SIMULATION OF THE FLOW OVER ELLIPTIC AIRFOILS OSCILLATING AT LARGE ANGLES OF ATTACK

The incompressible, viscous flow over two-dimensional elliptic airfoils oscillating in pitch at large angles of attack, such that flow separation occurs, has been simulated numerically for a Reynolds number of 3000. A vortex method is used to solve the two-dimensional Navier–Stokes equations in vorticity/stream-function form using a time-marching approach. Using an operator-splitting method the convection and diffusion equations are solved sequentially at each time step. The convection equation is solved using a vortex-in-cell method, and the diffusion equation using a second-order ADI finite-difference scheme. Elliptic profiles are obtained by mapping a circle in a computational domain into the physical domain using a Joukowski transformation. The effects of several parameters on the flow field are considered, such as: frequency of oscillation, mean angle of attack, location of pitch-axis and the thickness ratio of the ellipse. The results obtained are shown to compare favourably with available experimental results.     

Flow field characteristics study of a flapping airfoil using computational fluid dynamics

The flow field of a flapping airfoil in Low Reynolds Number (LRN) flow regime is associated with complex nonlinear vortex shedding and viscous phenomena. The respective fluid dynamics of such a flow is investigated here through Computational Fluid Dynamics (CFD) based on the Finite Volume Method (FVM). The governing equations are the unsteady, incompressible two-dimensional Navier-Stokes (N-S) equations. The airfoil is a thin ellipsoidal geometry performing a modified figure-of-eight-like flapping pattern. The flow field and vortical patterns around the airfoil are examined in detail, and the effects of several unsteady flow and system parameters on the flow characteristics are explored. The investigated parameters are the amplitude of pitching oscillations, phase angle between pitching and plunging motions, mean angle of attack, Reynolds number (Re), Strouhal number (St) based on the translational amplitudes of oscillations, and the pitching axis location (x/c). It is shown that these parameters change the instantaneous force coefficients quantitatively and qualitatively. It is also observed that the strength, interaction, and convection of the vortical structures surrounding the airfoil are significantly affected by the variations of these parameters.     

Convection,diffusion and reaction inside a spherical porous pellet in the presence of oscillatory flow

The problem of convection, diffusion and reaction inside a spherical porous pellet is investigated analytically. Unsteady Stokes equation is used for the flow outside the porous pellet and Darcy’s law is used inside the pellet. A solenoidal decomposition method is employed for the hydrodynamic problem. Following the above findings, the convection–diffusion–reaction problem is formulated and solved analytically for a first order reaction. The behavior of the nutrient transport is discussed with respect to various parameters like Darcy number, Peclet number, frequency and Thiele modulus. Also the effectiveness factor corresponding to the first order reaction is computed.     

Effects of chemical reaction on mixed convection flow of a polar fluid through a porous medium in the presence of internal heat generation

This paper reports a detailed numerical investigation on mixed convection flow of a polar fluid through a porous medium due to the combined effects of thermal and mass diffusion. The energy equation accounts for heat generation or absorption, while the nth order homogeneous chemical reaction between the fluid and the diffusing species is included in the mass diffusion equation. The governing equations of the linear momentum, angular momentum, energy and concentration are obtained in a non-similar form by introducing a suitable group of transformations. The final set of non-similar coupled non-linear partial differential equations is solved using an implicit finite-difference scheme in combination with quasi-linearization technique. The effects of various parameters on the velocity, angular velocity, temperature and concentration fields are investigated. Numerical results for the skin friction coefficient, wall stress of angular velocity, Nusselt number and Sherwood number are also presented.     

Effects of Brownian Motion and Thermophoresis on the Mixed Convection of Nanofluid in a Porous Channel Including Flow Reversal

This article is devoted to combined convection heat transfer of nanofluids through a vertical channel filled with a homogeneous and isotropic porous medium. The flow is assumed to be fully developed and the “Brinkman extended Darcy” model is used for the flow in the porous media and “clear compatible” viscous dissipation model is considered. Also the model utilized for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The governing momentum, energy, and nanopartices volume fraction equations are solved both analytically and numerically. The effects of the influential dimensionless parameters such as Brownian and thermophoresis parameters, mixed convection parameter (Gr/Re), Brinkman, Darcy and Lewis numbers on dimensionless velocity and temperature distributions and pressure drop are studied. Also, the results of the Nusselt number for the both left and right walls are presented and discussed.     

