Showing papers on "Similarity solution published in 1997"

A Semianalytic Model for Supercritical Core Collapse: Self-Similar Evolution and the Approach to Protostar Formation

Abstract: We use a semianalytic model to examine the collapse of supercritical cores (i.e., cores with a mass-to-flux ratio exceeding a critical value). Recent numerical simulations of the formation and contraction of supercritical cores show that the inner solution tends toward self-similar evolution. We use this feature to develop analytic expressions for quantities such as the density, angular velocity, and magnetic field. All forces involved in the problem (gravitational, magnetic, thermal, and centrifugal) can be calculated analytically in the thin-disk geometry of the problem. The role of each force during the contraction is analyzed. We identify the key role of ambipolar diffusion in producing a departure from an exact similarity solution. The slow leakage of magnetic flux during the supercritical phase is enough to significantly accelerate an otherwise near-quasi-static contraction. This leads to dynamic collapse with supersonic infall speeds in the innermost region of the core by the time of protostar formation. We find a time-dependent semianalytic solution for the late supercritical phase, and asymptotic forms are obtained for important profiles at the moment that a central protostar is formed. We obtain estimates for the rotational velocity, infall velocity, and mass accretion rate at this moment. The mass accretion rate is significantly greater than the canonical C3/G (where C is the isothermal sound speed and G is the universal gravitational constant) at the moment of protostar formation, although we argue that it is time-dependent and will eventually decrease. Comparisons are made with the predictions of existing spherical similarity solutions.

Solar radiation assisted natural convection in uniform porous medium supported by a vertical flat plate

Abstract: Natural convection flow of an absorbing fluid up a uniform porous medium supported by a semi-infinite, ideally transparent, vertical flat plate due to solar radiation is considered. Boundary-layer equations are derived using the usual Boussinesq approximation and accounting for applied incident radiation flux. A convection type boundary condition is used at the plate surface. These equations exhibit no similarity solution. However, the local similarity method is employed for the solution of the present problem so as to allow comparisons with previously published work. The resulting approximate nonlinear ordinary differential equations are solved numerically by a standard implicit iterative finite-difference method. Graphical results for the velocity and temperature fields as well as the boundary friction and Nusselt number are presented and discussed.

The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution

Abstract: An approximate theory is given for the generation of internal gravity waves in a viscous Boussinesq fluid by the rectilinear vibrations of an elliptic cylinder. A parameter λ which is proportional to the square of the ratio of the thickness of the oscillatory boundary layer that surrounds the cylinder to a typical dimension of its cross-section is introduced. When λ[Lt ]1 (or equivalently when the Reynolds number R[Gt ]1), the viscous boundary condition at the surface of the cylinder may to first order in λ be replaced by the inviscid one. A viscous solution is proposed for the case λ[Lt ]1 in which the Fourier representation of the stream function found in Part 1 (Hurley 1997) is modified by including in the integrands a factor to account for viscous dissipation. In the limit λ→0 the proposed solution becomes the inviscid one at each point in the flow field.For ease of presentation the case of a circular cylinder of radius a is considered first and we take a to be the typical dimension of its cross-section in the definition of λ above. The accuracy of the proposed approximate solution is investigated both analytically and numerically and it is concluded that it is accurate throughout the flow field if λ is sufficiently small, except in a small region near where the characteristics touch the cylinder where viscous effects dominate.Computations indicate that the velocity on the centreline on a typical beam of waves, at a distance s along the beam from the centre of the cylinder, agrees, within about 1%, with the (constant) inviscid values provided λs/a is less than about 10−3. This result is interpreted as indicating that those viscous effects which originate from the characteristics that touch the cylinder (places where the inviscid velocity is singular) reach the centreline of the beam when λs/a is about 10−3. For larger values of s, viscous effects are significant throughout the beam and the velocity profile of the beam changes until it attains, within about 1% when λs/a is about 2, the value given by the similarity solution obtained by Thomas & Stevenson (1972). For larger values of λs/a, their similarity solution applies.In an important paper Makarov et al. (1990) give an approximate solution for the circular cylinder that is very similar to ours. However, it does not reduce to the inviscid one when the viscosity is taken to be zero.Finally it is shown that our results for a circular cylinder apply, after small modifications, to all elliptical cylinders.

