Non-slip Boundary Condition Research Articles

Natural Convection in a Concentric Annulus: A Lattice Boltzmann Method Study with Boundary Condition-Enforced Immersed Boundary Method

AbstractIn this paper, a boundary condition-enforced IBM is introduced into the LBM in order to satisfy the non-slip and temperature boundary conditions, and natural convections in a concentric isothermal annulus between a square outer cylinder and a circular inner cylinder are simulated. The obtained results show that the boundary condition-enforced method gives a better solution for the flow field and the complicated physics of the natural convections in the selected case is correctly captured. The calculated average Nusselt numbers agree well with the previous studies.

Investigation of the Non-Equilibrium Flow Phenomena in the Boundary Layer of the Scramjet Engine

High-speed and high-temperature are the characteristics of the flow field in scramjet engine; the regular non-slip wall boundary condition requires zero speed at wall; in the same time, the material temperature limit does not allow high wall temperature; therefore the velocity gradient and temperature gradient in the engine boundary layer are huge. If these gradients are too large, the traditional assumption of the local thermal equilibrium in the fluid will fail, the Navier-Stokes equations are no longer valid in the boundary layer. For the first time, the non-equilibrium flow phenomena in Scramjet engine is studied here. Appropriate turbulence model and fine grid are used to analyze the turbulent boundary layer of the Hyshot scramjet engine with three different operating conditions. The result of the CFD simulation shows that the local Knudsen number in the engine boundary layer is greater than the critical value with the operating conditions 40Km/Ma8 and 30Km/Ma8; they are non-equilibrium flow and the Navier-Stokes equations fails. Special treatment of the boundary conditions are needed for these kinds of flow. With the operating condition of 20Km/Ma6, the local thermal equilibrium condition is observed and conventional CFD method is valid.

How Electrostatics Influences Hydrodynamic Boundary Conditions: Poiseuille and Electro-osmostic Flows in Clay Nanopores.

We report on a molecular simulation study of the origin of nonslip or slip hydrodynamic boundary conditions in clay nanopores, focusing on the role of electrostatics. We simulate hydrodynamic and electro-osmotic flows and consider both charged (montmorillonite) and uncharged (pyrophyllite) clays. We further use two commonly used force fields to analyze the effect of local interactions, in particular, the effect of the polarity of the surface, in addition to the mere effect of the presence or absence of a net charge and counterions. For the 6 nm pore investigated here, the molecular velocity profile can be well described by continuum hydrodynamics only if (a) proper boundary conditions, with a slip or stagnation length determined from molecular simulation, are taken into account and (b) the ionic density profiles from MD simulations are used in the case of electro-osmotic flow, because the Poisson–Boltzmann equation fails to reproduce the ionic profiles, hence the force acting on the fluid. Among the considered force fields, only CLAYFF predicts a hydrophobic pyrophyllite and hydrophilic montmorillonite, as expected from experimental behavior. The nonslip or slip boundary conditions at clay surfaces strongly depend on electrostatic interactions of water molecules with the surface. The presence of a net charge results in an average electric field experienced by surface water molecules between the charged surface and the condensed layer counterions, which influences their orientation. The charge distribution inside the clay layer determines the polarity of the surface and hence the strength of hydrogen bonds donated by water molecules to surface oxygen atoms.

A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations

We construct a Poiseuille type flow which is a bounded entire solution of the nonstationary Navier-Stokes and the Stokes equations in a half space with non-slip boundary condition. Our result in particular implies that there is a nontrivial solution for the Liouville problem under the non-slip boundary condition. A review for cases of the whole space and a slip boundary condition is included.

Asymptotic solutions of differential rotation driven by convection in rapidly rotating fluid spheres with the non-slip boundary condition

A strong Coriolis force in a rapidly rotating planet and star not only enforces two-dimensionality of fluid motion driven by thermal instabilities but also generates strong differential rotation even in the vicinity of the onset of instabilities. We derive an asymptotic solution describing convection-driven differential rotation in rotating, self-gravitating Boussinesq fluid spheres with the no-slip boundary condition, taking into account full spherical curvature and being valid for asymptotically small Ekman numbers. For the purpose of validating the asymptotic solution, the corresponding numerical analysis valid for large or small Ekman numbers is also carried out, showing a satisfactory agreement between the asymptotic and numerical solutions.

