RECONSTRUCTION OF TURBULENT FLUCTUATIONS FOR HYBRID
RANS/LES SIMULATIONS USING A SYNTHETIC-EDDY METHOD
N. Jarrin1, R. Prosser1, J. Uribe1, S. Benhamadouch2 and D. Laurence1,2
1 School
of MACE, the University of Manchester, M601QD, UK
2EDF R&D, 6 Quai Watier, Chatou, France
dominique.laurence@manchester.ac.uk
Abstract
A coupling methodology between an upstream
RANS simulation and a LES further downstream is
presented. The focus of this work is on the RANSto-LES interface inside an attached turbulent
boundary layer, where an unsteady LES content has
to be explicitly generated from a steady RANS solution. The performance of the Synthetic-Eddy Method (SEM), which generates realistic synthetic eddies
at the inflow of the LES, is investigated on a wide
variety of turbulent flows, from simple channel and
duct flows to the flow over an airfoil trailing edge.
The SEM is compared to other existing methods of
generation of synthetic turbulence, and is shown to
reduce substantially the distance required to develop
realistic turbulence.
1
Introduction
The large number of grid points required to perform LES at high Reynolds numbers, complex geometries and large domains is the main obstacle to
the application of LES to flows of industrial relevance.
In the aeronautical or automotive industry, engineers are interested in LES because it provides an
unsteady turbulent flow field which allows to compute the aeroacoustic noise generated by the vehicle
or the airfoil. In practise only a small specific region of interest such as the trailing edge of an airfoil
or the rear view mirror of a car is needs to be computed with LES. The specification of the upstream
flow conditions for the embedded LES domain requires the simulation of the whole geometry, which
can be achieved using RANS techniques at a relatively cheap computational cost. The challenge is
then to generate a mature unsteady turbulent LES
solution from a steady RANS solution within as
short a distance as possible in order to achieve both
a reduction of the total computational cost of the
simulation by limiting the size of the embedded
LES domain, and a better accuracy of the simulation
by using a LES model in the region of interest.
The present investigation thus focuses on the
RANS-to-LES interface, for what is often referred
to in the literature as zonal hybrid RANS-LES
methods, where LES and RANS regions use sepa-
rate domains. In this case unsteady turbulent velocity fluctuations must be explicitly reconstructed and
prescribed at the inflow of the LES region from a
steady upstream RANS solution.
There exists an overwhelming variety of methods available to generate inflow boundary conditions for LES (see Keating et al. (2004) for a review). Although recycling methods as in Lund et al.
(1998) produce very realistic inflow data, they increase the cost of the computation and lack generality to be employed in complex industrial applications. Synthetic turbulence generation methods provide an alternative, even though they yield a transition region downstream of the inlet where the synthetic fluctuations imposed at the inlet evolve towards real turbulence (Keating et al., 2004). In
Keating et al. (2006), synthetic turbulence was successfully used in parallel with a controlled body
force to generate inflow conditions for hybrid
RANS-LES simulations of non-equilibrium boundary layers.
In this paper, the Synthetic-Eddy Method of Jarrin et al. (2006) is used to generate fluctuations at
the RANS-to-LES interface of several wall flows.
All of the input parameters of the method are calculated using only statistical data that is available
from the upstream RANS simulation. The SEM is
compared to (and found to perform better than) other existing methods of generation of synthetic turbulence. Cases simulated include simple channel
and duct flows, and the more challenging case of
the turbulent flow over an airfoil trailing edge.
2
Methodology
The governing equations are the incompressible
Navier-Stokes equations, filtered (in the LES region) or time averaged (in the RANS region). In
both regions an eddy viscosity mode is used to close
the equations. The SST model of Menter et al.
(1993) is used in the RANS region, while the standard Smagorinsky model with CS = 0.065 and VanDriest damping at the wall is used in the LES region. The RANS and LES equations are solved with
the collocated finite volume code Code_Saturne
(Archambeau et al., 2004).
The LES and the RANS simulation are run on
two different domains which overlap so that the
RANS region can provide statistics to the inlet
boundary faces of the LES domain. The statistics
available from the upstream SST solution are the
mean velocity Ui, the Reynolds stress tensor Rij, and
the dissipation rate per unit of kinetic energy ω.
