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# MCQs on Linearized Subsonic Flow

What is the surface boundary condition for a thin airfoil at a subsonic flow? (Where shape of the airfoil is represented as y = f(x))

a) ∂ϕ∂x = V∞ dfdx

b) ∂ϕ∂y=dfdy

c) ∂ϕ∂x = – V2∞dfdx

d) ∂ϕ∂x=dV∞dx

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a

Explanation: For an airfoil with x – component of velocity as V∞ + u‘ and y – component of the velocity as v‘, the surface boundary condition is

dfdx=v‘V∞+u‘ = tanθ

Since it is a thin airfoil, the perturbation vector u‘ is very small in comparison to the freestream velocity V∞, resulting in dfdx=v‘V∞ = θ (Where tanθ ~ θ for small angles). Expressing the perturbation v‘ in terms of velocity potential we get

v‘ = ∂ϕ∂x

Substituting this in the above equation:

dfdx=∂ϕ∂xV∞ = θ

∂ϕ∂x = V∞ dfdx

Which of these is the linearized perturbation velocity potential equation over a thin airfoil in a subsonic compressible flow?

a) β2(ϕxx + ϕyy) = 0

b) ϕxx + ϕyy = 0

c) β2ϕxx + ϕyy = 0

d) β2ϕxx + ϕxy = 0

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c

Explanation: For a compressible subsonic flow over a thin airfoil, the two dimensional linearized perturbation velocity potential equation is given by

β2ϕxx + ϕyy = 0

In this equation the perturbations are assumed to be small with the value of β = 1–M2∞−−−−−−√.

Which of the equations governs the linearized incompressible flow over an airfoil at subsonic velocity using transformed coordinate system?

a) Laplace’s equation

b) Euler’s equation

c) Navier – Stokes equation

d) Cauchy’s equation

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a

Explanation: The compressible linearized perturbation velocity potential equation is transformed into incompressible using a transformed coordinate system (ξ, η). The equation is given by

ϕξξ + ϕηη = 0

This is the Laplace equation representing the incompressible flow in a linearized form

The shape of the airfoil in both (x, y) and transformed (ξ, η) space are different.

a) True

b) False

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b

Explanation: The shape of the airfoil in (x, y) space is given by y = f(x) and in (ξ, η) space is given by η = q(ξ). Since dfdx=dqdξ hence the shape of the airfoil in both the spaces irrespective of the transformation remains same.

What does the Prandtl – Glauert rule relate?

a) Shape of airfoil in transformed spaces

b) Incompressible flow to the compressible flow for same airfoil

c) Coefficient of lift to coefficient of pressure

d) Coefficient of drag to coefficient of pressure

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b

Explanation: The Prandtl – Glauert equation is given by:

Cp = Cp01–M2∞√, Cl = Cl01–M2∞√, Cd = Cd01–M2∞√

This equation relates the pressure/lift/drag coefficient in incompressible flow Cp0 to the pressure/lift/drag coefficient in compressible flow (Cp) for a two – dimensional airfoil with the same profile.

Linearized theory is applicable for transonic regions as well.

a) True

b) False

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b

Explanation: According to the Prandtl – Glauert rule, as the limit of Mach number is increased to one, the aerodynamic forces – lift and drag becomes infinity which is practically impossible. Thus, this rule is only applicable for subsonic and supersonic regimes.

Cl = Cl01–M2∞√, Cd = Cd01–M2∞√

For a subsonic flow, how does the coefficient of pressure vary with increasing Mach number?

a) Increases

b) Decreases

c) Remains same

d) First increases, then decreases

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a

Explanation: For a subsonic flow, the linearized coefficient of pressure is given by the equation below according to which when the Mach number is increased, the coefficient of pressure increases. Although, one thing to note is that as this Mach number is increased to unity, the coefficient of pressure reaches infinity and thus, for transonic regions, this equation fails.

Cp ∝ 11–M2∞√

Up to which Mach number is Prandtl – Glauert rule applicable for subsonic flow?

a) 1

b) 0.5

c) 0.8

d) 0.65

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c

Explanation: For an increasing Mach number in a subsonic flow over a body, the coefficient of pressure also increases as a result of Prandtl – Glauert rule. But, after Mach number 0.8 the equation fails because the flow enters transonic regime where coefficient of pressure tends to infinity as Mach number tends to unity.