The energy spectrum is commonly used to describe the scale dependence of turbulent fluctuations in homogeneous isotropic turbulence. In contrast, one-point statistical quantities, such as the turbulent kinetic energy, are mainly employed for inhomogeneous turbulence models. Attempts have been made to describe the scale dependence of inhomogeneous turbulence using the second-order structure function and two-point velocity correlation. However, unlike the energy spectrum, expressions for the energy density in the scale space fail to satisfy the requirement of being non-negative. In this study, a new expression for the scale-space energy density based on filtered velocities is proposed to clarify the reasons behind the negative values of the energy density and to obtain a better understanding of inhomogeneous turbulence. The new expression consists of homogeneous and inhomogeneous parts; the former is always non-negative, while the latter can be negative because of the turbulence inhomogeneity. Direct numerical simulation data of homogeneous isotropic turbulence and a turbulent channel flow are used to evaluate the two parts of the energy density and turbulent energy. It was found that the inhomogeneous part of the turbulent energy shows non-zero values near the wall and at the centre of a channel flow. In particular, the inhomogeneous part of the energy density changes its sign depending on the scale. A concave profile of the filtered-velocity variance at the wall accounts for the negative value of the energy density in the region very close to the wall.

The scaling of different features of streamwise normal stress profiles $\langle uu\rangle ^+(y^+)$ in turbulent wall-bounded flows is the subject of a long-running debate. Particular points of contention are the scaling of the ‘inner’ and ‘outer’ peaks of $\langle uu\rangle ^+$ at $y^+\approxeq ~15$ and $y^+ ={O}(10^3)$, respectively, their infinite Reynolds number limit, and the rate of logarithmic decay in the outer part of the flow. Inspired by the thought-provoking paper of Chen & Sreenivasan (J. Fluid Mech., vol. 908, 2021, p. R3), two terms of an inner asymptotic expansion of $\langle uu\rangle ^+$ in the small parameter $Re_{\tau }^{-1/4}$ are constructed from a set of direct numerical simulations (DNS) of channel flow. This inner expansion is for the first time matched through an overlap layer to an outer expansion, which not only fits the same set of channel DNS within 1.5 % of the peak stress, but also provides a good match of laboratory data in pipes and the near-wall part of boundary layers, up to the highest $Re_{\tau }$ values of $10^5$. The salient features of the new composite expansion are first, an inner $\langle uu\rangle ^+$ peak, which saturates at 11.3 and decreases as $Re_{\tau }^{-1/4}$. This inner peak is followed by a short ‘wall log law’ with a slope that becomes positive for $Re_{\tau }$ beyond ${O}(10^4)$, leading up to an outer peak, followed by the logarithmic overlap layer with a negative slope going continuously to zero for $Re_{\tau }\to \infty$.

The wake of a notchback Ahmed body presenting a bi-stable nature is investigated by performing wind tunnel experiments and large-eddy simulations. Attention is confined to the Reynolds number ($Re$) influence on the wake state instability within $5\times 10^{4}\leq Re \leq 25\times 10^{4}$. Experimental observations suggest a wake bi-stability with low-frequency switches under low $Re$. The wake becomes ‘tri-stable’ with the increase of $Re$ with the introduction of a new symmetric state. The higher presence of the symmetric state can be considered as a symmetrization of the wake bi-stability with an increasing $Re$. The wake symmetry under high $Re$ attributed to the highly frequent switches of the wake is extremely sensitive to small yaw angles, showing the feature of bi-stable flows. The wake asymmetry is confirmed in numerical simulations with both low and high $Re$. The wake asymmetries are indicated by the wake separation, the reattachment and the wake dynamics identified by the proper orthogonal decomposition. However, the turbulence level is found to be significantly higher with a higher $Re$. This leads to a higher possibility to break the asymmetric state, resulting in highly frequent switches showing symmetry.

