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A Turbulent Low-Speed Preconditioner for Unsteady Flows About Wind Turbine Airfoils

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22nd AIAA Computational Fluid Dynamics Conference, 22-26 June 2015, Dallas, Texas
A Turbulent Low-Speed Preconditioner for UnsteadyFlows About Wind Turbine Airfoils
Reza Djeddi
∗
, Jason Howison
†
and Kivanc Ekici
‡
University of Tennessee, Knoxville, Tennessee 37996
A harmonic balance method is used to study the unsteady ﬂows at low-speed regimestypically seen for wind turbines. The frequency-domain technique is applied to the Reynolds-averaged Navier-Stokes governing system of equations with the Spalart-Allmaras turbu-lence model. The convergence, stability and accuracy of the compressible solver is im-proved via a fully coupled viscous preconditioning designed for low speed ﬂows that fall inthe essentially incompressible ﬂow regime. The viscous preconditioner couples the Navier-Stokes equations to the turbulence model through the correct implementation of the pre-conditioning matrix and the subsequently matrix-valued artiﬁcial dissipation term. Finally,the viscous low-speed preconditioning is enhanced with a mixed-mechanism that would in-crease the stability and improve the convergence of the harmonic balance solver used forthe simulation of unsteady periodic ﬂows.
I. Nomenclature
b
semi-chord length of the airfoil
c
speed of sound
E
total energy
F
,
G
ﬂux vectors
H
total enthalpy
k
reduced frequency,
k
=
b ω/U
∞
p
pressure
Pr
l
laminar Prandtl number
Pr
t
turbulent Prandtl number
S
t
source term for the turbulence model
t
time
U
vector of conservative variables
U
0
vector of primitive variables
u,v
Cartesian velocities
U
∞
free stream velocity
V
control volume
x,y
Cartesian coordinates
ξ,η
cross-ﬂow coordinates
γ
speciﬁc heat ratio
µ
l
,µ
t
laminar and turbulent dynamic viscosity
ν
kinematic viscosity˜
ν
working variable for Spalart-Allmaras turbulence model
ρ
density
τ
pseudo-time
∗
Graduate Student, Department of Mechanical, Aerospace and Biomedical Engineering, Student Member AIAA.
†
Graduate Student, Department of Mechanical, Aerospace and Biomedical Engineering, Currently Assistant Professor, TheCitadel School of Engineering, Student Member AIAA.
‡
Associate Professor, Department of Mechanical, Aerospace and Biomedical Engineering, Senior Member AIAA.1 of 22American Institute of Aeronautics and Astronautics
D o w n l o a d e d b y U N I V E R S I T Y O F T E N N E S S E E o n J u n e 2 3 , 2 0 1 5 | h t t p : / / a r c . a i a a . o r g | D O I : 1 0 . 2 5 1 4 / 6 . 2 0 1 5 - 3 2 0 2
22nd AIAA Computational Fluid Dynamics Conference 22-26 June 2015, Dallas, TX AIAA 2015-3202 Copyright © 2015 by Reza Djeddi, Jason Howison and Kivanc Ekici. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. AIAA Aviation
II. Introduction
The aeromechanical design of wind turbines is a complex and multidisciplinary task that requires not onlythe consideration of a very large number of operating regimes of very low to moderate Mach number ﬂows withlow and high turbulence intensities but also the applicability to conventional and prospective unconventionalnew designs subject to challenging oﬀ-design conditions. This necessitates a high-ﬁdelity, robust unsteadysolver that can cover all aspects of the present wind turbine designs while building a platform for an adjointdesign optimization tool to further enhance the state-of-the-art. Due to the fact that most of the fundamentalwind turbine unsteady problems are periodic in time, frequency-domain techniques can be utilized to avoidprohibitively costly computational eﬀort of time-domain RANS solvers. The application of a harmonicbalance (HB) method
1
to Navier-Stokes (NS) solvers has proven to be a promising tool for turbomachineryand wind turbine ﬂow problems.
