1 / 15

von Kárman Equation for flat plates ( dp e / dx ≠ 0)

Procedure: Mass and momentum balance of the following control volume:. . x. x + dx. dx. von Kárman Equation for flat plates ( dp e / dx ≠ 0). For laminar or turbulent flows: in the turbulent case we take time-average velocity and pressure. Steady Flow. dx. Flow rate:.

moeshe
Download Presentation

von Kárman Equation for flat plates ( dp e / dx ≠ 0)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. Steady Flow dx • Flow rate: • Flow rate : von Kárman Equation forflat plates (dpe/dx≠0) • Mass balance: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

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

  4. Steady flow • Difference : von Kárman Equation for flat plates (dpe/dx≠0) • x momentum balance: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  5. von Kárman Equation for flat plates (dpe/dx≠0) • x momentum balance: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  6. p+1/2dp p p+dp τ0 von Kárman Equation for flat plates (dpe/dx≠0) • Forces along x: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  7. von Kárman Equation for flat plates (dpe/dx≠0) • Final result: • Using the definition of d and δm: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

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

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

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

  11. Approximate solutions for laminar boundary layer for dpe/dx=0 • We have and • Remark: a and βdepend on the velocity profile, however δ/x, cfand CDdo not vary much with profile shape Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

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

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

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

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

More Related