We investigate turbulent Taylor–Couette flow between two concentric cylinders, where the inner cylinder of radius $r_i$ rotates while the outer one of radius $r_o$ remains stationary. Using direct numerical simulations, we examine how varying the radius ratio $\eta = r_i / r_o$ from $\eta = 0.714$ down to $0.0244$ affects the flow characteristics at low to moderate Reynolds numbers. Our results show significant changes in the flow structures and statistics in the limit of a vanishingly small inner radius. The turbulent kinetic energy, scaled with the friction velocity at the inner cylinder, does not exhibit a self-similar scaling; instead, it decreases with decreasing $\eta$. The turbulent kinetic energy budgets reveal that the locations of peak production and total dissipation are independent of $\eta$, whereas their amplitudes decrease as $\eta$ increases. The pressure–velocity correlation near the inner cylinder is large for small $\eta$ and its amplitude decreases with increasing $\eta$, while the turbulent transport term exhibits the opposite trend. Numerical simulations for $\eta \leqslant 0.5$ show that, for our specific set-up, a rather good collapse of the distribution of the normalised torque versus the Taylor number ($ \textit{Ta}$) is obtained when the latter is defined according to Chandrasekhar (Hydrodynamic and Hydromagnetic Stability, Oxford Univ. Press, 1961), with a tendency towards a $ \textit{Ta}^{1/3}$ regime at sufficiently large $ \textit{Ta}$.

We study the dynamic interaction of two viscous gravity currents in a confined porous layer using laboratory experiments in a vertically placed bead-packed Hele-Shaw cell. By varying the injection rate, along with the density and viscosity of the injecting and ambient fluids, these experiments cover three exact and eight approximate regimes of gravity current interaction, as identified based on the one-dimensional sharp-interface model. By superimposing the theoretically predicted profile shapes and time-dependent frontal locations, a verification is obtained in the different asymptotic regimes of viscous current interaction. Overall, fairly good agreement has been observed between the time-dependent numerical solutions and laboratory measurements on the profile shapes, particularly in the bulk region, where the aspect ratio of the interface shape is fairly large. Such an observation indicates the applicability of the sharp-interface model of viscous current interaction, including the very interesting dynamics of overriding and coflowing. However, the self-similar solutions in some of the exact regimes fail to make reasonable predictions in these experiments, presumably due to the influence of unfinished time transition. We have also observed some degree of disagreement in the frontal regions, which is likely due to the influence of fluid mixing, two-dimensional flow, local heterogeneity and the development of hydrodynamic instabilities for the viscously unstable experiments. The theoretical predictions of the model problem, along with the laboratory experimental observations, offer useful insights into the potential application of, e.g. the technology of co-flooding CO$_2$ and water in oil fields for the co-profits of geological CO$_2$ sequestration and enhanced oil recovery.

This article derives analytical expressions fully describing laminar flow through concentric pipe-within-pipe set-ups, focusing on scenarios where one tube is pressure driven, and the other serves as a lubricant. Both fluid zones are axially unbounded, therefore excluding recirculation, and are connected along longitudinal infinite slits situated on the inner pipe wall, representing fluid–fluid interfaces. Crucially, the viscous interaction along these interfaces is captured by means of a local slip length, for which explicit formulae are provided, allowing a straightforward evaluation. With that, these models provide a full description of the velocity field for slippery concentric pipes, taking into account the viscosity ratio of both fluids and the overall geometry, therefore extending beyond the common assumption of perfect slip applied to superhydrophobic surfaces. Thereby, they enable a precise analysis of the flow, offering important tools to decipher the intricate dynamics of the two coupled fluids within such set-ups. As a result, the insights acquired contribute to the design and optimisation of superhydrophobic and liquid-infused surfaces, with implications for numerous engineering applications such as microfluidic contactors or drag reduction. The analytical models are in excellent agreement with numerical simulations, thus confirming the selected approach. Therefore, our study further illustrates an effective methodology to derive additional analytical models through the presented mathematical techniques, which can serve as a useful template for modelling such surfaces.

