Showing papers on "Similarity solution published in 2003"

Convergence rate for compressible Euler equations with damping and vacuum

Abstract: We study the asymptotic behavior of L∞ weak-entropy solutions to the compressible Euler equations with damping and vacuum. Previous works on this topic are mainly concerned with the case away from the vacuum and small initial data. In the present paper, we prove that the entropy-weak solution strongly converges to the similarity solution of the porous media equations in Lp(R) (2≤p<∞) with decay rates. The initial data can contain vacuum and can be arbitrary large. A new approach is introduced to control the singularity near vacuum for the desired estimates.

Generalized three‐dimensional flow due to a stretching sheet

Abstract: The three-dimensional steady, laminar flow due to stretching of a sheet is considered. The governing equations of motion admit a similarity solution. By drawing an analogy with the flow due to a rotating disk it is demonstrated that the resulting boundary value problem admits a solution in terms of series of exponentially decaying functions. The solution is highly efficient and very accurate, as it eliminates the need of replacing numerical infinity in the range of integration by a finite value. Two perturbation solutions are also developed when the primary flow corresponds to the two-dimensional case and the axisymmetric case. In each case the expansion is obtained up to the second degree term of the perturbation parameter. Finally, an approximate solution is derived, which is very simple and highly accurate.

Diffusion of chemically reactive species of a non-Newtonian fluid immersed in a porous medium over a stretching sheet

Abstract: The influence of reaction rate on the transfer of chemically reactive species in the laminar visco-elastic fluid flow immersed in a porous medium over a stretching sheet is considered. The flow is caused solely by the linearly stretching sheet and the reactive species is emitted from this sheet and undergoes an isothermal and homogeneous one-stage reaction as it diffuses into the surrounding fluid. A similarity transformation is introduced, which reduces the concentration conservation equation to an ordinary differential equation. An exact analytical solution due to Siddappa and Abel (Z. Angew. Math. Phys. 36 (1985) 890) is adopted for velocity, where as the concentration equation is obtained numerically for higher-order reactions. The numerical computations show that the effect of destructive chemical reaction is to reduce the thickness of concentration boundary layer and increase the mass transfer rate from the sheet to the surrounding fluid. This effect is more effective for zero- and first-order reaction than second- and third-order reactions.

Nonlinear internal gravity wave beams

Abstract: Based on linear inviscid theory, a two-dimensional source oscillating with frequency , however, the transient evolution of nearly vertical beams takes place on a slower time scale than that of oblique beams; this is shown to account for the discrepancies between the steady-state similarity solution of Gordon & Stevenson (1972) and their experimental observations. Finally, the present asymptotic theory is used to study the refraction of nearly vertical nonlinear beams in the presence of background shear and variations in the Brunt–Vaisala frequency.

Integral Solution for the Mean Flow Profiles of Turbulent Jets, Plumes, and Wakes

Abstract: Integral methods are used to derive similarity solutions for several quantities of interest including the cross-stream velocity, Reynolds stress, the dominant turbulent kinetic energy production term, and eddy diffusivities of momentum and heat for axisymmetric and planar turbulent jets, plumes, and wakes. A universal constant is evaluated for axisymmetric and planar plumes

Exponential vs Algebraic Growth and Transition Prediction in Boundary Layer Flow

Abstract: For applications regarding transition prediction, wing design and control of boundary layers, the fundamental understanding of disturbance growth in the flat-plate boundary layer is an important ...

Real-Gas Effects on Mean Flow and Temporal Stability of Binary-Species Mixing Layers

Abstract: Real-gas effects on the mean flow and inviscid stability of temporal mixing layers are examined for supercritical heptane/nitrogen and oxygen/hydrogen mixtures. The analysis is based on the compressible Navier–Stokes equations for conservation of mass, momentum, total energy, and species mass, with heat and species-mass fluxes derived from fluctuation-dissipation theory and incorporating Soret and Dufour effects. An approximate form of the equations is used to obtain a system of similarity equations for the streamwise velocity, the temperature, and the mass fraction. The similarity profiles show important real-gas nonideal-mixture effects, particularly for the temperature, in departing from the incompressible error-function similarity solution. Realistic Schmidt and Prandtl numbers were found to be important to the similarity profiles. A linear, inviscid stability analysis is then performed using the similarity profile, as well as analytical error-function profiles, as its basic flow. The stability analysis shows that the similarity profile has larger growth rates at a given wavelength and a shorter more unstable wavelength than the error-function profiles and than an incompressible flow. The similarity profile also has a larger range of unstable wavelengths than the error-function profiles.

