# Reynolds number dependence of length scales governing turbulent flow
separation with application to wall-modeled large-eddy simulations

Rahul Agrawal 111Ph.D. candidate, Department of Mechanical Engineering.
Student Member AIAA. (Email<EMAIL_ADDRESS> Center for Turbulence
Research, Stanford University  Sanjeeb T. Bose 222Distinguished Engineer,
Cadence Design Systems and Adjunct Professor, Institute for Computational and
Mathematical Engineering, Stanford University. Senior Member AIAA.  Cadence
Design Systems, Inc.  Parviz Moin 333Franklin P. and Caroline M. Johnson
Professor, Dept. of Mechanical Engineering, Stanford University. Director of
the Center for Turbulence Research. Fellow AIAA.  Center for Turbulence
Research, Stanford University

###### Abstract

This work proposes a Reynolds number scaling of the required number of grid
points to perform wall-modeled LES of turbulent flows encountering separation
off a solid surface. Based on comparisons between the various time scales in a
non-equilibrium turbulent boundary layer (due to the action of an external
pressure gradient), a simple definition of the near-wall “under-equilibrium"
and “out-of-equilibrium" scales is put forward (where “under-equilibrium"
scales are governed by a quasi-balance between the viscous and the pressure
gradient terms). It is shown that this “under-equilibrium" characteristic
length scale varies with Reynolds number as $l_{p}\sim Re^{-2/3}$. The same
scaling is obtained from a simplified Green’s function solution of the Poisson
equation in the vicinity of the separation point. A-priori analysis is used to
demonstrate that the resolution required to reasonably predict the wall-shear
stress (for example, errors lower than approximately $10-15\%$ in the entire
domain) in several nonequilibrium flows is at least $\mathcal{O}(10)$$l_{p}$
irrespective of the Reynolds number and the Clauser parameter. Further, a
series of _a-posteriori_ validation studies are performed to determine the
accuracy of this scaling including the flow over the Boeing speed bump, Song-
Eaton diffuser, Notre-Dame Ramp, and the backward-facing step. The results
suggest that for these flows, scaling the computational grids ($\Delta$) such
that $\Delta/l_{p}$ is independent of the Reynolds number results in accurate
predictions of flow separation at the same “nominal" grid resolution across
different Reynolds numbers. Finally, it is suggested that atleast locally, in
the vicinity of the separation and reattachment points, the grid-point
requirements for wall-modeled large eddy simulations may scale as $Re^{4/3}$,
which is more restrictive than the previously proposed flat-plate boundary
layer-based estimates ($\sim Re^{1}$) of Choi and Moin, _Phys. Fluids, 2012_
and Yang and Griffin, _Phys. Fluids, 2021_.

##  1 Nomenclature

$\alpha$ = | angle of tapering
---|---
$\rho$ = | density
$e$ = | resolved internal energy
$E$ = | sum of resolved internal and kinetic energies
$T$ = | temperature
$p$ = | pressure
$u_{i}$ = | velocity components
$S^{d}_{ij}$ = | deviatoric component of the strain-rate tensor
$\tau^{sgs}_{ij}$ = | subgrid-scale stress tensor
$Q^{sgs}_{j}$ = | subgrid heat-flux
$l_{p}$ = | pressure-gradient imposed viscous length scale
$y^{+}$ = | resolution based on friction velocity
$U_{\infty}$ = | freestream velocity
$\delta$ = | boundary layer thickness
$\theta$ = | momentum thickness
$\Delta$ = | size of computational grid
$u_{\tau}$ = | friction velocity
$u_{p}$ = | pressure-gradient based velocity
$Pr$ = | Molecular Prandtl number
$Pr_{t}$ = | Turbulent Prandtl number
$Re_{L},\;Re_{H}$ = | Reference Reynolds number based on a geometric length scales
$Re_{\tau}$ = | Friction Reynolds number
$Re_{\theta}^{ref}$ = | Reference Reynolds number based on momentum thickness
$t_{p}$ = | time-scale based on balance pressure-gradient and viscous terms
$t_{v}$ = | time-scale based on near-wall viscous effects
$t_{iv}$ = | time-scale based on local-shear
$Mcv$ = | Million Control Volumes
$LES$ = | Large-eddy simulation

##  2 Introduction

In complex engineering flows, the boundary layers over solid surfaces often
experience a combination of favorable and adverse pressure gradients. A
favorable pressure gradient reduces the turbulent normal stresses in the near-
wall flow, making it more organized [1]. In the presence of an adverse
pressure gradient, however, these normal stresses near the wall become
stronger [2]. In the latter case, these non-equilibrium flows can even exhibit
a boundary layer separation from the surface before finally reattaching
downstream. The structure of turbulent separation bubbles has remained a
matter of active research [3, 4, 2, 5, 6, 7, 8]. Even in canonical problems of
boundary layers experiencing pressure gradients without separating off the
surface, integrated streamwise history effects have been reported [9] to
determine the downstream state of a boundary layer. Devenport and Lowe [10]
have suggested that the flow history effects may even persist up to 50 local
boundary-layer thicknesses downstream. Recently, Agrawal et al. [11] developed
an extension of the method of Thwaites which also serves as a quantitative
analysis tool to account for the flow history.

Detailed knowledge of the near-wall structure of equilibrium boundary layers
has been leveraged to establish resolution requirements for both wall-resolved
and wall-modeled large-eddy simulations (LES). For example, a resolution
requirement of $\Delta u_{\tau}/\nu\sim O(1)$ (where $\Delta$ is the size of
the near-wall grid-cell, and $u_{\tau}$ is the friction-velocity) is
recommended for wall-resolved LES or the validity of the law of the wall in
the logarithmic regions of the velocity profiles in wall modeled LES. Despite
the augmentation of the structure of the turbulent boundary layer in the
presence of strong pressure gradients, existing grid point estimates still
rely on the phenomenology of equilibrium boundary layers [12, 13, 14]. These
studies suggest that the grid point requirements for wall-modeled LES scale
linearly with the Reynolds number if resolution with respect to the boundary
layer remains fixed. There are two shortcomings of this approach. First, the
conclusions regarding the Reynolds number scaling may be incorrect in the
presence of pressure gradients. Second, and perhaps more importantly, the
existing grid point estimates do not provide estimates on the _a-posteriori_
errors in quantities of interest given a specified outer layer resolution. For
instance, Lozano-Durán et al. [15] show that errors in the prediction of mean
velocity profiles greatly increase in regions of adverse pressure gradients
compared to zero pressure gradient regions even if the outer layer resolution
is fixed. This is especially true in separating boundary layers, where
significantly higher outer layer resolution has been required for accurate
simulations of flow exhibiting both smooth body separation [16] and large,
three-dimensional separation patterns [17, 18, 19, 20]. It is likely that for
such flows, the distribution of the pressure gradient may affect the
convergence rate of wall-modeled LES toward reference quantities of interest.
An example is the flow over an aircraft model, where for different angles of
attack, the pressure gradient distribution changes due to a change in the
effective body shape. Under milder pressure gradients (at low angles of
attack), the flow exhibits smooth body separation over the trailing edge of
the flaps, and for stronger pressure gradients (at high angles of attack), the
flow can exhibit strong unsteadiness and large separation patterns emanating
from the leading edges. It has been reported [19, 20] that converging the
lift, pitching moment and integrated drag for different angles of attacks
requires different grid resolutions for the same nominal Reynolds number flow.

This article aims to address the two aforementioned shortcomings,
specifically, in the context of separating flows. A characteristic length
scale that governs the near-wall flow equilibrium (a quasi-balance between the
viscous and the pressure gradient terms) is used to determine the resolution
required for accurately predicting quantities of interest for flows (such as
the surface pressure) that exhibit a smooth body separation across a range of
Reynolds numbers. To test the validity of this scaling, we use the charLES
flow solver [21] to perform wall-modeled LES. The rest of the article is
organized as follows. In section III we describe the governing equations being
solved in charLES solver and the choice of the subgrid-scale and the wall
model employed in this work. Section IV discusses the physical length scale
that governs the onset of flow separation. Section V discusses _a-posteriori_
wall-modeled LES results across multiple flows. Finally, some conclusions are
drawn in Section VI.

