TURBULENCE TREATMENT IN STEADY AND UNSTEADY

V European Conference on Computational Fluid Dynamics
ECCOMAS CFD 2010
J. C. F. Pereira and A. Sequeira (Eds)
Lisbon, Portugal,14-17 June 2010
TURBULENCE TREATMENT IN STEADY AND UNSTEADY
TURBOMACHINERY FLOWS
Martin Franke∗ , Thomas Röber† , Edmund Kügeler† and Graham Ashcroft†
∗ Institute
of Propulsion Technology, German Aerospace Center (DLR),
Müller-Breslau-Str. 8, 10623 Berlin, Germany
e-mail: martin.franke@dlr.de
† Institute of Propulsion Technology, German Aerospace Center (DLR)
Linder Höhe, 51147 Cologne, Germany
e-mail: {thomas.roeber,edmund.kuegeler,graham.ashcroft}@dlr.de
Key words: Turbomachinery, Aerodynamics, CFD, Turbulence Modelling
Abstract. The adequate representation of turbulence is crucial for the predictive accuracy
of the computational method used in the turbomachinery design process. While unsteady
simulations are of ever-increasing importance, steady Reynolds-Averaged Navier-Stokes
computations will remain the workhorse of industrial CFD for some time. Turbulence
treatments for both cases are presented.
Since standard linear eddy-viscosity models exhibit some well-known deficiencies, a twofold strategy is pursued in the DLR-code TRACE: While the inclusion of turbomachineryspecific extensions to two-equation eddy-viscosity models forms the backbone of the turbulence modelling effort, Explicit Algebraic Reynolds Stress Models are currently validated
for turbomachinery flows.
With the continued increase in available computer power, unsteady computations are no
longer beyond the realm of industrial CFD. In particular for flow features such as largescale separation and wakes, hybrid methods become increasingly popular. Here, DetachedEddy Simulation is used, where a RANS treatment is retained for attached boundary layers
while the separated areas of the flow are treated in an LES-type fashion.
The results demonstrate the improved predictive accuracy as well as the feasibility of
the presented approaches.
1
Martin Franke, Thomas Röber, Edmund Kügeler, Graham Ashcroft
1
INTRODUCTION
Turbomachinery are characterised by alternating moving (rotating) and non-moving
parts in the flow path. Therefore, flows in multistage turbomachinery are inherently
unsteady. Even in the stable areas of the operation map, a number of unsteady flow
features exist. These can roughly be grouped by their length scale into phenomena considerably larger than, about the same size as, and considerably smaller than one blade
pitch 1 . Examples for the first group are flutter, rotating stall or surge; the second group
comprises wake and potential field interactions. The third group is made up of small-scale
disturbances like turbulence and transition.
In order to resolve the first group of unsteady phenomena, time-accurate simulations
have to be carried out. Even though models for some aspects of the second group exist, reliable simulations taking into account all unsteady blade row interactions are still
generally carried out in unsteady mode. For the prediction of the third group of unsteady phenomena, generally, modelling assumptions of some kind are made. This work
addresses the last kind of unsteady flow features.
However, while unsteady simulations are of ever-increasing importance, steady Reynolds-Averaged Navier-Stokes (RANS) computations will remain the workhorse of industrial CFD for some time 2 , where the adequate representation of turbulence is crucial for
the predictive accuracy of the computational method used in the design process. Standard
linear eddy-viscosity models (EVM) exhibit some well-known deficiencies resolving intricate flow features especially in off-design conditions. Since the application of ReynoldsStress Transport Models (RSTM) is currently still in a development stage, present efforts
try to adopt a more feasible solution. A two-fold strategy is pursued here: While the inclusion of turbomachinery-specific extensions to two-equation EVM forms the backbone
of the turbulence modelling effort, Explicit Algebraic Reynolds Stress Models (EARSM)
are currently validated for turbomachinery flows.
This paper aims at investigating the potential of state-of-the-art turbulence representations for realistic turbomachinery computations. To this end, the above-mentioned
approaches are outlined and applied to various configurations ranging from simple, rather
academic cases to full multi-stage applications. The contribution is organized as follows: First, the modelling practices used for steady computations will be discussed before
presenting the work on unsteady flows. Afterwards, results will be presented for both.
Finally, conclusions will be drawn and future work will be identified.
2
NUMERICAL METHOD
TRACE 3 is the CFD solver for internal flows at German Aerospace Center (DLR),
especially for turbomachinery flows. It is developed at the Institute of Propulsion Technology since the end of the Eighties. Today it is a hybrid code for structured as well as
unstructured meshes. The code solves the URANS equations in the time domain, the
linearized URANS equations in the frequency domain and has an adjoint solver for the
2
Martin Franke, Thomas Röber, Edmund Kügeler, Graham Ashcroft
linearized RANS equations. All included models are at least second order accurate in
space and time. TRACE is a coupled multiphysics solver, which includes aerodynamics,
aeroelasticity with structural damping, aeroacoustics and aerothermics with structural
heat conduction.
