A Consistent Dual-Mesh Framework for Hybrid LES/RANS Modeling Heng Xiao∗, Patrick Jenny∗∗ Institute of Fluid Dynamics, ETH Z¨ urich, 8092 Zurich, Switzerland
Abstract In this work, we propose a consistent turbulence-modeling framework for hybrid LES/RANS modeling. In this framework, the filtered and Reynolds averaged Navier-Stokes (RANS) equations are solved simultaneously in the whole domain on their respective meshes. Consistency between the two solutions is achieved in terms of velocity, pressure, and turbulent quantities through additional drift terms in the corresponding equations. This approach leads to clean conditions at the LES/RANS interfaces. Note that this general framework does not depend on the specific choice of LES and RANS models. A hybrid LES/RANS solver is developed within this framework and used to simulate the flow in a plane channel and that in a channel with periodic hills. The results demonstrate that the hybrid solver leads to significantly improved results with moderate computational overhead compared to traditional LES, making it a promising candidate for industrial flow simulations. Keywords: hybrid LES/RANS, turbulent flow, wall-bounded flow
∗
Corresponding author. Tel: +41 44 632 5189; Fax: +41 44 632 1147 Primary corresponding author Email addresses:
[email protected] (Heng Xiao),
[email protected] (Patrick Jenny) ∗∗
Preprint submitted to Journal of Computational Physics
November 14, 2011
1
1. Introduction
2
Large-Eddy Simulation (LES) has gained popularity and successes in the
3
past decades, particularly for free-shear flows, where the computational cost
4
of LES is only weakly dependent on the Reynolds number (∼ Re0.4 ) [1, 2].
5
In wall-bounded flows, however, to resolve the near-wall region (inner layer),
6
the computational cost of LES scales as Re1.8 , similar to Direct Numerical
7
Simulation (DNS), which scales as Re9/4 . Therefore, the high computational
8
cost in wall-bounded flows is still a major hurdle for the application of LES in
9
industrial and practical flows. In many LES conducted for wall-bounded flows
10
in practice, wall modeling based on logarithmic profile or other theoretical
11
models is used to avoid the high resolution [3]. However, in complex flows
12
the validity of these theoretical models is in question.
13
To overcome this difficulty, various hybrid LES/RANS approaches have been
14
developed, where RANS (Reynolds-averaged Navier Stokes) equations are
15
solved in the near-wall region instead of using wall functions. A recent re-
16
view by Fr¨ohlich and von Terzi [3] presented an excellent survey and proposed
17
a detailed classification of these approaches. They divided current hybrid
18
LES/RANS methods into the following categories: (1) blending turbulence
19
models, (2) interfacing turbulence models, (3) segregated turbulence models,
20
and (4) second generation unsteady RANS models (2G-URANS), includ-
21
ing partially averaged Navier-Stokes (PANS) and scale-adaptive simulation
22
(SAS), etc.
23
The segregated models do not mainly target the difficulty related to wall-
24
bounded flows mentioned above and are not further discussed here. An im-
25
portant feature of the 2G-URANS models is that the turbulent stress models
26
do not explicitly depend on the grid spacing, and thus they do not revert
27
to DNS no matter how fine the grid is. According to the definition in the
28
review [3], they do not strictly qualify as a hybrid LES/RANS method, since 2
29
they do not have the LES component (i.e., with a turbulence model depend-
30
ing on grid spacing and reverting to DNS with refined grids), and thus they
31
are referred to as 2G-URANS. In addition, most models in this category are
32
still young and only preliminary results are available at present. Another
33
approach that is omitted in the review of Fr¨ohlich and von Terzi [3] is the
34
method based on hybrid-filtered Navier Stokes (HFNS) equations, proposed
35
by Germano [4]. More recently, Rajamani and Kim [5] conducted some a
36
priori tests and preliminary simulations using this method. This method is
37
also in its early stage of development.
38
The introduction here focuses on two categories of hybrid LES/RANS meth-
39
ods: blending models and interfacing models, because at present the most
40
widely used and mature hybrid LES/RANS models belong to one of the two
41
categories. The blending turbulence models take advantage of the structural
42
similarity of LES and RANS equations. That is, the LES and RANS equa-
43
tions have similar momentum equations, and the only difference lies in how
44
turbulent stresses are modeled. In fact, many subgrid scale (SGS) turbulence
45
models are developed from the corresponding RANS turbulence models. In
46
the blending approach, the turbulent stress tensor is computed as follows: hybrid
τij
= f τijRANS + (1 − f )τijLES ,
(1)
47
where the blending factor f is a smooth function ranging from 0 to 1, de-
48
pending on the spatial location and possibly time as well. The work by Uribe
49
et al. [6] is an example that blends turbulent stresses. Furthermore, if both
50
the LES and the RANS models that are blended belong to the class of the
51
eddy viscosity models, which is often the case in practice, one could directly
52
blend the turbulent eddy viscosity in the same way as in Equation (1). The
53
blending functions are often chosen in an ad-hoc way and the coefficients are
54
then calibrated empirically. This feature leaves a lot of freedom for tuning
55
and thus impairs its predictive capabilities, although sometimes very good
56
results can be obtained with the tuning. 3
57
The interfacing approach is the limiting case of the blending approach with
58
vanishing blending region, or equivalently using a step function as blending
59
function. It thus avoids the ad hoc choice of blending functions. However,
60
the two categories of hybrid models share most other limitations, which will
61
be detailed below. Notable examples of the interfacing models include the
62
detached-eddy simulation (DES) method [7] and its improved variants such as
63
delayed DES [8], which are among the most widely used hybrid LES/RANS
64
methods. However, the DES is based on the Spalart-Allmaras RANS model,
65
which is a one-equation model mostly tuned for external aerodynamic flows
66
at high Reynolds numbers for applications such as flow around airfoils. Its
67
performance on internal wall-bounded flows is not clear in general, although
68
much research is going on [e.g. 9].
69
In general, the interfacing models suffer from the fundamental inconsistency
70
between LES and RANS. The blending models have the same issue, although
71
to a lesser extent. Specifically, even though the LES and RANS equations are
72
similar in structure, the quantities (velocity, pressure, etc.) in the LES equa-
73
tions are filtered quantities and in RANS equations ensemble- or Reynolds-
74
averaged quantities. Consequently, it is difficult for the LES to sustain proper
75
fluctuations near the interfaces (in the blending models) or for RANS to pro-
76
vide boundary conditions to the LES (in the interfacing models). This in-
77
consistency has important physical implications. For example, the blending
78
can lead to artificial “super-streaks” near walls due to the “modeled stress
79
depletion (MSD)”, and consequently spurious buffer layers can be observed
80
in the mean flow profiles near the LES/RANS interfaces [e.g. 3, 10].
81
Additional forcing near the interface is necessary to sustain the velocity fluc-
82
tuations in the LES region. Various attempts have been made to address this
83
problem. For example, Piomelli et al. [11] applied a stochastic forcing term to
84
the momentum equations. Davidson and Dahlstr¨om [12] added fluctuations
85
to the momentum equations according to a DNS of a generic boundary layer. 4
86
Davidson [13] used the backscatter from a scale-similarity SGS stress model
87
as the forcing. Despite some successes, many difficulties remain, which are
88
not detailed here.
89
In addition to the fundamental difficulty discussed above, the blending and
90
interfacing approaches also have the drawbacks of relying upon the existence
91
of clear LES/RANS interfaces, which are difficult to specify in complex ge-
92
ometries. In practical simulations, it is more desirable to dynamically and
93
individually define each cell to be an LES cell or a RANS cell depending on
94
certain criteria, that is, a non-zonal approach is preferred.
