2021

[1]Christian Morsbach

[1]Institute of Propulsion Technology, German Aerospace Center (DLR), Linder
Höhe, 51147 Cologne, Germany 2]Institute of Test and Simulation for Gas
Turbines, German Aerospace Center (DLR), Am Technologiezentrum 5, 86159
Augsburg, Germany 3]MTU Aero Engines AG, Dachauer Str. 665, 80995 Munich,
Germany

# Large eddy simulation of a low-pressure turbine cascade with turbulent end
wall boundary layers

<EMAIL_ADDRESS>Michael Bergmann  Adem Tosun  Bjoern F. Klose
Edmund Kügeler  Matthias Franke  *  [  [

###### Abstract

We present results of implicit large eddy simulation (LES) and different
Reynolds-averaged Navier-Stokes (RANS) models of the MTU 161 low pressure
turbine at an exit Reynolds number of 90,000 and exit Mach number of 0.6. The
LES results are based on a high order discontinuous Galerkin method and the
RANS is computed using a classical finite-volume approach. The paper discusses
the steps taken to create realistic inflow boundary conditions in terms of end
wall boundary layer thickness and free stream turbulence intensity. This is
achieved by tailoring the input distribution of total pressure and
temperature, Reynolds stresses and turbulent length scale to a Fourier series
based synthetic turbulence generator. With this procedure, excellent agreement
with the experiment can be achieved in terms of blade loading at midspan and
wake total pressure losses at midspan and over the channel height. Based on
the validated setup, we focus on the discussion of secondary flow structures
emerging due to the interaction of the incoming boundary layer and the turbine
blade and compare the LES to two commonly used RANS models. Since we are able
to create consistent setups for both LES and RANS, all discrepancies can be
directly attributed to physical modelling problems. We show that the both a
linear eddy viscosity model and a differential Reynolds stress model coupled
with a state-of-the-art correlation-based transition model fail, in this case,
to predict the separation induced transition process around midspan. Moreover,
their prediction of secondary flow losses leaves room for improvement as shown
by a detailed discussion turbulence kinetic energy and anisotropy fields.

###### keywords:

large eddy simulation, low pressure turbine, discontinuous Galerkin method,
secondary flows

##  1 Introduction

In the aviation industry’s path towards sustainability, engines and their
efficiency play a crucial role. It is well known, that one way of increasing
the efficiency of a turbofan engine is to increase its bypass ratio, i.e. to
increase the amount of mass flow which does not pass through the core engine
with its compressor, combustion chamber and turbine. Since the outer engine
diameter is often limited by the aircraft design, this can be achieved by
reducing the core engine size. It leads, however, to relatively larger
secondary flow regions, which e.g. develop at intersections of blades and end
walls or in rotor blade tip gaps and contribute significantly to the overall
aerodynamic losses of the machine Denton1993 . Hence, understanding these
losses and the capability of lower fidelity Computational Fluid Dynamics (CFD)
approaches such as Reynolds-averaged Navier-Stokes (RANS) to predict them is
essential for further development in the field.

There exist several review papers discussing secondary flows in compressors
Gbadebo2005 ; Taylor2016  and turbines Langston2001  in detail, mainly relying
on experimental results or RANS based CFD. To extend the existing knowledge
numerically, methods are required which have little to no modelling
uncertainty. Large Eddy Simulation (LES) or Direct Numerical Simulation (DNS)
can be used to this end since they resolve the relevant turbulent length and
time scales and, therefore, offer a detailed picture of the unsteady flow
physics. Due to the unfavourable scaling of the computational costs with the
Reynolds number of roughly $\mathrm{Re}^{3.5}$ for DNS and $\mathrm{Re}^{2.5}$
for wall-resolved LES Choi2012 , these methods have only become more widely
used in the recent years with increasing availability of computational
resources and advancement in high-order discretisation methods. Many LES
studies of turbomachinery flows focus on the low-pressure turbine (LPT) due to
its low Reynolds number regime of $10^{5}$. To further reduce the simulation
costs, the assumption of a statistically 2D flow at the midspan of a turbine
blade is often made by applying periodic boundary conditions in the spanwise
direction, e.g. Sandberg2015 . These simulations have been used to improve the
understanding of midspan flow physics and investigate the performance of RANS
approaches Michelassi2015 . In view of the above argument, the next logical
step is to conduct 3D simulations, including the end wall boundary layers.

Cui et al. Cui2017  conducted an LES study on the influence of the inflow
boundary layer state on the secondary flow system in the T106A LPT with
parallel end walls in incompressible conditions at a Reynolds number of
160,000. They compare a laminar and a turbulent end wall boundary layer with
laminar free stream as well as unsteady wakes with secondary flows generated
by an upstream blade row and provide a detailed discussion of the secondary
vortices with focus on loss generation mechanisms. The same blade profile was
studied using LES by Pichler et al. Pichler2019  at a lower Reynolds number of
120,000 and a Mach number of 0.59. This paper also discusses the effect of
different turbulent boundary layer shapes prescribed at the inflow but at
freestream turbulence levels of roughly 5%. They found that while the boundary
layer shape has an effect on the penetration depth of the secondary flows into
the free stream, the midspan flow was largely unaffected by the difference.
Marconcini et al. Marconcini2019  use the same LES dataset to engage in a more
detailed discussion on the secondary flow system and its representation by
state-of-the-art industrial RANS. At midspan, their RANS results show a
significantly too narrow and too deep wake leading to an overall
underestimation of the wake losses, which can be explained by an
underprediction of the shear stress. They state that, in general, RANS
predicts a similar secondary flow system with vortices showing a greater
spanwise penetration compared with the LES. The peak losses associated with
the vortex cores are larger for RANS but in a pitch-averaged comparison, both
methods are comparable.

The MTU T161, considered in this work, is representative of high-lift LPT
airfoils used in modern jet engines Gier2007 . In contrast to the extensively
investigated T106 cascade, it features diverging end walls as found in real
engines, such that the flow cannot be studied using a simple spanwise periodic
setup. Its geometry and boundary conditions are in the public domain and it
has been the subject of both experimental Martinstetter2010  and numerical
Mueller-Schindewolffs2017 ; Rasquin2021 ; Iyer2021  investigations. The
numerical studies have focused on operating points with a Mach number of 0.6
and Reynolds numbers of 90,000 and 200,000 based on isentropic exit
conditions. Müller-Schindewolffs et al. Mueller-Schindewolffs2017  performed a
direct numerical simulation of a section of the profile in which the effect of
the diverging end walls was modelled using inviscid walls (termed quasi 3D,
Q3D). Recently, results including the end wall boundary layers and obtained
with a second order Finite Volume (FV) code were presented, but the analysis
was focused on the flow physics in the mid-section Afshar2022 . Various
computations of the full 3D configuration were conducted using high-order
codes during the EU project TILDA Rasquin2021 ; Iyer2021 . However, due to the
specification of laminar end wall boundary layers and no freestream turbulence
at the inflow, no satisfactory agreement with the experiment could be
obtained. Recently, Rosenzweig et al. Rosenzweig2022  presented a numerical
study of the T161 obtained with a high-order Finite Difference (FD) solver. It
features freestream turbulence and a Blasius boundary layer profile at the
inflow plane with sufficient development length to transition to turbulence
before reaching the turbine blade.

