Applied Mathematics GIDP
18th US National Congress for Theoretical and Applied Mechanics
June 4-9, 2018
The receptivity of high-‐speed compressible boundary layers to kinetic fluctuations (KF) is considered within the framework of fluctuating hydrodynamics. The formulation is based on the idea that KF induced dissipative fluxes may lead to the generation of unstable modes in the boundary layer. Fedorov and Tumin (AIAA J., 2017) solved the receptivity problem using an asymptotic matching approach which utilized a resonant inner solution in the vicinity of the neutral point of the second Mack mode. Here we adopt a slightly more general inhomogeneous multiple scales (IMS) approach, based on a WKB ansatz, which requires fewer assumptions about the locus of primary excitation. The approach is modeled after the one taken by Luchini (AIAA J., 2017) to study low speed incompressible boundary layers over a swept wing. The new framework is used to study examples of high-‐speed, high-‐enthalpy, flat plate boundary layers whose spectra exhibit nuanced behavior near the generation point, such as first Mack mode instabilities and near-‐neutral evolution over moderate length scales.
Abstract for Lay Audience
This research is motivated by the desire of various organizations to develop aircraft capable of traveling at extremely high speeds. DARPA (the Defense Advanced Research Projects Agency), for example, hopes to develop an aircraft which could travel between any two points on the globe in under 2 hours. Critical to the design of these aircraft is a detailed understanding of how the air would flow over the surface of such a vehicle.
Towards the front of the aircraft, the air tends to flow in a smooth and predictable manner, a desirable state referred to as ‘laminar flow’. Farther back on the vehicle body, the flow transitions to a highly disordered state known as ‘turbulence’. Turbulence is undesirable because it causes extreme heating on the surface of the vehicle which necessitates heavy and expensive heat shielding. Transition to this undesirable state is often inevitable, and thus it is of utmost importance from a vehicle design perspective to predict where the onset of turbulence occurs on the body of the aircraft. This is known as laminar-‐turbulent transition prediction.
Transition to turbulence is typically caused by external disturbances present in the air near the body of the vehicle. For example, turbulence can be induced by particles of atmospheric debris colliding with the vehicle surface. Our recent work is devoted to understanding how a much smaller and subtler disturbance might similarly lead to laminar-‐turbulent transition. In particular, we show how even the seemingly insignificant ‘jiggling’ of molecules in the air near the aircraft may be enough to cause a breakdown of laminar flow into disordered turbulence.
The presentation to be given at this conference describes our work on a new mathematical approach to studying this process. It was developed to facilitate the study of laminar-‐turbulent transition for a broad range of vehicle types, some of which cannot be studied using other methods. The talk will detail the new mathematical formulation as well as the theoretical background necessary to understand the results.