This work investigates the time-averaged shape dynamics of a soft elastic capsule in two phase-shifted orthogonal ultrasonic standing waves. The capsule consists of an elastic membrane that encloses a viscous fluid and is immersed in another viscous fluid. Combining the acoustic perturbation theory of fluid dynamics with the thin-shell mechanics of capsule membrane deformation, two sets of equations are established to govern the ultrasonic propagation and the time-averaged response of the fluid–capsule system, respectively. These governing equations are solved numerically based on the finite element method. Numerical simulations show that the ultrasonic standing waves have pure elongation and pure rotation effects on the initially circular capsule when the phase difference is 0 and ${\rm \pi}/2$, respectively. By setting the phase difference between 0 and ${\rm \pi}/2$, it is found that the initially circular capsule exhibits a tank-treading motion due to the combined effect of the elongation and rotation. The capsule membrane elasticity and internal fluid viscosity have significant effects on the tank-treading behaviour of the initially circular capsule, including elongation deformation, inclination angle and tank-treading velocity. For the initially non-circular capsule, three types of dynamical states, including stable orientation, swinging and tumbling, are predicted by varying the phase difference and intensity of the ultrasonic standing waves, as well as the initial shape, membrane elasticity and internal fluid viscosity of the initially non-circular capsule. This work enriches the cell manipulation capabilities of ultrasonic standing wave micro-acoustofluidics and may inspire new biological applications.

The physics of electrospray has been subject to an intense debate for three decades regarding the ultimate electrokinetics that determines the electric current and the size of the emitted droplets in the steady Taylor cone-jet mode (TCJ). In order to solve with a high degree of accuracy the complete electrokinetic structure of the TCJ, in this work, we have used the full Poisson–Nernst–Planck model electrokinetic equations, which have been solved using a high accuracy numerical scheme. We consider a formulation with no interfacial adsorption of ions, as in Mori & Young (J. Fluid Mech., vol. 855, 2018, pp. 67–130). Our simulations corroborate Mori and Young's conclusion that the classical leaky dielectric model (LDM) recovers the electrodiffusion theory for weak electrolytes when disregarding ion adsorption at the interface. However, for strong electrolytes, our results differ drastically from those provided by the LDM. In this case, we observe that the ion distribution, and consequently the conductivity in the bulk, can be strongly non-homogeneous. Given the rather universal validity of the LDM experimentally observed so far, we postulate that ion interfacial adsorption must be considered in the case of strong, highly dissociated electrolytes to retrieve the LDM limit, mostly for a cone jet operating in the vicinity of the minimum flow rate.

We model linear, inviscid, internal tides generated by the interaction of a barotropic tide with one-dimensional topography. Starting from the body-forcing formulation of the hydrodynamic problem, we derive a coupled-mode system (CMS) using a local eigenfunction expansion of the stream function. For infinitesimal topography, we solve this CMS analytically, recovering the classical weak topography approximation (WTA) formula for the barotropic-to-baroclinic energy conversion rate. For arbitrary topographies, we solve this CMS numerically. The CMS enjoys faster convergence with respect to existing modal solutions and can be applied in the subcritical and supercritical regimes for both ridges and shelf profiles. We show that the non-uniform barotropic tide affects the baroclinic field locally over topographies with large slopes and we study the dependence of the radiated energy conversion rate on the criticality. We show that non-radiating or weakly radiating topographies are common in the subcritical regime. We also assess the region of validity of the WTA approximation for the commonly used Gaussian ridge and a compactly supported bump ridge studied here for the first time. Finally, we provide numerical evidence showing that in the strongly supercritical regime, the energy conversion rate for a ridge (respectively shelf) approaches the value obtained by the knife-edge (respectively step) topography.

