# Sensitivities of Free-Air RANS and DDES Methods on the High-Lift NASA CRM

Zauner M.111Aerospace Project Research Associate, JAXA Aviation Technology
Directorate, Aircraft Lifecycle Innovation Hub. Corresponding author,
<EMAIL_ADDRESS>Japan Aerospace Exploration Agency, Chofu, Tokyo,
182-8522 Japan  Sansica A.222Researcher, JAXA Aviation Technology Directorate,
Aircraft Lifecycle Innovation Hub<EMAIL_ADDRESS>Japan Aerospace
Exploration Agency, Chofu, Tokyo, 182-8522 Japan  Matsuzaki T.333Senior
Researcher, FMIC R&D INC., Engineering Department<EMAIL_ADDRESS>Japan Aerospace Exploration Agency, Chofu, Tokyo, 182-8522 Japan  Lusher D.
J.444Researcher, JAXA Aviation Technology Directorate, Aircraft Lifecycle
Innovation Hub<EMAIL_ADDRESS>Japan Aerospace Exploration Agency,
Chofu, Tokyo, 182-8522 Japan  and Hashimoto A.555Manager, JAXA Aviation
Technology Directorate, Aircraft Lifecycle Innovation Hub,
<EMAIL_ADDRESS>Japan Aerospace Exploration Agency, Chofu, Tokyo,
182-8522 Japan

###### Abstract

To reduce the time-to-market of future aircraft, it is crucial to predict the
flight envelope accurately before building prototypes for flight tests. The
High-Lift Prediction Workshop (HLPW) series aims to assess the numerical
prediction capability of current CFD technology considering NASA’s high-lift
version of the Common Research Model (CRM-HL). The present work contributes to
these collaborative efforts, quantifying sensitivities for RANS-based steady,
unsteady, and hybrid RANS/LES scale-resolving approaches. Uncertainties
associated with the choice of turbulence model, initialization strategies,
grid resolution, and iterative convergence at free-air conditions are covered.
Near stall, a large spread of RANS results was observed for different
turbulence models and initialization strategies, while iterative convergence
appeared less crucial for the present simulations. Steady and unsteady RANS
simulations were unable to predict the correct flow physics near $C_{L,max}$,
even for large grids. Delayed Detached Eddy Simulations (DDES), however,
showed good accuracy compared with wind-tunnel experiments and predicted
$C_{L,max}$ with an error of around $5\%$. Compared to steady RANS, the
computational cost of DDES was a factor of ten higher. Lessons learned and
potential best-practice strategies are shared to aid future studies. While
warm-started RANS simulations using SA models are recommended at lower angles
of attack, scale-resolving methods are required near stall.

## Nomenclature

(Nomenclature entries should have the units identified)

$\alpha$ = | Angle of Attack (AoA)
---|---
$C_{p}$ = | Wall-pressure coefficient
$C_{f}$, $C_{f,x}$ = | Skin-friction coefficient magnitude and x-component
$C_{L}$, $C_{D}$, $C_{M}$ = | Lift-, drag-, and pitching-moment coefficient
$\overline{q}$ = | Time-averaged quantities
$C_{L,max}$ = | Lift coefficient near stall
$M=U^{*}_{\infty}/a^{*}_{\infty}$ = | Mach number
$\Delta\tau=\Delta\tau^{*}c^{*}_{mac}/U^{*}_{\infty}$ = | time step
$St$ = | Strouhal number, $f^{*}c^{*}_{mac}/U^{*}_{\infty}$
$CTU$ = | Convective time unit, $c^{*}_{mac}/U^{*}_{\infty}$

##  1 Introduction

The majority of commercial airplane accidents happen during the low-speed
phases of the flight envelope (24% during final approach, 30% during landing,
and 18% during take-off and initial climb [1]). It is therefore crucial to
guarantee that sufficient lift force is generated during these low-altitude
flight phases, even at off-design conditions with high-lift devices engaged,
which increase the geometrical complexity of the aircraft significantly. The
lift generation at low speeds is typically limited by the maximum lift
coefficient ($C_{L,\textrm{max}}$) and its corresponding critical angle of
attack, beyond which stall-related phenomena detrimentally impact the
aerodynamic performance and handling of the aircraft. At present, around two
thirds of total test flights for certification are performed at low speeds.
With the aim of reducing the amount of expensive flight testing required, it
is essential to demonstrate that numerical tools are reliable enough to
predict the aerodynamic performance at low speeds and increased angles of
attack. As a measure of accuracy, Clark et al. [2] suggest that deviations of
aerodynamic coefficients between Computational Fluid Dynamics (CFD) and
experiment should satisfy $\Delta C_{L}\leq 0.03$ and $\Delta C_{D}\leq 0.001$
near $C_{L,max}$ for acceptable agreement. While meeting these criteria is
certainly a considerable challenge for current CFD solvers, substantial
efforts are being made in both academic and industrial settings to assess and
improve the accuracy of state-of-the-art CFD methods. The National Aeronautics
and Space Administration (NASA) and Boeing jointly proposed the CFD Vision
2030 [3, 4] and Certification by Analysis (CbA) [5] reports and the AIAA
Recommended Practice for supporting aircraft certification through simulations
[6], where areas of focus to improve state-of-the-art CFD solvers are
documented. While CFD tools are well calibrated in very small regions of the
flight envelope, their extension to other design and off-design conditions is
key for maximizing the effectiveness of numerical simulations during design
and certification phases.

One example of an international collaboration bringing together numerous
world-leading aerospace research institutions, universities, and commercial
companies for accurate CFD-based prediction of the low-speed flight regime is
the NASA led AIAA CFD High Lift Prediction Workshop (HLPW)
[https://hiliftpw.larc.nasa.gov/]. Across all workshop editions, widely-used
Reynolds-Averaged Navier-Stokes (RANS) methods have been demonstrated to fail
near $C_{L,\textrm{max}}$ [7]. Furthermore, due to experimental unknowns and
uncertainties as well as simplifications in the numerical setup (i.e.
selection of inflow/outflow and wall boundary conditions), the inclusion of
wind-tunnel walls [8] was also shown not to improve the quality of the RANS
results.

Recently, Kiris et al. [9] performed an extensive and broad parametric study,
employing both RANS methods and higher-fidelity approaches, such as wall-
modeled Large Eddy Simulations (WM-LES), which were shown $11-12$ times more
expensive than conventional RANS methods. Theoretically, hybrid RANS/Large
Eddy Simulation (HRLES) allow significantly larger time steps compared to WM-
LES and could deliver results of similar accuracy at cheaper costs. In the
extensive computational study of Browne et al. [10], it was clearly shown that
HRLES methods overcome some of the RANS shortcomings. However, the authors
also showed that HRLES can be rather sensitive to grid resolution, which often
leads to spurious flow separation reminiscent of conventional RANS solutions.
Grids consisting of approximately 400 million points (denoted as Level C grids
at HLPW4) are of great interest for practical application, but were shown to
be insufficient for the structured over-set grids used by [10]. While HRLES
performed on meshes containing 571 million points (Level D grids) yielded very
promising results, we have to acknowledge that their grid size can exceed
those required by WM-LES. As scale resolving simulations, such as HRLES, are
sparse for complex configurations like the CRM-HL, it is important to gain
more experience to improve best practises. Eventually, this will help to
better understand grid-requirements and numerical settings that lead to
reduced costs and uncertainties.

Mavriplis et al. [11] performed RANS and HRLES for the present test case
considering an adaptively refined Cartesian mesh. The latter approach was
limited to only two angles of attack. The extensive work of Duensing et al.
[8] and Browne et al. [10] considered structured over-set grids for RANS,
URANS, and HRLES methods. In the present contribution, we extend their study
using unstructured meshes. We are particularly interested to understand, if we
encounter for Delayed Detached-Eddy Simulations (DDES) using unstructured
meshes similar limitations as [10] for Zonal Detached-Eddy Simulations (ZDES)
using over-set grids. While [10] considered Spalart-Allmaras (SA) turbulence
models only, we also test a Shear Stress Transport (SST) model. Since previous
work considered time histories below $80$ Convective Time Units (CTUs), we
increase the simulation time for selected cases to $>160$ CTUs, which allows
us to better characterize low-frequency phenomena. For RANS and URANS
simulations, we assess turbulence models and initialisation strategies over
the widest range of parameters and sensitivities to date. Eventually, the
present study and learned lessons allow us to define recommended practises for
high-lift predictions of aircraft models.

The paper is structured as follows: geometry, flow conditions, and
experimental datasets are briefly outlined in section 2. The numerical solver,
methods, and setups are described in section 3, where an iterative convergence
criterion as well as different boundary condition and initialization
strategies are reported. Results obtained by RANS/URANS and HRLES simulations
are examined in section 4 and best-practices/lessons learned are discussed in
section 5. Final conclusions are summarized in section 6. A comprehensive
description of the database for RANS and URANS is given in the Appendix.

##  2 Geometry, Flow Conditions and Experimental Dataset

The geometry and flow conditions of interest are based on the $4^{th}$ AIAA
CFD High-Lift Prediction Workshop (https://hiliftpw.larc.nasa.gov/index-
workshop4.html). The aim of this workshop was to assess the capabilities of
state-of-the-art CFD solvers for medium-to-high-aspect ratio wings on full-
aircraft configurations at landing/take-off (high-lift) conditions. The High-
Lift configuration of the NASA Common Research Model (CRM-HL) is analysed at
representative corrected (accounting for wind-tunnel confinement effects)
angles of attack of $\alpha=7.05^{\circ}$, $17.05^{\circ}$, $19.57^{\circ}$,
and $21.47^{\circ}$. Reference Mach and Reynolds numbers (based on mean
aerodynamic chord and free-stream quantities) are $M_{\infty}=0.2$ and
$Re=5.49$ million, respectively. The main components of the CRM-HL model
considered here are shown in Fig 1(a) and consist of: a semi-span wing with
extracted flaps and slats, flap track fairings and slat brackets, and a
nacelle/pylon equipped with a chine. Only nominal $40^{\circ}/37^{\circ}$
inboard/outboard trailing-edge flap and $30^{\circ}$ slats deflections are
considered. A large experimental database is provided for comparisons by Evans
et al. [12], which includes polars of aerodynamic forces, oil flow
visualizations, and sectional pressure distributions at several spanwise wing
cross-sections denoted by letters $A$-$H$ in Fig 1(b).

a) | b)
---|---
|
Figure 1:  (a) Side-view of the CRM-HL configuration. (b) Top-down view with
indication of cross-sections equipped with pressure tabs taken from the HLPW-4
webpage.

