Kolmogorov introduced dissipative scales based on the mean dissipation $\langle {\it\varepsilon}\rangle$ and the viscosity ${\it\nu}$, namely the Kolmogorov length ${\it\eta}=({\it\nu}^{3}/\langle {\it\varepsilon}\rangle )^{1/4}$ and the velocity $u_{{\it\eta}}=({\it\nu}\langle {\it\varepsilon}\rangle )^{1/4}$. However, the existence of smaller scales has been discussed in the literature based on phenomenological intermittency models. Here, we introduce exact dissipative scales for the even-order longitudinal structure functions. The derivation is based on exact relations between even-order moments of the longitudinal velocity gradient $(\partial u_{1}/\partial x_{1})^{2m}$ and the dissipation $\langle {\it\varepsilon}^{m}\rangle$. We then find a new length scale ${\it\eta}_{C,m}=({\it\nu}^{3}/\langle {\it\varepsilon}^{m/2}\rangle ^{2/m})^{1/4}$ and $u_{C,m}=({\it\nu}\langle {\it\varepsilon}^{m/2}\rangle ^{2/m})^{1/4}$, i.e. the dissipative scales depend rather on the moments of the dissipation $\langle {\it\varepsilon}^{m/2}\rangle$ and thus the full probability density function (p.d.f.) $P({\it\varepsilon})$ instead of powers of the mean $\langle {\it\varepsilon}\rangle ^{m/2}$. The results presented here are exact for longitudinal even-ordered structure functions under the assumptions of (local) isotropy, (local) homogeneity and incompressibility, and we find them to hold empirically also for the mixed and transverse as well as odd orders. We use direct numerical simulations (DNS) with Reynolds numbers from $Re_{{\it\lambda}}=88$ up to $Re_{{\it\lambda}}=754$ to compare the different scalings. We find that indeed $P({\it\varepsilon})$ or, more precisely, the scaling of $\langle {\it\varepsilon}^{m/2}\rangle /\langle {\it\varepsilon}\rangle ^{m/2}$ as a function of the Reynolds number is a key parameter, as it determines the ratio ${\it\eta}_{C,m}/{\it\eta}$ as well as the scaling of the moments of the velocity gradient p.d.f. As ${\it\eta}_{C,m}$ is smaller than ${\it\eta}$, this leads to a modification of the estimate of grid points required for DNS.

The nonlinear interaction between the mean flow and a coherent disturbance in a two-dimensional turbulent mixing layer is addressed. Based on considerations from stability theory, previous experimental results, in particular the modification of the mean velocity profile, the peculiar growth of the forced shear-layer thickness and the spatial growth of the disturbance amplitude, are explained. A model that assumes a quasi-parallel mean flow having a self-similar mean velocity profile is developed. The model is capable of predicting the downstream evolution of turbulent mixing layers subjected to external excitations.

The problem of capillary–gravity waves generated by certain moving oscillatory surface-pressure distributions is investigated. The main difficulty of the problem lies in finding the real roots of the modified frequency equations. This is dealt with by the use of certain geometric considerations. The critical condition that results from the formation of double roots of the modified frequency equations is represented as a surface. This surface divides the whole space into several distinct regions. For points in different regions the propagation of waves is different. The waves are determined in all cases.

In addition to conserving energy and enstrophy, the nonlinear terms of the two-dimensional incompressible Navier–Stokes equation are well known to conserve the global integral of any continuously differentiable function of the scalar vorticity field. However, the phenomenological role of these additional inviscid invariants, including the issue as to whether they cascade to large or small scales, is an open question. In this work, well-resolved implicitly dealiased pseudospectral simulations suggest that the fourth power of the vorticity cascades to small scales.