The object of investigation in this paper is the nonlinear equations of motion for two-dimensional inviscid water flows with piecewise constant density stratification in a three-layer fluid with a flat bottom, a free surface and two interfaces. We establish a Hamiltonian formulation for the nonlinear governing equations in this set-up. The Hamiltonian of the system and the equations of motion of the surface and of the interfaces are expressed with the help of the Dirichlet–Neumann (DN) operators, which are introduced for each of the layers. Then the linear equations for small amplitudes of the elevation of the surface and of the interfaces in the leading order are derived, from which a bi-cubic equation for the dispersion relation is obtained, whose solutions are analysed. The six real solutions for the possible propagation speeds (three positive, related to right-moving waves, and three negative, related to left-moving waves) have magnitudes of different order. Upper and lower bounds for the previously mentioned roots are also given in terms of the coefficients of the equation. Subsequently, approximate formulas for the propagation speeds are derived. The importance of the DN operators is further illustrated in a separate analysis of the three-layer model with flat surface (rigid lid). The full nonlinear evolution equations are expressed again in terms of the DN operators, and the equations in the linear regime and the weakly nonlinear propagation regime (the Boussinesq approximation) are derived by a proper expansion of the DN operators. Limits to the two-layer free surface model are obtained as well. The obtained results are applicable to internal waves in lakes and in the ocean as well as to laboratory experiments with three superimposed fluid layers.

The elliptic approximation (EA) – rooted in Taylor’s frozen flow hypothesis, Kolmogorov’s theory of small-scale turbulence, and the Kraichnan–Tennekes random sweeping hypothesis – remains a foundational framework for modelling spatiotemporal velocity correlations in incompressible wall-bounded turbulence. This study revisits the model’s theoretical basis, and extends its applicability to velocity and temperature fluctuations in supersonic channel flows. First, we identify non-elliptic distortions in the viscous sublayer, and introduce a shear-induced acceleration that captures the observed deviation from the assumed constant convection velocity at large time separations. Next, we show that the inertial-range scalings underpinning the EA are not valid in regions where the model remains accurate; instead, its validity is supported by extended self-similarity between spatial and temporal structure functions. Finally, we conduct high-fidelity direct numerical simulations of compressible channel flows with fluctuating Mach numbers up to 0.8; our data confirm the robustness of the EA under supersonic conditions, and its effectiveness in characterising both velocity and temperature correlations. Together, these findings provide new theoretical insights into the spatiotemporal structure of wall-bounded turbulence, and broaden the operational envelope of the EA.

We consider the steady heat transfer between a collection of impermeable obstacles immersed in an incompressible two-dimensional (2-D) potential flow, when each obstacle has a prescribed boundary temperature distribution. Inside the fluid, the temperature satisfies a variable-coefficient elliptic partial differential equation (PDE), the solution of which usually requires expensive techniques. To solve this problem efficiently, we construct multiply connected conformal maps under which both the domain and governing equation are greatly simplified. In particular, each obstacle is mapped to a horizontal slit and the governing equation becomes a constant-coefficient elliptic PDE. We then develop a boundary integral approach in the mapped domain to solve for the temperature field when arbitrary Dirichlet temperature data are specified on the obstacles. The inverse conformal map is then used to compute the temperature field in the physical domain. We construct our multiply connected conformal maps by exploiting the flexible and highly accurate AAA-LS algorithm. In multiply connected domains and domains with non-constant boundary temperature data, we note similarities and key differences in the temperature fields and Nusselt number scalings as compared with the isothermal simply connected problem analysed by Choi et al. (J. Fluid Mech., vol. 536, 2005, pp. 155–184). In particular, we derive new asymptotic expressions for the Nusselt number in the case of arbitrary non-constant temperature data in singly connected domains at low Péclet number, and verify these scalings numerically. While our language focuses on the problem of conjugate heat transfer (the transfer of heat between objects in a flow), our methods and findings are equally applicable to the advection–diffusion of any passive scalar in a potential flow.