The inviscid limit of the incompressible anisotropic Navier–Stokes equations with the non-slip boundary condition

In this paper, we study the asymptotic behavior for the incompressible anisotropic Navier–Stokes equations with the non-slip boundary condition in a half space of \({\mathbb{R}^3}\) when the vertical viscosity goes to zero. Firstly, by multi-scale analysis, we formally deduce an asymptotic expansion of the solution to the problem with respect to the vertical viscosity, which shows that the boundary layer appears in the tangential velocity field and satisfies a nonlinear parabolic–elliptic coupled system. Also from the expansion, it is observed that away from the boundary the solution of the anisotropic Navier–Stokes equations formally converges to a solution of a degenerate incompressible Navier–Stokes equation. Secondly, we study the well-posedness of the problems for the boundary layer equations and then rigorously justify the asymptotic expansion by using the energy method. We obtain the convergence results of the vanishing vertical viscosity limit, that is, the solution to the incompressible anisotropic Navier–Stokes equations tends to the solution to degenerate incompressible Navier–Stokes equations away from the boundary, while near the boundary, it tends to the boundary layer profile, in both the energy space and the L∞ space.

Numerical Investigation of Hypersonic Boundary-Layer Stabilization by Porous Surfaces

The acoustic second-mode instability predicted by linear stability theory is compared with direct numerical simulation for a hypersonic flow over various porous walls. The damping effect of the micropores on the second mode is shown by comparison of the two different approaches. In addition to investigating the effect of pore size, the influence of the pore shape is studied by using spanwise grooves and cylindrical pores. Specifically, the comparability of different pore shapes by two definitions of hydraulic diameter is analyzed. The influence of rarefied gas behavior of the flow inside the pores is also investigated by comparing a slip boundary condition with finite Knudsen numbers with a nonslip boundary condition for different radii and pore depths.

Simulation of Incompressible Viscous Flows by Local DFD-Immersed Boundary Method

AbstractA local domain-free discretization-immersed boundary method (DFD-IBM) is presented in this paper to solve incompressible Navier-Stokes equations in the primitive variable form. Like the conventional immersed boundary method (IBM), the local DFD-IBM solves the governing equations in the whole domain including exterior and interior of the immersed object. The effect of immersed boundary to the surrounding fluids is through the evaluation of velocity at interior and exterior dependent points. To be specific, the velocity at interior dependent points is computed by approximate forms of solution and the velocity at exterior dependent points is set to the wall velocity. As compared to the conventional IBM, the present approach accurately implements the non-slip boundary condition. As a result, there is no flow penetration, which is often appeared in the conventional IBM results. The present approach is validated by its application to simulate incompressible viscous flows around a circular cylinder. The obtained numerical results agree very well with the data in the literature.

Simulation of Incompressible Viscous Flows by Local DFD-Immersed Boundary Method

A local domain-free discretization-immersed boundary method (DFD-IBM) is presented in this paper to solve incompressible Navier-Stokes equations in the primitive variable form. Like the conventional immersed boundary method (IBM), the local DFD-IBM solves the governing equations in the whole domain including exterior and interior of the immersed object. The effect of immersed boundary to the surrounding fluids is through the evaluation of velocity at interior and exterior dependent points. To be specific, the velocity at interior dependent points is computed by approximate forms of solution and the velocity at exterior dependent points is set to the wall velocity. As compared to the conventional IBM, the present approach accurately implements the non-slip boundary condition. As a result, there is no flow penetration, which is often appeared in the conventional IBM results. The present approach is validated by its application to simulate incompressible viscous flows around a circular cylinder. The obtained numerical results agree very well with the data in the literature.

Boundary layer separation of hydromagnetic flows

AbstractThe influence of the magnetic field on the boundary layer separation of a two‐dimensional incompressible flow of a conducting fluid subject to a non‐slip boundary condition is studied. It is found that the key factor differentiating this phenomenon from the hydrodynamic case is the angular velocity of the magnetic field at the boundary wall. The general equation governing the start of separation is illustrated by a class of flows generalizing the classical Hartmann one, where the physical meaning of the different terms is very clear.

Simulation of Thermal Flow Problems via a Hybrid Immersed Boundary‐Lattice Boltzmann Method

A hybrid immersed boundary‐lattice Boltzmann method (IB‐LBM) is presented in this work to simulate the thermal flow problems. In current approach, the flow field is resolved by using our recently developed boundary condition‐enforced IB‐LBM (Wu and Shu, (2009)). The nonslip boundary condition on the solid boundary is enforced in simulation. At the same time, to capture the temperature development, the conventional energy equation is resolved. To model the effect of immersed boundary on temperature field, the heat source term is introduced. Different from previous studies, the heat source term is set as unknown rather than predetermined. Inspired by the idea in (Wu and Shu, (2009)), the unknown is calculated in such a way that the temperature at the boundary interpolated from the corrected temperature field accurately satisfies the thermal boundary condition. In addition, based on the resolved temperature correction, an efficient way to compute the local and average Nusselt numbers is also proposed in this work. As compared with traditional implementation, no approximation for temperature gradients is required. To validate the present method, the numerical simulations of forced convection are carried out. The obtained results show good agreement with data in the literature.