Incoherent turbulent motion modulated by coherent large-scale motion contributes to second-order coherent stresses. The spatial distribution of wave-induced stress was measured in a jet whose cross-section had been distorted through controlled resonant interactions between two forced, helical waves spinning in opposite directions. The transfer of energy from the coherent motion to broadband turbulence is documented. Shape assumptions are examined by comparing radial distributions to predictions from linear, inviscid stability theory. Control over small-scale mixing is examined by demodulating the coherent envelope of small-scale turbulence and by correlating it with features of the coherent, large-scale motion. Coherent production is shown to be associated with the roll-up process and there is evidence of secondary, inflexional instabilities.

A weakly nonlinear theory is developed for convection in an infinite rigid horizontal rectangular channel uniformly heated from below. A combination of analytical and numerical techniques along and in the cross-section of the channel leads to the derivation of an amplitude equation governing the spatial and temporal evolution of the flow above the critical Rayleigh number. Results are obtained for general Prandtl numbers and a wide range of aspect ratios. Overall trends are confirmed by comparison with results for an idealized model with stress-free horizontal boundaries. For wide channels, where the aspect ratio is large, the limiting form of the amplitude equation is predicted by reference to the two-dimensional equation describing roll patterns in infinite layers. The connection with this well-developed theory is established for both rigid and stress-free horizontal boundary conditions.

A model equation is derived for the combined longitudinal and lateral dispersion of a buoyant contaminant in open-channel flow. The central hypotheses are that the water is shallow and that to a first approximation the effluent is vertically well mixed. The model includes allowance for the reduction in turbulent intensity due to weak vertical stratification, a buoyancy-driven secondary flow, and the redistribution of longitudinal momentum by the secondary flow. For a plume the theoretical results for the excess spreading due to buoyancy agree well with Prych's (1970) experimental results.

We explore the rotational stability of hovering flight in an idealized two-dimensional model. Our model is motivated by an experimental pyramid-shaped object (Weathers et al., J. Fluid Mech, vol. 650, 2010, pp. 415–425; Liu et al., Phys. Rev. Lett., vol. 108, 2012, 068103) and a computational $\wedge$-shaped analogue (Huang et al., Phys. Fluids, vol. 27 (6), 2015, 061706; Huang et al., J. Fluid Mech., vol. 804, 2016, pp. 531–549) hovering passively in oscillating airflows; both systems have been shown to maintain rotational balance during free flight. Here, we attach the $\wedge$-shaped flyer at its apex in oscillating flow, allowing it to rotate freely akin to a pendulum. We use computational vortex sheet methods and we develop a quasi-steady point-force model to analyse the rotational dynamics of the flyer. We find that the flyer exhibits stable concave-down ($\wedge$) and concave-up ($\vee$) behaviour. Importantly, the down and up configurations are bistable and co-exist for a range of background flow properties. We explain the aerodynamic origin of this bistability and compare it to the inertia-induced stability of an inverted pendulum oscillating at its base. We then allow the flyer to flap passively by introducing a rotational spring at its apex. For stiff springs, flexibility diminishes upward stability but as stiffness decreases, a new transition to upward stability is induced by flapping. We conclude by commenting on the implications of these findings for biological and man-made aircraft.

This paper presents a theoretical analysis of perfect gas flow over a convex corner of a rigid-body contour. It is assumed that the flow is subsonic before the corner. It accelerates around the corner to become supersonic, and then undergoes an additional acceleration in the expansion Prandtl–Meyer fan that forms in the supersonic part of the flow behind the corner. The entire process is described by a self-similar solution of the Kármán–Guderley equation. The latter shows that the boundary layer approaching the apex of the corner is exposed to a singular pressure gradient, ${\rm d} p / {\rm d} x \sim (-x)^{-3/5}$, where $x$ denotes the coordinate measured along the body surface from the corner apex. Under these conditions, the solution for the boundary layer also develops a singularity. In particular, the longitudinal velocity near the body surface behaves as $U \sim Y^{1/2}$. Here $Y$ is the normal coordinate scaled with the boundary-layer thickness $Re^{-1/2}$; $Re$ being the Reynolds number, assumed large in this theory.

