We extend the linear stability analysis of electroconvective instability in a quiescent Oldroyd-B viscoelastic fluid subjected to an electric field into the strongly elastic regime, up to Weissenberg numbers ($\textit{Wi}$) of order $10^{3}$. The oscillatory onset is governed by the interplay among destabilising Coulomb forcing, viscous dissipation and elastic stress relaxation. For the quiescent base state, the linearised constitutive relation allows the polymer stress to be represented through a complex effective viscosity. We then apply a single-mode Galerkin projection based on the Newtonian critical eigenfunction, obtaining a reduced dispersion relation and a closed-form prediction for the neutral oscillation frequency. In the limit of large $\textit{Wi}$, this analytical solution yields two asymptotic scaling laws, namely a frequency scaling law $\omega _{c} \propto \sqrt {(1-\beta )/\textit{Wi}}$ ($\omega _{c}$ is the critical frequency and $\beta$ is the solvent to total viscosity ratio) and a threshold scaling law $T_{c} = \beta T_{c}^{N} + a(\beta )/\textit{Wi}$ ($T_{c}$ is the critical electric Rayleigh number, $T_{c}^{N}$ is the Newtonian critical electric Rayleigh number and $a(\beta )$ is the coefficient related to $\beta$), which satisfy a self-consistency relation. Here, $T_{c}$ exhibits a three-stage evolution with $\textit{Wi}$, and a frozen-elasticity limit is identified at large $\textit{Wi}$. With a single geometric constant as input, the values of $\omega _{c}$ and $T_{c}$ obtained from the analytical solution and from high-resolution spectral computations agree to within 0.3 %, providing mutual corroboration.

We present a theory for the hydrodynamic forces in the drainage of an electrolyte film confined between charged curved surfaces, which applies to small gaps under local equilibrium across the film thickness. We derive a local Onsager formalism describing the coupled flow rate, charge and solute fluxes, which accounts for all degrees of overlap of the electrical double layers (EDLs) and can be adapted to a variety of electrical, chemical or hydrodynamic boundary conditions. We explore in detail the case of a monovalent electrolyte with constant surface charge and no-slip boundary conditions. A complete phase diagram is established for the electro–kinetic damping, including non-overlapping EDLs at both high and low Dukhin numbers, as well as all degrees of EDLs overlap including the Gouy–Chapman, Debye–Hückel and ideal-gas domains. We highlight a simple relation between the apparent hydrodynamic distance at high Dukhin number and the surface electrical conductance. Our theory also predicts a diffusio–kinetic contribution to the so-called electroviscous effect, which does not reduce to a damping but exhibits a conservative (stiffness) component. We show that this stiffness decays as an algebraic function of the gap between the surfaces and may dominate the stiffness due to the equilibrium Derjaguin–Landau–Verwey–Overbeek interaction force.

We investigate the dynamics of a rigid spherical particle in plane Couette flow with a viscosity that varies linearly across the channel, from the Stokes limit to non-zero Reynolds numbers, using analytical theory and immersed-boundary lattice Boltzmann simulations. In the Stokes limit, we find that the particle develops a constant, viscosity-gradient-induced translational slip, defined as the difference between the particle velocity and the undisturbed fluid velocity at its centre. The particle leads the flow when the viscosity increases towards the moving wall and lags when it decreases. Its angular velocity equals half the local, viscosity-gradient-modified fluid vorticity, experiencing no additional rotational slip. Furthermore, we show that cross-stream particle migration is strictly prohibited at any confinement ratio or gradient magnitude. At non-zero Reynolds numbers in a constant-viscosity fluid, increasing the channel Reynolds number beyond a critical value induces a supercritical pitchfork bifurcation that splits the stable centreline equilibrium into two symmetric off-centre equilibria. With a viscosity gradient, we find that for Reynolds numbers below this bifurcation point, the particle consistently settles at a single off-centre equilibrium in the less-viscous half. For Reynolds numbers exceeding this point, however, we identify a critical viscosity gradient threshold: below this limit, the particle can stabilise in either channel half depending on its initial placement; above it, the particle focuses exclusively in the less-viscous half regardless of its initial position. These findings enable predictive particle manipulation in viscosity-stratified microflows.