##  3 Numerical Methodology

###  3.1 Simulation Setup

For all simulations presented in this work, JAXA’s unstructured flow solver
FaSTAR [13, 14] is employed. The solver applies a node-centered finite volume
approach, with the Harten-Lax-van Leer-Einfeldt-Wada (HLLEW) scheme[15] being
used to accurately compute advective terms. Additionally, a
Green–Gauss/Weighted-Least-Squares (GLSQ) method[16] is used for gradient
computations. For stabilisation, a Hishida (van Leer-type) slope limiter and
U-MUSCL scheme [17] are applied. Steady solutions are obtained by using the
Lower/Upper Symmetric Gauss-Seidel (LU-SGS) [18] time-stepping scheme with a
Courant–Friedrichs–Lewy (CFL)-fixed local time step. All calculations were
performed with a CFL number of 10. For the unsteady calculations, dual-time
stepping [19] is used to improve the accuracy of the implicit time integration
method. The LU-SGS scheme is used for the pseudo-time sub-iterations, with the
physical time derivatives being approximated by a three-point backward
difference. For hybrid RANS/LES, a non-dimensional time step of
$\Delta\tau=\Delta\tau^{*}c^{*}_{mac}/U^{*}_{\infty}=0.00036$ CTUs has been
used for all simulations, where $U^{*}_{\infty}$ and $c^{*}_{mac}$ denote the
free-stream velocity and aerodynamic mean chord, respectively. For URANS
computations, the time-step was set to
$\Delta\tau=\Delta\tau^{*}c^{*}_{mac}/U^{*}_{\infty}=0.0002$ CTUs to ensure
numerical robustness on the given RANS grid. Details will be follow later.
FaSTAR has been validated over numerous years and on many different flow
configurations, and has shown excellent agreement with other comparable
industrial solvers [20, 21, 22, 23]. Both RANS and hybrid RANS/LES approaches
are used to assess the solver’s capabilities for different levels of fidelity.
The main modeling features of both approaches are described as follows:

  * •

RANS and Unsteady RANS:

  * Several turbulence models are available in FaSTAR to close the averaged Reynolds stresses. In addition to the classic Spalart-Allmaras turbulence model with zero $f_{t2}$ tripping term (SA-noft2) [24], variants considering rotation (SA-noft2-R) [25] and rotation-curvature (SA-noft2-RC) [26] corrections have also been assessed in this work. The effects of the quadratic constitutive relation 2000 version applied to the SA-noft2-R model (SA-noft2-R-QCR2000) [27] have also been investigated. The two-equation $k-\omega$ Shear Stress Transport turbulence model in its standard form (SST), as well as the 2003 updated version (SST-2003) [28] with controlled decay (SST-2003sust) [29] or with additional vorticity source terms (SST-V) have also been considered.

  * •

Hybrid RANS/LES:

  * In the context of this work, the RANS/URANS framework described above is referred to as a "low-fidelity" method, in contrast to the higher fidelity offered by the hybrid RANS/LES approach. In the present study, the SA-noft2-R based Delayed Detached Eddy Simulation (DDES)[30] hybrid method is used. The turbulence length scales are estimated by using the wall distance in the RANS branch and the local grid scale in the LES branch. The transition from the RANS region to the LES region is determined based on the local grid scale, wall distance, and the local state of the degree of turbulence through the eddy viscosity and the deformation tensor.

###  3.2 Numerical Domains and Grids

Different sets of numerical grids have been used to target low (RANS/URANS)
and high (DDES) fidelity calculations. Categorized by the fidelity of the
method, the numerical grids are available on the HLPW-4 website
(https://hiliftpw.larc.nasa.gov/Workshop4/grids_downloads.html) and are
further described below:

  * •

RANS/URANS grids: the JAXA committee-provided C-level grid (240-JAXA-
unstructured/240-Config_CRM-HL_40-37_Nominal/240.C) consists of $86$ million
nodes and $208$ million cells. It is selected as our ‘Standard RANS grid’. To
perform a refinement study, the standard RANS grid was uniformly refined by
reducing the grid spacing by $\approx 33\%$ in each direction, resulting in a
$D$-level grid comprised of $182$ million nodes. Both unstructured hybrid
grids are composed of hexahedra, prisms, pyramids, and tetrahedra and were
generated using an unstructured mesh generator named Mixed-Element Grid
Generator in 3 Dimensions (MEGG3D)[31, 32, 33].

  * •

Hybrid RANS/LES: The C-level ANSA grid family for free-air simulations
(103-ANSA-Unstructured-hiA-Yplus1/103-Config_CRM-HL_40-37_Nominal/103.C) has
been selected. The unstructured grid uses a quad dominant surface mesh with
hexa layers and hexa-dominant in the farfield. Constructed for hybrid RANS/LES
calculations, the grid has a $\Delta y^{+}$ at the wall of unity and wake
refinement up to angles of attack around $20^{\circ}$. In total, the mesh
contains $218$ million nodes and $276$ million cells.

###  3.3 Boundary Conditions and Initialization Strategies

No-slip velocity and adiabatic temperature conditions are enforced on the
aircraft walls. Far-field boundary conditions are employed at the lateral
boundaries, and the angle of attack $\alpha$ is applied at the inflow of the
numerical domain. Only half of the aircraft is simulated to reduce
computational cost, with a symmetry condition applied on the longitudinal
center plane. Regarding the initialization of the simulations, we can
distinguish between:

  1. 1.

Cold-start: Initializing the flow field with uniform free-stream conditions,
performing linear blending to enforce no-slip conditions at the wall.
Initializing the simulation in this manner may require for some cases reduced
CFL numbers or time steps for the first $\approx 10,\\!000$ iterations, before
resuming to the target set up described in §3.1.

  2. 2.

Warm-start: In this approach, the simulation is restarted from a previous RANS
solution at a lower angle of attack.

The increments for the angles of attack to be considered are selected
according to the experimental data provided for HLPW4, which are summarised in
table 2.

$\alpha$-step |  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---|---|---
angle of attack | $\alpha$ | $0.0^{\circ}$ | $2.78^{\circ}$ | $7.05^{\circ}$ | $11.29^{\circ}$ | $17.05^{\circ}$ | $19.57^{\circ}$ | $20.55^{\circ}$ | $21.47^{\circ}$
experiment [12] | $C_{L}$ | 1.16 | 1.36 | 1.78 | 2.15 | 2.50 | 2.51 | 2.46 | 2.32
| $C_{D}$ | 0.12 | 0.13 | 0.19 | 0.25 | 0.33 | 0.36 | 0.37 | 0.44
| $C_{M}$ | -0.40 | -0.39 | -0.37 | -0.34 | -0.29 | -0.24 | -0.23 | -0.30
Table 2: Angles of attack considered for advancing warm-started RANS
simulations.

It has been shown in previous studies (e.g. [34]) that warm-started
simulations tend to deliver increased lift, as flow separation can be delayed
by restarting the simulation from solutions containing attached flow. Starting
from a low angle of attack of $7.05^{\circ}$, it has been shown to be
sufficient to run simulations for only $10,\\!000$ iterations to obtain an
initial solution for a higher angle of attack (see figure 3 in [35]).
Interestingly, Zauner et al. [35] showed that further increasing the number of
iterations before raising the angle of attack can again lead to pre-mature
flow separation and decreased lift, highlighting the sensitivity of RANS-based
solvers and the need for assessment and publication of best practices in this
field.

###  3.4 Iterative Convergence of the RANS Solutions

Residuals of the conservative variables are often used as a measure of the
iterative convergence for RANS solutions. The maximum of the $L2$-norm in the
entire domain for density, momentum, total energy, and turbulence-model
specific conservative variables can be tracked individually at each iteration
and used to determine when a solution has sufficiently converged and the
simulation is completed. Ideally, we consider RANS solutions to be iteratively
converged when all residuals reach machine-precision zeros. In practice,
however, it can be extremely difficult to reach sufficiently low residuals
when complex geometries such as the CRM-HL are considered. Although increased
grid resolution and quality of the CAD models, longer simulation run-time, and
improved convergence methods can help in lowering the residuals, we may still
not reach sufficient convergence due to modelling limitations and errors (e.g.
flow unsteadiness). For industrial applications, however, machine-zero
precision convergence may not be needed as long as residuals are reasonably
low in key-regions of interest and aerodynamic coefficients exhibit minimal
variation. Alternative more relaxed criteria can therefore be decided a priori
to determine a satisfactory level of iterative convergence for the solution.

For the present CRM-HL test case, we are mainly interested in the aerodynamic
performance of the entire aircraft as a whole. The present measure for
convergence is inspired by NASA’s approach of using the Root Mean Squared
($rms$) values of $C_{D}$ fluctuations[8], but is extended to also include
both $C_{L}$ and $C_{M}$ normalised by their mean values. It is defined as

$CPA=\sqrt{\left(\frac{C_{L,rms}}{\overline{C_{L}}}\right)^{2}+\left(\frac{C_{D,rms}}{\overline{C_{D}}}\right)^{2}+\left(\frac{C_{M,rms}}{\overline{C_{M}}}\right)^{2}},$
(1)

where mean values of the considered aerodynamic coefficients are assumed to be
non-zero. This coefficient is calculated over the final $10,000$ iterations of
each simulation and is used as a measure of the iterative convergence of the
RANS solution. For completeness, we note that we typically use the logarithm
of this parameter, denoted as $\log CPA$. FaSTAR typically provides histories
of aerodynamic coefficients up to a precision of 5 significant digits, which
is sufficient for present simulations targeting $CPA<10^{-4}$.

##  4 Results

###  4.1 Sensitivities of Steady RANS Methods

In this section RANS simulations are performed to first assess sensitivities
to turbulence modelling, before studying the influence of different
initialisation strategies. Comments are also made on convergence properties
for different test cases, and grid sensitivities are reported at the end of
this sub-section.

####  4.1.1 Sensitivity to the Choice of Turbulence Model

a)

b)

c)

Figure 2: (a) $C_{D}$, (b) $C_{L}$, and (c) $C_{M}$ over $\alpha$ for selected
RANS turbulence models. Black symbols and curves respectively denote
experimental measurements and $\pm 5\%$ errors.

Fig 2 shows (a) drag, (b) lift, and (c) moment coefficients for selected
turbulence models as a function of the angle of attack. Black symbols denote
experimental measurements and black dashed lines correspond to $\pm 5\%$ error
margins. For $\alpha\leq 7.05^{\circ}$, all considered turbulence models
deliver consistent results, where lift and drag stay within $5\%$ error
compared to experiments. Most turbulence models overestimate the moment
coefficient already at reduced angles of attack. Further increasing the angle
of attack leads to all turbulence models significantly over-predicting moment
and drag coefficients, while lift is consistently under-predicted.

a)

b)

c)

d)

Figure 3:  Relative errors and (CPA) comparing RANS results with respect to
experimental data [12]. Colours correspond to lines in Fig 2.

Corresponding relative errors are quantified in Fig 3 for selected turbulence
models. Cyan, blue, and green lines correspond to simulations using Spalart-
Allmaras (SA) models with and without rotation correction, and with rotation
and QCR corrections, respectively. Red and orange curves denote our baseline
and 2003 versions of Menter’s Shear-Stress Transport (SST) models,
respectively. We can see peaks in relative errors with respect to experimental
measurements near $C_{L,max}$ for all aerodynamic coefficients shown in sub-
figures (a-c). The convergence parameter shown in Fig 3(d) indicates
convergence problems with increasing angels of attack. We remind the reader
that based on previous experience $\log CPA<-4$ would be favourable (§3.4).
This is satisfied at reduced angles of attack, where $CPA$ values compare well
with the standard deviation of $C_{D}$ divided by their mean value, reported
in [8]. However, for such complex geometries grid-convergence (i.e. obtaining
low $CPA$ values) is challenging. In particular, SST models and the SA models
with `QCR` corrections show problems with iterative convergence. We will
return to discussions on iterative convergence later in this sub-section.
First, we want to look at some flow visualisations to understand the origin of
differences in aerodynamic coefficients.

a) SA-noft2

b)

c)

d) SA-noft2-R-QCR

e)

f)

j) SST-2003

k)

l)

Figure 4:  $C_{f}$-contours for cold-started RANS using (a-c) SA-noft2, (d-f)
SA-noft2-R-QCR, and (g-i) SST-2003 turbulence models. Columns (left-to-right)
correspond to $\alpha=7.05^{\circ}$, $17.05^{\circ}$, and $19.57^{\circ}$.
Iso-curves denote $C_{f,x}=0$.

