Unsteady Loss Mechanisms in Low-Pressure Turbines With Diverging End-Walls Studied Via High-Fidelity Simulation

In order to maximize the potential of modern geared turbofans, the low-pressure turbine (LPT) commonly features higher rotational speeds leading to a reduction in core engine size. In such scenarios, secondary flow effects can have increased impact on blade aerodynamics and overall performance. Furthermore, unsteadiness due to periodically incoming wakes, representative of LPT stage cases, increase the complexity for accurate time-averaged and time-resolved loss predictions. Loss generation mechanisms must be understood and well-predicted to better guide the development of more efficient designs and configurations in LPT design cycles. Generally speaking, the understanding of secondary flow phenomena in LPTs is well-documented for steady-state flow conditions, e.g., by Sieverding [1] and Goldstein and Spores [2], and their relation to loss generation is reported, e.g., by Langston [3]. The unsteady nature of blade-to-blade interactions and mechanisms driving unsteady losses, on the other hand, is challenging to predict. Most commonly, low-fidelity computational fluid dynamics (CFD) analyses and laboratory experiments have been standard industrial tools to yield detailed physical understanding. On the one hand, numerical studies are predominately carried out using unsteady Reynolds-averaged Navier–Stokes (URANS) analyses and compared to experimental campaigns, e.g., in Ref. [4]. The accuracy of URANS, driven by underlying turbulence modeling assumptions, however, is notoriously challenged by the complex flow conditions within LPTs, including strong pressure gradients, separated flow, and intermittent laminar to turbulent transition [5]. Experimental cascade testing, on the other hand, is time-consuming, costly, and notoriously difficult to instrument. Even though unsteady rotor-like wakes can be generated using upstream moving bars, engine-relevant flow coefficients and reduced frequencies often cannot be reproduced due to mechanical constraints [6,7].

With recent advancements in massively parallel and accelerated high-performance computing facilities, high-fidelity simulations are a promising additional tool for the detailed analysis of turbomachinery flows to URANS and experiments. They are not bound by mechanical constraints and, depending on resolution, only model the dissipation region while the dynamically relevant fluid flow scales are fully resolved. Engine-relevant operating conditions can now be simulated. Highly resolved, large eddy simulations (LES) and direct numerical simulations (DNS) have been validated against experimental data in the past and have been able to shed light on the physical understanding of unsteady LPT flows [8,–11]. Michelassi et al. [9,12] conducted DNS and LES of LPTs with upstream moving bars, discussing their effect on aerodynamic performance of the highly loaded T106 LPT blade. The studies comprised a set of cases with varying flow coefficient and reduced frequencies showing that the wake-to-wake width could be correlated to loss generation in the mid-span wake. To minimize computational requirements, spanwise periodic boundary conditions were employed. Taking spanwise end-wall effects into account, Cui et al. [13] and Koschichow et al. [14] studied the same cascade using low- and high-fidelity methods. Both cases reported time-dependent variations of mid-span profile losses due to intermittent separation of the suction side boundary layer, while deviations in secondary losses remained small compared to profile losses. Robison and Gross [15] concluded a stronger dependence if incoming wakes are generated by upstream rotor blades instead of cylindrical bars at a Mach number of 0.1. By contrast, Lopes et al. [16] reported a 10% increase of secondary losses due to moving bars at engine-relevant Mach numbers.

However, in all of these cases, the spanwise end-walls were perfectly parallel. Modern LPT configurations, on the other hand, feature divergent gas paths. The effect of tapering can lead to a significant increase in secondary losses [17] over a range of Mach number and Reynolds number and must be taken into account for modern LPTs. On top of that, axial-velocity-density-ratio effects further lead to an additional pressure gradient in the mid-span region which cannot be simulated using spanwise periodic boundary conditions or spanwise parallel end-walls [18].

