In this work, a near-wall model, which couples the inverse of a recently developed compressible velocity transformation (Griffin et al., Proc. Natl Acad. Sci., vol. 118, 2021, p. 34) and an algebraic temperature–velocity relation, is developed for high-speed turbulent boundary layers. As input, the model requires the mean flow state at one wall-normal height in the inner layer of the boundary layer and at the boundary-layer edge. As output, the model can predict mean temperature and velocity profiles across the entire inner layer, as well as the wall shear stress and heat flux. The model is tested in an a priori sense using a wide database of direct numerical simulation high-Mach-number turbulent channel flows, pipe flows and boundary layers (48 cases, with edge Mach numbers in the range 0.77–11, and semi-local friction Reynolds numbers in the range 170–5700). The present model is significantly more accurate than the classical ordinary differential equation (ODE) model for all cases tested. The model is deployed as a wall model for large-eddy simulations in channel flows with bulk Mach numbers in the range 0.7–4 and friction Reynolds numbers in the range 320–1800. When compared to the classical framework, in the a posteriori sense, the present method greatly improves the predicted heat flux, wall stress, and temperature and velocity profiles, especially in cases with strong heat transfer. In addition, the present model solves one ODE instead of two, and has a computational cost and implementation complexity similar to that of the commonly used ODE model.

While there have been numerous applications of large eddy simulations (LES) to complex flows, their application to practical engineering configurations, such as full aircraft models, have been limited to date. Recently, however, advances in rapid, high quality mesh generation, low-dissipation numerical schemes and physics-based subgrid-scale and wall models have led to, for the first time, accurate simulations of a realistic aircraft in landing configuration (the Japanese Aerospace Exploration Agency Standard Model) in less than a day of turnaround time with modest resource requirements. In this paper, a systematic study of the predictive capability of LES across a range of angles of attack (including maximum lift and post-stall regimes), the robustness of the predictions to grid resolution and the incorporation of wind tunnel effects is carried out. Integrated engineering quantities of interest, such as lift, drag and pitching moment will be compared with experimental data, while sectional pressure forces will be used to corroborate the accuracy of the integrated quantities. Good agreement with experimental $C_L$ data is obtained across the lift curve with the coefficient of lift at maximum lift, $C_{L,max}$, consistently being predicted to within five lift counts of the experimental value. The grid point requirements to achieve this level of accuracy are reduced compared with recent estimates (even for wall modelled LES), with the solutions showing systematic improvement upon grid refinement, with the exception of the solution at the lowest angles of attack, which will be discussed later in the text. Simulations that include the wind tunnel walls and aircraft body mounting system are able to replicate important features of the flow field noted in the experiment that are absent from free air calculations of the same geometry, namely, the onset of inboard flow separation in the post-stall regime. Turnaround times of the order of a day are made possible in part by algorithmic advances made to leverage graphical processing units. The results presented herein suggest that this combined approach (meshing, numerical algorithms, modelling, efficient computer implementation) is on the threshold of readiness for industrial use in aeronautical design.

Breaking waves generate a distribution of bubble sizes that evolves over time. Knowledge of how this distribution evolves is of practical importance for maritime and climate studies. The analytical framework developed in Part 1 (Chan, Johnson & Moin, J. Fluid Mech., vol. 912, 2021, A42) examined how this evolution is governed by the bubble-mass flux from large- to small-bubble sizes which depends on the rate of break-up events and the distribution of child bubble sizes. These statistics are measured in Part 2 as ensemble-averaged functions of time by simulating ensembles of breaking waves, and identifying and tracking individual bubbles and their break-up events. The large-scale break-up dynamics is seen to be statistically unsteady, and two intervals with distinct characteristics were identified. In the first interval, the dissipation rate and bubble-mass flux are quasi-steady, and the theoretical analysis of Part 1 is supported by all observed statistics, including the expected $-10/3$ power-law exponent for the super-Hinze-scale size distribution. Strong locality is observed in the corresponding bubble-mass flux, supporting the presence of a super-Hinze-scale break-up cascade. In the second interval, the dissipation rate decays, and the bubble-mass flux increases as small- and intermediate-sized bubbles become more populous. This flux remains strongly local with cascade-like behaviour, but the dominant power-law exponent for the size distribution increases to $-8/3$ as small bubbles are also depleted more quickly. This suggests the emergence of different physical mechanisms during different phases of the breaking-wave evolution, although size-local break-up remains a dominant theme. Parts 1 and 2 present an analytical toolkit for population balance analysis in two-phase flows.

