Meng-Sing Liou

Glenn Research Center, Cleveland, Ohio

Ten Years in the Making—AUSM-family

NASA/TM—2001-210977

November 2001

AIAA–2001–2521
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NASA/TM—2001-210977

November 2001

National Aeronautics and

Space Administration

Glenn Research Center

Prepared for the

15th Computational Fluid Dynamics Conference

sponsored by the American Institute of Aeronautics and Astronautics

Anaheim, California, June 11–14, 2001

AIAA–2001–2521

Meng-Sing Liou

Glenn Research Center, Cleveland, Ohio

Ten Years in the Making—AUSM-family
Acknowledgments

Several people have kindly permitted and sent me their calculated results that are included in the present paper.

Thanks are due to Drs. G. Billet and I. Mary of ONERA, France, D. Darracq of CERFACS, France, H. Paillère of

CEA, France, and J. De Wilde of University of Ghent, Belgium. Dr. Y. Zheng of Taitech, Inc. helped

in the preparation of the manuscript.

Available from

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Available electronically at http://gltrs.grc.nasa.gov/GLTRS
National Aeronautics and Space Administration

Glenn Research Center

21000 Brookpark Road

Cleveland, Ohio 44135

NASA/TM—2001-210977 1

Refs. [1,4–10]
NASA/TM—2001-210977 2
Van Leer F.V.S.

p/p∞

0.0 2.0 4.0

θ (deg) 10.0

11.0

12.0

13.0

14.0

15.0

T/T∞

0.0 10.0

Van Leer/Hanel91  ̈

p/p∞

0.0 2.0 4.0

θ (deg) 10.0

11.0

12.0

13.0

14.0

15.0

T/T∞

0.0 10.0

AUSM(M-Splitting)

p/p∞

0.0 2.0 4.0

θ (deg) 10.0

11.0

12.0

13.0

14.0

15.0

T/T∞

0.0 10.0

Exact Riemann Solver

p/p∞

0.0 2.0 4.0

θ (deg) 10.0

11.0

12.0

13.0

14.0

15.0

T/T∞

0.0 10.0

NASA/TM—2001-210977 3
NASA/TM—2001-210977 4
NASA/TM—2001-210977 5
NASA/TM—2001-210977 6
10 20 30 40 50 60

time

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

thickness

AUSM t ri ad

Sandham & Reynolds

Mc=0.8 - Re=600 - a =0.0001 random noise

AUSM + triad

DNS

0

2

4

6

8

y

0 2 4 6 8 10 12

x

0

5

10

z

0 25 50 75 100

time

0

1

2

3

4

5

6

δw

0 0.5 1 1.5 2

x1

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

ρ

0 0.5 1 1.5 2

x1

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

ρ

NASA/TM—2001-210977 7
NASA/TM—2001-210977 8
NASA/TM—2001-210977 9
two-phase

initial conditions

pure gas

flow mixture

steady state

pure liquid

02468

x [m]

0

0.2

0.4

0.6

0.8

1

void fraction

Phase Separation Problem

initial profile

t=0.2 s

t=0.4 s

t=0.6 s

t=0.8 s

t=1 s

t=1.2 s

t=2 s

02468

x [m]

1e+05

1.1e+05

1.2e+05

1.3e+05

1.4e+05

pressure [Pa]

Phase Separation Problem

AUSM+, sigma=0, x1=0.003, x2=0.004, 100pts, CFL=0.05

open to atmosphere

Vo=2.1 m/s

0 5 10 15 20

x [m]

0

0.2

0.4

0.6

0.8

1

void fraction

Oscillating Manometer

0 5 10 15 20

x [m]

0

0.2

0.4

0.6

0.8

1

void fraction

Oscillating Manometer

AUSM+, 4−eq. model, sigma=0.0, 220 points

0 5 10 15 20

time [s]

−2.2

−1.2

−0.2

0.8

1.8

liquid velocity [m/s]

