BoundaryLayerFlatPlate

# BoundaryLayerFlatPlate - SOLUTION FOR THE BOUNDARY LAYER ON...

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1 SOLUTION FOR THE BOUNDARY LAYER ON A FLAT PLATE Consider the following scenario. 1. A steady potential flow has constant velocity U in the x direction. 1. An infinitely thin flat plate is placed into this flow so that the plate is parallel to the potential flow (0 angle of incidence ). Viscosity should retard the flow, thus creating a boundary layer on either side of the plate. Here only the boundary layer on one side of the plate is considered. The flow is assumed to be laminar. Boundary layer theory allows us to calculate the drag on the plate! x y δ U U u plate

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2 A STEADY RECTILINEAR POTENTIAL FLOW HAS ZERO PRESSURE GRADIENT EVERYWHERE x y δ U U u plate A steady, rectilinear potential flow in the x direction is described by the relations According to Bernoulli’s equation for potential flows, the dynamic pressure of the potential flow p pd is related to the velocity field as Between the above two equations, then, for this flow 0 y v , U x u , Ux = φ = = φ = = φ const ) v u ( 2 1 p 2 2 pd = + ρ + 0 y p x p pd pd = =
3 BOUNDARY LAYER EQUATIONS FOR A FLAT PLATE x y δ U U u plate For the case of a steady, laminar boundary layer on a flat plate at 0 angle of incidence, with vanishing imposed pressure gradient, the boundary layer equations and boundary conditions become (see Slide 15 of BoundaryLayerApprox.ppt with dp pds /dx = 0) 0 y v x u y u y u v x u u 2 2 = + ν = + U u , 0 v , 0 u y 0 y 0 y = = = = = = Tangential and normal velocities vanish at boundary: tangential velocity = free stream velocity far from plate 0 y v x u y u y u v x u u 2 2 pds = + ν + - = +

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4 NOMINAL BOUNDARY LAYER THICKNESS x y δ U U u plate Until now we have not given a precise definition for boundary layer thickness. Here we use δ to denote nominal boundary thickness , which is defined to be the value of y at which u = 0.99 U, i.e. U 99 . 0 ) y , x ( u y = δ = x y u U u = 0.99 U δ The choice 0.99 is arbitrary; we could have chosen 0.98 or 0.995 or whatever we find reasonable.
5 STREAMWISE VARIATION OF BOUNDARY LAYER THICKNESS Consider a plate of length L. Based on the estimate of Slide 11 of BoundaryLayerApprox.ppt, we can estimate δ as or thus where C is a constant. By the same arguments, the nominal boundary thickness up to any point x L on the plate should be given as ν = δ - UL , ) ( ~ L 2 / 1 Re Re 2 / 1 2 / 1 U L C or U L ~ ν = δ ν δ 2 / 1 2 / 1 U x C or U x ~ ν = δ ν δ x y δ U U u plate L

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6 SIMILARITY One triangle is similar to another triangle if it can be mapped onto the other triangle by means of a uniform stretching. The red triangles are similar to the blue triangle. The red triangles are not similar to the blue triangle.
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BoundaryLayerFlatPlate - SOLUTION FOR THE BOUNDARY LAYER ON...

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