With a full 3D LES including appropriate turbulent end wall boundary layers
and freestream turbulence obtained with a high order Discontinuous Galerkin
(DG) method, we aim to provide a high-quality reference dataset for this
configuration. Because RANS is still the primary CFD tool in the design
process of turbomachinery components, we compare our LES results to two
commonly used RANS setups and focus on the accuracy of the representation of
secondary flow features. The paper is structured as follows: we will first
provide a brief summary of the high-order DG method used to perform the LES.
This will be followed by a detailed discussion of how to set up both LES and
RANS simulations consistently and in accordance with experimental data in
terms of operating point, freestream turbulence and end wall boundary layer
development. Finally, after demonstrating very good agreement between LES and
experiment, we will discuss the performance of both Linear Eddy Viscosity
Model (LEVM) and more advanced Differential Reynolds Stress Model (DRSM) RANS
approaches in predicting the midspan and end wall flow features.

##  2 Numerical method

DLR’s flow solver for turbomachinery applications is employed for the
presented simulations of MTU’s T161 cascade. The LES is performed using the
high-order DG solver of TRACE Bergmann2018a ; Bergmann2020 . Here, the
implicitly filtered compressible Navier-Stokes equations are, first, mapped
into the reference system, ensuring free-stream preservation Kopriva2016 , and
are spatially discretised with the 6th-order Discontinuous Galerkin Spectral
Element Method (DGSEM) on collocated Legendre-Gauss-Lobatto nodes. The
collocation of the integration and solution nodes gives rise to a highly
efficient discretisation scheme, since many numerical operations can be
omitted by design. Adjacent elements are coupled via Roe’s approximate Riemann
solver Roe1981  for the advective part. The required stabilisation for
turbulent flows is achieved by using a split formulation of the scheme
following Gassner et al. Gassner2016 , applying the kinetic-energy-preserving
two-point volume flux of Kennedy and Gruber Kennedy2008 . The viscous terms
are discretised using the Bassi-Rebay 1 Bassi1997  scheme with a central
numerical flux at the element interfaces, which, in combination with a stable
advective discretisation, was found to be stable without requiring an
additional penalty term Gassner2018 . Building on the favourable dispersion
and dissipation characteristics of the high-order DG scheme Moura2017 ;
Moura2017a , the effect of the unresolved scales in the LES is modelled
implicitly by the numerical scheme itself and without applying an explicit
subgrid-scale model. To advance in time, the strong-stability preserving
third-order explicit Runge-Kutta scheme of Shu and Osher Shu1988  is used.
Resolved turbulent scales are injected at the inflow boundary using a
Synthetic Turbulence Generator (STG) method based on randomised Fourier modes
Shur2014 . The fluctuating velocity is introduced using a Riemann boundary
condition Leyh2020  and, in contrast to the inner faces, no numerical flux
function is used at the inflow faces. At the outflow plane, one-dimensional
non-reflecting boundary conditions following Schlüß et al. Schluess2016  are
applied.

For the RANS simulations, we use TRACE’s density-based, compressible FV solver
Morsbach2016 ; Geiser2019 , which employs a Monotonic Upstream-centered Scheme
for Conservation Laws (MUSCL) scheme Leer1979  with a Van Albada 1 limiter
Albada1982  in combination with Roe’s approximate Riemann solver Roe1981  to
discretise the convective fluxes and central derivatives for the viscous
fluxes to obtain second order accuracy in space. Steady state solutions are
obtained with an implicit dual time-stepping approach which solves the five
conservation equations in a fully coupled manner. The additional turbulence
model equations are solved in a conservative but segregated manner
Morsbach2012  where the implicit matrix is constructed only per equation.
Menter’s SST $k$-$\omega$ model is implemented in its version from 2003
Menter2003  but with the production term modified according to Kato and
Launder Kato1993 . Laminar-to-turbulent transition is accounted for using the
correlation-based two-equation $\gamma$-$\mathrm{Re}_{\mathrm{\theta}}$ model
Langtry2009 . The SSG/LRR-$\omega$ DRSM originally devised by Eisfeld is used
in its latest version Cecora2014 ; Morsbach2016 . It is coupled with the same
transition model with the intermittency $\gamma$ not only used to modify the
production and dissipation terms but also the pressure-strain term Nie2018 .
The remaining parts of the turbulence and transition model are left unchanged.
In order to achieve a setup consistent with LES, the same formulation of
inflow and outflow boundary conditions is used for RANS.

##  3 Setup and boundary conditions

###  3.1 Computational domain and mesh

In practical CFD, a significant part of the overall effort is spent obtaining
a consistent setup between different computations and the experimental setup
of the same physical configuration. This comprises the representation of
geometrical features, manufacturing tolerances, fluid properties and the
boundary conditions. In this particular case, the measurements of have been
obtained by mounting the turbine profile in a wind tunnel facility as a linear
cascade with seven blades in order to achieve a near-periodic condition on the
centre blade. The blades with a chord length of $C=0.069935\,\mathrm{m}$ and
an average aspect ratio of 2.65 are staggered at an angle of $61.72^{\circ}$.
The cascade is arranged with pitch to chord ratio of
$l_{\mathrm{pitch}}/C=0.956$. The diverging end walls start at
$x/C_{\mathrm{ax}}=-0.6786$ (blade leading edge (LE) at $x=0$) and extend the
upstream channel height of $H=0.16\,\mathrm{m}$ at an angle of $12^{\circ}$.
For the simulations, we simplify the geometry by choosing a subdomain of the
whole test rig and model only a single blade with periodic boundary conditions
in pitchwise direction as sketched in Fig. 1. Downstream of the blade, the
domain extends until the point where the diverging end walls in the experiment
end and the flow enters a larger volume. The upstream length of the domain and
turbulent inflow boundary conditions are set up to reproduce the end wall
boundary layer state and freestream turbulence decay found in the experiment.
In addition to the simulations of the MTU T161, we therefore conduct finite
length channel flow simulations with parallel end walls. This setup reflects
the experiments conducted to determine the inflow conditions where the clean
wind tunnel without the cascade was used. As long as it can be guaranteed in
the experiment that the inflow conditions with and without blade are
comparable, this approach is very beneficial in terms of computational
resources required for preliminary studies because of less severe restrictions
on mesh size and time step in the channel flow.

Figure 1: Sketch of the domain and boundary conditions for the precursor
channel flow simulation and the simulation of the MTU T161 cascade Figure 2:
Average solution point distances in streamwise, wall normal and spanwise
direction on the blade at midspan (right)

