# Investigation of transonic flows through an idealized ORC turbine vane using Delayed Detached Eddy simulations

Aurélien Bienner, Xavier Gloerfelt, Paola Cinnella

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Aurélien Bienner, Xavier Gloerfelt, Paola Cinnella. Investigation of transonic flows through an idealized ORC turbine vane using Delayed Detached Eddy simulations. Applied Thermal Engineering, 2025, REGAL-ORC, 261, pp.124951. 10.1016/j.applthermaleng.2024.124951. hal-05080750

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Submitted on 19 Jun 2025

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# Investigation of transonic flows through an idealized ORC turbine vane using Delayed Detached Eddy simulations

A. Bienner<sup>a,*</sup>, X. Gloerfelt<sup>a</sup>, P. Cinnella<sup>b</sup>

## ARTICLE INFO

Keywords:

ORC

Turbine

CFD

DDES

Vortex shedding

Losses

## ABSTRACT

An idealized transonic blade configuration is studied using Delayed Detached Eddy Simulations (DDES), with focus on the loss mechanisms associated with the base pressure and turbulent wake development. The working fluid is Novec649, which is commonly used in low temperature Organic Rankine Cycle (ORC) power plants. Due to the molecular complexity of the fluid, a lower pressure ratio and exit Mach number are achieved compared to an air flow through the same geometry, a phenomenon that becomes more pronounced when considering thermodynamic operating conditions that lead to stronger non-ideal gas behavior, and which is well captured by models of standard use in ORC design and analysis, such as Reynolds-Averaged Navier-Stokes (RANS) models. Overall, as the fluid isentropic exponent and the exit Mach number decrease, the losses tend to increase. When performing a breakdown of the losses, the wake region provides the dominant contribution. Unlike RANS models, the DDES resolves the largest turbulent scales in the wake region, allowing fine-detail analysis of the unsteady wake dynamics, namely vortex shedding, and its influence on the loss coefficients. In the low supersonic Mach number range considered ($M_2 \sim 1–2$), the vortex shedding exhibits a flapping motion in the reattachment region behind the trailing edge. The Strouhal number, classically formed with the diameter of the trailing edge, is found to increase with increasing outlet Mach number. However, when introducing a new scaling based on the neck width of the reattachment region, directly related to the shape of the recirculation bubble and the associated reattachment shocks, the Strouhal number collapses to a relatively constant value of about 0.3 in all cases. DDES also shows that, due to the large heat capacity of Novec649, temperature fluctuations in the wake are greatly reduced, leading to the suppression of the so-called energy separation phenomenon occurring in air flows. While RANS simulations capture satisfactorily the main features and global quantities of the flow, they fail in describing the flow around the trailing edge. This results in inaccurate estimates of the back pressure, which ultimately leads to an underestimation of the losses by about 20%. Unsteady RANS simulations can predict the wake unsteadiness, but do not capture the couplings between large coherent structures, the recirculation, and the shocks, also resulting in inaccurate performance predictions. Thus, the present result showcase the importance of using advanced scale-resolving simulations for improved analysis and design of ORC turbines.

## 1. Introduction

In most Organic Rankine Cycles (ORC), the working fluid, an organic compound, is compressed, heated, and then expanded in a turbine. The latter component can be critical for the cycle efficiency and it is of utmost importance to estimate the various loss mechanisms. Since ORC are intended to convert energy from relatively low-temperature sources, organic fluids are expanded in small-scale turbines, working in the transonic to supersonic regimes. Due to their molecularly complex structures, organic fluids operate in the so-called dense-gas regime, where the fundamental derivative of fluid dynamics $\Gamma = 1 + \frac{\rho}{c} \left( \frac{\partial c}{\partial \rho} \right)_s$

is lower than one, with $\rho$ the density, $c$ the sound speed and $\left(\frac{\partial \cdot}{\partial \sigma}\right)_s$ a derivative along an isentropic line. This results in an inverse behavior of the speed of sound compared to perfect gases: the speed of sound increases as the flow expands. Therefore, transonic turbines expanding dense vapors reach critical chocking conditions at lower Mach numbers than those characterized by $\Gamma > 1$, leading to a lower pressure ratio at iso-geometry [1]. For weakly non-ideal conditions, the gas dynamics and the associated losses are mainly driven by the isentropic exponent, defined as $\gamma_{pv} = -\frac{v}{p} \left(\frac{\partial p}{\partial v}\right)_s$, where $v = 1/\rho$ is the specific volume

<footnote><sup>a</sup> DynFluid Laboratory, Arts et Métiers Institute of Technology, 151 bd. de l'Hôpital, Paris, 75013, France</footnote>

<footnote><sup>b</sup> Jean Le Rond D'Alembert Institute, Sorbonne University, 4 place Jussieu, Paris, 75005, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: aurelien.bienner@ensam.eu (A. Bienner).</footnote>
Nomenclature

Roman symbols

c Sound speed

s Entropy

T Temperature

p Pressure

M Mach number

v Specific Volume

h Enthalpy

Re Reynolds number

ch Chord

L<sub>z</sub> Spanwise extent

N<sub>z</sub> Number of points in the spanwise direction

d Diameter

St<sub>d</sub> Strouhal number based on d

f Frequency

U Velocity magnitude

t Inter-blade distance/pitch

ω Throat width

p<sub>p</sub> Average base pressure

C<sub>p,b</sub> Base pressure coefficient

Acronyms

ORC Organic Rankine Cycle

LE Leading-Edge

TE Trailing-Edge

CFD Computational Fluid Dynamics

SRS Scale Resolving Simulation

DNS Direct Numerical Simulation

LES Large Eddy Simulation

DDES Delayed Detached Eddy Simulation

DDES-SLA DDES-Shear Layer Adapted

RANS Reynolds-Averaged Navier-Stokes

CFL Courant-Friedrichs-Lewy number

FST FreeStream Turbulence

BL Boundary Layer

Greek symbols

Γ Fundamental derivative of gas dynamics

ρ Density

γ Heat capacity ratio

γ<sub>pv</sub> Isentropic exponent

δ* Displacement thickness

θ Momentum thickness

χ<sub>f</sub> Filtering coefficient

χ Stagger angle

π Pressure ratio p<sub>0,1</sub>/p<sub>2</sub>

α<sub>0</sub> Opening angle of fishtail shocks

α<sub>tf</sub> Mean turning flow angle

β<sub>2</sub> Exit wake angle

Δν Variation of Prandtl-Meyer function

μ<sub>t</sub> Turbulent viscosity

ζ<sub>out</sub> Enthalpy loss coefficient

ζ<sub>pass</sub> Enthalpy losses in the passage

ζ<sub>TE</sub> Enthalpy losses after the TE

ζ<sub>Den</sub> Loss coefficient estimation of Denton

Subscripts

•<sub>1</sub> Inlet quantities

•<sub>0,1</sub> Total inlet quantities

•<sub>2</sub> Mixed-out quantities

•<sub>1s</sub> Isentropic quantities

•<sub>c</sub> Critical quantities

•<sub>rms</sub> Root mean square

Superscripts

• Average

•' Fluctuation

and *p* is the pressure. The experimental study of Ref. [2] showed that, for an exit Mach number *M*<sub>2</sub> greater than 0.9, the losses increase with decreasing isentropic exponent *γ*<sub>pv</sub> due to the combined effect of shock loss within the blade passage and mixing loss downstream of the blade trailing edge. Furthermore, as the non-ideality of the flow increases, for example in supercritical applications, the expansion can exhibit strong non-ideal effects, such as non-classical expansion and compression waves [3,4], and an increased sensitivity to off-design conditions [5].

