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Purpose The Reynolds-averaged Navier–Stokes (RANS) equations represent the computational workhorse for engineering design, despite their numerous flaws. Improving and quantifying the uncertainties associated with RANS models is particularly critical in view of the analysis and optimization of complex turbomachinery flows. Design/methodology/approach First, an efficient strategy is introduced for calibrating turbulence model coefficients from high-fidelity data. The results are highly sensitive to the flow configuration (called a calibration scenario) used to inform the coefficients. Second, the bias introduced by the choice of a specific turbulence model is reduced by constructing a mixture model by means of Bayesian model-scenario averaging (BMSA). The BMSA model makes predictions of flows not included in the calibration scenarios as a probability-weighted average of a set of competing turbulence models, each supplemented with multiple sets of closure coefficients inferred from alternative calibration scenarios. Findings Different choices for the scenario probabilities are assessed for the prediction of the NACA65 V103 cascade at off-design conditions. In all cases, BMSA improves the solution accuracy with respect to the baseline turbulence models, and the estimated uncertainty intervals encompass reasonably well the reference data. The BMSA results were found to be little sensitive to the user-defined scenario-weighting criterion, both in terms of average prediction and of estimated confidence intervals. Originality/value A delicate step in the BMSA is the selection of suitable scenario-weighting criteria, i.e. suitable prior probability mass functions (PMFs) for the calibration scenarios. The role of such PMFs is to assign higher probability to calibration scenarios more likely to provide an accurate estimate of model coefficients for the new flow. In this paper, three mixture models are constructed, based on alternative choices of the scenario probabilities. The authors then compare the capabilities of three different criteria.

The present contribution reports the outcome of an experimental and numerical investigation of the behavior of a constant-temperature hot-wire anemometer in the high subsonic flow up to a Mach number of 0.7 of the organic vapor Novec 649 at pressure and temperature levels of typical organic Rankine cycle (ORC) turbine applications. The experiments were carried out in the calibration section of a closed-loop organic vapor wind tunnel test facility enabling the independent variation of Reynolds numbers, Mach numbers, and total temperature within a certain range. It was found that the calibration and the determination of the sensitivity coefficients can be done in a way as proposed by de Souza and Tavoularis for air. The sensitivity coefficients for velocity and density were essentially equal for higher overheat ratios which support the interpretation of the signals and data reduction for turbulent quantities. Computational fluid dynamics (CFD) provided additional insight into potential real gas effects for hot-wire performance.

A hypersonic, spatially evolving turbulent boundary layer at Mach 12.48 with a cooled wall is analysed by means of direct numerical simulations. At the selected conditions, massive kinetic-to-internal energy conversion triggers thermal and chemical non-equilibrium phenomena. Air is assumed to behave as a five-species reacting mixture, and a two-temperature model is adopted to account for vibrational non-equilibrium. Wall cooling partly counteracts the effects of friction heating, and the temperature rise in the boundary layer excites vibrational energy modes while inducing mild chemical dissociation of oxygen. Vibrational non-equilibrium is mostly driven by molecular nitrogen, characterized by slower relaxation rates than the other molecules in the mixture. The results reveal that thermal non-equilibrium is sustained by turbulent mixing: sweep and ejection events efficiently redistribute the gas, contributing to the generation of a vibrationally under-excited state close to the wall, and an over-excited state in the outer region of the boundary layer. The tight coupling between turbulence and thermal effects is quantified by defining an interaction indicator. A modelling strategy for the vibrational energy turbulent flux is proposed, based on the definition of a vibrational turbulent Prandtl number. The validity of the strong Reynolds analogy under thermal non-equilibrium is also evaluated. Strong compressibility effects promote the translational–vibrational energy exchange, but no preferential correlation was detected between expansions/compressions and vibrational over-/under-excitation, as opposed to what has been observed for unconfined turbulent configurations.

Reynolds-stress models (EARSM) from high-fidelity data is developed building on the frozen-training SpaRTA algorithm of [1]. Corrections for the Reynolds stress tensor and the production of transported turbulent quantities of a baseline linear eddy viscosity model (LEVM) are expressed as functions of tensor polynomials selected from a library of candidate functions. The CFD-driven training consists in solving a blackbox optimization problem in which the fitness of candidate EARSM models is evaluated by running RANS simulations. The procedure enables training models against any target quantity of interest, computable as an output of the CFD model. Unlike the frozen-training approach, the proposed methodology is not restricted to data sets for which full fields of high-fidelity data, including second flow order statistics, are available. However, the solution of a high-dimensional expensive blackbox function optimization problem is required. Several steps are then undertaken to reduce the associated computational burden. First, a sensitivity analysis is used to identify the most influential terms and to reduce the dimensionality of the search space. Afterwards, the Constrained Optimization using Response Surface (CORS) algorithm, which approximates the black-box cost function using a response surface constructed from a limited number of CFD solves, is used to find the optimal model parameters. Model discovery and cross-validation is performed for three configurations of 2D turbulent separated flows in channels of variable section using different sets of training data to show the flexibility of the method. The discovered models are then applied to the prediction of an unseen 2D separated flow with higher Reynolds number and different geometry. The predictions of the discovered models for the new case are shown to be not only more accurate than the baseline LEVM, but also of a multi-purpose EARSM model derived from purely physical arguments. The proposed deterministic symbolic identification approach constitutes a promising candidate for building accurate and robust RANS models customized for a given class of flows at moderate computational cost. ©2022 Elsevier Inc. All rights reserved.

