Couette Flow Problem Research Articles

Laplace transform solution of the time-dependent annular Couette flow with dynamic wall slip

The annular Couette flow has several industrial applications, particularly for the characterization of the fluid flow and deformation behavior of fluids. The inclusion of the dynamic wall slip into the flow boundary conditions seems to be necessary for an efficient continuum description of motion of nanofluidics as it reflects the importance of fluid–structure interface related phenomena. Dynamic wall slip introduces a dissipative boundary condition and thus increases the difficulties of finding solutions to related problems. In the present work we investigate the behavior of fluid flow between two infinitely long coaxial circular cylinders, when the inner cylinder is axially moving due to sudden constant velocity, while the outer cylinder is held stationary. The boundary condition on the outer cylinder is that of dynamic wall slip, in addition to the usual Navier slip. The medium considered here is a Newtonian viscous fluid. The solution of the governing equations, initial and boundary conditions for this flow is obtained using the Laplace transform technique and inversion by Laguerre polynomials. This method may be useful, when applied in conjunction with perturbation methods, to solve nonlinear Couette flow problems involving temperature changes. Numerical results are presented and discussed.

Volume of fluid based modeling of thermocapillary flow applied to a free surface lattice Boltzmann method

Thermocapillary flow, also termed as thermal Marangoni convection, is an important material transport mechanism in many technical processes. Thus, it has to be considered in the corresponding simulations. In this work, a novel numerical model for the implementation of thermocapillary flow into a volume of fluid based free surface lattice Boltzmann method is presented. It is based on the idea of discretizing the surface normal vector in accordance to the discretization scheme of the velocity space. This approach is used to interpolate the temperature at the surface and to calculate the local surface gradients.The result is a simple and efficient numerical model, which is able to calculate the Marangoni stress along a liquid surface with sufficient accuracy and without the need to explicitly reconstruct the interface. We show that the Marangoni stress can readily be incorporated into a free surface lattice Boltzmann model by adjusting the boundary condition at the surface. In doing so, no additional forcing scheme has to be applied.In order to verify the presented model, it is tested using several simulation setups. As the lattice Boltzmann method is particularly suitable for complex geometries, a special attention is paid to the behavior at curved surfaces. Additionally, a simple test case for thermocapillary flow that can be regarded as a modified version of the Couette flow problem, is presented.

Nonlinear inviscid damping and shear‐buoyancy instability in the two‐dimensional Boussinesq equations

AbstractWe investigate the long‐time properties of the two‐dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size ε. Under the classical Miles‐Howard stability condition on the Richardson number, we prove that the system experiences a shear‐buoyancy instability: the density variation and velocity undergo an inviscid damping while the vorticity and density gradient grow as . The result holds at least until the natural, nonlinear timescale . Notice that the density behaves very differently from a passive scalar, as can be seen from the inviscid damping and slower gradient growth. The proof relies on several ingredients: (A) a suitable symmetrization that makes the linear terms amenable to energy methods and takes into account the classical Miles‐Howard spectral stability condition; (B) a variation of the Fourier time‐dependent energy method introduced for the inviscid, homogeneous Couette flow problem developed on a toy model adapted to the Boussinesq equations, that is, tracking the potential nonlinear echo chains in the symmetrized variables despite the vorticity growth.

Development of a scattering model for diatomic gas–solid surface interactions by an unsupervised machine learning approach

This work proposes a new stochastic gas–solid scattering model for diatomic gas molecules constructed based on the collisional data obtained from molecular dynamics (MD) simulations. The Gaussian mixture (GM) approach, which is an unsupervised machine learning approach, is applied to H2 and N2 gases interacting with Ni surfaces in a two-parallel wall system under rarefied conditions. The main advantage of this approach is that the entire translational and rotational velocity components of the gas molecules before and after colliding with the surface can be utilized for training the GM model. This creates the possibility to study also highly nonequilibrium systems and accurately capture the energy exchange between the different molecular modes that cannot be captured by the classical scattering kernels. Considering the MD results as the reference solutions, the performance of the GM-driven scattering model is assessed in comparison with the Cercignani–Lampis–Lord (CLL) scattering model in different benchmarking systems: the Fourier thermal problem, the Couette flow problem, and a combined Fourier–Couette flow problem. This assessment is performed in terms of the distribution of the velocity components and energy modes, as well as accommodation coefficients. It is shown that the predicted results by the GM model are in better agreement with the original MD data. Especially, for H2 gas the GM model outperforms the CLL model. The results for N2 molecules are relatively less affected by changing the thermal and flow properties of the system, which is caused by the presence of a stronger adsorption layer.

