Free-slip Boundary Research Articles

Two-dimensional and three-dimensional computational studies of hydrodynamics in the transition from bubbling to circulating fluidised bed

This paper applies two-fluid modelling (TFM) to a two-dimensional and three-dimensional circulating fluidised bed (CFB). An energy minimisation multiscale (EMMS) based drag model is compared with a classical drag model, namely the Gidaspow model in the light of experimental data from the CFB. The axial particle velocities and the radial volume fraction at different heights are considered. The specularity coefficient responsible for the tangential solid velocities at the walls is varied to study the effect on the downflow of particles at the wall. The work is further extended to explore the effects of velocity variation on the flow distribution showing the transition from a bubbling to a fast fluidising regime. Furthermore, the diameters of the bubbles observed within the bubbling regime are compared with the Davidson’s bubble diameter model for a range of particle diameters. Varying the specularity coefficient showed that a free slip boundary condition underpredicted the downflow of particles at the wall and to add slight roughness to the wall gave a closer representation. The predictions for the 2D and 3D CFB axial velocities were in good agreement with the experimental data but the 2D results slightly overpredicted the core velocity. The transition from a bubbling to a fast fluidising regime as expected occurred once the inlet velocity exceeded the terminal velocity. The equivalent bubble diameter from the simulations agreed well with the calculated bubble diameter from the model.

Asymptotic analysis of the differences between the Stokes–Darcy system with different interface conditions and the Stokes–Brinkman system

We consider the coupling of the Stokes and Darcy systems with different choices for the interface conditions. We show that, comparing results with those for the Stokes–Brinkman equations, the solutions of Stokes–Darcy equations with the Beavers–Joseph interface condition in the one-dimensional and quasi-two-dimensional (periodic) cases are more accurate than are those obtained using the Beavers–Joseph–Saffman–Jones interface condition and that both of these are more accurate than solutions obtained using a zero tangential velocity interface condition. The zero tangential velocity interface condition is in turn more accurate than the free-slip interface boundary condition. We also prove that the summation of the quasi-two-dimensional solutions converge so that the conclusions are also valid for the two-dimensional case.

Viscous effects in two-layer, unidirectional hydraulic flow

Hydraulic equations are derived for a stratified (two-layer) flow in which the horizontal velocity varies continuously in the vertical. Viscosity is included in the governing equations, and the effect of friction in hydraulically controlled flows is examined. The analysis yields Froude numbers which depend upon the integrated inverse square of velocity but reduce to the original layered Froude numbers when velocity is constant with depth. The Froude numbers reveal a critical condition for hydraulic control, which equates to the arrest of internal gravity waves.Solutions are presented for the case of unidirectional flow through a lateral constriction, both with and without bottom drag. In the free-slip lower boundary case, viscosity transports momentum from the faster to the slower layer, thereby shifting the control point downstream and reducing the flux through the constriction. However, while the velocity shear at the interface between the two layers is reduced, the top-to-bottom velocity difference of the controlled solution is increased for larger values of viscosity. This counter-intuitive result is due to the restrictions placed on the flow at the hydraulic control point. When bottom drag is included in the model, the total flux may increase, in some cases exceeding that of the inviscid solution.

Effective Rotational Viscosity of Vertical Alignment Nematic Liquid Crystal Cells

An effective rotational viscosity of vertical alignment (homeotropic) nematic liquid crystal (NLC) cells with negative dielectric anisotropy is derived from a theory of NLCs in which flow effects under the free-slip boundary condition are taken into account. The effective rotational viscosity is a function of the Leslie viscosity coefficients and is much smaller than the rotational viscosity at the initial stage of the director reorientation induced by external electric field to the NLC cells. This is the origin of fast response of vertical alignment of NLC cells.

