Couette Flow Problem Research Articles

Transient motion of and heat transfer in a rarefied gas between plane parallel walls with different surface properties

Transient motion of and heat transfer in a rarefied gas between plane parallel walls with different surface properties are studied based on kinetic theory. It is assumed that one wall is a diffuse reflection boundary and the other wall is a Maxwell-type boundary, and the transient behavior of the gas caused by a sudden heating of one of the walls is studied. The linearized Boltzmann equation for a hard-sphere molecular gas is numerically studied using the modified hybrid scheme of the characteristic coordinate and finite difference methods, to correctly describe the discontinuities in the velocity distribution function. The transient motion of the gas from an early time stage to the final time-independent state is studied over a wide range of the mean free path and the accommodation coefficient of the boundary. Between the two transient flows caused by the heating of the respective walls, the values of the heat flow on the heated wall are different, whereas those on the unheated wall coincide identically. This property, which is a consequence of the symmetric relation of the linearized Boltzmann equation, is numerically confirmed over a wide range of the mean free path. The long time behavior of the heat flow on the walls is quite similar to that of the shear stress in the Couette flow problem, whereas a distinct wavy behavior is observed in an early time stage.

A high-performance finite-volume algorithm for solving partial differential equations governing compressible viscous flows on structured grids

This work focuses on the development of a high-performance fourth-order finite-volume method to solve the nonlinear partial differential equations governing the compressible Navier–Stokes equations on a Cartesian grid with adaptive mesh refinement. The novelty of the present study is to introduce the loop chaining concept to this complex fourth-order fluid dynamics algorithm for significant improvement in code performance on parallel machines. Specific operations involved in the algorithm include the finite-volume formulation of fourth-order spatial discretization stencils and optimal inter-loop parallelization strategies. Numerical fluxes of the Navier–Stokes equations comprise the hyperbolic (inviscid) and elliptic (viscous) ​components. The hyperbolic flux is evaluated using high-resolution Godunov’s method and the elliptic flux is based on fourth-order centered-difference methods everywhere in the computational domain. The use of centered-difference methods everywhere supports the idea of fusing modular codes to achieve high efficiency on modern computers. Temporal discretization is performed using the standard fourth-order ​Runge–Kutta method. The fourth-order accuracy of solution in space and time is verified with a transient Couette flow problem. The algorithm is applied to solve the Sod’s shock tube and the transient flat-plate boundary layer flow. The numerical predictions are validated by comparing to the analytical solutions. The performance of the baseline code is compared to that of the fused scheme which fuses modular codes via loop chaining concept and a significant improvement in execution time is observed.

Coupling heterogeneous continuum-particle fields to simulate non-isothermal microscale gas flows

This paper extends the hybrid computational method proposed by Docherty et al. (2014) for simulating non-isothermal rarefied gas flows at the microscale. Coupling a continuum fluid description to a direct simulation Monte Carlo (DSMC) solver, the original methodology considered the transfer of heat only, with validation performed on 1D micro Fourier flow. Here, the coupling strategy is extended to consider the transport of mass, momentum, and heat, and validation in 1D is performed on the high-speed micro Couette flow problem. Sufficient micro resolution in the hybrid method enables good agreement with an equivalent pure DSMC simulation, but the method offers no computational speed-up for this 1D problem. However, considerable speed-up is achieved for a 2D problem: gas flowing through a microscale crack is modelled as a microchannel with a high-aspect-ratio cross-section. With a temperature difference imposed between the walls of the cross-section, the hybrid method predicts the velocity and temperature variation over the cross-section very accurately; an accurate mass flow rate prediction is also obtained.

