Numerical solution of RANS equations. Numerical solution of RANS equations

Transcription

1 Convection linearization - Newton (Quasi-Newton) - Picard Discretization - Finite-differences - Finite-volume - Galerkin (Finite-elements,...) Solution of the algebraic systems of equations - Segregated - Coupled Linear solver - Usually iterative - Gauss-Seidel, Bi-CGSTAB, GMReS - Multigrid 2

2 Flow over a flat plate Re L = k-ω BSL,2 2,3 k-ω Wilcox 2,3 p= /h 3 Flow over a flat plate 2.8 Re L = k-ω BSL 2,3 2,3 k-ω Wilcox 2,3 p= /h 4

3 Flow over a flat plate Re L = k-ω BSL 2,3 2,3 k-ω Wilcox p= 2.0 p= /h 5 Flow over a flat plate Re L = ,3 p=.3 k-ω TNT, /h 6

4 Flow over a flat plate Re L = p=.8 p=.3 k-ω TNT 2, /h 7 Flow over a flat plate.6.58 Re L = p=.9 p=.3 k-ω TNT 2, /h 8

5 Flow over a flat plate Re L =0 7 Spalart & Allmaras p=.3 p=.7 Mν t p=.4 p=. KSKL p=.6 p= /h 9 Flow over a flat plate Re L = Spalart & Allmaras p=.5 p=.9 Mν t,2,2 KSKL p=.4 p= /h 0

6 Flow over a flat plate.58 Re L = Spalart & Allmaras p=.4 p=.9 Mν t p=.0 p=.4 KSKL p=.6 p= /h Flow over a flat plate Modelo C f 20 (ν t ) inlet =0.0ν (ν t ) inlet =ν (ν t ) inlet =0ν Blasius C f =0.370(log 0 Re x ) C f =(2log 0 Re x -0.65) -2.3 C f =0.455/(log(0.06Re x )) 2 Experimental, k =0.009U 2 Experimental, k =0.03U 2 Experimental, k =0.06U Re x 2

7 Flow over a flat plate ITTC 57 Schoenherr Grigson Katsui et al. SST k-ω log(rn) 3 4

8 5 6

9 7 3.6 k-ω ah+bh 2 Model Rn=4, /h 8

10 0.8 k-ω Model Rn=4,6 0 6 C P /h 9.55 Model Rn=2, p= /h 20

11 0.3 k-ω p=.6 Model Rn=2, C P /h 2 Cp: Model Rn=4,6 0 6 Cp: Rn=2,

12 Rn=4,6 0 6 Model Rn=2, U x : Model Z/Lpp Rn=4, Y/Lpp 24

13 U x : Model Z/Lpp Rn=2, Y/Lpp U Model z/l PP Rn=4, y/l PP 26

14 0-0.0 U Model z/l PP Rn=2, y/l PP 27 U x U x z= l PP z=-0.029l PP Model U x 0.5 z= l PP U x U x U x Experimental G G2 G3 G4 z= l PP z= l PP z=-0.058l PP y/l PP Rn=4,

15 0.6 k-ω ah 2 +bh Model 0.56 W f /h Wake fraction Rn=4, k-ω p= Model W f Rn=2, /h Wake fraction 30

16 3 in turbulent flow + Hydrodynamically smooth hr < 5 Fully rough hr+ = uτ hr ν hr+ > C f hr x hr 0.9 RL 0.7 RL = L L 2 L 32

17 Model testing is performed with hydrodynamically smooth surfaces Full scale ships have rough surfaces Extrapolation procedures include the roughness effects in the correlation allowance 33 Non-uniform roughness of a given hull is converted to an equivalent sand-grain roughness (defined by a single parameter, ) Such conversion is a fundamental unsolved problem that we have not addressed 34

18 Empirical correlations use the average roughness height, h M, (also single parameter) to quantify roughness effects (50µm recommended for clean ships) According to Schultz (2007) h M =50µm =30µm 0.2h M 35 Several proposals available in the open literature for the inclusion of sand-grain roughness effects in eddy-viscosity turbulence models Present research performed for the SST two-equation k-ω model 36

