This is the first of a two-part paper. We formulate a data-driven method for constructing finite-volume discretizations of an arbitrary dynamical system's underlying Liouville/Fokker–Planck equation. A method is employed that allows for flexibility in partitioning state space, generalizes to function spaces, applies to arbitrarily long sequences of time-series data, is robust to noise and quantifies uncertainty with respect to finite sample effects. After applying the method, one is left with Markov states (cell centres) and a random matrix approximation to the generator. When used in tandem, they emulate the statistics of the underlying system. We illustrate the method on the Lorenz equations (a three-dimensional ordinary differential equation) saving a fluid dynamical application for Part 2 (Souza, J. Fluid Mech., vol. 997, 2024, A2).

This is the second part of a two-part paper. We apply the methodology of the first paper (Souza, J. Fluid Mech., vol. 997, 2024, A1) to construct a data-driven finite-volume discretization of the Liouville/Fokker–Planck equation of a high-dimensional dynamical system, i.e. the compressible Euler equations with gravity and rotation evolved on a thin spherical shell. We show that the method recovers a subset of the statistical properties of the underlying system, steady-state distributions of observables and autocorrelations of particular observables, as well as revealing the global Koopman modes of the system. We employ two different strategies for the partitioning of a high-dimensional state space, and explore their consequences.

Rotating Rayleigh–Bénard convection denotes the convection between a warm plate and a cold plate in a rotating environment. It is a classic model for understanding convective vortices in the atmosphere and ocean. The influence of background rotation on fluid inertia breaks the symmetry between cyclones and anticyclones. Such a symmetry breaking could be represented by vorticity skewness, which still lacks a systematic theory. Rapidly rotating convection with stress-free boundaries and unit Prandtl number is a convenient starting point. The investigation starts from the convective onset stage, where the vortices grow stationarily. Asymptotic analysis shows that the volumetric vorticity skewness $S$ is produced by the interaction between the $n=0,1$ and $n=1,2$ vertical eigenmodes. The $n=0$ (barotropic) mode contributes positively to $S$ mainly by stretching the vertical relative vorticity, an ageostrophic effect. The $n=2$ mode makes a minor negative contribution to $S$ by preferentially intensifying the outflow over the inflow, a non-hydrostatic effect. The theory predicts $S$ to be proportional to the global Rossby number defined with the volumetric standard deviation of vorticity, ${Ro_g}$. The proportional factor does not depend on the Rayleigh and Ekman numbers, agreeing with direct numerical simulations. Then the system enters the equilibrium stage. The stretching of vertical vorticity still contributes to $S$ dominantly. At ${Ro_g}\gtrsim 0.5$, the emergent unsteady flow significantly suppresses the asymmetry between the inflow and outflow strength, and weakens its influence on $S$.

A previous paper of the authors (Duck & Stephen, J. Fluid Mech., vol. 917, 2021, A56) considered the effect of three-dimensional, temporally periodic, linear and incompressible disturbances on a Blasius boundary layer, in particular when the disturbance wavelength is both comparable to and longer than the boundary-layer thickness. This previous study revealed that, unlike the two-dimensional counterpart, a mode exists that exhibits regimes of downstream spatial growth. In this paper we extend the analysis to the compressible regime, based on the boundary-region equations methodology. The aforementioned unstable mode is seen to persist into the compressible regime, and is studied using a combination of numerical and asymptotic methods. The paper adopts several approaches. First is a numerical approach in which the spatial development of the disturbances is assessed. This then leads to a consideration of the far-downstream behaviour, using (several) asymptotic limits. Of some note, in addition to unstable modes found in the incompressible case, is the existence of a further class of instability, not found in the incompressible case (which is also analysed asymptotically), corresponding to what amounts to an inviscid instability. The far-downstream analysis enables a (sub-)classification into entropy and non-entropy modes. The former, according to this analysis, are spatially damped, with one caveat, as revealed by our marching procedure, which highlights how spatial development of disturbances can be important.

