16100lectre30_cj

# 16100lectre30_cj - 5 3 at 99 β = β² Ξ f x V vx vx V vx V...

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Solutions of the Laminar Boundary Layer Equations The boundary layer equations for incompressible steady flow, i.e., 2 2 0 y u dx dp y u v x u u y v x u e + = + = + µ ρ have been solved for a handful of important cases. We will look at the results for a flat plate and a family of solutions called Falkner-Skan Solutions. Flat Plate (Laminar): Blasius Solution For a flat plate, constant = p p e 0 = dx dp e Blasius was able to show that the boundary later equations could be rewritten to only depend on a parameter, vx V y 2 η and its derivatives The resulting solution has been tabulated and compared to experiments on the following page. Note: = = = = = V vx f V y u d df f f V y x u y w / 2 ) 0 ( where ) ( ) , ( 0 τ These values from the solution of ) ( f can be used to find: = V y x u y 99 . 0 ) , ( % 99 which at location δ Note: since ) ( set we , 0 x p p y p e = = , i.e. the boundary layer edge pressure.

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Solutions of the Laminar Boundary Layer Equations 16.100 2002 2 From the table,
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Unformatted text preview: 5 . 3 at 99 . ) ( β = β² Ξ· f : x V vx vx V vx V y 2 5 . 3 2 5 . 3 2 % 99 % 99 as grows layer boundary β = β = = β β β Ξ΄ Typically, this result is written βnon dimensionallyβ as: x v x V x x x on based number s Reynold' where β β‘ = Re Re . 5 % 99 We can also find: x w f x x V C x x Re 664 . 2 1 Re 664 . Re 7208 . 1 2 * = = = = β β Ο Ο ΞΈ Comment: β At leading edge of a flat plate β β β f C x gives this and ! β In reality, the leading edge of an infinitely thin plate would have very large, but not infinite f C . β The problem is that near the leading edge of a thin plate, the boundary layer equations are not correct and the Navier-Stokes equations are needed. Question: Why did the boundary layer approximation fail at β x ?...
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16100lectre30_cj - 5 3 at 99 β = β² Ξ f x V vx vx V vx V...

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