# LES/DNS of flow past T106 LPT cascade using a higher-order LB model

N. H. Maruthi111Technical Lead, Aerospace and Automotive group, AIAA Member,
Email address<EMAIL_ADDRESS>SankhyaSutra Labs Ltd.,
Bengaluru, Karnataka, India, 560047  Chakradhar Thantanapally 222Technical
Manager, Aerospace and Automotive group, AIAA Member, Email address:
<EMAIL_ADDRESS>SankhyaSutra Labs Ltd., Bengaluru, Karnataka,
India, 560047  Manjusha Namburi 333Former PhD student, Email address:
<EMAIL_ADDRESS>Jawaharlal Nehru Centre for Advanced Scientific
Research (JNCASR), Bengaluru, Karnataka, India, 560064  V. Kumaran
444Technical Advisor, also professor at the Indian Institute of Science,
Bengaluru, Karnataka, India, 560012, Email address:
<EMAIL_ADDRESS>SankhyaSutra Labs Ltd., Bengaluru, Karnataka,
India, 560047  Santosh Ansumali555Chief Technology Officer (CTO), also
professor at the Jawaharlal Nehru Centre for Advanced Scientific Research
(JNCASR), Bengaluru, Karnataka, India, 560064, Email address:
<EMAIL_ADDRESS>SankhyaSutra Labs Ltd., Bengaluru, Karnataka,
India, 560047

###### Abstract

The main objective of the present work is to assess higher-order Entropic
Lattice Boltzmann Method (ELBM) for separated and transitional flows without
the use of any explicit turbulence model. For this, we chose to simulate two
cases of T106 Low-Pressure Turbine (LPT) cascade —T106A and T106C
—representing incompressible and compressible flow regimes respectively. These
results are obtained using our company’s in-house higher-order ELBM transonic
solver. We have carried out two sets of simulations for both the test cases.
One with a clean inlet and the other with an inlet disturbance given by white
Gaussian noise superimposed on the inlet velocity. For the clean inlet case,
the flow remains laminar on the entire blade surface for both the test cases.
It undergoes transition on the suction side for the inlet disturbance case.
The pressure coefficient for the T106A and the isentropic Mach number on the
blade surface for the T106C matches well with the experimental results. Also,
the qualitative comparison of flow features in terms of the Q-criterion is in
good agreement with the earlier computational results reported in literature.

##  1 Introduction

The flow through a gas turbine is known to be very complicated and, in
particular, on the blades it is often called a flow zoo. This is because it
exhibits a rich diversity of flow phenomena such as separation, transition,
relaminarization and effects of the high curvature of the blade on the
boundary layer, heat transfer, etc. RANS models which are routinely used in
industry for design are inadequate in accurately predicting these features in
turbulent flows. Because of these and other reasons, the gas turbine engine is
known to be one of the most complex pieces of mechanical engineering. An
accurate understanding of these flow features requires DNS/LES like
simulations on a well-resolved mesh. To carry out these simulations, we use
our in-house Entropic Lattice Boltzmann Method (ELBM) solver. The details of
the model are tabulated in Table 1. LBM is a mesoscopic method that solves the
Boltzmann equation using an appropriate probability distribution function.
Macroscopic properties such as density, velocity, pressure, temperature, etc.
are derived as moments of the distribution function. LBM is becoming an
increasingly popular alternative to traditional CFD methods owing to its
simplicity and scalability on parallel computing architectures [1, 2].
However, the standard LB methods were unstable due to the lack of the discrete
H-theorem, especially when the mesh is under- resolved [3, 4]. A class of new
LB methods called Entropic Lattice Boltzmann Models (ELBM) are developed to
overcome these stability issues [5, 6, 7, 8, 9]. The recently introduced
crystallographic LBM further improves the capability of LBM by optimally
sampling the domain on a Body-Centred Cubic (BCC) lattice instead of the
standard Simple Cubic (SC) lattice [10]. The BCC lattice also improves the
discretization of geometries in the computational domain. Furthermore, higher-
order LB methods have been introduced to extend the applicability over a wider
range of Mach and Knudsen numbers. These higher-order models are characterized
by improved stability, accuracy, and thermal properties [11, 12]. The use of
ELBM ensures a seamless transition from LES to DNS based only on the
increasing the grid resolution without any empirical modelling for small
scales. This ensures that the flow features near the wall are accurately
resolved by performing DNS near the wall when the fine resolution is used.
Under-resolved simulations give results similar to that of LES.

