PART II. Pick up the correct answer from the in each part given four alternatives:-

1. The expression in a viscous flow is the average

angular velocity Component in a. X direction b. Y direction c. Z direction d. none of above

2. . The equation of continuity in compressible flow field is given by

a.

b.

c.

d. none of above

3. The velocity distribution in a simple couette flow is given by

a.

b.

c.

d.

4. A stream line in a steady incompressible flow represents a. magnitude of the velocity of the particle b. direction of velocity c .quality of the flow field d. nature of the velocity

5. In a barotropic fluid, density depends upona. Viscosityb. pressurec. kinematic viscosityd.Temperature

6. The Kutta condition in aerodynamics of airfoils and wings determinesa. vorticity in a lifting flowb .angular velocity in a lifting flowc. circulation in a lifting flowd. constancy of circulation in a lifting flow

7. In an inviscid compressible fluid low Cp has maximum value of

a. Cp =1b. Cp > 1c. Cp< 1d. Cp>1

8. The area velocity Mach number relationship in an isentropic flow is

a.

b.

c.

d.

9. Skin friction coefficient in a laminar boundary layer flow over a flat plate at Is given by

a.

b.

c.

d.

10. The maximum C1 that can be generated for a circulatory flow around a circularCylinder is

a.

b.

c.

d.

PART ІІ

II. a. Given a velocity field with components: Determine the translational and rotational velocity of the flow field

b. A 3-D flow is described by the velocity field . Obtain the equation for the streamlines passing through the point (1,0,1)

III. a. The y-component of velocity, for a flow field in x-y plane is given by . .Determine a possible x component for steady incompressible flow and comment on the validity of the solution.

b. A velocity field is given by .Obtain the acceleration vector at x=3, y=1. Also find the components of acceleration parallel and normal to

IV. a. Consider a potential function given as . (with usual

nomenclature in aerodynamics).Interpret each term on RHS and explain the physical situation thus created .Can you obtain lift and drag in this case?

b. An airspeed indicator of an airplane calibrated without taking compressibility effects registers 900 kmph at 10 km altitude (density =0.4127kg/m3). Determine TAS, Ts and

V. a. what is understood by the Kuttta-Joukowaski transformation ? Transform a circle to a symmetrical air foil and hence to circular arc boundary. What do you conclude?

b. A flat plate airfoil 0.25m chord is inclined at .its TE flap of 0.2C is deflected down by .Work out the C1. if the flap card is increased to 0.5C ,comment on the result.

VI. a. Consider flow over a half diamond wedge of included angle , placed in a supersonic stream of M=1.8. Work out the flow field properties and plot the pressure variation along the chord of air foil.

b. A normal shock moves at a constant speed of 500m/s into still air and 7atm. Determine the static and stagnation condition present in air after the passage of the wave.

VII. a. Illustrate with sketches and plots shock polar or representing oblique shock properties. Hence plot and describe the advantages of a dimensionless shock polar.

b. Describe with sketches and aerothermodynamics, choice between 1) a normal shock Diffuser and 2) a conical spike VG diffuser for a supersonic military airplane.

PART ІІІ

VIII. Elaborate the following with sketches and plots:-

a. Mach wave and shock waves are patching lines in supersonic flows. b. Laminar flows are dependent upon Reynolds number while Turbulent flows are weakly dependent upon Reynolds number c. Blowing is a convenient method of BLC as compared with suction. d. In a supersonic flow ,increases in the wall deflection angle beyond results in a detached shock. e. Laminar airfoil have better aerodynamic characterricts than that of NACA 4 digit series airfoils. f. Flow aft of an oblique shock is supersonic where as aft of a normal shock it is subsonic.1 .Explain the following in not more than 10 lines apart from sketches and plots; a. The diverging portion of a supersonic nozzle is always longer than the converging part. b. There is requirement of a second throat in a supersonic wind tunnel. c. Multiple slotted flaps are more efficient than a plain flap. d. Compression is also achieved by turning a supersonic flow.

