Two-dimensional Incompressible Fluid Flow Research Articles

Unsteady convetion flow of micropolar fluids past a vertical porous plate embedded in a porous medium

The unsteady two-dimensional laminar flow of a viscous incompressible micropolar fluid past a semi-infinite porous plate embebbed in a porous medium is studied. The free stream velocity follows an exponentially increasing or decreasing small perturbation law. The porous surface absorbs the micropolar fluid with a time varying suction velocity which has a small amplitude. The effects of material parameters on the velocity and temperature fields across the boundary layer are investigated. Numerical results of velocity distribution of micropolar fluids are compared with the corresponding flow problems for a Newtonian fluid. Results show that for the case of a surface cooling by natural convection the skin friction on the porous plate shows an increasing nature up to the critical value of ciscosity ratio. And the surface heat transfer tends to decrease slightly by increasing the magnitude of suction velocity with a given permeablity parameter, and given Prandtl number. However, for a surface heating case, the surface skin friction shows an opposite nature as compared with a surface cooling case.

Optimal Boundary Control of Steady-State Flow of a Viscous Inhomogeneous Incompressible Fluid

We study the problem of optimal boundary control of two-dimensional steady-state flow of a viscous inhomogeneous incompressible fluid. The role of control is played by the values of the velocity on a part of the boundary of the domain considered. On the remaining part of the boundary, the vector of flow velocity and the fluid density are given. We seek the fluid density as a scalar function (determined by the initial data) of the stream function, study the solvability of the problem, and obtain necessary optimality conditions.

Natural convection flow of a viscous fluid about a truncated cone with temperature dependent viscosity

The two-dimensional natural convection flow of a viscous incompressible fluid with temperature dependent viscosity about a truncated cone is considered. The governing equations for the flow are obtained by using suitable transformations and solved by using an implicit finite difference method. Perturbation solutions are obtained near the leading edge and in the downstream regime, and the results are obtained in terms of the local skin friction coefficient and the local Nusselt number. Perturbation solutions are compared with the finite difference solutions and are found to be in excellent agreement. The dimensionless velocity profiles and viscosity distribution are also presented graphically for various values of the viscosity variation parameter, ɛ, for small values of the Prandtl number appropriate for liquid metals.

AN IMPROVED SIMPLE METHOD ON A COLLOCATED GRID

A pressure-based method characterized by the SIMPLE algorithm is developed on a nonorthogonal collocated grid for solving two-dimensional incompressible fluid flow problems, using a cell-centered finite-volume approximation. The concept of artificial density is combined with the pressure Poisson equation that provokes density perturbations, assisting the transformation between primitive and conservative variables. A nonlinear explicit flux correction is utilized at the cell face in discretizing the continuity equation, which functions effectively in suppressing pressure oscillations. The pressure-correction equation principally consolidates a triplicate-time approach when the Courant number CFL > 1. A rotational matrix, accounting for the flow directionality in the upwinding, is introduced to evaluate the convective flux. The numerical experiments in reference to a few familiar laminar flows demonstrate that the entire contrivance executes a residual smoothing enhancement, facilitating an avoidance of the pressure underrelaxation. Consequently, included benefits are the use of larger Courant numbers, enhanced robustness, and improved overall damping properties of the unfactored pseudo-time integration procedure.

Lattice boltzmann method on composite grids

A composite block-structured lattice Boltzmann method is proposed for the simulation of two-dimensional incompressible fluid flows. The grid structure is composed of a coarse base grid and one or several fine grid(s). The former covers the entire physical domain; the latter are placed at regions where local grid refinement is desirable. The simulation is first carried out on the base grid level at a smaller relaxation time, allowing a rapid propagation of boundary information throughout the entire domain. Thus large-scale flow features can be resolved efficiently at a relatively low cost. At a later time, fine grid variables are initiated. The dependent variables on both grid levels are, then, advanced in time simultaneously with the fine grid boundary conditions obtained from the base grid solution at the grid interface. As a demonstration, the lid-driven cavity flow is selected for study. The results show good agreement with benchmark numerical data and those calculated from the finite-volume U2RANS model. The proposed method is able to produce accurate solutions on fine grids, with a considerable saving in CPU time.

Flow of a viscous incompressible fluid of temperature dependent viscosity past a permeable wedge with uniform surface heat flux

The effect of uniform suction on the steady two-dimensional laminar forced flow of a viscous incompressible fluid of temperature dependent viscosity past a wedge with uniform surface heat flux is considered. The governing equations for the flow are obtained by using suitable transformations and are solved by using an implicit finite difference method. Perturbation solutions are also obtained near the leading edge and in the downstream regime. The results are obtained in terms of the local skin friction coefficient and the rate of heat transfer for various values of the pertinent parameters, such as the Prandtl number, Pr, the velocity gradient parameter, m, the local suction parameter, ξ, and the viscosity variation parameter, ɛ. Perturbation solutions are compared with the finite difference solutions and are found to be in excellent agreement. The effect of ξ, m and ɛ on the dimensionless velocity profiles and viscosity distribution are also presented graphically for Pr = 0.7 and 7.0, which are the appropriate values for gases and water respectively.