Due to the combination of various mechanisms with a somehow contradictory influence, experimental and numerical databases are critical for improving turbine efficiency. Few facilities exist for dense gases and consist essentially in blowdown or suckdown tunnels, where only short-time statistics are possible [2,6]. A continuously running pressurized closed-loop wind tunnel (CLOWT) [7] using Novec649, an environmentally friendly working fluid, in the high subsonic speed range, has been built at the University of Applied Sciences Muenster. Due to the high pressures and the high-density of Novec649, the experimental investigation of the transonic linear cascade requires high compressor power. As a result, the tunnel test section has relatively small dimensions and measurements for a linear cascade are therefore challenging [8]. One way out of this is the use of an idealized vane configuration. Instead of an entire blade cascade, the test set-up is

limited to a single, geometrically simple blade with side-walls designed to produce pseudo-periodic flow conditions. This idea has been used previously to study the transonic flow around a blade trailing edge (TE) and the associated profile losses (e.g. [9,10]). An idealized vane configuration with low-curvature blades has been designed in [11] (see Fig. 1), inspired by the study of [12]. The experiments were conducted with air as the working fluid in the outlet Mach number range 0.8 to 1.7, and included Schlieren visualizations, blade pressure distribution and traversing probe measurements. While the profile lacks the typical curvature of most turbine blade profiles, it can be used as a test case to investigate fundamental phenomena such as tip gap flows in [13] or loss mechanisms with a dense gas in the present study.

In recent years, so-called scale resolving simulations (SRS), i.e. simulations resolving part of the turbulent flow dynamics, have gained more and more attention in turbomachinery analysis because of their improved ability to bring insights into the intricate fluid dynamics and its connection to the generation of losses [14] with respect to lower fidelity models. These approaches, which include direct numerical simulations (DNS) or large-eddy simulations (LES), remain extremely challenging for ORC turbines due to their computational cost, although they have recently contributed to important advances in the understanding of turbulent flows of non-ideal gases [15]. Typical ORC turbines can exhibit outlet Reynolds numbers as high as
<figure><img src="image_4.png" /><figcaption>**Fig. 1.** View of the experimental setup.
Source: From [11].</figcaption></figure>

10 millions (which is the case for the present configuration). Under such conditions, a wall-resolved LES of the full blade vane configuration would require a computational mesh of approximately 100 billions points, which is currently out of reach even with advanced high-performance computing resources. On the other hand, Computational Fluid Dynamics (CFD) tools commonly used in ORC design are mostly based on steady Reynolds-averaged Navier-Stokes (RANS) models, even if RANS is notoriously inaccurate for the prediction of several phenomena occurring in turbomachinery, such as laminar-turbulent transition, shock/boundary layer interaction, flow separation, and unsteady wakes. Little is known about the laminar-turbulent transition over ORC turbine blades and designers generally assume a fully turbulent boundary-layer state straight downstream of blade leading edge. Considering the high level of freestream turbulence (FST) intensity encountered in turbine environments, FST-induced transition is generally the principal mechanism of laminar-turbulent transition for turbine blades [14,16]. To gain knowledge on FST-induced transition in dense-gas boundary layers (BL), a study was realized for a zero-pressure-gradient Novec649 BL by means of wall-resolved LES [17]. It was found that the large-scale FST affects not only the location of transition but also its mechanism. Ref. [18] investigated the effect of a finite thickness leading edge and of the favorable pressure gradient on FST-induced transition by performing wall-resolved LES of the present idealized turbine blade nose. It was shown that, for this relatively flat geometry, transition occurs early on the blade (approximately within 6% of the chord) and that the laminar-streak breakdown is highly disrupted by large vortical structures impinging the leading edge.

Another flow region generally poorly modeled by RANS simulations is the wake region, which can account for a large part of the losses [19]. Durá Galiana and Wheeler [6] were the first to report simulations of ORC turbine cascades using higher-fidelity methods, with the aim to explore TE loss mechanisms with a combined experimental and numerical study. Despite the rather coarse grid, LES captured total pressure profiles across the wake much more accurately than RANS, leading to significant differences in losses estimations. Coarse wall-resolved LES of a turbine stator using an organic working fluid (the fluorocarbon PP11) were presented in [3] for various thermodynamic conditions and pressure ratios. The exit Reynolds number was chosen in the range $1.13 \times 10^6 \div 2.26 \times 10^6$ to reduce the computational cost. In all cases, the flow was subjected to a very strong favorable pressure gradient, contributing to maintain a laminar boundary layer over most of the blade surface. Boundary layer state played a dominant role in the complex shock/boundary layer interactions occurring in the trailing edge region, and could only be captured by LES, while RANS predicted excessively thick fully turbulent boundary layers. Such behavior was shown to have a profound effect on the predicted loss coefficients.

Recently, the experimental investigation of Baumgärtner et al. [2] on a realistic ORC turbine blade was numerically reproduced in [20,21]. To limit the computational effort, a simplified configuration corresponding to a 2D section extracted at the turbine shroud and extruded in the spanwise direction was investigated in [20] using wall-resolved LES. Due to the highly twisted 3D geometry, the 2D simulations showed discrepancies with the existing experimental measurements. However, the LES again highlighted the highly transitional nature of the blade boundary layers, despite the high exit Reynolds number of about 2 millions. As a consequence, fully turbulent RANS simulations yielded overly optimistic estimates of the loss coefficients. In [21], the CPU cost to reproduce Baumgärtner's experiment [2] was reduced by considering lower exit Reynolds number and by using wall-modeled LES. This allowed the full configuration to be simulated including the upper and bottom walls, and the influence of the secondary vortices on the transition location and overall flow topology to be taken into account. The RANS calculations lead to higher losses than LES by ~10%.

