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

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

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

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.

∂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

Approximate with;
• Finite Diﬀerence DISCRETE LOCAL
• Finite Volume DISCRETE LOCAL
• Finite Element FUNCTIONAL LOCAL
• Spectral Methods FUNCTIONAL GLOBAL
• Spectral Element Methods HYBRID

∂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.

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,

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+

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

Fourier Approximation & Differentiation
The approximation of a real, integrable periodic function with
truncated Fourier series;
uK (x) =
K
k=−K
ûkeikx
via orthogonality;
2π
0
eikx
e−ilx
dx =
2π if k = l
0 if k = l
coefficients;
ûk =
1
2π
2π
0
u(x)e−ikx
dx, k = 0, ±1, ±2, . . .

Nth order truncated Fourier series
PNu(x) =
N/2−1
k=−N/2
ûkeikx
For 2π (not mapped) periodic, the collocation grid points;
xj =
2πj
N
, j = 0, . . . , N − 1
the coefficients (again via orthogonality);
ûk =
1
N
N−1
j=0
u(xj)e−ikxj
, k = −N/2, . . . , N/2 − 1

1D Burgers Equation;
∂u
∂t
= v
∂2u
∂x2
− u
∂u
∂x
The discretized version;
1
∆t
(ûn+1
k − ûn
k ) = −vk2
ûn+1
k − ûn
k ikûn
k

Figure: 1D Burgers equation solved with increasing modes

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

’Collocated’ on the Gauss-Lobatto points;
u(xi ) = uN(xi ) =
N
k=0
ûkTk(xi ) =
N
k=0
ûk cos(
k π i
N
), i = 0, . . . , N
via orthogonality the coefficients are;
ûk =
2
ck N
N
i=0
1
ci
ui cos(
k π i
N
), k = 0, . . . , N

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

Figure: 1D Advection-Diﬀusion solutions with various Chebyshev modes

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

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

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

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

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)

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

AD-NR
UNIFORM FORCE DISTRIBUTION & NO ROTATION
LESS TIME CONSUMING

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

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

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
...
...
...
...
...
...
...

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

AD-NR
Brief Results
−6 0 4 9
0
1
V
Vo
−6 0 4 9
−0.2
0
P
x
D
Pressure
Velocity

AD-NR
Brief Results

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

AD-NR
Brief Results

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)

AD-R
Validation
Power Curve
Measurements
ACL−Troldborg
AD−R
Figure: Comparison of measured and computed power coeﬃcient for the
Tjaereborg wind turbine

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

AD-R
Validation
(3dvo)

AD-R
3D Vorticity Field
(vorﬁeld)

AD-NR & AD-R Simulations
Simulations 1 & 2 & 3 & 4

AD-NR & AD-R COMPARISON
(adrcompnew)

Figure: Contours of the streamwise velocity. Comparison of AD-NR
(Top), AD-R (Bottom)

¯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

Figure: Contours of the streamwise turbulence intensity, σu
¯u . Comparison
of AD-NR (Top), AD-R (Bottom)

Reynolds Number Eﬀect , AD-R
(Re500)

Figure: Time averaged stream-wise velocity, ¯u
Vo ([∼]), at the middle
vertical plane. Comparison Re-500 (Top), Re-2000 (Bottom)

Figure: Contours of stream-wise turbulence intensity,σu
¯u ,at the middle
vertical plane. Comparison Re-500 (Top), Re-2000 (Bottom)

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)

Mixing Eﬀect , 2 Turbines, Laminar&Disturbed
(2TurbineYESTurb)

Vo [], at the middle vertical
plane. Comparison Perturbed Inﬂow (Top), Laminar Inﬂow (Bottom)

¯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
Vo [], vertical profiles at
various downstream positions. Comparison Perturbed Inflow & LaminarEmre Barlas DTU-Wind Energy

Vo [], at the middle vertical
plane. Comparison Perturbed Inﬂow (Top), Laminar Inﬂow (Bottom)

Staggered
(Staggered)

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

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.

The End
QUESTIONS & REMARKS

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

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