Showing papers on "Similarity solution published in 1991"

Self-similar jets

Abstract: General arguments and numerical calculations are used to show that the flow caused by a supersonic gas jet is self-similar under certain conditions. If we assume that the jet has a high initial Mach number and is generated in a region small compared to its length, then the type of similarity solution depends on the density distribution of the gas through which the jet propagates. If this density decreases faster than 1/R 2 , where R is the distance from the source, then the length of the jet increases linearly with time and it may evolve into a classical double if it subsequently encounters a region of higher density.

Two-dimensional flow of a viscous fluid in a channel with porous walls

Abstract: We consider the flow of a viscous incompressible fluid in a parallel-walled channel, driven by steady uniform suction through the porous channel walls. A similarity transformation reduces the Navier-Stokes equations to a single partial differential equation (PDE) for the stream function, with two-point boundary conditions. We discuss the bifurcations of the steady solutions first, and show how a pitchfork bifurcation is unfolded when a symmetry of the problem is broken.Then we describe time-dependent solutions of the governing PDE, which we calculate numerically. We analyse these unsteady solutions when there is a high rate of suction through one wall, and the other wall is impermeable: there is a limit cycle composed of an explosive phase of inviscid growth, and a slow viscous decay. The inviscid phase ‘almost’ has a finite-time singularity. We discuss whether solutions of the governing PDE, which are exact solutions of the Navier-Stokes equations, may develop mathematical singularities in a finite time.When the rates of suction at the two walls are equal so that the problem is symmetrical, there is an abrupt transition to chaos, a ‘homoclinic explosion’, in the time-dependent solutions as the Reynolds number is increased. We unfold this transition by perturbing the symmetry, and compare direct numerical integrations of the governing PDE with a recent theory for ‘Lorenz-like’ dynamical systems. The chaos is found to be very sensitive to symmetry breaking.

Application of the velocity‐dissipation probability density function model to inhomogeneous turbulent flows

Abstract: Recently Pope and Chen [Phys. Fluids A 2, 1437 (1990)] developed a turbulence model based on the one‐point Eulerian joint probability density function (pdf) of velocity and dissipation. The modeling is performed by constructing stochastic processes for the velocity and dissipation following fluid particles. In the original work, these models were constructed by reference to the known statistics of homogenous turbulence, and the applicability of the model was restricted to this narrow class of flows. In this paper the model is extended to inhomogeneous flows, and calculations are presented to demonstrate aspects of the model’s performance. The model equation admits a similarity solution corresponding to the log‐law region of the turbulent boundary layer, and the principal statistics obtained from this solution are in good agreement with experimental data. Application of the model to the momentumless wake and to the plane mixing layer demonstrate its ability to represent turbulent/nonturbulent intermittency in these free flows: in the intermittent regions, the pdf of dissipation is bimodal, with a spike at zero corresponding to nonturbulent fluid. Fluid‐particle paths in the turbulent mixing layer obtained from the model correspond to large‐scale coherent motions, rather than to the small‐scale incoherent motion characteristic of diffusive transport.

The magnetohydrodynamic effects on a fluid film squeezed between two rotating surfaces

Abstract: Considers the motion of a fluid film squeezed between two rotating parallel plane surfaces in the presence of a magnetic field applied perpendicular to the surfaces. Attention is given to the case where a similarity solution can be obtained. Approximate analytical solutions are given, and a numerical solution to the resulting nonlinear ordinary differential equations is presented. The combined effects of the magnetic forces and the centrifugal inertial forces on the velocity profiles, the load capacity and the torques that the fluid exerts on the surfaces are studied. In general, the results show that these two forces have opposite effects.

Three-dimensional flow in a porous channel

Abstract: We describe three-dimensional flow of a viscous incompressible fluid driven along a channel by uniform suction through parallel porous walls, generalizing recent work on two-dimensional flow. The Navier-Stokes equations are reduced to two nonlinear diffusion equations with time and the coordinate normal to the walls as independent variables, by use of a generalization of the Hiemenz similarity solution

Analysis of steady flow in a channel with one porous wall, or with accelerating walls

Abstract: A similarity solution of the Navier-Stokes equations that describes the two-dimensional flow of a viscous incompressible fluid through a channel with one porous wall is analysed. The analysis, which determines the number and character of solutions in various cases according to the higher-order boundary conditions at the impermeable wall of the channel, agrees with asymptotic results in the limits of small and large Reynolds numbers. Flow in a channel with one or two accelerating walls is similarly analysed.

Mixed Convection to Power-Law Type Non-Newtonian Fluids from a Vertical Wall

Abstract: The mixed convection boundary layer flow on a vertical stationary or moving plate to a power-law non-Newtonian fluid is analyzed. An exact similarity solution is derived for the case when the surface temperature is inversely proportional to the distance from the leading edge of the plate. A discussion is provided on the effects of the flow index and buoyancy parameter on the velocity and temperature fields.

