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TECHNICAL PAPERS

# Curvature Law of the Wall for Swirling Axial Flows in Rotating Machinery

[+] Author and Article Information
Jinxiang Xi, P. Worth Longest

Department of Mechanical Engineering, Virginia Commonwealth University, Richmond, VA 23284

Xiuhua Si

Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX 77005

Department of Mechanical Engineering, Virginia Commonwealth University, Richmond, VA 23284gadelhak@vcu.edu

1

Corresponding author.

J. Fluids Eng 129(2), 169-178 (Jul 07, 2006) (10 pages) doi:10.1115/1.2409360 History: Received February 24, 2006; Revised July 07, 2006

## Abstract

A new law of the wall accounting for curvature effects in swirling axial flows is derived. The influence of the curvature on the turbulence mixing lengths in both axial and tangential directions is examined theoretically using the Reynolds stress transport equations. For equilibrium flows with weak curvature, identical mixing lengths are derived for the axial and tangential directions. Additionally, the effect of finite local curvature and shear stress ratio on the near-wall velocities is systematically explored. It is found that the curvature effect in swirling axial flows is suppressed by a factor of $1∕(1+σw2)$ compared to that in curved channel flows, where $σw$ is the ratio of the axial to swirl shear stress. For a given curvature radius, the maximum velocity deviation occurs when the axial-to-swirl shear stress ratio is zero. Finally, the performance of the new curvature law is evaluated by implementing it as a wall function in a well-established CFD code. The new wall function provides improved agreement for swirl velocity distributions inside labyrinth cavities in comparison with existing experimental laser Doppler anemometry measurements.

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## Figures

Figure 7

Comparison of numerical results of the new wall law with measurements for convex curvature (see Ref. 8); R+=30,000, σ=0, α=4.5

Figure 8

Comparison of numerical results of the new wall law with measurements for concave curvature (see Ref. 6); R+=25,000, σ=0, α=9.0

Figure 9

Comparison of the new curvature law and classical log-law with measurements (see Ref. 36) near the stator of a labyrinth seal (Re=24,000, R+=60,000, and σ=2.0)

Figure 10

Schematic configuration of the labyrinth seal in Morrison’s (Ref. 37) experimental apparatus

Figure 11

Predicted swirl profile inside a labyrinth seal from the new wall function versus measurement by Morrison (Ref. 37)

Figure 1

Schematic illustration of the effects of various parameters on the law of the wall in turbulent boundary layers

Figure 2

Typical curvature flows in turbomachinery: (a) curved channel flow; (b) axial flow; (c) swirling axial flow

Figure 3

Numerical results of the law of the wall in both axial (u+) and circumferential (w+) directions. The perturbation variables w0+, w1+, w2+ and ε are presented: (a) convex surface; (b) concave surface

Figure 4

Semi-logarithmic plots of near-wall velocities versus y+ with wτ* and uτ* (Eq. 42) as the velocity scales (σ=0.1, R+=50,000)

Figure 5

Semi-logarithmic plots of near-wall swirl velocity w+ versus y+ for different curvature radii R+: (a) convex surface; (b) concave surface

Figure 6

Semi-logarithmic plots of near-wall swirl velocity w+ versus y+ for different axial-to-tangential shear stress ratios σ: (a) convex surface; (b) concave surface

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