The effects of initial conditions on the evolution of the Richtmyer–Meshkov instability (RMI) at early to intermediate times are analysed, using numerical simulations of an idealised version of recent shock tube experiments performed at the University of Arizona (Sewell et al., J. Fluid Mech., vol. 917, 2021, A41). The experimental results are bracketed by performing both implicit large-eddy simulations of the high-Reynolds-number limit as well as direct numerical simulations (DNS) at Reynolds numbers lower than those observed in the experiments. Various measures of the mixing layer width $h$, known to scale as ${\sim }t^\theta$ at late time, based on both the plane-averaged turbulent kinetic energy and volume fraction profiles are used to explore the effects of initial conditions on $\theta$ and are compared with the experimental results. The decay rate $n$ of the total fluctuating kinetic energy is also used to estimate $\theta$ based on a relationship that assumes self-similar growth of the mixing layer. The estimates for $\theta$ range between 0.44 and 0.52 for each of the broadband perturbations considered and are in good agreement with the experimental results. Decomposing the mixing layer width into separate bubble and spike heights $h_b$ and $h_s$ shows that, while the bubbles and spikes initially grow at different rates, their growth rates $\theta _b$ and $\theta _s$ have equalised by the end of the simulations. Overall, the results demonstrate important differences between broadband and narrowband surface perturbations, as well as persistent effects of finite bandwidth on the growth rate of mixing layers evolving from broadband perturbations. Good agreement is obtained with the experiments for the different quantities considered; however, the results also show that care must be taken when using measurements based on the velocity field to infer properties of the concentration field.

Flow and turbulent structures generated by the interaction of forced incoming oscillatory flow with a circular, vertical cylinder placed in an open channel with a horizontal bed are investigated using eddy-resolving simulations. Validation simulations performed with a Keulegan–Carpenter (KC) number of 20 and multiple-mode forcing corresponding to the laboratory experiment of Sumer et al. (J. Fluid Mech., vol. 332, 1997, pp. 41–70) show that detached eddy simulation (DES) predicts more accurately the amplification of the bed shear stress beneath the horseshoe vortex system (upstream side of the cylinder) and the maximum magnitude of the bed shear stress at the downstream (wake) side of the cylinder compared with unsteady Reynolds-averaged Navier–Stokes simulations. High-Reynolds-number DES simulations are then conducted with 1.5 ≤ KC ≤ 30.8 and one-mode sinusoidal forcing of the streamwise velocity in the approaching flow to investigate the changes in the wake vortex-flow regimes, the coherence of the horseshoe vortices and the generation of other near-bed coherent structures in the wake during the oscillatory cycle. The flow is periodic, no horseshoe vortices form and no vortices are shed in the wake for KC = 1.5. By contrast, for KC ≥ 8 horseshoe vortices are present over part of the oscillatory cycle and up to three wake vortices are shed over each half-cycle as KC is increased to 30.8. For an intermediate range of KC numbers, one (KC = 15.4) or two (KC = 8) of the vortices forming at the back of the cylinder during each half-cycle are washed around it when the flow reverses. The main horseshoe vortex and other horizontal near-bed vortices have a large capacity to amplify the bed shear stresses when the incoming velocity magnitude is significantly less than its peak value. Assuming the depth-averaged velocity in the incoming (undisturbed) oscillatory flow is the same in simulations conducted with different KC numbers, the peak values of the sediment entrainment potential measured by the mean (cycle-averaged) volumetric flux of sediment entrained from the bed over one oscillatory cycle occur for 8 ≤ KC ≤ 15.4. For all KC numbers, the in-line force variation over the oscillatory cycle is fairly well approximated by the Morison equation. For KC = 1.5, the in-line force is only due to inertia effects. For KC = 30.8, the maximum and minimum values of the phase-averaged in-line force are approximatively in phase with those of the incoming flow velocity. For KC ≥ 15, the phase-averaged in-line force coefficients vary between 0.8 and 1.1 during most of the oscillatory cycle (e.g. when the incoming flow velocity is not close to zero). This is different from cases with KC ≤ 8 where the in-line force coefficient is equal to zero twice during the oscillatory cycle as the in-line force becomes equal to zero for non-zero values of the incoming velocity. The largest cycle-to-cycle variations of the in-line force coefficient and in-line force are observed around KC = 8. For KC = 8 and 15.4, the cylinder is subject to relatively large phase-averaged spanwise drag forces that are comparable to the peak phase-averaged streamwise drag forces. As KC is increased to 30.8, the phase-averaged spanwise drag force becomes zero over the whole oscillatory cycle but the cylinder is still subject to large instantaneous spanwise forces over part of the oscillatory cycle.

