Yuanwei Bin, George I<EMAIL_ADDRESS>Yu Lv, Xiang I.A. Yang

1Department of Mechanical Engineering, The Pennsylvania State University,
University Park, Pennsylvania, 16802 2State Key Laboratory for Turbulence and
Complex Systems, Peking University, Beijing, China, 100871 3Mechanical
Engineering and Applied Mechanics, The University of Pennsylvania,
Philadelphia, Pennsylvania, 19104 4The State Key Lab of Nonlinear Mechanics,
Institute of Mechanics, Chinese Academy of Sciences, Beijing, China, 100190

# Large-eddy simulation of separated flows on unconventionally coarse grids

###### Abstract

We examine and benchmark the emerging idea of applying the large-eddy
simulation (LES) formalism to unconventionally coarse grids where RANS would
be considered more appropriate at first glance. We distinguish this idea from
very-large-eddy-simulation (VLES) and detached-eddy-simulation (DES), which
require switching between RANS and LES formalism. LES on RANS grid is
appealing because first, it requires minimal changes to a production code;
second, it is more cost-effective than LES; third, it converges to LES; and
most importantly, it accurately predicts flows with separation. This work
quantifies the benefit of LES on RANS-like grids as compared to RANS on the
same grids. Three canonical cases are considered: periodic hill, backward-
facing step, and jet in cross flow. We conduct direct numerical simulation
(DNS), proper LES on LES grids, LES on RANS-quality grids, and RANS. We show
that while the LES solutions on the RANS-quality grids are not grid converged,
they are twice as accurate as the RANS on the same grids.

###### keywords:

Turbulence Modeling, Numerical Simulations, Computational Fluid Dynamics

Roman letters $L$domain size $R_{ij}$Reynolds stress $S_{ij}$strain rate
tensor $u_{i}$instantaneous velocity vector $x,y,z$three Cartesian coordinates

Greek letters $\Delta x,y,z$grid spacing in the three directions
$\eta$Kolmogorov scale $\nu$kinematic viscosity

Dimensionless groups ReReynolds number

##  1 Introduction

Reynolds-averaged Navier Stokes (RANS) has been used extensively in
engineering since the 1990s. It solves for the mean flow directly while the
entirety of turbulence is modeled [1], and therefore is not a scale-resolving
tool. Large-eddy simulation (LES) is a scale resolving tool, and it is seeing
more use in fluids engineering [2, 3, 4, 5]. Nonetheless, RANS is the go-to
tool for engineering design work.

In general, scale-resolving techniques are more accurate than non-scale-
resolving ones [6]. However, the above statement applies to properly grid-
converged results obtained with the two respective approaches. Grid-converged
results is not always obtainable with scale-resolving tools, especially at
high Reynolds number applications, due to the their high computational cost
[7, 8]. Here, we briefly review the convergence trends in RANS and LES. The
turbulence models deployed in RANS are designed and meant to be responsive to
variations at the scales of the mean flow only. They are diffusive, tending to
suppress the generation of scales smaller than the mean flow scales. As a
result, the grid convergence in RANS takes place much earlier compared to LES.
This converged solution would correspond to the exact solution of the model
equations rather than the Navier-Stokes equations. Consequently, model errors
in RANS do not diminish with (or respond to) grid refinements, in contrast to
LES subgrid models, which converge to the Navier Stokes equation with
sufficiently fine grid resolution.

Recent LES-based studies of high-Reynolds number external aerodynamics
configurations produced interesting observations in this context. Park and
Moin employed a very coarse grid when computing a flow over the NASA common
research model (CRM) with wall-modeled LES [9]. The grid used had $12\times
10^{6}$ locally isotropic cells, resolving the wing boundary layer on average
with up to 5 cells only. The grid used was more a RANS grid than a proper LES
grid, and this calculation was not grid-converged. Despite these apparent
inadequacies, the lift and drag forces were predicted with reasonable
accuracy. Goc et al. [2] made similar observations in their wall-modeled LES
of flow over the JAXA Stanford Model (JSM) in a high-lift configuration. They
were able to get a reasonably accurate lift and drag using $9\times 10^{6}$
grid cells at AOA from 5 to 22 degrees (at pre and post stall conditions). Goc
et al. further noted that coarse grid LESs give more accurate results than
RANSs on the same coarse grid [10]. These calculations can effectively be
viewed as LES calculations conducted on grids that are coarse enough to be
more suitable for RANS but produce better results than the usual RANS. This
brings us to the topic of this work, LES on RANS-quality grid, the concept of
which was discussed earlier from the industrial perspective in Ref. [11],
where the purpose was to make use of existing RANS grids that no longer offer
improvements in RANS predictions with further grid refinement.

