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
August 14th, 2012
Motivation
• Explict CFD methods resolve all relevant time scales of the
wave dynamics (~ 1 nano sec).
Cavitating flow in a
micro channel
Skoda, Iben, Morozov,
Mihatsch, Schmidt, Adams:
Warwick, UK, 2011
Liquid Volume fraction
• 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, …)
Pressure
Distance of Wave
travel at CFL = 1
The smallest cell in the domain
dictates the overall time step
• Is it really necessary to resolve all time scales?
We would like to increase the time step systematically and
therefore need an implicit method.
Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012
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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
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Non-Cavitating Riemann problem (CFL = 1)
• Temporal pressure development for 100 bar / 1 bar
Explicit
2. Order
p [Pa]
Implicit
2. Order in Space
1. Order in Time
1.2E+07
p [Pa]
1.2E+07
1.0E+07
1.0E+07
8.0E+06
8.0E+06
6.0E+06
6.0E+06
4.0E+06
4.0E+06
2.0E+06
2.0E+06
0.0E+00
0.00 0.02 0.04 0.06 0.08 0.10
Time instant
1
x [m]
2
3
0.0E+00
0.00 0.02 0.04 0.06 0.08 0.10
x [m]
4
With the Implicit method, we can reproduce the Explicit method results.
Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012
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Cavitating Riemann problem (CFL = 1)
• Temporal pressure development for 1 bar / 0.073 bar
Explicit
2. Order
p [Pa]
Implicit
2. Order in Space
1. Order in Time
1.2E+05
p [Pa]
1.2E+05
1.0E+05
1.0E+05
8.0E+04
8.0E+04
6.0E+04
4.0E+04
Twophase
flow
6.0E+04
4.0E+04
2.0E+04
2.0E+04
0.0E+00
0.00 0.02 0.04 0.06 0.08 0.10
Time instant
1
x [m]
2
3
0.0E+00
0.00 0.02 0.04 0.06 0.08 0.10
x [m]
4
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.
Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012
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NACA profile
• Computational setup
– Re = 1.56 e5
– a = 6°
S>0
S=0
y
x
Instantaneous results
Pressure
Vapour Volume
Fraction
Shock
wave
Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012
Periodic
shedding and
re-entrant jet
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Explicit vs implicit method at CFL = 2
Temporal development
of the wall pressure
1.E+07
Erosion probability
2.0E-03
THR =
5 bar
PP [-]
p [Pa]
Statistical
evaluation
(threshold)
8.E+06
6.E+06
1.5E-03
1.0E-03
Explicit
Implicit
4.E+06
5.0E-04
2.E+06
0.E+00
0.15
0.0E+00
0.20
0.25
t [ms]
0.30
0.0
0.5
1.0
10*s [m]
Analysis interval
1.5 Co-ordinate s
along suction
surface
s
The Explicit and Implicit
methods yield similar results.
Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012
s=0
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Increase of the CFL number
• Integral Vapour Volume Fraction
Integral vapour VF [-]
0.006
CFL = 2
0.006
CFL = 20
0.006
CFL = 200
0.006
CFL = 2000
0.004
0.004
0.004
0.004
0.002
0.002
0.002
0.002
0.000
0.000
0.3 0.1
0.000
0.3 0.1
0.000
0.3 0.1
0.1
0.2
t [sec]
0.2
t [sec]
0.2
t [sec]
0.2
t [sec]
0.3
No significant influence of the CFL number.
Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012
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Maximum pressure at suction surface
• Maximum pressure on the
suction suface in the
analysis time interval
1.E+08
pMax [Pa]
1.E+07
1.E+06
CFL = 2
CFL = 20
CFL = 200
1.E+05
CFL = 2000
0.0
0.5
1.0
1.5
10*s [m]
Co-ordinate s along suction side
Pressure peaks get lower with increasing CFL number.
Conclusion: the threshold for the statistical evaluation must not be too high.
Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012
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Wall load at suction surface
• Erosion probability
6.E-02
PP [-]
CFL = 2
CFL = 20
CFL = 200
CFL = 2000
THR =
1.5 bar
4.E-02
2.E-02
0.E+00
0.0
0.5
1.0
1.5
10*s [m]
Co-ordinate s
along suction side
For higher CFL-number, the solution tendency is maintained.
Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012
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Application to a micro channel flow
•
•
•
•
Hight: 100 mm
Length: 1000 mm
Inlet pressure pin = 300 bar
Variation of the outlet pressure
pout = 80 bar
pout = 125 bar
pout = 160 bar
Explicit CFL = 1
0.12
Pp [-]
0.10
THR =
250 bar
0.08
Erosion
probability
Erosion
probability
Pp [-]
Implicit CFL = 100
0.06
0.04
0.12
0.08
0.06
0.04
0.02
0.02
0.00
0.00
0.0
0.2
0.4
0.6
0.8
THR =
250 bar
0.10
1.0
Channel length [-]
Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012
0.0
0.2
0.4
0.6
0.8
1.0
Channel length [-]
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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 nearwall 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 densitybased methods.
Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012
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