comparison schemes benchmark problem

7/29/2019 Comparison Schemes Benchmark Problem

1/20

Comparison of numerical schemes for arealistic computational aeroacoustics

benchmark problem

R. HixonMechanical, Industrial, and Manufacturing Engineering Department, University of Toledo, Toledo,

OH 43606, [email protected]

M. NallasamyQSS Group, Inc., NASA Glenn Research Center, Cleveland, OH 44135,

[email protected]

S. SawyerMechanical Engineering Department, The University of Akron, Akron, OH 44325,

[email protected]

R. DysonAcoustics Branch, NASA Glenn Research Center, Cleveland, OH 44135,

[email protected]

ABSTRACT

In this work, a nonlinear block-structured CAA solver, the NASAGlenn Research Center BASS

code, is tested on a realistic CAA benchmark problem in order to ascertain what effect the high-

accuracy solution methods used in CAA have on a realistic test problem. In this test, the non linear

2-D compressible Euler equations are solved on a fully curvilinear grid from a commercial grid

generator. The solutions are obtained using several finite-difference methods on an identical grid

to determine the relative performance of these spatial differencing schemes on this benchmark

problem.

1. INTRODUCTIONThe field of Computational Aeroacoustics (CAA) is concerned with the time-accurate

calculation of unsteady flow fields. In order to accurately propagate the unsteady

acoustic, vortical, and entropy waves that are present in these flows, high-accuracy

numerical differencing schemes have been developed which require very few grid

points per disturbance wavelength to calculate an accurate value of the spatial derivative

aeroacoustics volume 3 number 4 2005 pages 379 397 379

Paper 3(4)-01-Hixon.Qxd 24-2-05 4:41 pm Page 379


2/20

(see Refs. 1 and 2 for an overview of CAA developments). These schemes have been

extended for use in nonlinear flow calculations, and have produced very good results

(e.g., Refs. 3-5).

However, for realistic flow calculations using curvilinear grids, it is not clear if these

high-accuracy schemes retain the advantages that they show for model problems.

Previous work has indicated that the grid generator has an effect on the attainable

accuracy of a numerical scheme6, even with a very smooth grid from a commercial grid

generator.7

In this work, the NASA BASS code is applied to the CAABenchmark problem of avortical gust impinging on a loaded 2D cascade.8 The BASS code has four spatial

differencing options available to the user: explicit 2nd order, explicit 6th order, optimized

DRP9, and prefactored compact 6th order.10 While it is expected that the three high-

accuracy schemes will perform adequately, the question is whether they will perform

better than the low-order scheme on a realistic problem.

It must be noted at this point that this test problem may well be weighted in favor of

the 2nd order explicit scheme because the wavelength of the vortical gust is very long

and the computational boundaries are very close. Thus, if the high-accuracy schemes

provide a measurably better answer, this will be a strong indication that high-accuracy

schemes are useful for traditional unsteady CFD problems as well as for CAA

applications.

2. GOVERNING EQUATIONS AND NUMERICAL METHODIn this work, the Euler equations are solved. The 2D nonlinear Euler equations may be

written in Cartesian form as:

(1)

The NASA Glenn Research Center BASS code was used to solve this equation.4-6,11-12

The BASS code uses optimized explicit time marching combined with high-accuracy

finite-differences to accurately compute the unsteady flow. The code is parallel, and

uses a block-structured curvilinear grid to represent the physical flow domain.

A constant-coefficient artificial dissipation model13 is used to remove unresolved high-

frequency modes from the computed solution.

The BASS code solves the Euler equations using the nonconservative chain-rule

formulation; previous experience has indicated that the formal lack of conservation is

offset by the increased numerical accuracy of the transformed equations.3-5

The chain-rule form of the Euler equations is:

(2)

For time marching, the optimized low-storage RK56 scheme of Stanescu and

Habashi14 was used for all spatial differencing schemes.

