Free-slip Boundary Conditions Research Articles

CFD simulation of the high shear mixing process using kinetic theory of granular flow and frictional stress models

In order to enhance process understanding and to develop predictive process models in high shear granulation, there is an ongoing search for simulation tools and experimental methods to model and measure the velocity and shear fields in the mixer. In this study, the Eulerian–Eulerian approach to model multiphase flows has been used to simulate the mixer flow. Experimental velocity profiles for the solid phase at the wall in the mixer have been obtained using a high speed camera following the experimental procedure as described by Darelius et al. [2007a. Measurement of the velocity field and frictional properties of wet masses in a high shear mixer. Chemical Engineering Science, 62, 2366–2374]. The governing equations for modelling the dense mixer flow have been closed by using closure relations from the kinetic theory of granular flow (KTGF) combined with frictional stress models. The free slip and partial slip boundary conditions for the solid phase velocity at the vessel wall have been utilized. The partial slip model originally developed for dilute flows by Tu and Fletcher [1995. Numerical computation of turbulent gas–solid particle flow in a 90 ∘ bend. A.I.Ch.E. Journal, 41, 2187–2197] has been employed. It was found that the bed height could be well predicted by implementing the partial slip model, whereas the free slip model could not capture the experimentally found bed height satisfactorily. In the simulation, the swirling motion of the rotating torus formed was over-predicted and the tangential wall velocity was under-predicted, probably due to the fact that the frictional stress model needs to be further developed, e.g. to tackle cohesive particles in dense flow. The advantage of using the Eulerian–Eulerian approach compared to discrete element methods is that there is no computational limitation on the number of particles being modelled, and thus manufacturing scale granulators can be modelled as well.

Hydrodynamic description of the long-time tails of the linear and rotational velocity autocorrelation functions of a particle in a confined geometry

We report a hydrodynamic analysis of the long-time behavior of the linear and angular velocity autocorrelation functions of an isolated colloid particle constrained to have quasi-two-dimensional motion, and compare the predicted behavior with the results of lattice-Boltzmann simulations. Our analysis uses the singularity method to characterize unsteady linear motion of an incompressible fluid. For bounded fluids we construct an image system with a discrete set of fundamental solutions of the Stokes equation from which we extract the long-time decay of the velocity. For the case that there are free slip boundary conditions at walls separated by H particle diameters, the time evolution of the parallel linear velocity and the perpendicular rotational velocity following impulsive excitation both correspond to the time evolution of a two-dimensional (2D) fluid with effective density rho_(2D)=rhoH. For the case that there are no slip boundary conditions at the walls, the same types of motion correspond to 2D fluid motions with a coefficient of friction xi=pi(2)nu/H(2) modulo a prefactor of order 1, with nu the kinematic viscosity. The linear particle motion perpendicular to the walls also experiences an effective frictional force, but the time dependence is proportional to t(-2) , which cannot be related to either pure 3D or pure 2D fluid motion. Our incompressible fluid model predicts correct self-diffusion constants but it does not capture all of the effects of the fluid confinement on the particle motion. In particular, the linear motion of a particle perpendicular to the walls is influenced by coupling between the density flux and the velocity field, which leads to damped velocity oscillations whose frequency is proportional to c_(s)/H , with c_(s) the velocity of sound. For particle motion parallel to no slip walls there is a slowing down of a density flux that spreads diffusively, which generates a long-time decay proportional to t(-1) .

Linear stability of the flow past a spheroidal bubble

The linear stability of the axisymmetric flow past a fixed-shape spheroid with free-slip boundary conditions is studied numerically to gain some insight into the path instability of bubbles rising in liquids. Qualitatively, the results are similar to those for a solid sphere. The m = 1 mode gives rise to a double-threaded wake and proves to be the most unstable mode, with a first regular bifurcation followed by a Hopf bifurcation. The importance of the base-flow vorticity is highlighted by a stability analysis of the axisymmetric base flow ‘frozen’ before reaching steady state. Setting viscosity to zero in the perturbation equations results in a faster growth of the primary instability, which indicates its root in inertial effects.