Compressible wall-bounded turbulent flows exhibit complex mean profiles because of the pronounced compressibility effects and heat transfer. We propose a hybrid transformation framework to collapse compressible mean velocity and temperature profiles onto incompressible forms through scaling each layer by its effective transformation, with the underlying mapping functions discovered via a physics-informed symbolic regression (PISR) method. The hybrid velocity transformation incorporates an intrinsic compressibility correction for the buffer layer and a PISR-derived mapping function for the logarithmic layer. For temperature, we introduce a hybrid transformation that integrates the Mach-invariant-type transformation in the viscous sublayer and a novel PISR-derived scaling in the logarithmic layer. The performance of these transformations is evaluated across compressible turbulent boundary layers with free-stream Mach numbers ranging from 0.5 to 8 and wall-to-recovery-temperature ratios ranging from 0.25 to 1. The hybrid velocity transformation outperforms Griffin–Fu–Moin transformation for the transformed mean velocity profiles, with the mean integrated percent error across the dataset decreasing from 1.67 % to 0.96 %. The hybrid temperature transformation performs better than the Mach-invariant-type and Trettel–Larsson-type transformations for mean temperature profiles. Moreover, the inverse hybrid velocity and temperature transformations can effectively predict the compressible mean velocity and temperature profiles with only wall conditions.

Although stably stratified shear flows, where the base velocity shear is quasi-continuously forced externally, arise in many geophysically and environmentally relevant circumstances, the emergent dynamics of their ensuing statistically steady stratified turbulence is still an open question. We address this phenomenon in a series of three-dimensional direct numerical simulations using spectral element methods. We consider a forced, stably stratified shear flow with an initial bulk Reynolds number $\textit{Re}_{0} = 50$, an initial bulk Richardson number $\textit{Ri}_{0} = 1/80$ (also corresponding to the initial minimum gradient Richardson number $\textit{Ri}_{{g}}$) and a fluid of Prandtl number ${\textit{Pr}} = 1$ in horizontally extended domains. Although the initial configuration is unstable to a primary Kelvin–Helmholtz instability, the ensuing turbulence is sustained by continuously relaxing the resulting flow back towards the initial profiles of streamwise velocity and buoyancy. We study statistical as well as structural aspects of the final statistically steady flows, including the flux coefficient $\varGamma _{\chi }$ and dynamically emergent length scales $\varLambda$ associated with the large-scale dynamics, respectively. Despite the ongoing stirring and mixing, we find that the shear layer half-depth converges to a finite value of $d \approx 8$ (i.e. $\varLambda _{z} \approx 16$) once the horizontal extent of the domain $L_{{h}} \gtrsim 96$. While this implies a final ${{Re}} \approx 400$ and ${Ri} \approx 0.1$, we hypothesise that such forced flows ‘tune’ themselves eventually to a state of a gradient Richardson number $\textit{Ri}_{{g}} \lesssim 0.2$, consistently with several previous studies. Moreover, provided sufficiently extended domains, we observe the emergence of large-scale flow structures with spanwise $\varLambda _{\!y} \approx 50$ and streamwise $\varLambda _{x} \lesssim 115$. Clearly, these observations demonstrate the marked anisotropy of characteristic emergent length scales, even for such ‘weakly stratified’ forced shear flows. We conjecture that the actual emergent streamwise structures are a vestigial ‘imprint’ in the sheared turbulent flow of the primary linear instability of the converged deepened turbulent shear layer.

