The refraction of surface gravity waves by currents leads to spatial modulations in the wave field and, in particular, in the significant wave height. We examine this phenomenon in the case of waves scattered by a localised current feature, assuming (i) the smallness of the ratio between current velocity and wave group speed, and (ii) a swell-like, highly directional wave spectrum. We apply matched asymptotics to the equation governing the conservation of wave action in the four-dimensional position–wavenumber space. The resulting explicit formulas show that the modulations in wave action and significant wave height past the localised current are controlled by the vorticity of the current integrated along the primary direction of the swell. We assess the asymptotic predictions against numerical simulations using WAVEWATCH III for a Gaussian vortex. We also consider vortex dipoles to demonstrate the possibility of ‘vortex cloaking’ whereby certain currents have (asymptotically) no impact on the significant wave height. We discuss the role of the ratio of the two small parameters characterising assumptions (i) and (ii) above, and show that caustics are significant only for unrealistically large values of this ratio, corresponding to unrealistically narrow directional spectra.

We propose a constitutive equation for flows of gases in high-rate regimes where the Navier–Stokes theory breaks down. The model generalizes the Navier–Stokes relation and agrees well with that model in all lower rate flows examined. Our proposed constitutive relation is calibrated with the method of objective molecular dynamics (OMD) using families of compressible and incompressible flows of Lennard-Jones argon. The constitutive relation makes use of the higher-order objective strain rates due to Rivlin and Ericksen (J. Rat. Mech. Anal., vol. 4, 1955, pp. 323–425). The constitutive relation is fully frame-indifferent, and the macroscopic flows corresponding to the OMD simulations are exact solutions for the proposed model. The model is shown to agree with atomistic results much better than the Navier–Stokes equations in the transition regime. The success of our model indicates that it is not higher gradients that become important in the high-rate regime, but rather higher rates of change of the strain rate tensor. While somewhat more complicated to implement than the Navier–Stokes relation, the proposed model is expected to be compatible with existing methods of computational fluid dynamics and may extend those methods to higher rate regimes, while preserving their ability to handle large spatial scales.

We carry out direct numerical simulation (DNS) of turbulent flow in rough pipes. Two types of irregular roughness are investigated, namely a grit-blasted and a graphite surface. A wide range of Reynolds numbers is tested, from the laminar up to the fully rough regime, attempting to replicate Nikuradse's pioneering study. Despite the large relative roughness, outer-layer similarity is achieved at high Reynolds number as hypothesised by Townsend, with deviations from the smooth wall case of 4 % for the grit-blasted surface and 13 % for the graphite surface. This makes it possible to define a roughness function and the equivalent sand-grain roughness. The results are compared with those obtained in plane channels, with small differences pointing to the residual influence of the duct cross-sectional shape in the presence of relatively large roughness. The computed friction factors behave similar to those Nikuradse's chart, with differences in terms of the friction factor in the laminar region and of the critical Reynolds number, which are partly absorbed by using the hydraulic radius as reference length scale. The distributions of the velocity fluctuations intensities point to a isotropisation of turbulence in the near-wall region resulting from the roughness, with influence of the roughness geometry. Comparison of the computed equivalent sand-grain roughness height suggest that existing correlations suffer from poor predictive power, at least for surfaces with large relative roughness.

We investigate steady and oscillatory flow through a hexagonal close-packed arrangement of spheres in the framework of the volume-averaged momentum equation. We quantify the friction and pressure drag based on a direct numerical simulation dataset. Using the pressure decomposition of Graham (J. Fluid Mech., vol. 881, 2019), the pressure drag can be further split up into an accelerative, a viscous and a convective contribution. For the accelerative pressure, a closed-form expression can be given in terms of the potential flow solution. We investigate the contributions of the different drag components to the volume-averaged momentum budget and their Reynolds number scaling. For steady flow, we find that the friction and viscous pressure drag are proportional to $Re$ at low Reynolds numbers and scale with $Re^{1.4}$ for high Reynolds numbers. This is close to the steady laminar boundary layer scaling. For the convective pressure drag, we find a cubic scaling at low and a quadratic scaling at high Reynolds numbers. The Reynolds stresses have a minor contribution to the momentum budget. For oscillatory flow at low and medium Womersley numbers, the amplitudes of the drag components are similar to the steady cases at the same Reynolds number. At high Womersley numbers, the drag components behave quite differently and the friction and viscous pressure drag are relatively insignificant. The drag components are not in phase with the forcing and the superficial velocity; the phase lag increases with the Womersley number. This suggests that new models beyond the current quasisteady approaches need to be developed.

