Showing papers on "Similarity solution published in 1979"

Numerical aspects of the approach to a Maxwellian distribution

Abstract: It has been conjectured, for the Boltzmann equation, that an arbitrary initial state tends first to relax towards a state characterized by the similarity solution. We present here a simple model where the possible validity of the statement can be studied numerically. The results of extensive numerical computation for the first time give support for the validity of this conjecture.

The hydrodynamic stability of ionization-shock fronts. Linear theory.

Abstract: The hydrodynamic stability of an ionization-shock front is investigated in the case where all regions are taken as isothermal and self-gravity is neglected. The evolving, unperturbed state is described by a similarity solution. A technique is developed which reduces the stability problem from a system of partial differential equations and associated boundary conditions to a coupled set of ordinary differential equations with time-dependent coefficients. This approach is shown to be valid as long as the perturbation wavelength is much greater than the thickness of the ionization-shock front. Perturbations are considered to arise from density inhomogeneities in the ambient medium. Numerical results along with an approximate analytic solution are given and show a new instability whereby all wavelengths greater than several recombination lengths grow without bound in an oscillatory manner. However, the wavelength with the fastest growth rate increases as the system evolves. A short discussion on the physical mechanism involved and several observational aspects, including a comparison with the morphology of the California nebula, is presented. The results suggest that this instability can produce irregular structures similar to the bright rims and elephant trunks seen in many diffuse nebulae.

Non-linear propagation of long Rossby waves

Abstract: An analysis suggests that away from western boundary currents, the westward propagation of long internal Rossby waves is governed by a simple non-linear wave equation. The non-linearity is shown to arise from variation in stratification rather than from non-linear momentum terms. The result is initially deduced for a reduced gravity model, but it is later generalized to a similarity solution of the thermocline equations. Numerical experimentation is used to confirm the analysis in the case of the reduced gravity mode.

Some qualitative properties of solutions of a generalised diffusion equation

Abstract: Atkinson and Peletier (2,3) have considered similarity solutions of the differential equation where the function k ( s ) is defined, real and continuous for s ≥ 0 and k ( s ) > 0 if s > 0 (in (2) k (0) = 0 is also assumed). In particular they look for similarity solutions of the form u ( x, t ) = f (η) where η = x ( t +l) −½ with boundary conditions f (0) = A and . They show that if k ( s ) satisfies the condition then for any A > 0 there is a unique similarity solution which is non-negative and has compact support in [0, ∞). They also show in (2) that is a necessary condition for the solution to have compact support. In (3) they prove existence of similarity solutions when and show that in this case the similarity solution has the property that f (η) > 0 for all η > 0.

A similarity solution to a nonlinear diffusion equation of the singular type: a uniformly valid solution by perturbations

