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

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.