The performance of several methods of generation of inflow conditions for LES is investigated.
The simplest method, referred to as the random
method, generates uncorrelated random numbers for
each component of the velocity at each point of the
inlet mesh, and at each iteration. The random numbers are then transformed using the Cholesky decomposition of the Reynolds stress tensor to create
one-point cross-correlations between the velocity
components (see Appendix B of Lund et al. (1998)).
The method of Batten et al. (2004) involves the
summation of sines and cosines with random amplitudes and phases. In all simulations presented here
we used 2000 random modes. Further details of the
method can be found in Batten et al. (2004).
The main focus of this paper is on the applications of the SEM to RANS-LES coupling. In Jarrin
et al. (2006), the input parameters of the SEM were
derived from a precursor LES or from ad-hoc formulae. In this paper, all of the input parameters of
the method are calculated using only statistics
available from the upstream SST simulation.
The LES inflow plane on which we want to generate synthetic velocity fluctuations with the SEM is
a finite set of points S = {x1,x2, ··· ,xs}. The first step
is to create a box of eddies B surrounding S which is
going to contain the synthetic eddies. Its minimum
and maximum coordinates are defined by
where σ is a characteristic length scale of the
flow whose computations from RANS statistics will
be detailed later. In order to ensure that the density
of eddies inside of the box of eddies is constant, the
number of eddies is set as N = max( VB / σ3 ), where
VB is the volume of the box of eddies.
The SEM decomposes a turbulent flow field in a
finite sum of eddies. The velocity fluctuations generated by N eddies have the representation
where the xk are the locations of the eddies, the εkj
are their respective intensities and aij is the Cholesky decomposition of the Reynolds stress tensor.
fσ(x−xk) is the velocity distribution of the eddy located at xk . We assume that the differences in the
distributions between the eddies depend only on the
length scale σ and define fσ by
In all our simulation f is a simple tent function,
and σ is a parameter that controls the size of the
structures. It is taken as
where Δ = max(Δx,Δy,Δz), ε = Cμ k ω is the rate of
dissipation, and δ is the thickness of the boundary
layer considered. The position xk and the intensity
εkj of each eddy are independent random variables.
At the first iteration, xk is taken from a uniform distribution over the box of eddies B and εkj = ±1, with
equal probability to take one value or the other.
The eddies are convected through the box of eddies B with a constant velocity Uc characteristic of
the flow. In our case it is straight forward to compute Uc as the averaged mean RANS velocity over
the LES inflow plane. At each iteration, the new position of eddy k is given by
where dt is the time step of the simulation. If an eddy k is convected out of the box through face F of
B, then it is immediately regenerated randomly on
the inlet face of B facing F with a new independent
random intensity vector εkj still taken from the same
distribution.
The method generates a stochastic signal with
prescribed mean velocity, Reynolds stresses, and
length and time scale distributions. Although the
SEM involves the summation of a large number of
eddies for each grid point on the inflow, the CPU
time required to generate the inflow data at each iteration did not exceed 1% of the total CPU time per
iteration of the LES simulation.
3
Results
3.1 Spatially developing channel flow
Hybrid RANS-LES simulations of the turbulent
flow in a plane channel are performed at Reτ = 395.
The RANS equations are solved for x/δ < 0. The
RANS grid is one-dimensional, and only uses one
cell with periodic boundary conditions in the
streamwise and spanwise directions. At x/δ = 0 the
LES domain, of dimensions 10πδ × 2δ × πδ, begins.
The grid spacing in wall units are Δx+ ≤ 50, Δz+ ≤
15, Δy+ = 2 at the wall and Δy = 0.1δ. The wallnormal grid resolution is the same as in the RANS
and in the LES to avoid interpolation of the RANS
data onto the LES grid. Several methods of generation of inflow conditions for LES are tested and the
simulations performed are summarized now.
Figure 1: Velocity vectors of LES inlet conditions
for hybrid simulations of channel flow. From top to
bottom: precursor LES, SEM, Batten's method and
random method.
