Showing papers on "Similarity solution published in 1986"

An exact solution of the Navier-Stokes equation which describes non-orthogonal stagnation-point flow in two dimensions

Abstract: A similarity solution is found which describes the flow impinging on a flat wall at an arbitrary angle of incidence. The technique is similar to a method used by Jeffery (1915) and discussed more recently by Peregrine (1981).

Hydrodynamics of exploding foil x‐ray lasers

Abstract: An accurate simple model for the hydrodynamics of laser heated exploding foils is presented. Particular emphasis is given to applications in the design of soft x‐ray lasers. The model predicts the conditions in the foil plasma (e.g., temperature, density, and scale length), given the experimental parameters (e.g., optical laser intensity, laser pulse duration, target thickness, and target composition). The simple model is based on an isothermal, homogeneous expansion similarity solution of the ideal hydrodynamic equations. Both analytical and numerical solutions of the similarity equations are studied. The numerical solutions agree closely with computational hydrodynamic simulations at times of interest—after the laser burns through the foil. Analytic solutions for constant intensity laser irradiation provide useful power‐law scaling relations between the input laser and target parameters and the plasma variables. The simple model is a powerful design tool that reproduces the essential results of more expensive and time‐consuming simulations over a large and important range of parameter space.

Thermals in extremely viscous fluids, including the effects of temperature-dependent viscosity

Abstract: The flow induced by injection of a given amount of buoyancy or hot fluid from a localized source in a viscous fluid is investigated for conditions under which the Reynolds number Re is small compared with one, and the dimensionless buoyancy or Rayleigh number Ra is large compared with one. Laboratory experiments show that the buoyant fluid rises in the form of an extremely viscous ‘thermal’ which enlarges with time as a result of entrainment of surrounding fluid. The formation of a stable ‘chemical ring’ or torus of passive tracer similar in appearance to high Reynolds-number vortex rings is a notable feature of the creeping flow for high Rayleigh numbers. The possibility of large variations of viscosity due to temperature differences is included. A self-similar model is developed based on a boundary-layer analysis of a thin diffusive layer surrounding a spherical thermal for which the flow field is given by the exact solution for non-diffusive Stokes’ flow. Experiments at 2.5 × 102 < Ra < 2.5 × 104 and Re < 10−2 demonstrate the nature of extremely viscous thermals, support the similarity solution and enable evaluation of a proportionality constant. Possible applications of the results to dispersion by viscous drops and particularly to thermal convection in the Earth's solid mantle are mentioned.

Flow past a suddenly heated vertical plate in a porous medium

Abstract: An analysis is presented for the steady free convection flow about a semi-infinite vertical flat plate that is embedded in a saturated porous medium at high Rayleigh numbers. Similarity solutions are obtained for a class of problems where the wall temperature varies as a power of the distance from the leading edge of the plate. The existence and uniqueness of the solutions are considered. The approach to this steady-state solution is also considered by investigating the temporal development of the flow when the temperature of the plate is impulsively increased from that of the surroundings. A numerical solution is presented that matches the small and large time solutions. For some temperature distributions on the plate it is found that the velocity achieves its maximum value within the boundary layer. For these the disturbance from the leading edge of the plate travels fastest within the boundary layer. An asymptotic solution valid at large times is presented and the approach of the numerical solution to this asymptotic solution is illustrated. For the situation in which the plate is impulsively heated to a constant temperature an analysis is presented for the early stages of the departure from the one-dimensional solution.

Melting from a flat plate embedded in a porous medium in the presence of steady natural convection

Abstract: The problem of melting from a flat plate embedded in a porous medium is studied. The main focus is to determine the effect of natural convection flow in the liquid phase on the melting phenomenon. Two configurations an considered and modeled mathematically: a vertical plate and a horizontal plate. The final similarity equations art integrated numerically by use of the fourth-order Runge-Kutta method. Systematic “shooting” is required to satisfy the boundary conditions at infinity. Results are reported for the temperature and flow fields in the melt region. The melting phenomenon decreases the local Nusselt number at the solid-liquid interface.

Lie group invariance properties of radiation hydrodynamics equations and their associated similarity solutions

Abstract: The Lie group invariance properties of the one‐dimensional radiation hydrodynamic equations with the equilibrium diffusion approximation, a local thermodynamical equilibrium assumption, and an arbitrary material equation of state are derived. These properties are used systematically to generate similarity solutions of these equations for a given form of the equation of state. A comprehensive list of allowed similarity solutions for a perfect gas is presented. Several special cases that have been found previously by other authors appear in the list. Many other cases not reported previously are also presented. An example numerical solution is given for a piston‐driven shock with a thermal precursor.