The aim of this paper is to investigate transonic flow through an idealized ORC turbine vane configuration at realistic Reynolds number using scale-resolving simulations. Results obtained for Novec649 are contrasted with those calculated for air, to highlight the role of the organic fluid complexity. To mitigate the computational burden of LES at the very high Reynolds number at stake, hybrid RANS/LES simulations are considered, in particular Delayed Detached Eddy Simulation (DDES). The latter uses RANS to model attached boundary layers and automatically switches to LES in regions with sufficient grid resolution. Although the RANS region prevents an accurate resolution of boundary layer transition, this is not expected to be critical for the configuration at stake, as most of the boundary layer is expected to be fully turbulent [5]. However, the LES mode of DDES is activated in the shear layers and wake region downstream of the trailing edge, which dramatically improves the description of the wake dynamics and its effect on vane losses. The simulations are compared with steady and unsteady RANS computations, showing significant differences in the computed loss coefficients. The paper is organized as follows. Numerical methods and simulation set-up are presented in Section 2. Numerical results for various flow conditions are discussed in Section 3. Finally, a detailed analysis of the blade losses and their decomposition into different contributions is performed in Section 4.

## 2. Numerical methods and set-up

### 2.1. Flow solver

The numerical simulations are based on the in-house finite-difference flow solver MUSICAA. The latter provides high-order accurate solutions of the compressible Navier-Stokes equations for multi-block curvilinear structured domains by using generalized coordinate transformations. The governing equations are complemented by equations of state and transport equation laws suitable for complex organic fluids. In particular, the thermodynamic behavior of Novec649 is modeled using the Peng-Robinson-Stryjek-Vera equation of state [22], and the Chung-Lee-Starling models [23] are used to describe the variations of the transport properties with temperature and density. For air flows, we simply use the usual perfect gas equation of state and the Sutherland law for viscosity. Turbulence effects are modeled by using either a RANS formulation together with the Spalart-Allmaras turbulence model, or a DDES hybrid RANS/LES formulation, namely the Delayed DES-Shear Layer Adapted (DDES-SLA) approach of [24]. The latter is based on a modified formulation of the Spalart-Allmaras model specifically designed to accelerate the transition to resolved turbulence in separated shear layers. In [25] it was shown that DDLES-SLA improves wake predictions compared to RANS.

In the following RANS and DDES-SLA simulations, the inviscid fluxes of the mean flow equations are discretized using 4th-order or 10th-order central finite differences, respectively, while viscous fluxes
are discretized using a 4th-order finite differences in all cases. Fourth- or tenth-order selective filtering is applied to eliminate spurious grid- to-grid oscillations, respectively for RANS and DDES-SLA. The filter is equipped with a low-order shock capturing term activated locally by a combination of Jameson's shock sensor and Ducros' dilatation/vorticity sensor. As the repeated application of the filter at each time step can lead to an overdissipation when using small time steps (or CFL numbers), the filtering coefficient $\chi_f$ is replaced by $\min(\chi_f, \text{CFL})$ as recommended in [26], so that the shocks are sharply captured in steady computations. The RANS transport equations are discretized using a 5-point third-order scheme in all cases.

For unsteady RANS and DDES, the time integration is performed with a low-storage four-stage Runge-Kutta (RK4) scheme supplemented with the fourth-order-accurate implicit residual smoothing (IRS) operator of [27], extended to multiblock parallel computations on general curvilinear grids in [28], to extend its stability domain, allowing an increase of the maximum allowable CFL by a factor of 5 compared to the explicit time integration for an overcost of ~40% per iteration. A less expensive second-order IRS operator is used for steady-state RANS simulations.

Boundary conditions based on the Riemann invariants are prescribed at the inlet/outlet boundaries of the computational domain. Specifically, the stagnation pressure and temperature are imposed at the inlet, along with the flow angle and an eddy viscosity value equal to 10 times the physical viscosity, while a back-pressure condition is used at the outlet. As the study configurations are characterized by a supersonic outlet, the outlet boundary condition corresponds to a direct extrapolation of the solution from the interior points. A sponge zone combining controlled grid stretching and a Laplacian filter is added at the outlet to avoid spurious reflections. Adiabatic no-slip wall conditions are imposed at all solid walls.

## 2.2. Idealized blade vane geometry

We investigate flows through the idealized turbine geometry of [11], where it was used to study tip gap effects on losses in the CLOWT facility (using air as the working fluid) [7]. The blade profile is designed as a combination of circular arcs and straight lines. The suction side is inclined at an angle of 4.5° with respect to the pressure side. The chord length of the profile is $ch = 61.2$ mm and the thick TE shape is circular, with a diameter of $d_{TE}/ch = 3.92\%$. The inter inter-blade distance $t$, or pitch, is $t/ch = 65.36\%$. A previous LES study of the laminar-turbulent transition occurring in the blade leading edge (LE) region [18] showed that the original blade geometry of [11] caused incipient flow separation due to the non-smooth curvature distribution [29]. The LE shape was then smoothed to ensure that the laminar-turbulent transition was induced by FST, the most likely mechanism in high-pressure turbine cascades [14]. The modified LE geometry consists of two super-ellipses defined by $(a-x/a)^n + (y/b)^n = 1$ [30], where $n = 2.3$, $a/ch = 18.47\%$, $b/ch = 5.81\%$ for the suction side super-ellipse and $n = 2.4$, $a/ch = 26.70\%$, $b/ch = 4.18\%$ for the pressure side. The upper super-ellipse is connected to the rear part of the suction side by fitting a Bézier curve. The rest of the blade, and especially the TE of the modified geometry, is kept equal to the original one. In Fig. 2, the original and smoothed nose geometries are compared, the smoothed one being closer to the typical shape of turbine stator noses. Note that the pressure distribution on the blade is little affected by LE modification. Analyses of the LES results on the smoothed geometry can be found in [18]. An important conclusion is that the FST-induced transition occurs rapidly at approximately 6% of the chord, despite the favorable pressure gradient that tends to delay the transition location [16]. For this reason, no transition model is used in the following analyses.

<figure><img src="image_5.png" /><figcaption>Fig. 2. Comparison of the original [11] (—) and smoothed (—) geometries, along with the profile curvature.</figcaption></figure>

## 2.3. Operating conditions

The numerical simulations are performed with Novec649 for two sets of inlet thermodynamic conditions: Case Novec-1 corresponds to conditions representative of the CLOWT facility, with a total pressure of $p_{0,1} = 3.8$ bars and a total temperature of $T_{0,1} = 335$ K. With regards to the critical pressure $p_c$ and temperature $T_c$, this corresponds to relatively dilute conditions with $p_{0,1}/p_c = 0.201$ and $T_{0,1}/T_c = 0.76$ (see Table 1). For the Novec-2 case, the inlet stagnation conditions are chosen such that the fundamental derivative of gas dynamics $\Gamma$ is equal to 0.41, which is close to the minimum value achievable for Novec649 in the dry vapor region ($\Gamma_{min} = 0.35$).

