The Stability of the Laminar Boundary Layer

The present papcr is a continuation of a theoretical investigation of the stability of the laminar boundary layer in a compressible fluid. An approximate estimate for the minimum critical Reynolds number Re[sub]cr[sub-sub]min, or stability limit, is obtained in terms of the distribution of the kinematic viscosity and the product of the mean density [rho][super][bar]* and mean vorticity [formula] across the boundary layer. With the help of this estimate for Re[sub]cr[sub-sub]min it is shown that withdrawing heat from the fluid through the solid surface increases RRe[sub]cr[sub-sub]min and stabilizes the flow, as compared with the flow over an insulated surface at the same Mach number. Conduction of heat to the fluid through the solid surface has exactly the opposite effect. The value of Re[sub]cr[sub-sub]min for the insulated surface decreases as the Mach number increases for the case of a uniform free-stream velocity. These general conclusions are supplemented by detailed calculations of the curves of wave number (inverse wave length) against Reynolds number for the neutral disturbances for 10 representative cases of insulated and noninsulated surfaces. So far as laminar stability is concerned, an important difference exists between the case of a subsonic and supersonic free-stream velocity outside the boundary layer. The neutral boundary-layer disturbances that are significant for laminar stability die out exponentially with distance from the solid surface; therefore, the phase velocity c* of these disturbances is subsonic relative to the free-stream velocity [symbol] or [symbol], [symbol] where is the local sonic velocity. When [symbol]<1, (where M[sub]0 is free-stream Mach number), it follows that [inequalities] and any laminar boundary-1ayer flow is ultimately unstable at sufficiently high Reynolds numbers because of the destabilizing action of viscosity near the solid surface, as explained by Prandtl for the incompressible fluid. When M[sub]0 >1, however, [inequalities]. If the quantity [forumla] is large enough negatively, the rate at which energy passes from the disturbance to the mean flow, which is proportional to [formula], can always be large enough to counterbalance the rate at which energy passes from the mean flow to the disturbance because of the destabilizing action of viscosity near the solid surface. In that case only damped disturbances exist and the laminar boundary layer is completely stable at all Reynolds numbers. This condition occurs when the rate at which heat is withdrawn from the fluid through the solid surface reaches or exceeds a critical value that depends only on the Mach number and the properties of the gas. Calculations show that for M[sub]0 > 3 (approx.) the laminar boundary-layer flow for thermal equilibrium -- where the heat conduction through the solid surface balances the heat radiated from the surface -- is completely stable at all Reynolds numbers under free-flight conditions if the free-stream velocity is uniform. The results of the analysis of the stability of the laminar boundary layer must be applied with care to discussions of transition; however, withdrawing heat from the fluid through the solid surface, for example, not only increases Re[sub]cr[sub-sub]min but also decreases the initial rate of amplification of the self-excited disturbances, which is roughly proportional fo 1/[sqrt]Re[sub]cr[sub-sub]min. Thus, the effect of the thermal conditions at the solid sufice on the transition Reynolds number Re[sub]tt, is similar to the effect on Re[sub]cr[sub-sub]min. A comparison between this conclusion and experimental investigations of the effect of surface heating on transition at low speeds shows that the results of the present paper give the proper direction of this effect. The extension of the results of the stability analysis to laminar boundary-layer gas flows with a pressure gradient in the direction of the free stream is discussed

A Discussion of the Application of the Prandtl-Glauert Method to Subsonic Compressible Flow over a Slender Body of Revolution

The Prandtl-Glauert method for subsonic potential flow of a compressible fluid has generally been believed to lead to an increase in the pressures over a slender body of revolution by a factor 1/([sqrt](1-M[sub]1^2)) (where M[sub]1 is Mach number in undisturbed flow) as compared with the pressures in incompressible flow. Recent German work on this problem has indicated, however, that the factor 1/([sqrt](1-M[sub]1^2)) is not applicable in this case. In the present discussion a more careful application of the Prandtl-Glauert method to three-dimensional flow gives the following results: The Prandtl-Glauert method does not lead to a universal velocity or pressure correction formula that is independent of the shape of the body. The factor 1/([sqrt](1-M[sub]1^2)) is applicable only to the case of two-dimensional flow. The increase with Mach number of the pressures over a slender body of revolution is much less rapid than for a two-dimensional airfoil. An approximate formula from which the increase can be estimated is derived theoretically. The increase with Mach number of the maximum axial interference velocity on a slender body of revolution in a closed wind tunnel is given approximately by the factor 1/((1-M[sub]1^2)^-3/2), rather than by the factor 1/([sqrt](1-M[sub]1^2)) previously obtained by Goldstein and Young and by Tsien and Lees

What are the factors which contribute to level one social work students failing to progress or achieving low grades?

