Shear Flow Problem Research Articles

Numerical discretisation of hyperbolic systems of moment equations describing sedimentation in suspensions of rod-like particles

We present a numerical discretisation of the coupled moment systems, previously introduced in Dahm and Helzel [1], which approximate the kinetic-fluid model by Helzel and Tzavaras [2] for sedimentation in suspensions of rod-like particles for a two-dimensional flow problem and a shear flow problem. We use a splitting ansatz which, during each time step, separately computes the update of the macroscopic flow equation and of the moment system. The proof of the hyperbolicity of the moment systems in [1] suggests solving the moment systems with standard numerical methods for hyperbolic problems, like LeVeque's Wave Propagation Algorithm [3]. The number of moment equations used in the hyperbolic moment system can be adapted to locally varying flow features. An error analysis is proposed, which compares the approximation with 2N+1 moment equations to an approximation with 2N+3 moment equations. This analysis suggests an error indicator which can be computed from the numerical approximation of the moment system with 2N+1 moment equations. In order to use moment approximations with a different number of moment equations in different parts of the computational domain, we consider an interface coupling of moment systems with different resolution. Finally, we derive a conservative high-resolution Wave Propagation Algorithm for solving moment systems with different numbers of moment equations.

A shear flow problem for compressible viscous micropolar fluid: Uniqueness of a generalized solution

In this paper, we consider the nonstationary shear flow of a compressible, viscous, and heat‐conducting micropolar fluid. The mathematical model is set up in Lagrangian description in the form of initial‐boundary problem with inhomogeneous boundary conditions for velocity and standard homogeneous boundary conditions for microrotation and heat flux. Under the assumptions that this problem has a generalized solution and that the initial mass density, temperature, the velocity, and microrotation vectors are smooth enough functions, we prove that this solution is unique.

A shear flow problem for compressible viscous micropolar fluid: Derivation of the model and numerical solution

In this paper we consider the nonstationary shear flow between two parallel solid and thermoinsulated horizontal plates with the upper one moving irrotationally. The fluid is compressible, micropolar, viscous and heat-conducting, as well as in the thermodynamical sense perfect and polytropic. We assume that, given a Cartesian coordinate system x, y and z, solutions of corresponding problem are x-dependent only. Mathematical model is derived in the Lagrangian description. By using the Faedo–Galerkin method, as well as homogenization of boundary conditions, we derive an approximate system, which we use to obtain a numerical solution to the given problem.

A solid-liquid local thermal non-equilibrium lattice Boltzmann model for heat transfer in nanofluids. Part I: Model development, shear flow and heat conduction in a nanofluid

A solid-liquid local thermal non-equilibrium lattice Boltzmann model for hydrodynamics and heat transfer in a nanofluid is developed in this paper. In this proposed model, interactions between fluid and solid nanoparticles, random motion of nanoparticles as well as heat transfer between nanoparticles and the base fluid are taken into consideration. This novel model is applied to two simple nanofluids problems: isothermal shear flow between two parallel plates moving at different velocities, and heat conduction between two parallel plates at different temperatures. For the problem of shear flow, it is found that nanoparticles random motion causes the instantaneous velocity distribution of a nanofluid to become non-linear in the shear flow between two parallel plates at microscopic level. At macroscopic level, it is found that as nanoparticles volume fraction increases, effects of nanoparticles random motion are enlarged and the time-average velocity distribution of a nanofluid deviates more from a linear distribution in a shear flow. For the problem of heat conduction between parallel plates at different temperatures, it is found that random motion of nanoparticles together with their high thermal diffusivity and thermal conductivity, causing the instantaneous temperature distribution in the nanofluid to become non-linear between two parallel plates at different temperatures. Temperature inside a nanoparticle is shown to be non-uniform, and temperature gradients in the fluid near a nanoparticles are elevated, thus heat transfer rate is enhanced near a nanoparticle. Time-average temperature gradients of a nanofluid at hot/cooled wall are higher than those of a pure fluid in heat conduction between parallel plates at different temperatures. Comparisons of calculated dynamic viscosity and thermal conductivity based on this novel model are found in good agreement with existing correlation equations. Thus, the accuracy of this novel solid–liquid local thermal non-equilibrium model for a nanofluid is validated.

