# High-fidelity numerical investigation of a real gas annular cascade with experimental validation

A. Giauque, Dominik Schuster, Christophe Corre

► To cite this version:

A. Giauque, Dominik Schuster, Christophe Corre. High-fidelity numerical investigation of a real gas annular cascade with experimental validation. Physics of Fluids, 2023, 35 (12), 10.1063/5.0174230. hal-04928352

HAL Id: hal-04928352

https://hal.science/hal-04928352v1

Submitted on 4 Feb 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
High fidelity numerical investigation of a real gas annular cascade
with experimental validation

Alexis Giauque,¹ Dominik Schuster,¹ and Christophe Corre¹

Univ Lyon, Ecole Centrale de Lyon, CNRS, Univ Claude Bernard Lyon 1, INSA Lyon, LMFA, UMR 5509, Ecully, 69130, France

(*Electronic mail: alexis.giauque@ec-lyon.fr)

(Dated: 1 December 2023)

This study aims at investigating real gas flow in the complex geometry of the Cambridge
University annular turbine cascade using numerical simulations. The objectives include
validating the numerical approach and understanding the loss mechanisms in this config-
uration. The numerical results are compared to experimental measurements obtained at
various locations in the domain. Two turbulence modeling techniques, Large Eddy Simu-
lation (LES) and Reynolds-Averaged Navier-Stokes (RANS), are employed to assess the
influence of turbulence models and inlet turbulence levels.

The results show good agreement between numerical simulations and experimental mea-
surements in regions upstream of the trailing edge. However, discrepancies arise in the
transition region of the suction side boundary layer, and RANS results are influenced by
the choice of turbulence injection. In the wake of the blade, both RANS and LES accu-
rately predict the stagnation pressure ratio, with some slight differences in shock positions
and total pressure levels.

The analysis reveals that large vortical structures at the hub contribute significantly to the
overall losses in this annular configuration. The study quantifies losses due to boundary
layers, the wake, and vortical structures using a loss coefficient, with RANS and LES pro-
ducing slightly different results. These differences while calling for further experimental
measurements, also hint at the possible inaccuracy of the present turbulence models in the
context of real gas flows for which a dedicated modeling effort is required.
# I. INTRODUCTION

The study of Non Ideal Compressible Fluid Dynamics (NICFD) is a strongly expanding research field supported by a dynamic community¹⁻⁴. Among the topics of interest for NICFD, one finds the flows of real and dense gas vapors in the vicinity of their thermodynamic critical point. These dense gases are characterized by a compressibility factor Z different from unity. Their specific properties are strongly connected to a fundamental derivative of gas dynamics, defined as $\Gamma = 1 + \frac{\rho}{c} \left( \frac{\partial c}{\partial \rho} \right)_s$, taking values smaller than one⁵. This research effort is motivated by real and dense gases being one of the cornerstone elements of fatal heat recovery and renewable energy systems when used in Organic Rankine Cycles (ORCs). In ORC power systems, the design of turbines involves major technical challenges as compressibility and non-ideal gas effects combine with compactness and flexibility requirements. The design of ORC turbines heavily relies on advanced aerodynamic models, the validation of which is crucial but still limited due to a lack of relevant experimental data.

Numerous experimental test benches for real and dense gas flows exist in Europe such as TROVA⁶⁻⁸ at Politecnico di Milano, ORCHID⁹,¹⁰ at TU Delft, the Imperial ORC blowdown facility¹¹,¹² or CLOWT¹³,¹⁴ at Muenster University of Applied Sciences. The experiments carried out in these facilities provide useful reference data for Verification & Validation (V&V) purpose but have been focused up to recently either on inviscid features¹⁵ such as shock/expansion waves produced over a diamond-shape airfoil or isentropic expansion in a converging-diverging nozzle or else on other academic configurations involving viscous effects and considered as elementary bricks representative of specific flow regions of a full turbine configuration. For instance, Matar *et al.*¹⁶ recently investigated the non-ideal gas (Novec™649) flows around a circular cylinder comparing RANS and Unsteady-RANS simulations with experimental measurements in the CLOWT facility, considering the wake dynamics downstream of the cylinder as representative of the viscous wake of bluff trailing edges of turbine blades. This focus on academic flows cases lead Otero *et al.*¹⁷ to point out the need to validate their design using experimental measurements in their 2021 numerical analysis of stator-rotor interactions in high-expansion ORC turbines using 3D URANS simulations.

To answer this need, an experimental campaign on a supersonic linear cascade representative of an ORC stator vane is in progress at Politecnico di Milano¹⁸, which targets the measurement of total pressure losses through the cascade with a relatively high spatial resolution. In 2015 already,
HIFI investigation of a real gas annular cascade

Dura Galiana, Wheeler and Ong¹⁹ studied the trailing-edge losses in ORC turbines using both simulations and a specifically designed experiment performed in a Ludwieg tube facility located at Cambridge University’s Whittle Laboratory: a wake flow was produced by a plate embedded into an axisymmetric nozzle with three different working fluids (air, CO₂ and SF₆). More recently, the working section of the facility was modified by Baumgärtner, Otter and Wheeler²⁰ to include an annular cascade with air, CO₂ and R134a used as working fluids. This first experimental campaign of cascade testing for real gas flows provides a useful set of reference data measurements, with which the present study intends to compare high-fidelity LES and RANS approaches.

Regarding the numerical simulation effort directed towards the analysis and understanding of turbulent flows in ORC turbine configurations, numerous Reynolds Averaged Navier-Stokes (RANS) simulations (see for instance Obert et al.²¹, Klonowicz et al.²² or Otero et al.¹⁷) have been performed using high-order Equation of State (EoS) for an accurate thermodynamic description and conventional one-equation or two-equation turbulence models. Note that these simulations are also instrumental for the development and calibration of experimental measurement techniques to overcome for example the unavailability of Mach number measurements upstream of probes allowing to calculate shock losses at the probe tip in order to retrieve the pre-shock total pressure and evaluate cascade losses²⁰. On the other hand, Direct Numerical Simulations (DNS) have been carried out over the last decade²³⁻²⁵ to characterize the turbulence properties of dense and real gases and identify their possible differences with the properties observed for turbulent flows of perfect gases. These studies, recently reviewed by Cinnella and Gloerfelt²⁶, overall suggest a peculiar behavior of turbulence in dense gases especially due to the decoupling between internal and kinetic energies. Even though very informative at the smallest scales, those simulations fail to characterize flows of interest to the industry due to the limitations induced by the available computational power.

As a compromise between the near perfect accuracy at a large cost of DNS requiring no modeling and the low-cost simulations of RANS yet requiring a large amount of modeling, Large Eddy Simulation (LES) now represents a numerical technique largely used by researchers and companies to better understand unsteady aspects of flows in complex geometries. Its most interesting property lies in its capability of directly capturing most of the turbulent kinetic energy present in the flow, only the smallest vortical structures being modeled²⁷⁻²⁹. Because the models used in LES have all been developed in the context of perfect gases, their relevance for real and dense gases can be questioned, especially following the DNS studies which established the specific features of
HIFI investigation of a real gas annular cascade

turbulence in dense gases. Yet, in the context of dense and real gas flows, the development of SubGrid-Scale (SGS) closure models adapted to these specific flows remains an open research topic$^{30,31}$.

To the best of the authors knowledge, no study confronts experimental and high-fidelity numerical results to assess the accuracy of turbulence closure models in dense gas flows with complex geometry. In a recent work$^{32}$, Hoarau et al. perform LES of the transonic flow of PP11 through the VKI LS89, a geometry extensively studied for air flows; qualitative conclusions can be drawn by the authors regarding the limitations of lower-fidelity RANS simulations with respect to LES results but no quantitative analysis of the LES accuracy can be provided for lack of dense gas experimental results. The experimental results obtained by Dura Galiana et al.$^{19}$ for the wake losses downstream of a flat plate are compared with some RANS and LES numerical simulations performed using a commercial solver. For the annular cascade configuration studied by Baumgartner et al.$^{20}$, the authors provide comparisons between some of the experimental results and RANS simulations using the Spalart-Allmaras model such as implemented in a commercial Computational Fluid Dynamics (CFD) software.

