Showing papers on "Similarity solution published in 2009"

An Introduction to Gravity Currents and Intrusions

Abstract: Introduction Classification The Navier-Stokes equations Non-stratified ambient currents Shallow-water (SW) formulation for high-Re flows Motion of the interface and the continuity equation One-layer model A useful transformation The full behavior by numerical solution Dam-break stage Similarity solution The validity of the inviscid approximation The steady-state current and nose jump conditions Benjamin's analysis Jump condition Box models for 2D geometry Fixed volume current with inertial-buoyancy balance Inflow volume change Two-layer SW model Introduction The governing equations Boussinesq system in dimensionless form Jumps of interface for H < 2 Energy and work in a two-layer model Axisymmetric currents, SW formulation Governing equations A useful transformation The full behavior by numerical solution Dam-break stage Similarity solution The validity of the inviscid approximation Some comparisons Box models for axisymmetric geometry Fixed volume current with inertial-buoyancy balance Inflow volume change Effects of rotation Axisymmetric case Rotating channel Buoyancy decays: particle-driven, porous boundary, and entrainment Particle-driven currents Axisymmetric particle-driven current Extensions of particle-driven solutions Current over a porous bottom Axisymmetric current over a porous bottom Entrainment Non-Boussinesq systems Introduction Formulation Dam-break and initial slumping motion The transition and self-similar stages Summary Lubrication theory formulation for viscous currents 2D geometry Axisymmetric current Current in a porous medium II Stratified ambient currents and intrusions Continuous density transition Introduction The SW formulation SW results and comparisons with experiments and simulations Dam break Critical speed and nose-wave interaction Similarity solution The validity of the inviscid approximation Axisymmetric and rotating cases SW formulation SW and NS finite-difference results The validity of the inviscid approximation The steady-state current Steady-state flow pattern Results Comparisons and conclusions Intrusions in 2D geometry Introduction Two-layer stratification Linear transition layer Rectangular lock configurations Cylindrical lock in a fully linearly-stratified tank Similarity solution Non-symmetric intrusions Intrusions in axisymmetric geometry Introduction Two-layer stratification Fully linearly-stratified tank, part-depth locks Box models for 2D geometry Fixed volume and inertial-buoyancy balance S = 1, inflow volume change Box models for axisymmetric geometry Fixed volume and inertial-buoyancy balance S = 1, inflow volume change Lubrication theory for viscous currents with S = 1 2D geometry Axisymmetric geometry Energy Introduction 2D geometry Axisymmetric geometry SW equations: characteristics and finite-difference schemes Characteristics Numerical solution of the SW equations Navier-Stokes numerical simulations Formulation A finite-difference code Other codes Some useful formulas Leibniz's Theorem Vectors and coordinate systems

Buoyancy-dominated displacement flows in near-horizontal channels: the viscous limit

Abstract: We consider the viscous limit of a plane channel miscible displacement flow of two generalized Newtonian fluids when buoyancy is significant. The channel is inclined close to horizontal. A lubrication/thin-film approximation is used to simplify the governing equations and a semi-analytical solution is found for the flux functions. We show that there are no steady travelling wave solutions to the interface propagation equation. At short times the diffusive effects of the interface slope are dominant and there is a flow reversal, relative to the mean flow. We are able to find a short-time similarity solution governing this initial counter-current flow. At longer times the solution behaviour can be predicted from the associated hyperbolic problem (where diffusive effects are set to zero). Each solution consists of a number N ≥ 1 of steadily propagating fronts of differing speeds, joined together by segments of interface that are stretched between the fronts. Diffusive effects are always present in the propagating fronts. We explore the effects of viscosity ratio, inclinations and other rheological properties on the front height and front velocity. Depending on the competition of viscosity, buoyancy and other rheological effects, it is possible to have single or multiple fronts. More efficient displacements are generally obtained with a more viscous displacing fluid and modest improvements may also be gained with slight positive inclination in the direction of the density difference. Fluids that are considerably shear-thinning may be displaced at high efficiencies by more viscous fluids. Generally, a yield stress in the displacing fluid increases the displacement efficiency and yield stress in the displaced fluid decreases the displacement efficiency, eventually leading to completely static residual wall layers of displaced fluid. The maximal layer thickness of these static layers can be directly computed from a one-dimensional momentum balance and indicates the thickness of static layer found at long times.