From a practical point of view, computational affordability and short
turnaround time is as important as accuracy. The trade-off between the two
heavily affects the choice between LES and RANS. Take the NASA CRM as an
example: the size of a typical RANS grid is O(10) millions [9, 12], and the
size of a proper (wall-modeled) LES grid is O(1) billion [9, 2]. For an
engineer with access to a few hundred CPU cores, an O(10) million grid-point
calculation is quite affordable, but an O(1) billion grid-point calculation is
not. On the other hand, if one sticks with RANS, the results stop improving
before exhausting the computing resources, but if proper LES is desired, there
are not enough computational resources. The practical question is: given an
affordable grid that does not offer further improvements from RANS with mesh
refinements, would it be better to solve LES equations on this grid? This
concept of LES on a RANS-quality grid in a cost-driven industry is attractive.
The computational costs of a RANS and an LES on a RANS-quality grid are
comparable—LES on a RANS-quality grid would be slightly more costly since the
instantaneous flow fields require time averaging.

The purpose of this work is to benchmark this idea of LES on a RANS-quality
grid. The benchmark here focuses on separated flows, which remains a challenge
to predictive CFD [13]. Three canonical cases are considered, namely, periodic
hill, backward-facing step, and jet in cross flow. Flow separates as a result
of the surface geometry in the first two cases and as a result of cross flows
in the third case. The geometry of the periodic hill gives rise to smooth-body
separation, and the geometry of the backward-facing step gives rise to bluff-
body separation. Among the three cases, the periodic hill case and the
backward-facing step case are widely used for RANS and LES validation and
verification [14, 15, 16, 17, 18]. We conduct direct numerical simulations
(DNSs), LESs, LESs on RANS-quality grids, and RANSs. The simulation details
are presented in Section 2. An emphasis is put on grid construction. The
simulation results are presented in Section 3, and we provide a explanation of
why LES on RANS-quality grids are successful in Section 4. Lastly, we conclude
in Section 5.

##  2 Computational details

###  2.1 Principles for meshing and complications

We must define DNS grids, LES grids, and RANS grids. Following the discussion
in Ref. [1], we review the basic grid resolution requirements for RANS, LES,
and DNS. A RANS grid should resolve the mean flow and its gradients, requiring
the grid spacing to scale with the mean flow scales $\Delta_{M}$. An LES grid
should resolve the large-scale turbulent eddies. Ideally, the grid spacing
$\Delta_{I}$ should be in the inertial range. A DNS grid should resolve the
viscous scales, and ideally, the grid spacing $\Delta_{V}$ should scale with
the Kolmogorov length scale (2 to 4 Kolmogorov length scales). Figure 1 is a
sketch of a generic turbulent energy spectrum, and we have also sketched the
mean flow scale $\Delta_{M}$, the inertial range scale $\Delta_{I}$, and the
viscous scale, $\Delta_{V}$. If the Reynolds number is sufficiently high, we
have: First, the mean flow scale is much larger than any turbulence length
scale. Second, turbulence is roughly isotropic at sufficiently small scales.
Third, the inertial range scale is much larger than the Kolmogorov length
scale. It follows that, at sufficiently high Reynolds numbers, a RANS grid is
much coarser than an LES grid, and an LES grid is much coarser than a DNS
grid.

Figure 1: A sketch of a generic turbulent energy spectrum and the various
length scales in a turbulent flow. At sufficiently high Reynolds numbers, the
mean flow scale, the inertial range scale(s), and the viscous scale(s) are
well separated.

