Cross-stream Diffusion Research Articles

Computation of incompressible and weakly-compressible viscoelastic liquids flow: finite element/volume schemes

In this study, we analyse viscoelastic numerical solution for an Oldroyd-B model under incompressible and weakly-compressible liquid flow conditions. We consider flow through a planar four-to-one contraction, as a standard benchmark, throughout a range of Weissenberg numbers up to critical levels. At the same time, inertial and creeping flow settings are also addressed. Within our scheme, we compare and contrast, two forms of stress discretisation, both embedded within a high-order pressure-correction time-marching formulation based on triangles. This encompasses a parent-cell finite element/SUPG scheme, with quadratic stress interpolation and recovery of velocity gradients. The second scheme involves a sub-cell finite volume implementation, a hybrid fe/fv-scheme for the full system. A new feature of this study is that both numerical configurations are able to accommodate incompressible, and low to vanishing Mach number compressible liquid flows. This is of some interest within industrial application areas. We are able to provide parity between the numerical solutions across schemes for any given flow setting. Close examination of flow patterns and vortex trends indicates the broad differences anticipated between incompressible and weakly-compressible solutions. Vortex reduction with increasing Weissenberg number is a common feature throughout. Compressible solutions provide larger vortices (salient and lip) than their incompressible counterparts, and larger stress patterns in the re-entrant corner neighbourhood. Inertia tends to reduce such phenomena in all instances. The hybrid fe/fv-scheme proves more robust, in that it captures the stress singularity more tightly than the fe-form at comparable Weissenberg numbers, reaching higher critical levels. The sub-cell structure, the handling of cross-stream numerical diffusion, and corner discontinuity capturing features of the hybrid fe/fv-scheme, are all perceived as attractive additional benefits that give preference to this choice of scheme.

Controlling Turbulence in a Rearward-Facing Step Combustor Using Countercurrent Shear

The present work describes the application of countercurrent shear flow control to the nonreacting flow in a novel step combustor. The countercurrent shear control employs a suction based approach, which induces counterflow through a gap at the sudden expansion plane. Peak turbulent fluctuation levels, cross-stream averaged turbulent kinetic energy, and cross-stream momentum diffusion increased with applied suction. The control downstream of the step operates via two mechanisms: enhanced global recirculation and near field control of the separated shear layer. The use of counterflow also enhances three dimensionality, a feature that is expected to be beneficial under burning conditions.

Mass-transport enhancement in regions bounded by rigid walls

The mass transport into a fluid bounded by stationary rigid walls in the limit of large Peclet number, Pe, is examined analytically. Two model systems are considered in detail: a stationary cavity and a model involving two concentric rotating cylinders. A macroscopic gradient is imposed between the top and bottom surfaces. It is demonstrated that mass transport into the fluid is enhanced owing to a recirculation zone which is connected to the solid boundary through a boundary layer of thickness O(Pe−1/3) in which cross-stream molecular diffusion is balanced by convection. The associated enhancement is large and scales as Pe1/3. Our asymptotic analysis is found to be in good agreement with numerical solutions of the full transport equation.

Flow structure and seasonalityin the Hebridean slope current

The intensity, structure and variability of the slope current have been determined from 16 months of observations with Acoustic Doppler Current Profilers (ADCP) and conventional current meters on a cross-slope section at the Hebridean shelf edge during the Shelf Edge Study (SES) programme. After removal of the tidal signals, the mean flow over the upper slope is found to be closely parallel to the topography with speeds of ≈ 20 cm·s –1. The flow extends down to a depth of 500 m and is predominantly barotropic, especially in winter when the flow is practically uniform between 350 m and the surface. In summer, there is a significant baroclinic component with a pronounced maximum in current at a depth of about 200 m but more than 80% of the kinetic energy is in the barotropic component. Flow in the core of the current is highly persistent with the Neumann’s steadiness St > 0.8 in summer. In winter the flow is generally more energetic and variable and extends onto the adjacent shelf. The cross-slope profile of sea surface elevation, computed from the mean barotropic currents, shows a consistent relation to seabed topography through the seasonal cycle. Long-term averages of the cross-slope components are generally small (≈ 2 cm·s –1) with some indication of persistent down-slope flow in the bottom Ekman layer. Measurements with shipboard ADCP on sections at intervals along the slope show a high degree of continuity in the structure of the flow. The core of the flow appears to be related to a weak positive salinity anomaly and a depression of the 9.5 °C isotherm near the shelf, but there is no strong correlation between the core of the slope-current and the core of the salinity anomaly. It is proposed that this may be due to differences in the cross-stream diffusion of salt and momentum which have different boundary conditions at the slope. The observed cross-stream structure of the current supports the hypothesis that JEBAR is the principal forcing mechanism but the result cannot be regarded as conclusive since a uniform potential vorticity model of the flow produces a similar cross-sectional structure of the current.