Fig 4 shows skin-friction contours for cold-started RANS simulations using
(a-c) SA-noft2, (d-f) SA-noft2-R-QCR, and (g-i) SST-2003 turbulence models.
Columns correspond to angles of attack of $7.05^{\circ}$ (left-hand-side),
$17.05^{\circ}$ (middle), and $19.57^{\circ}$ (right-hand-side), where
$C_{L,max}$ is expected just before the latter of these angles of attack.
While cold-started simulations indeed suggest maximum lift around these
angles, we can already observe for $\alpha=17.05^{\circ}$ significant regions
of separated flow that are not present in the experimental measurements. While
such non-physical large coherent separation regions appear mainly at the mid-
chord location for $\alpha=17.05^{\circ}$, they can also be observed for the
baseline SA model near the wing tip. At $\alpha=19.57^{\circ}$, flow
separation appears for all turbulence models also in the outboard wing region.
Regarding the observed differences in the moment coefficient, it is also
important to note significant regions of flow separation on the wing flaps
even at low angles of attack ($\alpha=7.05^{\circ}$). It should be noted that
due to the strong flap deflection with respect to the free-stream, the flaps
experience high aerodynamic loads. At increased angles of attack, flow
separation along the flaps weakens and is mainly pronounced near the wing
bend. Near the wing root, all turbulence models appear to produce consistent
behavior, while the SST model predicts reduced flow separation on top of the
nacelle. As we will see when warm-starting RANS simulations by incrementally
increasing the angle of attack, significant changes in these regions of
separated flow occur, which also impact the aerodynamic coefficients.

####  4.1.2 Sensitivity to Initialisation Strategies

For the sake of conciseness, a subset of three representative turbulence
models are selected here to show sensitivities to initialization strategies:
the baseline turbulence model SA-noft2, which predicted $C_{L,max}$ closest to
experimental measurements, and the SA-noft2-R-QCR and SST-2003 models (which
showed rather poor agreement for cold starts). A complete list of aerodynamic
coefficients including warm-started simulations using other turbulence models
can be found in the appendix in tables 11 and 12.

a)

b)

c)

Figure 5:  (a) $C_{D}$, (b) $C_{L}$, and (c) $C_{M}$ over $\alpha$ for warm-
and cold-started RANS simulations. Black symbols and curves respectively
denote experimental measurements and $\pm 5\%$ errors.

Fig 5 shows (a) drag-, (b) lift-, and (c) moment-coefficients as functions of
$\alpha$ for the selected cold- (blue, cyan, and pink curves) and warm-started
(red, orange, and magenta curves) simulations. Experimental measurements and
the $\pm 5\%$ error margin are denoted by black solid and dashed curves,
respectively. For $7.05^{\circ}<\alpha<21.47^{\circ}$ warm-started simulations
using SA-noft2 (red curves) and SA-noft2-R-QCR (orange curves) turbulence
models show significant improvement for the main aerodynamic quantities. Also
the warm-started simulations using the SST-2003 model perform well in terms of
aerodynamic coefficients. This tendency can be confirmed in figures 6 (a-c),
showing relative errors with respect to experimental measurements. However, at
post-stall conditions, we experience significant convergence problems for the
warm-started simulation using SA-noft2 (denoted by the red curve), which leads
to absence of the pitch-break in Fig 5(c). The SA-noft2-R-QCR turbulence model
shows slightly better convergence characteristics and leads to aerodynamic
coefficients which are in fair agreement with experiments. However, as we
shall see later, these promising trends are rather misleading, as they are a
result of error-cancellation.

a)

b)

c)

d)

Figure 6:  Relative errors and (CPA) comparing RANS results with respect to
experimental data [12]. Colours correspond to lines in Fig 5

a) SA-noft2

b)

c)

d) SA-noft2-R-QCR

e)

f)

g) SST-2003

h)

i)

Figure 7:  $C_{f}$-contours for warm-started RANS using (a-c) SA-noft2, (d-f)
SA-noft2-R-QCR, and (g-i) SST-2003 turbulence models. Columns (left to right)
correspond to $\alpha=17.05^{\circ}$, $19.57^{\circ}$, and $21.47^{\circ}$.
Iso-curves denote $C_{f,x}=0$.

Fig 7 shows warm-started RANS simulation results for the selected turbulence
models over a range of angles of attack of $17.05^{\circ}$ (left-hand-side
plots), $19.57^{\circ}$ (center plots), and $21.47^{\circ}$ (right-hand-side
plots). While skin-friction contour plots (a-c) display results using our
baseline SA-noft2 turbulence model, plots (d-f) show results with rotation and
QCR corrections applied. Plots (g-i) correspond to the SST-2003 model.
Firstly, it should be noted that below $17.05^{\circ}$, we do not observe
significant differences between cold- and warm-started CFD solutions (see Fig
5). Comparing the two SA turbulence models, we obtain good agreement up to
$\alpha=19.57^{\circ}$. Although we observe large outboard flow separation
appearing already at $\alpha=17.05^{\circ}$, we do not observe the massive
secondary flow separation around mid-span (upstream of the out-board flaps),
which is evident in cold-started simulation results in Fig 4. For further
increases of angle of attack, the SA-noft2 turbulence model shows similar flow
topology to cold-started simulations. The SA-noft2-R-QCR turbulence model, on
the other hand, shows massive non-physical flow separation behind the nacelle.
It is interesting to note that the flow-separation on flaps is partly
suppressed by these massively separated flow regions upstream. This is also
the reason why the SA-noft2-R-QCR turbulence model (showing reduced flap
separation), leads to increased values of pitch-up moment. At
$\alpha=17.05^{\circ}$ and $19.57^{\circ}$, RANS results obtained using the
SST-2003 model are also similar to the SA results in Fig 7, and those of
present cold-started simulations. At post-stall condition at
$\alpha=21.47^{\circ}$, however, the region of non-physical flow separation
around mid-span appears for SST results somewhere between those observed for
the SA models shown in Fig 7. This - potentially random - appearance of large
bulks of separated flow near mid span may well be due to convergence problems
at increased angles of attack (i.e. post-stall conditions). However, as we
will show in the following sub-section, improved convergence (despite not
machine-precision converged) has only a limited effect when compared to
uncertainties associated with the choice of turbulence model. While the
simulations using QCR corrections reported in [8] agree well with present
results at $\alpha=19.57^{\circ}$, their baseline SA model and SST models
deliver qualitatively different results to ours. Rather than a large bulk
region of flow separation near the tip, they show more localised triangular-
shaped patches of flow separation. The simulations of [8] also show more
sensitivity of the wing-root region to the turbulence models applied, whereas
the present results are more consistent in that regard. For all turbulence
models at $\alpha\leq 19.57^{\circ}$, [8] shows large flow separation on the
nacelle similar to Fig 7(f), which also differs from current observations. At
$\alpha=21.47^{\circ}$, simulations using QCR corrections in [8] also show
massive flow separation near the wing root, which is not observed in present
simulation results. Based on this comparison of simulation results using grids
of similar size (Type C grids), there seems to be significant sensitivity of
turbulence models to the grid topology and local details of resolution.

####  4.1.3 Assessment of Iterative Convergence

Up until this point, we have only looked at the convergence parameter $CPA$,
which is based on fluctuations of the aerodynamic coefficients. At increased
angles of attack we have found some cases that had a less than optimal
$CPA>10^{-4}$, which raises questions about the trustworthiness of the results
in that regime.

a) $\alpha=7.05^{\circ}$

b) $\alpha=17.05^{\circ}$

c) $\alpha=19.57^{\circ}$

d) $\alpha=21.47^{\circ}$

Figure 8:  Coloured lines show $C_{M}$ as a function of scaled iterations.
Dark and light grey curves showing residuals of $u$-velocity and turbulent
terms, respectively.

Coloured curves in Fig 8 correspond to the left-hand-side $y$-axis and show
the moment coefficient as a function of iterations normalised by the total
number of iterations of each corresponding RANS simulation at different angles
of attack. Dark and light grey curves correspond to the right-hand-side scale
showing residuals of the u-velocity component (representative of convective
terms) and turbulent terms, respectively. For brevity, we show only results
obtained with SA turbulence models. While we observe significant differences
in the $C_{M}$ curves and CPA values (corresponding values are summarised in
the tables 11 and 12 of the appendix), residuals reach a similar order of
magnitude for all simulations. An attempt to increase the CFL number lead to
the sudden increase in residuals of the u-velocity component in Fig 8(a) at
scaled iteration of approximately $0.65$. After resuming with $CFL=10$, the
residuals dropped to their original level after scaled iterations of
approximately $0.80$. For $CPA>10^{-4}$, oscillations of the $C_{M}$ curves in
Fig 8 are clearly evident. However, despite these oscillations, the mean
values remain largely the same. Therefore, it is questionable whether improved
convergence would correct the non-physical phenomena present in our results.

In order to address this question directly by improving the convergence
characteristics, we applied a Selective Frequency Damping (SFD) technique [36,
37, 38] to the steady RANS simulations using the SA-noft2 turbulence model.
Results are summarised in table 3 showing that even though the oscillations of
aerodynamic coefficients decreased significantly for results using SFD, their
mean values changed only marginally. It should also be noted that the
effectiveness of the present SFD approach depends on several parameters and
may not always be successfully in improving the convergence characteristics.
Additionally, while the oscillation amplitudes of aerodynamic coefficients and
residuals are relatively low, SFD solutions do not reach machine-precision
convergence. Therefore, we cannot conclude entirely whether RANS results would
(theoretically) change when reaching machine precision, but we have
demonstrated that they are only marginally affected by the better convergence
levels provided by SFD. Further detailed analysis of the effect SFD has on
steady RANS simulations is out of scope of the present contribution and will
not be considered any further.

| SA-noft2 | SA-noft2 with SFD
---|---|---
AoA | $\log({CPA})$ | $\log(R_{turb})$ | $\overline{C_{L}}$ | $\overline{C_{D}}$ | $\overline{C_{M}}$ | $\log({CPA})$ | $\log(R_{turb})$ | $\overline{C_{L}}$ | $\overline{C_{D}}$ | $\overline{C_{M}}$
07.05 | -5.69 | -5.72 | 1.79 | 0.188 | -0.362 | -5.89 | -5.72 | 1.79 | 0.188 | -0.363
17.05 | -3.91 | -5.41 | 2.31 | 0.369 | -0.143 | -5.87 | -5.39 | 2.31 | 0.368 | -0.146
19.57 | -4.32 | -5.33 | 2.30 | 0.420 | -0.063 | -5.68 | -5.31 | 2.30 | 0.419 | -0.066
21.47 | -2.95 | -5.30 | 2.14 | 0.453 | -0.026 | -5.61 | -5.28 | 2.20 | 0.444 | -0.057
Table 3: Summary of cold-started RANS results using SA-noft2 turbulence model
with and without Selective Frequency Damping (SFD).

####  4.1.4 Grid-convergence Study

Before concluding this sub-section, it is also important to briefly address
sensitivity to the grid resolution for the cold-started RANS simulations.
Using the SA-noft2 turbulence model, additional simulations were carried out
on refined grids at $\alpha=7.05^{\circ}$ and $17.05^{\circ}$. According to
the HLPW-4 guidelines, our ‘Standard RANS grid’ is classified as a $C$-level
grid, while a ‘Refined RANS grid’ is denoted as a $D$-level RANS grid. Even
though the ‘DDES grid’ (provided by ANSA for the HLPW-4) is classified as a
$C$-level grid, it is significantly larger compared to the RANS grids and will
be also used later for unsteady hybrid RANS simulations in §4.3.