It is therefore the main goal of this study to compare the experimental results obtained by Baumgartner et al.$^{20}$ for a supersonic real gas flow configuration of industrial interest with a high-fidelity LES. Carefully designed RANS simulations are also performed and both LES and RANS results are systematically compared with measurements. The numerically predicted flow features are also analyzed in details from a qualitative and quantitative viewpoint to determine the qualities and limitations of both modelling strategies. The objective of this study is therefore two-fold. First, by comparing experimental and numerical results from high-fidelity simulations, the authors intend to explore the limitations of current turbulence models in the context of real gas transonic flows. Secondly, this study provides a first step towards the development of more accurate turbulence closure models for real and dense gas flows of interest for ORC applications. It indeed represents a benchmark case to which future simulations including newly proposed SGS models for LES will be compared.

Section II reviews the equations and numerical methods used in this study. Section III displays verification arguments supporting the good accuracy and temporal convergence of the undertaken simulations. Section IV is two-fold: it first compares the LES and RANS numerical results to the available experimental data and proceeds next to extend the analysis of the computed flow to regions unavailable in the experiment.
## II. GOVERNING EQUATIONS AND NUMERICAL METHODS

### A. Conservation laws

The conservative equations for density $ρ$, momentum $ρu_i$ and density weighted total energy $ρE$ are solved in the case of compressible flows and write as follows:

$$ \frac{\partial \rho}{\partial t} + \frac{\partial \rho u_j}{\partial x_j} = 0, \qquad (1) $$

$$ \frac{\partial \rho u_i}{\partial t} + \frac{\partial \rho u_i u_j}{\partial x_j} = -\frac{\partial p}{\partial x_i} + \frac{\partial \tau_{ij}}{\partial x_j}, \qquad (2) $$

$$ \frac{\partial \rho E}{\partial t} + \frac{\partial \rho E u_j}{\partial x_j} = -\frac{\partial p u_j}{\partial x_j} + \frac{\partial \tau_{ij} u_i}{\partial x_j} - \frac{\partial q_j}{\partial x_j}, \qquad (3) $$

where $p$ is the pressure, $τ_{ij} = μ(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} - \frac{2}{3} \frac{\partial u_k}{\partial x_k} δ_{ij})$ is the viscous stress tensor (with $μ$ the dynamic viscosity), $E = E_{int} + \frac{1}{2} u_i u_i$, the specific total energy (with $E_{int}$ the internal energy) and $q_j = -κ \frac{\partial T}{\partial x_j}$ is the heat flux given by Fourier's law (with $κ$ the thermal conductivity).

Regardless of the numerical method, the formal methodology used in this study consists in solving a time-averaged (for RANS) or spatially filtered (for LES) version of the set of Eqs 1-3. Assuming commutativity between the filtering/averaging operator and temporal derivative or spatial gradient operators, leads to the following set of equations:

$$ \frac{\partial \bar{\rho}}{\partial t} + \frac{\partial \bar{\rho} \tilde{u}_j}{\partial x_j} = 0, \qquad (4) $$

$$ \frac{\partial \bar{\rho} \tilde{u}_i}{\partial t} + \frac{\partial \bar{\rho} \tilde{u}_i \tilde{u}_j}{\partial x_j} = -\frac{\partial \bar{\rho}}{\partial x_i} + \frac{\partial \bar{\tau}_{ij}}{\partial x_j}, \qquad (5) $$

$$ \frac{\partial \bar{\rho} \tilde{E}}{\partial t} + \frac{\partial \bar{\rho} \tilde{E} u_j}{\partial x_j} = -\frac{\partial \bar{p} u_j}{\partial x_j} + \frac{\partial \bar{\tau}_{ij} u_i}{\partial x_j} - \frac{\partial \bar{q}_j}{\partial x_j}, \qquad (6) $$

where $\tilde{f}$ represents the Favre filtered quantity $\frac{\bar{\rho} \tilde{f}}{\bar{\rho}}$. Some terms (such as $\tilde{E}u_j$) are unknown in equations (4)-(6) because they involve correlations, the filtered pressure ($\bar{p}$) or temperature ($\bar{T}$).
## B. Equations of state and transport coefficients

These equations need to be complemented by complex caloric and thermal EoS because of the peculiar thermodynamic behavior of real gases. In this study, the Martin-Hou (MH) thermal and caloric EoS³³,³⁴ are retained because they provide an accurate representation of the complex real gas thermodynamic behavior³⁵:

$$p = \frac{RT}{v-b} + \sum_{i=2}^{5} \frac{A_i + B_i T + C_i e^{-kT/T_c}}{(v-b)^i}, \qquad (7)$$

$$E_{int} = E_{int,ref} + \int_{T_{ref}}^{T} c_{v}(T')dT' + \sum_{i=2}^{5} \frac{A_i + C_i(1 + kT/T_c)e^{-kT/T_c}}{(i-1)(v-b)^{i-1}}, \qquad (8)$$

where $(.)_{ref}$ denotes a reference state, $b = v_c(1 - (-31,883Z_c + 20.533)/15)$, $k = 5.475$ and the coefficients $A_i$, $B_i$ and $C_i$ are numerical constants determined in³³,³⁴ from physical parameters summarized in Table I for the R134a gas considered in the present study. $v_c$ and $T_c$ denote respectively the critical specific volume and the critical temperature.

Chung's constitutive equations³⁶ provide the viscosity ($\mu$) and conductivity ($\kappa$) from the quantities $\rho u_i$, and $\rho E$.

<table><caption>TABLE I. Fluid properties: molar mass $M$, critical temperature $T_c$, critical pressure $P_c$, critical density $\rho_c$, acentric factor $\omega$, boiling temperature $T_b$,</caption><thead><tr><th></th><th><i>M</i> (g/mol)</th><th><i>T</i><sub>c</sub> (K)</th><th><i>P</i><sub>c</sub> (Pa)</th><th><i>Z</i><sub>c</sub></th><th><i>ω</i></th><th><i>T</i><sub>b</sub> (K)</th><th><i>c</i><sub>v∞</sub><sup>c</sup> / <i>R</i></th><th><i>n</i></th></tr></thead><tbody><tr><td>R134a</td><td>102</td><td>374.2</td><td>4.06·10<sup>6</sup></td><td>0.260</td><td>0.3268</td><td>247.1</td><td>10.82</td><td>0.695</td></tr></tbody></table>

The averaged/filtered pressure $\bar{p}$ and temperature $\bar{T}$ are indeed unknown quantities because there is no commutativity between the averaging/filtering operator and the thermal or caloric Martin-Hou EoS (equations (7)-(8)). In the following, the computable temperature and pressure are respectively denoted $\hat{T}$ and $\hat{P}$ (equations (11)-(12)): they represent the temperature and the
HIFI investigation of a real gas annular cascade

pressure computed from the averaged/filtered conservative variables.

$$ \bar{T} = \overline{\mathrm{MH}_C(\rho, \rho u_i, \rho E)}, \qquad (9) $$

$$ \bar{P} = \overline{\mathrm{MH}_T(T, \rho)}, \qquad (10) $$

$$ \hat{T} = \mathrm{MH}_C(\bar{\rho}, \bar{\rho} \tilde{u}_i, \bar{\rho} \tilde{E}) \neq \bar{T}, \qquad (11) $$

$$ \hat{P} = \mathrm{MH}_T(\bar{\rho}, \hat{T}) \neq \bar{P}, \qquad (12) $$

where $\mathrm{MH}_T$ and $\mathrm{MH}_C$ refer respectively to the thermal and caloric Martin-Hou EoS.

The transport coefficients are also impacted by the averaging/filtering process. $\hat{\kappa}$ and $\hat{\mu}$ denote respectively the conductivity and the viscosity computed from the filtered density $\bar{\rho}$ and the available temperature $\hat{T}$.