International audience; Large eddy simulations are carried out for flows in a channel with streamwise-periodic constrictions, a well-documented benchmark case to study turbulent flow separation from a curved surface. Resolution criteria such as wall units are restricted to attached flows and enhanced criteria, such as energy spectra or two-point correlations, are used to evaluate the effective scale separation in the present large eddy simulations. A detailed analysis of the separation above the hill crest and of the early shear layer development shows that the delicate flow details in this region may be hardly resolved on coarse grids already at Re = 10595, possibly leading to a non monotonic convergence with mesh refinement. The intricate coupling between numerical and modeling errors is studied by means of various discretization schemes and subgrid models. It is shown that numerical schemes maximizing the resolution capabilities are a key ingredient for obtaining high-quality solutions while using a reduced number of grid points. On this respect, the introduction of a sharp enough filter is an essential condition for separating accurately the resolved scales from the subfilter scales and for removing ill-resolved structures. The high-resolution approach is seen to provide solutions in very good overall agreement with the available experimental data for a range of Reynolds numbers (up to 37000) without need for significant grid refinement.

International audience; A Bayesian approach is developed to quantify uncertainties associated with the thermodynamic models used for the simulation of dense gas flows, i.e. flows of gases characterized by complex molecules of moderate to high molecular weight, in thermodynamic conditions of the general order of magnitude of the liquid/vapor critical point. The thermodynamic behaviour of dense gases can be modelled through equations of state with various mathematical structures, all involving a set of material-dependent coefficients. For several organic fluids of industrial interest abundant and high-quality thermodynamic data required to specify such coefficients are hardly available, leading to undetermined levels of uncertainty of the equation output. Additionally, the best choice for the kind of equation of state (mathematical form) to be used is not always easy to determine and it is often based on expert opinion. In other terms, equations of state introduce both parametric and model-form uncertainties, which need to be quantified to make reliable predictions of the flow field. In this paper we propose a statistical inference methodology for estimating both kinds of uncertainties simultaneously. Our approach consists of a calibration step and a prediction step. The former allows to infer on the parameters to be input to the equation of state, based on the observation of aerodynamic quantities like pressure measurements at some locations in the dense gas flow. The subsequent prediction step allows to predict unobserved flow configurations based on the inferred posterior distributions of the coefficients. Model-form uncertainties are incorporated in the prediction step by using a Bayesian model averaging (BMA) approach. This consists in constructing an average of the predictions of various competing models weighted by the posterior model probabilities. Bayesian averaging also provides a useful tool for making robust predictions from a set of alternative calibration scenarios (Bayesian model-scenario averaging or BMSA). The proposed methodology is assessed for a class of dense gas flows, namely transonic flows around an isolated airfoil, at various free-stream thermodynamic conditions in the dense-gas region.

International audience; Transonic flows of a molecularly complex organic fluid through a stator cascade were investigated by means of large eddy simulations (LESs). The selected configuration was considered as representative of the high-pressure stages of high-temperature Organic Rankine Cycle (ORC) axial turbines, which may exhibit significant non-ideal gas effects. A heavy fluorocarbon, perhydrophenanthrene (PP11), was selected as the working fluid to exacerbate deviations from the ideal flow behavior. The LESs were carried out at various operating conditions (pressure ratio and total conditions at inlet), and their influence on compressibility and viscous effects is discussed. The complex thermodynamic behavior of the fluid generates highly non-ideal shock systems at the blade trailing edge. These are shown to undergo complex interactions with the transitional viscous boundary layers and wakes, with an impact on the loss mechanisms and predicted loss coefficients compared to lower-fidelity models relying on the Reynolds-averaged Navier–Stokes (RANS) equations. View Full-Text

High-speed turbulent flows are encountered in most space-related applications (including exploration, tourism and defense fields) and represent a subject of growing interest in the last decades. A major challenge in performing high-fidelity simulations of such flows resides in the stringent requirements for the numerical schemes to be used. These must be robust enough to handle strong, unsteady discontinuities, while ensuring low amounts of intrinsic dissipation in smooth flow regions. Furthermore, the wide range of temporal and spatial active scales leads to concurrent needs for numerical stabilization and accurate representation of the smallest resolved flow scales in cases of under-resolved configurations. In this paper, we present a finite-difference high-order shock-capturing technique based on Jameson’s artificial diffusivity methodology. The resulting scheme is ninth-order-accurate far from discontinuities and relies on the addition of artificial dissipation close to large gradient flow regions. The shock detector is slightly revised to enhance its selectivity and avoid spurious activations of the shock-capturing term. A suite of test cases ranging from 1D to 3D configurations (namely, perfect-gas and chemically reacting shock tubes, Shu–Osher problem, isentropic vortex advection, under-expanded jet, compressible Taylor–Green Vortex, supersonic and hypersonic turbulent boundary layers) is analyzed in order to test the capability of the proposed numerical strategy to handle a large variety of problems, ranging from calorically-perfect air to multi-species reactive flows. Results obtained on under-resolved grids are also considered to test the applicability of the proposed strategy in the context of implicit Large-Eddy Simulations.

A robust design optimization (RDO) methodology for Organic Rankine Cycles (ORC) is presented, allowing to ensure an improved, stable performance over a large range of operating conditions. In contrast with classical ORC design methods, whereby all modeling hypotheses and operating conditions are considered as perfectly known, i.e. deterministic, the RDO approach allows to account for the manifold sources of uncertainty affecting the system. For geothermal ORC, the latter are related on one hand with the ill-known properties of the geothermal source and, on the other, with intrinsically random parameters, such as the condensation temperature. The proposed RDO approach selects values of the design parameters that maximize the expected (average) performance while minimizing its variance under uncertain nominal operating conditions. The optimal design delivered by the proposed strategy outperforms the one derived from the standard deterministic approach: specifically, the expected power output is increased by 1.5%, while its standard deviation is reduced by 8.5% and the surface of the heat exchangers by 34%.

A study of dense-gas effects on the laminar, transitional and turbulent characteristics of boundary layer flows is conducted. The laminar similarity solution shows that temperature variations are small due to the high specific heats of dense gases, leading to velocity profiles close to the incompressible ones. Nevertheless, the complex thermodynamics of the base flow has a major impact on unstable modes, which bear similarities with those obtained for a strongly cooled wall. Numerical simulations of spatially developing boundary layers yield turbulent statistics for the dense gas flow that remain closer to the incompressible regime than perfect gas ones despite the presence of strongly compressible structures.

