Reynolds number dependence of the mean flow in a circular pipe

An experimental investigation was conducted to determine the Mean-Flow scaling in a Fully-Developed, smooth pipe flow. Measurements of the mean velocity profiles and friction factors were performed over a large range of Reynolds numbers (31 × 10³ to 35 × 10⁶ based on average velocity and diameter). The results show that the friction factor data are significantly different (over 5 %) than predicted by the relation proposed by Prandtl. A new friction factor relation is proposed which is similar to Prandtl’s, but has different constants and includes an additional term to account for the Near-Wall velocity profile. Analysis of the mean velocity profiles indicate two types of overlap regions: one which scales as a Power-Law and one which scales as a Log-Law. The Log-Law is only evident if the Reynolds number is greater than approximately 300 × 10³. A new theory was developed to explain the scaling in both overlap regions. This theory requires a velocity scale for the outer region such that the ratio of the outer velocity scale to the inner velocity scale (the friction velocity) is a function of Reynolds number at low Reynolds numbers, and approaches a constant value at high Reynolds numbers. The proposed velocity scale was used to normalize the velocity profiles in the outer region and was found to give significantly better agreement between different Reynolds numbers than the friction velocity.