### C. Set of equations for LES

The previous observations eventually lead to the following set of equations where all terms needing modeling are in bold type and labeled:

$$ \frac{\partial \bar{\rho}}{\partial t} + \frac{\partial \bar{\rho} \tilde{u}_j}{\partial x_j} = 0, \qquad (13) $$

$$ \begin{aligned} \frac{\partial \bar{\rho} \tilde{u}_i}{\partial t} + \frac{\partial \bar{\rho} \tilde{u}_i \tilde{u}_j}{\partial x_j} = & -\frac{\partial \hat{p}}{\partial x_i} + \frac{\partial \hat{\tau}_{ij}}{\partial x_j} - \underbrace{\frac{\partial [\bar{p} - \hat{p}]}{\partial x_i}}_{\text{M1: Pressure}} + \underbrace{\frac{\partial [\hat{\tau}_{ij} - \tau_{ij}]}{\partial x_j}}_{\text{M2: Viscous}} \\ & - \underbrace{\frac{\partial \bar{\rho} [\tilde{u}_i \tilde{u}_j - \tilde{u}_i \tilde{u}_j]}{\partial x_j}}_{\text{M3: Turbulent stress}} \end{aligned} \qquad (14) $$

$$ \begin{aligned} \frac{\partial \bar{\rho} \tilde{E}}{\partial t} + \frac{\partial \bar{\rho} \tilde{E} \tilde{u}_j}{\partial x_j} = & -\frac{\partial \hat{p} \tilde{u}_j}{\partial x_j} + \frac{\partial \hat{\tau}_{ij} \tilde{u}_i}{\partial x_j} - \underbrace{\frac{\partial \hat{q}_j}{\partial x_j}}_{\text{E1: Pressure work}} - \frac{\partial [\bar{p} \tilde{u}_j - \hat{p} \tilde{u}_j]}{\partial x_j} \\ & + \underbrace{\frac{\partial [\hat{\tau}_{ij} \tilde{u}_i - \tau_{ij} \tilde{u}_i]}{\partial x_j}}_{\text{E2: Viscous work}} - \underbrace{\frac{\partial [\bar{q}_j - \hat{q}_j]}{\partial x_j}}_{\text{E3: Heat Flux}} - \underbrace{\frac{\partial \bar{\rho} [\tilde{E} u_j - \hat{E} \tilde{u}_j]}{\partial x_j}}, \end{aligned} \qquad (15) $$

Note that $\hat{\tau}_{ij} = \hat{\mu} \left( \frac{\partial \tilde{u}_i}{\partial x_j} + \frac{\partial \tilde{u}_j}{\partial x_i} - \frac{2}{3} \frac{\partial \tilde{u}_k}{\partial x_k} \delta_{ij} \right)$ and $\hat{q}_i = -\hat{\kappa} \frac{\partial \hat{T}}{\partial x_i}$. The set of equations (13)-(15) is the equivalent for real gases of the "system II" used in Vreman et al.³⁷ for perfect gases. All terms within brackets are a priori unknown and need to be modeled.
## D. Common modelling hypothesis

In a previous publication focused on LES³⁸, the authors have shown that the most important SGS terms in Eq 14 are M1 (pressure) and M3 (turbulent stress) in the inertial range of the turbulent kinetic energy spectrum. Regarding the conservation of total energy (Eq 15), they have shown that the most important terms are E4 (energy flux) and to a lesser degree E1 (pressure work). Vadrot further shows³⁹ that the pressure term M1 in the momentum equation, although of significant amplitude in the turbulent flow, has only a limited effect on turbulence because of the alignment of the SGS pressure gradient with the resolved density gradient which leads to a negligible SGS baroclinic term ($\vec{V}p_{SGS} \times \vec{V}\bar{\rho}$).

As a consequence, the authors concentrate their work on the modelling of the turbulent stress (M3), energy flux (E4) and pressure work (E1).

## E. Subgrid-scale Turbulence closure

In this study the turbulent stress term (M3) is modeled using the Boussinesq assumption such that

$$[\tilde{u}_i\tilde{u}_j - \tilde{u}_i\tilde{u}_j] = -2\nu_{sgs}\tilde{S}_{ij}$$

The wall-adapting local eddy-viscosity (WALE) model proposed by Nicoud & Ducros⁴⁰ is used in this study. It has the property that near solid surfaces in plane shear, the eddy-viscosity vanishes. The model reads

$$\nu_{sgs} = (C\omega\Delta)^2 (W_{ij}^d W_{ij}^d)^{3/2} / [(\bar{S}_{pq}\bar{S}_{pq})^{5/2} + (W_{ij}^d W_{ij}^d)^{5/4}],$$

ăwhere $W_{ij}^d$ is the traceless part of the tensor $W_{ij} = \bar{S}_{ik}\bar{S}_{kj} + \bar{\Omega}_{ik}\bar{\Omega}_{kj}$, and the model can be shown to lead to vanishing viscosity in the limit of planar flow.

The energy flux (E4) and pressure work (E1) are modeled together using the gradient of the computable temperature and the weak Reynolds analogy implied by a constant SGS Prandtl number so that

$$\bar{\rho} [\tilde{E}u_j - \tilde{E}\tilde{u}_j] + [\overline{pu}_j - \overline{p}\tilde{u}_j] = - \frac{\bar{\rho} \tilde{C}_p \nu_{sgs}}{Pr_{sgs}} \frac{\partial \hat{T}}{\partial x_j}$$

where $\tilde{C}_p$ is the computable heat capacity at constant pressure and $Pr_{sgs} = 0.6$. This value implies the hypothesis that the turbulent dissipation of heat and momentum are of comparable strength at
HIFI investigation of a real gas annular cascade

small scales in the flow.

## F. Wall modelling

Because the cost of large-eddy simulations (LES) of wall-bounded flows that resolve all the important eddies depends strongly on the Reynolds number, methods to bypass the wall layer are required to perform high-Reynolds-number LES at a reasonable cost. In this study a turbulent law of the wall is imposed at the hub, at the casing and on the blade surfaces. The model is such that the shear stress $\tau_w$ is assumed to be constant between the wall and the first point and is linked to the shear velocity such that:

$$ \tau_w = \rho_w * u^{*2} $$

where

$$ u^* = \begin{cases} \sqrt{u\mu_w/(y_w\rho_w)} & \text{Inner region: } y^+ \le 11.445 \\ (0.41u)/\log(9.2y^+) & \text{Outer region: } y^+ > 11.445 \end{cases} $$

## G. LES numerical method

Large Eddy Simulations (LES) are performed using the numerical solver AVBP$^{41}$. This parallel code solves the full compressible Navier-Stokes equations (see Eqs 13)-(15) using a two-step time-explicit Taylor-Galerkin scheme TTGC for the hyperbolic terms based on a cell-vertex formulation$^{42}$. This scheme provide high spectral resolution as well as low numerical dissipation and dispersion. It also ensures third-order accuracy in space and time$^{42}$. This scheme has been extensively validated in the context of Large Eddy Simulation of turbomachines$^{43-45}$.

A shock capturing methodology is used to preserve the positivity of the solution in regions where very strong gradients exist. To this end, an additional dissipation term proportional to the local velocity divergence is dynamically computed using the approach of Cook and Cabot$^{46}$ and added as a source term to the momentum equation. This numerical strategy can be interpreted as a localized modification of the viscous stress tensor $\tau_{ij}$ in which a local hyperviscosity $\beta(\vec{x})$ is added:

$$ \tau_{ij,\text{modified}} = \left( \beta(\vec{x}) - \frac{2}{3}\mu \right) \frac{\partial u_k}{\partial x_k} \delta_{ij} + \mu \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) $$
## H. RANS models and numerical methods

The open source code SU2<sup>47</sup> has been used to perform the Reynolds-averaged NavierStokes (RANS) simulations analyzed in the present study. The compressible RANS equations are classically solved using Roe's approximate Riemann solver to discretize the convective fluxes, extended to second order with a MUSCL reconstruction and Venkatakrishnan limiter to ensure the TVD property. Fast convergence to the steady-state is obtained thanks to an implicit formulation (see<sup>47</sup> for more details on these standard features). SU2 has been extended by Vitale et al.<sup>48</sup> and Pini et al.<sup>49</sup> to the domain of non-ideal compressible fluid dynamics (NICFD) by integrating complex thermodynamics laws and carefully adjusting the numerical treatments to real gas flows (in particular the well-established Vinokur-Montagne approach<sup>50</sup> has been implemented to compute Roe averages). The NICFD extension of SU2 has been carefully validated by comparison with experimental results (see Gori et al.<sup>15</sup> for instance). In order to perform in the present work a meaningful comparison between the AVBP LES simulations and the SU2 RANS calculations, the Martin-Hou thermal and caloric EoS and the Chung model for transport coefficients have been implemented in the SU2 framework, thus achieving an equivalent thermo-physical accuracy for both approaches.
Two popular turbulence closure models have been used in this study: the one equation SpalartAllmaras (SA) model<sup>51</sup> and the Menter shear stress transport (SST) $k-\omega$ model<sup>52</sup>. Numerical experiments have also been performed<sup>53</sup> using the compressibility (Gulyaev et al.<sup>54</sup> and Spalart<sup>55</sup> for SA and Wilcox<sup>56</sup> for $k-\omega$) and density corrections (see Catris et al.<sup>57</sup> and Otero et al.<sup>58</sup>) proposed in the literature. The impact of such corrections was observed to be negligible for the annular cascade configuration under study so that the results presented in the present work correspond to the baseline versions of both turbulence models.

