We present a new spectral method for the direct numerical simulation of magnetohydrodynamic turbulence at low magnetic Reynolds number. The originality of our approach is that instead of using traditional bases of functions, it relies on the basis of eigenmodes of the dissipation operator, which represents viscous and Joule dissipation. We apply this idea to the simple case of a periodic domain in the three directions of space, with a homogeneous magnetic field in the ez direction. The basis is then still a subset of the Fourier space, but ordered by growing linear decay rate |λ| (i.e. according to the least dissipative modes). We show that because the lines of constant energy tend to follow those of constant |λ| in the Fourier space, the scaling for the smallest scales |λmax| in a forced flow can be expressed, using this single parameter, as a function of the Reynolds number as , where kf is the forcing wavelength, or as a function of the Grashof number Gf, which gives a non-dimensional measure of the forcing, as |λmax|1/2/(2πkf) ≃ 0.47Gf0.20. This scaling is also found to be consistent with heuristic scalings derived by Alemany et al. (J. Mec., vol. 18, 1979, pp. 277–313) and Pothérat & Alboussière (Phys. Fluids, vol. 15, 2003, pp. 3170–3180) for interaction parameter S ≳ 1, and which we are able to numerically quantify as k⊥max/kf ≃ 0.5Re1/2 and kzmax/kf ≃ 0.8kfRe/Ha. Finally, we show that the set of least dissipative modes gives a relevant prediction for the scale of the first three-dimensional structure to appear in a forced, initially two-dimensional turbulent flow. This completes our numerical demonstration that the least dissipative modes can be used to simulate both two- and three-dimensional low-Rm magnetohydrodynamic (MHD) flows.

The irreversible mixing efficiency is studied using large-eddy simulations (LES) of stratified turbulence, where three different subgrid-scale (SGS) parameterizations are employed. For comparison, direct numerical simulations (DNS) and hyperviscosity simulations are also performed. In the regime of stratified turbulence where $Fr_{v}\sim 1$, the irreversible mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ in LES scales like $1/(1+2Pr_{t})$, where $Fr_{v}$ and $Pr_{t}$ are the vertical Froude number and turbulent Prandtl number, respectively. Assuming a unit scaling coefficient and $Pr_{t}=1$, $\unicode[STIX]{x1D6FE}_{i}$ goes to a constant value $1/3$, in agreement with DNS results. In addition, our results show that the irreversible mixing efficiency in LES, while consistent with this prediction, depends on SGS parameterizations and the grid spacing $\unicode[STIX]{x1D6E5}$. Overall, the LES approach can reproduce mixing efficiency results similar to those from the DNS approach if $\unicode[STIX]{x1D6E5}\lesssim L_{o}$, where $L_{o}$ is the Ozmidov scale. In this situation, the computational costs of numerical simulations are significantly reduced because LES runs require much smaller computational resources in comparison with expensive DNS runs.

The simplest model of isotropic compressible turbulence consists of the Euler equations augmented by the equation for the turbulent energy. This model can also be viewed as the Euler equations for a continuum with two independent entropies. One of them is a conventional thermodynamic entropy, and the other is associated with the turbulent energy. The shock relations for this model are examined. It is shown that the turbulent entropy cannot exceed some critical value. We propose a closed set of Rankine–Hugoniot relations for the description of shock waves in such a medium based on this estimation.

The interaction between internal waves (IW) and wind waves (WW) is studied. Three types of interaction are considered: spontaneous IW generation by a random field of WWs, and two feedback mechanisms - modulation and friction.

The latter mechanism has not been studied before. Its influence on the IW-WW coupling is of primary importance. The modulation and friction mechanisms result in exponential attenuation of the IWs. Attenuation of IWs propagating against wind is the strongest. The IW attenuation has a dimensionless decrement of order 10-3, whereas for storm winds it attains the value of 10-2. Joint action of the spontaneous generation of IWs and their attenuation due to feedback mechanisms permits a stationary ‘wind-IW’ spectrum to exist. For strong winds the ‘wind-IW’ energy is of order 105 erg cm-2. The effect of IWs on currents in the ocean's upper layer is considered. Momentum and energy lost by IWs due to their interaction with WWs generates inertial oscillations. Under the attenuation of intensive IWs, the amplitude of inertial oscillations may be compared with the background Ekman current.