Applying these gridding principles to real-world applications is not always
straightforward. Firstly, we usually do not know $\Delta_{M}$, $\Delta_{I}$,
$\Delta_{V}$ a priori. Secondly, all real-world flows are at finite Reynolds
numbers, and the three scales are not always well separated. For example,
complex geometries give rise to mean flow scales that are spatially varying.
These mean flow scales are comparable to turbulence scales in places where the
local Reynolds number is not high. In this situation, the grid spacing in RANS
would be comparable to that in LES, as in Ref. [19]. Thirdly, empiricism plays
an important role. For example, the viscous sublayer is well-resolved if the
grid is such that $\Delta x^{+}\times\Delta y^{+}\times\Delta z^{+}\approx
12\times 0.05\times 6$. Here, $\Delta x$, $\Delta y$, and $\Delta z$ are the
grid spacings in the streamwise, wall-normal, and spanwise directions, and the
superscript $+$ denotes normalization by the viscous wall units. The
Kolmogorov length scale is $\eta^{+}\approx 1$ at the wall. The grid spacing
in the streamwise and the spanwise directions, i.e., $\Delta x^{+}=12$ and
$\Delta z^{+}=6$, do not resolve the Kolmogorov length scale. Nonetheless,
$\Delta x^{+}=12$, $\Delta z^{+}=6$ still constitutes a DNS grid, which is an
empiricism that does not conform to the general principle. Fourth, the grid
and the turbulence model interact, leading to strong sensitivity of critical
flow events to the grid content. The grid-caused separation is such an
example, as Ref.[20], where slight changes in the grid may result in large
changes in the resulting RANS solution.

Considering these real-world complications, we choose to design the grids to
be used for the present study carefully (instead of using openly available
existing grids) so that we could stay as close to the basic gridding
principles as possible in arriving at a general conclusion about LES on RANS-
quality grids.

###  2.2 Meshing strategy

Figure 2: (a) A sketch of the general meshing strategy. Prism layers are
employed near solid boundaries. The viscous sublayer is resolved. The wall-
normal grid spacing is stretched until the grid aspect ratio is about unit. A
Voronoi grid is used away from solid boundaries, where the grid spacing scales
locally per the requirement of DNS, LES, and RANS. (b) A sketch of a Voronoi
grid. The grid is the median dual mesh of the primary (dashed) triangular
mesh.

Figure 2 (a) shows the general meshing strategy. The grid is a Voronoi grid at
a distance from solid boundaries. Voronoi grids are generated from point
clouds, as shown in Fig.2 (b). By controlling the clustering of these seeding
points, one can precisely control the grid spacing. For instance, the
resulting grid will be a RANS, LES, or DNS grid if we cluster the points such
that their local density is $1/\Delta_{M}^{3}$, $1/\Delta_{I}^{3}$, and
$1/\Delta_{V}^{3}$. In addition to allowing us to precisely control the grid
spacing, a Voronoi grid enables more accurate central reconstruction and
better numerical stability than, say, triangular unstructured meshes [21]. The
reader is directed to Ref. [22, 10, 23] for further details about Voronoi
grids and its recent applications. A different strategy is employed near the
solid boundaries, where we place anisotropic prism layers. The wall-normal
grid spacing is small at the wall and increases away from the wall until the
grid aspect ratio is about unity, and the grid transitions smoothly to a
locally isotropic Voronoi grid. The exact transition location varies from one
grid to another but usually is at about 20% of the local boundary layer
thickness. In our work, a Delaunay-based mesh-generation code is employed [24,
25] to generate the point cloud and a triangular mesh. The Voronoi mesh is the
median dual mesh of the triangular mesh.

###  2.3 Equations

In DNS, the incompressible Navier-Stokes and continuity equations are solved
on a sufficiently fine grid without any modeled terms:

$\displaystyle\frac{\partial u_{j}}{\partial x_{j}}$ $\displaystyle=$
$\displaystyle 0,$ (1) $\displaystyle\frac{\partial u_{i}}{\partial
t}+\frac{\partial}{\partial x_{j}}\left(u_{i}u_{j}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{\rho}\frac{\partial p}{\partial
x_{i}}+\nu\frac{\partial^{2}u_{i}}{\partial x_{j}^{2}},$ (2)

where $u_{i}$ is the velocity in $i$th Cartesian direction, $\rho$ is the
fluid density, $p$ is the pressure, and $\nu$ is the kinematic viscosity.

In LES, the following filtered conservation equations are solved on a grid
that resolves part of the turbulence scales

$\displaystyle\frac{\partial\bar{u}_{j}}{\partial x_{j}}$ $\displaystyle=$
$\displaystyle 0,$ (3) $\displaystyle\frac{\partial\bar{u}_{i}}{\partial
t}+\frac{\partial}{\partial x_{j}}\left(\bar{u}_{i}\bar{u}_{j}\right)$
$\displaystyle=$ $\displaystyle-\frac{1}{\rho}\frac{\partial\bar{p}}{\partial
x_{i}}+\nu\frac{\partial^{2}\bar{u}_{i}}{\partial
x_{j}^{2}}-\frac{\partial\tau_{ij}}{\partial x_{j}},$ (4)

where $\bar{\cdot}$ denotes a filtered quantity,
$\tau_{ij}\equiv\overline{u_{i}u_{j}}-\bar{u}_{i}\bar{u}_{j}$ is the subgrid-
scale (SGS) stress. Invoking the eddy-viscosity model, the anisotropic part of
the SGS stress is modeled as

