Showing papers on "Similarity solution published in 1999"

Self-similarity and internal structure of turbulence induced by Rayleigh Taylor instability

Abstract: This paper describes an experimental investigation of mixing due to Rayleigh–Taylor instability between two miscible fluids. Attention is focused on the gravitationally driven instability between a layer of salt water and a layer of fresh water with particular emphasis on the internal structure within the mixing zone. Three-dimensional numerical simulations of the same flow are used to give extra insight into the behaviour found in the experiments.The two layers are initially separated by a rigid barrier which is removed at the start of the experiment. The removal process injects vorticity into the flow and creates a small but significant initial disturbance. A novel aspect of the numerical investigation is that the measured velocity field for the start of the experiments has been used to initialize the simulations, achieving substantially improved agreement with experiment when compared with simulations using idealized initial conditions. It is shown that the spatial structure of these initial conditions is more important than their amplitude for the subsequent growth of the mixing region between the two layers. Simple measures of the growth of the instability are shown to be inappropriate due to the spatial structure of the initial conditions which continues to influence the flow throughout its evolution. As a result the mixing zone does not follow the classical quadratic time dependence predicted from similarity considerations. Direct comparison of external measures of the growth show the necessity to capture the gross features of the initial conditions while detailed measures of the internal structure show a rapid loss of memory of the finer details of the initial conditions.Image processing techniques are employed to provide a detailed study of the internal structure and statistics of the concentration field. These measurements demonstrate that, at scales small compared with the confining geometry, the flow rapidly adopts self-similar turbulent behaviour with the influence of the barrier-induced perturbation confined to the larger length scales. Concentration power spectra and the fractal dimension of iso-concentration contours are found to be representative of fully developed turbulence and there is close agreement between the experiments and simulations. Other statistics of the mixing zone show a reasonable level of agreement, the discrepancies mainly being due to experimental noise and the finite resolution of the simulations.

Evolution of a wave packet into vortex loops in a laminar separation bubble

Abstract: A laminar boundary layer develops in a favourable pressure gradient where the velocity profiles asymptote to the Falkner & Skan similarity solution. Flying-hot-wire measurements show that the layer separates just downstream of a subsequent region of adverse pressure gradient, leading to the formation of a thin separation bubble. In an effort to gain insight into the nature of the instability mechanisms, a small-magnitude impulsive disturbance is introduced through a hole in the test surface at the pressure minimum. The facility and all operating procedures are totally automated and phase-averaged data are acquired on unprecedently large and spatially dense measurement grids. The evolution of the disturbance is tracked all the way into the reattachment region and beyond into the fully turbulent boundary layer. The spatial resolution of the data provides a level of detail that is usually associated with computations.Initially, a wave packet develops which maintains the same bounded shape and form, while the amplitude decays exponentially with streamwise distance. Following separation, the rate of decay diminishes and a point of minimum amplitude is reached, where the wave packet begins to exhibit dispersive characteristics. The amplitude then grows exponentially and there is an increase in the number of waves within the packet. The region leading up to and including the reattachment has been measured with a cross-wire probe and contours of spanwise vorticity in the centreline plane clearly show that the wave packet is associated with the cat's eye pattern that is a characteristic of Kelvin–Helmholtz instability. Further streamwise development leads to the formation of roll-ups and contour surfaces of vorticity magnitude show that they are three-dimensional. Beyond this point, the behaviour is nonlinear and the roll-ups evolve into a group of large-scale vortex loops in the vicinity of the reattachment. Closely spaced cross-wire measurements are continued in the downstream turbulent boundary layer and Taylor's hypothesis is applied to data on spanwise planes to generate three-dimensional velocity fields. The derived vorticity magnitude distribution demonstrates that the second vortex loop, which emerges in the reattachment region, retains its identity in the turbulent boundary layer and it persists until the end of the test section.

Stability of Self-similar Solutions for Van der Waals Driven Thin Film Rupture

Abstract: Recent studies of pinch-off of filaments and rupture in thin films have found infinite sets of first-type similarity solutions. Of these, the dynamically stable similarity solutions produce observable rupture behavior as localized, finite-time singularities in the models of the flow. In this letter we describe a systematic technique for calculating such solutions and determining their linear stability. For the problem of axisymmetric van der Waals driven rupture (recently studied by Zhang and Lister), we identify the unique stable similarity solution for point rupture of a thin film and an alternative mode of singularity formation corresponding to annular “ring rupture.”