Fault zones have the potential to act as leakage pathways through low permeability structural seals in geological reservoirs. Faults may facilitate migration of groundwater contaminants and stored anthropogenic carbon dioxide (CO$_2$), where the waste fluids would otherwise remain securely trapped. We present an analytical model that describes the dynamics of leakage through a fault zone cutting multiple aquifers and seals. Current analytical models for a buoyant plume in a semi-infinite porous media are combined with models for a leaking gravity current and a new model motivated by experimental observation, to account for increased pressure gradients within the fault due to an increase in Darcy velocity directly above the fault. In contrast to previous analytical fault models, we verify our results using a series of analogous porous medium tank experiments, with good matching of observed leakage rates and fluid distribution. We demonstrate the utility of the model for the assessment of CO$_2$ storage security, by application to a naturally occurring CO$_2$ reservoir, showing the dependence of the leakage rates and fluid distribution on the fault/aquifer permeability contrast. The framework developed within this study can be used for quick assessment of fluid leakage through fault zones, given a set of input parameters relating to properties of the fault, aquifer and fluids, and can be incorporated into basin-scale models to improve computational efficiency. The results show the utility of using analytical methods and reduced-order modelling in complex geological systems, as well as the value of laboratory porous medium experiments to verify results.

Fully developed turbulent flow in channels with mild to strong longitudinal curvature is studied by direct numerical simulations. The Reynolds based on the bulk mean velocity and channel half-width $\delta$ is fixed at $20\,000$, resulting in a friction Reynolds number of approximately 1000. Four cases are considered with curvature varying from $\gamma = 2\delta /r_c = 0.033$ to 0.333, where $r_c$ is the curvature radius at the channel centre. Substantial differences between the mean wall shear stress on the convex and concave walls are already observed for $\gamma = 0.033$. A log-law region is absent and a region with nearly constant mean angular momentum develops in the channel centre for strong curvatures. Spanwise and wall-normal velocity fluctuations are strongly amplified by curvature in the outer region of the concave channel side. Only near the walls, where curvature effects are relatively weak, do the mean velocity and velocity fluctuation profiles approximately collapse when scaled by wall units based on the local friction velocity. Budgets of the streamwise and wall-normal Reynolds-stress equations are presented and turbulence structures are investigated through visualizations and spectra. In the case with strongest curvature, the flow relaminarizes locally near the convex wall. On the concave channel side, large elongated streamwise vortices reminiscent of Taylor–Görtler vortices develop for all curvatures considered. The maximum in the premultiplied two-dimensional wall-normal energy spectrum and co-spectrum shifts towards larger scales with increasing curvature. The large scales substantially contribute to the wall-normal velocity fluctuations and momentum transport on the concave channel side.

The instability of an incompressible boundary-layer flow over an infinite swept wing in the presence of disc-type roughness elements and free-stream turbulence (FST) has been investigated by means of direct numerical simulations. Our study corresponds to the experiments by Örlü et al. (Tech. Rep., KTH Royal Institute of Technology, 2021, http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-291874). Here, different dimensions of the roughness elements and levels of FST have been considered. The aim of the present work is to investigate the experimentally observed sensitivity of the transition to the FST intensity. In the absence of FST, flow behind the roughness elements with a height above a certain value immediately undergoes transition to turbulence. Impulse–response analyses of the steady flow have been performed to identify the mechanism behind the observed flow instability. For subcritical roughness, the generated wave packet experiences a weak transient growth behind the roughness and then its amplitude decays as it is advected out of the computational domain. In the supercritical case, in which the flow transitions to turbulence, flow as expected exhibits an absolute instability. The presence of FST is found to have a significant impact on the transition behind the roughness, in particular in the case of a subcritical roughness height. For a height corresponding to a roughness Reynolds number $Re_{hh}=461$, in the absence of FST the flow reaches a steady laminar state, while a very low FST intensity of $Tu =0.03\,\%$ causes the appearance of turbulence spots in the wake of the roughness. These randomly generated spots are advected out of the computational domain. For a higher FST level of $Tu=0.3\,\%$, a turbulent wake is clearly visible behind the element, similar to that for the globally unstable case. The presented results confirm the experimental observations and explain the mechanisms behind the observed laminar–turbulent transition and its sensitivity to FST.

