Ionizing argon boundary layers. Part 1. Quasi-steady flat-plate laminar boundary-layer flows

The compressible laminar boundary-layer flows of a dilute gas-particle mixture over a semi-infinite flat plate are investigated analytically. The governing equations are presented in a general form where more reasonable relations for the two-phase interaction and the gas viscosity are included. The detailed flow structures of the gas and particle phases are given in three distinct regions: the large-slip region near the leading edge, the moderate-slip region and the small-slip region far downstream. The asymptotic solutions for the two limiting regions are obtained by using a series-expansion method. The finite-difference solutions along the whole length of the plate are obtained by using implicit four-point and six-point schemes. The results from these two methods are compared and very good agreement is achieved. The characteristic quantities of the boundary layer are calculated and the effects on the flow produced by the particles are discussed. It is found that in the case of laminar boundary-layer flows, the skin friction and wall heat-transfer are higher and the displacement thickness is lower than in the pure-gas case alone. The results indicate that the Stokes-interaction relation is reasonable qualitatively but not correct quantitatively and a relevant non-Stokes relation of the interaction between the two phases should be specified when the particle Reynolds number is higher than unity.

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Predicting Transition on Concave Surfaces

Volume 3: Heat Transfer, Parts A and B ◽

10.1115/gt2006-90455 ◽

2006 ◽

Author(s):

Mark W. Johnson

Keyword(s):

Boundary Layer ◽

Flat Plate ◽

Boundary Layers ◽

Laminar Boundary Layer ◽

Rapid Development ◽

Current Work ◽

Concave Surface ◽

Transition Model ◽

Surface Boundary ◽

Flat Plates

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 stabilise 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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Predicting Transition on Concave Surfaces

Journal of Turbomachinery ◽

10.1115/1.2720865 ◽

2006 ◽

Vol 129 (4) ◽

pp. 750-755 ◽

Cited By ~ 1

Author(s):

Mark W. Johnson

Keyword(s):

Boundary Layer ◽

Flat Plate ◽

Boundary Layers ◽

Laminar Boundary Layer ◽

Rapid Development ◽

Current Work ◽

Concave Surface ◽

Transition Model ◽

Surface Boundary ◽

Flat Plates

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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Analysis of Unsteady Laminar Boundary Layer Flow by an Integral Method

Journal of Fluids Engineering ◽

10.1115/1.3447001 ◽

1973 ◽

Vol 95 (2) ◽

pp. 237-247 ◽

Cited By ~ 1

Author(s):

R. W. Miller ◽

L. S. Han

Keyword(s):

Boundary Layer ◽

Flat Plate ◽

Laminar Boundary Layer ◽

Integral Method ◽

Momentum Equation ◽

Asymptotic Solutions ◽

Mean Flow ◽

Boundary Layer Flows ◽

Approximate Integral ◽

Layer Flow

An approximate integral method of analysis is developed for unsteady laminar boundary layer flows. The case of a flat plate in a free stream with small harmonic velocity oscillations about a steady mean is used to formulate the method. Results are compared with the available experimental data. The essence of the method is to: First obtain asymptotic solutions for limiting cases of the flow under consideration. Then, assume velocity profiles with sufficient generality to include the asymptotic solutions. The profile form functions are determined by applying integral relations (velocity weighted averages of the momentum equation) and compatibility conditions (normal derivatives of the momentum equation evaluated at the boundary). As an example of the method, the second-order mean flow correction is determined for oscillating flow over a flat plate.

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