We present a theoretical stability analysis for an expanding accretion shock that does not involve a rarefaction wave behind it. The dispersion equation that determines the eigenvalues of the problem and the explicit formulae for the corresponding eigenfunction profiles are presented for an arbitrary equation of state and finite-strength shocks. For spherically and cylindrically expanding steady shock waves, we demonstrate the possibility of instability in a literal sense, a power-law growth of shock-front perturbations with time, in the range of $h_c< h<1+2 {\mathcal {M}}_2$, where $h$ is the D'yakov-Kontorovich parameter, $h_c$ is its critical value corresponding to the onset of the instability and ${\mathcal {M}}_2$ is the downstream Mach number. Shock divergence is a stabilizing factor and, therefore, instability is found for high angular mode numbers. As the parameter $h$ increases from $h_c$ to $1+2 {\mathcal {M}}_2$, the instability power index grows from zero to infinity. This result contrasts with the classic theory applicable to planar isolated shocks, which predicts spontaneous acoustic emission associated with constant-amplitude oscillations of the perturbed shock in the range $h_c< h<1+2 {\mathcal {M}}_2$. Examples are given for three different equations of state: ideal gas, van der Waals gas and three-terms constitutive equation for simple metals.

We use the recent frame-indifferent theory of diffusive momentum transport to identify internal barriers in wall-bounded turbulence. Formed by the invariant manifolds of the Laplacian of the velocity field, the barriers block the viscous part of the instantaneous momentum flux in the flow. We employ the level sets of single-trajectory Lagrangian diagnostic tools, the trajectory rotation average and trajectory stretching exponent, to approximate both vortical and internal wall-parallel momentum transport barrier (MTB) interfaces. These interfaces provide frame-indifferent alternatives to classic velocity-gradient-based vortices and high-shear boundaries between uniform momentum zones (UMZs). Indeed, we find that these elliptic manifold approximations and MTBs outperform standard vortices and UMZ interfaces in blocking diffusive momentum transport, which suggests our momentum barriers are physical features that may be the cause of coherence signatures in statistical and non-objective diagnostics. We also introduce normalized trajectory metrics that provide unprecedented visualizations of objective coherent structures by avoiding strong turbulence biases.

Spectral proper orthogonal decomposition (SPOD) is an increasingly popular modal analysis method in the field of fluid dynamics due to its specific properties: a linear system forced with white noise should have SPOD modes identical to response modes from resolvent analysis. The SPOD, coupled with the Welch method for spectral estimation, may require long time-resolved datasets. In this work, a linearised Ginzburg–Landau model is considered in order to study the method's convergence. Spectral proper orthogonal decomposition modes of the white-noise forced equation are computed and compared with corresponding response resolvent modes. The quantified error is shown to be related to the time length of Welch blocks (spectral window size) normalised by a convective time. Subsequently, an algorithm based on a temporal data shift is devised to further improve SPOD convergence and is applied to the Ginzburg–Landau system. Next, its efficacy is demonstrated in a numerical database of a boundary layer subject to bypass transition. The proposed approach achieves substantial improvement in mode convergence with smaller spectral window sizes with respect to the standard method. Furthermore, SPOD modes display growing wall-normal and spanwise velocity components along the streamwise direction, a feature which had not yet been observed and is also predicted by a global resolvent calculation. The shifting algorithm for the SPOD opens the possibility for using the method on datasets with time series of moderate duration, often produced by large simulations.

We use experiments and direct numerical simulations to probe the phase space of low-curvature Taylor–Couette flow in the vicinity of the ultimate regime. The cylinder radius ratio is fixed at $\eta =r_i/r_o=0.91$, where $r_i \, (r_o)$ is the inner (outer) cylinder radius. Non-dimensional shear drivings (Taylor numbers ${\textit {Ta}}$) in the range $10^7\leq {\textit {Ta}}\leq 10^{11}$ are explored for both co- and counter-rotating configurations. In the ${\textit {Ta}}$ range $10^8\leq {\textit {Ta}}\leq 10^{10}$, we observe two local maxima of the angular momentum transport as a function of the cylinder rotation ratio, which can be described as either ‘co-’ or ‘counter-rotating’ due to their location or as ‘broad’ or ‘narrow’ due to their shape. We confirm that the broad peak is accompanied by the strengthening of the large-scale structures, and that the narrow peak appears once the driving (Ta) is strong enough. As first evidenced in numerical simulations by Brauckmann et al. (J. Fluid Mech., vol. 790, 2016, pp. 419–452), the broad peak is produced by centrifugal instabilities and that the narrow peak is a consequence of shear instabilities. We describe how the peaks change with ${\textit {Ta}}$ as the flow becomes more turbulent. Close to the transition to the ultimate regime when the boundary layers (BLs) become turbulent, the usual structure of counter-rotating Taylor vortex pairs breaks down and stable unpaired rolls appear locally. We attribute this state to changes in the underlying roll characteristics during the transition to the ultimate regime. Further changes in the flow structure around ${\textit {Ta}}\approx 10^{10}$ cause the broad peak to disappear completely and the narrow peak to move. This second transition is caused when the regions inside the BLs which are locally smooth regions disappear and the whole boundary layer becomes active.