##  3 Numerical Solver and Governing equations

The simulations presented in this work are performed using charLES, an
explicit, unstructured, finite-volume solver for the compressible Navier-
Stokes equations. This code is formally 2nd-order accurate in space, and 3rd-
order accurate in time, and utilizes grids based on Voronoi diagrams. More
details of the solver and validation cases can be found in Agrawal et al.
[22], Brès et al. [21], Goc et al. [23], Fu et al. [24]. Formally skew-
symmetric operators are employed to conserve kinetic energy, these discrete
operators also approximately preserve entropy in the inviscid, adiabatic
limit.

A brief description of the previously mentioned compressible Navier-Stokes
equations for LES is provided as follows. If the filtered, large-scale fields
(such as velocity and pressure) in LES are denoted by $\overline{f}$, and
their corresponding Favre average is denoted by $\widetilde{f}$, then the
resulting equations of a compressible flow [of internal energy $e$, density
$\rho$, temperature $T$, viscosity $\mu(T)$ and thermal conductivity
$\kappa(T)$] and velocity vector $\vec{u}\;=\;\\{u_{1},\;u_{2},\;u_{3}\\}$ are
given as,

$\frac{\partial\overline{\rho}}{\partial
t}+\frac{\partial(\overline{\rho}\;\widetilde{u}_{i})}{\partial x_{i}}=0$ (1)

$\frac{\partial(\overline{\rho}\;\widetilde{u}_{i})}{\partial
t}+\frac{\partial(\overline{\rho}\;\widetilde{u}_{j}\;\widetilde{u}_{i})}{\partial
x_{j}}=-\frac{\partial\overline{p}}{\partial
x_{i}}+\frac{\partial(\mu\widetilde{S^{d}}_{ij})}{\partial
x_{j}}-\frac{\partial\tau^{sgs}_{ij}}{\partial x_{j}},$ (2)

and

$\frac{\partial\overline{E}}{\partial
t}+\frac{\partial(\overline{E}\;\widetilde{u}_{j})}{\partial
x_{j}}=-\frac{\partial(\overline{p}\;\widetilde{u_{i}})}{\partial
x_{i}}+\frac{\partial(\mu\widetilde{S^{d}_{ij}}\widetilde{u_{i}})}{\partial
x_{j}}-\frac{\partial(\tau^{sgs}_{ij}\widetilde{u_{i}})}{\partial
x_{j}}-\frac{\partial Q_{j}^{sgs}}{\partial x_{j}}+\frac{\partial}{\partial
x_{j}}(\kappa\frac{\partial\overline{T}}{\partial x_{j}}),$ (3)

where
$\overline{E}=\overline{\rho}\widetilde{e}+0.5\;\overline{\rho}\widetilde{u_{i}}\widetilde{u_{i}}$
is the sum of the resolved internal and kinetic energies.
$\widetilde{S^{d}}_{ij}$ is the deviatoric part of the resolved strain-rate
tensor. The relationship between the temperature and the molecular viscosity
is assumed to follow a power law with an exponent of 0.75. A constant
molecular Prandtl number approximation ($Pr=0.7$) allows computation of the
thermal conductivity. Two additional terms, $\tau^{sgs}_{ij}$ and
$Q^{sgs}_{j}$ require modeling closure. The subgrid stress tensor,
$\tau^{sgs}_{ij}$ is defined as
$\tau^{sgs}_{ij}=\overline{\rho}(\widetilde{u_{i}u_{j}}-\widetilde{u}_{i}\widetilde{u}_{j})$.
Similarly,
$Q_{j}^{sgs}=\overline{\rho}(\widetilde{eu_{j}}-\widetilde{e}\widetilde{u_{j}})$
is the subgrid heat flux. In this work, the isotropic component of the subgrid
stress is absorbed into pressure, leading to a pseudo-pressure field. The
dynamic, tensorial coefficient subgrid-scale model [25] is used in the
entirety of this work to model the unresolved scales of motion. Note that the
subgrid heat flux is modeled using the constant turbulent Prandtl number
approximation ($Pr_{t}=0.9$) applied to the dissipative component of the
subgrid-stress tensor.

In typical wall-modeled LES, a shear stress, and a heat flux boundary value is
supplied to the LES solver to close the discrete system of governing
equations. In this work, an algebraic form of the equilibrium wall-stress
model (EQWM) is used, in which the assumed mean velocity profile is a $C^{1}$
continuous piecewise fit of the viscous sublayer and the logarithmic layer.
The details of this wall model can be found in Lehmkuhl et al. [26]. It is
highlighted that a first-point matching approach is used since this solver has
not shown any evidence of a “log-layer mismatch" [27, 28] in the simulation of
turbulent channel flows in the range, $1000\leq Re_{\tau}\leq 4200$ with
typical wall-modeled LES resolutions.

##  4 A characteristic length scale for separated flows

The viscously dominated near-wall scales in flows experiencing pressure
gradients can go out of equilibrium (for example, not have a linear velocity
scaling in viscous units) if the time scale governed by the pressure gradient
($t_{p}\sim\rho u_{\tau}/dP/dx$) is faster than the viscous time scale
($t_{v}\sim\nu/u_{\tau}^{2}$) where $u_{\tau}$ is the local skin-friction
velocity. The time scale for eddies within a logarithmic region is governed by
the local shear ($t_{iv}\sim h/u_{\tau}$ where $h$ is the wall-normal height
of the flow scale of interest). The inner scales thus go out of equilibrium if

$t_{p}<t_{v}\;\mathrm{or}\;{u^{3}_{\tau}}<\frac{\nu}{\rho}\frac{dP}{dx}=(u_{p})^{3}\implies
u_{\tau}<u_{p}.$ (4)

Similarly, both the inner and outer scales respond at the same time scale if

$t_{p}<t_{v}\sim
t_{iv}\;\mathrm{or}\;\frac{u_{p}}{u_{\tau}}>1\;\mathrm{and}\;\frac{u_{\tau}h}{\nu}\sim
1.$ (5)

Near the onset of flow separation, these conditions can be met simultaneously.
For a spatially developing boundary layer, these relations also suggest that
the flow at a given wall-normal location $h$ is bound to be in non-equilibrium
if

$\frac{u_{p}h}{\nu}>1\;\mathrm{or}\;\frac{h}{l_{p}}>1\;\mathrm{where}\;l_{p}=\frac{\nu}{u_{p}}.$
(6)

Thus, the only scales that remain in equilibrium for a boundary layer before
the onset of a separation bubble are scaled such that $\frac{h}{l_{p}}\sim 1$.
This suggests that an equilibrium approximation of the near-wall flow
structure would only remain valid for matching locations at $h\lesssim\alpha
l_{p}$ (for some $\mathcal{O}(1)$ constant, $\alpha$).

This can be further reasoned from the analysis of a time-averaged Poisson
equation for pressure near the separation point, written as

$\langle\rho\frac{\partial\bar{u}_{i}}{\partial
x_{j}}\frac{\partial\bar{u}_{j}}{\partial
x_{i}}\rangle=-\nabla^{2}\langle\bar{p}\rangle.$ (7)

where $\bar{\cdot}$ and $\langle\cdot\rangle$ denote the filtered LES fields
and a time average operator, respectively, and homogeneous Neumann boundary
conditions are applied on the pressure field at the solid wall. Consider, a
two-dimensional boundary layer that undergoes a separation over a surface such
that $\delta/R\ll 1$ where $R$ is the local radius of curvature. (This
assumption may not strictly be true near the separation point over sharp
corners, but is convenient for analysis of smooth body separation over gently
curved surfaces). Let $(x_{sep},0)$ be the coordinates of the separation
point. In a semi-infinite domain, the Green’s function that satisfies Equation
7 and its boundary conditions is given as

$G(x,y,x_{0},y_{0})=\frac{1}{4\pi}ln([(x-x_{0})^{2}+(y-y_{0})^{2}][(x-x_{0})^{2}+(y+y_{0})^{2}]).$
(8)

for $y\geq 0$.

