Flow Development in the Entrance Region of Slender Converging Pipes

Abstract

This work presents an analytical investigation of the hydrodynamic entrance region in laminar flows through slender converging pipes. Extending previous analyses for straight pipes, the model radially divides the flow into a viscous wall region and a central core where both inertia and viscous effects are important. The study analyzes the impact of the inlet Reynolds number and convergence angle on the velocity profile and pressure drop. Results show that a converging geometry, which imposes a favorable pressure gradient, significantly shortens the hydrodynamic entrance length compared to a straight pipe. Analytical solutions show good agreement with numerical simulations.

Figures (4)

• Figure 1: Schematic representation of a converging slender pipe.
• Figure 2: Comparison between developing (solid lines) and fully developed (dashed lines) solutions at selected inlet angles for $Re_0 = 250.0$. (a) Radial scaled axial velocity profiles, and (b) Axial scaled pressure gradient profiles.
• Figure 3: Axial velocity profiles for $Re_0 = 250.0$. (a) Centerline profiles at selected inlet angles, (b) Profiles at selected axial positions for $\varphi_0 = -1°$, and (c) Relative axial velocity profiles at selected axial positions and inlet angles. Markers $\times$ denote the entrance length $\ell_e/Re_0$ for each case.
• Figure 4: Analytical (solid lines) and numerical (dashed lines) axial velocity solutions for $Re_0 = 500.0$. (a) Variation in the axial direction for selected $\varphi_0$ (markers represent entrance lengths), and (b) Profiles at selected axial positions for $\varphi_0 = \qty{-1}{\degree}$.