Grid | Grid nodes | AoA | $\log({CPA})$ | $\log(R_{turb})$ | $\overline{C_{L}}$ | $\overline{C_{D}}$ | $\overline{C_{M}}$
---|---|---|---|---|---|---|---
Standard RANS grid | $85,\\!969,\\!733$ | $7.05^{\circ}$ | -5.69 | -5.72 | 1.79 | 0.188 | -0.362
Refined RANS grid | $181,\\!692,\\!364$ | $7.05^{\circ}$ | -4.73 | -5.44 | 1.79 | 0.188 | -0.361
DDES grid | $218,\\!077,\\!184$ | $7.05^{\circ}$ | -4.26 | -5.67 | 1.78 | 0.187 | -0.361
Standard RANS grid | $85,\\!969,\\!733$ | $17.05^{\circ}$ | -3.91 | -5.41 | 2.31 | 0.369 | -0.143
Refined RANS grid | $181,\\!692,\\!364$ | $17.05^{\circ}$ | -4.70 | -5.55 | 2.31 | 0.376 | -0.140
DDES grid | $218,\\!077,\\!184$ | $17.05^{\circ}$ | -3.97 | -4.92 | 2.47 | 0.357 | -0.246
Table 4: Summary of cold-started RANS results using SA-noft2 turbulence model
for standard and refined grids.

Aerodynamic coefficients and convergence characteristics are summarised for
all three grids in table 4, considering two angles of attack. Using JAXA’s in-
house tool MEGG3D[31, 33], the ‘Standard RANS grid’ was uniformly refined in
each direction by about $33\%$. Despite the ‘Refined RANS grid’ containing
twice as many grid nodes compared to the standard RANS grid, we do not observe
significant differences in the solution for both angles of attack.

When considering steady RANS results using the DDES grid, however, we observe
significant differences in the results at $\alpha=17.05^{\circ}$, leading to
aerodynamic coefficients close to experimental measurements.

a)

b)

Figure 9: Contours showing skin-friction coefficient $C_{f}$ for cold-started
RANS on the (a) refined RANS grid and (b) a DDES grid at
$\alpha=17.05^{\circ}$, using SA-noft2. Black and red lines indicate locations
of cross-sections plotted in figures 23 and 23 of the appendix §6.2

Fig 9 shows the cold-started RANS results at $\alpha=17.05^{\circ}$ for the
(a) ‘Refined RANS grid’ and (b) ‘DDES grid’. While we observe for both grids
large non-physical flow separation near the wing tips, the flow is notably
different near the nacelle and near the end of the out-board flaps, where we
do not observe the second large region of out-board separation. For brevity,
figures showing details of the grids at cross-sections corresponding to black
(Slat region) and red (upper crest of nacelle) lines are added to the appendix
in §6.2. Generally speaking, despite different types of grid topologies being
applied, the differences in grid resolutions between the two grids are minor.
However, the leading edge of the slat does have higher resolution in the
streamwise direction for the DDES grid compared to the refined RANS grid. This
affects the evolution of the boundary layer and, subsequently, may lead to the
observed differences in separation phenomena on the main wing (where grid
resolutions are similar). For the nacelle, finer mesh resolution is present
for both the standard and refined RANS grids compared to the DDES grid, which
may also lead to the observed differences in $C_{f}$ contours. The large flow
separation on the nacelle in Fig 9(a) is reminiscent of the results of [8]
using overset grids. It is remarkable how well results for the DDES grid
compare with warm-started RANS results on the standard RANS grid shown before
in Fig 7. Given the relatively minor sensitivity of RANS solutions to
initialization strategies using over-set grids in [8], there is a high chance
of the sensitivity to initial solutions being grid dependent. To strengthen
present conclusions, future studies will apply adaptive mesh refinement
methods to build upon the work in this section in the future.

####  4.1.5 Quantification of Sensitivities

In this section we quantify the assessed sensitivities in terms of variations
of absolute values of aerodynamic coefficients. Details of all present
simulations are provided in the appendix 6.1 in tables 11 and 12. Table 5
lists differences in $C_{L}$, $C_{D}$, and $C_{M}$ for selected angles of
attack of $\alpha=7.05^{\circ}$, $17.05^{\circ}$, $19.57^{\circ}$, and
$21.47^{\circ}$, where $C_{L,max}$ is expected near $\alpha=19.57^{\circ}$.
The first segments of $\Delta$-values shows the spread across different
turbulence models. Considering present results using Spalart-Allmaras
turbulence model (values in brackets in table 5), we can see even at
$\alpha=17.05^{\circ}$ a moderate spread of results. However, we have seen in
previous sub-sections that looking only at popular integral quantities such as
aerodynamic coefficients can be misleading, when concluding on the physical
correctness of results. This becomes particularly evident at
$\alpha=21.47^{\circ}$, where simulation results using SA turbulence models
agree well with each other, but we have shown that they are far from accurate.
Near $C_{L,max}$, we observe a similar spread with respect to turbulence
models and initialization strategies of $\Delta C_{L}\approx 0.25$
corresponding to a variability of $10\%$ with respect to experimental
measurements. In the worst case that means an error of approximately $20\%$,
assuming grid convergence and sufficient iterative convergence. At reduced
angles of attack, however, steady RANS solutions appear fairly consistent in
terms of the choice of turbulence models, while variations remain high for the
choice of initialization. At $\alpha=7.05^{\circ}$, steady RANS predictions
show good convergence characteristics and agree very well among each other as
well as with experimental measurements.

| Turbulence Model | Initialisation | Convergence
---|---|---|---
AoA | $\Delta\overline{C_{L}}$ | $\Delta\overline{C_{D}}$ | $\Delta\overline{C_{M}}$ | $\Delta\overline{C_{L}}$ | $\Delta\overline{C_{D}}$ | $\Delta\overline{C_{M}}$ | $\Delta\overline{C_{L}}$ | $\Delta\overline{C_{D}}$ | $\Delta\overline{C_{M}}$
07.05 | 0.08 (0.07) | 0.01 (0.01) | 0.04 (0.04) | 0.03 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00
17.05 | 0.10 (0.08) | 0.01 (0.00) | 0.10 (0.03) | 0.22 | 0.02 | 0.12 | 0.00 | 0.00 | 0.01
19.57 | 0.27 (0.19) | 0.02 (0.01) | 0.10 (0.04) | 0.24 | 0.04 | 0.15 | 0.00 | 0.00 | 0.01
21.47 | 0.38 (0.05) | 0.06 (0.03) | 0.16 (0.05) | 0.18 | 0.01 | 0.14 | 0.06 | 0.01 | 0.03
Table 5: Sensitivities of RANS results to the choice of turbulence model,
initialisation strategy, and level of iterative convergence.

At reduced angles of attack, we observe remarkably good agreement across
results using different grids in terms of size as well as topology.
Considering §4.1.4 and reference literature, the present work shows that even
careful manual grid refinement struggles to overcome the appearance of non-
physical flow phenomena near $C_{L,max}$. However, given the present results,
it seems that targeted grid refinement can reduce sensitivities to initial
solutions. In the light of [39] on the present test case, adaptive mesh
refinement may reduce sensitivities to initialization strategies and grid
resolution in the future, but uncertainties associated with turbulence models
for physically accurate $C_{L,max}$ prediction remain. Results obtained by
adaptive mesh refinement across different participants of the HLPW-4 appear
fairly consistent, but always lead to questionable ‘pizza slice’ patterns of
flow separation near the wing tip. Further increasing the grid size and the
number of refinement cycles may help to improve the solutions, but compromise
the cost-effectiveness of RANS.

###  4.2 Sensitivities of Unsteady RANS Methods

In this sub-section, we want to determine if unsteady RANS simulations yield
improved results, considering comparable computational resources to steady
RANS simulations. We are particularly interested in the sensitivities of URANS
simulations to initialization strategies and SA turbulence model choices.
However, before we will assess sensitivities to the numerical URANS setup.

####  4.2.1 Sensitivity Study to Time-integration Parameters

The baseline parameter set-up for production runs of the present URANS study
consists of a CFL number of $10$, a time step of $\Delta\tau=0.0002$ CTUs, and
$n_{i}=5$ inner iterations.

CFL | Inner iterations | Time step | $\overline{C_{L}}$ | $C^{\prime}_{L,rms}$ | $\overline{C_{D}}$ | $\overline{C_{M}}$
---|---|---|---|---|---|---
1 | 5 | 0.0002 | 2.54 | 1.03e-3 | 0.389 | -0.207
10 | 5 | 0.0002 | 2.54 | 4.64e-3 | 0.387 | -0.215
50 | 5 | 0.0002 | 2.54 | 4.08e-3 | 0.388 | -0.210
50 | 5 | 0.0002 | 2.54 | 4.08e-3 | 0.388 | -0.210
50 | 5 | 0.0004 | 2.54 | 3.31e-3 | 0.388 | -0.205
50 | 5 | 0.0002 | 2.54 | 4.08e-3 | 0.388 | -0.210
50 | 10 | 0.0002 | 2.55 | 2.53e-3 | 0.386 | -0.218
50 | 20 | 0.0002 | 2.54 | 5.39e-3 | 0.385 | -0.211
50 | 20 | 0.0020 | 2.54 | 1.01e-3 | 0.389 | -0.205
Table 6: Summary of cold-started URANS results using the SA-noft2 turbulence
model for different parameter settings at $\alpha=19.57^{\circ}$. Mean values
are averaged over the final $30,\\!000$ iterations.

Table 6 shows the results of the time-stepping sensitivity study, varying the
CFL number, the maximum number of inner iterations $n_{i}$, as well as the
time step $\Delta\tau$, using the baseline Spalart-Allmaras SA-noft2 model at
$\alpha=19.57^{\circ}$. Given the trend of decaying unsteadiness (as shown
later in Fig 10), simulations in this sensitivity study are only run for
approximately $60,\\!000$ iterations (corresponding to approximately $12-24$
CTUs, depending on the time step). Although oscillation amplitudes vary
between the cases tested, mean values collected for the last $30,\\!000$
iterations ($\approx 6-12$ CTUs) agree very well within a parameter space of
$1<CFL<50$, $5<n_{i}<20$, and $0.0002<\Delta\tau<0.0004$. For the present
standard RANS grid, $\Delta\tau>0.0004$ or $CFL>50$ lead to numerical
instabilities for $n_{i}=5$. Increasing the maximum number of inner iterations
would allow for larger time steps to be taken, but even though mean results do
not seem to be significantly affected, oscillation amplitudes decay even
faster. For the example in table 6, we observe for $n_{i}=20$ and
$\Delta\tau=0.002$ significantly lower values for $C_{L,rms}^{\prime}$
compared to our baseline setup using $n_{i}=5$ and $\Delta\tau=0.0002$.

Based on these sensitivity tests, a baseline set-up for production runs was
selected according to the simulation with maximum unsteadiness (i.e. maximum
$C^{\prime}_{L,rms}$ values), using a CFL number of $10$, a time step of
$\Delta\tau=0.0002$ CTUs, and $n_{i}=5$ inner iterations. Given the increased
cost of URANS relative to RANS without showing significant improvement in
accuracy, the URANS study is limited to the parameters shown here.