1–4
The aforementioned HB-NS solver can be enhanced with a one-equationSpalart-Allmaras
5
turbulence model that incorporates a rotation correction that reduces the eddy viscosityin the vicinity of regions where the vorticity surpasses the strain rate.
6,7
The spatial discretization of theconvective and diﬀusive terms includes the central diﬀerence formula that can be stabilized and furtherenhanced with an artiﬁcial or numerical dissipation that uses a blend of second and fourth diﬀerences.
8–10
Amain diﬀerence between wind turbine ﬂows and other aerodynamic problems for which compressible harmonicbalance RANS solvers have been used is that the ﬂow-ﬁeld for wind turbines falls in the incompressible regime.This entails the use of a robust low-speed preconditioning to ensure the accuracy of the solutions in a widerange of ﬂow speeds while guaranteeing a rapid convergence at each sub-time level of the harmonic balanceRANS solver.In this paper, we develop a viscous low-speed preconditioning that incorporates the Spalart-Allmarasturbulence model. This approach not only improves the accuracy of the numerical solution but also alle-viates the convergence problems that are inherited from applying a compressible solver to an essentiallyincompressible ﬂow. The preconditioner couples the turbulence model to the rest of the governing equationsto ensure the correct scaling of the turbulence equation and its respective artiﬁcial dissipation term thatimproves the convergence rate and also the accuracy of the compressible solver in low-subsonic ﬂow regimes.The preconditioning is further enhanced with a mixed-mechanism that deals with the unsteadiness of theﬂuid ﬂow through the correct modiﬁcation of the preconditioner matrix for the harmonic balance imple-mentation. We believe that explicit handling of all primitive variables including the working variable of theturbulence model for the improved preconditioner is presented for the ﬁrst time in this work for unsteadyanalysis of wind turbine ﬂows using a harmonic balance RANS solver.
III. Governing Equations and Mathematical Formulation
Unsteady ﬂow past the wind turbine airfoils is governed by the Reynolds-averaged Navier-Stokes equationswhich include modiﬁed Spalart-Allmaras turbulence model. The two-dimensional conservation form of thegoverning equations for a moving grid then becomes:
∂
U
∂t
+
∂
F
∂x
+
∂
G
∂y
=
S
(1)where the vector of conservative variables is
U
= [
ρ,ρu,ρv,ρE,ρ
˜
ν
]
T
and the vectors of inviscid and viscousﬂuxes in
x
- and
y
-directions,
F
and
G
, and the vector of source terms,
S
, are deﬁned as:
F
=
ρu
−
ρu
g
ρu
2
+
p
−
τ
xx
−
ρuu
g
ρuv
−
τ
xy
−
ρvu
g
ρuH
−
τ
xH
−
ρEu
g
ρu
˜
ν
−
τ
xν
−
ρ
˜
νu
g
,
G
=
ρv
−
ρv
g
ρuv
−
τ
yx
−
ρuv
g
ρv
2
+
p
−
τ
yy
−
ρvv
g
ρvH
−
τ
yH
−
ρEv
g
ρv
˜
ν
−
τ
yν
−
ρ
˜
νv
g
,
S
=
0000
S
t
In general, the unsteady grid motion velocity is incorporated into the governing equations by
u
g
and
v
g
in
x
and
y
directions, respectively. The pressure
p
and enthalpy
H
are deﬁned in terms of the conservativevariables based on the assumption of an ideal gas with a constant speciﬁc heat ratio as:
2 of 22American Institute of Aeronautics and Astronautics
D o w n l o a d e d b y U N I V E R S I T Y O F T E N N E S S E E o n J u n e 2 3 , 2 0 1 5 | h t t p : / / a r c . a i a a . o r g | D O I : 1 0 . 2 5 1 4 / 6 . 2 0 1 5 - 3 2 0 2
p
= (
γ
−
1)
ρ
E
−
12(
u
2
+
v
2
)
H
=
ρE
+
pρ
=
γ γ
−
1
pρ
+ 12(
u
2
+
v
2
)While the inviscid ﬂuxes depend on the conservative variables
U
, viscous ﬂuxes depend on the gradientsof the ﬂow velocities, temperature and the working variable of the turbulence model as deﬁned below for theshear stress terms of the Navier-Stokes equations:
τ
xx
= (
µ
l
+
µ
t
)
43
∂u∂x
−
23
∂v∂y
τ
yy
= (
µ
l
+
µ
t
)
43
∂v∂y
−
23
∂u∂x
τ
xy
=
τ
yx
= (
µ
l
+
µ
t
)
∂u∂y
−
∂v∂x
τ
xH
=
uτ
xx
+
vτ
xy
+
µ
l
Pr
l
+
µ
t
Pr
t
∂H ∂xτ
yH
=
uτ
xy
+
vτ
yy
+
µ
l
Pr
l
+
µ
t
Pr
t
∂H ∂y
In the above equations,
µ
l
and
µ
t
are the laminar and eddy viscosities, respectively, where the former isdeﬁned based on the Sutherland’s law and the latter is determined using the Spalart-Allmaras turbulencemodel,
5–7
which also deﬁnes the respective source term,
S
t
, in Equation (1).
