Showing papers on "Similarity solution published in 2004"

Heat transfer over an unsteady stretching surface

Abstract: Similarity solution of the laminar boundary layer equations corresponding to an unsteady stretching surface have been studied. The governing time-dependent boundary layer are transformed to ordinary differential equations containg Prandtl number and unsteadiness parameter. The effect of various govern-ing parameters such as Prandtl number and unsteadiness param-eter which determine the velocity and temperature profiles and heat transfer coefficient are studied.

Long-Time Asymptotics of Kinetic Models of Granular Flows

Abstract: We analyze the long-time asymptotics of certain one-dimensional kinetic models of granular flows, which have been recently introduced in [22] in connection with the quasi-elastic limit of a model Boltzmann equation with dissipative collisions and variable coefficient of restitution. These nonlinear equations, classified as nonlinear friction equations, split naturally into two classes, depending on whether or not the temperature of their similarity solutions (homogeneous cooling states) reduce to zero in finite time. For both classes, we show uniqueness of the solution by proving decay to zero in the Wasserstein metric of any two solutions with the same mass and mean velocity. Furthermore, if the temperature of the similarity solution decays to zero in finite time, we prove, by computing explicitly upper bounds for the lifetime of the solution in terms of the length of the support, that the temperature of any other solution with initially bounded support must also decay to zero in finite time.

Separation criterion for turbulent boundary layers via similarity analysis

Abstract: By using the RANS boundary layer equations, it will be shown that the outer part of an adverse pressure gradient turbulent boundary layer tends to remain in equilibrium similarity, even near and past separation. Such boundary layers are characterized by a single and constant pressure gradient parameter, and its value appears to be the same for all adverse pressure gradient flows, including those with eventual separation

Envelope expansion with core collapse – I. Spherical isothermal similarity solutions

Abstract: We investigate self-similar dynamical processes in an isothermal self-gravitational fluid with spherical symmetry. With reference to the earlier complementary solution results of Larson, Penston, Shu, Hunter and Whitworth & Summers, we further explore the 'semi-complete solution space' from an initial instant t → 0 + to a final stage t → +∞. These similarity solutions can describe and accommodate physical processes of radial inflow, core collapse, oscillations and envelope expansion (namely, outflow or wind) or contraction as well as shocks. In particular, we present new classes of self-similar solutions, referred to as 'envelope expansion with core collapse' (EECC) solutions, that feature concurrent interior core collapse and exterior envelope expansion. The interior collapse towards the central core approaches a free-fall state as the radius r → 0, while the exterior envelope expansion gradually approaches a constant radial flow speed as r → +∞. There exists at least one spherical stagnation surface of zero flow speed that separates the core collapse and the envelope outflow and that travels outward at constant speed, either subsonically or supersonically, in a self-similar manner. Without crossing the sonic critical line where the travel speed of non-linear disturbances relative to the radial flow is equal to the sound speed, there exists a continuous band of infinitely many EECC solutions with only one supersonic stagnation point as well as a continuous band of infinitely many similarity solutions for 'envelope contraction with core collapse' (ECCC) without stagnation point. Crossing the sonic critical line twice analytically, there are infinitely many discrete EECC solutions with one or more subsonic stagnation points. Such discrete EECC similarity solutions generally allow radial oscillations in the subsonic region between the central core collapse and the outer envelope expansion. In addition, we obtained complementary discrete ECCC similarity solutions that cross the sonic critical line twice with subsonic oscillations. In all these discrete solutions, subsonic spherical stagnation surfaces resulting from similarity oscillations travel outward at constant yet different speeds in a self-similar manner. With specified initial boundary or shock conditions, it is possible to construct an infinite number of such EECC similarity solutions, which are conceptually applicable to various astrophysical problems involving gravitational collapses and outflows. We mention potential applications of EECC similarity solutions to the formation process of protoplanetary nebulae connecting the asymptotic giant branch phase and the planetary nebula phase to H II clouds surrounding star formation regions, and to a certain evolution phase of galaxy clusters.