Considering channel flow at Reynolds numbers below the linear stability threshold of the laminar profile as a generic example system showing a subcritical transition to turbulence connected with the existence of simple invariant solutions, we here discuss issues that arise in the application of linear feedback control of invariant solutions of the Navier–Stokes equations. We focus on the simplest possible problem, that is, travelling waves with one unstable direction. In view of potential experimental applicability we construct a pressure-based feedback strategy and study its effect on the stable, marginal and unstable directions of these solutions in different periodic cells. Even though the original instability can be removed, new instabilities emerge as the feedback procedure affects not only the unstable but also the stable directions. We quantify these adverse effects and discuss their implications for the design of successful control strategies. In order to highlight the challenges that arise in the application of feedback control methods in principle and concerning potential applications in the search for simple invariant solutions of the Navier–Stokes equations in particular, we consider an explicitly constructed analogue to closed-loop linear optimal control that leaves the stable directions unaffected.

We propose a model and numerical method for the propulsion of rectangular and rhombic lattices of flapping plates at $O$(10–100) Reynolds numbers in incompressible flow. The numerical method uses an adaptive mesh to mitigate singularities at the plates’ edges. We establish convergence rates and find good numerical accuracy in a test problem: Laplace's equation in the region exterior to a plate. We then use the method to establish benchmark results for a single flapping plate, including vortex wake characteristics and Froude efficiency over ranges of flapping amplitude, frequency and Strouhal number. As a prelude to a study of propulsive efficiency in Part 2 (J. Fluid Mech., vol. 915, 2021, A21), we study a key ingredient: the time-averaged input power in lattices of plates. Scaling laws for the mean input power are estimated in the limits of small and large streamwise spacings, using steady flow models with small-gap and Poiseuille-like flows between the plates respectively in the two limits. For both lattice types, the mean input power saturates as the lateral spacing becomes large (and thrust occurs). At small lateral spacings, the rhombic lattices’ input power becomes much larger when the plates overlap. The time-averaged input power in flapping lattices agrees qualitatively with the steady models.

This paper presents a detailed investigation of the role played by the wave steepness in connection with the statistical properties of the surface elevation and fluid kinematics in irregular, directionally spread, deep-water wave fields initially given by a JONSWAP spectrum. Using ensembles of large wave fields obtained from fully nonlinear simulations, we first consider the statistical properties of the surface elevation. In that connection we determine the probability density functions (PDFs) of the surface and crest elevations for wave fields of relatively small to unprecedentedly large steepness, and compare them with theoretical results from the literature in order to establish the latter's accuracy. We then consider certain statistical aspects of the fluid kinematics found at the surface and of the fluid kinematics accompanying large crests, which to our knowledge marks the first investigation of these properties in the literature. We first determine the PDFs of the horizontal fluid velocities and accelerations as well as the vertical fluid acceleration at the surface. Next, we investigate the joint PDF of the surface elevation and each of the velocities and accelerations at the surface, and use it to determine the surface elevations for which the velocities and accelerations at the surface are large. We then present an analysis of the largest fluid velocities and accelerations found in the vicinity of large crests, and compute the PDFs of these quantities. Finally, we consider the PDFs of the location at which the largest velocities and accelerations occur relative to the crest.

A direct numerical simulation is performed for a supersonic turbulent boundary layer interacting with a compression/expansion ramp at an angle $\alpha =24^{\circ }$, matching the same operating conditions of the direct numerical simulation by Priebe & Martín (J. Fluid Mech., vol. 699, 2012, pp. 1–49). The adopted numerical method relies on the high-order spectral difference scheme coupled with a bulk-based, low-dissipative, artificial viscosity for shock-capturing purposes (Tonicello et al., Comput. Fluids, vol. 197, 2020, 104357). Filtered and averaged fields are evaluated to study total kinetic energy transfers in the presence of non-negligible compressibility effects. The compression motions are shown to promote forward transfer of kinetic energy down the energy cascade, whereas expansion regions are more likely to experience backscatter of kinetic energy. A standard decomposition of the subgrid scale tensor in deviatoric and spherical parts is proposed to study the compressible and incompressible contributions in the total kinetic energy transfers across scales. On average, the correlation between subgrid scale dissipation and large-scale dilatation is shown to be caused entirely by the spherical part of the Reynolds stresses (i.e. the turbulent kinetic energy). On the other hand, subtracting the spherical contribution, a mild correlation is still noticeable in the filtered fields. For compressible flows, it seems reasonable to assume that the eddy-viscosity approximation can be a suitable model for the deviatoric part of the subgrid scale tensor, which is exclusively causing forward kinetic energy cascade on average. Instead, more complex models are likely to be needed for the spherical part, which, even in statistical average, provides an important mechanism for backscatter.