Small-scale shear layers in temporally developing turbulent boundary layers are investigated using direct numerical simulations at momentum-thickness Reynolds numbers from 2000 to 13 000. Flow statistics are evaluated in a local coordinate system aligned with each shear layer identified by the shear intensity. Conditional mean fields show that shear layers are embedded in a biaxial strain. Near the wall, these conditional fields further reveal two distinct regions of intense rigid-body rotation beneath the shear layer, leading to momentum transfer through a mechanism consistent with counter-rotating vortex pairs. Two distinct scalings are found for the shear layers. In the logarithmic and outer layers, the mean shear-related velocity jump $\Delta u_3$ and its width $\delta _3$ scale with the Kolmogorov velocity $u_{\eta T}$ and length $\eta _T$ evaluated in the turbulent region. In the buffer layer, $\delta _3$ scales with the viscous length $\delta _\nu =\nu /u_\tau$ ($\nu$ is the kinematic viscosity and $u_\tau$ the friction velocity), and $\Delta u_3$ scales with $\delta _\nu /\tau _{\eta T}$, consistent with the predominance of shear in energy dissipation, where $\tau _{\eta T}$ is the Kolmogorov time scale. Velocity jumps associated with the biaxial strain exhibit Kolmogorov-based scaling in the logarithmic and outer layers. In the buffer layer, their widths scale with $\delta _\nu$, but their velocity jumps do not scale with $\delta _\nu /\tau _{\eta T}$ because strain contributes little to dissipation. Comparisons with turbulent jets indicate weak wall influence in the logarithmic and outer layers. These scalings provide a basis for predicting shear-layer properties, which is crucial for developing flow-control strategies that leverage small-scale shear instability.

Large-eddy simulations of open-channel flow are conducted to investigate how the channel aspect ratio (), defined as the width ($B$) to depth ($h$) ratio (), affects secondary currents (SCs), large-scale motions (LSMs) and very-large-scale motions (VLSMs). Four cases spanning are considered. For wide-channel flows (), certain flow features dominate depending on the distance from the sidewall, which are: (i) secondary current instability (SCI) within ${\sim} 0.6h$ of the sidewalls; (ii) VLSMs and LSMs from distance $2.4h$ onwards from the sidewalls; and (iii) prevalence of LSMs between (i) and (ii). In the SCI-dominated region (i), two time-averaged SC cells appear, one near the bottom wall–bed corner (BV) and another near the surface (SV). The size and position of the BV and SV oscillate quasi-periodically with a wavelength ${\sim} 6h$, reflecting the instability of the secondary currents. In the VLSM/LSM-dominated region (ii), pre-multiplied spectra of the streamwise velocity fluctuations peak at ${\sim} 3h$ and ${\sim} 20h$, consistent with the presence of LSMs and VLSMs. In the LSM-dominated region (iii), velocity spectra exhibit a LSM prevalent peak at ${\sim} 3h$. The near-sidewall SCI region (i) and central VLSM/LSM region (ii) are phase independent in their oscillations, likely due to a barrier effect of LSM-dominated region (iii) between them. For , the VLSM/LSM region (ii) disappears, while the other two regions exhibit distinct alterations. At , the SCI-dominated region resembles that for , while VLSMs in the LSM-dominated region have a remarkably shorter wavelength (${\sim} 10h$). At a lower (), VLSMs are completely absent in the LSM-dominated region. Based on these results, a schematic framework of the interrelations between SCs and VLSMs across different channel is proposed.

We consider the dynamics of two equal, spherical liquid drops of Newtonian fluid, brought into contact at a negligible speed, and merging driven by surface tension. The effects of an outer fluid or that of gravity are disregarded, so that the entire motion is determined uniquely by the Ohnesorge number $\textit{Oh }= \eta /\sqrt {\rho \gamma R}$, where $\eta$ is the viscosity, $\rho$ the density, $\gamma$ the surface tension and $R$ the drop radius. Using methods of matched asymptotics, in the limit that the minimum radius $r_0$ of the bridge connecting the drops is much smaller than the initial drop radius, we compute the dynamics for $r_0(t)$, as well as the shape of the gap between the drops. For very early times, the translational motion of the drops along the axis of symmetry is arrested by inertia, which affects the shape of the scaling function describing the gap on the scale of $r_0$. For $r_0/R \approx \sqrt {\ln \textit{Oh}}/\textit{Oh}$, the motion crosses over to an entirely viscous similarity solution, passing through a sequence of time-dependent scaling functions, which we compute. Our results agree well with full numerical simulations of the Navier–Stokes equation, as well as experiment.

