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Title

ASYMPTOTIC SIMILARITY IN TURBULENT BOUNDARY LAYERS

Creator

Duncan, Richard D.

Date

2011-05-10, 2011-05

Description

The turbulent boundary layer is one of the most fundamental and important applications of fluid mechanics. Despite great practical interest...
Show moreThe turbulent boundary layer is one of the most fundamental and important applications of fluid mechanics. Despite great practical interest and its direct impact on frictional drag among its many important consequences, no theory absent of significant inference or assumption exists. Numerical simulations and empirical guidance are used to produce models and adequate predictions, but even minor improvements in modeling parameters or physical understanding could translate into significant improvements in the efficiency of aerodynamic and hydrodynamic vehicles. Classically, turbulent boundary layers and fully-developed turbulent channels and pipes are considered members of the same “family,” with similar “inner” versus “outer” descriptions. However, recent advances in experiments, simulations, and data processing have questioned this, and, as a result, their fundamental physics. To address a full range of pressure gradient boundary layers, a new approach to the governing equations and physical description of wall-bounded flows is formulated, using a two variable similarity approach and many of the tools of the classical method with slight but significant variations. A new set of similarity requirements for the characteristic scales of the problem is found, and when these requirements are applied to the classical “inner” and “outer” scales, a “similarity map” is developed providing a clear prediction of what flow conditions should result in self-similar forms. An empirical model with a small number of parameters and a form reminiscent of Coles’ “wall plus wake” is developed for the streamwise Reynolds stress, and shown to fit experimental and numerical data from a number of turbulent boundary layers as well as other wall-bounded flows. It appears from this model and its scaling using the free-stream velocity that the true asymptotic form of u′2 may not become self-evident until Re ≈ 275, 000 or δ+ ≈ 105, if not higher. A perturbation expansion made possible by the novel inclusion of the scaled streamwise coordinate is used to make an excellent prediction of the shear Reynolds stress in zero pressure gradient boundary layers and channel flows, requiring only a streamwise mean velocity profile and the new similarity map. Extension to other flows is promising, though more information about the normal Reynolds stresses is needed. This expansion is further used to infer a three layer structure in the turbulent boundary layer, and modified two layer structure in fully-developed flows, by using the classical inner and logarithmic profiles to determine which portions of the boundary layer are dominated by viscosity, inertia, or turbulence. A new inner function for U+ is developed, based on the three layer description, providing a much more simplified representative form of the streamwise mean velocity nearest the wall.
Ph.D. in Mechanical and Aerospace Engineering, May 2011
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