Publisher’s Note: “Comment on ‘Turbulent diffusion of momentum and suspended particles: A finite-mixing-length theory’” [Phys. Fluids 17, 079101 (2005)]

2005 ◽  
Vol 17 (10) ◽  
pp. 109901 ◽  
Author(s):  
Rafik Absi
Keyword(s):  
Turbulent Diffusion ◽  
Suspended Particles ◽  
Mixing Length ◽  
Mixing Length Theory

Turbulent buoyant convection from a maintained source of buoyancy in a narrow vertical tank

2012 ◽  
Vol 701 ◽  
pp. 278-303 ◽  
Author(s):  
Daan D. J. A. van Sommeren ◽  
C. P. Caulfield ◽  
Andrew W. Woods
Keyword(s):  
Diffusion Coefficient ◽  
Density Gradient ◽  
Turbulent Diffusion ◽  
Buoyancy Flux ◽  
Mixing Length ◽  
Reduced Gravity ◽  
Turbulent Diffusion Coefficient ◽  
Small Constant ◽  
Before And After ◽  
Mixing Length Theory

AbstractWe describe new experiments to examine the buoyancy-induced mixing which results from the injection of a small constant volume flux of fluid of density ${\rho }_{s} $ at the top of a long narrow vertical tank with square cross-section which is filled with fluid of density ${\rho }_{0} \lt {\rho }_{s} $. The injected fluid vigorously mixes with the less dense fluid which initially occupies the tank, such that a dense mixed region of turbulent fluid propagates downwards during the initial mixing phase of the experiment. For an ideal source of constant buoyancy flux ${B}_{s} $, we show that the height of the mixed region grows as $h\ensuremath{\sim} { B}_{s}^{1/ 6} {d}^{1/ 3} {t}^{1/ 2} $ and that the horizontally averaged reduced gravity $ \overline{{g}^{\ensuremath{\prime} } } = g( \overline{\rho } \ensuremath{-} {\rho }_{0} )/ {\rho }_{0} $ at the top of tank increases as $ \overline{{g}^{\ensuremath{\prime} } } (0)\ensuremath{\sim} { B}_{s}^{5/ 6} {d}^{\ensuremath{-} 7/ 3} {t}^{1/ 2} $, where $d$ is the width of the tank. Once the mixed region reaches the bottom of the tank, the turbulent mixing continues in an intermediate mixing phase, and we demonstrate that the reduced gravity at each height increases approximately linearly with time. This suggests that the buoyancy flux is uniformly distributed over the full height of the tank. The overall density gradient between the top and bottom of the mixed region is hence time-independent for both the mixing phases before and after the mixed region has reached the bottom of the tank. Our results are consistent with previous models developed for the mixing of an unstable density gradient in a confined geometry, based on Prandtl’s mixing length theory, which suggest that the turbulent diffusion coefficient and the magnitude of the local turbulent flux are given by the nonlinear relations ${ \kappa }_{T}^{\mathit{nl}} = {\lambda }^{2} {d}^{2} \mathop{ (\partial \overline{{g}^{\ensuremath{\prime} } } / \partial z)}\nolimits ^{1/ 2} $ and ${J}^{\mathit{nl}} = {\lambda }^{2} {d}^{2} \mathop{ (\partial \overline{{g}^{\ensuremath{\prime} } } / \partial z)}\nolimits ^{3/ 2} $, respectively. The $O(1)$ constant $\lambda $ relates the width of the tank to the characteristic mixing length of the turbulent eddies. Since the mixed region is characterized by a time-independent overall density gradient, we also tested the predictions based on a linear model in which the turbulent diffusion coefficient is approximated by a constant ${ \kappa }_{T}^{l} $. We solve the corresponding nonlinear and linear turbulent diffusion equations for both mixing phases, and show a good agreement with experimental profiles measured by a dye attenuation technique, in particular for the solutions based on the nonlinear model.

Friction Factor Comparison Between Mixing Length Theory and Empirical Correlation

2022 ◽  
Author(s):  
M Prasad
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Friction Factor ◽  
High Reynolds Number ◽  
Mathematical Representation ◽  
Mixing Length ◽  
Series Form ◽  
High Reynolds ◽  
Mixing Length Theory ◽  
Grain Roughness

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.

scholarly journals Turbulent Convection: A New Model

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.

scholarly journals Test of a New Theory for Stellar Convection using Helioseismology

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.

Sign in / Sign up

Export Citation Format

Share Document