## III. VERIFICATION FOR RANS AND LES ANALYSIS

### A. Real gas annular cascade configuration

The configuration under study is an annular cascade tested at Cambridge University by Baumgärtner, Otter and Wheeler<sup>59</sup>. This cascade is composed of eight stator blades. As shown in Figure 1, the numerical configuration comprises a single sector of angle 45°.

The radius of the hub is 15mm and the radius of the casing is 20mm which leads to a blade span of 5mm. In the simulation, the inlet is set at approximately one chord upstream of the leading edge
<figure><img src="image_2.png" /><figcaption>FIG. 1. Annular cascade configuration</figcaption></figure>

and the outlet is set at approximately two chords downstream of the trailing edge in the axial direction. Non reflecting boundary conditions are used at the inlet and outlet$^{60}$. Without knowledge regarding the inlet turbulence levels (not measured in the experiment), it has been decided to run the LES without turbulence injection and to assess the impact of this inlet condition in the RANS simulations. The fluid is R134a, a real gas commonly used in industrial systems as a working fluid (see Table I).

## B. Mesh resolution

The LES mesh grid (generated using CENTAUR Software from CentaurSoft) must be designed to capture most of the turbulent activity at the walls and in the domain. To this end, the mesh resolution at the walls (blade, hub and casing) is estimated *a priori* using the data presented in Table II and assuming $y^+ = 10$. This target value aims at reducing the influence of the law-of-wall model on the LES results.

Figure 2 (a) shows the hybrid mesh in the trailing edge region at mid-span with the estimated wall

<table><caption>TABLE II. Reference values used during the mesh design</caption><thead><tr><th>Freestream<br>velocity [m/s]</th><th>Density<br>[kg/m³]</th><th>Dynamic<br>viscosity [kg/m/s]</th><th>Characteristic<br>length [m]</th><th>Reynolds<br>number</th></tr></thead><tbody><tr><td>100</td><td>21</td><td>1.2.10<sup>-4</sup></td><td>0.01</td><td>1.75.10<sup>+5</sup></td></tr></tbody></table>
HIFI investigation of a real gas annular cascade

grid size $\Delta y = 1.10^{-5} m$. Figure 2 (b) illustrates the instantaneous value of $y^+$ on the casing and on the blade showing values well below 60 almost everywhere. In the neighborhood of the walls, 20 layers of prisms are imposed with a stretching ratio of 6% in the normal direction.

Away from the walls, tetrahedral elements are used. In this region of the flow, the grid resolution is set equal to $\Delta x = \mathcal{L}_i/5 = 5.10^{-5} m$ assuming the turbulent integral length $\mathcal{L}_i$ to be 5% of the blade span. Given the Reynolds number, this grid resolution lies in the inertial region of the turbulent kinetic energy spectrum where LES turbulence closure should operate. These constraints lead to a grid comprising 400M cells. Computations are distributed with Message Passing Interface (MPI) on 15360 cores using the IRENE High Performance Computing (HPC) facilities at Commissariat à l'Energie Atomique - Très Grand Centre de Calcul (CEA-TGCC). Figure 3 shows an isosurface of the instantaneous Q-criterion$^{61}$ colored by the Mach number. The development of the turbulent boundary layer on the suction side of the blade is identified, along with its strong interaction with the shock wave that form close to the trailing edge.

<figure><img src="image_3.png" /><figcaption>FIG. 2. LES – Mesh resolution: (a) Cylindrical mesh cut in the vicinity of the trailing edge (b) Casing and blade : Instantaneous $y^+$</figcaption></figure>

A usual verification step has been performed for RANS simulations so as to identify the grid resolution offering the best trade-off between accuracy and computational efficiency. Three levels of refinement in space are considered through a coarse (31M elements), a medium (47M elements) and a fine (116M elements) grid. Regardless of the mesh size, the flow is resolved at the wall ($y^+ = 1$). Figure 4 displays the isentropic Mach number distribution along the centerline of the computational domain, at the casing. This line follows the main direction of the flow at equal distance between the suction side of a blade and the pressure side of the neighbouring blade. All simulations are computed using the SST model on this series of increasingly refined grid. This comparison suggests that the medium grid is fine enough to achieve convergence in space. This
<figure><img src="image_4.png" /><figcaption>FIG. 3. LES – Suction side of the stator: Isosurface of the instantaneous Q-criterion ($Q * \Delta_x^2 = 250$) colored by the Mach number</figcaption></figure>

mesh is therefore retained for all the RANS calculations analyzed in the present work.

<figure><img src="image_5.png" /><figcaption>FIG. 4. Grid convergence study for RANS simulations. Isentropic Mach number $\frac{P_{\text{inlet}}}{P} = \left(1 + \frac{\gamma-1}{2}M_{is}^2\right)^{\frac{\gamma}{\gamma-1}}$ along the centerline of the annular cascade computed using the SST model and 3 increasingly refined grids: coarse (31M elements), reference/medium (47M elements), fine (116M elements).</figcaption></figure>

### C. Temporal convergence

The numerical convergence of the LES calculation is addressed using the method proposed by Boudet *et al.*$^{62}$, which aims at finding the simulation time interval beyond which the statistical mo-
HIFI investigation of a real gas annular cascade

ments (average and standard deviation) of the pressure do not change significantly anymore. This time interval is usually referred to as the numerical convergence time. At regular time intervals during the simulation, the pressure signal is cut into four equal pieces, and the statistic estimates are compared for the last three quarters. To measure the convergence, the following function is used

$$ \zeta(P) = 100 * \max_{i,j=2,4} \left| \frac{\phi(P)_i - \phi(P)_j}{\phi(P)_4} \right| $$

where $\phi(P)_i$ is the statistical quantity of interest (mean or standard deviation) computed for the quarter $i$. The evolution of this quantity is presented in Figure 5(a) for the mean and standard deviation of the pressure at the trailing edge of the stator blade. We consider that numerical convergence is reached when $\zeta$ stabilizes below 5%. From Figure 5(a), numerical convergence is reached for the mean pressure at $t^+ = 1$ where $t^+ = t/t_c$ is time normalized by the characteristic time ($t_c$) defined as the chord (≈10mm) divided by a reference velocity (100 m.s$^{-1}$). As the standard deviation, strongly related to turbulent phenomena, is a higher order statistical quantity, it requires at least 10 non-dimensional times to reach numerical convergence. Strongly related to this analysis, Figure 5(b) presents the statistical convergence of the mean and rms pressure at the trailing edge for the full temporal signal. It can be observed that the full temporal signal of length 50$t^+$ is adequate to obtain statistical fields well converged in time.

<figure><img src="image_6.png" /><figcaption>FIG. 5. Temporal convergence analysis: (a) Numerical convergence using the method proposed by Boudet et al.⁶², (b) Statistical convergence over the entire temporal signal.</figcaption></figure>
# IV. LES AND RANS RESULTS

## A. Validation of the numerical results

To validate the numerical approach followed by the authors to study the real gas flow in the complex geometry of the Cambridge University annular turbine cascade, RANS and LES results are compared to experimental measurements obtained at the casing and in the wake of the blade at mid-span⁵⁹. Figure 6 illustrates and localizes on an arbitrary axial cut in the domain the available measurements to which the numerical results are compared. The azimuthal pitch later used to display the comparisons and taking into account the periodicity of the domain is also defined.

In the following, $\sigma$ represents the experimental uncertainty applied to the different measured quantities. To mimic this uncertainty and have a sense of the dynamics around the converged mean quantities obtained in the LES, the RMS is also displayed for each quantity at each location.

Further tests have been performed using RANS in order to assess the influence of (i) the turbulence model and of (ii) the inlet turbulence level which is unknown in the experiment. Table III summarizes the turbulence models and inlet turbulence levels used in each simulation.

