Virtual Nikuradse

Abstract

In this paper we derive an accurate composite friction factor versus Reynolds number correlation formula for laminar, transition and turbulent flow in smooth and rough pipes. The correlation is given as a rational fraction of rational fractions of power laws which is systematically generated by smoothly connecting linear splines in log-log coordinates with a logistic dose function algorithm. We convert Nikuradse's (1933) (J. Nikuradse, 1933 Stromungsgesetz in rauhren rohren, vDI Forschungshefte 361. (English translation: Laws of flow in rough pipes). Technical report, NACA Technical Memorandum 1292. National Advisory Commission for Aeronautics (1950), Washington, DC.) data for six values of roughness into a single correlation formula relating the friction factor to the Reynolds number for all values of roughness. Correlation formulas differ from curve fitting in that they predict as well as describe. Our correlation formula describes the experimental data of Nikuradse's (1932, 1933) (J. Nikuradse, Laws of turbulent flow in smooth pipes (English translation), NASA (1932) TT F-10: 359 (1966).) and McKeon et al. (2004) (B.J. McKeon, C.J. Swanson, M.V. Zaragola, R.J. Donnelly, and J.A. Smits, Friction factors for smooth pipe flow, J. Fluid Mech. 511 (2004), 41–44.) but it also predicts the values of friction factor versus Reynolds number for the continuum of sand-grain roughness between and beyond those given in experiments. Of particular interest is the connection of Nikuradse's (1933) data for flow in artificial rough pipes to the data for flow in smooth pipes presented by Nikuradse (1932) and McKeon et al. (2004) and for flow in effectively smooth pipes. This kind of correlation seeks the most accurate representation of the data independent of any input from theories arising from the researchers ideas about the underlying fluid mechanics. As such, these correlations provide an objective metric against which observations and other theoretical correlations may be applied. Our main hypothesis is that the data for flow in rough pipes terminates on the data for smooth and effectively smooth pipes at a definite Reynolds number R _σ(σ); if λ = f(Re, σ) is the friction factor in a pipe of roughness parameter σ then λ = f(R _σ(σ), σ) is the friction factor at the connection point. An analytic formula giving R _σ(σ) is obtained here for the first time.

Keywords:

• friction factor
• sand-grain roughness
• artificial rough pipe
• Nikuradse's data
• smooth and effectively smooth pipes

Acknowledgement

This work was supported by the NSF/CTS under grant 0076648. We have profited from conversations with Grigory I. Barenblatt, Nigel Goldenfeld, Ivan Marusic, Beverley McKeon and Alex Smits about topics discussed in this paper. The research reported in this paper is taken from the Ph.D. thesis of Bobby (Haoping) Yang.