Breaking waves entrain gas beneath the surface. The wave-breaking process energizes turbulent fluctuations that break bubbles in quick succession to generate a wide range of bubble sizes. Understanding this generation mechanism paves the way towards the development of predictive models for large-scale maritime and climate simulations. Garrett et al. (J. Phys. Oceanogr., vol. 30, 2000, pp. 2163–2171) suggested that super-Hinze-scale turbulent break-up transfers entrained gas from large- to small-bubble sizes in the manner of a cascade. We provide a theoretical basis for this bubble-mass cascade by appealing to how energy is transferred from large to small scales in the energy cascade central to single-phase turbulence theories. A bubble break-up cascade requires that break-up events predominantly transfer bubble mass from a certain bubble size to a slightly smaller size on average. This property is called locality. In this paper, we analytically quantify locality by extending the population balance equation in conservative form to derive the bubble-mass-transfer rate from large to small sizes. Using our proposed measures of locality, we show that scalings relevant to turbulent bubbly flows, including those postulated by Garrett et al. (J. Phys. Oceanogr., vol. 30, 2000, pp. 2163–2171) and observed in breaking-wave experiments and simulations, are consistent with a strongly local transfer rate, where the influence of non-local contributions decays in a power-law fashion. These theoretical predictions are confirmed using numerical simulations in Part 2 (Chan et al., J. Fluid. Mech. vol. 912, 2021, A43), revealing key physical aspects of the bubble break-up cascade phenomenology. Locality supports the universality of turbulent small-bubble break-up, which simplifies the development of cascade-based subgrid-scale models to predict oceanic small-bubble statistics of practical importance.

A novel mechanism for the amplification of distortions to the material interface of liquid jets is identified. The mechanism is independent of the exponential instability of the flow and can intensify small perturbations to the material interface by several orders of magnitude. Depending on the parameters, it can amplify interfacial distortions at a faster pace than modal mechanisms such as the Kelvin–Helmholtz instability. The study is based on spatial linear stability theory in a two-fluid formulation that accounts for the effects of both viscosity and surface tension. The analysis of the mechanism is cast into an optimization problem in the surface tension energy of the interface distortion and discounts the trivial redistribution of perturbation kinetic energy. The identified mechanism is related to the Orr mechanism, and amplifies distortions to the material interface via a reorientation of perturbations by the mean shear. Analyses of the linearized energy budgets show that energy is extracted from the mean shear by the production term of the streamwise perturbation velocity component and subsequently transferred to the radial perturbation velocity component, where it is absorbed by the surface tension potential of the interface. The gain in surface tension energy attributable to the mechanism is shown to scale linearly with the Reynolds number. A critical Weber number is identified as a lower bound beyond which the mechanism becomes active, and a power-law relation to the Reynolds number is established. Nonlinear simulations based on the full two-fluid Navier–Stokes equations substantiate the observability and realizability of the mechanism.

The interaction between an incident shock wave and a Mach-6 undisturbed hypersonic laminar boundary layer over a cold wall is addressed using direct numerical simulations (DNS) and wall-modelled large-eddy simulations (WMLES) at different angles of incidence. At sufficiently high shock-incidence angles, the boundary layer transitions to turbulence via breakdown of near-wall streaks shortly downstream of the shock impingement, without the need of any inflow free-stream disturbances. The transition causes a localized significant increase in the Stanton number and skin-friction coefficient, with high incidence angles augmenting the peak thermomechanical loads in an approximately linear way. Statistical analyses of the boundary layer downstream of the interaction for each case are provided that quantify streamwise spatial variations of the Reynolds analogy factors and indicate a breakdown of the Morkovin's hypothesis near the wall, where velocity and temperature become correlated. A modified strong Reynolds analogy with a fixed turbulent Prandtl number is observed to perform best. Conventional transformations fail at collapsing the mean velocity profiles on the incompressible log law. The WMLES prompts transition and peak heating, delays separation and advances reattachment, thereby shortening the separation bubble. When the shock leads to transition, WMLES provides predictions of DNS peak thermomechanical loads within $\pm 10\,\%$ at a computational cost lower than DNS by two orders of magnitude. Downstream of the interaction, in the turbulent boundary layer, the WMLES agrees well with DNS results for the Reynolds analogy factor, the mean profiles of velocity and temperature, including the temperature peak, and the temperature/velocity correlation.