We report the existence of two new limiting turbulent regimes in horizontal convection (HC) using direct numerical simulations at intermediate to low Prandtl numbers. In our simulations, the flow is driven by a step-wise buoyancy profile imposed at the surface, with free-slip, no-flux conditions along all other boundaries, except along the spanwise direction, where periodicity is assumed. The flow is shown to transition to turbulence in the plume and the core, modifying the rate of heat and momentum transport. These transitions set a sequence of scaling laws that combine theoretical arguments from Shishkina, Grossmann and Lohse (SGL) and Hughes, Griffiths, Mullarney and Peterson (HGMP). The parameter range extends through Rayleigh numbers in the range [$6.4\times 10^5, 1.92\times 10^{15}$] and Prandtl numbers in the range [$2\times 10^{-3},2$]. At low Prandtl numbers and intermediate Rayleigh numbers, a core-driven regime is shown to follow a Nusselt-number scaling with $Ra^{1/6}Pr^{7/24}$. For Rayleigh numbers larger than $10^{14}$, the Nusselt number scales with $Ra^{0.225}Pr^{0.417}$. For these particular regimes, the Reynolds number is found to scale as $Ra^{2/5}Pr^{-3/5}$ for the low-Prandtl-number regime and $Ra^{1/3}Pr^{1}$ for Rayleigh numbers larger than $10^{14}$. These results embed the HGMP model in the SGL theory and extend the known regime diagram of HC at high Rayleigh numbers. In particular, we show that HC and Rayleigh–Bénard share similar turbulent characteristics at low Prandtl numbers, where HC is shown to be ruled by its core dynamics and turbulent boundary layers. This new scenario confirms that fully turbulent HC enhances the transport of heat and momentum with respect to previously reported regimes at high Rayleigh numbers. This work provides new insights into the applicability of HC for geophysical flows such as overturning circulations found in the atmosphere, the oceans, and flows near the Earth's inner core.

Horizontal convection at large Rayleigh and Prandtl numbers is studied experimentally in a regime up to seven orders of magnitude larger in terms of Rayleigh numbers than previously achieved. To reach Rayleigh numbers up to $10^{17}$, the horizontal density gradient is generated using differential solutal convection by a differential input of salt and fresh water controlled by diffusion in a novel experiment in which the zero-net mass flux of water is ensured through permeable membranes. This set-up allows us to accurately measure the Nusselt number in solutal convection by carefully controlling the amount of salt water exchanged through the membranes. Combined measurements of density and velocity across more than five orders of magnitude in Rayleigh numbers show that the flow transitions from the Beardsley & Festa (J. Phys. Oceanogr., vol. 2, issue 4, 1972, pp. 444–455), Shishkina & Wagner (Phys. Rev. Lett., vol. 116, issue 2, 2016, 024302) regime to the Chiu-Webster et al. (J. Fluid Mech., vol. 611, 2008, pp. 395–426) regime and frames the present results within the scope of Shishkina et al. (Geophys. Res. Lett., vol. 43, issue 3, 2016, pp. 1219–1225), and the theory of Part 1 (Passaggia & Scotti, vol. 997, 2024, J. Fluid Mech., A5). In particular, we show that, even for large Prandtl numbers, the circulation eventually clusters underneath the forcing horizontal boundary, leaving a stratified core without motion. Finally, the previous regime diagrams (Hughes & Griffiths, Annu. Rev. Fluid Mech., vol. 40, 2008, pp. 185–208; Shishkina et al., Geophys. Res. Lett., vol. 43, issue 3, 2016, pp. 1219–1225) are extended by combining the present results at high Prandtl numbers, the results at low Prandtl numbers of Part 1, together with previous results from the literature. This work sets a new picture of the transition landscape of horizontal convection over six orders of magnitude in Prandtl number and sixteen orders of magnitude in Rayleigh number.

Standing acoustic waves have been known to generate Eulerian time-mean ‘streaming’ flows at least since the seminal investigation of Lord Rayleigh in the 1880s. Nevertheless, a recent body of numerical and experimental evidence has shown that inhomogeneities in the ambient density distribution lead to much faster flows than arise in classical Rayleigh streaming. The emergence of these unusually strong flows creates new opportunities to enhance heat transfer in systems in which convective cooling cannot otherwise be easily achieved. To assess this possibility, a theoretical study of acoustic streaming in an ideal gas confined in a rectangular channel with top and bottom walls maintained at fixed but differing temperatures is performed. A two time scale system of equations is utilized to efficiently capture the coupling between the fast acoustic waves and the slowly evolving streaming flow, enabling strongly nonlinear regimes to be accessed. A large suite of numerical simulations is carried out to probe the streaming dynamics, to highlight the critical role played by baroclinically generated wave vorticity and to quantify the additional heat flux induced by the standing acoustic wave. Proper treatment of the two-way coupling between the waves and mean flow is found to be essential for convergence to a self-consistent steady state, and the variation of the resulting acoustically enhanced steady-state heat flux with both the amplitude of the acoustic wave and the $O(1)$ aspect ratio of the channel is documented. For certain parameters, heat fluxes almost two orders of magnitude larger than those realizable by conduction alone can be attained.

