17th Annual AeSI CFD Symposium, 11th -12th August, 2015, Bangalore, India
LES of flows over backward facing steps using digitally filtered inflow conditions
V Kalyana Chakravarthy, Konark Arora, Debasis Chakraborty
Scientists, Directorate of Computational Dynamics. DRDL, Hyderabad
Email: v_kalyana_chakravarthy@yahoo.co.in
Keywords: Large eddy simulation, inflow boundary conditions, separated flows
1. Introduction
Accuracy and efficiency of large eddy simulations (LES) depend not only on the numerical scheme and the closure
model but also on the boundary conditions used. Except in cases where laminar inflow conditions are used with the intent of
simulating transition to turbulence, the unsteady inflow conditions in LES should correspond to a state of fully developed
turbulence. The only way to ensure this is to use a precursor direct or large eddy simulation to supply such inflow
conditions. Though this is the most reliable approach, it is not often used due to additional substantial computational cost.
Any other method of inflow specification constitutes an approximation. Realistic turbulence can, therefore, be expected
beyond a transition zone that starts at the inflow boundary. With a good approximation, this transition zone is short while a
bad one leads to a very long transition zone and in some cases eventual relaminarization. Inflow boundary conditions should
be constructed with the goal of minimizing this transition zone.
Use of white noise for fluctuations constitutes a poor approximation for turbulence. Even if the random numbers
are scaled to result in desired single point variances of flow quantities (i.e., RMS values), the lack of appropriate spatial or
temporal correlations results in rapid decay of fluctuations with in a short distance from the inflow boundary. The inherent
instability of the mean flow triggered by the residual fluctuations has to then lead to a transition to turbulence. Several
synthetic turbulence approaches have been developed for generating efficient inflow boundary conditions which reduce the
transition zone significantly when compared to a white noise approach. Gao and Mashayek [1] developed a simple method
of generating inflow turbulence with given second moments and it has been used successfully in a few studies. In this
approach, though three different time scales are used for correlating the three velocity components, the correlation is only
along the flow direction. The temporal correlation at the inlet translates into a spatial correlation along the flow direction.
There is no spatial structure/correlation in any direction perpendicular to the inflow plane. The same is true with the
turbulence forcing scheme introduced by Volavy and coworkers [2]. The digital filtering approach of Klein and coworkers
[3], on the other hand, provides a way of correlating the fluctuations in the cross stream directions also. It has also been
extended to compressible flows by Touber and Sandham [4] by invoking the strong Reynolds analogy. It requires
specification of the second moments and integral length scales for the velocity field in all three directions.
Large eddy simulations of both subsonic and supersonic flows over backward facing steps are performed in this
study to verify the effectiveness of the digital filtering approach. The subsonic simulation corresponds to the experiment of
Jovic and Driver [5]. This flow is considered a good test case for LES. Results from a direct numerical simulation [6] and
several LES [7-9] are available for comparison. For supersonic testing, the LES reported by Ayyalasomayajula et al. [10] is
considered here.
2. Numerical method
A hybrid finite volume solver [11] which combines a shock capturing flux splitting scheme and a non-dissipative
central scheme is used for the simulations here. Although the flux splitting scheme is not necessary for simulating subsonic
flows, it is retained here. This is because the hybridization procedure is designed not only to facilitate capturing of shocks
and contact lines with out generating of numerical oscillations, but also to suppress Gibbs phenomenon (two-point
numerical oscillations) elsewhere. Two-point oscillations are sometimes generated when using central schemes even for
subsonic flows as evident from LES results of Panjwani et al. [7] for the very test case considered here.
3. Digital filtering approach
To generate each component of the velocity field, a random field is generated on the inflow plane. Spatial
correlations are built-in by filtering the random field in the two cross-stream directions. Instead of a Gaussian functional
form in the original work of Klein and coworkers [3], a more appropriate exponential functional form prescribed by Xie and
Castro [12] for the auto-correlation is used here. The cross-stream integral length scale determine the characteristic widths
of the the filters in the two directions on the inflow plane. Filtering results in random numbers (u”) that are spatially
correlated on the inflow plane. A weighted average of these and corresponding fluctuations at the previous time step
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provides a new field (u') that is spatial temporally correlated as required.
 t  ''
 t 
'
u '  uold
exp 
  u 1  exp 

