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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST von Kárman Equation for flat plates (dp e /dx≠0) u For laminar or turbulent flows: in the turbulent case we take time-average velocity and pressure. u Procedure: Mass and momentum balance of the following control volume: dx x x+dx

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST dx u Mass balance: Steady Flow o Flow rate: von Kárman Equation forflat plates (dp e /dx≠0)

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST dx o x Momentum flow rate through y=δ: u Mass balance : von Kárman Equation for flat plates (dp e /dx≠0)

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST o Difference : u x momentum balance: Steady flow von Kárman Equation for flat plates (dp e /dx≠0)

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u x momentum balance: von Kárman Equation for flat plates (dp e /dx≠0)

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u Forces along x: p+dpp p+ 1/2 dp τ0τ0 von Kárman Equation for flat plates (dp e /dx≠0)

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u Final result: u Using the definition of d and δ m : von Kárman Equation for flat plates (dp e /dx≠0)

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u When dp e /dx=0 (dU/dx=0): von Kárman Equation for flat plates (dp e /dx≠0) u When dp e /dx=0 (dU/dx=0) we have m =a (a takes different values in laminar and turbulent flow): Boundary layer grows faster when C f is higher

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Approximate solutions for laminar boundary layer for dp e /dx=0 u Blasius solution shows that with and a – constant along the BD

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Approximate solutions for laminar boundary layer for dp e /dx=0 u von Kárman Equation: but β - constant Integrating

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Approximate solutions for laminar boundary layer for dp e /dx=0 u Remark: a and β depend on the velocity profile, however δ/x, c f and C D do not vary much with profile shape u We have and

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Approximate solutions for laminar boundary layer for dp e /dx=0

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u Contents: –von Kármàn Equation; –Simplification for ; –Approximate solutions for laminar Boundary Layers with zero pressure gradient. Blasius Solution for Laminar Boundary Layer Equation over a flat plate with dp e /dx=0

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u Recommended Study Elements: –Sabersky – Fluid Flow: 8.6, 8.7 –White – Fluid Mechanics: 7.3, 7.4 Von Kárman Equation for a flat plate

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Problem on the Von Kármàn Equation

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