We investigate the transport and adsorption of solutes within graded porous filters characterised by a spatially varying microstructure. While classical homogenisation theory typically assumes periodic media, we employ the method of multiple scales to derive an effective macroscopic model for ‘near-periodic’ geometries where the porosity varies slowly over the longitudinal coordinate. A key novelty of this work is the departure from the standard solenoidal constraint; instead, we introduce a modified incompressibility condition derived from non-equilibrium thermodynamics that accounts for the coupling between the solute concentration and the solvent velocity. This leads to a generalised Darcy-scale description where the fluid velocity field is non-solenoidal within the porous domain. Through asymptotic analysis, we determine the leading-order concentration profiles and quantify first-order corrections that capture the interplay between the porosity gradient and the mixture composition. We evaluate filter performance across several metrics, including outflux concentration and total adsorption rate, under both fixed-flow and fixed-pressure-drop operating conditions. Our results demonstrate that the porosity gradient and the coupling parameter significantly influence the filtration efficiency, particularly as the medium approaches the clogging limit. The analysis reveals that the optimal filter design is highly sensitive to the chosen performance metric, highlighting the necessity of physically consistent boundary conditions and mixture dynamics in the design of high-efficiency graded filters.

This study investigates how the impact of an internal heat source and anisotropic thermal diffusivity influences convective motion in a horizontally inclined porous medium saturated with fluid, assuming an adiabatic bottom boundary and an isothermal top boundary. Here, the momentum equation is formulated using the Darcy–Forchheimer model and assumes that the viscosity is a linear function of the temperature. Linear and nonlinear stability analyses have been conducted to determine the stability by calculating the critical Rayleigh number and the associated wavenumber. For nonlinear stability, conditional stability has been established for all initial data with higher-order norms, showing that the $L^2$ theory is insufficient to achieve similar stability, where $L^2$  represents Hilbert space over the domain  $\omega$. It has also been shown that the conditional stability bound can be attained by restricting the internal heat source parameter Q using higher-order norms. The Chebyshev tau-QZ algorithm is used to solve the generalised eigenvalue problem, which is obtained by employing the normal mode technique on the disturbance equations. The effect of various dimensionless parameters, such as anisotropic thermal diffusivity parameters $(\xi )$, Rayleigh number (Ra), angle of inclination ($\phi$) and viscosity parameter $(\varGamma )$, on the stability of the medium is investigated, and streamlines are also shown by contour plots to depict the flow pattern. The results show that the system is stable when $\xi \gt 1$ and unstable when $\xi \lt 1$, favouring convection. It is also observed that the convection is delayed with rising values of $\phi$, and the region of subcritical instability increases uniformly with rising values of $\phi$.

A body-forcing strategy is proposed to model the effects of surface roughness on heat transfer in a turbulent flow. The model introduces a source term into the temperature transport equation that depends on a characteristic velocity scale, the local wall-relative temperature and the roughness density. We test it in direct numerical simulation (DNS) of a turbulent open channel at a constant Prandtl number of 0.71, and through extensive comparison with roughness-resolved DNS data, we show that the instantaneous local friction velocity is the appropriate velocity scale. Despite its simple formulation, it accurately captures the salient physics of rough-wall heat transfer, such as the breakdown of the Reynolds analogy, the characteristic plateau of the temperature roughness function and the fully rough behaviour of the ratio of roughness lengths of momentum and heat transfer. An appropriate decomposition of the total heat flux reveals that the scaling of the conductive and body-forcing Stanton numbers is similar to that of the exposed and sheltered Stanton numbers from fully resolved roughness simulations, respectively. This suggests that the primary effect of roughness on heat transfer can be effectively captured so long as the surface heat flux is modelled correctly in a statistical sense, even if the precise contribution from individual roughness elements is not accurately resolved. Finally, the body-forcing model can be used over regular no-slip smooth walls, bypassing the need to resolve the exact details of the roughness geometry.

