Aerodynamics - Linearized and Conical Flows Quiz(MCQ)

A)

Enthalpy equation

B)

Momentum equation

C)

Velocity potential equation

D)

Perturbation equation

Correct Answer : Velocity potential equation

A)

Transonic flow is excluded

B)

Subsonic flow is excluded

C)

Hypersonic flow is excluded

D)

Small perturbations are there

Correct Answer : Subsonic flow is excluded

Explanation : In order to obtain the linearized perturbation velocity potential equation, there are few assumptions made. The perturbations u‘, v‘, w‘ are assumed to be small in comparison to the free stream velocity. Apart from this the equation is not applicable for transonic flow with Mach number between 0.8 and 1.2 and for hypersonic flow with Mach number greater than 5.

A)

Is invalid

B)

Becomes 1

C)

Becomes zero

D)

Becomes infinity

Correct Answer : Is invalid

Explanation : The linearized velocity potential equation becomes invalid for flows at higher Mach number with limit tends to infinity. It is also not valid for flows over airfoil at higher angle of attack and high thickness to chard ratio.

A)

Energy equation

B)

Euler’s equation

C)

Momentum equation

D)

Laplace equation

Correct Answer : Laplace equation

A)

Increases

B)

Decreases

C)

Remains same

D)

First increases, then decreases

Correct Answer : Increases

Explanation : The linearized coefficient of pressure for a subsonic flow varies with respect to the Mach number as follows:

According to this proportionality, as the Mach number increases, the coefficient of pressure also increases.

A)

Increases

B)

Decreases

C)

Remains same

D)

First increases, then decreases

Correct Answer : Increases

Explanation : In the supersonic regime, according to the linearized coefficient of pressure is directly proportional to the local inclination and inversely proportional to

Due to this relation, as the Mach number decreases, the coefficient of pressure increases to a point where Mach number reaches transonic regime (M = 1) where the coefficient of pressure becomes infinity.

A)

True

B)

False

C)

Can Not Say

D)

None of the above

Correct Answer : True

Explanation : The linearized pressure distribution is usually inaccurate for higher deflection angles beyond 4 degrees. But when they are integrated to obtain the linearized coefficient of lift and drag, the inaccuracies are approximately compensated when adding the upper and lower surface.

A)

0.01

B)

0.12

C)

0.52

D)

0.85

Correct Answer : 0.12

Explanation : Given, M_{∞} = 2, ∝ = 3 = 0.052 rad

According to the linearized theory, the coefficient of lift over a flat plate is given by:

A)

?_{xx} + ?_{yy} = 0

B)

β^{2}(?_{xx} + ?_{yy}) = 0

C)

β^{2}?_{xx} + ?_{xy} = 0

D)

β^{2}?_{xx} + ?_{yy} = 0

Correct Answer : β^{2}?_{xx} + ?_{yy} = 0

Explaination : For a compressible subsonic flow over a thin airfoil, the two dimensional linearized perturbation velocity potential equation is given by**β ^{2}?_{xx} + ?_{yy} = 0**

In this equation the perturbations are assumed to be small with the value of

A)

Euler's equation

B)

Cauchy's equation

C)

Laplace's equation

D)

Navier – Stokes equation

Correct Answer : Laplace's equation

Explanation : The compressible linearized perturbation velocity potential equation is transformed into incompressible using a transformed coordinate system **(ξ, η)**. The equation is given by**? _{ξξ} + ?_{ηη} = 0**

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

A)

Shape of airfoil in transformed spaces

B)

Coefficient of lift to coefficient of pressure

C)

Coefficient of drag to coefficient of pressure

D)

Incompressible flow to the compressible flow for same airfoil

Correct Answer : Incompressible flow to the compressible flow for same airfoil

Explanation : The Prandtl – Glauert equation is given by:

This equation relates the pressure/lift/drag coefficient in incompressible flow **C _{p0}** to the pressure/lift/drag coefficient in compressible flow (

A)

1

B)

0.8

C)

0.65

D)

0.5

Correct Answer : 0.8

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.

A)

Increases

B)

Decreases

C)

Remains same

D)

First increases, then decreases

Correct Answer : Increases

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.

A)

?xx + ?yy = 0

B)

λ^{2}(?_{xx} + ?_{yy}) = 0

C)

λ^{2}?_{xx} + ?_{yy} = 0

D)

λ^{2}?_{xx} + ?_{xy} = 0

Correct Answer : λ^{2}?_{xx} + ?_{yy} = 0

Explaination : For a supersonic flow over a thin airfoil, the two dimensional linearized perturbation velocity potential equation is given by**λ ^{2}?_{xx} + ?_{yy} = 0**

In this equation the perturbations are assumed to be small with the value of

A)

Linear

B)

Elliptic

C)

Hyperbolic

D)

Parabolic

Correct Answer : Hyperbolic

Explanation : The linearized perturbation velocity potential equation for supersonic flow differs with the subsonic flow in the type of partial differential equation formed. The equation for supersonic flow given as below which is in the form of hyperbolic partial differential equation.**λ ^{2}?_{xx} + ?_{yy} = 0**

A)

Increases

B)

Decreases

C)

Remains same

D)

First increases, then decreases

Correct Answer : Decreases

Explanation : Based on the linearized coefficient of pressure derived for supersonic flow over an airfoil, the coefficient of pressure is inversely proportional to

Thus, with an increase in Mach number, the pressure coefficient decreases.

