Notes on Mixing Length Theory

Abstract Equivalent sand grain roughness is required for estimating friction factor for engineering applications from empirical relation via Haalands equation. The real surfaces are different from the sand grain profile. The correlations for friction factor were derived from use of discrete roughness elements with regular shapes such as cones, bars etc. The purpose of the paper is to derive analytical expression of friction factor for a 2 dimensional semi-cylindrical roughness (not exactly a 3 dimensional sand grain but for the circular profile of cross- section) using Navier Stoke equation and mixing length theory. This is compared with the modified series mathematical representation of Haalands equation for friction factor in terms of equivalent sand grain roughness. The comparison is valid for high Reynolds number where the velocity profile is almost flat beyond boundary layer and approximately linear all throughout the boundary layer. The high Reynolds number approximation for Haalands equation is derived and the series form of the friction factor compares approximately with the series form derived from first principles, where in the exponents of the series expansion are close.

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Turbulent Convection: A New Model

International Astronomical Union Colloquium ◽

10.1017/s0252921100079355 ◽

1991 ◽

Vol 130 ◽

pp. 27-32

Author(s):

V. M. Canuto

Keyword(s):

Observational Data ◽

Turbulent Convection ◽

Mixing Length ◽

Convective Flux ◽

New Model ◽

Mixing Length Theory

AbstractWe use the latest models of turbulence to compute a new expression for the turbulent convective flux, Fc. The new values of Fc are up to ten times larger than those given by the mixing length theory, MLT. Astrophysical considerations indicate that the new model fares better with observational data than the MLT.

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Test of a New Theory for Stellar Convection using Helioseismology

International Astronomical Union Colloquium ◽

10.1017/s0252921100017462 ◽

1993 ◽

Vol 137 ◽

pp. 63-65

Author(s):

L. Paternó ◽

R. Ventura ◽

V.M. Canuto ◽

I. Mazzitelli

Keyword(s):

Reaction Rates ◽

Turbulent Convection ◽

Standard Theory ◽

Evolutionary Models ◽

Mixing Length ◽

Surface Layers ◽

Stellar Convection ◽

The Standard Model ◽

Convection Model ◽

Mixing Length Theory

AbstractTwo evolutionary models of the Sun have been tested using helioseismological data. The two models use the same input micro-physics (nuclear reaction rates, opacity, equation of state) and the same numerical evolutionary code, but differ in the treatment of turbulent convection. The first model employs the standard mixing - length theory of convection, while the second one employs a new turbulent convection model which overcomes some basic inconsistencies of the standard theory of convection.The test rests on the calculation of p-mode eigenfrequencies and on the comparison with the helioseismological data.The comparison shows an overall improvement of the eigenfrequencies calculated with the new model with respect to those calculated with the standard model, although it appears that both models still suffer from inaccuracies especially in the treatment of the surface layers.

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A Unified View of the Law of the Wall Using Mixing-Length Theory

Aeronautical Quarterly ◽

10.1017/s0001925900006429 ◽

1973 ◽

Vol 24 (1) ◽

pp. 55-70 ◽

Cited By ~ 34

Author(s):

V C Patel

Keyword(s):

Friction Velocity ◽

Critical Value ◽

Wall Region ◽

Pressure Gradients ◽

Mixing Length ◽

Stress Distributions ◽

General Validity ◽

Law Of The Wall ◽

Unified View ◽

Mixing Length Theory

SummaryIt is shown that, if the well-known mixing-length formula is regarded simply as a relationship between the velocity and the stress distributions in the wall region of a turbulent flow, then a truly universal distribution of mixing length is sufficient to describe the experimentally observed departures of the velocity distribution from the usual law of the wall as a result of severe pressure gradients and transverse surface curvature. Comparisons have been made with a wide variety of experimental data to demonstrate the general validity of the mixing-length model in describing the flow close to a smooth wall.An extension of the re-laminarisation criterion of Patel and Head, and some experimental evidence, suggest that the thick axisymmetric boundary layer on a slender cylinder placed axially in a uniform stream cannot be maintained in a fully turbulent state for values of the Reynolds number, based on friction velocity and cylinder radius, below a certain critical value.

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