The diffusiophoresis of charged hydrophobic nanoparticles (NPs) governed by an imposed ionic concentration gradient is analysed. The main objective is to elucidate the impact of the laterally mobile adsorbed surface ions at the interface on the propulsion of the hydrophobic NPs in diffusiophoresis. In addition, the dielectric polarization due to the difference in dielectric constant between the NPs and the suspension medium is also considered. The mobile surface ions create a friction as well as an electric force at the hydrophobic surface, which leads to a modification of the slip velocity condition and the slip length. We obtain an exact numerical solution of the governing electrokinetic equations in their full form by adopting a control volume formulation. The numerical model is supplemented by analytical solutions derived based on the Debye–Hückel linearization. We find that the lateral mobility of the surface ions obstruct the coions to diffuse from the higher concentration side to the lower concentration side, which results in a repulsive force to the particle leading to the occurrence of a negative mobility. Based on the numerical results and analytical solutions, we have shown that for a fully mobile surface charge, the diffusiophoresis of a hydrophobic NP is identical to the diffusiophoresis of a liquid droplet whose viscosity is related to the slip length of the hydrophobic particle. We establish that the dielectric polarization enhances the velocity of a hydrophobic particle, which has potential applications in the practical context.

The shoreline hazard posed by ocean long waves such as tsunamis and meteotsunamis critically depends on the fraction of energy transmitted across the shallow near-shore shelf. In linear setting, bathymetric inhomogeneities of length comparable to the incident wavelength act as a protective high-pass filter, reflecting long waves and allowing only shorter waves to pass through. Here, we show that, for weakly nonlinear waves, the transmitted energy flux fraction can significantly depend on the amplitude of the incoming wave. The basis of this mechanism is the formation of dispersive shock waves (DSWs), a salient feature of nonlinear evolution of long water waves, often observed in tidal bores and tsunami/meteotsunami evolution. Within the framework of the Boussinesq equations, we show that the DSWs efficiently transfer wave energy into the high wavenumber band, where reflection is negligible. This is phenomenologically similar to self-induced transparency in nonlinear optics: small amplitude long waves are reflected by the bathymetric inhomogeneity, while larger amplitude waves that develop DSWs blueshift into the transparency regime and pass through. We investigate this mechanism in a simplified setting that retains only the key processes of DSW disintegration and reflection, while the effects such as bottom dissipation and breaking are ignored. The results suggests that the phenomenon is a robust, order-one effect. In contrast, the increased transmission due to the growth of bound harmonics associated with the steepening of the wave is weak. The results of the simplified modelling are validated by simulations with the FUNWAVE-TVD Boussinesq model.

The coupling between advection and diffusion in position space can often lead to enhanced mass transport compared with diffusion without flow. An important framework used to characterize the long-time diffusive transport in position space is the generalized Taylor dispersion theory. In contrast, the dynamics and transport in orientation space remains less developed. In this work we develop a rotational Taylor dispersion theory that characterizes the long-time orientational transport of a spheroidal particle in linear flows that is constrained to rotate in the velocity-gradient plane. Similar to Taylor dispersion in position space, the orientational distribution of axisymmetric particles in linear flows at long times satisfies an effective advection–diffusion equation in orientation space. Using this framework, we then calculate the long-time average angular velocity and dispersion coefficient for both simple shear and extensional flows. Analytic expressions for the transport coefficients are derived in several asymptotic limits including nearly spherical particles, weak flow and strong flow. Our analysis shows that at long times the effective rotational dispersion is enhanced in simple shear and suppressed in extensional flow. The asymptotic solutions agree with full numerical solutions of the derived macrotransport equations and results from Brownian dynamics simulations. Our results show that the interplay between flow-induced rotations and Brownian diffusion can fundamentally change the long-time transport dynamics.

