scholarly journals DERIVING THE SEMI-EMPIRICAL FORMULA TO COMPUTE THE FRICTION FACTOR λ FOR TURBULENT FLOW IN PIPE

2010 ◽  
Vol 13 (2) ◽  
pp. 48-58
Author(s):  
Duc Van Le
Keyword(s):  
Turbulent Flow ◽  
Friction Factor ◽  
Research Result ◽  
Shear Velocity ◽  
Point Of View ◽  
Mixing Length ◽  
Smooth Pipe ◽  
Wetted Area ◽  
Set Up ◽  
Mixing Length Theory

Based on law of shear stress in turbulent flow. Prandd's mixing length theory, and Bakhmeteflfs point of view on "wall velocity", turbulent velocity distribution u on wetted area can be derived for smooth pipe and complete turbulence, rough pipe. Discharge Q and average velocity y are obtained, after the integration, Q= ∫∫wu.dw is done. Relying on the properties of uniform flow, relationship between V, friction factor λ, and shear velocity u is set up. After eliminating u*. velocity V is obtained as a function of Reynolds number Re or relative roughness e/D. Finally, the value of friction factor z can be derived as a function of Re or e/D for the two above-mentioned cases. These formations of z formulas are almost same as the experimental ones introduced by Nikuradse with minor deviations in the factors and their relative errors do not exceed 1% for smooth pipe, and 2% for complete turbulence, rough pipe. Through this research result, the rightness of Prandtl's mixing length theory is almost asserted.

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 Closed Form Solution of Plane-Parallel Turbulent Flow Along an Unbounded Plane Surface

2021 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Energy Dissipation ◽  
Turbulent Flow ◽  
Closed Form ◽  
Scaling Law ◽  
Closed Form Solution ◽  
Form Solution ◽  
Analytic Description ◽  
Mixing Length ◽  
Mean Velocity ◽  
Mixing Length Theory

In this letter, a century-old problem is studied; namely, to find a unified analytic description of the non-uniform distribution of mean velocity across the entire domain of turbulent flow for all Reynolds numbers within the framework of the Prandtl mixing length theory. Considering the Prandtl mixing length model, a closed form solution of the mean velocity profile of plane turbulent flow is obtained. The profiles of several useful quantities are given, such as turbulent viscosity, Reynolds turbulent stress, Kolmogorov's scaling law, and energy dissipation density. It is shown that the energy dissipation density at the surface is finite, whereas Landau's energy dissipation density is infinite. The closed form solution reveals that the universality of the turbulent velocity logarithmic profile no longer holds, but the von K\'arm\'an constant is still universal. The closed form solution is validated by both direct numerical simulation and experiments. The studies confirm that the van Driest mixing length theory is suitable for smooth walls, and the Prandtl mixing length theory is suitable for rough walls. Furthermore, a new formulation of the resistance coefficient of turbulent flow in pipes is given in implicit form.

scholarly journals Closed Form Solution of Plane-Parallel Turbulent Flow Along an Unbounded Plane Surface

2021 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Energy Dissipation ◽  
Turbulent Flow ◽  
Closed Form ◽  
Scaling Law ◽  
Closed Form Solution ◽  
Form Solution ◽  
Analytic Description ◽  
Mixing Length ◽  
Mean Velocity ◽  
Mixing Length Theory

In this letter, a century-old problem is studied; namely, to find a unified analytic description of the non-uniform distribution of mean velocity across the entire domain of turbulent flow for all Reynolds numbers within the framework of the Prandtl mixing length theory. Considering the Prandtl mixing length model, a closed form solution of the mean velocity profile of plane turbulent flow is obtained, and approximate analytical solution of the van Driest mixing length theory is proposed. The profiles of several useful quantities are given, such as turbulent viscosity, Reynolds turbulent stress, Kolmogorov's scaling law, and energy dissipation density. It is shown that the energy dissipation density at the surface is finite, whereas Landau's energy dissipation density is infinite. The closed form solution reveals that the universality of the turbulent velocity logarithmic profile no longer holds, but the von K\'arm\'an constant is still universal. Furthermore, a new formulation of the resistance coefficient of turbulent flow in pipes is given in implicit form.

