Motivated by subsurface carbon sequestration, an experimental investigation of dissolution-driven Rayleigh–Darcy convection using two miscible fluids in a Hele-Shaw cell is conducted. A thin horizontal layer of circular impermeable discs is inserted to create an environment with heterogeneous and anisotropic permeability. The Sherwood number that measures the convective mass transfer rate between two fluids at the interface is linked to different parameters of the disc layer, including the disc size, spacing, layer permeability and its relative height with respect to the fluid interface. It is surprising that the convective mass transfer rate in our configuration is dominated by the disc spacing, but almost independent of either the disc size or the mean permeability of the layer. To explain this dependence, the convective mass transfer rate is decomposed into the number, velocity and density contrast of fingers travelling through the disc layer. Both the number and density contrast of fingers show dependences on the disc layer permeability, even though the product of them, the mass transfer rate, does not. In addition, the density contrast also shows a non-monotonic dependence on the disc spacing. The transition point is at a spacing that is close to the finger width. Based on this observation, a simple model based on mixing and scale competition is proposed, and it shows an excellent agreement with the experimental results.

The statistical mechanical description of two-dimensional inviscid fluid turbulence is reconsidered. Using this description, we make predictions about turbulent flow in a rapidly rotating laboratory annulus. Measurements on the continuously forced, weakly dissipative flow reveal coherent vortices in a mean zonal flow. Statistical mechanics has two crucial requirements for equilibrium: statistical independence of macro-cells (subsystems) and additivity of invariants of macro-cells. We investigate these requirements in the context of the annulus experiment. The energy invariant, an extensive quantity, should thus be additive, i.e. the interaction energy between a macro-cell and the rest of the system (reservoir) should be small, and this is verified experimentally. Similarly, we use additivity to select the appropriate Casimir invariants from the infinite set available in vortex dynamics, and we do this in such a way that the exchange of micro-cells within a macro-cell does not alter an invariant of a macro-cell. A novel feature of the present study is our choice of macro-cells, which are continuous phase-space curves based on mean values of the streamfunction. Quantities such as energy and enstrophy can be defined on each curve, and these lead to a local canonical distribution that is also defined on each curve. The distribution obtained describes the anisotropic and inhomogeneous properties of a flow. Our approach leads to the prediction that on a mean streamfunction curve there should be a linear relation between the ensemble-averaged potential vorticity and the time-averaged streamfunction, and our laboratory data are in good accord with this prediction. Further, the approach predicts that although the probability distribution function for potential vorticity in the entire system is non-Gaussian, the distribution function of micro-cells should be Gaussian on the macro-cells, i.e. for curves defined by mean values of the streamfunction. This prediction is also supported by the data. While the statistical mechanics approach used was motivated by and applied to experiments on turbulence in a rotating annulus, the approach is quite general and is applicable to a large class of Hamiltonian systems, including drift-wave plasma models, Vlasov–Poisson dynamics, and kinetic theories of stellar dynamics.

This paper is concerned with the boundary-layer separation in subsonic and transonic flows caused by a two-dimensional isolated wall roughness. The process of the separation is analysed by means of two approaches: the direct numerical simulation (DNS) of the flow using the Navier–Stokes equations, and the numerical solution of the triple-deck equations. Since the triple-deck theory relies on the assumption that the Reynolds number ($\mathit{Re}$) is large, we performed the Navier–Stokes calculations at $\mathit{Re}=4\times 10^{5}$ based on the distance of the roughness element from the leading edge of the flat plate. This $\mathit{Re}$ is also relevant for aeronautical applications. Two sets of calculation were conducted with the free-stream Mach number $\mathit{Ma}_{\infty }=0.5$ and $\mathit{Ma}_{\infty }=0.87$. We used different roughness element heights, some of which were large enough to cause a well-developed separation region behind the roughness. We found that the two approaches generally compare well with one another in terms of wall shear stress, longitudinal pressure gradient and detachment/reattachment points of the separation bubbles (when present). The main differences were found in proximity to the centre of the roughness element, where the wall shear stress and longitudinal pressure gradient predicted by the triple-deck theory are noticeably different from those predicted by DNS. In addition, DNS predicts slightly longer separation regions.