Maxwell fluid suction flow in a channel

Abstract: Two-dimensional, steady and incompressible suction flow of the upper-convected Maxwell fluid in a porous surface channel has been studied The combined effects of viscoelasticity and inertia are considered A similarity solution is assumed, resulting in a nonlinear system of ODEs that describes the relations between the two velocity components, the three deviatoric stresses and the pressure gradient This system is solved using two methods: an analytical solution, based on a power series method in terms of the transverse coordinate across the channel, and a fourth-order Runge–Kutta numerical integration scheme We first find the existing Newtonian flow solutions for suction and injection For the Maxwell fluid, the solutions of the power series and the numerical integration are in complete agreement in the range of Reynolds and Deborah numbers 0 ≤ Re ≤ 30 and 0 ≤ De ≤ 03 They show that the suction flow exhibits a flattening of the longitudinal velocity profile near the centerline and the establishment of boundary layers near the porous surfaces as Reynolds number increases It is also observed that when Deborah number increases, with a fixed Reynolds number, viscoelasticity affects the velocity profiles in the same way as inertia in a Newtonian fluid The application of the self-similar solution to the injection flow of the Maxwell fluid is also discussed

Curing simulation of thermoset composites

Abstract: This paper deals with the modelling and simulation of resin flow, heat transfer and the curing of multilayer thermoset composite laminates during processing in an autoclave. Darcy's Law and Stokes’ slow-flow equations are used for the flow model and, for approximately isothermal flows, a similarity solution is developed. This permits the decoupling of the velocity and thermal fields. A two-dimensional convection–diffusion heat equation with an internal heat generation term is then solved numerically, together with the equation for the rate of cure, using a finite difference scheme on a moving grid. The simulations are performed with varying composite thicknesses, and a comparison of numerical results with known experimental data confirms the approximate validity of the model.

Thermophoresis in natural convection with variable properties

Abstract: The present paper deals with thermophoresis in natural convection with variable properties for a laminar flow over a cold vertical flat plate. Variation of properties like density, viscosity and thermal conductivity with temperature is included in the formulation of the problem. Selection of components for the property ratio is made by fitting the property values between the desired temperature limits. For a selected fluid, Prandtl number variation with temperature is neglected and the Prandtl number corresponding to film temperature is used for the analysis. Solution is carried out by finite difference method. Variation of wall concentration and wall flux along the length of plate is studied. The effect of thermophoretic coefficient on wall concentration is also studied. Results are presented in the form of graphs. The result is compared with similarity solution by Runge-Kutta method and found to be accurate upto second decimal place.

Particle-driven gravity currents down planar slopes

Abstract: Particle-driven gravity currents, as exemplified by either turbidity currents in the ocean or ignimbrite flows in the atmosphere, are buoyancy-driven flows due to a suspension of dense particles in an ambient fluid. We present a theoretical study on the dynamics of and deposition from a turbulent current flowing down a uniform planar slope from a constant-flux point source of particle-laden fluid. The flow is modelled using the shallow-water equations, including the effects of bottom friction and entrainment of ambient fluid, coupled to an equation for the transport and settling of the particles. Two flow regimes are identified. Near the source and for mild slopes, the flow is dominated by a balance between buoyancy and bottom friction. Further downstream and for steeper slopes, entrainment also affects the behaviour of the current. Similarity solutions are also developed for the simple cases of homogeneous gravity currents with no settling of particles in the friction-dominated and entrainment-dominated regimes. Estimates of the width and length of the deposit from a monodisperse particle-driven gravity current with settling are derived from scaling analysis for each regime, and the contours of the depositional patterns are determined from numerical solution of the governing equations.

Drag reduction of a nonnewtonian fluid by fluid injection on a moving wall

Abstract: A boundary layer problem of a nonnewtonian fluid flow with fluid injection on a semi-infinite flat plate whose surface moves with a constant velocity in the opposite direction to that of the uniform mainstream is analyzed. Concluding similarity equations are solved numerically to show the dependence of the problem to the velocity ratio λ of the plate to uniform flow and to the injection velocity parameter C. The critical values of λ and C for each nonnewtonian power-law index n are obtained, and their significance in drag reduction is discussed.