Abstract: The one-dimensional nonlinear diffusion equation is solved by a perturbation technique. It is assumed that the diffusivity varies as a nonnegative power of the concentration, while the concentration at the supply surface varies as another power of time. The resulting similarity solution that has been derived via a perturbation scheme remains valid for all times and all distances. Explicit series formulae are also derived for the location of the concentration front. Since diffusivity vanishes at zero concentration, the study here pertains to a singular problem. Introduction. The nonlinear diffusion equation, and certain other types of parabolic equations associated with it, are appearing with increasing frequency in many problems of the physical and environmental sciences. A special case of the following problem, admitting of a similarity solution, will be solved in the next sections: _8_ 8x , 0 < 0 < 1, / > 0, 0 < * < F(t), F(t) unknown, 8t D(0) = 0, D(l) = 1, F(0) = 0; * = 0, 9 = /(/); t = 0, 0 = 0; x > F(t), 9 = 0 and D(6) = 0. Here x = F(t) is the distance of the advancing \"concentration or diffusing front\". Throughout this work, it is assumed that all variables and functions have been rendered dimensionless by appropriate normalizations, x and t are respectively the space and time variables. 9 may be identified with the concentration of a solution, moisture content of a partially saturated soil, temperature of a conducting medium, etc. The relation D(0) = 0 is an important one here, and mention must be made of its role in structuring a solution to this problem. It will be noticed that D(6) is the (nonlinear) coefficient of the highest derivative (828/8x2) in the differential equation under study. It is well known that the points at which such coefficients vanish are generally associated with \"singular\" points of the solution function 8(x, t). Shampine [1] investigated in detail a problem of this type. In his paper, the special boundary conditions x = 0, 0 1, x 0 = 0 led to an ordinary differential equation in the similarity variable .? = (x/Jt). He used the term \"singular problem\" to denote the cases that arise out of the special condition D(0) = 0. * Received December 27, 1977; revised version received August 4, 1978. Work on this paper was supported by the National Science Foundation under grants number ENG 7702013, ENG 7724979. 12 BABU AND VAN GENUCHTEN The mathematical pecularities, as well as the physical characteristics, of the resulting solution profiles 9{x, t) are worth mentioning at this stage. These are: (a) excluding any unbounded solutions, there still exists the possibility of singularities in the derivatives of 6; (b) although the problem could be formulated, as usual, in the semi-infinite domain 0 < x < co, 9 -> 0 as x -> oo, the (strong) possibility exists that the profile 6(x, t) actually terminates at a finite distance, called the diffusing front; (c) to begin with, the location of this front at F(t) is unknown; (d) it is highly likely that the gradient at the front is unbounded: x = F(t), 86/dx = — so that the \"sharp front\" occurrence is observed; and (e) in many applications, the function D(d) is an extremely rapidly varying function of 6. It is needless to state here that accurate profile evaluation and numerical computations would be rendered more difficult by each of the above pecularities (a)-(e). A basic purpose of the present investigation is to evolve a constructive technique that is capable of generating an accurate solution profile, in spite of the above complications. Indeed, it will be seen in later sections of this paper that the complexity (e) is turned around to advantage, and the construction of the solution specifically depends on the use of rapid variation in D(6). Since D(6) is allowed to vanish at 8 = 0 in the above equation, it is expected that the profile \"6 versus x\" is finite in extent, terminating abruptly at the diffusing front (Shampine [1]). In many applications, D(6) is known to be a rapidly varying function of 6. In soil moisture problems, D{6) is frequently assumed to be an exponential function of 6 (Reichardt et al. [2]). Often, it is assumed that /(?) = 1; the solution then will depend on a similarity variable rj = (x/Jt). Assuming D{6) = 0\", Pattle [3] constructed an instantaneous point-source type of solution for the nonlinear equation (see note at the end). Aronson [4] investigated certain regularity properties of solutions with finite profile termination. Solutions exhibiting \"square root of time\" dependence, and their asymptotic behavior, have been studied by Peleiter [5], Atkinson and Jones [6], and Pimbeley [7], Most of these studies deal with the existence and uniqueness aspects of the solutions. Horizontal absorption of water by dry and partially dry soils is extensively modelled on the diffusion equation. Survey work by Philip [8, 9] gives an idea of the importance attached to equations of this type in soil water transport problems. The high nonlinearities inherent in these equations have naturally generated a vast body of computer solutions based upon numerical schemes. Numerical solutions of the above equations, subject to general initial and boundary conditions, have been obtained by several authors. In general, finite difference and finite element schemes have gained wide acceptance in practice. Numerous references may be found in the articles by Ashcroft et al. [10], Neumann [11], Hayhoe [12] and van Genuchten [13]. By contrast, analytic and quasianalytic solutions of a constructive nature are few in number. Brutsaert [14] presented a solution with finitely terminating profiles, referring to an earlier work of Heaslet and Alksne [15]. Parlange [16] obtained elegant results by a method of iterations resembling the Picard iterates. Babu [17, 18, 19] applied perturbation techniques to achieve a constructive solution that is capable of yielding quantitative information in a straightforward and reliable fashion, for the special case of f(t) = 1. In a subsequent joint work, Parlange and Babu [20, 21] exhibited the essential identity of the solutions constructed via perturbation, iteration and optimization techniques. In what follows, it will be assumed that /(;) = tm and D(d) = 6\" (n > 0). The solution mechanism presented below is valid for values of m and t in the ranges —1/(« + 1) < m < + oo, m ^ 0; 0 < t < oo. The case m = 0 requires only minor adjustments in the computational detail. However, the case ofO 0 (1) A L Bl 8x1 dxJ subject to the conditions 88_ 8t 9 = tm, x = 0, m > 0, (2a) 0= 0, t = 0, (2b) 8= 0,x> F(t), (2c) 6\" ■ £ = 0, x > F(t). (2d) Condition (2c) states that the 6(x, t) solution profile \"hits\" the x-axis at (an unknown) distance F(t), forming a \"diffusing front\" there. And, since the flux also vanishes according to (2d), it follows that no disturbances are propagated into the medium beyond this front. Part of the solution process will thus be devoted to the determination of the unknown function F(t). Since d = 0 at x = F(t), condition (2d) may in fact be an identity, unless | 86/dx \\ = Hence, from a computational point of view, this condition may not often yield much useful information. However, an integration of Eq. (1), utilizing (2a-(2d), leads to the relation: r r *■£(-£) •* i. nn se = x-en —— 8x (n+ 1) where 6 = \\/n + 1. One final integration yields rFn) (p de 0 J 0 OX J 0 = 0 tm d CF(t) — , so that x-6{x, t)-dx = etm/(, o ut j 0 f x-6(x, t) ■ dx = (3) m + e Next, elementary estimates, based upon (3), lead to an explicit determination of the parameter of perturbation for this problem. This parameter is t = 1 /(n + 1). Neglecting the time dependency for the moment, it follows from (3) that x2 = 0(t2), showing that the entire phenomenon is of a boundary-layer nature, if e is small in some sense. Some general comments on the implications of (3) are to be found in the Appendix. The following transformations reduce (1) to an ordinary differential equation: e = 1 /(n + 1); ?? = (x/t) ■ (m/(t, m))v2 ■ (l/?mn+1)1/2; (4) 6 = tm • [V(m e; m)](;