$\small\tau_{ij}-\frac{1}{3}\tau_{kk}\delta_{ij}=-2\nu_{sgs}\bar{S}_{ij},$ (5)

where $\delta_{ij}$ is the Kronecker delta,
$\bar{S}_{ij}\equiv(\partial\bar{u}_{i}/\partial
x_{j}+\partial\bar{u}_{j}/\partial x_{i})/2$ is the filtered rate-of-strain
tensor, $\nu_{sgs}$ is the SGS eddy viscosity modeled with the WALE (Wall-
Adapting Local Eddy-viscosity) model. The reader is directed to Ref. [26] for
the details of the WALE model.

Last, the following Reynolds-averaged equations are solved on a grid that
resolves the mean flow

$\small\centering\begin{split}\small\frac{\partial\langle
u_{j}\rangle}{\partial x_{j}}&=0,\\\ \frac{\partial\langle
u_{i}\rangle}{\partial t}+\frac{\partial}{\partial x_{j}}\left(\langle
u_{i}\rangle\langle u_{j}\rangle\right)&=-\frac{1}{\rho}\frac{\partial\langle
p\rangle}{\partial x_{i}}+\nu\frac{\partial^{2}\langle u_{i}\rangle}{\partial
x_{j}^{2}}-\frac{\partial R_{ij}}{\partial x_{j}}.\end{split}\@add@centering$
(6)

where $\langle\cdot\rangle$ denotes Reynolds averaging, $R_{ij}$ is the
Reynolds stress and, here, is modeled according to

$\small R_{ij}-\frac{1}{3}R_{kk}\delta_{ij}=-2\nu_{t}\langle S_{ij}\rangle,$
(7)

where $\nu_{t}$ is the eddy viscosity and is modeled per the $k$-$\omega$ SST
model [27] or the SA model [28], and $\langle S_{ij}\rangle$ is mean strain
rate tensor.

###  2.4 Benchmark cases: computational setup

Figure 3: Schematics of (a) periodic hill (b) backward-facing step, and (c)
jet in cross flow.

The cases considered here are periodic hill, backward-facing step, and jet in
cross flow. The geometries, boundary conditions, and grids used are described
first in the forthcoming sections. The simulation results will be presented
separately in section 3.

Figure 3 (a) shows a schematic of the periodic hill, the geometry of which is
described in detail in Ref. [29]. The size of the domain is $L_{x}\times
L_{y}\times L_{z}=9h\times 3.0356h\times 4.5h$, where $h$ is the height of the
hill. The flow is doubly periodic in the streamwise ($x$) and the spanwise
($z$) directions, while it is statistically homogeneous only in the $z$
direction. The top and bottom boundaries are the no-slip walls. The flow is
driven by a body force in the $x$ direction. The bulk Reynolds number is
$Re_{b}=U_{b}h/\nu=5600$, where $U_{b}$ is the bulk velocity, and $\nu$ is the
kinematic viscosity. We will consider a higher Reynolds number in Section 4.

Figure 3 (b) is a schematic of the backward-facing step. The step is located
at $x=0$, and its height is $h$. The size of the domain is $L_{x}\times
L_{y}\times L_{z}=16h\times 2h\times 2\pi h$. The computational inlet is
located at $x=-h$ and the outlet at $x=15h$. The instantaneous fluctuating
inflow condition required in LES and DNS is extracted from a separate
precursor simulation of a fully developed channel flow at $Re_{\tau}=180$.
Figure 4 (a) shows the mean velocity profile at $x=-h$. The pressure is fixed
at the outlet, and the zero Neumann condition is imposed for the velocity at
the outlet. The pressure at the inlet is computed instead of imposed.
Periodicity is imposed in the $z$ direction. Both the bottom and top
boundaries are no-slip walls. The bulk Reynolds number is
$Re_{b}=U_{b}h/\nu=5641$.

Figure 4:  The velocity profiles (a) at the inlet of the back facing step case
and (b) at $x=-4h$ of the jet-in-cross-flow case.