3
STEADY COMPUTATIONS
Solving the RANS equations, a closure is needed for the Reynolds stresses. A variety
of RANS methods exist, ranging from simple algebraic models up to RSTM. To date,
however, transport-equation EVM remain the backbone of turbulent viscous computations in the industrial design process. These models, however, have difficulties predicting
e.g. pressure-induced separation and recirculation, non-primary effects such as secondary
flows, shock-induced separation and shock/boundary-layer interaction. This motivates
the work on improved turbulence models.
3.1
Turbomachinery-specific extensions
The starting point is the original formulation of the k-ω turbulence model by Wilcox 4 :
·
¸
∂ρk ∂ρkUj
∂
∂k
∗
+
= Pk − β ρωk +
(1)
(µ + σk µt )
∂t
∂xj
∂xj
∂xj
·
¸
∂ρω ∂ρωUj
ω
∂
∂ω
2
+
= α Pk − βρω +
(µ + σω µt )
∂t
∂xj
k
∂xj
∂xj
where ρ is the density, µ the molecular viscosity, Uj is the mean velocity vector, k the
turbulent kinetic energy and ω the specific dissipation rate of k. The eddy viscosity is
defined as
k
µt = ρ
(2)
ω
Setting the strain- and rotation-rate tensors to
µ
¶
µ
¶
1 ∂Ui ∂Uj
2 ∂Uk
1 ∂Ui ∂Uj
∗
∗
Sij ≡
+
−
δij
and Ωij ≡
−
,
2 ∂xj
∂xi
3 ∂xk
2 ∂xj
∂xi
(3)
the production of k and the Reynolds stresses can be written as
i
k
Pk = τij ∂U
= 2µt |S ∗ |2 − 23 ρk ∂U
δ , τij = −ρui uj = 2µt Sij∗ − 23 ρkδij ,
∂xj
∂xk ij
(4)
respectively. The inclusion of the divergence in the definition of Sij∗ is made in order to
have a consistent formulation in compressible flows. The constants are
α = 5/9, β = 3/40, β ∗ = 9/100, σk = σω = 0.5
(5)
The prediction of viscous effects is improved by successively implementing a fix for spurious production of turbulent energy as well as measures to account for the effects of
compressibility and rotation, respectively.
3
Martin Franke, Thomas Röber, Edmund Kügeler, Graham Ashcroft
The production of turbulent kinetic energy is overestimated by a linear EVM in regions
of large strain rates. This behaviour mostly occurs at the stagnation point, hence it has
been addressed by the term ”stagnation point anomaly”. Since the flow at the stagnation
point affects the complete boundary layer development downstream, it is of great importance to fix this problem, particularly if transition and heat transfer are to be simulated.
The production term of the k-equation, Pk , is thus modified, as proposed by Kato and
Launder 5 , where |S ∗ |2 is replaced by |S ∗ Ω∗ | in the Pk -equation (4 left). Note that, strictly
speaking, this modification violates the Boussinesq assumption and the conservation of
energy, while on the other hand it effectively suppresses the unphysical overproduction of
turbulent quantities at the stagnation point.
For compressible flows, in addition to pressure and momentum fluctuations, density
and temperature fluctuations have to be accounted for as well in averaging the equations
of motion. Similar to Reynolds averaging, this so-called Favre averaging leads to the
appearance of an additional stress tensor, i.e. the Favre-averaged Reynolds-stress tensor.
Compared to the incompressible Reynolds-stress tensor, several additional effects have to
be taken into account in modelling the compressible Reynolds stresses.
In order to account for pressure dilatation and pressure diffusion, additional source
terms depending on the turbulent Mach number Mt2 = 2k/a2 are inserted into the kequation following a proposal by Sarkar 6 :
i
Pd = α2 Mt τij ∂U
+ α3 β ∗ Mt2 ρωk, α2 = 0.15, α3 = 0.2
∂xj
(6)
Furthermore, the turbulence model has been adapted to improve the predictive accuracy in compressible turbulent mixing layers by adjusting the destruction terms’ closure
coefficients 7 :
£
¡
¢¤
£
¡
¢¤
1
1
,0
,0
βc∗ = β ∗ 1 + 32 max Mt2 − 16
βc = β − β ∗ 32 max Mt2 − 16
(7)
Finally, a new destruction term has been added to the ω-equation to model the effect
of streamline curvature and rotation on the dissipation of turbulent kinetic energy as
suggested by Bardina et al. 8 :
Dr = −α4 ρω|Ω|, α4 = 0.15
3.2
(8)
Explicit Algebraic Reynolds Stress Models
Another approach is the application of modelling practices beyond the Boussinesq approximation. The limitations of the linear eddy-viscosity modelling practice are mainly
due to the facts that the representation of turbulence by its kinetic energy cannot properly
represent turbulence anisotropy and that its linear constitutive relation leads to a rigid
stress-strain coupling. A general approach would arguably be based on a second-moment
closure (i.e. RSTM). Unfortunately, the applicability of RSTM is restrained by the necessity to solve seven additional, strongly coupled equations and the lack of a stabilizing
4
Martin Franke, Thomas Röber, Edmund Kügeler, Graham Ashcroft
eddy viscosity in the momentum and energy equations. Thus, present efforts try to adopt
a computationally more feasible solution.