Oscillating Manometer

AUSM+, 4−eq. model, sigma=0.0, 220 points

analytical

AUSM+

NASA/TM—2001-210977 10
0

2

4

6

8

10

12

14

z

Y X

Z 7.0E-02

6.9E-02

6.8E-02

6.7E-02

6.6E-02

6.5E-02

6.4E-02

6.3E-02

6.2E-02

6.1E-02

6.0E-02

5.9E-02

5.8E-02

5.7E-02

5.6E-02

5.5E-02

5.4E-02

5.3E-02

5.2E-02

5.1E-02

5.0E-02

4.9E-02

4.8E-02

4.7E-02

4.6E-02

4.5E-02

4.4E-02

4.3E-02

4.2E-02

4.1E-02

4.0E-02

3.9E-02

3.8E-02

3.7E-02

3.6E-02

3.5E-02

3.4E-02

3.3E-02

3.2E-02

3.1E-02

3.0E-02

Starting field: 0 sec 0

2

4

6

8

10

12

14

z

Y X

Z 7.0E-02

6.9E-02

6.8E-02

6.7E-02

6.6E-02

6.5E-02

6.4E-02

6.3E-02

6.2E-02

6.1E-02

6.0E-02

5.9E-02

5.8E-02

5.7E-02

5.6E-02

5.5E-02

5.4E-02

5.3E-02

5.2E-02

5.1E-02

5.0E-02

4.9E-02

4.8E-02

4.7E-02

4.6E-02

4.5E-02

4.4E-02

4.3E-02

4.2E-02

4.1E-02

4.0E-02

3.9E-02

3.8E-02

3.7E-02

3.6E-02

3.5E-02

3.4E-02

3.3E-02

3.2E-02

3.1E-02

3.0E-02

0.6 sec 0

2

4

6

8

10

12

14

z

Y X

Z 7.1E-02

7.0E-02

6.9E-02

6.8E-02

6.7E-02

6.6E-02

6.5E-02

6.4E-02

6.3E-02

6.2E-02

6.1E-02

6.0E-02

5.9E-02

5.8E-02

5.7E-02

5.6E-02

5.5E-02

5.4E-02

5.3E-02

5.2E-02

5.1E-02

5.0E-02

4.9E-02

4.8E-02

4.7E-02

4.6E-02

4.5E-02

4.4E-02

4.3E-02

4.2E-02

4.1E-02

4.0E-02

3.9E-02

3.8E-02

1.6 sec 0

2

4

6

8

10

12

14

z

Y X

Z 7.1E-02

7.0E-02

6.9E-02

6.8E-02

6.7E-02

6.6E-02

6.5E-02

6.4E-02

6.3E-02

6.2E-02

6.1E-02

6.0E-02

5.9E-02

5.8E-02

5.7E-02

5.6E-02

5.5E-02

5.4E-02

5.3E-02

5.2E-02

5.1E-02

5.0E-02

4.9E-02

4.8E-02

4.7E-02

4.6E-02

4.5E-02

4.4E-02

4.3E-02

4.2E-02

4.1E-02

4.0E-02

3.9E-02

3.8E-02

1.8 sec

0

2

4

6

8

10

12

14

z

Y X

Z

1.27E-01

1.25E-01

1.23E-01

1.21E-01

1.19E-01

1.17E-01

1.15E-01

1.13E-01

1.11E-01

1.09E-01

1.07E-01

1.05E-01

1.03E-01

1.01E-01

9.90E-02

9.70E-02

9.50E-02

9.30E-02

9.10E-02

8.90E-02

8.70E-02

8.50E-02

8.30E-02

8.10E-02

7.90E-02

7.70E-02

7.50E-02

7.30E-02

7.10E-02

6.90E-02

6.70E-02

6.50E-02

6.30E-02

6.10E-02

5.90E-02

5.70E-02

5.50E-02

5.30E-02

5.10E-02

4.90E-02

4.70E-02

4.50E-02

4.30E-02

3.2 sec 0

2

4

6

8

10

12

14

z

Y X

Z

1.27E-01

1.25E-01

1.23E-01

1.21E-01

1.19E-01

1.17E-01

1.15E-01

1.13E-01

1.11E-01

1.09E-01

1.07E-01

1.05E-01

1.03E-01

1.01E-01

9.90E-02

9.70E-02

9.50E-02

9.30E-02

9.10E-02

8.90E-02

8.70E-02

8.50E-02

8.30E-02

8.10E-02

7.90E-02

7.70E-02

7.50E-02

7.30E-02

7.10E-02

6.90E-02

6.70E-02

6.50E-02

6.30E-02

6.10E-02

5.90E-02

5.70E-02

5.50E-02

