Similarity solution

About: Similarity solution is a research topic. Over the lifetime, 2074 publications have been published within this topic receiving 59790 citations.

On axisymmetric intrusive gravity currents in a stratified ambient : shallow-water theory and numerical results

Abstract: The intrusion of a fixed volume of fluid, which is released from rest and then propagates horizontally-radially at the neutral buoyancy level in a stratified ambient fluid, in a cylindrical geometry with a vertical axis (either fully axisymmetric or a wedge), is investigated. It is assumed that the density change of the ambient fluid is linear with height. A closed one-layer shallow-water Boussinesq inviscid formulation is presented. In general, the solution of the resulting hyperbolic system is obtained by a finite-difference scheme. However, for the large-time developed motion an analytical similarity solutions is derived. The self-similar result indicates radial expansion with t 1 / 3 but the shape is peculiar: the intruding fluid propagates like a ring with a fixed ratio of inner to outer radii; the inner domain (between the axis and the inner radius of the ring) contains clear ambient fluid. It is verified that the initial-value lock-release finite-difference solution indeed approaches the similarity predictions after an initial spread of the outer radius to about 2.5 times the initial radius. The shallow-water results are corroborated by numerical solutions of the full axisymmetric Navier–Stokes formulation. It is concluded that the shallow-water model is a versatile and accurate predictive tool, and that the peculiar ring-shape prediction reproduces an interesting physical property of the axisymmetric intrusion. The interaction between the internal gravity waves and the head is less significant than in the two-dimensional geometry. However, a practical limitation on the applicability of the inviscid model is imposed by the prediction that the ratio of viscous to inertia forces increases like r N 7 (where r N is the radius of propagation, scaled with the initial value).

Continua of states in boundary-layer flows

Abstract: We consider a class of three-dimensional boundary-layer flows, which may be viewed as an extension of the Falkner–Skan similarity form, to include a cross-flow velocity component, about a plane of symmetry. In general, this provides a range of three- dimensional boundary-layer solutions, parameterized by a Falkner–Skan similarity parameter, n, together with a further parameter, [Psi][infty infinity], which is associated with a cross-flow velocity component in the external flow. In this work two particular cases are of special interest: for n = 0 the similarity equations possess a family of solutions related to the Blasius boundary layer; for n = 1 the similarity solution provides an exact reduction of the Navier–Stokes equations corresponding to the flow near a saddle point of attachment. It is known from the work of Davey (1961) that in this latter class of flow, a continuum of solutions can be found. The continuum arises (in general) because it is possible to find states with an algebraic, rather than exponential, behaviour in the far field. In this work we provide a detailed overview of the continuum states, and show that a discrete infinity of ‘exponential modes’ are smoothly embedded within the ‘algebraic modes’ of the continuum. At a critical value of the cross-flow, these exponential modes appear as a cascade of eigensolutions to the far-field equations, which arise in a manner analogous to the energy eigenstates found in quantum mechanical problems described by the Schrodinger equation. The presence of a discrete infinity of exponential modes is shown to be a generic property of the similarity equations derived for a general n. Furthermore, we show that there may also exist non-uniqueness of the continuum; that is, more than one continuum of states can exist, that are isolated for fixed n and [Psi][infty infinity], but which are connected through an unfolded transcritical bifurcation at a critical value of the cross-flow parameter, [Psi][infty infinity]. The multiplicity of states raises the question of solution selection, which is addressed using two stability analyses that assume the same basic symmetry properties as the base flow. In one case we consider a steady, algebraic form in the ‘streamwise’ direction, whilst in the other a temporal form is assumed. In both cases it is possible to extend the analysis to consider a continuous spectrum of disturbances that decay algebraically in the wall-normal direction. We note some obvious parallels that exist between such stability analyses and the approach to the continua of states described earlier in the paper. We also discuss the appearance of analogous non-unique states to the Falkner–Skan equation in the presence of an adverse pressure gradient (i.e. n < 0) in an appendix.

Post-breakup solutions of Navier-Stokes and Stokes threads

Abstract: We consider the breakup of a fluid thread, neglecting the effect of the outside fluid (or air). After breakup, the solution of the fluid equations consists of two threads, receding rapidly from the point of breakup. We show that the bulk of each thread is described by a similarity solution of slender geometry (which we call the thread solution), but which breaks down near the tip. Near the tip of the thread the thread solution can be matched to a solution of Stokes' equation, which consists of a finger of constant spatial radius, rounded at the end. Very close to breakup, the thread solution balances inertia, viscosity, and surface tension (Navier-Stokes case). If however the fluid viscosity is large (as measured by the dimensionless Ohnesorge number), some time after breakup the thread solution consists of a balance of surface tension and viscosity only (Stokes case), and the thread profile can be described analytically.

Natural Convection Induced by a Heated Vertical Plate Embedded in a Porous Medium with Transpiration: Local Thermal Non-equilibrium Similarity Solutions

Abstract: This paper is concerned with the thermal non-equilibrium free convection boundary layer, which is induced by a vertical heated plate embedded in a saturated porous medium. The effect of suction or injection on the free convection boundary layer is also studied. The plate is assumed to have a linear temperature distribution, which yields a boundary layer of constant thickness. On assuming Darcy flow, similarity solutions are obtained for governing the steady laminar boundary layer equations. The reduced Nusselt numbers for both the solid and fluid phases are calculated for a wide range of parameters, and compared with asymptotic analyses.

Vortex flow over a flat surface with suction

Abstract: Introduction B the use of momentum integral methods, Taylor and Cooke calculated the laminar boundary-layer development on the interior surface of a frustum of a cone, with a potential vortex as the outer flow. Taylor's swirl atomizer problem has also been examined by means of similarity transformations of the boundary-layer equations by Moore, Mager, and Rott and Lewellen, and in Ref. 3 it is concluded that no valid solution can be found. However, in Ref. 4 it is noted that the valid solution may be found if the velocity profile of secondary flow changes sign somewhere in the boundary layer. Contrary to this, in Ref. 5 the existence of similarity solution is denied by a simple and clear consideration. Here, for simplicity we restrict our attention to the vortex flow over an infinite flat surface, i.e., to the case when the vertex angle of a cone is equal to 180°. For this problem, even the complete Navier-Stokes equations have a similarity solution only for a limited range of the Reynolds number." In this Note, we attempt to solve the boundary-layer equations under the boundary condition of distributed suction, and show without any speculation to the singularity on the vortex axis that a formal similarity solution can be obtained for suction parameter greater than a certain value.