In this work, we present the stability theory for inhomogeneous fluids subjected to standing acoustic fields. Starting from the first principles, the stability criterion is established for two fluids of different acoustic impedance (product of density and speed of sound of the fluid) separated by a plane interface. Through stability theory and numerical simulations, we show that, in the presence of interfacial tension, the relocation of high-impedance fluid from the pressure anti-node to the pressure node occurs when the acoustic force overcomes the interfacial tension force, which is in agreement with recent acoustic relocation experiments in the microchannel. Furthermore, we establish an acoustic Bond number that characterizes stable ($Bo_{a}<1$) and relocation ($Bo_{a}>1$) regimes. Remarkably, it is found that the critical acoustic energy density required for relocation can be significantly reduced by increasing the height of the channel which could help in designing acoustofluidic devices that handle immiscible fluids.

Results from direct numerical simulations of a vertical natural convection boundary layer (NCBL) with $Pr=0.71$ reveal that the turbulence development of such a thermally driven convective flow has two distinct stages: at relatively low Grashof number, the bulk flow is turbulent while the near-wall region is laminar-like or weakly turbulent; at sufficiently high Grashof number, the entire flow becomes turbulent in the sense of von Kármán (cf. Grossmann & Lohse, J. Fluid Mech., vol. 407, 2000, pp. 27–56, for the ultimate turbulent regime). Investigations on the turbulence statistics show that the near-wall Reynolds shear stress is negligible in the weakly turbulent regime but will grow in magnitude as the flow transitions to the ultimate regime at higher Grashof number. Similar behaviour is also seen in the streamwise turbulence intensity, where it develops from a mono-peak profile into a dual-peak structure as the Grashof number increases. At higher Grashof number, the near-wall energetic site is shown to have an energy distribution similar to that of a canonical wall-bounded turbulence (e.g. Hutchins & Marusic, Phil. Trans. R. Soc. A, vol. 365, 2007, pp. 647–664), with a peak centred at fixed location and wavelength ($y^+=18$ and $\lambda ^+_x=1000$) in viscous coordinates. Investigation on the spanwise spectra also suggests that the turbulent near-wall streaks emerge only at sufficiently high Grashof number, with constant spacing $\lambda _z^+\approx 130$. The extent of the weakly turbulent regime is identified using the maximum velocity location $\delta_m$ and a laminar length scale $\delta_u$. The development of near-wall turbulence is also investigated by examining the turbulence kinetic energy budget. In the weakly turbulent regime, the near-wall turbulence is sustained predominantly by the pressure transport in addition to the shear production. At higher Grashof number, the flow becomes fully turbulent, and both turbulent transport and shear production become stronger, while the pressure transport is decreased. These results also reveal that the production–dissipation ratio $P/\varepsilon$ of the NCBL would follow a fundamentally different trend to the canonical wall-bounded flows, which further supports that the near-wall turbulence generation is affected by the bulk flow.

Hypersonic flow over a two-dimensional compression corner is investigated using computational fluid dynamics and global resolvent analysis in this study, which is a continuation of the work done by Hao et al. (J. Fluid Mech., vol. 919, 2021, p. A4). The same baseline free-stream conditions with a Mach number of 7.7 and a unit Reynolds number of 4.2 × 106 m−1 are considered. The ramp angles range from 0° (equivalent to a flat plate) to 12° (slightly below the critical angle of global instability). During this process, the base flow evolves from no separation to incipient separation to large separation. The resolvent analysis reveals that the optimal response to upstream disturbances localised near the leading edge is in the form of steady streamwise streaks for all interaction strengths, which arise from transient growth in the flat-plate boundary layer due to the lift-up mechanism and significantly grow near the corner due to the Görtler instability. The most amplified spanwise wavelength decreases as the ramp angle is increased and scales with the incoming boundary-layer thickness.

