Numerical Investigation on Marginal Stability and Convection with and without Magnetic Field in a Mushy Layer

In this article, an investigation is conducted to analyze the marginal stability with and without magnetic field in a mushy layer. During alloy solidification, such mushy layer, which is adjacent to the solidification front and composed of solid dendrites and liquid, is known to produce vertical chimneys. Here, we carry out numerical investigation for particular range of parameter values, which cover those of available experimental studies, to determine the convective flow at the onset of motion. The governing coupled non-linear partial differential equations are non-dimensionalised and solved to get the steady basic-state solution. The thickness of the mushy layer is determined as a part of the solution. Using multiple shooting technique, we determine the steady-state solutions in a range of critical Rayleigh number. We analyse the effect of several parameters, Chandrasekhar number Q, and Robert’s number τ on the problem. It was found that an increase in Q has a stabilizing effect on solidification and the critical Rayleigh number increases on increasing Q. It was also found that for moderate or small values of Robert’s number τ the critical Rayleigh number is mostly insensitive.     

Structure of a bathtub vortex: importance of the bottom boundary layer

A bathtub vortex in a cylindrical tank rotating at a constant angular velocity Ω is studied by means of a laboratory experiment, a numerical experiment and a boundary layer theory. The laboratory and numerical experiments show that two regimes of vortices in the steady-state can occur depending on Ω and the volume flux Q through the drain hole: when Q is large and Ω is small, a potential vortex is formed in which angular momentum outside the vortex core is constant in the non-rotating frame. However, when Q is small or Ω is large, a vortex is generated in which the angular momentum decreases with decreasing radius. Boundary layer theory shows that the vortex regimes strongly depend on the theoretical radial volume flux through the bottom boundary layer under a potential vortex : when the ratio of Q to the theoretical boundary-layer radial volume flux Q _b (scaled by ${2\pi R^2 ( \Omega \nu )^\frac{1}{2}}$ ) at the outer rim of the vortex core is larger than a critical value (of order 1), the radial flow in the interior exists at all radii and Regime I is realized, where R is the inner radius of the tank and ν the kinematic viscosity. When the ratio is less than the critical value, the radial flow in the interior nearly vanishes inside a critical radius and almost all of the radial volume flux occurs only in the boundary layer, resulting in Regime II in which the angular momentum is not constant with radius. This criterion is found to explain the results of the laboratory and numerical experiments very well.     

Boundary element solution of unsteady magnetohydrodynamic duct flow with differential quadrature time integration scheme

A numerical scheme which is a combination of the dual reciprocity boundary element method (DRBEM) and the differential quadrature method (DQM), is proposed for the solution of unsteady magnetohydrodynamic (MHD) flow problem in a rectangular duct with insulating walls. The coupled MHD equations in velocity and induced magnetic field are transformed first into the decoupled time‐dependent convection–diffusion‐type equations. These equations are solved by using DRBEM which treats the time and the space derivatives as nonhomogeneity and then by using DQM for the resulting system of initial value problems. The resulting linear system of equations is overdetermined due to the imposition of both boundary and initial conditions. Employing the least square method to this system the solution is obtained directly at any time level without the need of step‐by‐step computation with respect to time. Computations have been carried out for moderate values of Hartmann number (M?50) at transient and the steady‐state levels. As M increases boundary layers are formed for both the velocity and the induced magnetic field and the velocity becomes uniform at the centre of the duct. Also, the higher the value of M is the smaller the value of time for reaching steady‐state solution. Copyright © 2005 John Wiley & Sons, Ltd.     

Using a piecewise linear bottom to fit the bed variation in a laterally averaged,z‐co‐ordinate hydrodynamic model