5.30E-02

5.10E-02

4.90E-02

4.70E-02

4.50E-02

4.30E-02

4.1 sec 0

2

4

6

8

10

12

14

z

Y X

Z

1.030E-01

1.015E-01

1.000E-01

9.850E-02

9.700E-02

9.550E-02

9.400E-02

9.250E-02

9.100E-02

8.950E-02

8.800E-02

8.650E-02

8.500E-02

8.350E-02

8.200E-02

8.050E-02

7.900E-02

7.750E-02

7.600E-02

7.450E-02

7.300E-02

7.150E-02

7.000E-02

6.850E-02

6.700E-02

6.550E-02

6.400E-02

6.250E-02

6.100E-02

5.950E-02

5.800E-02

5.650E-02

5.500E-02

5.350E-02

5.200E-02

5.050E-02

4.900E-02

4.750E-02

4.600E-02

4.450E-02

4.300E-02

4.150E-02

4.000E-02

3.850E-02

3.700E-02

3.550E-02

3.400E-02

3.250E-02

4.8 sec 0

2

4

6

8

10

12

14

z

Y X

Z

1.030E-01

1.015E-01

1.000E-01

9.850E-02

9.700E-02

9.550E-02

9.400E-02

9.250E-02

9.100E-02

8.950E-02

8.800E-02

8.650E-02

8.500E-02

8.350E-02

8.200E-02

8.050E-02

7.900E-02

7.750E-02

7.600E-02

7.450E-02

7.300E-02

7.150E-02

7.000E-02

6.850E-02

6.700E-02

6.550E-02

6.400E-02

6.250E-02

6.100E-02

5.950E-02

5.800E-02

5.650E-02

5.500E-02

5.350E-02

5.200E-02

5.050E-02

4.900E-02

4.750E-02

4.600E-02

4.450E-02

4.300E-02

4.150E-02

4.000E-02

3.850E-02

3.700E-02

3.550E-02

3.400E-02

3.250E-02

5.5 sec

0

2

4

6

8

10

12

14

z

Y X

Z 7.0E-02

6.9E-02

6.8E-02

6.7E-02

6.6E-02

6.5E-02

6.4E-02

6.3E-02

6.2E-02

6.1E-02

6.0E-02

5.9E-02

5.8E-02

5.7E-02

5.6E-02

5.5E-02

5.4E-02

5.3E-02

5.2E-02

5.1E-02

5.0E-02

4.9E-02

4.8E-02

4.7E-02

4.6E-02

4.5E-02

4.4E-02

4.3E-02

4.2E-02

4.1E-02

4.0E-02

3.9E-02

3.8E-02

3.7E-02

3.6E-02

3.5E-02

3.4E-02

3.3E-02

3.2E-02

3.1E-02

3.0E-02

2.9E-02

2.8E-02

2.7E-02

2.6E-02

2.5E-02

6.0 sec 0

2

4

6

8

10

12

14

z

Y X

Z 7.0E-02

6.9E-02

6.8E-02

6.7E-02

6.6E-02

6.5E-02

6.4E-02

6.3E-02

6.2E-02

6.1E-02

6.0E-02

5.9E-02

5.8E-02

5.7E-02

5.6E-02

5.5E-02

5.4E-02

5.3E-02

5.2E-02

5.1E-02

5.0E-02

4.9E-02

4.8E-02

4.7E-02

4.6E-02

4.5E-02

4.4E-02

4.3E-02

4.2E-02

4.1E-02

4.0E-02

3.9E-02

3.8E-02

3.7E-02

3.6E-02

3.5E-02

3.4E-02

3.3E-02

3.2E-02

3.1E-02

3.0E-02

2.9E-02

2.8E-02

2.7E-02

2.6E-02

2.5E-02

6.4 sec

0

2

4

6

8

10

12

14 z (m) Y X

Z

7.4E-02

7.3E-02

7.2E-02

7.1E-02

7.0E-02

6.9E-02

6.8E-02

6.7E-02

6.6E-02

6.5E-02

6.4E-02

6.3E-02

6.2E-02

6.1E-02

6.0E-02

5.9E-02

5.8E-02

5.7E-02

5.6E-02

5.5E-02

5.4E-02

5.3E-02

5.2E-02

5.1E-02

5.0E-02

4.9E-02

4.8E-02

4.7E-02

4.6E-02

4.5E-02

4.4E-02

4.3E-02

4.2E-02

4.1E-02

4.0E-02

3.9E-02

Time-averaged over 1 cycle.

