Showing papers on "Similarity solution published in 2001"

Similarity Analysis for Turbulent Boundary Layer with Pressure Gradient: Outer Flow

Abstract: The equilibrium-type similarity analysis of George and Castillo for the outer part of zero pressure gradient boundary layers has been extended to include boundary layers with pressure gradient. The constancy of a single new pressure gradient parameter is all that is necessary to characterize these new equilibrium turbulent boundary layers. Three major results are obtained: First, most pressure gradient boundary experiments appear to be equilibrium flows (by the new definition), and nonequilibrium flows appear to be the exception. Second, there appear to be only three values of the pressure gradient parameter: one for adverse pressure gradients, one for favorable pressure gradients, and one for zero pressure gradients. Third, correspondingly, there appear to be only three normalized velocity deficit profiles, exactly as suggested by the theory

Similarity solutions for hydromagnetic simultaneous heat and mass transfer by natural convection from an inclined plate with internal heat generation or absorption

Abstract: The problem of coupled heat and mass transfer by natural convection from a semi-infinite inclined flat plate in the presence of an external magnetic field and internal heat generation or absorption effects is formulated. The plate surface has a power-law variation of both wall temperature and concentration and is permeable to allow for possible fluid wall suction or blowing. The resulting governing equations are transformed using a similarity transformation and then solved numerically by an implicit, iterative, finite-difference scheme. Comparisons with previously published work are performed and good agreement is obtained. A parametric study of all involved parameters is conducted and a representative set of numerical results for the velocity and temperature profiles as well as the skin-friction parameter, average Nusselt number, and the average Sherwood number is illustrated graphically to show typical trends of the solutions.

Optimal linear growth in swept boundary layers

Abstract: Optimal perturbations for the family of three-dimensional boundary layers described by the Falkner{Skan{Cooke similarity solution are obtained using a variational technique in the temporal framework. The disturbances experiencing the most growth take the form of vortices almost aligned with the external streamline at inception and evolve into streaks. In subcritical flows these can attain about twice the transient amplication observed in comparably forced two-dimensional flows. Possible connections between optimal perturbations and exponentially amplied crossflow vortices are explored. The laminar{turbulent transition process in boundary layers is generally studied for the limiting case of a two-dimensional base flow. While the complexity of the transition phenomenon warrants such simplication, it happens that three-dimensional boundary layers predominate in a wide spectrum of engineering applications. The flow over a swept wing is a canonical example of a three-dimensional boundary layer which is of obvious interest to the aerodynamicist. A spanwise pressure gradient on the wing causes a crossflow velocity component to develop inside the boundary layer, altering the flow’s stability characteristics as a consequence. In a two-dimensional boundary layer a favourable pressure gradient delays transition by changing the shape of the mean velocity prole. Since laminar flow incurs signicantly less viscous drag than turbulent flow, modern wing sections are often designed to ensure a favourable pressure gradient over a large part of the wing chord. It is precisely in this environment, however, that crossflow instability is likely to arise. Crossflow instability, which came to light in a series of flight tests on swept-wing aircraft by Gray (1952), diers fundamentally from the viscous Tollmien{Schlichting instability occurring in two-dimensional boundary layers, as explained by Gregory, Stuart & Walker (1955). It is an inviscid instability resulting from the inflectional crossflow velocity prole, and manifests itself primarily in the form of steady crossflow vortices almost aligned with the free stream upon which travelling disturbances may be superposed. A plausible scenario for crossflow-induced transition in the low-disturbance environment likely to be encountered by a cruising commercial airliner can be pieced

Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion

Abstract: The velocity field of generalized second order fluid with fractional anomalous diiusion caused by a plate moving impulsively in its own plane is investigated and the anomalous diffusion problems of the stress field and vortex sheet caused by this process are studied. Many previous and classical results can be considered as particular cases of this paper, such as the solutions of the fractional diffusion equations obtained by Wyss; the classical Rayleigh’s time-space similarity solution; the relationship between stress field and velocity field obtained by Bagley and co-worker and Podlubny’s results on the fractional motion equation of a plate. In addition, a lot of significant results also are obtained. For example, the necessary condition for causing the vortex sheet is that the time fractional diffusion index β must be greater than that of generalized second order fluid α; the establiihment of the vorticity distribution function depends on the time history of the velocity profile at a given point, and the time history can be described by the fractional calculus.

