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# Development of a Pseudo-Spectral 3D Navier Stokes Solver for Wind Turbine Applications

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M.Sc. Thesis Defense

Details in the thesis document:
http://www.emrebarlas.com/msc/msthesis.pdf

code in:
https://github.com/emrebar/Spectral_NavierStokes

videos in:
http://www.emrebarlas.com/msc/masters.html

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### Development of a Pseudo-Spectral 3D Navier Stokes Solver for Wind Turbine Applications

1. 1. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Development of a Pseudo-Spectral 3D Navier Stokes Solver for Wind Turbine Wakes Emre Barlas Technical University of Denmark s110988@student.dtu.dk May 26, 2014 Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 1/57
2. 2. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Overview 1 Introduction 2 Fourier & Chebyshev SM 3 3D Navier-Stokes Time & Spatial Discretization 4 Comp. Domain & W.T. Representation Computational Domain Wind Turbine Representation Methods 5 Validation AD-NR AD-R 6 Simulations 7 Conclusions & Future Work Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 2/57
3. 3. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Introduction ∂ρ ∂t + · (ρu) = 0 + ρ ∂u ∂t + u · u = − p + µ 2 u + f Approximate with; • Finite Diﬀerence FD • Finite Volume FVM • Finite Element FEM • Spectral Methods FUNCTIONAL • Spectral Element Methods SEM • Vortex/Particle VP etc. Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 3/57
4. 4. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl ∂ρ ∂t + · (ρu) = 0 + ρ ∂u ∂t + u · u = − p + µ 2 u + f Approximate with; • Finite Diﬀerence DISCRETE • Finite Volume DISCRETE • Finite Element FUNCTIONAL • Spectral Methods FUNCTIONAL • Spectral Element Methods HYBRID Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 4/57
5. 5. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Approximate with; • Finite Diﬀerence DISCRETE LOCAL • Finite Volume DISCRETE LOCAL • Finite Element FUNCTIONAL LOCAL • Spectral Methods FUNCTIONAL GLOBAL • Spectral Element Methods HYBRID Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 5/57
6. 6. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl ∂ρ ∂t + · (ρu) = 0 + ρ ∂u ∂t + u · u = − p + µ 2 u + f Approximate with; • Finite Difference FD • Finite Volume FVM • Finite Element FEM • Spectral Methods GLOBAL+FUNCTIONAL • Spectral Element Methods (maybe later) etc. Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 6/57
7. 7. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Polynomial Approximation Replacing; u(x) = uN(x) = N k=0 ˆuk φk results with a residual/error; Lu(x) = f (x) ⇒ RN(x) = LuN(x) − f (x) = 0 minimize the residual/error; (RN)w := Ω RN(x)ω(x)dx = 0, Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 7/57
8. 8. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Spectral Collocation RN(xk) = LuN(xk) − f (xk), 1 ≤ k ≤ N − 1 uN(x0) = g−, uN(xN) = g+ Plug this; uN(x) = N j=0 uN(xj)hj(x) into the above equations results; N j=0 [Lhj(xk)]uN(xj) = f (xk), 1 ≤ k ≤ N − 1 N j=0 [hj(x0)]uN(xj) = g−, N j=0 [hj(xN)]uN(xj) = g+ Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 8/57
9. 9. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Spectral Collocation um N (xk) = N j=0 dm kj uN(xj), where dm kj = hm j (xk) The matrix Dm = (dm kj )k,j=0,...,N Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 9/57
10. 10. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Fourier Approximation & Diﬀerentiation The approximation of a real, integrable periodic function with truncated Fourier series; uK (x) = K k=−K ˆukeikx via orthogonality; 2π 0 eikx e−ilx dx = 2π if k = l 0 if k = l coeﬃcients; ˆuk = 1 2π 2π 0 u(x)e−ikx dx, k = 0, ±1, ±2, . . . Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 10/57
11. 11. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Fourier Approximation & Diﬀerentiation Nth order truncated Fourier series PNu(x) = N/2−1 k=−N/2 ˆukeikx For 2π (not mapped) periodic, the collocation grid points; xj = 2πj N , j = 0, . . . , N − 1 the coeﬃcients (again via orthogonality); ˆuk = 1 N N−1 j=0 u(xj)e−ikxj , k = −N/2, . . . , N/2 − 1 Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 11/57
12. 12. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Fourier Approximation & Diﬀerentiation 1D Burgers Equation; ∂u ∂t = v ∂2u ∂x2 − u ∂u ∂x The discretized version; 1 ∆t (ˆun+1 k − ˆun k ) = −vk2 ˆun+1 k − ˆun k ikˆun k Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 12/57