A baseline simulation was performed as a comparison point for all other cases (run P1). Time series of instantaneous velocity planes were extracted
from a periodic LES (performed on a shorter domain but with the same grid refinement and the
same numerical options) and imposed at the inlet of
the present LES domain. In all other simulations,
methods of generation of synthetic turbulence are
used to prescribe inlet conditions for the LES region. Three hybrid calculations were conducted, using the SEM (run S1), Batten’s method (run B1)
and the random method (run R1).
Figure 1 shows instantaneous velocity fluctuations
prescribed at the inlet of the LES domain. Although
the SEM does not reproduce completely the complex structure of the near-wall turbulence observed
in the periodic LES, the length scale and the magnitude of the fluctuations are realistically reproduced
by the SEM. The velocity fluctuations generated using the method of Batten et al. (2004) exhibit surprising features. In the near-wall region, the fluctuations seem to be uncorrelated in space. In the centre
the fluctuations are correlated in the spanwise direction but seem decorrelated in the wall-normal direction. The reason for these phenomena is the decomposition into Fourier modes used in Batten’s method. The frequencies and wavelengths of the cosine
and sine functions are allowed to vary in the direction of non-homogeneity of the flow (in the present
case the wall-normal direction). The velocities at
two points separated even by an infinitesimal distance in the wall-normal direction will thus oscillate
at different frequencies, and therefore be completely
decorrelated from each other. In the direction of
homogeneity of the flow however (the spanwise direction in the present case), this problem does not
occur since the frequencies and wavelengths are
constant. Thus although the method of Batten et al.
(2004) might appear to be capable of generating
non-homogeneous turbulence by allowing the wavelengths to vary in space, it does so at the expense of
destroying the spatial correlations in the nonhomogeneous directions.
The development of the prescribed fluctuations
downstream of the inlet will now be studied. Figure
2 shows the downstream development of the coefficient of friction. The horizontal dashed line represents the value of the coefficient of friction in the
periodic LES and will be used as a reference point
for the present RANS-LES simulations. Run P1 has
a coefficient of friction in very good agreement with
the periodic LES over the whole domain. As expected, all three of the other simulations using synthetic turbulence exhibit a transient downstream of
the inlet. When the random method is used, the coefficient of friction drops continuously downstream
of the inlet, which indicates that the flow laminarizes. The decay of the coefficient of friction is also
quite important downstream of the inlet when the
method of Batten et al. (2004) is used. However the
coefficient of friction reaches a minimum after
about 8δ, before slowly recovering towards its fully
developed value about 25δ downstream of the inlet.
With the SEM, the coefficient of friction decays
downstream of the inlet to reach a minimum about
3δ downstream of the inlet (where it has only lost
15% of its initial value), and recovers its fully de-
veloped value only after 10δ downstream of the inlet.
corner bisectors, and the mean streamwise velocity
distribution (see Figure 5 (a)) is in excellent agreement with the one from the reference LES.
Figure 2: Coefficient of friction for hybrid simulations
of channel flow at Reτ = 395. Inflow conditions are
generated using a precursor LES (О), the SEM (___ ),
Batten's method ( _ _ _ ) and the random method( .... ).
The performance of the SEM is now tested at
two higher Reynolds number (Reτ=590 and
Reτ=950). Different grids are used for each Reynolds number, but the grid refinement in wall units
always satisfies the constraints Δx+ ≤ 50, Δz+ ≤ 15
and Δy+ = 2 at the wall. Figure 3 shows that the development of the coefficient of friction downstream
of the inlet is the same for the three Reynolds numbers considered when expressed as a function of x uτ /
υ. Analysis of other flow statistics not presented in
this paper confirm that in the near-wall region, the
length of the transition region scales approximately
as x+ ~ 3, 000.
3.2 Spatially developing duct flow
The SEM is now compared to Batten’s method
and to the random method in the case of a turbulent
flow through a square duct at Reτ = 600 (Huser and
Biringen, 1993). The computational set-up is similar
to the one used in the case of the channel flow. The
RANS domain is positioned upstream of the LES
domain. The upstream SST simulation uses periodic
boundary conditions in the streamwise direction. As
expected the SST solution does not exhibit any secondary motion. The ability of the SEM, the method
of Batten et al. (2004), and the random method to
yield, after a short development distance, a secondary motion in the LES region is investigated.