Inviscid spatial stability of a compressible mixing layer. Part 3. Effect of thermodynamics

Abstract: The results of a comprehensive comparative study of the inviscid spatial stability of a parallel compressible mixing layer using various models for the mean flow are reported. The models are: (1) the hyperbolic tangent profile for the mean speed and the Crocco relation for the mean temperature, with the Chapman viscosity-temperature relation and a Prandtl number of one; (2) the Lock profile for the mean speed and the Crocco relation for the mean temperature, with the Chapman viscosity-temperature relation and a Prandtl number of one; and (3) the similarity solution for the coupled velocity and temperature equations using the Sutherland viscosity temperature relation and arbitrary but constant Prandtl number. The purpose was to determine the sensitivity of the stability characteristics of the compressible mixing layer to the assumed thermodynamic properties of the fluid. It is shown that the quantative features of the stability characteristics are quite similiar for all models but that there are quantitative differences resulting from the difference in the thermodynamic models. In particular, it is shown that the stability characteristics are sensitive to the value of the Prandtl number.

Laminar channel flow driven by accelerating walls

Abstract: The two-dimensional flow of a viscous incompressible fluid in a channel with accelerating walls is analysed by use of Hiemenz's similarity solution. Steady flows and their instabilities are calculated, and unsteady flows are computed by solving the initial-value problem for the governing partial-differential system. Thereby, these exact solutions of the Navier–Stokes equations are found to exhibit turning points, pitchfork bifurcations, Hopf bifurcations and Takens–Bogdanov bifurcations along the route to chaos. The substantial physical result is that the chaos previously found for flows with symmetrically accelerating walls is easily destroyed by a little asymmetry.

Similarity solutions of the thermocline equations

Abstract: We apply symmetry group methods to find the group of transformations of the dependent and independent variables that leave the thermocline equations unchanged, These transformations lead to an optimal subset of sixteen forms of similarity solution, Each form obeys an equation with one fewer dependent variable than the original thermocline equations. Previously obtained similarity solutions, which are based solely upon scaling symmetries, are special cases of just three of these forms. Two of the sixteen forms lead to linear, two-dimensional, advection-diffusion equations for the temperature, Bernoulli functional or potential vorticity. Simple exact solutions contain \"internal boundary layers\" that resemble the thermocline in subtropical gyres.

Mixed convection similarity solutions of a non-Newtonian fluid on a horizontal surface

Abstract: The mixed convection boundary layer flow on a horizontal stationary or moving plate to a power-law non-Newtonian fluid is analyzed. An exact similarity solution is given for the case of a wall temperature that is inversely proportional to the power-law of the distance from the leading edge of the plate. It is shown that such a solution does not exist if the boundary parameter is smaller than a certain critical value. The effects of the flow index, buoyancy parameter and modified Prandtl number on the velocity and temperature profiles as well as on the global heat transfer and displacement thickness are discussed.

Self-similar radial two-phase flows

Abstract: A general mathematical formulation is developed, appropriate to both single- and two-phase conditions in a porous medium. A new similarity solution, generalizing the well known Theis solution, is derived for radial flow to a well in a region initially containing a two-phase mixture of steam and water, in which either steam or water is immobile. This generalized Theis solution follows from a new Ricatti equation for mass flow, which includes an additional nonlinear effect resulting from quadratic pressure gradient terms. Existing results for the saturation profile are extended by inclusion of nonlinear contributions, which are shown to be necessary for accurate descriptions of the saturation profile. A boundary-layer analysis is developed for flow about the well, where both mass flux and flowing enthaply are almost constant, which enables both the pressure and saturation profiles to be determined analytically. An analysis of two-phase self-similar shocks is given, together with the associated entropy conditions constraining the existence of shocks. Finally, numerical examples are discussed showing the agreement between theory and numerical simulations.

Fluid film sprayed on a stretching surface

Abstract: A liquid film, due to spraying, is deposited on a stretching surface. The physical problem is governed by a nondimensional parameter α, representing the relative importance of spraying rate to stretching rate. After a similarity transform, the Navier-Stokes equations reduce to a nonlinear ordinary differential equation which is solved by perturbation and numerical integration. The results show nonuniqueness and nonexistence of a steady state solution for certain values of α. Velocity profiles and heat transfer characteristics are determined.

Stress-assisted diffusion: a free boundary problem

Abstract: A Stefan problem is formulated for a diffusion equation where the Fickian flux is supplemented by a stress gradient term. The stress in turn obeys a concentration-dependent evolution equation that includes impulsive and relaxation effects. The speed of the free boundary is taken to be proportional to the flux at the front. For short times, the front position is found to be proportional to $t^{1/2} $. Using singular perturbation and numerical methods, three types of “long-time” behavior are found, depending on relationships between the coefficient functions and the parameters in the problem: (i) the front position is proportional to $t^{1/2} $; (ii) the front position is proportional to t; or (iii) the front stops in a finite time and the solution becomes singular (infinite gradients) at that point.