13. 13. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Fourier Approximation & Diﬀerentiation Figure: 1D Burgers equation solved with increasing modes Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 13/57
14. 14. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Chebyshev Approximation & Diﬀerentiation Basically Substituted ’Cosine’ functions; Tk = cos(kz) where x = cos(z) Possible FFT usage,or DCT Suitable for Non-Periodic B.C. Common collocation points; Gauss (Chebyshev zero points) xi = cos (i + 1 2 )π k , i = 0, . . . , k − 1 Gauss-Lobatto points(Chebyshev extreme points) xi = cos iπ k , i = 0, . . . , k Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 14/57
15. 15. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Chebyshev Approximation & Diﬀerentiation ’Collocated’ on the Gauss-Lobatto points; u(xi ) = uN(xi ) = N k=0 ˆukTk(xi ) = N k=0 ˆuk cos( k π i N ), i = 0, . . . , N via orthogonality the coeﬃcients are; ˆuk = 2 ck N N i=0 1 ci ui cos( k π i N ), k = 0, . . . , N Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 15/57
16. 16. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Chebyshev Approximation & Diﬀerentiation 1D Advection-Diﬀusion; ∂u ∂t + u ∂u ∂x = v ∂2u ∂x2 Discretized version; 3 2∆t ut+1 − 2 ∆t ut + 1 2∆t ut−1 − 2ut Dut + 2ut−1 Dut−1 = vD2 ut+1 + RHSt+1 Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 16/57
17. 17. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Chebyshev Approximation & Diﬀerentiation Figure: 1D Advection-Diﬀusion solutions with various Chebyshev modes Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 17/57
18. 18. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Chebyshev Approximation & Diﬀerentiation 10 1 10 2 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 N Error Figure: Convergence of Chebyshev method for 1D advection-diﬀusion equation Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 18/57
19. 19. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Pause & Recap Introduced & Assessed the methods → SM,FD,FEM,FVM Picked the method → S.M. - Collocation Manipulated the methods individually → Fourier & Chebyshev Next → Navier Stokes Implementation Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 19/57
20. 20. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Time & Spatial Discretization Splitting Algorithm Velocity Step 1 2∆t 3un+1/2 − 4un + un−1 − 1 Re ∆un+1/2 = 2hn − hn−1 + f un+1/2 = gn+1/2 where h = u · u Pressure Step 1 ∆t (un+1 − un+1/2 ) + pn+1 = 0 · un+1 = 0 Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 20/57
21. 21. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Time & Spatial Discretization Staggered Grid Figure: Staggered grid - [Canuto et.al.,Springer,2007] yj = cos jπ Ny , j = 0, ......, Ny (GL) yj+1 2 = cos (j + 1 2 ) π Ny , j = 0, ......, Ny − 1 (G)Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 21/57
22. 22. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Time & Spatial Discretization Velocity & Pressure Representation u(x, y, z, t) = Nx /2−1 kx =−Nx /2 Ny m=0 Nz /2−1 kz =−Nz /2 ˆukx,m,kz ¯Tm(y)e2πi(kx x/Lx+kz z/Lz) p(x, y, z, t) = Nx /2−1 kx =−Nx /2 Ny−1 m=0 Nz /2−1 kz =−Nz /2 ˆpkx,m,kz ¯Tm(y)e2πi(kx x/Lx+kz z/Lz) Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 22/57
23. 23. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Time & Spatial Discretization Pseudo-Code 1.Pre-Process Grid,Re,Diﬀ. Matrices,Operators 2.Turbine Preparation Nodes,Airfoils,Indexing 3.Solver 1: for Time Marching do 2: Non-Linear Terms(Pseudo-Spectrally) 3: for each mode in X do 4: for each mode in Z do 5: Solve the system for intermediate vel where the forces are fed in.Then solve for new pressure via that ﬁnd the divergence free velocity 6: end for 7: end for 8: Check Continuity 9: Transform to the real space & update the forces 10: end for Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 23/57
24. 24. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Computational Domain Computational Domain Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 24/57
25. 25. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Wind Turbine Representation Methods AD-NR UNIFORM FORCE DISTRIBUTION & NO ROTATION LESS TIME CONSUMING Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 25/57