Using Green’s identities and the symmetric property of Green’s function, the
pressure in the upper-half domain is given as

$\langle p^{*}_{sep}\rangle=\langle
p^{*}(x_{sep},0)\rangle=-\int^{\infty}_{0}\int^{\infty}_{-\infty}\frac{1}{2\pi}ln[(x^{*}-x^{*}_{sep})^{2}+(y^{*})^{2}]\langle\rho\frac{\partial
u^{*}_{i}}{\partial x^{*}_{j}}\frac{\partial u^{*}_{j}}{\partial
x^{*}_{i}}\rangle dx^{*}dy^{*}.$ (9)

where $(...)^{*}$ denotes a non-dimensional variable. Let the relevant non-
dimensionalization scales be $l_{min}=min(l_{p}(x))$ and the viscous velocity
scale that corresponds to $l_{min}$, equal to $u_{max}\approx\nu/l_{min}$.
Under this normalization, let $L^{M*}_{p}$ be the scale of the region that
dominantly influences the pressure at the separation point. For this region to
be of a finite length scale, the integrand must decrease eventually, or

$ln(L^{M*}_{p})^{2}\langle\rho\frac{\partial u^{*}_{i}}{\partial
x^{*}_{j}}\frac{\partial u^{*}_{j}}{\partial x^{*}_{i}}\rangle\leq\Lambda^{2}$
(10)

for some finite $\Lambda$. For a large-eddy simulation, one of the aims is to
find the minimum $L^{M*}_{p}$ that needs to be resolved on the computational
grid to accurately predict separation. The minimum $L^{M}_{p}$ that would
always satisfy the above inequality is one that also satisfies

$(L^{M*}_{p})^{2}\leq 1+\frac{\Lambda^{2}}{max\\{\langle\rho\frac{\partial
u^{*}_{i}}{\partial x^{*}_{j}}\frac{\partial u^{*}_{j}}{\partial
x^{*}_{i}}\rangle\\}}.$ (11)

where the logarithm has been Taylor expanded under the expectation that the
extent of the dominant upstream region that affects the occurrence of
separation is not too dissimilar from the length scale imposed by the
strongest adverse pressure gradient. Stratford [29] suggested that at the
points where the skin friction crosses zero, the near-wall velocity profile
(up to $y_{p}=\frac{u_{p}y}{\nu}=5-10$) is given as

$\frac{u}{u_{p}}=A+B\sqrt{y_{p}}=A+B\sqrt{\frac{yu_{p}}{\nu}}.$ (12)

In the context of turbulent flows exhibiting a smooth-body separation due to
mild pressure gradients, we assume that the maximum pressure gradient (to
which $l_{min}$ corresponds to) is an $\mathcal{O}(1)$ multiple of the
pressure gradient at the separation point. This assumption was also
numerically verified to hold in the flows considered in the _a-posteriori_
sections of this article. Further assuming that near the separation point, the
scales of motion in the streamwise and wall-normal directions scale similarly,
the continuity equation imposes that the corresponding velocities must also
scale similarly. Then, using Equation 12, we arrive at

$\max\\{\langle\rho\frac{du^{*}_{i}}{dx^{*}_{j}}\frac{du^{*}_{j}}{dx^{*}_{i}}\rangle\\}\sim\mathcal{O}(1).$
(13)

for the near wall region, as the flow nears separation. This implies that

$L^{M}_{p}\sim\chi\;l_{min}\;\mathrm{where}\;\chi\sim\Lambda.$ (14)

or that a reasonable length scale that may affect a flow nearing separation is
the one that is viscously scaled by the strongest adverse pressure gradient in
the vicinity of the separation.

Figure 1: A two-dimensional (spanwise and time-averaged) contour of
$\sqrt{\langle\rho\frac{du_{i}}{dx_{j}}\frac{du_{j}}{dx_{i}}\rangle}$ in the
vicinity of the separation point ($x/L=0.1$) for the flow over the Boeing
speed bump.

As a numerical example of the analysis above, we consider the flow over the
Boeing speed bump [30]. A highly resolved wall-modeled LES (with the matching
location as low as $\Delta x^{+}=\Delta y^{+}=\Delta z^{+}=5$ at the wall with
an equilibrium wall model) was performed by Whitmore et al. [16]. They also
showed that the skin friction and the surface pressure coefficients were
accurately predicted on this grid in comparison to the quasi-DNS reference
data [30]. The resulting distribution of the product,
$\sqrt{\langle\rho\frac{du_{i}}{dx_{j}}\frac{du_{j}}{dx_{i}}\rangle}$ is
plotted in Fig. 1. The maximal value is $\approx 0.4$, implying
$L^{M}_{p}\sim\mathcal{O}(1)$.

The value of $\chi$ may vary depending on the flow; however, this will not
affect the scaling $L^{M}_{p}\sim l_{min}$. Some reasonable estimates of
$\Lambda$ are explored through _a-priori_ analysis in the next subsection. A
further implication of this fact can be that the response of the near-wall
flow in the vicinity of a separation bubble can also be governed by a stronger
adverse pressure gradient upstream (locally close to the point of separation,
however) of this region.

###  4.1 _A-priori_ analysis

In this section, _a-priori_ analysis is performed to obtain an estimate for
the expected values of $\Lambda$ in non-equilibrium boundary layers. It is
highlighted that a value of $\Lambda\leq 1$ does not necessarily imply the
limit of “wall resolved LES", however, for practical wall-modeled LES,
$\Lambda\geq 1$ is useful. Hence, for this analysis, the objective is to find
reasonable values of $\Lambda$ that provide monotonically converging, and
fairly accurate (for example, $\leq 10-15\%$ error) estimates of the wall
shear stress for non-equilibrium flows. The equilibrium wall model is used for
providing the estimates of the modeled wall stress. For this purpose, a high-
fidelity database consisting of the adverse pressure gradient boundary layers
of Bobke et al. [9] and the flow over NACA airfoils [31, 32] is examined. A
momentum-thickness-based Reynolds number ($Re_{\theta}$) up to approximately
4000 and a Clauser parameter value ($\beta=\delta^{*}/\rho
u^{2}_{\tau}\;dP/dx$) of up to 6 is observed in these flows.

Figure 2: The distribution of the (a) location of the matching point in wall
units ($y^{+}_{w}$) and (b) the corresponding percentage error in the
prediction of the wall shear stress from the equilibrium wall model for
$\chi=25$ as a function of the Clauser parameter and the momentum thickness
Reynolds numbers. The Roman numerals denote the data points from different
flows: Data series I to V correspond to different streamwise stations in the
flat-plate boundary layers simulated in Bobke et al. [9]. Data series VI and
VII correspond to different streamwise stations in NACA 0012 and 4412 airfoils
studied by Tanarro et al. [31] and Vinuesa et al. [32] respectively. The
starred point, A, is a sample point chosen to aid the interpretability of this
plot. This data point corresponds to a flat-plate boundary layer flow at
$Re_{\theta}\sim 3200\;\mathrm{and}\;\beta=4.2$. The matching location for
this point is roughly, $y_{w}^{+}\approx 40$ [in subfigure (a)] and the
percentage error in the shear-stress prediction is approximately 20% [and
hence the contour is saturated in subfigure (b)].

In Figures 2 and 3, the value of $\chi$ decreased from 25 to 6. For both these
figures, subfigure (a) denotes the location of the matching point
($y^{+}_{w}$) in wall units with the ordinate and the abscissa denoting the
local Clauser parameter and the momentum thickness-based Reynolds number
respectively. Similarly, subfigure (b) presents the percentage error in the
prediction of the shear stress using the equilibrium wall model. As an
example, the starred point, A, in Fig. 2 corresponds to a flat-plate boundary
layer flow at $Re_{\theta}\sim 3200\;\mathrm{and}\;\beta=4.2$ with
$y_{w}^{+}\approx 40$ (in Fig. 2(a)) and the percentage error in the shear-
stress prediction is approximately 20% (in Fig. 2(b)). Generally, in Fig. 2,
the error in the prediction of the shear stress, $\tau_{w}$, reaches up to
approximately 30% with the error generally increasing for a higher $y^{+}_{w}$
except when the Reynolds number is low. At a given $Re_{\theta}$, the error in
the shear-stress predictions increases with increasing $\beta$. This is
expected as for a larger $\beta$, a large range of the flow scales within the
boundary layer are under non-equilibrium. A turbulent channel flow analog of
this argument can be found in Lozano-Durán et al. [33]. At a given $\beta$,
the increase in $Re_{\theta}$ generally leads to a lower error as the scale
separation between the equilibrium and non-equilibrium regions of the boundary
layer increases. Upon refinement in Fig. 3, by reducing $\chi$ to 6
(equivalent to quadrupling the resolution), the errors are lower than those at
$\chi=25$. The errors for all the streamwise stations in the considered flows
are less than $10\%$ in this limit. Given the relatively modest $Re_{\theta}$
of these flows, the corresponding $y^{+}_{w}$ values are also small. However,
based on these _a-priori_ results, it is hypothesized that a value of
$\chi\in[5,10]$ may be necessary to accurately predict quantities of interest
in wall-modeled LES of non-equilibrium flows, atleast while leveraging the
equilibrium wall model.