The inlet values are calculated one chord upstream of the LE and the exit values are evaluated along the line of measure (plotted in Fig. 4), which is defined to be parallel to a line passing through the trailing edges and which corresponds to a distance of $0.25 \times ch / \cos(\chi)$, where $\chi = 60^\circ$ denotes the stagger angle. The extent of the measure line is taken equal to the pitch $t$ and centered along the trailing-edge $y$ position. A uniform mixed-out final state was assumed and a constant area mixing calculation was performed to determine the exit values. Based on the conservation of mass, momentum, and energy, and using the non-uniform flow field along the line of measure as the initial state, the method matches this input state to a mixed-out state where the flow properties are uniform [31]. The exit Reynolds number $Re_{M_2}$, based on the chord and the exit Mach number $M_2$, is equal to $8.5 \times 10^6$. The chord is kept constant between the Novec-1 and Novec-2 case, leading to a different outlet Reynolds number $Re_{M_2}$ of $12.5 \times 10^6$. However, we verified (via RANS simulations) that this parameter plays a negligible role on comparisons between the Novec-1 and the Novec-2 cases.

For comparison, a simulation with air is also performed, at $p_{0,1} = 4$ bars and $T_{0,1} = 308$ K. While keeping the same proportions, the chord is rescaled to 246 mm in order to match the outlet Reynolds number of the Novec-1 simulation (see Table 1).

## 2.4. Numerical set-up

The simulations are performed on structured grids with multi-block topology. A grid convergence study is performed on the Novec-1 case for 2D RANS simulations in Appendix A. Overall, the results between the three different grids collapse perfectly and the medium grid, composed of 306k points with 1020 points along the blade, is retained to perform all the RANS simulations in the different cases. The RANS methodology has been validated for the air case using the experimental data of [11] (see Appendix B). The present DDES-SLA approach has also been previously validated for the well-known VKI LS59 turbine cascade using the experimental data of [32] and LES [25], the results showed that DDES-SLA captures the correct vortex shedding regime in the unsteady turbulent wake behind the blades.

The present DDES-SLA simulations are performed with a 3D mesh of 102 million points. First, the 2D vane is discretized with a mesh of
<figure><img src="image_6.png" /><figcaption>Fig. 4. Schlieren-like visualization of the idealized blade vane configuration in the Novec-1 DDES case: density gradient in y-direction (levels ±0.5 × 10⁴); - - -: line of measure with orientation in terms of stagger angle γ; identification of regions of interest in the flow (I, II and III) and of pressure (PS) and suction side (SS). The crosses denote the locations at which the spectra are performed in Fig. 12.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5. Isentropic Mach number distribution around the blade in the different cases.</figcaption></figure>

produced as the flows from either side of the blade meet and turn in a shared direction [12]. The turning angle of the flow at the suction (resp., pressure) side, which corresponds to the angle between the suction (resp., pressure) side and the shear layer, depends strongly on the thermodynamic state of the fluid [5,36]. The mean turning flow angle $\alpha_{tf}$, defined as the average between the suction and pressure side, for the different simulations is reported in Table 2. According to the present DDES results, the three flows exhibits similar mean turning flow angle. However, because of a different post-expansion, the expansion does not start at the same Mach number for the three cases and therefore the different cases cannot be directly compared. The variation of the Prandtl–Meyer function $\Delta v$ associated with the expansion at the suction side is evaluated in order to get some insights of how the non-ideal flow behavior affects the turning flow angle. For Novec649, we use the expression of the Prandtl–Meyer function for non-ideal gases as given in [36]. The latter relies on the assumption that the working fluid behaves as a polytropic perfect gas with an equivalent isentropic exponent $\gamma_{pv}$, and that the fluid remains outside the thermodynamic region where nonclassical effects are present, which is the case in the present configuration. In Fig. 7 we plot the variation of the Prandtl–Meyer function $\Delta v$ as a function of the Mach number at the beginning of the supersonic expansion $M$ for perfect gases with isentropic exponents corresponding to cases Novec-1 and Novec-2. As a comparison, $\Delta v$ is also evaluated for a perfect diatomic gas ($\gamma = 1.4$). As expected, starting with the same initial Mach number, more non-ideal gases lead to significantly greater values of $\Delta v$ for a given turning angle, hence to higher Mach numbers at the end of the expansion. In practice however,

<figure><img src="image_8.png" /><figcaption>Fig. 6. Averaged Mach number field in the trailing edge region in the RANS (top) and DDES (bottom) case. 1: separation shock; 2: trailing edge shocks; 3: free shear layers. - - - - - denotes the contour of the recirculation bubble with the criteria $u = 0$. •••••• denotes the contour of the sonic line with the criteria $M = 0$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7. Variation of Prandtl–Meyer function with Mach number for the TE expansion prior to separation at the suction side in the DDES.</figcaption></figure>

the upstream Mach number decreases when moving to Novec-1 and then to Novec-2, which leads as a consequence the effective overall variation of $\Delta v$ being approximately the same for all cases.

## 3.2. Unsteady wake dynamics

Boundary layer separation occurring at the TE leads to the formation of large coherent structures, due to the roll-up of the shear
<figure><img src="image_10.png" /><figcaption>**Fig. 8.** Instantaneous Mach number field in the trailing edge region in the DDES (top) and 2D URANS (bottom) Novec-1 case. —— denotes the contour of the turbulent viscosity line $\mu_t/\mu_l = 10$.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Strouhal number $St_d$ evaluation in the DDES, along with experimental results from the literature [37,38].</figcaption></figure>

layers, as depicted in Fig. 8 (top) for the Novec-1 DDES. In DDES-SLA simulations, the RANS mode is disabled downstream of the TE, the turbulent viscosity $\mu_t$ is greatly reduced (see isoccontour — —) and the flow rapidly transitions to resolved turbulence. Because vortex shedding is a low-frequency phenomena, unsteady RANS computations comply with the spectral gap requirement and unsteady RANS simulations should be predictive [19]. To verify this, an instantaneous visualization of a 2D unsteady RANS simulation in Novec-1 conditions is reported in Fig. 8 (bottom). The flow is characterized by large, coherent 2D vortex shedding which are in sharp contrast to the DDES. In addition, the increased momentum mixing in the DDES simulations leads to a much shorter recirculation bubble behind the blade TE compared to RANS. This results in larger values of the turning angle $\alpha_{tf}$ in DDES than in RANS. In addition, we observed that the unsteadiness of the wake in URANS is very sensitive to the approximation scheme and computational mesh (not shown for brevity), which gives little confidence in the robustness of the method and further invalidates the ability of URANS to correctly capture the unsteady wake.