95
To avoid the fundamental inconsistencies mentioned above, we propose a con-
96
sistent framework for turbulence modeling, where the filtered and Reynolds-
97
averaged equations are solved simultaneously in the entire domain. That
98
is, both LES and RANS simulation are conducted, which results in some
99
redundancy. To ensure consistency between the two solutions in terms of
100
velocity, pressure, and turbulent quantities, additional drift terms are added
101
to the corresponding equations. This approach leads to very clean condi-
102
tions at the LES/RANS interfaces. By solving two sets of equations (i.e.
103
filtered and RANS) separately, the difficulty of “modeled stress depletion” is
104
avoided. There is no need to add random fluctuations into the LES. On the
105
other hand, we acknowledge that MSD is only one of the many difficulties
106
faced by hybrid methods. Other notable difficulties include the prediction of
107
laminar-turbulence transition and the generation of inflow boundary condi-
108
tions. Admittedly, in these aspects a hybrid solver per se (this one included)
109
does not gain any advantage compared to its individual components. This
110
framework does not specifically deal with these issues. Readers are referred
111
to the recent work of Sagaut and co-workers [14] for an example of such ef-
112
forts. Finally, we emphasize that this general framework does not depend on
113
the specific choice of LES and RANS models.
5
114
The rest of the paper is organized as follows. The details of the framework
115
and the algorithm are presented in Section 2. A hybrid LES/RANS solver
116
is developed as a specific implementation of this general framework. The
117
detailed implementation of the solver, the turbulence models, and the nu-
118
merical methods used in the simulations are presented in Section 3. The
119
developed solver is used to simulate the flow in a plane channel and a chan-
120
nel with periodic hills. The results from numerical simulations are presented
121
in Section 4. Additional issues and further research related to this framework
122
are discussed in Section 5. Section 6 concludes the paper.
123
2. Proposed Framework and Algorithm
124
2.1. Model Equations
125
For incompressible flows with constant density, in the context of LES the
126
filtered momentum and pressure equations can be written as follows: ∂τijsgs ∂ U¯i ∂ U¯i U¯j 1 ∂ p¯ ∂ 2 U¯i + =− +ν − + QLi ∂t ∂xj ρ ∂xi ∂xj ∂xj ∂xj ∂QLi ∂2 1 ∂ 2 p¯ U¯i U¯j + τijsgs + =− , and ρ ∂xi ∂xi ∂xi ∂xj ∂xi
(2a) (2b)
127
where U¯i , p¯, and τijsgs are filtered velocity, filtered pressure, and residual
128
stresses, respectively; t and xi are time and space coordinates, respectively;
129
ν is the kinematic viscosity; ρ is the fluid density, which is assumed constant;
130
QLi is the drift force in the filtered equations to ensure consistency between
131
LES and RANS and will be defined later in Equation (10).
132
Similarly, the Reynolds-averaged momentum and pressure equations are writ-
6
133
ten as: ∂hUi i ∂ (hUi ihUj i) 1 ∂hpi ∂ 2 hUi i ∂hui uj i + =− +ν − + QRi ∂t ∂xj ρ ∂xi ∂xj ∂xj ∂xj 2 2 1 ∂ hpi ∂ ∂QRi and =− (hUi ihUj i + hui uj i) + , ρ ∂xi ∂xi ∂xi ∂xj ∂xi
(3a) (3b)
134
where hUi i, hpi, and hui uj i are filtered velocity, filtered pressure, and Reynolds
135
stresses, respectively; QR i is the drift force in the RANS equations to ensure
136
consistency and will be defined later in Equation (15).
137
As a clarification of the terminology: the ensemble- or Reynolds-averaged
138
Navier-Stokes equations in the context of most hybrid LES/RANS studies
139
are generally unsteady. However, usually they are still referred to as RANS
140
instead of URANS equations. This convention of terminology is followed in
141
this paper.
142
2.2. Consistency and Drift Terms
143
To discuss the consistency issue, we first define an exponentially weighted
144
averaging operator, h•iAVG , applied on a time-dependent quantity φ(t): Z t 1 0 AVG hφi (t) = φi (t0 ) W (t − t0 )dt0 with W (t − t0 ) = e−(t−t )/T , (4) T −∞
145
where t is the current time, t0 denotes earlier times over which the averaging
146
is performed, and T is the time scale for the averaging. The exponential
147
weighting function ensures that the more recent time is weighted stronger
148
than earlier times. Exponential function is chosen because is particularly
149
convenient for implementation (see Equation (18) in Section 2.4). With this
150
definition, the exponentially weighted average velocity, turbulent stresses,
7
151
and dissipation rate in the LES are Z t 1 0 AVG hU¯i i (t) = U¯i (t0 )e−(t−t )/T dt0 , (5a) T −∞ Z 1 t 00 0 00 0 0 AVG hτij i (t) = ui (t )uj (t ) + τijsgs (t0 ) e−(t−t )/T dt0 , (5b) T −∞ Z 1 t ¯ 0 ¯ 0 0 AVG 2ν Sij (t )Sij (t ) − τijsgs (t0 )S¯ij (t0 ) e−(t−t )/T dt0 , (5c) and hεi (t) = T −∞
152
respectively, where u00i = U¯i − hUi iAVG is the fluctuating velocity with respect
153
to the exponentially weighted average. In Equation (5b), the total turbulent
154
stress in the LES is defined as τij = u00i u00j + τijsgs ,
(6)
156
which includes the resolved part and the residual part. Similarly in Equation (5c), the total dissipation rate in LES, ε = 2ν S¯ij S¯ij −τ sgs S¯ij , also includes
157
the resolved part and the residual part. The resolved rate-of-strain tensor is
158
defined as
155
ij
1 S¯ij = 2 159 160
¯ ∂ Ui ∂ U¯j + . ∂xj ∂xi
(7)
One can think of the “exponentially weighted averaging operator” h(·)iAVG (t) = R 0 1 (·)(t0 )e−(t−t )/T dt0 as a linear operator, which operates on any instantaT −∞
161
neous field/quantity, and results in an corresponding exponentially weighted
162
average field/quantity. In Eq (5a), it operates on the velocity field and ob-
163
tains the EWA velocity; in (5b) and (5c) it operates on turbulent stresses
164
and on dissipation, obtaining EWA turbulent stresses and EWA dissipation,
165
respectively.
166
The legitimacy of the LES/RANS coupling shown above relies on the as-
167
sumption that the velocity (and other quantities) can be decomposed into the
168
following components: the mean flow, the resolved fluctuating flow, and the
169
unresolved fluctuating flow. This is a special case of the general framework
170
of multilevel turbulence decomposition [15, 16]. In addition, we assume that 8
171
the exponentially weighted averaging is approximately the same as ensemble-
172
or Reynolds-averaging, i.e., hφiAVG ≈ hφi.
(8)
173
This assumption in Eq. (8) is theoretically rigorous only if the filtering is re-
174
stricted to homogeneous direction of the flow with constant grid spacing [16],
175
which would certainly be violated in practice. However, compared to many
176
of the current hybrid methods where essentially the filtered quantities are re-
177
garded equal to the Reynolds averaged quantities at the interfaces or in the
178
blending regions, this approximation is a significant improvement in terms
179
of consistency.