Semi-Analytical Model of Mixed Electroosmotic/Pressure Driven Two Immiscible Fluids with Curved Interface

This study is motivated by the need to develop a semi analytical model for predicting the stratified two-fluid flow with a curved interface in a rectangular microchannel under the combined effect of pressure and electroosmosis. With the non-slip boundary conditions at the wall and the matching condition at the curved interface, the fully developed Navier-Stokes equation and Poisson-Boltzmann equation are solved using separate variable method. Part of parameters in the distributions of velocity and electric potential is calculated using the least-square method. Details of the analytical treatment of the two-fluid flow with curved interface are presented. The results show that the analysis can be employed for concave, convex and planar interface. The validity of the two-fluid model with curved interface is evaluated by comparing its prediction with available numerical data and with the results of exact analytical solutions for laminar flows with planar interface, comparison of the electric potential distribution and velocity distributions shows excellent agreement with data in the literature. Finally, the effects of interface shape on the electric potential distribution, electroosmotic velocity distribution, and flow rates are discussed, the results show that the interface shape influences the two-fluid flow in microchannel significantly. Keywords: Curved interface, microchannel, electroosmotic flow, two - fluid stratified flow, Lab-on-a-chip devices, EDLs, Electric Potential Distributions, Velocity, Pressure Gradient, Electroosmotic Force

Vanishing heat conductivity limit for the 2D Cahn-Hilliard-Boussinesq system

This article studies the vanishing heat conductivity limit for the 2D Cahn-Hilliard-boussinesq system in a bounded domain with non-slip boundary condition. The result has been proved globally in time.

A stream function–vorticity formulation‐based immersed boundary method and its applications

SUMMARYA new stream function–vorticity formulation‐based immersed boundary method is presented in this paper. Different from the conventional immersed boundary method, the main feature of the present model is to accurately satisfy both governing equations and boundary conditions through velocity correction and vorticity correction procedures. The velocity correction process is performed implicitly based on the requirement that velocity at the immersed boundary interpolated from the corrected velocity field accurately satisfies the nonslip boundary condition. The vorticity correction is made through the stream function formulation rather than the vorticity transport equation. It is evaluated from the firstorder derivatives of velocity correction. Two simple and efficient ways are presented for approximation of velocity‐correction derivatives. One is based on finite difference approximation, while the other is based on derivative expressions of Dirac delta function and velocity correction. It was found that both ways can work very well. The main advantage of the proposed method lies in its simple concept, easy implementation, and robustness in stability. Numerical experiments for both stationary and moving boundary problems were conducted to validate the capability and efficiency of the present method. Good agreements with available data in the literature were achieved. Copyright © 2011 John Wiley & Sons, Ltd.

Stokes flows of a Maxwell fluid with wall slip condition

Stokes flows of a Maxwell fluid produced by the motion of a wall are analyzed under the slip condition at the boundary. The wall is assumed to be translated in its plane with a given velocity. The relative velocity between the fluid at the wall and the wall is assumed to be proportional to the shear rate at the wall. The exact expressions for the velocity and shear stress are determined by means of a Laplace transform. The velocity fields corresponding to both slip and nonslip conditions for Maxwell and viscous Newtonian fluids are obtained. Two particular cases, namely sinusoidal oscillations and translation with a constant velocity of the wall, are studied. In the case of flows of a Maxwell fluid with a nonslip boundary condition, the velocity is discontinuous across a vortex sheet; this situation does not appear for flows with slip conditions. In this case, the velocity is always continuous. Because the exact expression for the velocity is rather complicated, two small-time and large-time expressions of the velocity are derived. Results for Maxwell fluids are compared with those of viscous Newtonian fluids in both cases of the flow with slip and nonslip conditions. Also, the exact and approximate solutions are compared and good agreement is found. In addition, the influence of the slip coefficient on the velocity and on the relative velocity is studied.

An immersed boundary method based on discrete stream function formulation for two- and three-dimensional incompressible flows

An immersed boundary method is proposed in the framework of discrete stream function formulation for incompressible flows. In order to impose the non-slip boundary condition, the forcing term is determined implicitly by solving a linear system. The number of unknowns of the linear system is the same as that of the Lagrangian points representing the body surface. Thus the extra cost in force calculation is negligible if compared with that in the basic flow solver. In order to handle three-dimensional flows at moderate Reynolds numbers, a parallelized flow solver based on the present method is developed using the domain decomposition strategy. To verify the accuracy of the immersed-boundary method proposed in this work, flow problems of different complexity (decaying vortices, flows over stationary and oscillating cylinders and a stationary sphere, and flow over low-aspect-ratio flat-plate) are simulated and the results are in good agreement with the experimental or computational data in previously published literatures.