We present findings on scalar flashes, turbulent spots and transition statistics in a direct numerical simulation of a 500 radius-long pipe flow with a radial-mode inlet disturbance. Transitional spots are found to contain two different types of eddies. The wall region consists primarily of ‘reverse hairpin vortices’. This unusual structure is related to the high-speed streaks arising from the prescribed inlet perturbation. The core region is populated by the normally observed ‘forward’ hairpin vortices. Number density of the reverse hairpins is quantified indirectly and conservatively by measuring the number density of negative skin-friction patches. During the late stages of transition, second-order statistics such as Reynolds stresses and the rate of dissipation of turbulent kinetic energy exhibit substantial overshoot. This is associated with mid-to-high frequency content in the energy spectra that exceeds the corresponding levels in fully developed turbulence. Flow visualizations reveal bursts of small-scale vortex motions, including the reverse hairpins, that probably account for the enhanced mid-to-high frequency spectral content. A passive scalar is injected at the centreline of the inlet plane, mimicking laboratory injection of dye through a needle, to investigate the mysterious phenomenon of turbulent scalar patches residing in fully developed turbulent pipe flow. At several hundred radii downstream of transition where the velocity field is genuine fully developed turbulence, the scalar patches retain persistent memory of events far upstream. Comparing the present flow with a similar pipe flow disturbed by a significantly different inlet condition suggests that the foregoing observations are insensitive to the form of the disturbances.

In wall-bounded turbulent flows, the wall-normal gradient in turbulence intensity causes inertial particles to move preferentially toward the wall, leading to elevated concentration levels in the viscous sublayer. At first glance, wall-modelled large-eddy simulations may seem ill suited for accurately simulating this behaviour, given that the sharp gradients and coherent structures in the viscous sublayer and buffer region are unresolved in this approach. In this paper, a detailed inspection of conservation equations describing the influence of turbophoresis and near-wall structures on particle concentration profiles reveals a more nuanced view depending on the friction Stokes number. The dynamics of low and moderate Stokes number particles indeed depends strongly on the complex spatio-temporal details of streaks, ejections, and sweeps in the near-wall region. This significantly impacts the near-wall particle concentration through a biased sampling effect which provides a net force away from the wall on the particle ensemble caused by the tendency of inertial particles to accumulate in low-speed ejection regions. At higher Stokes numbers, however, this biased sampling is of minimal importance, and the particle concentration becomes inversely proportional to the wall-normal particle velocity variance at a given distance from the wall. As a result, wall-modelled large-eddy simulations can predict concentration profiles with more accuracy in the high Stokes number regime than low Stokes numbers simply by modifying the interpolation scheme for particles between the first grid point and the boundary. However, accurate representation of low and moderate Stokes number particles depends critically on information not present in standard wall-modelled large-eddy simulations.

Non-equilibrium wall turbulence with mean-flow three-dimensionality is ubiquitous in geophysical and engineering flows. Under these conditions, turbulence may experience a counter-intuitive depletion of the turbulent stresses, which has important implications for modelling and control. Yet, current turbulence theories have been established mainly for statistically two-dimensional equilibrium flows and are unable to predict the reduction in the Reynolds stress magnitude. In the present work, we propose a multiscale model that captures the response of non-equilibrium wall-bounded turbulence under the imposition of three-dimensional strain. The analysis is performed via direct numerical simulation of transient three-dimensional turbulent channels subjected to a sudden lateral pressure gradient at friction Reynolds numbers up to 1000. We show that the flow regimes and scaling properties of the Reynolds stress are consistent with a model comprising momentum-carrying eddies with sizes and time scales proportional to their distance to the wall. We further demonstrate that the reduction in Reynolds stress follows a spatially and temporally self-similar evolution caused by the relative horizontal displacement between the core of the momentum-carrying eddies and the flow layer underneath. Inspection of the flow energetics reveals that this mechanism is associated with lower levels of pressure–strain correlation, which ultimately inhibits the generation of Reynolds stress, consistent with previous works. Finally, we assess the ability of the state-of-the-art wall-modelled large-eddy simulation to predict non-equilibrium three-dimensional flows.

Wall modelling in large-eddy simulation (LES) is necessary to overcome the prohibitive near-wall resolution requirements in high-Reynolds-number turbulent flows. Most existing wall models rely on assumptions about the state of the boundary layer and require a priori prescription of tunable coefficients. They also impose the predicted wall stress by replacing the no-slip boundary condition at the wall with a Neumann boundary condition in the wall-parallel directions while maintaining the no-transpiration condition in the wall-normal direction. In the present study, we first motivate and analyse the Robin (slip) boundary condition with transpiration (non-zero wall-normal velocity) in the context of wall-modelled LES. The effect of the slip boundary condition on the one-point statistics of the flow is investigated in LES of turbulent channel flow and a flat-plate turbulent boundary layer. It is shown that the slip condition provides a framework to compensate for the deficit or excess of mean momentum at the wall. Moreover, the resulting non-zero stress at the wall alleviates the well-known problem of the wall-stress under-estimation by current subgrid-scale (SGS) models (Jiménez & Moser, AIAA J., vol. 38 (4), 2000, pp. 605–612). Second, we discuss the requirements for the slip condition to be used in conjunction with wall models and derive the equation that connects the slip boundary condition with the stress at the wall. Finally, a dynamic procedure for the slip coefficients is formulated, providing a dynamic slip wall model free of a priori specified coefficients. The performance of the proposed dynamic wall model is tested in a series of LES of turbulent channel flow at varying Reynolds numbers, non-equilibrium three-dimensional transient channel flow and a zero-pressure-gradient flat-plate turbulent boundary layer. The results show that the dynamic wall model is able to accurately predict one-point turbulence statistics for various flow configurations, Reynolds numbers and grid resolutions.

