A yield stress is added to Taylor's (1952, Proc. Royal Soc. A, 211, 225-239) model of a microscopic organism with a wavy cylindrical tail swimming through a viscous fluid. Viscoplastic slender-body theory is employed for the task, generalising existing results for Bingham fluid to the Herschel–Bulkley constitutive model. Numerical solutions are provided over a range of the two key parameters of the problem: the wave amplitude relative to the wavelength, and a Bingham number which describes the strength of the yield stress. Numerical solutions are supplemented with discussions of various limits of the problem in which analytical progress is possible. If the wave amplitude is sufficiently small, the yield stress of the material inevitably dominates the flow; the resulting ‘plastic locomotion’ results in swimming speeds that depend strongly on the swimming gait, and can, in some cases, even be negative. Conversely, when the yield stress is large, swimming becomes possible at the wave speed, with the swimmer sliding or burrowing along its centreline with a relatively high efficiency.

The reflection of rightward moving shocks (RMSs) belonging to the first and second families, over an initially steady oblique shock wave (SOSW) produced by a wedge, is studied in this paper. Various possible combinations of primary reflection (reflection at the intersection point of the RMS and the SOSW) and secondary reflection (reflection, on the wedge, of reflected shock waves of the primary reflection) are identified and the transition conditions are studied. For an RMS of the first family, the shock reflection problem can be shown to be equivalent to a shock interference problem. If the wedge angle is large, then the problem is equivalent to a shock interaction problem with two incident shock waves of the same family so that we have type VI, type V and type IV shock interferences. Interestingly, when the wedge angle is small enough, deflection angle reversal is observed for the SOSW so that the right part of the SOSW can no longer be regarded as one incident shock wave. It is now the left part of the SOSW that becomes one incident shock wave. As a result, for a small wedge angle, type I or type II shock interference is observed. If the RMS belongs to the second family, then the primary reflection may have regular and Mach reflections, and one reflected shock of this primary reflection reflects over the wall as another pseudo-steady shock reflection, while the other reflected shock wave may be smooth or have a kink or a triple point, as in single, transitional and double Mach reflection of pseudo-steady shock reflection.

Geophysical turbulent flows, characterized by rapid rotation, quantified by small Rossby number, and stable stratification, often self-organize into a collection of coherent vortices, referred to as a vortex gas. The lowest-order asymptotic expansion in Rossby number is quasigeostrophy, which has purely horizontal velocities and cyclone–anticyclone antisymmetry. Ageostrophic effects are important components of many geophysical flows and, as such, these phenomena are not well modelled by quasigeostrophy. The next-order correction in Rossby number, which includes ageostrophic effects, is the so-called balanced dynamics. Balanced dynamics includes ageostrophic vertical velocity and breaks the geostrophic cyclone–anticyclone antisymmetry. Point-vortex solutions are well known in two-dimensional and quasigeostrophic dynamics and are useful for studying the vortex-gas regime of geophysical turbulence. Here, we find point-vortex solutions in fully three-dimensional continuously stratified QG$^{+1}$ dynamics, a particular formulation of balanced dynamics. Simulations of QG$^{+1}$ point vortices show several interesting features not captured by quasigeostrophic point vortices including significant vertical transport on long time scales. The ageostrophic component of QG$^{+1}$ point vortex point-vortex dynamics renders them useful in modelling flows where quasigeostrophy filters out important physical processes.

In the context of three-dimensional oceanic internal waves, taking topographic effects into account, a modified Benney–Luke equation is proposed for describing internal wave–wave interactions on a sloping bottom. The derived equation is characterised by isotropy and bi-directional propagation, which are absent in the widely used Kadomtsev–Petviashvili equation. Indeed, these disparities are confirmed by numerical results of the diffraction of a truncated internal solitary wave and the evolution of a partially bent solitary wave. However, a good agreement between the numerical results of the modified Benney–Luke equation and those of the primitive equations confirms the validity of our simplified model. Because the stratification in a realistic ocean environment is usually continuous, in contrast to the assumption of a sharp density discontinuity used here, to maintain the kinematical equivalence, a layering scheme for determining the density and thickness of each layer from a continuous stratification is proposed. In addition, the occasionally observed but rarely examined X-shaped internal wave–wave interactions are shown to feature novel wave patterns, where topographic effects modulate the propagation speed, amplitude and waveform.

