Non-Newtonian Natural Convection Along a Vertical Heated Wavy Surface Using a Modified Power-Law Viscosity Model

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
M. M. Molla ◽  
L. S. Yao
Keyword(s):  
Boundary Layer ◽  
Natural Convection ◽  
Shear Rate ◽  
Power Law ◽  
Leading Edge ◽  
Numerical Results ◽  
Wavy Surface ◽  
Viscosity Model ◽  
Length Scale ◽  
Modified Power Law

Natural convection of non-Newtonian fluids along a vertical wavy surface with uniform surface temperature has been investigated using a modified power-law viscosity model. An important parameter of the problem is the ratio of the length scale introduced by the power-law and the wavelength of the wavy surface. In this model there are no physically unrealistic limits in the boundary-layer formulation for power-law, non-Newtonian fluids. The governing equations are transformed into parabolic coordinates and the singularity of the leading edge removed; hence, the boundary-layer equations can be solved straightforwardly by marching downstream from the leading edge. Numerical results are presented for the case of shear-thinning as well as shear-thickening fluid in terms of the viscosity, velocity, and temperature distribution, and for important physical properties, namely, the wall shear stress and heat transfer rates in terms of the local skin-friction coefficient and the local Nusselt number, respectively. Also results are presented for the variation in surface amplitude and the ratio of length scale to surface wavelength. The numerical results demonstrate that a Newtonian-like solution for natural convection exists near the leading edge where the shear-rate is not large enough to trigger non-Newtonian effects. After the shear-rate increases beyond a threshold value, non-Newtonian effects start to develop.

Non-Newtonian Natural Convection Along a Vertical Flat Plate With Uniform Surface Temperature

2009 ◽  
Vol 131 (6) ◽  
Author(s):  
S. Ghosh Moulic ◽  
L. S. Yao
Keyword(s):  
Boundary Layer ◽  
Natural Convection ◽  
Shear Rate ◽  
Newtonian Fluid ◽  
Power Law ◽  
Similarity Solution ◽  
Leading Edge ◽  
Boundary Layer Equations ◽  
Shear Thinning Fluid ◽  
Vertical Flat Plate

Natural-convection boundary-layer flow of a non-Newtonian fluid along a heated semi-infinite vertical flat plate with uniform surface temperature has been investigated using a four-parameter modified power-law viscosity model. In this model, there are no physically unrealistic limits of zero or infinite viscosity that are encountered in the boundary-layer formulation for two-parameter Ostwald–de Waele power-law fluids. The leading-edge singularity is removed using a coordinate transformation. The boundary-layer equations are solved by an implicit finite-difference marching technique. Numerical results are presented for the case of a shear-thinning fluid. The results indicate that a similarity solution exists locally in a region near the leading edge of the plate, where the shear rate is not large enough to induce non-Newtonian effects; this similarity solution is identical to the similarity solution for a Newtonian fluid. The size of this region depends on the Prandtl number. Downstream of this region, the solution of the boundary-layer equations is nonsimilar. As the shear rate increases beyond a threshold value, the viscosity of the shear-thinning fluid is reduced. This leads to a decrease in the wall shear stress compared with that for a Newtonian fluid. The reduction in the viscosity accelerates the fluid in the region close to the wall, resulting in an increase in the local heat transfer rate compared with the case of a Newtonian fluid.

Viscous boundary layers in reversing flow

1976 ◽  
Vol 74 (1) ◽  
pp. 59-79 ◽  
Author(s):  
T. J. Pedley
Keyword(s):  
Boundary Layer ◽  
Shear Rate ◽  
Leading Edge ◽  
Wall Shear Rate ◽  
Viscous Boundary Layer ◽  
Large Molecules ◽  
Displacement Thickness ◽  
The Mean ◽  
Wall Shear ◽  
Viscous Boundary

The viscous boundary layer on a finite flat plate in a stream which reverses its direction once (at t = 0) is analysed using an improved version of the approximate method described earlier (Pedley 1975). Long before reversal (t < −t1), the flow at a point on the plate will be quasi-steady; long after reversal (t > t2), the flow will again be quasi-steady, but with the leading edge at the other end of the plate. In between (−t1 < t < t2) the flow is governed approximately by the diffusion equation, and we choose a simple solution of that equation which ensures that the displacement thickness of the boundary layer remains constant at t = −t1. The results of the theory, in the form of the wall shear rate at a point as a function of time, are given both for a uniformly decelerating stream, and for a sinusoidally oscillating stream which reverses its direction twice every cycle. The theory is further modified to cover streams which do not reverse, but for which the quasi-steady solution breaks down because the velocity becomes very small. The analysis is also applied to predict the wall shear rate at the entrance to a straight pipe when the core velocity varies with time as in a dog's aorta. The results show positive and negative peak values of shear very much larger than the mean. They suggest that, if wall shear is implicated in the generation of atherosclerosis because it alters the permeability of the wall to large molecules, then an appropriate index of wall shear at a point is more likely to be the r.m.s. value than the mean.