High-speed turbulent boundary layers of a dense gas (PP11) and a perfect gas (air) over flat plates are investigated by means of direct numerical simulations and large eddy simulations. The thermodynamic conditions of the incoming flow are chosen to highlight dense gas effects, and laminar-to-turbulent transition is triggered by suction and blowing. In the paper, the behavior of the fully developed turbulent flow region is investigated. Due to the low characteristic Eckert number of dense gas flows ( $$\hbox {Ec}=U_\infty ^2/c_{p,\infty }T_\infty$$ ), the mean velocity profiles are largely insensitive to the Mach number and very close to the incompressible case even at high speeds. Second-order velocity statistics are also weakly affected by the flow Mach number and the velocity spectra are characterized by a secondary peak in the outer region of the boundary layer because of the higher local friction Reynolds number. Despite the incompressible-like velocity and Reynolds-stress profiles, the strongly non-ideal thermodynamic and transport-property behavior of the dense gas results in unconventional distributions of the fluctuating thermo-physical quantities. Specifically, density and viscosity fluctuations reach a peak close to the wall, instead of vanishing as in perfect gas flows. Additionally, dense gas boundary layers exhibit higher values of the fluctuating Mach number and velocity divergence and a larger dilatational-to-solenoidal dissipation ratio in the near-wall region, which represents a major deviation from high-Mach-number perfect gas boundary layers. Other significant deviations are represented by the more symmetric probability distributions of fluctuating quantities such as the density and velocity divergence, due to the more balanced occurrence of strong expansion and compression events.

High-speed flows of gases with complex thermodynamic behavior (often referred to as “real gases”) have been paid growing interest from the scientific community due to the manifold applications in aerospace and power generation systems. In this work, we focus on so-called dense-gas flows found in several engineering applications, ranging from energy production to high-speed wind tunnels. Dense gases are single-phase fluids of complex molecules, at pressure and temperature conditions of the same order of their thermodynamic critical point. These fluids, in particular those belonging to the Bethe–Zel’dovich–Thompson family, may exhibit nonclassical phenomena in the transonic and supersonic regimes such as expansion shocks or mixed waves; these nonclassical effects may be characterized by means of the fundamental derivative of gas dynamics \(\varGamma \) [8].

International audience; A recently developed fourth-order accurate implicit residual smoothing scheme (IRS4) is investigated for the large eddy simulation of turbomachinery flows, characterized by moderate to high Reynolds numbers and subject to severe constraints on the maximum allowable time step if an explicit scheme is used. For structured multi-block meshes, the proposed approach leads to the inversion of a scalar pentadiagonal system by mesh direction, which can be done very efficiently. On the other hand, applying IRS4 at each stage of an explicit Runge–Kutta time scheme allows to increase the time step by a factor 5 to 10, leading to substantial savings in terms of overall computational time. With respect to standard second-order fully implicit approaches, the IRS4 does not require approximate linearization and factorization procedures nor inner Newton-Raphson subiterations. As a consequence, it represents a better cost-accuracy compromise for the numerical simulations of turbulent flows where the maximum time step is controlled by the lifetime of the smallest resolved turbulent structures. Numerical results for the well-documented high-pressure VKI LS-89 planar turbine cascade illustrate the potential of IRS4 for significantly reducing the overall cost of turbomachinery large eddy simulations, while preserving an accuracy similar to the explicit solver even for sensitive quantities like the heat transfer coefficient and the turbulent kinetic energy field.

Dense gas effects, encountered in many engineering applications, lead to unconventional variations of the thermodynamic and transport properties in the supersonic flow regime, which in turn are responsible for considerable modifications of turbulent flow behavior with respect to perfect gases. The most striking differences for wall-bounded turbulence are the decoupling of dynamic and thermal effects for gases with high specific heats, the liquid-like behavior of the viscosity and thermal conductivity, which tend to decrease away from the wall, and the increase of density fluctuations in the near wall region. The present work represents a first attempt of quantifying the influence of such dense gas effects on modeling assumptions employed for the closure of the Reynolds-averaged Navier–Stokes equations, with focus on the eddy viscosity and turbulent Prandtl number models. For that purpose, we use recent direct numerical simulation results for supersonic turbulent channel flows of PP11 (a heavy fluorocarbon representative of dense gases) at various bulk Mach and Reynolds numbers to carry out a priori tests of the validity of some currently-used models for the turbulent stresses and heat flux. More specifically, we examine the behavior of the modeled eddy viscosity for some low-Reynolds variants of the $k-\varepsilon $ model and compare the results with those found for a perfect gas at similar conditions. We also investigate the behavior of the turbulent Prandtl number in dense gas flow and compare the results with the predictions of two well-established turbulent Prandtl number models.

