Finite difference analysis of unsteady natural convection flow along an inclined plate with variable surface temperature and mass diffusion

An analysis is performed to study the transient laminar natural convection flows along an inclined semi-infinite flat plate in which the wall temperatureT _w ^′ and species concentration on the wallC _w ^′ vary as the power of the axial co-ordinate in the formT _w ^′ (x)=T _∞ ^′ +ax ⁿ andC _w ^′ =C _∞ ^′ +bx ^m respectively. The dimensionless governing equations considered here are unsteady, two-dimensional, coupled and non-linear integro-differential equations. A finite difference scheme of Crank-Nicolson type is employed to solve the problem. The velocity, temperature, concentration, skin friction, Nusselt number and Sherwood number are studied in detail for various sets of values of parameters. Correlation equations are also established for Nusselt number and Sherwood number in terms of parameters.     

Impuls- und Wärmeübertragung bei variablen Stoffwerten für die laminare Plattenströmung

The laminar two-dimensional flow along a flat plate of constant temperature at low velocity is investigated where the temperature dependence of the fluid properties is taken into account. The functions describing the temperature dependence of the fluid properties are expanded as Taylor series at the reference state (preferred free-stream state, but wall state also possible), whose coefficients are dimensionless fluid properties like the Prandtl number, but are not specified for particular fluids. Shear stress and heat flux at the wall are given for arbitrary temperature dependence of the fluid properties as universal power series of the parameter ε=(T_w?T_∞)/T_∞. For all fluids it is shown, how the exponents in the propertyratio method and the reference temperature in the referencetemperature method depend on the fluid properties. Hence, these two methods are not empirical ones any more.     

Natural convection-radiation interaction on boundary layer flow along a thin vertical cylinder

A free convertion flow of an optically dense viscous incompressible fluid along a vertical thin circular cylinder has been studied with effect of radiation when the surface temperature is uniform. With appropriate transformations, the boundary layer equations governing the flow are reduced to local nonsimilarity equations. Solutions of the governing equations are obtained employing the implicit finite difference methods together with Keller box scheme as well the local nonsimilarity method with second order truncation for all ξ (nondimensional transverse curvature parameter) in the interval [0,10] and are expressed in terms of local Nusselt number for a range of values of the pertinent parameters. Effects of pertinent parameters, such as, the radiation parameter, R _d , the surface temperature parameter, θ _w , taking Prandtl number, Pr, equals 0.7 on the velocity and temperature field are also presented graphically. From the solution it is seen that increase of R _d , or θ _w leads to increase in the local rate of heat transfer coefficients. Results obtained by both the methods are obtained in excellent agreement between each other upto ξ = 10.     

Combined forced and natural convection in laminar film flow

A mathematical model for the flow and heat transfer in a gravity-driven liquid film is presented, in which the strict Boussinesq approximation is adopted to account for buoyancy. A similarity transformation reduces the governing equations to a coupled set of ordinary differential equations. The resulting two-parameter problem is solved numerically for Prandtl numbers ranging from 1 to 1000. Favourable buoyancy arises when the temperatureT _w of the isothermal surface is lower than the temperatureT ₀ of the incoming fluid, and the principal effects of the aiding buoyancy are to increase the wall shear and heat transfer rate. For unfavourable buoyancy (T _w>T ₀), the buoyancy force and gravity act in opposite directions and the flow in the film boundary layer decelerates, whereas the friction and heat transfer are reduced. The observed effects of buoyancy diminish appreciably for higher Prandtl numbers.     

Turbulent film boiling from a vertical non-isothermal surface

The turbulent film boiling from a vertical non-isothermal surface is formulated with due consideration to thermal radiation from its lateral face. It is observed that the application of Reynolds analogy together with thermal conduction in the test surface has yielded a conjugate solution from which the case of an isothermal condition can be generated as a special case. The analysis has further paved the way in establishing a functional relation between the Nusselt numberNu, radiation parameterN _R , fin parameterM, temperature ratio termT _s /(T _w,0?T _s ), and a product of characteristic modified Grashof, Prandtl and superheating parameter defined as (Gr ² Pr S). In a fully developed turbulent film boiling i.e., modified Grashof number being greater than 10¹⁰, the temperature ratio term accounts for the non-linearities arising due to the inclusion of radiation from the lateral face of the fin. The results are in good agreement with experimental data over a wide range of system conditions.     