In summary, the available literature is scarce when engine-like geometries are of interest at engine-relevant conditions subject to varying incoming wakes. The present investigation aims to close this gap, complementing existing knowledge and providing insights into unsteady loss generation mechanisms studied via high-fidelity simulations. The MTU-T161 LPT cascade featuring spanwise divergent end-walls at a tapering angle of $12$ deg is chosen as a numerical test rig representative of modern LPT cascades. Its steady flow conditions have been widely discussed using high-fidelity simulations in the past, i.e., in Refs. [19–23]. A brief summary of key aerodynamic parameters, numerically determined in the absence of incoming wakes [23], and geometrical parameters is given in Table 1. At engine-relevant conditions of Reynolds number of 90,000, based on isentropic exit velocity and chord length, and Mach number of 0.6, the turbine blade features a closed separation bubble at the mid-span and significant secondary flow structures. This configuration was also considered by Morsbach et al. [22], who compared steady and unsteady inflow conditions by means of upstream moving bars, although only at a single operating point. The time-averaged results revealed that the incoming wakes are able to completely suppress the suction side separation bubble. The total pressure losses downstream of the blade increased over the whole span taking into account the losses due to the mixing behind the moving bars.

In the present study, we complement the insights from Morsbach et al. by performing a parametric sweep study. A systematic variation of flow coefficient and reduced frequency is presented via high-fidelity methods comprising a set of eight highly resolved large eddy simulations. The data-rich results establish a database for three-dimensional losses given wake-induced losses to elucidate the driving mechanisms behind periodically incoming, rotor-like wakes impinging on the stator blade.

The paper is structured in the following way. First, the numerical setup and parameter space is outlined. Second, a grid convergence study is presented and validation against experimental data benchmarks the present high-fidelity simulations. Third, focus is put on the mid-span section highlighting the importance of additional pressure gradients due to the spanwise divergent end-walls on the unsteady (profile) losses. Afterwards, end-wall associated losses are considered and phase-averaged results are presented. Lastly, insights and understanding of the data-rich results are summarized.

The flow solver of choice is the in-house CFD solver high-performance solver for turbulence and aeroacoustic research (HiPSTAR), which has specifically been developed to undertake highly resolved LES and DNS efficiently on the latest high-performance computing architectures, with a detailed description given in Ref. [8]. In the context of turbomachinery flows, HiPSTAR has been extensively validated for compressor, LPT, and high-pressure turbine blades [5,24–26]. Spatial discretization is achieved using a fourth-order accurate, finite-difference scheme. To advance the solution in time, an ultra-low storage, frequency optimized, fourth-order explicit Runge–Kutta method is used. For the present LES, the wall-adaptive, local eddy model is used as the subgrid-scale model [27]. Non-reflecting, zonal characteristic boundary conditions are employed at the outlet [28]. Periodicity is assumed in the pitchwise direction. The mesh configuration is based on a stable overset technique [29] with O-grids around the blades and bars to resolve the boundary layers adequately. These O-grids are embedded into two background H-grids which resolve the broader turbulent flow field as shown in Fig. 1 for multiple bar configurations. Each bar has a diameter of 2 mm to simulate rotor-like wakes. The sliding interface condition between the H-grids is based on the message passing interface halo exchange and subsequent fourth-order accurate Lagrange interpolation in the pitchwise direction. At the domain inlet, Riemann boundary conditions are prescribed with turbulent fluctuations based on a digital filter method [30] and have been calibrated in pre-curser simulations to match the decay of turbulent intensity in mid-span according to experimental data [31]. However, the experimentally determined end-wall boundary layer state cannot be reproduced numerically due to the nature of the test facility and the turbulent grid characteristics used in the measurement campaigns [32]. As the boundary layer state can influence the end-wall loss generation significantly, the simulations in this study use the same base inflow profile as Profile A in Ref. [23], which was found to result in good agreement with the experiments in the end-wall region of the blade wake.

