L.S. Yao

Free and forced convection in the entry region of a heated vertical channel

Abstract An analytical solution for the fluid flow and heat transfer in the entry region of a heated vertical channel is presented. The conditions of constant wall temperature and constant wall heat flux are studied. Different axial length scales are revealed by the analytical solution. These scales distinguish the regions of different convective mechanisms that a developing flow has to pass through before reaching its fully-developed state. The solution also indicates that natural convection eventually becomes the dominant heat transfer mode if Gr > Re for constant wall temperature, and Gr2 > Re for constant wall heat flux. If natural convection is a dominant mode, the evidence suggests that moving periodic and recirculating cells are generated. This provides an explanation of why reverse transition is observed, and laminar flow is maintained in a heated vertical tube for Re

Is a fully-developed and non-isothermal flow possible in a vertical pipe?

Abstract The results of a linear-stability analysis of the fully-developed flow in a heated vertical pipe are presented. They confirm the experimental observations that flow in a heated vertical pipe is supercritically unstable. The bifurcated new equilibrium laminar flow is likely to be a double spiral flow. Mixing induced by this spiral flow can cause a substantial increase in the heat-transfer rate and even delay transition to turbulence, as has been observed experimentally.

Natural convection in a heated annulus

Abstract Natural convection in a heated vertical concentric annulus is studied. A constant heat-flux is applied to the inner cylinder and the outer cylinder is insulated. Under these conditions, the mean temperature of the fluid increases linearly while, at the same time, heat diffuses from the heated surface into the fluid, resulting in a temperature distribution on the cross-section. Subtracting this steadily increasing temperature from the total temperature results in a steady temperature stratification on the cross-section which drives fluid motion. The mathematical form of the scaled problem is shown to be identical to that of a fluid in an annulus with uniformly distributed heat sources, with the inner cylinder maintained at constant temperature and the outer cylinder insulated. At low heat addition rates, the fluid motion is steady and parallel, and heat is transferred by conduction between the fluid layers. As the rate of heating increases, the flow becomes unstable and recirculating eddies appear, which transfer heat by convection. The onset of convection is determined by linear-instability analysis of the basic-state. The results demonstrate that when the Prandtl number is small, the dominant instability obtained energy primarily from shear production. On the other hand, when the Prandtl number is large, an instability that obtains kinetic energy from buoyant production is pre-eminent. Weakly nonlinear instability theory is used to analyze finite-amplitude effects. The results show that both types of linear instabilities are supercritical.

Non-Newtonian Fluid Flow on a Flat Plate Part 2: Heat Transfer

Forced convective heat transfer of non-Newtonian fluids on a flat plate is investigated using a recently proposed modified power-law model. For a shear-thinning fluid, non-Newtonian effects are illustrated via local temperature distributions, heat transfer rate, and surface temperature distribution. Most significant effects occur near the leading edge, gradually tailing off far downstream.

Finite-amplitude instability of mixed-convection in a heated vertical pipe

Abstract The instability of flow in a heated vertical pipe is studied using weakly nonlinear instability theory for both stably and unstably stratified cases. It is found that the dominant instability for stably stratified flow is a thermal-buoyant instability, while that of the unstably stratified case is a Rayleigh-Taylor instability. The weakly nonlinear theory predicts supercritical instability for the stably stratified case, in agreement with experimental observations. In this case, it is found that a wide band of wave numbers are linearly unstable soon after the onset of the initial instability. This limits the range for which the weakly nonlinear results are accurate in this case since the theory considers the growth of a single dominant wave. The results of the weakly nonlinear calculations for unstably stratified flow indicate that the flow is potentially subcritically unstable, again in agreement with the experimental observations. On the other hand, the theory predicts that a large amplitude disturbance will be necessary to initiate subcritical instability, while the amplitude of a supercritical disturbance will grow quickly as the magnitude of Ra increases. Therefore, another possible flow transition that is consistent with experimental observations involves rapid growth of the first azimuthal mode of a supercritical Rayleigh Taylor instability, followed by secondary instabilities that lead quickly to turbulence. Analysis of energy transfer in the fundamental wave demonstrates that the thermal-buoyant instability is supercritical because increases in the viscous dissipation rate and the rate of transfer of energy from the fundamental wave back into the mean flow overcome the destabilizing effect of an increase in the rate of buoyant production. Subcritical instability occurs with the Rayleigh-Taylor mode when the disturbance amplitude increases to the point that the combined destabilizing effects caused by a change in the shape of the fundamental wave induced by nonlinear effects become larger than the stabilizing effects due to the production of the harmonic wave and the distortion of the mean-flow. The increase in heat transfer rates due to instability predicted by the weakly nonlinear theory is smaller than the experimental observations. However, it is demonstrated that experimentally observed increases in Nu are predicted if the effects of additional waves are included in an approximate manner.