IV. Harmonic Balance RANS Solver
Wind turbine aerodynamics consist of unsteady but temporally periodic ﬂows usually due to the pitchingblades or yawed wind regimes. In these cases, the conservative variables for which the governing equationsare solved, can be written in terms of a truncated Fourier series up to a predeﬁned number of harmonics by:
U
∗
(
x,y,t
i
) =
A
0
(
x,y
) +
N
n
=1
[
A
n
(
x,y
)cos(
ωnt
i
) +
B
n
(
x,y
)sin(
ωnt
i
)] ;
i
= 1 : 2
N
+ 1 (2)where
ω
is the fundamental frequency of excitation and
A
0
,
A
n
and
B
n
are the Fourier series coeﬃcientsdeﬁned in terms of the conservative variables. The Fourier series is truncated in a way that the ﬂow variablesare computed and stored at 2
N
+1 equally-spaced sub-time levels over a single period. Based on Equation (2)and using a discrete Fourier transform, the Fourier series coeﬃcients can be determined from the solutionsstored at each sub-time level as shown below:
U
∗
=
E
ˆ
U
(3)In a similar fashion, the conservative variables at the sub-time levels can be determined using the inversediscrete Fourier transform:ˆ
U
=
E
−
1
U
∗
(4)It is worth mentioning that the discrete Fourier transform,
E
, and its inverse,
E
−
1
are both squarematrices with a size corresponding to 2
N
+1 times
N
+
N
+1 where 2
N
+1 is the number of sub-time levelsand there exists
N
sine coeﬃcients,
N
cosine coeﬃcients and one zeroth harmonic coeﬃcient. Looking backat the unsteady governing equations in the time-domain, we have:
ddt
(
V
U
) +
Q
C
(
U
) +
Q
V
(
U
)
−
Q
D
(
U
) = 0 (5)
3 of 22American Institute of Aeronautics and Astronautics
D o w n l o a d e d b y U N I V E R S I T Y O F T E N N E S S E E o n J u n e 2 3 , 2 0 1 5 | h t t p : / / a r c . a i a a . o r g | D O I : 1 0 . 2 5 1 4 / 6 . 2 0 1 5 - 3 2 0 2
where
V
is the control volume. Here, the convective terms,
Q
C
(
U
), are evaluated at the cell centers byaveraging the conservative variables at the corresponding cell vertex while a compact stencil central diﬀerenceformulation
11
is used for the viscous terms,
Q
V
(
U
), which slightly reduces the odd-even decoupling in thesolution ﬁeld. However, the ﬂux balance at each vertex still requires a robust artiﬁcial dissipation term,
Q
D
(
U
). The srcinal Jameson-Schmidt-Turkel (JST) formulation
8
is utilized to obtain the “scalar” artiﬁcialdissipation, which blends second and fourth diﬀerences of the conservation variables where the latter providesthe background dissipation that is essential for stability and convergence toward steady state. However, theuse of a low-speed preconditioning mechanism, which will be discussed in the following section, makes itinevitable to have a “matrix-valued” numerical viscosity that is scaled by the preconditioning matrix in afully coupled fashion. It is worth mentioning that the dissipation or smoothing coeﬃcients for the secondand fourth diﬀerences in the artiﬁcial dissipation model are deﬁned using a pressure switch
8
which is furthermodiﬁed
9
to ensure total variation diminishing (TVD) properties. Finally, the system of semi-discretizedinitial-value ODEs is marched temporally toward a steady state solution with an explicit multistage Runge-Kutta scheme in which the viscous terms are frozen at the ﬁrst stage and the artiﬁcial dissipation terms areevaluated only at the odd stages.Using the harmonic balance method, as explained herein, the governing equations of (1) are then rewrittensimilar to equation (5) in the semi-discrete form for all sub-time levels, i.e.