Symmetry-breaking in the near-wake of a wall-proximity sphere is a claussic problem. Yet a clear understanding of how the gap ratio ($G/D$) governs the evolution and suppression of spanwise flow asymmetry across different Reynolds numbers ($Re$) is still lacking. Here, direct numerical simulations combined with proper orthogonal decomposition and vorticity transport analysis are employed over $420 \leqslant Re \leqslant 600$ and $0.1 \leqslant G/D \leqslant 1.5$ to elucidate the wall-induced suppression of spanwise asymmetry. As $G/D$ decreases, four regimes are identified: asymmetric unsteady II, asymmetric unsteady I, symmetric unsteady and symmetric steady. Notably, the spanwise wake deflection reverses direction between the two asymmetric regimes, signalling a structural transition regulated by vortical competition. This corresponds to the transition from the alternating shedding of upper and lower hairpin vortices to the formation of longitudinal vortices (LV) that interact with wall-secondary vortices (WSV). Symmetry restoration is associated with merging and the emergence of the second harmonic of the dominant shedding frequency. Crucially, the wall effect homogenises the streamwise pressure gradient and velocity-gradients in the spanwise direction. This gradient homogenisation attenuates the asymmetric tilting of spanwise vorticity, thereby reducing the feedback that maintains the spanwise asymmetry of the shear layer. In asymmetric regimes, wake deflection is sustained by dominant spanwise-asymmetric vorticity tilting into streamwise and wall-normal components. In the symmetric regime, suppression of the spanwise deflection of the shear layer supports wake-symmetry recovery. These findings provide insights into symmetry-breaking in wall-proximity bluff-body wakes, and offer guidance for active flow control.

Wall-bounded turbulent flows are chaotic and multiscale, rendering fast prediction at high Reynolds numbers computationally prohibitive in applications such as wind farms. Classical data assimilation is based on repeated solutions of the governing equations and, thus, inherits this cost. Generative models learn the probability distribution of flow states, enabling scalable probabilistic reconstruction. Our generative framework couples a variational autoencoder with a diffusion transformer to generate four-dimensional spatiotemporal samples. Bayesian conditioning enables data assimilation without retraining and allows statistical constraints to be imposed through sampling. The framework is applied to a subdomain of turbulent plane Couette flow, where the corresponding direct numerical simulations in this generation region requires $O(10^6)$ spatial degrees of freedom. Using $O(10)$ latent spatial degrees of freedom, the model achieves a compression ratio of $O(10^5)$, which is one to two orders of magnitude above prior reports. It reproduces single-point statistics up to fourth order and the energy spectra, as well as the intermittency and phase-sensitive structure captured by velocity-increment and $Q$–$R$ statistics. Two assimilation scenarios demonstrate that, when observations are statistically consistent with the prior, conditional diffusion models with the proposed sampling strategy preserve complex turbulent statistics in the posterior. However, enforcing these constraints while preserving physical fidelity and sample diversity introduces an inherent trade-off. Excessive conditioning can distort the learned prior, paralleling limitations of classical ensemble-based data assimilation, where this can likewise degrade the prior covariance. These results highlight both the promise of diffusion models as probabilistic surrogates and the challenges of conditioning them.

The Maxey–Riley–Gatignol equation for the flow around a sphere at low particle Reynolds number tells us that the fluid–particle interaction force decomposes into a contribution from the local flow disturbance caused by the particle’s boundary – consisting of the drag, virtual-mass and history forces, and their Faxén corrections – and another contribution from the stress of the background flow, termed generalized-buoyancy force. There is also a consensus that, for general disperse two-phase flow, the interfacial force density, coupling the average fluid phase and dispersed-phase momentum balances, decomposes in a like manner. However, there has been a long-standing controversy about the physical closure separating the generalized buoyancy from the interfacial force density, especially whether or not pseudostresses, such as the Reynolds stress, should be attributed to the background flow. Furthermore, most existing propositions for this closure involve small-particle approximations. Here, we show that all existing buoyancy closures are inconsistent with particle-resolving numerical simulations and/or at least one of two simple thought experiments designed to determine the roles of pseudostresses and small-particle approximations. We then derive the unique consistent closure. It requires no approximation and implies that all stresses and pseudostresses in the average fluid phase momentum balance, except the Reynolds stress, fully contribute to the background flow responsible for buoyancy. Remarkably, it exhibits a low-pass filter property, attenuating buoyancy at short wavelengths, that prevents it from causing Hadamard instabilities, constituting a first-principles-based solution to the long-standing ill-posedness problem of two-fluid models. When employing the derived closure, even simplistic two-fluid models are linearly well-posed.