The mechanism underlying the coherent hairpin process in wall-bounded shear flows is studied. An algorithm for the identification and analysis of flow structures based on morphological operations is presented. The method distils the topology of the flow field into a discrete data set and enables the time-resolved sampling of coherent flow processes across multiple scales. Application to direct simulation data of transitional and turbulent boundary layers at moderate Reynolds number sheds light on the flow physics of the hairpin process. The analysis links the hairpin to an exponential instability which is amplified in the flow distorted by a negative perturbation in the streamwise velocity component. Linear analyses substantiate the connection to an inviscid instability mechanism of varicose type. The formation of packets of hairpins is related to a self-similar process which originates from a single patch of low-speed fluid and describes a recurrence of the dynamics that leads to the formation of an individual hairpin. Analysis of the evolution of several thousand turbulent hairpins provides a statistical characterization of the principal dynamics and yields a time-resolved average of the hairpin process. Comparisons with the transitional hairpin show qualitatively consistent trends and thus support earlier hypotheses of their equivalence. In terms of the causality of events, the results suggest that the hairpin is a manifestation of the varicose instability and as such is a consequence rather than a cause of the primary perturbations of the flow.

Algebraic disturbance growth in spatially developing boundary-layer flows is investigated using an optimization approach. The methodology builds on the framework of the parabolized stability equations and avoids some of the limitations associated with adjoint-based schemes. In the Blasius boundary layer, non-parallel effects are shown to significantly enhance the energy gain due to algebraic growth mechanisms. In contrast to parallel flow, the most energetic perturbations have finite frequency and are generated by the simultaneous activity of the Orr and lift-up mechanisms. The highest amplification occurs in a limited region of the parameter space that is characterized by a linear relation between the wavenumber and frequency of the disturbances. The frequency of the most highly amplified perturbations decreases with Reynolds number. Adverse streamwise pressure gradient further enhances the amplification of disturbances while preserving the linear trend between the wavenumber and frequency of the most energetic perturbations.

We present direct numerical simulation (DNS) of a stationary turbulent hydraulic jump with inflow Froude number of 2, Weber number of 1820 and density ratio of 831, consistent with ambient water–air systems, all based on the inlet height and inlet velocity. A non-dissipative geometric volume of fluid (VOF) method is used to track the detailed interactions between turbulent flow structures and the nonlinear interface dynamics. Level set equations are also solved concurrent with VOF in order to calculate the interface curvature and surface tension forces. The mesh resolution is set to resolve a wide range of interfacial scales including the Hinze scale. Calculations are compared against experimental data of void fraction and interfacial scales indicating, reasonable agreement despite a Reynolds number mismatch. Multiple calculations are performed confirming weak sensitivity of low-order statistics and void fraction on the Reynolds number. The presented results provide, for the first time, a comprehensive quantitative data for a wide range of phenomena in a turbulent breaking wave using DNS. These include mean velocity fields, Reynolds stresses, turbulence production and dissipation, velocity spectra and air entrainment data. In addition, we present the energy budget as a function of streamwise location by keeping track of various energy exchange processes in the wake of the jump. The kinetic energy is mostly transferred to pressure work, potential energy and dissipation while surface energy plays a less significant role. Our results indicate that the rate associated with various energy exchange processes peak at different streamwise locations, with exchange to pressure work flux peaking first, followed by potential energy flux and then dissipation. The energy exchange process spans a streamwise length of order ${\sim}10$ jump heights. Furthermore, we report statistics associated with bubble transport downstream of the jump. The bubble formation is found to have a periodic nature. Meaning that the bubbles are generated in patches with a specific frequency associated with the roll-up frequency of the roller at the toe of the jump, with its footprint apparent in the velocity energy spectrum. Our study also provides the ensemble-averaged statistics of the flow which we present in this paper. These results are useful for the development and validation of reduced-order models such as dissipation models in wave dynamics simulations, Reynolds-averaged Navier–Stokes models and air entrainment models.