Figure 3: The distribution of the (a) location of the matching point in wall
units ($y^{+}_{w}$) and (b) the corresponding percentage error in the
prediction of the wall shear stress from the equilibrium wall model for
$\chi=25$ as a function of the Clauser parameter and the momentum thickness
Reynolds numbers. The Roman numerals denote the data points from different
flows: Data series I to V correspond to different streamwise stations in the
flat-plate boundary layers simulated in Bobke et al. [9]. Data series VI and
VII correspond to different streamwise stations in NACA 0012 and 4412 airfoils
studied by Tanarro et al. [31] and Vinuesa et al. [32] respectively. The
starred point, A, is a sample point chosen to aid the interpretability of this
plot. This data point corresponds to a flat-plate boundary layer flow at
$Re_{\theta}\sim 3200\;\mathrm{and}\;\beta=4.2$. The matching location for
this point is roughly, $y_{w}^{+}\approx 10$ [in subfigure (a)] and the
percentage error in the shear-stress prediction is approximately 6% [in
subfigure (b)].

###  4.2 Reynolds number scaling

The currently accepted zero-pressure-gradient flow-based estimates [13, 14]
suggest that the number of grid points ($N_{cv}$) required to perform wall-
modeled LES of a boundary layer scale as $N_{cv}\sim Re^{1}$. Instead, in this
work, we propose an alternate grid-point scaling for non-equilibrium flows,
specifically those that exhibit a turbulent separation. These estimates will
be tested in _a-posteriori_ calculations in subsequent sections. The
assumption in this work is that a nearly constant value of $\chi$, or a
viscously driven length scale, can be chosen to predict the quantities of
interest as the Reynolds number is increased. The value of $L^{M}_{p}$ scales
as

$L^{M}_{p}\sim\chi l_{min}\sim\chi
Re^{-2/3}\max\left(\frac{1}{\rho}\frac{dP}{dx}\right)^{-1/3}.$ (15)

or that as the Reynolds number is increased, $L^{M}_{p}$ decreases as
$Re^{-2/3}$. Thus, for a nested grid where the number of points in the wall-
normal direction with respect to $L^{M}_{p}$ is fixed, the number of grid
points required to sufficiently resolve the non-equilibrium effects scales as
$N_{cv}\sim Re^{4/3}$. This prediction is more stringent than the equilibrium
flat-plate boundary layer wall-modeled LES estimates but grows slower than
$N_{cv}\sim Re^{13/7}$ scaling of wall-resolved LES. The assumption of
requiring a fixed $\chi$ across a range of Reynolds numbers may be negated
once improved wall models consistently and accurately account for the non-
equilibrium and the flow history effects at coarse resolutions. In Appendix I,
the performance of some existing non-equilibrium wall models is reported,
which suggests that present non-equilibrium models do not affect the
resolution requirements significantly. Following the works of Chapman [12],
Choi and Moin [13], Yang and Griffin [14] Appendix II also provides some _a-
priori_ estimates of the number of points required for wall-modeled LES of
non-equilibrium boundary layers assuming a fixed outer resolution
($\Delta/\delta=\mathrm{const.}$). However, it is noted that these _a-priori_
estimates may not be realized in _a-posteriori_ simulations owing to the
errors in the subgrid-scale and wall models.

##  5 _A-posteriori_ analysis

###  5.1 Wall-modeled LES of the Boeing speed bump

In this section, _a-posteriori_ results from wall-modeled LES of the flow over
the Boeing speed bump [34, 35, 36] are presented. This geometry was proposed
by the Boeing Company and Williams et al. [34] for validation of CFD methods
in the prediction of turbulent flow separation. Zhou et al. [37] recently
showed a large sensitivity in the prediction of the separation bubble to the
choice of the subgrid-scale and wall models. The simplified, spanwise-periodic
bump surface is defined by an analytical expression, $h(x)$, written as

$y\left(\frac{x}{L}\right)=0.085\,exp\left[-\left(\frac{x}{0.195L}\right)^{2}\right].$
(16)

Figure 4: A schematic of the flow over the simplified, spanwise periodic
Boeing speed bump geometry. The flow develops over a Gaussian-shaped bump
experiencing a favorable and an adverse pressure gradient before eventually
separating at $x/L\approx 0.1$.

where $x$ is the streamwise coordinate. The Reynolds number, $Re_{L}$ for this
flow is defined in terms of the freestream velocity $U_{\infty}$ and the bump
width $L$. It has been previously shown that the surface pressure ($C_{p}$)
and the skin friction ($C_{f}$) for the simplified geometry approximately
match those at the midspan of the experimental configuration [35]. In this
work, we first perform computations at a Reynolds number, $Re_{L}=2\times
10^{6}$, to match the prior quasi-DNS [30] and for comparison with experiments
[34, 35] at $Re_{L}=3.41\times 10^{6}$ and $Re_{L}=4\times 10^{6}$. The flow
over the speed bump experiences a strong favorable pressure gradient followed
by a strong adverse pressure gradient on the fore and aft sections of the bump
respectively. For $Re_{L}\geq 1.8\times 10^{6}$, these pressure gradients lead
to the formation of a turbulent separation bubble.

Mesh |  $N_{cv}$ |  max $\Delta/L$ |  min $\Delta/L$
---|---|---|---
Coarse |  $3$ Mil. |  $0.01$ |  $1.3\times 10^{-3}$
Medium |  $12$ Mil. |  $0.01$ |  $6.3\times 10^{-4}$
Fine |  $52$ Mil. |  $0.01$ |  $3.1\times 10^{-4}$
Table 2: Mesh parameters for the $Re_{L}=2\times 10^{6}$ case of the flow over
the spanwise-periodic Boeing speed bump. For subsequently higher Reynolds
numbers, the grid is refined by modifying the background resolution (equal to
max. $\Delta/L$) by a factor of $Re^{2/3}$.

Fig. 4 represents a schematic of the simulation setup. The inlet is located at
$x/L=-1.0$ as a plug-flow profile, and the flow then undergoes a transition to
turbulence over the region, $-1.0<x/L<-0.6$. This simple choice stems from the
observations of Agrawal et al. [11] who showed that the separation tendency of
this flow is approximately independent of the inlet boundary condition as long
as the inlet is sufficiently far upstream. This has also been reported in _a-
posteriori_ simulations by Whitmore et al. [16]. Free-stream velocities with a
non-reflective boundary condition are set at the top boundary. A
characteristic boundary condition (NSCBC) with constant pressure is applied at
the outlet ($x/L=2.5$). The computational meshing approaches are the same as
in Agrawal et al. [25]; some details are provided in Table 2. A schematic of
the computational mesh as a function of the streamwise coordinate is provided
in Appendix III.

The experiments of Williams et al. [34] and Gray et al. [35] have suggested an
approximate Reynolds number independence in the surface pressure distribution
over the bump surface for $Re_{L}\geq 2\times 10^{6}$. In light of this, in
this work, we perform numerical experiments up to $Re_{L}=10\times 10^{6}$,
assuming the invariance of the pressure coefficient profile holds at the
increased Reynolds numbers. Only comparisons of the surface pressure
distribution are made as it is expected that the skin-friction distribution in
the upstream region (nearly a zero pressure gradient flat-plate flow) is
expected to be a function of the Reynolds number. In a prior work [25], the
authors showed that for the Reynolds number, $Re_{L}=2\times 10^{6}$ case, the
skin friction compares well with the experiments. The surface pressure
coefficient ($C_{p}$) is mathematically defined as

$\frac{C_{p}}{2}=\frac{p-p_{\rm ref}}{\rho_{\infty}U^{2}_{\infty}}.$ (17)

where $U_{\infty}$, $p$, and $p_{\rm ref}$ are the mean free-stream velocity,
wall pressure, and the upstream reference pressure in the zero-pressure region
of the flow respectively.