This paper investigates the heat propagation process in a gas from concentrated energy sources with deposition times, $t'_d$, of the order of the characteristic acoustic time, $t'_a$, across the region where the temperature will be increased by a factor of order unity. Heat propagation takes place by two different mechanisms that act separately in two different neatly defined spatial regions of comparable size. Around the source, we find a conductive region of very high temperature where the spatial pressure variations are negligible. The edge of the resulting strongly heated low-density region appears as a contact surface that acts as a piston for the outer flow, where the pressure disturbances, of order of the ambient pressure in the distinguished regime $t'_d \,{\sim}\, t'_a$ considered here, generate a shock wave that heats up the outer gas as it propagates outwards. The mass and energy balances for the conductive region provide a differential equation linking its pressure with the velocity of its bounding contact surface, which is used, together with the jump conditions across the shock, when integrating the Euler equations for the outer compressible flow. Solutions for the heating history are obtained for point, line and planar sources for different values of the ratio $t'_d/t'_a$, including weak sources with $t'_d \,{\gg}\, t'_a$ and very intense sources with $t'_d \,{\ll}\, t'_a$. The solution determines in particular the temperature profile emerging as the pressure perturbations become negligible for times much larger than the acoustic time.

The ability of the helicity decomposition to describe compactly the dynamics of three-dimensional incompressible fluids is invoked to obtain new descriptions of both homogeneous and inhomogeneous turbulence. We first use this decomposition to derive four coupled nonlinear equations that describe an arbitrary three-dimensional turbulence, whether anisotropic and/or non-mirror-symmetric. We then use the decomposition to treat the inhomogeneous turbulence of a channel flow bounded by two parallel free-slip boundaries with almost the ease with which the homogeneous case has heretofore received treatment. However, this ease arises from the foundation of a random-phase hypothesis, which we introduce and motivate, that supersedes the translational invariance of a turbulence that is hypothesized to be homogeneous. For the description of this channel turbulence, we find that the three-dimensional modes and the two-dimensional modes having wave vectors parallel to the boundaries each couple precisely as in a homogeneous turbulence of the corresponding dimension. The anisotropy and inhomogeneity is in large part a feature incorporated into the solenoidal basis vectors used to describe an arbitrary solenoidal free-slip flow within the channel. We invoke the random-phase hypothesis, a feature of the dynamics, with closures, such as Kraichnan's direct-interaction approximation and his test-field model, in addition to the one most utilized in this manuscript, the eddy-damped quasi-normal Markovian (EDQNM) closure.

Turbulence characteristics inside a turbulent spot in plane Poiseuille flow are investigated by analysing a database obtained from a direct numerical simulation. The spot is found to consist of two distinct regions — a turbulent area and a wave area. The flow inside the turbulent area has a strong resemblance to that found in the fully developed turbulent channel. Suitably defined mean and r.m.s. fluctuations as well as the internal shear-layer structures are found to be similar to the turbulent counterpart. In the wave area the inflexional mean spanwise profiles cause a rapid growth of oblique waves, which break down to turbulence. The breakdown process of the oblique waves is reminiscent of the secondary instability observed during transition to turbulence in channel and boundary-layer flows. Other detailed characteristics associated with the Poiseuille spot are presented and are compared with experimental results.

Whereas the flow of simple single-phase Newtonian fluids tends to become more complex as the characteristic length scale in the problem (and hence the Reynolds number) increases, for complex elastic fluids such as dilute polymer solutions the opposite holds true. Thus small-scale, so-called ‘microfluidic’ flows of complex fluids can exhibit rich dynamics in situations where the ‘equivalent’ flow of Newtonian fluids remains linear and predictable. In the recent study of Qin et al. (J. Fluid Mech., vol. 864, 2019, R2) of the flow of a dilute polymeric fluid past a $50~\unicode[STIX]{x03BC}\text{m}$ cylinder (in a $100\times 60~\unicode[STIX]{x03BC}\text{m}$ channel), a novel 3-D holographic particle velocimetry technique reveals the underlying complexity of the flow, including inherent three-dimensionality and symmetry breaking as well as strong upstream propagation effects via elastic waves.

In this paper the methods of Parts 1 and 2 are extended to the case when the elliptic cylinder is executing angular oscillations about its centreline. At large distances from the cylinder the solution for a beam of waves tends to a similarity solution that decays more rapidly with distance than does the similarity solution for rectilinear oscillations described in Thomas & Stevenson (1972). Figure 4 shows in a remarkable way how the predicted wave profiles change with distance from the inviscid solution to the similarity one.

In the latter part of the paper the predictions of Parts 1 and 2 and other theories are compared with recent experimental observations. The results of the experiments are in good agreement with the predictions of Parts 1 and 2.

Remarkable differences in dispersion of a tracer material injected into turbulent pipe flows of water and water containing as little as 2·5 parts per million by weight of a soluble high-molecular-weight drag-reducing polyoxyethylene additive have been measured. Analysis of the tracer response curves in terms of a simple one-parameter model shows that the observed results are compatible with a drag-reduction mechanism based on thickening of the viscous sublayer adjoining the wall. Other experiments, reported briefly, suggest that polymer adsorption on to the wall is responsible for this thickening.