We prove the first rigorous bound on the heat transfer for three-dimensional Rayleigh–Bénard convection of finite-Prandtl-number fluids between free-slip boundaries with an imposed heat flux. Using the auxiliary functional method with a quadratic functional, which is equivalent to the background method, we prove that the Nusselt number $\mathit{Nu}$ is bounded by $\mathit{Nu}\leqslant 0.5999\mathit{R}^{1/3}$ uniformly in the Prandtl number, where $\mathit{R}$ is the Rayleigh number based on the imposed heat flux. In terms of the Rayleigh number based on the mean vertical temperature drop, $\mathit{Ra}$, we obtain $\mathit{Nu}\leqslant 0.4646\mathit{Ra}^{1/2}$. The scaling with Rayleigh number is the same as that of bounds obtained with no-slip isothermal, free-slip isothermal and no-slip fixed-flux boundaries, and numerical optimisation of the bound suggests that it cannot be improved within our bounding framework. Contrary to the two-dimensional case, therefore, the $\mathit{Ra}$-dependence of rigorous upper bounds on the heat transfer obtained with the background method for three-dimensional Rayleigh–Bénard convection is insensitive to both the thermal and the velocity boundary conditions.

The micro-convection caused by the ponderomotive forces of the self-magnetic field in a magnetic fluid is studied here both numerically and experimentally. The theoretical approach based on the general Brinkman model substantially improves the description with respect to the previously proposed Darcy model. The predictions of both models are here compared to finely controlled experiments. The Brinkman model, in contrast to the Darcy model, allows us to describe the formation of mushrooms on the plumes of the micro-convective flow and the width of the fingers. In the Brinkman approach, excellent quantitative agreement is also obtained for the finger velocity dynamics and the velocity maximal values as a function of the magnetic Rayleigh number. The diffusion coefficient of particles of the water-based magnetic colloid determined by the threshold field strength value of the micro-convection is significantly larger than the diffusion coefficient of individual particles. This result is confirmed by independent measurements of the diffusion coefficient at the smearing of the diffusion front.

We experimentally investigated streamwise variations of turbulent dynamics in drag-reducing turbulent boundary layer flows following the injection of non-ionic surfactant solutions, which mainly consisted of oleyldimethylamine oxide. We focus on the comparison of turbulence statistics between injected (i.e. heterogeneous) and premixed (i.e. homogeneous) surfactant solutions, in which the maximum drag reduction ratio of 50 % is the same at the most downstream position for both cases. The wall-normal profiles of turbulence statistics, such as streamwise and wall-normal turbulence intensities, seem to be noticeably different between heterogeneous and homogeneous surfactant solutions. However, streamwise variations in these maxima and the wall-normal locations are essentially similar to one another, except for the maximum of streamwise turbulence intensity, which is not arranged by the amount of drag reduction and is also dependent on the normalization due to outer and inner variables. Such complex behaviour of streamwise turbulence intensity would be caused by the formation of near-wall layered structures that are parallel to the wall. For both heterogeneous and homogeneous surfactant solutions, the streamwise variation in the drag reduction ratio corresponds well to those of the mean velocity in wall units and the wall-normal locations of maxima of streamwise and wall-normal turbulence intensities with both outer and inner scaling. Unlike the Reynolds shear stress, the correlation coefficient of the streamwise and wall-normal turbulent fluctuations is correlated well with the drag reduction ratio. We present plausible pictures of the development of turbulence structures such as hairpin vortices and low-speed streaks for the drag-reducing turbulent boundary layer in heterogeneous and homogeneous surfactant solutions, which are comprehensively derived from the present set of experimental measurements such as flow visualization, planar laser-induced fluorescence, two-component laser-Doppler velocimetry and particle image velocimetry on the streamwise and wall-normal plane and the streamwise and spanwise plane.

Identifying the convective/absolute instability nature of a local base flow requires an analysis of its linear impulse response. One must find the appropriate singularity in the eigenvalue problem with complex frequencies and wavenumbers and prove causality. One way to do so is to show that the appropriate integration contour of this response, a steepest decent path through the relevant singularity, exists. Due to the inherent difficulties of such a proof, one often verifies instead whether this singularity satisfies the collision criterion. In other words, one must show that the branches involved in the formation of this singularity come from distinct halves of the complex wavenumber plane. However, this graphical search is computationally intensive in a single plane and essentially prohibitive in two planes. A significant computational cost reduction can be achieved when root finding procedures are applied instead of graphical ones to search for singularities. They focus on locating these points, with causality being verified graphically a posteriori for a small parametric sample size. The use of root-finding procedures require auxiliary equations, often derived by applying the zero group velocity conditions to the dispersion relation. This relation, in turn, is derived by applying matrix forming to the differential eigenvalue problem and taking the determinant of the resulting system of algebraic equations. Taking the derivative of the dispersion relation with respect to the wavenumbers generates the auxiliary equations. If the algebraic system is decoupled, this derivation is straightforward. However, its computational cost is often prohibitive when the algebraic system is coupled. Other methods exist, but often they can also be too costly and/or not reliable for two wavenumber plane searches. This paper describes an alternative methodology based on sensitivity analysis and adjoints that allow the zero group velocity conditions to be applied directly to the differential eigenvalue problem. In doing so, the direct and auxiliary differential eigenvalue problems can be solved simultaneously using standard shooting methods to directly locate singularities. Auxiliary dispersion relations no longer have to be derived, although it is shown that they are the algebraic equivalent of the auxiliary differential eigenvalue problems obtained by this alternative methodology. Using the latter dramatically reduces computational costs. The search for arbitrary singularities is then not only accelerated in single wavenumber planes but it also becomes viable in two wavenumber planes. Finally, the new method also allows group velocity calculations, greatly facilitating the verification of causality. Several test cases are presented to illustrate the capabilities of this new method.