All LES configurations computed for the current paper are summarised in Tab.
1. The mesh for the turbulent channel flow is constructed as a single block
from the rectangular end wall surface with equally sized rectangular faces
over the complete domain by an extrusion with the respective wall normal
element size stretching. This results in 83,790 hexahedral elements with
geometry polynomial order of $q=4$ to allow for intra-element stretching of
solution point distances Hindenlang2014 . For the current flow conditions, we
obtain average axial, wall normal and pitchwise element sizes
$(\overline{\Delta\xi^{+}},\overline{\Delta\eta^{+}},\overline{\Delta\zeta^{+}})=(32.6,2.0,33.1)$
normalised by the solution polynomial order of $N=5$. For the MTU T161, the
mesh is also constructed as extrusion from the end wall surfaces. These are
meshed as a structured multi-block topology with an O-block around the blade
to allow for a high quality boundary layer mesh. Again, we use a polynomial
order of $q=4$ to accurately represent the curved geometry of the blade
surface. The O-block is wrapped by a C-block while the rest of the domain is
filled with H-blocks putting special focus on the resolution in the blade
passage and wake region where the secondary flow structures develop. The mesh
consisting of 876,960 hexahedral elements with a solution polynomial order of
$N=5$ (189.4M DoF in total) was designed to meet widely accepted criteria for
wall resolution required for LES Georgiadis2010 . This is demonstrated in Fig.
2, which shows the streamwise, wall normal and spanwise element sizes on the
blade surface at midspan with averages of
$(\overline{\Delta\xi^{+}},\overline{\Delta\eta^{+}},\overline{\Delta\zeta^{+}})=(23.1,0.742,15.0)$,
normalised by $N$. The resolution in the free stream was ensured by the ratio
of average solution point distance and estimated Kolmogorov scale below 6
along a mid-passage streamline to meet the requirement of a very well-resolved
LES Froehlich2005 . On the end walls, we determine the streamwise and cross-
stream directions using the local wall shear stress vector
$\mathbf{\tau}_{\mathrm{w}}$ and the wall normal vector $\mathbf{n}$ as
$\mathbf{e}_{\parallel}=\mathbf{\tau}_{\mathrm{w}}/\|\mathbf{\tau}_{\mathrm{w}}\|$
and $\mathbf{e}_{\perp}=\mathbf{e}_{\parallel}\times\mathbf{n}$, respectively.
We choose the cell length which is most aligned with the wall shear stress
vector as the streamwise cell length $\Delta\xi$ and choose $\Delta\zeta$
accordingly. Using this procedure, the arithmetic averages are
$(\overline{\Delta\xi^{+}},\overline{\Delta\eta^{+}},\overline{\Delta\zeta^{+}})=(23.2,0.809,18.8)$
with maximal values of
$(\max{\Delta\xi^{+}},\max{\Delta\eta^{+}},\max{\Delta\zeta^{+}})=(65.2,1.71,60.3)$.
To be able to assess mesh dependence, results from a preliminary study with a
significantly coarser mesh in spanwise direction with 312,424 elements ($q=4$,
$N=5$) and 67.9M DoF are also included.

Table 1: Overview of numerical setups. The number of degrees of freedom can be obtained by $\mathrm{nDoF}=n_{xy}\cdot n_{z}\cdot(N+1)^{3}$. Configuration | $n_{xy}$ | $n_{z}$ | nDoF / $10^{6}$ | nMPI | $\mathrm{CPUh}/t_{\mathrm{c}}$ | $\Delta t/10^{-6}t_{\mathrm{c}}$ | $t_{\mathrm{avg}}/t_{\mathrm{c}}$
---|---|---|---|---|---|---|---
Channel | 1710 | 49 | $18.1$ | 960 | 131 | $221.9$ | $351$
Coarse | 3188 | 98 | $67.9$ | 2560 | 5861 | $18.19$ | $31$
Fine | 5220 | 168 | $189.4$ | 5120 | 25852 | $13.65$ | $100$

###  3.2 Definition of operating point

For LPT cascade cases, the operating point is often specified in terms of the
inflow angle $\alpha_{1}$ and isentropic Mach and Reynolds numbers. The latter
are defined using upstream total pressure $p_{\mathrm{t}1,\mathrm{ref}}$ and
temperature $T_{\mathrm{t}1,\mathrm{ref}}$, downstream static pressure
$p_{2,\mathrm{ref}}$ and a reference chord length $C$ as

$\displaystyle\mathrm{Ma}_{2,\mathrm{th}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{\gamma-1}\left(\left(\frac{p_{\mathrm{t}1,\mathrm{ref}}}{p_{2,\mathrm{ref}}}\right)^{\frac{\gamma-1}{\gamma}}-1\right)}$
(1) $\displaystyle\mathrm{Re}_{2,\mathrm{th}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\gamma}{RT_{2,\mathrm{th}}}}\frac{\mathrm{Ma}_{2,\mathrm{th}}p_{2,\mathrm{ref}}C}{\mu_{2,\mathrm{th}}}$
(2)

using the ideal gas law with constant $R$, isentropic relations with a
coefficient $\gamma$ and Sutherland’s law for the viscosity

$\displaystyle\mu_{{2,\mathrm{th}}}$ $\displaystyle=$
$\displaystyle\mu_{\mathrm{ref}}\frac{T_{\mathrm{ref}}+S}{T_{2,\mathrm{th}}+S}\left(\frac{T_{2,\mathrm{th}}}{T_{\mathrm{ref}}}\right)^{\frac{3}{2}}$
(3) $\displaystyle T_{{2,\mathrm{th}}}$ $\displaystyle=$ $\displaystyle
T_{\mathrm{t}1,\mathrm{ref}}\left(1+\frac{\gamma-1}{2}\mathrm{Ma}_{2,\mathrm{th}}^{2}\right)^{-1}$
(4)

with Sutherland’s constants $\mu_{\mathrm{ref}}$, $T_{\mathrm{ref}}$ and $S$
for air. The reference quantities are taken to be pitch-averaged centerline
conditions. For the current case, the nominal values are
$\alpha_{1}=41^{\circ}$, $p_{\mathrm{t}1,\mathrm{ref}}=11636\,\mathrm{Pa}$,
$T_{\mathrm{t}1,\mathrm{ref}}=303.25\,\mathrm{K}$ at a Reynolds number of
$\mathrm{Re}_{2,\mathrm{th}}=90000$ and a Mach number of
$\mathrm{Ma}_{2,\mathrm{th}}=0.6$. The resulting static outflow pressure is
$p_{2,\mathrm{ref}}=9122.7\,\mathrm{Pa}$. These conditions can be easily
matched if the flow solver allows to specify the inflow boundary conditions in
terms of total pressure, total temperature and flow angle as usually done in
turbomachinery CFD. Complications can arise, however, at the intersection of
inflow panels and solid walls. Here, the velocity needs to vanish due to the
no-slip condition, which could be easily guaranteed for if the inflow
condition was specified using the velocity itself. If the total pressure is
used instead, the inflow velocity close to the wall is a result of the
specified total pressure and the static pressure within the domain, which
itself is only a result of the simulation determined by the specified outflow
pressure and flow through the domain. Therefore, the specification of boundary
layer profiles makes the proper setup of flows with end walls considerably
more difficult than simply setting of a free-stream state defined by
$\alpha_{1}$, $p_{\mathrm{t}1,\mathrm{ref}}$ and
$T_{\mathrm{t}1,\mathrm{ref}}$ away from walls, as it may require adaptation
while the simulation progresses.

###  3.3 Turbulent inflow

In addition to reproducing the overall operating point, it is required to
match the momentum thickness development of the incoming end wall boundary
layers as well as the decay of freestream turbulence given by measurement data
at a number of axial stations upstream of the blade. Since the measurement
data is not sufficient to be used as inflow data directly and, more
importantly, since synthetically generated turbulence does require a certain
adaptation length to develop Keating2004 , an indirect approach has to be
chosen.