In this paper, we propose an efficient method for generating turbulent inflow conditions based on deep neural networks. We utilise the combination of a multiscale convolutional auto-encoder with a subpixel convolution layer (${\rm MSC}_{\rm {SP}}$-AE) and a long short-term memory (LSTM) model. Physical constraints represented by the flow gradient, Reynolds stress tensor and spectral content of the flow are embedded in the loss function of the ${\rm MSC}_{\rm {SP}}$-AE to enable the model to generate realistic turbulent inflow conditions with accurate statistics and spectra, as compared with the ground truth data. Direct numerical simulation (DNS) data of turbulent channel flow at two friction Reynolds numbers $Re_{\tau } = 180$ and 550 are used to assess the performance of the model obtained from the combination of the ${\rm MSC}_{\rm {SP}}$-AE and the LSTM model. The model exhibits a commendable ability to predict instantaneous flow fields with detailed fluctuations and produces turbulence statistics and spectral content similar to those obtained from the DNS. The effects of changing various salient components in the model are thoroughly investigated. Furthermore, the impact of performing transfer learning (TL) using different amounts of training data on the training process and the model performance is examined by using the weights of the model trained on data of the flow at $Re_{\tau } = 180$ to initialise the weights for training the model with data of the flow at $Re_{\tau } = 550$. The results show that by using only 25% of the full training data, the time that is required for successful training can be reduced by a factor of approximately 80% without affecting the performance of the model for the spanwise velocity, wall-normal velocity and pressure, and with an improvement of the model performance for the streamwise velocity. The results also indicate that using physics-guided deep-learning-based models can be efficient in terms of predicting the dynamics of turbulent flows with relatively low computational cost.

Mechanical spark is an important ignition source in various industrial processes involving combustible mixtures and it may cause serious safety issues. In this work, we analysed the ignition induced by hot particles in a combustible mixture. In Semenov's transient ignition criterion, we introduced a hypothetical heat loss coefficient accounting for the temperature inhomogeneity and obtained a revised ignition criterion which allows us to calculate the ignition delay time. Explicit expressions for the critical ignition temperature were derived and used to demonstrate the primary impacts of temperature inhomogeneity on the ignition process. Consistent with experimental and numerical results, the temperature inhomogeneity is intensified by either reducing the particle size or convective heat transfer at the particle surface, resulting in an increase of the critical ignition temperature. For flow separation on the particle surface, the boundary layer problem was solved based on a Blasius series. A temperature gradient for ignition was defined at the location of flow separation to reproduce the experimentally observed phenomenon that ignition prefers to occur first near the flow separation position. It is shown that the unsteadiness of particle cooling makes negligible contribution to the ignition process because of the exceedingly large density ratio between the particle and the ambient gas. In addition, the finite residence time of ignition for a fluid parcel due to its elevation from the particle surface leads to additional growth in the critical ignition temperature. However, such a correction appears to be inconsequential because the ignition of the fluid parcel restricted to the Frank–Kamenetskii region is close to the particle surface.

We analyse the pressure-driven flow of the Oldroyd-B fluid in slowly varying arbitrarily shaped, narrow channels and present a theoretical framework for calculating the relationship between the flow rate $q$ and pressure drop $\Delta p$. We first identify the characteristic scales and dimensionless parameters governing the flow in the lubrication limit. Employing a perturbation expansion in powers of the Deborah number ($De$), we provide analytical expressions for the velocity, stress and the $q$–$\Delta p$ relation in the weakly viscoelastic limit up to $O(De^2)$. Furthermore, we exploit the reciprocal theorem derived by Boyko $\&$ Stone (Phys. Rev. Fluids, vol. 6, 2021, L081301) to obtain the $q$–$\Delta p$ relation at the next order, $O(De^3)$, using only the velocity and stress fields at the previous orders. We validate our analytical results with two-dimensional numerical simulations in the case of a hyperbolic, symmetric contracting channel and find excellent agreement. While the velocity remains approximately Newtonian in the weakly viscoelastic limit (i.e. the theorem of Tanner and Pipkin), we reveal that the pressure drop strongly depends on the viscoelastic effects and decreases with $De$. We elucidate the relative importance of different terms in the momentum equation contributing to the pressure drop along the symmetry line and identify that a pressure drop reduction for narrow contracting geometries is primarily due to gradients in the viscoelastic shear stresses. We further show that, although for narrow geometries the viscoelastic axial stresses are negligible along the symmetry line, they are comparable or larger than shear stresses in the rest of the domain.