In this study, we investigate the dynamic behaviour of reconfigurable circular plates under acceleration as a model problem to understand the interplay between kinematics and shape deformation in biological propulsion. A high-resolution force transducer and time-resolved particle image velocimetry were employed to simultaneously capture both hydrodynamic forces and vortex dynamics. The results reveal that, unlike rigid plates that exhibit Reynolds number independence, the force evolution of reconfigurable plates is governed by the dimensionless bending stiffness ${\textit{EI}}^*$. A distinct load-shifting phenomenon is observed – characterized by a reduction in peak force amplitude and an elevation of the postpeak force trough, contrasting with the ‘peak-valley’ behaviour typical of rigid plates. Based on ${\textit{EI}}^*$, reconfigurable plates are classified into three regimes: extra-flexible (${\textit{EI}}^* \lt 2.28 \times 10^{-3}$), flexible ($2.28 \times 10^{-3} \leqslant {\textit{EI}}^* \leqslant 0.143$) and rigid (${\textit{EI}}^* \gt 0.143$). Notably, only plates within the flexible regime exhibit the load-shifting phenomenon. Flow visualizations show that the flexible plates, due to their shape reconfiguration, produce flow fields with two distinct features: initially, the formation of three-dimensional, non-axisymmetric vortex rings; subsequently, vortex breakdown occurs due to instability. By applying the vorticity moment theorem, force generation is accurately estimated from the flow field. Using a vortex-based low-order force model, the radial distribution of vorticity is identified as the key mechanism underlying the load-shifting phenomenon. This finding suggests that biological morphing structures in real propulsion scenarios can reduce force fluctuations without compromising average thrust by ‘load-shifting’, offering insights into efficient propulsion strategies.

We report experimental evidence of an Eulerian-mean flow, $\overline {u}(z)$, created by the interaction of surface waves and tailored ambient sub-surface turbulence, which partly cancels the Stokes drift, $u_s(z)$, and present supporting theory. Water-side turbulent velocity fields and Eulerian-mean flows were measured with particle image velocimetry before vs after the passage of a wave group, and with vs without the presence of regular waves. We compare different wavelengths, steepnesses and turbulent intensities. In all cases, a significant change in the Eulerian-mean current is observed, strongly focused near the surface, where it opposes the Stokes drift. The observations support the picture that, when waves encounter ambient sub-surface turbulence, the flow undergoes a transition during which Eulerian-mean momentum is redistributed vertically (without changing the depth-integrated mass transport) until a new equilibrium state is reached, wherein the near-surface ratio between $|{\rm d}\overline {u}/{\rm d}z|$ and $|{\rm d}u_s/{\rm d} z|$ approximately equals the ratio between the streamwise and vertical Reynolds normal stresses. This accords with a simple statistical theory derived here and holds regardless of the absolute turbulence level, whereas stronger turbulence means faster growth of the Eulerian-mean current. We present a model based on Rapid distortion theory which describes the generation of the Eulerian-mean flow as a consequence of the action of the Stokes drift on the background turbulence. Predictions are in qualitative, and reasonable quantitative, agreement with experiments on wave groups, where equilibrium has not yet been reached. Our results could have substantial consequences for predicting the transport of water-borne material in the oceans.

In this study, direct numerical simulation of a turbulent flame–wall interaction (FWI) has been done for premixed H$_2/$air and NH$_3/$H$_2/$air flames in a fully developed channel flow at Re$_\tau$ $\approx$ 300. Both isothermal and adiabatic walls are considered. The results contribute to further clarification of the underlying mechanisms of FWIs. First, the underlying mechanism for the rapid increase of chemical flame thickness near the wall is found to be the zero-flux boundary condition for diffusion. Effects of wall heat loss and wall turbulence are minor. Then, a ridge-based flame surface identification method is proposed to track the flame front, which is found to be more accurate than an isosurface of $C$ (the progress variable), especially during FWIs. Using this technique, the near-wall flame geometry and orientation are correctly captured. It is found that the flames are laminarised near the wall and almost parallel to the isothermal wall shortly before quenching. Flame–vortex interactions lead to entrained flame pockets for H$_2$ as a fuel and to a distributed reaction zone for the case of NH$_3/$H$_2$. Finally, the turbulent combustion regime is investigated by checking wall-distance-dependent Reynolds number and Karlovitz number. It is found that the flames enter the laminar flame regime shortly before wall quenching, instead of the broken reaction regime suggested in previous studies. To support the analysis, the turbulent flame dynamics, including turbulent burning rate, turbulent flame surface area, flame stretch factor, local displacement speed, flame dilatation, flame strain rate (both tangential and normal) and flame alignment with the principal strain rate are quantified, providing a full picture of near-wall turbulent flames for the considered conditions.