Gravitational Collapse of Isothermal Magnetized Clouds: The Universality of Self‐similar Collapse

Abstract: We study the dynamical collapse of isothermal magnetized clouds with two-dimensional axisymmetric numerical simulations. As a model of a cloud, we consider an infinitely long filamentary cloud with a longitudinal magnetic field. An initial model is constructed by adding an axisymmetric perturbation to an equilibrium model. Because of gravitational instability, it fragments into magnetically supercritical cloud cores, which collapse to form dynamically contracting disks keeping nearly quasi-static equilibrium in the vertical direction. The disk contraction is followed until the central density increases by a factor of more than 107. The disk collapses self-similarly while oscillating with appreciable amplitude; the structure of the disk at the different times is similar, except for the scale. In each cycle of the oscillation, MHD fast and slow shock waves form. This oscillation is essentially the same as that during the collapse of an isothermal rotating cloud. We also follow the evolution of various models, changing the cloud mass and magnetic field strength. The disk evolution depends only weakly on the initial condition. Taking account of the magnetic pressure and tension, we refine the similarity solution for a magnetized thin disk obtained by Nakamura, Hanawa, & Nakano. We find that the refined similarity solution can reproduce the main features of our simulations. We also apply the similarity solution to the collapse of a magnetically subcritical cloud. We confirm that the dynamical collapse phase of the subcritical cloud can also be well approximated by the similarity solution. The structure of a dynamically collapsing magnetized disk is essentially similar irrespective of whether the initial cloud is supercritical or subcritical. It indicates that the similarity collapse is a universal characteristic of the dynamical collapse of magnetized clouds.

Heat transfer characteristics of the separation boundary flow induced by a continuous stretching surface

Abstract: The self-similar boundary flow with identically vanishing skin friction (`separation flow') induced by a continuous plane surface with temperature distribution Tw(x) = T+Cx and stretching velocity Uw(x) = Bx-1/3 is considered. The exact analytical solution for the velocity and temperature field is given in terms of hyperbolic and hypergeometric functions, respectively. For particular values of several closed-form solutions are written down. On this background the heat transfer characteristics of the separation flow are analysed in detail.

Capillary instability and breakup of a viscous thread

Abstract: Papageorgiou derived a similarity solution that describes the asymptotic behavior of a thinning viscous thread suspended in vacuum, near the critical time and around the location of breakup. The motion is driven by surface tension, and the fluid inertia is neglected throughout the evolution. To assess the physical relevance of the similarity solution, the evolution of an infinite thread immersed in an ambient fluid with arbitrary viscosity, subject to periodic axisymmetrtic perturbations is simulated through solution of the equations of Stokes flow by a boundary integral method. The results show that when the thread is suspended in vacuum, the similarity solution accurately describes the process of thinning over an extended length of the thread between the developing bulges, and captures the late stages of breakup for a broad range of initial conditions. But a small amount of ambient fluid viscosity, as small as 0.05 times the viscosity of the thread fluid, drastically alters the nature of the motion by shifting the location of the breakup points toward the bases of developing bulges, and causing the thread to develop locally asymmetric shapes.

Suction and injection effects on a similar flow between parallel plates

Abstract: A similarity solution has been obtained for a flow between two parallel plates (rectangular or circular) approaching or receding from each other with suction or injection at the porous plate. Approximate analytic solutions based on the regular perturbation technique are given and a numerical solution of the resulting nonlinear differential equations is presented. The effects of injection and suction on the velocity profiles, pressure distribution and shearing stress on the wall are studied. In the case of the flow between circular plates the effects on load capacity due to mass transfer are investigated.

Time-dependent simulations of point explosions with heat conduction

Abstract: A hydrodynamic-diffusion code is used to simulate a point explosion. The gas motion is governed by both hydrodynamics and nonlinear heat conduction and is a combination of the well-known, self-similar Taylor–Sedov spherically expanding shock wave and the spherically expanding thermal wave. Two problems are discussed. In the first problem, a similarity solution exists if the diffusion coefficient is given in terms of powers of density and temperature which also define the ambient spatial density profile. If the initial explosion energy is small, the diffusive effect is limited to a region behind the shock. However, if the explosion energy is large, the thermal front precedes the hydrodynamic front, which is then an isothermal shock. In the second problem, the initial density is constant and the diffusion coefficient depends on only a power of the temperature. In this case, the solution is not self-similar; in early times, heat conduction dominates; in late times—hydrodynamics. The problems were previously ...