APPLICATION OF CONVECTION-DIFFUSION EQUATION TO THEANALYSES OF CONTAMINATION BETWEEN BATCHESIN MULTI-PRODUCTS PIPELINE TRANSPORT

I.IntroductionIntilenearfuture,ourcountry'sproductspipelinetransportwillhavebeendevelopedgreatly.Themeansbywhichmanyrefinedpetroleumproductsaretransportedthroughasinglepipelineiscalledbatchingtransport.Thismeansispossessedofadvantagesthatavarietyofrefinedpetroleumproductscanbetrallsportedthroughasinglepipelineandinvestmentcanbereduced.[-lowever,tile1ila-jorproblemresultedfronlthebatchingtrallsportisthecontamillationbetweenbatcllcs,whichmaydeterioratethequalityofproductstransported.Therefore,i…     

Heat transfer in a radiating, nonsteady, three-dimensional stagnation flow

A similar solution has been obtained to the problem of simultaneous radiation and convection for nonsteady stagnation point flow over a three-dimensional blunt body with both boundary layer suction and injection. The diffusion approximation is used to characterize the radiative heat flux. The three-dimensional, time-dependent equations of motion and the energy equation have been transformed into a set of ordinary differential equations by the similarity transformation and the resulting ordinary differential equations have been solved numerically. The effects of accelerating and decelerating flow, the three-dimensional geometry, injection and suction, hot and cold wall conditions, and the conduction-to-radiation parameter on the temperature distribution within the flow have been investigated.     

Hydromagnetic flow in a viscoelastic fluid due to the oscillatory stretching surface

An analysis is carried out to study the unsteady magnetohydrodynamic (MHD) two-dimensional boundary layer flow of a second grade viscoelastic fluid over an oscillatory stretching surface. The flow is induced due to an infinite elastic sheet which is stretched back and forth in its own plane. For the investigated problem, the governing equations are reduced to a non-linear partial differential equation by means of similarity transformations. This equation is solved both by a newly developed analytic technique, namely homotopy analysis method (HAM) and by a numerical method employing the finite difference scheme, in which a coordinate transformation is employed to transform the semi-infinite physical space to a bounded computational domain. The results obtained by means of both methods are then compared and show an excellent agreement. The effects of various parameters like visco-elastic parameter, the Hartman number and the relative frequency amplitude of the oscillatory sheet to the stretching rate on the velocity field are graphically illustrated and analysed. The values of wall shear stress for these parameters are also tabulated and discussed.     

Steady suspension flows into two-dimensional horizontal and inclined channels

In this study a model which was developed previously is used to theoretically investigate the steady flow of a particulate suspension into two-dimensional horizontal and inclined channels. The continuity equation for the fluid and the simplified two-dimensional Navier-Stokes equations are then solved together with a particle concentration equation. This latter equation is formulated by considering the balance between the particle flux due to gravity with the corresponding particle fluxes due to convection and shear-induced diffusion. The resulting coupled system of equations is solved numerically using a specialised finite-difference method. It is found, for the parameter range for flows of proppants in hydraulic fractures, that when the suspension enters the channel with a uniform velocity profile it almost instantaneously becomes parabolic. In addition, the effects of particle sedimentation are most dominant in the entrance region, but further downstream such effects are balanced as shear-induced particle diffusion becomes more important. It is also shown that the suspension flow depends critically on the choice of the parameters used, e.g. the ratio of the particle radius to the height of the channel.     

Thermal radiation effect on mixed convection heat and mass transfer of a non-Newtonian fluid over a vertical surface embedded in a porous medium in the presence of thermal diffusion and diffusion-thermo effects

Thermal radiation, thermal diffusion, and diffusion-thermo effects on heat and mass transfer by mixed convection of non-Newtonian power-law fluids over a vertical permeable surface embedded in a saturated porous medium are investigated. The governing equations describing the problem are non-dimensionalized and transformed into a non-similar form. The transformed equations are solved by using the local non-similarity method combined with the shooting technique. The effects of the physical parameters of the problem on the fluid temperature and concentration are illustrated graphically and analyzed. Also, the effects of the pertinent parameters on the local Nusselt number and the local Sherwood number are presented.     

Unsteady double diffusive mixed convection flow over a vertically stretching sheet in the presence of suction/injection

An unsteady double diffusive mixed convection boundary layer flow over a vertically stretching sheet in the presence of suction/injection is investigated in this paper. The governing partial differential equations are reduced by applying suitable transformations to a set of nonlinear ordinary differential equations, which is solved by the Keller box method. The influence of various flow parameters on the velocity, temperature, and species concentration profiles of the fluid is studied. The effect of some problem parameters on the skin friction coefficient in the presence of suction/injection is considered.     