Boundary layer development on a continuous moving surface with a parallel free stream due to impulsive motion

The development of the momentum and thermal boundary layers over a semi-infinite flat plate has been studied when the external stream as well as the plate are impulsively moved with constant velocities. At the same time the temperature of the wall is suddenly raised from T_∞, the temperature of the surrounding fluid, toT _w and maintained at this temperature. The problem has been formulated in a new system of scaled coordinates such that fort?=0 it reduces to Rayleigh type of equation and fort? → ∞ it reduces to Blasius or Sakiadis type of equation. A new scale of time has been used which reduces the region of integration from an infinite region to a finite region which reduces the computational time considerably. The governing singular parabolic partial differential equations have been solved numerically using an implicit finite difference scheme. For some particular cases, analytical solutions have been obtained. The results show that there is a smooth transition from Rayleigh solution to Blasius or Sakiadis solution as the dimensionless timeξ increases from zero to one. The shear stress at the wall is negative for the friction parameterλ<0.5, positive forλ>0.5 and zero forλ=0.5. The zero shear stress at the wall does not imply separation but corresponds to the parallel flow. The surface heat transfer is strongly dependent on the Prandtl numberPr and increases with it. Also forPr<Pr ₀, the surface heat transfer is enhanced as the friction parameterλ increases, but forPr>Pr ₀ it get reduced.     

Combined radiation and convection heat transfer in simultaneously developing flow in ducts with semi-circular and right triangular cross sections

In this paper, the interaction of radiation and forced convection in simultaneously developing laminar flow through semicircular and right triangular ducts with isothermal non-black wall is investigated. The three dimensional momentum and energy equations are discretized by the method of lines and solved numerically by the marching method. The method of momentum is employed to consider the radiation contribution which models the radiation in the partial differential equation, instead of the partial integrodifferential equation. The effects of three major parameters, radiation-conduction parameter,N, optical thickness, τ _b , and wall emissivity, ε _w , in the entry region of these irregular geometry ducts are discussed. The numerical results in terms of the variation of the bulk temperature and the mean Nusselt number indicate that the thermal radiation not only enhances the heat transfer rate, but also changes the characteristics of the convective heat transfer. Furthermore, the results compare very well with the available data published in the open literature for the pure convective case.     

Mixed convection cooling of a heated, continuously stretching surface

 An numerical study of the flow and heat transfer characteristics associated with a heated, continuously stretching surface being cooled by a mixed convection flow has been carried out. The relevant heat transfer mechanisms are of interest in a wide variety of practical applications, such as hot rolling, continuous casting, extrusion, and drawing. The surface velocity of the continuously stretching sheet was assumed to vary according to a power-law form, that is, u _w (x)=Cx ^p . Two conditions of surface heating were considered, a variable wall temperature (VWT) in the form T _w (x)−T _∞=Ax ⁿ and a variable surface heat flux (VHF) in the form q _w (x)=Bx ^m . The governing differential equations are transformed by introducing proper nonsimilarity variables and solved numerically using a procedure based on finite difference approximations. Results for the local Nusselt number and the local friction coefficient are obtained for a wide range of governing parameters, such as the surface velocity parameter p, the wall temperature exponent n, the surface heat flux exponent m, the buoyancy force parameters (ξ for the VWT case and χ for the VHF case), and Prandtl number of the fluid. It is found that the local Nusselt number is increased with increasing the velocity exponent parameter p for the VWT case, while the opposite trend is observed for the VHF case. The local friction coefficient is increased for a decelerated stretching surface, while it is decreased for an accelerated stretching surface. Also, appreciable effects of the buoyancy force on the local Nusselt number and the local friction coefficient are observed for both VWT and VHF cases, as expected. Received on 11 January 1999     

