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

Experimental and theoretical studies have been carried out to examine the thermophysical properties of cryogenic insulators. Experimental studies were carried out in a cryogen canister with foam insulation. Liquid nitrogen is used as the cryogen. Chromel- alumel thermocouples were used for measuring the temperatures. Heat leak was measured from the boil-off loss of liquid nitrogen. Thermal conductivity of the foam insulation is deduced from the Fourier's phenomenological law. The encouraging results generated through these classical tests lead to say that using the same concept more profitable experiments can be carried out for the thermophysical property evaluation of insulators for the cryogenic upper stages.

A Classical Evaluation of the Thermophysical Properties of Insulators for Cryogenic Rockets

A wall jet, produced when fluid is blown tangentially across a flat surface, occurs in various applications The linear instability of this flow has been studied under the assumption of locally parallel flow [2,4] and weakly non-parallel flow, ie, using the parabolized stability equations (PSE) [7] but to our knowledge, its global stability has not

Global Instabilities in Wall Jets

This paper is a review of some of our recent work on the effect of wall heating on the stability of laminar flow in a channel. The summary of our results, some of them unexpected, is as follows. Viscosity stratification has very little effect on transient growth, whereas it results in linear mode stabilising or destabilising by an order of magnitude. It has hitherto been accepted that heat diffusivity does not affect stability. This is however true only for linear instability, transient growth is affected by an order of magnitude. Unusually, the growth is spanwise-independent and not in the form of streamwise vortices. It is also shown that flow is destabilised by secondary modes as the viscosity ratio increases. However, the viscosity ratio has no role in the selection of the pattern of Λ vortices.

Some indications from instability results about the effectiveness of wall heating as a control option for channel flow

Taylor–Couette flow is a canonical flow to study Taylor–Görtler (TG) instability or centrifugal instability and the associated vortices. TG instability has been traditionally associated with flow over curved surfaces or geometries. In the computational study, we confirm the presence of TG-like near-wall vortical structures in two lid-driven flow systems, the Vogel–Escudier (VE) and the lid-driven cavity (LDC) flows. The VE flow is generated inside a circular cylinder by a rotating lid (top lid in the present study), while the LDC flow is generated inside a square or rectangular cavity by the linear movement of the lid. We look at the emergence of these vortical structures through reconstructed phase space diagrams and find that the TG-like vortices are seen in the chaotic regimes in both flows. In the VE flow, these vortices are seen when the side-wall boundary layer instability sets in at large Re. The VE flow is observed to go to a chaotic state in a sequence of events from a steady state at low Re. In contrast to VE flows, in the LDC flow with no curved boundaries, TG-like vortices are seen at the emergence of unsteadiness when the flow exhibits a limit cycle. The LDC flow is observed to have transitioned to chaos from the steady state through a periodic oscillatory state. Various aspect ratio cavities are examined in both flows for the presence of TG-like vortices. This article is part of the theme issue ‘Taylor–Couette and related flows on the centennial of Taylor’s seminal Philosophical transactions paper (Part 2)’.

Near-wall vortical structures in domains with and without curved surfaces

Modal and non-modal analysis of superfluid He-II is carried out in this paper. Landau's two fluid model for superfluid flow is assumed and stability of both normal fluid and superfluid are considered separately by taking the effect of the other through mutual friction. Stability of normal fluid, with a numerically calculated base flow, shows Tollmien-Schlichting mode of instability similar to classical fluids. Non-modal analysis shows the transient growth of perturbations at sub-critical Reynolds numbers. Mutual friction is found to have a stabilizing effect on normal fluid. Superfluid, with an inviscid uniform profile is found to be linearly stable for all wave numbers. Non-modal analysis rules out the possibility of transient growth of perturbations and sub-critical transition. The presence of turbulence in superfluid brings in a possibility that an initial uniform base flow is getting altered due to the action of mutual friction and this altered profile is likely to become unstable.

A. Sameen

Papers

A Classical Evaluation of the Thermophysical Properties of Insulators for Cryogenic Rockets

Global Instabilities in Wall Jets

Some indications from instability results about the effectiveness of wall heating as a control option for channel flow

Near-wall vortical structures in domains with and without curved surfaces

Instabilities in superfluid plane Poiseuille flow