Q Q Q + E + E F + F =t t t x x y y+ + + 0

Q + E + Ft x y = 0

380 Comparison of numerical schemes for a realistic computational aeroacoustics



3/20

aeroacoustics volume 3 number 4 2005 381

In this work, four spatial differencing schemes were used: an explicit 2nd order

central differencing scheme, an explicit 6th order central differencing scheme, the 7-point

optimized DRP scheme of Tam and Webb9, and the prefactored sixth-order compact

differencing scheme of Hixon10. These four schemes are all user options in the BASS

code; each was coded for maximum performance.

3. THEORETICAL PERFORMANCE OF SPATIAL DIFFERENCINGSCHEMESIn an unsteady flow solver, there are two types of errors that are encountered: dispersion

error and dissipation error. The dispersion error is an error in the propagation speed of an

unsteady disturbance, while the dissipation error is an error in the amplitude of the

unsteady disturbance.

To quantify these errors, it is customary to compare the unsteady performance of

spatial differencing schemes by analyzing the wave propagation performance of these

methods. To accomplish this, we consider the 1-D linear advection equation:

(3)

For this analysis, the solution is simple harmonic:

(4)

In this analysis, the errors from the time marching scheme are neglected. Instead,

only the spatial derivative ofu is considered. Analytically,

(5)

The procedure is illustrated using the 2nd order central differencing method, defined as:

(6)

Substituting in the analytic solution at the neighboring grid points, we obtain the

numerical derivative:

(7)

By comparing equations (5) and (7), we define the numerical wavenumberfor the

2nd order central differencing scheme as:

(8)k x = k x ( ) ( )* sin

u =i

xk x ux j j|

sin ( )( )

uu u

xx j

j+ j| =

1 1

2

u = iku = ix

k x ux j j j|

( )

u x, t = Aei kx t

( )( )

u +cu =t x 0



4/20

It should be noted that the numerical wavenumber of a finite-differencing scheme

can have both real and imaginary parts. In general, the numerical wavenumber of a central

difference scheme will have only a real component, while the numerical wavenumber of

a biased stencil will have both real and imaginary components.

In a similar way to the previous analysis, the numerical wavenumbers can be determined

for all four schemes. Figure 1 compares the real part of the numerical wavenumbers. Again,

because all four schemes have central differencing stencils, the numerical wavenumbers

of these schemes have only real components.

The real part of the numerical wavenumber affects the wave propagation performanceof the numerical solver by changing the propagation speed of the waves. The numerical

propagation speed is:

(9)

Figure 2 shows the error in the propagation speed for each scheme as a function of

grid points per wavelength.

Note that, for the three maximum-order schemes, the numerical propagation speed

is always lower than the actual propagation speed, while for the optimized DRP scheme,

c

c=

k x

k x

*

( )*


Exact

Explicit 2nd order

Explicit DRPExplicit 6th order

Compact 6th order

3

2

1

(kdx)*

00 1 2 3

kdx

Figure 1. Comparison of numerical wavenumbers for spatial differencing schemes.



5/20

the numerical propagation speed is higher than the actual propagation speed for a

portion of the wavenumber spectrum.

This analysis shows only the best-case performance of the schemes in that only the

performance of the interior stencil is represented. The schemes all require special

stencils for use at and near the boundaries, and the performance of these stencils will

have an impact on the solution stability and accuracy at the inflow, outflow, and wall

boundaries. Due to the presence of the grid boundary and the size of the interior

differencing stencil (ranging from 3 points for the explicit 2nd order scheme to 7 points

for the explicit 6th order and DRP schemes), a series of special stencils is required asthe grid boundary is approached.

Because the interior stencils would require data outside the computational grid, these

special boundary stencils are biased to use points in the interior of the computational

domain. Because of this bias towards the interior, the boundary stencils have a different

dispersion error than that of the interior stencil; in addition, these biased stencils also

introduce dissipation errors.

Due to the changes in stencil performance and its adverse effects on numerical

stability, it is desirable to minimize the number of boundary stencils used. However, it

is also desirable to retain accuracy and numerical conservation if possible. Three of the


Exact

Explicit 2nd order

Explicit DRP

Explicit 6th order

Compact 6th order

1

0.510

Points per wavelength

c/c

Figure 2. Comparison of numerical propagation speeds for spatial differencing

schemes.



