Heat and mass transfer of a fuel droplet evaporating in oscillatory flow

A numerical study of the heat and mass transfer from an evaporating fuel droplet in oscillatory flow was performed. The flow was assumed to be laminar and axisymmetric, and the droplet was assumed to maintain its spherical shape during its lifetime. Based on these assumptions, the conservation equations in a general curvilinear coordinate were solved numerically. The behaviors of droplet evaporation in the oscillatory flow were investigated by analyzing the effects of flow oscillation on the evaporation process of a n-heptane fuel droplet at high pressure.The response of the time history of the square of droplet diameter and space-averaged Nusselt numbers to the main flow oscillation were investigated in frequency band of 1–75 Hz with various oscillation amplitudes. Results showed that, depending on the frequency and amplitude of the oscillation, there are different modes of response of the evaporation process to the flow oscillation. One response mode is synchronous with the main flow oscillation, and thus the quasi-steady condition is attained. Another mode is asynchronous with the flow oscillation and is highly unsteady. As for the evaporation rate, however, in all conditions is more greatly enhanced in oscillatory flow than in quiescent air.To quantify the conditions of the transition from quasi-steady to unsteady, the response of the boundary layer around the droplet surface to the flow oscillation was investigated. The results led to including the oscillation Strouhal number as a criteria for the transition. The numerical results showed that at a low Strouhal number, a quasi-steady boundary layer is formed in response to the flow oscillation, whereas by increasing the oscillation Strouhal number, the phenomena become unsteady.     

On solute dispersion in pulsatile flow through a channel with absorbing walls

The paper presents the longitudinal dispersion of passive tracer molecules released in an incompressible viscous fluid flowing through a channel with reactive walls under the action of a periodic pressure gradient. A finite-difference implicit scheme is adopted to solve the unsteady advection-diffusion equation based on the Aris-Barton method of moments for all time period. Here it is shown how the spreading of tracers is influenced by the shear flow, lateral diffusion about its mean position due to the action of absorption at both the walls. The analysis has been performed for three different cases: steady, periodic and the combined effect of steady and periodic currents, separately. The results show that for all cases the dispersion coefficient asymptotically reaches a stationary state after a certain critical time and it achieves a stationary state at earlier instant of time, when absorption at the walls increases. The axial distributions of mean concentration are determined from the first four central moments by using Hermite polynomial representation for all three different flow velocities.     

Unsteady flow and heat transfer of porous media sandwiched between viscous fluids

The problem of unsteady oscillatory flow and heat transfer of porous medin sandwiched between viscous fluids has been considered through a horizontal channel with isothermal wall temperatures. The flow in the porous medium is modeled using the Brinkman equation. The governing partial differential equations are transformed to ordinary differential equations by collecting the non-periodic and periodic terms. Closed-form solutions for each region are found after applying the boundary and interface conditions. The influence of physical parameters, such as the porous parameter, the frequency parameter, the periodic frequency parameter, the viscosity ratios, the conductivity ratios, and the Prandtl number, on the velocity and temperature fields is computed numerically and presented graphically. In addition, the numerical values of the Nusselt number at the top and bottom walls are derived and tabulated.     

Oscillatory flow of second grade fluid in cylindrical tube

The unsteady oscillatory flow of an incompressible second grade fluid in a cylindrical tube with large wall suction is studied analytically. Flow in the tube is due to uniform suction at the permeable walls, and the oscillations in the velocity field are due to small amplitude time harmonic pressure waves. The physical quantities of interest are the velocity field, the amplitude of oscillation, and the penetration depth of the oscillatory wave. The analytical solution of the governing boundary value problem is obtained, and the effects of second grade fluid parameters are analyzed and discussed.     

Stability Analysis of a Vertical Curved Channel Flow with a Radial Temperature Gradient

The linear stability analysis of a Newtonian incompressible fluid in a vertical curved channel formed by two coaxial cylindrical surfaces with a radial temperature gradient and an azimuthal pressure gradient shows that critical modes are oscillatory and non-axisymmetric. We have derived a generalized Rayleigh discriminant which includes both the curvature and buoyancy effects. Centrifugal buoyancy induces weak asymmetry of the dependence of the control parameter critical values on the sign of the temperature gradient. The critical parameters depend on the temperature gradient, the radius ratio and the nature of the fluid. For a wide curvature channel flow, there are two critical modes: oscillatory Dean modes for small temperature gradients and oscillatory centrifugal-thermal modes for relatively large temperature gradients. Received 14 November 2001 and accepted 29 March 2002 Published online: 2 October 2002 Communicated by H.J.S. Fernando     

