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# MCQs on Kelvin’s Circulation Theorem and the Starting Vortex – 1

1 - Question

Which of the following ensures flow smoothly leaving the trailing edge given the right value of circulation?

a) Kutta Condition

b) Momentum Theorem

c) Angle of Attack

d) The Shape of the Airfoil

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aExplanation: According to the Kutta Condition, for a given angle of attack, the value of circulation around the airfoil is such that the flow leaves the trailing edge smoothly. The velocity at the trailing edge is dependent on the shape of the airfoil.

2 - Question

It is possible to have lift without friction (i.e. in an inviscid medium).

a) True

b) False

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bExplanation: Nature enforces Kutta Condition through the means of the boundary layer (friction). If there were no friction Kutta condition would not be achieved and there will be no lift.

3 - Question

For an arbitrary inviscid and incompressible flow, with all the body forces zero, what is best described by the given sketch?

Kelvin’s circulation theorem in closed curves

a) Boundary Layer Formation

b) Kelvin’s Circulation Theorem

c) Kutta Condition

d) Generation of Lift

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bExplanation: For the same fluid elements in closed curves C1 and C2, the circulation remains constant with time as the fluid elements move downstream. This is essentially Kelvin’s circulation theorem.

4 - Question

Mathematically, what is meant by Kelvin’s Circulation Theorem for an inviscid and incompressible flow? (For the same set of fluid elements moving in a closed curve along with the fluid).

a) DΓ/Dt = 0

b) γ1 ≥ Γ2, where 1 is the upstream direction

c) Γ = -∮C1V.ds

d) Γ = 0

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aExplanation: Kelvin’s theorem states that the time rate of change of circulation for a set of fluid elements in a closed curve moving along the fluid is zero. In other words, circulation remains constant, not zero. Γ = -∮C1V.ds is the definition of circulation.

5 - Question

For which of the following Kelvin’s theorem is applicable?

a) Flow with Viscous Stresses

b) Compressible Flow

c) Inviscid, Compressible Barotropic Flow

d) Flow with Non-Conservative Body Forces

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cExplanation: Kelvin’s Theorem is applicable for the special case of barotropic flow while dealing with inviscid, compressible flows.

6 - Question

Which type of the following flow is characterized by density being a single-valued function of pressure only?

a) Viscous Flow

b) Barotropic Flow

c) Inviscid Flow

d) Baroclinic Flow

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bExplanation: A barotropic flow is a fluid where density is a function of pressure only, i.e. ρ = ρ(p). Baroclinic flow is the fluid which is not only dependent on the pressure but on other factors also. Viscous and inviscid flows are not necessarily dependent on pressure only.

7 - Question

A vortex sheet in the incompressible, inviscid fluid dies after some time.

a) True

b) False

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bExplanation: According to Kelvin’s theorem vortex sheet cannot die since circulation has to remain constant with time. It says a vortex sheet stays forever, in the ideal case.

8 - Question

For a fluid initially at rest, the formation of starting vortex implies ______

a) generation of lift

b) generation of circulation

c) generation of lift and circulation

d) no lift is produced

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cExplanation: From Kelvin’s Theorem, circulation remains constant with time. So for initial zero circulation, the formation of starting vortex means there has to be equal and opposite circulation in the form of lift.

9 - Question

During the formation of starting vortex, for an airfoil starting from rest, which is the correct sequence of events? (TE: Trailing Edge)

a) Velocity becomes infinite at the TE > Unstable vortex sheet formed due to very high vorticity > High velocity gradient formed at TE which is pushed downstream > Flow starts to curl at the TE > Unstable vortex sheet curls to form point vortex

b) Velocity becomes infinite at the TE > High velocity gradient formed at TE which is pushed downstream > Unstable vortex sheet formed due to very high vorticity > Flow starts to curl at the TE > Unstable vortex sheet curls to form point vortex

c) Velocity becomes infinite at the TE > Flow starts to curl at the TE > Unstable vortex sheet formed due to very high vorticity > Unstable vortex sheet curls to form point vortex > High velocity gradient formed at TE which is pushed downstream >

d) Flow starts to curl at the TE > Velocity becomes infinite at the TE > High velocity gradient formed at TE which is pushed downstream > Unstable vortex sheet formed due to very high vorticity > Unstable vortex sheet curls to form point vortex

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dExplanation: As the motion starts, the Kutta theorem starts enforcing itself due to which flow starts curling at the trailing edge and the high velocity gradient formed is pushed downwards so that the flow leaves the trailing edge smoothly. Then as the vortex sheet formed due to high velocity and consequently high vorticity is unstable, it curls itself to form what is called the starting vortex.

10 - Question

Generation of lift is accompanied by a starting vortex at the trailing edge. If the flow is inviscid, this will not happen. What reason can best describe this?

a) There is no boundary layer formation, hence no vorticity

b) Kutta Condition is enforced

c) Kelvin’s Theorem is violated

d) Starting Vortex dies off instantly

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aExplanation: For inviscid flows, the boundary layer is not formed. Therefore, in the regions of high velocity, high viscosity is not there and hence no vortex can form. Thus, there is no lift produced. Starting vortex cannot form in inviscid medium and in the viscous medium it dies due to viscosity.

11 - Question

Which of these is a result of Kelvin’s Theorem is essentially?

a) Frozen Vortex Lines

b) Vorticity

c) Circulation

d) Lift

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aExplanation: Kelvin’s theorem can be used to prove Helmholtz theorems, one of which says ‘vortex lines move with the fluid’ which is what is known as “frozen vortex lines”.

12 - Question

In reality, the starting vortex dies out. Why?

a) Lift becomes zero

b) At later times, Kelvin’s theorem is not applicable

c) Due to Viscosity

d) This assumption is wrong. Starting vortex never dies

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cExplanation: Starting vortex cannot form in inviscid medium. It can form only in a viscous medium. In a viscous medium, it dies instantly due to viscous effects.

13 - Question

Generation of lift over an airfoil and formation of starting vortex is correctly explained by which of these?

a) Kutta-Joukowski Theorem

b) Kutta Condition and Kelvin’s Theorem

c) Kutta-Joukowski Theorem and Kelvin’s Theorem

d) Kutta Condition and Helmholtz Theorem

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bExplanation: Kutta condition enforces smooth flow at the trailing edge. In doing so high velocity gradients formed at the trailing edge generates vorticity and hence circulation is there. From Kelvin’s circulation theorem starting vortex is formed to conserve circulation.