A theoretical model is presented for the steady multi-layered flow induced by a plane vertically distributed buoyancy source producing a turbulent wall plume in a ventilated box. While aspects of the stratification and rate of fluid exchange between box and exterior have been studied previously, the streamline pattern and velocity field have not been considered until now, despite having potentially important practical implications for achieving comfort in naturally ventilated buildings and for the indoor spread of airborne contagions. The boundary condition at the wall for each layer is established by deducing the turbulent entrainment rate. Using conformal mapping techniques and Poisson's integral theorem, closed-form solutions for the streamfunction of the induced flow in each layer are established. While the flow near the ceiling was overlooked in the classic model for the multi-layered stratification, after considering the possible flow scenarios, the stratification is re-evaluated herein by incorporating an entraining ceiling current. With a markedly thinner top layer, the refined stratification matches well with the available experimental observations, the restrictions we place on the applicability of the model overcoming the previous over-prediction in the number of interfaces. The magnitude of the dimensionless flow velocity, independent of the wall buoyancy flux and physical scale of the box, decreases significantly with the number of layers. Three types of layer, each with a distinct induced flow pattern, are distinguished and their implications for room airflow considered. Notably, the flow in the base layer represents a continual and smooth flushing of air between the inlet opening and the wall plume, whereas an intermediate layer is almost entirely comprised of near-stagnant air.

Upscaling the effect of heterogeneities in porous media is crucial for macroscopic flow predictions, with numerous applications in energy and environmental settings. In this study, we derive simple semi-analytical expressions for the upscaling of multiphase flow in a porous medium with a range of vertical heterogeneities. We use this upscaling to give insight into how the flow transitions between a viscous flow regime, in which macroscopic pressure gradients dominate over heterogeneity-driven capillary forces, and a capillary flow regime, in which these capillary forces dominate and set the saturation distribution of the flow. In particular, by studying the dynamics of flow in an aquifer, we demonstrate that different regions lie within the viscous and capillary flow regimes whilst other regions lie in between these regimes. By modifying the classic Buckley–Leverett problem for fluid displacement we demonstrate where and when the flow transitions between these regimes and how this affects flooding speeds. Then, we discuss the implications of these results in the case of carbon dioxide sequestration, making comparisons with field data.

Turbulent convection systems are known to give rise to prominent large-scale circulation. At the same time, the ‘background’ (or ‘small-scale’) turbulence is also highly relevant and e.g. carries the majority of the heat transport in the bulk of the flow. Here, we investigate how the small-scale turbulence is interlinked with the large-scale flow organization of Rayleigh–Bénard convection. Our results are based on a numerical simulation at Rayleigh number $Ra = 10^8$ in a large aspect ratio ($\varGamma =32$) cell to ensure a distinct scale separation. We extract local magnitudes and wavenumbers of small-scale turbulence and find significant correlation of large-scale variations in these quantities with the large-scale signal. Most notably, we find stronger temperature fluctuations and increased small-scale transport (by about 10 % of the global Nusselt number $Nu$) in plume impacting regions and opposite trends in the plume emitting counterparts. This concerns wall distances up to $2\delta _\theta$ (thermal boundary layer thickness). Local wavenumbers are generally found to be higher on the plume emitting side compared to the impacting one. A second independent approach by means of conditional averages confirmed these findings and yields additional insight into the large-scale variation of small-scale properties. Our results have implications for the modelling of small-scale turbulence.

