publications

2025

1. 

   First-order buoyancy correction of modal instabilities in stratified boundary layers

   P. C. Boldini,  R. Hirai,  B. Bugeat, and 1 more author

   arXiv preprint arXiv:2511.21130, 2025

   Abs HTML

   We present a perturbation-based framework that captures buoyancy effects on modal instabilities in stratified boundary-layer flows within the fully compressible, non-Oberbeck–Boussinesq formulation. Treating the Richardson number as a small parameter and recasting the stability problem into an adjoint-residual form, we derive a first-order correction for the eigenvalues using only the neutrally buoyant eigenvalue problem. This eliminates the need to re-solve the eigenvalue problem at each stratification level. For ideal-gas boundary layers, the framework accurately predicts how stable and unstable stratification modifies Tollmien–Schlichting waves, from growth rates and eigenfunctions to N-factors, holding across a wide range of Prandtl numbers, temperature ratios, and Mach numbers. Notably, the buoyancy sensitivity varies strongly with Prandtl number, revealing that for a given Richardson number, buoyancy can switch from destabilising to stabilising depending on the fluid. Beyond ideal-gas conditions, we apply the first-order buoyancy correction to strongly stratified boundary layers with supercritical fluids, where the phase relationship between density and velocity perturbations determines whether buoyancy stabilises or destabilises the underlying instability. The resulting N-factors demonstrate, for the first time, that buoyancy significantly affects transition predictions under pseudo-boiling conditions.

2. 

   Direct numerical simulation of complete transition to turbulence with a fluid at supercritical pressure

   P. C. Boldini,  B. Bugeat,  J. W. R. Peeters, and 2 more authors

   Journal of Fluid Mechanics, 2025

   Abs DOI

   The objective of this work is to investigate the unexplored laminar-to-turbulent transition of a heated flat-plate boundary layer with a fluid at supercritical pressure. Two temperature ranges are considered: a subcritical case, where the fluid remains entirely in the liquid-like regime, and a transcritical case, where the pseudo-critical (Widom) line is crossed and pseudo-boiling occurs. Fully compressible direct numerical simulations are used to study (i) the linear and nonlinear instabilities, (ii) the breakdown to turbulence, and (iii) the fully developed turbulent boundary layer. In the transcritical regime, two-dimensional forcing generates not only a train of billow-like structures around the Widom line, resembling Kelvin–Helmholtz instability, but also near-wall travelling regions of flow reversal. These spanwise-oriented billows dominate the early nonlinear stage. When high-amplitude subharmonic three-dimensional forcing is applied, staggered Λ-vortices emerge more abruptly than in the subcritical case. However, unlike the classic H-type breakdown under zero pressure gradient observed in ideal-gas and subcritical regimes, the H-type breakdown is triggered by strong shear layers caused by flow reversals – similar to that observed in adverse pressure gradient boundary layers. Without oblique wave forcing, transition is only slightly delayed and follows a naturally selected fundamental breakdown (K-type) scenario. Hence in the transcritical regime, it is possible to trigger nonlinearities and achieve transition to turbulence relatively early using only a single two-dimensional wave that strongly amplifies background noise. In the fully turbulent region, we demonstrate that variable-property scaling accurately predicts turbulent skin-friction and heat-transfer coefficients.

3. 

   CUBENS: A GPU-accelerated high-order solver for wall-bounded flows with non-ideal fluids