First, a grid convergence and validation study is performed by comparing against experimental data of Entlesberger et al. [33] to justify the numerical setup and subsequent simulations. In order to minimize computational costs, the convergence study is based on a single blade with one upstream moving bar. The grid resolution of the steady configurations reported in Ref. [23] are taken as a baseline case. A further refined case, labeled fine, yields DNS resolution in the spanwise direction, necessary to converge the skin friction coefficient on the blade suction side of LPTs [18]. The final grid counts are given in Table 2. For reference purposes, the hours required to simulate one convective time unit $tc=C/‖u¯‖2is$ on graphical processing units (GPUs), based on chord length $C$ and isentropic exit velocity $‖u¯‖2is$⁠, is appended alongside the total number of GPUs per simulation (NGPUs). The point spacings around the blade normalized on the local viscous length scale $δν=ν¯/uτ¯$ are denoted as $Δn+$ in the wall normal direction, $Δs+$ in the blade tangential direction, and $Δz+$ in the spanwise direction. Here, the wall friction velocity $uτ=τw¯/ρ¯$ is based on the wall shear stress $τw=sgn(τw,x¯)τw,x¯2+τw,y¯2$ and density $ρ¯$⁠. Looking at the two top graphs of Fig. 2, $Δn+<1.0$ is satisfied and $Δs+<20.0$ ensures high quality streamwise resolution along the pressure side with $s/Cax<0.0$ and suction side $s/Cax>0.0$⁠. The third graph outlines the differences between the two grids and reveals that the fine grid maintains DNS resolution of $Δz+<10$ over the entire blade. In terms of velocity gradients at the wall, the skin friction coefficient $cf¯=τw¯/(pt1−p2)$⁠, based on total pressure at station 1, $pt1$⁠, and static pressure at station 2, $p2$⁠, shows grid sensitivity in the separation bubble region, where $cf¯<0.0$⁠, which is expected and shown in the bottom plot. An adequate DNS-like resolution in the spanwise direction is therefore necessary to fully resolve the velocity gradients in the separated region along the blade [18]. The flow coefficient is commonly defined as

Lastly, experimentally traversing the bars along the entire span through slots results in a leakage flow at the spanwise end-walls altering the experimental operating conditions. The numerical operating conditions are therefore iteratively approximated via the baseline grid. The calibrated flow coefficient and reduced frequency are determined as $Φcalibrated≈‖u‖∞ax/ubar=2.49$ and $Fred,calibrated=0.20$⁠, respectively. The resulting blade loading and total pressure deficit in the wake are shown in Fig. 3. In the top graph, the blade pressure $p¯$ at the wall is non-dimensionalized using a reference pressure taken upstream of the moving bars in accordance with experimental data. Both grids are able to accurately capture the experimentally obtained data, giving confidence in the chosen numerical setup and subsequent parametric sweeps.

The present study investigates the effect of varying reduced frequency and flow coefficient on the aerodynamic performance of a high-lift LPT blade. Changing the bar speed results in a change of the flow coefficient $ϕ$⁠. In the simulations reported here, the flow coefficient approximating that of the experiment is labeled as 1u, and is subsequently doubled, 2u, and tripled 3u. For fixed flow coefficients, the reduced frequency can then be modified by varying the bar-to-blade count. Numerical setups with one, two, and four bars are named 1b, 2b, and 4b, respectively. The resulting combinations of flow coefficient and reduced frequency are summarized in Table 3.

The resulting wake-to-wake spacing non-dimensionalized by chord length, $Gw/C$⁠, evenly covers the value range reported in Ref. [12] for the mid-span section of a T106A profile. Small values of $Gw/C$ are associated with wake fogging, a phenomenon where the incoming wakes merge before entering the blade passage and the incoming disturbances, as a result, loose their discreteness and resemble more closely inflow turbulence with an elevated level [12]. On the other hand, large values of $Gw/c$ are associated with discrete wakes entering the passage leading to profound unsteadiness around the blade. In summary, the chosen parameter space covers engine-relevant conditions for LPT designs which are typically around $Fred≈0.6,ϕ≈0.75$ with respect to the definitions given above spanning the simulations 1b3u, 2b1u, 2b2u, and 2b3u [6]. Less engine-realistic conditions achieved by extreme values of $ϕ$ and $Fred$⁠, such as 1b1u, 4b1u, and 4b2u, are necessary to determine trends as a function of $Gw/C$ and validate the simulations against the experimental data approximated with 1b1u. In contrast to Ref. [22], the total pressure drop and flow turning associated with the moving bars are not corrected for, which is consistent with previous parametric sweep studies [12]. The resulting change of inlet flow angle behind the bars, $α1$⁠, reported in Table 3, shows little deviation with increasing slide speed and bar count for almost all simulations, except the extreme case of four bars and the doubled slide speed (4b2u). Here, the absolute flow angle reduces about 6 deg from the nominal case (1b1u) and the isentropic exit Reynolds number $Re2is$ decreases by less than 3% which is expected. However, for this case, due to the very small value of $Gw/C$⁠, we expect significant wake fogging and therefore less contribution of the unsteady losses. On the other hand, the outlet flow angle $α2$ and isentropic exit Mach number $Ma2is$ remain nearly constant for all cases. The resulting drift of operating conditions and inflow angle can therefore be assumed to have negligible effects. At the spanwise end-walls, the moving bars distort the prescribed boundary layer profile resulting in a drop of momentum thickness Reynolds number $Reθ1,z=0=θρBLEuBLE/μBLE$ at Sec. 1. With more incoming wakes, i.e., lower values of $Gw/C$⁠, the end-wall boundary layers get disturbed more drastically resulting in a further drop of $Reθ1,z=0$ summarized in the last row of Table 3.