<table><caption>TABLE III. Turbulence models and inlet turbulence treatment in the performed simulations</caption><tr><td>Simulation</td><td>LES+RMS</td><td>SA[ν̃<sub>∞</sub> = 3.0]</td><td>SA[ν̃<sub>∞</sub> = 0.1]</td></tr><tr><td>Turbulence model</td><td>SGS modelling</td><td>Spalart-Almaras</td><td>Spalart-Almaras</td></tr><tr><td>Inlet turbulence</td><td>None</td><td>ν̃<sub>∞</sub> = 3.0</td><td>ν̃<sub>∞</sub> = 0.1</td></tr></table>

<table><tr><td>Simulation</td><td>SST[TI<sub>∞</sub> = 5%]</td><td>SST[TI<sub>∞</sub> = 0%]</td></tr><tr><td>Turbulence model</td><td>k - ω - SST</td><td>k - ω - SST</td></tr><tr><td>Inlet turbulence</td><td>TI<sub>∞</sub> = 5%</td><td>None</td></tr></table>

## 1. Comparison with wall measurements

Figures 7 and 8 compare the evolution of the static pressure at the casing as a function of the pitch in the experiment to the data obtained by RANS and LES. Overall, the pressure ratio is expected to decrease as one moves to larger z values following the expansion of the flow along the stator blade. This expansion is also linked to the evolution of the local velocity so that it does not
<figure><img src="image_7.png" /><figcaption>FIG. 6. Sketch of an arbitrary axial cut in the domain featuring the boundary conditions, the location of the available measurements and the pitch definition.</figcaption></figure>

occur evenly as a function of the pitch. The flow accelerates more quickly on the suction side (on the left of the bladed regions) so that the pressure is expected to decrease more sharply on that side of the blade. From Figure 7, it can be seen that upstream of the trailing edge (located at $z \approx 9$mm), a very good agreement is reached between experimental measurements and numerical results. As one gets closer to the transition region of the suction side boundary layer ($z = 7.2$mm: bottom right figure), RANS results obtained with inlet turbulence injection start to diverge from those obtained without. RANS results obtained without turbulence injection are closer to the experimental measurements and LES results.

This phenomenon is further amplified downstream as clearly shown by Figure 8. Overall, numerical profiles are slightly above experimental ones. Yet results obtained without inlet turbulence injection are slightly closer to the experiment. More specifically, at $z = 8.1$mm pressure at the suction side (left side of the curve) does not drop in the simulations as much as it does in the experiment. It suggests that the mean tangential velocity in this region is underestimated in the numerics hinting to some degree of error in the prediction of the turbulent boundary layer profiles. Yet LES and RANS results remain very close from each other at this location. Away from the blade (beyond a pitch of 0.5), LES and RANS obtained without inlet turbulence injection (ITI)
HIFI investigation of a real gas annular cascade

are closer to the experimental measurements. At $z = 9mm$, one sits at the trailing edge. At this location the sharp decrease of the static pressure in the wake is missed by the experiment. Again, RANS and LES obtained without ITI perform better and agree very well with each other. The same comments can be given downstream of the blade at $z = 9.5mm$ and $z = 10.9mm$. RANS and LES results without ITI only depart from each other in the free stream for pitch values comprised between 0.8 and 1. In this region the scattering of the experimental results is large making it difficult to determine whether LES or RANS without ITI performs better.

In order to better quantify the quality of the different simulations, the evolution of the L2 norm of the difference between the mean experimental distribution of the static pressure ratio and the RANS and mean LES predictions (normalized by the experimental results) are computed for the 8 available axial locations and reported in Figure 9. The trends followed by the L2 errors are the same for all simulations. Before the boundary layer transition (i.e. before $z = 8.1mm$), all methods manage to recover the experimental results with an error lower than 8%. As previously described, the sharp drop of pressure observed in the experiment close to the blade is not captured at $z = 8.1mm$ which leads to an increase of the error (reaching 32 to 35%). The last three locations that stand at or beyond the trailing edge tend to show that RANS with turbulence injection performs worse (L2 error in the range of 20%) than RANS without inlet turbulence injection and LES (L2 error in the range of 11 to 14%). Both RANS without ITI and LES reach very close levels of accuracy at $z = 10.9mm$.

## 2. Comparison with wake measurements

In the wake of the blade, probes measuring the stagnation pressure induce a bow shock²⁰, which has to be corrected for in the post-processing of the simulation results by computing the shock corrected total pressure:

$$p_{t,\text{shock}} = p_t \frac{(\gamma+1)\text{M}^2}{(\gamma-1)\text{M}^2 + 2} \frac{\frac{1}{\gamma-1}}{2\gamma\text{M}^2 - (\gamma-1)} \frac{\frac{1}{\gamma-1}}{\gamma-1} \quad (16)$$

where a constant value for $\gamma = 1.07$ is chosen to match the treatment done in the experiment. Note that the assumption of a normal bow shock is only valid when the probes are aligned with the flow direction. Fig. 10 shows that all simulations without turbulence injection were able to predict the azimuthal location and total pressure ratio of the blade wake within the experimental uncertainty. However, there is a slight displacement between numerics and experiment in the position of the
HIFI investigation of a real gas annular cascade

<figure><img src="image_8.png" /><figcaption>FIG. 7. Experimental measurements of static pressure ratio at the casing (R=20mm), upstream of the trailing edge compared to RANS and LES results. Grayed areas correspond to the bladed regions.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>FIG. 8. Experimental measurements of static pressure ratio at the casing (R=20mm), at the trailing edge and downstream compared to RANS and LES results. Grayed areas correspond to the bladed regions.</figcaption></figure>

shock at pitch≈ 0.25 as well as a discrepancy on the total pressure level in the freestream region of the flow. This might be due to either or both of the following explanation arguments: (i) the inability of the numerical models to accurately describe the interaction of the oblique shock with the boundary layer which leads to a slight misprediction in the trailing edge shock angle or (ii) a slight misalignment of the probes in the experimental measurements.
<figure><img src="image_10.png" /><figcaption>FIG. 9. Evolution of the L2 errors, normalized by the experimental results, as a function of the axial location for LES and RANS with and without inlet turbulence injection. RANS results are obtained using the $k-\omega SST$ turbulence model.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>FIG. 10. Experimental measurements of shock corrected total pressure ratio in the wake region compared to RANS and LES results</figcaption></figure>

In summary, probes measuring stagnation pressure in the wake of the blade induce a bow shock, which, when corrected in post-processing using a shock corrected total pressure formula, is found to match fairly well with numerical simulations. Slight discrepancies in shock position and total pressure levels may be attributed to challenges in modeling oblique shock / boundary layer interactions or a slight misalignment of experimental probes.

Overall, RANS and LES results obtained without ITI agree fairly well with the experimental measurements especially before the boundary layers transition, in the wake and even more specifically on the pressure side of the wake where velocities are slightly lower leading to a smaller drop of
## B. Analysis of the 3D flow using numerical results

The overall agreement gives enough confidence in the numerical tools to further analyze the origin of the losses observed in this configuration. It will be shown that a significant part of the losses generated in this annular configuration originates from large vortical structures forming in the flow at the hub.

Figures 11 (a) to (c) respectively present a comparison of the pressure ratio obtained in LES and RANS simulations at the casing, at midspan and at the hub. Again, the agreement between both approaches is very good. The strong acceleration of the flow at the suction side is well captured. The bump in the profiles that arises at 50%, 60% and 70% of the chord at the casing, at midspan and at the hub corresponds to the impact of the next blade's trailing edge shock. This leads to a transition to turbulence of the boundary layer best seen by the downstream flat regions starting respectively for the casing, midspan and hub at 70%, 75% and 80% of the chord. It is in this turbulent region of the boundary layer that results from LES and RANS tend to depart from each other. In the next paragraph, to better understand the differences between the two numerical simulation techniques, one further analyzes this region where LES tends to predict smaller levels of the pressure ratio, especially at the hub.

