Four existing integral models of unsteady turbulent plumes are revisited. We demonstrate that none of these published models is ideal for general descriptions of unsteady behaviour and put forward a modified model. We show that the most recent (top-hat) plume model (Scase et al. J. Fluid Mech., vol. 563, 2006, p. 443), and the earlier (Gaussian) plume models (Delichatsios J. Fluid Mech., vol. 93, 1979, p. 241; Yu Trans. ASME, vol. 112, 1990, p.186), are all ill-posed. This ill-posedness arises from the downstream growth of short-scale waves, which have an unbounded downstream growth rate. We show that both the top-hat and the Gaussian (Yu) models can be regularized, rendering them well-posed, by the inclusion of a velocity diffusion term. The effect of including this diffusive mechanism is to include a vertical structure in the model that can be interpreted as representing the vertical extent of an eddy. The effects of this additional mechanism are small for steady applications, and cases where the plume forcing can be considered to follow a power law (both of which have been studied extensively). However, the inclusion of diffusion is shown to be crucial to the general initial-value problem for unsteady models.

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We consider the temporal evolution of a viscous incompressible fluid in a torus of finite curvature; a problem first investigated by Madden & Mullin (J. Fluid Mech., vol. 265, 1994, pp. 265-217). The system is initially in a state of rigid-body rotation (about the axis of rotational symmetry) and the container's rotation rate is then changed impulsively. We describe the transient flow that is induced at small values of the Ekman number, over a time scale that is comparable to one complete rotation of the container. We show that (rotationally symmetric) eruptive singularities (of the boundary layer) occur at the inner or outer bend of the pipe for a decrease or an increase in rotation rate respectively. Moreover, on allowing for a change in direction of rotation, there is a (negative) ratio of initial-to-final rotation frequencies for which eruptive singularities can occur at both the inner and outer bend simultaneously. We also demonstrate that the flow is susceptible to a combination of axisymmetric centrifugal and non-axisymmetric inflectional instabilities. The inflectional instability arises as a consequence of the developing eruption and is shown to be in qualitative agreement with the experimental observations of Madden & Mullin (1994). Throughout our work, detailed quantitative comparisons are made between asymptotic predictions and finite-(but small-) Ekman-number Navier-Stokes computations using a finite-element method. We find that the boundary-layer results correctly capture the (finite-Ekman-number) rotationally symmetric flow and its global stability to linearised perturbations.

Streaks are a common feature of disturbed boundary-layer flows. They play a central role in transient growth mechanisms and are a building block of self-sustained structures. Most theoretical work has focused on streaks that are periodic in the spanwise direction, but in this work we consider a single spatially localised streak embedded into a Blasius boundary layer. For small streak amplitudes, we show the perturbation can be described in terms of a set of eigenmodes that correspond to an isolated streak/roll structure. These modes are new, and arise from a bi-global eigenvalue calculation; they decay algebraically downstream and may be viewed as the natural three-dimensional extension of the classical two-dimensional Libby & Fox (J. Fluid Mech., vol. 17 (3), 1963, pp. 433–449) solutions. Despite their bi-global nature, we show that a subset of these eigenmodes (including the slowest decaying) is fundamentally related to the solutions first presented by Luchini (J. Fluid Mech., vol. 327, 1996, pp. 101–116), as derived for spanwise-periodic disturbances (at small spanwise wavenumber). This surprising connection is made by an analysis of the far-field decay of the bi-global state. We also address the fully non-parallel downstream development of nonlinear streaks, confirming that the aforementioned eigenmodes are recovered as the streak/roll decays downstream. Encouraging comparisons are made with available experimental data.

We explore the relevance of the idealized Jeffery-Hamel similarity solution to the practical problem of flow in a diverging channel of finite (but large) streamwise extent. Numerical results are presented for the two-dimensional flow in a wedge of separation angle 2α, bounded by circular arcs at the inlet/outlet and for a net radial outflow of fluid. In particular, we show that in a finite domain there is a sequence of nested neutral curves in the (Re, α) plane, each corresponding to a midplane symmetry-breaking (pitchfork) bifurcation, where Re is a Reynolds number based on the radial mass flux. For small wedge angles we demonstrate that the first pitchfork bifurcation in the finite domain occurs at a critical Reynolds number that is in agreement with the only pitchfork bifurcation in the infinite-domain similarity solution, but that the criticality of the bifurcation differs (in general). We explain this apparent contradiction by demonstrating that, for α 1, superposition of two (infinite-domain) eigenmodes can be used to construct a leading-order finite-domain eigenmode. These constructed modes accurately predict the multiple symmetry-breaking bifurcations of the finite-domain flow without recourse to computation of the full field equations. Our computational results also indicate that temporally stable, isolated, steady solutions may exist. These states are finite-domain analogues of the steady waves recently presented by Kerswell, Tutty, & Drazin (J. Fluid Mech., vol. 501, 2004, pp. 231-250) for an infinite domain. Moreover, we demonstrate that there is non-uniqueness of stable solutions in certain parameter regimes. Our numerical results tie together, in a consistent framework, the disparate results in the existing literature.