The probability density function (PDF) of the $cf$ time-series is additionally indicated by the gray-scale map ranging from light-gray (low values of the PDF) to black (high values of the PDF) around the time-averaged mean. Across all simulations, the PDF shows high concentration upstream of the separation onset (i.e., small variation around the mean) while downstream of the separation a much broader spectrum of gray scales is observed. Here, positive and negative areas of $cf$ are found, indicating the intermittent opening and closing of the separation bubble due to the incoming wakes impinging on the blade.

The time-averaged size of the separation bubble is computed as $lbubble=(sreattachment−sseparation)$ where $s$ denotes the blade path length. The bottom right graph in Fig. 5 plots $lbubble/C$ over the wake-to-wake width $Gw/C$ across the eight LES cases. The circle color represents the simulations titled in their respective $cf$ graph, e.g., orange denotes the 1b2u case. One can observe that the bubble length as a function of $Gw/C$ collapses onto a trend line indicated by the dashed black line with a local minimum around $Gw/C≈0.4$⁠. This trend, first reported for spanwise periodic setups [9,12], therefore also appears to hold true in the presence of an additional axial pressure gradient introduced by the divergent case path and for a section (T161) with a different loading than the T106A.

A detailed discussion of the evolving secondary flow phenomena have been provided in previous work, for steady conditions in Ref. [23] and for a single set of $Φ$ and $Fred$ in Ref. [22], and will therefore not be repeated here. This paper focuses on the evolution of those structures with varying bar wake-to-wake width $Gw/C$⁠. Looking at phase-averaged results for a fixed sliding speed and various bar configurations, Fig. 8 reveals multiple phase-dependent mechanisms. For the three bar configurations shown, four phases have been chosen from the interval $ϕ∈[0;2π]$ spanning one single bar wake-to-wake phase. At mid-span, the phase-averaged turbulent kinetic energy $k~=1/2(ui′ui′~)=1/2((ui−ui~)(ui−ui~)~)$ highlights the bar wakes convecting through the passage. All surfaces are colored by the skin friction coefficient and streaklines are represented by a line integral convolution of the wall shear stress vector. White iso-lines of $cf~=0$ further indicate separated flow regions. Toward the spanwise end-walls, three axial constant slices show the phase-averaged streamwise vorticity $ωsw~=u⋅ω/‖u‖~$ in Cartesian coordinates $ω$⁠.

The passage vortex (PV), originating from the pressure-sided leg of the horse-shoe vortex, is the concentrated region of blue color in the -marked graph. The trailing shed vortex (TSV) is colored in red emerging from the trailing edge and rotating in the opposite direction to the passage vortex. Looking at phase $ϕ=π/2$ of the one bar configuration, the bar wake is close to the trailing edge. As a result, the separation bubble decreases in size visible by the smaller enclosed region of the white iso-lines. Toward the spanwise end-wall, the wake mixes with the emerging passage vortex.

The next phase, $ϕ=π$⁠, shows locations of fully attached flow which is likely a result of spanwise-varying turbulent spots [22]. Compared to , the separation line of the passage vortex moves downstream, indicated by . Its overall structure remains intact despite the high level of fluctuations from the unsteady wakes.

At $ϕ=3/2π$⁠, the wake has convected through the passage. In its absence, the separation bubble widens and the passage vortex recovers close to the trailing edge. Comparing at the same phase to the case with two bars shows a smaller separation region for the latter as the overall level of unsteadiness increases in the passage region.

