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The stability properties of a natural convection boundary layer adjacent to an isothermally heated vertical wall, with Prandtl number 0.71, are numerically investigated in the configuration of a temporally evolving parallel flow. The instantaneous linear stability of the flow is first investigated by solving the eigenvalue problem with a quasi-steady assumption, whereby the unsteady base flow is frozen in time. Temporal responses of the discrete perturbation modes are numerically obtained by solving the two-dimensional linearized disturbance equations using a ‘frozen’ base flow as an initial-value problem at various
is the Grashof number based on the velocity integral boundary layer thickness
. The resultant amplification rates of the discrete modes are compared with the quasi-steady eigenvalue analysis, and both two-dimensional and three-dimensional direct numerical simulations (DNS) of the temporally evolving flow. The amplification rate predicted by the linear theory compares well with the result of direct numerical simulation up to a transition point. The extent of the linear regime where the perturbations linearly interact with the base flow is thus identified. The value of the transition
, according to the three-dimensional DNS results, is dependent on the initial perturbation amplitude. Beyond the transition point, the DNS results diverge from the linear stability predictions as nonlinear mechanisms become important.
Experimental evidence for previously unreported fountain behaviour is presented. It has been found that the first unstable mode of a three-dimensional round fountain is a laminar flapping motion that can grow to a circling or multimodal flapping motion. With increasing Froude and Reynolds numbers, fountain behaviour becomes more disorderly, exhibiting a laminar bobbing motion. The transition between steady behaviour, the initial flapping modes and the laminar bobbing flow can be approximately described by a function FrRe2/3=C. The transition to turbulence occurs at Re > 120, independent of Froude number, and the flow appears to be fully turbulent at Re≈2000. For Fr > 10 and Re≲120, sinuous shear-driven instabilities have been observed in the rising fluid column. For Re≳120 these instabilities cause the fountain to intermittently breakdown into turbulent jet-like flow. For Fr≲10 buoyancy forces begin to dominate the flow and pulsing behaviour is observed. A regime map of the fountain behaviour for 0.7≲Fr≲100 and 15≲Re≲1900 is presented and the underlying mechanisms for the observed behaviour are proposed. Movies are available with the online version of the paper.
The stability of the buoyancy layer on a uniformly heated vertical wall in a stratified fluid is investigated using both semi-analytical and direct numerical methods. As in the related problem in which the excess temperature of the wall is specified, the basic laminar flow is steady and one-dimensional. Here flows varying in time and with height are considered, the behaviour being determined by the fluid's Prandtl number and a Reynolds number proportional to the ratio of two temperature gradients: the horizontal one imposed at the wall and the vertical one existing in the far field. For low Reynolds numbers, the flow is stable with variation only in the wall-normal direction. For Reynolds numbers greater than a critical value, depending on the Prandtl number, the flow is unstableand supports two-dimensional travelling waves. The critical Reynolds number and other properties have been obtained via linearized stability analysis and are shown to accuratelypredict the behaviour of the full nonlinear solution obtained numerically for Prandtl number 7. The stability analysis employs a novel Laguerre collocation scheme while the direct numerical simulations use a second-order finite volume method.
A simple lumped hydraulic model of knee drainage following arthroplasty is developed incorporating a pressure-volume equation of state for the knee capsule and a wound healing rate dynamically retarded by the blood flow-induced shear stress. The resulting second-order nonlinear ordinary differential system is examined numerically and qualitatively to map the parameter space. In the model, moderate suction or a slight back-pressure promotes gradual drainage and healing whereas excessive suction can lead to a bifurcation in which healing is retarded or even prevented. Guided, then, by the model, the literature, and experience, continuous drainage with a small constant back-pressure appeared beneficial so we prospectively evaluated a series of ten patients. The results are consistent with the model and promising.
It is shown how to decompose a three-dimensional field periodic in two Cartesian coordinates into five parts, three of which are identically divergence-free and the other two orthogonal to all divergence-free fields. The three divergence-free parts coincide with the mean, poloidal and toroidal fields of Schmitt and Wahl; the present work, therefore, extends their decomposition from divergence-free fields to fields of arbitrary divergence. For the representation of known and unknown fields, each of the five subspaces is characterised by both a projection and a scalar representation. Use of Fourier components and wave coordinates reduces poloidal fields to the sum of two-dimensional poloidal fields, and toroidal fields to the sum of unidirectional toroidal fields.
We continue our study of the adaptation from spherical to doubly periodic slot domains of the poloidal-toroidal representation of vector fields. Building on the successful construction of an orthogonal quinquepartite decomposition of doubly periodic vector fields of arbitrary divergence with integral representations for the projections of known vector fields and equivalent scalar representations for unknown vector fields (Part 1), we now present a decomposition of vector field equations into an equivalent set of scalar field equations. The Stokes equations for slow viscous incompressible fluid flow in an arbitrary force field are treated as an example, and for them the application of the decomposition uncouples the conservation of momentum equation from the conservation of mass constraint. The resulting scalar equations are then solved by elementary methods. The extension to generalised Stokes equations resulting from the application of various time discretisation schemes to the Navier-Stokes equations is also solved.
Natural convection in horizontally heated spherical fluid-filled cavities is considered in
the low Grashof number limit. The equations governing the asymptotic expansion are
derived for all orders. At each order a Stokes problem must be solved for the momentum
correction. The general solution of the Stokes problem in a sphere with arbitrary
smooth body force is derived and used to obtain the zeroth-order (creeping) flow and
the first-order corrections due to inertia and buoyancy. The solutions illustrate the two
mechanisms adduced by Mallinson & de Vahl Davis (1973, 1977) for spanwise flow in
horizontally heated cuboids. Further, the analytical derivations and expressions clarify
these mechanisms and the conditions under which they do not operate. The inertia
and buoyancy effects vanish with the Grashof and Rayleigh numbers, respectively,
and both vanish if the flow is purely vertical, as in a very tall and narrow cuboid.
The fully developed flow in a vertical cavity or duct subject to horizontal heating
is considered. Solutions of the Boussinesq equations are obtained for rectangular
and elliptic sections, in terms of Fourier series and polynomials, respectively. Both
generalize the familiar odd-symmetric cubic profile of the plane cavity. Uniqueness is
demonstrated under the restriction that the flow is independent of height. For cavities
with rectangular sections, it is predicted and verified that the flow in the plane of
spanwise symmetry is practically independent of the span if this exceeds 1.7 times the
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