Boundary Conditions For Navier-Stokes Equations Research Articles

Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains

We present here different boundary conditions for the Navier-Stokes equations inbounded Lipschitz domains in $\mathbb{R}^3$, such as Dirichlet, Neumann or Hodgeboundary conditions.We first study the linear Stokes operator associated to the boundary conditions. Thenwe show how the properties of the operator lead to local solutions or global solutions forsmall initial data.

Implicit application of non-reflective boundary conditions for Navier–Stokes equations in generalized coordinates

The non-re∞ective boundary conditions for Navier-Stokes equations originally suggested by Poinsot and Lele[1] in Cartesian coordinates are extended to generalized coordinates. The boundary conditions are implicitly coupled with the Navier-Stokes ∞ow solver in the inner domain. The calculations are conducted for a subsonic vortex propagating ∞ow and the steady and unsteady transonic inlet-difiuser ∞ows. The results indicate that the present method is accurate and robust, and the non-re∞ective boundary conditions are essential for unsteady ∞ow calculations.

On the Boundary Conditions for Navier-Stokes Equations in Stream Function-Vorticity Variables in Simulation of a Flow around a System of Bodies

A method of determining the boundary conditions for the Navier-Stokes equations in stream function-vorticity variables, used for simulation of a nonstationary, asymmetric laminar flow of an incompressible viscous fluid around bodies, has been proposed. Universal relations for desired functions on surfaces around which the stream flows, independent of the method of spatial discretization, have been obtained.

Vorticity boundary conditions for Navier-Stokes equations

A numerical study of the streamfunction-vorticity formulation of the two-dimensional Navier-Stokes equations is developed using a boundary integral condition for the vorticity. The method is appropriate for steady viscous channel flows and flows past rigid bodies where vorticity is generated by the viscous shear, and concentrated near the no-slip boundaries. The boundary integral method is presented along with the discussion of the kinetic and kinematic aspects of the problem. An iterative method is required to reduce the integral boundary condition to a sequence of Dirichlet boundary conditions. The iterative process is incorporated into time marching and nonlinear iterations algorithm used for the vorticity transport equation. The convergence of the iterative process is very sensitive to relaxation parameters. A convergence analysis of the method for the steady Stokes problem leads to the selection of these parameters: the technique is applied to the Navier-Stokes equations. The vorticity transport equation is discretized in space using a Galerkin/Least-squares finite element formulation: this leads to a system of ordinary differential equations, which is solved using a predictor-corrector algorithm.

Motion of line of contact of three phases on a solid: thermodynamics and asymtotic theory

A thermodynamics approach to the description of the dynamic boundary angle is proposed. An energy equation for the moving line of contact of three media is derived taking into account viscous flow. The energy equation is used to substantiate the boundary condition of the asymptotic theory for the free boundary slope angle at a small distance from the solid. Second-order asymptotic relations of the dynamic wetting theory are developed assuming a low capillary constant. Boundary conditions for Navier-Stokes equations over a moving three-phase contact line for a general flow of viscous liquid are derived. A solution procedure for the case of low Reynolds numbers is considered. Assuming a stationary movement of the interface in a capillary, a method for establishing the second-order terms is proposed. Exact asymptotic solutions are obtained. Symmetry of theasymptotic relation for the dynamic boundary angle in the principal approximation is described.

Approximate nonreflecting inflow-outflow boundary conditions for the calculation of navier-stokes equations in internal flows

In this paper, the nonreflecting boundary conditions based upon fundamental ideas of the linear analysis are developed for gas dynamic equations, and the modified boundary conditions for Navier-Stokes equations are proposed as a substitute of the nonreflecting boundary conditions inside boundary layers near rigid walls. These derived boundary conditions are then applied to calculations both for the Euler equations and the Navier-Stokes equations to determine if they can produce acceptable results for the subsonic flows in channels. The numerical results obtained by an implicit second-order upwind difference scheme show the effectiveness and generality of the boundary conditions. Furthermore, the formulae and the analysis performed here may be extended to three dimensional problems.