 2 
  
The τ in the above equation is the integral time scale along flow direction. The normalized fluctuations for the three
components of velocity are then scaled according to the following equation to provide instantaneous velocities that satisfy
the desired first and second moments.
 R11
0



 u1  u1 
2
 R21

u   u    R21
R22  

 2  2 
R11
R11 



u3  u3 
 
0
 0

where ui is the velocity field
0 
 ' 
  u1 
0  u2' 
 ' 
 u3 
R33 

The temperature fluctuations are determined by invoking the strong Reynolds analogy. Though this is not strictly valid as
per many simulation studies, Touber and Sandham [4] justified using it since it provide a good estimate for the variance of
the temperature field. The density field is computed from temperature field by assuming constant pressure on the inflow
plane.
4. Results
Jovic and Driver's flow over a backward facing step experiment with an expansion ratio of 1.2 and a Reynolds
number of 5100 (based on step height and inlet velocity outside the boundary layer) is considered a good test case for LES.
Figure 1: Domain for LES of subsonic flow
for backward facing step.
Figure 2: Variation of coefficient of friction on the lower wall
downstream of the step. ‘h’ is the step height, Velocities are
normalized by free stream velocity.
A schematic of the flow geometry is shown in Figure 1. No slip boundary and isothermal conditions are used for
velocity and temperature respectively at the solid walls. Periodic boundary conditions are used in the spanwise direction (zdirection). A slip condition is used at the top boundary. The simulation actually corresponds to one half of the experiment
The top boundary here is actually the centerline in the experiment where there is a potential core and no turbulence. The
inlet Mach number is about 0.02 and so the compressibility effects are minimal in this flow. In addition to experimental data
and these studies, direct numerical simulation (DNS) data is also available for this flow. The DNS was performed on a
768x192x64 mesh while in the present case, a 144x72x32 mesh similar to one used by Panjwani et al. [7] is used. In DNS,
the inflow boundary is 10 step heights ahead of the step and the velocity field was initialized using a prescribed energy
spectrum. Due to use of randomized phase angles statistical characteristics were found to quickly drift away and then slowly
approach their corresponding targets. The inlet boundary in the present study is only 2.5 step heights ahead of the step just
as in the LES of Panjwani et al. The predicted stream wise variations of the skin friction coefficient are shown in figure 2.
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The reattachment point for the primary recirculation zone predicted with digital filtering turns out to be at 6.04h
which is close to experimental value of 6.07h and DNS prediction of 6.23h. This flow has a combination of large scale
structures generated due to instability of the shear layer formed at the corner and small scale turbulence. Both features need
to be captured adequately for quantitative accuracy. When random noise is used for inlet velocity fluctuations, the
uncorrelated high wave number noise is easily wiped out by the numerics or viscous terms. Not much energy is present in
the most unstable eigen modes of the shear layer and so the triggering of the shear layer instability is delayed. The flow
physics seen in experiment at a given point is shifted downstream in the simulation. The reattachment point is near or more
than 7h distance away from the step. The same is true in some of past LES studies [7,8]. In past LES where large eddies
that can trigger the instability are present either due to use of a pre-simulation or a synthetic turbulence, the predictions of
the reattachment length are nearly as accurate as in the case of digital filtering based simulation here. The key difference is
that all of these studies had the inflow plane at least 10 step heights ahead of the step location.
Figure 3: Mean stream-wise velocity profiles at various
axial locations
Figure 4: Stream-wise velocity fluctuations at various
axial locations.
The profiles of first and second moments of the stream-wise velocity at several axial locations are shown in figures
3 and 4. The first location is within the recirculation zone, second is at the reattachment point while the last two are
downstream of the recirculation zone. LES predictions are quite accurate at the first two locations while the mean velocity
profiles deviate from experimental data in the recovery zone. The stream-wise grid spacing is very coarse compared to the
DNS. At the last location, is is nearly equal to the step height. Better results may be obtained if the grid stretching in the
stream-wise direction is reduced to result in finer grid spacing at these two locations. The fluctuation levels predicted by
LES, surprisingly, match with corresponding DNS results.
Figure 5: Instantaneous density field in LES based on digital
filtering technique for inflow specification
