Numerical simulation of the horizontal Bridgman growth. Part III: Calculation of the interface

We study the transient motion of the solidification front during the growth of semiconductor crystals in the horizontal Bridgman geometry. The calculation is based on a two-dimensional flow. We use finite elements which deform with the motion of the interface. The energy equation is coupled with the isothermal constraint of the interface in an implicit transient algorithm. Several examples show the oscillatory motion of the interface caused by the periodic flow of the melt, and they reveal the importance of the growth rate on the shape of the interface.     

Numerical simulation of the horizontal Bridgman growth. Part I: Two-dimensional flow

We study the generation of periodic velocity and temperature fields in a plane horizontal crucible of molten metal under the action of a horizontal temperature gradient. The geometry and the boundary conditions are similar to those encountered in the Bridgman growth process of semiconductor crystals, although the present paper is limited to two-dimensional flows. We use transient finite difference and finite element algorithms which lead to identical results. We demonstrate the oscillatory mechanism in two different geometries.     

Three-dimensional disturbances of a plane-parallel two-layer flow of a viscous, heat-conducting fluid

The stability of the Marangoni-Poiseuille flow in an inclined channel against three-dimensional disturbances is studied. It is shown that the nature of instability is determined by wall heating conditions and the system orientation with respect to its horizontal position. With variation in the angle of inclination of the system spiral disturbances leading to the system crisis can develop.     

用LDA测量U形水槽内的振荡流

本文介绍LDA用于测量U形水槽内的振荡流。水槽直管段为正方形,弯管段内侧弯面及外侧弯面是专门计算设计的。对于水平直管段、竖管段及弯管段共七个截面,测量了边界层外振荡流速的振幅分布。对于水平直管段上底面及下底面各三个截面,测量了边界层內振荡流速的振幅分布及相位分布。测量结果和无限长槽道的理论计算结果作了比较,二者基本符合。     

Transonic Rotor/Stator Interaction Using a 3-D Linearised Unsteady Viscous Flow Model

The paper first summarizes the forced response problem in turbomachinery and reviews various numerical methods for the simulation of unsteady flows. A particular technique, based on the linearisation of the unsteady Favre-averaged Navier-Stokes equations on three-dimensional mixed-element grids of tetrahedra, hexahedra and wedges, is described in some detail. The methodology was applied to a NGV/rotor interaction benchmark case for which detailed steady and unsteady flow measurements are available. The steady-state flow, calculated using a non-linear viscous representation, was described in detail with emphasis on features such as separation, horseshoe and passage vortices, tip leakage and shock structure since these are likely to influence the unsteady flow. The sources of unsteadiness on the rotor passage were evaluated from the steady-state solution at the NGV outlet. The disturbances were split into vortical, entropic and potential waves, the Fourier components of which were considered separately. The summation of the vortical and entropic waves was used as a rotor inlet boundary condition in order to assess the wake/rotor unsteady interaction. Similarly, potential waves were used to study the potential/rotor interaction. The results obtained from these two types of unsteady interactions were superimposed and compared with experimental data. Good qualitative and, in most cases, quantitative agreement was obtained, a finding which suggests that the unsteady flowfield generated by the relative blade motion can be considered to be a quasi-linear phenomenon for the particular HP turbine studied. Finally, the mechanisms of wake/rotor and potential/rotor interactions were studied in some detail and it was concluded that the former was strong in the crown of the blade while the latter was dominant in the leading edge region.     

Three-dimensional disturbances of planar viscometric flow

This paper examines three-dimensional disturbances of a plane steady shear flow of simple fluids with short memory. Under the assumption of nearly-viscometric flow, constitutive equations are derived and then a general form of the Reynolds-Orr energy equation is obtained. With the aid of this derived energy formula, sufficient conditions are generated for the stability of three-dimensional disturbances of the planar viscometric flow. These conditions are analysed and a comparison is made with the corresponding two-dimensional stability problem. There is a strong indication that the basic flow is less stable against three-dimensional disturbances than against two-dimensional ones.     

Secondary instability modes generated by a Tollmien-Schlichting wave scattering from a bump

The plane finite-amplitude Tollmien-Schlichting wave interaction with a three-dimensional bump on a wall is considered for plane channel flow. The scattering of this wave leads to the production of unsteady three-dimensional disturbances which transform into growing secondary instability modes. The generation of such modes is studied assuming the three-dimensional disturbances to be small in comparison with the primary plane instability wave. The solution predicts that secondary disturbance amplification takes place only within a narrow wedge downstream of the bump. The qualitative comparison of results with experiments on turbulent wedge origination at an isolated roughness in a boundary layer is presented.     

Linear evolution of controlled disturbances in the supersonic boundary layer on a swept wing

The linear development of controlled disturbances in the three-dimensional supersonic boundary layer on a swept model wing with a sharp leading edge is experimentally investigated at the Mach number 2. The spatial-temporal and spectral-wave characteristics of the wave train of unstable disturbances are obtained. The asymmetry of these characteristics, due to the secondary flow in the three-dimensional boundary layer, is confirmed.     