On the formulation of sea-ice models. Part 2: Lessons from multi-year adjoint sea-ice export sensitivities through the Canadian Arctic Archipelago

The adjoint of an ocean general circulation model is at the heart of the ocean state estimation system of the Estimating the Circulation and Climate of the Ocean (ECCO) project. As part of an ongoing effort to extend ECCO to a coupled ocean/sea-ice estimation system, a dynamic and thermodynamic sea-ice model has been developed for the Massachusetts Institute of Technology general circulation model (MITgcm). One key requirement is the ability to generate, by means of automatic differentiation (AD), tangent linear (TLM) and adjoint (ADM) model code for the coupled MITgcm ocean/sea-ice system. This second part of a two-part paper describes aspects of the adjoint model. The adjoint ocean and sea-ice model is used to calculate transient sensitivities of solid (ice and snow) freshwater export through Lancaster Sound in the Canadian Arctic Archipelago (CAA). The adjoint state provides a complementary view of the dynamics. In particular, the transient, multi-year sensitivity patterns reflect dominant pathways and propagation timescales through the CAA as resolved by the model, thus shedding light on causal relationships, in the model, across the Archipelago. The computational cost of inferring such causal relationships from forward model diagnostics alone would be prohibitive. The role of the exact model trajectory around which the adjoint is calculated (and therefore of the exactness of the adjoint) is exposed through calculations using free-slip vs no-slip lateral boundary conditions. Effective ice thickness, sea surface temperature, and precipitation sensitivities, are discussed in detail as examples of the coupled sea-ice/ocean and atmospheric forcing control space. To test the reliability of the adjoint, finite-difference perturbation experiments were performed for each of these elements and the cost perturbations were compared to those “predicted” by the adjoint. Overall, remarkable qualitative and quantitative agreement is found. In particular, the adjoint correctly “predicts” a seasonal sign change in precipitation sensitivities. A physical mechanism for this sign change is presented. The availability of the coupled adjoint opens up the prospect for adjoint-based coupled ocean/sea-ice state estimation.

A STUDY ON IMPROVING THE NUMERICAL STABILITY IN REVERSE SIMULATION

Nowadays, reverse simulation seems a promising topic for environment research. Mathematically, a reverse simulation is the solution of a transport equation in negative time. The process of a transport equation is equivalent to that of positive time with negative convection and diffusion. However, there is a numerical instability problem with solving the diffusion term. This research introduces a method to improve numerical stability in reverse simulation by applying a low-pass filter. This is a Gaussian filter that is well established in LES, with a free-slip boundary condition for filter operation. Two examples are given in this paper. In the first, a simple planar wind flow is considered and pollutant is emitted uniformly over time. In the second example, a wind flow around a single building is considered.

Fluid dynamics simulation of the high shear mixing process

The Eulerian–Eulerian two-fluid approach for modelling multiphase flows is used to simulate the flow in a high shear mixer. The results are compared with experimental velocity profiles for the solids phase at the wall in the mixer obtained using a high speed camera (Darelius et al. Chem. Eng. Sci. 62 (2007) 2366). The governing equations are closed using relations from the Kinetic Theory of Granular Flow (KTGF) combined with a frictional stress model due to Johnson and Jackson and Schaeffer and inter-phase drag due to Wen and Yu. In addition, calculations are presented for a model with a constant particle phase viscosity (CPV). Free slip and partial slip boundary conditions for the solid phase velocity at the vessel wall and the impeller have been utilized. The results show that the bed height could be well predicted by the partial slip model, whereas the free slip model could not capture the experimentally found bed height satisfactorily. For the KTGF model, the swirling motion of the rotating torus that is formed by the moving powder bed 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 CPV model gave predictions in good agreement with the experiments for a solids viscosity of 0.1 Pa s and a wall slip parameter of 0.005 m/Pa s. However, for a very low or very high value of the particle phase viscosity and for a high value of the wall slip parameter the agreement with experiments was poor. Interestingly, values of the viscosity that are commonly employed for fluidized beds seem applicable also in the present case.