Stress distribution within subducting slabs and their deformation in the transition zone

The mechanical behavior of slabs in the mantle transition zone is a poorly understood phenomenon. To obtain a better understanding of the stress and deformation pattern inside subducting slabs, we performed 2D numerical simulations of the subduction process in a purely viscous, non-linear rheological model including diffusion creep, dislocation creep and a stress limiting mechanism. The model calculations are used to investigate the effects of yield stress, plate age and surface boundary condition (free-slip versus given plate velocity) on the stress development in and around a subducting slab and on the deformation of the subducting plate in the transition zone. The deformation and morphology of subducted slabs may differ between free-slip and kinematic boundary condition models in some cases, especially for young and relatively weak slabs. Even though kinematically driven subduction morphology may look realistic in most cases, our results show that stress distributions can differ significantly if the discrepancy between free-slip and kinematic subduction velocity is large, and in some model cases even influence the ability of the plate to penetrate into the lower mantle. Stress distributions such as the ones presented in this study can be a valuable tool in understanding the complex and enigmatic interaction between subducting slabs and the transition zone.

Empirical and stochastic formulations of western boundary conditions

High-frequency radar observations of surface current at 100 m resolution are used to stochastically determine the vorticity variability in the boundary layer between the Florida Current and the Florida coast. Using the empirical data, several formulations of the stochastic boundary condition (SBC) are then developed and evaluated for the parameterization of sub-grid scale variability along the western boundaries of an ocean general circulation model (OGCM). These SBCs are compared to both no-slip and free-slip boundary conditions along the western boundary of a channel model simulating separation of the boundary current at a cape, as well as simulations of the classic, wind-forced double-gyre circulation. Our studies indicate that simulated circulation patterns are sensitive to both the functional form of the SBC and the parameters used in a given formulation. In particular, a relatively simple, auto-regressive formulation of SBC can be applied to a coarse-resolution, eddy-permitting OGCM to increase flow turbulence and to affect the flow patterns in ways distinct from the standard, deterministic no-slip and free-slip boundary conditions. Correlation time scales of a few days to weeks, orders of magnitude larger than a typical time step of OGCM, are needed in the regression model for SBC. This formulation of SBC is shown to be effective in both quasi-geostrophic and shallow-water dynamics.

Structure of Turbulent Channel Flow under Spanwise System Rotation

Direct numerical simulation of turbulent channel flow under spanwise system rotation is performed. When system rotation is imposed on turbulent Poiseuille flow, the Reynolds number decreases in one side of a channel, while it increases in the other side. Hence, the effects of the Reynolds number as well as spanwise system rotation cannot be negligible in turbulent Poiseuille flow. To exclude the Reynolds-number effects, we performed the numerical simulation of open channel flow where the free-slip boundary condition is imposed on one wall, and the Reynolds number defined by the friction velocity and channel width can be always constant. When system rotation is imposed to enhance the spanwise vorticity of the mean shear, large-scale turbulence is attenuated and the mean velocity gradient increases over an entire channel. In contrast, when system rotation is imposed in the opposite sense to the mean spanwise vorticity, the mean velocity is leveled and small-scale turbulence is markedly decreased. We also investigate the effects of system rotation on the longitudinal vortical structure typically observed in near-wall turbulence.

Numerical study of the flow in a three-dimensional thermally driven cavity

Solutions for the fully compressible Navier–Stokes equations are presented for the flow and temperature fields in a cubic cavity with large horizontal temperature differences. The ideal-gas approximation for air is assumed and viscosity is computed using Sutherland's law. The three-dimensional case forms an extension of previous studies performed on a two-dimensional square cavity. The influence of imposed boundary conditions in the third dimension is investigated as a numerical experiment. Comparison is made between convergence rates in case of periodic and free-slip boundary conditions. Results with no-slip boundary conditions are presented as well. The effect of the Rayleigh number is studied. Results are computed using a finite volume method on a structured, collocated grid. An explicit third-order discretization for the convective part and an implicit central discretization for the acoustic part and for the diffusive part are used. To stabilize the scheme an artificial dissipation term for the pressure and the temperature is introduced. The discrete equations are solved using a time-marching method with restrictions on the timestep corresponding to the explicit parts of the solver. Multigrid is used as acceleration technique.