Large eddy simulation of hydrodynamic and magnetohydrodynamic channel flows with a collocated finite-volume scheme and improved subgrid-scale modeling

We study hydrodynamic and magnetohydrodynamic channel flows by means of large eddy simulations (LES) on the basis of a second-order finite-volume scheme in collocated variable arrangement with a focus on the impact of numerical diffusion on the subgrid-scale (SGS) modeling. It is found that a mixed SGS model, which is based on the velocity increment tensor and an eddy viscosity model, performs best and is able to capture near-wall regions of energy backscatter from small to larger scales. Thereby, it improves the accuracy of the LES computations significantly. Our studies suggest that the mixed SGS model is thus applicable for a wide class for shear flow problems in liquid metal flows where finite-volume methods with collocated grids are applied.

Onsager-theory-based dynamic model for nematic phases of bent-core molecules and star molecules

We construct the molecular model and the tensor model for the dynamics of the nematic phases of bent-core molecules and star molecules in incompressible fluid. We start from the molecular interaction and the molecule–fluid friction, and write down a general formulation based on the molecular shape and the free energy. Then we incorporate an Onsager-theory-based static tensor model to obtain the dynamic molecular model fully determined by the molecular architecture. The tensor model is obtained by adopting the quasi-equilibrium approximation that maintains energy dissipation. For bent-core molecules and star molecules that have the same molecular symmetry, the form of the model is identical. The molecular architecture is differentiated by the coefficients that are derived as functions of molecular parameters. Numerical simulation is carried out for the shear flow problem using both the molecular and the tensor models, focusing on the effect of altering molecular architecture. When the equilibrium phase is biaxial, novel flow modes are found, and the flow mode sequences show delicate dependence on the molecular architecture. The tensor model proves to exhibit all the flow modes found in the molecular model.

Modelling non-equilibrium thermodynamic systems from the speed-gradient principle.

The application of the speed-gradient (SG) principle to the non-equilibrium distribution systems far away from thermodynamic equilibrium is investigated. The options for applying the SG principle to describe the non-equilibrium transport processes in real-world environments are discussed. Investigation of a non-equilibrium system's evolution at different scale levels via the SG principle allows for a fresh look at the thermodynamics problems associated with the behaviour of the system entropy. Generalized dynamic equations for finite and infinite number of constraints are proposed. It is shown that the stationary solution to the equations, resulting from the SG principle, entirely coincides with the locally equilibrium distribution function obtained by Zubarev. A new approach to describe time evolution of systems far from equilibrium is proposed based on application of the SG principle at the intermediate scale level of the system's internal structure. The problem of the high-rate shear flow of viscous fluid near the rigid plane plate is discussed. It is shown that the SG principle allows closed mathematical models of non-equilibrium processes to be constructed.This article is part of the themed issue 'Horizons of cybernetical physics'.

Numerical modeling of anisotropic drag for a perforated plate with cylindrical holes

Direct numerical simulations of an incompressible laminar flow past a perforated plate are carried out at the pore scale. With the aim of evaluating the anisotropic drag of a perforated plate, two simple flow tests have been conducted: i) transpiration flow through a perforated plate with a regular array of cylindrical holes and ii) a linear shear flow over the plate. In the transpiration flow test, the dependence of the pressure drop on the transpiration velocity is discussed for porosities of 0.1–0.4 and hole depth to diameter ratios of 1–3 at pore-level Reynolds numbers of up to 25. The linear shear flow problem is then investigated along with a discussion of the effects of porosity, plate thickness, and pore distribution on the slip velocity at a clear fluid/perforated plate interface. Simple correlations of directional permeabilities and non-Darcy coefficients are also proposed for the perforated plate, with their application to a lid-driven cavity flow.

