Flow Past a Source (Part - 2) | Additional Documents & Tests for Civil Engineering (CE) PDF Download

Lift and Drag for Flow About a Rotating Cylinder
The pressure at large distances from the cylinder is uniform and given by p₀.

 Deploying Bernoulli's equation between the points at infinity and on the boundary of the cylinder,
   (23.9)

Hence,
   (23.10)

From Eqs (23.9) and (23.10) we can write
   (23.11)

The lift may calculated as


The drag force , which includes the multiplication by cosθ (and integration over 2π) is zero.

• Thus the inviscid flow also demonstrates lift.

•  lift becomes a simple formula involving only the density of the medium, free stream velocity and circulation.

•  in two dimensional incompressible steady flow about a boundary of any shape, the lift is always a product of these three quantities.----- Kutta- Joukowski theorem

Aerofoil Theory
Aerofoils are streamline shaped wings which are used in airplanes and turbo machinery. These shapes are such that the drag force is a very small fraction of the lift. The following nomenclatures are used for defining an aerofoil

              Fig 23.4      Aerofoil Section
 

• The chord (C) is the distance between the leading edge and trailing edge.
• The length of an aerofoil, normal to the cross-section (i.e., normal to the plane of a paper) is called the span of a aerofoil.
• The camber line represents the mean profile of the aerofoil. Some important geometrical parameters for an aerofoil are the ratio of maximum thickness to chord (t/C) and the ratio of maximum camber to chord (h/C). When these ratios are small, an aerofoil can be considered to be thin. For the analysis of flow, a thin aerofoil is represented by its camber.

The theory of thick cambered aerofoils uses a complex-variable mapping which transforms the inviscid flow across a rotating cylinder into the flow about an aerofoil shape with circulation


Flow Around a Thin Aerofoil

• Thin aerofoil theory is based upon the superposition of uniform flow at infinity and a continuous distribution of clockwise free vortex on the camber line having circulation density y(s) per unit length . 
• The circulation density y(s) should be such that the resultant flow is tangent to the camber line at every point.
• Since the slope of the camber line is assumed to be small, y(s)ds = y(n)dη. The total circulation around the profile is given by
     (23.13)

Fig 23.5    Flow Around Thin Aerofoil

A vortical motion of strength    develops a velocity at the point p which may be expressed as


The total induced velocity in the upward direction at point p due to the entire vortex distribution along the camber line is

For a small camber (having small α), this expression is identically valid for the induced velocity at point p' due to the vortex sheet of variable strength y(s) on the camber line. The resultant velocity due to    and v(x) must be tangential to the camber line so that the slope of a camber line may be expressed as
   (23.15)

From Eqs (23.14) and (23.15) we can write

Consider an element ds on the camber line. Consider a small rectangle (drawn with dotted line) around ds. The upper and lower sides of the rectangle are very close to each other and these are parallel to the camber line. The other two sides are normal to the camber line. The circulation along the rectangle is measured in clockwise direction as
[normal component of velocity at the camber line should be 
If the mean velocity in the tangential direction at the camber line is given by    it can be rewritten as


if v is very small    becomes equal to . The difference in velocity across the camber line brought about by the vortex sheet of variable strength  y(s) causes pressure difference and generates lift force.

 

Generation of Vortices Around a Wing

• The lift around an aerofoil is generated following Kutta-Joukowski theorem . Lift is a product of ρ ,  and the circulation   .
  

• When the motion of a wing starts from rest, vortices are formed at the trailing edge.
• At the start, there is a velocity discontinuity at the trailing edge. This is eventual because near the trailing edge, the velocity at the bottom surface is higher than that at the top surface. This discrepancy in velocity culminates in the formation of vortices at the trailing edge. 
• Figure 23.6(a) depicts the formation of starting vortex by impulsively moving aerofoil. However, the starting vortices induce a counter circulation as shown in Figure 23.6(b). The circulation around a path (ABCD) enclosing the wing and just shed (starting) vortex must be zero. Here we refer to Kelvin's theorem once again.
  
  
   
• Initially, the flow starts with the zero circulation around the closed path. Thereafter, due to the change in angle of attack or flow velocity, if a fresh starting vortex is shed, the circulation around the wing will adjust itself so that a net zero vorticity is set around the closed path. 
• Real wings have finite span or finite aspect ratio (AR) λ , defined as
     (23.16)
   

where b is the span length, A_s is the plan form area as seen from the top.. 

• For a wing of finite span, the end conditions affect both the lift and the drag. In the leading edge region, pressure at the bottom surface of a wing is higher than that at the top surface. The longitudinal vortices are generated at the edges of finite wing owing to pressure differences between the bottom surface directly facing the flow and the top surface