Conically similar swirling flows at high Reynolds numbers

Abstract: A new class of both inviscid and boundary layer self-similar solutions for conical swirling flows at high Reynolds numbers is analysed. For the case of one-cell solutions, the flow consists of an inviscid, but in general rotational, core with a velocity field, in spherical polar coordinates, of the form u = r m-2 V(θ), where m is any real number. Due to the known existence of two integrals to Euler's equations, the vector function V(θ) is obtained by the integration of a second-order ordinary differential equation, containing the two integration constants K and K 1 associated with the intensities of the swirl and the meridional motion, respectively. This inviscid flow is, however, singular at the axis and must be regularized through a thin viscous layer, which also has self-similar structure. A variety of flow regimes are obtained for different ranges of m, all of them exhaustively analysed. In particular, for 0 < m < 2, the solution to the near-axis boundary layer equations has the interesting property of losing existence when a certain inviscid swirl parameter, D ∼ K 1 /K 4 , is either larger or smaller than a critical value, depending on m

Similarity solution of free convective boundary-layer behavior at a stretching surface

Abstract: In the present study, free convection and heat transfer behavior of electrically conducting fluid in the boundary layer over a vertical continuously stretching surface is investigated. The effects of free convection, magnetic field, suction/blowing at the surface and the stretching speed of the surface on the flow and heat transfer characteristics are considered. By applying one-parametric group theory to analysis of the problem, a similarity solution is found. The governing equations of continuity, momentum and energy are solved numerically by a fourth-order Runge-Kutta scheme. The numerical results, which are obtained for the flow and heat transfer characteristics, reveal the influences of the parameters.

A similarity solution to a problem in nonlinear ion transport with a nonlocal condition

Abstract: We derive an initial–boundary value problem with a nonlocal condition to model mass and charge transport in a controlled potential experiment called chronoamperometry. The charge is carried by ions set in motion by a change in the applied electrode potential. The resulting electric current is called the current response. When the charge flux arises from diffusion alone, the model reduces to an initial–boundary value problem for the heat equation. We consider diffusion and migration (motion under the influence of an electric field). This latter component of the total flux gives rise to a nonlinearity in the transport equations. Experimentalists have noted the current response to be proportional to the square root of the time with or without migration. It is shown that the initial–boundary value problem has a unique similarity solution that accounts for this observation.

Unsteady compressible flow in the stagnation region of two-dimensional and axi-symmetric bodies

Abstract: This paper deals with a new similarity solution of unsteady laminar compressible two-dimensional and axi-symmetric boundary layers. It has been shown that a self-similar solution is possible when the free stream velocity varies inversely with time. The two-point boundary value problems governed by self-similar equations have been solved numerically using an implicit finite difference scheme in combination with the quasi-linearization technique. It is observed that the effect of the acceleration parameter (A) in the free stream velocity on the skin friction is more pronounced compared to the heat transfer. For certain values of the acceleration parameter and the total enthalpy at the wall, the surface shear stress (skin friction) vanishes. The skin friction and heat transfer increase due to suction, and the effect of injection is found to be just opposite. Velocity profiles are presented with reverse flow and without reverse flow depending on the values of toal enthalpy at the wall and the acceleration parameter.

Self-similar dynamics of a viscous wedge of fluid

Abstract: The 2D (two dimensional) motion of two symmetric wedges of viscous fluid is determined. The evolution of the fluid is governed by the Stokes equations plus the interfacial boundary conditions including surface tension. A similarity solution is determined and these equations are solved numerically by using a boundary integral method. Solutions for different wedge angles are found.

Solutions for the flow of a conducting power-law fluid in a transverse magnetic field and with a pressure gradient using Crocco variables

Abstract: The steady two-dimensional incompressible flow of a conducting power-law fluid past a flat plate in the presence of a transverse magnetic field of the formH0x(m−1)/2 and under the influence of a pressure gradient is considered. The resulting similarity equation is first converted into a different form using Crocco variables and then solved by choosing a suitable profile for the dependent variable. The results are compared with those obtained by direct numerical integration of the original differential equation. The energy equation for a special case for which similarity solutions exist is also considered.