Acoustic Field Effect on Laminar Turbulent Transition on a Swept Wing in the Favorable Pressure Gradient Region

The development of disturbances in a three-dimensional boundary layer on a swept wing model is studied both under natural conditions and for artificial excitation of traveling waves by an acoustic field. It is found that steady-state streamwise structures are formed in the three-dimensional boundary layer; under natural conditions a wave packet leading to turbulence is detected. When the flow is exposed to the action of an acoustic field at a frequency from the wave packet, disturbances whose velocity along the streamwise structures is equal to 0.55 of the oncoming flow velocity are formed, while the laminar-turbulent transition is displaced upstream.     

Development of three-dimensional disturbances in a boundary layer

In the region of transition from a two-dimensional laminar boundary layer to a turbulent one, three-dimensional flow occurs [1–3]. It has been proposed that this flow is formed as the result of nonlinear interaction of two-dimensional and three-dimensional disturbances predicted by linear hydrodynamic stability theory. Using many simplifications, [4, 5] performed a calculation of this interaction for a free boundary layer and a boundary layer on a wall with a very coarse approximation of the velocity profile. The results showed some argreement with experiment. On the other hand, it is known that disturbances of the Tollmin—Schlichting wave type can be observed at sufficiently high amplitude. This present study will use the method of successive linearization to calculate the primary two- and three-dimensional disturbances, and also the average secondary flow occurring because of nonlinear interaction of the primary disturbances. The method of calculation used is close to that of [4, 5], the disturbance parameters being calculated on the basis of a Blazius velocity profile. A detailed comparison of results with experimental data [1] is made. It developed that at large disturbance amplitude the amplitude growth rate differs from that of linear theory, while the spatial distribution of disturbances agree s well with the distribution given by the natural functions and their nonlinear interaction. In calculating the secondary flow an experimental correction was made to the amplitude growth rate.     

On a family of Stokes flows

A family of simple Stokes flows involving sliding surfaces adjacent to surfaces at rest is considered. Principally, two specific flow configurations are investigated: (i) that arising when parts of the boundary of an infinitely long circular cylinder are rotating about the axis while other parts of the boundary are at rest, and (ii) the flow produced when a cap of a sphere is held at rest while the remainder of the sphere rotates about the symmetry axis. In each case computer plots of streamlines or constant velocity lines are presented to give a general impression of the resulting flow pattern.     

Generalized Wall Functions for Rough Walls Based on Data Assimilation

We present a systematic identification method for the development of equivalent boundary conditions for turbulent flow simulation over rough walls in engineering applications where the typical roughness involved is known for a given application. After parameterizing the roughness and sampling of the parameter space, numerical simulations or experiences are used to generate a database which is used to identify the coefficients of a polynomial reduced-order model. These laws are parameterized by an a priori length scale for which we propose a new a posteriori criterion.     

Particle imaging velocimetry of a wet aqueous foam with an underlying liquid film

 We investigate the utility of particle imaging velocimetry (PIV) for performing kinematic measurements in wet aqueous foam with a liquid film beneath it. The flow velocities are measured near the walls of a square cross-section horizontal duct. The flow velocities are useful for validating the rheological models. We show that there is a discrepancy between the velocity profiles in the wet foam and the Bingham plastic model of flow. The velocity measurements reveal a more complex flow pattern, which may be analysed following three different regimes: a plug flow, a shear flow in a vertical plane and a three-dimensional shear flow. The transition between the plug flow and the shear flows may be explained by a shear-induced migration of bubbles. Received: 25 April 2000 / Accepted: 26 February 2001     

Three-dimensional problem of exponentially stratified flow over bottom roughness in the case of finite depth

The three-dimensional problem of the flow of an exponentially stratified fluid of finite depth over bottom roughness is considered in the rigid roof approximation and in the presence of a free surface. In the rigid roof approximation the solution is obtained in the form of a Fourier series in the vertical Lagrangian coordinate, and the series coefficients are expressed in terms of single integrals outside a horizontal strip whose sides are parallel to the flow axis and tangential to the projection of the support of the function describing the bottom roughness. This makes it possible to investigate the near field in regions not considered in [1, 2]. The presence of a small parameter in the boundary condition at the free surface makes it possible to find, in the first approximation, the wave motions and nonwave disturbances at the free surface in the near and far fields. In the near field the width of the wave zone is of the order of the flow depth, expands with distance from the bottom and is broadest at the free surface. As distinct from the annular disturbances within the fluid, the pattern of the nonwave disturbances at the free surface depends on the polar angle. The law of similarity for the diverging waves at the free surface is also obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 101–111, May–June, 1990.The authors are grateful to É. V. Teodorovich for discussing the formulation of the problem.     

