of LEE instabilities

Disadvantages Large global matrix assembly Linear single frequency solutions

Unable to predict nonlinear effects Unable to predict truly broadband sound propagation

Time domain solutions have a complementary set of advantages and disadvantages

Frequency Domain Methods DG method

For high order accurate solutions, Adaptive refinement on unstructured mesh Prohibitive memory requirements for elliptic

problems (Rao & Morris). Finite difference method

Lower memory requirement. Unsuitable for adaptive unstructured mesh.

Continuous Galerkin methods Lower memory requirement – higher order

solution can be sought. Suitable for unstructured mesh.

Traditional Galerkin Method

1,2,3

{ , , , , }

for rr

T

i rx

u v w p

qq A Cq s

q

Approximate q with basis set,4

1

( , , ) en nq x y z q

In each element,

( ) 0 1,2,3,4 for e

ek r

r

iω d kx

q

q A Cq s

Galerkin method is consistent, but unstable for convection dominated problems.

Produces oscillations in regions with high convection speed, that pollute the whole solution.

Problem historically dealt by adding artificial dissipation.

Improved methods such as the Streamline Upwind Petrov Galerkin method (SUPG) have been proposed.

Continuous Galerkin Methods

Element Reynolds number:

12

ha

R ee

Lxuxudx

ud

dx

duae

,1;0,0

02

2

SUPG Method The trial functions are chosen to be functions of

derivatives of test functions.

Zero diffusion in the direction normal to convection – equivalent to a streamline upwinding.

e is the element diffusion parameter – a function of element Reynolds number and the element size.

Adds a symmetric stabilizing term to the weak form of the conventional Galerkin formulation.

dsdFu ktk )(

1

1

N

jjjuu

kekk u

F

sFut

SUPG Methodq q

q A Cq τ A q A Cq

τ A

e e

e e

e ee e e e T e e e

k r k rr k r

e e e T ek k

k

i d i dx x x

s d dx

where,e

e r

r

hmax

τ I

In discretized matrix form

e e eP q f

ePis a 20x20 matrix and and are 20x1 column vectorseq ef

Currently using Buffer Zone Type BC’s:

0 0

0 0

0 0

2

0

( , ) ( ; ) ,

( ; ) ,

( ),

1 ( )( )

1b

p x t p x t

p x kx k c

x x

xx

exp -i where

exp i

replaced by i where

exp and are positive constants

exp

Non-Reflecting Boundary Conditions

A

C

B

X

Y

Z

2D Benchmark Problem Using the SUPG

method, the instability waves are suppressed on the coarse grid.

Solution obtained using MATLAB® in a minute- compared to hours for the time domain solution, that results in the instability waves.

Plot: the instantaneous pressure at y=15 for the 89x49 grid.

2D Benchmark Problem

89 x 49 grid 177 x 97 grid4 minutes on 1GHzPentium IV (512 MB RAM)

Current Activities

Extension of analysis to three dimensions Decided not to continue with 2-d formulation

because of expected problems with 3d implementation

Rewriting of 2D code for 3D problem – major undertaking

Mean Flow Solution Grid generation: 3D unstructured

tetrahedron grid generated by Gridgen V15.0 (Pointwise Inc.)

Benchmark Problem

Mean Flow Solution

Using STAR-CD – a commercial CFD software package Unstructured grid 2nd order in space and time Various turbulence models Collaboration with CD-adapco

Sample Mean Flow Solutions

Baseline Geometry

Mach Number Contours

Matrix Size/Computing Cost

For a sequential test problem (point source in the center of a cubic domain). Single Pentium-4 processor with 2Gb memory Number of nodes=4854 Number of tetrahedral elements=23942 Number of boundary elements=3200 Global matrix size=24270 x 24270 Matrix sparseness, number of nonzero

values=518269 ~ 0.08% Harwell-Boeing format used (Sparsekit, Univ. of

Minnesota) 2 hours on single processor

Sample grid snapshot – half of volume along axisymmetric plane

Sample Acoustic Grid

Matrix Size/Computing Cost

For a more realistic problem Number of nodes=102572 Number of tetrahedral

elements=550897 Number of boundary elements=43810 Global matrix size=512860 x 512860 Matrix sparseness, number of nonzero

values ~ 0.08%

SuperLU Solver

Open source code from Lawrence Berkeley National Laboratories (Dr. Xiaoye Li)

Able to solve complex matrix problem Sequential (for test problem) Parallel MPI version for prototype and

production acoustic solutions Estimated computer time for sample grid

problem: ~5 hours on 25 processors?

Future Activities

Finish code debugging – model problem Optimization of boundary buffer zone parameters

Grid number Exponential decay rate

Artificial parameter in SUPG method Obtain mean flow solution Interpolation from CFD to acoustic grid Benchmark comparisons (Eversman & Okunbor) 3D visualization (Fieldview by Intelligent Light)

e