Radiation effects on combined convection over a vertical flat plate embedded in a porous medium of variable porosity

The present paper is concerned with the study of radiation effects on the combined (forced-free) convection flow of an optically dense viscous incompressible fluid over a vertical surface embedded in a fluid saturated porous medium of variable porosity with heat generation or absorption. The effects of radiation heat transfer from a porous wall on convection flow are very important in high temperature processes. The inclusion of radiation effects in the energy equation leads to a highly non-linear partial differential equations which are transformed to a system of ordinary differential equations using non-similarity transformation. These equations are then solved numerically using implicit finite-difference method subject to appropriate boundary and matching conditions. A parametric study of the physical parameters such as the particle diameter-based Reynolds number, the flow based Reynolds number, the Grashof number, the heat generation or absorption co-efficient and radiation parameter is conducted on temperature distribution. The effects of radiation and other physical parameters on the local skin friction and on local Nusselt number are shown graphically. It is interesting to observe that the momentum and thermal boundary layer thickness increases with the radiation and decrease with increase in the Prandtl number.     

Evaporation phenomenon past a rotating hydrocarbon droplet of ternary components

A numerical study of heat and mass transfer from an evaporating fuel droplet rotating around its vertical axis was performed in forced convection only on the side opposite to the flow. The flow was assumed to be laminar, and the droplet was assumed to maintain its spherical shape during its lifetime. Based on the abovementioned assumption, the conservation equations in a general curvilinear coordinate were solved numerically. The behavior of rotating droplet evaporation in the forced convection flow can be investigated by analyzing the effects of the rotation of the droplet on the evaporation process of multi-component hydrocarbons droplet. The droplet is simulated to behave as a hard sphere. The transfer equations are discretized using an implicit finite difference method. Thomas algorithm is used to solve the system of algebraic equations. Moreover, dimensionless parameters of heat and mass transfer phenomena around a rotating hydrocarbon droplet were determined. The thickness of the boundary layer is unknown for this model and therefore, it was determined in function of time. Additionally, the study concerns “Dgheim dimensionless number” which is the ratio of the rotation forces over the viscosity forces. Dgheim dimensionless number is correlated to Nusselt and Sherwood numbers for multi-component hydrocarbon droplets in evaporation by taking into account the effect of heat and mass Spalding, Prandtl and Schmidt numbers respectively. Also, correlations for Nusselt and Sherwood numbers in terms of Reynolds, Prandtl and Schmidt numbers are proposed. These correlations consider the rotation phenomenon and advance the variation of the thermophysical and transport properties in the vapor phase of multi-component blends.     

Asymptotic theory of neutral stability of the Couette flow of a vibrationally excited gas

An asymptotic theory of the neutral stability curve for a supersonic plane Couette flow of a vibrationally excited gas is developed. The initial mathematical model consists of equations of two-temperature viscous gas dynamics, which are used to derive a spectral problem for a linear system of eighth-order ordinary differential equations within the framework of the classical linear stability theory. Unified transformations of the system for all shear flows are performed in accordance with the classical Lin scheme. The problem is reduced to an algebraic secular equation with separation into the “inviscid” and “viscous” parts, which is solved numerically. It is shown that the thus-calculated neutral stability curves agree well with the previously obtained results of the direct numerical solution of the original spectral problem. In particular, the critical Reynolds number increases with excitation enhancement, and the neutral stability curve is shifted toward the domain of higher wave numbers. This is also confirmed by means of solving an asymptotic equation for the critical Reynolds number at the Mach number M ≤ 4.     

Free Convection of Non-Newtonian Nanofluids in Porous media with Gyrotactic Microorganisms

The free convection of non-Newtonian nanofluids along a vertical plate in porous medium is investigated numerically. It is assumed that the medium contains gyrotactic microorganisms along with nanoparticles and the plate is subjected to prescribed temperature, concentration of nanoparticles and density of motile microorganisms. It is further assumed that the plate is impermeable. The governing partial differential equations are reduced to nonlinear ordinary differential equations using similarity transformations. The nonlinear ordinary differential equations are then solved by a finite difference numerical method. The effects of controlling parameters on several dimensionless quantities and numbers of our interest are investigated. The numerical results are compared with the published data and an excellent agreement has been found. It is found that nanofluid and bioconvection parameters have strong effects on local Nusselt, Sherwood and density numbers.     