A continuation method is used to study Rayleigh–Bénard convection in a non-Newtonian fluid inside a parallelepiped cavity. The cavity has its length equal to twice the side of the square cross-section. Shear-thinning and shear-thickening Carreau fluids are considered. The focus is put on the two stable branches which exist in the Newtonian case, a stable primary branch of transverse rolls $B_1$ and a primary branch of longitudinal roll $B_2$, stabilized beyond a secondary bifurcation point $S_2$. Although the primary bifurcation points are unchanged, the non-Newtonian properties strongly modify the bifurcation diagram. Indeed, for a shear-thinning fluid, the stable solutions can exist at much smaller Rayleigh numbers ${Ra}$, on subcritical branches beyond saddle-node points $SN_1$ and $SN_2$, and small perturbations can be sufficient to reach them. In agreement with Bouteraa et al. (J. Fluid Mech., vol. 767, 2015, pp. 696–734), the change of the primary bifurcations from supercritical to subcritical occurs at given values of what they define as the degree of shear-thinning parameter $\alpha$. Moreover, the value of the Rayleigh number at the saddle-node points can be approximated by a simple expression, as proposed by Jenny et al. (J. Non-Newtonian Fluid Mech., vol. 219, 2015, pp. 19–34). In the case of a shear-thickening fluid, the branches remain supercritical, but the secondary point $S_2$ is strongly moved towards larger ${Ra}$, making it more difficult to reach the longitudinal roll solution. Energy analyses at the bifurcations $SN_1$, $SN_2$ and $S_2$ show that the changes of the corresponding critical thresholds ${Ra}_c$ are connected with the changes of the viscous properties, but also with changes of the buoyancy effect.

This is a theoretical and laboratory study of stationary internal hydraulic jumps. These jumps are rapid transitions between internally supercritical flow, generated by placing a sill on the bed of a horizontal rectangular channel, and internally subcritical flow, generated by installing a downstream contraction. This contraction generates an approximately uniform flow downstream of the jump; thus mimicking barotropically driven two-layer flows, as found in tidally driven flows over underwater sills, and flows over mountain ranges driven by large-scale pressure gradients. Upstream of the jump a train of Kelvin–Helmholtz billows forms on the interface between the layers. Upper layer fluid is entrained into these billows, which are subsequently advected into the lower portion of the jump. These billows are broken down by the turbulence of the jump, and the entrained upper layer fluid is mixed with lower layer fluid. Downstream of the jump the upper layer remains homogeneous, the density step at the interface is weakened, the upper portion of the lower layer is approximately linearly stratified, and the lower portion of the lower layer is undisturbed. This altered density profile is the downstream conjugate state of the jump. When the contraction is narrowed the jump moves upstream and ‘drowns’ part of the train of billows, reducing the amount of entrainment. Thus, while the jump is responsible for mixing fluid from the upper layer into the lower layer, it is the position of the jump relative to the upstream train of billows that determines the amount of entrainment.

The dynamics of a droplet on an inhomogeneously cooled liquid substrate is investigated numerically. The longwave approximation is applied. It is shown that spatial temperature modulation leads to the droplet's motion towards the region of lower temperature, which is accompanied by the change of the droplet shape. An intensive cooling from below can lead to periodic or quasiperiodic oscillations or the droplet's decomposition. A spatial temperature modulation can suppress the oscillatory instability.