6/20

four schemes (the DRP, compact 6th order, and explicit 2nd order) use boundary stencils

that retain the formal accuracy of the scheme but do not provide full conservation near

the boundaries. The explicit 6th order scheme, however, retains full conservation to the

boundary at the cost of using more boundary stencils with lower accuracy. At the

boundary, the explicit 6th order scheme uses a 4th order one-sided difference, theoretically

returning 5th order global accuracy.

In addition to the propagation speed error from the spatial differencing scheme, the

artificial dissipation adds a dissipation error to the solution. In this work, two constant-

coefficient artificial dissipation schemes were employed: an explicit 4th order dampingfor the 2nd order differencing scheme, and an explicit 10th order damping for the other

schemes. Since both artificial dissipation schemes employ centered stencils, the

addition of artificial dissipation does not affect the propagation speed errors except near

the boundaries of the domain.

Figure 3 shows the damping effect of each method as a function of grid points per

wavelength of the disturbance, compared to the wavespeed error of the spatial

differencing schemes. Note that, since the explicit 2nd order scheme has much higher

wavespeed errors compared to the high-accuracy schemes, a higher dissipation rate is

required to remove the erroneous waves from the computed solution. It should be noted

that all spatial differencing schemes used were central differencing; because of this,


Explicit 2nd order1

0.1

0.01

0.001

WavespeedError

0.0001

le-05

le-061 10 100

DissipationRate

Points per Wavelength

Explicit DRP

Explicit 6th order

Compact 6th order

Explicit 4th dissipation

Explicit 10th dissipation

Figure 3. Dissipation rate compared to the wavespeed errors of the spatial

differencing methods.



7/20

the schemes have no inherent dissipation. Also, the artificial dissipation schemes used

are designed to have no inherent dispersion errors.

Again, this analysis is presented for the relatively simple case of a linear 1-D wave

propagation problem solved on a uniform grid; thus, the results can be viewed as the

best-caseperformance for each scheme. The focus of this work is to solve an unsteady

benchmark problem with realistic flow and geometry to determine if the theoretical

performance advantages of high-accuracy schemes are realized in real-world CAA

calculations.

4.TEST CASEIn this work, the 4th CAA Workshop benchmark cascade problem given in Ref. 8 is

computed using the NASA GRC BASS code. This benchmark consists of a loaded 2-D

cascade which has vortical wakes from an upstream rotor impinging upon it, creating

unsteady flow and noise. These vortical wakes are at the blade passing frequency (BPF),

and its first two harmonics (2xBPF and 3xBPF).

The grid used is a complex structured multiblock curvilinear grid; Figure 4 shows

one passage of the grid. For this problem, there are 11 rotor blades and 27 stator blades;

only the flow about the stator blades is directly calculated.

The BASS code was validated on this problem for the 4th CAA Workshop15, using

this grid with the 6th order prefactored compact differencing scheme. During that work,

a grid density study was conducted to ensure a grid-converged solution.

The gust wavelengths in this problem are very long, with approximately 35 points

per wavelength for the highest frequency gust at the inflow boundary. Such a large


1

0.5

0

0.51 0 1

Y

X

Figure 4. Curvilinear grid about one blade of the 2D cascade.



8/20

number of points per wavelength is excessive for the high-accuracy schemes; however,

this grid was used for the original validation of the high-accuracy schemes.

For all four differencing schemes, the same grid, same boundary conditions, same

time stepping scheme, and same time step were used. The only differences in the

calculations were the spatial differencing method and the artificial dissipation used.

5. NUMERICAL RESULTSTo compare the results from the four schemes, several metrics are used. First, the

calculated mean flows from each method are compared. Figure 5 shows the meanpressure on the cascade blade surface. All four solvers are obtaining good results on

the blade surface. The mean flow at the inflow and outflow were also compared, with

the three high-accuracy schemes getting very similar results. The explicit 2nd order

method obtained slightly different results for the mean flow; however, the mean flow

results were still within 0.1% of the desired values.