EARSM are a class of non-linear EVM derived by a systematic approximation to the
second moment closure. Unlike other, more empirical approaches to non-linear modelling,
EARSM are rational approximations to RSTM and thus yield clear prospects with respect
to model validity. The model investigated here is the Hellsten EARSM k-ω model 9 , which
is based on the self-consistent approach of Wallin & Johannson, optionally including a
curvature correction. This EARSM is based on a recalibrated LRR-RSTM, from which
a fully explicit and self-consistent algebraic relation is derived. Other topics addressed
by the approach are the model behaviour at turbulent-laminar edges, its sensitivity to
pressure gradients and the model coefficient calibration.
The approach replaces the eddy-viscosity assumption by a more general constitutive
relation for the Reynolds stress anisotropy in terms of the strain- and rotation-rate tensors. The model is formulated based on an effective turbulent viscosity µt and an extra
(ex)
anisotropy aij :
2
(ex)
−ρui uj = 2µt Sij∗ − ρkδij − ρkaij
(9)
3
The effective turbulent viscosity and the extra anisotropy are given by
1
µt = − (β1 + IIΩ β6 ) ρkτ
2
and
¡
¢
a(ex) = β3 Ω2 − 13 IIΩ I +
β4 (SΩ
− ΩS) +
¡
¢
β6 ¡SΩ2 + Ω2 S − II¢Ω S − 23 IV I +
β9 ΩSΩ2 − Ω2 SΩ
(10)
(11)
To this end, the strain and vorticity tensors are normalized by the turbulent
time scale,
³
´
p
incorporating the Durbin time-scale realizability limitation, τ = max k/ε, Cτ µ/ρε
with Cτ = 6.0 and ε being the dissipation rate of k:
S = Sij = τ Sij∗
and Ω = Ωij = τ Ω∗ij
(12)
The invariants are
© ª
© ª
©
ª
IIS ≡ tr S 2 , IIΩ ≡ tr Ω2 , IV ≡ tr SΩ2
(13)
The β coefficients are given by
β1 = −
β4 = −
N (2N 2 −7IIΩ )
,
Q
2(N 2 −2IIΩ )
,
Q
−1 IV
β3 = − 12NQ
β6 = − 6N
, β9 =
Q
5
6
Q
(14)
(15)
Martin Franke, Thomas Röber, Edmund Kügeler, Graham Ashcroft
where the denominator Q reads
¢¡
¢
5¡ 2
Q=
N − 2IIΩ 2N 2 − IIΩ
6
The function N is given by
√ ¢1/3 ¡
√ ¢1/3
( C10 ¡
+
P
+
P
+
P
−
P2µ
1
2
1
3
µ
N=
1/6
C10
+ 2 (P12 − P2 ) cos 13 arccos √ P21
3
P1 −P2
where
³
P1 =
C10 2
27
+
9
II
20 S
´
³ 02
C
− 23 IIΩ C10 , P2 = P12 − 91 +
and
C10 =
β (eq) = − 56
(16)
¶¶ , P2 ≥ 0
, P2 < 0
9
II
10 S
+ 23 IIΩ
´3
³
´
9 9
(eq)
+ Cdiff max 1 + β1 IIS , 0
5 4
N (eq)
((N (eq) )2 −2IIΩ )
, N (eq) = 81/20, Cdiff = 2.2
(17)
(18)
(19)
(20)
It should be remarked that in 3D mean flows the relation for N does not exactly imply
self consistency, which is only strictly fulfilled in the 2D limit. In fully three-dimensional
cases, N is represented by a sixth-order equation, to which no explicit solution can be
found.