A compressible lattice Boltzmann finite volume model for high subsonic and transonic flows on regular lattices

A multi-dimensional double distribution function thermal lattice Boltzmann model has been developed to simulate fully compressible flows at moderate Mach number. The lattice Boltzmann equation is temporally and spatially discretizated by an asymptotic preserving finite volume scheme. The micro-velocities discretization is adopted on regular low-symmetry lattices (D1Q3, D2Q9, D3Q15, D3Q19, D3Q27). The third-order Hermite polynomial density distribution function on low-symmetry lattices is used to solve the flow field, while a second-order energy distribution is employed to compute the temperature field. The fully compressible Navier–Stokes equations are recovered by standard order Gauss–Hermite polynomial expansions of Maxwell distribution with cubic correction terms, which are added by an external force expressed in orthogonal polynomials form. The proposed model is validated considering several benchmark cases, namely the Sod shock tube, thermal Couette flow and two-dimensional Riemann problem. The numerical results are in very good agreement with both analytical solution and reference results.

Frequency staircases in narrow-gap spherical Couette flow

Recent studies of plane parallel flows have emphasised the importance of finite-amplitude self-sustaining processes for the existence of alternative non-trivial solutions. The idea behind these mechanisms is that the motion is composed of distinct structures that interact to self-sustain. These solutions are not unique and their totality form a skeleton about which the actual realised motion is attracted. Related features can be found in spherical Couette flow between two rotating spheres in the limit of narrow-gap width. At lowest order the onset of instability is manifested by Taylor vortices localised in the vicinity of the equator. By approximating the spheres by their tangent cylinders at the equator, a critical Taylor number based on the ensuing cylindrical Couette flow problem would appear to provide a lowest order approximation to the true critical Taylor number. At next order, the latitudinal modulation of their amplitude a satisfies the complex Ginzburg-Landau equation (CGLe)where is latitude scaled on the modulation length scale, t is time and is proportional to the excess Taylor number. The amplitude a governed by our CGLe is linearly stable for all but possesses non-decaying nonlinear solutions at finite , directly analogous to plane Couette flow. Furthermore, whereas the important balance suggests that the Taylor vortices ought to propagate as waves towards the equator with frequency proportional to latitude, the realised solutions are found to exist as pulses, each locked to a discrete frequency, of spatially modulated Taylor vortices. Collectively they form a pulse train. Thus the expected continuous spatial variation of the frequency is broken into steps (forming a staircase) on which motion is dominated by the local pulse. A wealth of solutions of our CGLe have been found and some may be stable. Nevertheless, when higher-order terms are reinstated, solutions are modulated on a yet longer length scale and must evolve. So, whereas there is an underlying pulse structure in the small but finite gap limit, motion is likely to be always weakly chaotic. Our CGLe and its solution provides a paradigm for many geophysical and astrophysical flows capturing in minimalistic form interaction of phase mixing , diffusion and nonlinearity .

Atomistic-Continuum Hybrid Simulation of Heat Transfer Between Argon Flow and Copper Plates

A simulation work aiming to study heat transfer coefficient between argon fluid flow and copper plate is carried out based on atomistic-continuum hybrid method. Navier–Stokes equations for continuum domain are solved through the pressure implicit with splitting of operators (PISO) algorithm, and the atom evolution in molecular domain is solved through the Verlet algorithm. The solver is validated by solving Couette flow and heat conduction problems. With both momentum and energy coupling method applied, simulations on convection of argon flows between two parallel plates are performed. The top plate is kept as a constant velocity and has higher temperature, while the lower one, which is modeled with FCC copper lattices, is also fixed but has lower temperature. It is found that the heat transfer between argon fluid flow and copper plate in this situation is much higher than that at macroscopic when the flow is fully developed.