Surface water waves are considered propagating over highly variable non-smooth topographies. For this three-dimensional problem a Dirichlet-to-Neumann (DtN) operator is constructed reducing the numerical modelling and evolution to the two-dimensional free surface. The corresponding discrete Fourier integral operator is defined through a matrix decomposition. The topographic component of the decomposition requires special care, and a Galerkin method is provided accordingly. One-dimensional numerical simulations, along the free surface, validate the DtN formulation in the presence of a large-amplitude rapidly varying topography. An alternative conformal-mapping-based method is used for benchmarking. A two-dimensional simulation in the presence of a Luneburg lens (a particular submerged mound) illustrates the accurate performance of the three-dimensional DtN operator.

Earth’s magnetic field is generated in its fluid metallic core through motional induction in a process termed the geodynamo. Fluid flow is heavily influenced by a combination of rapid rotation (Coriolis forces), Lorentz forces (from the interaction of electrical currents and magnetic fields) and buoyancy; it is believed that the inertial force and the viscous force are negligible. Direct approaches to this regime are far beyond the reach of modern high-performance computing power, hence an alternative ‘reduced’ approach may be beneficial. Taylor (Proc. R. Soc. Lond. A, vol. 274 (1357), 1963, pp. 274–283) studied an inertia-free and viscosity-free model as an asymptotic limit of such a rapidly rotating system. In this theoretical limit, the velocity and the magnetic field organize themselves in a special manner, such that the Lorentz torque acting on every geostrophic cylinder is zero, a property referred to as Taylor’s constraint. Moreover, the flow is instantaneously and uniquely determined by the buoyancy and the magnetic field. In order to find solutions to this mathematical system of equations in a full sphere, we use methods of optimal control to ensure that the required conditions on the geostrophic cylinders are satisfied at all times, through a conventional time-stepping procedure that implements the constraints at the end of each time step. A derivative-based approach is used to discover the correct geostrophic flow required so that the constraints are always satisfied. We report a new quantity, termed the Taylicity, that measures the adherence to Taylor’s constraint by analysing squared Lorentz torques, normalized by the squared energy in the magnetic field, over the entire core. Neglecting buoyancy, we solve the equations in a full sphere and seek axisymmetric solutions to the equations; we invoke $\unicode[STIX]{x1D6FC}$- and $\unicode[STIX]{x1D714}$-effects in order to sidestep Cowling’s anti-dynamo theorem so that the dynamo system possesses non-trivial solutions. Our methodology draws heavily on the use of fully spectral expansions for all divergenceless vector fields. We employ five special Galerkin polynomial bases in radius such that the boundary conditions are honoured by each member of the basis set, whilst satisfying an orthogonality relation defined in terms of energies. We demonstrate via numerous examples that there are stable solutions to the equations that possess a rapidly decreasing spectrum and are thus well-converged. Classic distributions for the $\unicode[STIX]{x1D6FC}$- and $\unicode[STIX]{x1D714}$-effects are invoked, as well as new distributions. One such new $\unicode[STIX]{x1D6FC}$-effect model possesses oscillatory solutions for the magnetic field, rarely before seen. By comparing our Taylor state model with one that allows torsional oscillations to develop and decay, we show the equilibrium state of both configurations to be coincident. In all our models, the geostrophic flow dominates the ageostrophic flow. Our work corroborates some results previously reported byWu & Roberts (Geophys. Astrophys. Fluid Dyn., vol. 109 (1), 2015, pp. 84–110), as well as presenting new results; it sets the stage for a three-dimensional implementation where the system is driven by, for example, thermal convection.

