Prandtl number scaling of the unsteady natural convection boundary layer adjacent to a vertical flat plate for Pr>1 subject to ramp surface heat flux

It is found in the literature that the existing scaling results for the boundary layer thickness, velocity and steady state time for the natural convection flow over an evenly heated plate provide a very poor prediction of the Prandtl number dependency of the flow. However, those scalings provide a good prediction of two other governing parameters’ dependency, the Rayleigh number and the aspect ratio. Therefore, an improved scaling analysis using a triple-layer integral approach and direct numerical simulations have been performed for the natural convection boundary layer along a semi-infinite flat plate with uniform surface heat flux. This heat flux is a ramp function of time, where the temperature gradient on the surface increases with time up to some specific time and then remains constant. The growth of the boundary layer strongly depends on the ramp time. If the ramp time is sufficiently long, the boundary layer reaches a quasi-steady mode before the growth of the temperature gradient is completed. In this mode, the thermal boundary layer at first grows in thickness and then contracts with increasing time. However, if the ramp time is sufficiently short, the boundary layer develops differently, but after the wall temperature gradient growth is completed, the boundary layer develops as though the startup had been instantaneous.     

An improved boundary layer scaling with ramp heating on a sloping plate

A scaling analysis for the natural convection boundary layer adjacent to an inclined semi-infinite plate subject to a non-instantaneous heating in the form of an imposed wall temperature which increases linearly up to a prescribed steady value over a prescribed time is reported. The development of the boundary layer flow from start-up to a steady-state has been described based on scaling analyses and verified by numerical simulations. The analysis reveals that, if the period of temperature growth on the wall is sufficiently long, the boundary layer reaches a quasi-steady mode before the growth of the temperature is completed. In this mode the thermal boundary layer at first grows in thickness and then contracts with increasing time. However, if the imposed wall temperature growth period is sufficiently short, the boundary layer develops differently, but after the wall temperature growth is completed, the boundary layer develops as though the startup had been instantaneous. The steady state values of the boundary layer for both cases are ultimately the same.     

Scaling of natural convection of an inclined flat plate: Ramp cooling condition

A scaling analysis is performed for the transient boundary layer established adjacent to an inclined flat plate following a ramp cooling boundary condition. The imposed wall temperature decreases linearly up to a specific value over a specific time. It is revealed that if the ramp time is sufficiently large then the boundary layer reaches quasi-steady mode before the growth of the temperature is finished. However, if the ramp time is shorter then the steady state of the boundary layer may be reached after the growth of the temperature is completed. In this case, the ultimate steady state is the same as if the start up had been instantaneous. Note that the cold boundary layer adjacent to the plate is potentially unstable to Rayleigh–Bénard instability if the Rayleigh number exceeds a certain critical value for this cooling case. The onset of instability may set in at different stages of the boundary layer development. A proper identification of the time when the instability may set in is discussed. A numerical verification of the time for the onset of instability is presented in this study. Different flow regimes based on the stability of the boundary layer have also been discussed with numerical results.     

Scaling of unsteady natural convection boundary layers with a non-instantaneous initiation

A scaling analysis is presented for the transient boundary layer established on a vertical wall following non-instantaneous heating in the form of an imposed wall temperature which increases linearly up to a prescribed steady value over a prescribed time. The scaling analysis is verified by comparison with numerical solutions of the full equations of motion and energy. The analysis reveals that, if the period of temperature growth on the wall is sufficiently long, the boundary layer reaches a quasi-steady state before the temperature growth is completed. In this mode the thermal boundary layer at first grows in thickness and then contracts with increasing time, and the fluid acceleration also changes character. If the wall temperature growth period is sufficiently short, the boundary layer commences differently, but after the wall temperature growth is completed, the boundary layer develops as though the start up had been instantaneous. In both cases, the ultimate steady state is the same as if the start up had been instantaneous; however the different transient nature may have implications for the stability of the boundary layer or of the subsequent development of the flow in a cavity of which the wall is one boundary.     

On impulsively started convection: The case of stagnation point flow

Transient convection in incompressible planar and axisymmetric point flow is analyzed numerically in this work, and the thermal boundary layer response to surface sudden heating and cooling in the two settings is presented and compared over a range of Prandtl number between 0.5 and 100. A comparison between surface sudden cooling and heating is performed and different criteria are established as to when surface sudden heating and cooling are equivalent in terms of the transition time. With no initial thermal boundary layer (surface and fluid are at the same temperature), the transition time from the initial steady state to the final steady state upon surface sudden cooling or heating is found to be a constant regardless of the surface heating or cooling extent above or below the initial surface temperature, and is dependent only on the Prandtl number. With the existence of an initial thermal boundary layer, the transition time is dependent upon the heating or cooling extent, the initial surface temperature, the Prandtl number and whether heating/cooling is towards building-up or demolishing the thermal boundary later. It takes longer time when surface sudden heating or cooling is towards demolishing the thermal boundary layer than building it up. With symmetric surface sudden cooling or heating above or below the far-field fluid temperature, the transition time is independent on the surface cooling or heating extent and is a function of only the Prandtl number. A considerable difference in the thermal boundary layer response in the two settings is found. The transition time from the initial to the final steady state in axisymmetric stagnation point flow is less than that in plane stagnation flow under the same conditions.     