Direct numerical simulations and linear stability analysis are carried out to study mixed convection in a horizontal duct with constant-rate heating applied at the bottom and an imposed transverse horizontal magnetic field. A two-dimensional approximation corresponding to the asymptotic limit of a very strong magnetic field effect is validated and applied, together with full three-dimensional analysis, to investigate the flow's behaviour in the previously unexplored range of control parameters corresponding to typical conditions of a liquid metal blanket of a nuclear fusion reactor (Hartmann numbers up to $10^4$ and Grashof numbers up to $10^{10}$). It is found that the instability to quasi-two-dimensional rolls parallel to the magnetic field discovered at smaller Hartmann and Grashof numbers in earlier studies also occurs in this parameter range. Transport of the rolls by the mean flow leads to magnetoconvective temperature fluctuations of exceptionally high amplitudes. It is also demonstrated that quasi-two-dimensional structure of flows at very high Hartmann numbers does not guarantee accuracy of the classical two-dimensional approximation. The accuracy deteriorates at the highest Grashof numbers considered in the study.

The interface between two immiscible fluids can become unstable under the effect of an imposed tangential electric field along with a stagnation point flow. This canonical situation, which arises in a wide range of electrohydrodynamic systems including at the equator of electrified droplets, can result in unstable interface deflections where the perturbed interface gets drawn along the extensional axis of the flow while experiencing strong charge build-up. Here, we present analytical and numerical analyses of the stability of a planar interface separating two immiscible fluid layers subject to a tangential electric field and a stagnation point flow. The interfacial charge dynamics is captured by a conservation equation accounting for Ohmic conduction, advection by the flow and finite charge relaxation. Using this model, we perform a local linear stability analysis in the vicinity of the stagnation point to study the behaviour of the system in terms of the relevant dimensionless groups of the problem. The local theory is complemented with a numerical normal-mode linear stability analysis based on the full system of equations and boundary conditions using the boundary element method. Our analysis demonstrates the subtle interplay of charge convection and conduction in the dynamics of the system, which oppose one another in the dominant unstable eigenmode. Finally, numerical simulations of the full nonlinear problem demonstrate how the coupling of flow and interfacial charge dynamics can give rise to nonlinear phenomena such as tip formation and the growth of charge density shocks.

The phenomena of microlayer formation and its dynamic characteristics during the nucleate pool boiling regime have been widely investigated in the past. However, experimental works on real-time microlayer dynamics during nucleate flow boiling conditions are highly scarce. The present work is an attempt to address this lacuna and is concerned with developing a fundamental understanding of microlayer dynamics during the growth process of a single vapour bubble under nucleate flow boiling conditions. Boiling experiments have been conducted under subcooled conditions in a vertical rectangular channel with water as the working fluid. Thin-film interferometry combined with high-speed cinematography have been adopted to simultaneously capture the dynamic behaviour of the microlayer along with the bubble growth process. Transients associated with the microlayer have been recorded in the form of interferometric fringe patterns, which clearly reveal the evolution of the microlayer beneath the growing vapour bubble, the movement of the triple contact line and the growth of the dryspot region during the bubble growth process. While symmetric growth of the microlayer was confirmed in the early growth phase, the bulk flow-induced bubble deformation rendered asymmetry to its profile during the later stages of the bubble growth process. The recorded fringe patterns have been quantitatively analysed to obtain microlayer thickness profiles at different stages of the bubble growth process. For Re = 3600, the maximum thickness of the almost wedge-shaped microlayer was obtained as δ ~ 3.5 μm for a vapour bubble of diameter 1.6 mm. Similarly, for Re = 6000, a maximum microlayer thickness of δ ~ 2.5 μm was obtained for a bubble of diameter 1.1 mm.