An Experimental Study of Vigorous Transient Natural Convection

1970 ◽  
Vol 92 (4) ◽  
pp. 628-634 ◽  
Author(s):  
J. C. Mollendorf ◽  
B. Gebhart
Keyword(s):  
Boundary Layer ◽  
Natural Convection ◽  
Moving Boundary ◽  
Leading Edge ◽  
Linear Stability Theory ◽  
Extreme Condition ◽  
Transient Effects ◽  
Theory Analysis ◽  
Main Observation ◽  
The Stability

External natural convection transient response leading to transition and established turbulent flow is determined experimentally and compared with the laminar double-integral theory predictions for processes wherein all transient effects are important. The theory is shown to give very accurate predictions during the laminar portion of the transient, and temperature overshool is not observed experimentally. In addition, several unexpected and very interesting observations were made concerning the stability of the flow as it proceeds to turbulence. The first main observation is that the propagating leading edge effect serves as a very effective moving boundary layer trip and triggers the resulting turbulence. Also for the less extreme condition (less vigorous transient) there is a relaminarization of the boundary layer. Explanations of these observations are proposed in the light of recently acquired results of linear stability theory analysis for small disturbances.

Lift on a steady 2-D symmetric airfoil in viscous uniform shear flow

2017 ◽  
Vol 837 ◽  
Author(s):  
Patrick R. Hammer ◽  
Miguel R. Visbal ◽  
Ahmed M. Naguib ◽  
Manoochehr M. Koochesfahani
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Shear Rate ◽  
Lift Force ◽  
Computational Study ◽  
Leading Edge ◽  
Boundary Layer Separation ◽  
Uniform Shear ◽  
Uniform Shear Flow ◽  
Boundary Layer Development

We present an investigation into the influence of upstream shear on the viscous flow around a steady two-dimensional (2-D) symmetric airfoil at zero angle of attack, and the corresponding loads. In this computational study, we consider the NACA 0012 airfoil at a chord Reynolds number $1.2\times 10^{4}$ in an approach flow with uniform positive shear with non-dimensional shear rate varying in the range 0.0–1.0. Results show that the lift force is negative, in the opposite direction to the prediction from Tsien’s inviscid theory for lift generation in the presence of positive shear. A hypothesis is presented to explain the observed sign of the lift force on the basis of the asymmetry in boundary layer development on the upper and lower surfaces of the airfoil, which creates an effective airfoil shape with negative camber. The resulting scaling of the viscous effect with shear rate and Reynolds number is provided. The location of the leading edge stagnation point moves increasingly farther back along the airfoil’s upper surface with increased shear rate, a behaviour consistent with a negatively cambered airfoil. Furthermore, the symmetry in the location of the boundary layer separation point on the airfoil’s upper and lower surfaces in uniform flow is broken under the imposed shear, and the wake vortical structures exhibit more asymmetry with increasing shear rate.

Natural Convection Heat Transfer from the Upper Plate of a Colinear, Separated Pair of Vertical Plates

1980 ◽  
Vol 102 (4) ◽  
pp. 623-629 ◽  
Author(s):  
E. M. Sparrow ◽  
M. Faghri
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Natural Convection ◽  
Vertical Plate ◽  
Convection Heat Transfer ◽  
Leading Edge ◽  
Lower Plate ◽  
Natural Convection Heat Transfer ◽  
Convection Heat ◽  
Plate Heat

The effect of a buoyant boundary layer spawned by a heated vertical plate on the natural convection heat transfer from an upper colinear vertical plate has been determined analytically. The interplate spacing was varied parametrically, as were the relative temperatures and relative lengths of the two plates; the Prandtl number was equal to 0.7 for all cases. Heat transfer at the upper plate was found to be affected both by the preheating and by the finite velocity imparted to the fluid by the first plate, respectively tending to degrade and to enhance the heat transfer. The upper-plate heat transfer was compared to that of an otherwise identical vertical plate, but with the lower plate absent. When the temperatures of the upper and lower plates are the same, the overall upper-plate heat transfer is less than that of its single-plate counterpart for small interplate spacings, with the opposite relationship at larger spacings. If the temperature of the upper plate is substantially below that of the lower plate, the overall heat transfer is degraded. On the other hand, heat transfer enhancement generally occurs when the upper plate is relatively hot. In general, the heat transfer from relatively short upper plates is very sensitive to the presence of the lower plate, with a lessening sensitivity with increasing plate length. The computed temperature and velocity profiles demonstrated that near the leading edge of the upper plate, a new boundary layer develops within the already existing boundary layer spawned by the first plate.