Kinetic and thermodynamic examinations for the unsteady couette flow problem of a plasma using the BGK cylindrical model

We aim to investigate the kinetic and irreversible thermodynamic treatment of the Couette flow of gaseous plasma (GP) confined between two coaxial rotating circular rigid cylinders. The Bhatnagar-Gross-Krook (BGK) model of the Boltzmann kinetic equation (BKE) is solved by applying the perturbation method coupled with the moments' approach with a two-stream Maxwellian distribution function (MDF). Nonlinear partial differential equations are precisely solved. As an actual application, we are considering the laboratory Argon GP. An interesting comparison between the non-equilibrium electron velocity distribution function (EVDF) and the equilibrium EVDF is made carefully with 3-Dimensional graphics in various time values. We found that the system goes to an equilibrium state (ES) with time compatible with Le Chatelier's principles. Moreover, we determined with very high precision the value of the system equilibrium time, which is tequ.≅2.4. That great advantage cannot be reached by solving other macroscopic magnetohydrodynamic models. The relations between the various macroscopic variables of the GP are studied. The irreversible non-equilibrium thermodynamics (NT) properties of the system are presented. We present the extended Gibbs equation (EGE) for the internal energy variation (IEV) in cylindrical geometry for GP for the first time, to the best of our knowledge. We have shown that our model is compatible with the H-theorem of Boltzmann and the Onsager-Casimir relations. The various participations in IEV are introduced. The significance of this search is due to its vast implementations in various fields, such as in GP physics, electrical engineering, medicine, biological systems, and numerous commercial and industrial deployments.

High-Order Spectral/-Based Solver for Compressible Navier–Stokes Equations

High-order spectral element methods are becoming more established for solving problems in computational fluid dynamics. In this paper, we introduce new stabilization parameters for the solution of compressible Navier–Stokes equations in conjunction with high-order spectral element methods. We previously defined a new set of stabilization parameters for Euler equations in a high-order spectral/ framework. In this paper, we examine the definition of these stabilization parameters in the context of solutions of full Navier–Stokes equations, and we report good agreement with previously published results in the literature. We establish spectral convergence of errors for Kovasznay flow. We then validate the definition of the stabilization parameter for Couette flow, flat plate, and compression corner problems. In a later example, we solve the flow past a cylinder and benchmark results with previously published results obtained with low-order methods and other approaches in compressible flow computations. We further solve hypersonic flow past a wedge and supersonic flow past a Flight Investigation of Re-Entry program (FIRE) entry capsule. We explore the definition of the shock-capturing parameter based on the parameter. The introduced definitions are found to provide excellent results for the full Navier–Stokes equations.

Size-dependent steady creeping microfluid flow based on the boundary element method

In order to investigate size-dependent creeping plane microfluidic flow, a boundary element method is implemented that involves the calculation of interior quantities and multi-domain problems. The governing equations are formulated using the skew-symmetric character of the couple-stress tensor and its energy conjugate mean-curvature tensor, as established in consistent couple stress theory. This theoretical formulation includes one characteristic material length scale parameter l that represents the size-dependency of the problem. Here, we present the boundary integral representation and numerical implementation for two-dimensional size-dependent steady state creeping incompressible flow, in which velocities, angular velocities, force- and couple-tractions are the primary variables. Beyond the previously known singular fundamental solution kernels for point force and point couple in the velocity and angular velocity integral equations, here we present the integral equations for calculating internal mean curvatures and couple-stresses, along with their corresponding kernels. The formulation is then applied to solve some computational problems both to investigate the size effects on the response resulting from the theory, and to validate the strength of the numerical method. For this purpose, we study the important effect of different boundary conditions on the flow pattern and consider a multi-fluid domain Couette flow problem.