The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution

Abstract: An approximate theory is given for the generation of internal gravity waves in a viscous Boussinesq fluid by the rectilinear vibrations of an elliptic cylinder. A parameter k which is proportional to the square of the ratio of the thickness of the oscillatory boundary layer that surrounds the cylinder to a typical dimension of its cross-section is introduced. When k ’ 1 (or equivalently when the Reynolds number R ( 1), the viscous boundary condition at the surface of the cylinder may to first order in k be replaced by the inviscid one. A viscous solution is proposed for the case k ’ 1 in which the Fourier representation of the stream function found in Part 1 (Hurley 1997) is modified by including in the integrands a factor to account for viscous dissipation. In the limit k U 0 the proposed solution becomes the inviscid one at each point in the flow field. For ease of presentation the case of a circular cylinder of radius a is considered first and we take a to be the typical dimension of its cross-section in the definition of k above. The accuracy of the proposed approximate solution is investigated both analytically and numerically and it is concluded that it is accurate throughout the flow field if k is suciently small, except in a small region near where the characteristics touch the cylinder where viscous eects dominate. Computations indicate that the velocity on the centreline on a typical beam of waves, at a distance s along the beam from the centre of the cylinder, agrees, within about 1%, with the (constant) inviscid values provided ks}a is less than about 10’$. This result is interpreted as indicating that those viscous eects which originate from the characteristics that touch the cylinder (places where the inviscid velocity is singular) reach the centreline of the beam when ks}a is about 10’$. For larger values of s, viscous eects are significant throughout the beam and the velocity profile of the beam changes until it attains, within about 1% when ks}a is about 2, the value given by the similarity solution obtained by Thomas & Stevenson (1972). For larger values of ks}a, their similarity solution applies. In an important paper Makarov et al. (1990) give an approximate solution for the circular cylinder that is very similar to ours. However, it does not reduce to the inviscid one when the viscosity is taken to be zero. Finally it is shown that our results for a circular cylinder apply, after small modifications, to all elliptical cylinders.

On the Application of the Briggs' and Steepest‐Descent Methods to a Boundary‐Layer Flow

Abstract: The evolution of wave packets generated by impulsive disturbances of the rotating-disk boundary-layer flow is studied. It is shown, by comparison with Briggs' method, that there are certain difficulties associated with the steepest-descent time-asymptotic method of evaluating the impulse response. The rotating-disk boundary layer provides examples of saddle points through which the steepest-descent path can and cannot be made to pass. It is shown that it is critically important to establish, from the topography of the phase function, whether the steepest-descent path passes through particular saddle points. If this procedure is not carried out, calculations of the wave-packet evolution can be majorly flawed and incorrect conclusions can be drawn about the stability of the flow, i.e., whether it is convectively or absolutely unstable.

On the asymptotic solution of a high–order nonlinear ordinary differential equation

Abstract: We examine Berman's similarity solution for steady flow of a viscous incompressible fluid in a channel, driven by suction or injection of the fluid through the channel walls. We convert the two–point boundary–value problem for the similarity function into an initial–value problem in which the position of the upper channel wall and the Reynolds number are determined as part of the solution, and consider two cases. In the first case equal suction or injection takes place at each wall and solutions are sought that are symmetrical about the mid–line of the channel. In the second case fluid can pass through only the upper channel wall. In each case analysis of a single initial–value problem yields two distinct solutions, one corresponding to injection, the other to suction, and our analysis is valid in the limit of large suction or injection. The injection solutions are well known. For the first (symmetrical) case the suction solution has previously been given only incompletely, while for the second case the asymptotic structure of the solution for large suction has been an open problem. We complement our asymptotics with numerical integrations of the ordinary differential equation governing the similarity function and find excellent agreement between the two.