Figure 5: The streamwise distribution of the surface pressure coefficient,
$C_{p}$ for the (a) coarse grid, (b) medium grid, and (c) fine grid across
$2\times 10^{6}\leq Re_{L}\leq 10\times 10^{6}$. Note that across different
Reynolds numbers, the nominal “coarse", “medium" and “fine" grids are scaled
in their resolution by a factor of $Re_{L}^{2/3}$. Subfigure (d) shows the
$L_{\infty}$ norm of the error in $C_{p}$ for all the Reynolds number flows.
The vertical bars in the simulations in subfigures (a)-(c) represent the root
mean square error around the mean value of $C_{p}$.

In Fig. 5, the predictions of the surface pressure as a function of the
Reynolds number are plotted. Subfigures (a)-(c) provide these predictions at a
given nominal grid for different $Re_{L}$, where at each level (e.g., coarse,
medium, fine), grids are refined based on the proposed Reynolds number scaling
($Re^{2/3}_{L}$). When the resolution of the calculation with respect to
$l_{p}$ is held constant, it is apparent that predictions of wall-modeled LES
at different Reynolds numbers are similar. For all the considered cases, the
finest grid is required to accurately predict $C_{p}$, specifically, the flow
separation ($0.1\leq x/L\leq 0.4$). To quantify these errors, Fig. 5(d)
presents the $L_{\infty}$ norm of the error in $C_{p}$. This norm is chosen in
the entirety of this work to specifically highlight the largest differences in
wall-modeled LES and reference values; which occur primarily due to incorrect
prediction of the extent of the separation bubble (since the distribution of
$C_{p}$ sufficiently far upstream and downstream of the separation bubble can
be described mostly from inviscid flow arguments). The error values are
expected to reach an asymptotic value on very coarse grids when the prediction
of $C_{p}$ reaches the inviscid limit (the flow remains attached and the
boundary layers are sufficiently thin). The error convergence on the fine grid
corresponds to a $\Delta/min(l_{p})\sim 12$, which is comparable to the values
from the _a-priori_ analysis in the preceding sections. This is also
encouraging because the preceding analysis does not account for numerical and
subgrid-scale modeling errors. Appendix IV provides a discussion on the
limited sensitivity in the convergence behavior of wall-modeled LES on the
choice of the subgrid-scale model between the dynamic Smagorinksy model [38]
and the dynamic tensor coefficient Smagorinsky model [25]. Although not shown,
it was observed that the minimum $y^{+}$ values before the separation point
are approximately $y^{+}_{w}\approx 15-20$, implying the simulations are not
wall-resolved. An additional simulation on a “very-fine" grid (twice as
refined in each direction than the “fine” grid, not shown) for the
$Re_{L}=5\times 10^{6}$ case confirmed that the results are grid-converged.

###  5.2 Wall-modeled LES of the Song-Eaton Diffuser

Song and Eaton [39] performed detailed experiments on the Reynolds number
sensitivities of flow separation over a circular arc-shaped diffuser. The
experiments recorded a small separation bubble with nearly Reynolds number
invariant $C_{p}$ distribution across a range of $3400\leq
Re^{ref}_{\theta}\leq 20100$ where $Re^{ref}_{\theta}$ denotes the momentum
thickness based Reynolds number at a reference station ($x/L=-2$ where $L$ is
the length of the arc of the diffuser) in the flat-plate region of the
diffuser. Fig. 6 presents a schematic of the simulation setup. In this flow,
the unsteady separation process occurs over a large portion of the arc surface
leading to the formation of intermittently attached and separated boundary
layers. Radhakrishnan et al. [40] performed a detached-eddy simulation of this
flow with a stochastic forcing [41] in the “transition-region" between quasi-
steady RANS near-wall flow to the unsteady LES outer flow; this results in an
_ad hoc_ breakdown of the streamwise coherence of the eddies. Their results
reasonably captured the separation bubble for the $Re^{ref}_{\theta}=13200$
flow, however, they employed the conventional RANS type stretched grids in the
wall-normal direction. The near-wall cells had an approximate aspect ratio of
$300:1$ and were refined down to $y^{+}=1$.

Figure 6: A schematic of the simulation setup for the flow over the arc-shaped
diffuser studied by Song and Eaton [39]. The flow develops over a flat plate
upstream of the arc before experiencing an adverse pressure gradient that
leads to a flow separation at $x/L\approx 0.79$. The vertical dashed line at
$x/L=0$ marks the beginning of the diffuser’s arc.

Table 3 presents details of the computational grids in this work. A schematic
of the computational mesh as a function of the streamwise coordinate can be
found in Appendix III. The simulation setup for this flow is very similar to
the Boeing speed bump. A plug-flow is fed at the inlet ($x/L=-5$) which
transitions into an equilibrium turbulent boundary layer, as it develops over
a long flat-plate region before encountering the diffuser. The outlet boundary
in the simulations was extended up to $x/L=10$ to let the flow fully recover
post-reattachment. The top boundary is placed at $y/L=2.1$; both the top and
the bottom walls are treated using the equilibrium wall model. The outlet is
treated with the characteristic boundary condition [42]. Unlike Radhakrishnan
et al. [40], the grid resolution is isotropic in all three spatial directions
near the wall. Periodic boundary conditions are used along the spanwise
direction with a period of $10\;\delta^{ref}_{99}$ where $\delta^{ref}_{99}$
is the experimental boundary layer thickness at the reference upstream station
($x/L=-2$).

Mesh |  $N_{cv}$ |  max. $\Delta/L$ |  min. $\Delta/L$
---|---|---|---
Coarse |  $0.4$ Mil. |  $0.03$ |  $7.6\times 10^{-3}$
Medium |  $1.3$ Mil. |  $0.03$ |  $3.8\times 10^{-3}$
Fine |  $5.2$ Mil. |  $0.03$ |  $1.9\times 10^{-3}$
Table 3: Mesh parameters for the Reynolds number, $Re^{ref}_{\theta}=3400$
(based on the momentum thickness at the upstream station, $x/L=-2$) case of
the flow over the circular arc-shaped diffuser studied by Song and Eaton [39].
For subsequently higher Reynolds numbers, the grid is refined by modifying the
background resolution (equal to max. $\Delta/L$) by a factor of $Re^{2/3}$.
Figure 7: The prediction of surface pressure coefficient $C_{p}$ for the flow
over the Song-Eaton diffuser at (a) $Re^{ref}_{\theta}=3400$, (b)
$Re^{ref}_{\theta}=13200$, and (c) $Re^{ref}_{\theta}=20100$ respectively. The
flow separates at $x/L\approx 0.79$ and reattaches at $x/L\approx 1.2$.
Subfigure (d) provides the error convergence plot upon grid-refinement for
different Reynolds numbers. The vertical bars in the simulations and the
experiment in subfigures (a)-(c) represent the uncertainty in the mean values.
In the simulations, the root mean square error was used to quantify this
uncertainty. The reference $C_{p}$ in $Re^{ref}_{\theta}=20100$ simulations is
taken from $Re^{ref}_{\theta}=13200$ experiments due to lack of the
availability of experimental data at $Re^{ref}_{\theta}=20100$ in Song and
Eaton [39].

We performed wall-modeled LES of all three Reynolds numbers,
$Re^{ref}_{\theta}=3400,\;13200,\;20100$ considered in the experiments. Fig. 7
shows the $C_{p}$ distribution for these cases. Subfigures (a)-(c) correspond
to the three Reynolds numbers respectively. The $C_{p}$ distribution is nearly
identical for all Reynolds numbers, with the flow experiencing an acceleration
slightly upstream of the arc (as $x/L\rightarrow 0$). Then, as the boundary
layer experiences an adverse pressure gradient over the arc, a small
separation occurs at $x/L\approx 0.79$. For all Reynolds numbers, the fine
grid accurately captures the separation bubble and the flow recovery regions.
Fig. 7 also shows that at $Re_{\theta}=20100$, a fine grid simulation with no-
slip boundary conditions on the bottom walls produces the inviscid $C_{p}$
solution and does not predict any flow separation which confirms the necessity
of a wall model to simulate this flow at the finest grid resolution. For this
flow, the performance of wall-modeled LES for coarse and medium grids varies
between the lowest and the other two Reynolds numbers similar to the Boeing
speed bump. Fig. 7(d) presents the $L_{\infty}$ norm of the error in the
$C_{p}$; with the errors becoming “small" only for the finest grid. Selected
sensitivity tests (not shown) to the choice of the subgrid-scale model
suggested no differences in the grid-point scaling with the Reynolds number.