180
With the two assumptions above, if both filtered and RANS equations are
181
solved for the same flow, the consistency requirements suggest that hU¯i iAVG ≈ hUi i,
(9a)
hτij iAVG ≈ hui uj i,
(9b)
and hεiAVG ≈ εR ,
(9c)
182
where εR is the modeled dissipation rate as in k–ε two-equation turbulence
183
models [17].
184
weighted average quantities, instead of the filtered quantities themselves,
185
are made consistent with their RANS counterparts. This is fundamentally
186
different from the LES/RANS coupling in previous hybrid methods. For this
It is emphasized that in this framework the exponentially
188
reason, we shall also refer to the Exponentially Weighted Average quantities as averaged quantities or EWA quantities for simplicity, i.e., hU¯i iAVG , hτij iAVG ,
189
and hεiAVG are called EWA velocity, EWA turbulent stress tensor, and EWA
190
dissipation rate, respectively. For statistically stationary flows, the consis-
191
tency is rigorous by using a large averaging time scale, although a finite value
192
is usually taken for convergence reasons. For transient flows with coherent
193
structures, however, the averaging time scale should be smaller than the time
194
scale of these structures. See Section 4.4 for more discussions.
187
9
195 196
To fulfill the consistency requirements above, the drift force in the filtered equations is formulated as (hUi i − hU¯i iAVG )/τl + Gij (hU¯j iAVG − U¯j )/τg L Qi = 0
in RANS regions in LES regions, (10)
197
where Gij =
198
hτij iAVG − hui uj i . hτkk iAVG + huk uk i
Equation (10) can be equivalently written as QLi = QL,u + QL,g i i
199
(11)
in RANS regions,
(12)
where ≡ (hUi i − hUi iAVG )/τl QL,u i and QL,g ≡ Gij (hUj iAVG − U¯j )/τg . i
(13a) (13b)
200
We will interpret QL,u and QL,g shortly. LES and RANS regions refer to i i
201
the sub-domains which are well resolved and under-resolved for LES, respec-
202
tively; τl and τg are relaxation time scales related to the EWA velocity and
203
EWA turbulent stresses, respectively. The drift term QLi is only active in the
204
RANS region, where the LES is under-resolved. The first part, QL,u i , lets the
205
average filtered velocity relax towards the RANS velocity. The second part,
206 207
QL,g i , lets the turbulent stresses relax towards the RANS modeled turbulent stresses. Note that the term QL,u only changes the average of U¯i and not the i
QL,g i
208
fluctuations, while
209
The term QL,g can be interpreted as follows. First, it is a linear function i
210
of the tensor Gij , defined as the difference between hτij iAVG and hui uj i (nor-
211 212 213
only changes the fluctuations and not the average.
malized by the sum of their traces). This serves to let hτij iAVG relax towards hui uj i, which is achieved by scaling the fluctuation u00j = U¯j − hUj iAVG of the filtered velocity with respect to the averaged velocity. 10
214
For many RANS models, the reconstructed Reynolds stresses may not be
215
reliable, particularly for separate flows. For example, in the standard k–ε
216
model, the reconstructed Reynolds stresses do not account for the anisotropy
217
near the wall. In these scenarios, it may be better to only require consistency
218
between total turbulent kinetic energy in LES and RANS, that is hτii /2iAVG ≈
219
k R , instead of requiring the full turbulent stress tensors to be consistent
220
(k R being the turbulent kinetic energy in RANS; usually one of the solved
221
quantities). Therefore, QL,g is reformulated as i QL,g = G(hUi iAVG − U¯i )/τg , i where G =
222
223
(14)
R
hτii /2i −k AVG hτii /2i + kR AVG
is a scalar instead of a tensor as in Equation (11). Similarly, the drift force in the RANS equations is: (hUi iAVG − hUi i)/τr in LES regions R Qi = 0 in RANS region,
(15)
224
where τr is the relaxation time scale for the RANS velocity. Also note that
225
QR i is only active in the LES region, where the LES is well resolved. Drift
226
terms are also added in a similar way to RANS model equations for the
227
turbulent quantities. For example, in k–ε turbulence models, the drift terms
228
for k- and ε- equation are Qk = hτii /2iAVG − k R /τr and Qε = hεiAVG − εR /τr .
(16a) (16b)
229
Both source terms are active only in the LES region. Here, the same re-
230
laxation time scale τr as in the velocity equations is adopted, although it is
231
possible to use different time scales.
232
It can be seen that there are several algorithmic parameters in this formu-
233
lation, i.e., T , τl , τg , and τr . It will be discussed in Section 4.4 how these 11
234
algorithmic parameters should be chosen and that the computational results
235
are not very sensitive on these parameters, as long as they are within a rea-
236
sonable range. The determination of whether a region is well resolved or not
237
in LES can be done on a cell-by-cell basis. Therefore, a clear segregation of
238
the LES and RANS region is not essential for the proposed framework.
239
2.3. Dual-mesh approach
240
In this framework, both the filtered equations and the RANS equations are
241
solved in the whole domain. While one could use the same mesh for the two
242
solvers, it is concluded, however, that using different and separate meshes is
243
in general more efficient, due to the following reasons:
244
(1) The RANS mesh should be refined in the near-wall region in the wall-
245
normal direction. If a low Reynolds number turbulence model is used
246
(without using wall functions), the first grid point should be no more
247
than one wall-unit away from the walls. On the other hand, the LES
248
mesh should well resolve the free shear flow region away from the walls.
249
It does not need to resolve the near wall region, since this would be too
250
expensive; this region would be resolved by RANS mesh. This near-wall
251
region becomes the RANS region, and thus the average quantities in the
252
LES relax towards the RANS quantities. Clearly, a dual-mesh approach
253
is optimal since the different resolution requirements for RANS and LES
254
can be honored.
255
(2) For the RANS mesh, particularly in relatively simple flows or in simple
256
blocks of a complex geometry, the resolutions in streamwise and spanwise
257
directions are less important than they are in wall-normal direction, and
258
the overall required resolution only weakly depends on the Reynolds
259
number (∼ log(Re), see Ref [18]). On the contrary, the LES mesh needs
260
to resolve the streamwise and spanwise directions, and the required grid
261
spacing is proportional to the wall-units (and thus strongly depends on 12
262
the Reynolds number). The different scaling behaviors of RANS and
263
LES meshes justify different meshes in the cases mentioned above.
264
An example of such a dual-mesh is shown in Figure 3. With the dual-mesh
265
setup, when calculating the drift terms, for example QLi in Equation (10),
267
the RANS velocity hUi i needs to be interpolated from the RANS mesh to the LES mesh, and vice versa for hU¯i iAVG in Equation (15), which has to
268
be interpolated from the LES mesh to the RANS mesh for the calculation
269
of QR i . It is emphasized that the interpolated quantities are only used to
270
compute the drift terms, which are eventually divided by relaxation time
271
scales. On the other hand, the primary variables (e.g., velocity, pressure,
272
turbulent kinetic energy) are never directly manipulated or overwritten by
273
these interpolated quantities.
274
2.4. Solution Algorithm
275
Summarizing the presentation of the modeling framework, the overall algo-
276
rithm of a hybrid LES/RANS solver is presented in Algorithm 1.
277
For the computation of exponentially weighted average quantities in Equa-
278
tion (5), time integration is needed. To reduce storage and to facilitate
279
numerical computation, it is recognized that Equation (4) is the solution of
280
following equation:
281
dhφiAVG 1 = (φ − hφiAVG ) . (17) dt T Hence, the exponentially weighted averaging can be approximated up to first
282
order as [19]
266
hφiAVG, n =(1 − α)φn + αhφiAVG, n−1 1 with α = , 1 + ∆t/T
13
(18) (19)
283
where n indicates the current time level, and (1 − α) is the weight of the
284
current value φn . With this implementation, the fields from previous steps
285
earlier than (n − 1) do not need to be stored.