Figure 3 (c) is a schematic of the jet-in-cross-flow case. A fully developed
turbulent channel encounters a 2D vertical plane jet at $x=0$. The size of the
channel is $L_{x}\times L_{y}\times L_{z}=40h\times 2h\times 4\pi h/3$, and
the size of the feed is $L_{x}^{\prime}\times L_{y}^{\prime}\times
L_{z}^{\prime}=d\times 6h\times 4\pi h/3$, where $d=0.5h$. The inlet is
located at $x=-20h$, and the outlet at $x=20h$. A fully developed channel flow
is prescribed at the inlet ($Re_{\tau}=180$). Fluctuating inflow turbulence is
synthesized at the inlet when needed (in LES and DNS) using the approach in
Ref. [30].

Figure 5: Velocity profiles at $x/h=0$, 1, 2, 3, 4, 5, 6, 7 and $8$. We
compare our DNS and that in [31] (KKW18 in the figure). Figure 6: Velocity
profiles at $x/h=0,1,2,3,4,5,6,7,8,$ and $9$. We compare our DNS and that in
[32] (BKAP09 in the figure).

Figure 4 (b) shows the mean velocity profile at $x=-4h$. The jet velocity is
$U_{j}=U_{b}$ at $y=-6h$, corresponding to a blowing ratio of 1. Here, $U_{j}$
is the jet velocity, and $U_{b}$ is the bulk velocity of the incoming channel
flow.

###  2.5 Benchmark cases: DNS results

We present the DNS results of the benchmark cases here. The purposes of the
DNSs are first to validate our code, second to obtain the Kolmogorov length
scale distribution to guide the grid generation, and third to serve as the
reference data for LESs, RANSs, and LESs on RANS-quality grids. We would scale
the LES and RANS grid spacings such that they are multiples of the local
Kolmogorov length scale, $\eta=(\nu^{3}/\epsilon)^{0.25}$, where $\epsilon$ is
the dissipation rate and is available only in DNSs. Here, we follow Ref. [31,
32] and repeat their DNSs. DNS of jet in cross flow can be found in Xu et al.
[33].

Figure 5 shows the velocity profiles at 9 streamwise locations from the
periodic hill calculation. We compare our DNS result to that in Ref. [31],
where a very good agreement is found. Figure 6 shows the velocity at 10
streamwise locations downstream of the backward-facing step, which is compared
to the DNS in Ref. [32] and shows excellent agreement.

###  2.6 Benchmark cases: grids

We scale the Voronoi grid spacing such that they are 7.5, 15, and 25 times the
local Kolmogorov length scale. These grids are generated by employing the
meshing strategy outlined in Section 2.2. We refer to the three grids as the
L-grid, the LR-grid, and R-grid, where the L-grid is LES-quality, and the
R-grid is a fairly coarse RANS grid. RANS and LES equations are solved on all
three grids. LES on the L-grid is a proper LES. LES on the R-grid is LES on
RANS-quality grid. LES on the RANS-quality grid does not require the grid
spacing to be proportional to the local Kolmogorov length scale. The purpose
of scaling the grid spacing to the local Kolmogorov length scale is to
systematically control the grid quality.

Figure 7 shows the L-grid, LR-grid, and R-grid for the periodic hill. Figure 7
also shows the contours of the local Kolmogorov length scale. The presence of
the hill gives rise to smaller turbulence scales in the bottom half of the
channel than the top half of the channel, with the smallest scales found in
the shear layer downstream of the hill. The L-grid, LR-grid, and R-grid have
about $21\times 10^{3}$, $6\times 10^{3}$, and $2\times 10^{3}$ grid points in
the x-y plane. We compare these numbers to the ones in the literature: the LES
grid in Ref. [34] has $39\times 10^{3}$ grid points in the plane, and the RANS
grid in Ref. [35] contains about $3\times 10^{3}$ grid points in the plane.

Figure 7: (a) L-grid, (b) LR-grid, and (c) R-grid for the periodic hill. The
contours show the Kolmogorov length scale computed from our DNS.

Figure 8: Same as Fig.7, but for the back facing step.

Figure 9: Same as Fig.7 but for jet in cross flow.

Figure 8 shows the L-grid, LR-grid, and R-grid for the backward-facing step
and contours of the local Kolmogorov length scale (made available by our
DNSs). The smallest turbulence scales are found in the shear layer near
$0<x<h$, $y/h=1$. Turbulence dissipation is small in the immediate wake of the
step, leading to large turbulence scales there. We require the grid spacing in
the L-grid, LR-grid, and R-grid to be smaller than $0.05h$, $0.1h$, and
$0.2h$. These spacings translate to no fewer than 20, 10, and 5 grid points
across the height of the $Re_{\tau}=180$ inlet channel. The purpose of the
limit is to prevent an overly under-resolved wake. There are about $18\times
10^{3}$, $5\times 10^{3}$, and $2\times 10^{3}$ grid points in the L-grid, LR-
grid, and R-grid between $x=-h$ and $x=10h$ in the $x$-$y$ plane. The numbers
are $10\times 10^{3}$ in Juste et al.’s [36] LESs and $6\times 10^{3}$ in Wu
et al.’s [37] RANSs. Again, the R-grid is a coarse RANS grid.