A hybrid multiscale dissipative particle dynamics method coupling particle and continuum for complex fluid

A hybrid multiscale dissipative particle dynamics (HMDPD) method is proposed for simulating complex fluid, in which dissipative particle dynamics (DPD) is used in the mesoscale flow region and continuum Navier–Stokes equations in the macroscale one. In particular, the variational multiscale method is introduced to overcome the numerical instabilities of the finite element method for solving the Navier–Stokes equations. The spatial coupling between continuum equations and DPD is achieved through constrained dynamics in an overlap region. The Couette flow problem is simulated to verify the proposed HMDPD method. Then, the method is applied to simulate extremely dilute polymer solution in Couette flow. In order to more accurately imitate the polymer chain in the dilute polymer solution, we develop a modified finitely extensible nonlinear elastic chain model, which perfectly describes both the elastic tension and the elastic repulsion between the adjacent beads with bond length as the equilibrium length of one segment. It is found that, using the proposed HMDPD method, not only the macroscopic flow velocity can be calculated, but also the mesoscopic molecular details such as conformational changes as well as dynamic behaviors of polymer chain under the influence of shear flow field can be observed.

Couette flow and heat transfer between parallel plates in a rarefied gas

The effectiveness of the self-similar interpolation method in its simplest form is demonstrated by solving the problem of a planar Couette flow in a rarefied gas between plates with different temperatures. An analytic representation of the solution is compared with the results obtained by the DSMC method. The most interesting result is the nonmonotonicity of the heat flow and the change in its sign accompanying the modification in the gas rarefaction, i.e., in the Knudsen number Kn.

Molecular Dynamics–Continuum Hybrid Simulation for the Impingement of Droplet on a Liquid Film

A molecular dynamics–continuum hybrid method is used to study the droplet impingement process on a liquid film. The hybrid code is validated by simulating the sudden-start Couette flow and unsteady heat transfer problem. The impingement process is strongly affected by Weber number. At low Weber number, the evolution of the crown after droplet impingement is stable, while at high Weber number the secondary droplets emerge and the splash phenomenon occurs. The effect of liquid film thickness on the evolution of crown diameter is also investigated.

In an attempt to generalize wall slip in fluid flows using a series expansion of the wall shear stress: Case of non-Newtonian [Phan–Thien–Tanner fluid

In this paper, we address a new wall slip formulation based on a series expansion of both differential and exponential forms of wall shear stress. As this is initially considered as an infinite series, we use a truncated version of it to get a slip law having three slip coefficients, where we have considered the exponential form only. We use our new truncated triple-slip-coefficients wall slip law to analyze a particular class of viscoelastic non-Newtonian fluid. This special case follows the Phan–Thien–Tanner fluid model. Moreover, for this fluid, we identify Weissenberg number as one salient parameter affecting slip. In particular, our new model is tested on a familiar planar Couette and planar Poiseuille flow problem, where two parallel and infinitely long plates have been used. Furthermore, slip-dependent analytical expressions for field quantities such as velocity and shear stress at the walls and beyond, are derived and numerically analyzed for both problems. In addition, the pressure and flow-rate are derived for various slip coefficients for Poiseuille flow case.Our results obtained prove reasonable as represented by physically realistic plots herein. This feasibility is especially dictated by the velocity profile across channel width for both application problems. Thus, we can infer from all these, that this new model has the potential of providing results which can match experimental data, that is, if the three slip-coefficients are properly chosen.

The development of thermal lattice Boltzmann models in incompressible limit

In this paper, an incompressible two-dimensional (2-D) and three-dimensional (3-D) thermohydrodynamics for thelattice Boltzmann scheme are developed. The basic idea is to solve the velocity field and the temperature field usingtwo different distribution functions. A derivation of the lattice Boltzmann scheme from the continuous Boltzmannequation for 2-D is discussed in detail. By using the same procedure as in the derivation of the discretised densitydistribution function, we found that new lattice of four-velocity (2-D) and eight-velocity (3-D) models for internal energydensity distribution function can be developed where the viscous and compressive heating effects are negligible.These models are validated by the numerical simulation of the 2-D porous plate Couette flow problem where theanalytical solution exists and the natural convection flows in a cubic cavity.