0.95

0.9 E-6

C-6, DRP

E-2

Mean Surface Pressure

0.85

0.8

0.5 0.25 0

X

E

0.25

Figure 5. Mean pressure on the blade surface.



9/20

Figures 6-9 show the pressure mode amplitudes at the inflow and outflow boundaries

for the four schemes. In these figures, (a) is the compact 6th order scheme, (b) is the

DRP scheme, (c) is the explicit 6th order scheme, and (d) is the explicit 2nd order

scheme.

For the 2xBPF frequency, the allowable modes are given by:

(10)

The allowable modes for the 3xBPF frequency are:

(11)M = n

n =

33 27

0, 1, 2, 3, ...

+

M = n

n =

22 27

0, 1, 2, 3, ...

+


86 59 32 5 22

Spatial mode order (m)

PressureAmplitude

0.00025

0.0003

0.00035

0.0002

0.00015

0.0001

5E-05

049 76 86 59 32 5 22


P

ressureAmplitude

0.00025

0.0003

0.00035

0.0002

0.00015

0.0001

5E-05

049 76

86 59 32 5 22


PressureAmplitude

0.00025

0.0003

0.00035

0.0002

0.00015

0.0001

5E-05

049 76 86 59 32 5 22


PressureAmplitude

0.00025

0.0003

0.00035

0.0002

0.00015

0.0001

5E-05

049 76

(a) (b)

(c) (d)

Figure 6. 2xBPF mode amplitudes at inflow boundary (x/c = 1.5).



10/20

In Figures 6-9, the allowable modes are labeled. All other modes are spurious.

In Figures 6 and 7, the compact 6th order and explicit DRP schemes are getting very

comparable solutions for the 2xBPF frequency. The explicit 6th order scheme also

obtains a similar solution, with the addition of a spurious m = 0 mode. This spurious

mode was unexpected, particularly since the compact 6th order and explicit DRP

schemes did not exhibit this behavior. The obvious difference in the three schemes is at

the grid boundaries; the explicit 6th order scheme has much less accurate boundary

stencils than the compact 6th and explicit DRP schemes.

The explicit 2nd order obtains a very different solution from the three high-accuracy

schemes, predicting the input gust mode (m = 22) to be dominant at the inflow

boundary. Analysis showed that the incoming vortical gust was generating a spurious

incoming acoustic wave for this scheme.

Figures 7 and 8 compare the results for the 3xBPF frequency. The performance of

the schemes is similar for this frequency, with the explicit 2nd order scheme again


86 59 32 5 22


P

ressureAmplitude

0.00025

0.0003

0.00035

0.0002

0.00015

0.0001

5E-05

049 76

(a)86 59 32 5 22


P

ressureAmplitude

0.00025

0.0003

0.0002

0.00015

0.0001

5E-05

049 76

(b)

86 59 32 5 22


PressureAmplitude

0.00025

0.0003

0.0002

0.00015

0.0001

5E-05

049 76

(c)86 59 32 5 22


PressureAmplitude

0.00025

0.0003

0.00035

0.0002

0.00015

0.0001

5E-05

049 76

(d)

Figure 7. 2xBPF mode amplitudes at the outflow boundary (x/c = 1.5).



11/20

predicting the incoming gust mode (m = 33) to be dominant. At the outflow boundary,

both the explicit 6th order scheme and the explicit 2nd order scheme are predicting a

number of spurious modes.

The low performance of the explicit 2nd order scheme on this problem was

unexpected; particularly the low mode amplitudes predicted. The wavelengths

associated with these waves, particularly at 2xBPF, should be resolved by the 2nd order

scheme. Two mechanisms existed for the low mode amplitude: either the wavespeed for

the explicit 2nd order scheme was so incorrect that the modes were not forming, or the

damping from the 4th order dissipation was removing the waves from the solution.

The effect of the 4th order dissipation on the performance of the 2nd order explicit

scheme is shown in Figure 10. In Figure 10, the spatial amplitudes of the v velocity gust

components for the DRP and the explicit 2nd order scheme are compared. The rapid

damping of the incoming gust is shown, with the 3xBPF gust showing the highest

damping rate.