Figure 9 shows the L-grid, LR-grid, and R-grid for the jet in cross flow case
and the Kolmogorov length scale. The jet gives rise to strong turbulence
downstream of its injection and small turbulence length scale near the bottom
wall. Again, we require the grid spacing to be smaller than 0.1$h$, 0.2$h$,
and 0.4$h$ in the L-grid, LR-grid, and R-grid. These grid spacings translate
to 20, 10, and 5 grid points across the channel height. The L-grid, LR-grid,
and R-grid have $6.5\times 10^{3}$, $2.5\times 10^{3}$, and $1.6\times 10^{3}$
grid points from $x=-h$ to $x=5h$.

Table 1 shows the further details of the grids. The nomenclature is
[Geometry][grid], where Geometry is PH, i.e., periodic hill, BFS, i.e., back
facing step, or JCF, i.e., jet in cross flow, We list the approximate number
of grids in the $x$-$y$ plane, the grid spacing, its cutoff, and the number of
prism layers.

Table 1:  Further details of the grids. Here, $n_{c}$ is the number of grid points in the x-y plane, $\Delta/\eta$ is the grid resolution, $\Delta_{\rm max}$ is the largest grid spacing (grid spacing cutoff), $h_{\rm prism}$ is the height of the prism layer, and $n_{y}$ is the number of grids across the prism layer. There are two numbers for the JCF case denoting the height of the prism layer upstream and downstream of the leakage jet. Case | $n_{c}$ | $\Delta/\eta$ | $\Delta_{\rm max}$ | $h_{\rm prism}$ | $n_{y}$
---|---|---|---|---|---
PH-L | 20678 | 7.5 | N/A | $0.125h$ | 19
PH-LR | 5760 | 15 | N/A | $0.125h$ | 12
PH-R | 2301 | 25 | N/A | $0.125h$ | 9
BFS-L | 25563 | 7.5 | $0.05h$ | $0.125h$ | 16
BFS-LR | 7227 | 15 | $0.1h$ | $0.125h$ | 11
BFS-R | 2936 | 25 | $0.2h$ | $0.125h$ | 9
JCF-L | 25509 | 7.5 | $0.1h$ |  $0.25h$, $0.125h$ | 14
JCF-LR | 8136 | 15 | $0.2h$ |  $0.25h$, $0.125h$ | 10
JCF-R | 4038 | 25 | $0.4h$ |  $0.25h$, $0.125h$ | 7

##  3 Results

We present the periodic hill, backward-facing step, and jet in cross flow
results in Secs. 3.1, 3.2, and 3.3, respectively. We examine velocity profiles
at multiple streamwise locations, the skin friction and its error, with the
latter more clearly showing differences among the solutions than the former.
We shall see that for a RANS grid that no longer offers much improvement for
RANS with further grid refinement, solving the LES equations on that grid is
beneficial, even though the LES on the RANS-quality grid is not grid
converged. Specifically, LESs on RANS-quality grids are considerably more
accurate than RANS on RANS-quality grids in terms of their predictions of the
skin friction coefficient.

###  3.1 Periodic Hill

Figure 10:  Profiles of the $x$ direction velocity. (a) L-grid results. (b)
LR-grid results. (c) R-grid results.

We compare the LES and RANS solutions on the three grids. Figure 10 shows the
velocity profiles at several streamwise locations, and we compare the
solutions on the L-grid, LR-grid, and R-grid. We see that, for a given
turbulence modeling approach, LES or RANS, the solutions do not differ across
the L-grid, LR-grid, and R-grid. The LES solution, i.e., WALE in Fig.10,
follows the DNS solution more closely than the two RANS solutions.

Figure 11: The skin friction coefficients on (a) the L-grid, (b) the LR-grid,
and (c) the R-grid. Figure 12:  The errors in $C_{f}$ as measured by Eq. 9 as
a function of the grid resolution. The three data points on each line are the
L-grid, LR-grid, and R-grid results, respectively, from left to right.