The topology of the mean flow is studied in a
cross-section at x/D=15 (roughly x + = 9,000) downstream of the RANS-to-LES interface. The simulation using the SEM exhibits two mean streamwise
counter-rotating vortices in the corner of the duct,
as shown in Figure 4 (a). Their centre location and
topology are in very good agreement with those
from the reference fully developed LES. Due to the
action of the secondary motion, momentum is convected from the central region to the walls along the
Figure 3: Development of the coefficient of friction
(normalized by the coefficient of fiction obtained in
the periodic LES) as a function x uτ /υ for hybrid
simulations of channel flow at Reτ = 395 (◊), Reτ =
590 (□), and Reτ = 950(О).
The simulation using the random method does not
exhibit any secondary motion (see Figure 5 (c)).
With Batten’s method, two very weak streamwise
corner vortices can be observed, but their weak intensity does not alter the mean streamwise velocity
distribution in the correct manner as shown in Figure 5 (b). We saw that Batten’s method destroys spatial velocity correlations in the direction of nonhomogeneity of the flow. In the present case, the upstream k and Ω profiles extracted from the SST solution and transmitted to Batten’s method are non homogeneous in the two transverse directions. Consequently Batten’s method does not generate any twopoint velocity correlations in the inlet plane. The better results obtained than when using the random
method can be explained by the better time correlation of the inflow data generated using Batten’s
method.
Figure 6: Sketch of the hybrid RANS-LES simulations of the airfoil trailing edge.
Figure 4: Transverse velocity vectors at x/D = 15
for hybrid simulations of square duct flow with (a)
the SEM, (b) Batten’s method, (c) the random method and (d) the reference periodic LES.
Figure 5: Mean streamwise velocity distribution
normalized by bulk velocity at x/D=15 for hybrid
simulations with (a) the SEM, (b) Batten’s method,
(c) the random method and (d) the reference periodic LES. Contours lines are evenly space between
0.3, 0.4, ..., 1.2.
3.3 Turbulent flow over an airfoil trailing edge
The airfoil considered is a two-dimensional flat
strut with a circular leading edge and an asymmetric
beveled trailing edge with a 25 o tip angle. The geometry of the airfoil and the flow conditions are described in details by Blake (1975) and by Wang and
Moin (2000).
As shown in Figure 6, the RANS domain encloses the entire airfoil and only the rear part of the
trailing edge and the near wake are simulated with
LES (the non-equilibrium region of the flow). The
RANS simulation is conducted on a C-grid domain
using only 0.1M cells. The LES domain begins at x
/ h = -4 and x / h = -2 on the low- and high-pressure
side of the airfoil, respectively. For the LES mesh,
644 cells are uniformly distributed along the upper
surface, and 384 along the lower surface. This gives
a grid spacing in wall units at the inlet of Δx+ = 41
and Δx+ = 34 on the upper and lower surfaces, respectively. 150 cells are non-uniformly distributed
along the wake line. The wall-normal grid spacing
increases as the upper and lower walls are approached. 64 cells are used and the near-wall grid
spacing is at a minimum at the walls, with Δy+ ~ 2.
In the spanwise direction, 64 cells are uniformly
distributed. The grid spacing in wall units on the
upper surface is around 26 at the inlet of the LES
domain. In total, the LES mesh has about 3.0×106
cells.
At the inlet plane of the LES domain, data are
extracted from the SST solution, interpolated onto
the LES grid, and used for the generation of synthetic turbulence. Results on the embedded LES
domain using inflow data generated with the SEM,
Batten’s method and the random method are compared with the finely resolved LES of Wang and
Moin (2000).
Profiles of mean velocity magnitude (U2+V2)1/2
and streamwise velocity fluctuations u’ on the lowpressure side of the airfoil are shown in Figure 7.
The hybrid simulations using the random method
and Batten’s method initially laminarize (as expected from previous computations), and consequently show very early separation at x/h = −2.4
and x/h = −2.1, respectively. When the SEM is used
the mean velocity profiles are in much better
agreement with the reference LES. The hybrid simulation using the SEM detaches at x/h = −0.75,
slightly after the reference LES (x/h = −1.17).
The early detachment observed with the random
method and with Batten’s method is caused by the
lack of coherence of the prescribed inlet fluctuations. As a result there is a lack of near-wall turbulent structures close to the inlet, and the fluctuations
are underestimated at the first station. Further
downstream the presence of a large separation bubble in run B3 and R3 produces larger levels of fluc-
tuations in the recirculation region (see Figure 7 at
x/h = −1.125 and at x/h = −0.625).