The exponential decay of homogeneous turbulence

Abstract: Similarity equations of the spectral equations for decaying homogeneous turbulence are considered for which the similarity length scale is not allowed to grow. Two types of solutions are found: an inviscid solution and one involving viscosity. For the former, the energy decays asymptotically as t(-2), while for the latter the energy decays exponentially and the ratio of integral scale to Taylor microscale is constant. For both the spectra for fixed initial conditions collapse during decay with simply the energy and a single length scale. The exponentially decaying solution appears to provide an excellent description of the turbulence generated in recent space-filling fractal grid experiments.

Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic modelwith large data and vacuum

Abstract: In this paper, a one-dimensional bipolar hydrodynamic model is considered. This system takes the form of Euler-Poisson with electric field and frictional damping added to the momentum equations. The large time behavior of L ∞ entropy solutions of the bipolar hydrodynamic model is firstly studied. Previous works on this topic are mainly concerned with the smooth solution in which no vacuum occurs and the initial data is small. It is proved in this paper that any bounded entropy solution strongly converges to the similarity solution of the porous media equation or the heat equation in L 2(R) with time decay rate. The initial data can contain vacuum and can be arbitrarily large. The method is also applied to improve the convergence rate of [F.Huang, R.Pan, Arch. Rational Mech. Anal.,166(2003),359-376] for compressible Euler equations with damping. As a by product, it is shown that the bounded L ∞ entropy solution of the bipolar hydrodynamic model converges to the entropy solution of Euler equations with damping as $t\rightarrow\infty$.

Boundary-Layer Similarity Under an Axisymmetric, Gradient Wind Vortex

Abstract: We present similarity solutions for the mean boundary-layer profiles under an axisymmetric vortex that is in gradient wind balance; the similarity model includes the nonlinear momentum advection and curvature terms. These solutions are a generalization of the Ekman layer mean flow, which is the canonical boundary-layer basic state under a uniform, geostrophically-balanced flow. Near-surface properties such as inflow angle, surface wind factor, diffusive transport of kinetic energy into the surface layer and dissipational heating are calculated and shown to be sensitive to the choice of turbulence parameterization.

A similarity solution for a dual moving boundary problem associated with a coastal-plain depositional system

Abstract: Assuming that the sediment flux in the Exner equation can be linearly related to the local bed slope, we establish a one-dimensional model for the bed-load transport of sediment in a coastal-plain depositional system, such as a delta and a continental margin. The domain of this model is defined by two moving boundaries: the shoreline and the alluvial–bedrock transition. These boundaries represent fundamental transitions in surface morphology and sediment transport regime, and their trajectories in time and space define the evolution of the shape of the sedimentary prism. Under the assumptions of fixed bedrock slope and sea level the model admits a closed-form similarity solution for the movements of these boundaries. A mapping of the solution space, relevant to field scales, shows two domains controlled by the relative slopes of the bedrock and fluvial surface: one in which changes in environmental parameters are mainly recorded in the upstream boundary and another in which these changes are mainly recorded in the shoreline. We also find good agreement between the analytical solution and laboratory flume experiments for the movements of the alluvial–bedrock transition and the shoreline.

Mixed convection flow of a micropolar fluid near a non‐orthogonal stagnation‐point on a stretching vertical sheet

Abstract: Purpose – The purpose of this paper is to study theoretically the steady two‐dimensional mixed convection flow of a micropolar fluid impinging obliquely on a stretching vertical sheet. The flow consists of a stagnation‐point flow and a uniform shear flow parallel to the surface of the sheet. The sheet is stretching with a velocity proportional to the distance from the stagnation point while the surface temperature is assumed to vary linearly. The paper attempts also to show that a similarity solution of this problem can be obtained.Design/methodology/approach – Using a similarity transformation, the basic partial differential equations are first reduced to ordinary differential equations which are then solved numerically using the Keller box method for some values of the governing parameters. Both assisting and opposing flows are considered. The results are also obtained for both strong and weak concentration cases.Findings – These results provide information about the effect of a/c (ratio of the stagnati...