Similarity solutions for a moving-flat plate thermal boundary layer

Abstract: In this paper, the thermal boundary-layer problem of a semi-infinite flat plate moving in a constant velocity free stream is studied. The similarity equations with viscous dissipation for the thermal boundary-layer are derived and solved by numerical techniques. Under some specific conditions, the thermal boundary-layer similarity equation can be integrated analytically. The results are analyzed for very small Eckert number case and large Eckert number case. It is found that, for the two cases, wall heat fluxes will increase with the increase of the velocity ratio λ. With increasing Eckert number, the viscous dissipation heating will become dominant. However, for the Prandtl number when the Eckert number is small, it is found that wall heat fluxes will increase with increasing Prandtl number only for a certain range of velocity ratio λ. For the other range, the wall heat fluxes will have a maximum at a certain Prandtl number, and, when the Prandtl number is larger than the critical value, wall heat fluxes will decrease with increasing Prandtl number. Some examples of the lower solution branch are also presented to compare with the upper solution branch. It is found that the lower solution branch will result in lower heat fluxes at the wall.

A model for diffusion-controlled solidification of ternary alloys in mushy layers

Abstract: We describe a model for non-convecting diffusion-controlled solidification of a ternary (three-component) alloy cooled from below at a planar boundary. The modelling extends previous theory for binary alloy solidification by including a conservation equation for the additional solute component and coupling the conservation equations for heat and species to equilibrium relations from the ternary phase diagram. We focus on growth conditions under which the solidification path (liquid line of descent) through the ternary phase diagram gives rise to two distinct mushy layers. A primary mushy layer, which corresponds to solidification along a liquidus surface in the ternary phase diagram, forms above a secondary (or cotectic) mushy layer, which corresponds to solidification along a cotectic line in the ternary phase diagram. These two mushy layers are bounded above by a liquid layer and below by a eutectic solid layer. We obtain a one-dimensional similarity solution and investigate numerically the role of the control parameters in the growth characteristics. In the special case of zero solute diffusion and zero latent heat an analytical solution can be obtained. We compare our predictions with previous experimental results and with theoretical results from a related model based on global conservation laws described in the Appendix. Finally, we discuss the potentially rich convective behaviour anticipated for other growth conditions.

On flow of binary alloys during crystal growth

Abstract: The topic of this review article is about some aspect of flow during alloy solidification and the corresponding research development that has been accomplished in the recent years to determine the features of such flows, which obey either linear or weakly nonlinear system of the relevant governing equations and the boundary conditions for the associated fluid mechanics and heat or mass transfer. Modeling efforts, the associated analyses and the numerical computations are reviewed in the absence or presence of rotation both for single-layer case, where a mushy layer of composed solid dendrites and liquid melt exists above the solid portion of the alloy, and for double-layer case, where, in addition to the mushy layer, a liquid layer lies over the mushy layer. INTRODUCTION AND DISCUSSION In the last two decades or so, there has been a number of research works on the problem of flow in a mushy layer of mixed solid and liquid phases (Hills et al., 1983; Huppert and Worster, 1985; Fowler, 1985; Worster, 1985, 1992, 1997; Huppert, 1990; Amberg and Homsy, 1993; Chen et al., 1994; Sayre and Riahi, 1995, 1996, 1997; Anderson and Worster, 1995, 1996; Chung and Chen, 2000a, 2000b, 2003; Guba, 2001; Riahi, 2002, 2003a, 2003b, 2003c). Hills, Loper and Roberts (1983) developed a set of thermodynamic equations for a mushy zone and solved a much reduced set of such equations approximately for the constrained growth of a binary alloy. Later, Fowler (1985) provided a more complete solution. Huppert and Worster (1985) formulated a simple model of the mushy layer based on the consideration of global conservation relationship. The prediction based on their model agreed well with the observation by them of ice growing at a plane boundary from aqueous solutions of various kinds of salt. Worster (1986) presented a mathematical model for the region of the mushy layer and treated the mushy zone as a continuum whose properties vary with the local volume fraction of solid. The equations governing the Worster’s model conserve both heat and solute locally but on a macroscale larger than the typical pore-size of the mush. Worster was then able to compute local bulk properties of the mush and computed the local solid fraction as a function of distance from the cooled boundary, which gave some indications of the morphology of the growing solid. His predictions for the growth rate of the mushy layer, based on his similarity solution, agreed well with the existing experimental measurements of ice growing from aqueous salt solutions (Huppert and Worster, 1986). As will be reviewed in more details in the next section, Worster (1992) investigated linear instabilities of the flow in the liquid and mushy regions during solidification of alloy. He discovered two stationary modes of buoyancy driven compositional convection

