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Turbulent Flow

Turbulent Flow

Turbulent flow is characterized by unpredictable fluctuations in velocity and pressure, which result in a chaotic fluid movement distinct from the orderly patterns of laminar flow. While laminar flow is governed by smooth, parallel layers with minimal mixing, turbulent flow exhibits highly irregular, three-dimensional patterns. This behavior arises due to instabilities in the fluid's velocity profile, and amplifies as the flow velocity increases. Minor disturbances, known as turbulent spots,...

Steady, Laminar Flow in Circular Tubes

Steady, Laminar Flow in Circular Tubes

Hagen-Poiseuille flow describes a viscous fluid's steady, incompressible flow through a cylindrical tube with a constant radius R. This flow profile is often applied to understand fluid transport in narrow channels, such as capillaries. It serves as a foundational example of laminar flow. In this model, cylindrical coordinates (r,θ,z) are used to describe the radial (r), angular (θ), and axial (z) dimensions within the tube. For Hagen-Poiseuille flow, the velocity profile is purely axial,...

Laminar and Turbulent Flow

Laminar and Turbulent Flow

Fluid dynamics is the study of fluids in motion. Velocity vectors are often used to illustrate fluid motion in applications like meteorology. For example, wind—the fluid motion of air in the atmosphere—can be represented by vectors indicating the speed and direction of the wind at any given point on a map. Another method for representing fluid motion is a streamline. A streamline represents the path of a small volume of fluid as it flows. When the flow pattern changes with time, the streamlines...

Laminar Flow

Laminar Flow

Laminar flow represents a smooth, orderly fluid motion where particles move along parallel paths, resulting in minimal mixing between layers. Streamlined particle paths characterize this flow regime and occur under conditions where viscous forces dominate over inertial forces. The distinction between laminar, transitional, and turbulent flow is primarily determined by the Reynolds number, a dimensionless quantity calculated as:

Poiseuille's Law and Reynolds Number

Poiseuille's Law and Reynolds Number

Any fluid in a horizontal tube can flow due to pressure differences—fluid flows from high to low pressure. The flow rate (Q) is the ratio of pressure difference and resistance through a horizontal tube. The greater the pressure difference, the higher the flow rate. The flow resistance is expressed as:

Irrotational Flow

Irrotational Flow

Irrotational flow is characterized by fluid motion where particles do not rotate around their axes, resulting in zero vorticity. For a flow to be irrotational, the curl of the velocity field must be zero. This imposes specific conditions on velocity gradients. For instance, to maintain zero rotation about the z-axis, the gradient condition:

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Is there a universal log law for turbulent wall-bounded flows?

William K George¹

¹Department of Applied Mechanics, Chalmers University of Technology, 412 96 Gothenburg, Sweden. wkgeorge@chalmers.se

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences

|January 25, 2007

Summary

A universal log law for turbulent wall-bounded flows is not supported by theory or data. While pipe and channel flows show logarithmic profiles, flat plate boundary layers align better with power law theories.

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Area of Science:

Fluid Dynamics
Turbulence Theory

Background:

The concept of a universal logarithmic law for turbulent wall-bounded flows has been a long-standing topic in fluid dynamics.
Previous justifications based on constant Reynolds stress layer arguments have been found to be insufficient.

Purpose of the Study:

To critically review the theoretical basis and empirical evidence for a universal logarithmic law in turbulent wall-bounded flows.
To compare the applicability of logarithmic and power law theories to different flow configurations.

Main Methods:

Theoretical analysis of the foundations of the log law, including constant Reynolds stress arguments.
Comparison of theoretical predictions with experimental data for pipe, channel, and boundary layer flows.
Empirical curve fitting to velocity and friction data to determine constants like kappa.

Main Results:

The theoretical justification for a universal log law based on constant Reynolds stress is deficient.
Logarithmic friction laws and velocity profiles are well-supported for pipe and channel flows.
For flat plate boundary layers, power law theories appear more logically consistent, although empirical logarithmic fits are indistinguishable.
The universal constant kappa differs between pipe flow (approx. 0.43) and boundary layer flow (approx. 0.38).

Conclusions:

The idea of a universal logarithmic law for all turbulent wall-bounded flows is not supported by current theoretical understanding or empirical data.
Different flow geometries exhibit distinct scaling laws, necessitating distinct theoretical approaches.