Numerical simulations of the geodynamo (and other planetary dynamos) have made significant progress in recent years. As computing power has advanced, some new models claim to be ever more appropriate for understanding Earth's core dynamics. One measure of the success of such models is the ability to replicate the expected balance between forces operating within the core; Coriolis and Lorentz forces are predicted to be most important. The picture is complicated for an incompressible flow by the existence of the pressure gradient force which renders the gradient parts of all other forces dynamically unimportant. This can confuse the situation, especially when the scale dependence of forces is considered. In this work we investigate force balances through the alternative approach of eliminating gradient parts of each force to form ‘solenoidal force balances’. We perform a length-scale-dependent analysis for several spherical simulations and find that removal of gradient parts offers an alternative picture of the force balance compared with looking at traditional forces alone. Solenoidal force balances provide some agreement with the results of previous studies but also significant differences. They offer a cleaner overall picture of the dynamics and introduce differences at smaller scales. This has implications for geodynamo models purporting to have reached Earth-like regimes: in order to achieve a meaningful comparison of forces, only the solenoidal part of forces should be considered.

In this work, the modelling of collisional bedload transport is investigated with a focus on the continuum modelling of the granular flow. For this purpose, a frictional–collisional approach, combining a Coulomb model with the kinetic theory of granular flows, is developed. The methodology is based on a comparison with coupled fluid–discrete simulations that the classical kinetic theory model fails to reproduce. This inaccuracy may be explained by the assumptions of negligible interparticle friction and the absence of a saltation model in the continuum approach. In order to provide guidelines for the modelling, the fluctuating energy balance is computed in the discrete simulations and systematically compared with the kinetic theory laws. Interparticle friction is shown to affect the radial distribution function and to increase the energy dissipation, in agreement with previous observations. In addition, a saltation regime is identified, leading to departure from the viscosity and pseudo-thermal diffusivity laws of the kinetic theory in the dilute regime. Based on these observations, modifications to account for interparticle friction are included in the two-fluid model, and the kinetic theory is coupled with a saltation model. The results show that for inelastic frictional particles, interparticle friction controls energy dissipation, and the macroscopic behaviour of the granular flow does not depend on the microscopic particle properties. The proposed model reproduces the $\mu (I)$ rheology in the dense regime of the granular flow. Finally, the model is evaluated with experiments, showing significant improvements concerning the granular flow modelling.

In the present study, the influence of a uniformly moving impinging shock on the resulting shock wave–turbulent boundary layer interaction is numerically investigated. The relative Mach number of the shock front travelling above a flat-plate model varied between 0 and 2.3, while the quasi-steady inflow conditions remained constant with a Mach number of 3. To quantitatively evaluate the effect of shock travelling speed, a well-known scaling method for interaction length in quasi-steady flows was applied as a reference after significant improvements in modelling the effects of Reynolds number and wall temperature using new and existing data. Moreover, previously obtained experimental results for a limited range of travelling speeds were employed to validate the obtained numerical results. Three ranges of shock travelling speeds with distinctly different properties were extracted and quantitatively described using a developed correlation-based approach built on the extended quasi-stationary scaling law. In the first range, the scaled interaction length reaches its maximum for the given interaction strength and can be directly described by the scaling law obtained for quasi-stationary interactions. In the second travelling-speed range, the dependence of the interaction length on the interaction strength is explicitly influenced by the shock movement. With increasing shock travelling speed, the scaled interaction length here decreases significantly faster than in the quasi-stationary reference case. The end of this speed range is reached when the absolute shock front speed has caught up with the maximum speed of sound in the interaction zone, and thus the interaction length has fallen to zero. This travelling-speed limit signifies the transition to the third range, where upstream influence is no longer possible.