286
3. Implementation and Numerical Methods
287
The framework and algorithm presented in Section 2.4 do not depend on
288
specific LES and RANS models or on specific flow solvers. In this section,
289
we present an implementation of the framework and algorithm in an in-
290
compressible turbulent flow solver. This implementation is presented in a
291
separate section in order to emphasize the fact that the framework and al-
292
gorithm of the previous section are general. Similarly, one could implement
293
this algorithm in a compressible solver or in other simulation platforms. The
294
implementation details are presented in Section 3.1. The turbulence mod-
295
els and the numerical methods used for the simulations in this study are
296
discussed in Sections 3.2 and 3.3, respectively.
297
3.1. Solver Implementation
298
According to the framework and algorithm proposed above, a hybrid solver
299
for incompressible turbulent flows has been implemented based on the open
300
source CFD platform OpenFOAM [20, 21], taking advantage of its existing
301
LES and RANS solvers as well as its field operation capabilities. The finite
302
volume method with unstructured meshes is used for both LES and RANS
303
solvers, but LES and RANS may use different meshes, which is motivated
304
in Section 2.3. The PISO (Pressure Implicit with Splitting of Operators)
305
algorithm [22] is used to solve the coupled momentum and pressure equations
306
in both LES and RANS solvers. Collocated grids are used and the Rhie and
307
Chow interpolation is used to prevent the pressure–velocity decoupling [23]. 14
308
At each time step, the PISO algorithm as used in OpenFOAM can be sum-
309
marized as follows [24]:
310
(1) Momentum prediction: solve the fluid momentum equation using the
311
312 313
314
pressure field and the external forcing from the previous step. (2) Pressure solution: solve the pressure equation for the updated pressure field. (3) Velocity correction: correct the velocity field using the updated pressure
315
field with the momentum equation.
316
Repeat steps 2 and 3 until convergence.
317
Parallel processing is a built-in capability of OpenFOAM through domain
318
decomposition, where the mesh (and all the fields associated with the mesh,
319
the same hereafter) are decomposed and distributed to all processors. The
320
processors communicate with each other using MPI (Message Passing Inter-
321
face). Since a dual-mesh approach is used in this study, the LES and the
322
RANS meshes are decomposed separately, both meshes are distributed to all
323
processors. Thus, each processor holds part of the LES mesh and part of the
324
RANS mesh. For the convenience of implementation and to minimize com-
325
munication, we have ensured that the LES and RANS meshes on the same
326
processor approximately cover the same domain. This is easily achieved be-
327
cause we assume that the LES and RANS meshes cover the same domain,
328
although the meshes are different. This implementation decision is made for
329
coding convenience and computational efficiency only. It is not an essential
330
part of the algorithm, and one could certainly implement differently.
331
3.2. Turbulence Models
332
In the filtered momentum equation (2a), the residual stress term τijsgs needs
333
to be modeled. Various SGS turbulence models exist. In this study, a one-
334
equation eddy (OEE) viscosity model is used, where the equation for the 15
335
SGS kinetic energy ksgs is solved [25, 26], i.e., ∂ksgs ∂(ksgs U¯i ) ∂ ∂ksgs sgs ¯ + = − τij Sij + νsgs − εsgs , ∂t ∂xi ∂xi ∂xi 2 1 with τijsgs − ksgs δij = − 2νsgs (S¯ij − S¯kk δij ), 3 3 1/2 νsgs = Ck ksgs ∆ , 3/2 −1 ∆ , and εsgs = Cε ksgs
(20a) (20b) (20c) (20d)
336
where ∆ is the filter size. The rate-of-strain tensor S¯ij is defined in Equa-
337
tion (7); Ck and Cε are model constants; εsgs is the SGS dissipation rate;
338
νsgs is the subgrid scale eddy viscosity. Yoshizawa [25] observed that the
339
standard Smagorinsky model would be recovered from this model if produc-
340
tion equals dissipation, where the correspondence between model constants
341
in Equation (20) and the Smagorinsky constant is Cs = Ck /Cε . The stan-
342
dard coefficients are used, Ck = 0.094 and Cε = 1.048, which corresponds to
343
Cs = 0.167. Finally, van-Driest damping l = Cs ∆(1 − exp(−y + /25)3 )0.5 is
344
applied on the length scale near the wall.
345
Similarly, in the RANS momentum equation (3a), the Reynolds stress term
346
hui uj i also needs to be closed. The low-Reynolds number k–ε turbulence
347
model by Launder and Sharma is used [27]. Details of the model can be
348
found, for example, in reference [17]. The same standard coefficients as in
349
that reference are employed here. Since a damping function is included in
350
the modeling equations, no wall functions are used, but such low-Reynolds
351
number models require that the first grid points are located approximately
352
within one wall-unit away from the wall.
353
3.3. Numerical Methods
354
In the LES and the RANS simulations, second order central schemes are used
355
for both convective and diffusion terms. A second-order backward scheme is
3/4
16
1/4
356
used for the time integration of all variables including filtered velocities U¯i ,
357
SGS kinetic energy ksgs , RANS velocities hUi i, the turbulent kinetic energy
358
k R , and the dissipation rate εR . A multi-grid method is used to solve the
359
linear equations obtained from the finite volume discretizations.
360
4. Numerical Simulations
361
The developed hybrid LES/RANS solver as described in Section 3 is used
362
to simulate two representative cases: (1) the plane channel flow and (2) the
363
flow in a channel with periodic hills. The first case is a classical benchmark
364
case of wall-bounded flows with simple geometry. The second case features
365
a massive separation due to the wall curvature. It has been reported that
366
RANS models, even with second-order closure, have difficulties predicting
367
the mean flow profiles correctly [9]. On the other hand, properly resolved
368
LES shows much better performance.
369
Before presenting the results, the general issues are explained in Section 4.1,
370
including guidelines for choosing mesh resolutions, the objectives of the sim-
371
ulations, and the methods used to assess the results. The simulation results
372
are shown in Sections 4.2 and 4.3. Finally, the sensitivity of the results on
373
the algorithmic parameters is studied in Section 4.4.
374
4.1. General remarks on the simulations
375
In this study, the LES meshes are designed to be adequate to resolve the
376
free-shear region, but with little refinement near the wall. As a result, in the
377
near-wall region the mesh is coarse in all directions in terms of wall-units. It is
378
expected that the mesh in the near-wall region is at least fine enough to allow
379
for enough fluctuations, if forced by the term QL,g as in Equation (12) in order i
380
to achieve consistency with the TKE in RANS. On the other hand, the RANS 17
381
meshes are designed such that the first grid point is located approximately
382
one wall unit from the wall. The grids in other directions, particularly for
383
the homogeneous directions, are very coarse. The LES and RANS meshes for
384
the test case are shown in Figure 3. More discussion on grid spacing follows
385
in Section 2.3.
386
Compared to the fine meshes used in previous studies, the LES mesh adopted
387
here is very coarse. For example, for the periodic hill case of Re=10595, ap-
388
proximately five million cells are used for a properly revolved LES and one
389
million cells are used for DES (see reference [9]). Simulations presented here
390
employ a grid with less than 105 cells. However, it is fine enough to resolve the
391
free shear region. The purpose of this study is not to obtain the best results
392
possible or to provide benchmark solutions. Instead, we intend to demon-
393
strate that the proposed method can lead to improved results compared to
394
standalone LES on the relatively coarse grids with moderate computational
395
overhead. Therefore, it is reasonable to assess the performance of the hy-
396
brid method by examining quality of the results against those obtained with
397
standalone LES on the same mesh.
