Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances (Asymptotic Methods i)
Venue: CISM, Udine
Event Date/Time: Sep 21, 2009 | End Date/Time: Sep 25, 2009 |
Registration Date: Sep 18, 2009 | |
Early Registration Date: Jul 27, 2009 |
Description
most powerful methods used in fluid mechanics are the method of matched asymptotic expansions and multiple scales methods.
Matched asymptotic expansions are based on the idea of Prandtlâ€™s boundary layer theory. In case of high Reynolds number flows the flow field can be approximated by an inviscid flow with the exception of a thin boundary layer along the wall where the viscosity has to be taken into account. Both approximations have to match in an intermediate region.
In some cases the inviscid flow and the viscous flow in a sub-layer have to be determined simultaneously. Thus one speaks of â€œInteracting boundary layersâ€.An introduction to triple deck problems and recent applications to internal flows, external sub- or super sonic flows, thermal flows and free surface flows will be presented.
Another fruitful application is the theory of separated laminar incompressible flows. Various examples of fluid flows involving separation will be considered, including self-induced separation of the boundary layer in supersonic gas flows, and incompressible flow separation at the leading edge of an aerofoil.
A characteristic feature of multiple scales problem is that the solution exhibits almost periodic structures whose properties vary on a large scale. Recently multiple scales methods have been applied to problem in meteorology. Thus well established ad hoc approximations have been verified by applying the method of multiple scales to the basic equations of fluid flow in the atmosphere. It will be demonstrated how a large collection of well-established models of theoretical meteorology can be recovered systematically, how new insight into scale interaction processes is gained, and how the asymptotic analyses provide hints for the construction of accurate and efficient numerical methods. The known limitations of the approach will be also discussed.
Many problems in fluid mechanics involve asymptotic expansions in the form of power series. Such expansions necessarily fail to find terms which are exponentially smaller than all terms in the series. Although small, these missing terms are often of physical importance. How to find such exponentially small terms, using as the main tool matched asymptotic expansions in the complex plane and Borel summation will be discussed.