Interscale energy transfer in wall turbulence has been intensively studied in recent years, and both forward (i.e. from larger to smaller scales) and reversed transfers of turbulent energy have been found, although their corresponding physical phenomena have not been revealed. In the present study, we perform direct numerical simulations of turbulent plane Couette flow with reduced-size computational domains, where either the streamwise or the spanwise domain size is reduced to its minimal length, aiming to elucidate the role of scale interactions in each direction. Our computational results with the streamwise-minimal domain suggest that the interplays between streamwise-elongated streaks and vortices smaller than the streamwise minimal length are the essential scale interactions for both the inner and the outer structures. We further show that these streamwise-independent and smaller-scale structures exchange energy through forward and reversed interscale energy transfers, and the reversed energy transfer maintains the energy production at larger scales. Based on the resemblance between the observed Reynolds-stress transport and the scenario of the self-sustaining cycle, we conjecture that the forward and reversed energy transfers mainly represent the streak instabilities and regeneration of streamwise vortices, respectively. Furthermore, the computation with the spanwise-minimal domain indicates that the interscale energy transfers observed by one-dimensional spanwise spectral analysis are likely related to the individual dynamics of each inner and outer structure, rather than representing their interactions.

This paper solves an approximate form of conservation of mass and momentum for a turbine in a wind farm array. The solution is a fairly simple explicit relationship that predicts the streamwise velocity distribution within a wind farm with an arbitrary layout. As this model is obtained by solving flow-governing equations directly for a turbine that is subject to upwind turbine wakes, no ad hoc superposition technique is needed to predict wind farm flows. A suite of large-eddy simulations (LES) of wind farm arrays is used to examine self-similarity as well as validity of the so-called conservation of momentum deficit for turbine wakes in wind farms. The simulations are performed with and without the presence of some specific turbines in the wind farm. This allows us to systematically study some of the assumptions made to develop the analytical model. A modified version of the conservation of momentum deficit is also proposed to provide slightly better results at short downwind distances, as well as in the far wake of turbines deep inside a wind farm. Model predictions are validated against the LES data for turbines in both full-wake and partial-wake conditions. While our results highlight the limitation in capturing the flow speed-up between adjacent turbine columns, the model is overall able to acceptably predict flow distributions for a moderately sized wind farm. Finally, the paper employs the new model to provide insights on the accuracy of common wake superposition methods.

This work provides evidence that anisotropic drainage in sheared foam is at the origin of convective instability in very long foam channels. Convective instability occurs in foam under forced drainage when a critical liquid fraction is exceeded. Liquid spontaneously accumulates at one side of the channel. The weight imbalance induces convection rolls in the foam. Experiments in a very long vertical foam channel demonstrate that the critical liquid fraction is smaller than in previous findings by a factor of five. The critical liquid fraction depends on both the channel length and the inhomogeneity of the liquid feed. Well below the critical liquid fraction, a static, elastic shear deformation of the foam structure occurs. At the critical liquid fraction, initial steady convection rolls are located at the lower region of the channel and expand as the liquid fraction further increases. Combining the drainage equation with both the elastic response of the foam and a model for anisotropic drainage, a critical liquid fraction for the growth of an initial liquid imbalance is derived analytically, which corresponds very well to experimental findings. Numerical simulations of the drainage equation and the elastic response of the foam reproduce these experimental and analytical findings.

Numerical simulations are used to probe Rayleigh–Darcy convection in fluid-saturated porous media towards the ultimate regime. The present three-dimensional dataset, up to Rayleigh–Darcy number $\textit {Ra}=8\times 10^4$, suggests that the appropriate scaling of the Nusselt number is $\textit {Nu}=0.0081\textit {Ra}+0.067\textit {Ra}^{0.61}$, fitting the computed data for $\textit {Ra}\gtrsim 10^3$. Extrapolation of current predictions to the ultimate linear regime yields the asymptotic law $\textit {Nu}=0.0081 \textit {Ra}$, about $16\,\%$ less than indicated in previous studies. Upon examination of the flow structures near the boundaries, we confirm previous indications of small flow cells hierarchically nesting into supercells, and we show evidence that the supercells at the boundary are the footprints of the megaplumes that dominate the interior part of the flow. The present findings pave the way for more accurate modelling of geophysical systems, with special reference to geological $\textrm {CO}_2$ sequestration.