A comparison of different numerical schemes in spherical Couette flow simulation

We compare the performance of line Gauss–Seidel (LGS), point Gauss–Seidel (PGS), and alternating direction implicit (ADI) linear solvers used in the artificial compressibility method for the numerical solution of the three-dimensional incompressible Navier–Stokes equations. Spatial discretization is carried out using a fifth-order WENO scheme for the convective terms and a second-order central difference scheme for the viscous terms. A comparison is made by simulating the spherical Couette flow problem, with only the inner sphere rotating and the outer one fixed. OpenMP is used for numerical computation in parallel for the three schemes. First, we compare the numerical efficiency of the solvers by computing 0-vortex flow for a medium-gap σ=R2−R1/R1=0.25. Second, we make a residual comparison for a steady-state flow calculation based on the CFL number and artificial compressibility factor. Finally, we compare the three solvers for unsteady flow computations based on the artificial compressibility factor. The results show that the LGS solver is more reliable than the PGS and the ADI solvers. To show the accuracy of the LGS scheme, we compute different flow modes for an intermediate-gap clearance ratio σ=R2−R1/R1=0.14. The computed results have good agreement with the existing numerical results.

On the entropic property of the Ellipsoidal Statistical model with the prandtl number below 2/3

Entropic property of the Ellipsoidal Statistical model with the Prandtl number Pr below 2/3 is discussed. Although 2/3 is the lower bound of Pr for the H theorem to hold unconditionally, it is shown that the theorem still holds even for $\mathrm{Pr}<2/3$, provided that anisotropy of stress tensor satisfies a certain criterion. The practical tolerance of that criterion is assessed numerically by the strong normal shock wave and the Couette flow problems. A couple of moving plate tests are also conducted.

Formation and Topology of vortices in Couette Flow over open cavities

The present study investigates the planar Couette flow problem for low Reynolds numbers inside a rectangular duct with a morphing cavity serving as a vortex formation promoter. A finite element code implemented in COMSOL Multiphysics is employed to analyze the effects of the cavity aspect ratio and variations of the Reynolds number on formation and topology of the vortices within the embedded cavity. The obtained results indicate that the cavity height is influential in the number of vortices. It is shown by increasing the Reynolds number, a single vortex tends to move towards the outlet. In addition, streamlines demonstrate that small vortices in vicinity of the cavity corner tend to be enlarged with increase of the Reynolds number. The developed numerical model can be extended to the flow structure of natural systems such as an embayment subjected to parallel-to-shore currents.

Energy redistribution dynamics in coupled Couette–Poiseuille flows using large-Eddy simulation

The problem of turbulent Couette flow driven by a statistically steady external wind is studied in the framework of spatially filtered Navier–Stokes equations. The phenomenon of wind-driven flow of water is represented by a layer of air modeled as Poiseuille flow (air sub-domain), coupled to a layer of water modeled as Couette flow (water sub-domain). We focus on changes in the statistics in either the air or the water sub-domain, due to the coupling with the other sub-domain. We also highlight dynamic flow structures forming near the air-water interface. Simulations based on different Reynolds numbers in the air and the water sub-domains are compared to computationally less demanding simulations with equal Reynolds numbers. Results of these simulations indicate strong similarities, i.e., the flow is well approximated by simulating air and water at the same Reynolds numbers. Further analysis shows that the flow in the water domain shares important features with classical Couette flows. The horizontal turbulent mixing renders a thinner boundary layer in the water sub-domain. Moreover, an increased intermittency in the flow velocities is observed, which may be linked to so-called splat events near the air-water interface. These splats characterize the interaction of coherent structures across the interface, being stronger in the water phase. An analysis of the pressure-strain correlation near the air-water interface on the water side shows that such splats are responsible for redistributing energy from the streamwise and spanwise directions, to the vertical direction. This behavior, although qualitatively similar to wall-bounded flows, differ mainly on the fact that most of the energy drained comes from the streamwise direction: in wall-bounded the main contributor is the spanwise direction. The boundary layers near the air-water interface show inclined vortical structures. Unlike in coupled Couette–Couette flow, the peak in the Reynolds stress is displaced from the channel’s center into the buffer region of the water sub-domain.