Boundary layer similarity flow driven by power-law shear

Abstract: Similarity solution of the Prandtl boundary layer equations describing wallbounded flows and symmetric free-shear flows driven by rotational velocitiesU(y)=βyα are determined for a range of exponents α and amplitudes β Asymptotic analysis of the equations shows that for α −2/3 the shear stressf″(0) parameter is determined as a function of α and β Symmetric free-shear flow solutions become singular as α→α0 = −1/2 and no solutions are found in the range −1 −1/2 the centerline velocityf′(0) is determined as a function of α and β An asymptotic analysis of the singular behavior of these two problems as α→α0, given in a separate Appendix, shows excellent comparison with the numerical results Similarity solutions at the critical values α0 have exponential decay in the far field and correspond to the Glauert wall jet for wall-bounded flow and to the Schlichting/Bickley planar jet for symmetric free-shear flow

On the similarity solution of the fragmentation equation

Abstract: An analysis of the similarity solution of the linear, homogeneous, fragmentation equation is given for a general volume-conserving daughter-fragment distribution. For the special case of a polynomial daughter-distribution of degree p, an exact, basic similarity solution for the time evolution of the particle-volume distribution is derived. The solution is proportional to the Meijer G-function which may be represented as a linear combination of generalized hypergeometric functions. The properties of generalized hypergeometric functions and G-functions are given in the special function literature and include the continuation of the solution to large values of the similarity variable which is given here for special cases.

Experiments on Stability and Transition at Mach 3

Abstract: The results of an experimental study of stability, receptivity and transition of the flat-plate laminar boundary layer at Mach 3 are discussed. With a relatively low free-stream disturbance level (∼0.1%), spectra, growth rates and amplitude distributions of naturally occurring boundary layer waves were measured using hot wires. Physical (mass-flux) amplitudes in the boundary layer and free stream are reported and provide stability and receptivity results against which predictions can be directly compared. Comparisons are made between measurements of growth rates of unstable high-frequency waves and theoretical predictions based on a non-parallel, mode-averaging stability theory and receptivity assumptions; good agreement is found. In contrast, it was found that linear stability theory does not account for the measured growth of low-frequency disturbances. A detailed investigation of the disturbance fields in the free stream and on the nozzle walls provides the basis for a discussion of the source and the development of the measured boundary layer waves. Attention is drawn to the close matching in streamwise wavelengths for instability waves and the free-stream acoustic disturbances. It was also found that a calibration of the hot wire in the free stream yields a double-peak boundary layer disturbance amplitude distribution, as has been found by previous investigators, which is not consistent with the predictions of linear stability theory. This double peak was found to be an experimental anomaly which resulted from assumptions that are frequently made in the free-stream calibration procedure. A single-peak amplitude distribution across the boundary layer was established only when the hot-wire voltage was calibrated against the mean boundary layer profile. Finally, the late stages of transition, at a higher Reynolds number with a higher free-stream disturbance level, were explored. Calibrated amplitude levels are provided at locations where nonlinearities are first detected and where the mean boundary layer profile is first observed to depart from the laminar similarity solution. A qualitative discussion of the character of ensuing nonlinearities is also included.

Diffusion flame in a two-dimensional, accelerating mixing layer

Abstract: A theoretical and computational study of a laminar, two-dimensional, compressible, mixing, reacting layer with a pressure gradient that accelerates the flow in the direction of the primary stream is performed. One objective is to analyze the problems of a new technology related to combustion occurring in an accelerating transonic flow. Potential exists for reduction in nitric oxide formation and improvement in engine efficiency and/or power/weight. A similarity solution is found that reduces the partial differential equations to a system of ordinary differential equations. The solution is found in terms of a new acceleration parameter for compressible flows. The parameter is also useful for nonreacting mixing layers and for reacting or nonreacting wall boundary layers. For a low Mach number, the parameter reduces to the classical incompressible parameter. A numerical solution to these equations was performed. In the presence of exothermic reaction and accelerating pressure gradients, there are nonmonotoni...

A similarity solution for viscous source flow on a vertical plane

Abstract: A thin-film approximation is used in an analysis of the flow of a thin trickle of viscous fluid down a near-vertical plane. An approximate similarity solution is obtained, representing essentially a source (or sink) flow. Several interpretations of the solution are discussed. Problems concerning the ‘draining’ of viscous films down inclined surfaces have received much attention in the literature. In this paper we use a thin-film approximation to study the steady spreading or contraction of viscous liquid supplied (at a prescribed rate) on a near-vertical plane; gravity may be considered the ‘main’ driving force, but surface tension cannot be neglected. The assumption that the flow changes only slowly down the plane then leads to an approximate similarity solution for this three-dimensional viscous free-surface flow. Several interpretations of this solution are discussed. Only steady flows are considered, so that, in particular, any contact lines are fixed; diculties associated with moving contact lines are thereby avoided. (Cf. Davis, 1983, and the many references therein.) Unsteady flows have been considered (within a thin-film theory) by, for example, Huppert (1982), Schwartz (1989), Lister (1992) and Moriarty et al. (1991).

