New Closed-Form Thermal Boundary Layer Solutions in Shear Flow With Power-Law Velocity

2019 ◽  
Vol 141 (6) ◽  
Author(s):  
Chang Yi Wang ◽  
Chien-Cheng Chang
Keyword(s):  
Boundary Layer ◽  
Shear Flow ◽  
Power Law ◽  
Thermal Boundary Layer ◽  
Mass Diffusion ◽  
Gamma Functions ◽  
Thermal Boundary Conditions ◽  
Flux Change ◽  
Convection Problem ◽  
Parallel Shear Flow

The forced convection problem for a developing thermal boundary layer in a parallel shear flow is studied. If the shear flow has a power-law velocity profile, exact similarity thermal boundary layer solutions in terms of Gamma functions can be found. Specifically, three types of thermal boundary conditions are considered: a step temperature change, a step flux change, and a concentrated heat source. The latter is also analogous to mass diffusion form an isolated source. The mixing index for mass diffusion is found exactly.

The effect of a shear flow on convection in a layer heated non-uniformly from below

1985 ◽  
Vol 154 ◽  
pp. 303-319 ◽  
Author(s):  
I. C. Walton
Keyword(s):  
Boundary Layer ◽  
Shear Flow ◽  
Fluid Layer ◽  
Heated Layer ◽  
Vertical Temperature ◽  
Onset Of Convection ◽  
A Value ◽  
Blasius Boundary Layer ◽  
Convection Problem

In an earlier paper (Walton 1982) we showed that, when a fluid layer is heated non-uniformly from below in such a way that the vertical temperature difference maintained across the layer is a slowly varying monotonic function of a horizontal coordinate x, then convection occurs for x > xc, where xc is the point where the local Rayleigh number is equal to the critical value for a uniformly heated layer. Furthermore, the amplitude of the convection increases smoothly from exponentially small values for x [Lt ] xc and asymptotes to a value given by Stuart–Watson theory for x [Gt ] xc.At the present time no solutions of this kind are available for a class of problems in which the onset of instability is heavily influenced by a shear flow (e.g. Görtler vortices in a boundary layer on a curved wall, convection in a heated Blasius boundary layer). In a first step to bridge the gap between these problems and in order to elucidate the difficulties associated with the presence of a shear flow, we investigate the effect of a (weak) shear flow on our earlier convection problem. We show that the onset of convection is delayed and that it appears more suddenly, but still smoothly. The role of horizontal diffusion is shown to be of paramount importance in enabling a solution of this kind to be found, and the implications of these results for instabilities in higher-speed shear flows are discussed.

Experimental studies of the viscous boundary layer properties in turbulent Rayleigh–Bénard convection

2008 ◽  
Vol 605 ◽  
pp. 79-113 ◽  
Author(s):  
CHAO SUN ◽  
YIN-HAR CHEUNG ◽  
KE-QING XIA
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Shear Stress ◽  
Rayleigh Number ◽  
Power Law ◽  
Thermal Boundary Layer ◽  
Viscous Sublayer ◽  
Viscous Boundary Layer ◽  
Velocity Boundary Layer ◽  
Viscous Boundary