Unsteady forces on an accelerating plate and application to hovering insect flight

Abstract: The aerodynamic forces on a flat plate accelerating from rest at fixed incidence in two-dimensional power-law flow are studied analytically and numerically. An inviscid approximation is made in which separation at the two plate edges is modelled by growing spiral vortex sheets, whose evolution is determined by the Birkhoff– Rott equation. A solution based on a similarity expansion is developed, valid when the scale of the separated vortex is much smaller than the plate dimension. The leading order is given by the well-known similarity growth of a vortex sheet from a semi-infinite flat plate, while equations at the second order describe the asymmetric sweeping effect of that component of the free-stream parallel to the plate. Owing to subtle cancellation, the unsteady vortex force exerted on the plate during the starting motion is independent of the sweeping effect and is determined by the similarity solution, to the order calculated. This gives a mechanism for dynamic stall based on a combination of unsteady vortex lift and pure added mass; the incidence angle for maximum vortex lift is arccos √ 3/8 ≈ 52.2 ◦ independent of the acceleration profile. Circulation on the flat plate makes no direct contribution. Both lift and drag force predictions from the unsteady inviscid theory are compared with those obtained from numerical solutions of the two-dimensional unsteady Navier–Stokes equations for an ellipse of high aspect ratio, and with predictions of Wagner’s classical theory. There is good agreement with numerical results at high incidence and moderate Reynolds number. The force per unit span predicted by the vortex theory is evaluated for parameters typical of insect wings and is found to be in reasonable agreement with numerical simulations. Estimates for the shed circulation and the size of the start-up vortices are also obtained. The significance of this flow as a mechanism for insect hovering flight is discussed.

Thermal-diffusion and diffusion-thermo effects on mixed free-forced convective flow and mass transfer over an accelerating surface with a heat source in the presence of suction and blowing in the case of variable viscosity

Abstract: An analysis has been carried out to obtain the thermal-diffusion and the diffusion-thermo effects on the mixed free-forced convective and mass transfer steady laminar boundary-layer flow over an accelerating surface with a heat source in the presence of suction and blowing. The fluid viscosity is assumed to vary as an inverse linear function of temperature. The partial differential equations governing the problem under consideration have been transformed by a similarity transformation into a system of ordinary differential equations which is solved numerically by applying the shooting method. The results for an impermeable accelerating surface are discussed. The effects of the variable viscosity parameter θr, the thermal diffusion parameter Sr, the diffusion-thermo parameter Df, suction or blowing parameter m, heat flux parameter s and Schmidt number Sc have been examined on the flow field of a hydrogen-air mixture as a non-chemical reacting fluid pair. The effects of varying these parameters are studied in the case of a surface with prescribed wall temperature and a surface with prescribed wall heat flux.

Combined radiation and mixed convection from a vertical wall with suction/injection in a non–Darcy porous medium

Abstract: Mixed convection flow of an absorbing fluid up a uniform non–Darcy porous medium supported by a semi-infinite ideally transparent vertical flat plate due to solar radiation is considered. The external flow field is assumed to be uniform, the effect of the radiation parameter in the boundary layer adjacent to the vertical flat plate with fluid suction/injection through it is analyzed in both aiding and opposing flow situations. It is observed that the similarity solution is possible only when the fluid suction/injection velocity profile varies as x −1/2. The velocity and temperature profiles in the boundary layer and the heat transfer coefficient are presented for selected values of the parameters. It is observed that the Nusselt number increases with the increase in the radiation parameter and also when the value of the surface mass flux parameter moves from the injection to the suction region.