Energetically efficient proportional-integral-differential (PID) control of wake vortices behind a circular cylinder

The control of vortex shedding behind a circular cylinder is a precursor to a wide range of external shear flow problems in engineering, in particular the flow-induced vibrations. In the present study, numerical simulation of an energetically efficient active flow control strategy is proposed, for the control of wake vortices behind a circular cylinder at a low Reynolds number of 100. The fluid is assumed to be incompressible and Newtonian with negligible variation in properties. Reflectionally symmetric controllers are designed such that, they are located on a small sector of the cylinder over which, tangential sliding mode control is imparted. In the field of modern controls, proportional (P), integral (I) and differential (D) control strategies and their numerous combinations are extremely popular in industrial practice. To impart suitable control actuation, the vertically varying lift force on the circular cylinder, is synthesised for the construction of an error term. Four different types of controllers considered in the present study are, P, I, PI and PID. These controllers are evaluated for their energetic efficiency and performance. A linear quadratic optimal control problem is formulated, to minimise the cost functional. By performing detailed simulations, it was observed that, the system is energetically efficient, even when the twin eddies are still persisting behind the circular cylinder. To assess the adaptability of the controllers, the actuators were switched on and off to study their dynamic response.

Numerical simulation on weld line development of injection molding in mold cavity with inserts

A gas-liquid two-phase model for a viscoelastic fluid is proposed and used to simulate and predict the behavior of melt welding in injection molding process, in which the extended pom-pom (XPP) model and cross-WLF viscosity model combined with Tait state equation are used to describe the constitutive relationship and viscosity change of the viscoelastic melt in this paper, respectively. Meanwhile, the coupled level-set and volume-of-fluid (CLSVOF) method is employed to capture the melt front, and the immersed boundary method is applied to the simulation of the polymer melt flows with the aid of a shaped level-set function to describe and treat the irregular mold cavities. A finite volume method on non-staggered grid is used to solve the mass, momentum, and energy conservation equations. Firstly, the benchmark problem of the single shear flow is simulated to verify the validity of the CLSVOF method. Then, the non-isothermal filling process of the viscoelastic fluid based on the XPP model in a mold with square inset is simulated, and the behavior of the weld line devolopment in the filling process is shown and compared with the experimental result. Finally, it is to simulate the evolution processes of the melt front interface and weld line in a mold with the circular notched inset; and the linear stress-optical rule is adopted to calculate the flow-induced birefringence. Numerical results show that the numerical model proposed in this paper can be employed to simulate the non-isothermal filling process in complex mold cavity and to capture the weld line automatically. Because of the complexity of polymer melt flows, the flow-induced stress increases quickly near the weld line region and then decreases gradually until reaching the mold cavity wall. The maximum value of the flow-induced stress appears at some point after the insert. The distributions of physical quantities, such as pressure and temperature in the mold, are given during the mold filling process. Moreover, it is also discussed the influence of melt and mold temperatures on the solidified layer thickness. The higher the melt or mold temperature, the thinner the solidified layer is. Thus, raising the melt or the mold temperature will improve or remove the weld line in melt filling process.

Effective slip lengths for longitudinal shear flow over partial-slip circular bubble mattresses

The problem of longitudinal shear flow over a circular bubble mattress with partial slip and protrusion angle 90o is solved in a quasi-analytical fashion by a novel transform scheme recently devised by the author. The general approach can be readily adapted to other mixed boundary value problems. From the analysis explicit approximations for the effective slip lengths are found as a function of the Navier-slip parameter and the area fraction of the surface covered by protrusions. These new approximation formulas for the slip lengths both unify and extend those based on empirical polynomial fits to numerical data given recently by Ng and Wang (2011 Fluid Dyn. Res. 43 065504).

On the bounds for the complex wave velocity for growing perturbations

For the instability problem of shear flow in sea straits of arbitrary cross section known as extended Taylor-Goldstein problem, we obtained an instability region for the complex wave velocity c = cr+ici with positive imaginary part in the upper half of the (cr, ci) plane. Furthermore, we derived a necessary condition for instability and upper bound for wave number. Mathematics Subject Classification: 76E05

Comparison of boundary slip for two variants of immersed boundary method in lattice Boltzmann framework

In this paper, the Immersed Moving Boundary-Lattice Boltzmann (IMB-LB) method is compared with the single relaxation time and multiple-relaxation-time versions of the Immersed Boundary-Lattice Boltzmann (IB-LB) method in terms of the amount of numerical velocity slip produced on solid boundaries. The comparisons are performed for both straight and curved boundaries based on the effects of thickness of virtual domain used in the IB method for the first time, and relaxation time parameter(s) of the LB method. For the straight boundaries, a shear flow problem is studied while for the curved boundaries, a falling circular cylinder in an infinite channel is investigated. First, sensitivities of the IB-LB and IMB-LB methods to the relaxation time parameter(s) and thickness of virtual domain are presented. Then, the interaction of two solid particles in a shear flow is studied to assess the validity of arguments presented in solving a typical practical problem.