Figures 12 (a) and (b) focus on the suction side and trailing edge regions. In these figures, the flow comes from the right and goes toward the top-left of the image. Streamlines originating from the two visible white lines along the hub and suction side of the blade are also displayed and colored by the magnitude of the velocity. In Figure 12 (a), it can be seen that in the RANS results, streamlines originating from the suction side quickly join those from the hub leading to a single well defined stream-tube. Streamlines then tend to rotate along the longitudinal axis without much dispersion. These observations are consistent with the presence of a vortical structure originating at the corner between the hub and suction side of the blade. In the LES results (Figure 12 (b)), streamlines originating from the suction side quickly move away from the wall and display large velocity values. In parallel those originating from the hub also move away from the wall. Both streamline systems keep significantly different velocities but intertwine with each other above the blade-hub corner in the region characterized by the boundary layer - shock wave interaction. Once the trailing edge is passed, they progressively tend to form a single structure that can be evidenced
HIFI investigation of a real gas annular cascade

<figure><img src="image_12.png" /><figcaption>FIG. 11. Comparison of RANS and LES pressure ratio along the normalized chord at : (a) casing, (b) mid-span, (c) hub. "ss" stands for "suction side" and "ps" stands for "pressure side".</figcaption></figure>

by the computation of the Q-criterion in the flow.

Figures 13 (a) and (b) present isosurfaces of the Q-criterion colored by the velocity magnitude. Planar cuts perpendicular to the bulk flow are also shown, colored by the entropy. In Figure 13 (a), it can be seen that in the RANS results, the flow is mostly characterized by a single large structure aligned with the bulk flow and finding its origin in the presence of a strong corner separation. In the LES prediction, Figure 13 (b) actually shows the formation of two distinct large vortical structures at the blade/hub corner. The vortex closest to the blade has a larger velocity whereas the second one is associated to lower velocity values. Downstream, those two structures tend to progressively merge in coherence with the information provided by the streamlines. As expected, entropy levels are significant in the boundary layers and vortical structures only. Once the trailing edge is passed, entropy reaches large levels in the wake and in the large vortical structures formed
<figure><img src="image_13.png" /><figcaption>FIG. 12. Visualization of the corner separation at the hub: Streamlines colored by the velocity magnitude in (a) RANS without inlet turbulence injection and (b) LES. The bulk flow goes from right to left.</figcaption></figure>

at the hub/blade corner separation.

### C. Analysis of losses using numerical results

To further understand the loss mechanisms in this stator flow, Figure 14 compares the growth rates of the mean entropy (integrated over the entire section) along the axial direction obtained from RANS and LES. The first observation is that because of the large intrinsic value of the entropy in the flow at the inlet, the entropy only rises by less than 1% over the entire length of the system. Although very small, this growth is consistently obtained by both RANS and LES. After
<figure><img src="image_14.png" /><figcaption>FIG. 13. Visualization of the corner separation at the hub: Isosurface of the Q criterion ($Q * \Delta_x^2 = 25$) colored by the velocity magnitude and plane cuts colored by the entropy in (a) RANS without inlet turbulence injection and (b) LES. The bulk flow goes from right to left.</figcaption></figure>

the trailing edge, both methods also agree and predict a reduction by a factor of two for the entropy growth rate in the wake region. The spikes observed at the trailing edge in both RANS and LES results are merely numerical artefact that might be related to the very strong gradients occurring in this region.

To more precisely quantify the amount of losses due to the boundary layers, the wake and the

<figure><img src="image_15.png" /><figcaption>FIG. 14. Visualization of the entropy rise: Mean entropy (compared to the inlet) as a function of the axial direction in RANS and LES.</figcaption></figure>
HIFI investigation of a real gas annular cascade

vortical structures related to the corner separation, the following loss coefficient is introduced:

$$ \zeta_{\text{loss}}(x) = \left\langle \frac{H - H_s}{H_0 - H_s} \right\rangle_x, $$

where $H$ is the specific enthalpy, $H_0$ is the specific total enthalpy, $H_s$ is the specific isentropic enthalpy and $< \cdot >_x$ stands for the averaging operator at section $x$. Since $(H - H_s)(x) = \int_{ref}^x (Tds)$ where $s$ is the entropy, this ratio compares the share of enthalpy related to entropy to the sum of itself with the available kinetic energy. Appendix A describes the precise implementation of $\zeta_{\text{loss}}(x)$. In regions where the cross section does not change, it can only grow and accounts both for the entropy generation at the hub and casing walls and in the wake.

Table IV presents the values obtained by RANS and LES for $\zeta_{\text{loss}}$ at some sections of interest. To determine the influence of the corner separation, $\zeta_{\text{loss}}(x)$ is both averaged over the full section or at the mid-span radius only. Regardless of the extraction location, levels gathered at mid-span

<table><caption>TABLE IV. Comparison of the loss coefficient $\zeta_{\text{loss}}$ at the trailing edge and at the outlet obtained by RANS and LES.</caption><thead><tr><th>Numerical method<br>(extraction location)</th><th>ζ<sub>loss</sub> at<br>trailing edge</th><th>ζ<sub>loss</sub> at<br>outlet</th><th>% of ζ<sub>loss</sub><br>after blade</th></tr></thead><tbody><tr><td>RANS (whole span)</td><td>0.2831</td><td>0.3815</td><td>34.75</td></tr><tr><td>LES (whole span)</td><td>0.2574</td><td>0.3446</td><td>33.88</td></tr><tr><td>RANS (midspan)</td><td>0.2546</td><td>0.3153</td><td>23.84</td></tr><tr><td>LES (midspan)</td><td>0.2269</td><td>0.2905</td><td>28.03</td></tr></tbody></table>

are systematically 15% lower as they ignore losses at the hub and casing. RANS tends to predict loss levels 10% larger than LES. Yet when the whole span is considered, both methods agree on the share, equal to roughly 33%, associated to the unbladed region of the flow after the trailing edge. This agreement may be incidental given the large difference in the turbulence modeling used in both methods and RANS does not fully agree with LES on the share associated with the wake but places it around 25%. Even though the computation method for $\zeta_{\text{loss}}$ is slightly different, it is interesting to note however that these results confirm and extend the original findings of Baumgartner *et al.*²⁰.

From these results, one can see that even if local discrepancies still remain with LES, RANS
HIFI investigation of a real gas annular cascade

manages to provide meaningful results at a much lower cost especially for the overall share of the
losses due to the unbladed region of the flow.

V. CONCLUSION

Numerical simulations were conducted in this study using RANS and LES approaches to ana-
lyze the flow of a real gas in a complex geometry (annular cascade). The results of these simula-
tions are compared to experimental measurements taken at the casing and in the wake of a blade at
mid-span in order to validate the numerical approaches. The results show good agreement between
the experimental pressure measurements and the numerical results upstream of the trailing edge.
However, as one moves closer to the transition on the suction side boundary layer, RANS results
with inlet turbulence injection begin to diverge from those without. Overall, the numerical profiles
are slightly above the experimental ones, with the LES and RANS without inlet turbulence injec-
tion performing better (Relative L2 error with experimental results below 15%). All simulations
without turbulence injection are able to accurately predict the azimuthal location of the wake and
its total pressure ratio within the experimental uncertainty. There is a slight displacement between
the numerics and the experiment in the position of the shock at a pitch of approximately 0.25 and
the total pressure in the freestream region of the flow, which may be due (i) to the inability of the
numerical models to accurately describe the interaction of the oblique shock with the boundary
layer or/and (ii) to a slight longitudinal offset in the experimental setup.

With this fair agreement between experimental measurements and numerical results, the study then
shows that a strong corner separation is probably present at the foot of the blade on the suction
side. This flow feature remains to be experimentally verified but influences the loss coefficients
in the cascade. These loss coefficients are found to be slightly different between RANS and LES,
most probably because of the strong difference in the turbulence modeling used for both methods.
This discrepancy also indicates the need for further research in turbulence modeling for real gas
flows. The transition to turbulence is a particularly complex aspect that requires more investigation
in order to improve the accuracy of numerical simulations in real or dense gas flows.

Given the importance of accurately predicting turbulent flow characteristics in real gas flows of
practical interest, the authors are currently using the results of this study as a detailed database
to continue their research into improving turbulence closure models for RANS and LES using
machine learning techniques and in particular Artificial Neural Networks (ANN)⁶³.
## VI. ACKNOWLEDGEMENTS

The version of SU2 used in this study (including Martin-Hou thermal and caloric EoS and the Chung model for transport coefficients) is available upon reasonable request. This work has been supported by the JCJC ANR EDGES project, grant #ANR-17-CE06-0014-01 of the French Agence Nationale de la Recherche. Simulations have been carried out using HPC resources at CINES under the project grants #A0102A07564 and #A0122A07564. We also acknowledge PRACE for awarding us access to the IRENE HPC facility @ TGCC, CEA, France. The authors would like to thank Prof. Andrew Wheeler and Dr. John Otter for providing them with the experimental results. The authors would also like to thank the CFD team (especially Florent Duchaine and Gabriel Staffelbach) at CERFACS for their support and for giving access to their solver AVBP during the course of the PRACE project EDGES. Part of the present results have been presented at the 4th NICFD conference held in December 2022 at London City University.

## DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## Appendix A: Computation of the loss coefficient

If one further considers (i) the inlet as a common reference for all z and (ii) that the isentropic enthalpy and enthalpy are arbitrarily the same at the inlet ($H_s(z_{in}) = H(z_{in})$), one can write

$$H(z) = H(z_{in}) + \Delta_{in,z}H$$

$$H_s(z) = H_s(z_{in}) + \Delta_{in,z}H_s = H(z_{in}) + \Delta_{in,z}H_s$$

$$H_0 = \text{constant} = H(z_{in}) + \frac{V_{in}^2}{2}$$

$$\zeta_{\text{loss}}(z) = \left\langle \frac{\Delta_{in,z}H - \Delta_{in,z}H_s}{\frac{V_{in}^2}{2} - \Delta_{in,z}H_s} \right\rangle_z$$
REFERENCES

¹A. Guardone, P. Colonna, A. Wheeler, and C. Giorgia, "1st international seminar on non-ideal compressible-fluid dynamics for propulsion and power," Journal of Physics: Conference Series **821**, 011001 (2017).

²F. di Mare, A. Spinelli, and M. Pini, *Non-Ideal Compressible Fluid Dynamics for Propulsion and Power: Selected Contributions from the 2nd International Seminar on Non-Ideal Compressible Fluid Dynamics for Propulsion & Power*, NICFD 2018, October 4-5, 2018, Bochum, Germany (Springer Nature, 2020).

³M. Pini, C. De Servi, A. Spinelli, F. di Mare, and A. Guardone, *Proceedings of the 3rd International Seminar on Non-Ideal Compressible Fluid Dynamics for Propulsion and Power: NICFD 2020*, Vol. 28 (Springer Nature, 2021).

⁴M. White, T. El Samad, I. Karathanassis, A. Sayma, M. Pini, and A. Guardone, *Proceedings of the 4th International Seminar on Non-Ideal Compressible Fluid Dynamics for Propulsion and Power: NICFD 2022* (Springer Cham, 2023).

⁵P. A. Thompson, "A fundamental derivative in gasdynamics," The Physics of Fluids **14**, 1843-1849 (1971).

⁶G. Cammi, C. C. Conti, A. Spinelli, and A. Guardone, "Experimental characterization of nozzle flow expansions of siloxane mm for orc turbines applications," Energy **218**, 119249 (2021).

⁷M. Manfredi, G. Persico, A. Spinelli, P. Gaetani, and V. Dossena, "Design of experiments for supersonic orc nozzles in linear cascade configuration," in *Proceedings of the 6th International Seminar on ORC Power Systems*, edited by T. U. of Munich, Technical University of Munich (Technical University of Munich, 2021) pp. 1-12.

⁸C. C. Conti, A. Fusetti, A. Spinelli, and A. Guardone, "Shock losses and pitot tube measurements in non-ideal supersonic and subsonic flows of organic vapors," Energy **265**, 126087 (2023).

⁹C. M. De Servi, M. Burigana, M. Pini, and P. Colonna, "Design method and performance prediction for radial-inflow turbines of high-temperature mini-organic rankine cycle power systems," Journal of Engineering for Gas Turbines and Power **141** (2019).

¹⁰F. Beltrame, A. Head, C. De Servi, M. Pini, F. Schrijer, and P. Colonna, "First experiments and commissioning of the orchid nozzle test section," (Springer, 2021) Chap. 1, pp. 169-178.
<footnote n="11">¹¹M. Robertson, P. Newton, T. Chen, A. Costall, and R. Martinez-Botas, “Experimental and numerical study of supersonic non-ideal flows for organic rankine cycle applications,” Journal of Engineering for Gas Turbines and Power **142** (2020).</footnote>

<footnote n="12">¹²M. C. Robertson, P. J. Newton, T. Chen, and R. F. Martinez-Botas, “Development and commissioning of a blowdown facility for dense gas vapours,” in *Turbo Expo: Power for Land, Sea, and Air*, Vol. 3 (2019).</footnote>

<footnote n="13">¹³R. Felix, Y. K. Eugeny, P. Max, and a. Stefan, “Closed loop organic wind tunnel (clowt): Design, components and control system,” Energy Procedia **129**, 200–207 (2017).</footnote>

<footnote n="14">¹⁴*CLOWT: A Multifunctional Test Facility for the Investigation of Organic Vapor Flows*, Fluids Engineering Division Summer Meeting, Vol. 2 (2018).</footnote>

<footnote n="15">¹⁵G. Gori, M. Zocca, G. Cammi, A. Spinelli, and A. Guardone, “Experimental assessment of the open-source SU2 CFD suite for ORC applications,” Energy Procedia **129**, 256–263 (2017).</footnote>

<footnote n="16">¹⁶C. Matar, P. Cinnella, X. Gloerfelt, F. Reinker, and S. aus der Wiesche, “Investigation of non-ideal gas flows around a circular cylinder,” *Energy* **268**, 1265–63 (2023).</footnote>

<footnote n="17">¹⁷G. J. Otero R, S. H. Smit, and R. Pecnik, “Three-dimensional unsteady stator-rotor interactions in high-expansion organic Rankine cycle turbines,” *Energy* **217**, 1193–39 (2021).</footnote>

<footnote n="18">¹⁸M. Manfredi, G. Persico, A. Spinelli, P. Gaetani, and V. Dossena, “Design and commissioning of experiments for supersonic ORC nozzles in linear cascade configuration,” Applied Thermal Engineering **224**, 1199–96 (2023).</footnote>

<footnote n="19">¹⁹F. J. Dura Galiana, A. P. S. Wheeler, and J. Ong, “A study of trailing-edge losses in organic rankine cycle turbines,” in *ASME Turbo Expo* (2015) pp. 1–12.</footnote>

<footnote n="20">²⁰D. Baumgärtner, J. J. Otter, and A. P. Wheeler, “The effect of isentropic exponent on transonic turbine performance,” *Journal of Turbomachinery* **142** (2020).</footnote>

<footnote n="21">²¹B. Obert and P. Cinnella, “Comparison of steady and unsteady rans cfd simulation of a supersonic orc turbine,” *Energy Procedia* **129**, 1063–1070 (2017).</footnote>

<footnote n="22">²²P. Klonowicz, P. Lampart, T. Suchocki, D. Zaniewski, P. Klimaszewski, et al., “Optimization of an axial turbine for a small scale orc waste heat recovery system,” *Energy* **205**, 1180–5 (2020).</footnote>

<footnote n="23">²³L. Sciacovelli, P. Cinnella, and X. Gloerfelt, “Direct numerical simulations of supersonic turbulent channel flows of dense gases,” *Journal of Fluid Mechanics* **821**, 153–199 (2017).</footnote>

<footnote n="24">²⁴A. Vadrot, A. Giauque, and C. Corre, “Direct numerical simulations of temporal compressible mixing layers in a bethe–zel’dovich–thompson dense gas: influence of the convective mach number,” *Journal of Fluid Mechanics* **922** (2021).</footnote>
²⁵X. Wang, J. Wang, and S. Chen, "Compressibility effects on statistics and coherent structures of compressible turbulent mixing layers," Journal of Fluid Mechanics **947**, A38 (2022).

²⁶P. Cinnella and X. Gloerfelt, "Insights into the turbulent flow of dense gases through high-fidelity simulations," Computers & Fluids **267**, 106067 (2023).

²⁷M. Harnieh, M. Thomas, R. Bizzari, J. Dombard, F. Duchaine, and L. Gicquel, "Assessment of a coolant injection model on cooled high-pressure vanes with large-eddy simulation," Flow, Turbulence and Combustion **104**, 643–672 (2020).

