4 Momentum Integral Equation. Exact solutions of the boundary-layer equations are possible only in simple cases. In more complicated problems, approximate ...
058:0160 Jianming Yang
Chapter 7 14
Fall 2012
4 Momentum Integral Equation Exact solutions of the boundary-layer equations are possible only in simple cases. In more complicated problems, approximate methods satisfy only an integral of the boundary-layer equations across the layer thickness. When this integration is performed, the resulting ordinary differential equation involves the boundary layer’s displacement and momentum thicknesses, and its wall shear stress. 4.1 Momentum integral equation Momentum equation: The pressure gradient is evaluated form the outer potential flow using Bernoulli equation
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058:0160 Jianming Yang
Chapter 7 15
Fall 2012
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von Karman boundary-layer momentum integral equation, which is valid for steady laminar boundary layers and for time-averaged flow in turbulent boundary layers. It is a single ordinary differential equation that relates three unknowns , , and , so additional assumptions must be made or correlations provided to obtain solutions for these parameters. ( (
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058:0160 Jianming Yang
Chapter 7 16
Fall 2012
Historically two approaches for solving the momentum integral equation for specified potential flow ( ): 1. Guessed Profiles 2. Empirical Correlations Best approach is to use empirical correlations to get integral parameters ( , , , , , ) after which use these to get velocity profile . 4.2 Thwaites Method Multiply momentum integral equation by ( LHS and
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are dimensionless and can be correlated with pressure gradient parameter as shear and shape-factor correlations ( )
Note
Substitute above into momentum integral equation ( ) [ ( )
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058:0160 Jianming Yang
Chapter 7 17
Fall 2012
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based on AFD and EFD
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Thwaites (1949), 𝑚
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Overall, the accuracy of Thwaites’ method is or so for favorable pressure gradients, and for adverse pressure gradients but perhaps slightly worse near boundary-layer separation. The great strength of Thwaites’ method is that it involves only one parameter ( ) and requires only a single integration. This simplicity makes it ideal for preliminary engineering calculations that are likely to be followed by more formal computations or experiments.