Permeability is a nonlinear and non-local function of the intimate coupling between pore fluid flow and solid deformation in porous media. A class of related problems involves the effect of fluid injection or withdrawal on the transport properties of geomaterials. This paper presents an analytical solution for the nonlinear and non-local problem of fluid flow through a disk-shaped porous elastic inclusion below the free surface of an elastic half-space. The solution accounts for the following nonlinear mechanisms: (i) variations in the permeability coefficient due to the flow-induced deformation of the inclusion; (ii) the inertial losses of the pore fluid flow. The former and latter mechanisms are formulated using the Green's function for a dilatation centre in an elastic half-space and the Darcy–Forchheimer equation for fluid flow through porous media, respectively. An analytical perturbation solution to the considered problem is developed and validated against the numerical finite element solution to the same problem. The described nonlinear mechanisms are represented by two dimensionless parameter groups. The extreme values of these dimensionless groups govern the solution asymptotic behaviours mimicking the special-case solutions in which either mechanism is forced to vanish. The applied aspects of the solution are demonstrated through the wellbore performance index parameter that quantifies the subsurface rock ability to deliver the pore fluid toward or away from a wellbore in a porous reservoir. Unlike the linear models, the presented nonlinear solution captures the observed dependence of the performance index on the wellbore flow rate.

The leakage from a fracture network to a surrounding medium during drainage, or backflow, driven by elastic relaxation, is considered. A network model is extended to include the effects of permeable boundaries, with the permeation through the wall assumed to be proportional to the local pressure. The regimes in which leakage is dominant relative to the parallel flow along the channel are evaluated at different times. Results show that, when the aperture of the channel is large enough, the parallel flow is greater than the permeation through the wall, and the channel thickness decreases in time, $t$, with a $t^{-1/3}$ behaviour, as reported previously. However, when the aperture is small, the channel thickness decreases exponentially in time. An asymptotic investigation of the solution for a single fracture is performed and extended to network systems. The study provides insight into the influence leakage may have on squeezing-induced flows, which is relevant to natural and engineering systems.

We present results from open-channel, Euler–Lagrange (EL) simulations of turbulent flow over an erodible particle bed at a shear Reynolds number of $Re_{\tau }=180$. Upon space and time averaging, our simulations correctly reproduce the Wong & Parker (ASCE J. Hydraul. Engng, vol. 132, issue 11, 2006, pp. 1159–1168) bedload transport relation (WP). However, local and instantaneous sediment flux shows orders of magnitude scatter around the WP prediction. Visualization of the vortical structures using swirling strength shows the existence of bed-penetrating Kelvin–Helmholtz (KH) vortex packets, which coupled with particle inertia are primarily responsible for the large scatter. The results also show that Euler–Euler (EE) simulations, where the individual sediment grains are not distinguished, are still able to reliably capture the turbulent shear stress variation, however, they do not capture the wide distribution of sediment flux indicative of saltating transport. The KH vortices induce non-zero streamwise and bed-normal velocities at the upper surface of the bed, which must be considered in EE simulations.

Regulation of fluid flow by deformations of the surrounding elastic structure is observed in many natural and artificial system, such as in the cardiovascular system. As the first step to study the regulation of oscillating flows, we consider synchronization of vortex shedding past a cylinder within an elastic structure with a sinusoidal external forcing. We use phase-reduction theory to evaluate the synchronization characteristics of the oscillating fluid–structure coupled dynamics. We find that the phase-sensitivity function, which characterizes the phase-response of the oscillation, is significantly affected by the Cauchy number and slightly affected by the fluid-to-structure density ratio and Poisson's ratio of the structure material, for fixed model configuration and Reynolds number. The predicted synchronization characteristics are in close agreement with results from direct numerical simulations. The synchronization region is maximized when the sinusoidal perturbation is applied near the downstream end of the cylinder. These findings open further possibility for the utilization of phase-reduction theory to characterize synchronization in other practical problems exhibiting fluid–structure coupled dynamics, such as in biological systems and the control of microfluidics.

