This paper develops scaling laws for wall-pressure root mean square and the streamwise turbulence intensity peak, accounting for both variable-property and intrinsic compressibility effects – those associated with changes in fluid volume due to pressure variations. To develop such scaling laws, we express the target quantities as an expansion series in powers of an appropriately defined Mach number. The leading-order term is represented using the scaling relations developed for incompressible flows, but with an effective Reynolds number. Higher-order terms capture intrinsic compressibility effects and are modelled as constant coefficients, calibrated using flow cases specifically designed to isolate these effects. The resulting scaling relations are shown to be accurate for a wide range of turbulent channel flows and boundary layers.

It is difficult to envision an industrial application where turbulent flows interacting with solid walls do not play a critical role. While understanding these flows at low speeds is already challenging, the complexity increases significantly when the flow speed exceeds the speed of sound or when heat transfer through the walls is intense. These so-called compressible flows are at the core of many engineering applications including aerospace vehicles, combustors, high-speed propulsion systems, gas turbines and other power-generating technologies. Understanding the physics governing these flows is essential for developing accurate predictive models, which in turn enable the improved design of engineering systems.

Compressible wall-bounded turbulent flows involve two distinct effects: those related to heat transfer, commonly referred to as variable-property effects, and those arising from density changes of fluid elements in response to changes in pressure, termed intrinsic compressibility (IC) effects. While the former can occur across all flow speeds, the latter becomes significant only at high Mach numbers. In the past, variable-property effects have been extensively studied; in contrast, the influence of intrinsic compressibility has received limited attention. This gap is largely attributed to Morkovin’s hypothesis, which asserts that IC effects can be neglected in wall-bounded flows under certain conditions. The present work revisits this assumption and directly addresses the question posed by Otto Zeman in 1993: “are the (intrinsic) compressibility effects significant in reality, and can they be isolated in experiments and verified?” To isolate such effects, we perform direct numerical simulations (DNS) of fully developed high-Mach-number channel flows, in which the energy equation is augmented with an external heat source to maintain approximately constant mean thermophysical properties, thereby eliminating variable-property effects.

This thesis is divided in two parts. The first part uses these tailored flow cases to investigate the physics associated with IC effects. We demonstrate that IC effects significantly influence various turbulence statistics—an influence previously misattributed to variable-property effects. The underlying mechanism is as follows: pressure-induced expansions and contractions of the near-wall fluid oppose sweeps and ejections, leading to a weakening of quasi-streamwise vortices. The weakened vortices reduce the energy transferred from the streamwise to the wall-normal velocity components, thereby modulating turbulence statistics.

The second part builds on these insights to develop scaling laws and predictive models applicable to a wide range of channel flows and zero-pressure-gradient boundary layers. Specifically, we derive scaling laws for wall pressure fluctuations, the peak of streamwise turbulent stress, and the mean velocity profile—accounting for both variable-property and intrinsic compressibility effects. The mean velocity scaling is then further exploited to derive predictive models that estimate skin friction and heat transfer coefficients, and to propose compressibility corrections for Reynolds-averaged Navier- Stokes (RANS) turbulence models. These corrected models demonstrate significantly improved accuracy over the state-of-the-art and hold strong potential for enhancing the modeling of complex, real-world engineering systems.

This chapter focuses on modeling compressible wall-bounded turbulent flows. We first make the distinction between variable-property effects and intrinsic compressibility effects. Variable-property effects are associated with changes in density, viscosity, etc., primarily due to heat transfer, while intrinsic compressibility effects are associated with volume changes induced by pressure fluctuations. Both phenomena significantly impact turbulence and thus complicate turbulence modeling. The focus is on adapting Reynolds-averaged Navier-Stokes (RANS) models to account for these effects, which can also be applied to wall-modeled large eddy simulations (WMLES). Canonical flows are examined to illustrate the necessary modifications needed to incorporate compressibility effects into general RANS models. These improvements are applicable to many turbulence models and for complex flow geometries.

We introduce a novel approach to derive compressibility corrections for Reynolds-averaged Navier–Stokes (RANS) models. Using this approach, we derive variable-property corrections for wall-bounded flows that take into account the distinct scaling characteristics of the inner and outer layers, extending the earlier work of Otero Rodriguez et al. (Intl J. Heat Fluid Flow, 73, 2018, 114–123). We also propose modifying the eddy viscosity to account for changes in the near-wall damping of turbulence due to intrinsic compressibility effects. The resulting corrections are consistent with our recently proposed velocity transformation (Hasan et al. Phys. Rev. Fluids, 8, 2023, L112601) in the inner layer and the Van Driest velocity transformation in the outer layer. Furthermore, we address some important aspects related to the modelling of the energy equation, primarily focusing on the turbulent Prandtl number and the modelling of the source terms. Compared with the existing state-of-the-art compressibility corrections, the present corrections, combined with accurate modelling of the energy equation, lead to a significant improvement in the results for a wide range of turbulent boundary layers and channel flows. The proposed corrections have the potential to enhance modelling across a range of applications, involving low-speed flows with strong heat transfer, fluids at supercritical pressures, and supersonic and hypersonic flows.

The impact of intrinsic compressibility effects – changes in fluid volume due to pressure variations – on high-speed wall-bounded turbulence has often been overlooked or incorrectly attributed to mean property variations. To quantify these intrinsic compressibility effects unambiguously, we perform direct numerical simulations of compressible turbulent channel flows with nearly uniform mean properties. Our simulations reveal that intrinsic compressibility effects yield a significant upward shift in the logarithmic mean velocity profile that can be attributed to the reduction in the turbulent shear stress. This reduction stems from the weakening of the near-wall quasi-streamwise vortices. In turn, we attribute this weakening to the spontaneous opposition of sweeps and ejections from the near-wall expansions and contractions of the fluid, and provide a theoretical explanation for this mechanism. Our results also demonstrate that intrinsic compressibility effects play a crucial role in the increase in inner-scaled streamwise turbulence intensity in compressible flows, as compared with incompressible flows, which was previously regarded to be an effect of mean property variations alone.

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.

A transformation that relates a compressible wall-bounded turbulent flow with nonuniform fluid properties to an equivalent incompressible flow with uniform fluid properties is derived and validated. The transformation accounts for both variable-property and intrinsic compressibility effects, the latter being the key improvement over the current state of the art. The importance of intrinsic compressibility effects contradicts the renowned Morkovin's hypothesis.