Periodic Travelling Wave Selection by Dirichlet Boundary Conditions in Oscillatory Reaction-Diffusion Systems

Abstract: Periodic travelling waves are a fundamental solution form in oscillatory reaction-diffusion equations. Here I discuss the generation of periodic travelling waves in a reaction-diffusion system of the generic $\lambda$-$\omega$ form. I present numerical results suggesting that when this system is solved on a semi-infinite domain subject to Dirichlet boundary conditions in which the variables are fixed at zero, periodic travelling waves develop in the domain. The amplitude and speed of these waves are independent of the initial conditions, which I generate randomly in numerical simulations. Using a combination of numerical and analytical methods, I investigate the mechanism of periodic travelling wave selection. By looking for an appropriate similarity solution, I reduce the problem to an ODE system. Using this, I derive a formula for the selected speed and amplitude as a function of parameters. Finally, I discuss applications of this work to ecology.

Similarity analysis in magnetohydrodynamics: Hall effects on free convection flow and mass transfer past a semi-infinite vertical flat plate

Abstract: Heat and mass transfer along a semi-infinite vertical flat plate under the combined buoyancy force effects of thermal and species diffusion is investigated in the presence of a strong non-uniform magnetic field and the Hall currents are taken into account. The induced magnetic field due to the motion of the electrically conducting fluid is negligible. This assumption is valid for a small magnetic Reynolds number. The similarity solutions are obtained using the scale group of transformations. These are the only symmetry transformations admitted by the field equations. The non-linear boundary layer equations with the boundary conditions are transferred to a system of non-linear ordinary differential equations with the appropriate boundary conditions. Furthermore, the similarity equations are solved numerically by using a fourth order Runge–Kutta scheme with the shooting method. Numerical results for the velocity profiles, the temperature profiles and the concentration profiles are presented graphically for various values of the magnetic parameter M in the range of 0–1 with the Hall parameter m taking the values 0.5, 1, 2, and 3.

Propagation of strong converging shock waves in a gas of variable density

Abstract: The problem of a strong converging spherical (or cylindrical) shock collapsing at the centre (or axis) of symmetry is extended to take into account the inhomogeneity of a gaseous medium, the density of which is decreasing towards the centre (or axis) according to a power law. The perturbative approach used in this paper provides a global solution to the implosion problem yielding accurately the results of Guderley's similarity solution, which is valid only in the vicinity of the center/axis of implosion. The analysis yields refined values of the leading similarity parameter along with higher-order terms in Guderley's asymptotic solution near the center/axis of convergence. Computations of the flow field and shock trajectory in the region extending from the piston to the center/axis of collapse have been performed for different values of the adiabatic coefficient and the ambient density exponent.

Blow-up in a fourth-order semilinear parabolic equation from explosion-convection theory

Abstract: with a parameter β 0, which is a model equation from explosion-convection theory Unlike the classical Frank-Kamenetskii equation ut = uxx +e u (a solid fuel model), by using analytical and numerical evidence, we show that the generic blow-up in this fourth-order problem is described by a similarity solution u∗(x, t )= − ln(T − t )+ f1(x/(T − t) 1/4 )( T> 0 is the blowup time), with a non-trivial profile f1 0 Numerical solution of the PDE shows convergence to the self-similar solution with the profile f1 from a wide variety of initial data We also construct a countable subset of other, not self-similar, blow-up patterns by using a spectral analysis of an associated linearized operator and matching with similarity solutions of a first-order Hamilton–Jacobi equation

On the Similarity Solutions of the Calogero–Degasperis–Fokas Modified KdV Equation via Symmetry Method

Abstract: Using the symmetry method, we analyze the Calogero–Degasperis–Fokas modified KdV equation u t + u x x x - a u x 3 - f ( u ) u x =0; a ∈ R , where the function f solves f '''( u )=8 a f '( u ). The ...

A similarity solution describing the collision of two planar premixed flames

Abstract: The problem of the head-on collision of two planar premixed flame fronts is considered using the idealized model of a single step reaction controlled by the deficient species and an Arrhenius reaction law. Density changes are neglected. It is shown that a similarity solution exists if the deficient species and temperature are equidiffusive (Lewis number unity). The similarity solution, derived using activation energy asymptotics, is valid in the intermediate region when the flames are close enough that their pre-heat zones overlap but their reaction zones may be considered to be well separated. A one–dimensional numerical simulation shows good agreement with the analytical solution.