Direct numerical simulations with two-way coupled Lagrangian tracking are carried out to study the bubble preferential concentration and the flow field modification. Simulations are conducted in an upward vertical turbulent channel driven by a constant pressure gradient, corresponding to a friction Reynolds number $Re_{\tau 0}=180$. Micro-sized bubbles with diameters ranging from 0.72 to 1.43 wall units are considered. Competition between lift force and wall-lift force in the wall-normal direction leads to significant near-wall bubble accumulation and directly results in distinct preferential concentration patterns across the channel. Below (above) the peak concentration height, the wall-lift (lift) force dominates, driving bubbles to accumulate in regions of high-speed sweep (low-speed ejection) events. In the vicinity of the wall, the wall-normal lift force exhibits a strong correlation with the local streamwise flow velocity, further reinforcing the preferential concentration of bubbles in high-speed regions. Additionally, bubbles show a strong preference for the low-enstrophy and high-dissipation nodal topologies. Furthermore, small bubbles primarily accumulate in the vicinity of the wall, reducing the work done on the flow and leading to a decrease in bulk velocity and turbulence statistics. In contrast, the turbulence statistics of large bubbles are nearly identical to those of the unladen flow. The impact of large bubbles on the flow field primarily manifests as an effective increase in the mean pressure gradient. These findings demonstrate that bubbles in the upward vertical channel flow exhibit strong preferential concentration behaviours, whereas their ability to modulate turbulence remains limited.

Exact mathematical expressions are derived to predict the exponent $p$ observed in non-equilibrium turbulence, where the classical dissipation law is replaced by a new dissipation scaling law $C_{\varepsilon } \sim \textit{Re}_{\lambda }^p$. Here, $ \textit{Re}_{\lambda }$ is the Taylor-based Reynolds number and $C_{\varepsilon } = \varepsilon L_{11} / u^{\prime 3}$ is the non-dimensional dissipation rate, defined by the viscous dissipation rate, $\varepsilon$, longitudinal integral scale, $L_{11}$, and root-mean-square of the velocity fluctuations $u^{\prime} = \sqrt {\overline {u^{\prime 2}}}$ (Vassilicos, Annu. Rev. Fluid Mech., vol. 47, 2015, pp. 95–114). Assuming homogeneous and isotropic turbulence, it is shown that the exact value of $p$ involves only first-order derivatives of these variables; however, at very high Reynolds numbers, and under particularly strong changes in the power input of the external forcing (without changing the shape of the forcing spectrum), the exact expression simplifies to $p = 3\pi / 4\alpha L_{110} - 5 / 2$, where $L_{110}$ is the initial value of the longitudinal integral scale and $\alpha$ represents an effective forcing wavenumber. Thus, the main finding is that only large-scale effects are involved in the imposition of the non-equilibrium dissipation scaling law. The results are compared with direct numerical simulation (DNS) results of isotropic turbulence under abruptly changing forcing conditions and with experimental data of non-equilibrium decaying isotropic turbulence, showing consistent results.

An oscillating body floating at the water surface produces a field of self-generated waves. When the oscillation induces a difference in fore–aft wave amplitude squared, these self-generated waves can be used as a mechanism to propel the body horizontally across the surface (Longuet-Higgins 1977 Proc. R. Soc. Lond. A, vol. 352, no. 1671, pp. 463–480). The optimisation of this wave-driven propulsion is the interest of this work. To study the conditions necessary to produce optimal thrust we will consider a shallow water set-up where a periodically oscillating pressure source acts as the body. In this framework, an expression for the thrust is derived by relation to the difference in fore–aft amplitude squared. The conditions on the source for maximal thrust are explored both analytically and numerically in two optimal control problems. The first case is where a bound is imposed on the norm of the control function to regularise it. Secondly, a more physically motivated case is studied where the power injected by the source is bounded. The body is permitted to have a drift velocity $U$. When scaled with the wave speed $c$, the dimensionless velocity $v=U/c$ divides the study into subcritical, critical and supercritical regimes and the optimal conditions are presented for each. The result in the bounded power case is then used to demonstrate how the modulation of power injected can slowly change the cruising velocity from rest to supercritical velocities.