In developing a 3D or laterally averaged 2D model for free‐surface flows using the finite difference method, the water depth is generally discretized either with the z‐co‐ordinate (z‐levels) or a transformed co‐ordinate (e.g. the so‐called σ‐co‐ordinate or σ‐levels). In a z‐level model, the water depth is discretized without any transformation, while in a σ‐level model, the water depth is discretized after a so‐called σ‐transformation that converts the water column to a unit, so that the free surface will be 0 (or 1) and the bottom will be ‐1 (or 0) in the stretched co‐ordinate system. Both discretization methods have their own advantages and drawbacks. It is generally not conclusive that one discretization method always works better than the other. The biggest problem for the z‐level model normally stems from the fact that it cannot fit the topography properly, while a σ‐level model does not have this kind of a topography‐fitting problem. To solve the topography‐fitting problem in a laterally averaged, 2D model using z‐levels, a piecewise linear bottom is proposed in this paper. Since the resulting computational cells are not necessarily rectangular looking at the x–z plane, flux‐based finite difference equations are used in the model to solve the governing equations. In addition to the piecewise linear bottom, the model can also be run with full cells or partial cells (both full cell and partial cell options yield a staircase bottom that does not fit the real bottom topography). Two frictionless wave cases were chosen to evaluate the responses of the model to different treatments of the topography. One wave case is a boundary value problem, while the other is an initial value problem. To verify that the piecewise linear bottom does not cause increased diffusions for areas with steep bottom slopes, a barotropic case in a symmetric triangular basin was tested. The model was also applied to a real estuary using various topography treatments. The model results demonstrate that fitting the topography is important for the initial value problem. For the boundary value problem, topography‐fitting may not be very critical if the vertical spacing is appropriate. Copyright © 2004 John Wiley & Sons, Ltd.     

An asymptotic iteration method for the numerical analysis of near-critical free-surface flows

Satisfying the boundary conditions at the free surface may impose severe difficulties to the computation of turbulent open-channel flows with finite-volume or finite-element methods, in particular, when the flow conditions are nearly critical. It is proposed to apply an iteration procedure that is based on an asymptotic expansion for large Reynolds numbers and Froude numbers close to the critical value 1.The iteration procedure starts by prescribing a first approximation for the free surface as it is obtained from solving an ODE that has been derived previously by means of an asymptotic expansion (Grillhofer and Schneider, 2003). The numerical solution of the full equations of motion then gives a surface pressure distribution that differs from the constant value required by the dynamic boundary condition. To determine a correction to the elevation of the free surface we next solve an ODE that is obtained from the asymptotic analysis of the flow with a prescribed pressure disturbance at the free surface. The full equations of motion are then solved for the corrected surface, and the procedure is repeated until criteria of accuracy for surface elevation and surface pressure, respectively, are satisfied.The method is applied to an undular hydraulic jump as a test case.     

A generating function for the yield criterion of isotropic and anisotropic polycrystalline materials

Summary A yield criterion for elastic pure-plastic polycrystalline materials is generated under simplified conditions by assuming that for yielding a certain fraction Q _c of the total number of slip planes in the material has to be active. This fraction Q _c is called the critical active quantity. We suppose Q _c to be independent of the state of stress. The yield criterion is mathematically expressed as an integral, which is a function of Q _c. This criterion can also be used for anisotropic materials.For isotropic materials the ratio (r) of the yield stress in torsion to that in tension is calculated as a function of Q _c. We find 0.5r0.61.The value r=0.5 (Tresca's criterion) is obtained for Q _c=0 and Q _c=1. The value r=0.577 (von Mises criterion) is obtained for Q _c=0.34 and Q _c=0.79. The difference between two criteria with the same r is the magnitude of the yield stress. We think the value Q _c=0.79 corresponds to the experiments for f.c.c. materials, since a rough estimation gives Q _c>0.75 for yielding.The independence of Q _c on the state of stress brings on that r>0.5 is more probable. This is caused by the slower increase to Q _c in torsion compared with the case of tension.From the theory follows that in the general case (Q _c0) the middle principal stress has influence on yielding.In this paper we don't determine Q _c, but adapt its value to the experimental results. However, a rough estimation of Q _c is given for isotropic materials.     

A boundary element formulation using the simple method

A new boundary element method is described for calculation of the steady incompressible laminar flows. The method is based on the well-known SIMPLE algorithm. The new boundary element method allows one to find the fields of the pressure and velocity corrections without inner iterations, thus reducing the computational time drastically. This makes it different from the method developed by Patankar and Spalding.³² However, the new method demands a much larger computer strorage. The boundary integral equations are discretized with the help of constant boundary elements and constant cells. The values of the integrals along the boundary elements and the cells for the two-dimensional domain are found analytically. To preserve the stability in the iteration process, under-relaxation for the convection terms is used. This paper gives the results of calculations of the flows between two plane parallel plates at Re = 20 and Re = 200, the flows in a square cavity with a moving upper lid at Re = 1 and Re = 100 and the flow in a plane channel with sudden symmetric expansion at Re =46·6.     