Model-based active flow control requires predictive models that are accurate, stable and fast enough for real-time optimisation. In controlled wake flows, this is often achieved through reduced-order models (ROMs) that first compress high-dimensional velocity snapshots into a latent space and then learn a time-stepping predictor for the dynamics in the latent space. Here, we study how the choice of the spatial encoder affects the predictability of the resulting latent coordinates for wake flows under control inputs. Using two actuated two-dimensional wake configurations, a simplified truck wake and the fluidic pinball, we compare proper orthogonal decomposition (POD) against nonlinear convolutional autoencoders (CAEs) and two types of variational autoencoders for compression, and evaluate several temporal predictors based on long short-term memory networks. The CAEs achieve higher compression efficiency and sharper short-term reconstructions, but they produce a latent dynamics that is more irregular and with broadband spectral content. As a consequence, long-horizon forecasts degrade faster and show a higher probability of catastrophic divergence than POD-based models. The POD yields smoother latent trajectories that are easier to learn and extrapolate, leading to more reliable predictions beyond the short-term regime. These results reveal a clear trade-off between compactness and forecast accuracy, and suggest that the stability of the latent dynamics prediction can outweigh maximal compression. This is particularly relevant for control strategies rooted in forecasts of the dynamics, such as model predictive control and reinforcement learning. The findings provide practical guidance for designing actuation-aware, hardware-feasible predictive ROMs for real-time flow control.

Mean flow and instability measurements of a nominally axisymmetric, transitional separation bubble arising from a flare-induced shock-wave/boundary-layer interaction were conducted on a cone–flare geometry in hypersonic flow. Simultaneous infrared thermography and high-speed, high-resolution background-oriented schlieren measurements provided a global, time-resolved characterisation of both the mean and unsteady flow. Experiments were performed in the AFOSR–Notre Dame Large Mach-6 Quiet Tunnel under conventional noise at free-stream unit Reynolds numbers ranging from $5.8\times 10^6$ to $12.3\times 10^6$ m$^{-1}$. The model comprised a $7^\circ$ half-angle circular cone, a $20^\circ$ half-angle flare and three nose radii. Complementary laminar and turbulent base flows, convective and global stability analyses (GSAs) and direct numerical simulations (DNSs) were carried out at the experimental conditions. With inflow forcing of disturbances, the DNS reproduces the experimentally observed flare-heating trends. The flow exhibits both convective and global instabilities, with the latter giving rise to stationary and oscillatory three-dimensional disturbances within the recirculation bubble. Thermal streaks observed in global heating measurements provide experimental evidence of these instabilities, which is further corroborated by the onset of three-dimensionality in DNS. Quantitative agreement of streak wavelengths and their spatial prevalence across experiments, GSAs and DNSs confirms the recirculation bubble’s global instability is the primary driver of the reattachment streaks in this flow, providing direct experimental evidence of this mechanism.

Due to attractive inter-particle forces, cohesive particles suspended in turbulence undergo a complex process of aggregation, breakup and restructuring. Despite a growing body of knowledge on the `flocculation’ of cohesive granular materials suspended in homogeneous isotropic turbulence, little focus has so far been placed on wall-bounded flows where turbulence and shear are inhomogeneous. This study presents a first investigation of a fully developed wall-bounded flow of resolved cohesive particles. Five direct numerical simulations of turbulent channel flows laden with finite-sized particles at successively increasing cohesive strength are performed. A population balance equation (PBE) framework is used to analyse aggregate dynamics. When integrated over the full domain, the PBE is closed by aggregation and breakup alone. However, this balance is found to not hold locally in the wall-normal direction, where regions of net aggregate production and depletion are identified. This imbalance is shown to be compensated by the size-dependent wall-normal transport of aggregates, revealing a mean circulation: larger aggregates are preferentially produced in the channel centre and migrate toward the wall where they break, while smaller aggregates are transported away from the wall, grow and re-enter the cycle.