High angle of attack flows over swept three-dimensional wings based on the NACA 0015 profile are studied numerically at low Reynolds numbers. Linear stability analysis is used to compute instability and receptivity of the flow via the respective three-dimensional (triglobal) direct and adjoint eigenmodes. The magnitude of the adjoint eigenvectors is used to identify regions of maximum flow receptivity to momentum forcing. It is found that such regions are located above the primary three-dimensional separation line, their spanwise position varying with wing sweep. The wavemaker region corresponding to the leading global eigenmode is computed and found to lie inside the laminar separation bubble (LSB) at the spanwise location of peak recirculation. Increasing the Reynolds number leads to the wavemaker becoming more compact in the spanwise direction, and concentrated in the top and bottom shear layers of the LSB. As sweep is introduced, the wavemaker moves towards the wing tip, following the spanwise displacement of maximum recirculation. Flow modifications resulting from application of different types of forcing are studied by direct numerical simulation initialised with insights gained from stability analysis. Periodic forcing at the regions of maximum receptivity to momentum forcing results in greater departure from the baseline case compared to same (low, linear) amplitude forcing applied elsewhere, underlining the potential of linear stability analysis to identify optimal regions for actuator positioning.

Flow resistance reduction, quantified as a change in flow rate with respect to a reference isothermal flow driven by the same pressure gradient, is realizable in a channel flow using a thermal wave applied on the bounding wall. Countercurrent waves provide a resistance-reducing effect at any wave velocity, Reynolds number and wavenumber considered. Cocurrent waves can reduce resistance only if the wave velocity is lower than a certain threshold, and the Reynolds number is larger than a certain threshold, otherwise, such waves increase resistance. The increase of the wave amplitude increases resistance reduction and resistance increase up to a specific limit. It is possible to reduce resistance up to 20 times compared with the isothermal channel using proper waves. It is shown that the same effect is achieved regardless of the waves applied at the upper and lower walls. The wave-modified flows are shown to be stable for the conditions used in this study.

This work examines the control of cross-flow instabilities (CFIs) and laminar–turbulent transition on a swept wing, through the plasma-based base flow modification (BFM) technique. The effect of experimentally derived plasma body forces on the steady boundary layer base flow is explored through numerical simulations. Linear stability theory is subsequently used to predict the net BFM effect on CFIs. Based on these preliminary predictions, experiments are conducted in a low-turbulence wind tunnel where a spanwise-invariant plasma actuator is installed near the wing leading edge and operated at constant input voltage and frequency. Various flow parameters governing the plasma-based BFM technique are investigated, namely the Reynolds number, angle of attack and wavelength of excited stationary CFI modes. Stationary and travelling CFIs are quantified by planar particle image velocimetry while the transition topology and location are recorded by infrared thermography. The results confirm the stabilising effect of BFM on the swept-wing boundary layer. However, the plasma-based BFM is found to render the boundary layer more susceptible to travelling CFIs. In the presence of both net BFM effect and intrinsic plasma unsteady perturbations, the plasma-based BFM technique achieves transition delay with specific combinations of Reynolds number, angle of attack and wavelength of excited stationary CFI modes. The present findings provide insights into the fundamental principles of operating plasma actuators within the context of BFM control.

Scale dependence of local shearing motion is investigated experimentally in decaying homogeneous isotropic turbulence generated through multiple-jet interaction. The turbulent Reynolds number, based on the Taylor microscale, is between approximately 900 and 400. Velocity fields, measured using particle image velocimetry, are analysed through the triple decomposition of a low-pass filtered velocity gradient tensor, which quantifies the intensities of shear and rigid-body rotation at a given scale. These motions manifest predominantly as layer and tubular vortical structures, respectively. The scale dependence of the moments of velocity increments, associated with shear and rigid-body rotation, exhibits power-law behaviours. The scaling exponents for shear are in quantitative alignment with the anomalous scaling of the velocity structure functions, suggesting that velocity increments are influenced predominantly by shearing motion. In contrast, the exponents for rigid-body rotation are markedly smaller than those predicted by Kolmogorov scaling, reflecting the high intermittency of rigid-body rotation. The mean flow structure associated with shear at intermediate scales is investigated with conditional averages around locally intense shear regions in the filtered velocity field. The averaged flow field exhibits a shear layer structure with aspect ratio approximately 4.5, surrounded by rotating motion. The analysis at different scales reveals the existence of self-similar structures of shearing motion across various scales. The mean velocity jump across the shear layer increases with the layer thickness. This relationship is well predicted by Kolmogorov's second similarity hypothesis, which is useful in predicting the mean characteristics of shear layers across a wide range of scales.