The settling velocity of porous particles in linear stratification is affected by the diffusive exchange between interstitial and ambient water. The extent to which buoyancy and interstitial mass adaptation alters the settling velocity depends on the ratio of the diffusive and viscous time scales. We conducted schlieren experiments and lattice Boltzmann simulations for highly porous (95 %) but impermeable spheres settling in linear stratification. For a parameter range that resembles marine porous particles, ‘marine aggregates’, i.e. low Reynolds numbers ($0.05\leq \textit {Re}\leq 10$), intermediate Froude numbers ($0.1\leq \textit {Fr}\leq 100$) and Schmidt number of salt ($\textit {Sc}=700$), we observe delayed mass adaptation of the interstitial fluid due to lower-density fluid being dragged by a particle that forms a density boundary layer around the particle. The boundary layer buffers the diffusive exchange of stratifying agent with the ambient fluid, leading to an enhanced density contrast of the interstitial pore fluid. Stratification-related drag enhancement by means of additional buoyancy of dragging lighter fluid and buoyancy-induced vorticity resembles earlier findings for solid spheres. However, the exchange between density boundary layer and pore fluid substantially increases stratification drag for small $\textit {Fr}$. To estimate the effect of stratification on marine aggregates settling in the ocean, we derived scaling laws and show that small particles ($\leq$0.5 mm) experience enhanced drag which increases retention times by 10 % while larger porous particle (>0.5 mm) settling is dominated by delayed mass adaptation that diminishes settling velocity by 10 % up to almost 100 %. The derived relationships facilitate the integration of stratification-dependent settling velocities into biogeochemical models.

In the coastal ocean, interactions of waves and currents with large roughness elements, similar in size to wave orbital excursions, generate drag and dissipate energy. These boundary layer dynamics differ significantly from well-studied small-scale roughness. To address this problem, we derived spatially and phase-averaged momentum equations for combined wave–current flows over rough bottoms, including the canopy layer containing obstacles. These equations were decomposed into steady and oscillatory parts to investigate the effects of waves on currents, and currents on waves. We applied this framework to analyse large-eddy simulations of combined oscillatory and steady flows over hemisphere arrays (diameter $D$), in which current ($U_c$), wave velocity ($U_w$) and period ($T$) were varied. In the steady momentum budget, waves increase drag on the current, and this is balanced by the total stress at the canopy top. Dispersive stresses from oscillatory flow around obstacles are increasingly important as $U_w/U_c$ increases. In the oscillatory momentum budget, acceleration in the canopy is balanced by pressure gradient, added-mass and form drag forces; stress gradients are small compared to other terms. Form drag is increasingly important as the Keulegan–Carpenter number $KC=U_wT/D$ and $U_c/U_w$ increase. Decomposing the drag term illustrates that a quadratic relationship predicts the observed dependences of steady and oscillatory drag on $U_c/U_w$ and $KC$. For large roughness elements, bottom friction is well represented by a friction factor ($f_w$) defined using combined wave and current velocities in the canopy layer, which is proportional to drag coefficient and frontal area per unit plan area, and increases with $KC$ and $U_c/U_w$.

Modulation of fluid temperature fluctuations by particles due to thermal interaction in homogeneous isotropic turbulence is studied. For simplicity, only thermal coupling between the fluid and particles is considered, and momentum coupling is neglected. Application of the statistical theory used in cloud turbulence research leads to the prediction that modulation of the intensity of fluid temperature fluctuations by particles is expressed as a function of the Damköhler number, which is defined as the ratio of the turbulence large-eddy turnover time to the fluid thermal relaxation time. Direct numerical simulations are conducted for two-way thermal coupling between the fluid temperature field and point particles in homogeneous isotropic turbulence. The simulation results are shown to agree well with the theoretical predictions.

Tandem configurations of two self-propelled flexible flappers of finite span are explored by means of numerical simulations. The same sinusoidal vertical motion is imposed on the leading edge of both flappers, but with a phase shift ($\phi$). In addition, a vertical offset, $H$, is prescribed between the flappers. The configurations that emerge are characterized in terms of their hydrodynamic performance and topology. The flappers reach a stable configuration with a constant mean propulsive speed and a mean equilibrium horizontal distance. Depending on $H$ and $\phi$, two different tandem configurations are observed, namely compact and regular configurations. The performance of the upstream flapper (i.e. the leader) is virtually equal to the performance of an isolated flapper, except in the compact configuration, where the close interaction with the downstream flapper (i.e. the follower) results in higher power requirements and propulsive speed than an isolated flapper. Conversely, the follower's performance is significantly affected by the wake of the leader in both regular and compact configurations. The analysis of the flow shows that the follower's performance is influenced by the interaction with the vertical jet induced by the vortex rings shed by the leader. This interaction can be beneficial or detrimental for the follower's performance, depending on the alignment of the jet velocity with the follower's vertical motion. Finally, a qualitative prediction of the performance of a hypothetical follower is presented. The model is semi-empirical, and it uses the flow field of an isolated flapper.