International audience; High-order extensions of the Vorticity Confinement (VC) method are developed for the accurate computation of vortical flows, following the VC2 conservative formulation of Steinhoff. First, a high-order formulation of VC is presented for the case of the linear transport equation for decoupled schemes in space and time. A spectral analysis shows that the new nonlinear schemes have improved dispersive and dissipative properties compared to their linear counterparts at all orders of accuracy. For the Euler and Navier–Stokes equations, the original VC method is extended to 3 rd - and 5 th -order of accuracy, with the goal of developing a VC formulation that maintains the vorticity preserving capability of the original 1 st -order method and is suitable for application to high-order numerical simulations. The high-order extensions remain both independent of the choice of baseline numerical scheme and rotationally invariant since they are based on the Laplace operator. Numerical tests validate the increased order of accuracy, vorticity-preserving capability and compatibility of the VC extensions with high-order methods.Numerical tests validate the increased order of accuracy, vorticity-preserving capability and compatibility of the VC extensions with high-order methods.; Des extensions d'ordre élevé de la méthode de confinement de vorticité (Vorticity Confinement, VC) ont été développées pour le calcul précis des écoulements tourbillonnaires, en suivant la formulation conservative VC2 de Steinhoff. Tout d'abord, une formulation de la méthode VC d'ordre élevé est présentée pour le cas de l'équation de transport linéaire basée sur des schémas découplés en espace et en temps. Une analyse spectrale démontre que les nouveaux schémas non linéaires ont des propriétés dispersives et dissipatives améliorées par rapport à leurs équivalents linéaires. Pour les équations d'Euler et de Navier-Stokes, la méthode VC originale est étendue au 3ème et 5ème ordre de précision, dans le but de développer une formulation de la méthode VC qui maintient la capacité de préservation de la vorticité de la méthode initiale et est applicable dans des simulations numériques d'ordre élevé. Les extensions d'ordre élevé restent indépendantes du choix du schéma numérique de base et sont invariantes par rotation car elles sont basées sur l'opérateur de Laplace. Les tests numériques effectués valident l'augmentation de l'ordre de précision, la capacité de conservation de la vorticité et la compatibilité des extensions VC avec des méthodes d'ordre élevé.

This article presents computational fluid dynamics (CFD) simulations of a supersonic flow inside the first stage of an Organic Rankine Cycle (ORC) turbine. The expander considered in this work is the first stage of a three-stage axial turbine that uses hexamethyldisiloxane (MM) as the working fluid. The thermodynamic properties of the working fluid are modelled by a Helmoltz free energy-based equation of state during both the design and simulation steps to accurately account for the non-ideal nature of the fluid under the considered conditions. The high expansion ratio of the turbine leads to a supersonic flow in the stator and rotor blade passages. The design of the stator blade shape is carried out by means of the generalized method of characteristics (MOC). The CFD code used in this work is the commercial ANSYS CFX solver and the simulations are based on the two-dimensional Reynolds-Averaged Navier-Stokes (RANS) equations supplemented by a k-ω turbulence model. Results provided by steady mixing plane simulations are compared to those of unsteady sliding mesh simulations in order to understand the implications of stator/rotor flow interaction on blade loading, torque, entropy generation and flow structure.

A robust parametric optimization of Organic Rankine Cycles (ORC) is carried out by taking into account uncertainties in the cycle input parameters. A typical engine driving duty cycle is considered and sampled in order to construct a suitable probability distribution describing the variability of the exhaust gases mass-flow and temperature. Besides, the environmental and condensing temperatures, specific heat ratio of the waste gases, turbine and pump efficiencies are also considered as uncertain. The parameter combination that maximizes the mean cycle efficiency and minimizes its variance is sought as optimal. Six organic fluids have been taken into account, namely R245fa, R245ca, R134a, R11, R113 and Novec649, with the best results provided by R11 and R113.

After the extraordinary diffusion that we have observed over the last ten years, Organic Rankine Cycles (ORCs) are nowadays widely recognized as “the unrivalled technical solution for generating electricity from low-medium temperature heat sources of limited capacity” [1]. Despite the high level of confidence and know-how reached about ORCs, they still remain a delicate technology, hiding a great amount of technical difficulties which sometimes still make them a risky investment. Most of these complexities are originated from manifold sources of uncertainty which impact on almost the whole life of the ORC project, from their design to the commissioning and operation steps, with heavy consequences in terms of performance and costs. In this work we present the proof of concept assessing and validating an innovative technique for the robust design optimization (RDO) of ORC under uncertainty. The approach allows to deal with both aleatory and epistemic uncertainty in order to avoid an over-optimization of the system that can result in a high sensitivity to small changes. Because of the large number of sources of uncertainty, the design problem must be solved in a highly multi-dimensional space, spanned by the uncertain and design variables. In such a situation, the “brute-force” Monte-carlo approach [2] is not a viable technique, since it is limited to cheap and excessively simplified models. Consequently, in the present work we consider a more efficient design methodology relying on two nested Bayesian Kriging surrogates.

A high-order wall treatment is investigated for a family of high-order Godunov-type finite volume schemes based on k-exact polynomial reconstructions of the primitive variables in each cell via a successive-corrections procedure of the derivatives. We focus more specifically on the 1-exact and 2-exact schemes, which offer a good trade-off between accuracy and computational efficiency. The modifications are threefold. First, wall curvature is taken into account by introducing a new model of the wall surfaces based on bicubic Bezier patches. A low-Mach correction is then introduced in the Riemann solver to retrieve accuracy in near-wall regions characterized by low velocity. Finally, the reconstruction stencil is enriched at the boundaries. The benefits of these numerical treatments are studied for two inviscid test cases, namely the flow past a Gaussian bump and the subsonic flow around a circular cylinder.

In this work we investigate the influence of dense gas effects on compressible wall-bounded turbulence. Turbulent flows of dense gases represent a research field of great importance for a wide range of applications in engineering. Dense gases are single-phase fluids with a molecular complexity such that the fundamental derivative of gas dynamics [1], which measures the rate of change of the sound speed in isentropic transformations, is less than one in a range of thermodynamic conditions close to the saturation curve. In such conditions, the speed of sound increases in isentropic expansions and decreases in isentropic compressions, unlike the case of perfect gases. For dense gases, the perfect gas model is no longer valid, and more complex equations of state must be used to account for their peculiar thermodynamic behavior. Moreover, in the dense gas regime, the dynamic viscosity μ and the thermal conductivity λ depend on temperature and pressure through complex relationships. Similarly, the approximation of nearly constant Prandtl number Pr= μ c p / λ is no longer valid. Numerical simulations of turbulent dense gas flows of engineering interest are based on the (Reynolds-Averaged Navier–Stokes) RANS equations, which need to be supplemented by a model for the Reynolds stress tensor and turbulent heat flux. The accuracy of RANS models for dense-gas flows has not been properly assessed up to date, due to the lack of both experimental and numerical reference data. DNS databases [2, 3] are then needed to quantify the deficiencies of existing turbulence models and to develop and calibrate improved ones. In this work we first summarize some recent direct numerical simulation (DNS) results [4] for supersonic turbulent channel flows (TCF) of PP11, a heavy fluorocarbon representative of dense gases, at various bulk Mach and Reynolds numbers. The most relevant effects are represented by non-conventional variations of the fluctuating thermodynamic quantities, compared to perfect gases and a strong decoupling between thermal and dynamic effects almost everywhere in the flow, except in the immediate vicinity of the solid wall. Preliminary considerations about the validity of some currently-used models for the turbulent stresses and heat flux are carried out based on a priori comparisons between the exact terms computed from the DNS and their modeled counterparts.