Falling plumes in bacterial bioconvection

Abstract: Experiments by Kessler on bioconvection in laboratory suspensions of bacteria (Bacillus subtilis), contained in a deep chamber, reveal the development of a thin upper boundary layer of cell-rich fluid which becomes unstable, leading to the formation of falling plumes. We use the continuum description of such a suspension developed by Hillesdon et al. (1995) as the basis for a theoretical model of the boundary layer and an axisymmetric plume. If the boundary layer has dimensionless thickness λ [Lt ] 1, the plume has width λ1/2. A similarity solution is found for the plume in which the cell flux and volume flux can be matched to those in the boundary layer and in the bulk of the suspension outside both regions. The corresponding model for a two-dimensional plume fails to give a self-consistent solution.

Prandtl number effects for marangoni convection over a flat surface

Abstract: This paper presents a similarity solution for Marangoni flow over a flat surface for both the momentum equations and the energy equation assuming a developing boundary layer along a surface The analysis also shows how the heat transfer and flow rate vary with Prandtl number Since the Predicted boundary layer thickness would be less than the diameter of a typical Vapor bubble during nucleate boiling, the curvature effects can be neglected and this analysis can be used as a first estimate of the effect of Marangoni flow around a vapor bubble

Local non-similarity solutions for the flow of a non-Newtonian fluid over a wedge

Abstract: The boundary layer and heat transfer equations for a non-Newtonian fluid, represented by a power-law model, over a porous wedge is studied. The free stream velocity, the surface temperature variations, and the injection velocity at the surface are assumed variables. Similar and non-similar solutions are presented and the restrictions for these cases are studied. The results are presented for velocity and temperature profiles for various values of the dimensionless numbers. The effects of the different parameters on the skin friction co-efficient and the local heat transfer co-efficient are also studied.

Water Entry of a Wedge with Finite Deadrise Angle

Abstract: Water entry of a rigid wedge is analyzed by matched asymptotic expansions. The water is incompressible and the flow irrotational. The method is simple and robust. A jet domain, inner domains at the spray roots and an outer domain are defined. The important difference from the matched asymptotic expansions solution by Armand & Cointe (1987) is that the deadrise angle β is assumed finite when the outer and inner domain solutions are found. It is the matching that requires β to be small. However, the method predicts satisfactory details such as jet thickness, energy, and mass flux into the jet for finite β. This is verified by comparing with the similarity solution by Dobrovol'skaya (1969). The solution by Armand & Cointe gives only satisfactory predictions of these variables for very small β values. The procedure is applicable to gravity-free water entry of a general 2-D body shape. It is discussed how the inner domain solution at the spray root can be used in further developments of numerical methods for nonlinear wave-induced motions and loads on a ship. Also discussed is how the results can be used to predict sloshing damping due to tank roof impact.

The transition from inertia- to bottom-drag-dominated motion of turbulent gravity currents

Abstract: The influence of drag on the motion of gravity currents over rigid horizontal surfaces is considered analytically using a Chezy model of boundary shear stress. Although the initial motion is governed by a balance between the buoyancy forces and fluid inertia, drag gradually influences the flow. The length and time scales at which these effects become significant are identified. A perturbation series, valid at early times, is constructed to analyse the changes to the velocity and height of the evolving current due to drag. At much later times, a new class of similarity solutions is developed to model the motion which is now governed by a balance between buoyancy and drag. The transition in the dominant forces which govern the dynamics of the flow is examined by numerically integrating the equations of motion for flows generated by a constant flux of relatively dense fluid. The numerical results confirm both the perturbation solution, valid at early times, and the new similarity solution valid at late times. The transition between the two may involve the formation of a discontinuity (bore). Finally particle-driven currents, which exhibit different dynamical behaviour due to the progressive reduction of their density arising from particle sedimentation, are investigated.