Unsteady natural convection boundary-layer flow along a vertical isothermal plate in a linearly stratified fluid with Pr>1

In this study, the unsteady natural convection boundary-layer flow along an impulsively heated vertical isothermal plate immersed in a stably stratified semi-infinite ambient fluid is explored using scaling analysis and direct numerical simulation. Scaling relations are obtained for the thermal and velocity boundary layer thicknesses, the boundary layer velocity, the development time and the Nusselt number, in terms of the Rayleigh and Prandtl numbers and the stratification parameter. The scaling results are validated using the numerical simulations.     

Scaling for the Prandtl number of the natural convection boundary layer of an inclined flat plate under uniform surface heat flux

An improved scaling analysis and direct numerical simulations are performed for the unsteady natural convection boundary layer adjacent to a downward facing inclined plate with uniform heat flux. The development of the thermal or viscous boundary layers may be classified into three distinct stages: a start-up stage, a transitional stage and a steady stage, which can be clearly identified in the analytical as well as the numerical results. Previous scaling shows that the existing scaling laws of the boundary layer thickness, velocity and steady state time scale for the natural convection flow on a heated plate of uniform heat flux provide a very poor prediction of the Prandtl number dependency of the flow. However, those scalings perform very well with Rayleigh number and aspect ratio dependency. In this study, a modified Prandtl number scaling is developed using a triple-layer integral approach for Pr > 1. It is seen that in comparison to the direct numerical simulations, the modified scaling performs considerably better than the previous scaling.     

Free convection from a vertical plate in a porous media subjected to a sudden change in surface temperature

The unsteady heat transfer process involved in free convection flow along a vertical surface embedded in a porous medium is investigated. An analytical solution has been obtained for the temperature/velocity field for small times in which the transport effects are confined within an inner layer adjacent to the plate. Then, a numerical solution of the full boundary-layer equation is obtained for the whole transient from the initial unsteady state to the final steady state. Detailed results of the effect of the temperature inputs on the transient process are given.     

Thermal receptivity of free convective flow from a heated vertical surface: Linear waves

Numerical techniques are used to study the receptivity to small-amplitude thermal disturbances of the boundary layer flow of air which is induced by a heated vertical flat plate. The fully elliptic nonlinear, time-dependent Navier–Stokes and energy equations are first solved to determine the steady state boundary-layer flow, while a linearised version of the same code is used to determine the stability characteristics. In particular we investigate (i) the ultimate fate of a localised thermal disturbance placed in the region near the leading edge and (ii) the effect of small-scale surface temperature oscillations as means of understanding the stability characteristics of the boundary layer. We show that there is a favoured frequency of excitation for the time-periodic disturbance which maximises the local response in terms of the local rate of heat transfer. However the magnitude of the favoured frequency depends on precisely how far from the leading edge the local response is measured. We also find that the instability is advective in nature and that the response of the boundary layer consists of a starting transient which eventually leaves the computational domain, leaving behind the large-time time-periodic asymptotic state. Our detailed numerical results are compared with those obtained using parallel flow theory.     

Mixed convection boundary layer flow adjacent to a vertical surface embedded in a stable stratified medium

The steady mixed convection boundary layer flow through a stable stratified medium adjacent to a vertical surface is investigated. The velocity outside the boundary layer and the surface temperature are assumed to vary linearly from the leading edge of the surface. The transformed ordinary differential equations are solved numerically by the Keller-box method. It is found that dual solutions exist, and the thermal stratification delays the boundary layer separation.     

Laminar free-convection in glycerol with variable physical properties adjacent to a vertical plate with uniform heat flux

The steady laminar boundary layer flow of glycerol along a vertical stationary plate with uniform heat flux is studied in this paper. The density, thermal conductivity and heat capacity of this liquid are linear functions of temperature but dynamic viscosity is a strong, almost exponential, function of temperature. The results are obtained with the numerical solution of the boundary layer equations. Both upward flow (plate heating) and downward flow (plate cooling) is considered. The variation of μ with temperature has significant influence on wall heat transfer and much stronger influence on wall shear stress. It was also found that the similarity exponent, which is equal to 0.20 for the classical problem with constant properties, is lower than 0.20 in the upward flow and higher than 0.20 in the downward flow.     

Effects of buoyancy assisting and opposing flows on mixed convection boundary layer flow over a permeable vertical surface

The present study deals with new similarity solution of steady mixed convection boundary layer flow over a permeable surface for convective boundary condition. It has been shown that a self similar solution is possible when the mass transfer velocity at the surface of the plate varies like x^−1/2, where x is the distance from the leading edge of the solid surface. Two point boundary value problem governed by non-linear coupled ordinary differential equations have been solved numerically using implicit finite difference scheme in combination with the quasi-linearization technique. It is interesting to note that dual solutions exist for buoyancy assisting flow, besides that usually reported in literature for buoyancy opposing flow. Further, the buoyancy assisting force causes considerable overshoot in the velocity profile and the Prandtl number strongly affects the thermal boundary layer thickness including the surface heat transfer rate.     