Scaling of square-prism shear layers

2018 ◽  
Vol 849 ◽  
pp. 1096-1119 ◽  
Author(s):  
D. C. Lander ◽  
D. M. Moore ◽  
C. W. Letchford ◽  
M. Amitay
Keyword(s):  
Boundary Layer ◽  
Shear Layer ◽  
Power Law ◽  
Frequency Ratio ◽  
Shear Layers ◽  
Nonlinear Dependence ◽  
Front Face ◽  
Length Scale ◽  
Square Prism ◽  
Ratio Scaling

Scaling characteristics, essential to the mechanisms of transition in square-prism shear layers, were explored experimentally. In particular, the evolution of the dominant instability modes as a function of Reynolds number were reported in the range $1.5\times 10^{4}\lesssim Re_{D}\lesssim 7.5\times 10^{4}$. It was found that the ratio between the shear layer frequency and the shedding frequency obeys a power-law scaling relation. Adherence to the power-law relationship, which was derived from hot-wire measurements, has been supported by two additional and independent scaling considerations, namely, by particle image velocimetry measurements to observe the evolution of length and velocity scales in the shear layer during transition, and by comparison to direct numerical simulations to illuminate the properties of the front-face boundary layer. The nonlinear dependence of the shear layer instability frequency is sustained by the influence of $Re_{D}$ on the thickness of the laminar front-face boundary layer. In corroboration with the original scaling argument for the circular cylinder, the length scale of the shear layer was the only source of nonlinearity in the frequency ratio scaling, within the range of Reynolds numbers reported. The frequency ratio scaling may therefore be understood by the influence of $Re_{D}$ on the appropriate length scale of the shear layer. This length scale was observed to be the momentum thickness evaluated at a transition point, defined where the Kelvin–Helmholtz instability saturates.

Laminar Natural Convection Between a Vertical Surface With Uniform Heat Flux and Pseudoplastic and Dilatant Fluids

2014 ◽  
Vol 136 (9) ◽  
Author(s):  
Massimo Capobianchi ◽  
A. Aziz
Keyword(s):  
Heat Flux ◽  
Nusselt Number ◽  
Natural Convection ◽  
Shear Rate ◽  
Power Law ◽  
Average Nusselt Number ◽  
Vertical Surface ◽  
Flux Boundary Condition ◽  
Shear Rates ◽  
Laminar Natural Convection

This paper reports the average Nusselt number for steady, laminar natural convection between a vertical surface and otherwise quiescent pseudoplastic and dilatant fluids under a constant and uniform surface heat flux boundary condition. Models for the fluids' apparent viscosity were utilized that are valid in all five regions of the flow curve. The results are thus applicable for whatever shear rates may exist within the flow field and a dimensionless shear rate parameter was identified that quantifies the shear rate region where the given system is operating. The data indicate that the average Nusselt numbers approach the corresponding Newtonian values when the shear rates are predominantly in either the zero or the infinite shear rate Newtonian regions. However, power law values are approached only when both of the following two conditions are met: (1) the shear rates are principally in the power law region and (2) the fluid's limiting zero and infinite shear rate Newtonian viscosities differ sufficiently, by approximately 4 orders of magnitude or more. For all other cases, the average Nusselt number was found to reside between the Newtonian and the power law asymptotes. Results are provided in both graphical and tabular form over a broad range of system parameters.

Long-range wall perturbations in dense granular flows

2014 ◽  
Vol 764 ◽  
pp. 171-192 ◽  
Author(s):  
Pierre G. Rognon ◽  
Thomas Miller ◽  
Bloen Metzger ◽  
Itai Einav
Keyword(s):  
Shear Rate ◽  
Power Law ◽  
Eddy Viscosity ◽  
Granular Flows ◽  
Length Scale ◽  
Local Models ◽  
Plane Shear ◽  
Eddy Viscosity Model ◽  
Non Local ◽  
Dense Granular Flows

AbstractWe explore how the rheology of dense granular flows is affected by the presence of sidewalls. The study is based on discrete element method simulations of plane-shear flows between two rough walls, prescribing both the normal stress and the shear rate. Results confirm previous observations for different systems: large layers near the walls develop where the local viscosity is not constant, but decreases when approaching the walls. The size of these layers can reach several dozen grain diameters, and is found to increase when the flow decelerates, as a power law of the inertial number. Two non-local models are found to adequately explain such features, namely the kinetic elasto-plastic fluidity (KEP) model and the eddy viscosity model (EV). The analysis of the internal kinematics further shows that the vorticity and its associated length scale may be a key component of these non-local behaviours.

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