Compressible turbulent plane Couette flows are studied via direct numerical simulation for wall Reynolds numbers up to $Re_w=10\ 000$ and wall Mach numbers up to $M_w=5$. Various turbulence statistics are compared with their incompressible counterparts at comparable semilocal Reynolds numbers $Re^*_{\tau,c}$. The skin friction coefficient $C_f$, which decreases with $Re_w$, only weakly depends on $M_w$. On the other hand, the thermodynamic properties (mean temperature, density and others) strongly vary with $M_w$. Under proper scaling transformations, the mean velocity profiles for the compressible and incompressible cases collapse well and show a logarithmic region with the Kárman constant $\kappa =0.41$. Compared with wall units, the semilocal units yield a better collapse for the profiles of the Reynolds stresses. While the wall-normal and spanwise Reynolds stress components slightly decrease in the near-wall region, the inner peak of the streamwise component notably increases with increasing $M_w$ – indicating that flow becomes more anisotropic when compressible. In addition, the near-wall turbulence production decreases as $M_w$ increases – due to rapid wall-normal changes of viscosity caused by viscous heating. The streamwise and spanwise energy spectra show that the length scale of near-wall coherent structures does not vary with $M_w$ in semilocal units. Consistent with those in incompressible flows, the superstructures (the large-scale streamwise rollers) with a typical spanwise scale of $\lambda _z/h\approx 1.5{\rm \pi}$ become stronger with increasing $Re_w$. For the highest $Re_w$ studied, they contribute about $40\,\%$ of the Reynolds shear stress at the channel centre. Interestingly, flow visualization and correlation analysis show that the streamwise coherence of these structures degrades with increasing $M_w$. In addition, at comparable $Re^*_{\tau,c}$, the amplitude modulation of these structures on the near-wall small scales is quite similar between incompressible and compressible cases – but much stronger than that in plane Poiseuille flows.

In this study, for the first time, we consider longitudinal motion of the walls during peristalsis in a distensible tube and how this affects backward (or retrograde) flow, i.e. peristaltic reflux. Building on the analytical model developed by Shapiro et al. (J. Fluid Mech., vol. 37, no. 4, 1969, pp. 799–825) based on lubrication theory, we model peristalsis as a two-dimensional infinite sinusoidal wavetrain. We develop an objective function with high mechanical pumping efficiency and low reflux to find optimal peristalsis conditions. We show that optimal wall longitudinal motion contributes substantially to limiting reflux during peristalsis. The results suggest that the optimal form of wall longitudinal velocity is a linear function of the wall transverse coordinate, moving forward with the wave when the tube is distended and retracting when contracted. Our results are in general agreement with clinical observations of ureteral peristalsis.

We study the influence of fringing magnetic fields on turbulent thermal convection in a horizontally extended rectangular domain. The magnetic field is created in the gap between two semi-infinite planar magnetic poles, with the convection layer located near the edge of the gap. We employ direct numerical simulations in this set-up for fixed Rayleigh and small Prandtl numbers, but vary the fringe width by controlling the gap between the magnetic poles and the convection cell. The magnetic field generated by the magnets is strong enough to cease the flow in the high magnetic flux region of the convection cell. We observe that as the local vertical magnetic field strength increases, the large-scale structures become thinner and align themselves perpendicular to the longitudinal sidewalls. We determine the local Nusselt and Reynolds numbers as functions of the local Hartmann number (based on the vertical component of the magnetic field), and estimate the global heat and momentum transport. We show that the global heat transport decreases with increasing fringe width for strong magnetic fields but increases with increasing fringe width for weak magnetic fields. In the regions of large vertical magnetic fields, the convective motion becomes confined to the vicinity of the sidewalls. The amplitudes of these wall modes show a non-monotonic dependence on the fringe width.