Optionally, for flows in which streamline curvature plays an important role, the use of
a curvature correction can be beneficial. In this case, Ω is replaced by
µ ¶
e = Ω − τ Ω(r)
Ω
(21)
A0
where the correction term is defined as follows:
(r)
Ωij = −²ijk Bkm Spr
DSrq
²pqm
Dt
(22)
with
Bkm =
© ª
IIS2 δkm + 12IIIS Skm + 6IIS Skl Slm
, IIIS ≡ tr S 3 , A0 = −0.72
3
2
2IIS − 12IIIS
(23)
The two-equation model base for the Hellsten EARSM k-ω is a modified Menter BSL
approach specifically recalibrated to be used in conjunction with the nonlinear constitutive
relation. The transport equations are given by eq. (1), where an additional cross-diffusion
term is added to the ω-equation:
µ
¶
∂k ∂ω
ρ
,0
(24)
Pxdiff = σd max
ω
∂xj ∂xj
6
Martin Franke, Thomas Röber, Edmund Kügeler, Graham Ashcroft
The model coefficients α, β, σk , σω , σd are blended in space according to
C = fmix C1 + (1 − fmix )C2 ,
(25)
where the blending function is constructed as
fmix = tanh(1.5Γ4 ), Γ = min (max (Γ1 , Γ2 ) , Γ3 )
Γ1 =
√
k
,
β ∗ ωy
Γ2 =
500µ
,
ρωy 2
Γ3 =
(26)
20k
max(y 2 (∇k∇ω)/ω,200k∞ )
The k-ω transport equations from the background model are extended with the EARSM
relations for the eddy viscosity and Reynolds stresses. When using the k-ω base, the
turbulent dissipation rate is given by ε = β ∗ ωk. For the Reynolds stresses and the
turbulent viscosity, the expressions given in (9) and (10) are used. The production term
Pk is no longer modelled along the second expression in (4 left) and thus exact. The
model coefficients are given in Table 1.
Set
1
2
α
β
0.518 0.747
0.44 0.828
σk
1.1
1.1
σω
σd
0.53 1.0
1.0 0.4
Table 1: Model coefficients for the Hellsten EARSM k-ω
4
UNSTEADY COMPUTATIONS
A natural choice for the computation of unsteady flows would be Large-Eddy Simulation (LES), in which the large, energy-bearing eddies are resolved and only the small,
isotropic eddies need modelling, thus obviating the necessity of a distinct spectral gap between flow unsteadiness and turbulence. Additionally, the underlying assumption of LES,
i.e. that small-scale eddies possess a more universal character than large-scale ones, allows
the use of relatively simple turbulence models. On the other hand, the required refinements in spatial and temporal resolution are quite large, especially in attached boundary
layers, where the turbulent scales are very small compared to the geometrical scale. It
is assumed that the computational resources to fully resolve the energy-bearing eddies
in standard industrial applications will not be available for several decades to come 10 .
Thus, to overcome well-known deficiencies of eddy-viscosity models regarding the prediction of separated flow, while at the same time retaining their undisputed performance for
attached boundary layers 11 , hybrid RANS/LES approaches have been proposed.
Several hybrid methods have been suggested over the past decade 12 , with DetachedEddy Simulation (DES) representing a prominent example. The applications presented
in this paper are based on an adaptation of a ”classical” DES approach to two-equation
models, see Strelets 13 . In the model, the turbulence length scale of a RANS type eddyviscosity model, in terms of a k-ω-model,
√
k
,
(27)
`kω =
ω
7
Martin Franke, Thomas Röber, Edmund Kügeler, Graham Ashcroft
is complemented by an LES-type filter which may, essentially, be any valid LES filter. As
in many cases, the mesh resolution ∆ = max (∆i , ∆j , ∆k ) is used as filter width, so that
the length scale of the turbulence model is replaced by
` = min (`kω , CDES ∆) .
(28)
The constant CDES corresponds to the Smagorinsky constant of standard LES subgrid
stress models. It has to be calibrated to reflect the dissipativeness of the numerical scheme
used in the particular flow solver and is usually in the order CDES ≈ 10−1 . The original
model’s destruction term in eq. (1) can be rewritten as
3
β ∗k 2
ε = β kω =
,
`
∗
(29)
with ` being set according to eq. (28) for DES computations and according to eq. (27) for
RANS computations.
5
5.1
RESULTS AND DISCUSSION
Turbomachinery-specific extensions
The use of these extensions is demonstrated computing the performance map of a 3.5stage intermediate-pressure compressor, which is taken from the RB199 engine 14 . The
mesh topology is chosen to match meshing techniques used in typical industrial applications. A multiblock grid featuring a OH topology incorporating additional H blocks in the
tip-gap regions with a total of 3.8 million nodes is employed. While the blade surfaces are
meshed fine enough to apply low-Re boundary conditions (y + < 2), the resolution on the
hub and casing walls remains rather coarse and thus necessitates wall-function boundary
conditions (y + ≈ 60). The performance map is given in Figure 1, where an excellent
agreement can be seen. This demonstrates the capability of the standard approach used
in TRACE to address modern industrial turbomachinery applications.