WELL-POSEDNESS OF THE INFINITE PRANDTL NUMBER MODEL FOR CONVECTION WITH TEMPERATURE-DEPENDENT VISCOSITY

We establish the well-posedness of the infinite Prandtl number model for convection with temperature-dependent viscosity, free-slip boundary condition and zero horizontal fluxes.

Nusselt–Rayleigh number scaling for spherical shell Earth mantle simulation up to a Rayleigh number of [formula omitted

An investigation of the power law relationship between Nusselt number (Nu) and Rayleigh number (Ra) for Earth’s convecting mantle is presented. The Nu(Ra) relationship was calculated from the results of a model with three dimensional spherical geometry and free slip boundary conditions. Both basally and internally heated convection has been examined. For Nu ( Ra ) = a Ra β , β was found to be 0.294 ± 0.004 for basally heated systems, which is lower than the value of 1/3 suggested by conventional boundary layer theory. The exponent β = 0.337 ± 0.009 for internally heated systems, when the internally heated Ra is converted to a basally heated equivalent for comparison. The influence of the method used to calculate β was also considered, with particular attention paid to high Ra. As an example of the significance of β = 0.29 rather than 1/3, a Ra of 1 0 9 results in a surface heat flux which is ≈ 32% lower. Within the range of Ra used in this study, there is no evidence that β changes at high Ra. Therefore, that mechanism cannot be used to moderate mantle temperature when projecting back to early Earth conditions. The differing planform of basally and internally heated models was shown to result in different scaling relationships between root mean square surface velocity and Ra for the two modes of heating, in particular, a much lower surface velocity for internally heated cases relative to equivalent Ra basally heated cases.

Lee-Wave Resonances over Double Bell-Shaped Obstacles

Abstract Lee-wave resonance over double bell-shaped obstacles is investigated through a series of idealized high-resolution numerical simulations with the nonhydrostatic Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS) model using a free-slip lower boundary condition. The profiles of wind speed and stability as well as terrain derive from observations of lee-wave events over the Sierra Nevada and Inyo Mountains from the recently completed Terrain-Induced Rotor Experiment (T-REX). Numerical experiments show that double bell-shaped obstacles promote trapped lee waves that are in general shorter than those excited by an isolated ridge. While the permissible trapped lee-wave modes are determined by the upstream atmospheric structure, primarily vertical wind shear, the selected lee-wave wavelengths for two obstacles that are close or equal in height are dictated by the discrete terrain spectrum and correspond to higher harmonics of the primary orographic wavelength, which is equal to the ridge separation distance. The exception is the smallest ridge separation distance examined, one that corresponds to the Owens Valley width and is closest to the wavelength determined by the given upstream atmospheric structure, for which the primary lee-wave and orographic wavelengths were found to nearly coincide. The influence two mountains exert on the overall lee-wave field is found to persist at very large ridge separation distances. For the nonlinear nonhydrostatic waves examined, the ridge separation distance is found to exert a much stronger control over the lee-wave wavelengths than the mountain half-width. Positive and negative interferences of lee waves, which can be detected through their imprint on wave drag and wave amplitudes, were found to produce appreciable differences in the flow structure mainly over the downstream peak, with negative interference characterized by a highly symmetric flow pattern leading to a low drag state.

