Download - The Prediction of Fan Exhaust Noise_YER
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The Prediction of Fan Exhaust Noise
Philip J. Morris and Yuan ZhaoDepartment of Aerospace
EngineeringPenn State University
AARC Year End ReviewNovember 18-19, 2004
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Outline
Background The basic problem Existing approaches Problems to be overcome
Present Approach General analysis Implementation
Current Status Future Activities
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High Bypass Ratio Turbofan Engines
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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
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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
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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.)
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4th CAA Benchmark Problems
Time domain solution
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Time Domain: Realistic Configuration
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4th CAA Benchmark Problems
Frequency domain solution
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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
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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
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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.
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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
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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
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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
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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
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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
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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.
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2D Benchmark Problem
89 x 49 grid 177 x 97 grid4 minutes on 1GHzPentium IV (512 MB RAM)
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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
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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
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Sample Mean Flow Solutions
Baseline Geometry
Mach Number Contours
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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
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Sample grid snapshot – half of volume along axisymmetric plane
Sample Acoustic Grid
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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%
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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?
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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