Blasius_Boundary.nb

# Blasius_Boundary.nb - B uy @ h D , 8 h , 0, h Big 2...

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Blasius Boundary Layer APPH 4200 Physics of Fluids Columbia University Similarity Equation In[1]:= eq = f @ h D D @ f @ h D , 8 h , 2 <D ê 2 + D @ f @ h D , 8 h , 3 <D ã 0 Out[1]= 1 2 f @ h D f ££ @ h D + f H 3 L @ h D ã 0 In[2]:= h Big = 25.0; In[3]:= bc = 8 f' @ h Big D ã 1, f @ 0 D ã 0, f' @ 0 D ã 0 < Out[3]= 8 f £ @ 25. D ã 1, f @ 0 D ã 0, f £ @ 0 D ã 0 < Finding the Solution In[4]:= sol = NDSolve @8 eq < ~ Join ~ bc, f, 8 h , 0, h Big <D Out[4]= 88 f Ø InterpolatingFunction @88 0., 25. << , <> D<< Graph the Solution In[5]:= fBlasius = f ê . First @ sol D Out[5]= InterpolatingFunction @88 0., 25. << , <> D In[6]:= gBlasius @ h _ D = D @ fBlasius @ h D , h D Out[6]= InterpolatingFunction @88 0., 25. << , <> D@ h D In[7]:= Plot @ fBlasius @ h D , 8 h , 0, h Big < , PlotLabel Ø "f @ h D " D Out[7]= 5 10 15 20 25 5 10 15 20 f @ h D

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In[8]:= Plot @ gBlasius @ h D , 8 h , 0, h Big ê 2 < , PlotLabel Ø "g @ h D ", PlotRange Ø All, AxesLabel Ø 8 " h ", "u ê U " <D Out[8]= 2 4 6 8 10 12 h 0.2 0.4 0.6 0.8 1.0 u ê U g @ h D Cross Velocity Normalized to the Re … In[9]:= uy @ h _ D = H 1 ê 2 L H h gBlasius @ h D - fBlasius @ h DL Out[9]= 1 2 H - InterpolatingFunction @88 0., 25. << , <> D@ h D + h InterpolatingFunction @88 0., 25. << , <> D@ h DL In[10]:= Plot
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Unformatted text preview: B uy @ h D , 8 h , 0, h Big 2 &lt; , PlotLabel &quot;Normalized u y @ h D &quot;, PlotRange All, AxesLabel : &quot; h &quot;, &quot; H u y U L Re &quot; &gt;F Out[10]= 2 4 6 8 10 12 h 0.2 0.4 0.6 0.8 H u y U L Re Normalized u y @ h D Normalized Shear Stress At the plate surface In[11]:= D @ gBlasius @ h D , h D . h Out[11]= 0.332057 2 Blasius_Boundary.nb Summary Blasius provided the mathematically rigorous &quot;similiarity&quot; solution to the boundary layer thickness for uniform flow along a fixed plate. The similiarity transformation creates an nonlinear ordinary differential equation that can be easily integrated. Blasius_Boundary.nb 3...
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## This document was uploaded on 10/18/2011.

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Blasius_Boundary.nb - B uy @ h D , 8 h , 0, h Big 2...

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