In this paper we report on an experimental study focusing on the manifestation and dynamics of the large-scale circulation (LSC) in turbulent liquid metal convection. The experiments are performed inside a cylinder of aspect ratio $\varGamma = 0.5$ filled with the ternary alloy GaInSn, which has a Prandtl number of $Pr = 0.03$. The large-scale flow structures are classified and characterized at Rayleigh numbers of ${Ra} = 9.33 \times 10^6, 5.31 \times 10^7$ and $6.02 \times 10^8$ by means of the contactless inductive flow tomography which enables the full reconstruction of the three-dimensional (3-D) flow structures in the entire convection cell. This is complemented with the multi-thermal-probe method for capturing the azimuthal temperature variation induced by the LSC at the sidewall. We use proper orthogonal decomposition (POD) to identify the dominating modes of the turbulent convection. The analysis reveals that a single-roll structure of the LSC alternates in short succession with double-roll structures or a three-roll structure. This is accompanied by dramatic fluctuations of the Reynolds number, whose instantaneous values can deviate by more than 50 % from the time-average value. No coherent oscillations are observed, whereas a correlation analysis indicates a residual contribution of the torsion and sloshing modes. Results of the POD analysis suggest a stabilization of the single-roll LSC with increasing $Ra$ at the expense of flow structures with multiple rolls. Moreover, the relative lifetime of all identified flow states, measured in units of free-fall times, increases with rising $Ra$.

A one-dimensional mechanism of deflagration to detonation transition is identified and investigated by an asymptotic analysis in the double limit of large activation energy and small Mach number of the laminar flame velocity. The unsteady analysis concerns the self-accelerating tip of an elongated flame in a smooth walled tube. The flame on the tip, considered as plane and orthogonal to the tube axis, is pushed from behind by the longitudinal flow resulting from the cumulative effect of the radial flows of burned gas issued from the lateral flame of the finger-like front (called backflow in the following). The analysis of the one-dimensional dynamics is performed by coupling the flame structure with the downstream-running compression waves propagating in the external flows. A critical elongation is identified from which the slightest increase in elongation leads to a pressure runaway producing the flame blow-off. The dynamics of the inner structure of the laminar flame on the tip which is accelerated by the self-induced backflow is characterized by a finite-time singularity of the reacting flow in the form of a dynamical saddle-node bifurcation.

A turbulent circular disk wake database (Chongsiripinyo & Sarkar, J. Fluid Mech., vol. 885, 2020) at Reynolds number $Re = U_\infty D/\nu = 5 \times 10^{4}$ is interrogated to identify the presence of large-scale streaks – coherent elongated regions of streamwise velocity. The unprecedented streamwise length – until $x/D \approx 120$ – of the simulation enables investigation of the near and far wakes. The near wake is dominated by the vortex shedding (VS) mode residing at azimuthal wavenumber $m=1$ and Strouhal number $St = 0.135$. After filtering out the VS structure, conclusive evidence of large-scale streaks with frequency $St \rightarrow 0$, equivalently streamwise wavenumber $k_x \rightarrow 0$ in the wake, becomes apparent in visualizations and spectra. These streaky structures are found throughout the simulation domain beyond $x/D \approx 10$. Conditionally averaged streamwise vorticity fields reveal that the lift-up mechanism is active in the near as well as the far wake, and that ejections contribute more than sweeps to events of intense $-u'_xu'_r$. Spectral proper orthogonal decomposition is employed to extract the energy and the spatiotemporal features of the large-scale streaks. The streak energy is concentrated in the $m=2$ azimuthal mode over the entire domain. Finally, bispectral mode decomposition is conducted to reveal strong interaction between $m=1$ and $St = \pm 0.135$ modes to give the $m=2$, $St \rightarrow 0$ streak mode. Our results indicate that the self-interaction of the VS mode generates the $m=2$, $St \rightarrow 0$ streamwise vortices, which leads to streak formation through the lift-up process. To the authors’ knowledge, this is the first study that reports and characterizes large-scale low-frequency streaks and the associated lift-up mechanism in a turbulent wake.

The present paper finds that the coexistence of multiple primary instability waves may cause a non-trivial nonlinear interaction and breakdown process, which has not been reported before. In the considered Mach 6 flat-plate boundary layer, a global resolvent analysis reports three optimal disturbances (local maxima): a high-frequency planar wave, a low-frequency oblique wave and a stationary streak. For the dominant planar and oblique waves, a parabolised stability equation analysis identifies the initial non-modal transient growth and downstream modal growth. Initiated by these two optimal disturbances jointly, the complete linear and nonlinear instability processes until breakdown to turbulence are shown with direct numerical simulation. Owing to the transient growth, the oblique wave may be more significant than the planar wave in the breakdown. The oblique wave and scales of nonlinear interactions are pronounced in the outer layer, whose significance may not be comprehensively characterised by the wall pressure measurement. Fourier modes characterising the oblique-wave oblique breakdown, the planar-wave fundamental resonance, the planar-wave subharmonic resonance and the combination resonance related to a detuned mode are observed successively. The detuned mode seems to dominate the near-wall dynamics in the late nonlinear stage, characterised by $\varLambda$-like structures. Meanwhile, the existence of this detuned mode is independent of the initial amplitude ratio and the absolute amplitude of the oblique and planar waves. Weakly nonlinear stability analyses demonstrate that the detuned mode is mainly a consequence of the secondary instability under the combination of planar and oblique primary waves. Wave vector plots reveal the resonant state of multiple triads. Energy budget and amplitude-correction analyses provide a clear physical image of energy transfer.