International audience; A new 3rd-order Vorticity Confinement scheme is presented as an extension of the original VC2 scheme developed by Steinhofffor the resolution of the fluid dynamic equations. The theoretical developments are explained, and the method is tested. The results obtained show that the new scheme combines the accuracy of the underlying higher order scheme and the confinement capability of the original VC2 method.

Efficient Robust Design Optimization (RDO) strategies coupling a parsimonious uncertainty quantification (UQ) method with a surrogate-based multi-objective genetic algorithm (SMOGA) are investigated for a test problem in computational fluid dynamics (CFD), namely the inverse robust design of an expansion nozzle. The low-order statistics (mean and variance) of the stochastic cost function are computed through either a gradient-enhanced kriging (GEK) surrogate or through the less expensive, lower fidelity, first-order method of moments (MoM). Both the continuous (non-intrusive) and discrete (intrusive) adjoint methods are evaluated for computing the gradients required for GEK and MoM. In all cases, the results are assessed against a reference kriging UQ surrogate not using gradient information. Subsequently, the GEK and MoM UQ solvers are fused together to build a multi-fidelity surrogate with adaptive infill enrichment for the SMOGA optimizer. The resulting hybrid multi-fidelity SMOGA RDO strategy ensures a good tradeoff between cost and accuracy, thus representing an efficient approach for complex RDO problems.

The Reynolds-Averaged Navier-Stokes (RANS) equations represent the computational workhorse for engineering design, despite their numerous flaws. Improving and quantifying the uncertainties associated with RANS models is particularly critical in view of the analysis and optimization of complex turbomachinery flows. In this work, we use Bayesian inference for assimilating data into RANS models for the following purposes: (i) updating the model closure coefficients for a class of turbomachinery flows, namely a compressor cascade; (ii) quantifying the parametric uncertainty associated with closure coefficients of RANS models and (iii) quantifying the uncertainty associated with the model structure and the choice of the calibration dataset based on an ensemble of concurrent models and calibration scenarios. Inference of the coefficients of three widely employed RANS models is carried out from high-fidelity LES data for the NACA65 V103 compressor cascade [1, 2]. Posterior probability distributions of the model coefficients are collected for various calibration scenarios, corresponding to different values of the flow angle at inlet. The Maximum A Posteriori estimates of the coefficients differ from the nominal values and depend on the scenario. A recently proposed Bayesian mixture approach, namely, Bayesian Model-Scenario Averaging (BMSA) [3, 4], is used to build a prediction model that takes into account uncertainties associated with alternative model forms and with sensitivity to the calibration scenario. Stochastic predictions are presented for the turbulent flow around the NACA65 V103 cascade at mildly and severe off-design conditions. The results show that BMSA generally yields more accurate solutions than the baseline RANS models and succeeds well in providing an estimate for the predictive uncertainty intervals, provided that a sufficient diversity of scenarios and models is included in the mixture.

A fast preliminary design methodology for supersonic organic Rankine cycle (ORC) stator and rotor axial turbine blades with low degree of reaction is presented. First, the stator and rotor blade mean-line profiles are designed by using the two-dimensional (2D) method of characteristics (MOC), extended to gases governed by general equations of state (EOS). We focus more specifically on working fluids with medium to high molecular complexity, operating at thermodynamic conditions such that the fundamental derivative of gas dynamics Γ is lower than one in a significant portion of the flow field. For rotor blades, MOC is combined with a free-vortex method to achieve a smooth deflection of the supersonic incoming flow. A numerical approach is developed for solving the unique incidence problem in the case of gases governed by general EOS. Both stator and rotor blade geometries designed according to the inviscid MOC model are subsequently corrected to account for the development of viscous boundary layers by solving the compressible integral boundary layer equations extended to dense gases. The resulting blade designs are assessed by means of computational fluid dynamics (CFD) simulations based on a high-order finite volume solver equipped with advanced thermodynamic and transport-property models. Properly accounting for dense gas and viscous effects at an early design stage is found to improve the expected performance of ORC turbine rows significantly and delivers valuable baseline profiles for any further optimization.

International audience; The lack of an universal modelling approach for turbulence in Reynolds-Averaged Navier–Stokes simulations creates the need for quantifying the modelling error without additional validation data. Bayesian Model-Scenario Averaging (BMSA), which exploits the variability on model closure coefficients across several flow scenarios and multiple models, gives a stochastic, a posteriori estimate of a quantity of interest. The full BMSA requires the propagation of the posterior probability distribution of the closure coefficients through a CFD code, which makes the approach infeasible for industrial relevant flow cases. By using maximum a posteriori (MAP) estimates on the posterior distribution, we drastically reduce the computational costs. The approach is applied to turbulent flow in a pipe at Re= 44,000 over 2D periodic hills at Re=5600, and finally over a generic falcon jet test case (Industrial challenge IC-03 of the UMRIDA project).

