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Turbulent Boundary Layer on a flat plate ( dp e / dx =0)

Turbulent Boundary Layer on a flat plate ( dp e / dx =0). Contents : 1/7 velocity law; Equations for the turbulent boundary layer with zero pressure gradient ( dp e /dx=0) ; Virtual origin of the boundary layer; Hydraulically smooth and fully rough flat plates.

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Turbulent Boundary Layer on a flat plate ( dp e / dx =0)

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  1. Turbulent Boundary Layer on a flat plate (dpe/dx=0) • Contents: • 1/7 velocity law; • Equations for the turbulent boundary layer with zero pressure gradient (dpe/dx=0); • Virtual origin of the boundary layer; • Hydraulically smooth and fully rough flat plates. Fluid Mechanics II Prof. António Sarmento - DEM/IST

  2. x – Distance to the leading edge Laminar flow • Beginning of the BL very large • Sufficiently long plate : Re increases Critical Re (5105) decreases Transition to turbulent Boundary Layer Introdution • Transition from laminar to turbulent regime: Fluid Mechanics II Prof. António Sarmento - DEM/IST

  3. Boundary Layer Introdution • Turbulent regions of the BL: • Linear sub-layer (no turbulence); • Transition layer; • Central region – logaritmic profile zone (turbulence not affected by the wall); • External zone(turbulent vortices mixed with non-turbulent outside flow). Fluid Mechanics II Prof. António Sarmento - DEM/IST

  4. Law of the wall • Experimental results from the law of the wall Fluid Mechanics II Prof. António Sarmento - DEM/IST

  5. Central region • Transition layer Law of the wall • Characteristics of the velocity profile u*=f(y*): • Linear, laminar or viscous sub-layer Fluid Mechanics II Prof. António Sarmento - DEM/IST

  6. Take for any y • Take - less reliable approximation, but easier to apply; • does not allow to calculate the shear stress in the wall. Other approxmations for u=u(y) Fluid Mechanics II Prof. António Sarmento - DEM/IST

  7. (Flat plates and ReL107) – Velocity profile (empirical): Turbulent Boundary Layer on a flat plate (dpe/dx=0) • Bases: –Von Kárman equation: Note 1: the velocity profile in the BL follows the law of the wall , but this law has a less convenient form. Note 2: as we saw in the laminar case, the integral parameters of the BL are little affected by the shape of the velocity profile Fluid Mechanics II Prof. António Sarmento - DEM/IST

  8. Turbulent Boundary Layer on a flat plate (dpe/dx=0) • Shear stress on the wall: Note: this expression relates 0 with  (still unknown). Fluid Mechanics II Prof. António Sarmento - DEM/IST

  9. a =7/72 • Conclusion: 7/72=0,0972<0,133 (Laminar BL) Turbulent Boundary Layer on a flat plate (dpe/dx=0) • As we saw: Fluid Mechanics II Prof. António Sarmento - DEM/IST

  10. Turbulent Boundary Layer on a flat plate (dpe/dx=0) • On the other hand: • Form Factor: Laminar BL => 2,59 The fuller the velocity profile is, closer to 1 the Form Factor is. Fluid Mechanics II Prof. António Sarmento - DEM/IST

  11. Von Kármàn Equation: Equation to 0: Turbulent Boundary Layer on a flat plate (dpe/dx=0) Note: xo is the point where =0. In general we choose xo to be in the beginning of the turbulent BL. Fluid Mechanics II Prof. António Sarmento - DEM/IST

  12. Transtion zone Laminar BL c 0 xc x0 (Rec5,5105) Turbulent Boundary Layer on a flat plate (dpe/dx=0) • BL evolution on the flat plate: Turbulent BL Fluid Mechanics II Prof. António Sarmento - DEM/IST

  13. Turbulent Boundary Layer on a flat plate (dpe/dx=0) • Case 1 – the section of interest is very far away from the critical section (x>>xc): the BL is assumed to be turbulent from the beginning of the plate (x0=0=0). Valid if L>>xc (or ReL>>Rec). L is the plate lenght Fluid Mechanics II Prof. António Sarmento - DEM/IST

