This paper presents an approximate method for solving Oseen's linearized equations for a two-dimensional steady flow of incompressible viscous fluid past arbitrary cylindrical bodies at low Reynolds numbers. The formulation is based on a discrete singularity method with a least squares criterion for satisfying the no-slip boundary condition. That is, sets of Oseenlets, sinks, sources and vortices are discretely distributed in the interior of the body, and then the least squares criterion attempts to minimize the integrated squares of velocities along the body contour, thus leading to a system of simultaneous algebraic equations. Complex-variable arithmetic, usually available on modern computers, makes the computation algorithm very simple. Furthermore, the method is applicable to cases that cannot be solved by classical analytical approaches. As examples of application, we computed the forces acting on a single circular cylinder, two circular cylinders of equal radius separated by a distance, an inclined elliptic cylinder and an inclined square cylinder all of which are immersed in uniform flow fields. The computed results agree very well with those of classical analytical methods.

We study the influence of shock and boundary layer interactions in transonic flutter of an aeroelastic system using a Reynolds-averaged Navier–Stokes (RANS) solver together with the Spalart–Allmaras turbulence model. We show that the transonic flutter boundary computed using a viscous flow solver can be divided into three distinct regimes: a low transonic Mach number range wherein viscosity mimics increasing airfoil thickness thereby mildly influencing the flutter boundary; an intermediate region of drastic change in the flutter boundary due to shock-induced separation; and a high transonic Mach number zone of no viscous effects when the shock moves close to the trailing edge. Inviscid transonic flutter simulations are a very good approximation of the aeroelastic system in predicting flutter in the first and third regions: that is when the shock is not strong enough to cause separation, and in regions where the shock-induced separated region is confined to a small region near the trailing edge of the airfoil. However, in the second interval of intermediate transonic Mach numbers, the power distribution on the airfoil surface is significantly influenced by shock-induced flow separation on the upper and lower surfaces leading to oscillations about a new equilibrium position. Though power contribution by viscous forces are three orders of magnitude less than the power due to pressure forces, these viscous effects manipulate the flow by influencing the strength and location of the shock such that the power contribution by pressure forces change significantly. Multiple flutter points that were part of the inviscid solution in this regime are now eliminated by viscous effects. Shock motion on the airfoil, shock reversal due to separation, and separation and reattachment of flow on the airfoil upper surface, also lead to multiple aerodynamic forcing frequencies. These flow features make the flutter boundary quantitatively sensitive to the turbulence model and numerical method adopted, but qualitatively they capture the essence of the physical phenomena.

Receptivity of a laminar boundary layer to the interaction of time-harmonic free-stream disturbances with a three-dimensional roughness element is studied. The three-dimensional nonlinear triple–deck equations are solved numerically to provide the basic steady-state motion. At high Reynolds numbers, the governing equations for the unsteady motion are the unsteady linearized three-dimensional triple-deck equations. These equations can only be solved numerically. In the absence of any roughness element, the free-stream disturbances, to the first order, produce the classical Stokes flow, in the thin Stokes layer near the wall (on the order of our lower deck). However, with the introduction of a small three-dimensional roughness element, the interaction between the hump and the Stokes flow introduces a spectrum of all spatial disturbances inside the boundary layer. For supercritical values of the scaled Strouhal number, S0 > 2, these Tollmien–Schlichting waves are amplified in a wedge-shaped region, 15° to 18° to the basic-flow direction, extending downstream of the hump. The amplification rate approaches a value slightly higher than that of two-dimensional Tollmien–Schlichting waves, as calculated by the linearized analysis, far downstream of the roughness element.

We investigate the time-harmonic small-amplitude motion of the mechanical system that consists of water and a body freely floating in it; water occupies a half-space, whereas the body is either surface-piercing or totally submerged. As a mathematical model of this coupled motion, we consider a spectral problem (the spectral parameter is the frequency of oscillations), for which the following results are obtained. The total energy of the water motion is finite and the equipartition of energy holds for the whole system. For any value of frequency, infinitely many eigensolutions are constructed and each of them consists of a non-trivial velocity potential and the zero vector describing the motion of the body; the latter means that trapping bodies (infinitely many of them are found) are motionless although they float freely. They are surface-piercing, have axisymmetric submerged parts and are obtained by virtue of the so-called semi-inverse procedure. We also prove that certain restrictions on the body geometry (which are violated for the constructed trapping bodies) guarantee that the problem has only a trivial solution for frequencies that are sufficiently large being measured in terms of a certain dimensionless quantity.