###  5.3 Wall-modeled LES of the Notre-Dame Ramp

Figure 8: A schematic of the flow over the smooth ramp studied by Simmons
[43]. The flow develops over a flat plate upstream of the ramp before
separating at $x/L\approx 0.5$. The dotted lines are only plotted for marking
the reference coordinates and are not a part of the simulation domain. The
curved portion of the upper surface in the upstream region ($x/L<0$) is
replicated from the experimental geometry to provide a weak favorable pressure
gradient to accelerate the incoming flow.

In this section, we further test the proposed
$\Delta/min(l_{p})\sim\mathcal{O}(10)$ resolution requirements for predicting
turbulent separation accurately on the flow over a smoothly curved ramp.
Simmons [43] performed a series of experiments on a smooth ramp over which a
high Reynolds number boundary layer (upstream reference $Re_{\theta}=11000$ at
$x/L=-0.75$, or, $Re_{H}=U_{\infty}H^{r}/\nu=8.4\times 10^{5}$ where $H^{r}$
is the height of the ramp) exhibits a smooth-body separation; this flow is a
candidate for the “High-Fidelity CFD Verification" Workshop at AIAA SciTech,
2024. The top wall of the experimental setup was slightly tapered to create a
pressure gradient, with the tapering angle having a significant impact on the
extent of the separation [43]. Fig. 8 provides a schematic of the simulation
setup that nearly reproduces the experimental wind tunnel configuration. The
ramp geometry ($0<x/L<1$ on the bottom wall) is obtained from a fifth-order
polynomial that smoothly joins the upstream and downstream flat plates. Three
tapering angles, $\alpha=3.2^{o},\;5.6^{o},\;7.7^{o}$ were studied
experimentally, with the smallest angle producing the largest separation
bubble, and the largest angle producing a fully attached flow. In this work,
we consider the $\alpha=5.6^{o}$, expecting that this angle will be the most
challenging condition in regard to the prediction of the separation bubble.

Mesh |  $N_{cv}$ |  max. $\Delta/L$ |  min. $\Delta/L$
---|---|---|---
Coarse |  $16$ Mil. |  $0.02$ |  $2.5\times 10^{-3}$
Medium |  $64$ Mil. |  $0.02$ |  $1.3\times 10^{-3}$
Fine |  $252$ Mil. |  $0.02$ |  $6.25\times 10^{-4}$
Table 4: Mesh parameters for the $Re^{ref}_{\theta}=11000$ (Reynolds number
based on the momentum thickness at $x/L=-0.75$) or $Re_{H}=8.4\times 10^{5}$
(based on ramp-height) case of the flow over the smooth ramp studied by
Simmons [43].

In our computations, the inlet is fed as a plug flow located at $x/L=-2.82$.
To approximately match the experimental boundary conditions, both the top and
the bottom walls are treated inviscidly in the region $-2.82\leq x/L\leq-1.6$.
Downstream of this region, the bottom wall flat plate and the ramp are treated
with the equilibrium wall model. The top wall is located at the same height as
in the experiments, the equilibrium wall model closure is also used at this
boundary. The smooth curvature of the top wall (between $-1.6\leq x/L\leq-1.2$
and $0\leq x/L\leq 1.4$) is replicated from the experiments to provide a weak
favorable pressure gradient to the flow. In the experiments, after the flow
recovers from the separation, it leaves the test section at $x/L\approx 1.4$
to enter a diffuser region. In our simulations, the top wall is flattened
after $x/L=1.4$ and the flow is allowed to recover downstream in a nominally
zero-pressure gradient region with the outlet located at $x/L=5.4$. The outlet
boundary condition is a non-reflective characteristic boundary condition [42].
The domain is $\approx 25\delta^{ref}_{99}$ wide in the spanwise direction.
Details of the grid distribution and resolutions are provided in Table 4. The
schematics of the computational mesh are provided in Appendix III.

Fig. 9(a) shows the $C_{p}$ distribution across the grid refinement sweep. As
the flow encounters the ramp, a favorable pressure gradient is experienced
initially due to the tapering of the top wall that creates a flow
acceleration. However, at $x/L\approx 0.2$, the adverse pressure gradient due
to the ramp surface dominates and leads to the formation of a separation
bubble at $x/L\approx 0.5$. In the current wall-modeled LES results, on the
coarsest grid, the flow does not separate. As the grid is refined, the flow
starts separating, and on the fine grid, the solution is nearly identical to
the experimental measurement of $C_{p}$. Table 4 and Fig. 9(b) suggest that
for the fine grid, the resolution in terms of the pressure-scaled units is
approximately the same as that required on the Boeing speed bump, and the
Song-Eaton diffuser flow.

Figure 9: The (a) surface pressure distribution and the (b) error convergence
upon grid-refinement plots for the flow over the smooth-ramp designed and
studied by Simmons [43]. The flow develops over a flat plate upstream of the
ramp before separating at $x/L\approx 0.5$, and later reattaching with the
bottom wall at $x/L\approx 0.90$. The vertical bars in the experiment in
subfigure (a) represent the uncertainty in the mean values.

###  5.4 Wall-modeled LES of the Backward-Facing step

The flow over the backward-facing step has been extensively studied, both
experimentally [44, 45, 2] and computationally [46, 47, 48] to describe the
structure of a turbulent separating boundary layer. Jovic and Driver [44]
observed that the reattachment length of the separated flow past the step is a
weak function of the Reynolds number. The separation over the step is a
geometric constraint in this flow, but the prediction of the reattachment
length is dependent on the induced pressure gradient due to the extent of the
separation bubble. Unlike the other cases considered in this work, the flow
past the separation point for the backward-facing step experiences the
strongest pressure gradient post-separation. It is highlighted that given its
geometrically imposed separation point, this flow allows solely focusing on
examining the resolution requirements for accurate prediction of the
reattachment point. Jovic and Driver [44] also reported a weak Reynolds number
dependence of the separated flow for different expansion ratios, $E_{r}$
(equal to the ratio of the height of the region downstream of the step to the
region upstream of it).

Figure 10: A schematic of the flow over a backward-facing step studied by
Driver and Seegmiller [45]. The flow develops over a flat plate upstream of
the step before separating at the corner. The flow eventually reattaches
downstream with its reattachment length shown to be a function of the Reynolds
number in Jovic and Driver [44].

The experimental configuration of $E_{r}\approx 1.1$ and
$Re_{H}=U_{\infty}H/\nu=37500$ (where $H$ is the step height) from Driver and
Seegmiller [45] is simulated in this work. Fig. 10 provides a schematic of the
exact simulation setup with the inlet extended significantly upstream (at
$x/H=-110$) to develop a realistic turbulent boundary layer that matches the
experimental $Re_{\theta}$ at the upstream reference location of $x/H=-4$. The
inviscid, top boundary is placed at $y/H=9$ to match the expansion ratio; an
equilibrium wall model closure is applied on the bottom wall. A separate set
of experiments in Jovic and Driver [44] previously suggested only limited
differences in the pressure distribution associated with a double-sided
expansion and a single-sided expansion at the same expansion ratio. The outlet
is placed at $x/H=50$ with non-reflective boundary conditions. Table 5
provides the details of the gridding approach for this flow. A schematic of
the employed computational mesh is available in Appendix III.