Despite extensive research on vortex shedding in incompressible flows behind bluff bodies, supersonic unsteady wake complex dynamics and aeroacoustic processes have received limited attention, and the results available in the literature lead to contradictory conclusions. On the one hand, studies [39–41] in the high supersonic Mach number range ($M \ge 2$) showed that the characteristic frequency in the wake of a flat plate is independent of Mach and Reynolds numbers and is equal to 0.3. On the other hand, studies in the same Mach number range of supersonic cylinders reported an influence of the Reynolds number on the Strouhal number. Moreover, studies of transonic turbine wakes [19, 37,38] ($M \sim 1-1.3$) show an increase of the Strouhal number with Mach number, mainly due to the strong modification of the trailing edge shock system. Using the velocity power density spectra extracted at the measurement line in the wake (such as the one in Fig. 3), the Strouhal number $St_d = f \times d_{TE}/U_{2,ls}$, based on the trailing edge thickness and exit velocity and corresponding to the vortex shedding frequency, is evaluated for the DDES simulations, and the results are shown in Fig. 9 along with experimental results for the Strouhal number in supersonic turbine TE wakes available in the literature. Despite the scatter in the results, we observe a clear trend for $St_d$ to increase with Mach number in the range considered. Interestingly, despite a lower Mach number, the Novec-2 flow has a similar Strouhal number very close to that of the Novec-1 case, probably due to the higher Reynolds number of this flow. Note that, according to the unsteady RANS simulations performed for the Novec-1 case (Fig. 8, bottom), the Strouhal number is estimated to be $St = 0.18$, which is well below the values predicted by DDES or found in the literature, and which justifies the use of DDES to capture the unsteady behavior of the wake.

In order to understand this increase in the Strouhal number, the vortex shedding mechanism is examined in detail. Humble et al. [42] demonstrated that around Mach 1.46, the wake behind a symmetric double wedge was characterized by a vertical flapping motion in the reattachment region (end of free shear layers in Fig. 6). This flapping is depicted in the Novec-1 DDES in Fig. 10, where the wake is animated by a vertical motion in the reattachment region which interacts with the recompression shocks. This motion is constrained by the shock system. The separation locations (beginning of free shear layers in Fig. 6) are also moving a little around the trailing edge. This point is more easily seen in the movies available as supplementary materials.

This flapping motion marks the streamwise and vertical turbulence intensity distribution in the Novec-1 DDES in Fig. 11 (top), similarly to what is shown in [42]. The interaction between the two shear layers in the reattachment region appears on the streamwise turbulence intensity as two distinct bands. Moreover, the motion of the separation locations marks the streamwise turbulence intensity along the shear layers. In [42], the authors showed that, as the Mach number was increased from 1.46 to 2.27 and the TE shock system became more rigid, this flapping predominant global mode evolved into a pulsating motion aligned with the wake axis, and that their intermediate case at $M = 1.78$ exhibited both modes. In the present study, mainly due to higher exit Mach number in air, we observe that the span of the flapping motion, which corresponds to the neck width $N$ reported on Fig. 11, is indeed reduced in the air DDES compared to Novec-1. Moreover, the less intense $u_{rms}/U_2$ in the shear layers close to the wall indicates that the motion of the separation locations are greatly reduced in air compared to Novec-1. This is also revealed in the movies available as supplementary materials.

To account for the differences in the physical mechanisms in the supersonic wake compared to a subsonic flow, several authors [43,44] proposed to modify the definition of the Strouhal number and to replace the commonly used diameter by another more meaningful characteristic length of the flow. Studying supersonic cylinder wakes at $M = 4$, Schmidt & Sheperd [43] based the Strouhal number on the sliplines length $L_{sl}$ and obtained values of $St_{sl}$ around 0.5 (values between 0.4–0.6). On the contrary, the authors showed that a scaling based on the neck width $N$ did not work in their case. On a supersonic
<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Observation of the flapping motion (left to right) on the instantaneous Mach number field evolution during one vortex shedding period in the Novec-1 DDES. Each picture is separated by 4000 temporal iterations.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 11.** Streamwise ($u_{rms}/U_2$, left) and vertical ($v_{rms}/U_2$, right) turbulence intensity averaged field in the wake of the Novec-1 (top) and air (bottom) DDES. —— denotes the isocontour $u_{rms}/U_2 = 0.1$. The neck width $N$ and the shear-layer length $L_s$ are reported in black and purple, respectively.</figcaption></figure>

<caption>**Table 3**</caption>

Formation of Strouhal numbers based on the shear-layer length $L_s$ or the neck width $N$, using an averaged speed velocity $U_{TE}$ between the pressure side and the suction side upstream of the TE separations: $St_s = f L_s/U_{TE}$ and $St_N = f N/U_{TE}$. The standard definition of the Strouhal number $St_d = f d_{TE}/U_{2,ls}$ is reported as a comparison.

<table><thead><tr><th>Flow</th><th><i>U</i><sub>TE</sub><br>[m/s]</th><th><i>L</i><sub>s</sub>/<i>d</i><sub>TE</sub></th><th><i>N</i>/<i>d</i><sub>TE</sub></th><th><i>St</i><sub>d</sub></th><th><i>St</i><sub>s</sub></th><th><i>St</i><sub>N</sub></th></tr></thead><tbody><tr><td>Air</td><td>425</td><td>0.85</td><td>0.39</td><td>0.65</td><td>0.62</td><td>0.29</td></tr><tr><td>Novec-1</td><td>116</td><td>0.75</td><td>0.46</td><td>0.55</td><td>0.50</td><td>0.31</td></tr><tr><td>Novec-2</td><td>111</td><td>0.79</td><td>0.49</td><td>0.55</td><td>0.53</td><td>0.32</td></tr></tbody></table>

cylinder at $M = 3$, replacing the slipline by the shear-layer lengths $L_s$, Awasthi et al. [44] recovered similar values, with $St_s$ lying between 0.4 and 0.6. The same scaling is applied in the present case, as shown in Table 3. To form the Strouhal number, the choice is made to use, rather than $U_{2,ls}$ as in the definition of $St_d$, the velocity upstream of the TE separations and averaged between the pressure and the suction side. Concerning $St_s$, similar values to [43,44] are found but the Strouhal numbers $St_s$ evaluated are still relatively different between air and Novec cases. In [43,44], this scaling was justified by the fact that the dominant global feature is streamwise pulsations of the reattachment region [42]. In our case, the physical mechanism is a flapping motion constrained by the shock system and the neck width $N$ seems to be

<figure><img src="image_14.png" /><figcaption>**Fig. 12.** Energy spectra of the velocity component $v$ in the wake at the locations indicated by the crosses in Fig. 4. ···········: Strouhal number and its two first harmonics.</figcaption></figure>

the natural dimension to use. The Strouhal numbers scaled on the neck width $N$ in the different DDES are reported in Table 3. The values obtained, approximately equal to 0.3, are in a good agreement, suggesting that the neck width is indeed the relevant length scale in the considered range of Mach. Thus, the increase of $St_d$ with the Mach number observed in Fig. 9 is mainly a direct consequence of the neck width reduction as shock strength increases.