Error estimation of prediction-correction legendre spectral approximation to incompressible fluid flow

The initial-boundary value problem of two-dimensional incompressible fluid flow in stream function form is considered. A prediction-correction Legendre spectral scheme is presented, which is easy to be performed. It is strictly proved that the numerical solution possesses the accuracy of second-order in time and higher order in space.

Flow of viscous incompressible fluid with temperature dependent viscosity and thermal conductivity past a permeable wedge with uniform surface heat flux

We consider a steady two-dimensional laminar forced flow and heat transfer of a viscous incompressible fluid having temperature dependent viscosity and thermal conductivity past a wedge with a uniform surface heat flux. The governing equations, reduced to local nonsimilarity boundary layer equations using suitable transformations, have been integrated employing an implicit finite difference method. Perturbation techniques are employed to obtain the solutions near the leading edge as well as far from it. The perturbation solutions are compared with the finite difference solutions and found to be in excellent agreement. The results are presented in terms of local skin friction coefficient and rate of heat transfer for various values of the governing parameters, such as the Prandtl number Pr , the pressure gradient parameter m , the viscosity variation parameter ε and thermal conductivity variation parameter γ , against the local permeability parameter ξ . The effect of variations in ξ , ε and γ on the dimensionless velocity, viscosity and thermal conductivity distributions are also depicted graphically for Pr=0.7 .

Simulation of transient cavity flows driven by buoyancy and shear

The unsteady two-dimensional flow of a viscous incompressible fluid in a rectangular cavity, driven by effects of shear and/or buoyancy, is modelled numerically. This physical problem is studied from the viewpoint of obtaining accurate and efficient numerical solutions to the Navier-Stokes equations, applicable to the simulation of convective exchange processes in stratified water bodies, where flows are driven by forced (wind-induced) and/or natural (buoyancy-induced) convection, e.g. in lakes, ponds and reservoirs. The hydrodynamic model, based on the vorticity-stream function formulation, was found to be highly accurate and economical in obtaining iterative solutions to nonlinear coupled systems for a wide range of Reynolds/ Grashof numbers and height/width cavity ratios that can be encountered in natural environments.

Expansion Functions for Two-Dimensional Incompressible Fluid Flow in Arbitrary Domains

Expansion functions are presented for two-dimensional incompressible fluid flow in arbitrary domains that optimally conserve the 2D structure of vortex dynamics. This is obtained by conformal mapping of the domain onto a circle and by constructing orthogonal radial polynomials and angular harmonics on the new domain such that the kinetic energy is diagonal and the separate components satisfy all of the required physical boundary conditions.

Mixed convection flow from a vertical flat plate with temperature dependent viscosity

A two-dimensional mixed convection flow of a viscous incompressible fluid of temperature dependent viscosity past a vertical impermeable fluid is considered. The governing equations for the flow are transformed for the regions appropriate to the forced convection, free convection and forced-free convection regimes. Solutions of the reduced equation appropriate in the forced convection and free convection regime are obtained using the perturbation technique treating ξ, the buoyancy parameter, as the perturbation parameter and those for the forced-free convection regime are obtained by the implicit finite difference method. Numerical results thus obtained are presented in terms of the local shear stress and local surface heat-flux. Effect of the viscosity variation parameter, ε, on the surface shear stress and the surface heat-flux for the fluid appropriate for Prandtl number ranging from 0.02 to 100 is shown. The perturbation solutions obtained for small and large values of ξ are found in excellent agreement with the finite difference solutions for the entire ξ regime.

On two-dimensional unsteady incompressible fluid flow in an infinite strip

In this paper, we study two-dimensional incompressible fluid flow in an infinite strip. The stream function form of Navier–Stokes equation is considered, which keeps the physical boundary condition and avoids some difficulties in numerical simulations. The existence and uniqueness of global solution are proved. Some results on the regularity of solution are obtained. Copyright © 2000 John Wiley & Sons, Ltd.

Steady flow in a diverging symmetrical channel: numerical study of bifurcation by analytic continuation

The two-dimensional steady flow of a viscous incompressible fluid in a diverging symmetrical channel is examined. The paper exploits a new series summation and improvement technique (i.e. Drazin and Tourigny, 1996). The solutions are expanded into Taylor series with respect to the corresponding Reynolds number and the bifurcation study is perfomed. Parameter ranges for the Reynolds number, where no, one or two solutions of the given type exist, are computed.

Modelling of two-dimensional laminar flow using finite element method

In this paper, a Galerkin weighted residual finite element numerical solution method, with velocity material time derivative discretisation, is applied to solve for a classical fluid mechanics system of partial differential equations modelling two-dimensional stationary incompressible Newtonian fluid flow. Classical examples of driven cavity laminar flow and laminar flow past a cylinder are presented. Numerical results are compared with data found in the literature. Copyright © 1999 John Wiley & Sons, Ltd.