An Approach to Transport in Heterogeneous Porous Media Using the Truncated Temporal Moment Equations: Theory and Numerical Validation

In the last decade, the characterization of transport in porous media has benefited largely from numerical advances in applied mathematics and from the increasing power of computers. However, the resolution of a transport problem often remains cumbersome, mostly because of the time-dependence of the equations and the numerical stability constraints imposed by their discretization. To avoid these difficulties, another approach is proposed based on the calculation of the temporal moments of a curve of concentration versus time. The transformation into the Laplace domain of the transport equations makes it possible to develop partial derivative equations for the calculation of complete moments or truncated moments between two finite times, and for any point of a bounded domain. The temporal moment equations are stationary equations, independent of time, and with weaker constraints on their stability and diffusion errors compared to the classical advection–dispersion equation, even with simple discrete numerical schemes. Following the complete theoretical development of these equations, they are compared firstly with analytical solutions for simple cases of transport and secondly with a well-performing transport model for advective–dispersive transport in a heterogeneous medium with rate-limited mass transfer between the free water and an immobile phase. Temporal moment equations have a common parametrization with transport equations in terms of their parameters and their spatial distribution on a grid of discretization. Therefore, they can be used to replace the transport equations and thus accelerate the achievement of studies in which a large number of simulations must be carried out, such as the inverse problem conditioned with transport data or for forecasting pollution hazards.     

Transient mixed convection flow arising due to thermal and mass diffusion over porous sensor surface inside squeezing horizontal channel

The double diffusion effect on the mixed convection flow over a horizontal porous sensor surface placed inside a horizontal channel is analyzed.With the appropriate transformations,the unsteady equations governing the flow are reduced to non-similar boundary layer equations which are solved numerically for the time-dependent mixed convection parameter.The asymptotic solutions are obtained for small and large values of the time-dependent mixed convection parameter.The results are discussed in terms of the skin friction,the heat transfer coefficient,the mass transfer coefficient,and the velocity,temperature,and concentration profiles for different values of the Prandtl number,the Schmidt number,the squeezing index,and the mixed convection parameter.     

The onset of double diffusive convection in a viscoelastic fluid layer

The onset of double diffusive convection in a viscoelastic fluid layer is studied using a linear and a weak nonlinear stability analyses. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. There is a competition between the processes of thermal diffusion, solute diffusion and viscoelasticity that causes the convection to set in through oscillatory mode rather than stationary. The effect of Deborah number, retardation parameter, solutal Rayleigh number, Prandtl number, Lewis number on the stability of the system is investigated. It is shown that the critical frequency increases with Deborah number and solutal Rayleigh number while it decreases with retardation parameter and Lewis number. The nonlinear theory based on the truncated representation of Fourier series method is used to find the heat and mass transfers. The transient behaviour of the Nusselt number and Sherwood number is investigated by solving the finite amplitude equations using Runge-Kutta method. The effect of viscoelastic parameters on heat and mass transfer is brought out.     

Natural convection in a rotating anisotropic porous layer with internal heat generation

In this article, linear and nonlinear thermal instability in a rotating anisotropic porous layer with heat source has been investigated. The extended Darcy model, which includes the time derivative and Coriolis term has been employed in the momentum equation. The linear theory has been performed by using normal mode technique, while nonlinear analysis is based on minimal representation of the truncated Fourier series having only two terms. The criteria for both stationary and oscillatory convection is derived analytically. The rotation inhibits the onset of convection in both stationary and oscillatory modes. Effects of parameters on critical Rayleigh number has also been investigated. A weak nonlinear analysis based on the truncated representation of Fourier series method has been used to find the Nusselt number. The transient behavior of the Nusselt number has also been investigated by solving the finite amplitude equations using a numerical method. Steady and unsteady streamlines, and isotherms have been drawn to determine the nature of flow pattern. The results obtained during the analysis have been presented graphically.     