26. 26. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Wind Turbine Representation Methods AD-R The velocity components at the rotor disc are extracted from the ﬂow solver, Vx ,Vy ,Vz (for 3D Cartesian coordinates) The inﬂow angle was calculated considering the angular velocity of the turbine,Vθ (projected from Vy & Vz) and Vx φ = tan−1 Vx ωr + Vθ The local twist and pitch is subtracted from inﬂow angle in order to ﬁnd the angle of attack. α = φ − γ Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 26/57
27. 27. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Wind Turbine Representation Methods AD-R Via the look up tables the lift and drag coeﬃcients (Cl (α, Re), Cd (α, Re)) are stored. The forces acting on the rotor disc are found , by considering an annular area of diﬀerential size dA = 2πrdr. The resulting force per unit rotor area is; dF dA = ρV 2 rel 2 Bc 2πr (Cl el + Cd ed ) FORCE DISTRIBUTION W.R.T. LOCAL CHARACTERISTICS & WITH ROTATION MORE TIME CONSUMING & STILL A DISC NOT ACL Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 27/57
28. 28. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl AD-NR Validation Grid Independency Simulations Nx Ny Nz Disk Nodes Ct Re a,induction 192 28 64 39 0.6 1000 0.162 256 28 64 39 0.6 1000 0.158 192 42 64 61 0.6 1000 0.165 256 42 64 61 0.6 1000 0.167 300 42 64 61 0.6 1000 0.166 192 56 64 75 0.6 1000 0.183 256 56 64 75 0.6 1000 0.186 300 56 64 75 0.6 1000 0.186 ... ... ... ... ... ... ... Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 28/57
29. 29. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl AD-NR ValidationAxialInductionFactor(a) Disk Resolution (N) 20 40 60 80 100 120 140 160 0.28 0.3 0.32 0.34 Re 1000 Re 3000 Axial Induction Factor (a) Ct&Cp 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 Ct-Theory Cp-Theory Computed Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 29/57
30. 30. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl AD-NR Brief Results −6 0 4 9 0 1 V Vo −6 0 4 9 −0.2 0 P x D Pressure Velocity Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 30/57
31. 31. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl AD-NR Brief Results Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 31/57
32. 32. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl AD-NR Brief Results V V o 0.5 1 0 1 2 3 4 X D =-2 y D 0.5 1 X D =0 y D 0.5 1 X D =1 y D 0.5 1 X D =2 y D 0.5 1 X D =4 y D 0.5 1 0 1 2 3 4 X D =6 y D 0.5 1 X D =10 y D 0.5 1 X D =14 y D 0.5 1 X D =18 y D 0.5 1 X D =20 y D Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 32/57
33. 33. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl AD-NR Brief Results Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 33/57
34. 34. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl AD-NR Brief Results Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 34/57
35. 35. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl AD-NR Reynolds Number Eﬀect Figure: Contour of the streamwise velocity component at diﬀerent Reynolds Numbers; 500, 1000, 2500, 5000, 10000 (from top to bottom) Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 35/57
36. 36. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl AD-R Validation Power Curve Measurements ACL−Troldborg AD−R Figure: Comparison of measured and computed power coeﬃcient for the Tjaereborg wind turbine Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 36/57
37. 37. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl AD-R Validation 0 0.1 0.2 r/R AxialInduction 0 0.2 0.4 0.6 0.8 1 0 0.1 r/R Circulation Figure: Loading (1- V Vo ) and Circulation ( Γ RVo ) along the blade Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 37/57
38. 38. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl AD-R Validation (3dvo) Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 38/57
39. 39. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl AD-R 3D Vorticity Field (vorﬁeld) Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 39/57
40. 40. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl AD-NR & AD-R Simulations Simulations 1 & 2 & 3 & 4 Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 40/57
41. 41. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl AD-NR & AD-R COMPARISON (adrcompnew) Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 41/57
42. 42. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl AD-NR & AD-R COMPARISON Figure: Contours of the streamwise velocity. Comparison of AD-NR (Top), AD-R (Bottom) Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 42/57
43. 43. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl AD-NR & AD-R COMPARISON ¯u V o 0.6 1 X D =15 y D 0.6 1 X D =12 y D 0.6 1 X D =10 y D 0.6 1 X D =7 y D 0.6 1 0 1 2 3 4 X D =4 y D 0.6 1 X D =3 y D 0.6 1 X D =2 y D 0.6 1 X D =1 y D 0.6 1 X D =0 y D 0.6 1 0 1 2 3 4 X D =-3 y D AD-NR AD-R Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 43/57