We report an idealized numerical study of a melting and freezing solid adjacent to a turbulent, buoyancy-affected shear flow, in order to improve our understanding of topography generation by phase changes in the environment. We use the phase-field method to dynamically couple the heat equation for the solid with the Navier–Stokes equations for the fluid. We investigate the evolution of an initially flat and horizontal solid boundary overlying a pressure-driven turbulent flow. We assume a linear equation of state for the fluid and change the sign of the thermal expansion coefficient, such that the background density stratification is either stable, neutral or unstable. We find that channels aligned with the direction of the mean flow are generated spontaneously by phase changes at the fluid–solid interface. Streamwise vortices in the fluid, the interface topography and the temperature field in the solid influence each other and adjust until a statistical steady state is obtained. The crest-to-trough amplitude of the channels is larger than approximately 10$\delta _{\nu }$ in all cases, with $\delta _{\nu }$ the viscous length scale, but is much larger and more persistent for an unstable stratification than for a neutral or stable stratification. This happens because a stable stratification makes the cool melt fluid buoyant such that it shields the channel from further melting, whereas an unstable stratification makes the cool melt fluid sink, inducing further melting by rising hot plumes. The statistics of flow velocities and melt rates are investigated, and we find that channels and keels emerging in our simulations do not significantly change the mean drag coefficient.

An understanding of pressure propagation mechanisms is crucial in finding suitable conditions for the temporary storage, transmission and attenuation of a large pressure signal in a multi-plug gelled pipeline. The current work investigates the transient flow start in a multi-plug gelled pipeline, filled with a weakly compressible, thixotropic waxy gelled crude oil. A multi-plug gel can form either naturally or artificially. We solve time-dependent mass and momentum conservation equations for the gas phase and conservation equations together with a shear-history-dependent constitutive model for the gel phase to capture the transient phenomena. An advection equation traces the motion of the gel–gas interface using the volume of fluid method. The results of this work show that pressure propagation in the first gel plug compresses it. The compressed first gel plug further compresses the gas pocket, which delays pressure propagation in the downstream gel. The delay in the pressure propagation through the second gel plug creates a high-pressure gradient in the first gel plug. A high-pressure gradient in the first gel results in faster degradation of the first gel. Degraded gel offers a substantially lower resistance to flow, allowing a high-pressure gradient in the second gel. A steeper pressure gradient assists the breakage of the next gel plug and results in a sequential gel breakage. This sequential gel breakage allows flow restart in a long pipeline. The results also indicate that the compressibility of the gas pocket dominates the overall compressibility of the system, controlling both pressure propagation and flow-restart time.

We develop a continuous adjoint formulation for controlling the deformation of clean, neutrally buoyant droplets in Stokes flow. The focus is on surface tension-driven flows where the interface is deformed by the local fluid velocity. We apply results from shape optimization to rigorously derive the optimality conditions for a range of interfacial problems. In the cases of interest, we make use of boundary integral methods as a natural choice for the numerical discretization of the flow variables. In the static case, our methodology is tested on a tracking-type cost functional and corresponds to classic shape optimization problems. We show agreement with black-box finite difference-based gradients and accurate minimization of the cost functionals. Finally, we demonstrate the methodology on the control of unsteady droplet deformation through external forcing.