398
In addition, particular emphasis is placed on the internal consistency of the
399
LES and RANS solutions, which forms the foundation of the proposed frame-
400
work. For the case presented here, it is challenging due to the transient co-
401
herent structures, and thus smaller relaxation time scales are used. Detailed
402
consistency study is presented.
403
As demonstrated by the studies of Davidson [28, 29], evaluating whether the
404
resolution of an LES is adequate is a challenging task. Considering the fact
405
that determining appropriate LES/RANS regions is a separate issue from
406
the coupling algorithm itself, for simplicity, pre-specified LES/RANS regions
407
according to the cell distances to the nearest wall are used for the simulations
408
presented here. This issue is further discussed in Section 5.2. 18
409
Compared to the LES on the same grid, the hybrid method also needs to
410
compute the drift terms and to solve the RANS equations additionally (usu-
411
ally on a coarser grid). The overhead due to these operations is approxi-
412
mately 30% to 50% of the total computational cost for the simulations pre-
413
sented here. However, since the RANS meshes are almost independent of the
414
Reynolds number, the fraction of the overhead becomes negligible for high
415
Reynolds number flows encountered in practice. Future development will
416
consider larger time-steps for the RANS solver (compared to the small time-
417
steps in LES), and the computational overhead compared to standalone LES
418
would be further reduced. For high Reynolds number flows, theoretically the
419
computational cost of the hybrid method should depend on the Reynolds
420
number in the same way as LES does for free-shear flows, probably with a
421
slightly larger proportionality constant.
422
4.2. Flow in a plane channel
423
In this case, the fully developed turbulent flow in a plane channel is simulated
424
using the hybrid method. The domain size, meshes, and resolutions are pre-
426
sented in Table 1. The nominal Reynolds number based on friction velocity p uτ and half channel width δ is Reτ = uτ δ/ν = 395, where uτ = τw /ρ, and
427
τw is the wall shear stress. The actual Reτ is a result of the computation.
428
The streamwise (x) and the spanwise (z) directions have periodic boundary
429
conditions. Non-slip boundary conditions are applied at the wall (and y
430
coordinate is aligned with the wall-normal direction). A pressure gradient is
431
applied on the whole domain to keep the mass flux constant in x direction.
432
The flow-through time is thus defined as Tthr = 2πδ/Ub , and the algorithmic
433
parameters are: T = 11δ/Ub , τl = 1.3δ/Ub , τr = 1.3δ/Ub , and τg = 0.67δ/Ub .
434
The RANS region consists of all cells with distance smaller than D = 0.2δ to
435
the nearest wall. To achieve stable coupling, a linear ramp function F (t) is
425
19
436
multiplied on all drift terms during an initial simulation period of 2T , where
437
F (0) = 0 and F (t ≥ 2T ) = 1.
438
The computed Reτ , which is indicative of the prediction of wall shear stress,
439
is presented in Table 2. It can be seen that the predicted wall shear stress is
440
significantly improved in the hybrid method compared to the standalone LES
441
result on the same mesh (The LES mesh is intentionally made uniform in
442
all directions without attempting to resolve the wall). The wall shear stress
443
prediction is improved because the averaged velocity relaxes towards the
444
RANS velocity in the near-wall region. The mean velocity obtained through
445
averaging in time and spanwise/streamwise directions are shown in Figure 1.
446
The improvement over LES is appreciable.
447
4.3. Flow in a channel with periodic hills
448
The benchmark case of the flow in a channel with periodic hills is based
449
on the original experiment by Almeida et al. [30]. Recently this case has
450
been modified to make it more convenient for numerical simulations. New
451
experiments and benchmark numerical simulations (including LES and DNS)
452
have been conducted for a wide range of Reynolds numbers within the French-
453
German research group on Large-Eddy Simulation of Complex Flows [9, 31].
454
In this work, the results from the high Reynolds number case Re = 10595 is
455
presented.
456
The geometry of the computational domain is shown in Figure 2 and spec-
457
ifications are given in Table 1. The exact shape of the hill is described by
458
piecewise continuous polynomials [32]. Similar to the case above, periodic
459
boundary conditions are used in the streamwise and spanwise directions, and
460
a pressure gradient is applied on the whole domain to keep the mass flux con-
461
stant. The Reynolds number based on crest height and bulk velocity (at the
462
crest) is 10595. 20
463
Meshes with resolutions, simulation time-span, and time step size are shown
464
in Table 1. In this case, all lengths and times are normalized with H and
465
H/Ub , respectively. The flow-through time is defined as Tthr = Lx /Ub =
466
9H/Ub , and the algorithmic parameters are: T = 2.2H/Ub , τl = 0.28H/Ub ,
467
τr = 0.28H/Ub , and τg = 0.07H/Ub . The RANS region consists of all cells
468
with distance smaller than D = 0.2H from the nearest wall. The same linear
469
ramp function as above is applied.
471
The internal consistency between LES and RANS is investigated. Time series of the filtered velocity component U¯1 , the exponentially weighted average
472
velocity component hU¯1 iAVG , and the RANS velocity component hU1 iAVG are
473
depicted in Figure 4 at four different locations. The same comparison for k,
474
hτii /2iAVG , and k R is presented in Figure 5 at the same four locations.
475
From Figures 4 and 5, it can be observed that the filtered signals fluctuate
476
with high frequencies, while the averaged LES and the RANS quantities
477
are much smoother. The internal consistency, i.e., the agreement between
478
exponentially weighted average quantities and RANS quantities, is achieved
479
reasonably well. The fluctuations of RANS quantities are still smaller than
480
the EWA quantities (which are obtained by averaging the LES quantities;
481
see Equation (5)). This is true even when a much larger averaging time-
482
scale is used and is due to the very coarse RANS mesh, and probably more
483
important, due to the high dissipation from the RANS turbulence model. The
484
differences in the amount of fluctuations that can be sustained by RANS and
485
LES lead to some transient inconsistencies, e.g., t/Tthr = 14 at points (b) and
486
(c), but the durations are rather short. Since the consistency is enforced by
487
relaxation models, if so desired, the consistency can be further improved by
488
decreasing relaxation time scales. As can be seen from Figure 6(a), the mean
489
streamwise velocity profiles of the RANS and those of the LES agree very
490
well at all locations. For comparison, the same velocity profiles obtained by
491
standalone LES and RANS solvers are shown in Figure 6(b), which displays
470
21
492
a much larger discrepancy. This comparison confirms the effectiveness of the
493
drift terms.