44. 44. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl AD-NR & AD-R COMPARISON Figure: Contours of the streamwise turbulence intensity, σu ¯u . Comparison of AD-NR (Top), AD-R (Bottom) Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 44/57
45. 45. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl AD-NR & AD-R COMPARISON Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 45/57
46. 46. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Reynolds Number Eﬀect , AD-R (Re500) Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 46/57
47. 47. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Reynolds Number Eﬀect , AD-R Figure: Time averaged stream-wise velocity, ¯u Vo ([∼]), at the middle vertical plane. Comparison Re-500 (Top), Re-2000 (Bottom) Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 47/57
48. 48. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Reynolds Number Eﬀect , AD-R Figure: Contours of stream-wise turbulence intensity,σu ¯u ,at the middle vertical plane. Comparison Re-500 (Top), Re-2000 (Bottom) Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 48/57
49. 49. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Reynolds Number Eﬀect , AD-R Figure: Contours of turbulence kinetic energy (u )2+(v )2+(w )2 2 at the middle vertical plane. Comparison Re-500 (Top), Re-2000 (Bottom) Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 49/57
50. 50. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Mixing Eﬀect , 2 Turbines, Laminar&Disturbed (2TurbineYESTurb) Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 50/57
51. 51. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Mixing Eﬀect , 2 Turbines, Laminar&Disturbed Figure: Time averaged stream-wise velocity, ¯u Vo [], at the middle vertical plane. Comparison Perturbed Inﬂow (Top), Laminar Inﬂow (Bottom) Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 51/57
52. 52. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Mixing Eﬀect , 2 Turbines, Laminar&Disturbed ¯u V o 0.6 1 X D =17 y D 0.6 1 X D =14 y D 0.6 1 X D =12y D 0.6 1 X D =9 y D 0.6 1 0 1 2 3 4 X D =6 y D 0.6 1 X D =3 y D 0.6 1 X D =2 y D 0.6 1 X D =1 y D 0.6 1 X D =0 y D 0.6 1 0 1 2 3 4 X D =-3 y D Turb-Inﬂow Uni-Inﬂow Figure: Time averaged stream-wise velocity, ¯u Vo [], vertical proﬁles at various downstream positions. Comparison Perturbed Inﬂow & LaminarEmre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 52/57
53. 53. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Mixing Eﬀect , 3 Turbines, Laminar&Disturbed Figure: Time averaged stream-wise velocity, ¯u Vo [], at the middle vertical plane. Comparison Perturbed Inﬂow (Top), Laminar Inﬂow (Bottom) Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 53/57
54. 54. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Staggered (Staggered) Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 54/57
55. 55. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Conclusions Spectral methods are very convenient for such ﬂows. Fast codes can be developed Chebyshev grid distribution was not very suitable for a case where the boundaries are not of paramount interest Under uniform inﬂow conditions both models perform similarly, in terms of wake modelling apart from the near wake region Axis symmetric wake development were captured TI-The tips are the regions where the highest turbulence occurs. If the tower was modelled this might have been valid for only ’upper-side’ Reynolds number’s role for wake development for laminar and perturbed inﬂow are diﬀerent Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 55/57
56. 56. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl Future Work In order to reach more realistic atmospheric Reynolds numbers with reasonable grid points, it is required to implement a turbulence model to this code. The boundary layer inﬂow. Taking energy equation into account in order to see the atmospheric stability eﬀect in the wind turbine/farm wakes. Chebyshev grid issue. Tip correction for more detailed loading investigations ACL,sacriﬁcing from computational time Continuous and controlled turbulence should be provided to the ﬂow. Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 56/57
57. 57. Introduction Fourier & Chebyshev SM 3D Navier-Stokes Comp. Domain & W.T. Representation Validation Simulations Concl The End QUESTIONS & REMARKS Emre Barlas DTU-Wind Energy Spectral 3D Navier Stokes Solver for WT Wakes 57/57