Flow and heat transfer on a continuous flat surface moving in a parallel free stream with variable fluid properties

Abstract: Boundary layer flow and heat transfer on a continuous flat surface moving in a parallel free stream with variable fluid properties are investigated. The similarity solution is used to transform the problem under consideration into a boundary value problem of coupled ordinary differential equations. Numerical results are carried out for various values of the dimensionless parameters of the problem. The results have demonstrated that the assumption of constant properties may introduce severe errors in the prediction of surface friction factor and heat transfer rate. For the same Reynolds numbers, Prandtl numbers, heating parameter, temperature exponents for viscosity and thermal conductivity parameters, and the same velocity difference |U w - U∞|, larger skin friction, and heat transfer coefficient results for U w > U∞ than for U w < U∞.

Internal Boundary Layer Scaling in ``Two Layer'' Solutions of the Thermocline Equations

Abstract: The diffusivity dependence of internal boundary layers in solutions of the continuously stratified, diffusive thermocline equations is revisited. If a solution exists that approaches a two-layer solution of the ideal thermocline equations in the limit of small vertical diffusivity κυ, it must contain an internal boundary layer that collapses to a discontinuity as κυ → 0. An asymptotic internal boundary layer equation is derived for this case, and the associated boundary layer thickness is proportional to κ1/2υ. In general, the boundary layer remains three-dimensional and the thermodynamic equation does not reduce to a vertical advective–diffusive balance even as the boundary layer thickness becomes arbitrarily small. If the vertical convergence varies sufficiently slowly with horizontal position, a one-dimensional boundary layer equation does arise, and an explicit example is given for this case. The same one-dimensional equation arose previously in a related analysis of a similarity solution tha...

The local flow in a wedge between a rigid wall and a surface of constant shear stress

Abstract: The viscous incompressible flow in a wedge between a rigid plane and a surface of constant shear stress is calculated by use of the Mellin transform. For wedge angles below a critical value the asymptotic solution near the vertex is given by a local similarity solution. The respective stream function grows quadratically with the distance from the origin. For supercritical wedge angles the similarity solution breaks down and the leading order solution for the stream function grows with a power law having an exponent less than two. At the critical angle logarithmic terms appear in the stream function. The asymptotic dependence of the stream function found here is the same as for the 'hinged plate' problem. It is shown that the validity of the Stokes flow assumption is restricted to a vanishingly small distance from the vertex when the wedge angle is above critical and when the region of nonzero constant shear stress is extended to infinity. The relevance of the present result for technical flow systems is pointed out by comparison with the numerically calculated flow in a thermocapillary liquid bridge.

Heat transfer from moving surfaces in a micropolar fluid

Abstract: A boundary layer analysis is presented for the forced convection problem of a surface moving continuously in a flowing stream of a micropolar fluid. Two cases are considered, one corresponding to a plane surface moving in parallel with the free stream and the other, a surface moving in the opposite direction to the free stream. A similarity solution to the governing momentum, angular momentum, and energy equations is derived.These equations were solved numerically and the flow and heat transfer characteristics of the micropolar fluid are presented. PACS No.: 61.00

Semianalytical self-similar solution of bent-over jet in cross-flow

Abstract: The flow and scalar field in the cross section of a bent-over momentum jet in cross-flow has been investigated via a 2D numerical solution of the governing equations with similarity transformation. The semianalytical model employs a free shear layer model for turbulent closure and the assumption of a constant velocity in the direction of the cross-flow; the entire solution depends solely on a dimensionless turbulent mixing parameter λ that measures the relative importance of advection and diffusion. The computed streamlines and vorticity field clearly indicate a vortex-pair flow for all λ. The shape of the scalar distribution, however, grows from a circular cell to a kidney-shaped double peak structure. Using a value of λ = 55, the computed jet trajectory and spreading and the cross-sectional shape are in good agreement with the experiments, although the dilution is somewhat overpredicted. The results suggest a free shear layer model of e = 0.0272Wml*, where e, Wm, and l* are the eddy viscosity, maximum v...