494
In Figure 7(a), the mean streamwise velocities (time- and spanwise-averaged
495
filtered velocity) are presented at nine locations (from x/H = 0 to 8, with an
496
interval of 1); plotted at the corresponding locations in the domain. The hy-
497
brid results are compared with the standalone LES (the same grid is used as
498
for the hybrid LES solver) and the benchmark solutions [31]. The improve-
499
ment of the hybrid results compared to LES is significant, particularly from
500
y/H = 1.8 to y/H = 2.8 and at the bottom of the channel. The shaded re-
501
gion covering the recirculation and reattachment regions is enlarged to show
502
more details in Figure 7(b). The improvement in the near-wall region is due
503
to the good wall-resolution of RANS mesh, which in turn provides guidance
504
to the LES mean velocity via the drift term. It is noted that the standalone
505
RANS performs poorly in this region, partly because the spanwise velocity
506
fluctuations are not well represented in RANS [9]. The RANS results are
507
shown in Figure 4(b). From our experience with this study, in pure RANS
508
simulations the fluctuations gradually damp out and the solutions eventually
509
approach steady state after a few flow-through times. However, when cou-
510
pled with the LES solver, the hybrid RANS solutions are made consistent
511
with the exponentially weighted average filtered velocities (in the LES re-
512
gion). Consequently, the hybrid RANS simulations are unsteady in general.
513
This explains why the RANS solver predictions are poor, but that RANS
514
can still help to improve the LES results in the hybrid solver.
515
To further illustrate the improvements of the hybrid simulation results over
516
the pure LES, the mean turbulent kinetic energy are presented in Figure 8
517
for nine locations along the channel (same as above). It is evident from Fig-
518
ure 8(a) that the mean TKE in the free-shear region (between y/H = 0.5
519
and y/H = 2) is better predicted by the hybrid solver, particularly down-
520
stream of x/H = 3. The shaded region is zoomed in and highlighted in 22
521
Figure 8(b), which suggest that the improvement in the near-wall region is
522
equally good. The comparison of turbulent shear stress component u0 v 0 is
523
shown in Figure 9 for four locations (x/H = 1, 3, 5, 7). The hybrid simulation
524
results again show better agreement with the benchmark solution than the
525
pure LES results.
526
The friction coefficient Cf at the bottom wall predicted by standalone LES
527
and RANS are compared to benchmark results in Figure 10(a). The hybrid
528
solutions are presented in Figure 10(b), which show significant improvement
529
in the Cf prediction, compared to standalone LES and RANS. It can be seen
530
that in the whole region the prediction of Cf by pure LES is too low in mag-
531
nitude compared to the reference results, while RANS simulation predicts a
532
wrong reattachment point and thus a different overall behavior. In summary,
533
compared to the standalone LES and RANS solvers, the hybrid method leads
534
to improved wall-shear stress predictions at the bottom wall, which can be
535
explained similarly as above.
536
4.4. Parameter choices and sensitivity study
537
In this study, we have three groups of algorithmic parameters: (1) the lo-
538
cation D of the LES/RANS interface; (2) the averaging time scale T ; and
539
(3) the relaxation time scales for the average filtered velocities (τl ), for the
540
RANS velocity (τr ), and for the fluctuations in the filtered velocity (τg ).
541
In the future LES and RANS regions shall be detected dynamically based on
542
cell quantities, and thus this free parameter eventually will be eliminated. In
543
addition, clear guidelines have been proposed for the choice of this parameter
544
in the hybrid methods such as DES [33]. Therefore, choosing this parameter
545
is not of concern here, although studies of the sensitivity on this parameter
546
are presented.
23
547
The averaging time scale T and the relaxation time scales τl , τl , and τg are
548
specific to this framework, and will thus be discussed in more detail. The
549
purpose of the averaging on filtered quantities is to obtain appropriate quan-
550
tities that can be correctly made consistent with the corresponding RANS
551
quantities (since the filtered quantities in LES are fundamentally different
552
from the RANS quantities). Therefore, the averaging should smooth out all
553
the small-scale turbulent structures, while keeping the coherent structure.
554
Consequently, the time scale T should be larger than the turnover time of
555
most eddies, but smaller than the time scale of the large coherent structures.
556
Based on the same reasoning, the time scales τl and τr should be fractions
557
of T (e.g., between 1/4 and 1/20). Values of τl and τr comparable to or
558
even larger than T may lead to large internal inconsistencies. On the other
559
hand, using too small relaxation time scales (for example, two or more orders
560
of magnitude smaller) may cause convergence difficulties and deteriorated
561
results. The time scale τg is related to the QL,g term, which adjusts the i
562
fluctuation levels of the filter velocity to achieve consistency with the modeled
563
turbulent stresses in the RANS calculations. The time scale of these actions
564
has to be comparable to the smallest resolved eddies. Hence, τg should be
565
smaller than τl and τr . Using a too large τg would lead to large inconsistencies
566
in the Reynolds stresses or TKE, while a too small τg may cause convergence
567
difficulties. From our experience, if the time scales are chosen based on the
568
reasoning discussed above, good results are usually observed.
569
Parametric studies are conducted to assess the influence of the averaging
570
and relaxation time-scales and of the LES/RANS interface location on the
571
computational results. The simulation presented in Section 4.3 is considered
572
as base case and the parameters are varied as follows:
573
(a) the relaxation time scales in the base case are scaled by factors of 0.75,
574 575
1.5, and 2; (b) the averaging time scale is varied, i.e., T Ub /H is set to 1.7, 2.8, and 4.5; 24
576
577 578
and (c) the LES/RANS interface location is placed at 0.1H, 0.15H, and 0.25H away from the walls.
579
Simulations are conducted with the varied parameters (nine cases in total, in
580
addition to the base case), and the results are presented in Figures 11(a)–(c),
581
respectively, and are summarized in Figure 11(d). In all the plots, the results
582
from the standalone LES are shown as a reference to quantify the variations of
583
the results due to perturbed parameters. It can be seen that the results seem
584
to be relatively more sensitively to the perturbation of LES/RANS location
585
D, as shown in Figure 11(c). However, overall the differences between the
586
results with the perturbed parameters are small compared to the difference
587
to the standalone LES results.
588
Another parameter of interest, although not intrinsic to the hybrid frame-
589
work, is the Smagorinsky constant Cs . The sensitivity of the hybrid simula-
590
tion results to the perturbation of Cs is studied. In Figure 12, the mean
591
velocity at four locations, (x/H = 1, 3, 5, 7) with standard and reduced
592
Smagorinsky constants (Cs = 0.167 and 0.1, respectively) are compared.
593
The results obtained from pure LES with Cs = 0.167 are also presented as
594
above. It can be seen that differences due to perturbed Cs are relatively small
595
compared to the differences between hybrid simulations results and pure LES
596
results. This observation further demonstrates the robustness of the hybrid
597
framework.
598
5. Discussions
599
In this study, we propose a novel hybrid framework and algorithm for tur-
600
bulent flow simulations. At this stage, we focus on the most fundamental
601
aspects of the method to investigate and demonstrate its feasibility and po25
602
tential. Other non-essential features, possible improvements, variants, and
603
extensions of the algorithm are left for future research, but are briefly dis-
604
cussed in this section.
605
5.1. Using other turbulence models
606
As emphasized in Section 3, the hybrid solver used in this study is one pos-
607
sible implementation. Considering the purpose of this study as mentioned
608
above, we choose to use simple LES and RANS models, which are not nec-
609
essarily state-of-the-art according to the most recent literature. By using
610
modern subgrid-scale models for LES (e.g., dynamic models, scale-similarity
611
models, or mixed SGS models) and/or RANS models with better perfor-
612
mance near the wall (e.g., k–ω [17] or Reynolds stress models with elliptic
613
relaxation [34]), better performance can be expected from the hybrid method.
614
Other models will be implemented and investigated in future studies.