Buoyant jets or forced plumes are discharged into a turbulent ambient in many natural and engineering applications. The background turbulence generally affects the mixing characteristics of the buoyant jet, and the extent of the influence depends on the characteristics of both the jet discharge and ambient. Previous studies focused on the experimental investigation of the problem (for pure jets or plumes), but the findings were difficult to generalize because suitable scales for normalization of results were not known. A model to predict the buoyant jet mixing in the presence of background turbulence, which is essential in many applications, is also hitherto not available even for a background of homogeneous and isotropic turbulence (HIT). We carried out experimental and theoretical investigations of a buoyant jet discharging into background HIT. Buoyant jets were designed to be in the range of $1<z/l_{M}<5$, where $l_{M}=M_{o}^{3/4}/F_{o}^{1/2}$ is the momentum length scale, with $z/l_{M}<\sim 1$ and $z/l_{M}>\sim 6$ representing the asymptotic cases of pure jets and plumes, respectively. The background turbulence was generated using a random synthetic jet array, which produced a region of approximately isotropic and homogeneous field of turbulence to be used in the experiments. The velocity scale of the jet was initially much higher, and the length scale smaller, than that of the background turbulence, which is typical in most applications. Comprehensive measurements of the buoyant jet mixing characteristics were performed up to the distance where jet breakup occurred. Based on the experimental findings, a critical length scale $l_{c}$ was identified to be an appropriate normalizing scale. The momentum flux of the buoyant jet in background HIT was found to be conserved only if the second-order turbulence statistics of the jet were accounted for. A general integral jet model including the background HIT was then proposed based on the conservation of mass (using the entrainment assumption), total momentum and buoyancy fluxes, and the decay function of the jet mean momentum downstream. Predictions of jet mixing characteristics from the new model were compared with experimental observation, and found to be generally in agreement with each other.

Fully developed turbulent free shear layers exhibit a high degree of order, characterized by large-scale coherent structures in the form of spanwise vortex rollers. Extensive experimental investigations show that such organized motions bear remarkable resemblance to instability waves, and their main characteristics, including the length scales, propagation speeds and transverse structures, are reasonably well predicted by linear stability analysis of the mean flow. In this paper, we present a mathematical theory to describe the nonlinear dynamics of coherent structures. The formulation is based on the triple decomposition of the instantaneous flow into a mean field, coherent fluctuations and small-scale turbulence but with the mean-flow distortion induced by nonlinear interactions of coherent fluctuations being treated as part of the organized motion. The system is closed by employing a gradient type of model for the time- and phase-averaged Reynolds stresses of fine-scale turbulence. In the high-Reynolds-number limit, the nonlinear non-equilibrium critical-layer theory for laminar-flow instabilities is adapted to turbulent shear layers by accounting for (1) the enhanced non-parallelism associated with fast spreading of the mean flow, and (2) the influence of small-scale turbulence on coherent structures. The combination of these factors with nonlinearity leads to an interesting evolution system, consisting of coupled amplitude and vorticity equations, in which non-parallelism contributes the so-called translating critical-layer effect. Numerical solutions of the evolution system capture vortex roll-up, which is the hallmark of a turbulent mixing layer, and the predicted amplitude development mimics the qualitative feature of oscillatory saturation that has been observed in a number of experiments. A fair degree of quantitative agreement is obtained with one set of experimental data.

In fully developed single-phase turbulent flow in straight pipes, it is known that mean motions can occur in the plane of the pipe cross-section, when the cross-section is non-circular, or when the wall roughness is non-uniform around the circumference of a circular pipe. This phenomenon is known as secondary flow of the second kind and is associated with the anisotropy in the Reynolds stress tensor in the pipe cross-section. In this work, we show, using careful laser Doppler anemometry experiments, that secondary flow of the second kind can also be promoted by a non-uniform non-axisymmetric particle-forcing, in a fully developed turbulent flow in a smooth circular pipe. In order to isolate the particle-forcing from other phenomena, and to prevent the occurrence of mean particle-forcing in the pipe cross-section, which could promote a different type of secondary flow (secondary flow of the first kind), we consider a simplified well-defined situation: a non-uniform distribution of particles, kept at fixed positions in the ‘bottom’ part of the pipe, mimicking, in a way, the particle or droplet distribution in horizontal pipe flows. Our results show that the particles modify the turbulence through ‘direct’ effects (associated with the wake of the particles) and ‘indirect’ effects (associated with the global balance of momentum and the turbulence dynamics). The resulting anisotropy in the Reynolds stress tensor is shown to promote four secondary flow cells in the pipe cross-section. We show that the secondary flow is determined by the projection of the Reynolds stress tensor onto the pipe cross-section. In particular, we show that the direction of the secondary flow is dictated by the gradients of the normal Reynolds stresses in the pipe cross-section, and . Finally, a scaling law is proposed, showing that the particle-driven secondary flow scales with the root of the mean particle-forcing in the axial direction, allowing us to estimate the magnitude of the secondary flow.