####  4.2.2 Sensitivity of URANS to Initialization Strategies and Turbulence
Models

a)

b)

c)

Figure 10:  $C_{L}$ for URANS simulations using (a) SA-noft2, (b) SA-
noft2-R-QCR2000, and (c) SST turbulence models. Solid and dashed lines denote
cold-started and warm-started simulations (only in (b)), respectively.

Fig 10 shows histories of lift coefficient for URANS simulations using (a) SA-
noft2, (b) SA-noft2-R-QCR2000, and (c) SST turbulence models. The bottom and
top horizontal axes correspond to convective time units and iterations
respectively, to allow for cost comparison to RANS666Note that each (time)
iteration requires 5 sub-iterations.. Solid lines denote cold-started
simulations, while dashed lines in (b) correspond to warm-started simulations.
We obtain very good agreement between the cold-started test cases. Simulations
which are warm-started from cold-started RANS solutions in Fig 10(b) (denoted
by dotted curves) seem to be locked on the initial solution and show similar
discrepancies when compared to the RANS study of the previous sub-section. Due
to their poor performance, warm-started URANS simulations have not been
carried out for the other turbulence models. For the remainder of this sub-
section, we will focus on cold-started URANS simulations only. The baseline
Spalart-Allmaras turbulence model `SA-noft2` performs best at pre-stall
conditions based on the cases shown in table 7, whereas the R-QCR correction
leads to increased deviations of $C_{M}$ compared to experiments near
$C_{L,max}$. However, for all present URANS results we shall see that we
encounter similar problems as reported for our best-practice steady RANS.

$\alpha$ | Type | Turbulence model | Initialization | $\overline{C_{L}}/C_{L,exp}$ | $\overline{C_{M}}/C_{M,exp}$ | $\overline{C_{L}}$ | $\overline{C_{D}}$ | $\overline{C_{M}}$
---|---|---|---|---|---|---|---|---
$07.05^{\circ}$ | URANS | SA-noft2 | cold-started | 1.00 | 0.97 | 1.78 | 0.187 | -0.357
$07.05^{\circ}$ | URANS | SA-noft2-R-QCR | cold-started | 0.99 | 0.92 | 1.76 | 0.184 | -0.339
$07.05^{\circ}$ | URANS | SST | cold-started | 0.97 | 0.89 | 1.73 | 0.181 | -0.330
$07.05^{\circ}$ | URANS | SA-noft2-R-QCR | warm-started | 0.97 | 0.89 | 1.73 | 0.180 | -0.331
$17.05^{\circ}$ | URANS | SA-noft2 | cold-started | 0.99 | 0.90 | 2.48 | 0.355 | -0.259
$17.05^{\circ}$ | URANS | SA-noft2-R-QCR | cold-started | 0.98 | 0.83 | 2.45 | 0.350 | -0.240
$17.05^{\circ}$ | URANS | SST | cold-started | 0.97 | 0.80 | 2.43 | 0.345 | -0.234
$17.05^{\circ}$ | URANS | SA-noft2-R-QCR | warm-started | 0.88 | 0.31 | 2.21 | 0.373 | -0.095
$19.57^{\circ}$ | URANS | SA-noft2 | cold-started | 1.01 | 0.86 | 2.54 | 0.386 | -0.207
$19.57^{\circ}$ | URANS | SA-noft2-R-QCR | cold-started | 0.99 | 0.74 | 2.49 | 0.382 | -0.176
$19.57^{\circ}$ | URANS | SST | cold-started | 0.98 | 0.61 | 2.46 | 0.379 | -0.144
$19.57^{\circ}$ | URANS | SA-noft2-R-QCR | warm-started | 0.90 | 0.12 | 2.26 | 0.421 | -0.029
$21.47^{\circ}$ | URANS | SA-noft2 | cold-started | 1.08 | 0.49 | 2.51 | 0.407 | -0.150
$21.47^{\circ}$ | URANS | SA-noft2-R-QCR | cold-started | 1.03 | 0.36 | 2.39 | 0.413 | -0.100
$21.47^{\circ}$ | URANS | SST | cold-started | 0.94 | 0.03 | 2.18 | 0.437 | -0.012
$21.47^{\circ}$ | URANS | SA-noft2-R-QCR | warm-started | 0.85 | 0.23 | 1.98 | 0.442 | -0.066
Table 7: Summary of URANS results for selected turbulence models. Mean values
are averaged over the last 20 CTUs.

a)

b)

Figure 11: Contours showing skin-friction coefficient $C_{f}$ for (a) current
best-practice unsteady (cold-started) and (b) steady (warm-started) RANS
simulations at $\alpha=19.57^{\circ}$, using SA-noft2.

Fig 11 shows a comparison of skin-friction contours for (a) cold-started URANS
(instantaneous snapshot777No statistical data has been collected for present
URANS simulations. Even though oscillations are expected in the wake, due to
the mild temporal variations it is sufficient to show a representative
instantaneous snapshot of the wing surface.) and (b) warm-started RANS
simulations (using our best-practices for each of the respective methods),
with the `SA-noft2` turbulence model at $\alpha=19.57^{\circ}$. Both methods
exhibit non-physical large separation regions at the out-board region near
stall conditions. While URANS is significantly more expensive compared to the
steady RANS, it does not show improvement in resolving the issue of
nonphysical separation. It is interesting to note that steady RANS delivers
similar results (especially near the wing root and nacelle) as the present
URANS approach, for significantly lower costs and complexity.

We have analysed URANS simulations to understand the sensitivity they exhibit
to choice of numerical parameters (see table 6). Present results agree well
with the study of [8], who similarly only reported mild temporal variations of
aerodynamic loads over time. Furthermore, in a similar manner to the present
contribution, the authors of [8] also highlighted the close similarities
between URANS and steady RANS results. Based on these findings, we recommend
cold-started URANS using the SA-noft2 turbulence model as the most reliable
and accurate method, and should be used as a best practice for future studies
on similar configurations to those simulated here. With regards to comparison
between the choice of using either RANS or URANS, unsteady RANS do not lead to
significant improvements compared to steady RANS. While some of the URANS
cases in this study show mild variations in time, detailed assessment of
unsteady flow phenomena will instead be examined in the next section in the
context of hybrid RANS/LES methods where they are more prominent.

###  4.3 Sensitivities of Hybrid RANS/LES (DDES) Methods

Having demonstrated the significant limitations of pure RANS-based methods for
simulating the CRM-HL configuration near stall conditions ($C_{L,max}$), we
now extend the sensitivity study to scale-resolving unsteady methods using
Delayed Detached-Eddy Simulations (DDES), which is a hybrid RANS/LES approach.
Unless stated otherwise, all simulations in this section are performed with
the SA-noft2-R turbulence model and a C-level grid provided by ANSA for HLPW4
(103-Case2a888https://hiliftpw.larc.nasa.gov/Workshop4/grids_downloads.html)
consisting of 218 million nodes (276 million cells). As a first step, we will
investigate unsteady characteristics observed during cold-started DDES. For a
representative test case at an angle of attack of $\alpha=17.05^{\circ}$, we
will compare between cold- and warm-started simulations and comment on the
statistical convergence. We will then focus on sensitivities of the mean-flow
characteristics to initial conditions, and compare results to those presented
in the previous sub-sections and experiments. Eventually, we will investigate
differences between DDES results using SA- or SST-based turbulence models.

$\alpha$ | Init. | Turb. | Sampling Interval | $\overline{C_{L}}/C_{L,exp}$ | $\overline{C_{M}}/C_{M,exp}$ | $\overline{C_{L}}$ | $C^{\prime}_{L,rms}$ | $\overline{C_{D}}$ | $\overline{C_{M}}$
---|---|---|---|---|---|---|---|---|---
7.05 | cold | SA | $10.9<t<47.1$ | 0.92 | 0.81 | 1.64 | 2.97e-3 | 0.167 | -0.299
7.05 | warm | SA | $10.9<t<47.1$ | 0.92 | 0.81 | 1.64 | 4.87e-3 | 0.168 | -0.300
17.05 | cold | SA | $36.3<t<72.5$ | 0.96 | 0.88 | 2.39 | 1.02e-2 | 0.339 | -0.253
17.05 | warm | SA | $36.3<t<72.5$ | 0.97 | 0.87 | 2.42 | 7.03e-3 | 0.336 | -0.250
17.05 | warm | SA | $116.0<t<170.4$ | 0.97 | 0.87 | 2.419 | 6.47e-3 | 0.336 | -0.250
19.57 | cold | SA | $36.3<t<72.5$ | 0.93 | 1.00 | 2.35 | 1.69e-2 | 0.371 | -0.245
19.57 | warm | SA | $36.3<t<72.5$ | 0.94 | 1.01 | 2.36 | 2.49e-2 | 0.373 | -0.248
19.57 | cold | SST | $36.3<t<72.5$ | 0.91 | 0.97 | 2.30 | 2.47e-2 | 0.373 | -0.238
21.47 | cold | SA | $36.3<t<72.5$ | 0.94 | 1.17 | 2.14 | 3.28e-2 | 0.453 | -0.355
21.47 | warm | SA | $36.3<t<72.5$ | 0.95 | 1.12 | 2.17 | 7.88e-2 | 0.454 | -0.340
Table 8: Summary of cold- and warm-started DDES results.

a)

b)

Figure 12: (a) $C_{L}$ histories of cold-started DDES and (b) corresponding
$PSD$ for $0.1<St<1000$, considering a sampling interval starting at the
vertical dash-dotted lines.

Table 8 summarizes simulation details of all DDES carried out in the scope of
this work. For clarity, we show in Fig 12(a) only $C_{L}$ histories of cold-
started DDES at different angles of attack. Comparison to warm-started
histories will be shown later in Fig 13(a) only near $C_{L,max}$ at
$\alpha=17.05^{\circ}$, where main differences occur. For cold-started
simulations in Fig 12(a), we observe the duration of the initial transients
changing with increasing angle of attack. While the initial transient at
$\alpha=7.05^{\circ}$ is relatively short ($\Delta t_{0}\approx 10$ CTUs), it
increases at higher $\alpha$ to $\Delta t_{0}\approx 36$ CTUs. Due to the
presence of large-scale unsteadiness associated with large time scales
relative to the length of the present time signal, it is not clear whether the
flow at $\alpha=21.47^{\circ}$ is already fully developed. It is interesting
to note that simulations of [11] using a different grid show significantly
weaker oscillation amplitudes of aerodynamic coefficients, acknowledging a
limited runtime of $30$ CTUs though. However, as pre-stall conditions are the
primary focus of this work, further discussion of post-stall predictions is
omitted for conciseness.