We perform a precursor channel flow simulation to determine the distance from
the inlet $d_{\mathrm{inlet}}$ required for the boundary layer to develop and
meet the specifications in the reference plane as sketched in Fig. 1
(arbitrary position due to confidentiality). The boundary conditions are then
set by spanwise varying profiles of $p_{\mathrm{t},1}(z)$ and
$T_{\mathrm{t},1}(z)$ with a constant $\alpha_{1}$, as well as a spanwise
varying Reynolds stress tensor $\overline{u_{i}^{\prime}u_{j}^{\prime}}(z)$
and turbulent length scale $L_{\mathrm{T}}(z)$. At midspan ($z=0$), the
profiles must yield the reference values defining the operating point as
described above. All these profiles will be obtained by scaling a boundary
layer profile in wall units at $\mathrm{Re}_{\mathrm{\theta}}=670$ from DNS
data (https://www.mech.kth.se/~pschlatt/DATA/ Schlatter2010 ) and combining it
with freestream turbulence values.

Starting from the above centerline inflow conditions, we use the isentropic
relations

$\frac{T_{\mathrm{t}}(z)}{T_{1}}=\left(\frac{p_{\mathrm{t}}(z)}{p_{1}}\right)^{\frac{\gamma-1}{\gamma}}=1+\frac{\gamma-1}{2}\mathrm{Ma}(z)^{2}$
(5)

to obtain a spanwise variation. Here, we assume a constant static temperature
$T_{1}$ and pressure $p_{1}$ throughout the boundary layer. These two
quantities can be computed from (5) by introducing a centerline Mach number
$\mathrm{Ma}_{1,\mathrm{c}}$, which essentially controls the value of the
$T_{\mathrm{t}}$ and $p_{\mathrm{t}}$ at the end wall. As discussed above, it
has to be chosen such that the total pressure is always greater than the
static pressure resulting from the simulation to avoid backflow close to the
wall. An iteration might be necessary to satisfy this condition. It should be
noted that small adaptations of $\mathrm{Ma}_{1,\mathrm{c}}$ mainly influence
the total pressure profile close to the wall and do not have a significant
impact on the quantities of interest in the simulation. For this specific
case, we chose $\mathrm{Ma}_{1,\mathrm{c}}=0.362$. The Mach number at the
centerline, however, remains a result of the simulation.

The Mach number profile $\mathrm{Ma}(z)$ is computed based on the discrete DNS
velocity profile $u_{i,\mathrm{DNS}}^{+}$ and the corresponding distances to
the wall $z^{+}_{i,\mathrm{DNS}}$ simply using the definitions of $u^{+}$ and
$z^{+}$:

$\displaystyle\mathrm{Ma}(z_{i})$ $\displaystyle=$ $\displaystyle
u^{+}_{i,\mathrm{DNS}}\cdot\frac{u_{\tau}}{a_{1}},$ (6) $\displaystyle z_{i}$
$\displaystyle=$ $\displaystyle
z^{+}_{i,\mathrm{DNS}}\cdot\frac{\nu_{1}}{u_{\tau}}$ (7)

with

$u_{\tau}=\frac{a_{1}\cdot\mathrm{Ma}_{1,\mathrm{c}}}{u_{\infty,\mathrm{DNS}}^{+}}$
(8)

and the DNS freestream velocity $u^{+}_{\infty,\mathrm{DNS}}$. The speed of
sound $a_{1}$ and viscosity $\nu_{1}$ are computed from $T_{1}$ and $p_{1}$
using ideal gas and Sutherland laws, respectively. This allows to describe the
spanwise variation of $T_{\mathrm{t},1}(z)$ and $p_{\mathrm{t},1}(z)$ using
(5). The non-dimensional turbulent stress tensor is scaled analogously via

$\overline{u_{j}^{\prime}u_{k}^{\prime}}(z_{i})=\overline{u_{j}^{\prime}u_{k}^{\prime}}^{+}|_{i,\mathrm{DNS}}\cdot
u_{\tau}^{2}.$ (9)

Finally, the integral turbulent length scale in the boundary layer is
estimated from the turbulent kinetic energy and dissipation rate and
dimensionalised with $\frac{\nu_{1}}{u_{\tau}}$, as it is done in (7):

$L_{\mathrm{T},\mathrm{BL}}(z_{i})=\frac{(k^{+}_{i,\mathrm{DNS}})^{3/2}}{\epsilon^{+}_{i,\mathrm{DNS}}}\cdot\frac{\nu_{1}}{u_{\tau}}.$
(10)

Optimally, the freestream Reynolds stresses and turbulent length scale should
be chosen such that the measured turbulent decay is matched. However, in this
case, this would result in a length scale in the order of the cascade pitch.
This could, in principle, be accommodated for by simulating more than one
blade at the respective expense of computational effort. Instead, we chose to
decrease the turbulent length scale and scale the non-zero components of the
Reynolds stress tensor at the inflow of the domain to reproduce the turbulence
intensity in the blade LE plane according to the experiment. This leads to a
stronger decay of turbulent structures and has to be considered when assessing
the quality of the results. Turbulence anisotropy, however, is specified as
found by hot wire measurements.

The boundary layer profile and freestream values of the turbulence quantities
are combined where they intersect at the edge of the boundary layer. Because
of the large freestream turbulent length scale compared to the smaller length
scale in the boundary layer, we use

$L_{\mathrm{T}}=\max(L_{\mathrm{T},\mathrm{BL}},L_{\mathrm{T},\mathrm{freestream}})$
(11)

for distances to the wall greater than $\delta_{99}$. Finally, the Reynolds
stress tensor is rotated from the streamline-aligned to a Cartesian coordinate
system.

For the RANS simulations, we use the same boundary layer profile as for the
LES. The freestream turbulence, however, is treated differently depending on
the model type with the goal of reproducing the turbulence decay of the LES
and not the experiment. The turbulence intensity along with a turbulent length
scale are specified in the freestream for the LEVM setups to determine
boundary conditions for the turbulent kinetic energy $k$ and its specific
dissipation rate $\omega$. For the DRSM setup, on the other hand, we specify
the complete Reynolds stress tensor $\overline{u_{i}^{\prime}u_{j}^{\prime}}$
and $\omega$ directly to closely match the turbulence anisotropy and decay
produced by the STG.

In the following we will characterise the inflow boundary layer and freestream
turbulence obtained with the approach described above both in terms of the
precursor channel flow LES and the final simulation with blade on the fine
mesh. Fig. 3 (left) shows the development of the momentum thickness Reynolds
number

$\mathrm{Re}_{\mathrm{\theta}}=\frac{\|\overline{\mathbf{u}}_{\infty}\|\theta}{\nu_{\mathrm{wall}}},\quad\theta=\int_{0}^{\delta_{99}}\frac{\|\overline{\mathbf{u}}(z)\|}{\|\overline{\mathbf{u}}_{\infty}\|}\left(1-\frac{\|\overline{\mathbf{u}}(z)\|}{\|\overline{\mathbf{u}}_{\infty}\|}\right)\mathrm{d}z$
(12)

in comparison with a fit of an exponential function proportional to $x^{4/5}$
to the experimental data over the axial distance from the LE of the blade
$x/C_{\mathrm{ax}}$. We note that the absolute values cannot be shown here due
to confidentiality restrictions and that experimental data available for the
fit is scarce. Given these limitations, the precursor channel flow LES agrees
reasonably well with the experiments. Because the boundary layer of the
simulation with blade (LES fine) is heavily influenced by the diverging end
walls and the potential field of the blade, we show the diffusing section of
the channel and the extent of the blade as light and dark grey, respectively,
in this and all following plots. Nevertheless, the development of the boundary
layer thickness is well matched far upstream from the blade. Fig. 3 (right)
shows normalised boundary layer velocity profiles extracted at
$x/C_{\mathrm{ax}}=-0.99$ for reference. The upstream effect of the blade and
diffuser is most significant in the wake region of the boundary layer. Both
RANS models agree reasonably well with the LES in terms of boundary layer
shape and integral properties.