A)

Large camber

B)

Small angle of attack

C)

Two – dimensional flow

D)

Thin sharp edged airfoil

Correct Answer : Large camber

Explanation : Ackeret developed the linearized supersonic theory in which there were simple assumptions made. The airfoil was assumed to be sharp edged, kept at very small angle of attack having small camber in a two – dimensional supersonic flow.

A)

0.857

B)

0.0987

C)

1.231

D)

1.362

Correct Answer : 0.0987

Explanation : Given, M_{∞} = 3, ∝ = 4 = 0.0698 rad

According to the linearized theory, the coefficient of lift over a flat plate is given by:

A)

Form drag

B)

Wave drag

C)

Skin – friction drag

D)

Pressure drag

Correct Answer : Pressure drag

Explanation : When the flow exceeds critical Mach number, there is a formation of supersonic region which is followed by a shock wave. There is a pressure loss across the shock wave which results in large pressure drag.

A)

The flow around airfoil becomes sonic

B)

The flow around airfoil becomes subsonic

C)

The flow around airfoil becomes supersonic

D)

The flow around airfoil becomes hypersonic

Correct Answer : The flow around airfoil becomes supersonic

Explanation : Critical Mach number is of two kinds – lower and upper critical Mach number. When the flow around the airfoil is at the upper critical Mach number, then the flow around the entire airfoil reaches supersonic speed.

A)

Swept wing

B)

Increase camber

C)

Increasing the thickness of airfoil

D)

Decrease drag – divergence Mach number

Correct Answer : Swept wing

Explanation : There are two ways employed to increase the critical Mach number thereby increasing drag – divergence Mach number. First is to reduce the thickness of the airfoil, and second is to add sweep to the wing.

A)

1.15

B)

1.42

C)

1.53

D)

1.66

Correct Answer : 1.66

Explanation :

A)

Length of the airfoil

B)

Critical Mach number

C)

Freestream Mach number

D)

Chord/thickness ratio of airfoil

Correct Answer : Critical Mach number

Explanation : Coefficient of pressure at minimum pressure point which is present at the upper surface of the airfoil is given by the relation :

In this the value of gamma is constant depending on the medium. The only varying quantity is critical Mach number. Thus, the coefficient of pressure at minimum pressure point is a function of only critical Mach number.

A)

M_{drag – divergence} = M_{crit}

B)

M_{drag – divergence} < M_{crit}

C)

M_{drag – divergence} > M_{crit}

D)

M_{drag – divergence} × M_{crit} = 0

Correct Answer : M_{drag – divergence} > M_{crit}

Explaination : Drag – divergence Mach number is greater than critical Mach number. At this Mach speed, there is a region of local supersonic flow followed by a shock wave. This results in pressure drag eventually causing boundary layer separation.

A)

Axisymmetric flow

B)

One – dimensional flow

C)

Two – dimensional flow

D)

Flow symmetrical about x – z plane

Correct Answer : Axisymmetric flow

Explanation : The conical flow is symmetric about a particular axis hence its flow properties such as pressure, density remain the same in plane which passes through a symmetric line. The flow properties depend only on radius r and the axis hence it is known as axisymmetric flow.

A)

Flow over a wedge

B)

Flow over a cone

C)

Flow over a flat – plate

D)

Flow over a flat plate kept at 90 degrees

Correct Answer : Flow over a cone

Explanation : Since the pressure over the conical surface is not constant unlike the flow over the wedge, the streamlines tend to curve behind the shock wave. The 3 – dimensional nature of conical flow provides the streamlines with an extra space relieving any obstruction from the surface of the body. This is known as 3D relieving effect.

A)

Boeing A – 320

B)

Hot – air balloon

C)

Re-entry shuttle

D)

Flow over flat plate in wind – tunnel

Correct Answer : Re-entry shuttle

Explanation : Any object which has blunt conical nose at the front end travelling at high supersonic speeds such as re-entry vehicle, missiles have formation of shock waves. Thus, studying conical flow is of great significance.

A)

Normal shock

B)

Straight shock

C)

Conical shock

D)

Triangular shock

Correct Answer : Conical shock

Explanation : When a cone having semi vertex angle θc is kept in an incoming supersonic flow, then there is a formation of shock wave. This is an oblique shock wave which has the shape of the cone and the streamlines downstream of the shock are not immediately parallel to the surface.