Wall effects in non-Boussinesq density currents

We report on the results of a numerical study of nearly immiscible contrasted density currents aimed at shedding light on the influence of wall effects on current dynamics in the lock-exchange configuration. The numerical approach is an interface-capturing method which does not involve any explicit reconstruction of the interface. Navier–Stokes equations are solved on a fixed grid and a hyperbolic equation is used for the transport of the local volume fraction of one of the fluids. This allows us to describe the density currents for the complete range of density contrast 10−3≤ρL/ρH≤0.99 (ρL and ρH being the density of the light and heavy fluids) and a wide range of Reynolds number 70≤Re≤5×104 (based on the channel height and the viscosity of the heavy fluid). The use of free-slip vs. no-slip boundary conditions enables us to separate the dissipation at the interface from the dissipation at the boundaries. Present results reveal that wall effects play a significant role on the propagation of contrasted density currents, unlike dissipation at the interface. It is first shown that when wall friction can be neglected, theoretical models based on the inviscid shallow-water approximations and Benjamin's steady-state result describe fairly well the light and heavy front velocities of density currents for the complete range of density ratio. However, when wall friction cannot be neglected, the results depart significantly from the prediction of inviscid theories. It is observed that most of the dissipation in highly contrasted currents takes place at the bottom wall and is a maximum at the head of the heavy current. This dissipation is shown to be responsible for the decrease of the front velocity. We propose a simple model based on Benjamin's analysis that includes wall friction. Keeping in mind the simplicity and limitations of the present model, the prediction of the front velocity of both the heavy and light currents is observed to be in good agreement with the numerical results for the complete range of density contrast. This gives further support to the idea that wall effects are the crucial ingredient for accurately predicting the front velocity of highly contrasted density currents.

Effects of Fluid Flow on Electric-Field-Induced Director Reorientation in Homogeneous and Homeotropic Nematic Liquid Crystal Cells, Probed by Transient Current Measurements

Flow effects on the dynamic behavior of the director in homogeneous nematic liquid crystal (NLC) cells with positive dielectric anisotropy (Δε> 0) and homeotropic NLC cells with negative dielectric anisotropy (Δε< 0) have been studied by analyzing the transient currents. The transient currents of homogeneous and homeotropic NLC cells were excited by step voltage application. The measured transient currents were well reproduced by computer simulation, based on the dynamic theory of NLCs in which the flow effects are taken into account. It is found that the flow effects with the free-slip boundary condition accelerate the electric-field-induced director reorientation in homeotropic NLC cells with Δε< 0, while the flow effects are negligible in homogeneous NLC cells with Δε> 0.

Effects of viscosities on the transient current in homeotropic nematic liquid crystal cells

Effects of the Leslie viscosity coefficients α i ( i = 1, 2, …, 5) of nematic liquid crystals (NLCs) with negative dielectric anisotropy (Δ ε < 0) on the electric-field-induced director reorientation in homeotropic NLC cells have been studied from the analysis of the transient current induced by step voltage application. The transient current in a homeotropic NLC cell with Δ ε < 0 was well reproduced by computer simulation, based on the theory of NLCs in which the flow effects and the free-slip boundary condition are taken into account. It is found that the response time of vertical alignment NLC displays is dominantly governed by α 2 and α 4 + α 5 of NLCs with Δ ε < 0.

Measurement of viscosities from the transient current of nematic liquid crystals with negative dielectric anisotropy

A method for measuring the complete set of the Leslie viscosity coefficients of nematic liquid crystals (NLCs) with negative dielectric anisotropy (Δ ε < 0) is proposed. The method is based on the analysis of the transient current, induced by step voltage excitation, in homeotropic NLC cells. It has been shown that the transient current in homeotropic NLC cells with Δ ε < 0 can be explained by the theory of NLCs in which the flow effects and the free-slip boundary condition are taken into account [Y. Iwata et al., Jpn. J. Appl. Phys. 43 (2004) L 1588]. It is demonstrated that the complete set of the Leslie viscosity coefficients of NLCs with Δ ε < 0 can be simultaneously and accurately measured by fitting the transient current calculated by the theory to the experimental data with a differential correction method.

Unsteady flow separation on slip boundaries

We derive analytic criteria for the location and angle of unsteady particle separation and reattachment in two-dimensional flows with free-slip boundary conditions. Our wall-based criteria show that, in general, fluid breakaway from the boundary takes place at locations different from either instantaneous or averaged stagnation points. Indeed, for time-varying flows, separation does not occur along a free streamline or along an average free streamline. We apply the formula to transport in randomized Rayleigh–Bénard convection cells, as well as to boundary current separation and reattachment in high-frequency radar data collected in Monterey Bay, California.