Infinite-Prandtl-number convection. Part 2. A singular limit of upper bound theory

An upper bound on the heat flux for infinite-Prandtl-number convection between two parallel plates is determined for the cases of no-slip and free-slip boundary conditions. For no-slip the large-Rayleigh-number ( $\hbox{\it Ra}$ ) scaling for the Nusselt number is consistent with $\hbox{\it Nu}\,{ , as predicted by Chan (1971). However, his commonly accepted picture of an infinite hierarchy of multiple boundary layer solutions smoothly approaching this scaling is incorrect. Instead, we find a novel terminating sequence in which the optimal asymptotic scaling is achieved with a three-boundary-layer solution. In the case of free-slip, we find an asymptotic scaling of $\hbox{\it Nu}\,{ , corroborating the conservative estimate obtained in Plasting & Ierley (2005). Here the infinite hierarchy of multiple-boundary-layer solutions obtains, albeit with anomalous features not previously encountered. Thus for neither boundary condition does the optimal solution conform to the well-established models of finite-Prandtl-number convection (Busse 1969 b ), plane Couette flow, and plane or circular Poiseuille flow (Busse 1970). We reconcile these findings with a suitable continuation from no-slip to free-slip, discovering that the key distinction – finite versus geometric saturation – is entirely determined by the singularity, or not, of the initial, single-boundary-layer, solution. It is proposed that this selection principle applies to all upper bound problems.

Lattice–Boltzmann study of the transition from quasi-two-dimensional to three-dimensional one particle hydrodynamics

We report the results of a study designed to categorize the hydrodynamics of a quasi-two-dimensional system as either a 2D or 3D fluid. The characterization is based on the asymptotic decay of the velocity autocorrelation functions for different modes of motion and different boundary conditions at the enclosing walls. Our results show that for the case of no-slip boundary conditions the long time decay corresponds to neither 2D or 3D behaviour, nor anything in-between. The no-slip walls cause the long time tail of the velocity autocorrelation functions to have an exponential decay, more in agreement with Langevin model predictions. The free-slip boundary conditions create no-friction conditions for the tangential flow, and no-slip conditions for the perpendicular flow. The tangential component of the flow behaves like flow in 2D, but the perpendicular flow is hindered. As a result, the effect of the free-slip walls on the particle motion depends on the particular mode of motion. For perpendicular rotation where there is no flow perpendicular to the walls and for parallel translation where the perpendicular flow component is weak, we retrieve a 2D like decay for the autocorrelation functions. When the greatest part of the flow is in the perpendicular direction, the distinction between the no-slip and free-slip boundary is small, as in perpendicular translation. For the case of parallel rotation, the situation is more complicated as the perpendicular and tangential flows are more or less balanced, thus the no-slip boundary conditions slow down and speed up the autocorrelation decay. In the net result, the slowing down shows up in the increased diffusion coefficient, but the autocorrelation function shows a faster long time decay. As the plate separation decreases, the balance between the two contributions shifts and the diffusion coefficient begins to fall with decreasing separation.

β -plane turbulence in a basin with no-slip boundaries

On a domain enclosed by no-slip boundaries, two-dimensional, geostrophic flows have been studied by numerical simulations of the Navier-Stokes equation with the β-plane approximation at intermediate Reynolds numbers and a range of values for β. The β effect causes a refinement of the flow structures, and the presence of basin modes has been revealed by means of frequency spectra. The presence and apparent stability of basin modes on a domain enclosed by no-slip boundaries is a rather surprising observation, because these modes are solutions of the inviscid flow equations on a bounded domain (with free-slip boundaries). To understand the persistence of these basin modes, the viscous boundary layers near the no-slip walls have been investigated. The mean flow in forced simulations shows a zonal band structure, much unlike the regular Fofonoff-like solution observed when free-slip boundary conditions are used.

Incompressibility of the Leray-α model for wall-bounded flows

This study shows that the Leray-α model does not explicitly enforce a divergence-free field for the filtered velocity. While this condition is automatically satisfied in the absence of boundaries, bounded domains require extra attention. It is shown, both analytically and through simulations of Rayleigh–Bénard convection, that incompressibility of the filtered velocity field cannot be guaranteed in the current formulation. Several suggestions are made to restore the incompressibility of the filtered velocity, and it is shown that free-slip boundary conditions for the filtered velocity do guarantee incompressibility for the domain under consideration.