Understanding the hysteretic behaviour in fluid–fluid displacement processes in porous media is critical in many engineering applications. In this work, we study the quasi-static immiscible displacement process in two-dimensional porous media during cyclic injections in the context of carbon geosequestration. The role of wettability on the residual trapping of CO$_2$ is investigated numerically using an extended interface tracking algorithm. Despite that higher CO$_2$ saturation can be achieved in CO$_2$-wet porous media after the first CO$_2$ injection, the majority of CO$_2$ is found to be unstable and can be mobilised during subsequent water injection processes. An improvement in the residual trapping of CO$_2$ is observed as the number of injection cycles increases, which is associated with the dispersion of continuous CO$_2$ ganglia into numerous smaller blobs. Compared with either water-wet or CO$_2$-wet porous media, it is found that less CO$_2$ is trapped within the neutral-wet ones at equilibrium state after a sufficient number of injection cycles. The hysteretic behaviour of saturation between water/CO$_2$ injection cycles is found to follow an exponential decay, which eventually reaches a finite value. This process corresponds to the shift of the mobile region during displacement from typical capillary fingering to a less ramified regime, which ultimately converges towards main flow channels. This work highlights the hysteretic behaviour during cyclic injections, providing insights on the wettability impacts on multiphase flow in porous media, which is of great importance in applications such as carbon geosequestration and geological hydrogen storage.

The migration of dense fluid through a saturated, layered porous medium leads to two end-member examples of buoyancy-driven flow, namely plumes and gravity currents. Herein we develop an integrated theoretical model to study this scenario for the special case where the boundary between the permeable layers, in a two-layered porous medium, is inclined at an angle to the horizontal. Far from being a routine detail, the inclination of the permeability jump leads to a symmetry-breaking: up- and downdip flows have different volume fluxes and travel different distances, possibly substantially different distances, before becoming arrested at the point where plume inflow balances basal draining. Our model predicts these associated run-out lengths and the transient approach thereto. Predictions are validated with measurements from similitude laboratory experiments, in which the upper and lower layers are comprised of glass beads of different diameters. Experiments are conducted for a range of inclination angles and also a range of plume source conditions. The experimental data suggest a complicated structure for the gravity currents, whose boundaries are blurred by dispersion in a manner not captured by our (sharp interface) model. This observation has particular significance in predicting the lateral spread of contaminated fluid through real geological formations, particularly in instances where for example groundwater contamination is of particular concern.

Heating a quiescent fluid from below gives rise to cellular convective motion as the temperature gradient becomes sufficiently steep. Typically, this transition increases heat transfer. Differentially rotating spherical shells also generate a state of cellular motion, which in this case transports angular momentum. When both effects are present, it is often assumed that the fluid adopts a configuration which maximises the transfer of angular momentum and heat. Depending on how the equilibrium is reached, however, this maximisation may not always be achieved, with two different stable equilibria often co-existing for the same heating and rotation strengths. We want to understand why the fluid motion in a spherical shell is bistable, and how this scenario might arise. We consider a deep, highly viscous fluid layer, of relevance to the ice shells of Saturn's and Jupiter's moons. We find that bistability depends largely on the relative strength of heating and differential rotation, as characterised by the Rayleigh number $Ra$ and inner sphere Reynolds number $Re_1$, and that the nature of the transition between bistable states depends strongly on the ratio of momentum diffusivity $\nu$ to thermal diffusivity $\kappa$ defined by the Prandtl number ${\textit {Pr}} = \nu /\kappa$. In particular, we find that the transition between solutions at large ${\textit {Pr}}$, depends on the strength of thin thermal layers and can occur either due to the destabilisation of an equatorial jet by buoyancy forces, or alternatively of a polar thermal plume by differential rotation. Our results demonstrate that, although bistability in this system cannot be simply explained by the flow maximising its torque or heat transfer, the polar and equatorial regions are of particular significance.