We extend the Matsuno–Gill model, originally developed on the equatorial $\beta$-plane, to the surface of the sphere. While on the $\beta$-plane the non-dimensional model contains a single parameter, the damping rate $\gamma$, on a sphere the model contains a second parameter, the rotation rate $\epsilon ^{1/2}$ (Lamb number). By considering the different combinations of damping and rotation, we are able to characterize the solutions over the $(\gamma, \epsilon ^{1/2})$ plane. We find that the $\beta$-plane approximation is accurate only for fast rotation rates, where gravity waves traverse a fraction of the sphere's diameter in one rotation period. The particular solutions studied by Matsuno and Gill are accurate only for fast rotation and moderate damping rates, where the relaxation time is comparable to the time on which gravity waves traverse the sphere's diameter. Other regions of the parameter space can be described by different approximations, including radiative relaxation, geostrophic, weak temperature gradient and non-rotating approximations. The effect of the additional parameter introduced by the sphere is to alter the eigenmodes of the free system. Thus, unlike the solutions obtained by Matsuno and Gill, where the long-term response to a symmetric forcing consists solely of Kelvin and Rossby waves, the response on the sphere includes other waves as well, depending on the combination of $\gamma$ and $\epsilon ^{1/2}$. The particular solutions studied by Matsuno and Gill apply to Earth's oceans, while the more general $\beta$-plane solutions are only somewhat relevant to Earth's troposphere. In Earth's stratosphere, Venus and Titan, only the spherical solutions apply.

We present a theoretical and numerical stability analysis for a piston-driven planar shock against two-dimensional perturbations. The results agree with the well-established theory for isolated planar shocks: in the range of $h_c< h<1+2 {\mathcal {M}}_2$, where $h$ is the D'yakov–Kontorovich (DK) parameter related to the slope of the Rankine–Hugoniot curve, $h_c$ is its critical value corresponding to the onset of the spontaneous acoustic emission (SAE) and ${\mathcal {M}}_2$ is the downstream Mach number, non-decaying oscillations of shock-front ripples occur. The effect of the piston is manifested in the presence of additional frequencies occurring by the reflection of the sonic waves on the piston surface that can reach the shock. An unstable behaviour of the shock perturbation is found to be possible when there is an external excitation source affecting the shock, whose frequency coincides with the self-induced oscillation frequency in the SAE regime, thereby being limited to the range $h_c< h<1+2 {\mathcal {M}}_2$. An unstable evolution of the shock is also observed for planar shocks restricted to one-dimensional perturbations within the range $1< h<1+2 {\mathcal {M}}_2$. Both numerical integration of the Euler equations via the method of characteristics and theoretical analysis via Laplace transform are employed to cross-validate the results.

The dynamics of compound drops impacting on a flat substrate is numerically investigated using a ternary-fluid diffuse-interface method, with the aim of assessing the effect of a density difference between the inner and outer droplets (denoted by $\lambda$) on the evolution of the interfaces. With the help of numerical simulations, we find that, at the intermediate stage of drop impact, the inner droplet exhibits a self-similar deformation at $\lambda =1$ and relatively high Weber number, and experiences more or less a uniform acceleration for various $\lambda$. In particular, the acceleration magnitude at $\lambda \ne 1$ can be correlated with the acceleration at $\lambda =1$ and the Atwood number. When the inner droplet is denser than the outer one, a lamella occurs at the spreading front of the inner droplet. We present a scaling analysis of the thickness of the lamella, and the resultant theoretical prediction is in good agreement with numerical results. At the maximal spreading of the compound drop, a bulging structure is formed around the symmetry axis due to the presence of the inner droplet, thereby effectively reducing the liquid supply to the spreading front and leading to a decrease of maximal spreading ratio $\beta _{max}$ as compared with a pure drop. We proposed a corrected Weber number $We^*_\lambda$ by taking account of the combined effects of $\lambda$, volume fraction of the inner droplet, Weber number and morphology of the compound drop. Integrating $We^*_\lambda$ with the universal model of $\beta _{max}$ for impacting pure drops, we successfully build up a new model for predicting the maximal spreading ratio of impacting compound drops with various $\lambda$.