An Efficient Hybrid DSMC/MD Algorithm for Accurate Modeling of Micro Gas Flows

AbstractAiming at simulating micro gas flows with accurate boundary conditions, an efficient hybrid algorithm is developed by combining the molecular dynamics (MD) method with the direct simulation Monte Carlo (DSMC) method. The efficiency comes from the fact that the MD method is applied only within the gas-wall interaction layer, characterized by the cut-off distance of the gas-solid interaction potential, to resolve accurately the gas-wall interaction process, while the DSMC method is employed in the remaining portion of the flow field to efficiently simulate rarefied gas transport outside the gas-wall interaction layer. A unique feature about the present scheme is that the coupling between the two methods is realized by matching the molecular velocity distribution function at the DSMC/MD interface, hence there is no need for one-to-one mapping between a MD gas molecule and a DSMC simulation particle. Further improvement in efficiency is achieved by taking advantage of gas rarefaction inside the gas-wall interaction layer and by employing the “smart-wall model” proposed by Barisiket al.The developed hybrid algorithm is validated on two classical benchmarks namely 1-D Fourier thermal problem and Couette shear flow problem. Both the accuracy and efficiency of the hybrid algorithm are discussed. As an application, the hybrid algorithm is employed to simulate thermal transpiration coefficient in the free-molecule regime for a system with atomically smooth surface. Result is utilized to validate the coefficients calculated from the pure DSMC simulation with Maxwell and Cercignani-Lampis gas-wall interaction models.

Multiscale modeling with smoothed dissipative particle dynamics

In this work, we consider two issues related to the use of Smoothed Dissipative Particle Dynamics (SDPD) as an intermediate mesoscale model in a multiscale scheme for solution of flow problems when there are local parts of a macroscopic domain that require molecular resolution. The first is to demonstrate that SDPD with different levels of resolution can accurately represent the fluid properties from the continuum scale all the way to the molecular scale. Specifically, while the thermodynamic quantities such as temperature, pressure, and average density remain scale-invariant, we demonstrate that the dynamic properties are quantitatively consistent with an all-atom Lennard-Jones reference system when the SDPD resolution approaches the atomistic scale. This supports the idea that SDPD can serve as a natural bridge between molecular and continuum descriptions. In the second part, a simple multiscale methodology is proposed within the SDPD framework that allows several levels of resolution within a single domain. Each particle is characterized by a unique physical length scale called the smoothing length, which is inversely related to the local number density and can change on-the-fly. This multiscale methodology is shown to accurately reproduce fluid properties for the simple problem of steady and transient shear flow.

Lattice Boltzmann Finite Volume Formulation with Improved Stability

AbstractThe most severe limitation of the standard Lattice Boltzmann Method is the use of uniform Cartesian grids especially when there is a need for high resolutions near the body or the walls. Among the recent advances in lattice Boltzmann research to handle complex geometries, a particularly remarkable option is represented by changing the solution procedure from the original “stream and collide” to a finite volume technique. However, most of the presented schemes have stability problems. This paper presents a stable and accurate finite-volume lattice Boltzmann formulation based on a cell-centred scheme. To enhance stability, upwind second order pressure biasing factors are used as flux correctors on a D2Q9 lattice. The resulting model has been tested against a uniform flow past a cylinder and typical free shear flow problems at low and moderate Reynolds numbers: boundary layer, mixing layer and plane jet flows. The numerical results show a very good accuracy and agreement with the exact solution of the Navier-Stokes equation and previous numerical results and/or experimental data. Results in self-similar coordinates are also investigated and show that the time-averaged statistics for velocity and vorticity express self-similarity at low Reynolds numbers. Furthermore, the scheme is applied to simulate the flow around circular cylinder and the Reynolds number range is chosen in such a way that the flow is time dependent. The agreement of the numerical results with previous results is satisfactory.