  14. From the Von Kármán equation aL=0,133 (Blasius) aT=7/72 Turbulent Boundary Layer on a flat plate (dpe/dx=0) • Case 2 – the section of interest is not very faraway from the critical section: the transition zone is not considered • => m0mc andx0=xc. Fluid Mechanics II Prof. António Sarmento - DEM/IST

  15. Turbulent Boundary Layer on a flat plate (dpe/dx=0) • Virtual origin of the turbulent BL: xv xv xc=xo Would be as if the BL started turbulent from xvto reache 0 in x0. Fluid Mechanics II Prof. António Sarmento - DEM/IST

  16. xv xc=xo Turbulent Boundary Layer on a flat plate (dpe/dx=0) • Case 2: calculation of the drag on the plate. Fluid Mechanics II Prof. António Sarmento - DEM/IST

  17. Turbulent Boundary Layer on a flat plate (dpe/dx=0) • Correlations for higher Re: for Re109 for 3106 Re109 Fluid Mechanics II Teacher António Sarmento - DEM/IST

  18. Hidraulicaly fully rough plates if Turbulent Boundary Layer • Hidraulically smooth plates if All the contents studied before are for smooth plates Fluid Mechanics II Prof. António Sarmento - DEM/IST

  19. Turbulent Boundary Layer on a flat plate (dpe/dx=0) • Contents: • 1/7 Law of velocities; • Turbulent boundary layer expressions with dpe/dx null above a flat plate; • Virtual origin of the boundary layer; • Hydraulically smooth and fully rough plates. Fluid Mechanics II Teacher António Sarmento - DEM/IST

  20. Turbulent Boundary Layer on a flat plate (dpe/dx=0) • Sources: • Sabersky – Fluid Flow: 8.9 • White – Fluid Mechanics: 7.4 Fluid Mechanics II Teacher António Sarmento - DEM/IST

  21. Exercise • A plate is 6 m long and 3 m wide and is immersed in a water flow (=1000 kg/m3, =1,1310-6 m2/s) with na undisturbed velocity of 6 m/s parallel to the plate. Rec=106. Compute: • a) The thickness of the BL at x=0,25 m; • b) The thickness of the BL at x=1,9 m; • c) The total drag on the plate; • d) The maximum roughness on the plate for it to be hydraulically smooth. Fluid Mechanics II Prof. António Sarmento - DEM/IST

  22. Exercise: solution • L= 6 m; b=3 m; =1000 kg/m3; =1,1310-6 m2/s; U= 6 m/s; Rec=106. • a) Thickness of the BL at x1=0,25 m? If we had addmited that the BL grew turbulent from the beginning: In this case, the result would be significantly different Fluid Mechanics II Prof. António Sarmento - DEM/IST

  23. Exercise: solution • L= 6 m; b=3 m; =1000 kg/m3; =1,1310-6 m2/s; U= 6 m/s; Rec=106. • b) Thickness of the BL at x2=1,9 m? If we had addmited that the BL grew turbulent from the beginning: In this case, the result would have a much smaller difference Fluid Mechanics II Prof. António Sarmento - DEM/IST

  24. Exercise: solution • L= 6 m; b=3 m; =1000 kg/m3; =1,1310-6 m2/s; U= 6 m/s; Rec=106. • c) Total drag on the plate? For a 1/7 velocity law => a=7/72 => If we had addmited that the BL grew turbulent from the beginning: Difference of 2,5% Fluid Mechanics II Prof. António Sarmento - DEM/IST

  25. Exercise: solution • L= 6 m; b=3 m; =1000 kg/m3; =1,1310-6 m2/s; U= 6 m/s; Rec=106. • c) Total drag on the plate? Difference between computing D taking into account the laminar BL or assuming turbulent from the leading edge. Fluid Mechanics II Prof. António Sarmento - DEM/IST

  26. Exercise: solution • L= 6 m; b=3 m; =1000 kg/m3; =1,1310-6 m2/s; U= 6 m/s; Rec=106. • d) Maximum roughness on the plate to be hidraulically smooth? It is necessary that: with Where is 0 bigger? In the beginning of the turbulent BL Fluid Mechanics II Prof. António Sarmento - DEM/IST

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