Steady doublet-type flow takes place in a porous formation, where the log-transform $Y = \ln K$ of the spatially variable hydraulic conductivity $K$ is regarded as a stationary random field of two-point autocorrelation $\rho _Y$. A passive solute is injected at the source in the porous formation and we aim to quantify the resulting dispersion process between the two lines by means of spatial moments. The latter depend on the distance $\ell$ between the lines, the variance $\sigma ^2_Y$ of $Y$ and the (anisotropy) ratio $\lambda$ between the vertical and the horizontal integral scales of $Y$. A simple (analytical) solution to this difficult problem is obtained by adopting a few simplifying assumptions: (i) a perturbative solution, which regards $\sigma ^2_Y$ as a small parameter, of the velocity field is sought; (ii) pore-scale dispersion is neglected; and (iii) we deal with a highly anisotropic formation ($\lambda \lesssim 0.1$). We focus on the longitudinal spatial moment, as it is of most importance for the dispersion mechanism. A general expression is derived in terms of a single quadrature, which can be straightforwardly carried out once the shape of $\rho _Y$ is specified. Results permit one to grasp the main features of the dispersion processes as well as to assess the difference with similar mechanisms observed in other non-uniform flows. In particular, the dispersion in a doublet-type flow is observed to be larger than that generated by a single line. This effect is explained by noting that the advective velocity in a doublet, unlike that in source/line flows, is rapidly increasing in the far field owing to the presence there of the singularity. From the standpoint of the applications, it is shown that the solution pertaining to $\lambda \to 0$ (stratified formation) provides an upper bound for the dispersion mechanism. Such a bound can be used as a conservative limit when, in a remediation procedure, one has to select the strength as well as the distance $\ell$ of the doublet. Finally, the present study lends itself as a valuable tool for aquifer tests and to validate more involved numerical codes accounting for complex boundary conditions.

Direct numerical simulations have been performed for heat and momentum transfer in internally heated turbulent shear flow with constant bulk mean velocity and temperature, $u_{b}$ and $\theta _{b}$, between parallel, isothermal, no-slip and permeable walls. The wall-normal transpiration velocity on the walls $y=\pm h$ is assumed to be proportional to the local pressure fluctuations, i.e. $v=\pm \beta p/\rho$ (Jiménez et al., J. Fluid Mech., vol. 442, 2001, pp. 89–117). The temperature is supposed to be a passive scalar, and the Prandtl number is set to unity. Turbulent heat and momentum transfer in permeable-channel flow for the dimensionless permeability parameter $\beta u_b=0.5$ has been found to exhibit distinct states depending on the Reynolds number $Re_b=2h u_b/\nu$. At $Re_{b}\lesssim 10^4$, the classical Blasius law of the friction coefficient and its similarity to the Stanton number, $St\approx c_{f}\sim Re_{b}^{-1/4}$, are observed, whereas at $Re_{b}\gtrsim 10^4$, the so-called ultimate scaling, $St\sim Re_b^0$ and $c_{f}\sim Re_b^0$, is found. The ultimate state is attributed to the appearance of large-scale intense spanwise rolls with the length scale of $O(h)$ arising from the Kelvin–Helmholtz type of shear-layer instability over the permeable walls. The large-scale rolls can induce large-amplitude velocity fluctuations of $O(u_b)$ as in free shear layers, so that the Taylor dissipation law $\epsilon \sim u_{b}^{3}/h$ (or equivalently $c_{f}\sim Re_b^0$) holds. In spite of strong turbulence promotion there is no flow separation, and thus large-amplitude temperature fluctuations of $O(\theta _b)$ can also be induced similarly. As a consequence, the ultimate heat transfer is achieved, i.e. a wall heat flux scales with $u_{b}\theta _{b}$ (or equivalently $St\sim Re_b^0$) independent of thermal diffusivity, although the heat transfer on the walls is dominated by thermal conduction.

