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RESEARCH PAPERS

# Predicting Transition on Concave Surfaces

[+] Author and Article Information
Mark W. Johnson

Department of Engineering, University of Liverpool, Liverpool L69 3GH, UK

J. Turbomach 129(4), 750-755 (Aug 18, 2006) (6 pages) doi:10.1115/1.2720865 History: Received July 13, 2006; Revised August 18, 2006

## Abstract

Boundary layers on concave surfaces differ from those on flat plates due to the presence of Taylor-Goertler (T-G) vortices. These vortices cause momentum transfer normal to the blade’s surface and hence result in a more rapid development of the laminar boundary layer and a fuller profile than is typical of a flat plate. Transition of boundary layers on concave surfaces also occurs at a lower $Rex$ than on a flat plate. Concave surface transition correlations have been formulated previously from experimental data, but they are not comprehensive and are based on relatively sparse data. The purpose of the current work was to attempt to model the physics of both the laminar boundary layer development and transition process in order to produce a transition model suitable for concave surface boundary layers. The development of the laminar boundary layer on a concave surface was modeled by considering the profiles at the upwash and downwash locations separately. The profiles of the boundary layers at these two locations were then combined to successfully approximate the spanwise averaged profile. The ratio of the boundary layer thicknesses at the two locations was found to be as great as 50 and this leads to laminar boundary layer shape factors as low as 1.3 and skin friction coefficients up to 12 times the value for a flat plate laminar boundary layer. Boundary layers therefore grow much more rapidly on concave surfaces than on flat plates. The transition model assumed that transition commenced in the upwash location boundary layer at the same transition inception $Reθ$ observed on a flat plate. Transition at the downwash location then results from the growth of turbulent spots from the upwash location rather than through the initiation of spots. The model showed that initially curvature promotes transition because of the thickened upwash boundary layer, but for strong curvature the T-G vortices effectively stabilize the boundary layer and transition then occurs at a higher $Reθ$ than on a flat plate. Results from the transition model were in broad agreement with experimental observations. The current work therefore provides a basis for the modeling of transition on concave surfaces.

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## Figures

Figure 1

Typical laminar boundary layer profile on a concave surface

Figure 2

Square wave model for spanwise variation in boundary layer thickness

Figure 3

Skin friction coefficient and shape factors for laminar boundary layers on concave surfaces

Figure 4

Evolution of the proportion of the span occupied by the upwash profile

Figure 5

Evolution of the ratio of the boundary layer thickness at upwash and downwash locations

Figure 6

Correlation for ratio of upwash to downwash boundary layer thickness

Figure 7

Laminar boundary layer development for a range of blade radius Reynolds numbers

Figure 8

Laminar boundary layer development for ReR=800,000

Figure 9

Predicted transition inception on concave surfaces

Figure 10

Shape factor H variation through transition on constant curvature blades

Figure 11

Skin friction coefficient Cf variation through transition

Figure 12

Boundary layer development through transition

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