The unsteady separation on the cylinder surface has been linked by some authors [43,44] to an acoustic feedback loop occurring in the subsonic region between the separation points on the cylinder wall and the reattachment region. Schmidt & Sheperd [43] showed that the oscillations in supersonic flow were likely to be driven by acoustic signals propagating through the subsonic region. This hypothesis was reinforced later by Awasthi et al. [44]. In particular, the authors observed that the vortex shedding frequency, which marked spectra downstream of the reattachment region, did not appear on spectra located between the cylinder wall and the reattachment region. The spectra on several locations aligned with the flow (see crosses in Fig. 4) are depicted in Fig. 12 for the Novec-1 (left) and air (right) DDES. In particular, the spectra in blue (—) and purple (—) are located in the subsonic region and at the reattachment point, respectively. Independently of their location, all the spectra exhibit a peak around the Strouhal number position, which means a priori that it is not an acoustic feedback loop which is responsible for the unsteadiness of the wake in our case. This is coherent with the fact that the flapping motion switch to a pulsating motion in the high supersonic Mach number range ($M \ge 2.2$) [42] and that the global dynamic of the wake is therefore not exactly driven by the same physical mechanisms as at lower supersonic Mach numbers.

However, this flapping motion still leads to strong pressure waves downstream of the reattachment region, as visualized in Fig. 13 by means of contour plots of the fluctuating pressure. The acoustic waves strength, evaluated as $p_{rms}/\bar{p}$ at a location outside the wake highlighted by a symbol in Fig. 13, is increased by ~50% in Novec-1 case compared to air. This is primarily due to the flapping motion which is stronger in the Novec-1 case, as we saw earlier. It can also be linked to the fact
<figure><img src="image_15.png" /><figcaption>**Fig. 13.** Pressure fluctuations $p' = p - \bar{p}$ (levels ±0.025$\rho_{0,1}$) in the Novec-1 (top) and Air (bottom) DDES.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Fig. 14.** Root-mean square pressure (top left), root-mean square temperature (bottom left) and total temperature (right) profiles along the line of measure. The cross denotes the center of the wake.</figcaption></figure>

that, in organic vapors, the thermal mode is generally damped whereas the acoustic mode is reinforced, as demonstrated with the profiles of selected fluctuating quantities across the wake in Fig. 14. This is due to the high specific heat of Novec649, which greatly reduces temperature fluctuations in the wake compared to air, as depicted in the bottom left panel of Fig. 14, whereas the pressure fluctuations are reinforced in the dense gas, as illustrated by the profiles of fluctuating pressure $p_{rms}$ reported in the top left panel. The second peak on $p_{rms}$ exhibited in the Novec-2 curve around $l/t = -0.12$ is not linked to the wake but is rather due to the crossing of the shock by the line of measure in the Novec-2 case. As shown in the right panel of Fig. 14 (right), representing the profile of the total temperature $T_0$ across the wake, the high heat capacity of Novec649 also prevents the phenomenon of so-called energy separation [19], which corresponds to an increase of the total temperature at the borders of the wake due to the intermittent shedding of vortices.

The correct capture of the wake flow dynamic and the base region is crucial as it has a major impact on the back pressure and therefore on the estimation of the wake-related losses, which can represent a significant percentage of total losses [19]. The pressure distribution

<figure><img src="image_17.png" /><figcaption>**Fig. 15.** Pressure distribution around the trailing edge for the different DDES, along with RANS simulation in the Novec-1 case. ● denotes the separation locations, used as limits to calculate the average back pressure $\rho_b$, and $s - s_{TE}$ is the curvilinear abscissa relative to the trailing edge position.</figcaption></figure>

<table><caption>**Table 4**
Losses estimations for the different flows and simulations.</caption><thead><tr><th>Flow</th><th>Simu.</th><th>ζ<sub>out</sub><br>(%)</th><th>ζ<sub>pass</sub><br>(%)</th><th>ζ<sub>TE</sub><br>(%)</th><th>ζ<sub>Den</sub><br>(%)</th><th>C<sub>p,b</sub></th></tr></thead><tbody><tr><td rowspan="2">Air</td><td>DDES</td><td>6.1</td><td>1.7</td><td>4.4</td><td>8.0</td><td>-0.23</td></tr><tr><td>RANS</td><td>5.1</td><td>1.9</td><td>3.2</td><td>6.7</td><td>-0.16</td></tr><tr><td rowspan="2">Novec-1</td><td>DDES</td><td>7.5</td><td>2.1</td><td>5.4</td><td>8.1</td><td>-0.24</td></tr><tr><td>RANS</td><td>6.2</td><td>2.4</td><td>3.8</td><td>6.1</td><td>-0.12</td></tr><tr><td rowspan="2">Novec-2</td><td>DDES</td><td>8.1</td><td>2.1</td><td>6.0</td><td>7.5</td><td>-0.21</td></tr><tr><td>RANS</td><td>6.6</td><td>2.2</td><td>4.4</td><td>5.5</td><td>-0.09</td></tr></tbody></table>

around the TE is plotted in Fig. 15 for the three DDES simulations. In all cases, the TE base flow is characterized by a constant-pressure region comprised between the separation locations on both sides of the blade (the separation points are highlighted by the bullets). The back pressure increases with increasing outlet Mach number [12]. RANS results for the Novec-1 configuration are also reported in the figure. While the pre-separation expansion is similar for RANS and DDES, the RANS simulation exhibits a stronger separation shock, leading to a significantly higher back pressure. This results in an inaccurate evaluation of the blade losses, as discussed in the next section.