Quantization of the Low-Frequency Variability of the Double-Gyre Circulation

AbstractThe low-frequency dynamics of the double-gyre wind-driven circulation in large midlatitude oceanic basins is investigated. It is shown that for quasigeostrophic models linear (Rayleigh) friction is necessary to obtain realistic recirculation gyres and elongated jet streams with small meridional-to-zonal aspect ratio. It is also found that the use of either no-slip or free-slip boundary conditions does not change the drastic effects of bottom drag on the large scales. These long oceanic jets are alternatively destabilized and restabilized through successive (subcritical) supercritical symmetry-breaking bifurcations that are linked to the (non) existence of stationary Rossby waves. These waves are strongly localized along the oceanic front and are thus hardly affected by the basin geometry. Numerical and analytical results show that these waves are “quantized” with respect to the length of the jet, and an explicit dispersion relation is given. Numerical computations of branches of steady states, together with linear and nonlinear analysis, indicate that two classes of regimes characterize the low-frequency dynamics of the flow. The first class corresponds to supercritical regimes, which are associated with oceanic jets that have been destabilized by undamped stationary Rossby waves. In particular, these regimes allow the formation of gyre modes responsible for low-frequency relaxation oscillations of the jet. The second class corresponds to subcritical regimes, which are either quiescent or dominated by high-frequency instabilities and are characterized by jets that do not allow the formation of both stationary Rossby waves and gyre modes. Each of these regimes is characterized by typical spatial and time scales that are both quantized. The number n of “bumps” of the jet, which is related to the zonal wavenumber of the stationary Rossby waves, is used to distinguish between these regimes either in their supercritical or subcritical phase. For instance, the supercritical n = 2 regime is associated with a class of interdecadal gyre modes that extend up to 3000 km in the zonal direction. The quantization of the low-frequency dynamics and the existence of these regimes are also found to survive severe modifications of the basin geometry. These quantized regimes suggest that the low-frequency dynamics in turbulent regimes is likely to be autosimilar to the low-frequency dynamics found in a weakly nonlinear “ground” regime corresponding to n = 0.

Large eddy simulations of two-dimensional turbulent convection in a density-stratified fluid

Large eddy simulations (LES) of two-dimensional turbulent convection within the anelastic approximation are presented for Rayleigh number Ra = 109, Prandtl number Pr = 1 with free-slip boundary conditions. Various subgrid-scale (SGS) models are investigated such as a similarity model, a dynamic similarity model, a dynamic eddy-viscosity model, a hyperdiffusion model and a hybrid model (dynamic similarity hyperdiffusion model). To study the effects of density stratification on the models, we have also carried out simulations for a Boussinesq flow. The SGS models are compared to direct numerical simulation (DNS) data on the basis of kinetic energy and entropy variance spectra, mean entropy profiles, r.m.s. entropy profiles and r.m.s. kinetic energy density profiles. The results show that for the Boussinesq flow, all the SGS models agree fairly well with the high resolution DNS data. However, for the strongly density-stratified flow, only the hyperdiffusion and the hybrid model show good performance.

Secondary fluid flows driven electromagnetically in a two-dimensional extended duct

This study focuses on two-dimensional fluid flows in a straight duct with free-slip boundary conditions applied on the channel walls y =0 and y =2 πN with N >1. In this extended wall-bounded fluid motion problem, secondary fluid flow patterns resulting from steady-state and Hopf bifurcations are examined and shown to be dependent on the choice of longitudinal wave numbers. Some secondary steady-state flows appear at specific wave numbers, whereas at other wave numbers, both secondary steady-state and self-oscillation flows coexist. These results, derived through analytical arguments and truncation series approximation, are confirmed by simple numerical experiments supporting the findings observed from laboratory experiments.

Axisymmetric convective states of pure and binary liquids enclosed in a vertical cylinder and boundary conditions’ influence thereupon

The high Rayleigh number (Ra) axisymmetric convection regimes of Pr=1 pure and (Le=0.1, Ψ=−0.2) binary liquids are numerically investigated and compared. The fluids are enclosed in a vertical cylinder of aspect ratio height∕radius=2 and heated from below, with either no-slip or free-slip kinematic lateral boundary conditions. Branches of solutions and transitions between states that occur as Ra is varied up to O(105) are given, along with a description of the encountered bifurcations. When a free-slip condition is imposed along the circumference of the cell, pure fluid and binary liquid stationary flows are found to become identical at high Ra, an often reported feature. When the lateral boundary condition is set to no-slip, the high Ra steady flows of pure and binary liquid, although very similar, undergo different bifurcations. This is related with a locally quasiquiescent region present in both cases, the stability of which controls the flow regime in the whole fluid layer. A branch of resulting oscillatory states thus does not appear in the bifurcation diagram of the binary liquid.