Mesh |  $N_{cv}$ |  $max.\Delta/H$ |  $min.\Delta/H$ |  $x_{r}/H$
---|---|---|---|---
Coarse |  $2$ Mil. |  $1$ |  $1.2\times 10^{-1}$ |  5.04
Medium |  $8$ Mil. |  $1$ |  $6.2\times 10^{-2}$ |  5.56
Fine |  $32$ Mil. |  $1$ |  $3.1\times 10^{-2}$ |  6.24
Table 5: Mesh parameters for the $Re_{H}=37500$ case of the flow over the
backward-facing step studied by Driver and Seegmiller [45]. The last column
denotes the mean flow reattachment length as a function of the employed grid.
The reported experimental value is $x_{r}/H=6.26\pm 0.05$.  Figure 11: The (a)
bottom wall surface pressure for the flow past a backward-facing step studied
by Driver and Seegmiller [45] at a step-height Reynolds number,
$Re_{H}=37500$. Subfigure (b) provides the error convergence plot upon grid-
refinement at this Reynolds number. In the experiments, the flow separates
past the step and reattaches with the bottom wall at $x_{r}/H\approx 6.26$.
The scaling variable $l^{*}_{p}$ is the pressure gradient length scale imposed
by the flow upstream of the separation point ($x/H=0$) for consistency with
the other cases considered in this work. The vertical bars in the simulations
and the experiment represent the uncertainty around the mean. In the
simulations, the root mean square error is plotted for the same.  Figure 12:
The (a) skin-friction distribution and the (b) surface pressure distribution
at the top wall for the flow past a backward-facing step studied by Driver and
Seegmiller [45] at a step-height Reynolds number, $Re_{H}=37500$. The vertical
bars in the simulations and the experiment represent the uncertainty around
the mean. In the simulations, the root mean square error was used to quantify
this uncertainty. ‘

Fig. 11(a) compares the predictions of the surface pressure on the bottom wall
with the experiments [45]. The predictions of the surface pressure improve on
grid refinement with good agreement with the experiment resulting when using
the finest grid. In figure 11(b), the error in $C_{p}$ converges to $\approx
0.04-0.05$ when $\Delta/min(l_{p}^{*})\sim 7-15$. In Fig. 12(a), the skin-
friction distribution is compared against the experiments, and the finest grid
recovers the experimental reporting well in the separated, and recovery
regions. In Fig. 12(b), the pressure at the top wall is also captured well in
wall-modeled LES on the finest grid. Overall, for this flow, our results
suggest that the grid-resolution requirement for the accurate prediction of
the reattachment point ($x_{r}$), and the surface pressure ($C_{p}$) past a
step are similar to the resolution required to accurately capture the extent
of the separated flow region over smooth surfaces.

##  6 Concluding Remarks

In this work, the Reynolds number scaling of the required number of grid
points to perform wall-modeled LES of flow encountering separation is
examined. Based on the various time scales in a turbulent boundary layer
experiencing external pressure gradients, it is suggested that the Reynolds
number dependence of the “under-equilibrium" length scale (scales for which
the viscous and the pressure gradient effects approximately balance each
other) can be hypothesized to be $l_{p}\sim Re^{-2/3}$. The same scaling is
revealed based on a simplified Green’s function type solution of the pressure
equation around the separation point. _A-priori_ analysis suggests that the
minimum resolution required to reasonably predict the shear stress (errors
lower than approximately $10-15\%$ in the entire domain) with the equilibrium
wall model is at least $\mathcal{O}(10)$$l_{p}$ irrespective of the Reynolds
number and the Clauser parameter for several non-equilibrium flat-plate
boundary layers and airfoil flows. Several _a-posteriori_ calculations are
then performed to determine the accuracy of this scaling. Numerical
simulations are performed and compared with experimental data for the flow
over the Boeing speed bump, a diffuser, Notre-Dame Ramp, and the backward-
facing step. The results suggest that for these flows, scaling the grids
($\Delta$) to maintain a fixed $\Delta/l_{p}$ results in accurate predictions
of flow separation at the same “nominal" grid resolution across different
Reynolds numbers. Upon employing a dynamic subgrid-scale model and an
equilibrium wall modeling closure, the minimum resolution requirement for
capturing the flow separation was found to be
$\Delta/min(l_{p})\sim\mathcal{O}(10)$. We also find that selected flow cases
that use existing non-equilibrium wall models also require similar grid
resolutions to achieve accurate results (as detailed in Appendix I). Finally,
it is suggested that in more complex, three-dimensional flows, at least,
locally, the grid-point requirements to predict flow-separation may scale as
$Re^{4/3}$, which is more restrictive than the previously proposed flat-plate
boundary layer-based estimates for wall-modeled LES.

## Appendix I: Performance of existing non-equilibrium wall models

Several non-equilibrium wall models have been proposed [49, 50, 37, 51] to
account for the flow non-equilibrium by modeling the pressure gradient and
convective terms either using governing equations, or data-driven methods.
These studies have all reported some improvements in the _a-posteriori_ wall-
modeled LES predictions upon employing their respective non-equilibrium wall
models over a range of mildly separated flows. Table 6 summarizes the near-
wall, wall-normal grid resolution (non-dimensionalized by the viscous scale
imposed by the strongest pressure gradient) required when using these models
to predict the quantities of interest accurately. It is clear that for all
considered flows, the required resolution to predict the quantities of
interest accurately is comparable, and not significantly coarser, than the
resolution required with the equilibrium wall model as found in this work
[$\Delta/min(l_{p})\sim\mathcal{O}(10)$]. Further research may provide the
Reynolds number scaling of these models, and improvements in the $N_{cv}\sim
Re^{4/3}$ cost scaling, if any.

Model Description |  Flow |  $\Delta/min.(l_{p})$
---|---|---
PDE, non-equilibrium |  NASA Hump |  $\approx 16$
wall-model [50] |  |
PDE, non-equilibrium |  Airfoil |  $\approx 14$
wall-model [52] |  |
Reinforcement learning |  Periodic Hill |  $\approx 14$
wall-model [37] |  |
ODE, non-equilibrium |  Suction, blowing |  $\approx 15$
wall-model [51] |  boundary layer |
Table 6: The minimum wall normal grid resolutions reported for accurate
prediction of quantities of interest upon leveraging existing non-equilibrium
wall models for a range of mildly separated flows.

##  Appendix II: Grid estimates of wall-modeled LES based on fixed outer
resolution ($\frac{\Delta}{\delta}$)

Previous efforts [12, 13, 14] have provided grid-point estimates for wall-
modeled LES assuming that the grid resolutions scale in outer flow units
($\delta$). It is known that the mean local shear determines the dynamics of
turbulence at different wall-normal locations in both equilibrium and non-
equilibrium flows [53, 54]. As long as the pressure gradient is not too
strong, this shear rate also does not significantly vary across the outer
region of the boundary layer for non-equilibrium flows. Hence, a similar
assumption about requiring a fixed outer resolution ($\delta/\Delta$) of the
boundary layer may be motivated for non-equilibrium flows. It is highlighted,
however, that such a scaling relies on the assumption that the subgrid-scale
and wall models remain accurate in a range of non-equilibrium conditions.

For this purpose, we recall the relationships suggested in Agrawal et al. [11]
between the growth rates of the momentum thickness ($\theta$) and the boundary
layer thickness ($\delta_{99}$). For a turbulent boundary layer with
freestream velocity distribution, $U_{e}(x)$, the authors proposed that

$\frac{d\delta}{dx}\approx\frac{5}{200}+\frac{8}{200}m-\frac{U_{e}}{200\nu}\frac{d\theta^{2}}{dx}+\frac{d\delta{zpg,corr}}{dx}.$
(18)

where $\delta_{zpg,corr}/x=0.16Re^{-1/7}_{x}=0.16(U_{e}x/\nu)^{-1/7}$. $m$ is
the Holstein-Bohlen parameter given as, $m=\theta^{2}/\nu dU_{e}/dx$. The
expression for the growth of the momentum thickness is another ordinary
differential equation given as

$\frac{d}{dx}(U^{C_{m}}_{e}\theta^{2})\approx\nu
C_{c}U^{C_{m}-1}_{e}+C_{Re,\infty}{U^{C_{m}}_{e}\theta}.$ (19)

where $C_{c}=1.45$, $C_{m}=7.2$, and $C_{Re,\infty}=0.0024$. Consider, a
turbulent boundary layer experiencing mild pressure gradients, such that it
grows according to Equations 18-19. Let
$\eta=\frac{x^{*}}{U^{2}_{e}}\frac{dP}{\rho dx}$ be the parameter that governs
the streamwise distance over which the pressure gradient grows slowly (or
remains locally constant). Bernoulli’s principle implies that over the region
($0-x^{*}$)