Self-similar solution of the unsteady mixed convection flow in the stagnation point region of a rotating sphere

Abstract: An analysis is performed to present a new self-similar solution of unsteady mixed convection boundary layer flow in the forward stagnation point region of a rotating sphere where the free stream velocity and the angular velocity of the rotating sphere vary continuously with time. It is shown that a self-similar solution is possible when the free stream velocity varies inversely with time. Both constant wall temperature and constant heat flux conditions have been considered in the present study. The system of ordinary differential equations governing the flow have been solved numerically using an implicit finite difference scheme in combination with a quasilinearization technique. It is observed that the surface shear stresses and the surface heat transfer parameters increase with the acceleration and rotation parameters. For a certain value of the acceleration parameter, the surface shear stress in x-direction vanishes and due to further reduction in the value of the acceleration parameter, reverse flow occurs in the x–component of the velocity profiles. The effect of buoyancy parameter is to increase the surface heat transfer rate for buoyancy assisting flow and to decrease it for buoyancy opposing flow. For a fixed buoyancy force, heating by constant heat flux yields a higher value of surface heat transfer rate than heating by constant wall temperature.

Similarity solutions for breakup of jets of power law fluids

Abstract: We compute similarity solutions for the breakup of a jet of a power law liquid surrounded by a vacuum. As is known from the Newtonian case, such similarity solutions are fundamentally different depending on whether creeping flow or flow with inertia is considered. We shall investigate both cases. For flow with inertia, we start with Eggers’ solution for the Newtonian case, and we continue it to other values of the power law exponent. The jet profile corresponding to Eggers’ solution is highly asymmmetric. We find that the degree of asymmetry decreases as we deviate from the Newtonian case in either direction. For small as well as for large values of the power law exponent, we find new branches of symmetric solutions. These branches establish a connection between the similarity solutions with and without inertia.

The adhesive or slippery flat plate viscoplastic boundary layer for a shear-thinning power-law viscosity

Abstract: New viscoplastic boundary layer (VPBL) solutions around a flat plate are obtained for viscoplastic fluids in the limiting case of creeping flows with a dominant yield value. Previous studies in the field were mainly heuristic, since only Bingham behavior and no slip at the wall were considered. In fact, in most practical cases, shear-thinning behavior occurs together with slip at the wall. Moreover, together with the yield stress value, these are the main properties which dominate the fields at hand. Due consideration is paid to the permissible stress fields far from the obstacle, and to normalized Cauchy equations introduced in the flowing regions. New similarity solutions are derived for the boundary layer along a flat plate, and the VPBL properties are given in terms of velocity profiles, boundary layer thickness and total drag on the plate. In the case of slip at the wall, a new dimensionless slip parameter ψ is introduced. Total drag forces are found to be increasingly smaller when slip and the values of ψ increase progressively from zero. For values of ψ larger than 1, the material is unyielded and slips with a solid type friction at the wall, after having been cut at the leading edge of the plate.

Influences of fluid property variation on the boundary layers of a stretching surface

Abstract: In this work, the influences of temperature-dependent fluid properties on the boundary layers over a continuously stretching surface with constant temperature are investigated. Based on the boundary layer assumptions, the coupled similarity equations are obtained for special situations, in which the fluid density and heat capacity are assumed without dependence on the temperature. Those similarity equations are solved numerically. The influences of property variation on wall stresses and heat fluxes are discussed. It is found that the property variation can influence the distributions of both fluid velocity and temperature across the boundary layers. For the thermal boundary layer, using mean properties evaluated at the average temperature of wall and ambient fluid can give good results for the temperature distribution. However, for the momentum boundary layer, the difference of velocity distributions can be large.

A numerical model for the jet flow generated by water impact

Abstract: In this paper a numerical model is developed aimed at describing the jet flow caused by water impact. The study, carried out in the framework of a potential-flow assumption, exploits the shallowness of the jet region to significantly simplify the local representation of the velocity field. This numerical model is incorporated into a fully nonlinear boundary-element solver that describes the flow generated by the water entry of two-dimensional bodies. Attention is focused on the evaluation of the capability of the model to provide accurate free-surface shape and pressure distribution along the wetted part of the body contour, with particular regard to the jet region. After a careful verification, the proposed model is validated through comparisons with the similarity solution of the wedge impact with constant entry velocity. This similarity solution is derived with the help of an iterative procedure which solves the governing boundary-value problem written in self-similar variables.