The Onset of Double Diffusive Convection in a Couple Stress Fluid Saturated Anisotropic Porous Layer

The double diffusive convection in a horizontal couple stress fluid saturated anisotropic porous layer, which is heated and salted from below, is studied analytically. The modified Darcy equation that includes the time derivative term is used to model the momentum equation. The critical Rayleigh number, wavenumber for stationary and oscillatory modes, and frequency of oscillations are obtained analytically using linear theory. The effect of anisotropy parameter, solute Rayleigh number, Lewis number, couple stress parameter, and Vadasz number on the stationary, oscillatory, and finite amplitude convection is shown graphically. It is found that the thermal anisotropy parameter, couple stress parameter, and solute Rayleigh number have stabilizing effect on the stationary, oscillatory, and finite amplitude convection. The mechanical anisotropy parameter has destabilizing effect on stationary, oscillatory, and finite amplitude convection. The Lewis number has stabilizing effect in the case of stationary and finite amplitude modes, with dual effect in the case of oscillatory convection. Vadasz number advances the onset of oscillatory convection. The heat and mass transfer decrease with an increase in the values of couple stress parameter, while both increase with an increase in the value of solute Rayleigh number and mechanical anisotropy parameter. The thermal anisotropy parameter and Lewis number have contrasting effect on the heat mass transfer.     

A moment model for phonon transport at room temperature

Heat transfer in solids is modeled by deriving the macroscopic equations for phonon transport from the phonon-Boltzmann equation. In these equations, the Callaway model with frequency-dependent relaxation time is considered to describe the Resistive and Normal processes in the phonon interactions. Also, the Brillouin zone is considered to be a sphere, and its diameter depends on the temperature of the system. A simple model to describe phonon interaction with crystal boundary is employed to obtain macroscopic boundary conditions, where the reflection kernel is the superposition of diffusive reflection, specular reflection and isotropic scattering. Macroscopic moments are defined using a polynomial of the frequency and wave vector of phonons. As an example, a system of moment equations, consisting of three directional and seven frequency moments, i.e., 63 moments in total, is used to study one-dimensional heat transfer, as well as Poiseuille flow of phonons. Our results show the importance of frequency dependency in relaxation times and macroscopic moments to predict rarefaction effects. Good agreement with data reported in the literature is obtained.     

Local and global instabilities of flow in a flexible-walled channel

We consider laminar high-Reynolds-number flow through a long finite-length planar channel, where a segment of one wall is replaced by a massless membrane held under longitudinal tension. The flow is driven by a fixed pressure difference across the channel and is described using an integral form of the unsteady boundary-layer equations. The basic flow state, for which the channel has uniform width, exhibits static and oscillatory global instabilities, having distinct modal forms. In contrast, the corresponding local problem (neglecting boundary conditions associated with the rigid parts of the system) is found to be convectively, but not absolutely, unstable to small-amplitude disturbances in the absence of wall damping. We show how amplification of the primary global oscillatory instability can arise entirely from wave reflections with the rigid parts of the system, involving interacting travelling-wave flutter and static-divergence modes that are convectively stable; alteration of the mean flow by oscillations makes the onset of this primary instability subcritical. We also show how distinct mechanisms of energy transfer differentiate the primary global mode from other modes of oscillatory instability.     

Dispersion of reactive species with reversible and irreversible wall reactions

The longitudinal dispersion of a chemical species released in an oscillatory flow through an annular tube has been studied in presence of two kinds of first order reactions between the species and tube-wall. The species is supposed to undergo kinetic reversible phase exchange with the outer-wall material and irreversible absorption into the wall. Due to the variation of velocity across the tube section, the chemical species may spread out axially along the tube at a much faster rate than that produced by the molecular diffusion. A finite-difference implicit scheme has been adopted to solve the unsteady convection-diffusion equation for all time period based on the Aris method of moments. Axial distributions of mean concentration are determined from the first four central moments using Hermite polynomial representation for the periodic flow with and without non-zero mean flow. The study brings forward the coupled effects of reversible phase exchange and irreversible absorption on dispersion coefficient. Both the reversible and irreversible reactions are found to inhabit the dispersion process at early times, but at developed stage dispersion may be enhanced by the reversible phase exchange, provided the velocity comprises time invariant component. The decrease of peak of the mean concentration distribution with the increase of reaction rate is found irrespective of the nature of reaction.     

On Disturbance Propagation in Internal-Flow Boundary Layers

The unsteady interaction of plane-channel wall boundary layers with a supersonic inviscid flow is investigated. The flow regimes in which disturbances introduced by the boundary layer developing on one wall influence the boundary layer on the other wall are considered. The regime of relatively large pressure disturbance amplitudes generated near the nozzle outlet or by deforming the channel walls is studied. In these conditions, the interaction process is described by a system of Burgers equations with retarded arguments. Numerical solutions of this system are obtained for symmetric and antisymmetric perturbations of the channel walls.     

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.     