## 4. Analysis of blade losses

In this Section we examine the loss mechanisms in detail. First, we consider the enthalpy loss coefficient, defined as:

$$ \zeta_{out} = 1 - \frac{h_{0,1} - h_2}{h_{0,1} - h_{2,1s}} \qquad (1) $$

To remove the influence of the upper and lower walls and to extract the contribution of a single blade to the losses, the evaluation of $\zeta_{out}$ is based on the mixed-out state along the measurement line comprised between the inter-blade meridional lines, as shown in Fig. 16. The figure also reports typical iso-contours of the entropy deviation with respect to the inlet value, showing that most of the entropy generation is associated with the turbulent wake. The study of Ref. [2] showed that, for a fixed pressure ratio $\pi$, the losses increase as the isentropic exponent decreases $\gamma_{pu}$ and that, for a fixed $\gamma_{pu}$, losses increase when $\pi$ is decreased. The present results for $\zeta_{out}$, reported in Table 4, exhibit a similar trend, with both RANS and DDES predicting an increase of the loss coefficient as $\gamma_{pu}$ and $\pi$ decrease. However, RANS simulations systematically underestimate the loss coefficient by ~20% compared to DDES, mainly due to the overestimated back pressure in RANS.
<figure><img src="image_18.png" /><figcaption>Fig. 16. Entropy deviation ($\bar{s} - s_1$) / $s_1$ in the Novec-1 case, along with the control lines used for the estimation of the losses.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 17. Denton's loss decomposition in the Novec-1 case for the DDES and RANS simulations.</figcaption></figure>

The reasons for the observed discrepancies between DDES and RANS are further investigated by performing a loss breakdown analysis. The losses are decomposed into two contributions: a contribution corresponding to the losses in the passage vane, $\zeta_{pass}$, and a contribution from the flow downstream of the TE, $\zeta_{TE}$. A control line $L_{pass}$ aligned with the blades TEs is defined by connecting the separation points to the meridional lines (see Fig. 16). The global passage losses are then evaluated by introducing the loss coefficient:

$$ \zeta_{pass} = \frac{h_{pass} - h_{pass, is}}{h_{0,1} - h_{pass, is}} \qquad (2) $$

where $h_{pass}$ and $h_{pass, is}$ are the mass-averaged enthalpy and isentropic enthalpy along the control line $L_{pass}$, respectively. Then the trailing edge losses are evaluated as the difference between $\zeta_{out}$ and $\zeta_{pass}$. The values calculated for various cases are reported in Table 4. Based on the previous breakdown, passage losses account for only ~25 – 30% of the total losses in the various DDES simulations, and most of the losses are generated downstream of the TE. The percentage contributions of $\zeta_{pass}$ and $\zeta_{TE}$ to the total losses are about the same for the three flows and are little affected by the non-ideal conditions. As expected, the differences between DDES and RANS simulations are mainly due to the miscalculation of the TE losses in RANS, with an error of ~30%.

To further study how the base pressure affects the losses, the decomposition of the loss coefficient proposed by Denton [45] is considered. This coefficient is decomposed into several terms corresponding to the main physical mechanisms responsible for the losses:

$$ \zeta_{Den} = \underbrace{-\frac{C_{pb}d_{TE}}{w}}_{\zeta_{C_{p,b}}} + \underbrace{\frac{2\theta}{w}}_{\zeta_{\theta}} + \underbrace{(\frac{\delta^* + d_{TE}}{w})^2}_{\zeta_{\delta^*}} \qquad (3) $$

<figure><img src="image_20.png" /><figcaption>Fig. 18. Evolution of trailing-edge losses with the compressible pressure coefficient $C_{pb,comp}$ version proposed in [2]. The solid lines correspond to linear regression on the DDES (—) and RANS (—) values. Values from Baumgärtner et al. [2] are also reported.</figcaption></figure>

where $\theta$ and $\delta^*$ are the momentum and displacement thicknesses at TE upstream of the separation points, respectively, $C_{pb} = (p_b - p_2)/(\frac{1}{2}\rho_{2, is}U_{2, is})$ is the base pressure coefficient, $d_{TE}$ is the TE thickness and $w$ is the throat width. The terms $\zeta_{C_{p,b}}$, $\zeta_0$ and $\zeta_{\delta^*}$ correspond to the profile loss, the momentum loss and the blockage losses, respectively. Values of the total loss $\zeta_{Den}$, reported in Table 4, do not differ much from $\zeta_{out}$. However, different trends of the loss coefficients are predicted by the DDES and the RANS simulations. The contributions of the different terms in Eq. (3) are reported in Fig. 17 for the case Novec-1. DDES and RANS show similar momentum and blockage losses, but the losses associated with base pressure differ between the two simulations. As a result, the blockage losses, which were the main source of losses in RANS, become the second source of losses after the base pressure losses in DDES. This decomposition also highlights that the trailing edge losses $\zeta_{TE}$ are mostly related to the base pressure. Similar considerations apply to the other cases.

Baumgärtner et al. [2] introduced a compressible version of the base pressure coefficient, defined as:

$$ C_{pb,comp} = (p_b/p_{ref} - 1) \frac{2}{M_{ref}^2 \gamma_{pv}} \frac{d_{TE}}{t} \qquad (4) $$

Interestingly, the authors observed that the trailing edge losses $\zeta_{TE}$ collapsed on a line when plotted against $C_{pb,comp}$. To see if this behavior is recovered in our configuration, the compressible base pressure coefficient is evaluated, taking $M_{ref} = M_2$ and $p_{ref} = p_{0,1}$, and the evolution of $\zeta_{TE}$ w.r.t. $C_{pb,comp}$ in the RANS and DDES is reported in Fig. 18. This linear behavior is well recovered in both RANS and DDES. A linear regression of the DDES and RANS points is also plotted, mainly to serve as a visual indicator of the alignment of the points. The two lines are roughly parallel and the difference on $\zeta_{TE}$ between the RANS and DDES calculations remains relatively constant in the restricted range of conditions at stake. Moreover, the RANS values from [2], also reported on the figure, are in good agreement with our RANS simulations. The RANS simulations show good qualitative behavior and identify correctly the performance hierarchy between cases, despite the fact that they underestimate the losses.

## 5. Conclusions

The transonic flow of the dense organic vapor Novec649 through an idealized turbine vane configuration, representative of an Organic Rankine Cycle turbine, has been studied using high-fidelity numerical simulations based on a high-order accurate flow solver and a hybrid RANS/LES turbulence model. Specifically, the study is performed using the DDES-SLA methodology of Shur et al. [24], designed to facilitate
the transition to resolved turbulence in separated shear layers, such as
those occurring downstream of the blade trailing edge, while applying
RANS modeling in turbulent attached boundary layers. For the present
blade configuration, the transition occurs rather close to the leading
edge [18], making the use of a transition model unnecessary. Such a
simulation strategy allows to accurately resolve the most relevant of
the turbulent wake dynamics, which plays a crucial role on the overall
blade losses.