NUMERICAL STUDY OF EFFECTS OF PULSATILE AMPLITUDE ON UNSTEADY LAMINAR FLOWS IN RIGID PIPE WITH RING-TYPE CONSTRICTIONS

The effects of pulsatile amplitude on sinusoidal laminar flows through a rigid pipe with sharp-edged ring-type constrictions have been studied numerically. The parameters considered are: mean Reynolds number (Re) of the order of 100; Strouhal number (St) in the range 0·0–3·98; Womersley number (Nw) in the range 0·0–50·0. The pulsatile amplitude (A) varies in the range 0·0–2·0. The flow characteristics were studied through the pulsatile contours of streamline, vorticity, shear stress and isobars. Within a pulsatile cycle the relations between instantaneous flow rate (Q) and instantaneous pressure gradient (dp/dz) are observed to be elliptic. The relations between instantaneous flow rate (Q) and pressure loss (P_loss) are quadratic. Linear relations exist between instantaneous flow rate (Q) and maximum velocity, maximum vorticity and maximum shear stress. © by 1997 John Wiley & Sons, Ltd.     

Temperature distribution in a Newtonian fluid injected between two semi-infinite plates

The analysis of the temperature distribution in time and place of a hot heat-conducting Newtonian fluid injected between two cooled parallel plates is presented. The 2-dimensional flow has a free flow front moving with constant velocity. The kernel of the fluid remains almost at the inlet temperature, but at the walls boundary layers occur with steeply descending temperature. The inner solutions inside these boundary layers are determined. To this end, the total region is divided into three distinct regions: the region G_I far behind the flow front, the flow front region G_II, and the intermediate region G_III between G_I and G_II. The asymptotics owing to each region are presented. The fundamental small parameter here is the thickness-to-length ratio of the 2-dimensional flow region. In most of the cases, similarity solutions are found. In the flow front region, for the formulation of the inner solution a Wiener–Hopf technique is used. Via matching procedures, the separate boundary layers are linked to each other to form one global boundary layer for the whole front region. All calculations in this paper are performed by analytical means, and all results are in analytical form. Comparison of our results with numerical solutions shows perfect agreement.     

Convection development in a liquid layer with a free boundary

In certain calculations of the critical Rayleigh number for a liquid layer with free boundary which is heated from below, the linearization method has been used and it has been assumed that the temperature perturbations disappear at the undisturbed free boundary.Proper linearization shows that the temperature perturbation is proportional to the free surface perturbation, and the latter is proportional to the normal stress perturbation with the proportionality factor F=²/gh³ (g is the free-fall acceleration, is the kinematic viscosity, h is the liquid layer thickness). In §1 we present a formulation of the problem with account for the parameter F; in §2 we consider the linearized equations and the existence of a stability threshold is proved-a positive eigenvalue-and it is established that with an increase in the parameter F/P (P is the Prandtl number) the value of the critical Rayleigh number Ra_* decreases; §3 presents the results of a numerical calculation of Ra as a function of the parameter F/P.Convection development in a liquid layer with a free surface on which a given temperature is maintained was studied in [1, 2]. The value R_*=1100 found for the critical Rayleigh number agrees well with the experimental value. In the calculations made in [1, 2] the linearization method is used, and it is assumed that the temperature perturbations disappear at the undisturbed free boundary. Strictly speaking, this assumption is not correct.Correct linearization shows that the temperature perturbation is proportional to the perturbation of the free boundary, and the latter is proportional to the normal stress perturbation (see below (2.3)).The problem formulation is presented in §1; §2 deals with the linearized equations and the existence (Theorem 2.1) is demonstrated of a stability threshold—which is a simple positive eigenvalue; §3 presents the results of a numerical calculation of R_* as a function of the parameter =F/P.     

Iterative stabilization of the bilinear velocity–constant pressure element

Some finite element approximations of incompressible flows, such as those obtained with the bilinear velocity–constant pressure element (Q₁?P₀), are well known to be unstable in pressure while providing reasonable results for the velocity. We shall see that there exists a subspace of piecewise constant pressures that leads to a stable approximation. The main drawback associated with this subspace is the necessity of assembling groups of elements, the so-called ‘macro-elements’, which increases dramatically the bandwidth of the system. We study a variant of Uzawa's method which enables us to work in the desired subspace without increasing the bandwidth of the system. Numerical results show that this method is efficient and can be made to work at a low extra cost. The method can easily be generalized to other problems and is very attractive in three-dimensional cases.     

Uniqueness of free-boundary flows under gravity

A class of steady potential flows of an ideal fluid is considered in which the fluid flows between fixed boundaries and then emerges as a jet with one free boundary. Gravity acts on the fluid perpendicularly to the direction of the jet at infinity downstream. An inverse Froude number α is defined in terms of the flux Q and the depth d of the fluid at the separation point. It is proved that for each α>0 there is at most one flow which reaches to a supercritical uniform stream depth at infinity downstream. Monotonicity properties are proved for various flow parameters, and the behaviour of the flow as α → 0 is described.     