On the Diffusive Profiles for the System of Compressible Adiabatic Flow through Porous Media

Abstract: We study the Cauchy problem for the system of one dimensional compressible adiabatic flow through porous media and the related diffusive problem. We introduce a new approach which combines the usual energy methods with special L1 -estimates and use the weighted Sobolev norms to prove the global existence and large time behavior for the solutions of the problems. The asymptotic states for the solutions are given by either stationary solutions or similarity solutions depending on the behavior of the initial data when $|x|\rightarrow \infty$. Our estimates provide asymptotic time decay rates.

Flow along two dimensions of liquid pulses in foams: Experiment and theory

Abstract: Experiments on foam drainage have so far only been performed in essentially one-dimensional flow geometries aligned with the direction of gravity. Here a foam-filled Hele-Shaw cell is used to examine pulsed drainage, which is the flow of a finite liquid volume, both along and perpendicular to the direction of gravity. An exact similarity solution to the generalized foam drainage equation exists, and an asymptotic analysis is presented to elucidate the nonlinear dynamics of the model. Good qualitative and quantitative agreement between theory and experiments on aqueous foams made with SDS surfactant is found when the node-dominated foam drainage model is applied.

Asymptotic analysis of the steady-state and time-dependent Berman problem

Abstract: The Berman problem for two-dimensional flow of a viscous fluid through an infinite channel is studied. Fluid motion is driven by uniform suction (or injection) of fluid through the upper channel wall, and is characterised by a Reynolds number R; the lower wall is impermeable. A similarity solution in which the streamfunction takes the form ψ = −xF(y, t) is examined, where x and y are coordinates parallel to and normal to the channel walls, respectively. The function F satisfies the Riabouchinsky-Proudman-Johnson equation, a partial differential equation in y and t; steady flows satisfy an ordinary differential equation in y. The steady states are computed numerically and the asymptotics of these solutions described in the limits of small wall suction or injection, large wall injection and large wall suction, the last of these being given more concisely and more accurately than in previous treatments. In the time-dependent problem, the solution appears to be attracted to a limit cycle when R ≫ 1 (large wall suction). This solution has been computed numerically for e = 1/R down to 0·011, but the structure of the solution makes further numerical progress currently infeasible. The limit cycle consists of several phases, some with slow and others with very rapid evolution. During one of the rapid phases, the solution achieves a large amplitude, and this feature of the solution lies behind the practical difficulties encountered in numerical simulations. The profile of the solution is plotted during the various phases and corresponding asymptotic descriptions are given. An exact solution to the Riabouchinsky-Proudman-Johnson equation covers most of the phases, although separate discussion is required of the boundary layers near the two walls and an interior layer near a zero of F. Particular consideration is required when this zero approaches the upper channel wall.

A mathematical framework for the analysis of particle–driven gravity currents

Abstract: Recent studies have modelled the flow of particle–driven gravity currents over horizontal boundaries using either shallow–water equations or simple ‘box’ (integral) models. The shallow–water equations are typically integrated numerically, whereas box models admit analytical solutions. However, the theoretical validity of the latter models has not been fully established. In this paper a novel mathematical technique is developed which permits the derivation of analytical solutions to the shallow–water model for gravity current motion. These solutions, confirmed by comparison with the results of numerical integration, are in good agreement with experimental observations. They also indicate why the simplified box models have been so successful. Moreover, they reveal how the internal dynamics of particle–driven flows are different from gravity currents arising solely due to compositional density differences. While compositionally driven gravity currents, which have a fixed density difference between the intruding and ambient fluids, may be modelled using similarity solutions to the governing equations, particle–driven gravity currents do not possess such solutions because their density is progressively reduced by particle sedimentation. Instead the new analysis determines how their behaviour progressively diverges from the similarity solution. By a change of independent variables, it is possible to develop convergent series expansions for each of the dependent variables which characterize the motion. It is suggested that this approach may find application to a number of other problems in which the dynamics are initially governed by a simple dynamical balance which is progressively lost as extra physical effects begin to influence the motion.

