Boundary-Layer Theory

Boundary-Layer Theory - Boundary-Layer Theory Simplified...

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Boundary-Layer Theory Simplified Navier-Stokes Equations for a very Thin Layer of Flow Adjoining a Solid Boundary Laminar Boundary Layer Over Flat Plate 2-D Incompressible Flow, x-y coordinates, Navier-Stokes Equations: xyx x x uu vu p (u u ) += + ν + y y y y 0 xyy x x uv vv p (v v ) + ν + xy Boundary Layer Assumptions: ~O( ) δν , 1 x δ << ; Friction as well as inertia forces are important. Here, ν is the fluid viscosity. y, v x , u x O(1), u O(1), y O( ), === δ since 2 O( ) ν Consequently, 1 u O , v ⎛⎞ == = ⎜⎟ δ ⎝⎠ δ , here, v is the y-component of velocity. xx yy y yy 2 11 uO ( 1 ) , , v O ( 1 ) , vO δ δ . Here, the symbol “O( )” denotes “of the order of magnitude of ( ).” Perform an order-of-magnitude analysis for the Navier-Stokes equations and drop out small terms, using 2 2 1 δ<δ< < < δδ . 1
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Boundary-Layer Equations xyx uu vu p u += + ν y y 0 xy uv Must be supplied from inviscid- flow calculations or other means Assumption: . (x) x δ< < Therefore, streamlines remain nearly horizontal, i.e., streamlines have negligible curvature. Hence, the centrifugal force normal to the plate is negligible, so that p 0 y = . This result is also true for boundary layers on curved surfaces with 1 R(x) >> δ , where R(x) is the radius of surface curvature.
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Boundary-Layer Theory - Boundary-Layer Theory Simplified...

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