Figure 11 shows the skin friction coefficient $C_{f}$

$\small C_{f}\equiv\frac{\tau_{w}}{\frac{1}{2}\rho
U_{b}^{2}},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\tau_{w}\equiv\left.\mu\left(\frac{{\rm d}\langle U\rangle}{{\rm
d}n}\right)\right|_{w},$ (8)

as a function of the $x$ coordinate, where $\tau_{w}$ is the wall-shear
stress, $n$ is the wall-normal direction, and $w$ denotes the wall. Again, we
do not see significant differences among the three grids. The two RANS models
over-predict the size of the recirculation zone and under-predict the peak
skin friction value at the windward side of the hill on all three grids.
Figure 12 quantifies the error in the skin friction as a function of the grid
resolution:

$\small{\rm err}=\sqrt{\frac{\int(C_{\rm f,pred}-C_{\rm f,DNS})^{2}{\rm
d}x}{\int C^{2}_{\rm f,DNS}{\rm d}x}}\times 100,$ (9)

where $C_{\rm f,pred}$ is the predicted skin friction coefficient, $C_{\rm
f,DNS}$ is the DNS predicted skin friction coefficient. We see from Fig.12
that the RANS solution barely improves from the R-grid to the LR- and the
L-grids. The LES solution, i.e., WALE in Fig.12, outperforms the RANS solution
on all 3 grids. In fact, the LES solution is twice as accurate as of the RANS
solution on the R-grid: the error in the R-grid LES is about 17%, and the
error in the two R-grid RANSs is about 35%. It is worth noting that the R-grid
LES result is not grid-converged. This should be no surprise: we cannot expect
grid-converged LES on a RANS grid. Also important is that the RANS solution is
insensitive to the grid refinement, and extra grid resolution offered no
improvement in the solution.

###  3.2 Backward-Facing Step

Figure 13:  Same as Fig.10, but for back facing step.

Figure 14: Skin fiction coefficient as a function of the $x$ coordinate for
the back facing step case. (a) L-grid results. (b) LR-grid results. (c) R-grid
results. Figure 15: Errors in the LES and RANS predicted skin friction
coefficient as measured by Eq. 9.

Figure 13 shows the velocity profiles at a few $x$ locations in the RANSs and
the LESs on the three grids. SA is noticeably off, but both SST and WALE give
reasonably accurate results on the three grids. Figure 14 shows the skin
friction coefficient. Both RANS models predict a smaller skin friction
coefficient, although the $k$-$\omega$ SST model seems to be slightly more
accurate than the SA model. The LES solution improves from the R-grid to the
LR-grid and the L-grid, and it is more accurate than the RANS solution on all
3 grids. Figure 15 shows the error in the skin friction as a function of the
grid resolution. We observe the following. Firstly, refining the R-grid does
not offer much improvement for RANS. Secondly, the LES solution, i.e., WALE in
Fig.15, is more accurate than the RANS solutions on all 3 grids. Specifically,
on the R-grid, the errors in the LES solution and the SA, SST RANS solutions
are 32%, 62% and 45%, respectively. Thirdly, the LES solution is not grid
converged on the R-grid (RANS grid). Specifically, the errors of LES reduce
from 32% on the R grid to 22% and 6% on the L-grid and LR-grid, respectively.

###  3.3 Jet in cross flow

Figure 16:  Same as Fig.10, but for jet in cross flow.

Figure 17: Same as Fig.14 but for jet in cross flow.  Figure 18: Same as
Fig.15 but for jet in cross flow.

Figure 16 shows the streamwise velocity profiles at a few $x$ locations
downstream of the leakage jet on the three grids. The two RANS models predict
a much more extended separation bubble, whereas the WALE model agrees fairly
well with the DNS on all three grids. Figure 17 shows the skin friction
coefficient as a function of the $x$ coordinate for all 3 grids. The RANSs
have difficulty handling separation, which is expected. The SA model is
notably more accurate than the $k$-$\omega$ SST model, unlike the back-facing
step case, for which the $k$-$\omega$ SST model is more accurate. The LES
solution follows the DNS more closely than the two RANSs on all 3 grids, and
the results improve as the grid refines. On the contrary, the RANS results
seem to have grid converged in all grids, and grid refinement offers little
improvement in the RANS solution. Figure 18 shows the error as a function of
the grid resolution. The error in the SA RANS slightly decreases from the
R-grid to the LR-grid and stays constant between the LR-grid and the L-grid.
The error in the SST model increases as we refine the grid. We can attribute
this behavior to grid-induced separation. On the R-grid, the LES incurs half
as much error as the two RANS: the error in the LES is about 38%, and the
error in the two RANSs is about 70%. Again, the LES solution is not grid
converged on the R-grid, and the solution keeps improving as we refine the
grid. Nonetheless, this case is challenging, and even the L-grid LES incurs a
30% error.