This article was first published in the Wolverhampton Intellectual Repository and E-Theses (WIRE). There is no printed version.This study is a preliminary review of the possible reasons for low achievement among some level one social work undergraduates. These may be viewed as challenges to the individual, attempting to study in a particular social context, or as challenges to the institution in raising achievement and accommodating differing needs. Much of the literature is concerned with the experiences of students from particular social groups. In some studies, these concerns are integrated with the identification of individual strategies for success and/or institutional practices which foster or inhibit achievement

Self-protected electrodes limit field-emission current

One cathode includes array of square-shaped conductor columns. Columns are electrically interconnected by conducting plate on bottom. Each column provides field-emission current. Another cathode includes array of rodlike conductors. Layer is covered by control film made of conducting material. Film is isolated from each conductor

Big city life? Challenges and trade-offs for Auckland city

Priming Auckland for ambitious future growth requires change on many policy fronts. Auckland is starting to swell. Rapid population and income growth are occurring, but Auckland’s narrow geography – with harbours on both sides of the city – limits land availability. Without change Auckland cannot reap the benefits of a growing population. Expect lower living standards from longer commute times and rising housing costs. “The outcomes we want from Auckland, and the outcomes we should expect given Auckland’s narrow geography, are two very different things.” said Dr Kirdan Lees. “That heightens the need to get infrastructure and other urban policies right”. Policymakers have generally identified the right set of levers – including extending the urban boundary, reducing urban planning restrictions and improving productivity in the housing sector. But improving transport infrastructure to reduce the cost of land, making it cheaper to build and buy housing for families, is an option that looks a little underdone in the public debate. NZIER recommends: Taking a closer look at the benefits of transport infrastructure projects that reduce the cost of well-located land, reducing the cost of housing. Keep grinding away at construction productivity – a 15 percent improvement makes the largest gain to the welfare of Auckland families compared to other policies. Extend the urban boundary to include 25 percent more land by 2031. Our framework suggests that policy would make each family $860 a year better off. Implement change over a full range of policies: our work suggests leaving the heavy lifting to a single policy makes achieving the kinds of living standard untenable. The report, “Big city life? Challenges and trade-offs for Auckland city”, was funded by the Reserve Bank, New Zealand Treasury, the Ministry of Transport and NZIER’s public good programme, which supports research into areas of general interest to New Zealanders

Fighting fit? Assessing New Zealand’s fiscal sustainability

This report highlights the unprecedented fiscal challenges that New Zealand politicians will face in coming decades.The report recommends that:The government better manages expectations about what New Zealand can afford in the future by mapping out the types of spending and taxation policies that will change. Taxes and government spending are changed soon to start spreading the costs of the ageing adjustment over several years. Broadening the tax base and reorienting welfare back to those most in need are good starting points. Political parties seek a bipartisan agreement on addressing New Zealand’s future superannuation costs

Incoherence and incompatibility in planning law

Purpose– The purpose of this paper is to analyse the obligations imposing localism and the presumption in favour of sustainable development in English planning law.Design/methodology/approach– The paper uses doctrinal analysis to examine section 38 PCPA 2004 and the NPPF to assess whether the obligations are coherent when considered as stand-alone obligations, and whether they are compatible when combined. Case law and the statutory provisions are examined to assess this. Planning theory is also examined to bring a multidisciplinary focus to the analysis.Findings– The paper concludes that there are problems with these legal obligations when considered as stand-alone obligations. There is imprecision over the meaning of key terms; the “presumptions” established do not operate as true presumptions; and there is an ambiguity as to the hierarchy of norms and the allocation of decision-making control. When combined, the incoherence increases. It is argued that this occurs thanks to underlying disagreements in key concepts in planning theory.Originality/value– This paper examines the new structures of planning law introduced under the Localism Act 2011 and the NPPF and considers how this structure works in practice. It does so from the multidisciplinary viewpoint of planning law and planning theory and links these two approaches. This is not replicated elsewhere in the literature. It considers in detail the ensuing case law, and the contradictions that appear. Again, there is little surveying the overall framework of planning law in the UK.</jats:sec