Boundary conditions for direct simulations of compressible viscous flows

Procedures to define boundary conditions for Navier-Stokes equations are discussed. A new formulation using characteristic wave relations through boundaries is derived for the Euler equations and generalized to the Navier-Stokes equations. The emphasis is on deriving boundary conditions compatible with modern non-dissipative algorithms used for direct simulations of turbulent flows. These methods have very low dispersion errors and require precise boundary conditions to avoid numerical instabilities and to control spurious wave reflections at the computational boundaries. The present formulation is an attempt to provide such conditions. Reflecting and non-reflecting boundary condition treatments are presented. Examples of practical implementations for inlet and outlet boundaries as well as slip and no-slip walls are presented. The method applies to subsonic and supersonic flows. It is compared with a reference method based on extrapolation and partial use of Riemann invariants. Test cases described include a ducted shear layer, vortices propagating through boundaries, and Poiseuille flow. Although no mathematical proof of well-posedness is given, the method uses the correct number of boundary conditions required for well-posedness of the Navier-Stokes equations and the examples reveal that it provides a significant improvement over the reference method.

Stokes Roughness Effects on Hydrodynamic Lubrication. Part II—Effects Under Slip Flow Boundary Conditions

Stokes roughness effects on hydrodynamic lubrication are studied in the slip flow regime. Slip flow boundary conditions for Navier-Stokes equations are derived, assuming that the fluid on a surface slips due to the molecular mean free path along the surface, even if the surface is rough. The perturbation method for Navier-Stokes equations, which was derived in Part I of this report, is then applied. Slip flow effects on load carrying capacity and frictional force are numerically clarified for both Stokes and Reynolds roughnesses. In the slip flow regime, second-order quantities induced by Stokes effects, such as flow rate, load carrying capacity, and frictional force are in proportion to the wavenumber squared. This phenomenon relative to the quantities being proportional is also the same as that in the continuum flow regime. As a result of velocity slippage, the load carrying capacity in Stokes roughness is found to decrease more than in Reynolds roughness for incompressible films, while the relationship is reversed for compressible films having a high compressibility number. The simulation of random roughness, which is generated by numerical means, clarifies one important result: the average slip flow effects associated with random Stokes roughness become similar to the slip flow effects in deterministic sinusoidal Stokes roughness, whose wavelength and height are statistically equivalent to those of random roughness. Although attention should be given to the fact that Stokes effects on random roughness demonstrate considerable scattering with the continuum flow, such scattering diminishes with the slip flow.

Boundary conditions for Navier-Stokes equations in rarefied gas: R G Barantsey and M O Lutset,Dokl akad nauk SSSRJ,173 (5), 11 th April 1967, 1021–1023 ( in Russian)

A theoretical study of the first step in the reaction mechanism of the electrophilic addition reaction of protonation was done on the following two systems: CH2=CH2+HeHP+→CH3CH+2+He (1) CH3-CH=CH2+HeH+→(CH3)2CH+ or CH3-CH2-CH2++He (2) In this study, He is considered as the proton carrier. The energy-gradient method was used to optimize the molecular geometries for critical points along the reaction coordinate using the 3-2 IG basis set for the self-consistent-field (SCF) calculations. Single-point energies were determined with the 6-31G∗∗ basis set for all relevant geometries optimized at the 3-21G basis set level. The related potential curves of the reactions studied were investigated. The results obtained reveal that reaction (1) is a multistep reaction, where first the He-H bond is broken, forming an open cationic intermediate C2H5+·He. Further along the reaction coordinate, the product was found to be a “bridged structure” of C2H5+. It was also found that reaction (2) proceeds according to Markovnikov's rule, i.e. the H+ of HeH+ adds onto the double bond of the carbon atom having the greatest number of hydrogen atoms. This leads to the formation of the more stable secondary carbocation. The key intermediate in this reaction is an open isopropyi cation. Along the reaction coordinate, another product complex which has a semi-closed bridged structure is also formed. Using the topological theory of electronic density distribution suggested by Bader, the molecular graphs and the properties of the bond critical points of the reactants, transition states, intermediates and products were studied.