August 14th, 2012

Comparison of compressible explicit density-based and implicit pressure-based

CFD methods for the simulation of cavitating flows

Romuald Skoda Uwe Iben Martin Güntner,Rudolf Schilling

Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012 2

Pressure

Motivation

• Explict CFD methods resolve all relevant time scales of the wave dynamics (~ 1 nano sec).

• Problem: Due to the coupling of spatial and temporal resolution (accoustic CFL condition) explicit methods generate prohibitely long computation times in complex geometries (injection systems, pumps, …)

• Is it really necessary to resolve all time scales? We would like to increase the time step systematically and therefore need an implicit method.

Distance of Wave travel at CFL = 1

Cavitating flow in a micro channel

The smallest cell in the domain dictates the overall time step

Skoda, Iben, Morozov, Mihatsch, Schmidt, Adams: Warwick, UK, 2011

Liquid Volume fraction

Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012 3

Numerical method and Physical model

• To get an implict method we modify a compressible standard pressure-based algorithm (SIMPLE, 2. order in space and time)

1.) local underrelaxation (preconditioning of the matrix)2.) density- instead of pressure correction, pressure from

EOS

• Reference method: Explicit density-based code with CATUM flux functions (TU Munich) and time integration scheme (2. order)

• Homogenous model

• Neglect of the energy equation and use of a barotropic EOS

• inviscid flow

Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012 4

Non-Cavitating Riemann problem (CFL = 1)

• Temporal pressure development for 100 bar / 1 bar

0.0E+00

2.0E+06

4.0E+06

6.0E+06

8.0E+06

1.0E+07

1.2E+07

0.00 0.02 0.04 0.06 0.08 0.10

Explicit2. Order

0.0E+00

2.0E+06

4.0E+06

6.0E+06

8.0E+06

1.0E+07

1.2E+07

0.00 0.02 0.04 0.06 0.08 0.10

Implicit2. Order in Space1. Order in Time

1 2 3 4Time instant

With the Implicit method, we can reproduce the Explicit method results.

x [m]x [m]

p [Pa] p [Pa]

Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012 5

Cavitating Riemann problem (CFL = 1)

• Temporal pressure development for 1 bar / 0.073 bar

1 2 3 4Time instant

0.0E+00

2.0E+04

4.0E+04

6.0E+04

8.0E+04

1.0E+05

1.2E+05

0.00 0.02 0.04 0.06 0.08 0.100.0E+00

2.0E+04

4.0E+04

6.0E+04

8.0E+04

1.0E+05

1.2E+05

0.00 0.02 0.04 0.06 0.08 0.10x [m]x [m]

Two-phase flow

With the Implicit method, we can reproduce the Explicit method results. Conclusion: The Implicit scheme is feasible.For the next test case, we use a second order in time and space.

p [Pa]

Implicit2. Order in Space1. Order in Time

Explicit2. Order

p [Pa]

Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012 6

NACA profile

• Computational setup– Re = 1.56 e5– a = 6° S=0

S>0

x

y

Pressure

Vapour Volume Fraction

Instantaneous results

Shock wave

Periodic shedding and re-entrant jet

Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012 8

0.E+00

2.E+06

4.E+06

6.E+06

8.E+06

1.E+07

0.15 0.20 0.25 0.30

Explicit vs implicit method at CFL = 2

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

0.0 0.5 1.0 1.5

PP [-] THR = 5 bar

Erosion probability

10*s [m]

The Explicit and Implicit methods yield similar results.

Co-ordinate s along suction surface

Temporal development of the wall pressure

t [ms]

p [Pa]Statistical evaluation (threshold)

ExplicitImplicit

s=0

s

Analysis interval

Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012 9

Increase of the CFL number

• Integral Vapour Volume Fraction

0.000

0.002

0.004

0.006

0.1 0.2 0.30.000

0.002

0.004

0.006

0.1 0.2 0.30.000

0.002

0.004

0.006

0.1 0.2 0.30.000

0.002

0.004

0.006

0.1 0.2 0.3t [sec]

Inte

gral

vap

our V

F [-]

t [sec]t [sec]t [sec]

CFL = 2 CFL = 20 CFL = 200 CFL = 2000

No significant influence of the CFL number.

Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012 10

Maximum pressure at suction surface

• Maximum pressure on the suction suface in the analysis time interval

Pressure peaks get lower with increasing CFL number.Conclusion: the threshold for the statistical evaluation must not be too high.

1.E+05

1.E+06

1.E+07

1.E+08

0.0 0.5 1.0 1.5

pMax [Pa]

10*s [m]

Co-ordinate s along suction side

CFL = 2

CFL = 20

CFL = 200

CFL = 2000

Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012 11

• Erosion probability

Wall load at suction surface

For higher CFL-number, the solution tendency is maintained.

0.E+00

2.E-02

4.E-02

6.E-02

0.0 0.5 1.0 1.5

THR = 1.5 bar

PP [-]

CFL = 2CFL = 20CFL = 200CFL = 2000

10*s [m]

Co-ordinate s along suction side

Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012 12

Application to a micro channel flow

• Hight: 100 mm• Length: 1000 mm• Inlet pressure pin = 300 bar• Variation of the outlet pressure

pout = 80 barpout = 125 barpout = 160 bar

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.0 0.2 0.4 0.6 0.8 1.0

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.0 0.2 0.4 0.6 0.8 1.0

Explicit CFL = 1 Implicit CFL = 100Pp [-] THR =

250 bar

Channel length [-]Channel length [-]

Pp [-]THR = 250 bar

Ero

sion

pr

obab

ility

Ero

sion

pr

obab

ility

Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012 13

Conclusions

• For the prediction of the wall load which is the origin of cavitation erosion it is sufficient to use CFL ~ 100.

• Possible application: visous flow computations with a fine near-wall resolution.

• The pressure-based code has in total a much higher CPU time than the explicit code due to numerical issues. The cost per time step must be decreased.

• For future investigations we recommend to use implicit density-based methods.