A)

Shock angle

B)

Intensity of shock

C)

Normal shock properties

D)

Oblique shock properties

Correct Answer : Oblique shock properties

Explanation : Shock polar is graphical representation of all the properties of the oblique shock waves. It is the locus of all the possible velocities behind the shock wave. The downstream velocities are plotted on the y – axis and the upstream velocities are plotted on the x – axis.

A)

Along the axis

B)

Ray from a vertex

C)

Along the conical base

D)

Interior of the conical surface

Correct Answer : Ray from a vertex

Explanation : Flow properties such as the pressure and density remain constant along the ray originating from the vertex of the cone including on the surface of the cone. Although, the flow properties vary from one ray to another.

A)

∇s = 0

B)

∇ × s = 0

C)

(∇s) × s = 0

D)

(∇ × s).s = 0

Correct Answer : ∇s = 0

Explaination : The shock wave in the conical flow is assumed to be straight resulting in same increase in the entropy for all streamlines passing through the shock **(∇s = 0)**. This property implies that the conical flow is irrotational as per Crocco’s equation.

A)

Shock wave is curved

B)

Flow is axisymmetric

C)

Cone is placed at the zero angle of attack

D)

Flow properties along a ray of cone are constant

Correct Answer : Shock wave is curved

Explanation : There are certain assumptions made to determine the conical flow. As per Taylor – Maccoll, the flow is assumed to be axisymmetric about z – axis **∂/∂? = 0**, and the cone is assumed to be at zero angle of attack.

If it was kept at any other angle then there will be 3 – dimensional effects that will be hard to account for. Also, the flow properties along the ray of the cone is assumed to be constant **∂/∂r = 0** and the shock wave is straight.

A)

Simulation

B)

Graphically

C)

Analytically

D)

Numerically

Correct Answer : Numerically

Explanation : The supersonic flow over a cone was first obtained by A. Busemann in the year 1929 when the supersonic flow was not studied or achieved practically. Later in the year 1933, Taylor and Maccoll came up with a numerical solution for the supersonic conical flow. The equation obtained is a ordinary differential equation having no closed – form solution thus seeking a numerical solution.

A)

One

B)

Two

C)

Three

D)

Four

Correct Answer : One

Explanation : Taylor – Maccoll is a one – dimensional equation in which it is dependent on only one unknown. This term is the radial velocity. Thus the radial velocity is a function of angle θ.

V_{r} = f(θ)

A)

One – dimensional

B)

Three – dimensional

C)

Quasi two – dimensional

D)

Quasi three – dimensional

Correct Answer : Quasi two – dimensional

A)

No symmetry

B)

Z – axis

C)

Y – axis

D)

X – axis

Correct Answer : Z – axis

Explanation : The conical flow is obtained by keeping a wedge in a y – z plane which is rotated about z – axis. This results in an axisymmetric flow in which the flow properties remain constant along the ray in a cone and depend only on radius r and the axis.

A)

1

B)

2

C)

3

D)

4

Correct Answer : 2

Explanation : For a conical flow placed in a freestream Mach number M∞ with a particular cone angle θc, there exists two oblique shock waves which is a result of one weak and one strong solution. It is similar to the case of wedge where using θ – β – M graph we obtain two shock waves.

A)

Shock wave becomes detached

B)

Oblique shock formation does not occur

C)

There is formation of normal shock wave

D)

Shock wave is attached to the cone

Correct Answer : Shock wave is attached to the cone

Explanation : For the oblique shock wave to be attached to the vertex of the cone, the half cone angle **θ _{c}** must be less than the maximum cone angle

A)

Between the oblique shock and the sonic line

B)

Between the sonic line and the conical surface

C)

Flow is not supersonic beyond the oblique shock

D)

Between the oblique shock and the conical surface

Correct Answer : Between the oblique shock and the sonic line

Explanation : The incoming flow over a cone is mostly supersonic between the surface of the cone and the oblique shock wave. Except in some cases when the half cone angle is large, flow sometimes becomes subsonic and one of the rays originating the cone’s vertex acts at the sonic line. The flow between the sonic line and the oblique shock thus remains supersonic.

A)

(θ_{max})_{wedge} = (θ_{max})_{cone}

B)

(θ_{max})_{wedge} > (θ_{max})_{cone}

C)

(θ_{max})_{wedge} < (θ_{max})_{cone}

D)

(θ_{max})_{wedge} ≈ (θ_{max})_{cone}

Correct Answer : (θ_{max})_{wedge} > (θ_{max})_{cone}

Explaination : Due to the three-dimensional relieving effect, for a particular freestream Mach number, the maximum cone angle allowed is larger than the maximum wedge angle before the shock wave becomes detached.

A)

θ_{c} = 0

B)

θ_{c} = infinty

C)

θ_{c} = θ_{cmax}

D)

θ_{c} > θ_{cmax}

Correct Answer : θ_{c} > θ_{cmax}

Explaination : For a given freestream Mach number M_{∞}, there exists a maximum cone angle for which if we go beyond that value, the shock wave originating at the cone’s vertex gets detached. Thus, the condition for detached shock wave is θ_{c} > θ_{cmax}.