Ion transport in polyelectrolyte membranes (charged hydrogels) is of significant technological (and biological) importance, but little is known of how micro-structural inhomogeneity affects ionic conductivity. Whereas a uniform electric field drives uni-directional electro-migrative and electro-osmotic ion fluxes in perfectly uniform microstructures, this study considers the influence of spherical inclusions/cavities on the hydrodynamic and ion permeability of charged hydrogels. Such cavities have a high permeability, but they can bear a much lower conductivity due to the partitioning of counter-ions between the cavity and bulk hydrogel phases, also inducing micro-scale electro-osmotic flow. To understand these, perturbations from a nonlinear Poisson–Boltzmann equilibrium state are used to compute the velocity disturbances, and electrostatic and ion-concentration polarization. These furnish three independent Onsager coefficients: one of which is the effective hydrodynamic permeability, and all of which contribute to the two principal electrical conductivities (distinguished by electrode configuration). Cavities with diameters in the range $10$–$1000$ nm are found to be readily polarized, decreasing the effective conductivity of an otherwise uniform polyelectrolyte. In highly permeable hydrogels, however, electro-osmosis may enhance the electrical conductivity when flow is blocked by impenetrable electrodes. Explicit formulas for the hydrodynamic permeability are provided, complementing a simplified (Maxwell–Donnan) analysis of the conductivity, which neglects diffuse double-layer effects and ion-concentration perturbations.

External modulation on thermal convection has been studied extensively to achieve the control of flow structures and heat-transfer efficiency. In this paper, we carry out direct numerical simulations on Rayleigh–Bénard convection accounting for both the modulation of wall shear and roughness over the Rayleigh number range $1.0 \times 10^6 \le Ra \le 1.0 \times 10^8$, the wall shear Reynolds number range $0 \le Re_w \le 5000$, the aspect-ratio range $2 \le \varGamma \le 4{\rm \pi}$, and the dimensionless roughness height range $0 \le h \le 0.2$ at fixed Prandtl number $Pr = 1$. Under the combined actions of wall shear and roughness, with increasing $Re_w$, the heat flux is initially enhanced in the buoyancy-dominant regime, then has an abrupt transition near the critical shear Reynolds number $Re_{w,cr}$, and finally enters the purely diffusion regime dominated by shear. Based on the crossover of the kinetic energy production between the buoyancy-dominant and shear-dominant regimes, a physical model is proposed to predict the transitional scaling behaviour between $Re_{w,cr}$ and $Ra$, i.e. $Re_{w,cr} \sim Ra^{9/14}$, which agrees well with our numerical results. The reason for the observed heat-transport enhancement in the buoyancy-dominant regime is further explained by the fact that the moving rough plates introduce an external shear to strengthen the large-scale circulation (LSC) in the vertical direction and serve as a conveyor belt to increase the chances of the interaction between the LSC and secondary flows within cavities, which triggers more thermal plumes, efficiently transports the trapped hot (cold) fluids outside cavities.

The backward-facing step is perhaps the quintessential geometry used to study separated flow. Extensive previous research has quantified its detailed flow characteristics. However, often regions of separated flow do not exist in isolation; rather, interaction occurs between multiple regions. This motivated an experimental investigation into the time-averaged and dynamic flow features of a double backward-facing step, covering separations of zero to eight step heights between equal-height steps. Three flow regimes are identified. A single reattachment regime occurs for separations of less than four step heights, perhaps remarkable for the lack of variation in key flow characteristics from a single backward-facing step response. Next, an intermediate regime is identified for a separation of four step heights. In this case, the flow does not yet reattach on the first step, although significant differences in reattachment length, surface pressure on the vertical step faces and turbulence statistics occur. Finally, for greater step separations, a double reattachment regime, with reattachment on both steps, is identified. Downwash, induced by the first recirculation zone, reduces the reattachment length and turbulent fluctuations of the second recirculation zone. The surface pressure on the first-step vertical face is reduced, seemingly a result of an upstream influence due to the low pressure in the second-step recirculation zone. Detailed characterisation of the regimes offers insight into the fundamental interaction of regions of separated flow, revealing aspects of complex dynamics relevant to a broad range of practical scenarios.