   P. C. Boldini,  R. Hirai,  P. Costa, and 2 more authors

   Computer Physics Communications, 2025

   Abs DOI HTML

   We present a massively parallel GPU-accelerated solver for direct numerical simulations of transitional and turbulent flat-plate boundary layers and channel flows involving fluids in non-ideal thermodynamic states. While several high-fidelity solvers are currently available as open source, all of them are restricted to the ideal-gas region. In contrast, the CUBic Equation of state Navier-Stokes solver (CUBENS) can accurately model and simulate the non-ideal thermodynamics of single-phase compressible fluids in the vicinity of the vapor-liquid saturation line or the thermodynamic critical point. By employing high-order finite-difference schemes and convective terms in split, kinetic-energy-, and entropy-preserving form, the solver is numerically stable, and robust with minimal numerical dissipation, enabling it to capture the steep variations of non-ideal thermodynamic properties. For cost-effective high-fidelity simulations, in addition to MPI parallelization, CUBENS is GPU-accelerated using OpenACC directives for computation offloading, and asynchronous GPU-aware MPI for efficient GPU-GPU communication. Moreover, CUBENS is compatible with both NVIDIA and AMD GPU architectures, achieving significant performance results while ensuring energy-efficient simulations. For instance, using 64 NVIDIA A100 GPUs compared to 8192 CPUs at the same computational cost results in a speedup of approximately 130×. In multi-node and multi-GPU configurations ranging from 2 to 128 compute nodes (8 to 512 GPUs), a strong scaling efficiency of around 52% and a weak scaling efficiency of 0.88 with 10243 points per GPU, corresponding to approximately 5 billion degrees of freedom, are achieved. The CUBENS solver is validated against selected cases from the literature, covering transitional to turbulent ideal and non-ideal flows up to the transonic regime. In particular, we demonstrate the solver’s suitability and applicability for direct numerical simulations of transitional boundary layers with fluids at supercritical pressure and with buoyancy effects. The development of this high-fidelity solver offers the potential for future fundamental research in non-ideal compressible fluid dynamics.

2024

1. 

   Direct Numerical Simulations of K-type transition in a flat-plate boundary layer with supercritical fluids

   P. C. Boldini,  B. Bugeat,  J. W. R. Peeters, and 2 more authors

   2024

   Abs HTML

   We investigate the controlled K-type breakdown of a flat-plate boundary-layer with highly non-ideal supercritical fluid at a reduced pressure of p_r,∞=1.10. Direct numerical simulations are performed at a Mach number of M_∞=0.2 for one subcritical (liquid-like regime) temperature profile and one strongly-stratified transcritical (pseudo-boiling) temperature profile with slightly heated wall. In the subcritical case, the formation of aligned Λ-vortices is delayed compared to the reference ideal-gas case of Sayadi et al. (J. Fluid Mech. vol. 724, 2013, pp. 480–509), with steady longitudinal modes dominating the late-transitional stage. When the wall temperature exceeds the pseudo-boiling temperature, streak secondary instabilities lead to the simultaneous development of additional hairpin vortices and near-wall streaky structures near the legs of the primary aligned Λ-vortices. Nonetheless, transition to turbulence is not violent and is significantly delayed compared to the subcritical regime.

2. 

   Instability in strongly stratified plane Couette flow with application to supercritical fluids

   B. Bugeat,  P. C. Boldini,  A. M. Hasan, and 1 more author

   Journal of Fluid Mechanics, 2024

   Abs DOI HTML

   This paper addresses the stability of plane Couette flow in the presence of strong density and viscosity stratifications. It demonstrates the existence of a generalised inflection point that satisfies the generalised Fjørtoft criterion of instability when a minimum of kinematic viscosity is present in the base flow. The characteristic scales associated with this minimum are identified as the primary controlling parameters of the associated instability, regardless of the type of stratification. To support this finding, analytical stability models are derived in the long-wave approximation using piecewise linear base flows. Numerical stability calculations are carried out to validate these models and to provide further information on the production of disturbance vorticity. All instabilities are interpreted as arising from the interaction between two vorticity waves. Depending on the type of stratification, these two waves are produced by different physical mechanisms. When both strong density and viscosity stratifications are present, we show that they result from the concurrent action of shear and inertial baroclinic effects. The stability models developed for simple fluid models ultimately shed light on a recently observed unstable mode in supercritical fluids (Ren et al., J. Fluid Mech., vol. 871, 2019, pp. 831–864), providing a quantitative prediction of the stability diagram and identifying the dominant mechanisms at play. Furthermore, our study suggests that the minimum of kinematic viscosity reached at the Widom line in these fluids is the leading cause of their instability. The existence of similar instabilities in different fluids and flows (e.g. miscible fluids) is finally discussed.