Figure 3: Momentum thickness Reynolds number over axial distance from inlet
plane (left) and boundary layer velocity profile at $x/C_{\mathrm{ax}}=-0.99$
(right)

With the turbulent boundary layer properly defined, we shift our to focus to
the characterisation of the freestream turbulence. The data is obtained from
the time-resolved sampling of the flow field along a streamline at midspan
passing through the blade LE plane at $y/l_{\mathrm{pitch}}=0.5$. Fig. 4
(left) shows the development of freestream turbulence intensity $\mathrm{Tu}$
in comparison with a fit to the measured data. As discussed, the turbulence
decay is slightly steeper in our setup but the turbulence intensity at the
blade LE is well captured. Compared to the simulation of the MTU T161 (LES
fine), a much lower statistical error could be obtained for the precursor
channel flow simulation due to less severe time step constraints. Again, in
the upstream area with negligible potential flow effects, both simulations
agree very well, before they deviate due to the different geometry. Both types
of RANS models are able to closely match the turbulence decay of the LES.
Differences start to occur in the blade passage, where the Menter SST
$k$-$\omega$ model starts to underestimate the turbulence intensity shortly
behind the LE. The SSG/LRR-$\omega$ model, on the other hand, is able to
reproduce the proper turbulence levels throughout the passage before it starts
to deviate close to the trailing edge (TE). To achieve this, it was essential
to appropriately specify the turbulence anisotropy at the inflow (results with
different anisotropy but the same turbulence intensity not shown here).

Fig. 4 (right) shows the Reynolds stress anisotropy in terms of the
barycentric map Banerjee2007  on the same streamline. Again, both the channel
flow and the MTU T161 (LES fine) simulations are shown and the axial position
is colour-coded in grey scale with the LE marked in red and the TE marked in
purple. At the inflow, the injected turbulence has a two-component character
(as specified as input to the STG) and, after a short length of adaptation,
the channel flow shows a return-to-isotropy trajectory (towards the three-
component corner). For the simulation with blade, the effect of flow
acceleration on the turbulence structure can be observed. After an initial
decay towards isotropy, the vortices are stretched by the contraction in the
blade passage. In the barycentric triangle, we can observe a trend towards the
two-component axisymmetric corner (2C) from LE to TE along the disk-like
axisymmetric border Simonsen2005 . Upon leaving the blade passage, the
turbulence exhibits a trend towards the isotropic state. The results from the
SSG/LRR-$\omega$ model confirm that the turbulence anisotropy upstream of the
blade is consistent with the LES. Although slight deviations in the trajectory
become apparent between LE and TE, the representation of the freestream
turbulence anisotropy is qualitatively matched very well. Since we can only
specify isotropic turbulence for the Menter SST $k$-$\omega$ model, we
deliberately omit this comparison here.

Figure 4: Development of freestream turbulence intensity (left) and Reynolds
stress anisotropy tensor eigenvalues visualised in barycentric coordinates
with the axial position indicated by colour (right)

Finally, we analyse the spectral content of the streamwise, cross-stream and
wall-normal velocity components $(u_{\parallel},u_{\perp},u_{z})$. The
turbulent length scale $L_{\mathrm{T}}$ normalised with the length scale
specified to the STG $L_{\mathrm{T},0}$ is shown in Fig. 5 (left). It has been
obtained using the frozen turbulence hypothesis

$L_{\mathrm{T},i}=\|\overline{\mathbf{u}}\|T_{\mathrm{int},i}$ (13)

with the integral timescale computed as the integral over the normalised auto-
correlation of the time signal of the respective velocity component up to the
first zero-crossing at $\tau_{0}$ (no summation over $i$):

$T_{\mathrm{int},i}=\int_{0}^{\tau_{0}}\left(\frac{1}{t_{\mathrm{avg}}\cdot\overline{u_{i}^{\prime}u_{i}^{\prime}}}\int_{0}^{t_{\mathrm{avg}}}u_{i}^{\prime}(t)u_{i}^{\prime}(t-\tau)\mathrm{d}t\right)\mathrm{d}\tau.$
(14)

The axial development shows a relatively rapid decrease in turbulent length
scale down to roughly half the specification, which is in line with previous
experience with this particular STG for decaying isotropic turbulence Leyh2020
. In the blade passage, it can be observed that the length scale in the
directions perpendicular to the mean flow directions grows strongly in over
the rear part of the blade before the decay towards isotropy sets in
downstream. Fig. 5 (right) shows the power spectral density in the freestream
at $x/C_{\mathrm{ax}}=-0.99$. On the large scales, the turbulence is still
considerably anisotropic. On the smaller scales, however, it has developed an
inertial range as indicated by the line proportional to $f^{-5/3}$.

Figure 5: Development of freestream turbulence length scale (left) and PSD at
$x/C_{\mathrm{ax}}=-0.99$ for LES with blade (right) for streamwise, cross
stream and spanwise velocity components

In conclusion, we have presented both an LES setup consistent with available
experimental data and two RANS setups consistent with the LES.

##  4 Flow analysis

###  4.1 Midspan

After the characterisation of the incoming flow in the previous section, we
now analyse the flow through the MTU T161 cascade with a focus on secondary
flow structures and their representation by the RANS models. First, we briefly
discuss the consistency between the coarse and the fine mesh LES for the
quantities of interest and compare with experiments where applicable to build
up trust in the current LES dataset. After clearing the initial transient,
statistics on the fine and coarse mesh were sampled for 100 and 31 convective
time units $t_{\mathrm{c}}$ based on chord length and outflow velocity,
respectively, as indicated in Tab. 1. This difference in number of samples
will show in the 68% confidence intervals Bergmann2021 .

Figure 6: Midspan blade pressure distribution (left) and skin friction
coefficient (right); errorbars indicate 68% confidence intervals

LES fine

SSG/LRR-$\omega$

Menter SST $k$-$\omega$

Menter SST $k$-$\omega$$L_{\mathrm{T}}=\frac{L_{\mathrm{T},0}}{16.7}$

Figure 7: Midspan streamlines in the separation region of the blade

We start our analysis with the discussion of the blade loading and the wake
losses. Fig. 6 shows a comparison of the midspan blade surface pressure
distribution with the experiment (left) and the skin friction coefficient