We conducted an axisymmetric numerical study of drop impact on a thin film of the same liquid in order to generate maps identifying the fluid elements in the drop and film that are transferred to the corolla during impact. We find that mass contribution from the drop comes from a surprisingly thin surface layer on the drop, and furthermore, that the shape of this layer in the drop and the film scales with film thickness, not the Weber number and Reynolds number as one might expect. The maps could be used to tailor drop composition for applications such as coatings or encapsulations.

The present study investigates streamwise ($\overline {u^2}$) energy-transfer mechanisms in the inner and outer regions of turbulent boundary layers (TBLs). Particular focus is placed on the $\overline {u^2}$ production, its inter-component and wall-normal transport as well as dissipation, all of which become statistically significant in the outer region with increasing friction Reynolds number ($Re_{\tau }$). These properties are analysed using published data sets of zero, weak and moderately strong adverse-pressure-gradient (APG) TBLs across a decade of $Re_{\tau }$, revealing similarity in energy-transfer pathways for all these TBLs. It is found that both the inner and outer peaks of $\overline {u^2}$ are always associated with local maxima in the $\overline {u^2}$ production and its inter-component transport, and the regions below/above each of these peaks are always dominated by wall-ward/away-from-wall transport of $\overline {u^2}$, thereby classifying the $\overline {u^2}$ profiles into four distinct regimes. This classification reveals existence of phenomenologically similar energy-transfer mechanisms in the ‘inner’ and ‘outer’ regions of moderately strong APG TBLs, which meet at an intermediate location coinciding with the minimum in $\overline {u^2}$ profiles. Conditional averaging suggests existence of similar phenomena even in low $Re_{\tau }$ canonical and/or weak APG TBLs, albeit with the outer-region mechanisms weaker than those in the inner region. This explains the absence of their $\overline {u^2}$ outer peak and the dominance of $\overline {u^2}$ wall-normal transport away from the wall, which potentially originates from the inner region. Given that the wall-ward/away-from-wall transport of $\overline {u^2}$ is governed by the $Q_4$(sweeps)/$Q_2$(ejections) quadrants of the Reynolds shear stress, it is argued that the emergence of the $\overline {u^2}$ outer peak corresponds with the statistical dominance of $Q_4$ events in the outer region. Besides unravelling the dynamical significance of $Q_2$ and $Q_4$ events in the outer region of TBLs, the present analysis also proposes new phenomenological arguments for testing on canonical wall-turbulence data at very high $Re_{\tau }$.

We use multiphase direct numerical simulations to identify, analyse and quantify components of wall-normal heat flux distributions in evaporative vertical falling films with surface modifications at industrially relevant conditions. Previous experiments showed a potential increase of the heat transfer rate through the film by up to 100 % using various types of modifications. We show that the modifications induce significant advective heat transport and hypothesise that four synergistic mixing mechanisms are behind the heat transfer rate improvement. Additionally, we examine how the important surface topology parameters, pitch $\hat {p}$ (distance between modifications), height $\hat {h}$ and the liquid Prandtl number $\mathit {Pr}_l$, influence the mode of heat transport and the Nusselt number $\mathit {Nu}$. We show that $\hat {p}/\hat {h} \approx 10$ maximises $\mathit {Nu}$ and that the optimal pitch is related to the recirculation zone length $L_r$ behind the modification. We find that $L_r/\hat {h} \approx 3.5$ and that $\mathit {Nu} \propto \mathit {Pr}_l^{0.42}$ in the investigated parameter ranges. We also show that all our cases on both smooth and modified surfaces have $\mathit {Pe}_l \gg 1$ and collapse well on a line $\mathit {Nu} \propto (\mathit {Pe}_l/\mathit {Re})^{0.35}$. This relation suggests that $\mathit {Nu}$ is governed by the balance of film mixing, thermal resistance and diffusivity, and that the ratio $\mathit {Pe}_l/\mathit {Re}$ can be used to estimate $\mathit {Nu}$. Our methodology and findings extend the knowledge concerning the mechanisms behind the heat transfer improvement due to surface modifications and facilitate guidelines for designing more efficient modified surfaces in industrial evaporators.