Shells | Discrete Velocities($c_{i}$) |  Weight($w_{i}$)
---|---|---
SC - 1 | $\left(\pm 1,0,0\right),\left(0,\pm 1,0\right),\left(0,0,\pm 1\right)$ |  0.0111443971
SC - 2 | $\left(\pm 2,0,0\right),\left(0,\pm 2,0\right),\left(0,0,\pm 2\right)$ |  0.0009925820
SC - 3 | $\left(\pm 3,0,0\right),\left(0,\pm 3,0\right),\left(0,0,\pm 3\right)$ |  0.0004322964
FCC-1 | $\left(\pm 1,\pm 1,0\right),\left(0,\pm 1,\pm 1\right),\left(\pm 1,\pm 1,0\right)$ |  0.0078020595
FCC-2 | $\left(\pm 2,\pm 2,0\right),\left(0,\pm 2,\pm 2\right),\left(\pm 2,\pm 2,0\right)$ |  0.0008848260
FCC-3 | $\left(\pm 3,\pm 3,0\right),\left(0,\pm 3,\pm 3\right),\left(\pm 3,\pm 3,0\right)$ |  $6.6974\times 10^{-6}$
BCC-1 | $\left(\pm 1,\pm 1,\pm 1\right)$ |  0.0071479427
BCC-2 | $\left(\pm 2,\pm 2,\pm 2\right)$ |  0.0000121756
BCC-h | $\left(\pm 1/2,\pm 1/2,\pm 1/2\right)$ |  0.0447801239
BCC-3h | $\left(\pm 3/2,\pm 3/2,\pm 3/2\right)$ |  0.0026467301
BCC-5h | $\left(\pm 5/2,\pm 5/2,\pm 5/2\right)$ |  $4.8164\times 10^{-6}$
offdiagonal BCC- h |  $\left(\pm 1/2,\pm 1,\pm 1\right),$ $\left(\pm 1,\pm 1/2,\pm 1\right)$, $\left(\pm 1,\pm 1,\pm 1/2\right)$ |  0.0103828723
offdiagonal BCC-1 |  $\left(\pm 2,\pm 1,\pm 1\right),$ $\left(\pm 1,\pm 2,\pm 1\right)$, $\left(\pm 1,\pm 1,\pm 2\right)$ |  0.0002149798
offdiagonal FCC | $\left(\pm 2,\pm 1,0\right),\left(0,\pm 2,\pm 1\right),\left(\pm 2,\pm 1,0\right)$ |
| $\left(\pm 1,\pm 2,0\right),\left(0,\pm 1,\pm 2\right),\left(\pm 1,\pm 2,0\right)$ |  0.0041472523
Table 1: Energy shells and their corresponding velocities with weights for the
RD3Q167 model.

##  2 Computational setup

###  2.1 T106A

The Low-Pressure Turbine (LPT) blade geometry (T106) chosen for simulation is
of the Pratt and Whitney rotor named PW2037. The blade has a chord length of
198mm, an axial chord length of 170mm, and a pitch of 158mm [13]. The
computational setup consists of one T106 blade with periodicity in a pitch-
wise direction to represent the turbine passage. The inflow boundary is
located at $C_{ax}$ (axial chord length) upstream from the leading edge, and
the outflow boundary is located at $2C_{ax}$ downstream from the trailing edge
as shown in Fig. 1. The computational domain’s spanwise extent is chosen as
20% of the axial chord length. Uniform inflow and constant outflow boundary
conditions are used for the inlet and outlet. Periodic boundary conditions are
used in cross-stream and spanwise directions. The diffuse bounce back method
is applied on the blade surface [14]. The multi-resolution approach with a
finer resolution near the blade surface and coarser-resolution away from the
blade reduces the computational cost (see Fig 1). Flow parameters chosen for
simulations are similar to those of Kalitzin et al. [15]. The inflow Mach
number is $0.1$, the inlet flow angle is $37.7^{o}$, and the Reynolds number
based on axial chord length is $148,000$.

