A 2-D flow having velocity V = (x + 2y + 2)i + (4 - y)j will be
X component of the velocity = Vx = u = x+ 2y+ 2;
y component of the velocity = Vy = v = 4-y;
z component of the velocity = Vz = w =0
Continuity Equation for the incompressible flow must be satisfied which is given by—
∂u/∂x+∂v/∂y+∂w/∂z = 0
Therefore, (∂(x+2y+2))/∂x+(∂(4-y))/∂y+(∂(0))/∂z= 1 -1 +0 = 0
Hence, the flow is incompressible.
For flow to be irrotational,
Angular Velocity = ω ⃗=1/2 (curl ( v) ⃗ )
x- component of angular velocity -> ω ⃗_z=0.5 (∂v/∂x-∂u/∂y) = 0
Since the flow is 2-d and no z- component of velocity, therefore, x and y components of the velocity will be automatically zero.
Hence, the flow is incompressible and irrotational.
In a flow net
A stream function is
A stream function is defined by following characteristic:
The partial derivative of stream function w.r.t y will give velocity in x-direction.
The partial derivative of stream function w.r.t x will give velocity in negative y-direction.
It is valid for steady, incompressible flow since, is satisfies the continuity equation
The continuity equation for steady incompressible flow is expressed in vector notation as
In a converging steady flow there is
In two dimensional flow, the equation of a streamline is given as
The concept of stream function which is based on the principle of continuity is applicable to
Velocity potential function is valid for 3-dimensional flow while stream function is valid for 2 dimensional flow.
Which of the following velocity potentials satisfies continuity equation?
For the velocity potential function to satisfy continuity equation:
Where φ is velocity potential, φ = x2 - y2 satisfies this equation
In a two dimensional incompressible steady flow around an airfoil, the stream lines are 2 cm apart at a great distance from the airfoil, where the velocity is 30 m/sec. The velocity near the airfoil, where the stream lines are 1.5 cm apart, is
The velocity potential function for a source varies with distance r as
Velocity at any point r in the flow field of source is given by, Vr = q/2πr.