Analysis of turbulent flow patterns of soil water under field conditions using Burgers equation and porous suction-cup samplers

Abstract: This paper presents results using the Burgers equation in spherical coordinates for the investigation of soil water dynamics around porous cup samplers under field conditions. The analysis reveals that the yield (V) of the porous suction-cup sampler is a power function of inflow duration (T) and source moisture strength (A). It also shows that the shock front velocity and moving boundary of the front are both power functions of A and time. The analysis further demonstrates that the asymptotic shock front velocity derived from a similarity solution is the same for different versions of Burgers equation, indicating the validity of different versions of Burgers equation for the same flow patterns. The observation using time domain reflectometry shows that a moisture shock was induced around the sampler following application of a vacuum, which justifies the application of Burgers equation. The values of A were derived using field data to determine the location and the front velocity of the shock.

Symmetry groups and similarity solutions for a free convective boundary-layer problem

Abstract: The free convective boundary-layer problem due to the motion of an elastic surface into an electrically conducting fluid is studied with group-theoretical methods. The symmetry groups admitted by the corresponding boundary value problem are obtained. Particular attention is paid on the group of scaling which provides the similarity solution of the problem. Also, the admissible form of the data, in order to be conformed to the obtained symmetries, is provided. Finally, with the use of the entailed similarity solution the problem is transformed into a boundary value problem of ODEs and is solved numerically.

Imploding cylindrical and spherical shock waves in a non-ideal medium

Abstract: Converging shock waves in an almost ideal medium are considered. The kinematics of one-dimensional motion have been applied to construct an evolution equation for strong cylindrical and spherical shock waves propagating into a low density gas at rest. The approximate value of the similarity parameter obtained from there is compared with those derived from Whitham's Rule and the exact similarity solution at the instant of collapse of the shock wave. The above computation is carried out for different values of the parameter α, which depends on the internal volume of the gas molecules.

Similarity Analysis for Nonequilibrium Turbulent Boundary Layers

Abstract: was a constant and the velocity deficit profile normalized with the friction velocity, u * , was independent of the streamwise direction. Thus, the profiles should collapse into a single curve. However, most flows did not satisfy these conditions, especially flows near separation or separated, where the friction velocity, u * ,w as approaching zero. Clauser @1# further concluded that these equilibrium flows were a special type of flows which were difficult to generate and maintain in equilibrium, and therefore most flows were recognized as nonequilibrium flows. Bradshaw @3# showed that a necessary condition for a turbulent boundary layer to maintain equilibrium was that the contribution of the pressure gradient to the growth of the momentum deficit should be a constant multiple of the contribution from the surface shear stress, which was shown to be the same as Clauser’s pressure parameter, b. Townsend @4# developed a self-preserving theory which was more rigorous than the analysis by Clauser @1#. Unfortunately, Townsend @4# overconstrained the problem by assuming the existence of a single velocity scale. Rotta @5# studied the adverse pressure gradient ~APG! flow and showed that the length scale and the velocity scale were given as, d * U‘ /u * and u * , respectively. Obviously, these scalings also failed for flows with strong APG or near separation. Later on, a criterion given as U‘5a(x 2x 0) m with m,0 was used very often to predict the equilibrium adverse pressure gradient boundary layer by Townsend @6#, East ] ]

Stability criterion for self-similar solutions with a scalar field and those with a stiff fluid in general relativity

Abstract: A stability criterion is derived in general relativity for self-similar solutions with a scalar field and those with a stiff fluid, which is a perfect fluid with the equation of state P = ρ. A wide class of self-similar solutions turns out to be unstable against kink mode perturbation. According to the criterion, the Evans–Coleman stiff-fluid solution is unstable and cannot be a critical solution for the spherical collapse of a stiff fluid if we allow sufficiently small discontinuity in the density gradient field in the initial data sets. The self-similar scalar-field solution, which was recently found numerically by Brady et al (2002 Class. Quantum Grav. 19 6359), is also unstable. Both the flat Friedmann universe with a scalar field and that with a stiff fluid suffer from kink instability at the particle horizon scale.