Existence and non-uniqueness of similarity solutions of a boundary layer problem

Abstract: A Blasius boundary value problem with inhomogeneous lower boundary conditions f(0) = 0 and f'(0) = - lambda with lambda strictly positive was considered. The Crocco variable formulation of this problem has a key term which changes sign in the interval of interest. It is shown that solutions of the boundary value problem do not exist for values of lambda larger than a positive critical value lambda. The existence of solutions is proven for 0 lambda lambda by considering an equivalent initial value problem. It is found however that for 0 lambda lambda, solutions of the boundary value problem are nonunique. Physically, this nonuniqueness is related to multiple values of the skin friction.

Similarity solutions of three-dimensional boundary layer equations of non-Newtonian fluids

Abstract: An analysis of possibility of finding similaarity solutions to the three-dimensional, steady, incompressible, boundary layer equations in rectangular co-ordinates for a power law fluid has been discussed in the literature. In the present paper a similarity analysis is made of the steady, three-dimensional, incompressible, Iaminar, boundary layer flow of all time independent non-Newtonian fluids. The important conclusion drawn from this analysis in that for a non-Newtonian fiuid of any model, a similarity solution exists for the fluid for which shearing stress and rate of strain can be related by an arbitrary continuous function.

On similarity solutions for turbulent and heated round jets

Abstract: Commonly used empirical correlations for incompressible, heated round jets are shown to represent similarity solutions of the governing jet equations. These solutions give rise to self-similar eddy viscosities. Not all the similarity solutions are physically valid because some lead to zero eddy viscosities at the jet centerline. One physically valid solution is found to correlate best with round jet measurements and it gives a Gaussian error function description for the normalized mean velocity and temperature. Heat and momentum fluxes thus calculated are also in good agreement with measurements. Therefore, in addition to the classical similarity solution obtained by assuming constant eddy viscosity, another similarity solution to the jet equations is found where the eddy viscosity is self-similar.

Similarity analysis of energy transport in gas-driven fractures

Abstract: Gas-driven fracture propagation is important in a variety of blasting technologies. In most applications, the driving gases are very hot and so heat transfer from the gas to the fracture walls strongly influences the speed and the ultimate extent of fracture propagation. In this paper, similarity solutions are presented for the problem of a planar gas-driven fracture propagating into a brittle elastic solid. Local values of the fluid pressure, temperature and speed are computed from the one-dimensional mass, momentum and energy equations governing the turbulent flow of an ideal gas. Crack opening displacements and the stress intensity at the crack tip are calculated from the gas pressure distribution, in accordance with the quasi-steady integral relations from linear/elastic fracture mechanics. Four thermal models are considered in the analysis: isothermal, isentropic, adiabatic with frictional heating, and the more general case with heat transfer to a cold fracture wall. When heat transfer is very weak, the general solution approaches the adiabatic result; when heat transfer is very strong, the general result approaches the isothermal limit. The applicable range of each model is described. Example calculations are presented for a number of applications, including: underground nuclear testing, tailored-pulse well shooting, and near surface blasting.

A similarity solution for the axisymmetric viscous‐gravity jet

Abstract: A local similarity solution to the viscous‐gravity jet valid for large Reynolds number flows is given. The jet is divided into an inner core where axial gradients are small relative to the outer annular region. A similarity transformation is found for the outer region, reducing the resulting differential equations to a two‐point boundary value problem. This solution is matched to the inner solution through the boundary conditions. This approach effectively eliminates the mathematical difficulties associated with the unknown free surface and the stress singularity at the exit.

Eigenvalue Similarity Rules for Symmetric Cross-Ply Laminated Plates

Abstract: Symmetric cross-ply laminated plates are analyzed by employing a previously presented affine stretching process, some material constant definitions from Tsai (1964), and by gneralizing existing similarity rules. As examples, similarity rules are employed to find the frequency spectrum of a cross-ply symmetric plate from the corresponding isotropic solution, and a similarity solution is presented for the vibration of an axially loaded, simply supported cross-ply symmetric plate. Buckling similarity rules are also presented.

Analytical and Numerical Solutions for Natural Convection in a Corner

Abstract: Laminar natural convection flow in a corner delimited by a vertical heated semi-infinite plate and a second room-temperature plate forming an arbitrary angle is studied both analytically and numerically. Analytically, repeated use of the matched asymptotic expansions technique shows that the boundary layer along the vertical plate is described, to first order, by Polhausen's classical similarity solution; this buoyancy-induced flow drives an outer irrotational flow which, in turn, provides the motive force for a self-similar viscous boundary layer along the second plate. In the numerical approach, a variable-spaci ng, finite difference method with dynamic mesh modification permits a solution of the Navier-Stokes equations to be obtained in the whole field, confirming the validity of the approximate analytical model.