Due to its molecular complexity, Novec649 exhibits a much lower isentropic exponent than air, even at relatively dilute operating thermodynamic conditions, where gas behavior is only weakly nonideal. In addition, the fundamental derivative of the gas dynamics is less than one throughout the flow, implying an opposite behavior of the speed of sound during expansion than in a light perfect gas. These parameters essentially affect the achievable pressure ratio between the vane inlet and outlet sections, and the inclination and intensity of the shock waves generated at the blade trailing edge, but no significant non-ideal phenomena are observed, and the overall flow topology is similar to that obtained in air. The lower isentropic exponent of Novec649, which favors larger flow deflections, is compensated by the lower Mach number conditions, and similar trailing edge turning angles are obtained in the different flows.

The scale-resolving DDES model allows, for the first time in the context of non-ideal gas turbines, to characterize the wake development and unsteady behavior by means of time-averaged flow properties and velocity spectra. The vortex shedding is characterized by a vertical flapping motion in the reattachment region, which is constrained by the shock system. The latter exhibits a dominant frequency corresponding to the vortex shedding, with the flow Strouhal number formed on the trailing-edge diameter depending on both the Mach and Reynolds numbers achieved at vane exit. However, when scaled by the neck width of the reattachment region, which corresponds approximately to the vertical extent of the flapping motion, the Strouhal number remains relatively constant and equal to 0.3 for all cases. The large heat capacity of Novec649 results in a significant reduction of temperature fluctuations in the wake, thus leading to the suppression of the energy separation phenomenon typically encountered in air flows. On the other hand, pressure fluctuations are enhanced in Novec649, leading to the emission of intense acoustic waves not observed in air flow. Inspection of the enthalpy loss coefficient based on the mixed-out exit state shows that, as the non-ideality of the flow increases and the pressure ratio decreases, the loss grows. The breakdown of the losses between the passage vane and the trailing edge region reveals that most of the losses occur in the wake downstream of the trailing edge. Furthermore, the wake losses are mainly related to the base pressure according to Denton's loss decomposition [45].

It is noteworthy that steady-state RANS simulations (which represent the routine tool for industrial ORC turbine design) severely underestimate the base pressure coefficient by a factor of 2 compared to DDES, leading to an error of 30% in the estimation of the total trailing edge losses. The error is due to the significant overestimation of the size of the recirculation bubble downstream of the blade trailing edge. This flaw cannot be improved by using unsteady RANS simulations. While the latter comply with the spectral gap requirement, unsteady RANS cannot resolve the relevant coherent turbulent scales involved in the wake flapping motion, resulting in little or no improvement over steady RANS predictions.

The present results are not restricted to the Novec649 working fluid,
but generalize to other similar fluids used in ORC turbine technol-
ogy, operating in dense gas conditions. Specifically, one may expect
increased losses when moving to more non-ideal conditions, due to the
increased flow turning angle around the trailing edge, and subsequent
stronger trailing edge losses. On the other hand, the suppression of the
energy separation phenomenon in high specific capacity gases tends to
produce more homogeneous flow entering the subsequent row. Finally,
the complex flow dynamics behind the blade trailing edge cannot be

<figure><img src="image_21.png" /><figcaption>Fig. A.19. Isentropic Mach distribution around the blade (top) and pressure along the line of measure (bottom) in the RANS Novec-1 case for the coarse ($N_{pts}$ = 212 k), the medium (306 k) and the fine (416 k) grids.</figcaption></figure>

captured by low fidelity methods, independently on the nature of the
working fluid. Therefore, there is a strong industrial interest in moving
toward improved models, and include them efficiently in the design
process. The associated increase in computational cost can be strongly
alleviated by combining high-fidelity models with modern data-driven
or machine-learning assisted techniques (see [46] for a review). Some
work ongoing in our group will be presented in upcoming publications.

Declaration of competing interest

The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.

Acknowledgments

This research has received funding from the Agence Nationale de la
Recherche, France through the ANR-20-CE92-0019-02 project Regal-
ORC. This work was granted access to the HPC resources of IDRIS
and TGCC under the allocation A0132A01736 made by GENCI (Grand
Equipement National de Calcul Intensif).

Appendix A. Grid quality in RANS simulations

A mesh sensitivity analysis is performed for the Novec-1 case to
assess the quality of the computational grid in RANS simulations.
<figure><img src="image_22.png" /><figcaption>Fig. A.20. Mean velocity profile in the RANS Novec-1 case at $x/ch = 0.5$ at the suction side for the coarse ($N_{pr} = 212k$), the medium (306 k) and the fine (416 k) grids.</figcaption></figure>

The study uses three different grids composed of 212 k, 306 k and 416 k points, respectively. The grid with 306 k points, called hereafter medium grid, corresponds to the one used in the RANS simulation in this study. Between two successive grids, the mesh is refined in both the boundary layer region and in the wake region. A wall-normal resolution $\Delta n_w^+ \sim 1.0$ at the wall is used in all the grids, and the stretching increases as the grid is coarsened. The isentropic Mach distribution around the blade and pressure along the line of measure are shown in Fig. A.19. Results for the three grids are nearly superposed, giving good confidence in the quality of the grid. The near-wall grid resolution is also assessed in Fig. A.20 by inspecting the mean velocity profiles in the boundary layer, at a station $x/ch = 0.5$ on the suction side. Velocity profiles computed in the different grids are superposed, further confirming grid independency of the present RANS simulations. Due to the large heat capacity of Novec649, the velocity profile follows very well the incompressible law of the wall for zero-pressure-gradient, despite the supersonic conditions in the vane.

## Appendix B. Validation of the RANS strategy

The present RANS methodology is assessed against the experimental results of [11] (using the original leading-edge geometry). This configuration is characterized by an inlet Mach number $M_1 = 0.52$ and an outlet Mach number $M_2 = 1.62$, along with an outlet Reynolds number of $Re_{M_2} = 3.67 \times 10^6$. The 2D computational grid of 234 000 points has a resolution $\Delta n_w^+ \sim 1.0 - 1.2$ and $\Delta s/ch \sim 0.30\%$, with 720 points along the blade. The computational domain extends from 1.5 chords upstream of the leading edge to 2 chords downstream of

the trailing edge. At the inlet, a total pressure of $p_{0,1} = 4$ bars and a total temperature of $T_{0,1} = 308$ K are imposed. A comparison of the shock system in the TE region with an experimental Schlieren picture is proposed in Fig. B.21. The numerical simulation correctly reproduces the fishtail shock system and the position of the oblique shock waves. Moreover, the pressure distribution along the blade in Fig. B.21 (right) is also in good agreement with the experimental one. To check the grid convergence, RANS is also run on a finer mesh, with $\Delta s/ch \sim 0.22\%$ and 900 points along the blade. The pressure distribution perfectly matches the medium grid one, showing that mesh independency is achieved on the medium grid.

## Appendix C. Supplementary data

Supplementary movies are available online.

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.applthermaleng.2024.124951.

### Data availability

Data will be made available on request.

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