Effect of surfactants on free-surface turbulent flows

Abstract: In two earlier papers, we studied the statistical and mechanistic structure of the turbulent boundary layer under a stress-free (clean) free surface. Findings there, such as the presence of inner and outer surface layers, are very much the direct result of the absence of shear stresses at the surface. The latter condition is easily lost when the surface is contaminated and surface elasticity varies with space and time. In this paper we consider the effect of surfactant on features of the free-surface turbulent flow. We perform direct numerical simulations of the Navier–Stokes equations subject to surfactant-laden free-surface boundary conditions for varying Reynolds and Marangoni numbers and low Froude numbers. As expected, the Marangoni effect decreases the horizontal turbulence intensity and normal vorticity at the surface. The direct effect on the turbulent kinetic energy is an increase in the dissipation and viscous diffusion and a decrease in the production near the surface relative to the clean case. The most prominent effect of the presence (of even a small amount) of surfactant is the drastic reduction in the surface divergence and the associated sharp decrease of up- and downwelling at the surface which has direct implications to near-surface turbulent transport. The observed surfactant effects on turbulent kinetic energy budget can be attributed to the generation of Marangoni vorticity at the free surface by approaching hairpin vortices. The Marangoni effect has also a direct effect on the boundary-layer structure, causing an increase of the thickness of the boundary layer and in the maxima of the mean shear near the surface. For moderate values of the Marangoni number, up-/downwelling effectively vanishes and the flow approaches a state independent of the Marangoni number. Guided by these results and to obtain theoretical insight, we develop a similarity solution for the mean flow. The analytic solution agrees well with the numerical data and provides precise measures for the multi-layer structure of the boundary layer. Based on the theoretical model, we derive scaling laws for the thickness of the inner and the outer boundary layers, which are also confirmed by numerical simulations.

Self-similar solutions of the axisymmetric shallow-water equations governing converging inviscid gravity currents

Abstract: A phase-plane approach is used to determine similarity solutions of the axisymmetric shallow-water equations which represent inwardly propagating, inviscid gravity currents. A Froude number condition characterizes the movement of the front. The unique similarity exponent is found numerically as a function of the frontal Froude number and the height and velocity profiles are presented for three different Froude numbers. The fluid speed and height are seen to increase monotonically towards the front except very close to the front where the height decreases. The maxima in both height and speed increase as the Froude number increases, reflecting the change in ambient resistance.For the Froude number that has been obtained experimentally for lock-exchange Boussinesq flows (. This similarity solution describes the formation of a shock, as well as its initial propagation.

Boundary Layer Self-Similar Solution for the Hot Radiative Accretion onto a Rapidly Spinning Neutron Star

Abstract: We consider hot accretion onto a rapidly spinning neutron star (or any other compact object with a surface). A radiative hot settling flow was discovered at low accretion rates in the early work by Medvedev & Narayan, and an analytical solution was presented. It was shown later that this flow could match the external medium smoothly, thus enforcing its physical feasibility. Here we complete the study of the global structure of such hot accretion by presenting the analytical solution for the boundary layer, which forms between the bulk of the flow and the stellar surface. We confirm our results via a full numerical solution of height-integrated two-temperature hydrodynamic equations.

Compressibility Effects on the Extended Crocco Relation and the Thermal Recovery Factor in Laminar Boundary Layer Flow

Abstract: The relation between velocity and enthalpy in steady boundary layer flow is known as the Crocco relation. It describes that for an adiabatic wall the total enthalpy remains constant throughout the boundary layer, when the Prandtl number (Pr) is one, irrespective of pressure gradient and compressibility. A generalization of the Crocco relation for Pr near one is obtained from a perturbation approach. In the case of constant-property flow an analytic expression is found, representing a first-order extension of the standard Crocco relation and confirming the asymptotic validity of the square-root dependence of the recovery factor on Prandtl number. The particular subject of the present study is the effect of compressibility on the extended Crocco relation and, hence, on the thermal recovery in laminar flows. A perturbation analysis for constant Pr reveals two additional mechanisms of compressibility effects in the extended Crocco relation, which are related to the viscosity law and to the pressure gradient. Numerical solutions for (quasi-)self-similar as well as non-similar boundary layers are presented to evaluate these effects quantitatively