Fig 12(b) shows Power Spectral Densities (PSD) of the lift histories shown in
Fig 12(a). For all cold-started simulations, the last $36$ CTUs of the signals
are used for the PSD calculation (starting at the black dash-dotted lines in
Fig 12(a)), where statistics of aerodynamic coefficients are sufficiently well
converged. After first subtracting the mean value, all spectra are obtained by
Fast Fourier Transformation applying a single Hanning window to the raw data.
The high-frequency range (defined here as $St>1.0$) of the spectrum agrees
well for all simulations, where we observe spectral peaks at $St\approx 370$,
$170$, and $6.5$. Considering the acoustic analysis of [40]999For comparison,
frequencies in Hertz were computed according to $f=St\cdot
U^{*}_{\infty}/c^{*}_{mac}\approx St\cdot 66.3/0.7$. for a similar
configuration, frequencies at $St>100$ were attributed to the presence of slat
noise. In the present study, we cannot determine the exact physical phenomena
associated with frequencies at $St\approx 6.5$, but it is reasonable to expect
them being related to vortex shedding. The low-frequency part of the spectrum
(defined here as $St<1$) shows more significant variations between the
different angles of attack. At $\alpha>7.05^{\circ}$, we observe spectral
peaks around $St\approx 0.4-0.6$, which are in the range of frequencies
obtained by global stability analysis[41]. Such instabilities could be
associated with unsteady flow-phenomena appearing within the hollow nacelle.
While we observe for $\alpha=17.05^{\circ}$ a sharp spectral peak near
$St\approx 0.2$, it becomes comparatively less pronounced at
$\alpha=19.57^{\circ}$, and eventually becomes a bump for
$\alpha=21.47^{\circ}$. However, at increased angles of attack we observe also
fairly high PSD-levels near cut-off frequencies of $St<0.1$. As the signals of
the cold-started simulations are limited ($\Delta t\approx 36$ CTUs) compared
to the time scales of $\Delta t\geq 10$ associated with low-frequency peaks at
$St\leq 0.1$, it is worth analysing a warm-started DDES near $C_{L,max}$ with
longer runtime.

a)

b)

Figure 13: (a) $C_{L}$-histories of cold- (cyan curve) and warm-started DDES
(black curve) at $\alpha=17.05^{\circ}$. (b) $PSD$ for $0.1<St<1000$,
considering sampling intervals between $36.3<t<72.5$ (red curves) and
$116.0<t<170.4$ (blue curves).

Fig 13(a) shows histories of the lift coefficient $C_{L}$ of cold- (cyan
curve) and warm-started DDES (black curve) at $\alpha=17.05^{\circ}$. For a
comparison of warm- and cold-started results, we consider a time interval
between $36.3<t<72.5$, which is indicated by red vertical dash-dotted lines.
While $\overline{C_{L}}$ at $\alpha=17.05^{\circ}$ is about
$\Delta\overline{C_{L}}\approx 0.03$ lower for cold-started simulations,
oscillation amplitudes remain similar with $\Delta C_{L}\approx 0.04$. We
would like to emphasize that we observe the largest differences between warm-
and cold-started DDES at $\alpha=17.05^{\circ}$. To confirm the better
agreement between cold- and warm-started DDES in terms of statistical
quantities, numerical data is reported in table 8. Looking at spectra over the
same sampling interval in Fig 13(a), however, we note some differences in the
low frequency range. Peaks at $St\approx 0.6$ and $St\approx 0.2$ for the
cold-started DDES (cyan curve in Fig 13(b)) are instead found in the spectra
of the warm-started simulation histories (red curve in Fig 13(b)) at the
slightly lower frequencies of $St\approx 0.5$ and $St\approx 0.15$,
respectively. In the warm-started simulation results these peaks are also less
pronounced. Looking at the black curve in Fig 13(a), we can visually identify
a low-frequency phenomenon at a time scale of $\Delta t\approx 15$ being well
pronounced after $100$ CTUs. Considering a sampling interval covering about
$54$ CTUs indicated by the vertical blue dash-dotted lines, we obtain in Fig
13(b) (blue curve) a spectral peak between $St\approx 0.05-0.06$. We can also
observe at post-stall conditions (red curve in Fig 12(a)) low-frequency
content, even though these oscillations show intermittency and irregular
behaviour in terms of amplitude and occurrence. However, further analysis
(e.g. modal decomposition) is required to confirm the physical phenomena
associated with these spectral peaks, which will be the topic of future work.
Here we want to emphasise the need for sufficiently long run times to (a)
overcome the initial transients and (b) capture low-frequency oscillations
near $C_{L,max}$. Even though present simulations are run for long enough to
obtain well-converged mean values for aerodynamic coefficients (compare mean
values for different sampling windows at $\alpha=17.05$ in table 8), low-
frequency oscillations may still develop or become more pronounced for longer
run times. However, present run times of approximately $70$ CTUs may still be
sufficient for practical application to estimate the mean-flow properties near
$C_{L,max}$.

a)

b)

c)

Figure 14: (a) Drag, (b) lift, and (c) moment coefficient for warm- and cold-
started DDES as a function of the angle of attack. Black symbols denote
experimental measurements and black dashed lines correspond to $\pm 5\%$ error
margins.

Fig 14 shows now time-averaged (a) drag, (b) lift, and (c) moment coefficient
for warm- and cold-started DDES as a function of the angle of attack. Black
symbols denote experimental measurements and black dashed lines correspond to
$\pm 5\%$ error margins. At increased angles of attack, the relative
difference between simulation and experimental results remain within $5\%$
even for the moment coefficient. At low angles of attack, however, RANS
results on the same grid (see §4.1.4) appear in better agreement with
experimental data compared to the DDES results. It is important to note that
flaps experience high aerodynamic loads at reduced angles of attack and are
prone to separation, which also has significant effect on the moment
coefficient, and leads to notable deviations from experimental results in Fig
14.

a)

b)

Figure 15: Contours showing time-averaged skin-friction coefficient $C_{f}$
for cold-started DDES at (a) $\alpha=7.05^{\circ}$ and (b)
$\alpha=19.57^{\circ}$. The last 36 CTUs are considered for time-averaging.

Excessive flow separation along the flaps at $\alpha=7.05^{\circ}$ can be
confirmed in the skin-friction contours of Fig 15(a). Adaptively refined
Cartesian HRLES grids of [11] deliver higher lift at the same angle of attack.
Their results show reduced flow separation along the flaps, where their grid
is significantly refined. At an increased angle of attack, our results near
$C_{L,max}$ match well with experiments of [12]. Contrasting RANS solutions,
we do not observe non-physically large bulks of flow separation near the wing
tip in Fig 15(b). Comparison was made between skin-friction contours for both
cold- and warm-started DDES, however, they are omitted here for brevity as we
did not observe any significant differences.

a)

b)

c)

Figure 16: Red curves show wall-pressure $C_{p}$ distributions of cold-started
DDES at $\alpha=19.57^{\circ}$, taken at cross-sections (a) $A$ $\eta=0.15$,
(b) $E$ $\eta=0.55$, and (c) $G$ $\eta=0.82$ according to Fig 1(b). Black
symbols denote experimental data and error bars indicating $10\%$ deviation
from experimental measurements.

Fig 16 shows wall-pressure coefficient as a function of $x$ for cold-started
DDES at $\alpha=19.57^{\circ}$, taken at spanwise locations (a) $\eta=0.15$,
(b) $\eta=0.55$, and (c) $\eta=0.82$. The locations of these cross-sections
are respectively labelled in Fig 1(b) as $A$, $E$, and $G$. Black symbols
denote experimental data and a $\pm 5\%$ error bar. While the pressure side
agrees very well with experiments at all locations, we observe difference on
the suctions side over the slats as well as front sections of main wing and
flaps. Given the deficit in lift prediction, the flow acceleration over the
suction side is under-predicted. Potential reasons for this could be excessive
flow separation, over-predicted boundary-layer thickness, or even transition
effects. There may also be sensitivities of the shielding function, which
controls the switching between RANS and LES schemes based on a characteristic
length scale computed for each cell. In the hybrid RANS/LES study of Browne et
al. [10] using Zonal Detached Eddy Simulation (ZDES), significant grid-
sensitivity was shown near the out-board region (i.e. section G), where
present DDES results show better performance on a level C grid. Near the wing
root (i.e. section A), however, their results do not show significant grid-
dependency. In particular, they do not show a kink in the $C_{p}$
distributions that we observe in Fig 16(a) around $x\approx 1100$. Looking at
skin-friction contours of 15(b), we can observe a pocket of separated flow
behind the slat at $\eta=0.15$. Whether these differences are due to
variations in grid resolution/topology or numerical approach (e.g. shielding
function) is difficult to answer and requires further investigations.

Before summarizing the lessons learnt from this work and suggesting best
practises for future studies, sensitivities of DDES results to the turbulence
model are reported here at an angle of attack of $\alpha=19.57^{\circ}$,
representative of $C_{L,max}$ conditions. Table 8 shows that aerodynamic
coefficients are very similar for cold-started DDES using SA- and SST-based
turbulence models at $\alpha=19.57^{\circ}$.

a)

b)

Figure 17: (a) Histories of the lift coefficient $C_{L}$ of cold-started DDES
and (b) corresponding Power-Spectral-Density $PSD$ for Strouhal numbers
$0.1<St<1000$ using SA- (orange) and SST-based (black) turbulence models. A
single Hanning window is applied to the sampling interval after the end of an
initial transient indicated by vertical dash-dotted lines.

Time histories showing $C_{L}$ for cold-started DDES in Fig 17(a) confirm the
good agreement between SA- (orange curve) and SST-based (black curve) DDES
results. The lift coefficient drops for the SST-based DDES after $\approx 45$
CTUs, leading to an extended initial transient, indicated by the vertical
orange line. Therefore the overall run-time of the simulation has been
extended to maintain a sampling interval of $\approx 36$ CTUs to compute
spectra in Fig 17(b). For frequencies at $St>0.1$, we observe good agreement
between DDES results of both turbulence models. However, due to intermittent
or irregular phenomena in the lift signals of Fig 17(a) after the initial
transients, we observe increased PSD at $St<0.1$. Given the limited runtime of
the corresponding simulations, we cannot trust this part of the spectrum.
However, we have to acknowledge the possibility that for significantly longer
run-times, a low-frequency peak may develop at $St\approx 0.03$.

a)

b)

Figure 18: Contours showing time-averaged skin-friction coefficient $C_{f}$
for cold-started DDES at $\alpha=19.57^{\circ}$ using (a) SST and (b) SA
turbulence models. The last 36 CTUs are considered for time-averaging.

Finally, Fig 18 shows time-averaged $C_{f}$ contours for cold-started DDES at
$\alpha=19.57^{\circ}$ using (a) SST and (b) SA turbulence models. Again we
can see strong similarities between both results. We observe slight difference
in the separation region near the wing root, which may lead to the slight
decrease in $C_{L}$. In contrast to RANS and URANS results in previous
sections, DDES results seem very consistent for the considered flow
conditions, with far less sensitivity to initialization strategy and
turbulence model.

##  5 Recommended Practices and Lessons Learned

Before presenting final conclusions, this section first discusses
computational costs, before summarizing the main lessons learnt from this
study. This includes detailing recommended practices for steady, unsteady, and
hybrid scale-resolving RANS methods in the context of the CRM-HL
configuration.

a)

b)

c)

Figure 19: Contours showing skin-friction coefficient $C_{f}$ for best
practice of (a) RANS (warm-started), (b) URANS (cold-started), and (c) DDES
(cold-started) at $\alpha=19.57^{\circ}$

To emphasize that only scale-resolving hybrid RANS/LES simulations were
capable to show significant improvement in terms of flow physics, Fig 19 shows
skin-friction coefficient $C_{f}$ for best practice of (a) RANS, (b) URANS,
and (c) DDES near $C_{L,max}$ at $\alpha=19.57^{\circ}$, which can be compared
with experimental results in [12]. After commenting on the computational
costs, we will summarize best practises and lessons learned for each approach
individually.

###  5.1 Computational Costs

Simulations in this work were computed on the JAXA Supercomputing System
Generation 3 (JSS3) consisting of $5,\\!760$ standard nodes (Fujitsu PRIME-HPC
FX1000). Each node contains $48$ cores (Fujitsu A64FX) delivering a total peak
performance of $19.4$ PFLOPS. Costs are measured in core hours (coreh),
considering a typical runtime to generate simulation results according to
recommended practises discussed in previous sections. We consider the entire
simulation loop as well as I/O procedures, while pre- and post-processing have
not been included. The aim is to provide an overview of the costs and time
required to generate simulation results for given flow conditions using
different grids and methods presented in this work.