A zero pressure gradient turbulent boundary layer of $\textit {Re}_{\tau }=2500$ was perturbed by a single spanwise array of finite cylinders mounted on the bounding surface and extending through the logarithmic region. The cylinder height was $H/\delta =0.2$ ($H^{+}=500$), where $\delta $ is the boundary layer thickness, with an aspect ratio ($AR$) (height/diameter) of four. Streamwise–spanwise ($x\text {--}y$) planes of the flow were examined by particle image velocimetry (PIV) up to $7\delta $ downstream at a wall-normal location of $z^{+}=300$ for cylinder array spacings ranging from $0.2\delta $ to $0.8\delta $. Average streamwise velocity fields showed a splitting, then merging pattern of cylinder wakes which occurred further downstream as the cylinder spacing increased. Based on measurements at the furthest downstream location, both the spanwise variation of average streamwise velocity and the Fourier content in the instantaneous fields suggested that the case with $0.6\delta $ cylinder spacing, which matched the dominant spanwise scale in the unperturbed flow, yielded the most persistent downstream flow organization. A flying PIV method was implemented to track specific packet structures over a range $-2<x/\delta <7$ with respect to the cylinder array, corresponding to a time scale of $12.4\delta /U_{\infty }$. Packets approaching the $0.2\delta $ spacing array first lost their organization but then regained it a distance $2\delta $ downstream, suggesting that a persistent outer layer organization propagated inwards into the log region. For arrays with larger spanwise spacing, approaching packets were generally redirected into the spanwise location midway between cylinders and sometimes enhanced.

Viscous flow past a finite flat plate which is impulsively started in the direction normal to itself is studied numerically using a high-order mixed finite-difference and semi-Lagrangian scheme. The goal is to resolve the details of the vorticity generation, and to determine the dependence of the flow on time and Reynolds number. Vorticity contours, streaklines and streamlines are presented for a range of times $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}t\in [0.0002,5]$ and Reynolds numbers $\mathit{Re}\in [250,2000]$, non-dimensionalized with respect to the driving velocity and the plate length. At early times, the starting vortex is small relative to the plate length and is expected to grow as if an external length scale were absent. We identify three different types of scaling behaviours consistent with this premise. (i) At early times, solutions with different values of $\mathit{Re}$ are identical up to rescaling. (ii) The solution for fixed $\mathit{Re}$ satisfies a viscous similarity law in time, locally in space, as illustrated by the core vorticity maximum, the upstream boundary layer thickness, and the maximum speed, in three different regions of the flow. (iii) The vortex core trajectory and the shed circulation satisfy inviscid scaling laws for several decades in time, and are consequently essentially $\mathit{Re}$-independent at these times. In addition, the computed induced drag and tangential forces are found to follow approximate scaling laws that define their dependence on time and $\mathit{Re}$.

The Miles–Howard theorem states that a necessary condition for normal-mode instability in parallel, inviscid, steady stratified shear flows is that the minimum gradient Richardson number, $Ri_{g,min}$, is less than $1/4$ somewhere in the flow. However, the non-normality of the Navier–Stokes and buoyancy equations may allow for substantial perturbation energy growth at finite times. We calculate numerically the linear optimal perturbations which maximize the perturbation energy gain for a stably stratified shear layer consisting of a hyperbolic tangent velocity distribution with characteristic velocity $U_{0}^{\ast }$ and a uniform stratification with constant buoyancy frequency $N_{0}^{\ast }$. We vary the bulk Richardson number $Ri_{b}=N_{0}^{\ast 2}h^{\ast 2}/U_{0}^{\ast 2}$ (corresponding to $Ri_{g,min}$) between 0.20 and 0.50 and the Reynolds numbers $\mathit{Re}=U_{0}^{\ast }h^{\ast }/\unicode[STIX]{x1D708}^{\ast }$ between 1000 and 8000, with the Prandtl number held fixed at $\mathit{Pr}=1$. We find the transient growth of non-normal perturbations may be sufficient to trigger strongly nonlinear effects and breakdown into small-scale structures, thereby leading to enhanced dissipation and non-trivial modification of the background flow even in flows where $Ri_{g,min}>1/4$. We show that the effects of nonlinearity are more significant for flows with higher $\mathit{Re}$, lower $Ri_{b}$ and higher initial perturbation amplitude $E_{0}$. Enhanced kinetic energy dissipation is observed for higher-$Re$ and lower-$Ri_{b}$ flows, and the mixing efficiency, quantified here by $\unicode[STIX]{x1D700}_{p}/(\unicode[STIX]{x1D700}_{p}+\unicode[STIX]{x1D700}_{k})$ where $\unicode[STIX]{x1D700}_{p}$ is the dissipation rate of density variance and $\unicode[STIX]{x1D700}_{k}$ is the dissipation rate of kinetic energy, is found to be approximately 0.35 for the most strongly nonlinear cases.