Radiation-conduction interaction in mixed convection along rotating bodies

This paper investigates the interaction of the steady mixed convection boundary layer flow past a rotating impermeable body placed in a uniform stream moving opposite to the gravitational force and parallel to the axes of the body of revolution with uniform surface temperature and thermal radiation. The fluid considered here is a gray, absorbing-emitting but non-scattering medium, and the Rosseland approximation is used to describe the radiative heat flux in the analysis. The difficulty of having a unified mathematical treatment of this problem is due to the nonsimilarity nature of the governing equations arising from the buoyant force-field and the transverse curvature of the bodies. The important parameters of this problem are the radiation-conduction parameter R _d and the wall to free stream temperature ratio θ _w , for the case of a heated surface. Numerical simulations of the boundary layer equations are performed using the local nonsimilarity method as well as an implicit finite-difference method. The theory is applied to a rotating sphere for the gases with Prandtl number of 0.7. The results are shown graphically in terms of the local skin-friction coefficients and the local rate of heat transfer. Effects of the pertinent parameters R _d and θ _w are also shown on the components of the velocity distribution as well as on the temperature distribution in the boundary layer. Received on 14 January 1997     

Mixed convection boundary layer flow over a horizontal circular cylinder with Newtonian heating

The steady mixed convection boundary layer flow over a horizontal circular cylinder, generated by Newtonian heating in which the heat transfer from the surface is proportional to the local surface temperature, is considered in this study. The governing boundary layer equations are first transformed into a system of non-dimensional equations via the non-dimensional variables, and then into non-similar equations before they are solved numerically using a numerical scheme known as the Keller-box method. Numerical solutions are obtained for the skin friction coefficient Re ^1/2 C _f and the local wall temperature θ _w (x) as well as the velocity and temperature profiles with two parameters, namely the mixed convection parameter λ and the Prandtl number Pr.     

Viscous dissipation effects on thermal entrance region heat transfer in pipes with uniform wall heat flux

The analytical solution to Graetz problem with uniform wall heat flux is extended by including the viscous dissipation effect in the analysis. The analytical solution obtained reduces to that of Siegel, Sparrow and Hallman neglecting viscous dissipation as a limiting case. The sample developing temperature profiles, wall and bulk temperature distributions and the local Nusselt number variations are presented to illustrate the viscous dissipation effects. It is found that the role of viscous dissipation on thermal entrance region heat transfer is completely different for heating and cooling at wall. In the case of cooling at wall, a critical value of Brinkman number, Br _c=−11/24, exists beyond which (−11/24<Br<0) the fluid bulk temperature will always be less than the uniform entrance temperature indicating the predominance of cooling effect over the viscous heating effect. On the other hand, with Br < Br _c the bulk temperature T _b will approach the wall temperature T _w at some downstream position and from there onward the bulk temperature T _b becomes less than the wall temperature T _w with T _w > B _b > T ₀ indicating overall heating effect for the fluid. The numerical results for the case of cooling at wall Br < 0 are believed to be of some interest in the design of the proposed artctic oil pipeline.     

Slow gas motions and the negative drag of a strongly heated spherical particle

In the slow flows of a strongly and nonuniformly heated gas, in the continuum regime (Kn → 0) thermal stresses may be present. The theory of slow nonisothermal continuum gas flows with account for thermal stresses was developed in 1969–1974. The action of the thermal stresses on the gas results in certain paradoxical effects, including the reversal of the direction of the force exerted on a spherical particle in Stokes flow. The propulsion force effect is manifested at large but finite temperature differences between the particle and the gas. This study is devoted to the thermal-stress effect on the drag of a strongly heated spherical particle traveling slowly in a gas for small Knudsen numbers (M ～ Kn → 0), small but finite Reynolds numbers (Re ≤ 1), a linear temperature dependence of the transport coefficients µ ∝ T, and large but finite temperature differences ((T _w ? T _∞ )/T _M8 ～ 1). Two different systems of equations are solved numerically: the simplified Navier-Stokes equations and the modified Navier-Stokes equations with account for the thermal stresses.     