Dynamics of nonlinear three-dimensionalwaves on the interface between two fluids in a channel with low-sloping bottom and top

A combined approach is proposed to describe the transformation of three-dimensional disturbances of the interface between two incompressible immiscible fluids of different densities contained in a channel with fixed rigid top and bottom. It is assumed that the wavelengths are moderately large, the amplitudes are small but finite, the top and the bottom can be gently sloping, and capillary effects are small. The system of equations derived is applicable for modeling disturbances simultaneously scattering in arbitrary horizontal directions. Some typical wave problems are numerically solved and the effect of governing parameters is shown.     

Streamwise Algebraic Growth (and Breakdown) in Three-Dimensional Boundary Layers

We investigate the effect of flow perturbations on a broad class of three-dimensional boundary-layer flows applicable close to lines of symmetry. The base-flow class includes all the classical two-dimensional Falkner–Skan solution family, together with a three-dimensional solution family involving a crossflow velocity component. When considering perturbations to these flows (which take on the same crossflow form as the base states), linear, steady eigenstates which grow algebraically in the streamwise direction frequently exist (including in the case of Blasius flow). The natural question that arises is whether nonlinear effects can in turn provoke a transition process; this is the main focus of this paper. The methodology is fully rational, with, in particular, non-parallel effects being treated in an entirely proper manner. In the case of steady flow perturbations, two scenarios are found to occur in our numerical results: either the nonlinear response arising from the growth of linear eigenstates leads to a translation from one solution branch to another (one not possessing steady, linear, leading-edge eigenstates), or alternatively a nonlinear breakdown/singularity occurs at a finite downstream location, which may be interpreted physically as a burst of vorticity out of the boundary layer, linked to an early phase of the transition process. In the case of temporally periodic disturbances, these always lead to a breakdown; however, temporally impulsive-type disturbances lead to a wave-packet-like response downstream, which may or may not ultimately breakdown. Received 10 December 2001 and accepted 1 October 2002 Published online 5 February 2003 Communicated by P. Hall     

Direct numerical simulation of the laminar-turbulent transition in a thick spherical layer

The results of a numerical study of the laminar-turbulent transition in unsteady isothermal three-dimensional flows of viscous incompressible fluid in a thick spherical layer between counter-rotating spherical boundaries are presented. The calculations are performed for the governing parameters corresponding to the experimental data [1, 2]. The numerical investigations include both solving the complete system of Navier-Stokes equations and analyzing the linear stability of steady-state axisymmetric flows with respect to three-dimensional disturbances. A stochastic flow regime is calculated for the first time. The limits of existence of different flow regimes and the hysteresis regions are found. The spatial flow patterns and frequency characteristics are obtained, which makes it possible to extend and refine the existing experimental data.     

Associated disturbances around a vortex ring in a stratified fluid

A motion of a vortex ring in a stratified fluid is accompanied by associated disturbances which, in the schlieren visualization in the field of a horizontal density gradient, have the shape of a symmetric four-petal configuration. The criterion of the existence of the disturbances is the Froude number Fr based on the motion velocity and the vertical vortex size. On the range Fr > 1, the disturbances are stable with respect to the variation of themotion regime and the distortion of the vortex shape. For Fr < 1 the disturbances disappear. Computer processing of the schlieren photographs showed that the experimental spatial dependences of the disturbance amplitude are close to the functions describing the distribution of the vertical velocity component in the inviscid flow past a sphere.     

On Three-Dimensional Non-linear Buoyant Convection in Ternary Solidification

We consider the problem of three-dimensional non-linear buoyant convection in ternary solidification. Under the limit of large far-field temperature, the convective flow is modeled to be in a rectangular cube composing of a horizontal liquid layer above a primary mushy layer, which itself is over a secondary mushy layer. We first apply linear stability analysis to calculate the conditions at the onset of motion. Next, we carry out weakly non-linear analyses to determine solutions in the form of hexagons and their possible stability and to obtain information about tendency for chimney formation. We find that if the flow is driven either from both mushy layers with equal critical conditions at the onset of motion or only by the primary mushy layer, then the flow can be in the form of a double-cell structure vertically with down-hexagons below or above up-hexagons. There is tendency for vertically oriented chimney formation at different horizontal locations in each mushy layer. For the cases where only the critical conditions at the onset of motion are equal in both mushy layers and depending on the values of the mush Rayleigh numbers, the flow can be subcritical (or supercritical) in both mushy layers or mixed subcritical in one layer and supercritical in another layer.     

Propagation of weakly nonlinear disturbances of the interface between two layers of fluid

The dynamics of internal waves of small but finite amplitude in a two-layer fluid system bounded by rigid horizontal surfaces at bottom and top is investigated theoretically. For linear disturbances of the fluid interface the authors propose a polynomial approximation of the dispersion relation which has the same asymptotics as the exact formula in the limiting situations of very long and short waves. In the case of three-dimensional, weakly nonlinear disturbances of slowly varying shape (in the coordinate system moving with the wave) an equation like the wave equation is derived. This equation has Stokes solutions coinciding with the well-known results for infinitely deep layers. For fairly long disturbances solitary solutions of the model wave equation which fit the experimental data are determined. Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 125–131, January–February, 1994.