Viscous Dissipation and Thermal Radiation Effects on Mixed Convection from a Vertical Plate in a Non-Darcy Porous Medium

The paper presents an investigation of the influence of thermal radiation and viscous dissipation on the mixed convective flow due to a vertical plate immersed in a non-Darcy porous medium saturated with a Newtonian fluid. The physical properties of the fluid are assumed to be constant. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The governing partial differential equations are transformed into a system of ordinary differential equations and solved numerically using a shooting method. The results are analyzed for the effects of various physical parameters such as viscous dissipation, thermal radiation, mixed convection parameters, and the modified Reynolds number on dynamics. The heat transfer coefficient is also tabulated for different values of the physical parameters.     

Analytical formulation of 2-D aeroelastic model in weak ground effect

This paper deals with the aeroelastic modeling and analysis of a 2-D oscillating airfoil in ground effect, elastically constrained by linear and torsional springs and immersed in an incompressible potential flow (typical section) at a finite distance from the ground. This work aims to extend Theodorsen theory, valid in an unbounded flow domain, to the case of weak ground effect, i.e., for clearances above half the airfoil chord. The key point is the determination of the aerodynamic loads, first in the frequency domain and then in the time domain, accounting for their dependence on the ground distance. The method of images is exploited in order to comply with the impermeability condition on the ground. The new integral equation in the unknown vortex distribution along the chord and the wake is solved using asymptotic expansions in the perturbation parameter defined as the inverse of the non-dimensional ground clearance of the airfoil. The mathematical model describing the aeroelastic system is transformed from the frequency domain into the time domain and then in a pure differential form using a finite-state aerodynamic approximation (augmented states). The typical section, which the developed theory is applied to, is obtained as a reduced model of a wing box finite element representation, thus allowing comparison with the corresponding aeroelastic analysis carried out by a commercial solver based on a 3-D lifting surface aerodynamic model. Stability (flutter margins) and response of the airfoil both in frequency and time domains are then investigated. In particular, within the developed theory, the solution of the Wagner problem can be directly achieved confirming an asymptotic trend of the aerodynamic coefficients toward the steady-state conditions different from that relative to the unbounded domain case. The dependence of flutter speed and the frequency response functions on ground clearance is highlighted, showing the usefulness of this approach in efficiently and robustly accounting for the presence of the ground when unsteady analysis of elastic lifting surfaces in weak ground effect is required.     

Large eddy simulation of natural convection along a vertical isothermal surface

Large eddy simulations of natural convection along a vertical isothermal surface have been carried out using a parallel CFD code SMAFS (Smoke Movement And Flame Spread) developed by the first author to study the dynamics of the natural convection flow and the associated convective heat transfer, with sub-grid scale turbulence modeled using the Smagorinsky model. In the computation, the filtered governing equations are discretized using finite volume method, with the variables at the cell faces in the finite volume discrete equations approximated by a second order bounded QUICK scheme and the diffusion term computed based on central difference scheme. The computation was time marched explicitly, with momentum equations solved based on a second order fractional-step Adams–Bashford scheme and enthalpy computed using a second order Runge–Kutta scheme. The Poisson equation for pressure from the continuity equation was solved using a multi-grid solver. The results including the temperature and velocity profiles of the boundary layer and the local heat transfer rate are analyzed. Comparison is made with experimental data and good agreement is found.     

Effects of thermal dispersion and stratification on non-darcy mixed convection from a vertical plate in a porous medium

A boundary layer analysis is presented for the mixed convection from a vertical plate embedded in a porous medium. The effects of thermal dispersion and stratification on the flow and temperature fields are investigated. The conservation equations that govern the problem are reduced to a system of nonlinear ordinary differential equations. The resulting equations were solved on the basis of the local similarity approach. Received on 12 February 1998     

A multigrid method for steady incompressible Navier–Stokes equations based on partial flux splitting

Flux splitting is applied to the convective part of the steady Navier–Stokes equations for incompressible flow. Partial upwind differences are introduced in the split first-order part, while central differences are used in the second-order part. The discrete set of equations obtained is positive, so that it can be solved by collective variants of relaxation methods. The partial upwinding is optimized in the same way as for a scalar convection–diffusion equation, but involving several Peclet numbers. It is shown that with the optimum partial upwinding accurate results can be obtained. A full multigrid method in W-cycle form, using red–black successive under-relaxation, injection and bilinear interpolation, is described. The efficiency of this method is demonstrated.