ddt
(
V
∗
U
∗
) +
Q
C
(
U
∗
) +
Q
V
(
U
∗
)
−
Q
D
(
U
∗
) = 0 (6)where
V
∗
is the control volume and the rest of the left-hand-side terms are the convective, viscous andartiﬁcial dissipation terms evaluated at each sub-time level. Here, the time-derivative term in equation (6)is replaced by a pseudo-spectral operator
D
, that gives:
ω
D
(
V
∗
U
∗
) +
Q
C
(
U
∗
) +
Q
V
(
U
∗
)
−
Q
D
(
U
∗
) = 0 (7)In order to obtain the steady state solution at each sub-time level, a “pseudo-time” derivative term isadded to the harmonic balance equation of (7) which enables us to use the solution algorithm explainedearlier in this paper. This equation now reads:
∂ ∂τ
(
V
∗
U
∗
) +
ω
D
(
V
∗
U
∗
) +
Q
C
(
U
∗
) +
Q
V
(
U
∗
)
−
Q
D
(
U
∗
) = 0 (8)where
τ
is the pseudo time which enables us to march the above equation to steady state, vanishing thepseudo-time term. Thus, the pseudo-time harmonic balance equations of (8) can be solved in a similarfashion to the time-domain governing equations of (5).As shown earlier by Hall et al.,
1
an explicit treatment of the pseudo-spectral term in an explicit time-marching scheme can cause instabilities for the numerical solver. While non-reﬂecting far-ﬁeld boundariestypically remedy these stability issues for low reduced frequencies, extra care must be given to cases withhigher reduced frequencies or when too many harmonics are retained. In the present study, the CFL numberconstraint that was initially proposed by Van der Weide et al.
12
is used to stabilize the explicit solver for awide range of reduced frequencies and number of harmonics. The adapted stabilization technique introducesexcitation frequency and the number of harmonics (
N
) into the calculation of the time-step where∆
t
= CFL
V
λ
+
ωN
V
(9)where
λ
is taken to be the maximum of the spectral radius in all directions. The implementation of the present harmonic balance solver in wind turbine applications was presented by Howison and Ekici
3,4,13
along with the validating test cases and the deﬁnition of boundary conditions at the airfoil surface and atfar-ﬁeld. In this work, we develop a new low-speed preconditioner including the eﬀects of turbulence modeland implement it into our HB solver.