Soluble model for the analysis of stability in an imploding compressible liner

Abstract: A soluble model of the development of the linear pertubations about a time‐varying state of a compressible medium is presented. A Lagrangian description is employed to rederive the equations for the self‐similar motion of an ideal fluid and to obtain the linearized equations of motion for pertubations about a general time‐varying basic state. The resulting formalism is applied in cylindrical geometry to calculate the growth of flute‐like modes associated with a similarity solution modeling the implosion and expansion of a fluid liner. A complete solution is obtained for the perturbed motion. The only modes for which the perturbation amplitudes grow faster than the unperturbed liner radius during both implosion and expansion are divergence‐ and curl‐free. Numerical and analytical results are obtained for these and shown to reduce, in the short‐wavelength limit, to the Rayleigh–Taylor instability found previously for incompressible time‐independent basic states. In addition, a new kind of instability is found: a class of overstable internal modes (sound waves), which are ’’pumped up’’ in amplitude during implosion, but decay during the expansion phase.

Theoretical and Computational Approach to Modeling Flame Ignition.

Abstract: : In this paper, time-dependent results obtained from a simplified but nonlinear analytic similarity solution and a detailed numerical simulation are used to study the relations between the fundamental processes occurring in the very early stages of flame ignition in homogeneous premixed gases. The parameters which may be varied are the composition of the mixture, the initial radius of energy deposition R sub 0, the duration of the heating TAN SUB 0, and the total energy deposited in the system E sub 0. The similarity solution plus the ignition delay times Tau sub c for the fuel-oxidizer mixture as a function of temperature can be used to calculate whether or not a given energy source is adequate to ignite the system. This simple procedure may then be calibrated using the time-dependent NRL detailed reactive flow models. The models include the thermophysical properties of the mixture, a full chemical kinetics scheme, the nonlinear convection of self-consistent fluid dynamics and the matrix molecular diffusion coefficients for the individual species. Results are presented for a selected mixture of H2-O2-N2 for two values of R sub 0 which show that a model must be constructed for a quench volume in order to complete the similarity solution calibration. (Author)

A New Formulation For Buoyancy-Induced Flow Adjacent to a Vertical Uniform Flux Surface in Cold Water

Abstract: An analysis of buoyancy-induced flow adjacent to a vertical surface dissipating a uniform flux in cold pure and saline water is presented. For these conditions a simple yet very accurate density relationship proposed by Gebhart and Mollendorf is employed to estimate the buoyancy force. To obtain a similarity solution under these conditions, the quiescent medium is considered to be at the temperature condition at which the density extremum occurs. A modified Grashof number is defined, based on the surface heat flux. The fifth-order two-point boundary value problem is solved by a fourth-order Runge-Kutta method, starting from an asymptotic solution at a large distance from the surface. Solutions are obtained for different pressure and salinity conditions over a Prandtl number range of 8.0 to 13.0, The following heat transfer correlation is proposed for all values of q(s, p) from 1.5829 to 1.8948: Nux = 0.577[Ra* x]1/(4 + q).

Wake structure in high-speed flow of a rarefied plasma over a body

Abstract: In the study of qualitative features of flow of a rarefied plasma over bodies in ionospheric aerodynamics, the problem of flow behind a two-dimensional plate is often considered. The formulation of this problem and its relation to flow over real objects was considered in detail in [1], This model problem has been analyzed in a number of papers using two main approaches: description of the flow with the help of the similarity solution found in [2, 3], and numerical solution of the equations of plasma motion [4–7]. A review of the main results obtained by the two methods can be found in [1, 6]. This paper gives a numerical solution of the problem of transverse supersonic flow over a flat plate. The plasma is assumed to be collisionless and is described by the kinetic equation with a self-consistent field. The particle-in-cell method is used to solve the kinetic equation. In contrast with most numerical calculations previously performed [4–6], the present paper considers the case, of greater practical interest, of flow over a body whose dimension R is much greater than the Debye radius Di in the unperturbed plasma. Practically all the known results for this case have been obtained using the similarity solution [2, 3], which is not valid, however, in the entire region of unperturbed flow, and therefore does not give a complete solution to the problem. Individual numerical calculations (see [7]) do not add much to the similarity analysis, since they refer to a very narrow range of the flow parameters. The main emphasis in the present paper is the study of wake structure behind a flat plate and plasma instability in the wake. The computations were performed in a wide range of variation of the ratioβ=Te/Ti, and one can follow the processes of ion acceleration, interaction of the accelerated group of ions with the plasma, development of beam-type instability [1, 8], and formation and decay of the turbulent wake. The qualitative wake structure features discussed below are also found, of course, in plasma flow over actual three-dimensional bodies.