Magnetohydrodynamic oblique stagnation-point flow

Abstract: Laminar two-dimensional stagnation flow of a viscous and electrically conducting fluid obliquely impinging on a flat plate in the presence of a uniform applied magnetic field is formulated as a similarity solution of the Navier-Stokes equations. The relative importance of this flow is measured by the dimensionless strain rate and magnetohydrodynamic parameters γ and M. The viscous problem is reduced to a coupled pair of ordinary differential equations governed by γ and M. It is found that the parameter M causes a shift in the position of the point of zero skin friction along the wall.

Nonaxisymmetric Three-Dimensional Stagnation-Point Flow and Heat Transfer on a Flat Plate

Abstract: The existing solutions of Navier―Stokes and energy equations in the literature regarding the three-dimensional problem of stagnation-point flow either on a flat plate or on a cylinder are only for the case of axisymmetric formulation. The only exception is the study of three-dimensional stagnation-point flow on a flat plate by Howarth (1951, "The Boundary Layer in Three-Dimensional Flow―Part II: The Flow Near Stagnation Point, " Philos. Mag., 42, pp. 1433―1440), which is based on boundary layer theory approximation and zero pressure assumption in direction of normal to the surface. In our study the nonaxisymmetric three-dimensional steady viscous stagnation-point flow and heat transfer in the vicinity of a flat plate are investigated based on potential flow theory, which is the most general solution. An external fluid, along z-direction, with strain rate a impinges on this flat plate and produces a two-dimensional flow with different components of velocity on the plate. This situation may happen if the flow pattern on the plate is bounded from both sides in one of the directions, for example x-axis, because of any physical limitation. A similarity solution of the Navier―Stokes equations and energy equation is presented in this problem. A reduction in these equations is obtained by the use of appropriate similarity transformations. Velocity profiles and surface stress-tensors and temperature profiles along with pressure profile are presented for different values of velocity ratios, and Prandtl number.

Analytical solution of a two-dimensional stagnation flow in the vicinity of a shrinking sheet by means of the homotopy analysis method:

Abstract: In this article, stagnation flow in the vicinity of a shrinking sheet is studied. A similarity transformation is employed to reduce the Navier—Stokes equations to a set of non-linear ordinary differential equations. These equations are then solved analytically by means of the homotopy analysis method (HAM). The results obtained were shown to compare well with the numerical results available in the literature for the same problem. Close agreement between the two sets of results indicates the accuracy of the HAM. The method can predict the flow field in all vertical distances from the sheet, and is also able to control the convergence of the solution. The numerical solution of the similarity equations is also developed and the results are in good agreement with the analytical results based on the HAM.

A reliable treatment of the homotopy analysis method for viscous flow over a non-linearly stretching sheet in presence of a chemical reaction and under influence of a magnetic field

Abstract: The similarity solution for the steady two-dimensional flow of an incompressible viscous and electrically conducting fluid over a non-linearly semi-infinite stretching sheet in the presence of a chemical reaction and under the influence of a magnetic field gives a system of non-linear ordinary differential equations. These non-linear differential equations are analytically solved by applying a newly developed method, namely the Homotopy Analysis Method (HAM). The analytic solutions of the system of non-linear differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. Graphical results are presented to investigate the influence of the Schmidt number, magnetic parameter and chemical reaction parameter on the velocity and concentration fields. It is noted that the behavior of the HAM solution for concentration profiles is in good agreement with the numerical solution given in reference [A. Raptis, C. Perdikis, Int. J. Nonlinear Mech. 41, 527 (2006)].

Similarity solution of boundary layer flows for non-Newtonian fluids

Abstract: The present paper deals with the analysis of similarity solutions of the two-dimensional boundary layer flow of a power-law non-Newtonian fluid past a semi-infinite flat plate. The boundary value problem of the momentum equation is converted into an initial value problem using a Töpfer-like transformation. The dimensionless wall gradient is determined numerically. The existence of the power series solutions for the problem is presented and the convergence radius of the proposed solutions is estimated.

