Test: Vorticity - Civil Engineering (CE) MCQ

Vorticity(ξ):
It is defined as the ratio of limiting value of circulation and area of a closed contour. it measures the local rotation of a fluid parcel.  
vorticity = Circulation / Area
Vorticity is defined as the twice of the rotational component.
ξ  = 2ω,
But, 

Where  represents velocity field

So vorticity is also equal to the curl of the velocity factor.

Concept:
If the velocity vector is given as:

Vorticity in the z-direction is defined as:

Calculation:
Given:

Velocity Potential function: These function is defined as a function of space and time in a flow such that the negative derivation of this function with respect to any direction gives the velocity of fluid in that direction.
If velocity potential (ϕ) exist, there will be a flow.

Now, Angular velocity is given by:

Therefore,
ω_z = 0
i.e Flow is irrotational.
Therefore for existence of velocity potential function in a fluid flow, flow must be irrotational.
Vorticity is twice of angular velocity, hence it is also zero.

Concept:
Circulation = Vorticity × Area
where, 
Vorticity is defined as the value twice of the rotation.
ζ = 2ω
Rotation (ω) =  
Now,
∴ Vorticity = ζ =
Calculation:
Given,
Velocity field, u = x, v = x + y
∴  ∂u/∂y = 0, ∂v/∂x = 1
Rectangular Field x = 0, y = 0, x = 1, y = 1 ( i.e Dimension of rectangular are 1 × 1 )
Area of Recatangle = 1 × 1 = 1
Circulation = Vorticity × Area

Circulation= (1 – 0) × 1
∴ Circulation= 1 units

Vorticity is equal to twice of the rotation vector
i.e.  
But, 


Where represents velocity field

Concept:
Incompressible flow: When the density of fluid remains constant that it does not change with respect to time or space, then it is known as an incompressible flow.
General continuity equation:

For steady & incompressible flow:

Hence, it is incompressible flow.
Rotational flow: When the fluid particles rotate about their center of mass, then the flow is known as rotational flow.
For rotational flow, the following condition must be followed.

Here, u = (x + 2y + 2), v = (4 - y), w = 0

2 ≠ 0
Therefore, it is a rotational flow.

Vorticity(ξ):
It is defined as the ratio of limiting value of circulation and area of a closed contour. it measures the local rotation of a fluid parcel.  
vorticity = Circulation / Area
Vorticity is defined as the twice of the rotational component.
ξ  = 2ω,
But, 


Where  represents velocity field

So vorticity is also equal to the curl of the velocity factor.

Concept:
Angular velocity,

∴ ω­_xy = -3
Now,
Rate of shear strain

Angular velocity comes in the picture when the flow is rotational that is the flow which has both translational as well as rotational motion. It is the rate of change of angular displacement. It is denoted by omega and its SI unit id radian per second.

Strain is defined as the change in angle between the two lines in a flow field. Suppose Δθ₁ and Δθ₂ are the angles between the two lines in a flow field. Therefore, strain can be given by – Strain= Δθ₂ – Δθ₁.