Interpolation error in DNS simulations of turbulence: consequences for particle tracking

Transcription

1 Interpolation error in DNS simulations of turbulence: consequences for particle tracking Michel van Hinsberg Department of Physics, Eindhoven University of Technology, PO Box 513, 5600MB Eindhoven, The Netherlands J.H.M. ten Thije Boonkkamp F. Toschi H.J.H. Clercx

2 Content Introduction Interpolation - Performance - Consequences Summary and outlook Calzavarini, E., Kerscher, M., Lohse, D. and Toschi. F. Dimensionality and morphology of particle and bubble clusters in turbulent flow. Journal of Fluid Mechanics, 607:13-24., / name of department PAGE 1

3 Introduction Fluid Isotropic turbulence DNS simulations Pseudo spectral code Tri-periodic domain Particles Lagrangian tracking One-way coupling Small light spherical particles Maxey & Riley equations / name of department PAGE 2

4 Introduction Light particles Bed-load sediments - Pattern formation - turbidity currents Plankton aggregates Plankton / name of department PAGE 3

5 Criteria for interpolation methods High order of convergence High order of smoothness Small number of FFTs Small overall errors / name of department PAGE 4

6 Interpolation methods Order of convergence Order of smoothness FFT Overall errors comment Lagrange Interpolation N For even N N For odd N Spline interpolation N-2 (N-2)/2 1 - Only even N Hermite interpolation N-1 (N-2)/2 8 + Only even N B-spline interpolation N-1 N / name of department PAGE 5

7 B-spline functions / name of department PAGE 6

8 B-spline interpolation 2 steps: - transformation to B-spline basis - carry out the interpolation Transformation efficiently done in Fourier space Fast algorithm for interpolation / name of department PAGE 7

9 Relative interpolation error k x / name of department PAGE 8

10 Discretisation error Re=275 K max η=1.8 K max η=3.7 / name of department PAGE 9

11 Error interpolation / name of department PAGE 10

12 Comparison errors Error ratio = Interpolation error Discretisation error - Linear inter. (N=2): Lagrange inter. (N=4): Spline inter. (N=4): Hermite inter. (N=4): B-spline inter. (N=4): 0.08 Error ratio = / name of department PAGE 11

13 Lagrangian error Testing the theory Time evolvement of error Evolving tracers with different interpolation methods Comparison with a simulation of double grid resolution / name of department PAGE 12

14 Lagrangian error of tracers Linear N=2 B spline N=2 B spline N=4 L 4 L 2 L 1 error t / name of department PAGE 13

15 Lagrangian error of tracers Linear N=2 B spline N=2 B spline N=4 L 4 L 2 L 1 error t / name of department PAGE 14

16 Lagrangian error at Kolmogorov time scale 10 1 Lagrange Spline B spline Hermite 10 2 error N / name of department PAGE 15

17 Lagrangian error double grid resolution 10 1 Lagrange Spline 10 2 B spline Hermite 10 3 error N / name of department PAGE 16

18 Acceleration of particles Important for the calculation of forces on the particles Important for statistics / name of department PAGE 17

19 Acceleration of tracers a 5 linear N=2 10 Lagrange N=4 spline N=4 B spline N=4 Hermite N= t / name of department PAGE 18

20 Acceleration of tracers a linear N=2 10 Lagrange N=4 spline N=4 B spline N=4 Hermite N= t / name of department PAGE 19

21 Acceleration spectrum acc spectrum linear N=2 Lagrange N=4 spline N=4 B spline N=4 Hermite N=4 Spectral k / name of department PAGE 20

22 Acceleration spectrum acc spectrum linear N=2 Lagrange N=4 spline N=4 B spline N=4 Hermite N=4 Spectral k / name of department PAGE 21

23 Energy in spectrum 10 3 Lagrange Spline B spline Hermite 10 4 energy N / name of department PAGE 22

24 Energy in spectrum double grid resolution 10 3 Lagrange Spline 10 4 B spline Hermite 10 5 energy N / name of department PAGE 23

25 Optimal interpolation method N kmax = 3 N g 3 Lagrange 2 Spline B spline Hermite k η max / name of department PAGE 24

26 Conclusions Advantages B-spline interpolation - High order of convergence - High order of smoothness - Low number of FFTs needed - Errors comparable with Hermite interpolation M.A.T. van Hinsberg, J.H.M. ten Thije Boonkkamp, F. Toschi and H.J.H. Clercx. On the effiency and accuracy of interpolation methods for spectral codes. Submitted to SIAM J. Sci. Comput. Methods for estimation interpolation error - Compare interpolation error with discretisation error - Linear interpolation not sufficient for our purpose / name of department PAGE 25

27 Questions? / name of department PAGE 26

28 Optimal interpolation method Criteria: Error ratio < 1 6 N Lagrange 2 Spline B spline Hermite k η max k = 2 max 3 N g / name of department PAGE 27

29 real(u k ) F(U k ) k -1 0 k D 1 F(D) 1 -- x x real(u k D) 1 F(U k D) 1 -- x 0 0 k k x / name of department PAGE 28

30 real(u k D) 1 F(U k D) 1 -- x 0 0 k k x C 1 F(C) x real((u k D)*C) 1 F((U k D)*C) k k x -1 / name of department PAGE 29

31 3D interpolation Sequence of 1D interpolations: fast 2 properties needed Superposition M. van Aartrijk, Dispersion of inertial particles in stratified turbulence. PHD thesis, Eindhoven University of technology, / name of department PAGE 30

32 Maxey & Riley equations / name of department PAGE 31

33 Maxey & Riley equations M.A.T. van Hinsberg, J.H.M. ten Thije Boonkkamp, and H.J.H. Clercx. An efficient, second order method for the approximation of the Basset history force. J. Comput. Phys., 230: , / name of department PAGE 32

34 Interpolation High order interpolation vs linear interpolation Accuracy vs Speed Interpolation error vs discretisation error u o x 2 x 3 x 4 x / name of department PAGE 33

35 Error for B-spline interpolation error t e t e double res t K t K double res t e = after one eddy turnover time t k = at Kolmogorov time scale N / name of department PAGE 34