Analysis of A New Model for Unsaturated Flow in Porous Media Including Hysteresis and Dynamic Effects

Abstract: A new model for unsaturated flow in porous media, including capillary hysteresis and dynamic capillary effects, is analyzed. Existence and uniqueness of solutions are established and qualitative and quantitative properties of (particular) solutions are analyzed. Some results of numerical computations are given. The model under consideration incorporates simple ‘play’-type hysteresis and a dynamic term (time-derivative with respect to water content) in the capillary relation. Given an initial water content distribution, the model determines which parts of the flow domain are in drainage and which parts are in imbibition. The governing equations can be recast into an elliptic problem for fluid pressure and an evolution equation for water content. Standard methods are used to obtain numerical results. A comparison is given between J.R. Philip's semi-explicit similarity solution for horizontal redistribution in an infinite one-dimensional domain and solutions of the new model.

Ignition and Flame Studies for an Accelerating Transonic Mixing Layer

Abstract: Numerical solutions have been obtained for a diffusion e ame in the two-dimensional, laminar, steady, viscous, multicomponent, compressible mixing layer in the presence of a pressure gradient by using the boundary-layer approximationsand solvingthe x-momentum,energy,and speciesconservation equations. Thenumericalsolutions have been validated against similarity solutions and then are extended to cases where no similarity solution exists. The numerical solutions are used to study the ignition process and the e ame structure in an accelerating transonic mixing layer. It is shown how ignition length depends on initial temperature, initial pressure, initial velocity, viscosity, and pressure gradient. Ignition is found to occur on the high-temperature-air side. Oxidation kinetics and transport are both controlling in the upstream ignition region. Farther downstream, transport is controlling in the fully established e ame. The boundary-layer approximation is found to be valid everywhere including the upstream ignition region.

The falkner-skan wedge flows of power-law fluids embedded in a porous medium

Abstract: The non-Darcy flow characteristics of power-law non-Newtonian fluids past a wedge embedded in a porous medium have been studied. The governing equations are converted to a system of first-order ordinary differential equations by means of a local similarity transformation and have been solved numerically, for a number of parameter combinations of wedge angle parameter m, power-law index of the non-Newtonian fluids n, first-order resistance A and second-order resistance B, using a fourth-order Runge–Kutta integration scheme with the Newton–Raphson shooting method. Velocity and shear stress at the body surface are presented for a range of the above parameters. These results are also compared with the corresponding flow problems for a Newtonian fluid. Numerical results show that for the case of the constant wedge angle and material parameter A, the local skin friction coefficient is lower for a dilatant fluid as compared with the pseudo-plastic or Newtonian fluids.

Internal annular wall jets: Radial flow in a stirred tank

Abstract: Similarity solutions are derived for internal annular wall jets and compared with experimentally determined velocity profiles at the wall of a stirred tank with radial flow. The first similarity solution considers a geometry where the thickness of the wall jet is small relative to the diameter of the cylinder and the fluid flows relative to a free stream velocity of zero. In the second solution, the free stream velocity opposes the direction of the jet flow, as is observed in the recirculating flow in a stirred tank. The internal annular wall jet model agrees very well with the flow at the wall of a stirred tank agitated with a Rushton turbine. The free stream counter flow is driven by the wall jet, and a single velocity and length scale define the self-similar velocity profiles for the flow in the jet and in the recirculating flow. Both the analytical model and experimentally observed expansion rates are linear and the maximum velocity in the jet decays as [U ∝ (z/T)−0.5], where (z/T) is the dimensionless streamwise distance. This decay can be contrasted with the faster decay [U ∝ (z/T)−1] observed in the 3-D wall jets produced by axial impellers. These results provide a basis for validating computational fluid dynamic simulations in stirred tanks and for estimating the largest length scales of turbulence in the bulk of the tank.