19 Sand-grain roughness effects may be included with or without wall functions boundary conditions Wilcox (2006) proposal includes only a change in the ω wall boundary condition Extra damping function in the limiter of the eddy-viscosity, (Hellsten, 997) 37 Calculations performed for 0 300µm Three test cases:. Tanker (model and full scale) 2. Container ship (model and full scale) 3. Car carrier (full scale) Sets of 6 grids (4 0 6 to cells for =) 38

20 No gravity waves (double body) Selected convergence criteria guarantees that the iterative error is negligible when compared with the discretization error Numerical uncertainty estimated for functional quantities with a least squares version of the GCI 39 H-O grids Tanker Container ship 40

21 Numerical uncertainty =0 p=.8 /L=30µm p= 2.0 /L=50µm p * =2 /L=75µm p * =2 /L=00µm p=.9 X X X X X X X /h /L=50µm p= 0.9 /L=200µm p * =2 /L=250µm p * =2 /L=300µm p * =2 Tanker C P =0 p=.5 /L=30µm p=.5 /L=50µm p=.6 /L=75µm p=.6 /L=00µm p=.6 X + + X X /h /L=50µm p=.6 /L=200µm p=.6 /L=250µm p=.7 /L=300µm + X + X + X + X Friction resistance, Pressure resistance, C P Numerical uncertainty =0 ah+bh 2 /L=30µm p= 0.6 /L=50µm /L=75µm /L=00µm X X X /h /L=50µm p= 4.0 /L=200µm p * =2 /L=250µm p= 2.0 /L=300µm p=.9 + X Container ship X X X C P =0 ah+bh 2 /L=30µm ah+bh 2 /L=50µm ah+bh 2 /L=75µm ah+bh 2 /L=00µm ah+bh 2 X + + X X /h /L=50µm ah+bh 2 /L=200µm ah+bh 2 /L=250µm ah+bh 2 /L=300µm + X + X + X + X Friction resistance, Pressure resistance, C P 42

22 Reynolds number based on the roughness height, R hr 43 / ( =0).5 Tanker, Rn= Tanker, Rn= Container ship, Rn= Container ship, Rn= Car carrier, Rn= Friction resistance, / ( =0) Tanker, Rn= Tanker, Rn= Container ship, Rn= Container ship, Rn= Car carrier, Rn= h R ( µm) R = hru ν 44

23 Pressure resistance, C P.5 Tanker, Rn= Tanker, Rn= Container ship, Rn= Container ship, Rn= Car carrier, Rn= Tanker, Rn= Tanker, Rn= Container ship, Rn= Container ship, Rn= Car carrier, Rn= C P /C P ( =0).3.2. C P /C P ( =0) h R ( µm) R = hru ν 45 Wake fraction, W f W f /W f ( =0).5 Tanker, Rn= Tanker, Rn= Container ship, Rn= Container ship, Rn= Car carrier, Rn= W f /W f ( =0) Tanker, Rn= Tanker, Rn= Container ship, Rn= Container ship, Rn= Car carrier, Rn= ( µm) R = hru ν 46

24 Non-dimensional roughness height, + 47 Comparison with empirical correlations Bowden-Davison Townsin et al Himeno Wright C F C C C F F F 0 3 h = 05 L PP 3 3 h M 0 = 44 LPP 3 hm 0 8 RL L = PP R 3 L M hm B T 0 =.03 3 RL 3 T 48

25 Comparison with empirical correlations h M (µm) Bowden and Davison Townsin et al. Himeno Wright C V numerical, tanker C V numerical, container ship C V numerical, car carrier (µm) 49 Comparison with empirical correlations h M U /ν Bowden and Davison Townsin et al. Himeno Wright C V numerical, tanker C V numerical, container ship C V numerical, car carrier U /ν 50

26 Series 60 inviscid flow `h / L;Y/L;V3 viscous (model) `h / L;Y/L;V3 viscous (ship) `h / L;Y/L;V3 experiments (model) `h / L;Y/L;V3 y / L pp = z / L pp Dyna Tanker inviscid z;y/l;v3 flow viscous z;y/l;v3 (model) viscous z;y/l;v3 (ship) experiments z;y/l;v3 (model) y / L pp =

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