Periodic heat transfer by forced laminar boundary layer flow over a semi-infinite flat plate

The paper reports a study of periodic convection in a steady forced laminar boundary layer flow over a semi-infinite impermeable flat plate due to periodical variation of the wall heat flux. The Fourier transform based approach allows to obtain a transfer function for the boundary layer that can be used to solve also transient (non-periodic) heating problems, and examples are reported comparing with available studies in the open literature. The effect of periodic heating on the value of the the average heat transfer coefficient is analysed and it is found to be important for relatively high frequency fluctuations of the imposed heat flux, whereas fluctuation amplitude of the instantaneous heat transfer coefficient is non-negligible also for lower exciting frequency.     

Numerical study of a nuclear fuel element dissipating fission heat into its surrounding fluid medium

The objective of the present work is twofold – the first to establish the criterion for the boundary layer solution to be accurate enough in the study of conjugate heat transfer problem associated with a rectangular nuclear fuel element washed by upward moving coolant and the second to predict the critical thermal performance characteristics of the fuel element with uniform volumetric energy generation. Accordingly, employing stream function–vorticity formulation, equations governing the steady, two-dimensional flow and thermal fields in the coolant are solved simultaneously with the steady, two-dimensional heat conduction equation for the fuel element using second-order accurate finite difference schemes. Keeping the Prandtl number constant at 0.005 for liquid sodium as coolant, numerical results are presented for wide range of aspect ratio, conduction–convection parameter, energy generation parameter and Reynolds number. It is found that for all value of aspect ratio greater than 15, numerical prediction using the boundary layer approximation based model is quite accurate enough. It is also concluded that other parameters being kept constant, the increase in the maximum fuel element temperature due to increase in aspect ratio beyond 15 is negligible. Further, it is found that a relatively higher value of conduction–convection parameter reduces the coolant pumping power requirement to a large extent.     

Transient free convection near the lower stagnation point of a cylindrical surface subjected to a sudden change in surface temperature

This paper presents the development of the free convection boundary-layer flow of a viscous and incompressible fluid near the lower stagnation point of a cylindrical body which is subjected to a sudden change in surface temperature. Analytical solutions for both small (unsteady) and large (steady) values of time have been obtained for the boundary-layer equations. These equations are then integrated numerically.     

Study of buoyancy-induced flows subjected to partially heated sources on the left and bottom walls in a square enclosure

In this paper, numerical simulations of laminar, steady, two-dimensional natural convection flows in a square enclosure with discrete heat sources on the left and bottom walls are presented using a finite-volume method. Two different orientated wall boundary conditions are designed to investigate the natural convection features. The computational results are expressed in the form of streamlines and isothermal lines for Rayleigh numbers ranging from 10² to 10⁷ in the cavity. In the course of study, a combination of third-order and exponential interpolating profile based on the convective boundedness criterion is proposed and tested against the partially heated cavity flow up to the highest Rayleigh number 10⁷. The effects of thermal strength and heating length on the hydrodynamic and thermal fields inside the enclosure are also presented. Numerical results indicate that the average Nusselt number increases as Rayleigh number increases for both cases. Moreover, it is seen that the effect of the heat transfer rate due to the heating strength on the left wall is different from the one on the bottom. For the heater size effect, it is observed that by increasing the length of heat source segment, the heat transfer rate is gradually increased for both cases.     

Free convective stagnation-point flow in a porous medium using a thermal nonequilibrium model

An analysis is made for the steady free convection boundary-layer flow near the stagnation point of a two-dimensional body which is embedded in a porous medium by adopting a two temperature model of microscopic heat transfer. It is found that such a model modifies substantially the behaviour of the flow characteristics, particularly those of the heat transfer coefficients, and the region over which the thermal fields extend.     

Transient natural convection of non-Newtonian fluids about a vertical surface embedded in an anisotropic porous medium

An analytical method is carried out to investigate transient free convection boundary layer flow along a vertical surface embedded in an anisotropic porous medium saturated by a non-Newtonian fluid. The porous medium is anisotropic in permeability with its principal axes oriented in a direction that is non-coincident with the gravity force. A step increase in wall temperature or in surface heat flux is considered. On the basis of the modified Darcy power-law model proposed by Pascal [H. Pascal, Rheological behaviour effect of non-Newtonian fluids on steady and unsteady flow through porous media, Int. J. Numer. Anal. Methods in Geomech. 7 (1983) 207–224] and the generalized Darcy’s law described by Bear [J. Bear, Dynamics of fluids in porous media. Dover Publications, Elsevier, New York (1972)], boundary-layer equations are solved exactly by the method of characteristics. Scale analysis is applied to predict the order-of-magnitudes involved in the boundary layer regime. Analytical expressions are obtained for the limiting time required to reach steady-state, the boundary-layer thickness and the local Nusselt number in terms of the modified-Darcy Rayleigh number, the power-law index, the anisotropic permeability ratio, and the orientation angle of the principal axes. It is demonstrated that both the power-law index and the anisotropic properties have a strong influence on the heat transfer rate.