Computation of Hele-Shaw free boundary problems near obstacles

The time-dependent evolution of source-driven Hele-Shaw free boundary flows in the presence of an obstacle is computed numerically. The Baiocchi transformation is used to convert the Hele-Shaw Laplacian growth problem into a free boundary problem for a streamfunction-like variable u(x, y, t) governed by Poisson’s equation (with constant right-hand side) with the source becoming a point vortex of strength linearly dependent on time. On the free boundary, both u and its normal derivative vanish, and on the obstacle, the normal derivative of u vanishes. Interpreting u as a streamfunction, at a given time, the problem becomes that of finding a steady patch of uniform vorticity enclosing a point vortex of given strength such that the velocity vanishes on the free boundary and the tangential velocity vanishes on the obstacle. A combination of contour dynamics and Newton’s method is used to compute such equilibria. By varying the strength of the point vortex, these equilibria represent a sequence of source-driven growing blobs of fluid in a Hele-Shaw cell. The practicality and accuracy of the method is demonstrated by computing the evolution of Hele-Shaw flow driven by a source near a plane wall; a case for which there is a known exact solution. Other obstacles for which there are no known exact solutions are also considered, including a source both inside and outside a circular boundary, a source near a finite-length plate and the interaction of an infinite free boundary impinging on a circular disc.     

On the role of the adjoint problem in dissipative,nonconservative problems of elastic stability

Summary The asymptotic stability of linear isotropic Kelvin-Voigt solids subjected to external damping and noncon-servative surface tractions which are linear functions of displacements and displacement gradients is considered. The boundary value problem that is adjoint to the original system is derived, and a variational principle from which these two boundary value problems may be generated is stated. The variational principle serves as a basis for an approximate method (similar to the Ritz method) for solving nonconservative stability problems in which the effects of internal and external damping are present. The method is applied to determine the value of the critical load intensity Q_cr in a cantilever and a clamped-simply supported beam subjected to a linearly distributed tangential load as well as to internal and external damping forces. Plots of the variation of Q_cr with the two damping parameters are given, and it is shown the nature of the boundary conditions can have a significant effect on the manner in which Q_cr varies as a function of the damping parameters.

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Sommario Si considera la stabilità asintotica di solidi isotropi e lineari di Kelvin e Voigt sollecitati da forze smorzanti proporzionali alla velocità e carichi superficiali non conservativi, i quali sono funzioni lineari di spostamenti e derivate di spostamenti. Si deduce il problema ai limiti che è aggiunto al sistema originario di equazioni di campo, e si imposta un principio variazionale dal quale si generano questi due problemi ai limiti. Il principio variazionale serve come base per un procedimento approssimato (simile al metodo di Ritz) per risolvere problemi non conservativi di stabilità elastica nei quali sono presenti gli effetti di smorzamento interno ed esterno. Si applica questo procedimento per constatare il valore critico del parametro dell'intensità del carico, Q_cr, in una mensola e una trave con un estremo incastrato e l'altro incernierato, caricate da una distribuzione continua di forze di modulo che varia linearmente con la coordinata di posizione e retta d'azione solidale con la deformata del l'asse della trave. Si mostrano grafici dell'andamento di Q_cr con i due parametri di smorzamento, e si nota che il tipo di vincolo alle estremità della trave può avere un effetto significante sulla maniera in cui Q_cr varia come una funzione dei parametri di smorzamento.

     

Diffusion of macromolecular solutions in the turbulent boundary layer of a cylindrical pipe

This paper presents a model for the evolution of a transversal concentration profile of a macromolecular solution (PEO) injected into a cylindrical pipe at turbulent flow conditions (R 40000). This model, based on the diffusion of a scalar quantity emitted by two diametrically opposed point sinks, proves to be in good agreement with the experimental data. C concentration - C _w wall concentration - C _i initial concentration before injection - x downstream distance from the slot - y normal distance from the wall - characteristic height of diffusion, i.e. the value ofg at whichC/C _w = 0.5 - n characteristic exponent - R radius of pipe - D diameter of pipe - a, b constants - L _m mixing length, i.e. the value ofx at whichC _w /C _i =Q _i /Q _T - Q _i flow rate of injection - Q _T flow rate - f, g annex functions - n ₀ maximal value ofn     