As usual, the boundary layer splits up into two parts, a viscous near-wall sublayer and a locally inviscid main part of the boundary layer. The analysis of the displacement effect of the boundary layer shows that neither the viscous sublayer nor the main part determines the displacement thickness. Instead, the overlapping region situated between them proves to be responsible for the shape of the streamlines at the outer edge of the boundary layer. This leads to a significant simplification of the analysis of the flow behaviour in the viscous–inviscid interaction region that forms in a small vicinity of the corner. In order to describe the flow behaviour in this region, one has to solve the Kármán–Guderley equation for the inviscid part of the flow outside the boundary layer. The influence of the boundary layer is expressed through a boundary condition, that relates the streamline deflection angle $\vartheta $ at the outer edge of the boundary layer to the pressure gradient ${\rm d}p / {\rm d}x$ acting upon the boundary layer. The boundary-layer analysis leads to an analytical formula that relates $\vartheta $ and ${\rm d}p /{\rm d}x$ (unlike in previous studies of the viscous–inviscid interaction). The interaction problem was solved numerically to confirm that the solution develops a finite-distance singularity.

The decay of high-Reynolds-number inhomogeneous turbulence in an unbounded domain is considered. The turbulence may be initially localized in one to three spatial directions and the fluid is assumed to be at rest at infinity in those directions. Previous arguments used to determine the decay laws of homogeneous turbulence are extended to the decay of inhomogeneous turbulence by integrating the turbulence statistics over the inhomogeneous directions. Dimensional arguments based on the invariance or near-invariance of low-wavenumber spectral coefficients associated with the integrated mean-square velocity are used to determine asymptotic decay laws for inhomogeneous turbulence. These decay laws depend on the number of inhomogeneous directions of the flow field and reduce to the well-known decay laws of homogeneous turbulence when this number is zero. Different decay laws are determined depending on the spectral behaviour at low wavenumbers. Asymptotic similarity states of the spectrum during the decay and of the distribution of the mean-square velocity along the inhomogeneous directions are also determined. An analytical result for the decay of the mean-square velocity at the centre of the initial disturbance is found, and the decay proceeds more rapidly with increasing number of inhomogeneous directions due to the transport of energy along those directions.

Large-eddy simulations of decaying turbulence homogeneous in a plane and localized in a single direction are performed to test the theoretical scaling laws. The numerically determined asymptotic decay laws of the integrated mean-square velocity agree well with the theoretical predictions. A self-similar decay of the spectra and mean-square velocity distributions is also observed. The simulation results suggest that when the low-wavenumber spectral coefficient is an exact invariant, a unique similarity state depending only on the initial value of this invariant and independent of all other aspects of the initial conditions is attained asymptotically.

A solution is presented of the asymptotic flow due to a point source of momentum in a uniformly rotating unbounded environment. Under the condition that the relative swirl velocity of the jet is small compared with the ambient swirl velocity, the equations of motion reduce to a set of linear equations. These equations are expressed in terms of similarity variables and a single ordinary differential equation is derived in terms of the similarity stream function. The profiles of the flow are calculated numerically.

The jet is shown to have a narrow viscous core whose thickness increases with distance z from the virtual source of momentum as ($(vz|\Omega)^{\frac {1}{3}}$, where v is the kinematic viscosity and Ω the ambient angular velocity.

This paper considers the stability of the one-dimensional boundary layer generated by sudden heating of an infinite vertical wall. A quasi-steady approximation is used to analyse the asymptotic form of the lower branch of the neutral curve, corresponding to disturbances of wavelength much greater than the boundary-layer width. This leads to predictions of the critical wavenumber for neutral stability and the maximum phase speed of the travelling waves. Results are obtained for a range of Prandtl numbers and are compared with solutions of the full stability equations and with numerical simulations and experimental observations of cavity flows driven by sudden heating of the sidewalls.