##  4 Discussion

We explain why LESs on RANS-quality grids outperform RANSs on the same grids.
When the grid is fixed, the only difference between LES and RANS is how the
turbulent stresses are modeled. Figure 19 shows effective eddy viscosity in
DNS, LES, and RANS of the periodic hill on the R-grid. Here, the effective
eddy viscosity is

$\small\nu_{\rm eff}=\frac{b_{ij}\langle S_{ij}\rangle}{2\langle
S_{ij}\rangle\langle S_{ij}\rangle},$ (10)

where $b_{ij}$ is the anisotropic part of the Reynolds stress, and it is fully
resolved in DNS, partly resolved in LES, and entirely modeled in RANS. Per the
definition above, $b_{ij}$ in LES is the sum of the resolved Reynolds stress
and the modeled sub-grid scale Reynolds stress, $\nu_{\rm eff}$ in RANS is the
eddy viscosity $\nu_{t}$, and $\nu_{\rm eff}$ in DNS and LES is the effective
eddy viscosity that minimizes the difference between the $2\nu_{\rm
eff}\left<S_{ij}\right>$ and $b_{ij}$. We see that the LES resembles DNS,
whereas RANS significantly under-predicts the effective eddy viscosity. Being
able to capture the effective eddy viscosity is why LES outperforms RANS on
RANS-quality grids.

Figure 19: Time-averaged effective eddy viscosity in the periodic hill case on
the R-grid.

Next, we show that the Voronoi grid is not essential and also that LES on the
RANS grid outperforms RANS at higher Reynolds numbers as well. To show these,
we conduct LES and RANS of the periodic hill case at the Reynolds number
$Re_{h}=10595$. We use the structured grid in Ref. [35]. The size of the grid
is $74\times 37=2738$, which is slightly larger than our R-grid, as shown in
Figure 20. Figure 21 shows the skin friction coefficient. The result looks
essentially the same as the one in Fig. 11 (c), and WALE outperforms the two
RANS models. Further benchmarking at high Reynolds numbers is not pursued here
and is left for future work.

Figure 20: The RANS grid used in Ref. [35] for the periodic hill case. The
grid is a structured one and its size is $74\times 37$. Figure 21: Skin
friction prediction on the grid in Fig.20. The DNS results are reported in
[35].

##  5 Concluding remarks

We explore the concept of LES on the RANS-quality grid. LES on the RANS-
quality grid differs from DES, VLES, and hybrid RANS/LES in that it does not
try to match the grid and the equations. Rather, to conduct an LES on a RANS-
quality grid is to solve the LES equations on a RANS grid without switching
between the RANS and LES equations. The concept has seen successes [11, 9, 2]
but has not been formally benchmarked for canonical cases. This study
benchmarks LES on a RANS-quality grid for three cases, namely, periodic hill,
backing facing step, and jet in cross flow. The three flows feature flow
separation. Special attention is given to grid generation. We use prism layers
and Voronoi cells near and away from the wall, respectively. LES and RANS
grids are generated by systematically controlling the local grid spacing. We
show that for a RANS grid that no longer offers much improvement for RANS with
further grid refinement, solving LES equations on that grid is highly
beneficial. Specifically, LES on the RANS-quality grid is twice as accurate as
RANS in terms of predicting the skin friction coefficient. An implication of
this is that when conducting hybrid RANS LES, where RANS is used for attached
flow regions and LES is used for separated flow regions, one can get
reasonable results by using coarse grids in the LES part of the simulation,
particularly when it is too costly to employ fine grids in the LES part. The
success of LES on the RANS-quality grid is attributed to its ability to
capture effective eddy viscosity. The relative cost of an LES and a RANS on
the same grid roughly scales as the time steps required for the two. For the
three flows studied in this work, it takes about 7 flow-throughs to arrive at
a converged RANS solution and about 30 flow-throughs to arrive at a
statistically stationary state and get converged LES statistics. It follows
that LES on a RANS-quality grid is about 5 times as expensive as RANS on the
same grid, which is considerably more cost-effective than a proper LES.

## Acknowledgments

Bin acknowledges NNSFC grant number 91752202. Yang acknowledges NSF for
financial support.

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