Computation of time-periodic flows with the Time Spectral Method

Transcription

1 Computation of time-periodic flows with the Time Spectral Method Sabine Baumbach Volkswagen AG Component Development Prof. Dr. Norbert Kroll - Deutsches Zentrum für Luft- und Raumfahrt

2 1. Problem Aim: Development of an efficient tool to compute time-periodic, incompressible flows in the open source tool OpenFOAM Method: investigation of the application of Time Spectral Method (TSM) to incompressible fluids Implementation of TSM and suitable boundary conditions into OpenFOAM Important: Has to be fast and stable! page 2

3 2. What is TSM? Alternative method to regular unsteady Reynolds Averaged Navier Stokes (U-RANS) methods U-RANS equations are transformed by Fourier transformation into a system of stationary coupled equations Thus numerical methods for steady state computations can also be applied to unsteady problems page 3

4 2. What is TSM? 2-d Navier Stokes equation: W = dd + F Ndd = 0 Ω Ω ρ ρu 1 ρu 2 ρρ F = ρu i ρu 1 u i + δ 1i p + σ ii ρu 2 u i + δ 2i p + σ ii ρρu i + u i p + u j σ ii p = γ 1 ρ(e 1/2(u j u j )) ρ u 1, u 2 E λ δ ii μ density velocity internal energy adiabatic exponent Kronecker delta viscosity σ ii = μ u i x j + u j x i + λδ ii u k x k page 4

5 2. What is TSM? V + R W = 0 Integral equations are discretised W t = W k e ikωt k= Each variable can be presented as Fourier polynom R t = R e iiωt k k= The term R k yields non-linear terms in the frequency domain, therefore problematic to solve iiii W k + R k = 0 Thus the equation system can be derived in frequency domain page 5

6 2. What is TSM? N W n = W k e iωnδt k= N R W n + VD t W n = 0 D t W n N = d m W n+m m= N d m = π T 1 m+1 csc ( ππ 2N + 1 ) V n τ n + R W n + VD t W n = 0 Retransformation of the single terms into time domain by using a Fourier matrix (DFT) Whole equation system is transformed into time domain D t couples the single time steps with each other over all equations An additional pseudo time term is added to march the equations to convergence stationary equation system with coupled time source term page 6

7 3. U-RANS Starting from time t 0 the future times are computed using small intermediate time steps d t the soultion at time t 0 + d t can be determined by using iterative schemes amplitude A U-RANS time t dt V + R W = 0 page 7

8 3. TSM If the bse frequency of a problem is known, the flow solution can be determined at equidistant distributed times within a period (here: 3 time steps (t 0, t 1, t 2 )) All time steps are coupled with each other (e.g.to compute the solution at time t 0 the solutaions at the time steps t 1 and t 2 are also used The number of time steps for the TSM is smaller than using regular U-RANS methods amplitude A TSM t 0 dt t 1 time t t 2 coupled V W 0 + R W 0 + VD t (W 0 ) = 0 V W 1 + R W 1 + VD t (W 1 ) = 0 V W 2 + R W 2 + VD t (W 2 ) = 0 page 8

9 4. Testcase pitching wing Data: Inlet: v = 1 m/s p = = 0 Outlet: v = v = 0 p = 0 Pa Wing: ω= rotating around quarter chord c(chord length) = 1 Re = u c ν = 100 page 9

10 4. Testcases pitching wing Mesh Fig. 2: near field mesh Fig. 1: farfield mesh page 10

11 4. Testcases pitching wing Fig. 3: Pitching wing U-RANS computation page 11

12 4. Testcases pitching wing 0,075 0,065 U-RANS TSM1 Fig. 5: wing at t 0 Drag 0,055 0,045 0,035 Fig. 6: wing at t 1 0,025 0,011 0,012 0,013 0,014 0,015 Lift Fig. 7: wing at t 2 Fig. 4: wing position at all times page 12

13 4. Testcases pitching wing Drag 0,075 0,07 0,065 U-RANS CCM+ 0,06 U-RANS 0,055 TSM1 TSM2 0,05 TSM3 0,045 TSM4 0,04 0,035 0,03 0,025 0,011 0,012 0,013 0,014 0,015 Lift page 13

14 4. Testcases ball valve Data: f = 183,3 Hz u = 1 m/s hub = 0,4 mm o Fig. 8: U-RANS computation of ball valve page 14

15 4. Testcases ball valve 3,00 Druck auf Kugel pressure Druck in Pa in Pa 2,50 2,00 1,50 1,00 0, U-RANS TSM N=1 TSM N=2 TSM N=3 TSM N=4 0,00 0,000 0,002 0,004 0,006 Zeit in s time in s time: 0 s s s -S. Baumbach page 15

16 5. Conclusion Time Spectral Method has been successfully implemented into OpenFOAM and tested for easy laminar, incompressible cases Next steps: Include turbulence model Increase stability and decrease computation time page 16

17 6. Acknowledgments Special thanks to: Dr. Andreas Gitt-Gehrke (Volkswagen) Prof. Dr. Norbert Kroll (Deutsches Zentrum für Luftund Raumfahrt) Contact: Sabine Baumbach Volkswagen Salzgitter S.Baumbach page 17