PressureAmplitude

750

1E-05

2E-05

3E-05

5E-05

4E-05

6E-05

7E-05

8E-05

9E-05

0.0001

48 21 6 33 60 87

(a) Spatial mode order (m)

PressureAmplitude

750

1E-05

2E-05

3E-05

5E-05

4E-05

6E-05

7E-05

8E-05

9E-05

0.0001

48 21 6 33 60 87

(b)


PressureAmplitude

750

1E-05

2E-05

3E-05

5E-05

4E-05

6E-05

7E-05

8E-05

9E-05

0.0001

48 21 6 33 60 87

(c) Spatial mode order (m)

PressureAmplitude

750

1E-05

2E-05

3E-05

5E-05

4E-05

6E-05

7E-05

8E-05

9E-05

0.0001

48 21 6 33 60 87

(d)

Figure 8. 3xBPF mode amplitudes at the inflow boundary (x/c = 1.5).



12/20


3.5E-05

3E-05

2E-05

1E-05

2.5E-05

1.5E-05

5E-06

075 48 21 6


P

ressureAmplitude

33 60 87

(a)

3.5E-05

3E-05

2E-05

1E-05

2.5E-05

1.5E-05

5E-06

075 48 21 6


P

ressureAmplitude

33 60 87

(b)

3.5E-05

3E-05

2E-05

1E-05

2.5E-05

1.5E-05

5E-06

075 48 21 6


PressureAmplitude

33 60 87

(c)

3.5E-05

3E-05

2E-05

1E-05

2.5E-05

1.5E-05

5E-06

075 48 21 6


PressureAmplitude

33 60 87

(d)


0.01

0.001v

0.0001

1.4 1.2 1 0.8x/c

BPF (DRP)

BPF (E2/E4)

2xBPF (DRP)

2xBPF (E2/E4)

3xBPF (DRP)

3xBPF (E2/E4)

Figure 10. Axial distribution of disturbance velocity upstream of the cascade.



13/20

To test the effect of reducing the damping, the explicit 2nd order scheme was run

from the converged solution using the 10th order damping. It must be emphasized that

the 2nd order scheme required the 4th order damping to avoid code instability during the

initial transient calculation. The 10th order dissipation could only be used after a converged

solution was obtained with the 4th order dissipation.

Figures 11-14 compare the results for these four schemes: (a) is the compact 6th order

scheme, (b) is the explicit 6th order scheme, (c) is the explicit 2nd order scheme with

4th order dissipation, and (d) is the explicit 2nd order scheme with 10th order dissipation.

These results show that the explicit 4th order dissipation is magnifying the spuriousacoustic wave formation from the incoming vortical gusts. With the reduced dissipation,

however, the explicit 2nd order scheme is producing many spurious modes, particularly

at the 3xBPF frequency. While the amplitudes of the physical modes are close to that

predicted by the high-accuracy schemes, the presence of many spurious modes renders

the solution unacceptable.


Spatial mode order (m)86 59 32 5 22 49 76

0

5E-05

0.0001

0.00015

PressureAmplitude

0.0002

0.00025

0.0003

0.00035

(a) Spatial mode order (m)86 59 32 5 22 49 76

0

5E-05

0.0001

0.00015

PressureAmplitude

0.0002

0.00025

0.0003

0.00035

(b)

0.00035

0.0003

0.00025

0.0002

0.00015

0.0001

5E-05

086 59 32 5 22 49 76

PressureAmplitude

Spatial mode order (m)(c) Spatial mode order (m)86 59 32 5 22 49 76

0

5E-05

0.0001

0.00015

PressureAmplitude

0.0002

0.00025

0.0003

0.00035

(d)

Figure 11. 2xBPF mode amplitudes at inflow boundary (x/c = 1.5).