New one-dimensional model equations of magnetohydrodynamic turbulence

Abstract: New one-dimensional model equations of magnetohydrodynamic turbulence depending on the position x and the time t are proposed considering the x component of the velocity u(x,t) and the y and z components of the magnetic field by(x,t) and bz(x,t). It is found that the model equations conserve the total energy and the quantity similar to the cross helicity, and that they have a similarity solution, u(x,t)∼x/t and by(x,t), bz(x,t)∼C/t in the inviscid limit. A shock type solution is obtained where both the velocity and the magnetic fields have a jump in a narrow viscous or diffusive region. Numerical simulations are also carried out and the results show that the k−2 energy spectrum develops both for the velocity and the magnetic fields.

Intrusive Gravity Currents

Abstract: Intrusive gravity currents arise when a fluid of intermediate density intrudes into an ambient fluid. These intrusions may occur in both natural and human-made settings and may be the result of a sudden release of a fixed volume of fluid or the steady or time-dependent injection of such a fluid. In this article we analytically and numerically analyze intrusive gravity currents arising both from the sudden release of a fixed volume and the steady injection of fluid having a density that is intermediate between the densities of an upper layer bounded by a free surface and a heavier lower layer resting on a flat bottom. For the physical problems of interest we assume that the dynamics of the flow are dominated by a balance between inertial and buoyancy forces with viscous forces being negligible. The three-layer shallow-water equations used to model the two-dimensional flow regime include the effects of the surrounding fluid on the intrusive gravity current. These effects become more pronounced as the fraction of the total depth occupied by the intrusive current increases. To obtain some analytical information concerning the factors effecting bore formation we further reduce the complexity of our three-layer model by assuming small density differences among the different layers. This reduces the model equations from a 6×6 to a 4×4 system. The limit of applicability of this weakly stratified model for various ranges of density differences is examined numerically. Numerical results, in most instances, are obtained using MacCormack's method. It is found that the intrusive gravity current displays a wide range of flow behavior and that this behavior is a strong function of the fractional depth occupied by the release volume and any asymmetries in the density differences among the various layers. For example, in the initially symmetric sudden release problem it is found that an interior bore does not form when the fractional depth of the release volume is equal to or less than 50% of the total depth. The numerical simulations of fixed-volume releases of the intermediate layer for various density and initial depth ratios demonstrate that the intermediate layer quickly slumps from any isostatically uncompensated state to its Archimedean level thereby creating a wave of opposite sign ahead of the intrusion on the interface between the upper and lower layers. Similarity solutions are obtained for several cases that include both steady injection and sudden releases and these are in agreement with the numerical solutions of the shallow-water equations. The 4×4 weak stratification system is also subjected to a wavefront analysis to determine conditions for the initiation of leading-edge bores. These results also appear to be in agreement with numerical solutions of the shallow-water equations.

Similarity solution of mixed convection over a horizontal moving plate

Abstract: Investigation to the mixed convective heat and mass transfer over a horizontal plate has been carried out. By applying transformation group theory to analysis of the governing equations of continuity, momentum, energy and diffusion, we show the existence of similarity solution for the problem provided that the temperature and concentration at the wall are proportional to x 4/(7-5n) and that the moving speed of the plate is proportional to x (3-n)/(7-5n), and further obtain a similarity representation of the problem. The similarity equations have been solved numerically by a fourth-order Runge–Kutta scheme. The numerical results obtained for Pr=0.72 and various values of the parameters Sc, K 1, K 2 and K 3 reveals the influence of the parameters on the flow, heat and mass transfer behavior.