Chebyshev polynomials for generalized Couette flow of fractional Jeffrey nanofluid subjected to several thermochemical effects

The generalized Couette flow of Jeffrey nanofluid through porous medium, subjected to the oscillating pressure gradient and mixed convection, is numerically simulated using variable-order fractional calculus. The effect of several involving parameters such as chemical reactions, heat generation, thermophoresis, radiation, channel inclination, and Soret effect is also considered. To the best of the authors’ knowledge, the described general form of the Couette flow problem is not tackled by the researchers yet. The non-dimensional form of heat, mass, and momentum equations is solved as a coupled set. The effect of several parameters such as Grashof, Hartmann, Prandtl, Soret, and Schmidt numbers in addition to oscillation frequency, retardation time, radiation, heat absorption, and reaction rate are determined and presented graphically. An operational matrix method based on the second kind shifted Chebyshev polynomials is proposed to investigate the behavior of the interested problem. In fact, regarding the established method, the unknown solutions are expanded by the mentioned basis polynomials. Then, the operational matrix of the variable-order fractional derivative is utilized to transfer the problem into solving an algebraic system of equations. According to the obtained results, the growth of fractional order from 0 to 1 changes the skin friction coefficient, Nu and flow rate by $$-18.1$$ , 35.5, and $$10\%$$ , respectively.

Steady planar Couette flow of rarefied binary gaseous mixture based on kinetic modeling

The planar Couette flow of binary gaseous mixtures is studied on the basis of the kinetic model introduced by Kosuge (2009). The calculations were carried out for a wide range of the gas rarefaction, various values of the mole fraction and for two values of the wall speed corresponding to linear and non-linear Couette flow configurations. The kinetic model shear stress, number density, temperature and the mole fraction have been found to be in very good agreement with corresponding ones obtained by the DSMC method. However, the difference in the velocity profiles obtained by the Kosuge model and the DSMC method can be large, especially near the free molecular regime. Also, very good agreement between the Kosuge model results and the Chapman–Enskog solutions is observed. Moreover, the influence of the intermolecular potential is investigated by comparing the results of the Hard-Sphere model with those for a Realistic potential. It was shown that the shear stress significantly depends on the intermolecular potential when the wall speed is large. Finally, applying the equivalent single gas approach it is deduced that this concept can be applied in binary gas mixture Couette flows at any pressure when the molecular mass ratio approaches unity, but only at high pressure when the ratio is large. • The Couette flow problem is solved on the basis of the Kosuge model. • Very good agreement between Kosuge model and the DSMC method has been provided for the shear stress, number density, temperature and the mole fraction. • The differences in the velocity distributions of the individual species can be large. • The influence of the intermolecular potential is investigated. • The validity of the equivalent single gas approach is analyzed.

Numerical solutions of 2-D steady compressible natural convection using high-order flux reconstruction

Low Mach number flows are common and typical in industrial applications. When simulating these flows, performance of traditional compressible flow solvers can deteriorate in terms of both efficiency and accuracy. In this paper, a new high-order numerical method for two-dimensional (2-D) state low Mach number flows is proposed by combining flux reconstruction (FR) and preconditioning. Firstly, a Couette flow problem is used to assess the efficiency and accuracy of preconditioned FR. It is found that the FR scheme with preconditioning is much more efficient than the original FR scheme. Meanwhile, this improvement still preserves the numerical accuracy. Using this new method and without the Boussinesq assumption, classic natural convection is directly simulated for cases of small and large temperature differences. For the small temperature difference, a p and h refinement study is conducted to verify the grid convergence and accuracy. Then, the influence of the Rayleigh number (Ra) is analyzed. By comparing with the reference results, the numerical results of preconditioned FR is very close to that calculated by incompressible solvers. Furthermore, a large temperature difference test case is calculated and analyzed, indicating this method is not limited by the Boussinesq assumption and is also applicable to heat convection with large temperature differences.