A note on the mixing length theory of turbulent flow

AIChE Journal ◽  
1970 ◽  
Vol 16 (5) ◽  
pp. 885-888 ◽  
Author(s):  
Mahendra R. Doshi ◽  
William N. Gill
Keyword(s):  
Turbulent Flow ◽  
Mixing Length ◽  
Mixing Length Theory

scholarly journals Closed Form Solution of Plane-Parallel Turbulent Flow Along an Unbounded Plane Surface

2022 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Energy Dissipation ◽  
Turbulent Flow ◽  
Closed Form ◽  
Scaling Law ◽  
Closed Form Solution ◽  
Form Solution ◽  
Analytic Description ◽  
Mixing Length ◽  
Mean Velocity ◽  
Mixing Length Theory

In this paper, a century-old problem is solved; namely, to find a unified analytic description of the non-uniform distribution of mean velocity across the entire domain of turbulent flow for all Reynolds numbers within the framework of the Prandtl mixing length theory. This study obtains a closed form solution of the mean velocity profile of plane turbulent flow for the Prandtl theory, and as well an approximate analytical solution for the van Driest mixing length theory. The profiles of several useful quantities are given based the closed form solution, such as turbulent viscosity, Reynolds turbulent stress, Kolmogorov's scaling law, and energy dissipation density. The investigation shows that the energy dissipation density at the surface is finite, whereas Landau's energy dissipation density is infinite. Strictly speaking, the closed form solution reveals that the universality of the turbulent velocity logarithmic profile no longer holds, but the von K\'arm\'an constant is still universal. Furthermore, a new formulation of the resistance coefficient of turbulent flow in pipes is formulated in implicit form.

Applicability of Mixing Length Theory to Turbulent Boundary Layers Beneath Intense Vortices

1974 ◽  
Vol 41 (1) ◽  
pp. 15-19 ◽  
Author(s):  
S. W. Chi ◽  
W. J. Glowacki
Keyword(s):  
Boundary Layer ◽  
High Speed ◽  
Vortex Flow ◽  
Energy Equation ◽  
Shear Velocity ◽  
Turbulent Energy ◽  
Mixing Length ◽  
Boundary Layer Equations ◽  
Mixing Length Theory ◽  
Speed Computer

The ability of mixing length theory to correlate boundary-layer data beneath intense vortices is evaluated. The boundary-layer equations with turbulent eddy viscosity in terms of a mixing length and an appropriate shear velocity have been developed by consideration of a simplified turbulent energy equation. In the absence of quantitative knowledge of the mixing length for the vortex flow, an expression for the mixing length has been first derived from the flat-plate data and then modified for adaptation to the vortex flow. The full turbulent boundary-layer equations were finally solved on a high-speed computer; and the calculated results were compared with the available experiments under the same conditions. A conclusion was reached that the mixing length theory adequately represents the experimental data.

The Revolutionizing of the Revolution

2017 ◽  
Author(s):  
R. R. Palmer
Keyword(s):  
French Revolution ◽  
Point Of View ◽  
Strict Sense ◽  
French Government ◽  
Thing In Itself ◽  
Set Up ◽  
The Moment ◽  
The French Revolution ◽  
The Revolution

In 1792, the French Revolution became a thing in itself, an uncontrollable force that might eventually spend itself but which no one could direct or guide. The governments set up in Paris in the following years all faced the problem of holding together against forces more revolutionary than themselves. This chapter distinguishes two such forces for analytical purposes. There was a popular upheaval, an upsurge from below, sans-culottisme, which occurred only in France. Second, there was the “international” revolutionary agitation, which was not international in any strict sense, but only concurrent within the boundaries of various states as then organized. From the French point of view these were the “foreign” revolutionaries or sympathizers. The most radical of the “foreign” revolutionaries were seldom more than advanced political democrats. Repeatedly, however, from 1792 to 1799, these two forces tended to converge into one force in opposition to the French government of the moment.

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