NASA/TM—2001-210977 11
AUSM+

-u

p/p∞

0.0 2.0 4.0

θ (deg) 10.0

11.0

12.0

13.0

14.0

15.0

T/T∞

0.0 10.0

NASA/TM—2001-210977 12
AUSM+ CFL=0.50 N= 200

β=0.1250 α=0.1875

-1.0 -0.5 0.0 0.5 1.0

x

0.0

20.0

40.0

60.0

80.0 Pressure

Density

Temperature

AUSMDV : f(pL,pR)=pL,K=10 CFL=0.50 N= 200

-1.0 -0.5 0.0 0.5 1.0

x

0.0

20.0

40.0

60.0

80.0 Pressure

Density

Temperature

AUSM+-u CFL=0.50 N= 200

β=0.1250 α=0.1875

-1.0 -0.5 0.0 0.5 1.0

x

0.0

20.0

40.0

60.0

80.0 Pressure

Density

Temperature

AUSM+-up CFL=0.50 N= 200

-1.0 -0.5 0.0 0.5 1.0

x

0.0

20.0

40.0

60.0

80.0 Pressure

Density

Temperature

Roe CFL=0.50 N= 200

-1.0 -0.5 0.0 0.5 1.0

x

0.0

20.0

40.0

60.0

80.0 Pressure

Density

Temperature

Exact Riemann Solver CFL=0.50 N= 200

-1.0 -0.5 0.0 0.5 1.0

x

0.0

20.0

40.0

60.0

80.0 Pressure

Density

Temperature

AUSM+ CFL=0.50 N=4000

β=0.1250 α=0.1875

x

-1.0 -0.5 0.0 0.5 1.0 -1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

u-c : ∑(∆p -ρ ̄c ̄ ∆u)/(2c ̄ 2)

u+c : ∑(∆p +ρ ̄c ̄ ∆u)/(2c ̄ 2) x10

u : ∑(∆ρ-∆p/c ̄ 2) x10

AUSM+-u CFL=0.50 N=4000

x

-1.0 -0.5 0.0 0.5 1.0 -1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

u-c : ∑(∆p -ρ ̄c ̄ ∆u)/(2c ̄ 2)

u+c : ∑(∆p +ρ ̄c ̄ ∆u)/(2c ̄ 2) x10

u : ∑(∆ρ-∆p/c ̄ 2) x10

Roe

E-Fix CFL=0.50 N=4000

x

-1.0 -0.5 0.0 0.5 1.0 -1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

u-c : ∑(∆p -ρ ̄c ̄ ∆u)/(2c ̄ 2)

u+c : ∑(∆p +ρ ̄c ̄ ∆u)/(2c ̄ 2) x10

u : ∑(∆ρ-∆p/c ̄ 2) x10

Exact Riemann Solver CFL=0.50 N=4000

x

-1.0 -0.5 0.0 0.5 1.0 -1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

u-c : ∑(∆p -ρ ̄c ̄ ∆u)/(2c ̄ 2)

u+c : ∑(∆p +ρ ̄c ̄ ∆u)/(2c ̄ 2) x10

u : ∑(∆ρ-∆p/c ̄ 2) x10

320.0 340.0

X-axis(centerline)

Pressure

0.0

Y-axis(behind shock)

Density

0.0

2.0

4.0

6.0

8.0

GERROR = 0.10E-01

ITPG = 2.53

JMAX = 350

CFL = 0.50

Pr/Pl = 0.71E+01

Density Contours

AUSM+-u

NASA/TM—2001-210977 13
NASA/TM—2001-210977 14
NASA/TM—2001-210977 15
This publication is available from the NASA Center for AeroSpace Information, 301–621–0390.

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13. ABSTRACT (Maximum 200 words)

14. SUBJECT TERMS

17. SECURITY CLASSIFICATION

 OF REPORT

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Unclassified Unclassified

Technical Memorandum

Unclassified

National Aeronautics and Space Administration

John H. Glenn Research Center at Lewis Field

Cleveland, Ohio 44135–3191

1. AGENCY USE ONLY (Leave blank)

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 AGENCY REPORT NUMBER

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National Aeronautics and Space Administration

Washington, DC 20546–0001

Available electronically at http://gltrs.grc.nasa.gov/GLTRS

November 2001

NASA TM—2001-210977

AIAA–2001–2521

E–12831

WU–708–28–13–00

21

CFD; Upwind schemes; AUSM schemes; Low Mach-number flows; Multiphase flows

Unclassified -Unlimited

Subject Categories: 02 and 34 Distribution: Nonstandard

Prepared for the 15th Computational Fluid Dynamics Conference sponsored by the American Institute of Aeronautics and

Astronautics, Anaheim, California, June 11–14, 2001. Responsible person, Meng-Sing Liou, organization code 5880,

216–433–5855.

Ten Years in the Making—AUSM-family

Meng-Sing Liou

We begin by describing the motivations that gave birth to the original AUSM scheme and then focus on the ingredients

that has spurred its growth and acceptance by the world of computational fluid dynamics. As it has played out more in

the field, weaknesses have also surfaced. Hence, nutrients and supplements are prescribed to help it grow and stay

strong and robust. In this paper, we will describe the saga of efforts owing to researchers who have contributed to

building up the AUSM-family for the CFD community. It is hoped that a healthy scheme will contribute to the

accurate and robust solution of problems encountered in a wide range of disciplines. We analyze numerical mass

fluxes with an emphasis on their capability for accurately capturing shock and contact discontinuities. We will present

a new scheme for the pressure flux, along with results for a host of test problems.