The Lagrangian L for gravity waves of finite amplitude in an N-layer, stratified shear flow is constructed as a functional of the generalized coordinates qv(t)≡ {qnv(t)} of the N + 1 interfaces, where the qnv are the Fourier coefficients in the expansion of the interfacial displacement ηv(x, t) in a complete, orthogonal set {ψn(x)}. The explicit expansion of L is constructed through fourth-order in the qnν and $\dot{q}_{n}^{\nu}$. Progressive interfacial waves and Kelvin–Helmholtz instability in a two-layer fluid are examined, and the earlier results of Drazin (1970), Nayfeh & Saric (1972) and Weissman (1979) are extended to finite depth. It is found that the pitchfork bifurcation associated with the critical point for Kelvin–Helmholtz instability, which is supercritical for infinitely deep layers, may be subcritical (inverted) for finite depths. The evolution equations that govern Kelvin–Helmholtz waves in the parametric neighbourhood of this critical point are shown to be equivalent to those for a particle in a two-parameter, central force field. The effect of surface tension is examined in an Appendix. Finally, the wave motion forced by flow over a sinusoidal bottom (as in Thorpe's tilting tank) is examined and the corresponding resonance curves and Hopf bifurcations determined. Numerical integrations reveal that stable limit cycles exist in some parametric neighbourhoods of these bifurcations. Period doubling was observed but did not lead to chaotic motion.

Direct numerical simulations (DNS) are performed to analyse the global transport properties of turbulent Taylor–Couette flow with inner rough wall up to Taylor number$Ta=10^{10}$. The dimensionless torque $Nu_{\unicode[STIX]{x1D714}}$ shows an effective scaling of $Nu_{\unicode[STIX]{x1D714}}\propto Ta^{0.42\pm 0.01}$, which is steeper than the ultimate regime effective scaling $Nu_{\unicode[STIX]{x1D714}}\propto Ta^{0.38}$ seen for smooth inner and outer walls. It is found that at the inner rough wall, the dominant contribution to the torque comes from the pressure forces on the radial faces of the rough elements; while viscous shear stresses on the rough surfaces contribute little to $Nu_{\unicode[STIX]{x1D714}}$. Thus, the log layer close to the rough wall depends on the roughness length scale, rather than on the viscous length scale. We then separate the torque contributed from the smooth inner wall and the rough outer wall. It is found that the smooth wall torque scaling follows $Nu_{s}\propto Ta_{s}^{0.38\pm 0.01}$, in excellent agreement with the case where both walls are smooth. In contrast, the rough wall torque scaling follows $Nu_{r}\propto Ta_{r}^{0.47\pm 0.03}$, very close to the pure ultimate regime scaling $Nu_{\unicode[STIX]{x1D714}}\propto Ta^{1/2}$. The energy dissipation rate at the wall of an inner rough cylinder decreases significantly as a consequence of the wall shear stress reduction caused by the flow separation at the rough elements. On the other hand, the latter shed vortices in the bulk that are transported towards the outer cylinder and dissipated. Compared to the purely smooth case, the inner wall roughness renders the system more bulk dominated and thus increases the effective scaling exponent.

The global temporal stability of three-dimensional compressible flow about a yawed parabolic body of infinite span is investigated using an iterative eigenvalue technique in combination with direct numerical simulations. The computed global spectrum provides a comprehensive picture of the temporal perturbation dynamics of the flow, and a wide and rich variety of modes has been uncovered for the investigated parameter choices: stable and unstable boundary-layer modes, different types of stable and unstable acoustic modes, and stable wavepacket modes have been found. A parameter study varying the spanwise perturbation wavenumber and the sweep Reynolds number reproduced a preferred spanwise length scale and a critical Reynolds number for a boundary-layer or acoustic instability. Convex leading-edge curvature has been found to have a strongly stabilizing effect on boundary-layer modes but only a weakly stabilizing effect on acoustic modes. Furthermore, for certain parameter choices, the acoustic modes have been found to dominate the boundary-layer modes.