5.2
EARSM
The performance of the non-linear model is evaluated on four steady testcases, i.e. two
academic cases to determine the model capability to capture turbulence damping and
amplification by curvature, a well-documented low-speed compressor cascade featuring a
tip gap and a 5-stage low-pressure turbine plus IGV including transitional effects.
Influence of Curvature on Turbulence
The curved bend case of So and Mellor 15 addresses the stabilizing curvature effects on
turbulence at convex walls. Owing to the fact that the experiment was performed at an almost zero pressure gradient, the influence of the curvature becomes the dominant feature.
The geometry of the case is shown on the left-hand side of Figure 2 with every second grid
8
Martin Franke, Thomas Röber, Edmund Kügeler, Graham Ashcroft
Figure 1: Performance map of the intermediate-pressure 3.5-stage compressor
line displayed. The grid used consists of 129 · 161 points, the dimensionless wall distance
of the wall-adjacent cells is y + < 0.15. The inflow conditions are set to M = 0.063 and
Re = 1.42 · 106 per meter. The outer wall is treated with a symmetry boundary condition and shaped accordingly to match the measured pressure distribution 16 . Prescribed
velocity and turbulent profiles at the inlet ensure that the inflow conditions are identical
to the experiment.
0.006
cf
0.2
Y
U∞
0.005
0
0.004
-0.2
stabilizing curvature
0.003
-0.4
-0.6
0.002
no-slip wall
-0.8
0.001
symmetry
Experiment
Wilcox k-ω
EARSM
-1
0.4
0.6
0.8
1
1.2
1.4
0
0.5
1.6
1
1.5
2
s [m]
X
Figure 2: Geometry and skin-friction distribution of the So-Mellor curved wall
Skin friction comparisons of the Hellsten EARSM and Wilcox k-ω models are given
9
Martin Franke, Thomas Röber, Edmund Kügeler, Graham Ashcroft
on the right-hand side of Figure 2. It is apparent that the EARSM predictions agree
considerably better with the experiment, as it is able to better capture the damping effect
of the convex curvature on turbulence.
In contrast, the U-duct of Monson and Seegmiller 17 is dominated by the amplification
of turbulence on the concave outer wall. The geometry and the grid are displayed on the
left-hand side of Figure 3, again, every second grid line is shown. The grid is made up of
361 · 141 nodes and extends, like in the experiment, for a distance of 12 channel heights
upstream and downstream of the duct, thus allowing for a fully-developed turbulent inflow.
The Reynolds number based on the channel height of H = 3.81 cm and a bulk velocity
of Ub = 31.1 m/s is 1 · 106 . A y + ≈ 0.15 is guaranteed in the duct region.
Y
destabilizing curvature
0.015
0.05
Exp. inner
Exp. outer
Wilcox k-ω
CC-EARSM
cf
H
0.010
0
stabilizing curvature
0.005
U∞
0.000
-0.05
21.7<duct<24.8
-0.1
-0.05
0
0.05
X
-0.005
15
20
25
30
35 s/H 40
Figure 3: Geometry and skin-friction distribution of the Monson U-duct
The skin friction coefficient is given on the right-hand side of Figure 3. Attention is
drawn to the distribution on the outer wall, which for the EARSM is in excellent agreement
with the experiment, while the linear Wilcox model underpredicts the magnitude of cf .
It should be remarked that for this case, the curvature correction given in equation (22)
is necessary in conjunction with the EARSM for improved predictive accuracy.
Virginia-Tech Compressor
In the following, a low-speed subsonic compressor cascade incorporating tip-gap flow,
for which extensive measurement data are available 18,19 , is considered. The geometry and
the relevant measurement points are given in Figure 4. The cascade consists of GE rotor B
section blades. The inlet angle is 65.1◦ , the stagger angle is 56.9◦ , the chord length and the
blade height are both 254 mm. The axial chord ca length thus is 138.68 mm. The cascade
features a tip gap of 1.65% profile height, the Reynolds number based on the chord length
and an inflow velocity of U∞ = 24.5 m/s is set to Re = 403, 000. The computational
10
Martin Franke, Thomas Röber, Edmund Kügeler, Graham Ashcroft
grid consists of approx. 2.57 million cells, a y + < 1 is guaranteed on all solid walls. This
is especially important when employing a non-linear, anisotropy-resolving model, since
standard wall functions do not model turbulence anisotropy. The tip gap is discretized
with 34 cells between the blade tip and the casing.