A mass, energy, vorticity, and potential enstrophy conserving lateral fluid–land boundary scheme for the shallow water equations

A numerical scheme for treating fluid–land boundaries in inviscid shallow water flows is derived that conserves the domain-summed mass, energy, vorticity, and potential enstrophy in domains with arbitrarily shaped boundaries. The boundary scheme is derived from a previous scheme that conserves all four domain-summed quantities only in periodic domains without boundaries. It consists of a method for including land in the model along with evolution equations for the vorticity and extrapolation formulas for the depth at fluid–land boundaries. Proofs of mass, energy, vorticity, and potential enstrophy conservation are given. Numerical simulations are carried out demonstrating the conservation properties and accuracy of the boundary scheme for inviscid flows and comparing its performance with that of four alternative boundary schemes. The first of these alternatives extrapolates or finite-differences the velocity to obtain the vorticity at boundaries; the second enforces the free-slip boundary condition; the third enforces the super-slip condition; and the fourth enforces the no-slip condition. Comparisons of the conservation properties demonstrate that the new scheme is the only one of the five that conserves all four domain-summed quantities, and it is the only one that both prevents a spurious energy cascade to the smallest resolved scales and maintains the correct flow orientation with respect to an external forcing. Comparisons of the accuracy demonstrate that the new scheme generates vorticity fields that have smaller errors than those generated by any of the alternative schemes, and it generates depth and velocity fields that have errors about equal to those in the fields generated by the most accurate alternative scheme.

Modeling ocean circulation on unstructured meshes: comparison of two horizontal discretizations

Finite-element models on unstructured meshes are frequently formulated in terms of continuous linear elements, which suffer from pressure modes and require stabilization. Alternatively, horizontal velocities may be represented with linear nonconforming elements. While the latter formulation uses three times more degrees of freedom for the velocity, it does not support pressure modes. The effects of stabilization are estimated by comparing the performance of continuous linear and nonconforming versions of the finite-element ocean circulation model (FEOM) in two simple configurations: a Munk gyre and baroclinic turbulence in a zonally reentrant channel. It is shown that, outside the free slip boundary layers, the presence of stabilization does not lead to noticeable effects if its strength is kept within certain limits. In order to evaluate the performance of FEOM, the baroclinic turbulence test is repeated with the MIT general circulation model (MITgcm), which serves as a benchmark, and reasonable agreement between different model codes is found. The two versions of FEOM have a similar computational cost, but both are significantly slower (per node) than the regular-mesh MITgcm. The paper also provides a brief description of the implementation of the nonconforming version of FEOM.

A numerical study of the frontal region of gravity currents propagating on a free-slip boundary

We investigate numerically the three-dimensional flow near the head of gravity currents propagating along a free-slip boundary. The simulations show that two states are possible: a high-mixing state, where the flow departs from the analytic inviscid solution 0.5 channel heights downstream of the front location, and with characteristics similar to the ones observed for purely two-dimensional simulations; and a low-mixing state, where billows are weaker and appear further downstream. To access the high-mixing state, it is necessary to add a source of perturbation upstream of the head in the form of turbulence. At high values of the Reynolds number, the intensity of rms turbulent fluctuations necessary to switch to the high-mixing state is small (0.5% of the speed of propagation) and may explain why the low-mixing state has so far eluded experimental detection. In the low-mixing state, the flow becomes three-dimensional near the front due to centrifugal instabilities caused by the curved streamlines. This instability of the outer flow is coupled to overturning instabilities that develop within the heavy fluid in the head, and suppresses the formation of billows. This complex behavior, which feeds on the interplay of streamline curvature and stratification, should be considered a good model to understand how instabilities occur in other types of strongly nonlinear stratified flows.

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) .