超音速气流中受热曲壁板的非线性颤振特性

基于von Karman 大变形理论及带有曲率修正的一阶活塞理论, 用Galerkin方法建立了超音速气流中受热二维曲壁板的非线性气动弹性运动方程; 采用牛顿迭代法计算得到由静气动载荷和热载荷引起的静气动弹性变形; 根据李雅谱诺夫间接法分析了壁板初始曲率与温升对颤振边界的影响; 对二维曲壁板的非线性气动弹性方程组进行数值积分求解,分析了动压参数对受热二维曲壁板分岔特性的影响, 给出了典型状态下曲壁板非线性颤振响应的时程图与相图. 分析结果表明对小初始曲率的曲壁板, 温升对其静气动弹性变形影响较大, 且随着温升的增加其颤振临界动压急剧减小; 对具有较大初始曲率的曲壁板, 温升对其静气动弹性变形的影响较弱, 且随着温升的增加颤振临界动压基本保持不变. 初始几何曲率与气动热效应使得曲壁板具有复杂的动力学特性, 不再像平壁板一样, 经过倍周期分岔进入混沌, 而会出现由静变形状态直接进入混沌运动的现象, 且在混沌运动区域中还会出现静态稳定点或谐波运动, 在大曲率情况下, 曲壁板不会产生混沌运动, 而是幅值在一定范围内的极限带振荡.      

Propagation of Rayleigh waves on curved surfaces

The propagation and properties of Rayleigh waves on curved surfaces are investigated theoretically. The Rayleigh wave dispersion equation for propagation on a curved surface is derived as a parabolic equation, and its penetration depth is analyzed using the curved surface boundary. Reciprocity is introduced to model the diffracted Rayleigh wave beams. Simulations of Rayleigh waves on some canonical curved surfaces are carried out, and the results are used to quantify the influence of curvature. It is found that the velocity of the surface wave increases with greater concave surface curvature, and a Rayleigh wave no longer exists once the surface wave velocity exceeds the bulk shear wave velocity. Moreover, the predicted wave penetration depth indicates that the energy in the Rayleigh wave is transferred to other modes and cannot propagate on convex surfaces with large curvature. A strong directional dependence is observed for the propagation of Rayleigh waves in different directions on surfaces with complex curvatures. Thus, it is important to include dispersion effects when considering Rayleigh wave propagation on curved surfaces.     

Comparison of waveguide properties of curved versus straight planar elastic layers

Dispersion equations are solved for the in-plane and anti-plane wave propagation in planar elastic layer with constant curvature. The classical Lamé formulation of displacements via elastic potentials is applied and appropriate simplifications are employed. The dispersion diagrams in each case are compared with their counterparts for a straight layer, e.g., the classical Rayleigh–Lamb solution. The curvature-induced symmetry-breaking effects are investigated for layers with symmetric boundary conditions. The role of curvature is also investigated in the cases, when the boundary conditions are not symmetrical. The elementary Bernoulli–Euler theory is employed to analyze the wave guide properties of a curved planar elastic beam in its in-plane deformation. The validity range of the Bernoulli–Euler theory is assessed via comparison of dispersion diagrams.     

低雷诺数翼型蒙皮主动振动气动特性及流场结构数值研究

针对低雷诺数(Re)翼型气动性能差的特点,文章通过对翼型柔性蒙皮施加主动振动的方法,提高翼型低Re下的气动特性,改善其流场结构.采用带预处理技术的Roe方法求解非定常可压缩Navier-Stokes方程,对NACA4415翼型低Re流动展开数值模拟.通过时均化和非定常方法对比柔性蒙皮固定和振动两种状态下的升阻力气动特性和层流分离流动结构.初步研究工作表明在低Re下柔性蒙皮采用合适的振幅和频率,时均化升阻力特性显著提高,分离泡结构由后缘层流分离泡转变为近似的经典长层流分离泡,分离点后移,分离区缩小.在此基础上,文章更加细致研究了柔性蒙皮两种状态下单周期内的层流分离结构及壁面压力系数分布非定常特性和演化规律.蒙皮固定状态下分离区前部流场结构和压力分布基本保持稳定,表现为近似定常分离,仅在后缘位置出现类似于卡门涡街的非定常流动现象.柔性蒙皮振动时从分离点附近开始便产生分离涡,并不断向下游移动、脱落,表现为非定常分离并出现大范围的压力脉动.蒙皮振动使流体更加靠近壁面运动,大尺度的层流分离现象得到有效抑制.