615
5.2. Dynamic evaluation of resolution
616
Although in the current study the LES/RANS regions are pre-specified, the
617
framework in principle can accommodate for cell-based dynamic adaptation
618
of these regions. The main difficulty, however, is a general and reliable cri-
619
terion. Davidson conducted extensive studies on the evaluation of appropri-
620
ate resolution for LES of simple wall-bounded flows [28] and recirculating
621
flows [29]. It is concluded that none of the criteria found in the literature
622
consistently give reliable results and that the two-point correlations are con-
623
sidered to be the most reliable. However, the two-point correlations criterion
624
is not practical in the context of hybrid LES/RANS simulations because res-
625
olution evaluations need to be performed on the fly and based on individual
626
cell quantities. In addition, the numerical dissipation, which is significant
26
627
in solvers using low-order schemes [35, 36], further complicates the prob-
628
lem. This issue is further investigated and the results will be published in a
629
separate work.
630
6. Conclusion
631
In this work, we propose a novel consistent framework for hybrid LES/RANS
632
modeling, where the filtered and RANS equation are solved simultaneously in
633
the whole domain on their respective meshes. Consistency between the LES
634
and RANS solutions is enforced via drift terms in the corresponding equa-
635
tions. This framework leads to very clean conditions at the LES/RANS in-
636
terfaces and allows for individual cell-based determination of LES and RANS
637
regions.
638
As a specific implementation, a hybrid solver has been developed according
639
to the proposed framework and algorithm using the open-source CFD plat-
640
form OpenFOAM. The developed solver is used to simulate a representative
641
case of the flow in a plane channel and that in a channel with periodic hills.
642
Results demonstrate that internal consistency is honored faithfully and that
643
the improvements over standalone LES on the same grid are appreciable.
644
Parametric studies suggest that the sensitivity of the results on algorith-
645
mic parameters is minor. Therefore, the proposed framework is a promising
646
candidate for hybrid LES/RANS simulations.
647
Further improvements are likely by using dynamically adjusted averaging
648
time scales, automatically detected LES/RANS regions based on cell quan-
649
tities and more advanced turbulence models. This, however, is subject of
650
future investigations.
27
651
Acknowledgement
652
We acknowledge the financial support from the Swiss Commission for Tech-
653
nology and Innovation (CTI), and computational resources provided by ETH
654
Z¨ urich. We thank Dr. M. Breuer at the University of Erlangen-Nuremberg
655
for providing us the benchmark data for comparison. Helpful discussions
656
with Prof. L. Kleiser at ETH Z¨ urich are appreciated. The first author would
657
like to thank Michael Wild for the fruitful discussions on numerous occasions
658
throughout this work. His technical advice during the development of the
659
hybrid solver is gratefully acknowledged.
660
References
661 662
663 664
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ters in Heat and Mass Transfer 1 (2) (1974) 131–138.
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flow, in: Quality and Reliability of Large-Eddy Simulations II, Springer,
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2010, pp. 269–286.
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[30] G. P. Almeida, D. F. G. Durao, M. V. Heitor, Wake flows behind two-
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dimensional model hills, Experimental Thermal and Fluid Science 7 (1)
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[31] M. Breuer, N. Peller, C. Rapp, M. Manhart, Flow over periodic hills:
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Numerical and experimental study in a wide range of reynolds numbers,
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Computers & Fluids 38 (2) (2009) 433–457.
746 747
748 749
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[32] L. Temmerman, M. A. Leschziner, Flow over 2D periodic hills, http://cfd.mace.manchester.ac.uk/twiki/bin/view/CfdTm/TestCase014. [33] P. R. Spalart, Young person’s guide to detached-eddy simulation grids, Tech. Rep. CR-2001-211032, NASA (2001). [34] S. B. Pope, Turbulent Flows, Cambridge University Press, , Cambridge, 2000. 31
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tion of a forward-backward facing step for acoustic source identification,
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International Journal of Heat and Fluid Flow 24 (4) (2003) 562–571.
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[36] I. B. Celik, Z. N. Cehreli, I. Yavuz, Index of resolution quality for large eddy simulations, Journal of Fluids Engineering (2005) 949–958.
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[37] R. D. Moser, J. D. Kim, N. N. Mansour, Direct numerical simulation
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of turbulent channel flow up to Reτ =590, Physics of Fluids 11 (1999)
759
943–945.
32
1. Choose LES and RANS models and other parameters ; 2. Initialize the fields for LES and RANS ; 3. Initialize the fields for Exponentially Weighted Average (EWA) quantities ; 4. Initialize all drift terms to zeros ; for each time step do 1. For each grid cell determine whether it is well resolved (in LES), and assign it to LES or RANS region accordingly ; 2. Solve filtered momentum and pressure Poisson equations (2) for U¯i and p¯ ; 3. Solve averaged momentum and pressure Poisson equations (3) for hUi i and hpi ; 4. Update EWA quantities hU¯i iAVG , hτij iAVG and hεiAVG , according to Equation (5) ; 5. Interpolate the quantities needed for the calculation of drift terms ; 6. Update drift terms according to Equations (10), (15), and (16) ; end Algorithm 1: Overall algorithm of the hybrid solver as implemented in this work.
33
Table 1: Domain and mesh parameters for the test case. x, y, z are aligned with streamwise, wall-normal, and spanwise directions, respectively.
cases
plane channel
periodic hill
domain size (Lx × Ly × Lz )
2πδ × 2δ × πδ
9H × 3.036H × 4.5H
simulation time-span
800(≈ 120Tthr )
460Ub /H (≈ 50 Tthr )
Nx × Ny × Nz (LES)
50 × 60 × 30
74 × 37 × 36
Nx × Ny × Nz (RANS)
10 × 80 × 10
74 × 37 × 18
∆x × ∆y × ∆z in y + (LES) 50 × 12 × 41
76 × [30, 72] × 78
(1)
first grid point (RANS)
0.65y +
below 2y + at most areas
time-step size
6.68 × 10−3 Ub /δ
2.8 × 10−3 Ub /H
(2)
(1)Numbers in the brackets indicate the range of ∆y (smallest for the cells next to the wall and largest for those at the center line). (2) The wall unit is defined p as y + = ν/uτ = ν/ τ /ρ, where τ is the shear stress. Table 2: Comparison of computed Reτ (Reynolds number based on friction velocity and half channel height), which is indicative of the wall shear stress predictions.
Reτ nominal
395
DNS (reference [37])
392
pure LES
339
hybrid LES
371
34
25 20 U/uτ
15 10
hybrid LES pure LES DNS (Moser et al. 1999)
5 00.0
0.2
0.4
x/δ
0.6
0.8
1.0
Figure 1: Mean velocity profile of the hybrid method compared to the LES results on the same mesh and with DNS data [37]. The LES mesh is uniform in all directions with mesh density (in y + ) presented in Table 1.
Lx general flow direction
Ly y z
H x
Figure 2: Domain shape for the flow in the channel with periodic hills. The square and the circle denote the approximate locations of the separation and reattachment, respectively. x, y, and z coordinates are aligned with the streamwise, wall-normal, and spanwise directions, respectively. The dimensions of the domain are: Lx = 9H, Ly = 3.036H, and Lz = 4.5H.
35
Figure 3: The meshes used for (a) the LES and (b) the RANS in the simulation of the flow over periodic hills. The LES mesh is designed to resolve the free-shear region and with little stretching towards the wall. The RANS mesh is refined in the near-wall region in the wall-normal direction.