Z
Y
Y
0.22
X
plane x/ca=0.98
9b 9c
9p
U∞
0.20
tip gap
0.36
0.37
0.38
0.39
X
Figure 4: Geometry (left) and measurement positions (right) of the Virginia Tech low-speed compressor
cascade
Only few results can be reported here. The velocity components in x- and y-direction
in measurement plane 9 (x/ca = 1.0) are given in Figure 5. It becomes apparent that both
the turbomachinery-specific extensions as well as the EARSM yield superior predictions
compared to the standard Wilcox k-ω model. While there is no significant difference
between the two in the gap, the picture changes when looking at the magnitude of velocity
in the plane x/ca = 0.98, see Figure 6. The magnitude of the velocity is slightly too high
in the free stream, however, the size of the vortex and thus the blockage of the passage is
predicted well by all computations. Remarkably, only the EARSM is able to compute the
correct level of flow deceleration in the vortex, which is overpredicted by the linear model
both with and without the extensions. Also, the position of the vortex core, marked with
a yellow cross, is matched better by the EARSM.
5-Stage Low-Pressure Turbine
As a final test case, a 5-stage low-pressure turbine rig (designated as MTU-B 20 ) is
included for demonstration purposes. Computations are performed with both Wilcox k-ω
with extensions and the EARSM. In both cases, transition is treated with a multimode
transition model including quasi-unsteady wake modelling 21 . A grid with 5.5 million
nodes is employed. While the mesh resolution on the blades is fine enough to allow for
low-Re boundary conditions, both hub and tip are treated with wall functions. Thus,
11
Martin Franke, Thomas Röber, Edmund Kügeler, Graham Ashcroft
9c
0.004
9b
0.004
9p
0.005
0.004
0.002
0.001
z [m]
0.003
z [m]
z [m]
0.003
0.002
0.001
0.000
2
4
6
8
U [m/s]
10
12
0.000
0
9c
0.005
0.002
0.001
0.000
0
0.003
2
4
6
8
U [m/s]
10
12
0
9b
0.004
2
4
0.002
0.001
0.001
0.000
-4
z [m]
z [m]
z [m]
0.002
0.003
4
8 12
V [m/s]
16
20
24
12
14
9p
0.003
0.002
0.001
0.000
0
10
Experiment
Wilcox
Tm.-spec. ext.
EARSM
0.004
0.004
0.003
6
8
U [m/s]
0.000
0
4
8
12
V [m/s]
16
20
0
4
8
12
V [m/s]
16
20
Figure 5: Low-speed cascade: Velocity component in x- (top) and y-direction (bottom) in measurement
plane 9
Turbom.-spec. extensions
1.0
0.8
0.8
0.6
0.6
z/Ca
z/Ca
Experiment Muthanna
1.0
0.4
0.2
0.0
0.2
+
2.2
2.0
1.8
1.6
1.4
0.4
1.2
1.0
0.0
0.8
+
2.2
2.0
1.8
y/Ca
0.8
0.8
0.6
0.6
0.4
0.2
+
1.8
1.6
1.4
1.2
1.0
0.0
0.8
0.8
+
2.2
2.0
1.8
y/Ca
Uges/Uref: 0.15
1.0
0.4
0.2
2.0
1.2
Hellsten EARSM k-ω
1.0
z/Ca
z/Ca
Wilcox k-ω
2.2
1.4
y/Ca
1.0
0.0
1.6
1.6
1.4
1.2
1.0
0.8
y/Ca
0.25
0.35
0.45
0.55
0.65
0.75
Figure 6: Low-speed cascade: Velocity magnitude in measurement plane x/ca = 0.98
12
Martin Franke, Thomas Röber, Edmund Kügeler, Graham Ashcroft
the EARSM cannot deliver its full potential. Nevertheless, the results are included here
to demonstrate the capability of the EARSM to be applied to multi-stage machines in
conjunction with a transition model.
The geometry and the lapse rate are presented in Figure 7. Concerning the losses, it
can be seen that the EARSM results are very close to the experimental data whereas the
linear prediction is somewhat too high. These results need to be interpreted carefully,
though, as side geometries have not been included in the computation. Therefore, no final
conclusion can be drawn here, the results, however, look very promising, especially taking
into account that the computational overhead – less than 20% increase in computing time
and negligible additional memory requirements – remains limited.
ηis
δη=2%
Experiment
Turbom.-spec. extensions
Hellsten EARSM
1.0E5
2.0E5
3.0E5
Re
Figure 7: 5-stage low-pressure turbine: Geometry and lapse rate
5.3
DES
The DES scheme implemented in TRACE has been calibrated and applied to two
different test cases that are relevant to turbomachinery flows. While both applications
are characterized by separated flow areas, the first application focuses on the prediction of
unsteady transitional flow, whereas the second one is aimed at the simulation of broadband
noise in turbomachinery.