We study miscible displacements in long ducts in the dispersive limit of small $\varepsilon \hbox{\it Pe}$, where $\varepsilon \,{\ll}\, 1$ is the inverse aspect ratio and $\hbox{\it Pe}$ the Péclet number. We consider the class of generalized Newtonian fluids, with specified closure laws for the fluid properties of the concentration-dependent mixture. Regardless of viscosity ratio and the constitutive laws of the pure fluids, for sufficiently small $\varepsilon \hbox{\it Pe}$ these displacements are characterized by rapid cross-stream diffusion and slow streamwise dispersion, i.e. the concentration appears to be near-uniform across the duct and spreads slowly as it translates. Using the multiple-scales method we derive the leading-order asymptotic approximation to the average fluid concentration $\bar{c}_0$. We show that $\bar{c}_0$ evolves on the slow timescale $t \sim (\varepsilon \hbox{\it Pe})^{-1}$, and satisfies a nonlinear diffusion equation in a frame of reference moving with the mean speed of the flow. In the case that the two fluids have identical rheologies and the concentration represents a passive tracer, the diffusion equation is linear. For Newtonian fluids we recover the classical results of Taylor (l953), Aris (1956), and for power-law fluids those of Vartuli et al. (1995). In the case that the fluids differ and/or that mixing is non-passive, $\bar{c}_0$ satisfies a nonlinear diffusion equation in the moving frame of reference. Given a specific mixing/closure law for the rheological properties, we are able to compute the dispersive diffusivity $D_T(\bar{c}_0)$ and predict spreading along the channel. We show that $D_T(\bar{c}_0)$ can vary significantly with choice of mixing law and discuss why. This also opens the door to possibilities of controlling streamwise spreading by the rheological design of reactive mixtures, i.e. including chemical additives such that the rheology of the mixture behaves very differently to the rheology of either pure fluid. Computed examples illustrate the potential effects that might be achieved.

The stability of quasi-geostrophic three-layer stratified flow in a channel is examined. The mean zonal velocity $\overline{U}_i$ is uniform within each layer (i = 1, 2, 3). Thus, as in the two-layer model of Phillips (1954), the only source of energy for growing disturbances is the potential energy stored in the sloping interfaces. Attention is focused upon the case in which ε = Δρ2/Δρ1 [Lt ] 1 (Δρ1, Δρ2 are the changes in density across the upper and the lower interfaces). Two scales of instability are possible: long waves (wavenumber O(1)) associated with the upper interface and short waves (wave-number O(ε−½)) associated with the lower interface. It is found that short waves are unstable only when S (the ratio of the slope of the lower interface to that of the upper interface) is greater than one or less than zero, i.e. when the gradients of potential vorticity in the two lower layers have opposite signs. The short waves have the largest growth rates when S2ε (the ratio of the potential energy stored in the lower interface to that stored in the upper interface) [gsim ] 1. The results of this analysis are used in an accompanying paper to interpret some experiments with three-layer eddies.

In Jeffery–Hamel flow, the motion of a viscous incompressible fluid between rigid plane walls, unidirectional flow is impossible if the angle between the walls exceeds a critical value of 2α2 which depends on the Reynolds number. In this paper the nonlinear development of the flow near this critical value is studied through numerical solutions of the two-dimensional Navier–-Stokes equations for flow in divergent channels with piecewise straight walls. It is found that if the angle between the walls exceeds 2α2 then Jeffery–-Hamel flow does not occur, and the solution takes the form of a large-amplitude wave with eddies attached alternately to the upper and lower walls. When viewed in the appropriate coordinate system, far downstream the wave has constant wavelength and strength, although, physically, there is a linear increase in wavelength with distance downstream, i.e. the wavelength is proportional to the channel width. If the angle between the walls is less than 2α2, then the existence (or otherwise) of the wave depends on the conditions near the inlet, in particular the local geometry of the channel. Jeffery–-Hamel flow is obtained downstream of the inlet for angles well below 2α2, but close to but below the critical value, solutions have been obtained with the wave extending (infinitely) far downstream. The wavelengths obtained numerically were compared with those from linear theory with spatially developing steady modes. No agreement was found: the wavelengths from the steady Navier–-Stokes solutions are significantly larger than that predicted by the theory. However, in other important aspects the results of this study are consistent with those from previous studies of the development/existence of Jeffery–-Hamel flow, in particular as regards the importance of the upstream conditions and the subcritical nature of the spatial development of the flow near the critical boundary in the Reynolds number–wall angle parameter space.