The separation bubble grows and shrinks depending on the phase with locally fully attached flow at $ϕ=3/2π$ while the passage vortex separation line remains nearly unaffected. However, due to increased levels of turbulent kinetic energy, the separation line is mostly attached for the two bar case. This results in a weakening of the passage vortex and subsequently the trailing shed vortex. Comparing the axial constant slices at the trailing edge of the 1b2u and 2b2u cases highlights lower values of streamwise rotation across all phases for the two bar configuration confirming the weakening of secondary flow structures for cases with more bars. Unsurprisingly, the four bar configuration, associated with wake fogging, shows little to no variation over the different phases. The high background turbulence level resulting from the interaction and mixing of the incoming bar wakes, clearly visible in mid-span, suppresses the detachment of the passage vortex from the blade surface indicated by . Therefore, and due to strong mixing in the passage, the passage vortex and trailing shed vortex are less profound downstream of the wake.

The evolving secondary vortex patterns and their relation toward loss generation in a time-averaged manner can be visualized by contour plots of the total pressure loss coefficient (see Eq. (5)) across all eight LES cases in Fig. 9. In all subplots, the spanwise coordinate is normalized over the local spanwise extend on the x-axis and the normalized pitchwise coordinate is plotted on the y-axis. In the top left plot, the 1b1u further highlights the PV, TSV, and the end-wall separated region (ES) for clarity. Increasing the bar speed for cases with one bar is shown in the left column. Looking from 1b1u to 1b3u reveals a gradual decrease in the size of the trailing shed vortex. Simultaneously, the passage vortex decreases in strength and appears to merge with the trailing shed vortex. This is caused by increased perturbations of the secondary vortex system, which forms as a consequence of the spanwise end-walls, due to the presence of more wakes, i.e., with decreasing $Gw/C$⁠. A further decrease in $Gw/C$⁠, by increasing the bar count, leads to an amplified disturbance of the secondary vortex system and a further reduction in size of the trailing shed vortex. Interestingly, in case of wake fogging, which occurs for the 4b2u case, the passage vortex starts to grow in strength again which can be seen when comparing to the 1b3u or 2b3u cases. This can be explained by the fact that the passage vortex and trailing shed vortex start to spatially separate further due to high background turbulence levels rather than discrete wakes. On the other hand, the end-wall region showing non-zero $ζ$⁠, labeled ES, is continuously decreasing in size when decreasing $Gw/C$⁠, i.e., comparing 1b1u and 4b2u. This is a result of a decreased inflow boundary layer momentum thickness at the spanwise end-walls upstream of the blade (see Table 3) due to the mixing of the boundary layer with the moving bars. Simulations in the absence of incoming wakes have shown that decreased levels of incoming end-wall boundary layer momentum thickness directly correlate to the end-wall associated losses in the wake [23]. This holds true for the unsteady cases as more incoming wakes continuously disturb the freestream boundary layer resulting in a drop of $Reθ1,z=0$ and, hence, less total pressure drop in the wake.

The present study reports on highly resolved large eddy simulations of a linear low-pressure turbine cascade in a diverging gas path. Rotor-like wakes are generated by upstream moving bars and periodically impinge on the blade leading edge. A grid convergence study and bench-marking against available experimental data validate the present numerical setup. In total, eight LES comprise a set of systematically varying flow coefficient and reduced frequency such that the wake-to-wake width $Gw/C$ is evenly distributed over a wide range. While the total pressure loss and flow turning caused by the bar wakes are not corrected, their impact on operating conditions such as isentropic exit Reynolds number and isentropic exit Mach number remains small.

First, the mid-span section is analyzed. The separation bubble on the blade suction side is heavily influenced by the intermittent wakes impinging on the blade. Comparing phase-averaged results, high values of $Gw/C$ result in strong, intermittent lengthening and shortening of the separated region due to the high-amplitude unsteadiness of the wakes periodically disturbing the blade boundary layers. On the other hand, wake fogging and merging is associated with low values of $Gw/C$⁠. This leads to elevated levels of background turbulence in the passage, rather than discrete wakes, experienced by the blade boundary layers. The time-averaged bubble size therefore shows a minimum between the extreme cases of low and high wake-to-wake width. The total pressure loss coefficient in the wake downstream of the blade follows a similar trend as the separation bubble size with respect to $Gw/C$⁠. However, its local minimum is at a decreased $Gw/C$ value compared to the minimum of the bubble size. The trend line of the wake loss over $Gw/C$ is opposite to those in previous spanwise periodic studies on the T106A profile which is caused by two distinct phenomena. First, the time-averaged suction side separation bubble decreases in size when $Gw/C$ decreases, with a local minimum between the extreme values of $Gw/C$⁠. And, second, the additional axial pressure gradient in mid-span impacts the incoming wakes positively by reducing the total pressure loss at the wake onset of its suction side related branch.