$U_{e}(x)=U^{0}_{e}\sqrt{1-2\frac{x^{*}}{U^{2,0}_{e}}\frac{dP}{\rho
dx}}\approx U^{0}_{e}(1-\eta).$ (20)

Using this relation and the equations above, it can be shown that to the
leading order (in $\eta$)

$\frac{d\theta}{dx}\approx\left[\frac{C_{c}\nu}{2U^{0}_{e}\theta}+\frac{C_{Re,\infty}}{2}\right]+\left[\eta\left(\frac{C_{c}\nu}{2U^{0}_{e}\theta}+\frac{C_{m}\theta}{2x}\right)\right].$
(21)

Due to the proposed linear expansion in ($m,Re_{\theta}$) space for the growth
of the momentum thickness, the effect of the Reynolds number and the pressure
gradient on the growth of the boundary layer are explicitly separable. The
terms in the first parenthesis represent the growth rate of a zero-pressure
gradient boundary layer. Thus

$\frac{d\theta}{dx}\approx\frac{d\theta^{zpg}}{dx}+\left[\eta\left(\frac{C_{c}\nu}{2U^{0}_{e}\theta}+\frac{C_{m}\theta}{2x}\right)\right].$
(22)

If the inviscid pressure gradient distribution, $\eta=\eta(x)$, was known,
then Equation 22 could be integrated to give estimates for the growth of the
boundary layer. If we further assume that the pressure gradients are
sufficiently mild such that,
$\theta\approx\theta^{zpg}+\eta\frac{d\theta}{d\eta}\big{|}_{zpg}$, then the
leading order expansion in $\eta$ would yield

$\frac{d\theta}{ds}\approx(0.016+0.6\eta)Re^{-1/7}_{x}.$ (23)

Using Equations 23 and 18, an approximate relationship between the boundary
layer thickness, $\delta$ (evaluated using the method of Griffin et al. [55])
in ($\eta,Re_{x}$) space can be derived as

$\delta(x)\approx\delta^{zpg}+10^{-5}\eta xRe^{13/14}_{x}\approx
0.16xRe^{-6/7}_{x}+10^{-5}x[\frac{x}{U^{2,0}_{e}}\frac{dP}{\rho
dx}]Re^{13/14}_{x}.$ (24)

In the asymptotic limit of a large Reynolds number, the model predicts that
the boundary layer growth is determined linearly by the pressure-gradient, and
slightly sub-linearly by the Reynolds number. Following Choi and Moin [13],
Yang and Griffin [14], an integral relationship between $\delta$ and the total
number of points required for performing wall-modeled LES is written as

$N_{pts}^{wm}=N(x<x_{0})+\int^{L_{x}}_{x_{0}}\int^{L_{z}}_{0}\frac{n_{x}n_{y}n_{z}}{\delta^{2}}dxdz.$
(25)

where $n_{x},\;\mathrm{and}\;n_{z}$ are the number of points inside the
boundary layer in the streamwise, and spanwise directions respectively.
$x_{0},\;\mathrm{and}\;L_{x},$ are the limits of the streamwise integral, and
$L_{z}$ is the domain size in the spanwise direction. Thus, for a given
inviscid flow distribution, the grid-point requirements for a boundary layer
experiencing mild pressure gradients can be evaluated using Equations 24 and
25.

## Appendix III: Details of computational meshes

The computational grids employed in this work are based on a centroidal
Voronoi diagram generated from a hexagonally close-packed lattice of seed
points. While refining the grid, the control volumes are refined isotropically
by factors of two in the layers near the boundaries. Each refinement step
injects a thin layer (containing 10 cells) into the regions immediately
adjacent to the wall of twice as fine cells in each direction compared to the
previous mesh. Further, Lloyd iterations are performed to make the mesh
transitions between layers of different cell sizes smoother. Control volumes
far from domain boundaries and refinement transitions consist of tessellated
truncated octahedra. Although not required, the mesh arrangement was chosen to
be uniform in the streamwise and spanwise directions to properly evaluate the
predictive abilities of wall-modeled LES.

The schematics of the meshes corresponding to the “coarse grid" for each flow
case are now provided as follows. Fig. 13 corresponds to the mesh distribution
for the flow over the Boeing speed bump. Similarly, Figures 14, 15, and 16
correspond to the Song-Eaton diffuser, Notre-Dame smooth ramp, and the
backward-facing step cases respectively.

Figure 13: Cross section of the “coarse" grid for the spanwise-periodic
variant geometry of the Boeing speed bump [30] at $Re_{L}=2\times 10^{6}$. The
streamwise extent is $x/L\in[-0.3,0.3]$ showing both the fore and the aft
sides of the bump. Only the lower half of the vertical extent of the domain is
shown. The top half is meshed uniformly up to the top wall. Three layers of
isotropic refinement are visible adjacent to the bump surface. The qualitative
arrangement of the control volume remains identical to this arrangement while
simulating the higher Reynolds number flows at $Re_{L}=5,\;10\times 10^{6}$.
Figure 14: Cross section of the “coarse" grid for the Song-Eaton diffuser
geometry [39] at $Re^{ref}_{\theta}=3400$. The streamwise extent is
$x/L\in[-3,4.5]$ showing both the upstream (of the diffuser) and the
downstream regions. The entire vertical extent of the domain is shown. Two
layers of isotropic refinement are visible adjacent to the walls on both the
lower and the upper walls. The qualitative arrangement of the control volume
remains identical to this arrangement while simulating the higher Reynolds
number flows at $Re^{ref}_{\theta}=13200,\;20100$.  Figure 15: Cross section
of the “coarse" grid for the smooth-ramp geometry [43] at $Re_{H}=8.4\times
10^{5}$. The streamwise extent is $x/L\in[-0.5,1.5]$ showing both the upstream
(of the ramp) and the downstream flat-plate regions; the entire vertical
extent of the domain corresponding to the streamwise region,
$x/L\in[-0.5,1.5]$, is also shown. Three layers of isotropic refinement are
visible adjacent to the walls on both the lower and the upper walls.  Figure
16: Cross section of the “coarse" grid for the backward-facing step geometry
[44] at $Re_{H}=37500$. The streamwise extent is $x/L\in[-10,12]$ and the
entire vertical extent of the domain is shown. Three layers of isotropic
refinement are visible adjacent to the walls on both the lower and the upper
walls.

## Appendix IV: Effect of subgrid-scale model

Fig. 17 compares the predictions of $C_{p}$ upon leveraging the dynamic
Smagorinsky model [38] instead of the dynamic tensor-coefficient model [25].
As reported previously in Agrawal et al. [25], on the lower Reynolds number,
$Re_{L}=2\times 10^{6}$, the predictions of wall-modeled LES using the dynamic
Smagorinsky model converge towards the reference experiments non-
monotonically. However, at the $Re_{L}\uparrow$ up to $Re_{L}=5\times 10^{6}$,
the predictions approach the solution monotonically. For both Reynolds
numbers, the “fine" grid is required for wall-modeled LES to match the
experimental measurements, implying that the effect of the subgrid-scale model
is minimal in improving the Reynolds number scaling of grid points required in
wall-modeled LES of this flow.

Figure 17: The streamwise distribution of the surface pressure coefficient,
$C_{p}$ for (a) $Re_{L}=2\times 10^{6}$ and (b) $Re_{L}=5\times 10^{6}$ for
the flow over the Boeing speed bump using the dynamic Smagorinksy model and
the equilibrium wall model. Note that across the two Reynolds numbers, the
nominal “coarse", “medium" and “fine" grids are scaled in their resolution by
a factor of $Re_{L}^{2/3}$. The vertical bars in the simulations represent the
root mean square error around the mean value of $C_{p}$.

## Acknowledgments

This work was supported by NASA’s Transformational Tools and Technologies
project under grant number
#80NSSC20M0201. Computing resources were awarded through the Oak Ridge
Leadership Computing Facility (DoE ALCC). We thank Michael P. Whitmore for the
helpful comments on the initial version of this manuscript. We also
acknowledge fruitful discussions with Ahmed Elnahhas, Kevin Griffin, and
Christoffer Hansen.

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