We report high-resolution measurements of the properties of the velocity boundary layer in turbulent thermal convection using the particle image velocimetry (PIV) technique and measurements of the temperature profiles and the thermal boundary layer. Both velocity and temperature measurements were made near the lower conducting plate of a rectangular convection cell using water as the convecting fluid, with the Rayleigh number Ra varying from 109 to 1010 and the Prandtl number Pr fixed at 4.3. From the measured profiles of the horizontal velocity we obtain the viscous boundary layer thickness δυ. It is found that δυ follows the classical Blasius-like laminar boundary layer in the present range of Ra, and it scales with the Reynolds number Re as δυ/H = 0.64Re−0.50±0.03 (where H is the cell height). While the measured viscous shear stress and Reynolds shear stress show that the boundary layer is laminar for Ra < 2.0 × 1010, two independent extrapolations, one based on velocity measurements and the other on velocity and temperature measurements, both indicate that the boundary layer will become turbulent at Ra ~ 1013. Just above the thermal boundary layer but within the mixing zone, the measured temperature r.m.s. profiles σT(z) are found to follow either a power law or a logarithmic behaviour. The power-law fitting may be slightly favoured and its exponent is found to depend on Ra and varies from −0.6 to −0.77, which is much larger than the classical value of −1/3. In the same region, the measured profiles of the r.m.s. vertical velocity σw(z) exhibit a much smaller scaling range and are also consistent with either a power-law or a logarithmic behaviour. The Reynolds number dependence of several wall quantities is also measured directly. These are the wall shear stress τw ~ Re1.55, the viscous sublayer δw ~ Re−0.91, the friction velocity uτ ~ Re0.80, and the skin-friction coefficient cf ~ Re−0.34. All of these scaling properties are very close to those predicted for a classical Blasius-type laminar boundary layer, except that of cf. Similar to classical shear flows, a viscous sublayer is also found to exist in the present system despite the presence of a nested thermal boundary layer. However, velocity profiles normalized by wall units exhibit no obvious logarithmic region, which is likely to be a result of the very limited distance between the edge of the viscous sublayer and the position of the maximum velocity. Compared to traditional shear flows, the peak position of the wall-unit-normalized r.m.s. profiles is found to be closer to the plate (at z+ = z/δw ≃ 5). Our overall conclusion is that a Blasius-type laminar boundary condition is a good approximation for the velocity boundary layer in turbulent thermal convection for the present range of Rayleigh number and Prandtl number.

An analysis of the characteristics of the thermal boundary layer in power law fluid

2008 ◽  
Vol 17 (3) ◽  
pp. 233-237 ◽  
Author(s):  
Hao Zhang ◽  
Xinxin Zhang ◽  
Liancun Zheng
Keyword(s):  
Boundary Layer ◽  
Power Law ◽  
Thermal Boundary Layer ◽  
Power Law Fluid

An integral approach to the calculation of skin friction and heat transfer in the laminar flow of a high-Prandtl-number power-law non-Newtonian fluid past a wedge

1991 ◽  
Vol 69 (2) ◽  
pp. 83-89 ◽  
Author(s):  
G. Ramamurty ◽  
K. Narasimha Rao ◽  
K. N. Seetharamu
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Prandtl Number ◽  
Skin Friction ◽  
Layer Thickness ◽  
Power Law ◽  
Boundary Layer Thickness ◽  
Thermal Boundary Layer ◽  
Integral Approach ◽  
High Prandtl Number

An integral approach to the theoretical analysis for the skin friction of a non-Newtonian, power-law-fluid flow over a wedge is presented, when the inertia terms in the boundary-layer equations are small but need consideration. The method adopted for the solution of the equations considers an integrated average value of the inertia terms in the momentum equation. The values of the velocities and the boundary-layer thickness obtained from the hydrodynamic analysis are used for the calculation of the thermal-boundary-layer thickness. A linear velocity profile is assumed for the flow field within the thermal boundary layer as the fluids chosen for the analysis are high-Prandtl-number fluids. The results of the skin friction and the rates of the heat transfer are tabulated for a number of values of the flow behaviour index, n, varying from 0.05 to 5.0. This analysis is applicable to viscous polymer solutions having high Prandtl numbers.

Similarity Solution for Unsteady Free Convection From a Vertical Plate at Constant Temperature to Power Law Fluids

2012 ◽  
Vol 134 (10) ◽  
Author(s):  
J. Abolfazli Esfahani ◽  
B. Bagherian
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Free Convection ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Power Law ◽  
Thermal Boundary Layer ◽  
Theoretic Approach ◽  
Local Skin ◽  
Power Law Fluids

The transformation group theoretic approach is applied to perform an analysis of unsteady free convection flow over a vertical flat plate immersed in a power law fluid. The thermal boundary layer induced within a vertical semi-infinite layer of Boussinseq fluid. The system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions via two-parameter group theory. The obtained ordinary differential equations are solved numerically for velocity and temperature using the fourth order Runge-Kutta and shooting method. The effect of Prandtl number and viscosity index (n) on the thermal boundary-layer, velocity boundary-layer, local Nusselt number, and local skin-friction were studied.

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