Nonlinear oscillations of semigeostrophic Eady waves in the presence of diffusivity

Analyses are performed to examine the physical processes involved in nonlinear oscillations of Eady baroclinic waves obtained from viscous semigeostrophic models with two types of boundary conditions (freeslip and non-slip). By comparing with previous studies for the case of the free-slip boundary condition, it is shown that the nonlinear oscillations are produced mainly by the interaction between the baroclinic wave and zonal-mean state (total zonal-mean flow velocity and buoyancy stratification) but the timescale of the nonlinear oscillations is largely controlled by the diffusivity. When the boundary condition is non-slip, the nonlinear oscillations are further damped and slowed by the diffusive process. Since the free-slip (non-slip) boundary condition is the zero drag (infinite drag) limit of the more realistic drag boundary condition, the nonlinear oscillations obtained with the two types of boundary conditions are two extremes for more realistic nonlinear oscillations.

Numerical Study of Flows in a Cylindrical Container with Rotating Bottom and Top Flat Free Surface

Numerical study was conducted for steady swirling flows of viscous incompressible fluid confined in cylindrical containers driven by the bottom wall rotating at constant angular velocity with top free surface. The flow axisymmetry and the free-slip boundary condition is assumed at the top flat surface of the cylinder. Parametric computation was performed over a wide range of the Reynolds number Re and the radius to height aspect ratio h which covers parameter range not previously studied. Steady solutions disclosed a variety of flow patterns which include corner and surface bubbles with inner structures not noticed in the previous studies. Associated parameter regions are mapped on the parameter plane ( h , Re ) as a flow state diagram. It is shown that the bifurcation diagram is more complicated than previously believed. Comparison with the previous experimental and numerical studies is discussed in detail.

The flow in a cylindrical container with a rotating end wall at small but finite Reynolds number

Similarity solutions of the first kind are intermediate asymptotic solutions for the Stokes flow field and for the first-order inertial correction to the Stokes flow field in small aspect ratio geometries with both no-slip and free-slip boundary conditions opposite the rotating end wall. These results differ from the semi-infinite cylindrical container, where similarity solutions of the second kind are intermediate asymptotic representations of the Stokes and first-order flow fields. Lanczos factors are used to show that for Reynolds numbers less than 1, the boundary discontinuity has a limited influence on the flow field with a no-slip boundary condition opposite the rotating end wall but the boundary discontinuity is important in determining the flow field with a free-slip boundary condition opposite the rotating end wall.

A non-hydrostatic flat-bottom ocean model entirely based on Fourier expansion

We show how to implement free-slip and no-slip boundary conditions in a three dimensional Boussinesq flat-bottom ocean model based on Fourier expansion. Our method is inspired by the immersed or virtual boundary technique in which the effect of boundaries on the flow field is modeled by a virtual force field. Our method, however, explicitly depletes the velocity on the boundary induced by the pressure, while at the same time respecting the incompressibility of the flow field. Spurious spatial oscillations remain at a negligible level in the simulated flow field when using our technique and no filtering of the flow field is necessary. We furthermore show that by using the method presented here the residual velocities at the boundaries are easily reduced to a negligible value. This stands in contradistinction to previous calculations using the immersed or virtual boundary technique. The efficiency is demonstrated by simulating a Rayleigh impulsive flow, for which the time evolution of the simulated flow is compared to an analytic solution, and a three dimensional Boussinesq simulation of ocean convection. The second instance is taken form a well studied oceanographic context: A free slip boundary condition is applied on the upper surface, the modeled sea surface, and a no-slip boundary condition to the lower boundary, the modeled ocean floor. Convergence properties of the method are investigated by solving a two dimensional stationary problem at different spatial resolutions. The work presented here is restricted to a flat ocean floor. Extensions of our method to ocean models with a realistic topography are discussed.

Eulerian and Lagrangian studies in surface flow turbulence

Experimental and numerical studies of turbulent fluid motion in a free surface are presented. The flow is realized experimentally on the surface of a tank filled with water stirred well below the surface. Numerically, it is modelled by free-slip boundary conditions. The surface flow is unconventional: it is not incompressible and neither kinetic energy nor enstrophy is conserved in the limit of zero fluid viscosity and in the absence of external driving as is the case for incompressible two-dimensional turbulent flows. The dynamics of passive Lagrangian tracers that are advected in such flows are dominated by rapidly changing patches of the surface flow divergence. Owing to compressibility, particles form clusters within multifractal mass distributions. Also studied is the motion of pairs and triplets of particles. The mean square separation shows an extended range with a reduced scaling exponent in comparison with the classical Richardson value. Clustering is also manifest in strongly deformed triangles spanned within triplets of tracers.