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Person# Peter Monkewitz

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ME-608: Methods of asymptotic analysis in mechanics

The introduction to asymptotic analysis provides the basis for constructing many simplified analytical models in mechanics and for testing computations in limiting cases.

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Reynolds number

In fluid mechanics, the Reynolds number (Re) is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous force

Boundary layer

In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction wi

Turbulence

In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows

Further examination of the pressure measurement uncertainties in determining the pressure gradient along the fully-developed section of CICLoPE, yields a Karman constant based on the centerline velocity of kappa(CL) = 0.43 +/- 0.06. More significantly, we conclude that the CICLoPE facility requires a more accurate pressure scanning method for determining dp/dx; e.g., using a Scanivalve connected to a single more accurate pressure transducer.

The trinity of so-called "canonical" wall-bounded turbulent flows, comprising the zero pressure gradient turbulent boundary layer, abbreviated ZPG TBL, turbulent pipe flow, and channel/duct flows has continued to receive intense attention as new and more reliable experimental data have become available. Nevertheless, the debate on whether the logarithmic part of the mean velocity profile, in particular the Karman constant kappa, is identical for these three canonical flows or flow-dependent is still ongoing. In this paper, the asymptotic matching requirement of equal. in the logarithmic overlap layer, which links the inner and outer flow regions, and in the expression for the centerline/free-stream velocity is reiterated and shown to preclude a universal logarithmic overlap layer in the three canonical flows. However, the majority of pipe and channel flowstudies at friction Reynolds numbers Re-tau below approximate to 10(4) extract from near-wall profiles the same kappa of 0.38-0.39 as in the ZPG TBL. This apparent contradiction is resolved by a careful reanalysis of high-quality mean velocity profiles in the Princeton "Superpipe" and other pipes, channels, and ducts, which shows that the mean velocity in a near-wall region extending to around 700 "+" units in channels and ducts and 500 "+" units in pipes is the same as in the ZPG TBL. In other words, all the "canonical" flow profiles contain the lower end of the ZPG TBL log-region, which starts at a wall distance of 150-200 "+" units with a universal kappa of kappa(ZPG) approximate to 0.384. This interior log-region is followed by a second logarithmic region with a flow specific. > kappa(ZPG), which increases monotonically with pressure gradient. This second, exterior log-layer is the actual overlap layer matching up to the outer expansion, which implies equality of the exterior. and kappa(CL) obtained from the evolution of the respective centerline velocity with Reynolds number. The location of the switch-over point implies furthermore that this second log-layer only becomes clearly identifiable, i.e., separated from the wake region, for Re-tau well beyond 10(4) (see Fig. 1). This explains the discrepancies between the Karman constants of 0.38-0.39, extracted from near-wall pipe profiles below Re-tau approximate to 10(4) and the kappa's obtained from the evolution of the centerline velocity with Reynolds number. The same analysis is successfully applied to velocity profiles in channels and ducts even though experiments and numerical simulations have not yet reached Reynolds numbers where the different layers have even started to clearly separate.

Jiannong Fang, Marco Giovanni Giometto, Peter Monkewitz, Marc Parlange

The Nieuwstadt closed-form solution for the stationary Ekman layer is generalized for katabatic flows within the conceptual framework of the Prandtl model. The proposed solution is valid for spatially-varying eddy viscosity and diffusivity (O'Brien type) and constant Prandtl number (Pr). Variations in the velocity and buoyancy profiles are discussed as a function of the dimensionless model parameters and , where is the hydrodynamic roughness length, is the Brunt-Vaisala frequency, is the surface sloping angle, is the imposed surface buoyancy, and is a reference velocity scale used to define eddy diffusivities. Velocity and buoyancy profiles show significant variations in both phase and amplitude of extrema with respect to the classic constant model and with respect to a recent approximate analytic solution based on the Wentzel-Kramers-Brillouin theory. Near-wall regions are characterized by relatively stronger surface momentum and buoyancy gradients, whose magnitude is proportional to and to . In addition, slope-parallel momentum and buoyancy fluxes are reduced, the low-level jet is further displaced toward the wall, and its peak velocity depends on both and .