Analytical Solution of the Couette Flow Problem for Arbitrary Values of the Knudsen Number

An analytical (in the form of the Neumann series) solution of the Couette flow problem has been constructed with the use of the kinetic approach. For the basic equation the Williams equation was used, and for the boundary condition on the channel walls the mirror-diffuse model was applied. With account for the constructed distribution function the value of the gas mass flow per unit width of the channel through its upper half has been calculated and a viscous stress tensor component other than zero has been found. A comparison with analogous results obtained by numerical methods has been made.

Analytical solution of plane Couette flow in the transition regime and comparison with Direct Simulation Monte Carlo data

This paper provides analytical solution for the steady-state Couette flow problem in the transition flow regime, while capturing the non-linear Knudsen layer near the walls. Slope at the center obtained from Direct Simulation Monte Carlo (DSMC) data and inherent symmetry in the problem have been utilized for obtaining the solution. A detailed study of the solutions obtained from the linearized super-Burnett, augmented Burnett and R13 equations is presented. The analytical results are compared against DSMC data; good agreement between them is shown till Kn=10. These are among the first set of analytical solutions in the transition regime. The results indicate that the solution tends to become linear as the Knudsen number increases. The results have allowed formulation of a slip relationship, which can potentially yield more accurate slip velocity than Maxwell’s slip model in the transition regime. Our analysis suggests that the Knudsen number envelope over which the R13 and higher-order continuum equations can be employed is substantially extendable.

Linear stability of the Couette flow of a vibrationally excited gas. 1. Inviscid problem

Stability of the Couette flow of a vibrationally excited diatomic gas with a parabolic profile of static temperature is studied within the framework of the linear theory. A set of explicit asymptotic estimates are obtained for inviscid disturbances described by a system of linearized equations of two-temperature gas dynamics. It is shown that the first Rayleigh condition (theorem) is satisfied for unstable modes, and the classification of inviscid modes into even and odd modes is valid. A generalized condition of the presence of an inflection point on the velocity profile, which is necessary for disturbances to evolve, is obtained. The sufficient condition in Howard’s semi-circle theorem is refined. Complex phase velocities of two-dimensional even and odd inviscid modes are numerically calculated as functions of the Mach number, degree of excitation of vibrational levels of energy, and characteristic relaxation time. In the Couette flow problem, in contrast to the case of a free shear layer, the growth rate of the most unstable second mode increases with increasing Mach number and tends to a certain limit for which an asymptotic expression in the form of an ordinary differential equation is obtained. The calculated results show that the effect of reduction of the growth rate on the background of the relaxation process is clearly expressed in the range of flow parameters considered.

Couette Flow Problem for an Unsteady MHD Fourth-Grade Fluid with Hall Currents

In this work, we analyze Couette flow problem for an unsteady magnetohydrodynamic (MHD) fourth-grade fluid in presence of pressure gradient and Hall currents. The existing literature on the topic shows that the effect of Hall current on Couette flow of an unsteady MHD fourth-grade fluid with pressure gradient has not been investigated so far. The arising non-linear problem is solved by the homotopy analysis method (HAM) and the convergence of the obtained complex series solution is carefully analyzed. The influence of pressure number, Hartmann number, Hall parameter and fourth-grade material parameters on the unsteady velocity is discussed through plots and on local skin-friction coefficient discussed through numerical values presented in tabular form.

Investigating the Applicability of Combined Lattice Boltzmann-Smoothed Profile Methods in Particulate Systems

In this article, a combination of Lattice-Boltzmann method (LBM) and smoothed profile method (SPM) are used in simulation of one, two, and many particles’ motion in fluid flow. SPM is used as the simulation method for particles motion. A shear flow is produced using a) solid walls for which standard bounce-back (SBB) boundary condition is applied and b) Lees-Edwards boundary conditions (LEBC). Simulation of Couette flow problem of fluid-only system reveals the accuracy of both methods. LBM-SPM coupled with LEBC is applied for flow geometries of one and two particles in a shear flow. Comparison of results obtained from simulation of one and two interacting particles using moving walls with SBB with those in the literature shows good correspondence. In addition, for the case of many particles, the effective viscosity of a suspension of circular particles which are placed randomly between two parallel walls is investigated from dilute to dense regimes. Relative bulk viscosity is compared with theoretical and semi-empirical results of other investigators and satisfactory agreement is found. Finally, a combination of SPM-LBM is examined for sedimentation of one, two and many particles. Numerical results show suitability of LBM-SPM with LEBC for simulation of particles suspended in flow.