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Figure 6: Friction coefficient on the lower wall
The supersonic flow case has a free stream Mach number of 1.5 and a Reynolds number based on step height and
flow quantities outside the boundary layer of 2,80,000. The rest of the details are available in ref [10]. In the reference LES
[10], a precursor simulation is used to provide the inflow conditions 2 step height ahead of the step. The same is done here
except that the digital filtering is used instead for specifying the inflow conditions. The density field predicted using digital
filtering for inflow specification is shown in figure 5. The flow as in the experiment seems to reattach roughly three step
heights downstream of the step location. When uncorrelated random numbers (scaled to produce specified intensity profiles)
are used for specifying inflow turbulence, as in subsonic case, the reattachment point shifts downstream to a point more than
four step heights away from the corner. The predicted profiles of friction coefficient on the lower wall are compared in
figure 6. As in case of subsonic flow, the shear layer instability is delayed when random noise is used for inflow
specification. The digital filtering approach prediction of the reattachment point is roughly 3.02h (where “h” is the step
height) which is close to the experimental measurement of 3.0h. Comparison of the first and second statistical moments are
made at four different locations. First two are well with in the recirculation zone, third is at the reattachment point and the
fourth is further downstream. The mean velocity predictions are compared in figure 7. Results from present LES match well
with those of reference LES when digital filtering is used. The predicted density fluctuation levels are compared to
corresponding predictions from reference LES in figure 8. Except for the first axial location, the predictions match very
well. The deviation at the first location is due to propagation of disturbances generated at the inflow boundary along the
Mach line. Note that the fluctuations provided by the digital filter are not completely realistic. The unphysical disturbances
at the inflow plane travel along the Mach lines. So the expansion fan generated at the corner experiences unsteadiness as a
result of this artifact. In reality, density/pressure fields are gradient-free outside the boundary layer. A way to prevent
propagation of inflow plane disturbances along the Mach line atmost is needed to fix this problem.
Figure 7: Mean streamwise velocity profiles at various
locations.
Figure 8: Density fluctuation levels at various axial
locations.
5. Conclusions
Digital filtering technique developed originally by Klein [3] and extended to compressible flows by Touber and
Sandham [4] has been used to generate inflow conditions for both subsonic and supersonic flows over backward facing
steps. While Touber and Sandham [4] have shown that nearly length equaling 15 boundary layer widths is needed to
establish a realistic turbulent boundary layer, applying digitally filtered boundary conditions just 2 step heights upstream
seems to improve predictions substantially for the flows simulated here. The spatiotemporal correlations that result from
filtering lead to generation of well resolved eddies of various sizes including those that can trigger eigen modes of the
instability. Size of the recirculation zones and their internal dynamics are accurately predicted as a result. Establishing a
completely realistic boundary layer prior to the step does not seem necessary. It is sufficient to have a broad band spectrum
with sufficient energy in eddies that can trigger instability at the corner. Skipping the filtering step, on the other hand results
in high frequency oscillations that can be easily damped out by sub-grid stresses or numerical damping. Due to weaker
triggering of the instabilities, the shear layer dynamics are stretched out over a longer region of space. This results in longer
recirculation zones than in experiments. The over prediction of the primary recirculation bubble size is more drastic in case
of supersonic flow. The inflow turbulence seems more important for supersonic flows as there is no self similarity based on
coordinate transformation as in case of incompressible case because the reattachment shock length depends on the location
of the recirculation zone. Hence prediction of the size of this zone is very important. While the prediction of the size of the
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recirculation zone vastly improves by the use of the digital filtering technique in case of supersonic flow also, the
unsteadiness in the supersonic region of the inflow boundary creates unphysical noise propagated along the Mach lines.
This leads to the over prediction of the turbulence levels in the recirculation zone. Avoiding this problem would require
combining the digital filtering technique with characteristic boundary condition.
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