On the evolution of a singular vortex patch in a two-dimensional incompressible fluid flow

The angle evolution of a tangent-slope discontinuity on a singular vortex patch, i.e. a patch of vorticity with tangent discontinuities in its boundary, is studied from a numerical and theoretical point of view. Different numerical examples show that the angle shrinks for an initial angle less than 90°, the angle widens when the initial angle is greater than 90° or is approximately preserved for 90° for small time evolution. An asymptotic expansion of the initial velocity field near a singularity is performed for a class of singular vortex patches to reinforce analytically these results.

Unsteady Viscous Flow Past an Impulsively Started Oscillating and Translating Elliptic Cylinder

The unsteady two-dimensional flow of a viscous incompressible fluid past an impulsively started oscillating and translating elliptic cylinder has been investigated. The governing Navier-Stokes equations expressed in terms of a stream-function/vorticity formulation are solved numerically for the early stages of the flow for moderate to high Reynolds numbers. A boundary-layer type transformation was adopted to scale out the singular nature in the vorticity at the start of the motion. Some comparisons of the flow patterns with the impulsively started translating case have been included to illustrate the effect of oscillation. Wherever possible, comparisons with existing numerical results have been made and the agreement was found to be good.

Mixed convection flow of micropolar fluid over an isothermal plate with variable spin gradient viscosity

A steady two-dimensional mixed convection flow of viscous incompressible micropolar fluid past an isothermal horizotal heated plate with uniform free stream and variable spin-gradient viscosity is considered. With appropriate transformations the boundary layer equations are transformed into nonsimilar equations appropriate for three distinct regimes, namely, the forced convection regime, the free convection regime and the mixed convection regime. Solutions of the governing equations for these regimes are obtained by an implicit finite difference scheme developed for the present problem. Results are obtained for the pertinent parameters, such as the buoyancy parameter, ζ in the range of 0 to 10 and the vortex viscosity parameters, Δ=0.0, 1.0, 3.0, 5.0 and 10.0 for fluid with Prandtl number Pr=0.7 and are presented in terms of local shear-stress and the local rate of heat transfer. Effects of these parameters are also shown graphically on the velocity, temperature and the couple stress distributions. From the present analysis, it is observed that both the momentum boundary layer and the thermal boundary layer increase due to an increase in the vortex viscosity of the fluid.

Long series analysis of laminar flow through parallel and uniformly porous walls of different permeability

The steady two-dimensional laminar flow of a viscous incompressible fluid in a channel due to non-uniform suction or injection through its uniformly porous parallel walls is considered. A solution valid for large suction/injection Reynolds numbers comprising various cases is presented. Following Terrill [5], the governing equations reduce to nonlinear ordinary differential equation. The resulting equation is solved using a long series with polynomial coefficients. The analysis involves recasting the series representing physical parameters into new forms, series with Legendre polynomials as coefficients, divergent but asymptotic series, application of Pade' approximants, reversion of series, Euler transformation, etc. These techniques contribute considerably to obtaining larger regions of validity of the series.

Phase space structure and fractal trajectories in 1½ degree of freedom Hamiltonian systems whose time dependence is quasiperiodic

Abstract. We consider particle motion in nonautonomous 1 degree of freedom Hamiltonian systems for which H(p,q,t) depends on N periodic functions of t with incommensurable frequencies. It is shown that in near-integrable systems of this type, phase space is partitioned into nonintersecting regular and chaotic regions. In this respect there is no different between the N = 1 (periodic time dependence) and the N = 2, 3, ... (quasi-periodic time dependence) problems. An important consequence of this phase space structure is that the mechanism that leads to fractal properties of chaotic trajectories in systems with N = 1 also applies to the larger class of problems treated here. Implications of the results presented to studies of ray dynamics in two-dimensional incompressible fluid flows are discussed.

Heat transfer response of free convection flow from a vertical heated plate to an oscillating surface heat flux

An investigation is undertaken of the unsteady response of two-dimensional laminar free convection boundary layer flow of a viscous incompressible fluid along a semi-infinite vertical heated plate where the mean surface heat flux oscillates with a small amplitude about a steady profile. The buoyancy forces are favourable, resulting from a positive flux of heat from the surface of the plate into the fluid. The interaction of the time-periodic heat flux with the usual boundary-layer flow is examined by using a linearized theory. Solutions are obtained using three distinct methods, namely an extended series expansion method for low frequencies, an asymptotic series expansion method for high frequencies and a fully numerical finite difference method for general frequencies. Calculations have been carried out for a wide range of parameters to examine the solutions in terms of the amplitude and phase angle of the fluctuating parts of the surface shear stress and the surface temperature. It has been found that the amplitude and phase angle of both the shear stress and the surface temperature predicted by these three methods are in very good agreement in their respective ranges of validity.