3. 

   Transient growth in diabatic boundary layers with fluids at supercritical pressure

   P. C. Boldini,  B. Bugeat,  J. W. R. Peeters, and 2 more authors

   Physical Review Fluids, 2024

   Abs DOI HTML

   In the region close to the thermodynamic critical point and in the proximity of the pseudoboiling (Widom) line, strong property variations substantially alter the growth of modal instabilities, as revealed in Ren et al. [J. Fluid Mech. 871, 831 (2019)]. Here, we study nonmodal disturbances in the spatial framework using an eigenvector decomposition of the linearized Navier-Stokes equations under the assumption of locally parallel flow. To account for nonideality, a new energy norm is derived. Several heat transfer scenarios at supercritical pressure are investigated, which are of practical relevance in technical applications. The boundary layers with the fluid at supercritical pressure are heated or cooled by prescribing the wall and free-stream temperatures so that the temperature profile is either entirely subcritical (liquidlike), supercritical (gaslike), or transcritical (across the Widom line). The free-stream Mach number is set to 10^−3. In the nontranscritical regimes, the resulting streamwise-independent streaks originate from the lift-up effect. Wall cooling enhances the energy amplification for both subcritical and supercritical regimes. When the temperature profile is increased beyond the Widom line, a strong suboptimal growth is observed over very short streamwise distances due to the Orr mechanism. Due to the additional presence of transcritical Mode II, the optimal energy growth at large distances is found to arise from an interplay between lift-up and Orr mechanism. As a result, optimal disturbances are streamwise-modulated streaks with strong thermal components and with a propagation angle inversely proportional to the local Reynolds number. The nonmodal growth is put in perspective with modal growth by means of an N-factor comparison. In the nontranscritical regimes, modal stability dominates regardless of a wall-temperature variation. In contrast, in the transcritical regime, nonmodal N factors are found to resemble the imposition of an adverse pressure gradient in the ideal-gas regime. When cooling beyond the Widom line, optimal growth is greatly enhanced, yet strong inviscid instability prevails. When heating beyond the Widom line, optimal growth could be sufficiently large to favor transition, particularly with a high free-stream turbulence level.

2023

1. 

   Direct numerical simulation of H-type transition in a flat-plate boundary layer with supercritical fluids

   In Proceedings of 14th International ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements (ETMM14), Barcelona, Spain, September 6-8, 2023

   Abs HTML

   We investigate the laminar-to-turbulent transition of highly non-ideal supercritical fluids. The controlled H-type breakdown in a three-dimensional flat-plate boundary layer is chosen. Direct numerical simulations are performed at low Mach numbers, for isothermal and heated walls. We consider a fluid following the Van der Waals (VdW) equation of state (EoS) at a supercritical reduced pressure of p_r = 1.10. A newly developed GPU-accelerated code is first successfully validated against linear two-dimensional simulations using the VdW EoS, and transitional simulations using ideal gas. Subsequently, H-type breakdown of two subcritical (liquid-like only) and one strongly-stratified transcritical (pseudo-boiling) profiles are considered. As the wall temperature approaches the Widom line, the formation of staggered Λ-vortices, with hairpin-shaped vortices at their tips, is delayed. When the wall temperature is higher than the pseudo-boiling temperature, the transition scenario differs from the classical H-type breakdown. Patterns of Λ-structures are alternated by high-low-velocity and-density streaks before hairpin-shaped vortices form. Finally, the skin friction coefficient and Stanton number are analysed, shedding light on the thermodynamic-regime dependence of the transitional overshoot.

2022

1. 