Buoyant plume above concentrated heat source in stratified porous media.

Abstract: Buoyant plume arising from a point heat source and a line heat source in stratified porous media is investigated analytically and experimentally. First, a similarity solution, assuming the boundary layer approximation, is obtained for an inhomogeneous porous medium with variable permeability and thermal conductivity. Next, the analytical results are applied to the case of a porous system consisting of two horizontal porous sublayers with different permeabilities and thermal conductivities. Experiments are conducted on the temperature distributions and the variation in the width of the plume using a layered porous medium composed of water and two kinds of glass beads of different diameters. It is then found that the similarity solutions can considerably describe the development of the buoyant plume in the regions, except those near the interface. The vicinity of the interface is understood as a transition region of plume behaviors between the two similarity solutions.

Nonlinear waves on a coupled density front

Abstract: It is shown that the inclusion of the nonlinear terms in the equations of motion of a coupled density front of zero potential vorticity results in wave solutions which merely propagate with time. The linear theory, on the other hand, predicts an exponential temporal growth. The nonlinear equation admits steady solutions representing standing waves whereas if the nonlinear terms are omitted no steady solutions exist. The general initial value problem is difficult to solve numerically since the linear problem is ill posed. In addition we prove that the general similarity solution of the nonlinear equation tends to zero for large times, at any point in space, regardless of the initial condition.

Laminar thermal boundary layer on a continuous accelerated sheet extruded in an ambient fluid

Abstract: Exact boundary layer similarity solutions are developed for flow, friction and heat transfer on a continuously accelerated sheet extruded in an ambient fluid of a lower temperature.

Effects of large suction in natural convection boundary layer on a horizontal plate

Abstract: The natural convection boundary layer on a horizontal plate with suction is investigated. The similar solutions of the boundary layer equations require suction to vary as a power law. The similarity equations have been integrated for various values of suction parameter and Prandtl number σ=0.72 and 6.8. For non-similar flow situation, a large suction approximation has been developed using the asymptotic approach. It is shown that large suction approximation is surprisingly good even for moderate values of suction parameter.

A similarity solution for flow in a narrow channel of varying gap

Abstract: La surface inferieure peut glisser dans les deux directions et il peut y avoir aspiration ou injection a l'autre surface. Solutions asymptotique aux grands et aux petits nombres de Reynolds

Momentum-flux condition for Landau-Squire jet flow

Abstract: For a class of similarity solutions of the Navier-Stokes equations called general Slezkin flow, explicit expressions are derived for the components of the momentum-flux tensor. Remarkably enough some of the components depend only on the coefficients of the governing differential equation. Closely related is the introduction of a “jet condition”, that has to be satisfied a priori by a Landau-Squire jet flow. The new criterion is applied to some jet flows proposed in the literature.

A strong shock wave in a medium with exponentially varying density

Abstract: The effect of the disturbance behind a strong plane shock wave propagating in a medium with exponentially varying density in the presence of magnetic field is investigated. The similarity solution is used to find the two interaction terms at all points of the flow. A method has been developed to determine the similarity exponent α in magnetogasdynamics.

On the propagation of shear oscillations in inelastic power-law fluids

Abstract: A similarity solution is presented to the problem of oscillatory flow produced by an oscillating flat plate in a semi-infinite sea of an inelastic power-law fluid. The physical relevance of the similarity solution is discussed in relation to actual oscillatory flows.

Note on the similarity solutions of unsteady jets in rotating fluids

Abstract: The structure of time dependent jets in rotating fluids using similarity transformations is studied theoretically for which exact solutions are discussed. Approximate solution using a modified yon Mises transformation is also explored.

Similarity solutions for two-dimensional magnetogasdynamic boundary layer flow in a transverse magnetic field

Abstract: The conditions for the existence of the similarity solution of a two-dimensional boundary layer flow past a plane wall of an electrically conducting gas, in the presence of a transverse magnetic field at small magnetic Reynolds number, are studied. It is found that, in general, the similarity requirements involve as many as eleven parameters out of which eight are the same as these of nonmagnetic gasdynamic boundary layer flow. The solution is obtained for power law velocity distribution, in Illingworth variables, at the outer edge of the boundary layer and for the corresponding variable magnetic field. The Prandtl number of the fluid is taken as unity and the magnetic-field effects are confined to the boundary layer only.