36
Normalized velocity
1.2 1.1
(a) filtered velocity consistent velocity RANS velocity
(b) 1.2
1.0 0.9 0.8 0.7 0.6 0.5 0
5
10
t/T0
4 3 2 1 0 1 0 2 4 6 8 10 15 20
Normalized velocity
1.3
0.6 0.4 0
5
10
t/T0
4 3 2 1 0 1 0 2 4 6 8 10 15 20
(d) 0.6
0.0 0.1 0.2 0.3 0.4 0.5 5
10
t/T0
4 3 2 1 0 1 0 2 4 6 8 10 15 20
Normalized velocity
0.1 Normalized velocity
0.8
(c)
0.2
0.6 0
1.0
0.4 0.2 0.0 0.2 0
5
10
t/T0
4 3 2 1 0 1 0 2 4 6 8 10 15 20
Figure 4: Time series of filtered velocity, exponentially weighted average (EWA) velocity, and RANS velocity (all normalized by Ub ). The time series are presented for four locations: (a) x/H = 3, y/H = 1.9; (b) x/H = 7, y/H = 1.9; (c) x/H = 3, y/H = 0.1; and (d) x/H = 7, y/H = 1.9. The spanwise coordinates are z/H = 2 for all points. Plots (a) and (b) correspond to points located in the LES region, and plots (c) and (d) correspond to points located in the RANS region.
37
(a)
Normalized TKE
0.06
(b) 4 3 2 1 0 1 0 2 4 6 8 10
total TKE EWA TKE RANS TKE
0.05 0.04 0.03 0.02
0.06 0.04 0.02
0.01 0
4 3 2 1 0 1 0 2 4 6 8 10
0.08 Normalized TKE
0.07
5
10
t/T0
15
0.00 0
20
5
(c)
0.06 0.05 0.04 0.03
0.05 0.04 0.03 0.02
0.02 0.01 0
20
4 3 2 1 0 1 0 2 4 6 8 10
0.06
Normalized TKE
Normalized TKE
0.07
15
(d) 4 3 2 1 0 1 0 2 4 6 8 10
0.08
10
t/T0
5
10
t/T0
15
0.01 0
20
5
10
t/T0
15
20
Figure 5: Time series of total turbulent kinetic energy in LES (i.e., τii /2; see Equation (6)), exponentially weighted average (EWA) TKE hτii /2iAVG , and RANS turbulent kinetic energy k R . The coordinates of the four points are the same as in Figure 4.
38
y/H y/H
3.0 2.5 2.0 1.5 1.0 0.5 0.0 0
3.0 2.5 2.0 1.5 1.0 0.5 0.0 0
(a)
LES (hybrid)
RANS (hybrid)
outline of domain
2
4
LES (standalone)
6
x/H; 3U/Ub + x/H
(b)
8
outline of domain 10
12
RANS (standalone)
outline of domain
2
4
6
x/H; 3U/Ub + x/H
8
outline of domain 10
12
Figure 6: Consistency between the LES and the RANS: mean streamwise velocity from the LES and that from the RANS are compared. (a) LES and RANS velocities from the hybrid solver with coupling; (b) LES and RANS velocities from standalone LES and RANS solvers without coupling. For the LES results, the lines pass through all data points, but markers are only shown for every seventh points for clarity.
39
y/H
3.0 2.5 2.0 1.5 1.0 0.5 0.0 0
(a) outline of domain
4
2
6
DNS (Breuer et al.)
8
outline of domain 10
pure LES
12
hybrid
(b)
1.0 0.8 y/H
0.6 0.4 0.2 0.01
2
3
4
5
6
x/H; 3U/Ub + x/H
7
8
9
Figure 7: Comparison of mean streamwise velocities between the results from the benchmark simulations [31], the standalone LES, and the current hybrid method. The velocities are all normalized by the bulk velocity Ub (at the crest), presented for all locations between x/H = 0–8 with an interval of 1. Plot (a) shows the velocities plotted on the shape of the domain, at the corresponding locations. Plot (b) is the enlargement of the shaded region in plot (a).
40
(a)
3.0 2.5
y/H
2.0 1.5 1.0 0.5 0.0
0
2
x/H;
Breuer et al.
4
−20k/Ub2
+ x/H
6
8
pure LES
hybrid
(b)
3.0 2.9 y/H
2.8 2.7 2.6 2.5 2.43
4
5
6
−40k/Ub2
+ x/H
7
8
9
Figure 8: Comparison of mean turbulent kinetic energy (TKE) between the results from the benchmark solution [31], the standalone LES, and the current hybrid method. The energy is normalized by Ub2 , presented for all locations between x/H = 0–8 with an interval of 1. Plot (a) shows the TKE plotted on the shape of the domain, at the corresponding locations. Plot (b) is the enlargement of the shaded region in plot (a).
41
(a)
3.0 2.5
y/H
2.0 1.5 1.0 0.5 0.00
2
Breuer et al.
x/H;
4
−90
®
6
u0 v0 /Ub2 + x/H
pure LES
8
hybrid
Figure 9: Comparison of shear stress between the results from the benchmark solution [31], the standalone LES, and the current hybrid method. The shear stresses are normalized by Ub2 , presented for four locations (x/H = 1, 3, 5, 7).
42
(a) DNS (Breuer et al. 2009) pure LES Pure RANS
0.03
Cf
0.02 0.01 0.00 0.010
1
3
4
x/H
5
6
7
8
9
6
7
8
9
(b) DNS (Breuer et al. 2009) hybrid
0.03 0.02 Cf
2
0.01 0.00 0.010
1
2
3
4
x/H
5
Figure 10: Comparison of friction coefficients Cf = 2τw /(ρUb2 ) on the bottom wall, obtained from the benchmark simulation [31], pure LES, pure RANS, and the current hybrid method. Pure LES/RANS results are presented in separate plots from the hybrid results for clarity. (a) Friction coefficient obtained using pure LES and pure RANS, compared to benchmark results. (b) Friction coefficient obtained from the hybrid method. The RANS velocities are used for the calculation of wall shear stress since it is based on a finer near-wall mesh.
43
3.0 2.5 2.0 1.5 1.0 0.5 0.0
(a)
10
3.0 2.5 2.0 1.5 1.0 0.5 0.0
y/H
10
3.0 2.5 2.0 1.5 1.0 0.5 0.0
2
0
2
4
6
x/H; 3U/Ub + x/H
8
(c) y/H
y/H y/H
3.0 2.5 2.0 1.5 1.0 0.5 0.0
2
0
2
4
6
x/H; 3U/Ub + x/H
8
(b)
2
0
4
2
6
8
10
6
8
10
x/H; 3U/Ub + x/H
(d)
2
0
2
4
x/H; 3U/Ub + x/H
Figure 11: Parameter sensitivity study: mean streamwise velocity of the base case compared to the cases with perturbed algorithmic parameters. Legend:
pure LES; solid
lines of various colors (or gray scales): hybrid LES in the base case and in those with perturbed parameters. The plots show the scattering of the velocity profiles with (a) the relaxation times scaled by factors of 0.75, 1.5, and 2; (b) varied averaging time scales, T Ub /H = 1.7, 2.8, and 4.5; and (c) varied LES/RANS interface locations: D/H = 0.1, 0.15, and 0.25 away from the walls. Plot (d) summarizes all the profiles in (a)–(c). In each plot, a part of the profile for one location is enlarged. The base case result is shown on all the plots.
44
3.0 2.5
y/H
2.0 1.5
Pure LES Hybrid Hybrid, CS =0.1
1.0 0.5 0.00
2
4
6
6U/Ub + x/H
8
10
12
Figure 12: The results obtained from the current hybrid method using standard Smagorinsky constant (Cs = 0.167) and reduced value Cs = 0.1 are compared with the standalone LES results.
45
