[Paper Review] Roughness effects in laminar pipe flow

Utku Şentürk, Alexander J. Smits|arXiv (Cornell University)|May 29, 2019

Fluid Dynamics and Turbulent Flows14 references4 citations

TL;DR

This study presents a computational investigation of laminar pipe flow over square-ribbed wall roughness, demonstrating that the friction factor scales inversely with Reynolds number, quadratically with roughness height, and linearly with pitch. A unified correlation collapses all data across varying roughness geometries and Reynolds numbers, offering a predictive model for rough pipe flow in laminar regimes.

ABSTRACT

The impact of wall roughness on fully developed laminar pipe flow is investigated numerically. The roughness is comprised of square bars of varying size and pitch. Results show that the inverse relation between the friction factor and the Reynolds number in smooth pipes still persists in rough pipes, regardless of the rib height and pitch. At a given Reynolds number, the friction factor varies quadratically with roughness height and linearly with roughness pitch. We propose a single correlation for the friction factor that successfully collapses the data.

Motivation & Objective

• To investigate the influence of wall roughness on fully developed laminar pipe flow using numerical simulations.
• To determine how roughness height (k/D) and pitch (λ/w) affect the Darcy friction factor in laminar conditions.
• To develop a universal correlation that collapses friction factor data across varying Reynolds numbers, roughness heights, and spacings.
• To analyze the relative contributions of pressure and shear forces to the total friction factor in rough pipe flow.
• To validate the applicability of constricted flow concepts from channel flow to pipe flow with regular roughness elements.

Proposed method

• Numerical simulations using finite volume method with unstructured, non-uniform triangular grids clustered near walls.
• Application of translational periodicity to model fully developed flow over periodic roughness elements, enabling simulation of infinite periodic arrays.
• Use of modified SIMPLE algorithm to solve Navier-Stokes equations with an additional source term (β) representing the mean pressure gradient.
• Computation of friction factor via force balance on the control volume, separating pressure (Cp) and shear (Cv) contributions.
• Definition of effective friction factor as a weighted sum of pressure and shear coefficients, with geometry-dependent coefficients (α, γ, θ).
• Grid refinement study to ensure solution convergence and second-order accuracy for velocity and pressure fields.

Experimental results

Research questions

• RQ1 How does relative roughness height (k/D) influence the Darcy friction factor in laminar pipe flow?
• RQ2 What is the role of roughness pitch (λ/w) in determining the friction factor, and how does it interact with roughness height?
• RQ3 Does the inverse Reynolds number dependence of friction factor in smooth pipes persist in rough pipes with regular square ribs?
• RQ4 Can a single correlation collapse friction factor data across varying Reynolds numbers, roughness heights, and spacings?
• RQ5 How do pressure and shear stress contributions to the total friction factor vary with roughness geometry and Reynolds number?

Key findings

• The friction factor in rough pipes still scales inversely with Reynolds number, preserving the smooth-pipe trend despite roughness.
• The friction factor increases quadratically with relative roughness height (k/D), indicating a strong dependence on protrusion size.
• The friction factor decreases linearly with increasing roughness pitch (λ/w), showing a significant influence of spacing between ribs.
• A single correlation, f = 64/Re × (1 + c₁(k/D)² + c₂(λ/w)), successfully collapses all simulation data across the tested Reynolds numbers, roughness heights, and spacings.
• For d-type roughness (small pitch), shear stress dominates the friction factor, while for k-type roughness (large pitch), pressure forces become increasingly important at higher roughness heights.
• Vorticity and streamline analysis confirm that k-type roughness induces flow separation and vortex formation in grooves, with stronger vortices near corner regions as roughness height increases.