14/20

Figures 15-17 show the amplitude of the perturbation pressure on the surface of

the stator blades. Generally, the three high-accuracy schemes get similar solutions

for all three frequencies. The explicit 2nd order scheme obtains either a greatly reduced

amplitude (with the 4th order damping), or a highly oscillatory solution (with the

10th order damping). From these results, it is apparent that the 10th order damping is

not adequate for the 2nd order explicit scheme when applied to a nonlinear

problem.

Figure 18 compares the CPU time required per time step for the four schemes. The

figure shows that the explicit 2nd order scheme ran only 25% faster than the compact

6th order scheme per time step. In fairness, the higher stability bounds for the explicit

2nd order scheme would also allow a time step that is twice as large as that of the

compact 6th order scheme. If the minimum cell size does not change appreciably (i.e.,

the CFL condition remains constant for the smallest grid cell), the 2nd order scheme


0.00035

0.0003

0.00025

0.0002

0.00015

0.0001

5E-05

086 59 32 5 22 49 76

PressureAmplitude

Spatial mode order (m)(a)

0.0003

0.00025

0.0002

0.00015

0.0001

5E-05

086 59 32 5 22 49 76

Pre

ssureAmplitude

Spatial mode order (m)(b)

0.00035

0.0003

0.00025

0.0002

0.00015

0.0001

5E-05

086 59 32 5 22 49 76

PressureAmplitude

Spatial mode order (m)(c)

0.00035

0.0003

0.00025

0.0002

0.00015

0.0001

5E-05

086 59 32 5 22 49 76

PressureAmplitude

Spatial mode order (m)(d)Figure 12. 2xBPF mode amplitudes at the outflow boundary (x/c = 1.5).



15/20

could theoretically run about 2.6 times as many grid points in the same CPU time as the

compact 6th scheme. This would result in roughly a 62% grid refinement in each

coordinate direction. Based on the computed results, it is felt that such a small

amount of refinement would not appreciably improve the solution from the 2nd order

scheme.

6.CONCLUSIONSIn this work, a realistic unsteady nonlinear flow problem was solved using a high-

accuracy time marching method coupled with various spatial differencing schemes. The

results show the advantage of using high-accuracy spatial differencing for unsteady

flow calculations. While this is only one test case on one cascade geometry, it indicates

that the theoretical accuracy of the high-order schemes translates into improved

solutions for realistic unsteady flows about complex geometries.


0.0001

9E-05

8E-05

7E-05

6E-05

PressureAmplitude

5E-05

4E-05

3E-05

2E-05

1E-05

075 48 21 6

Spatial mode order (m)33 60 87

(a)

PressureAmplitude


0.0001

9E-05

8E-05

7E-05

6E-05

5E-05

4E-05

3E-05

2E-05

1E-05

075 48 21 6 33 60 87

(b)

0.0001

9E-05

8E-05

7E-05

6E-05

PressureAmplitude

5E-05

4E-05

3E-05

2E-05

1E-05

0


(c)

0.0001

9E-05

8E-05

7E-05

6E-05

PressureAmplitude

5E-05

4E-05

3E-05

2E-05

1E-05

0


(d)

Figure 13. 3xBPF mode amplitudes at the inflow boundary (x/c = 1.5).



16/20


3.5E-05

3E-05

2.5E-05

PressureAmplitude

2E-05

1.5E-05

1E-05

5E-06

075 48 21 6 33 60 87

Spatial mode order (m)(a)

3.5E-05

3E-05

2.5E-05

PressureAmplitude

2E-05

1.5E-05

1E-05

5E-06

075 48 21 6 33 60 87

Spatial mode order (m)(b)

3.5E-05

3E-05

2.5E-05

PressureAmplitude

2E-05

1.5E-05

1E-05

5E-06

075 48 21 6 33 60 87

Spatial mode order (m)(c)

3.5E-05

3E-05

2.5E-05

PressureAmplitude

2E-05

1.5E-05

1E-05

5E-06

075 48 21 6 33 60 87

Spatial mode order (m)(d)


0.50

0.001

0.002

0.003

0.004

BPF Harmonic Loading

C-6DRP

E-2 with E-10 Diss.

E-2

0.005

0.25 0 0.25 0.5

Pressure

X

Figure 15. RMS pressure distribution on the blade surface for BPF disturbance.