Rheotaxis and migration of cells in a flow field have been investigated intensively owing to their importance in biology, physiology and engineering. In this study, first, we report our experiments showing that the microalgae Chlamydomonas can orient against the channel flow and migrate to the channel centre. Second, by performing boundary element simulations, we demonstrate that the mechanism of the observed rheotaxis and migration has a physical origin. Last, using a simple analytical model, we reveal the novel physical mechanisms of rheotaxis and migration, specifically the interplay between cyclic body deformation and cyclic swimming velocity in the channel flow. The discovered mechanism can be as important as phototaxis and gravitaxis, and likely plays a role in the movement of other natural microswimmers and artificial microrobots with non-reciprocal body deformation.

Bounds are derived for rotating Rayleigh–Bénard convection with free slip boundaries as a function of the Rayleigh, Taylor and Prandtl numbers ${\textit {Ra}}$, ${\textit {Ta}}$ and ${\textit {Pr}}$. At infinite ${\textit {Pr}}$ and ${\textit {Ta}} > 130$, the Nusselt number ${\textit {Nu}}$ obeys ${\textit {Nu}} \leqslant \frac {7}{36} \left ({4}/{{\rm \pi} ^2} \right )^{1/3} {\textit {Ra}} {\textit {Ta}}^{-1/3}$, whereas the kinetic energy density $E_{kin}$ obeys $E_{kin} \leqslant ({7}/{72 {\rm \pi}}) \left ({4}/{{\rm \pi} } \right )^{1/3} {\textit {Ra}}^2 {\textit {Ta}}^{-2/3}$ in the frame of reference in which the total momentum is zero, and $E_{kin} \leqslant ({1}/{2{\rm \pi} ^2})({{\textit {Ra}}^2}/{{\textit {Ta}}})({\textit {Nu}}-1)$. These three bounds are derived from the momentum equation and the maximum principle for temperature and are extended to general ${\textit {Pr}}$. The extension to finite ${\textit {Pr}}$ is based on the fact that the maximal velocity in rotating convection at infinite ${\textit {Pr}}$ is bound by $1.23 {\textit {Ra}} {\textit {Ta}}^{-1/3}$.

Many environmental flows arise due to natural convection at a vertical surface, from flows in buildings to dissolving ice faces at marine-terminating glaciers. We use three-dimensional direct numerical simulations of a vertical channel with differentially heated walls to investigate such convective, turbulent boundary layers. Through the implementation of a multiple-resolution technique, we are able to perform simulations at a wide range of Prandtl numbers ${Pr}$. This allows us to distinguish the parameter dependences of the horizontal heat flux and the boundary layer widths in terms of the Rayleigh number $\mbox {{Ra}}$ and Prandtl number ${Pr}$. For the considered parameter range $1\leq {Pr} \leq 100$, $10^{6} \leq \mbox {{Ra}} \leq 10^{9}$, we find the flow to be consistent with a ‘buoyancy-controlled’ regime where the heat flux is independent of the wall separation. For given ${Pr}$, the heat flux is found to scale linearly with the friction velocity $V_\ast$. Finally, we discuss the implications of our results for the parameterisation of heat and salt fluxes at vertical ice–ocean interfaces.

Air exchange between people has emerged in the COVID-19 pandemic as the important vector for transmission of the SARS-CoV-2 virus. We study the airflow and exchange between two unmasked individuals conversing face-to-face at short range, which can potentially transfer a high dose of a pathogen, because the dilution is small when compared to long-range airborne transmission. We conduct flow visualization experiments and direct numerical simulations of colliding respiratory jets mimicking the initial phase of a conversation. The evolution and dynamics of the jets are affected by the vertical offset between the mouths of the speakers. At low offsets the head-on collision of jets results in a `blocking effect', temporarily shielding the susceptible speaker from the pathogen carrying jet, although, the lateral spread of the jets is enhanced. Sufficiently large offsets prevent the interaction of the jets. At intermediate offsets (8-10 cm for 1 m separation), jet entrainment and the inhaled breath assist the transport of the pathogen-loaded saliva droplets towards the susceptible speaker's mouth. Air exchange is expected, in spite of the blocking effect arising from the interaction of the respiratory jets from the two speakers.