Energy constraints in forced recirculating MHD flows

We are concerned here with forced steady recirculating flows which are laminar, two-dimensional and have a high Reynolds number. The body force is considered to be prescribed and independent of the flow, a situation which arises frequently in magnetohydrodynamics. Such flows are subject to a strong constraint. Specifically, the body force generates kinetic energy throughout the flow field, yet dissipation is confined to narrow singular regions such as boundary layers. If the flow is to achieve a steady state, then the kinetic energy which is continually generated within the bulk of the flow must find its way to the dissipative regions. Now the distribution of u2/2 is governed by a transport equation, in which the only cross-stream transport of energy is diffusion, v∇2 (u2/2). It follows that there are only two possible candidates for the transport of energy to the dissipative regions: the energy could be diffused to the shear layers, or else it could be convected to the shear layers through entrainment of the streamlines. We investigate both options and show that neither is a likely candidate at high Reynolds number. We then describe numerical experiments for a model problem designed to resolve these issues. We show that, at least for our model problem, no stable steady solution exists at high Reynolds number. Rather, as soon as the Reynolds number exceeds a modest value of around 10, the flow becomes unstable via a supercritical Hopf bifurcation.

A quasi-one-dimensional premixed flame model with cross-stream diffusion

A differential quasi-one-dimensional flame model which accounts for flame curvature, lateral flow expansion, and cross-stream diffusion is formulated. Expressions for the flame propagation speed (or displacement speed) are obtained by integration of the governing equations. The analysis shows that three mechanisms contribute to the enhancement of the displacement speed: a chemical mechanism associated with modifications to the reaction zone structure, lateral flow expansion, and cross-stream diffusion. An approach to study the sensitivity of the flame structure and propagation to curvature and strain rate is proposed based on the model formulation. In particular, a method for computing the curvature Markstein lengths from the solution of one-dimensional flames is described. Contributions to the Markstein length based on sensitivity analysis are shown to be associated primarily with processes in the reaction zone.

Cross-Stream Thermal Dispersion in a Staggered Tube Bundle With Crossflow

The investigation of thermal dispersion in tube bundles has shown two things. First, numerical simulations with the k-e turbulence model can accurately represent the molecular and turbulent thermal dispersion in a tube bundle when a turbulent Prandtl number of 0.63 is used. Second, porous body simulations can accurately represent the effects of cross-stream diffusion when an effective thermal conductivity model is used

An element-by-element BICGSTAB iterative method for three-dimensional steady Navier-Stokes equations

Construction of a stabilized Galerkin upwind finite element model for steady and incompressible Navier-Stokes equations in three dimensions is the main theme of this study. In the time-independent context, the weighted residuals statement is kept biased in favor of the upstream flow direction by adding an artificial damping term of physical plausibility to the Galerkin framework. This upwind approach has significant advantage of seeking solutions free from cross-stream diffusion error. Finite element solutions have been found by mixed formulation, implemented in quadratic cubic elements which are characterized as possessing the so-called LBB (Ladyzhenskaya-Babuška-Brezzi) condition. An element-by-element BICGSTAB solution solver is intended to alleviate difficulties regarding the asymmetry and indefiniteness arising from the use of a mixed formulation for incompressible fluid flows. The developed three-dimensional finite element code is first rectified by solving a problem amenable to analytic solution. A well-known lid-driven cavity flow problem in a cubical cavity is also studied.

Self-Similar Diffusion Flame Including Effects of Streamwise Diffusion

Abstract Using Lie group theory, a self-similar solution. representing a diffusion flame in which both cross stream and streamwise diffusion is included is obtained. The solution arises in problems related to conjugate heat transfer and pollutant dispersion applications that are governed by the same partial differential equation. Results including flame shape, flame height dependence on Peelet number and overall stoichiometry, are compared with other known solutions. A unique feature of the solution is the existence of a maximum in the size of the back diffusion region as a function of Peclet number. This could be important in very low speed burner applications.

Analysis of a Nonlinear Diffusive Amplitude Equation for Waves on Thin Films

This paper presents analytical and numerical solutions of a new amplitude equation governing long waves on thin films. At lowest order in the long-wave parameter, the equation is nondispersive and represents a balance between nonlinearity and cross-stream diffusion. Numerical solutions tracing the temporal evolution of an initially localized disturbance indicate that the aforementioned diffusion partly mitigates the tendency of the wave to break. We have also obtained a closed-form solution resembling an undular bore propagating in an oblique direction.

Effect of non-uniform currents and depth variations upon steady discharges in shallow water

When advection dominates diffusion, there are special directions (rays), distinct from but closely related to the contaminant flux vector, along which information is carried. The geometry of these ray paths is found to depend in a simple way upon the advection-diffusion vector ½u/D, where u is the flow velocity and D the cross-stream diffusivity. Simple models are used to show that for a steady point discharge the contaminant concentration is greatest in shallow water and towards the outside of bends.

A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows

A general, numerical, marching procedure is presented for the calculation of the transport processes in three-dimensional flows characterised by the presence of one coordinate in which physical influences are exerted in only one direction. Such flows give rise to parabolic differential equations and so can be called three-dimensional parabolic flows. The procedure can be regarded as a boundary-layer method, provided it is recognised that, unlike earlier published methods with this name, it takes full account of the cross-stream diffusion of momentum, etc., and of the pressure variation in the cross-stream plane. The pressure field is determined by: first calculating an intermediate velocity field based on an estimated pressure field; and then obtaining appropriate correction so as to satisfy the continuity equation. To illustrate the procedure, calculations are presented for the developing laminar flow and heat transfer in a square duct with a laterally-moving wall.