²⁸J. De Laborderie, F. Duchaine, L. Gicquel, O. Vermorel, G. Wang, and S. Moreau, "Numerical analysis of a high-order unstructured overset grid method for compressible les of turbomachinery," Journal of Computational Physics **363**, 371–398 (2018).

²⁹G. Wang, M. Sanjose, S. Moreau, D. Papadogiannis, F. Duchaine, and L. Gicquel, "Noise mechanisms in a transonic high-pressure turbine stage," International Journal of Aeroacoustics **15**, 144–161 (2016).

³⁰C. Zhang, Z. Yuan, L. Duan, Y. Wang, and J. Wang, "Dynamic iterative approximate deconvolution model for large-eddy simulation of dense gas compressible turbulence," Physics of Fluids **34**, 125103 (2022).

³¹A. Giauque, A. Vadrot, P. Errante, and C. Corre, "Towards subgrid-scale turbulence modeling in dense gas flows," in *International Seminar on Non-Ideal Compressible-Fluid Dynamics for Propulsion & Power* (Springer, 2021) pp. 71–77.

³²J.-C. Hoarau, P. Cinnella, and X. Gloerfelt, "Large eddy simulations of strongly non-ideal compressible flows through a transonic cascade," Energies **14**, 772 (2021).

³³J. J. Martin and Y.-c. Hou, "Development of an Equation of State for Gases," AIChE journal **2**, 142–151 (1955).

³⁴Y.-C. Hou and J. Martin, "Physical and thermodynamic properties of trifluoromethane," American Institute of Chemical Engineers **5**, 125–129 (1959).

³⁵A. Guardone, L. Vigevano, and B. Argrow, "Assessment of thermodynamic models for dense gas dynamics," Physics of Fluids **16**, 3878–3887 (2004).

³⁶T. H. Chung, M. Ajlan, L. L. Lee, and K. E. Starling, "Generalized multiparameter correlation for nonpolar and polar fluid transport properties," Industrial & engineering chemistry research **27**, 671–679 (1988).

³⁷B. Vreman, B. Geurts, and H. Kuerten, "A priori tests of large eddy simulation of the compressible plane mixing layer," Journal of engineering mathematics **29**, 299–327 (1995).
<sup>38</sup>A. Giauque, A. Vadrot, P. Errante, and C. Corre, “A priori analysis of subgrid-scale terms in compressible transcritical real gas flows,” Physics of Fluids <u>33</u>, 085126 (2021).

<sup>39</sup>A. Vadrot, *Numerical simulation and modeling of compressible turbulence in dense gas flows*, Ph.D. thesis, "Ecole Centrale de Lyon, ED MEGA" (2021).

<sup>40</sup>F. Nicoud and F. Ducros, “Subgrid-scale stress modelling based on the square of the velocity gradient tensor,” Flow, turbulence and Combustion <u>62</u>, 183–200 (1999).

<sup>41</sup>T. Schonfeld and M. Rudgyard, “Steady and unsteady flow simulations using the hybrid flow solver avbp,” AIAA journal <u>37</u>, 1378–1385 (1999).

<sup>42</sup>O. Colin and M. Rudgyard, “Development of high-order taylor–galerkin schemes for les,” Journal of computational physics <u>162</u>, 338–371 (2000).

<sup>43</sup>N. Odier, A. Thacker, M. Harnieh, G. Staffelbach, L. Gicquel, F. Duchaine, N. G. Rosa, and J.-D. Müller, “A mesh adaptation strategy for complex wall-modeled turbomachinery les,” Computers & Fluids <u>214</u>, 104766 (2021).

<sup>44</sup>D. Dupuy, L. Gicquel, N. Odier, F. Duchaine, and T. Arts, “Analysis of the effect of intermittency in a high-pressure turbine blade,” Physics of Fluids <u>32</u>, 095101 (2020).

<sup>45</sup>F. Duchaine, N. Maheu, V. Moureau, G. Balarac, and S. Moreau, “Large-eddy simulation and conjugate heat transfer around a low-mach turbine blade,” Journal of Turbomachinery <u>136</u>, 051015 (2014).

<sup>46</sup>A. W. Cook and W. H. Cabot, “A high-wavenumber viscosity for high-resolution numerical methods,” Journal of Computational Physics <u>195</u>, 594–601 (2004).

<sup>47</sup>T. D. Economon, F. Palacios, S. R. Copeland, T. W. Lukaczyk, and J. J. Alonso, “SU2: An Open-Source Suite for Multiphysics Simulation and Design,” AIAA Journal <u>54</u>, 828–846 (2016).

<sup>48</sup>S. Vitale, G. Gori, M. Pini, A. Guardone, T. D. Economon, F. Palacios, J. J. Alonso, and P. Colonna, “Extension of the SU2 open source CFD code to the simulation of turbulent flows of fluids modelled with complex thermophysical laws,” 22nd AIAA Computational Fluid Dynamics Conference June, 1–22 (2015).

<sup>49</sup>M. Pini, S. Vitale, P. Colonna, G. Gori, A. Guardone, T. Economon, J. Alonso, and F. Palacios, “SU2 : the Open-Source Software for Non-ideal Compressible Flows SU2 : the Open-Source Software for Non-ideal Compressible Flows,” Journal of Physics: Conference Series <u>821</u> (2017).

<sup>50</sup>M. Vinokur and J. L. Montagné, “Generalized flux-vector splitting and Roe average for an equilibrium real gas,” Journal of Computational Physics <u>89</u>, 276–300 (1990).
<footnote n="51">⁵¹P. R. Spalart, S. R. Allmaras, and A, “A one equation turbulence model for aerodynamic flows,” La Recherche Aérospatiale, 5–21 (1994).</footnote>

<footnote n="52">⁵²F. R. Menter, “Two-equation eddy-viscosity turbulence models for engineering applications,” AIAA Journal **32**, 1598–1605 (1994).</footnote>

<footnote n="53">⁵³D. Schuster, Y. Ince, A. Giauque, and C. Corre, “Assessment of Density and Compressibility Corrections for RANS Simulations of Real Gas Flows Using SU2,” ERCOFTAC Series **29**, 82–90 (2023).</footnote>

<footnote n="54">⁵⁴A. N. Gulyaev, V. E. Kozlov, and A. N. Sekundov, “A universal one-equation model for turbulent viscosity,” Fluid Dynamics **28**, 485–494 (1993).</footnote>

<footnote n="55">⁵⁵P. Spalart, “Trends in Turbulence Treatments,” in *AIAA Fluids 2000-2306* (2000) pp. 1–13.</footnote>

<footnote n="56">⁵⁶D. C. Wilcox, “Dilatation-dissipation corrections for advanced turbulence models,” AIAA Journal **30**, 2639–2646 (1992).</footnote>

<footnote n="57">⁵⁷S. Catris and B. Aupoix, “Density corrections for turbulence models,” Aerospace Science and Technology **4**, 1–11 (2000).</footnote>

<footnote n="58">⁵⁸G. J. Otero R., A. Patel, R. Diez S., and R. Pecnik, “Turbulence modelling for flows with strong variations in thermo-physical properties,” International Journal of Heat and Fluid Flow **73**, 114–123 (2018).</footnote>

<footnote n="59">⁵⁹D. Baumgärtner, J. J. Otter, and A. P. Wheeler, “The effect of compressibility factor on turbine performance,” in *Turbo Expo: Power for Land, Sea, and Air*, Vol. 84966 (American Society of Mechanical Engineers, 2021) p. V004T06A020.</footnote>

<footnote n="60">⁶⁰T. J. Poinsot and S. Lele, “Boundary conditions for direct simulations of compressible viscous flows,” Journal of computational physics **101**, 104–129 (1992).</footnote>

<footnote n="61">⁶¹J. Jeong and F. Hussain, “On the identification of a vortex,” Journal of fluid mechanics **285**, 69–94 (1995).</footnote>

<footnote n="62">⁶²J. Boudet, J.-F. Monier, and F. Gao, “Implementation of a roughness element to trip transition in large-eddy simulation,” Journal of Thermal Science **24**, 30–36 (2015).</footnote>

<footnote n="63">⁶³A. Giauque, A. Vadrot, and C. Corre, “Development, Validation and Application of an ANN-Based Large Eddy Simulation Subgrid-Scale Turbulence Model for Dense Gas Flows,” in *ER-COFTAC Series*, Vol. 29 (Springer Nature Switzerland, 2023) pp. 43–52.</footnote>