The wall cooling effect on the spectra and structures of thermodynamic variables are investigated in hypersonic turbulent boundary layers. The density and temperature can be divided into the acoustic and entropic modes based on the Kovasznay decomposition. The intensities of the pressure and the acoustic modes of density and temperature attain the maximum values near the wall, while those of the entropy and the entropic modes of density and temperature achieve their primary peaks near the edge of the boundary layer. In the near-wall region, the pressure and the acoustic modes of density and temperature are significantly enhanced when the wall is strongly cooled, which can be attributed to the appearance of the travelling-wave-like alternating positive and negative structures. Moreover, the intensities of the entropy and the entropic modes of density and temperature become stronger near the wall as the wall temperature decreases, due to the appearance of the streaky entropic structures (SES). The SES are mainly caused by the advection effect of the strong positive wall-normal gradient of the mean temperature associated with ejection and sweep events. It is also found that the profiles of the intensities of the entropy, density and temperature are similar to each other far from the wall, which is mainly due to the reason that the entropic modes are dominant in the fluctuating density and temperature in the far-wall region. The acoustic modes of density and temperature only have significant contributions in the near-wall region.

The objective of the present study is to provide a numerical database of thermal boundary layers and to contribute to the understanding of the dynamics of passive scalars at different Prandtl numbers. In this regard, a direct numerical simulation (DNS) of an incompressible zero-pressure-gradient turbulent boundary layer is performed with the Reynolds number based on momentum thickness ${\textit {Re}}_{\theta }$ ranging up to $1080$. Four passive scalars, characterized by the Prandtl numbers ${\textit {Pr}} = 1,2,4,6$ are simulated using the pseudo-spectral code SIMSON (Chevalier et al., SIMSON : a pseudo-spectral solver for incompressible boundary layer flows. Tech. Rep. TRITA-MEK 2007:07. KTH Mechanics, Stockholm, Sweden, 2007). To the best of our knowledge, the present DNS provides the thermal boundary layer with the highest Prandtl number available in the literature. It corresponds to that of water at ${\sim }24\,^{\circ }{\rm C}$, when the fluid temperature is considered as a passive scalar. Turbulence statistics for the flow and thermal fields are computed and compared with available numerical simulations at similar Reynolds numbers. The mean flow and scalar profiles, root-mean-squared velocity and scalar fluctuations, turbulent heat flux, turbulent Prandtl number and higher-order statistics agree well with the numerical data reported in the literature. Furthermore, the pre-multiplied two-dimensional spectra of the velocity and of the passive scalars are computed, providing a quantitative description of the energy distribution at the different length scales for various wall-normal locations. The energy distribution of the heat-flux fields at the wall is concentrated on longer temporal structures with increasing Prandtl number. This is due to the thinner thermal boundary layer as thermal diffusivity decreases and, thereby, the longer temporal structures exhibit a different footprint at the wall.

Self-propulsion of chemically active droplets and phoretic disks has been studied widely; however, most research overlooks the influence of disk shape on swimming dynamics. Inspired by experimentally observed prolate composite droplets and elliptical camphor disks, we employ simulations to investigate the phoretic dynamics of an elliptical disk that emits solutes uniformly in the creeping flow regime. By varying the disk's eccentricity $e$ and the Péclet number $Pe$, we distinguish five disk behaviours: stationary, steady, orbiting, periodic and chaotic. We perform a linear stability analysis (LSA) to predict the onset of instability and the most unstable eigenmode when a stationary disk transitions spontaneously to steady self-propulsion. In addition to the LSA, we use an alternative approach to determine the perturbation growth rate, illustrating the competing roles of advection and diffusion. The steady motion features a transition from a puller-type to a neutral-type swimmer as $Pe$ increases, which occurs as a bimodal concentration profile at the disk surface shifts to a polarized solute distribution, driven by convective solute transport. An elliptical disk achieves an orbiting motion through a chiral symmetry-breaking instability, wherein it repeatedly follows a circular path while simultaneously rotating. The periodic swinging motion, emerging from a steady motion via a supercritical Hopf bifurcation, is characterized by a wave-like trajectory. We uncover a transition from normal diffusion to superdiffusion as eccentricity $e$ increases, corresponding to a random-walking circular disk and a ballistically swimming elliptical counterpart, respectively.

The Earth's landscape hosts a variety of patterns resulting from the interaction between a sediment-carrying fluid and an erodible boundary. Here, the morphodynamics of river bifurcations is interpreted as a second-order phase transition. A consolidated one-dimensional bifurcation model is re-examined in the light of classical Landau theory of critical phenomena. The transition from a balanced to an unbalanced flux partition is described in terms of an order parameter. The equilibrium states of the system are shown to be minima of a morphodynamic potential function. Finally, the role of a generic external forcing is investigated. A threshold value of the forcing is shown to set bounds separating two different morphodynamic responses to allogenic and autogenic dynamics.