In computational fluid dynamics simulations of industrial flows, models based on the Reynolds-averaged Navier--Stokes (RANS) equations are expected to play an important role in decades to come. However, model uncertainties are still a major obstacle for the predictive capability of RANS simulations. This review examines both the parametric and structural uncertainties in turbulence models. We review recent literature on data-free (uncertainty propagation) and data-driven (statistical inference) approaches for quantifying and reducing model uncertainties in RANS simulations. Moreover, the fundamentals of uncertainty propagation and Bayesian inference are introduced in the context of RANS model uncertainty quantification. Finally, the literature on uncertainties in scale-resolving simulations is briefly reviewed with particular emphasis on large eddy simulations., submitted to Progress in Aerospace Sciences as an invited review article

International audience; This work presents developments towards a deterministic symbolic regression method toderive algebraic Reynolds-stress models for the Reynolds-Averaged Navier-Stokes (RANS)equations. The models are written as tensor polynomials, for which optimal coe cients arefound using Bayesian inversion. These coe cient fields are the targets for the symbolic regression.A method is presented based on a regularisation strategy in order to promote sparsityof the inferred models and is applied to high-fidelity data. By being data-driven the methodreduces the assumptions commonly made in the process of model development in order toincrease the predictive fidelity of algebraic models.

International audience; Improved wall boundary treatments are investigated for a family of high-order Godunovtypefinite volume schemes based on k-exact polynomial reconstructions in each cell of theprimitive variables, via a successive corrections procedure. We focus more particularly on the1-exact and 2-exact schemes which offer a good trade-off between accuracy and computationalefficiency. In both cases, the reconstruction stencil needs to be extended to the boundaries.Additionally, information about wall curvature has to be taken into account, which is doneby using a surface model based on bicubic Bézier patches for the walls. The performance ofthe proposed models is presented for two compressible cases, namely the inviscid flow past aGaussian bump and the viscous axisymmetric Couette flow.

The industrial aerodynamic design of helicopter rotor blades needs to consider the two typical flight conditions of hover and forward flight simultaneously. Here, this multi-objective design problem is tackled by using a genetic algorithm, coupled to rotor performance simulation tools. The turn-around time of an optimization loop is acceptable in an industrial design loop when using low-cost, low-fidelity tools such as the comprehensive rotorcraft code HOST, but becomes excessively high when employing high-fidelity models like CFD methods. To incorporate high-fidelity models into the optimization loop while maintaining a moderate computational cost, a Multi-Fidelity Optimization (MFO) strategy is proposed: as a preliminary step, a HOST-based genetic algorithm optimization is used to reduce the parameter space and select a set of blade geometries used for initializing the high-fidelity stage. Secondly, the selected blades are re-evaluated by CFD and used to construct a high-fidelity surrogate model. Finally, a Surrogate Based Optimization (SBO) is carried out and the Pareto optimal individuals according to the SBO are recomputed by CFD for final performance evaluation. The proposed strategy is validated step by step. It is shown that an industrially acceptable number of CFD-simulations is sufficient to obtain blade designs with a significantly higher performance than the baseline and then SBO results issued from a standard Latin-Hypercube-Sampling initialization. The proposed MFO strategy represents an efficient method for the simultaneous optimization of rotor blade geometries in hover and forward flight. Contrat industriel Eurocopter

The convergence of Fourier-based time methods applied to turbomachinery flows is assessed. The focus is on the harmonic balance method, which is a timedomain Fourier-based approach standing as an efficient alternative to classical time marching schemes for periodic flows. In the literature, no consensus exists concerning the number of harmonics needed to achieve convergence for turbomachinery stage configurations. In this paper it is shown that the convergence of Fourier-based methods is closely related to the impulsive nature of the flow solution, which in turbomachines is essentially governed by the characteristics of the passing wakes between adjacent rows. As a result of the proposed analysis, a priori estimates are provided for the minimum number of harmonics required to accurately compute a given turbomachinery configuration. Their application to several contra-rotating open-rotor configurations is assessed, demonstrating the practical interest of the proposed methodology. Contrat industriel CERFACS

International audience; In large-eddy simulations (LES), only the dynamics of large scales is computedand the effect of smaller scales is modelled. Scale separation is however difficult toestablish since the low-pass filtering arises from a complex combination of implicitfiltering by the grid and the discretization schemes. Evenwhen explicit filtering is performed,the implicit filtering due to the application of discretizationmethods changesthe shape of the filter function.The principal objective of the current study is to identify the sensitivity of the predictive accuracy to resolutionand modelling issues. Simulations were performed for six different finite-differenceschemes, six explicit filters and five subgrid-scalemodels on a coarse grid. It is shownin particular that the dissipative part of the discretization scheme plays a determinantrole and overwhelms the choice of a SGS closure.

International audience; In this work, Bayesian techniques are employed to quantify model-form and predictive uncertainty in the linear behavior of an elastically mounted airfoil undergoing pitching and plunging motions. The Bayesian model averaging approach is used to construct an adjusted stochastic model from different model classes for time-harmonic incompressible flows. From a set of deterministic function approximations, we construct different stochastic models, whose uncertain coefficients are calibrated using Bayesian inference with regard to the critical flutter velocity. Results show substantial reductions in the predictive uncertainties of the critical flutter speed compared to non-calibrated stochastic simulations. In particular, it is shown that an efficient adjusted model can be derived by considering a possible bias in the random error term on the posterior predictive distributions of the flutter index.