In this paper it is shown that a modal detuned instability of periodic near-wall streaks originates a large-scale structure in the bulk of the turbulent channel flow. The effect of incoherent turbulent fluctuations is included in the linear operator by means of an eddy viscosity. The base flow is an array of periodic two-dimensional streaks, extracted from numerical simulations in small domains, superposed to the turbulent mean profile. The stability problem for a large number of periodic units is efficiently solved using the block-circulant matrix method proposed by Schmid et al. (Phys. Rev. Fluids, vol. 2, 2017, 113902). For friction Reynolds numbers equal or higher than $590$, it is shown that an unstable branch is present in the eigenspectra. The most unstable eigenmodes display large-scale modulations whose characteristic wavelengths are compatible with the large-scale end of the premultiplied velocity fluctuation spectra reported in previous computational studies. The wall-normal location of the large-wavelength near-wall peak in the spanwise spectrum of the eigenmode exhibits a power-law dependence on the friction Reynolds number, similarly to that found in experiments of pipes and boundary layers. Lastly, the shape of the eigenmode in the streamwise-wall-normal plane is reminiscent of the superstructures reported in the recent experiments of Deshpande et al. (J. Fluid Mech., vol. 969, 2023, A10). Therefore, there is evidence that such large-wavelength instabilities generate large-scale motions in wall-bounded turbulent flows.

Many natural and industrial processes involve the flow of fluids made of solid particles suspended in non-Newtonian liquid matrices, which are challenging to control due to the fluid's nonlinear rheology. In the present work, a Taylor–Couette canonical system is used to investigate the flow of dilute to semi-dilute suspensions of neutrally buoyant spherical particles in highly elastic base polymer solutions. Friction measurement synchronized with direct flow visualization are combined to characterize the critical conditions for the onset of elasto-inertial instabilities, expected here as a direct transition to elasto-inertial turbulence (EIT). Adding a low particle volume fraction (${\leq }2\,\%$, dilute regime) does not affect the nature of the primary transition and reduces the critical Weissenberg number for the onset of EIT, despite a significant decrease in the apparent fluid elasticity. However, for particle volume fractions ${\geq }6\,\%$ (semi-dilute regime), EIT is no longer observed in the explored Reynolds range, suggesting an apparent relaminarization with yet not further decrease in fluid elasticity. Instead, a new regime, termed here elasto-inertial dissipative (EID), was uncovered. It originates from particle–particle interactions altering particle–polymer interactions and occurring under elasto-inertial conditions comparable to those of EIT. Increasing particle volume fraction in the semi-dilute regime and, in so, the particle contribution to the overall viscosity, delays the onset of EID similarly to what was observed previously for EIT in lower elasticity fluids. After this onset, a decrease in the pseudo-Nusselt number observed with increasing inertia and particle-to-polymer concentration ratio confirms a particle-induced alteration of energy transfer in the flow.

In the present work, the tangential (swirl) velocity component is superimposed at the intake of a narrow fluidic cylindrical pipe to achieve the desired mixing of inelastic non-Newtonian fluids/solutes at the outlet. We discuss an analytical method for obtaining the swirl velocity profile, considering the nonlinear viscous effects for both shear-thinning and shear-thickening fluids, represented by the power-law model. We numerically solve the species transport equation, coupled with the analytically derived swirl velocity, using our in-house developed code for the concentration distribution in the flow field. The results show that the inlet swirl and an increase in the shear-thinning fluid property improve advection-dominated mixing. Additionally, higher Reynolds numbers significantly enhance advection's dominance, as more rotation leads to the generation of vortices, resulting in an engulfment flow (chaotic convection) based mixing. We demonstrate that considering the increase in the shear-thinning fluid property with swirl intake reduces the amount of mixing time required in the convective regime.