Radiation effect on Darcy free convection flow along an inclined surface placed in porous media

The paper investigates the effect of radiation on Darcy's buoyancy induced flow of an optically dense viscous incompressible fluid along a heated inclined flat surface maintained at uniform temperature placed in a saturated porous medium with Rosseland diffusion approximation employing the implicit finite difference method together with Keller box elimination technique. Both the streamwise and normal components of the buoyancy force are retained in the momentum equations. The numerical results show that as the buoyancy parameter, ξ, increases the local Nusselt number increases. The results for the locally nonsimilar solutions are compared with the locally similar solutions for small angle of inclination and approximate similar solutions along vertical surface. The effect of the conduction-radiation parameter, R _d , and the surface temperature excess ration, θ _w , on the local Nusselt number, the tangential velocity distribution and the temperature distribution are also shown graphically.     

Computational study of combined effects of conduction-radiation and hydromagnetics on natural convection flow past magnetized permeable plate

The computational study of the combined effects of radiation and hydromagnetics on the natural convection flow of a viscous,incompressible,and electrically conducting fluid past a magnetized permeable vertical plate is presented.The governing non-similar equations are numerically solved by using a finite difference method for all values of the suction parameter ξ and the asymptotic solution for small and large values of ξ.The effects of varying the Prandtl number P r,the magnetic Prandtl number P r m,the magnetic force parameter S,the radiation parameter R d,and the surface temperature θ w on the coefficients of the skin friction,the rate of heat transfer,and the current density are shown graphically and in tables.An attempt is made to examine the effects of the above mentioned physical parameters on the velocity profile,the temperature distribution,and the transverse component of the magnetic field.     

An experimental investigation of the ELAC 1 configuration at supersonic speeds

Supersonic flight of aerospace planes is of marked interest since several flow regimes characterized by different local flow structures have to be flown through. This problem was investigated experimentally for the hypersonic research configuration ELAC 1. The aim of the study was to detect the influence of the rounded leading edge, of the thickness distribution prescribed, and of the Reynolds number, especially on the flow on the leeward side of the configuration. The experiments were carried out in the transonic wind tunnel of Aerodynamisches Institut of RWTH Aachen, at a freestream Mach number Ma _∞=2, a unit Reynolds number of Re _∞=13×10⁶, angles of attack between ?3°?α?10°, and in a wind tunnel of the Institute for Theoretical and Applied Mechanics of the Russian Academy of Sciences in Novosibirsk. The freestream Mach numbers covered in these experiments were varied between 2?Ma _∞?4, freestream Reynolds numbers per unit length between 25×10⁶?Re _∞?56×10⁶ and angles of attack between ?3°?α?10°. Flow visualization studies, measurements of surface pressure distributions and of aerodynamic forces were used to analyze the flow. The results, which will also be compared with numerical data, clearly indicate marked differences in the location of the separation and reattachment lines, and the formation of the primary, secondary and tertiary vortices, for the flow regimes investigated.     

Radiation-conduction interaction on mixed convection from a horizontal circular cylinder

A mixed convection flow of an optically dense viscous incompressible fluid along a horizontal circular cylinder has been studied with the effect of radiation when the surface temperature is uniform. Using appropriate transformations, the boundary layer equations governing the flow are reduced to local nonsimilarity form. Solutions of the governing equations are obtained employing the implicit finite difference method. Effects of varying the pertinent parameters, such as, the Planck number, R _w the surface temperature parameter, θ_w and the buoyancy parameter, α on the local skin-friction and local heat transfer coefficients are shown graphically as well as in tabular form against the curvature parameter ξ, while taking Prandtl number Pr = 1.0. It is found that an increase of R _d,θ_w or α leads to increases in the values of the local skin-friction and the local rate of heat transfer coefficients. At the stagnation point asymptotic solutions for large value of α are also obtained and the effect of the other pertinent parameters on the formation of the flow separation are studied. Received on 28 July 1998     

Heat and mass transfer characteristics of the self-similar boundary-layer flows induced by continuous surfaces stretched with rapidly decreasing velocities