The paper by Dritschel et al. (J. Fluid Mech., vol. 783, 2015, pp. 1–22) describes the long-time behaviour of inviscid two-dimensional fluid dynamics on the surface of a sphere. At issue is whether the flow settles down to an equilibrium or whether, for generic (random) initial conditions, the long-time solution is periodic, quasi-periodic or chaotic. While it might be surprising that this issue is not settled in the literature, it is important to keep in mind that the Euler equations form a dissipationless Hamiltonian system, hence the set of equations only redistributes the initial vorticity, generating smaller and smaller scales, while keeping kinetic energy, angular impulse and an infinite family of vorticity moments (Casimirs) intact. While special solutions that never settle down to an equilibrium state can be constructed using point vortices, vortex patches and other distributions, the fate of random initial conditions is a trickier problem. Previous statistical theories indicate that the long-time state should be a stationary large-scale distribution of vorticity. By carrying out careful numerical simulations using two different methods, the authors make a compelling case that the generic long-time state resembles a large-scale oscillating quadrupolar vorticity field, surrounded by persistent small-scale vortices. While numerical simulations can never conclusively settle this issue, the results might help guide future theories that seek to prove the existence of such an interesting dynamical long-time state.

We derive a dispersive model of shear shallow water flows which takes into account a non-uniform horizontal velocity. This model generalizes the Green–Naghdi model to the case of shear flows. Besides the classical dispersion term in the Green–Naghdi model related to the acceleration of the free surface, it also contains a new dispersion parameter related to the flow structure. This parameter is related to the second moment of the velocity fluctuation with respect to the vertical coordinate. The distinction between shearing and turbulence based on the scale of variation of the velocity fluctuation is proposed. In particular, an equation for the turbulence generation is derived. Solitary waves for this model are obtained in explicit form. Comparison of solitary wave profiles with experimental ones is also performed. The agreement is very good apart from the small region near the top of the wave.

The resonance modes in Mach 0.94 turbulent flow over a cavity having a length-to-depth ratio of five were explored using time-resolved particle image velocimetry (TR-PIV) and time-resolved pressure sensitive paint (TR-PSP). Mode switching was quantified in the velocity field simultaneous with the pressure field. As the mode number increased from one through three, the resonance activity moved from a region downstream within the recirculation region to areas further upstream in the shear layer, an observation consistent with linear stability analysis. The second and third modes contained organized structures associated with shear layer vortices. Coherent structures occurring in the velocity field during modes two and three exhibited a clear modulation in size with streamwise distance. The streamwise periodicity was attributable to the interference of downstream-propagating vortical disturbances with upstream-travelling acoustic waves. The coherent structure oscillations were approximately $180^{\circ }$ out of phase with the modal surface pressure fluctuations, analogous to a standing wave. Modal propagation (or phase) velocities, based on cross-correlations of bandpass-filtered velocity fields were found for each mode. The phase velocities also showed streamwise periodicity and were greatest at regions of maximum constructive interference where coherent structures were the largest. Overall, the phase velocities increased with modal frequency, which coincided with the modal activity residing at higher portions of the cavity where the local mean flow velocity was elevated. Together, the TR-PIV and TR-PSP provide unique details not only on the distribution of modal activity throughout the cavity, but also new understanding of the resonance mechanism as observed in the velocity field.

A simple approach is described for computing spatially extended, weakly nonlinear optimal disturbances, suitable for maintaining a disturbance-regeneration cycle in a simple shear flow. Weakly nonlinear optimals, computed over a short time interval for the expansion used to remain tenable, are oblique waves which display a shorter streamwise and a longer spanwise wavelength than their linear counterparts. Threshold values of the initial excitation energy, separating the region of damped waves from that where disturbances grow without bounds, are found. Weakly nonlinear optimal solutions of varying initial amplitudes are then fed as initial conditions into direct numerical simulations of the Navier–Stokes equations and it is shown that the weakly nonlinear model permits the identification of flow states which cause rapid breakdown to turbulence.