The study of a turbulent premixed flame often involves analysing quantities conditioned to different iso-surfaces of a reactive scalar field. Under the influence of turbulence, such a surface is deformed and translated. To track the surface motion, the displacement speed ($S_d$) of the scalar field respective to the local flow velocity is widely used and this quantity is currently receiving growing attention. Inspired by the apparent benefits from a simple decomposition of $S_d$ into contributions due to (i) curvature, (ii) normal diffusion and (iii) chemical reaction, this work aims at deriving and exploring new evolution equations for these three contributions averaged over the reaction surface. Together with a previously obtained $S_d$-evolution equation, the three new equations are presented in a form that emphasizes the decomposition of $S_d$ into three terms. This set of equations is also supplemented with a curvature-evolution equation, hence providing a new perspective to link the flame topology and its propagation characteristics. Using two direct numerical simulation databases obtained from constant-density and variable-density reaction waves, all the derived equations and the term-wise decomposition relations are demonstrated to hold numerically. Comparison of the simulated results indicates that the thermal expansion weakly affects the key terms in the considered evolution equations. Thermal expansion can cause variations in the averaged $S_d$ and its decomposed parts through multiple routes more than introducing a dilatation term. The flow plays a major role to influence the key terms in all equations except the curvature one, due to a cancellation between negatively and positively curved surface elements.

We use direct numerical simulation (DNS) to study drag reduction in a lubricated channel, a flow instance in which a thin layer of lubricating fluid is injected in the near-wall region so as to favour the transportation of a primary fluid. In the present configuration, the two fluids have equal density but different viscosity, so that a viscosity ratio $\lambda =\eta _1/\eta _2$ can be defined. To cover a meaningful range of possible situations, we consider five different $\lambda$ in the range $0.25 \le \lambda \le 4$. All DNS are run using the constant power input (CPI) approach, which prescribes that the flow rate is adjusted according to the actual pressure gradient so as to keep constant the power injected into the flow. The CPI approach has been purposely extended here for the first time to the case of multiphase flows. A phase-field method is used to describe the dynamics of the liquid–liquid interface. We unambiguously show that a significant drag reduction (DR) can be achieved for $\lambda \le 2.00$. Reportedly, the observed DR is a non-monotonic function of $\lambda$ and, in the present case, is maximum for $\lambda = 1.00$ (${\simeq }13\,\%$ flow-rate increase). Upon a detailed analysis of the energy budgets, we are able to show the existence of two different DR mechanisms. For $\lambda =1.00$ and $\lambda =2.00$, DR is purely due to the effect of the surface tension – a localized elasticity element that separates the two fluids – which, decoupling the wall-normal momentum transfer mechanisms between the primary and the lubricating layer, suppresses turbulence in the lubricating layer (laminarization) and reduces the overall drag. For $\lambda < 1.00$, turbulence can be sustained in the lubricating layer, because of the increased local Reynolds number. In this case, DR is simply due to the smaller viscosity of the lubricating layer that acts to decrease directly the corresponding wall friction. Finally, we show evidence that an upper bound for $\lambda$ exists, for which DR cannot be observed: for $\lambda =4.00$, we report a slight drag enhancement, thereby indicating that the turbulence suppression observed in the lubricating layer cannot completely balance the increased friction due to the larger viscosity.

Given a passive tracer distribution $f(x,y)$, what is the simplest unstirred pattern that may be reached under incompressible advection? This question is partially motivated by recent studies of three-dimensional (3-D) magnetic reconnection, in which the patterns of a topological invariant called the field line helicity greatly simplify until reaching a relaxed state. We test two approaches: a variational method with minimal constraints, and a magnetic relaxation scheme where the velocity is determined explicitly by the pattern of $f$. Both methods achieve similar convergence for simple test cases. However, the magnetic relaxation method guarantees a monotonic decrease in the Dirichlet seminorm of $f$, and is numerically more robust. We therefore apply the latter method to two complex mixed patterns modelled on the field line helicity of 3-D magnetic braids. The unstirring separates $f$ into a small number of large-scale regions determined by the initial topology, which is well preserved during the computation. Interestingly, the velocity field is found to have the same large-scale topology as $f$. Similarity to the simplification found empirically in 3-D magnetic reconnection simulations supports the idea that advection is an important principle for field line helicity evolution.

