External flow: drag and Lift CHAPTER 11 External flow: drag and Lift 11-5: Parallel flow over flat plates 11-6: Flow over cylinders and spheres

Physical Features Physical Features As with all external flows, the boundary layers develop freely without constraint. Boundary layer conditions may be entirely laminar, laminar and turbulent, or entirely turbulent. To determine the conditions, compute and compare with the critical Reynolds number for transition to turbulence,

As for discussion, we consider the fluid to consist of adjacent layers piled on top of each other as in this figure. velocity of the particles in the first fluid layer is ZERO because the non-slip condition. the motionless layer slows down due to friction between the fluid particles (of the two adjoining fluid layers. The fluid layers slow down the molecules of the next layer and so on. Resulting the x-component of fluid velocity, u, varies from 0 at y=0 to nearly V at y=δ. Flow above the plate bounded by δ due to viscous shearing forces by fluid viscosity, called as velocity boundary layer.

The hypothetical line of u=0 The hypothetical line of u=0.99V divides the flow over a plate into two regions: boundary layer region (viscous effects and velocity changes are significant) irrotational flow region (frictional effects and velocity chages are significant).

Four regions of turbulent boundary layers: Viscous sublayer Buffer layer Overlap layer Turbulent layer (outer)

Friction coefficient Laminar flat plate boundary layer. Average friction coefficient for the entire flat plate  Eq 11.19 & eq 11.20.

In some cases the flat plate is sufficiently long for the flow to become turbulent but not long enough to disregard the laminar flow region. Therefore, in such case, the average friction coefficient can be determined by performing integration in Eq.11.16 over two parts (as in Eq. 11.21). As such, the average friction coefficient over the entire plate is; Example 11.3: Flow over hot oil over a flat plate. Questions : 11.46C , 11.50, 11.53, 11.57

Friction coefficient for parallel flow over smooth and rough flat plates.

Example 11.3: Flow over hot oil over a flat plate.

Questions : 11.46C We are to discuss the fluid property responsible for the development of a boundary layer. Answer: The fluid viscosity is responsible for the development of the velocity boundary layer. Velocity forces the boundary layer closer to the wall. Therefore, the higher the velocity (and thus Reynolds number), the lower the thickness of the boundary layer.

Questions : 11.50 Wind is blowing parallel to the side wall of a house. The drag force acting on the wall is to be determined for two different wind velocities. 10 m 4 m Air 55 km/h Assumptions 1. The flow is steady and incompressible. 2.The critical Reynolds number is Recr = 5105. 3. Air is an ideal gas. 4. The wall surface is smooth (the actual wall surface is usually very rough). 5. The wind blows parallel to the wall. Properties The density and kinematic viscosity of air at 1 atm and 5C are r = 1.269 kg/m3 and  = 1.382×10–5 m2/s . Analysis The Reynolds number is

The Re obtained is greater than the critical Reynolds number The Re obtained is greater than the critical Reynolds number. Thus we have combined laminar and turbulent flow, and the friction coefficient is Noting that the pressure drag is zero and thus for a flat plate, the drag force acting on the wall surface is

(b) When the wind velocity is doubled to 110 km/h, the Reynolds number becomes which is greater than the critical Reynolds number. Thus we have combined laminar and turbulent flow, and the friction coefficient and the drag force become

How realistic is it to treat the flow over the side wall surfaces as flow over a flat plate? Treating flow over the side wall of a house as flow over a flat plate is not quite realistic. When flow hits a bluff body like a house, it separates at the sharp corner and a separation bubble exists over most of the side panels of the house. Therefore, flat plat boundary layer equations are not appropriate for this problem, and the entire house should considered in the solution instead. Discussion: Note that the actual drag will probably be much higher since the wall surfaces are typically very rough. Also, we can solve this problem using the turbulent flow relation (instead of the combined laminar-turbulent flow relation) without much loss in accuracy. Finally, the drag force nearly quadruples when the velocity is doubled. This is expected since the drag force is proportional to the square of the velocity, and the effect of velocity on the friction coefficient is small.

Try it on your own…! Questions : 11.53, 11.55 & 11.57

The Cylinder in Cross Flow Conditions depend on special features of boundary layer development, including onset at a stagnation point and separation, as well as transition to turbulence. Stagnation point: Location of zero velocity and maximum pressure. Followed by boundary layer development under a favorable pressure gradient and hence acceleration of the free stream flow . As the rear of the cylinder is approached, the pressure must begin to increase. Hence, there is a minimum in the pressure distribution, p(x), after which boundary layer development occurs under the influence of an adverse pressure gradient

Cylinder in Cross Flow (cont.) Separation occurs when the velocity gradient reduces to zero. and is accompanied by flow reversal and a downstream wake. Location of separation depends on boundary layer transition.

Cylinder in Cross Flow (cont.) What features differentiate boundary development for the flat plate in parallel flow from that for flow over a cylinder? Force imposed by the flow is due to the combination of friction and form drag. The dimensionless form of the drag force is

Effect of surface roughness

Example 11-4: Drag force acting on a pipe in a river

Question 11.61c We are to discuss why the drag coefficient suddenly drops when the flow becomes turbulent. Analysis Turbulence moves the fluid separation point further back on the rear of the body, reducing the size of the wake, and thus the magnitude of the pressure drag (which is the dominant mode of drag). As a result, the drag coefficient suddenly drops. In general, turbulence increases the drag coefficient for flat surfaces, but the drag coefficient usually remains constant at high Reynolds numbers when the flow is turbulent. Discussion The sudden drop in drag is sometimes referred to as the drag crisis.

Questions to solve… Questions : 11.65, 11.70