Effect of inertia on thermoelastic flow instability

Thermal effects induced by viscous heating cause thermoelastic flow instabilities in curvilinear shear flows of viscoelastic polymer solutions. These instabilities could be tracked experimentally by changing the fluid temperature T₀ to span the parameter space. In this work, the influence of T₀ on the stability boundary of the Taylor–Couette flow of an Oldroyd-B fluid is studied. The upper bound of the stability boundary in the Weissenberg number (We)–Nahme number (Na) space is given by the critical conditions corresponding to the extension of the time-dependent isothermal eigensolution. Initially, as T₀ is increased, the critical Weissenberg number, We_c, associated with this upper branch increases. Increasing T₀ beyond a certain value T^* causes the thermoelastic mode of instability to manifest. This occurs in the limit as We/Pe → 0, where Pe denotes the Péclet number. In this limit, the fluid relaxation time is much smaller than the time scale of thermal diffusion. T₀ = T^* represents a turning point in the We_c–Na_c curve. Consequently, the stability boundary is multi-valued for a wide range of Na values. Since the relaxation time and viscosity of the fluid decrease with increasing T₀, the elasticity number, defined as the ratio of the fluid relaxation time to the time scale of viscous diffusion, also decreases. Hence, O(10) values of the Reynolds number could be realized at the onset of instability if T₀ is sufficiently large. This sets limits for the temperature range that can be used in experiments if inertial effects are to be minimized.     

The unsteady heat transfer due to a heat source in an MHD stretching sheet flow

The time evolution in the temperature field resulting from the sudden introduction of a heat source into the already fully established steady MHD flow of an electrically conducting fluid past a linearly stretching isothermal surface is considered. The problem is shown to be fully described by two dimensionless parameters, a modified magnetic field strength ?? and a heat source strength Q. Numerical solutions of the initial-value problem show that there is a critical value Q _c of the parameter Q, dependent on ??, such that, for Q<Q _c , the solution approaches a steady state at large times and, for Q>Q _c , the solutions grows exponentially large as time increases. This growth rate is determined through an eigenvalue problem which also determines the critical value Q _c . The limits of Q _c for both small and large values of ?? are discussed.     

Numerical simulation of turbulent free surface flow with two‐equation k–ε eddy‐viscosity models

This paper presents a finite difference technique for solving incompressible turbulent free surface fluid flow problems. The closure of the time‐averaged Navier–Stokes equations is achieved by using the two‐equation eddy‐viscosity model: the high‐Reynolds k–ε (standard) model, with a time scale proposed by Durbin; and a low‐Reynolds number form of the standard k–ε model, similar to that proposed by Yang and Shih. In order to achieve an accurate discretization of the non‐linear terms, a second/third‐order upwinding technique is adopted. The computational method is validated by applying it to the flat plate boundary layer problem and to impinging jet flows. The method is then applied to a turbulent planar jet flow beneath and parallel to a free surface. Computations show that the high‐Reynolds k–ε model yields favourable predictions both of the zero‐pressure‐gradient turbulent boundary layer on a flat plate and jet impingement flows. However, the results using the low‐Reynolds number form of the k–ε model are somewhat unsatisfactory. Copyright © 2004 John Wiley & Sons, Ltd.     

Injection of ideal fluid from a slot into a free stream

A cold gas is injected from a slot into a free stream of hot gas. In a simple model this leads to a two-fluid free boundary problem with the jump relation |u^-|²–|u⁺|² = ( constant) on the free boundary {u=0}, where u is the stream function. We prove that for any (–1, ) there exists a unique solution (Q, u) where Q is the flux of the injected fluid. Various properties of the solution u and of the free boundary are established.     

A new outflow boundary condition

Boundary conditions come from Nature. Therefore these conditions exist at natural boundaries. Often, owing to limitations in computing power and means, large domains are truncated and confined between artificial synthetic boundaries. Then the required boundary conditions there cannot be provided naturally and there is a need to fabricate them by intuition, experience, asymptotic behaviour and numerical experimentation. In this work several kinds of outflow boundary conditions, including essential, natural and free boundar conditions, are evaluated for two flow and heat transfer model problems. A new outflow boundary condition, called hereafter the free boundary condition, is introduced and tested. This free boundary condition is equivalent to extending the validity of the weak form of the governing equations to the synthetic outflow instead of replacing them there with unknown essential or natural boundary conditions. In the limit of zero Reynolds number the free boundary condition minimizes the energy functional among all possible choices of outflow boundary conditions. A review of results from applications of the same boundary conditions to several other flow situations is also presented and discussed.