$\overline{c_{\mathrm{f}}}=\frac{\mathrm{sgn}(\overline{\tau_{\mathrm{w},x}})\sqrt{\overline{\tau_{\mathrm{w},x}}^{2}+\overline{\tau_{\mathrm{w},y}}^{2}}}{{p_{\mathrm{t}1,\mathrm{ref}}}-{p_{2,\mathrm{ref}}}}.$
(15)

for which no experimental data are available (right). Both LES produce the
same blade loading and show only a subtle difference in skin friction
downstream of the transition peak of the laminar separation bubble indicating
that the resolution on the coarse mesh is not entirely sufficient. While LES
and experiment agree very well on the pressure side, a difference in surface
pressure can be found on the suction side between $x/C_{\mathrm{ax}}=0.1$ and
0.6. Similar offsets have been found in previous studies of this
configuration, e.g. Mueller-Schindewolffs2017 . The laminar separation bubble
and subsequent turbulent reattachment indicated by the pressure plateau
between 0.7 and 0.9 and recovery of the base pressure, on the other hand, are
captured very well. Both the Menter SST $k$-$\omega$ and the SSG/LRR-$\omega$
models fail to replicate this characteristic pressure plateau. Since it seems
to be common practice to compensate for this problem by reducing the inflow
turbulent length scale in the RANS models, we include such a solution obtained
with the Menter SST $k$-$\omega$ model as well. Note, that this leads to a
turbulence intensity in the LE plane reduced by roughly 56% compared to the
results shown in Fig. 4. While this seems to improve the prediction of blade
loading at first sight, it can be observed that the TE pressure is not
recovered, indicating an open separation bubble. For further interpretation,
we evaluate the midspan skin friction coefficient in Fig. 6 (right) and
streamlines in Fig. 7. The LES exhibits the skin friction profile of a classic
laminar separation bubble (LSB) with weak reverse flow after separation
followed by a transition peak and subsequent reattachment. Both RANS models
with properly configured inflow turbulence levels, however, show a slightly
delayed separation and premature reattachment leading to a very thin
separation bubble. The simulation with the adapted turbulent length scale
separates early and has the general characteristics of a LSB but fails to
reattach before the TE. Although this separation bubble is at the wrong
streamwise position, its thickness is more comparable to that computed by LES.
The larger separation bubble can be explained by the behaviour of the
transition model, which keeps a larger part of the boundary layer prior to
separation in a laminar state by a value of $\gamma\ll 1$.

Figure 8: Midspan wake total pressure loss coefficient at
$x/C_{\mathrm{ax}}=1.4$

Fig. 8 shows the total pressure loss coefficient

$\overline{\omega}=\frac{p_{\mathrm{t}1,\mathrm{ref}}-p_{\mathrm{t}}(\overline{\rho},\overline{u},\overline{v},\overline{w},\overline{p})}{p_{\mathrm{t}1,\mathrm{ref}}-p_{2,\mathrm{ref}}},$
(16)

computed from time-averaged primitive variables in the wake plane at
$x/C_{\mathrm{ax}}=1.4$. The pitch coordinate $y$ is given with respect to the
point $y_{\mathrm{LE}}$ on the blade at its minimum axial coordinate
$x/C_{\mathrm{ax}}=0$. Both LES and the experimental data agree very well
within the statistical confidence intervals. None of the RANS setups, on the
other hand, is able to reproduce the wake shape accurately. Due to the very
thin separation bubble, both simulations with the appropriate turbulence decay
predict a too strong turning of the flow with mass averaged losses
underestimated by 43% and 36% for the Menter SST $k$-$\omega$ and
SSG/LRR-$\omega$, respectively. The RANS simulation with the tweaked turbulent
length scale profits from a lucky cancellation of errors and produces a mass
averaged loss underestimated by only 12%. Similar discrepancies between RANS
and LES in terms of wake development have been reported in the literature for
the MTU T161 Mueller-Schindewolffs2017 ; Afshar2022  and the T106A
Marconcini2019 .

Figure 9: Wake total pressure loss coefficient at $x/C_{\mathrm{ax}}=1.4$

###  4.2 Secondary flow system

As a final validation of the LES, we turn to the secondary flow system,
visualised as total pressure loss coefficient in the wake plane at
$x/C_{\mathrm{ax}}=1.4$ in Fig. 9. Note that the $z$-coordinate is given with
respect to the local channel height $h$. The LES on the fine mesh is able to
reproduce the general shape of the loss cores found in the experiment very
well. The differences in the region of the trailing shed vortex (TSV) (or
counter vortex Langston2001 ; Cui2017 ) and passage vortex (PV) are rather
subtle and are visually amplified by the much lower spatial resolution of the
experimental dataset. What can be deduced from the figure is that the loss
core associated with the PV is predicted slightly closer to the end wall by
the current LES than in the experiment. In the second panel of the figure, the
LES on the coarse mesh is compared to the one on the fine mesh now plotted as
contour lines. As already stated for the midspan results, both simulations
give consistent results despite the relatively low spanwise resolution of the
coarse LES.

The remaining three panels of Fig. 9 show the RANS results compared with the
LES on the fine mesh. A first observation that can be made here is that the
tweaking of the freestream turbulent length scale does not significantly
modify the secondary flow losses. Differences between the two RANS simulations
using the Menter SST $k$-$\omega$ model only become apparent around midspan
($z\approx\pm 0.2h$). We will, therefore, focus on the comparison of the two
different RANS model approaches (panel 3 and 4) from here on. The overall
shape of the secondary flow loss region depends only weakly on the chosen
model and resembles the one obtained with LES. It is noteworthy, however, that
RANS shows a drastically increased maximum loss coefficient in the region of
both the PV and the TSV. The position of the maximum is marked by an x in the
plot. With the Menter SST $k$-$\omega$, it is overestimated by 34% while the
SSG/LRR-$\omega$ model results in a 48% increase with the maximum loss in a
different peak compared to both Menter SST $k$-$\omega$ and the LES. This is
consistent with the results reported by Marconcini et al. Marconcini2019  with
the difference, that in our case, the RANS shows losses that are too large
even in a pitch-averaged sense.

Figure 10: Wake streamwise vorticity $\omega_{\mathrm{sw}}$ at
$x/C_{\mathrm{ax}}=1.4$

An explanation for the increased losses in the RANS results is offered by the
topology of the streamwise vorticity
$\omega_{\mathrm{sw}}=\mathbf{u}\cdot\mathbf{\omega}/\|\mathbf{u}\|$ shown in
Fig. 10. The plots are in the same plane at $x/C_{\mathrm{ax}}=1.4$ as the
ones in Fig. 9. The RANS models show increased streamwise vorticity very close
to the end wall, indicating that the corner vortex persists further downstream
than computed by the LES, cf. Marconcini2019 . Moreover, in both models, the
system of TSV and PV is stronger, located at a greater distance from the end
wall and slightly offset towards the suction side (positive $y$). While DRSMs
are often praised for being less diffusive in the representation of vortices,
we have a strong exaggeration of the compactness and strength of both the TSV
and PV by the SSG/LRR-$\omega$ model in this case. To gain further insights
into the origin of this discrepancy, we shift our focus further upstream in
the paragraphs below.