International audience; A high-order Vorticity Confinement (VC) method is applied to the calculation of compressible vortical flows. High-order extensions of the VC methodology have been developed for the Navier-Stokes equations using a methodology that remains independent of the baseline numerical scheme, as the original VC formulation of Steinhoff. The VC method is primarily oriented towards the accurate calculation of cases involving the advection of vortical structures (e.g. wakes), allowing their propagation over long distances with low numerical dissipation. The present work however mainly investigates their applicability to the simulation of turbulent flows, where the improvement in the preservation of vortical structures shows great interest. High-order VC schemes were found to be consistent with complex vortical flow dynamics without the need of special treatment to accommodate the vortex interaction mechanisms characteristic of turbulent flows. Furthermore, they were found to consistently improve the resolvability of baseline FE-MUSCL schemes for reasonable values of the confinement parameter ε, close to the value of the high-order artificial dissipation coefficient. Last, the dependence of the solution on the confinement parameter µ was found to be moderate for the values and cases investigated in this paper.; Une méthode de confinement de vorticité (VC) d'ordre élevé est appliquée dans des calculs des écoulements tourbillonnaires compressibles. Les extensions de la méthode VC pour les ordres élevés ont été développées par une méthodologie qui reste indépendante du schéma numérique de base, comme la formulation VC originale de Steinhoff. Le VC est principalement orienté vers le calcul précis des cas impliquant l'advection de structures tourbillonnaires (p. ex. sillages), permettant leur propagation sur de longues distances avec une faible dissipation numérique. Cependant, le présent travail étudie principalement leur applicabilité dans la simulation des écoulements turbulents, où une meilleure préservation des structures tourbillonnaires présente un grand intérêt. Les schémas VC d'ordre élevé se sont révélés être consistants avec la dynamique des écoulements tourbillonnaires complexes sans besoin de traitement spécial pour accommoder les mécanismes d'interactions tourbillonnaires caractéristiques des écoulements turbulents. De plus, ils permettent d'améliorer la résolvabilité des schémas numériques FE-MUSCL de base pour des valeurs raisonnables du paramètre de confinement epsilon, proche de la valeur du coefficient de dissipation artificielle d'ordre élevé. Enfin, la dépendance de la solution à la valeur du paramètre de confinement mu reste modérée pour les valeurs et cas-tests étudiés dans cet article.

The influence of dense-gas effects on compressible wall-bounded turbulence is investigated by means of direct numerical simulations of supersonic turbulent channel flows. Results are obtained for PP11, a heavy fluorocarbon representative of dense gases, the thermophysics properties of which are described by using a fifth-order virial equation of state and advanced models for the transport properties. In the dense-gas regime, the speed of sound varies non-monotonically in small perturbations and the dependency of the transport properties on the fluid density (in addition to the temperature) is no longer negligible. A parametric study is carried out by varying the bulk Mach and Reynolds numbers, and results are compared to those obtained for a perfect gas, namely air. Dense-gas flow exhibits almost negligible friction heating effects, since the high specific heat of the fluids leads to a loose coupling between thermal and kinetic fields, even at high Mach numbers. Despite negligible temperature variations across the channel, the mean viscosity tends to decrease from the channel walls to the centreline (liquid-like behaviour), due to its complex dependency on fluid density. On the other hand, strong density fluctuations are present, but due to the non-standard sound speed variation (opposite to the mean density evolution across the channel), the amplitude is maximal close to the channel wall, i.e. in the viscous sublayer instead of the buffer layer like in perfect gases. As a consequence, these fluctuations do not alter the turbulence structure significantly, and Morkovin’s hypothesis is well respected at any Mach number considered in the study. The preceding features make high Mach wall-bounded flows of dense gases similar to incompressible flows with variable properties, despite the significant fluctuations of density and speed of sound. Indeed, the semi-local scaling of Patel et al. (Phys. Fluids, vol. 27 (9), 2015, 095101) or Trettel & Larsson (Phys. Fluids, vol. 28 (2), 2016, 026102) is shown to be well adapted to compare results from existing surveys and with the well-documented incompressible limit. Additionally, for a dense gas the isothermal channel flow is also almost adiabatic, and the Van Driest transformation also performs reasonably well. The present observations open the way to the development of suitable models for dense-gas turbulent flows.

The influence of dense gas effects on compressible turbulence is investigated by means of numerical simulations of the decay of compressible homogeneous isotropic turbulence (CHIT) and of supersonic turbulent flows through a plane channel (TCF). For both configurations, a parametric study on the Mach and Reynolds numbers is carried out. The dense gas considered in these parametric studies is PP11, a heavy fluorocarbon. The results are systematically compared to those obtained for a diatomic perfect gas (air). In our computations, the thermodynamic behaviour of the dense gases is modelled by means of the Martin-Hou equation of state. For CHIT cases, initial turbulent Mach numbers up to 1 are analyzed using mesh resolutions up to 5123. For TCF, bulk Mach numbers up to 3 and bulk Reynolds numbers up to 12000 are investigated. Average profiles of the thermodynamic quantities exhibit significant differences with respect to perfect-gas solutions for both of the configurations. For high-Mach CHIT, compressible structures are modified with respect to air, with weaker eddy shocklets and stronger expansions. In TCF, the velocity profiles of dense gas flows are much less sensitive to the Mach number and collapse reasonably well in the logarithmic region without any special need for compressible scalings, unlike the case of air, and the overall flow behaviour is midway between that of a variable-property liquid and that of a gas.

International audience; For the purpose of estimating the epistemic model-form uncertainty in Reynolds-Averaged Navier-Stokes closures, we propose two transport equations to locally perturb the Reynolds stress tensor of a given baseline eddy-viscosity model. The spatial structure of the perturbations is determined by the proposed transport equations, and thus does not have to be inferred from full-field reference data. Depending on a small number of model parameters and the local flow conditions, a ’return to eddy viscosity’ is described, and the underlying baseline state can be recovered. In order to make predictions with quantified uncertainty, we identify two separate methods, i.e. a data-free and data-driven approach. In the former no reference data is required and computationally inexpensive intervals are computed. When reference data is available, Bayesian inference can be applied to obtained informed distributions of the model parameters and simulation output.