A Similarity Solution for Oil Lens Redistribution Including Capillary Forces and Oil Entrapment

Abstract: Redistribution of a LNAPL lens (oil) at the phreatic surface is described using a multi-phase flow model, with emphasis on the effect of oil entrapment by water. The flow process is analyzed under the assumption that the vertical capillary and gravitational forces balance. Vertical integration leads to explicit functions which approximate the relations between the free oil volume per unit lateral area and the vertically averaged oil relative permeability on the one hand and the vertical position of the interface between zones with either two or three phases on the other hand. A linear relation between the trapped and free oil volume per unit lateral area approximates the vertically integrated nonlinear expression for the trapped oil saturation. The resulting differential equation admits a similarity solution describing the lateral spreading of free oil and the amount and location of trapped oil. Comparison with illustrative numerical computations, which are based on the nonreduced flow model in a two-dimensional planar or axisymmetric domain, shows that the analytical solution provides a good approximation of the free oil distribution at all later times.

The Structure of Cooling Fronts in Accretion Disks

Abstract: Recent work has shown that the speed of the cooling front in soft X-ray transients may be an important clue in understanding the nature of accretion disk viscosity. In a previous paper (Vishniac & Wheeler), we derived the scaling law for the cooling front speed. Here we derive a similarity solution for the hot inner part of disks undergoing cooling. This solution is exact in the limit of a thin disk, power-law opacities, and a minimum hot state column density, which is an infinitesimal fraction of the maximum cold state density. For a disk of finite thickness, the largest error is in the ratio of the mass flow across the cooling front to the mass flow at small radii. Comparison to the numerical simulations of Cannizzo et al. indicates that the errors in other parameters do not exceed (csF/rFΩF)q, that is, the ratio of the sound speed at the disk midplane to its orbital velocity, evaluated at the cooling front, to the qth power. Here q ≈ ½. Its precise value is determined by the relevant hot state opacity law and the functional form of the dimensionless viscosity.

Vorticity Generation in Slow Cooling Flows

Abstract: We show that any generic non-adiabatic slow flow of ideal compressible fluid develops a significant vorticity. As an example, an initially irrotational conductive cooling flow is considered. A perturbation theory for the vorticity generation is developed that employs, as a zero order solution, a novel two-dimensional similarity solution. Full gasdynamic simulations of this CF demonstrate the vorticity generation and support the theory. The relevance of this problem to the experiments with the "hot channels" is discussed.

Nonlinear growth dynamics of Langmuir monolayers limited by both surface and bulk diffusion

Abstract: A theory of growth of Langmuir monolayers limited by both surface and bulk impurity diffusion is developed. It is shown that unlike the traditional situation where only surface diffusion is present---leading to similarity solution where a straight front advances with time as $\sqrt{t}$ which is at the basis of dendritic growth and ``fractal-like'' morphologies---bulk diffusion leads to the existence of a straight front moving at a constant speed. This is interpreted in terms of dimensional considerations: bulk diffusion introduces a new length scale, making this solution possible. As a consequence, the growth morphology must be dense. This is what is observed experimentally. An exact solution for a straight front and its stability is provided analytically. The straight front is unstable above a critical speed (or critical supersaturation). The nonlinear dynamics are tackled by means of a gauge-field-invariant geometrical formulation. It is shown that the existence of a straight front solution moving steadily also implies that a circular front solution moving at a constant speed exists as well. For a nearly straight geometry (but deformed) dynamics falls into a Kuramoto-Sivashinsky one where spatiotemporal chaos is expected. For a more curved front (such as the one generated initially from a circle instability), numerical analysis reveals a variety of compact patterns.

On Similarity Solutions for Three Dimensional Flow of Second Order Fluids

Abstract: The existence of similarity solutions for the steady three-dimensional boundary layer flow of a viscoelastic fluid has been investigated. It has been shown that similarity solutions exist for two mainstream flow patterns which correspond to the flow near a stagnation point. The non-linear boundary value problem governing one of these stagnation point flows has been solved numerically combined with a perturbation approach. It is found that the effects of non-Newtonian parameters are to increase the velocity in the boundary layer. Some relevant particular cases have also been discussed.

Similarity solution for fragmentation with volume change

Abstract: The similarity form of the solution of the linear homogeneous equation for fragmentation with volume change is given. The solution shows the time evolution of the particle-volume distribution when a particle splits into a polynomial distribution of fragment volumes. The limits for small and large values of the similarity variable are derived, the long-time limit for all values of the similarity variable is given and the characteristic time to approach the limit is identified. The solution shows the effect of volume change on the particle-volume distribution and on the time dependence of the moments of the distribution.