Method | Grid size | Turb. Model | Cores | Core-hours per $2\cdot 10^{5}$ iterations
---|---|---|---|---
RANS | $85,\\!969,\\!733$ | SA | $576$ | $32,\\!000$
RANS | $85,\\!969,\\!733$ | SST | $576$ | $37,\\!000$
Table 9: Computational costs of present steady simulations.

Table 9 summarizes simulation details and corresponding costs for the steady
RANS cases. A typical RANS simulation required $200,\\!000$ iterations to
obtain sufficient convergence (see §4.1). Considering the standard RANS grid
consisting of $85,\\!969,\\!733$ nodes and using $576$ cores ($12$ nodes) on
JSS3 requires for typical RANS jobs about $55$ hours elapsed real time
($\approx 32,\\!000$ coreh) using Spalart-Allmaras turbulence models, while
present SST models are $\approx 16\%$ more expensive. As shown in section
§4.1, iterative convergence can vary for different angles of attack,
turbulence models, or grids and eventually leads to differences in
computational costs, which are not detailed here. However, based on
information provided in table 9, one can scale projected costs accordingly.

Method | Grid size | Turb. Model | Time Step | Cores | Core-hours per $72.5$ CTU
---|---|---|---|---|---
URANS | $85,\\!969,\\!733$ | SA | 0.00020 | $576$ | $300,\\!000$
URANS | $85,\\!969,\\!733$ | SST | 0.00020 | $960$ | $360,\\!000$
DDES | $218,\\!077,\\!184$ | SA | 0.00036 | $1920$ | $320,\\!000$
DDES | $218,\\!077,\\!184$ | SST | 0.00036 | $1920$ | $340,\\!000$
Table 10: Computational costs of present unsteady simulations.

Table 10 summarises the numerical costs for the unsteady (URANS and DDES)
simulations using FaSTAR. For the unsteady methods, we measure the costs in
core-hours per $72.5$ convective time units, which is considered as a minimum
run time to capture unsteady phenomena near $C_{L,max}$. Using a time steps of
$\Delta\tau=0.00036$ and $\Delta\tau=0.0002$ for DDES and URANS, respectively,
run-times corresponds to approximately $200,\\!000$ and $360,\\!000$
iterations (For URANS $5$ sub-iterations per time step are considered). It was
shown in §4.3 that even longer run times may be required to collect sufficient
statistics associated with post-stall low-frequency phenomena at $St\approx
0.1$. The estimates of costs for URANS simulations are based on the numerical
experiment in §4.2 assuming a time step of $\Delta\tau=0.0002$. Based on the
URANS sensitivity study we could increase the time step cutting down costs,
but it has been shown that unsteadiness gets even more damped for increased
$\Delta\tau$ and numerical instabilities may occur. In any case, as shown in
§4.2, URANS simulations did not show significantly improved results compared
to RANS. Computational costs for the DDES are approximately one order of
magnitude higher compared to steady RANS, but lead to significantly improved
results due to the improved fidelity. Only marginal increase of costs is
observed for SST-based DDES. As reported in Fig 18, results between the two
turbulence models were very consistent for the grid tested. It may be
interesting to note that present DDES required only $0.6\%$ of the
computational resources of the JSS3 cluster. Using $5\%$ of the cluster, DDES
results for a given set of flow conditions can be produced within $24$ hours.
Therefore, we want to emphasize the technical feasibility of complementing
RANS simulations with scale-resolving simulations at conditions near stall on
modern clusters.

###  5.2 Steady RANS

Based on present steady RANS results for a given level C grid, it is hard to
conclude on the best choice of turbulence model. No correction of the Spalart-
Allmaras turbulence model lead to significant improvement of solutions. SST
models also showed similar deficiencies as SA models, but were more expensive
and sometimes even led to more difficulties in terms of convergence. As a
result, we consider the baseline Spalart-Allmaras turbulence model without
$f_{t2}$ term for our current best practice recommendations. For all
considered turbulence models, we could already see at $\alpha=17.05$ the
appearance of spurious patches of outboard separation. At increased angles of
attack, we observe additional non-physically large separation zones, which
show high sensitivity to the initial solutions. These simulations were
performed on the standard RANS grid, which has been hand-crafted according to
the HLPW-4 guidelines and best practices. For warm-started simulations, non-
physical flow separation could be delayed to higher angles of attack. Even
though larger grids may lead to reduced sensitivity to initial conditions, due
to their increased costs, such grids would not be practical for real-life
application. Despite increased computational costs, cold-started steady RANS
solutions of significantly refined grids seem to suffer from the same non-
physical flow separation near wing tips as those of warm-started solutions
computed on the standard grid. Therefore, we consider warm-started steady RANS
simulations performed on a $C$-level RANS grid as a recommended practice.
However, we reiterate that all our efforts for steady RANS computations lead
to the appearance of non-physical flow phenomena near $C_{L,max}$, despite
fair agreement with experiments in terms of aerodynamic coefficients.
Therefore, we cannot trust the state-of-the-art RANS methods for $C_{L,max}$
predictions over such a complex aircraft configuration. In the future, better
turbulence models or adaptive mesh refinement techniques may provide solutions
to the present deficiencies observed for steady RANS simulations.

Considering present state-of-the-art methods and turbulence models, RANS can
still be used for high-lift configurations at reduced angles of attack (i.e.
the linear part of the $C_{L}$-$\alpha$ curve). The challenge is to assess
(quantify) the trustworthiness (uncertainty) of results at a given angle of
attack. In other words, the challenge is to identify a certain threshold angle
of attack, beyond which higher-fidelity methods are needed to maintain
sufficient accuracy of results. As an additional indicator of trustworthiness,
one could test two turbulence models (e.g. standard SA with and without R-QCR
corrections) for their consistency. As shown in §4.1, results using different
turbulence models agree well (e.g. $\Delta\overline{C_{M}}<0.05$101010While
$C_{L}$ and $C_{D}$ are less sensitive due to error-compensation, $C_{M}$
appears more sensitive according to table 5.), when errors to experimental
measurements are low (i.e. less than $\pm 5\%$). At increased angles of
attack, where non-physical phenomena appear in RANS solutions, variations
between different turbulence models become significant (e.g.
$\Delta\overline{C_{L}}>0.05$).

###  5.3 Unsteady RANS

Cold-started URANS simulations lead to similar results compared to warm-
started steady RANS simulations. Warm-started URANS simulations, on the other
hand, appeared to be "locked" on their initial (cold-started) RANS solution.
In any case, present URANS simulations lead to similar problems encountered
for steady RANS simulations, while computational costs would notably increase.
While cold-started initialization for URANS is preferred, we recommend the use
of steady RANS methods.

###  5.4 Hybrid RANS/LES

The use of higher-fidelity DDES methods provide a potential solution to the
shortcomings of RANS methods discussed in this work. We observed reduced
sensitivities to the initial solution and turbulence model for present
simulations compared to RANS methods. While aerodynamic coefficients remain
within a $\pm 5\%$ error range with respect to experimental results, regions
of flow separation have been more localised and agree better with experimental
observations. Even though computational costs are a factor of $10$ higher
compared to steady RANS results111111Computational costs of present methods
are summarised in table 10., we still recommend scale resolving methods for
future best practise to predict $C_{L,max}$ more accurately. As SA-based DDES
are cheaper compared to SST-based approaches and deliver results of similar
accuracy, we recommend the SA-noft2-R turbulence model for current best
practise. Despite the promising results for Hybrid RANS/LES methods, there is
still additional work required to assess the sensitivity to grid resolution
and the choice of shielding function leading to excessive flap separation at
reduced angles of attack.

##  6 Conclusions

An extensive sensitivity study has been performed using steady and unsteady
RANS as well as hybrid RANS/LES methods to predict stall for NASA’s high-lift
configuration of the Common Research Model (CRM-HL) at free-air conditions.
The present contribution provides quantitative results for a large number of
simulations testing widely-used turbulence models and initialisation
strategies to the community, to further establish the current state-of-the-art
of RANS-based simulations in terms of accuracy and computational costs. The
present study extends the work of Duensing et al. [8] and Browne et al. [10],
who considered RANS, URANS and Hybrid RANS/LES for structured over-set to
unstructured grids. While [10] limited their ZDES study to SA-based turbulence
models, [8] considered one SST model for their RANS analysis. The present work
tests a significantly wider range of turbulence models including variations of
Menter’s Shear-Stress Transport model for steady and unsteady RANS as well as
DDES. In terms of unsteady flow characteristics, present simulation times are
extended up to $>160$ CTUs, which is more than twice as long as those in [10].
Eventually, the present study and learned learned allow us to define
recommended practises for high-lift predictions of aircraft models.

Near stall ($C_{L,max}\approx 2.5$) at an angle of attack of $\alpha=19.57$, a
large spread of RANS results was observed for different turbulence models
($\Delta C_{L}=0.27$) and initialization strategies ($\Delta C_{L}=0.24$).
Iterative convergence appeared less crucial for present simulations, but the
use of Selective Frequency Damping (SFD) did not deliver machine-precision
convergence of RANS solutions. Even though results agree well with
experimental measurements at low angles of attack, steady as well as unsteady
RANS simulations were unable to predict the correct flow physics at increased
$\alpha$. Non-physical flow separation was observed for all results near the
wing tips even for relatively fine grids. However, we found strong indications
that sensitivities to initialization strategies are grid dependent. Further
grid refinement or adaptive mesh refinement may reduce sensitivity to initial
solutions near $C_{L,max}$, but may not solve issues related to turbulence
models without compromising the cost-effectiveness of RANS[39].

Delayed Detached Eddy Simulations (DDES), showed good accuracy with wind-
tunnel measurements of approximately $\pm 5\%$ near $C_{L,max}$. At reduced
angles of attack, however, present DDES results showed excessive separation
along the flaps leading to discrepancies with experimental measurements
($\overline{C_{M}}/C_{M,exp}\approx 0.8$ at $\alpha=7.05^{\circ}$). Even
though [10] showed that $C_{L}$ approaches experimental values using D level
grids at low angles of attack, they still observe significant discrepancies in
$C_{M}$. Considering SST-based DDES instead of an SA-based approach does not
improve results, but leads to increased costs. Considering different
initialization strategies and turbulence models, it is remarkable how
consistent DDES results are for a given grid, compared to purely RANS-based
methods.

At lower angles of attack, warm-started RANS simulations using SA models are
recommended. Due to the limitations discussed in 4.1, we suggest to consider
scale-resolving methods - such as hybrid RANS/LES - near stall. Even though
the increased fidelity leads to ten times higher costs compared to RANS,
considering modern CPU clusters like JSS3, such simulations can still be
performed within $24$ hours using $5\%$ of the available computational
resources. Future studies will focus on wind-tunnel confinement effects and
the effect of shielding function formulation for hybrid RANS/LES.

## Appendix

###  6.1 Steady RANS

Tables 11 and 12 summarize aerodynamic coefficients and performance parameters
for all steady RANS simulations using Spalart-Allmaras and Shear-Stress-
Transport models, respectively. The convergence parameter CPA together with
the $C_{L}$ and $C_{M}$ ratios of simulation results over experimental values
are chosen as performance indicators. In addition, the aerodynamic
coefficients as well as the L2-norm of residuals of density and model
conservative variable are listed as well. Mean values are averaged over the
last $10,\\!000$ iterations. Figures 20, 21, and 22 show skin-friction contour
plots for all cold-started simulations using Spalart-Allmaras as well as
Shear-Stress Transport turbulence models. These figures complement §4.1 and
confirm that no turbulence model is capable of overcoming the appearance of
non-physical flow separation at increased angles of attack.

