The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Recent experimental, numerical and theoretical advances in turbulent Rayleigh-Benard convection are presented. Particular emphasis is given to the physics and structure of the thermal and velocity boundary layers which play a key role for the better understanding of the turbulent transport of heat and momentum in convection at high and very high Rayleigh numbers. We also discuss important extensions of Rayleigh-Benard convection such as non-Oberbeck-Boussinesq effects and convection with phase changes.

https://link.springer.com/content/pdf/10.1140%2Fepje%2Fi2012-12058-1.pdf

New perspectives in turbulent Rayleigh-Bénard convection.

This article reviews linear instability analysis of flows over or through complex two-dimensional (2D) and 3D geometries. In the three decades since it first appeared in the literature, global instability analysis, based on the solution of the multidimensional eigenvalue and/or initial value problem, is continuously broadening both in scope and in depth. To date it has dealt successfully with a wide range of applications arising in aerospace engineering, physiological flows, food processing, and nuclear-reactor safety. In recent years, nonmodal analysis has complemented the more traditional modal approach and increased knowledge of flow instability physics. Recent highlights delivered by the application of either modal or nonmodal global analysis are briefly discussed. A conscious effort is made to demystify both the tools currently utilized and the jargon employed to describe them, demonstrating the simplicity of the analysis. Hopefully this will provide new impulses for the creation of next-generation algorithms capable of coping with the main open research areas in which step-change progress can be expected by the application of the theory: instability analysis of fully inhomogeneous, 3D flows and control thereof.

Global Linear Instability

Hydrodynamic and hydromagnetic stability

This review highlights the profound and unexpected ways in which viscosity varying in space and time can affect flow. The most striking manifestations are through alterations of flow stability, as established in model shear flows and industrial applications. Future studies are needed to address the important effect of viscosity stratification in such diverse environments as Earth's core, the Sun, blood vessels, and the re-entry of spacecraft.

Instabilities in Viscosity-Stratified Flow

The effect of confinement on the decay of vortex ring is studied computationally using Lattice Boltzmann Method. An Initial vortex ring, introduced inside a wall bounded cubical domain, is let to evolve and its decay is noted in terms of maximum vorticity at the core and the total kinetic energy inside the domain. The study shows distinct regimes of decay in all cases of confinement ratios(ratio of vortex ring diameter to length of the cubical domain).

Effect of confinement on the decay of vortex rings

We perform a three-dimensional, short-wavelength stability analysis on the numerically simulated two-dimensional flow past a circular cylinder for Reynolds numbers in the range $50\le Re\le300$; here, $Re = U_{\infty}D/\nu$ with $U_\infty$, $D$ and $\nu$ being the free-stream velocity, the diameter of the cylinder and the kinematic viscosity of the fluid, respectively. For a given $Re$, inviscid local stability equations from the geometric optics approach are solved on three distinct closed fluid particle trajectories (denoted as orbits 1, 2 & 3) for purely transverse perturbations. The instability on orbits 1 & 2, which are symmetric counterparts of one another, is shown to undergo a bifurcation at $Re\approx50$, and two more bifurcations at $Re\approx250$, which is around the same value at which mode B secondary instability becomes dominant. This instability on orbits 1 & 2 is shown to be stable for all $Re\leq262$ after incorporating finite-wavenumber, finite-Reynolds number corrections on the computed local stability growth rates. The transition Reynolds number for the corrected growth rate $Re\approx 262$, above which the instability on orbits 1 & 2 is synchronous, is remarkably close to the mode B critical Reynolds number ($Re=259$) estimated from global stability analysis in an earlier study. Furthermore, for $Re= 280$, the corrected growth rate variation with the span-wise wavelength is shown to display great qualitative and quantitative agreement with the mode B instability growth rates from the global analysis. These results strongly suggest that the short-wavelength instability on orbit 1 & 2 is a possible mechanism for the birth of the mode B secondary instability.

Bifurcations of short-wavelength instabilities in the vortex shedding flow behind a two-dimensional cylinder

The effect of viscosity stratification on the different mechanism of transition to turbulence is not well understood. In this paper, a viscosity variation normal to the flow in a channel is investigated. The primary and secondary instability are computed, and the transient growth is analysed. It is found that viscosity stratification can have different effects in each case.

Effect of viscosity stratification on secondary and nonmodal instabilities

Gas-kinetic simulations of rarefied and compressible mixing layers are performed to characterize continuum breakdown and the effect on the Kelvin-Helmholtz instability. The unified gas-kinetic scheme (UGKS) is used to perform the simulations at different Mach and Knudsen numbers. The UGKS stress tensor and heat-flux vector fields are compared against those given by the Navier-Stokes-Fourier constitutive equations. The most significant difference is seen in the shear stress and transverse heat flux. The study demonstrates the existence of two distinct continuum breakdown regimes, one at low and the other at high convective Mach numbers. Overall, at low convective Mach numbers, the deviation from continuum stress and heat flux appears to scale exclusively with the micro-macro length scale ratio given by the Knudsen number. On the other hand, at high convective Mach numbers, the deviation depends on the global micro-macro timescale ratio given by the product of Mach and Knudsen numbers. We further demonstrate that, unlike shear stresses and transverse heat flux, the deviations in normal stresses and the streamwise heat flux depend separately on Knudsen and Mach numbers. A local parameter called the gradient Knudsen number is proposed to characterize the rarefaction effects on the local momentum and thermal transport. Noncontinuum aspects of gas-kinetic stress-tensor and heat-flux behavior that Grad's 13-moment equation model reasonably captures are identified.

Continuum breakdown in compressible mixing layers.

The flow past a heated cylinder is studied numeri-cally in the steady separated buoyancy aided flow regime (Reynolds number < 47) by using a hybrid finite element finite volume (FEM-FVM) solver. In this study we eluci-date the physics behind the Prandtl number effects on heat transfer from the cylinder surface

/pdf/effect-of-prandtl-number-on-heat-transfer-in-flow-past-a-54nu1fxc3d.pdf

A. Sameen

Papers

Effect of confinement on the decay of vortex rings

Bifurcations of short-wavelength instabilities in the vortex shedding flow behind a two-dimensional cylinder

Effect of viscosity stratification on secondary and nonmodal instabilities

Continuum breakdown in compressible mixing layers.

Effect of Prandtl number on Heat Transfer In Flow Past A heated cylinder