2004Fluid Mechanics II Prof. António Sarmento - DEM/IST u 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. Turbulent Boundary Layer on a flat plate (dp e /dx=0)

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST Boundary Layer Introdution u Transition from laminar to turbulent regime: x – Distance to the leading edge Beginning of the BL Laminar flow Sufficiently long plate :Re increases Critical Re (  5  10 5 ) Transition to turbulent very large decreases

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST Boundary Layer Introdution u 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).

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST Law of the wall u Experimental results from the law of the wall

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST Law of the wall u Characteristics of the velocity profile u * =f(y * ): o Linear, laminar or viscous sub-layer o Central region o Transition layer

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST Other approxmations for u=u(y) o Take for any y o Take - less reliable approximation, but easier to apply; does not allow to calculate the shear stress in the wall.

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST u Bases: –V on 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 – Velocity profile (empirical): (Flat plates and Re L  10 7 ) Turbulent Boundary Layer on a flat plate (dp e /dx=0)

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST u Shear stress on the wall: Note: this expression relates  0 with  (still unknown). Turbulent Boundary Layer on a flat plate (dp e /dx=0)

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST =7/72 u As we saw: a u Conclusion: Turbulent Boundary Layer on a flat plate (dp e /dx=0) 7/72=0,0972<0,133 (Laminar BL)

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST u On the other hand: u Form Factor: Turbulent Boundary Layer on a flat plate (dp e /dx=0) Laminar BL => 2,59 The fuller the velocity profile is, closer to 1 the Form Factor is.

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST Note: x o is the point where  =  0. In general we choose x o to be in the beginning of the turbulent BL. u Von Kármàn Equation: Equation to  0 : Turbulent Boundary Layer on a flat plate (dp e /dx=0)

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST u BL evolution on the flat plate: xcxc x0x0 Laminar BL Turbulent BL Transtion zone (Re c  5,5  10 5 ) 00 cc Turbulent Boundary Layer on a flat plate (dp e /dx=0)

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST u Case 1 – the section of interest is very far away from the critical section (x>>x c ): the BL is assumed to be turbulent from the beginning of the plate (x 0 =  0 =0). Valid if L>>x c (or Re L >>Re c ). L is the plate lenght Turbulent Boundary Layer on a flat plate (dp e /dx=0)

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST u Case 2 – the section of interest is not very faraway from the critical section: the transition zone is not considered =>  m0  mc and x 0 =x c. From the Von Kármán equation Turbulent Boundary Layer on a flat plate (dp e /dx=0) a L =0,133 (Blasius) a T =7/72

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST u Virtual origin of the turbulent BL: x v xvxv x c =x o Turbulent Boundary Layer on a flat plate (dp e /dx=0) Would be as if the BL started turbulent from x v to reache  0 in x 0.

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST u Case 2: calculation of the drag on the plate. Turbulent Boundary Layer on a flat plate (dp e /dx=0) xvxv x c =x o

2004Fluid Mechanics II Teacher António Sarmento - DEM/IST u Correlations for higher Re: for Re  10 9 for 3  10 6  Re  10 9 Turbulent Boundary Layer on a flat plate (dp e /dx=0)

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST u Hidraulically smooth plates if All the contents studied before are for smooth plates Turbulent Boundary Layer u Hidraulicaly fully rough plates if

2004Fluid Mechanics II Teacher António Sarmento - DEM/IST u Contents: –1/7 Law of velocities; –Turbulent boundary layer expressions with dp e /dx null above a flat plate; –Virtual origin of the boundary layer; –Hydraulically smooth and fully rough plates. Turbulent Boundary Layer on a flat plate (dp e /dx=0)

2004Fluid Mechanics II Teacher António Sarmento - DEM/IST u Sources: –Sabersky – Fluid Flow: 8.9 –White – Fluid Mechanics: 7.4 Turbulent Boundary Layer on a flat plate (dp e /dx=0)

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST u A plate is 6 m long and 3 m wide and is immersed in a water flow (  =1000 kg/m3, =1,13  m 2 /s) with na undisturbed velocity of 6 m/s parallel to the plate. Re c =10 6. Compute: u a) The thickness of the BL at x=0,25 m; u b) The thickness of the BL at x=1,9 m; u c) The total drag on the plate; u d) The maximum roughness on the plate for it to be hydraulically smooth. Exercise

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST u L= 6 m; b=3 m;  =1000 kg/m3; =1,13  m 2 /s; U= 6 m/s; Re c =10 6. u a) Thickness of the BL at x 1 =0,25 m? Exercise: solution If we had addmited that the BL grew turbulent from the beginning: In this case, the result would be significantly different

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST u L= 6 m; b=3 m;  =1000 kg/m3; =1,13  m 2 /s; U= 6 m/s; Re c =10 6. u b) Thickness of the BL at x 2 =1,9 m? Exercise: solution If we had addmited that the BL grew turbulent from the beginning: In this case, the result would have a much smaller difference

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST u L= 6 m; b=3 m;  =1000 kg/m3; =1,13  m 2 /s; U= 6 m/s; Re c =10 6. u c) Total drag on the plate? Exercise: solution 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%

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST u L= 6 m; b=3 m;  =1000 kg/m3; =1,13  m 2 /s; U= 6 m/s; Re c =10 6. u c) Total drag on the plate? Exercise: solution Difference between computing D taking into account the laminar BL or assuming turbulent from the leading edge.

2004Fluid Mechanics II Prof. António Sarmento - DEM/IST u L= 6 m; b=3 m;  =1000 kg/m3; =1,13  m 2 /s; U= 6 m/s; Re c =10 6. u d) Maximum roughness on the plate to be hidraulically smooth? Exercise: solution It is necessary that:with Where is  0 bigger? In the beginning of the turbulent BL