Empirical Results on Boundary Layer TheoryComparison of Empirical formulas of Laminar and Turbulent Boundary layer for flow over

smooth flat plate at zero incidencesS.No. Parameters Laminar Boundary Layer Turbulent Boundary Layer

1.Boundary Layer Thickness ()

2.Displacement Thickness (*)

3. Momentum Thickness (θ)

4.Energy Thickness (3)

5.

Skin Friction Drag (Df)For both sides of platel – length of plateb – width of plate

6.Reynolds Number Range

Note:Shape Factor H : This quantity is defined as the ratio of displacement thickness to momentum thickness For laminar flows - H varies between 2 and 3. It is 3.7 near separation point. * Thus excessively large values of H (above 3) indicate that the boundary layer is about to separate. * Separation is predicted in the laminar regime when negative skin friction occurs. For turbulent flows - H varies between 1.5 and 2. * When H exceeds 2.2, the flow is assumed to be separated and results should generally be regarded as meaningless.

Ref.: Schlichting, H. (1979): ‘Boundary Layer Theory’, 7th Edn, McGraw-Hill Book Company.

To the practicing engineer the velocity profile itself is of very little interest. The following parameters are more important.

1. Boundary Layer Thickness : This is defined as the y- location where u/ue reaches 0.99%, that is the u- velocity becomes 99% of the edge velocity.

2. Displacement Thickness * : This is a measure of the outward displacement of the streamlines from the solid surface as a result of the reduced u- velocity within the boundary layer. This quantity is defined as

where the subscript 'e' refers to the conditions at the boundary layer edge. This quantity is usually computed by numerical integration.

3. Momentum Thickness : This is a measure of the momentum loss within the boundary layer as a result of the reduced velocities within the boundary layer. It is defined as

and may be found by numerical integration of the velocity profile.

4. Shape Factor H : This quantity is defined as the ratio of displacement thickness to momentum thickness (*/. For laminar flows H varies between 2 and 3. It is 3.7 near separation point. Thus excessively large values of H (above 3) indicate that the boundary layer is about to separate. In turbulent flows, H varies between 1.5 and 2. It is given in NACA TM1379 by Truckenbrodt, turbulent separation occurs at the H value between 1.8 to 2.4. In general trailing edge turbulent separation starts at H=1.87.

5. Surface Shear Stress: The shear stress at the wall can be found from the definition of shear stress. It is given by,

6. Skin friction Coefficient cf: The derivative of u is computed numerically. This quantity is usually non-dimensionalized by the dynamic pressure at the boundary layer edge, giving the skin friction coefficient cf as

7. Skin Friction Drag, D : The shear stress may be numerically integrated over the entire solid surface to give the skin friction drag force along the x- axis:

8. Skin Friction Drag Coefficient Cd: The drag force is usually non-dimensionalized by the free stream dynamic pressure times the chord of the airfoil c, giving the skin friction drag coefficient along the x- axis, Cd.

Important: Note that all of the above definitions hold for laminar and turbulent, compressible and incompressible boundary layers!