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Bernoulli's equation

Euler's equation is given as dp=ρvdv
This equation applies to an incompressible and in-viscid flow where ρ=constant. In a streamline, in between two points 1 and 2, the above Euler equation can be integrated as p2p1dp=ρv2v1vdv
p2p1=ρ(v222v212)
p1+12ρv21=p2+12ρv22
This can be written asp+12ρv2=constant
for a streamline.For an rotational flow the value of constant is changing from streamline to another.For irrotational flow,the constant is same for all streamlines and p+12ρv2=constant
throughout the flow. Physical significance of Bernoulli's equation is that when the velocity increases, the pressure decreases  and when the velocity decreases, the pressure increases. ...

Non Lifting flow over a circular cylinder

When there is a superposition of a uniform flow with a doublet (Which is a source-sink pair of flows), a non-lifting flow over a circular cylinder can be analysed. Stream function ψ for a uniform flow is ψ=(Vrsinθ)(1R2r2)
velocity field is obtained by differentiating above equationVr=(1R2r2)Vcosθ
andVθ=(1+R2r2)Vsinθ
Stagnation points can be obtained by equating Vr and Vθ to zero.Considering an incompressible and inviscid flow pressure coefficient over a circular cylinder is Cp=14sin2θ
Example: Calculate locations on the surface of a cylinder where surface pressure equals to free-stream pressure considering a non-lifting flow. Solution: Pressure coefficient ...

Lifting flow over a cylinder

Lifting flow over a circular cylinder can be synthesised as superposition of a vortex flow of strength τ with non lifting flow over a cylinder. The stream function for the flow can be given asψ=(Vrsinθ)(1R2r2)+τ2πlnrR
and velocity components in the radial and tangential direction can be obtained as Vr=(1R2r2)Vcosθ
Vθ=(1+R2r2)Vsinθτ2πr
The velocity on the surface of the cylinder can be given as  V=2Vsinθτ2πR
and coefficient of lift as cl=τRV
Lift per unit span is given as L=ρVτ
Example: Calculate the peak coefficient of pressure for a...