Flow Past a Source
When a uniform flow is added to that due to a source -
Stream function;
Velocity Potential;
so
and
Fig 23.1 The streamlines of the flow past a line source for equal increments of 2πψ/q
The Plane coordinates are x/a, y/a where a=k/u
Explanation of Figure
Flow Past Vortex
when uniform flow is superimposed with a vortex flow -
Stream Function:
Velocity Potential:
so that;
Flow About a Rotating Cylinder
Magnus Effect
Flow about a rotating cylinder is equivalent to the combination of flow past a cylinder and a vortex.
As such in addition to superimposed uniform flow and a doublet, a vortex is thrown at the doublet centre which will simulate a rotating cylinder in uniform stream.
The pressure distribution will result in a force, a component of which will culminate in lift force
The phenomenon of generation of lift by a rotating object placed in a stream is known as Magnus effect.
Velocity Potential and Stream Function
The velocity potential and stream functions for the combination of doublet, vortex and uniform flow are
(clockwise rotation) (23.1)
(clockwise rotation) (23.2)
By making use of either the stream function or velocity potential function, the velocity components are (putting x= rcosθ and y= rsinθ )
(23.3)
(23.4)
Stagnation Points
At the stagnation points the velocity components must vanish. From Eq. (23.3), we get
(23.5)
Solution :
Eq. (23.4) depicts that a zero transverse velocity requires
(23.6)
At the stagnation point, both radial and transverse velocity components must be zero .
Thus the location of stagnation point occurs at
There will be two stagnation points since there are two angles for a given sine except for sin^{-1}(±1)
Determination of Stream Line
The streamline passing through these points may be determined by evaluating ψ at these points.
Substitution of the stagnation coordinate (r, θ) into the stream function (Eq. 23.2) yields
(23.7)
Equating the general expression for stream function to the above constant, we get
By rearranging we can write
(23.8)
All points along the circle satisfy Eq. (23.8) , since for this value of r, each quantity within parentheses in the equation is zero.
Considering the interior of the circle (on which ψ = 0) to be a solid cylinder, the outer streamline pattern is shown in Fig 23.2.
At the stagnation point
The limiting case arises for , where θ = sin^{ -1}(-1) = -90^{0} and two stagnation points meet at the bottom as shown in Fig. 23.3.
In the case of a circulatory flow past the cylinder, the streamlines are symmetric with respect to the y-axis. The presures at the points on the cylinder surface are symmetrical with respect to the y-axis. There is no symmetry with respect to the x-axis. Therefore a resultant force acts on the cylinder in the direction of the y-axis, and the resultant force in the direction of the x-axis is equal to zero as in the flow without circulation; that is, the D'Alembert paradox takes place here as well. Thus, in the presence of circulation, different flow patterns can take place and, therefore, it is necessary for the uniqueness of the solution, to specify the magnitude of circulation.
However, in all these cases the effects of the vortex and doublet become negligibly small as one moves a large distance from the cylinder.
The flow is assumed to be uniform at infinity.
We have already seen that the change in strength of the vortex changes the flow pattern, particularly the position of the stagnation points but the radius of the cylinder remains unchanged.
Lift and Drag for Flow About a Rotating Cylinder
The pressure at large distances from the cylinder is uniform and given by p_{0}.
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
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
(23.13)
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 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
[normal component of velocity at the camber line should be it can be rewritten as
f v is very small [v << U_{∞}.], V_{s }becomes equal to U_{∞}. 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
(23.16 )
where b is the span length, A_{s} is the plan form area as seen from the top..