The generation of internal waves by vibrating elliptic cylinders. Part 3. Angular oscillations and comparison of theory with recent experimental observations

Abstract: The methods of Parts 1 and 2 are extended to the case when the elliptic cylinder is executing angular oscillations about its centreline. At large distances from the cylinder the solution for a beam of waves tends to a similarity solution that decays more rapidly with distance than does the similarity solution for rectilinear oscillations described in Thomas & Stevenson

Linear stability of a gas boundary layer flowing past a thin liquid film over a flat plate

Abstract: The flow of a gas stream past a flat plate under the influence of rainfall is investigated. As raindrops sediment on the flat plate, they coalesce to form a water lm that flows under the action of shear from the surrounding gas stream. In the limit of (a) large Reynolds number, Re, in the gas phase, (b) small rainfall rate, _, compared to the free-stream velocity, U1, and (c) small lm thickness compared to the thickness of the boundary layer that surrounds it, a similarity solution is obtained that predicts growth of the liquid lm like x 3=4 ; x denotes dimensionless distance from the leading edge. The flow in the gas stream closely resembles the Blasius solution, whereas viscous dissipation dominates inside the lm. Local linear stability analysis is performed, assuming nearly parallel base flow in the two streams, and operating in the tripledeck regime. Two distinct families of eigenvalues are identied, one corresponding to the well-known Tollmien{Schlichting (TS) waves that originate in the gas stream, and the other corresponding to an interfacial instability. It is shown that, for the air{water system, the TS waves are convectively unstable whereas the interfacial waves exhibit a pocket of absolute instability, at the streamwise location of the applied disturbance. Moreover, it is found that as the inverse Weber number (We 1 ) increases, indicating the increasing eect of surface tension compared to inertia, the pocket of absolute instability is translated towards larger distances from the leading edge and the growth rate of unstable waves decreases, until a critical value is reached, We 1 We 1 c , beyond which the family of interfacial waves becomes convectively unstable. Increasing the inverse Froude number (Fr 1 ), indicating the increasing eect of gravity compared to inertia, results in the pocket of absolute instability shrinking until a critical value is reached, Fr 1 Fr 1 c , beyond which the family of interfacial waves becomes convectively unstable. As We 1 and Fr 1 are further increased, interfacial waves are eventually stabilized, as expected. In this context, increasing the rainfall rate or the free-stream velocity results in extending the region of absolute instability over most of the airfoil surface. Owing to this behaviour it is conjectured that a global mode that interacts with the boundary layer may arise at the interface and, eventually, lead to three-dimensional waves (rivulets), or, under extreme conditions, even premature separation.

Mass transfer effects on the non-Newtonian fluids past a vertical plate embedded in a porous medium with non-uniform surface heat flux

Abstract: The present study is devoted to investigate the influences of mass transfer on buoyancy induced flow over vertical flat plate embedded in a non-Newtonian fluid saturated porous medium. The Ostwald–de Waele power-law model is used to characterize the non-Newtonian fluid behavior. Similarity solution for the transformed governing equations is obtained with prescribed variable surface heat flux. Numerical results for the details of the velocity, temperature and concentration profiles are shown on graphs. Excess surface temperature as well as concentration gradient at the wall associated with heat flux distributions, which are entered in tables, have been presented for different values of the power-law index n, buoyancy ration B and the exponent λ as well as Lewis number Le.

Granular matter in a silo

Abstract: A dynamic model for slow accumulation of granular matter, with standing and rolling layers, is adapted to the geometry of a silo with vertical walls and bounded cross section. For sources constant in time the typical behavior of a rising level of matter with constant surface profile is described by a similarity solution. This solution can be characterized by a nonlinear boundary value problem. For the case of a circular silo with central point source and the corresponding 1D problem of an interval with central point source the profiles of the standing and rolling layers are explicitly computed. The computed shapes agree qualitatively with experimental observations and seem better justified than results based on the assumption of constant speed of the rolling layer.

Similarity solution of a penny-shaped fluid-driven fracture in a zero-toughness linear elastic solid

Abstract: This note deals with the problem of a penny-shaped hydraulic fracture propagating in an impermeable elastic solid. Growth of the fracture is driven by injection of an incompressible Newtonian fluid at the center of the fracture. The solution is restricted to the so-called viscosity-dominated regime where it can be assumed that the solid has zero toughness. The paper describes the construction of a semi-analytical similarity solution, which incorporates the known singularity of the fluid pressure at the center of the fracture and at the tip and which is based on series expansions of the fracture opening and fluid pressure in terms of Jacobi polynomials.