V. Low-Speed Preconditioning
Incompressible ﬂow regimes in which the density variations become small as the Mach number approacheszero, can be handled by a simpliﬁed incompressible form of the governing equations. However, in somesituations we have to stay with the compressible ﬂow solver since our simulations should ideally cover a
4 of 22American Institute of Aeronautics and Astronautics
D o w n l o a d e d b y U N I V E R S I T Y O F T E N N E S S E E o n J u n e 2 3 , 2 0 1 5 | h t t p : / / a r c . a i a a . o r g | D O I : 1 0 . 2 5 1 4 / 6 . 2 0 1 5 - 3 2 0 2
wider range of ﬂow regimes, which is the case for wind turbine applications. At low Mach numbers and ona ﬁxed grid, convective waves that are traveling at speed
u
become much slower than the acoustic wavesthat are traveling at
u
+
c
. What happens mathematically is that this disparity between the eigenvalues of the Jacobian matrices stiﬀens the governing equations, destabilizes the solution, slows down the convergenceand degrades the accuracy by resulting in unbalanced amounts of artiﬁcial dissipation. Also, when anexplicit time-marching method is used to obtain a steady state solution, the local time step depends onthe eigenvalues of the ﬂux Jacobian matrices and the aforementioned disparity in wave speeds substantiallyimpairs the convergence rate. In order to overcome these problems, low-speed preconditioning methods havebeen introduced which basically scale the acoustic wave speeds to be more on the order of the convective wavespeeds. Various eﬀorts have been made which are presented in the literature by pioneers such as Turkel,
14–16
Choi and Merkle,
17
Weiss and Smith,
18
and Van Leer
19
with varying degrees of success and a multitude of simpliﬁcations and limitations that are imposed along the way, especially in viscous cases.In simple words, almost all of these tools pre-multiply the time derivatives by a preconditioning matrixto overcome the stiﬀness problem of the system of equations. Also, more often, a set of primitive variablesis used to construct this matrix since it has been proven that it can ease up the derivation process. Thediﬀerences would then be in terms of the preconditioner matrix and the set of primitive variables used. In arecent eﬀort, Collin et al.
20
have studied a robust low-speed preconditioning tool for viscous/turbulent ﬂowsbased on the symmetric preconditioner of Weiss and Smith
18
and the viscous preconditioner of Choi andMerkle
17
in a combined so-called form of WSCM preconditioner. Also, Campobasso and Baba-Ahmadi
2
haveused a slightly diﬀerent viscous preconditioner that srcinated from works of Venkateswaran and Merkle
21
and applied it to unsteady ﬂow past horizontal axis wind turbine blades. More recently, Howison andEkici
3
used an inviscid preconditioner based on Turkel’s method
16
for unsteady harmonic balance analysis of wind turbine ﬂows. However, in all of these earlier works, either an inviscid preconditioner was used or theEuler/Navier-Stokes equations were preconditioned separately and therefore a robust viscous preconditioningof RANS equations including the turbulence model has remained untouched.
A. Preconditioning Matrix
In this paper, a fully coupled viscous preconditioning of RANS equations that incorporates the Spalart-Allmaras turbulence model is studied and developed to tackle the convergence, stability and accuracyproblems of solving an essentially incompressible ﬂow with a compressible ﬂow solver. First, consider thetwo-dimensional RANS equations in primitive variables on a ﬁxed grid:
∂
U
0
∂t
+
∂
F
0
∂x
+
∂
G
0
∂y
=
S
+
L
ν
(
U
0
) (10)where the primitive or non-conservation variables (pressure-velocity-entropy) are deﬁned as
U
0
= [
p,u,v,S,
˜
ν
]
T
and the inviscid ﬂux vectors in
x
- and
y
-directions are given as:
F
0
=
u
(
p
+
ρc
2
)
u
2
+
p/ρuvuS u
˜
ν
,
G
0
=
v
(
p
+
ρc
2
)
uvv
2
+
p/ρvS v
˜
ν
Also, the right hand side of equation (10) includes the vector of source terms,
S
, and
L
ν
is the diﬀerentialoperator for the viscous terms. Equation (10) can be rewritten in quasi-linear form as below:
∂
U
0
∂t
+
A
0
∂
U
0
∂x
+
B
0
∂
U
0
∂y
=
R
.
H
.
S
.
(11)where the Jacobian matrices of the ﬂux vectors are given by:
5 of 22American Institute of Aeronautics and Astronautics
D o w n l o a d e d b y U N I V E R S I T Y O F T E N N E S S E E o n J u n e 2 3 , 2 0 1 5 | h t t p : / / a r c . a i a a . o r g | D O I : 1 0 . 2 5 1 4 / 6 . 2 0 1 5 - 3 2 0 2

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