Particle-based fluid dynamics: Comparison of different Bhatnagar-Gross-Krook models and the direct simulation Monte Carlo method for hypersonic flows

The Bhatnagar-Gross-Krook (BGK) model and its extensions (ellipsoidal statistical BGK, Shakhov BGK, and unified BGK) are used in particle-based fluid dynamics and compared with the Direct Simulation Monte Carlo (DSMC) method. To this end, different methods are investigated that allow efficient sampling of the Shakhov and the unified target distribution functions. As a consequence, particle simulations based on the Shakhov BGK and unified BGK models are possible in an efficient way. Furthermore, different energy conservation schemes are tested for the BGK models. It is shown that the choice of the energy conservation scheme has a major impact on the quality of the results for each model. The models are verified with a Couette flow problem at different Knudsen numbers and wall velocities. Furthermore, the models are compared with the DSMC results of a hypersonic flow around a 70° blunted cone. It is shown that the unified BGK model is able to reproduce rarefied gas phenomena. Furthermore, it is shown that the difference in the reproduction of the shock structure is not significant between the ellipsoidal statistical BGK and Shakhov BGK models for the flow around a 70° blunted cone and significantly depends on the energy conservation method. The choice of the energy conservation method is especially crucial for the Shakhov model whereas the ellipsoidal statistical BGK model is much more robust concerning the energy conservation scheme. Additionally, a computational time study is performed to show the efficiency of BGK-based simulations for low Knudsen number flows compared with DSMC.

Simple methods for fluidic drag estimation during pipe installation via HDD

This paper proposes a series of simple methods for fluidic drag evaluation of non-Newtonian slurries flowing within a bore during Horizontal Directional Drilling (HDD) pipe installation operations. The methods are based on adapting the solutions of annular and planar Couette flow problems to the HDD pullback problem and are favorable for practical uses by the HDD industry. In comparison to the existing methods, the proposed method accounts for a wide range of parameters, including slurry rheology, bore geometry, and pullback rate. For verification purposes, verified fluidic drag forces for two actual HDD crossings have been used. Comparison of the forces obtained via different methods confirms the ability of the new simple methods to capture the fluidic drag changes accurately.

Multiscale simulation of polymeric fluids using the sparse grid combination technique

We present a computationally efficient sparse grid approach to allow for multiscale simulations of non-Newtonian polymeric fluids. Multiscale approaches for polymeric fluids often involve model equations of high dimensionality. A conventional numerical treatment of such equations leads to computing times in the order of months even on massively parallel computers.For a reduction of this enormous complexity, we propose the sparse grid combination technique. Compared to classical full grid approaches, the combination technique strongly reduces the computational complexity of a numerical scheme but only slightly decreases its accuracy.Here, we use the combination technique in a general formulation that balances not only different discretization errors but also considers the accuracy of the mathematical model. For an optimal weighting of these different problem dimensions, we employ a dimension-adaptive refinement strategy. We finally verify substantial cost reductions of our approach for simulations of non-Newtonian Couette and extensional flow problems.

A complex-valued first integral of Navier-Stokes equations: Unsteady Couette flow in a corrugated channel system

For a two-dimensional incompressible viscous flow, a first integral of the governing equations of motion is constructed based on a reformulation of the unsteady Navier-Stokes equations in terms of complex variables and the subsequent introduction of a complex potential field; complementary solid and free surface boundary conditions are formulated. The methodology is used to solve the challenging problem of unsteady Couette flow between two sinusoidally varying corrugated rigid surfaces utilising two modelling approaches to highlight the versatility of the first integral. In the Stokes flow limit, the results obtained in the case of steady flow are found to be in excellent agreement with corresponding investigations in the open literature. Similarly, for unsteady flow, the results are in accord with related investigations, exploring material transfer between trapped eddies and the associated bulk flow, and vice versa. It is shown how the work relates to the classical complex variable method for solving the biharmonic problem and perspectives are provided as to how the first integral may be further utilised to investigate other fluid flow features.

Analytic and Numerical Solutions of Couette Flow Problem: A Comparative Study

In this work, an incompressible viscous Couette flow is derived by simplifying the Navier-Stokes equations and the resulting one dimensional linear parabolic partial differential equation is solved numerically employing a second order finit difference Crank-Nicolson scheme. The numerical solution and the exact solution are presented graphically.Journal of the Institute of Engineering, 2016, 12(1): 105-113

Finite Difference Solver for Couette Flow with Applied Pressure Gradients

A finite difference scheme was used to obtain solutions to a two dimensional CouetteFlow problem with applied pressure gradients. The solution to the non dimensionalizedCouette Flow problem was then obtained via Implicit methods and the results were subsequently validated with the analytical solution of the same problem. Finite difference scheme used returned satisfactory results under varying input conditions and was successfully validated on the Couette Flow problem through comparison with the analytical solution.