2D Viscous Aerodynamic Shape Design Optimization for Turbine Blades Based on Adjoint Method

This paper presents an adjoint optimization technique and its application to the design of a transonic turbine cascade. Capable of a quick and exact sensitivity analysis and using little computational resources, the adjoint method has been a focus of research in aerodynamic shape design optimization. The goal of this work is to extend the adjoint method into turbomachinery design applications for viscous and compressible flow, and to further improve the aerodynamic performance. In the work, the minimization of the entropy generation rate with the mass flow rate constraint was considered as the cost function of the optimization, and was applied in the direct design process. The adjoint boundary conditions of the corresponding cost function were derived in detail, using the nonslip boundary condition on the blade wall, while the flow viscous effect on the cascade inlet and outlet was neglected. Numerical techniques used in Computational Fluid Dynamics (CFD) were employed to solve the adjoint linear partial difference equations. With the solved adjoint variables, the final expression of the cost function gradient with respect to the design variables was formulated. Combined with quasi-Newton algorithm, an aerodynamic design approach based on the adjoint method for turbine blades was presented, which was independent of the Navier–Stokes solver being used. Finally, to validate the present optimization algorithm, the aerodynamic design cases of a transonic turbine blade with and without mass flow rate restriction were performed and analyzed.

On fluid flows in precessing spheres in the mantle frame of reference

We investigate, through both asymptotic and numerical analysis, precessionally driven flows of a homogeneous fluid confined in a spherical container that rotates rapidly with angular velocity Ω and precesses slowly with angular velocity Ωp about an axis that is fixed in space. The precessionally driven flows are primarily characterized by two dimensionless parameters: the Ekman number E providing the measure of relative importance between the viscous force and the Coriolis force, and the Poincaré number Po quantifying the strength of the Poincaré forcing. When E is small but fixed and |Po| is sufficiently small, we derive a time-dependent asymptotic solution for the weakly precessing flow that satisfies the nonslip boundary condition in the mantle frame of reference. No prior assumption about the spatial-temporal structure of the precessing flow is made in the asymptotic analysis. A solvability condition is derived to determine the spatial structure of the precessing flow, via a selection from a complete spectrum of spherical inertial modes in the mantle frame. The weakly precessing flow within the bulk of the fluid is characterized by an inertial wave moving retrogradely. Direct numerical simulation of the same problem in the same frame of reference shows a satisfactory agreement between the time-dependent asymptotic solution and the nonlinear numerical simulation for sufficiently small Poincaré numbers.

A lattice Boltzmann model coupled with a Large Eddy Simulation model for flows around a groyne

This present paper proposes a two-dimensional lattice Boltzmann model coupled with a Large Eddy Simulation (LES) model and applies it to flows around a non-submerged groyne in a channel. The LES of shallow water equations is efficiently performed using the Lattice Boltzmann Method (LBM) and the turbulence can be taken into account in conjunction with the Smagorinsky Sub-Grid Stress (SGS) model. The bounce-back scheme of the non-equilibrium part of the distribution function is used to determine the unknown distribution functions at inflow boundary, the zero gradient of the distribution function is set normal to outflow boundary to obtain the unknown distribution functions here and the bounce-back scheme, which states that an incoming particle towards the boundary is bounced back into fluid, is applied to the solid wall to ensure non-slip boundary conditions. The initial flow field is defined firstly and then is used to calculate the local equilibrium distributions as initial conditions of the distribution functions. These coupled models successfully predict the flow characteristics, such as circulating flow, velocity and water depth distributions. The comparisons between the simulated results and the experimental data show that the model scheme has the capacity to solve the complex flows in shallow water with reasonable accuracy and reliability.

On fluid flows in precessing narrow annular channels: asymptotic analysis and numerical simulation

We consider a viscous, incompressible fluid confined in a narrow annular channel rotating rapidly about its axis of symmetry with angular velocity Ω that itself precesses slowly about an axis fixed in an inertial frame. The precessional problem is characterized by three parameters: the Ekman number E, the Poincaré number ε and the aspect ratio of the channel Γ. Dependent upon the size of Γ, precessionally driven flows can be either resonant or non-resonant with the Poincaré forcing. By assuming that it is the viscous effect, rather than the nonlinear effect, that plays an essential role at exact resonance, two asymptotic expressions for ε ≪ 1 and E ≪ 1 describing the single and double inertial-mode resonance are derived under the non-slip boundary condition. An asymptotic expression describing non-resonant precessing flows is also derived. Further studies based on numerical integrations, including two-dimensional linear analysis and direct three-dimensional nonlinear simulation, show a satisfactory quantitative agreement between the three asymptotic expressions and the fuller numerics for small and moderate Reynolds numbers at an asymptotically small E. The transition from two-dimensional precessing flow to three-dimensional small-scale turbulence for large Reynolds numbers is also investigated.