Developing flow of a power-law liquid film on an inclined plane

Abstract: Developing flow of a liquid film along a stationary inclined wall is analyzed for a power-law constitutive equation. For films with appreciable inertia and therefore small interfacial slopes, the boundary-layer approximation may be used. The boundary-layer equations are solved numerically through the von Mises transformation that gives a partial differential equation over a semi-infinite strip and approximately by the method of von Karman and Polhausen that gives an ordinary differential equation for the film thickness, called a film equation. Film equations derived from self-similar velocity profiles fail when the film thickens and the flow undergoes a supercritical to subcritical transition; a nonremovable singularity arises at the critical point, the location of the flow transition. A film equation is developed that accommodates this transition. Predictions exhibit a standing wave where hydrostatic pressure becomes important and opposes inertia. This thickening effect is accentuated for small angles of inclination at moderate Reynolds numbers. In the limit of small film thickness in which gravitational effects are negligible, the thickness profile is nonlinear in agreement with an independent and new similarity solution. This result contrasts with the established linear thickness profile for a Newtonian liquid. The circumstances in which the film equation gives results close to the full boundary layer equation are identified.

Plif temperature and velocity distributions in laminar hypersonic flat-plate flow

Abstract: Rotational temperature and velocity distributions have been measured across a hypersonic laminar flat-plate boundary layer, using planar laser-induced fluorescence. The measurements are compared to a finite-volume computation and a first-order boundary layer computation, assuming local similarity. Both computations produced similar temperature distributions and nearly identical velocity distributions. The disagreement between calculations is ascribed to the similarity solution not accounting for leading-edge displacement effects. The velocity measurements agreed to within the measurement uncertainty of 2% with both calculated distributions. The peak measured temperature was 200 K lower than the computed values. This discrepancy is tentatively ascribed to vibrational relaxation in the boundary layer.

Self Similarity of Cross‐Stream Droplet Momentum Displacement in Dispersed Multiphase Flow

Abstract: Analytical methods are developed and applied to droplet motion, as it relates to aircraft icing. Impinging droplets largely affect the heat balance at an iced aircraft surface, as well as the final ice shape. In this study, a similarity solution of the Eulerian droplet momentum equation is developed. Droplet motion near a flat plate is investigated with a similarity solution. By using scaling, sensitivity, order of magnitude and similarity methods, a momentum displacement of droplets (or particles) due to the presence of the solid surface is predicted. Self similarity of the droplet profiles is established, such that downstream propagation can be expressed in terms of a single independent coordinate. Limiting trends of momentum/drag induced and Blasius-diffusion profiles are found to identify the spatial range encompassing the droplet motion. The predicted results are successfully compared against the scaling requirements.

Flow Past a Swept Wing with a Compliant Surface: Stabilizing the Attachment‐Line Boundary Layer

Abstract: Many aquatic species such as dolphins and whales have fins, which can be modeled as swept wings. Some of these fins, such as the dorsal fin of a dolphin, are semi-rigid and therefore can be modeled as a rigid swept wing with a compliant surface. An understanding of the hydrodynamics of the flow past swept compliant surfaces is of great interest for understanding potential drag reduction mechanisms, especially since swept wings are widely used in hydrodynamic and aerodynamic design. In this paper, the flow past a swept wing with a compliant surface is modeled by an attachment-line boundary layer flow, which is an exact similarity solution of the Navier-Stokes equations, flowing past a compliant surface modeled as an elastic plate. The hydrodynamic stability of the coupled problem is studied using a new numerical framework based on exterior algebra. The basic instability of the attachment line boundary layer on a rigid surface is a traveling wave instability that propagates along the attachment line, and numerical results show that the compliance results in a substantial reduction in the instability region. Moreover, the results show that, although the flow-field is three-dimensional, the qualitative nature of the instability suppression is very similar to the qualitative reduction of instability of the two-dimensional Tollmien-Schlichting modes in the classical boundary-layer flow past a compliant surface.

A Contribution to the Integral Theory of a Turbulent Jet above a Point Heat Source of Constant Strength

Abstract: Consideration has been given to the integral model of an unsteady vertical convective jet above a point heat source of constant strength. It has been shown that this problem is reduced to self‐similar equations allowing the analytical solution. An algebraic invariant relating the parameters of the velocity and the temperature along the jet axis has been constructed. A comparison of the analytical solution and the existing experimental data on the propagation of the upper boundary of the convective jet has been made.