In this paper, we investigate an old classical free streamline problem, namely, the two-dimensional re-entrant jet cavity flow past an obstacle. It is well known that for the re-entrant jet model, the direction of the jet is a free parameter that can be specified arbitrarily. To fix this uncertainty, we make a complementary conjecture: the direction should be chosen so that the mean kinetic energy of the remote part of the jet is minimal. Considering cavity flows over an oblique flat plate as an example, we show numerically that the direction is almost opposite to the incident flow. In addition, we present an analytical confirmation of this conclusion that is independent of the obstacle shape. Further, considering again the oblique flat plate as an example, we give an answer to the following question: What happens when the angle of attack tends to zero and the cavity number is finite? We demonstrate that for the limiting configurations, the re-entrant jet vanishes, and the limit is a free-surface flow with a symmetric bubble above the plate and two stagnation points on the lower side of the plate. For this limit we construct a simple exact analytical solution.

The depth-integrated wave–current models developed by Yang & Liu (J. Fluid Mech., vol. 883, 2020, A4) are extended to investigate currents with an arbitrary vertical profile in the water column. In the present models, horizontal velocities are decomposed into two components. The first part deduces the prescribed current velocity when waves are absent. The second part is approximated in a polynomial form. The resulting depth-integrated wave–current models are obtained by applying the weighted residual method. In the absence of currents, the present models are identical to those in Yang & Liu (J. Fluid Mech., vol. 883, 2020, A4) and are validated with several three-dimensional (3D) benchmark laboratory experiments. A theoretical analysis is conducted to study the frequency dispersion relation of linear waves on currents with an exponential vertical profile and the results are compared with numerical solutions of the Rayleigh equation. Using the new models, validations and investigations are then conducted for periodic waves and solitary waves on currents with an arbitrary profile in one-dimensional horizontal (1DH) space. Furthermore, the new models are applied to wave–current interactions in two-dimensional horizontal (2DH) space. Two scenarios are considered: (1) wave propagation over a vortex-ring-like current and (2) obliquely incident wave propagation over a 3D sheared current on a varying bathymetry. The vertical and horizontal shear of the current have significant effects on modifying various wave properties, which are well captured by the present models. However, the time-averaged velocity under wave–current interaction shows small differences with the prescribed current velocity, except in the region between the wave trough and crest.

This work addresses the effects of different thermal sidewall boundary conditions on the formation of flow states and heat transport in two- and three-dimensional Rayleigh–Bénard convection (RBC) by means of direct numerical simulations and steady-state analysis for Rayleigh numbers ${\textit {Ra}}$ up to $4\times 10^{10}$ and Prandtl numbers ${\textit {Pr}}=0.1,1$ and $10$. We show that a linear temperature profile imposed at the conductive sidewall leads to a premature collapse of the single-roll state, whereas a sidewall maintained at a constant temperature enhances its stability. The collapse is caused by accelerated growth of the corner rolls with two distinct growth rate regimes determined by diffusion or convection for small or large ${\textit {Ra}}$, respectively. Above the collapse of the single-roll state, we find the emergence of a double-roll state in two-dimensional RBC and a double-toroidal state in three-dimensional cylindrical RBC. These states are most prominent in RBC with conductive sidewalls. The different states are reflected in the global heat transport, so that the different thermal conditions at the sidewall lead to significant differences in the Nusselt number for small to moderate ${\textit {Ra}}$. However, for larger ${\textit {Ra}}$, the heat transport and flow dynamics become increasingly alike for different sidewalls and are almost indistinguishable for ${\textit {Ra}}>10^{9}$. This suggests that the influence of imperfectly insulated sidewalls in RBC experiments is insignificant at very high ${\textit {Ra}}$ – provided that the mean sidewall temperature is controlled.

A generalised quasilinear (GQL) approximation (Marston et al., Phys. Rev. Lett., vol. 116, 2016, 104502) is applied to turbulent channel flow at $Re_\tau \simeq 1700$ ($Re_\tau$ is the friction Reynolds number), with emphasis on the energy transfer in the streamwise wavenumber space. The flow is decomposed into low- and high-streamwise-wavenumber groups, the former of which is solved by considering the full nonlinear equations whereas the latter is obtained from the linearised equations around the former. The performance of the GQL approximation is subsequently compared with that of a QL model (Thomas et al., Phys. Fluids, vol. 26, 2014, 105112), in which the low-wavenumber group only contains zero streamwise wavenumber. It is found that the QL model exhibits a considerably reduced multi-scale behaviour at the given moderately high Reynolds number. This is improved significantly by the GQL approximation which incorporates only a few more streamwise Fourier modes into the low-wavenumber group, and it reasonably well recovers the distance-from-the-wall scaling in the turbulence statistics and spectra. Finally, it is proposed that the energy transfer from the low- to the high-wavenumber group in the GQL approximation, referred to as the ‘scattering’ mechanism, depends on the neutrally stable leading Lyapunov spectrum of the linearised equations for the high-wavenumber group. In particular, it is shown that if the threshold wavenumber distinguishing the two groups is sufficiently high, the scattering mechanism can be completely absent due to the linear nature of the equations for the high-wavenumber group.