The profile losses calculated following Denton’s loss analysis are subtracted from the LES losses in order to isolate the wake distortion losses. For the current case, an identical trend of the wake distortion losses over wake-to-wake width is observed compared to the previous spanwise periodic studies on the T106A profile. The maximum wake distortion losses are associated with high unsteadiness of the blade boundary layer state along the suction side. In this scenario, the incoming discrete wakes are able to intermittently fully suppress the separated region. Yet, the wake-to-wake width is sufficiently high for the separation bubble to recover before the next wake affects it. On the other hand, the minimum of the wake distortion losses occurs at low unsteadiness around the separation bubble and the minimum of the total pressure loss coefficient. At low values of $Gw/C$⁠, the wake distortion losses are also decreased as the incoming wakes merge. In summary, the wake distortion losses are characterized by their interaction with the blade profile and the axial pressure gradient rather than the unsteady nature of the incoming wakes themselves.

The secondary flow system toward the spanwise end-walls is found to remain mostly intact for all cases considered. The incoming secondary structures of the bars are not able to fully suppress the formation of the horse-shoe vortex which feeds the passage vortex and trailing shed vortex. However, the passage vortex and trailing shed vortex significantly reduce in strength with decreasing wake-to-wake width and are therefore more sensitive toward elevated background turbulence levels in the passage rather than discrete incoming wakes. The end-wall losses caused by the end-wall boundary layer deficit are decreasing in a similar manner due to the increased mixing of the freestream boundary layer with the moving bars upstream of the blade and a subsequent reduction in end-wall boundary layer momentum thickness. The trend lines of the secondary vortex system toward loss generation are therefore steadily decreasing with decreasing values of $Gw/C$⁠. Comparing the total integrated loss coefficient across all simulations reveals a similar trend to those of the profile losses, with the profile losses being the predominant source of loss generation and highlighting the importance of mid-span blade design. However, for the $2b3u$ configuration showing the smallest total pressure loss coefficient, secondary losses amount to 50% of the total losses.

The bulk of the computing time has been provided by the Pawsey Supercomputing Center in Perth, Australia, awarded by the National Computational Merit Allocation Scheme. Furthermore, this research was undertaken using the LIEF HPC-GPGPU Facility hosted at the University of Melbourne. This Facility was established with the assistance of LIEF Grant LE170100200. Experimental data were provided under a non-disclosure agreement between MTU Aero Engines, A.G. and The University of Melbourne.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

$l$ =

length

$p$ =

pressure

$u$ =

velocity vector in Cartesian coordinates

$C$ =

chord

$cf$ =

skin friction coefficient

$cp$ =

pressure coefficient

$x,y,z$ =

Cartesian coordinates

$α$ =

absolute flow angle

$Δ$ =

wake distortion losses

$Δs+$ =

viscous point spacing in the blade tangential direction

$Δn+$ =

viscous point spacing in the blade normal direction

$Δz+$ =

viscous point spacing in the blade spanwise direction

$ζ$ =

total pressure loss coefficient

$ν$ =

kinematic viscosity

$ρ$ =

density

$τw$ =

wall shear stress

$uτ$ =

wall friction velocity

$Φ$ =

flow coefficient

$ϕ$ =

bar passing phase

$1$ =

quantity at section 1

$2$ =

quantity at section 2

$m$ =

mixed-out value

$t$ =

total

$ax$ =

axial

$is$ =

isentropic

BLE =

boundary layer edge

LE =

leading edge

TE =

trailing edge

$¯$ =

time-averaged value

$~$ =

phase-averaged value

$∞$ =

freestream value

$Fred$ =

reduced frequency

$Gw$ =

wake-to-wake width

Ma =

Mach number

Re =

Reynolds number

Zw =

Zweifel number