   On the new unstable mode in the boundary layer flow of supercritical fluids

   B. Bugeat,  P. C. Boldini, and  R. Pecnik

   In Proceedings of the 12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP-12), 2022

   Abs HTML

   Ren et al. (2019) recently studied the stability of the boundary layer flow over a flat plate for supercritical CO2. While only one unstable mode usually exists for boundary layer flows, the authors found an additional unstable mode, whose origin has so far not been identified. In the present work, we carry out a stability analysis in the general case ofa fluid following the Van der Waals equation of state and flowing over a heated flat plate in the limit of zero Eckert number.In this framework, the second unstable mode is also recovered, ruling out an acoustic origin. From the Rayleigh equation derived in the presence of density gradients, a generalised inflection point (GIP) criterion of instability exists, similar to that of fully compressible flows. Inviscid stability calculations confirm the existence of an unstable mode in the presence ofa GIP, which is linked to the additional second mode found at finite Reynolds numbers. A theoretical analysis is then carried out by approximating the momentum equation for a base flow exhibiting strong gradients of dynamic viscosity. It is shown that the origin of the GIP, and hence the additional unstable mode, is associated with a minimum of kinematic viscosity atthe Widom line. The universality of this result beyond supercritical fluids is eventually discussed.

2019

1. Research on Hypersonic Boundary-Layer Stability with High-Temperature Effects

   Xianliang Chen, Pietro Carlo Boldini, and Song Fu

   In The Proceedings of the 2018 Asia-Pacific International Symposium on Aerospace Technology (APISAT 2018), 2019

   Abs HTML

   High-temperature effects need to be considered for a better design of hypersonic and reentry vehicles. They affect both the boundary layer flow and its flow transition, whose primary stages can be investigated through modal stability analysis. In this work, physical and numerical tools for high-temperature flows are presented and the efficiency of the new developed in-house boundary layer and stability solvers is tested. Specially, we focus on the stability of a flat plate flow in thermochemical non-equilibrium through an investigation of growth rates under the influence of various flow parameters.

2018

1. Effects of Thermal Non-Equilibrium on Hypersonic Boundary-Layer Transition

   P.C. Boldini

   2018

   Master thesis

   Abs HTML

   Laminar-to-Turbulent Transition (LTT) in hypersonic boundary layers plays a key role in the design of hypersonic and reentry vehicles. In high-speed flows, high-temperature effects need to be considered as the Calorically Perfect Gas (CPG) assumption fails, leading to a wrong prediction of the transition onset. Therefore a better understanding of the influence of inter-energy exchange and chemical reactions on the flow instability is required. The present thesis deals with the effects of Thermal Non-Equilibrium (TNEQ) on hypersonic boundary-layer transition. Linear-Stability Theory (LST) is selected to investigate the stability of a laminar at plate flow in a chemically inert air mixture. Firstly, thermodynamic models for high- temperature effects and hypersonic LTT features are introduced with special regard to the influence of thermal relaxation processes. Secondly, the compressible Navier-Stokes equations for a TNEQ flow are set as the basis equations for the development of the linear-stability analysis. On one hand, with respect to the mean flow, TNEQ boundary-layer equations are derived after a coordinate transformation. On the other hand, TNEQ linearised disturbance equations are obtained under the ’locally parallel flow’ assumption. In order to solve the latter, a global and a local method are developed and compared for specific high-speed flow configurations. The new base-flow and stability solvers are validated over different literature sources, first of all by assuming a CPG flow and then by extending the governing equations to a TNEQ case. Finally, three different TNEQ cases are selected: a cold wall case with a free-stream Mach number of 6 and a real hypersonic case at an altitude of 30 km with isothermal and adiabatic wall. Mean-flow solutions are analysed with respect to the influence of TNEQ, for example it has been found that diatomic nitrogen is more in non-equilibrium than diatomic oxygen. For all configurations, linear-stability calculations are performed with two dimensional disturbances; subsequently, growth rates and amplitude distributions are extracted and investigated. Vibrational non-equilibrium destabilises more the second-mode instability and the supersonic mode region compared to the CPG assumption, vice versa for the first-mode instability. Furthermore, the following effects on the hypersonic TNEQ flow instability are investigated: vibrational relaxation, viscosity, disturbance boundary conditions, Reynolds number, temperature, three dimensional disturbances, rotational relaxation with or without carbon dioxide.