Non-Newtonian Fluid Flow on a Flat Plate Part 1: Boundary Layer

A modified power-law viscosity for non-Newtonian fluids based on actual measurements is proposed. This realistic model allows removal of the singularities at the leading edge of a flat-plate boundary layer for either shear-thinning or shear-thickening fluids. Under this condition, the boundary-layer equations can be solved numerically by simple finite difference methods that march downstream from the leading edge, as is usually done for Newtonian fluids. Numerical results are presented for the case of a shear-thinning fluid; applying the model to a shear-thickening fluid is straightforward. The effects of this new variable viscosity are explicitly demonstrated by comparing plots of isolines of viscosity and shear rate, the velocity distribution, and the wall shear stress for non-Newtonian and Newtonian fluids.

Natural convection effects in the continuous casting of a horizontal cylinder

Abstract The effects of natural convection along the interface between two phases in the continuous casting of a horizontal cylinder are studied. Due to natural convection, the solidification rate along the bottom of a horizontal cylinder is shown to be faster than that along the top. The solidification rate grows downstream as z 5 2 (z being the axial coordinate) due to the natural convection, and is not aximuthally uniform. This conclusion is an important extension over previous work which considered only conduction and showed a solidification rate which increased as z 5 2 .

The effect of mixed convection instability on heat transfer in a vertical annulus

Abstract The hydrodynamic stability of mixed convection in an annulus is studied. The linear stability limit for forced flow up a vertical annulus with a constant heat flux applied to the inner wall and the outer wall insulated is determined. The result indicates that the fully-developed flow is thermally unstable in most regions of an appropriate parameter space. The magnitudes of the finite amplitude disturbances in the unstable region are determined by utilizing Stuarts shape assumption. Distorted mean flow profiles are obtained and the increase in heat transfer rates due to these disturbances are calculated from the results and agree well with the experimental data.

A Weak collision of two natural-convection boundary layers

Abstract The collision of two natural-convection boundary layers at the tip of a vertical wedge is used to demonstrate that a double-deck flow structure provides a proper description of the heat-convection mechanism which is shared by many convection problems with sudden geometry changes. The present theory differs from previous work which indicate the existence ofrecirculating flow regions. This difference is due to a failure to recognize that recirculating flow structures can only exist for forced flows, but not for natural-convection boundary layers or wall jets. The present solution is obtained by a proper application of Prandtls transposition theorem for geometries with finite solid displacements and appropriately matches the upstream natural-convection boundary layers and the downstream thermal plume. The interaction of the local pressure development in the main deck (outer layer) and the displacement of the lower deck (inner layer) removes the singularity associated with the boundary-layer equations at the location where the viscous layers leave the solid surface.

Mixed convection over a vertical zircaloy plate in steam with simultaneous oxidation

Abstract An analytical study is made of the quasi-steady, laminar, binary boundary layer flow of steam and hydrogen over a vertical Zircaloy plate under conditions pertinent to the Three Mile Island nuclear reactor accident. The experimentally observed oxidation rate law is modified to account for the possible presence of hydrogen in steam outside the boundary layer and for variable plate temperatures. The oxidation of Zircaloy gives rise to a small suction velocity of the steam-hydrogen mixture at the cladding surface. Details of the longitudinal and transverse components of the velocity profiles, temperature, and hydrogen concentration profiles across the binary boundary layer are presented as well as longitudinal distributions of wall suction velocity, wall shear, and wall temperature. A parametric study of the effect of surface radiative heat loss was made, using recently reported data on the radiation heat transfer coefficient. Correlations are proposed for the local wall temperature rise and for the local Nusselt and Sherwood numbers.