A novel experimental finding of vortex splitting and merging (VSM) route to turbulence is reported in the counter-rotation regime of the Taylor–Couette flow of a Newtonian fluid, and the role of neutrally buoyant inertial particles on this transition pathway is analysed up to a particle volume fraction of $\phi = 0.3$ over a range of rotation ratios $-0.35\leqslant \varOmega =\omega _o/\omega _i\lt 0$ (where $\omega _i$ and $\omega _o$ are the angular speeds of the inner and outer cylinders, respectively). The onset of VSM is found to coincide with the transition of a quasi-periodic state (‘doubly’ modulated wavy vortex [DMWV], characterised by two incommensurate frequencies) to a chaotic state (chaotic wavy vortex [CWV], in the background of VSM) at $\phi \leqslant 0.15$, giving rise to a new state of ‘VSM + CWV’ in which the VSM lives within a distinct CWV state. With further increasing particle loading, however, the VSM appears via MWV$\to$DMWV$\to$CWV$_{\mathrm{B}}\to$VSM + CWV$\to$CWV at $\phi = 0.2$ and is found to be absent at $\phi = 0.3$; another variant (CWV$_{\mathrm{B}}$) of CWV, characterised by an additional broad low-frequency peak, bifurcates from the quasi-periodic (DMWV) state at $\phi \geqslant 0.2$. The VSM events are shown to be temporally uncorrelated, with a multimodal skewed probability density function for the interevent time interval, implying that the VSM-induced chaos (VSM + CWV) is stochastic. The final transition to turbulent Taylor vortex (TTV) is found to occur via a propagating state of turbulent wavy vortices (TWV). The power spectrum of TTV is flat and featureless with a broad noisy tail, whereas the flat spectrum of TWV contains a sharp peak that represents a large-scale wavy vortex mode, whose disappearance marks the onset of TTV. It is shown that the propagation frequency of the TWV mode decreases with increasing shear Reynolds number (${\textit{Re}}_s(\phi )$) and follows a power-law scaling in terms of ${\textit{Re}}_s(\phi )$ and the relative viscosity of the suspension. Our results establish that the Ruelle–Takens route to chaos (Ruelle & Takens, Commun. Math. Phys., 1971, vol. 20, pp. 167–192) holds in counter-rotating suspension Taylor–Couette flow over a range of particle loading $\phi \leqslant 0.3$ at $-0.35\leqslant \varOmega \lt 0$.

Transonic shock buffet is a typical flow instability induced by shock-wave/boundary-layer interaction. It exhibits sustained shock oscillations and lift fluctuations, thereby restricting the flight envelope of aircraft. Previous numerical studies have revealed that evident hysteresis envelopes exist between the onset and extinction boundaries of transonic buffet during angle-of-attack sweeps. Hence accurate characterisation of buffet boundaries under representative dynamic manoeuvres is critical for assessing flight safety and expanding the flight envelope. In this study, a time-varying autoregressive with exogenous input model is established by integrating reduced-order and time-varying modelling to predict the buffet boundary under nonlinear transonic dynamic conditions. Specifically, the nonlinear flow evolution during ascending and descending angle-of-attack sweeps is modelled via piecewise linearisation, and the evolutionary trajectory of system stability is derived from state-space representation to characterise when buffet occurs. The influences of pitching rate and sweep direction on the hysteresis characteristics of the buffet onset boundary are investigated. For rate-dependent hysteresis, a lower pitching rate advances the buffet onset boundary under ascending angle-of-attack sweep, and the boundary gradually converges as the pitching rate further decreases. For path-dependent hysteresis, the ascending sweep yields a higher buffet onset angle of attack than the descending sweep, and such discrepancy diminishes gradually with the reduction of pitching rate. In summary, lowering the pitching rate can effectively suppress both types of dynamic hysteresis characteristics. The proposed method offers engineering guidance for monitoring and early warning of time-varying stability in aerospace and mechanical vibration applications.