Calibration of DES model
For the calibration of LES subgrid stress models, the decay of isotropic homogeneous
turbulence 22 is simulated. The flow field is initialized with a turbulent fluctuation field,
which is then allowed to decay. As no walls are present — all boundaries are assumed to
13
Martin Franke, Thomas Röber, Edmund Kügeler, Graham Ashcroft
10
10
t=0.00 s
t=0.87 s
t=2.00 s
-1
10
Turbulent Kinetic Energy [-]
Turbulent Kinetic Energy [-]
10
0
10-2
-4
10
-5
10
-6
t=0.00 s
t=0.87 s
t=2.00 s
-1
10-2
10-3
10
0
10-3
100
101
102
Wavenumber [-]
10
-4
10
-5
10
-6
100
101
102
Wavenumber [-]
Figure 8: Turbulent spectra predicted by DES for meshes with 163 nodes (left) and 323 nodes (right)
with CDES = 0.156 at three different instants (lines) and comparison to experiment (squares).
be periodic —, the model is supposed to run entirely in LES mode. To prevent the model
from unwantedly switching into near-wall mode, the RANS mode has been disabled for
the calibration.
In order to ensure good predictions on meshes with varying levels of refinement, the
computational mesh was refined after calibration, and the computation was repeated.
The turbulence spectra predicted by the DES model with CDES = 0.156 on meshes with
163 and 323 nodes are shown in Figure 8. In the figure, the experimental spectra have
been indicated as symbols, while the predicted spectra are drawn as lines. It can be seen
that a greater part of the turbulent spectrum is resolved on the finer mesh, whereas a
great deal of the turbulence is contained in the subgrid stresses on the coarser mesh.
Low-Pressure Turbine Cascade
As a first application, the unsteady flow in a highly loaded low-pressure turbine cascade,
T106C 23 , is simulated. The Reynolds number based on exit values is Re2,is = 140, 000,
the exit Mach number M2,is ≈ 0.65, the pitch-to-chord ratio p/c = 0.95. The flow is
characterised by boundary layer transition on the blade suction side which is influenced
by the wakes of a cylindrical bar of diameter d = 2 mm located 0.58 chord lengths c
upstream of the leading edge. The bar pitch is the same as that of the turbine blade, to
simulate the effects of an upstream rotor row, the cylinder moves at v = 141 m/s.
In order to predict the three-dimensional vortex breakdown phenomena, the computational mesh was generated with a lateral width of 0.1c. The mesh consisted of a total
of ca. 550,000 nodes, with a lateral resolution of 11 nodes. The instantaneous flow field
at the center plane of the computational domain is shown in Figure 9. The vortex street
14
Martin Franke, Thomas Röber, Edmund Kügeler, Graham Ashcroft
Figure 9: Schematic of T106C configuration superimposed with an instantaneous snapshot of the computed vorticity magnitude: 20 levels below |Ω| = 50000s−1 .
developing behind the cylinder can very well be observed, as can the interaction between
wake and blade leading edge, and the wake stretching in the passage.
To preclude the development of turbulent “subgrid” stresses in the laminar region of
the boundary layer — which is typically in the “RANS region” of the flow field —, the
multimode transition model 21 was enabled. The behaviour of the boundary layer can by
summarized as follows 24 : Along the wake path on the suction side, transition is triggered
in wake-induced mode. Between two wakes, bypass and separated-flow transition occurs,
before the next wake hits the blade.
From inspecting the development of the suction side wall shear stress, Figure 10, it can
be seen that all these features are present in the computations. While both the URANS
and DES computations agree very well with the experiment, the calmed region can be
observed much better in the DES computation. Note that, in the computations, three
consecutive wake passages are shown, while in the experiment, a phase locked average is
presented. Therefore, in both computations, the frequency of the cylinder’s vortex street
can be observed in the distribution of the surface shear stress: alternating vortices with
opposing senses of rotation hit the blade and result in increased and decreased levels of
surface shear stress, respectively. Generally, the observed surface shear stress levels are of
the same order of magnitude in both computations. However, the maximum values tend
to be slightly larger in the DES computation than in the URANS computation, because
some vortical content from the vortex street that has been averaged out in the URANS
computation is still present in the DES even in the boundary layer.
The size of the separation bubble on the suction side is smaller in the DES than in the
URANS computation. However, in the DES, a small separation near the trailing edge is
visible spatially behind the wake path. The calmed region temporally behind the wake
path is visible in the DES computation.
15
Martin Franke, Thomas Röber, Edmund Kügeler, Graham Ashcroft
Figure 10: Surface shear stress τW over time for three consecutive wake passing periods and comparison
to pseudo wall shear stress from experiment. In the plots showing the CFD simulations, separation has
been indicated as a black line.