International audience; Turbulent flows of industrial interest are often dominated by large turbulent structures. A typical example is provided by launcher base flows, combining one or more extended and interacting separated regions with strong compressibility effects. Such flow features represent a challenge for CFD simulations, since, on the one hand, well established Reynolds-Averaged-Navier-Stokes (RANS) solvers cannot represent such highly unsteady and three-dimensional large scales and, on the other hand, Large Eddy Simulation (LES) approaches remain too expensive for routine production use in industry. In fact, LES methods can lead to inaccurate results if the mesh size and/or the scheme accuracy are not good enough to resolve correctly most of the relevant flow scales. In complex industrial applications, the energy spectrum is often ill defined and changes from one point to another of the simulation, so that it is difficult to warrant a sufficient resolution everywhere, unless extremely fine meshes are used. Mesh resolution requirements become particularly severe if industrial codes based on low order, low resolution discretization schemes are used. In thiswork we assess a recently developed hybrid RANS/LES strategy, combining a selfadaptive hybrid turbulence model [6] and a hybrid high-order finite volume scheme [7], for flow around industrial geometries, namely, the Ariane 6 and Ariane 5 space launchers. Increasingly complex geometrical details are included in the simulation and the results are compared with the experimental data available for the average and root mean square (rms) of the longitudinal distribution of the pressure coefficient, which represent a significant quantity of interest for launcher design.

A Godunov's type unstructured finite volume method suitable for highly compressible turbulent scale-resolving simulations around complex geometries is constructed by using a successive correction technique. First, a family of k-exact Godunov schemes is developed by recursively correcting the truncation error of the piecewise polynomial representation of the primitive variables. The keystone of the proposed approach is a quasi-Green gradient operator which ensures consistency on general meshes. In addition, a high-order single-point quadrature formula, based on high-order approximations of the successive derivatives of the solution, is developed for flux integration along cell faces. The proposed family of schemes is compact in the algorithmic sense, since it only involves communications between direct neighbors of the mesh cells. The numerical properties of the schemes up to fifth-order are investigated, with focus on their resolvability in terms of number of mesh points required to resolve a given wavelength accurately. Afterwards, in the aim of achieving the best possible trade-off between accuracy, computational cost and robustness in view of industrial flow computations, we focus more specifically on the third-order accurate scheme of the family, and modify locally its numerical flux in order to reduce the amount of numerical dissipation in vortex-dominated regions. This is achieved by switching from the upwind scheme, mostly applied in highly compressible regions, to a fourth-order centered one in vortex-dominated regions. An analytical switch function based on the local grid Reynolds number is adopted in order to warrant numerical stability of the recentering process. Numerical applications demonstrate the accuracy and robustness of the proposed methodology for compressible scale-resolving computations. In particular, supersonic RANS/LES computations of the flow over a cavity are presented to show the capability of the scheme to predict flows with shocks, vortical structures and complex geometries.

The wave propagation (spectral) properties of high-order Residual-Based compact (RBC) discretizations are analyzed to obtain information on the evolution of the Fourier modes supported on a grid of finite size. For these genuinely multidimensional and intrinsically dissipative schemes, a suitable procedure is used to identify the modified wave number associated to their spatial discretization operator, and their dispersive and dissipative behaviors are investigated as functions of a multidimensional wave number. For RBC schemes of higher orders (5 and 7), both dissipation and dispersion errors take very low values up to reduced wave numbers close to the grid resolvability limit, while higher frequencies are efficiently damped out. Thanks to their genuinely multidimensional formulation, RBC schemes conserve good dissipation and dispersion properties even for flow modes that are not aligned with the computational grid. Numerical tests support the theoretical results. Specifically, the study of a complex nonlinear problem dominated by energy transfer from large to small flow scales, the inviscid Taylor-Green vortex flow, confirms numerically the interest of a well-designed RBC dissipation to resolve accurately fine scale flow structures.

International audience; The Simplex-Stochastic Collocation (SSC) method is a robust tool used to propagate uncertain input distributions through a computer code. However, it becomes prohibitively expensive for problems with dimensions higher than 5. The main purpose of this paper is to identify bottlenecks, and to improve upon this bad scalability. In order to do so, we propose an alternative interpolation stencil technique based upon the Set-Covering problem, and we integrate the SSC method in the High-Dimensional Model-Reduction framework. In addition, we address the issue of ill-conditioned sample matrices, and we present an analytical map to facilitate uniformly-distributed simplex sampling.

International audience; Restrictions on the maximum allowable time step of explicit time integration methods for direct and large eddy simulations of compressible turbulent flows at high Reynolds numbers can be very severe, because of the extremely small space steps used close to solid walls to capture tiny and elongated boundary layer structures. A way of increasing stability limits is to use implicit time integration schemes. However, the price to pay is a higher computational cost per time step, higher discretization errors and lower parallel scalability. In quest for an implicit time scheme for scale-resolving simulations providing the best possible compromise between these opposite requirements, we develop a Runge–Kutta implicit residual smoothing (IRS) scheme of fourth-order accuracy, based on a bilaplacian operator. The implicit operator involves the inversion of scalar pentadiagonal systems, for which efficient parallel algorithms are available. The proposed method is assessed against two explicit and two implicit time integration techniques in terms of computational cost required to achieve a threshold level of accuracy. Precisely, the proposed time scheme is compared to four-stages and six-stages low-storage Runge–Kutta method, to the second-order IRS and to a second-order backward scheme solved by means of matrix-free quasi-exact Newton subiterations. Numerical results show that the proposed IRS scheme leads to reductions in computational time by a factor 3 to 5 for an accuracy comparable to that of the corresponding explicit Runge–Kutta scheme.

Residual-Based Compact (RBC) schemes are reviewed and applied to realistic compressible flow problems. The principle and advantages of high-order RBC schemes are first presented. Then, an implementation in the elsA code of the 3rd-order RBC scheme is used for RANS calculations of the transonic flow over the ONERA M-6 wing and of the flow in a CME2 compressor stage. Finally, the 7th-order RBC scheme is applied to the complete Euler equations for computing a planar aeroacoustic problem and the propagation of a spinning acoustic mode in an axisymetric aeroengine inlet. Numerical results demonstrate the accuracy and efficiency of the RBC approach.