Direct numerical simulations (DNS) of controlled H- and K-type transitions to turbulence in an $M = 0.2$ (where $M$ is the Mach number) nominally zero-pressure-gradient and spatially developing flat-plate boundary layer are considered. Sayadi, Hamman & Moin (J. Fluid Mech., vol. 724, 2013, pp. 480–509) showed that with the start of the transition process, the skin-friction profiles of these controlled transitions diverge abruptly from the laminar value and overshoot the turbulent estimation. The objective of this work is to identify the structures of dynamical importance throughout the transitional region. Dynamic mode decomposition (DMD) (Schmid, J. Fluid Mech., vol. 656, 2010, pp. 5–28) as an optimal phase-averaging process, together with triple decomposition (Reynolds & Hussain, J. Fluid Mech., vol. 54 (02), 1972, pp. 263–288), is employed to assess the contribution of each coherent structure to the total Reynolds shear stress. This analysis shows that low-frequency modes, corresponding to the legs of hairpin vortices, contribute most to the total Reynolds shear stress. The use of composite DMD of the vortical structures together with the skin-friction coefficient allows the assessment of the coupling between near-wall structures captured by the low-frequency modes and their contribution to the total skin-friction coefficient. We are able to show that the low-frequency modes provide an accurate estimate of the skin-friction coefficient through the transition process. This is of interest since large-eddy simulation (LES) of the same configuration fails to provide a good prediction of the rise to this overshoot. The reduced-order representation of the flow is used to compare the LES and the DNS results within this region. Application of this methodology to the LES of the H-type transition illustrates the effect of the grid resolution and the subgrid-scale model on the estimated shear stress of these low-frequency modes. The analysis shows that although the shapes and frequencies of the low-frequency modes are independent of the resolution, the amplitudes are underpredicted in the LES, resulting in underprediction of the Reynolds shear stress.

The onset and development of turbulence from controlled disturbances in compressible ($\mathit{Ma}= 0. 2$), flat-plate boundary layers is studied by direct numerical simulation. We have validated the initial disturbance development, confirmed that H- and K-regime transitions were reproduced and, from these starting points, we carried these simulations beyond breakdown, past the skin-friction maximum and to higher Reynolds numbers than investigated before to evaluate how these two flow regimes converge towards turbulence and what transitional flow structures embody the statistics and mean dynamics of developed turbulence. We show that H- and K-type breakdowns both relax toward the same statistical structure typical of developed turbulence at high Reynolds number immediately after the skin-friction maximum. This threshold marks the onset of self-sustaining mechanisms of near-wall turbulence. At this point, computed power spectra exhibit a decade of Kolmogorov inertial subrange; this is further evidence of convergence to equilibrium turbulence at the late stage of transition. Here, visualization of the instantaneous flow structure shows numerous, tightly packed hairpin vortices (Adrian, Phys. Fluids, vol. 19, 2007, 041301). Strongly organized coherent hairpin structures are less perceptible farther downstream (at higher Reynolds numbers), but the flow statistics and near-wall dynamics are the same. These structurally simple hairpin-packet solutions found in the very late stages of H- and K-type transitions obey the statistical measurements of higher-Reynolds-number turbulence. Comparison with the bypass transition of Wu & Moin (Phys. Fluids, vol. 22, 2010, pp. 85–105) extends these observations to a wider class of transitional flows. In contrast to bypass transition, the (time- and spanwise-averaged) skin-friction maximum in both H- and K-type transitions overshoots the turbulent correlation. Downstream of these friction maxima, all three skin-friction profiles collapse when plotted versus the momentum-thickness Reynolds number, ${\mathit{Re}}_{\theta } $. Mean velocities, turbulence intensities and integral parameters collapse generally beyond ${\mathit{Re}}_{\theta } = 900$ in each transition scenario. Skin-friction maxima, organized hairpin vortices and the onset of self-sustaining turbulence found in controlled H- and K-type transitions are, in many dynamically important respects, similar to development of turbulent spots seen by Park et al. (Phys. Fluids, vol. 24, 2012, 035105). A detailed statistical comparison demonstrates that each of these different transition scenarios evolve into a unique force balance characteristic of higher-Reynolds-number turbulence (Klewicki, Ebner & Wu, J. Fluid Mech., vol. 682, 2011, pp. 617–651). We postulate that these dynamics of late-stage transition as manifested by hairpin packets can serve as a reduced-order model of high-Reynolds-number turbulent boundary layers.