Rod-Airfoil Testcase
In order to assess the accuracy of the employed DES model for broadband-noise prediction, simulations of the rod-airfoil test case investigated by Jacob et al. 25 were carried
out. This test case comprises a symmetric NACA0012 airfoil located one chord length c
downstream of a rod of diameter d = 0.1c, cf. Figure 11. Rod and airfoil are submerged
in a uniform flow with a Mach number M ≈ 0.2 and a Reynolds number based on the
cylinder diameter of Red ≈ 48, 000.
At this Reynolds number, the rod wake comprises highly-periodic vortex structures
superimposed with more stochastic small-scale vortical fluctuations and is therefore analogous to a turbulent gust field seen by guide vanes downstream of a rotating blade row.
To accurately predict the noise radiated by the rod-airfoil configuration, a numerical
model must be capable of resolving directly at least the larger turbulent structures in
the wake of the cylinder and their interaction with the downstream airfoil. Therefore,
the computational domain was created with a span of three rod diameters. To enable a
portion of the acoustic near-field to be resolved, the computational domain was designed
to extend approximately twice the wavelength of the dominant tone away from the leading
edge of the airfoil in the x-y plane. In total, the mesh contained 3 million cells, with 30
cells in the spanwise direction.
In Figure 11, superimposed about the profiles of the rod and cylinder, are contours
of the computed spanwise vorticity along the centerline of the computational domain
(z = 0) at an arbitrary instant. Immediately behind the cylinder, vortical structures
begin to form. These large-scale vortical structures display a large degree of regularity and
may be described as quasi-periodic. Their subsequent interaction with the downstream
16
Martin Franke, Thomas Röber, Edmund Kügeler, Graham Ashcroft
Figure 11: Schematic of rod-airfoil configuration superimposed with an instantaneous snapshot of the
computed spanwise vorticity: 10 levels between Ω = ±20000s−1 (Negative contours are dashed).
airfoil gives rise to a strong acoustic tone at the cylinder vortex shedding frequency, as
can be seen in Figure 12. Here the computed and measured Power Spectral Densities
(PSD) are shown for the base line configuration for a point on the centerline of the airfoil
surface at 20% chord. It can be seen that the amplitude and frequency of the dominant
tone are largely well reproduced in the numerical simulation with the computed tonal
frequency giving a Strouhal number St = f d/U = 0.2, compared to the measured value
of St = 0.192, and computed amplitude of 114.7 dB that lies within 2 dB of the measured
value. In Figure 12 it can also be seen that, at least for higher frequencies, the broadband
component of the signal is also generally well predicted in the numerical simulation. The
poorer agreement for lower frequencies can be attributed to an insufficient sampling period
for the accurate statistical resolution of these components.
Figure 12: Comparison of computed and measured surface pressure power spectra at x/c = 0.2, z/c = 0.0:
Solid line - experiment, dashed line - computation.
The broadband components of the spectra in Figure 12 are the result of the turbulent
fluctuations in the cylinder wake. The level of agreement observed indicates that the
employed model is capable of resolving the turbulent structures responsible for broadband
noise generation. A snapshot of the computed acoustic near-field is shown in the left of
Figure 13. Here the real part of the Fourier transform of the computed pressure field is
shown at the cylinder vortex shedding frequency. The acoustic field exhibits a dipoletype directivity with the peak direction of radiation forming an angle of approximately
±80◦ to the mean flow direction. The computed far-field directivity is shown alongside
17
Martin Franke, Thomas Röber, Edmund Kügeler, Graham Ashcroft
Figure 13: Measured and computed acoustic fields: Left - computed acoustic near-field, right - comparison
of computed and measured acoustic far-fields (Dashed line - experiment, symbols - computation).
the experimental data in the right of Figure 13. Here the sound pressure level heard by
a limited number of far-field observers has been computed with the aid of an integral
radiation model 26 . For the purpose of evaluating these data an integration surface was
defined that enclosed both the rod and airfoil and was located approximately 2d from
both surfaces. As can be seen the far-field values are generally well predicted and, similar
to the near-field data, display a dipole directivity.
6
CONCLUSION
The current capabilities of the DLR-solver TRACE concerning turbulence modelling
have been presented. For steady computations, a two-equation model with turbomachinery-specific extensions is employed as a standard practice, allowing for the accurate prediction of multi-stage applications. To improve turbulence representation in complex flow
situations, a non-linear EARSM is currently validated, yielding very promising results
for the cases presented at a tolerable computational surplus. A DES approach has been
successfully implemented and validated on unsteady cases featuring complex flow physics.
Turbulence representation, however, remains a challenge for day-to-day industrial application of CFD methods. Future work will thus include refinement and further validation
of the methods presented. Additionally, a complete RSTM implementation is planned.
Improvement of the predictive accuracy at tolerable costs remains a prime goal, the effort
towards accurate and reliable viscous flow prediction methods will continue.
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