Settings | Performance indicators | Aerodyn. coefficients | Residuals
---|---|---|---
AoA | SA corr. | Initial | $\log({CPA})$ | $\overline{C_{L}}/C_{L,exp}$ | $\overline{C_{M}}/C_{M,exp}$ | $\overline{C_{L}}$ | $\overline{C_{D}}$ | $\overline{C_{M}}$ | $R_{\rho}$ | $R_{turb}$
7.05 | - | cold | -5.69 | 1.005 | 0.98 | 1.79 | 0.19 | -0.36 | 7.7e-10 | 1.9e-6
7.05 | - | warm | -5.49 | 1.005 | 0.98 | 1.79 | 0.19 | -0.36 | 7.7e-10 | 1.9e-6
7.05 | R-QCR | cold | -3.26 | 0.97 | 0.90 | 1.73 | 0.18 | -0.33 | 7.3e-11 | 2.6e-6
7.05 | R-QCR | warm | -5.31 | 0.99 | 0.91 | 1.76 | 0.18 | -0.34 | 7.0e-11 | 1.9e-6
7.05 | R | cold | -6.50 | 0.99 | 0.94 | 1.76 | 0.18 | -0.35 | 7.1e-11 | 4.0e-1
7.05 | RC | cold | -3.74 | 1.01 | 1.00 | 1.80 | 0.19 | -0.37 | 1.2e-10 | 2.7e-6
17.05 | - | cold | -3.91 | 0.92 | 0.50 | 2.31 | 0.37 | -0.14 | 7.4e-11 | 3.9e-6
17.05 | - | warm | -3.07 | 0.99 | 0.90 | 2.48 | 0.35 | -0.26 | 6.7e-11 | 5.5e-6
17.05 | R-QCR | cold | -3.24 | 0.89 | 0.38 | 2.22 | 0.37 | -0.11 | 7.5e-11 | 2.7e-6
17.05 | R-QCR | warm | -3.98 | 0.98 | 0.80 | 2.44 | 0.35 | -0.23 | 6.7e-11 | 2.6e-6
17.05 | R | cold | -3.61 | 0.92 | 0.45 | 2.30 | 0.37 | -0.13 | 7.3e-11 | 2.8e-6
17.05 | R | warm | -3.63 | 0.99 | 0.88 | 2.47 | 0.35 | -0.25 | 6.1e-11 | 2.5e-6
17.05 | RC | cold | -2.39 | 0.91 | 0.49 | 2.28 | 0.37 | -0.14 | 9.0e-10 | 1.3e-4
19.57 | - | cold | -4.32 | 0.91 | 0.26 | 2.30 | 0.42 | -0.06 | 8.3e-11 | 4.7e-6
19.57 | - | warm | -5.16 | 1.01 | 0.83 | 2.54 | 0.39 | -0.20 | 7.1e-11 | 4.1e-6
19.57 | R-QCR | cold | -2.35 | 0.90 | 0.12 | 2.26 | 0.42 | -0.03 | 8.9e-11 | 3.9e-6
19.57 | R-QCR | warm | -2.37 | 0.98 | 0.73 | 2.48 | 0.38 | -0.18 | 9.3e-11 | 8.1e-6
19.57 | R | cold | -4.27 | 0.92 | 0.29 | 2.32 | 0.41 | -0.07 | 7.2e-11 | 3.0e-6
19.57 | RC | cold | -1.62 | 0.85 | 0.20 | 2.13 | 0.42 | -0.05 | 1.1e-9 | 1.4e-4
21.47 | - | cold | -2.95 | 0.92 | 0.09 | 2.14 | 0.45 | -0.03 | 6.7e-11 | 5.0e-6
21.47 | - | warm | -0.90 | 0.99 | 0.76 | 2.31 | 0.44 | -0.01 | 7.9e-11 | 5.2e-5
21.47 | R-QCR | cold | -1.95 | 0.84 | 0.22 | 1.94 | 0.43 | -0.07 | 7.2e-11 | 3.9e-6
21.47 | R-QCR | warm | -3.35 | 0.91 | 0.70 | 2.12 | 0.43 | -0.21 | 8.0e-11 | 3.8e-6
21.47 | R | cold | -1.76 | 0.90 | 0.20 | 2.09 | 0.45 | -0.06 | 8.0e-11 | 5.0e-6
21.47 | RC | cold | -2.36 | 0.92 | 0.26 | 2.13 | 0.46 | -0.08 | 8.3e-10 | 1.3e-4
Table 11:  Summary of selected cold- and warm-started results for baseline SA-noft2 and SA-noft2-R-QCR2000 model.  Settings | Performance indicators | Aerodyn. coefficients | Residuals
---|---|---|---
AoA | SST corr. | Initial | $\log({CPA})$ | $\overline{C_{L}}/C_{L,exp}$ | $\overline{C_{M}}/C_{M,exp}$ | $\overline{C_{L}}$ | $\overline{C_{D}}$ | $\overline{C_{M}}$ | $R_{\rho}$ | $R_{turb}$
07.05 | - | cold | -3.41 | 0.97 | 0.89 | 1.72 | 0.18 | -0.33 | 6.9e-11 | 1.6e-9
07.05 | 2003 | cold | -4.39 | 0.97 | 0.89 | 1.72 | 0.18 | -0.33 | 7.8e-11 | 1.6e-9
07.05 | V | cold | -3.19 | 0.97 | 0.89 | 1.72 | 0.18 | -0.33 | 1.2e-10 | 1.6e-9
07.05 | sust | cold | -3.41 | 0.97 | 0.89 | 1.73 | 0.18 | -0.33 | 5.9e-11 | 1.6e-9
07.05 | V-sust | cold | -4.74 | 0.97 | 0.89 | 1.73 | 0.18 | -0.33 | 8.3e-11 | 1.6e-9
07.05 | 2003-sust | cold | -5.17 | 0.97 | 0.89 | 1.72 | 0.18 | -0.33 | 7.7e-11 | 1.6e-9
17.05 | - | cold | -2.33 | 0.91 | 0.42 | 2.25 | 0.36 | -0.12 | 6.1e-10 | 1.6e-9
17.05 | 2003 | cold | -3.52 | 0.89 | 0.35 | 2.20 | 0.37 | -0.10 | 6.6e-11 | 1.6e-9
17.05 | 2003 | warm | -3.65 | 0.98 | 0.85 | 2.44 | 0.34 | -0.25 | 6.8e-11 | 1.6e-9
17.05 | V | cold | -3.18 | 0.91 | 0.42 | 2.26 | 0.36 | -0.12 | 1.8e-10 | 1.6e-9
17.05 | sust | cold | -2.36 | 0.91 | 0.38 | 2.25 | 0.37 | -0.11 | 6.2e-10 | 1.6e-9
17.05 | V-sust | cold | -3.03 | 0.91 | 0.38 | 2.25 | 0.37 | -0.11 | 1.6e-10 | 1.6e-9
17.05 | 2003-sust | cold | -4.04 | 0.90 | 0.14 | 2.24 | 0.37 | -0.04 | 7.2e-11 | 1.6e-9
19.57 | - | cold | -1.75 | 0.88 | 0.12 | 2.22 | 0.41 | -0.03 | 7.6e-10 | 1.6e-9
19.57 | 2003 | cold | -2.10 | 0.85 | -0.04 | 2.13 | 0.43 | 0.01 | 8.2e-11 | 1.5e-9
19.57 | 2003 | warm | -2.35 | 0.98 | 0.70 | 2.48 | 0.38 | -0.17 | 2.3e-10 | 1.5e-9
19.57 | V | cold | -1.64 | 0.83 | 0.00 | 2.09 | 0.43 | 0.00 | 2.0e-10 | 1.5e-9
19.57 | sust | cold | -1.81 | 0.91 | 0.16 | 2.28 | 0.41 | -0.04 | 7.0e-10 | 1.6e-9
19.57 | V-sust | cold | -2.56 | 0.82 | -0.08 | 2.06 | 0.43 | 0.02 | 1.9e-10 | 1.5e-9
19.57 | 2003-sust | cold | -1.48 | 0.82 | -0.12 | 2.05 | 0.43 | 0.03 | 3.1e-10 | 1.5e-9
21.47 | - | cold | -2.44 | 0.79 | 0.23 | 1.84 | 0.413 | -0.07 | 5.4e-10 | 1.6e-9
21.47 | 2003 | cold | -2.32 | 0.80 | 0.26 | 1.86 | 0.42 | -0.08 | 8.1e-11 | 1.6e-9
21.47 | 2003 | warm | -1.08 | 0.91 | 0.16 | 2.13 | 0.45 | -0.06 | 2.3e-10 | 1.5e-9
21.47 | V | cold | -1.03 | 0.86 | -0.07 | 2.00 | 0.44 | 0.02 | 2.2e-10 | 1.5e-9
21.47 | sust | cold | -0.90 | 0.76 | 0.56 | 1.76 | 0.44 | -0.17 | 5.0e-10 | 1.6e-9
21.47 | V-sust | cold | -2.95 | 0.88 | -0.23 | 2.04 | 0.46 | 0.07 | 2.2e-10 | 1.5e-9
21.47 | 2003-sust | cold | -2.50 | 0.89 | -0.26 | 2.07 | 0.47 | 0.08 | 8.4e-11 | 1.5e-9
Table 12: Summary of selected cold-started results using SST models.

a)SA-noft2

b)

c)

d)SA-noft2-R

e)

f)

g)SA-noft2-R-QCR2000

h)

i)

j)SA-noft2-RC

k)

l)

Figure 20: Skin-friction coefficient contour plots for SA turbulence models

a) Menter SST

b)

c)

d)

e) Menter SST-V

f)

g) Menter SST-2003

h)

i)

Figure 21: Skin-friction coefficient contour plots for SST Turbulence models

j) Menter SST-sust

k)

l)

j) Menter SST-Vsust

k)

l)

j) Menter SST-2003sust

k)

l)

Figure 22: Continuation of skin-friction coefficient contour plots for SST
Turbulence models

###  6.2 Steady RANS Grid-refinement Study

Figures 23 and 24 show close-ups of the grids for cross-sections respectively
indicated by black and red lines in Fig 9. Fig. 23 compares refined RANS and
DDES grids near the leading edge of the slat in the wing outboard region.
Besides different grid topologies, main differences are observed in the
streamwise resolution. Fig. 24 shows comparison between standard RANS and DDES
grids around the nacelle, where differences are mainly observed in the
circumferential resultion.

a)

b)

Figure 23: Close-up of (a) refined RANS and (b) DDES grid at a slice indicated
by the black line in Fig 9.

a)

b)

Figure 24: Close-up of (a) standard RANS and (b) DDES grid at a slice
indicated by the red line in Fig 9.

## Acknowledgments

The authors would like to acknowledge Dr. Yasushi Ito (JAXA) for providing
grids that were used for RANS simulations. We would also like to thank Dr.
Yoimi Kojima (JAXA) and Dr. Kenji Hayashi (Ryoyu Systems) for their great
technical support and fruitful discussions. Simulations were carried out on
JAXA’s high-performance computing facility JSS3. We would like to thank JSS3
for providing resources and technical support.

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