On the continuous squeezing flow in a wedge

Abstract: The paper is concerned with the continuous squeezing flow of Oldroyd-type fluids in a two-dimensional wedge. The flow mimics the lubrication action in a squeezing flow and is important in that there exists a similarity solution for any simple fluid. We are only concerned with Oldroyd-type fluids, however. It is shown by using a parameter continuation method that the Oldroyd-B model has a limiting Weissenberg number. The Phan Thien/Tanner model does not have this limiting Weissenberg number.

Horizontal boundary-layer natural convection in a porous medium saturated with a gas

Abstract: Boundary-layer analysis is performed for free convection flow over a hot horizontal surface embedded in a porous medium saturated with a gas of variable properties. The variable gas properties are accounted for via the assumption that thermal conductivity and dynamic viscosity are proportional to temperature. A similarity solution is shown to exist for the case of constant surface temperature. Numerical results for the stream function, horizontal velocity, and temperature profiles within the boundary layer as well as for the mass of entrained gas, surface slip velocity, and heat transfer rate at different values of the wall-temperature parameter are presented. Asymptotic solutions for large heating are also available to support the numerical work.

Similarity solutions of the shallow water equations

Abstract: A one-dimensional shock (bore) reflection problem is discussed for the two-dimensional shallow water equations with cylindrical symmetry. The differential equations for a similarity solution are derived and solved numerically in conjunction with the Rankine-Hugoniot shock relations.

Similarity prediction of wall jets past axisymmetric bodies for power-law fluids

Abstract: The similarity solution is presented for laminar wall jets on bodies of revolution for power-law fluids. The functional dependence of the length and velocity similarity scales on a shape parameter and a flow behaviour index is determined. Discussion dealing with flow invariance, modified Mangler's transformation and with previously published results is carried out.

A class of similarity solutions for radial motions of compressible hyperelastic spheres and cylinders

Abstract: A class of similarity solutions is obtained for radial motions of spherical and cylindrical bodies made of a certain type of compressible hyperelastic materials. The equations satisfied by the infinitesimal generators of the symmetry group of the unified governing first order field equations for spheres and cylinders are found. It is shown that these equations admit a special class of solutions which generate a five-parameter group of transformations. The form of the strain energy function Σ corresponding to the resulting symmetry group is evaluated. The similarity variable is determined and ordinary differential equations satisfied by similarity solutions are obtained. Numerical solutions are given for a Ko material which falls into the class of admissible materials.

Self-similar solutions for displacement of non-Newtonian fluids through porous media

Abstract: This paper presents a class of self-similar solutions describing piston-like displacement (single-phase flow is included as a special case) of one slightly compressible non-Newtonian, power-law, dilatant fluid by another through a homogeneous, isotropic porous medium. These solutions can be used to evaluate the validity and accuracy of existing approximate solutions, such as the assumption of constant flow rate at each radial distance that Ikoku and Ramey use to linearize the partial differential equation for the flow of non-Newtonian, power-law fluid through a porous medium.

Shear flow over a wall with variable temperature

Abstract: The Leveque solution for forced convection due to shear flow over a plate is one of the basic solutions of convective heat transfer. The solution is exact if the fluid velocity is linear, such as that generated by the movement of parallel plates. The solution is asymptotic if the thermal boundary layer is much thinner than the velocity boundary layer, such that the local velocity profile is approximately linear. This occurs for developing thermal boundary layers. Although the author discussed the general case, Leveque originally considered shear flow over a plate with a sudden constant temperature change. The problem was subsequently extended to the constant heat flux case by Worsoe-Schmidt. In both instances exact similarity boundary layer solutions are obtained. In this note he considers the variable wall temperature case. Both analytic and numerical methods will be used.

A new solution for the nonlinear diffusion-convection equation

Abstract: The Cauchy problem for a one-dimensional nonlinear porous-media diffusion-convection differential equation is considered. A self-similar solution is derived that, after the Riccati transformation, is described in terms of the Airy functions. The asymptotic behavior of the solution for a small amount of species introduced into the porous media is investigated. (lb)~~~~~~~~~ A) = * Sn - Sn-m as ax

Similarity analysis of axisymmetric free convection on a horizontal infinite plate subjected to a mixed thermal boundary condition

Abstract: Similarity analysis of the problem of axisymmetric free convection on a horizontal infinite plate is considered assuming that the plate is subjected to a mixed thermal boundary condition. It is shown that the thermal boundary condition is characterized by a nonnegative parameterm and the two cases ofm=0 andm=1 correspond to prescribed plate temperature and prescribed surface heat flux respectively. If one has to compute the heat transfer coefficient for various values ofm, there is no need to solve the boundary value problem everytime; it is enough to solve a certain polynomial equation provided the solution is known for any particular value ofm.

Suction-driven flow in a finite wedge

Abstract: Many studies have been made of suction or injection through the walls of a porous channel. In most cases the flow is assumed to be two-dimensional and a similarity solution is exploited. In this paper we consider the effect of side walls on the axisymmetric flow between two parallel and porous discs with suction on both discs when the Reynolds number is large. We find that a threedimensional boundary layer is required on each of the bounding side walls, the results of which cast doubt on the notion of a simple axisymmetric flow in the inviscid part of the field.