APVBC

HSVPVD

Figure 11: Instantaneous flow field visualised with
$QC_{\mathrm{ax}}^{2}/\|\overline{\mathbf{u}_{2}}\|^{2}=500$ over the suction
side (left) and with
$QC_{\mathrm{ax}}^{2}/\|\overline{\mathbf{u}_{2}}\|^{2}=100$ over the end wall
(left), both coloured with velocity magnitude; time-averaged solution
visualised with surface streaklines and
$\overline{Q}C_{\mathrm{ax}}^{2}/\|\overline{\mathbf{u}_{2}}\|^{2}=1$ iso-
surface coloured with streamwise vorticity

Fig. 11 provides a three-dimensional view of the secondary flow system in the
blade passage near the end walls. It shows surface streaklines computed with
line integral convolution (LIC) using the time-averaged wall shear stress
vector and a semi-opaque time-averaged
$\overline{Q}C_{\mathrm{ax}}^{2}/\|\overline{\mathbf{u}_{2}}\|^{2}=1$
Haller2005  iso-surface coloured in blue-to-red with streamwise vorticity
$\omega_{\mathrm{sw}}$ to indicate the sense of rotation. Instantaneous vortex
structures are visualised as
$QC_{\mathrm{ax}}^{2}/\|\overline{\mathbf{u}_{2}}\|^{2}=500$ iso-surface over
the suction side of the blade (left) and as
$QC_{\mathrm{ax}}^{2}/\|\overline{\mathbf{u}_{2}}\|^{2}=100$ iso-surface in
the top view (right) allowing to inspect weaker vortex structures of the
incoming turbulent boundary layer. Both are coloured with the same map of
velocity magnitude. The structures of the freestream turbulence are too weak
to be visualised by this choice of $Q$. Towards mid-span, significant
turbulent structures are generated over the separation bubble
(\footnotesize{A}⃝) and form a major contribution to the wake turbulence
downstream. Increased turbulence activity can also be found within the PV and
within two separate vortex cores close to the blade surface
(\footnotesize{B}⃝), which originate in the same area. Cui et al. Cui2017
also report a separate vortex core above the PV, but state that it has an
opposite sense of rotation. In this case, all three vortices rotate in the
same direction. While the end wall boundary layer directly downstream of the
PV does not yield any significant instantaneous vortical flow structures,
strong turbulent flow begins to reappear downstream of a saddle point close to
the suction side TE (\footnotesize{C}⃝). The top view of the end wall with the
lower value of $Q$ confirms that the incoming turbulent boundary layer
(\footnotesize{D}⃝) is completely lifted off the wall by the PV resulting in
the downstream calmed region, cf. Cui2017 .

HSVPVLECVTSVA

SP2SP1

Figure 12: Surface streaklines on the end wall and vortices by $Q$ isosurface
coloured with streamwise vorticity for LES average (left), Menter SST
$k$-$\omega$ (middle) and SSG/LRR-$\omega$ (right)

A less obstructed view of the averaged flow can be found in Fig. 12, which
shows the same view as Fig. 11 (right), but without the instantaneous
structures. In addition to the LES (left), the figure shows results for Menter
SST $k$-$\omega$ (middle) and SSG/LRR-$\omega$ (right) for comparison. The
horse shoe vortex (HSV) develops due to a roll-up of the incoming boundary
layer. But while its suction side leg (red) dissipates well before the suction
peak, its pressure side leg (blue) follows the passage cross flow towards the
suction side of the next blade, lifts off the end wall and develops into the
PV. Behind the blade, a second large structure can be identified as the TSV,
which rotates in the opposite direction. Both vortices persist well downstream
of the blade and become visible as the two regions of strong pressure loss in
Fig. 9 and vorticity in Fig. 10. Another small but very distinct vortex
developing along the pressure side of the blade is the leading edge corner
vortex (LECV). The pressure side features a short separation bubble close to
the LE. Due to the diverging end walls, the backflow within the bubble is
driven towards the end walls where it rolls up, lifts off and mixes with the
newly developing boundary layer in the passage between the PV and the pressure
side of the blade (\footnotesize{A}⃝).

With the RANS models, the general topology of the secondary flow can be
reproduced quite reasonably. The most striking topological difference between
the models is the appearance of a second saddle point (SP2) between the LE and
the first saddle point (SP1) slightly upstream for the SSG/LRR-$\omega$ model.
Apart from that, the DRSM produces more compact vortex cores than the LEVM as
already observed in the downstream wake plane. Both models predict the HSV to
appear at nearly the same position and follow the same trajectory with the
pressure side leg following a nearly straight path. The LES, on the other
hand, shows the HSV to originate closer to the LE. Its suction side leg
follows the blade surface more closely and the pressure side leg curves
through the passage before reaching the adjacent blade. Another difference is
the vortex resulting from the pressure side (PS) separation
(\footnotesize{A}⃝), which is present in the LES, but visible only weakly for
the Menter SST $k$-$\omega$ model and almost impossible to identify for the
SSG/LRR-$\omega$ model. This is caused by a weaker PS separation, which for
the SSG/LRR-$\omega$ model, does not even extend over the entire blade span.

Figure 13: Turbulent kinetic energy $k$ in different cuts normal to blade
surface for LES on fine mesh (top), Menter SST $k$-$\omega$ (middle) and
SSG/LRR-$\omega$ (bottom); passage vortex core illustrated using
$\overline{Q}C_{\mathrm{ax}}^{2}/\|\overline{\mathbf{u}_{2}}\|^{2}=10$ contour
line

Following the overview of the secondary flow system, we now turn to a more
detailed analysis of the turbulence close to the suction side of the blade as
potential root cause for the discrepancies in the wake losses discussed
earlier. For the midspan region, this has already been discussed extensively
by Fard Afshar et al. Afshar2022  and our results are consistent with their
findings. We, therefore, focus on the end wall region. Fig. 13 shows the
turbulence kinetic energy $k$ in eight different wall-normal planes over the
suction side surface whose positions are sketched in the schematic on the
right. Note, that the $z$-coordinate is normalised by the inflow channel
height $H$ instead of the local channel height $h$ and that $\eta$ denotes the
distance from the blade wall. A single contour line of
$\overline{Q}C_{\mathrm{ax}}^{2}/\|\overline{\mathbf{u}_{2}}\|^{2}=10$ helps
identify vortical structures in a more quantitative manner, with the most
prominent being the PV. RANS predicts the vortex to approach the blade wall
more upstream, as can be seen in the cut at $x/C_{\mathrm{ax}}=0.50$.
Subsequently, it lifts off the end wall more quickly leading to the
discrepancy observed in the wake plane in Figs. 9 and 10. A similar
observation has been made by Marconcini et al. Marconcini2019  for the T106A
with parallel end walls. The vortex core coincides with elevated $k$, which is
qualitatively reproduced well by the Menter SST $k$-$\omega$ model with a
slight tendency to overpredict $k$ towards the trailing edge of the blade.
Curiously, the SSG/LRR-$\omega$ model fails to produce enough $k$ over the
development of the PV providing a possible explanation for the slower vortex
decay in comparison with the LES results. Two other vortex cores can be
observed above the PV in the LES at $x/C_{\mathrm{ax}}=0.75$
(\footnotesize{A}⃝, also see Fig. 11 \footnotesize{B}⃝). They are also
produced by the DRSM but again dissipate too slowly, as they can still be seen
at $x/C_{\mathrm{ax}}=0.96$. In the LES on the other hand, the vortices have
disappeared by $x/C_{\mathrm{ax}}=0.9$, where a patch of high turbulence
levels lifts off the blade surface (\footnotesize{B}⃝). Moreover, a small
corner vortex (\footnotesize{C}⃝) can be seen where the blade intersects the
end wall. It travels up the blade for a short distance before it is terminated
again by the increasing turbulence intensity in this area (\footnotesize{D}⃝).
Qualitatively, this can also be found in the RANS, again with significantly
lower levels of $k$. The largest discrepancy in terms of $k$ can be seen in
the region of essentially two-dimensional flow (\footnotesize{E}⃝). As
discussed above, this is caused by the severe underprediction of the
separation bubble by the RANS models but has no significant influence on the
discussion of the end wall flow.