The effect of sudden source buoyancy flux increases on turbulent plumes

Abstract: Building upon the recent experimentally verified modelling of turbulent plumes which are subject to decreases in their source strength (Scase et al., J. Fluid Mech., vol. 563, 2006b, p. 443), we consider the complementary case where the plume's source strength is increased. We consider the effect of increasing the source strength of an established plume and we also compare time-dependent plume model predictions for the behaviour of a starting plume to those of Turner (J. Fluid Mech., vol. 13, 1962, p. 356). Unlike the decreasing source strength problems considered previously, the relevant solution to the time-dependent plume equations is not a simple similarity solution. However, scaling laws are demonstrated which are shown to be applicable across a large number of orders of magnitude of source strength increase. It is shown that an established plume that is subjected to an increase in its source strength supports a self-similar ‘pulse’ structure propagating upwards. For a point source plume, in pure plume balance, subjected to an increase in the source buoyancy flux F0, the rise height of this pulse in terms of time t scales as t3/4 while the vertical extent of the pulse scales as t1/4. The volume of the pulse is shown to scale as t9/4. For plumes in pure plume balance that emanate from a distributed source it is shown that the same scaling laws apply far from the source, demonstrating an analogous convergence to pure plume balance as that which is well known in steady plumes. These scaling law predictions are compared to implicit large eddy simulations of the buoyancy increase problem and are shown to be in good agreement. We also compare the predictions of the time-dependent model to a starting plume in the limit where the source buoyancy flux is discontinuously increased from zero. The conventional model for a starting plume is well approximated by a rising turbulent, entraining, buoyant vortex ring which is fed from below by a ‘steady’ plume. However, the time-dependent plume equations have been defined for top-hat profiles assuming only horizontal entrainment. Therefore, this system cannot model either the internal dynamics of the starting plume's head or the extra entrainment of ambient fluid into the head due to the turbulent boundary of the vortex ring-like cap. We show that the lack of entrainment of ambient fluid through the head of the starting plume means that the time-dependent plume equations overestimate the rise height of a starting plume with time. However, by modifying the entrainment coefficient appropriately, we see that realistic predictions consistent with experiment can be attained.

Flow and Heat Transfer Over a Stretched Microsurface

Abstract: The convection heat transfer induced by a stretching flat plate has been studied. Similarity conditions are obtained for the boundary layer equations for a flat plate subjected to a power law temperature and velocity variations. It is found that a similarity solution exists only for a linearly stretching plate and only when the plate is isothermal. The analysis shows that three parameters control the flow and heat transfer characteristics of the problem. These parameters are the velocity slip parameter K 1 , the temperature slip parameter K 2 , and the Prandtl number. The effect of these parameters on the flow and heat transfer of the problem has been studied and presented. It is found that the slip velocity parameter affect both the flow and heat transfer characteristics of the problem. It is found that the skin friction coefficient decreases with increasing K 1 and most of the changes in the skin friction takes place in the range 0

Time dependence of advection-dominated accretion flow with a toroidal magnetic field

Abstract: The present study examines the self-similar evolution of advection-dominated accretion flow (ADAF) in the presence of a toroidal magnetic field. In this research, it was assumed that angular momentum transport is due to viscous turbulence and the α-prescription was used for the kinematic coefficient of viscosity. The flow does not have a good cooling efficiency and so a fraction of energy accretes along with matter on to the central object. The effect of a toroidal magnetic field on such a system with regard to the dynamical behaviour was investigated. In order to solve the integrated equations that govern the dynamical behaviour of the accretion flow, a self-similar solution was used. The solution provides some insights into the dynamics of quasi-spherical accretion flow, and avoids many of the strictures of steady self-similar solutions. The solutions show that the behaviour of the physical quantities in a dynamical ADAF is different from that for a steady accretion flow or a disc using a polytropic approach. The effect of the toroidal magnetic field is considered using additional variable β (= P mag /P gas , where P mag and P gas are the magnetic and gas pressure, respectively). Also, to consider the effect of advection in such systems, the advection parameter f, which stands for the fraction of energy that accretes by matter on to the central object, was introduced. The solution indicates a transonic point in the accretion flow for all selected values of f and β. Also, by increasing the strength of the magnetic field and the degree of advection, the radial thickness of the disc decreases and the disc compresses. The model implies that the flow has differential rotation and is sub-Keplerian at small radii and super-Keplerian at large radii, and that different results were obtained using a polytropic accretion flow. The β parameter obtained was a function of position, and increases with increasing radii. Also, the behaviour of ADAF in a large toroidal magnetic field implies that different results are obtained using steady self-similar models in large magnetic fields.