Radiative transfer in ocean turbulence and its effect on underwater light field

Turbulence in the upper ocean generates fluctuations in temperature and salinity, which result in variations in inherent optical properties (IOPs) and further change the underwater light field. A simulation‐based study is performed for the radiative transfer (RT) of natural light in the turbulent flows in the upper ocean. For a canonical problem of turbulent shear flow interacting with the sea surface with and without surface waves, large‐eddy simulations are performed for fluid motions and the transport of temperature and salinity. Based on the resolved turbulence temperature and salinity fields, IOP variations are quantified, and the inhomogeneous RT equation is then simulated using a Monte Carlo method. Through the simulations of a variety of cases with different flow, temperature, and salinity conditions, the statistics of downwelling irradiance are quantified and analyzed. It is found that the vertical profile of the mean downwelling irradiance is mainly determined by the vertical structure of the mean values of the IOPs; and turbulence effect is manifested in the horizontal variations of the downwelling irradiance. The magnitude of the irradiance variation is governed by the differences in the temperature and salinity between their values at the surface and in the deep region. In the presence of surface waves, the irradiance variation is enhanced due to the surface deformation, which is also largely affected by wave–turbulence interaction. The LES and inhomogeneous RT simulation may provide a useful tool for the characterization of upper‐ocean turbulence processes based on underwater RT measurements.

Boundary layer flow past a stretching/shrinking surface beneath an external uniform shear flow with a convective surface boundary condition in a nanofluid

The problem of a steady boundary layer shear flow over a stretching/shrinking sheet in a nanofluid is studied numerically. The governing partial differential equations are transformed into ordinary differential equations using a similarity transformation, before being solved numerically by a Runge-Kutta-Fehlberg method with shooting technique. Two types of nanofluids, namely, Cu-water and Ag-water are used. The effects of nanoparticle volume fraction, the type of nanoparticles, the convective parameter, and the thermal conductivity on the heat transfer characteristics are discussed. It is found that the heat transfer rate at the surface increases with increasing nanoparticle volume fraction while it decreases with the convective parameter. Moreover, the heat transfer rate at the surface of Cu-water nanofluid is higher than that at the surface of Ag-water nanofluid even though the thermal conductivity of Ag is higher than that of Cu.

Initial value problems for creeping flow of Maxwell fluids

We consider the flow of nonlinear Maxwell fluids in the unsteady quasistatic case, where the effect of inertia is neglected. We study the well-posedness of the resulting PDE initial-boundary value problem locally in time. This well-posedness depends on the unique solvability of an elliptic boundary value problem. We first present results for the 3D case with sufficiently small initial data and for a simple shear flow problem with arbitrary initial data; after that we extend our results to some 3D flow problems with large initial data. We solve our problem using an iteration between linear subproblems. The limit of the iteration provides the solution of our original problem.

A corrected smoothed particle hydrodynamics method for solving transient viscoelastic fluid flows

In this work, a corrected smoothed particle hydrodynamics (CSPH) method is proposed and extended to the numerical simulation of transient viscoelastic fluid flows due to that its approximation accuracy in solving the Navier–Stokes equations is higher than that of the smoothed particle hydrodynamics (SPH) method, especially near the boundary of the domain. The CSPH approach comes with the idea of combining the SPH approximation for the interior particles with the modified smoothed particle hydrodynamics (MSPH) method for the exterior particles, this is because that the later method has higher accuracy than the SPH method although it also needs more computational cost. In order to show the validity of CSPH method to simulate unsteady viscoelastic flows problems, the planar shear flow problems, including transient Poiseuille, Couette flow and transient combined Poiseuille and Couette flow for the Oldroyd-B fluid are solved and compared with the analytical and SPH results. Subsequently, the general viscoelastic fluid based on the eXtended Pom–Pom (XPP) model is numerically investigated and the viscoelastic free surface phenomena of impacting drop are simulated by the CSPH for its extended application and the purpose of illustrating the ability of the proposed method. The numerical results are presented and compared with available solutions, which shows a very good agreement. All the numerical results show the higher accuracy and better stability of the CSPH than the SPH, especially for larger Weissenberg numbers.