We derive an analytical solution for the acoustic streaming inside a rigid tube resulting from a pseudo-standing wave field, generated by two counterpropagating travelling waves. We solve the second-order axisymmetric problem that follows from the perturbation expansion of the governing equations. In the process, we impose no restriction on the diameter of the tube with respect to the thickness of the viscous boundary layer and acoustic wavelength. The derived solution is then used to study the evolution of streaming patterns inside the tube when geometrical and material parameters are varied. We show how the Schlichting streaming torus at the wall bounds the Rayleigh streaming near the axis of the tube. Decreasing the ratio ($\varXi$) of the tube radius to the viscous boundary layer thickness gradually expands the Schlichting streaming, suppressing the Rayleigh streaming. Considering the average mass transport velocity, the Rayleigh streaming vanishes at the critical ratio $\varXi _{{S}}^{{M}}=6.2$. The critical ratio is independent of fluid properties in the limit of large acoustic wavelength relative to the radius of the tube ($\varLambda$). When $\varLambda$ decreases towards unity, large-scale Eckart-like streaming develops near the axis, superseding the Rayleigh streaming, while the Schlichting streaming remains at the wall. In addition, we demonstrate the relevance of the compressibility of the streaming flow and of the full inclusion of the spatial variation of the Reynolds stress that acts as the streaming source. The study is especially relevant for microfluidic systems, wherein the viscous boundary layer can reach significant thicknesses.

We employ reconnection-capable, vortex filament methods and finite-volume, Navier–Stokes flow solvers to investigate the topological and helicity dynamics of vortex links for medium and high Reynolds numbers. Our vortex-dynamical model is based on discretization of vortex tubes into bundles of numerical analogues of vortex lines. Due to their nearly singular nature, the numerical vortex lines have topological writhe but not twist. By means of our reconnecting vortex tube model, it is shown that the helicity of a vortex link is conserved during the unknotting process. The dynamics of linking and writhe topological measures indicate that most of the initial linking becomes writhe during the post-reconnection evolution. The helicity spectra of the vortex link present alternating-sign helicity fluctuations at all (potential flow) scales up to the vortex core. At pre-reconnection times, these fluctuations are damped by Biot–Savart vortex stretching and helicity becomes single signed. The post-reconnection spectra indicate an inverse helicity cascade corresponding to the creation of a homogenized vortex blob, a process reminiscent of coherent structure formation in turbulence. An accompanying Navier–Stokes calculation of vortex link dynamics at Reynolds numbers $Re=1500$ confirms the post-reconnection transformation of linking into different topological measures, the pre-reconnection damping of helicity spectra fluctuations and the spectral shift to low wavenumbers at post-reconnection times. Due to viscous dissipation action, however, this shift is accompanied by progressive reduction of helicity peak values.

Spontaneous capillary flow of liquids in narrow spaces plays a key role in a plethora of applications including lab-on-a-chip devices, heat pipes, propellant management devices in spacecrafts and flexible printed electronics manufacturing. In this work we use a combination of theory and experiment to examine capillary-flow dynamics in open rectangular microchannels, which are often found in these applications. Scanning electron microscopy and profilometry are used to highlight the complexity of the free-surface morphology. We develop a self-similar lubrication-theory-based model accounting for this complexity and compare model predictions to those from the widely used modified Lucas–Washburn model, as well as experimental observations over a wide range of channel aspect ratios $\lambda$ and equilibrium contact angles $\theta _0$. We demonstrate that for large $\lambda$ the two model predictions are indistinguishable, whereas for smaller $\lambda$ the lubrication-theory-based model agrees better with experiments. The lubrication-theory-based model is also shown to have better agreement with experiments at smaller $\theta _0$, although as $\theta _0\rightarrow {\rm \pi}/4$ it fails to account for important axial curvature contributions to the free surface and the agreement worsens. Finally, we show that the lubrication-theory-based model also quantitatively predicts the dynamics of fingers that extend ahead of the meniscus. These findings elucidate the limitations of the modified Lucas–Washburn model and demonstrate the importance of accounting for the effects of complex free-surface morphology on capillary-flow dynamics in open rectangular microchannels.