Figure 1: Computational setup used in the simulations is shown along with
distribution of grid resolution with the finest near the surface to accurately
capture the boundary layer

###  2.2 T106C

This is one of the test cases chosen in International Workshop on High-Order
CFD Methods (HiOCFD) to assess the CFD solvers for their ability to resolve
separated and transitional flows[16]. This is a high-lift blade with higher
blade loading, leading to large separation on the suction side [17]. This
blade has a chord length of 93.01mm, an axial chord of 79.9mm and a pitch of
88.36mm. The computational domain and setup for T106C is similar to that of
T106A with the same domain length in all the three directions. Total pressure
and temperature based subsonic inflow boundary condition and static pressure
outlet is prescribed. Periodic boundary conditions are specified for the
cross-stream and pitch wise directions. For this test case, exit Reynolds
number based on chord length is $80,000$, Mach number is $0.65$ and inflow
angle is $32.7^{o}.$

##  3 Results and Discussion

###  3.1 T106A

Two sets of simulations with 1024 and 1536 grid points on the chord were
carried out respectively, for both the cases of clean inlet and inlet with
disturbance. In Fig. 2, the time-averaged coefficient of pressure
($C_{p}=\frac{2(p-p_{ref})}{\rho U^{2}}$) is shown on the blade surface. The
reference pressure is the inlet pressure averaged over the pitch. As can be
seen from the figures, the $C_{p}$ result with clean inlet matches well with
the experimental results, whereas the result with inlet disturbance
overpredicts surface pressure near the trailing edge. Inlet disturbance has no
effect on the pressure side.

Figure 2: Averaged Pressure coefficient ($C_{p}$) on the surface of blade at
its mid-span

Fig. 3 shows the contours of the magnitude of instantaneous velocity compared
with the reference solution [15]. The reference solution is obtained with
inlet turbulence, which induces bypass transition. In the turbulence-free
inlet case, ideally the flow undergoes a natural transition from laminar to
turbulent flow due to the instabilities arising at the separation bubble on
the suction side. However, in the present case, we do not observe bypass
transition but the flow does become turbulent on the suction side.

Figure 3: Instantaneous velocity magnitude (with inlet disturbance (b)) in
comparison with reference solution (a) which is obtained with grid turbulence.

To visualize the transition, we have plotted the y-component of velocity on
the blade surface in Fig. 4 along with the reference solution in which the
velocity component normal to the blade surface is shown (Fig. 4c). Fig. 4a is
obtained with clean inlet, and Fig. 4b is obtained with inlet disturbance.
These figures clearly show the effect of inflow noise in tripping the flow to
turbulence as inflow disturbances penetrate the boundary layer and trigger the
separated shear layer.

(a)

(b)

Figure 4:  The v component of instantaneous velocity on the blade surface to
visualise transition. (a) clean inlet, (b) with inlet disturbance and (c)
velocity normal to surface (clean inlet)

For better visualization of flow structures near the wall, isosurfaces of
instantaneous Q-criterion are plotted in Fig. 5 and 6 for clean inlet and for
inflow with noise respectively. As can be seen from these figures, for the
clean inlet, the flow remains laminar throughout the blade surface and ordered
$\Lambda$ (Lambda) type vortices can be seen near the trailing edge. For the
inlet case with noise, $\Lambda$ structures form hairpin vortices downstream
and further break down to form small scale structures. Although the flow
structures near the blade surface are drastically different for the two cases,
the pressure coefficient is similar as evidenced in Fig. 2. The comparisons
given herein are for the qualitative flow structures (see Fig. 4, 5 and 6),
more detailed quantitative analysis of the flow field is in progress.