The interaction of steady free-surface flows of viscoplastic material with a surface-piercing obstruction of square cross-section on an inclined plane is investigated theoretically. The flow thickness increases upstream of the obstruction and decreases in its lee. The flow depends on two dimensionless parameters: an aspect ratio that relates the flow thickness, the obstruction width and the plane inclination; and a Bingham number that quantifies the magnitude of the yield stress relative to the gravitationally induced stresses. Flows with a non-vanishing yield stress always form a static ‘dead’ zone in a neighbourhood of the upstream and downstream stagnation points. For relatively wide obstructions, a deep ‘ponded’ region develops upstream with a small dead zone, while the deflected flow reconnects over relatively long distances downstream. The depth of the upstream pond increases with both the dimensionless yield stress and width of the obstruction, while the unyielded dead zone varies primarily with the yield stress. Both are predicted asymptotically by balancing the volume flux of fluid into and out of the ponded region. When the obstruction is narrow, the perturbation to the depth of the oncoming flow is reduced. It exhibits fore–aft antisymmetry, while the dead zone is symmetric to leading order. Increasing the yield stress leads to larger dead zones that eventually encompass all of the upstream- and downstream-facing boundaries of the obstruction and fully divert the flow. Results for obstructions with circular and rhomboidal cross-sections are also presented and illustrate the effects of boundary shape on the properties of the steady flow.

Turbulent Taylor–Couette flow between two concentric independently rotating cylinders with a radius ratio of $\eta = 0.1$ is studied experimentally. While the scope is to study the counter-rotating cases between both cylinders, the radial and azimuthal velocity components are recorded at different horizontal planes with high-speed particle image velocimetry. The parametric study considered a set of different shear Reynolds numbers in the range of $20\,000 \leq Re_s \leq 1.31 \times 10^5$, with different rotation ratios of $-0.06 \leq \mu \leq +0.008$. The observed flow fields had a clear dependence on the rotation ratio, where flow patterns evolved with a more pronounced axial dependence. The angular momentum transport is computed as a result of the recorded flow fields and given by a quasi-Nusselt number. The dependence of the Nusselt number on the different rotation ratios shows a maximum for the low counter-rotating case and $\mu _{max}$ is found between $-0.011 < \mu _{max} < -0.0077$. The Nusselt number decreases for stronger counter-rotation until a minimum is reached, where it tends to increase again for higher counter-rotation rates. The space–time behaviour of the turbulent flow showed the existence of patterns propagating from the inner region towards the outer region for all studied counter-rotating cases. In addition, patterns have been found that tend to propagate from the outer region towards the inner region with a novel character at high counter-rotation cases. These patterns enhance the angular momentum transport where a second maximum in the transport mechanism has to be expected.

One of the most significant issues in hydrodynamic interfacial instabilities is the growth rate of the interfacial perturbations, which plays an important role in both scientific research (e.g. supernova explosion) and engineering applications (e.g. inertial confinement fusion). Yet the underlying mechanisms of such flow phenomena remain unclear or controversial. In this paper the decoupled mechanisms of two effects are found to dominate the interface growth of the single-mode Rayleigh–Taylor instability (RTI) and Richtmyer–Meshkov instability (RMI) via Layzer's potential-flow model. One is the inertial effect induced by the interfacial density gradient and the acceleration, and the other is the frontal distortion effect stemming from interface shape evolution. The former determines the dominant features of interface evolution, while the latter influences the local concavity and convexity of growth rate such as the overshoot phenomenon. These two effects can be approximated as a linearly decoupled analytical solution if their nonlinear interaction term is neglected. With the decoupled solution, the theoretical growth rates agree well with high-fidelity numerical simulation results. The present result indicates that the long-time evolution of fluid interface in both RTI and RMI at all density ratios can be accurately predicted if both inertia and frontal distortion effects are taken into account. Furthermore, the strong dependence of instability evolution on initial amplitude is quantified based on the effects of decoupling, which sheds light on the physical origin of the overshoot phenomenon.