1. August 14 th, 2012 Comparison of compressible explicit density-based and implicit pressure-based CFD methods for the simulation of cavitating flows Romuald SkodaUwe IbenMartin Güntner, Rudolf Schilling

2. 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

3. 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

4. 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 Explicit 2. Order Implicit 2. Order in Space 1. Order in Time 12 34 Time instant With the Implicit method, we can reproduce the Explicit method results. x [m] p [Pa]

5. 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 12 34 Time instant 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] Implicit 2. Order in Space 1. Order in Time Explicit 2. Order p [Pa]

6. Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012 6 NACA profile Computational setup –Re = 1.56 e5 –  = 6° Pressure Vapour Volume Fraction Instantaneous results Shock wave Periodic shedding and re-entrant jet

7. Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012 7 Convergence of the implicit method Equation / CFL2202002000 U5.64.23.42.9 V5.74.13.32.9 M4.63.42.51.7 Resiudual drop:  CFL = 2 and 20:good  CFL = 200feasible  CFL = 2000poor Average drop of residuums CFL = 2 CFL = 200

8. Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012 8 Explicit vs implicit method at CFL = 2 P P [-] 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) Explicit Implicit s=0 s Analysis interval

9. Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012 9 Increase of the CFL number Integral Vapour Volume Fraction t [sec] Integral vapour VF [-] t [sec] CFL = 2CFL = 20CFL = 200CFL = 2000 No significant influence of the CFL number.

10. 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. p Max [Pa] 10*s [m] Co-ordinate s along suction side CFL = 2 CFL = 20 CFL = 200 CFL = 2000

11. 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. THR = 1.5 bar P P [-] CFL = 2 CFL = 20 CFL = 200 CFL = 2000 10*s [m] Co-ordinate s along suction side

12. Implicit pressure-based CFD methods for Cavitatiing Flows | Romuald Skoda | 14.08.2012 12 Application to a micro channel flow Hight: 100  m Length: 1000  m Inlet pressure p in = 300 bar Variation of the outlet pressure p out = 80 bar p out = 125 bar p out = 160 bar Explicit CFL = 1Implicit CFL = 100 P p [-] THR = 250 bar Channel length [-] P p [-] THR = 250 bar Erosion probability

13. 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.