(a)

(b)

Figure 5: Isosurfaces of Q-criterion coloured with stream-wise velocity for
the inlet case

(a)

(b)

Figure 6: Isosurfaces of Q-criterion coloured with stream-wise velocity for
the inlet flow with noise

##  4 T106C

Both clean inlet and inlet disturbance cases are simulated with 2048 points on
the axial chord. Fig. 10(a) shows the isentropic Mach number on the airfoil
surface in comparison with experimental results. As can be seen from the
figure, the computed isentropic Mach number exactly matches with experimental
results on the pressure side, whereas it is over predicted on the suction side
near the trailing edge. This result is consistent with the earlier
computational results reported in the literature. This maybe due to the
discrepancy between experimental and computational setups, mainly the inflow
and stagger angle of the blades [18]. We can observe that the inlet
disturbance has a minor effect on the isentropic Mach number on the surface,
specifically towards the trailing edge on the suction side. Our results are
closer to Garai et al [19]., who carried out simulations with inlet turbulence
of $3.2\%$. Fig.7 shows the skin friction coefficient. Effect of inlet
disturbance can be seen on the suction side towards the trailing edge. Due to
the inlet disturbance, location of the flow separation changes. The flow
remains attached on the pressure side for both the clean inlet and inlet with
disturbances. For the clean inlet case, flow separates at $0.7C_{ax}$ and
reattaches at $0.89C_{ax}$. For the inlet disturbance case, flow separates at
$0.68C_{ax}$ and reattaches at $0.89C_{ax}$. However, Garai et al [19].,
reported a separation point of $0.67C_{ax}$ and $0.74C_{ax}$ for the clean
inlet and inlet turbulence case respectively with $0.96C_{ax}$ as a
reattachment point for inlet turbulence case.

(a)

(b)

Figure 7: Isentropic Mach number (a) and skin-friction coefficient (b) on the
surface of the blade

Fig. 8 shows the wake loss profiles in comparison with experimental and
simulation results reported in the literature. Pressure loss profiles are
extracted in the wake at $x=0.465C_{ax}$. As can be seen from figures, wake
loss profiles reported in the literature widely vary with respect to
experiments. One reason could be the discrepancy reported in the surface
pressure. Present results with clean inlet predicts the peak loss similar to
that of experiments. We have shifted the curve by $17\%$ of the pitch to match
peak and width of the wake loss measured from experiments. However, pressure
loss is under predicted for the inlet disturbance case for which the profile
has been shifted by $12\%$ to match the peak. This result is similar to that
of Mitra et al [20]., who have shifted the curve by $10\%$ and reported the
peak loss of 0.05 for both clean inlet and inlet turbulence case.

Figure 8: Total pressure loss in the wake of the blade

To understand the transition process we have plotted Q criterion coloured with
stream-wise velocity (see Fig. 9 and 10). As can be seen from figures, flow
remains attached and laminar until about $60$ to $70\%$ of the blade, beyond
which the flow separates due to adverse pressure gradient although remaining
laminar in the case of clean inlet. However, for the inlet disturbance case,
separated shear layer undergoes transition to turbulence (due to the
disturbances triggering the boundary layer), which can be clearly seen from
isosurfaes of the Q-criterion. The flow on the pressure side is laminar and
attached for both the cases.

(a)

(b)

Figure 9: Isosurfaces of Q (>0) criterion coloured with stream-wise velocity
obtained with clean inlet flow

(a)

(b)

Figure 10: Isosurfaces of Q (>0) criterion coloured with stream-wise velocity
obtained with inflow disturbance

##  5 Summary

Preliminary results on the incompressible and compressible flow past the T106
cascade have been reported. We have carried out simulations for both cases of
clean inlet and inlet with disturbances. It has been shown that the pressure
coefficient (for T106A) and isentropic Mach number (for T106C) on the surface
of the blades matches well with the experimental results. For both the turbine
blades with clean inlet, the flow remains laminar. However, with inlet
disturbances, the flow undergoes transition to turbulence. Isosurfaces of
Q-criterion show the transition zone for the inlet disturbance cases, which
qualitatively agree with the reference solutions. We have observed that with
clean inlet, the flow may not transition to turbulence. For the T106C cascade,
peak wake loss matches well with experimental data and for the case of clean
inlet while for the case of inlet disturbance, results are consistent with
those reported in the literature. Further analysis is being carried out to
study the transition and separation mechanisms over these blades.

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