Draft:Nikuradse roughness

In fluid mechanics, the friction between a flowing fluid and a pipe's inner surface contributes to energy losses, which are critical in designing efficient piping systems. Surface roughness affects the boundary layer and, consequently, the frictional resistance. Nikuradse's experiments built upon earlier work by Ludwig Prandtl and others, systematically studying the relationship between surface roughness and flow behaviour in turbulent flow.

Johann Nikuradse conducted his seminal experiments in the 1930s at the Kaiser Wilhelm Institute (now the Max Planck Institute) in Germany. He created artificial roughness by gluing uniformly sized sand grains to the inner surfaces of pipes, allowing precise control and measurement of the roughness parameter, known as the equivalent sand-grain roughness (denoted as k_s). Nikuradse measured the pressure drop along pipes with varying roughness levels across a range of Reynolds numbers to study both laminar and turbulent flows.

His key findings include:

• Smooth Pipe Flow: In smooth pipes, the friction factor depends solely on the Reynolds number and is unaffected by surface roughness at low roughness levels.
• Transitional Roughness: At intermediate roughness levels, the friction factor depends on both the Reynolds number and the roughness parameter.
• Fully Rough Flow: In highly rough pipes, the friction factor becomes independent of the Reynolds number and is determined solely by the relative roughness (k_s/D, where D is the pipe diameter).

Nikuradse's data were later incorporated into the Moody diagram, a graphical tool developed by Lewis Ferry Moody to represent the friction factor as a function of Reynolds number and relative roughness.

The equivalent sand-grain roughness (k_s) is a standardised measure introduced by Nikuradse to quantify surface roughness. It represents the diameter of hypothetical sand grains that, when uniformly glued to a smooth pipe, would produce the same frictional resistance as the actual surface. The relative roughness is defined as:

|1=, |, =)

where:

• k_s: Equivalent sand-grain roughness (typically in millimetres or metres)
• D: Pipe diameter (in the same units)

Typical values of k_s for common materials include:

• Smooth pipes (e.g., glass, plastic): k_s ≈ 0.0015–0.01 mm
• Steel pipes: k_s ≈ 0.015–0.05 mm
• Concrete pipes: k_s ≈ 0.3–3 mm

These values are used in engineering calculations to predict pressure losses.

Nikuradse's work on roughness is critical in several engineering applications, including:

• Pipeline Design: Calculating pressure losses in water, oil, and gas pipelines.
• Hydraulic Systems: Optimising flow in irrigation, HVAC, and industrial fluid systems.
• Aerodynamics: Understanding roughness effects on airfoils and other surfaces in turbulent flow.
• Environmental Engineering: Modelling flow in natural channels and rivers, where roughness varies due to sediment or vegetation.

The concept is embedded in empirical correlations like the Colebrook-White equation, which relates the friction factor to the Reynolds number and relative roughness:

frac{1}{sqrt{f = -2 log_{10} left( frac{k_s / D}{3.7} + frac{2.51}{text{Re} sqrt{f}} right)}}

where:

• f: Friction factor
• text{Re}: Reynolds number
• k_s/D: Relative roughness

Nikuradse's experiments provided a rigorous empirical basis for understanding surface roughness in fluid flow. His work influenced the development of the Moody diagram and modern computational fluid dynamics (CFD) models. While the sand-grain roughness model is an idealisation, it remains a practical standard for characterising surface roughness in engineering applications. Nikuradse's findings are taught in fluid mechanics courses worldwide and continue to inform the design of efficient fluid transport systems.

• Fluid dynamics
• Friction factor
• Moody diagram
• Reynolds number
• Colebrook-White equation
• Boundary layer

• Nikuradse, J. (1933). "Strömungsgesetze in rauhen Rohren" [Laws of Flow in Rough Pipes]. Forschungsheft (in German). 361. VDI-Verlag.
• Moody, L. F. (1944). "Friction Factors for Pipe Flow". Transactions of the ASME. 66 (8): 671–684.
• White, F. M. (2006). Fluid Mechanics. McGraw-Hill Education. ISBN 978-0072938449.

• Surface Roughness of Pipes - Engineering Toolbox