Simulation of transport processes in the Couette flow problem under incomplete accommodation of the tangential momentum of gas molecules by channel walls

The Couette flow problem has been used as an example for the proposed analytical method for calculating macro parameters of gas in channels, the thickness of which is comparable with the mean free path of gas molecules by means of simple numerical procedures. As a basic equation, the linearized BGK (Bhatnagar-Gross-Krook) model of the Boltzmann kinetic equation is used, and the boundary condition on the channel walls is taken to be the model of a specular-diffuse reflection. For different values of the channel thickness and the coefficient of accommodation of the tangential momentum of gas molecules by the channel walls, the profiles of the gas mass velocity in the channel have been constructed and the values of the nonzero component of the viscous stress tensor and the gas mass flow per unit channel width have been calculated. A comparison with similar results published in the press has been made.

Efficient Analytical Approaches For Motion Of A Spherical Solid Particle In Plane Couette Fluid Flow Using Nonlinear Methods

In this study, we approach a spherical particle in plane Couette fluid flow problem utilizing Adomian’s decomposition method (ADM) as well as variational iteration method (VIM) to find a rapidly convergent power series solution. Equation of particle’s motion in Couette flow considering the rotation and shear effects on lift force and neglecting gravity has been investigated by Vander Werff. The required time and distance for a spherical particle to reach terminal velocity trajectory of particle obtained which has application in transferring the medicine in blood in medical area or control of particles motion during spraying or injecting processes in industry. The precious contribution of the work is introducing a new fast and efficient solution of analytical methods in a spherical particle in plane Couette fluid flow over the previous numerical and analytical counterpart results in literature, while it is shown that both methods give approximations of a high degree of accuracy and least computational effort for studying particle motion in Couette fluid flow.

Analysis of steady flows in viscous fluid with heat/mass transfer and slip effects

The goal of present article is to discuss the effects of heat and mass transfer with slip on the Couette and generalized Couette flow in a homogeneous and thermodynamically compatible third grade non Newtonian viscous fluid. Exact solutions of velocity and temperature in Couette flow problem are derived. The nonlinear analysis for generalized Couette flow problem is performed by using spectral homotopy analysis method (SHAM). Convergence and residual error are shown explicitly. The obtained solutions satisfy the boundary conditions and the governing equations. The dimensionless Nusselt numbers at the walls for various involved parameters are also given in each case. Graphical illustrations are shown for certain embedded parameters of interest in the derived solutions.

Compressibility and rarefaction effects on entropy and entropy generation in micro/nano Couette flow using DSMC

In the present work, compressible flow of argon gas in the famous problem of Couette flow in micro/nano-scale is considered and numerically analyzed using the direct simulation Monte Carlo (DSMC) method. The effects of compressibility and rarefaction on entropy and entropy generation in terms of viscous dissipation and thermal diffusion are studied in a wide range of Mach and Knudsen numbers and the observed physics are discussed. In this regard, we computed entropy by using its kinetic theory formulation in a microscopic way while the entropy generation distribution is achieved by applying a semi-microscopic approach and thoroughly free from equilibrium assumptions. The results of our simulations demonstrated that the entropy profiles are in accordance with the temperature profiles. It is also illustrated that the increase of Mach number will result in non-uniform entropy profiles with increase in the vicinity of the central regions of the channel. Moreover, generation of entropy in all regions of the domain stages clear growth. By contrast, increasing the Knudsen number has inverse effects such as: uniform entropy profiles and a falling off in entropy generation amount throughout the channel.