The effect of the inflow data on the turbulent
structures will now be described. Streamwise velocity fluctuations along the upper surface of the airfoil
are shown in Figure 8. The simulation using Batten’s method - although leading to early separation
and weak magnitude fluctuations in the near wall
region - still shows features similar to the simulation using the SEM: the weak near-wall streaks are
elongated in the streamwise direction (due to the favorable pressure gradient experienced by the
boundary layer), followed by a rapid transition towards a more turbulent state (after the removal of
the pressure gradient), before finally separating
from the wall. With the random method, no turbulent structures are present in the near-wall region of
boundary layer downstream of the inlet, which also
leads to early separation. However in this case, the
separation is laminar and leads to the formation of
large scale two-dimensional Kelvin-Helmotz vortices in the subsequent shear layer.
Finally Figure 9 shows the frequency spectrum
of the v-fluctuations at x/h=4 downstream of the
trailing edge. A strong peak around f h/U0 = 0.6 can
be observed with the random method and Batten’s
method, indicating the presence in the flow of Kelvin-Helmotz vortices shedding in the wake of the
airfoil. On the contrary the frequency spectrum in
the case of the SEM does not exhibit any peak, in
agreement with observations of instantaneous fluctuations in the near-wake which did not exhibit any
clear vortex shedding. This is the physical behaviour of the flow observed in the reference LES of
Wang and Moin (2000).
Figure 7: Profiles of (a) the mean velocity magnitude and (b) the rms streamwise velocity fluctuations normalized by the edge velocity as a function
of vertical distance from the upper surface, at
x/h=−3.125, −2.125, −1.625, −1.125, −0.625:
___ , SEM;_ _ _ , Batten et al. (2004);
..., random method; О, LES Wang and Moin (2000)
Figure 8: Streamwise velocity fluctuations in a
plane parallel to the wall at y+ = 1 for hybrid
simulations of the trailing edge flow. From top to
bottom: SEM, method of Batten et al. (2004) and
random method.
Figure 9: Frequency spectrum of the v fluctuations
in the near wake at x/h = 4 and y/h = 0.5: ,SEM; ,
Batten et al. (2004); and , random method.
3
Conclusions
The SEM was used to generate inlet conditions
for a LES using only information available from an
upstream SST simulation. This hybrid RANS-LES
coupling strategy was first tested in the case of
channel and duct flows. The SEM was systematically compared to other existing methods of generation
of synthetic turbulence; the random method and the
method of Batten et al. (2004). With the SEM, the
development length of the eddies in the near wall
region was shown to be approximately 3,000 wall
units in both cases simulated. This offers significant
promise for the application of the method to high
Reynolds number flows of engineering interest.
With the random method, the velocity fluctuations
prescribed at the inlet were immediately dissipated
and the flow became laminar. With Batten’s method, the use of Fourier harmonics with spatially
varying wavelengths leads to a destruction of the
spatial correlations of the signal in the direction of
non-homogeneity of the flow.
Finally hybrid simulations of the flow over an
airfoil trailing edge were performed. With the SEM,
realistic turbulence is generated upstream of the
separation, and thus flow predictions downstream of
the separation are in good agreement with the reference data. The inlet was positioned only 3 boundary
layer thicknesses upstream of the location where the
boundary layer experiences maximum acceleration,
this without significant alteration of the results. The
length of the LES inlet sections used did not allow
either the random method or Batten’s method to
generate a realistic boundary layer upstream of the
region of interest. The lack of turbulent structures in
the upstream boundary layer lead to an early separation and hence a higher production of turbulent kinetic energy; leading to the growth (after separation) of quasi two-dimensional structures characteristic of transitional flows.
The use of small LES domains (without significant loss of accuracy compared to the reference data) in the hybrid simulations using the SEM led to
substantial savings in terms of number of cells used.
The reduction in terms of CPU time achieved with
the present LES domain is over 40% when compared to the domain used in the reference LES of
(Wang and Moin, 2000), and over 80% when compared to a full domain LES enclosing the entire airfoil.
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