17/20


0.002

0.00175

0.0015

0.00125

Pressure

0.001

0.00075

0.0005

0.00025

00.5 0.25 0

X

0.25 0.5

E-2 with E-10 Diss.

2BPF Harmonic Loading

E-6DRP

C-6

E-2

Figure 16. RMS pressure distribution on the blade surface for 2xBPF disturbance.

0.0004

0.0003

3BPF Harmonic Loading

E-2 with E-10 Diss.

DRP

C-6

E-2

E-6

0.0002

0.0001

00

X

Pressure

0.250.5 0.25

Figure 17. RMS pressure distribution on the blade surface for 3xBPF disturbance.



18/20

REFERENCES1) Tam, C. K. W., Computational Aeroacoustics: An Overview of Computational

Challenges and Applications, International Journal of Computational Fluid

Dynamics, Vol. 18, No. 6, 2004, pp. 547-567.

2) Lele, S. K., Computational Aeroacoustics: A Review, AIAA Paper 97-0018,

January 1997.

3) Hixon, R., Mankbadi, R. R., and Scott, J. R., Validation of a High-Order

Prefactored Compact Code on Nonlinear Flows with Complex Geometries,

AIAA Paper 2001-1103, Jan. 2001.

4) Nallasamy, M., Hixon, R., Sawyer, S., Dyson, R., and Koch, L., ATime Domain

Analysis of Gust-Cascade Interaction Noise, AIAAPaper 2003-3134, May 2003.

5) Sawyer, S., Nallasamy, M., Hixon, R., Dyson, R., and Koch, D., Computational

Aeroacoustic Prediction of Discrete-Frequency Noise Generated by a Rotor-

Stator Interaction, AIAA Paper 2003-3268, May 2003.

6) Hixon, R., Nallasamy, M., and Sawyer, S., Effect of Grid Singularities on the

Solution Accuracy of a CAA Code, AIAA Paper 2003-0879, Jan. 2003.


0.6

0.7

0.5

ComputingTime,Sec./timestep

0.4

0.3

0.2

0.1

0E-2 DRP E-6 C-6

Comparision of computing times

Figure 18. Comparison of CPU time per time step.



19/20

7) GridPro/az3000, Program Development Corporation, White Plains, NY.

8) Proceedings of the Fourth Computational Aeroacoustics Workshop on Benchmark

Problems, NASA, CP-2004-. See also website www.math.fsu.edu/caa4

9) Tam, C. K. W., and Webb, J. C., Dispersion-Relation-Preserving Finite-

Difference Schemes for Computational Acoustics, Journal of Computational

Physics, Vol. 107, 1993, pp. 262-281.

10) Hixon, R., Prefactored Small-Stencil Compact Schemes, Journal of

Computational Physics, Vol. 165, 2000, pp. 522-541.11) Hixon, R., Nallasamy, M., and Sawyer, S., Parallelization Strategy for an Explicit

Computational Aeroacoustics Code, AIAA Paper 2002-2583, July 2002.

12) Hixon, R., Nallasamy, M., Sawyer, S., and Dyson, R., Mean Flow Boundary

Conditions for Computational Aeroacoustics, AIAAPaper 2003-3299, May 2003.

13) Kennedy, C. A., and Carpenter, M. H., Several New Numerical Methods for

Compressible Shear-Layer Simulations,Applied Numerical Mathematics, Vol. 14,

1994, pp. 397-433.

14) Stanescu, D., and Habashi, W. G., 2N-Storage Low Dissipation and Dispersion

Runge- Kutta Schemes for Computational Acoustics,Journal of Computational

Physics, Vol. 143, No. 2, 1998, pp. 674-681.

15) Nallasamy, M., Hixon, R., Sawyer, S. D., and Dyson, R. W., Category 3: Sound

Generation by Interacting with a Gust, Problem 2 Cascade-Gust Interaction,

Proceedings of the Fourth Computa tional Aeroacoust ics Workshop for

Benchmark Problems, 2004.




20/20


comparison schemes benchmark problem

Documents