2016

1. Transmission probabilities of rarefied flows in the application of atmosphere-breathing electric propulsion

   T. Binder, P. C. Boldini, F. Romano, and 2 more authors

   AIP Conference Proceedings, Nov 2016

   Abs DOI HTML

   Atmosphere-Breathing Electric Propulsion systems (ABEP) are currently investigated to utilize the residual atmosphere as propellant for drag-compensating thrusters on spacecraft in (very) low orbits. The key concept for an efficient intake of such a system is to feed a large fraction of the incoming flow to the thruster by a high transmission probability Θ for the inflow while Θ for the backflow should be as low as possible. This is the case for rarefied flows through tube-like structures of arbitrary cross section when assuming diffuse wall reflections inside and after these ducts, and entrance velocities u larger than thermal velocities vth∝kBT/m. The theory of transmission for free molecular flow through cylinders is well known for u = 0, but less research results are available for u > 0.In this paper, the desired theoretical characteristics of intakes for ABEP are pointed out, a short review of transmission probabilities is given, and results of Monte Carlo simulations concerning Θ are presented. Based on simple algebraic relations, an intake can be optimized in terms of collection efficiency by choosing optimal ducts. It is shown that Θ depends only on non-dimensional values of the duct geometry combined with vth and u. The simulation results of a complete exemplary ABEP configuration illustrate the influence of modeling quality in terms of inflow conditions and inter-particle collisions.

2. Optimum Design of the Intake for an Atmosphere-Breathing Electric Propulsion System

   P.C. Boldini

   Nov 2016

   Bachelor thesis

   Abs HTML

   An atmosphere-breathing electric propulsion system (ABEP) uses rarefied atmosphere gases, collected by an intake, as propellant for an electric thruster. This would theoretically allow a spacecraft (S/C), flying at low orbit altitudes, to extend its orbit lifetime without carrying, in the best case, any propellant necessary to compensate the local drag force. With respect to the peculiar hyperthermal flow conditions, the design of an intake needs optimisation to increase its performance. The present thesis deals with the investigation of an optimum design of the intake for Earth’s and Mars’ atmospheres, and respective altitude ranges. In this study an inductive plasma thruster (IPT) is considered. Firstly, flow physics and features inside the intake are explained with special regard to transmission probabilities of different ducts as possible parts of an intake. Secondly, the general ABEP concept is introduced with the aid of a literature review, where focus lies on the current intake designs. Considerations on the atmospheres and on the IPT candidate thruster, the inductively heated plasma generator of the University of Stuttgart (IPG6-S), are made. Main parameters of the intake are derived and analysed, also in relation to the Balancing Model, an analytical model based on the flow balance inside the intake. Here, the use of frontal ducts is investigated with the help of sensitivity analyses on circular, hexagonal and annular ducts. Detailed results on two of the currently most advanced intake configurations are presented along with the definition of a new design suitable for the IPG6-S, the Enhanced Funnel Design (EFD). Its final geometry and altitude operation ranges are defined, having regard to collection efficiencies, compression ratios and intake drag compensation. The configuration with known IPG6-S exhaust velocity from literature is chosen as best-case design for its ideally intake full drag compensation. EFD has theoretically a collection efficiency of 43% for Earth, whereas 32% for Mars. In both cases full-drag compensation of the intake is estimated to be guaranteed. Finally, DSMC simulations on the full intake are compared to verify the improved Balancing Model applied for the EFD IPG6-S optimisation. There is good accuracy between the analytical model and the computational method with or without intermolecular collisions: the discrepancy is less than 6% in terms of intake collection efficiency..