The mechanical and thermal characteristics of the self-similar boundary-layer flows induced by continuous surfaces stretched with rapidly decreasing power-law velocities U _w∝x ^m , m<?1 are considered. Comparing to the well studied cases of the increasing stretching velocities (m>0) several new features of basic significance have been found. Thus: (i) for m<?1 the boundary layer equations admit self-similar solutions only if a lateral suction is applied; (ii) the dimensionless suction velocity f _w<0 must be strong enough, i.e. f _w<f _w,max(m) where f _w,max(m) depends on m so that its absolute maximum max (f _w,max(m))=?2.279 is reached for m→?∞, while for m→?1, f _w,max(m)→?∞; (iii) the case {m→?∞, f _w,max(m)=?2.279} of the flow boundary value problem is isomorphic to the stretching problems with exponentially decreasing velocities U _w∝e ^ax with arbitrary a<0; (iv) for any fixed m<?1 and f _w<f _w,max(m) the flow problem admits a non-denumerable infinity of multiple solutions corresponding to the values of the dimensionless skin friction f ^″(0)≡s belonging to a finite interval s∈ [s _min(f _w,m), s _max(f _w,m)]; (v) the solution is only unique for f _w=f _w,max(m) where s=s _min(f _w,m)= s _max(f _w,m) holds; (vi) to every one of the multiple solutions of the flow problem there corresponds a unique solution of the heat transfer problem with a wall temperature distribution T _w?T _∞∝x ⁿ and a well defined and distinct value of the dimensionless wall temperature gradient ?^′(0), except for the cases n=(|m|?1)/2 where ?^′(0) has the same value ?^′(0)=Pr·f _w for the whole class of flow solutions with s∈[s _min(f _w,m), s _max(f _w,m)]; (vii) for f _w→?∞ one obtains the `asymptotic suction profiles' corresponding to s=s _min(f _w,m)?f _w and ?^′(0)?Pr·f _w in an explicit analytic form. The paper includes several examples which illustrate the dependence of the heat and fluid flows induced by surfaces stretching with rapidly decreasing velocities on the physical parameters f _w, m, n and Pr.     

Heat transfer and equilibrium temperature of a sphere in a supersonic rarefied gas flow

Some results are presented of experimental studies of the equilibrium temperature and heat transfer of a sphere in a supersonic rarefied air flow.The notations D sphere diameter - u, , T,,l, freestream parameters (u is velocity, density, T the thermodynamic temperature,l the molecular mean free path, the viscosity coefficient, the thermal conductivity) - T₀ temperature of the adiabatically stagnated stream - T_e mean equilibrium temperature of the sphere - T_w surface temperature of the cold sphere (T_we) - mean heat transfer coefficient - _e air thermal conductivity at the temperature T_e - P Prandtl number - M Mach number     

Transient compressible boundary layer on a wedge impulsively set into motion

Summary The development of a compressible boundary layer over a wedge impulsively set into motion is studied in this paper. The initial motion is independent of the leading edge effect and the solutions are those of a Rayleigh-type problem. The motion tends to an ultimate steady state of Falkner-Skan type. The equations governing the transient boundary layer from the initial steady state to the terminal steady-state change their character after certain time due to the leading edge effect and thereafter solution depends on both the end conditions. Numerical solutions are obtained through the second-order accuracy upwind scheme. The effects of the Falkner-Skan parameter and the surface temperature on the transient flow and heat transfer are also studied. It has been found that the flow separation does not occur form–0.0707 when _w = 1.5 (hot wall), andm–0.118 when _0.5 (cold wall).     

The effect of suction or injection on the boundary layer flows induced by continuous surfaces stretched with prescribed skin friction

The boundary layer flow and heat transfer on a stretched surface moving with prescribed skin friction is studied for permeable surface. Three major cases are studied for isothermal surface (n=0) stretched corresponding to different dimensional skin friction boundary conditions namely; skin friction at the surface scales as (x ^?1/2) at m=0, constant skin friction at m=1/3 and skin friction scales as (x) at m=1. The constants m and n are the indices of the power law velocity and temperature exponent respectively. Similarity solutions are obtained for the boundary layer equations subject to power law temperature and velocity variation. The effect of various governing parameters, such as Prandtl number Pr, suction/injection parameter f _w , m and n are studied. The results show that for isothermal surface increasing m enhances the dimensionless heat transfer coefficient for fixed f _w at the suction case and the reverse is true at the injection case. Furthermore, for fixed m, as f _w increases the dimensionless heat transfer coefficient increases. Large enhancements are observed in the heat transfer coefficient as the temperature boundary condition along the surface changes from uniform to linear where the dimensional skin friction is of order (x) at m=1. This enhancement decreases as the suction increases.