Figure 14: Reynolds stress tensor anisotropy visualised as RGB colours in
different cuts normal to blade surface for LES on fine mesh (top), Menter SST
$k$-$\omega$ (middle) and SSG/LRR-$\omega$ (bottom); passage vortex core
illustrated using
$\overline{Q}C_{\mathrm{ax}}^{2}/\|\overline{\mathbf{u}_{2}}\|^{2}=10$ contour
line

We want to conclude our discussion of the secondary flow system with an
analysis of the turbulence topology in terms of Reynolds stress anisotropy to
add some facts to the common notion that DRSMs should outperform LEVMs in flow
regions with significant turbulence anisotropy Hanjalic2002 . This discussion
is facilitated by using the approach of Emory and Iaccarino Emory2014  to
spatially visualise the eigenvalues of the Reynolds stress anisotropy tensor
by colour coding them as RGB channels. Fig. 14 shows the same cut planes as
above now coloured according to the turbulence anisotropy. Each colour
corresponds to a set of anisotropy tensor eigenvalues expressed as position in
the barycentric triangle in the legend. Starting our discussion with the
freestream turbulence in the blade passage away from the end walls
(\footnotesize{A}⃝), the colour map shows that the fluctuations transform from
mixed two- and three-component turbulence towards the two-component (2C)
corner (see also Fig. 4). While this behaviour is reproduced reasonably well
by the DRSM, it cannot be expected from an LEVM as the turbulence anisotropy
equals the strain rate anisotropy through the linear Boussinesq approximation.
We nevertheless plot the results for the Menter SST $k$-$\omega$ model for the
sake of completeness. Close to the blade, in the statistically two-dimensional
part of the flow, the transitional separated shear layer can be observed in
the $Q$-contour (\footnotesize{B}⃝) featuring turbulence distinctively close
to the one-component limit, cf. Afshar2022 . After transition to turbulence,
this area can be found significantly closer to isotropic (3C) turbulence
(\footnotesize{C}⃝). The SSG/LRR-$\omega$ model fails to reproduce this
characteristic feature of the separation, and, as shown in Fig. 7, the
separation itself. It only shows a reasonable match after transition
(\footnotesize{C}⃝), yet, as discussed above, at much too low levels of
turbulence. Note, that the model is a high Reynolds number model with the
simpler LRR pressure strain model Launder1975  active close to solid walls.
While this choice contributes to the model’s greater numerical stability
compared to low Reynolds number approaches, it can be questioned in cases like
the current. A similar observation can be made for two more regions where near
wall 1C-turbulence is lifted. One is close to the end wall at the intersection
with the blade wall (\footnotesize{D}⃝) and one within the upwards cross flow
induced by the PV on the blade surface (\footnotesize{E}⃝), both producing
areas of increased turbulence (cf. Fig. 13 \footnotesize{D}⃝ and
\footnotesize{B}⃝). While it was easy to associate the dominant PV with
increased $k$ in Fig. 13, the picture is not so clear in terms of turbulence
anisotropy. Generally, the turbulence is in a state close to isotropy in its
area of influence, which is in agreement between LES and DRSM. The LES,
however, shows this area to be confined relatively close to the blade wall and
the region downstream of the PV is quickly filled with 2C-turbulence from the
freestream by its down-wash (\footnotesize{F}⃝). Conversely, the DRSM produces
a large area of stresses in a rod-like state, there (\footnotesize{G}⃝).

In conclusion, none of the RANS models is able to produce an overall
satisfying picture of the turbulence in the secondary flow region. But while
the Menter SST $k$-$\omega$ model is able to predict the levels of turbulence
intensity within the PV reasonably well, this is not the case with the
SSG/LRR-$\omega$ model and could be seen as a major shortcoming of this
approach. Neither model is able to accurately produce the high turbulence
areas where fluid is lifted from the walls. In terms of turbulence anisotropy,
the SSG/LRR-$\omega$ cannot reproduce many of the distinctive features of the
secondary flow system seen in the LES results, despite being able to model the
vortex stretching in the free stream.

##  5 Conclusions

We have presented a new well-resolved LES dataset of the MTU T161 at
$\mathrm{Ma}_{2,\mathrm{th}}=0.6$, $\mathrm{Re}_{2,\mathrm{th}}=90000$ and
$\alpha_{1}=41^{\circ}$ obtained with a high-order DG method with a focus on
the appropriate reproduction of inflow turbulent boundary layers and
freestream turbulence. Average blade loading and pressure loss distribution
attributed to secondary flow features computed by the LES agree well with the
experiment and underline the validity of the presented approach. However, the
case poses a challenge to RANS models due to the laminar-to-turbulent
transition at midspan and the complex secondary flow system at the end walls.
The analysis was conducted using an LEVM and a DRSM, both coupled with a
state-of-the-art topology-independent transition model. With the LES reference
at hand, boundary conditions could be reproduced consistently with the RANS
setup and all differences between the simulation approaches can be attributed
to physical modelling problems. At midspan, no appropriate prediction of the
separation-induced transition could be obtained leading to a too narrow wake
of the blade. Because the midspan flow solution is shown to only weakly effect
on the secondary flow system, we could still obtain a meaningful assessment of
the RANS model performance in this region. Both models overestimate local
total pressure losses in the wake due to the PV and TSV, which can be traced
back to the development of the vortex system beginning with the HSV at the
blade LE. Although the DRSM shows a more appropriate representation of
turbulence anisotropy throughout the flow field, it suffers from an
underprediction of turbulence associated with large scale vortices, leading to
too compact and persistent secondary flow structures. In summary, we were able
to identify important shortcomings in common RANS approaches, which can be
addressed in the future due to the improved consistency between LES and RANS
setup in contrast to the traditional comparisons with experiments.

## Declarations

  * •

Funding: No external funding was received for this work.

  * •

Conflict of interest: The authors declare that they have no conflict of
interest.

  * •

Ethical approval: Not applicable.

  * •

Informed consent: Not applicable.

  * •

Authors’ contributions: C.M. wrote the main manuscript text, prepared the
figures, performed post-processing and the RANS simulations. M.B. developed
the DG solver and wrote the part of the manuscript describing it. A.T.
performed the LES and post-processing. B.K. contributed to the post-
processing. M.F. contributed the experimental data. All authors reviewed the
manuscript.

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  *[CFD]: Computational Fluid Dynamics
  *[RANS]: Reynolds-averaged Navier-Stokes
  *[LES]: Large Eddy Simulation
  *[DNS]: Direct Numerical Simulation
  *[LPT]: low-pressure turbine
  *[FV]: Finite Volume
  *[FD]: Finite Difference
  *[DG]: Discontinuous Galerkin
  *[LEVM]: Linear Eddy Viscosity Model
  *[DRSM]: Differential Reynolds Stress Model
  *[DGSEM]: Discontinuous Galerkin Spectral Element Method
  *[STG]: Synthetic Turbulence Generator
  *[MUSCL]: Monotonic Upstream-centered Scheme for Conservation Laws
  *[LE]: leading edge
  *[TE]: trailing edge
  *[LSB]: laminar separation bubble
  *[TSV]: trailing shed vortex
  *[PV]: passage vortex
  *[DRSMs]: Differential Reynolds Stress Model
  *[LIC]: line integral convolution
  *[HSV]: horse shoe vortex
  *[LECV]: leading edge corner vortex
  *[PS]: pressure side
  *[LEVMs]: Linear Eddy Viscosity Model