We study stirring and dispersion in steady flow through two-dimensional heterogeneous porous media. A fluid is stirred by the porous medium structure that leads to spatial flow variability and the deformation of fluid elements. While dispersion measures the extension of a solute distribution, stretching underpins solute mixing, chemical reactions and biological activity. Despite this, much less is known about fluid stretching in porous media. Its relation to the dispersion process on the one hand and the flow and medium properties on the other remain open questions. To close this gap, we probe the heterogeneity mechanisms that are at the origin of stretching and dispersion of fluid elements, and quantify the upscaled laws that govern their dynamics. Thus we perform detailed numerical simulations of Darcy flow in two-dimensional porous media characterised by a spatially varying hydraulic conductivity. Dispersion is analysed in terms of first-passage time distributions of fluid elements and the temporal evolution of their displacement mean and variance, while deformation is studied in terms of the probability density function and average elongation of fluid elements. We find that the dispersion and stretching behaviours are closely related. The exponents that govern the decay of the first-passage time distributions and evolution of the displacement variances also determine the evolution of the mean elongation. This dynamics is quantified in a single upscaled theoretical framework that bridges between Eulerian flow structure, large-scale dispersion and stretching of fluid elements.

Flow separation in rocket nozzles generates intense side loads that can lead to structural damage affecting space flight safety. Recent studies have hypothesised that a screech-like resonance loop may be responsible for such oscillations in the free shock separation (FSS) regime. In this work, we aim to confirm if screech is indeed present in this flow. Five thrust-optimised parabolic nozzles with different length and area ratios were examined at a nozzle pressure ratio range where FSS was expected to occur. Spectral proper orthogonal decomposition (SPOD) revealed resonance for both $m=0$ and $\,m=1$ azimuthal modes. Further decomposition of the SPOD modes allowed for the identification of downstream- and upstream-travelling waves, whose characteristics match the description of the Kelvin–Helmholtz (KH) instability wave and the guided jet mode (GJM), respectively. Linear stability analysis performed on Reynolds-averaged Navier–Stokes mean-flow fields confirms the existence of these two waves for the conditions studied and provides insight into their dispersion relation. A well-validated screech predictive model was employed and accurately predicted the resonant frequencies. The accurate predictions, combined with the observation of the KH and GJM waves, confirm that screech is active in this flow.

Linear-model-based estimation provides an equation-based route to reconstruct turbulent fluctuations from limited measurements, but its performance in compressible turbulent boundary layers depends critically on both the inter-layer coherence and the modelling of nonlinear forcing. This study systematically extends and assesses linear-model-based estimation for velocity and temperature fluctuations in compressible turbulent boundary layers, using a logarithmic-region measurement plane at Mach numbers $M=2$, 4 and 6 under both adiabatic- and cold-wall conditions. With the direct numerical simulation databases, the linear coherence spectra are first analysed. Results show that the recoverable information is mainly confined to large scales especially when the estimation plane is located near the wall, whereas the coherent scale range broadens markedly as the estimation location moves towards the measurement plane. The temperature coherence exhibits stronger sensitivity to Mach number and wall thermal condition than the streamwise velocity coherence, especially in cold-wall cases where the non-monotonic mean temperature profile alters the inter-layer thermal correlation. Based on these coherence characteristics, large-scale velocity and temperature snapshots are reconstructed using three levels of linear-model-based estimation. A controlled hierarchy comprising the baseline linear model with white forcing, an eddy-viscosity-corrected operator and a combined eddy-viscosity and forcing model is used to distinguish the effects of operator correction and forcing closure. Finally, a single-plane-calibrated estimation of wall-normal fluctuation-energy profiles is performed using the response cross-spectral density estimated by the Lyapunov formulation. The principal physical insight is that inter-layer coherence determines which scales are recoverable, the eddy-viscosity correction primarily improves the reconstructed response structure and the forcing model modifies the fluctuation amplitude and wall-normal energy redistribution. These roles are distinct in instantaneous and statistical reconstruction and are particularly important for temperature fluctuations under cold-wall conditions.