In this paper, direct numerical simulations are performed for low-speed flows past a NACA0012 aerofoil at high incidence angles. The aim is to investigate the significance of quadrupole noise generated due to separated shear layers, in comparison to dipole noise emanating from the aerofoil surface. The two different noise components (dipole and quadrupole) are calculated by using the Ffowcs Williams & Hawkings method in two different approaches: one with a solid surface and another with a permeable surface. The quadrupole noise is then estimated approximately by taking the relative difference between the two. The current study provides detailed comparisons between the quadrupole and dipole noise components at various observer locations in a wide range of frequencies. The comparisons are also made in terms of Mach number scaling, which differs significantly from theoretical predictions and changes rapidly with frequency. Additionally, pre-, near- and full-stall conditions are cross-examined, which reveals significant differences in the quadrupole contributions, including changes in the major source locations and frequencies. It is found that the inclusion of the quadrupole sources gives rise to the predicted noise power level at all frequencies (varying between 2 and 10 dB for an observer above the aerofoil) compared to the dipole-only case. The quadrupole contribution is far from negligible even at the low subsonic speeds (Mach 0.3 and 0.4) when aerofoil stall occurs.

The Naval Research Laboratory ‘Mag Noh problem’, described in this paper, is a self-similar magnetized implosion flow, which contains a fast magnetohydrodynamic (MHD) outward propagating shock of constant velocity. We generalize the classic Noh (OSTI Tech. Rep. 577058, 1983) problem to include azimuthal and axial magnetic fields as well as rotation. Our family of ideal MHD solutions is five parametric, each solution having its own self-similarity index, gas gamma, magnetization, the ratio of axial to the azimuthal field and rotation. While the classic Noh problem must have a supersonic implosion velocity to create a shock, our solutions have an interesting three-parametric special case with zero initial velocity in which magnetic tension, instead of implosion flow, creates the shock at $t=0+$. Our self-similar solutions are indeed realized when we solve the initial value MHD problem with the finite volume MHD code Athena. We numerically investigated the stability of these solutions and found both stable and unstable regions in parameter space. Stable solutions can be used to test the accuracy of numerical codes. Unstable solutions have also been widely used to test how codes reproduce linear growth, transition to turbulence and the practically important effects of mixing. Now we offer a family of unstable solutions featuring all three elements relevant to magnetically driven implosions: convergent flow, magnetic field and a shock wave.

Dip coating consists of withdrawing a substrate from a bath to coat it with a thin liquid layer. This process is well understood for homogeneous fluids, but heterogeneities, such as particles dispersed in liquid, lead to more complex situations. Indeed, particles introduce a new length scale, their size, in addition to the thickness of the coating film. Recent studies have shown that, at first order, the thickness of the coating film for monodisperse particles can be captured by an effective capillary number based on the viscosity of the suspension, providing that the film is thicker than the particle diameter. However, suspensions involved in most practical applications are polydisperse, characterized by a wide range of particle sizes, introducing additional length scales. In this study, we investigate the dip coating of suspensions having a bimodal size distribution of particles. We show that the effective viscosity approach is still valid in the regime where the coating film is thicker than the diameter of the largest particles, although bidisperse suspensions are less viscous than monodisperse suspensions of the same solid fraction. We also characterize the intermediate regime that consists of a heterogeneous coating layer and where the composition of the film is different from the composition of the bath. A model to predict the probability of entraining the particles in the liquid film depending on their sizes is proposed and captures our measurements. In this regime, corresponding to a specific range of withdrawal velocities, capillarity filters the large particles out of the film.