2 . a. Prove that the vorticity vector is twice the angular velocity in a flow field. b. Define the term circulation in a flow field. The absolute value of velocity and the equation of potential function lines in a 2-D velocity field defined by (0,0), (2,0), (2,1) and (0, 1) in Cartesian system; are given as and . Evaluate and .

3. a. Prove that the stream lines and equipotential lines are orthogonal to each other. Hence plot the network for flow over a circular cylinder and an airfoil. b. State and prove the Kutta- Joukowaski law relating circulation and lift over an arbitrary object in a flow field. c. Demonstrate that the joukowaski transformation is a kind of conformal mapping of stream lines around a circle into an ellise.Bring home your agreement to the above statement.

4. a. Interpret and demonstrate the physical meaning of the equation of thin

airfoil theory given as , where is the vorticity distribution

b. An airspeed indicator of an airplane calibrated without taking compressibility effects registers 750 kmph at 6 km altitude( =0.6584kg/m3).Determine TAS,Ts and .

5. a. Define the term ‘shape factor’ in the boundary layer theory. If the velocity distribution

in laminar boundary layer is given by ,Calculate the shape

factor, where .

b. an approximate solution for a plane surface which suddenly starts moving in its

own plane with a constant velocity is assumed to be , where a

is an arbitrary constant . Calculate the boundary layer thickness .

c. explain the phenomenon of ‘separation’ of boundary layer. Illustrate with sketches for such such flows over an airfoil and that in a diverging channel.

6. Write full Navier-stokes equations in Cartesian co-ordinates. Hence verify for plane poiseuille flow that the shearing stress becomes zero in the plane of symmetry and

that .

7. a .Consider a normal shock and demonstrate that the downstream mach number M2 is

given in terms of the free stream Mach number M1 as

b. Consider flow over a diamond shaped airfoil section of the wedge angle ,kept in a supersonic stream of M= 2.5 at .Sketch the flow field and distribution over it . What are your comments?

8. a. Describe the method of characteristics of designing the diverging portion of a supersonic nozzle with an example of your choice.

b. Explain with sketches and plots a) strong shock, b) weak shock and c)detached shock.

Q1.(a) Define the following .

(i) Streak line, (2) compressibility

(b) State whether the statements are true or false (T/F) and explain why.

(i) In an oblique shock for fixed Mach number, as the deflection angle increases, pressure ratio (p2/p1) increases for weak shock and decreases for strong shock . (ii) In two dimensional, steady, boundary layer flow over a flat plate pressure gradient in the direction perpendicular to the plate decreases monotonically.

(iii) Velocity for a potential vortex is given in (r,) coordinate by Vr =0 and V=(- k/r), where k is constent,therefore stream lines are circular. (iv) Swept wing has a critical Mach number compared to a straight wing of same chord length. (v) A straight duct is fed by a C-D nozzle. Flow in the duct can be considered adiabatic, 1-D frictional flow and the nozzle is chocked . Following curves traced in the mollier diagram will lie on the same line: (a) for back pressure (pb) same as critical pressure (p*) at the duct exist (corresponding length L1), (b) Length L1 and Pb<P* and (c) L< L1.

Q2. Air is flowing at M=2.5 along a plane wall on which there is a two –dimensional circular arc as shown in figure below. Thickness (t) to chord (c) ratio is 0.1. Ignoring wave interactions plot pressure coefficient with length (take x/c=0,0.1,0.5,0.9 and 1.0) using thin air foil theory. Compare the results with shock expansion theory.

Q3. A converging-diverging nozzle is designed to operate with exit Mach number of 1.70 . The nozzle is supplied from an air reservoir at pressure 493608Pa (absolute).Ambient pressure is 100kpa. In an accident the nozzle edge is damaged and one-third portion of its diverging part is removed. Assume area in the diverging part is varying linearly with length. Find exit Mach number and pressure of the jet issuing out of the truncated nozzle. Determine the deflection angle of the jet boundary. Show qualitatively nature of the jet low field.

Q4. Consider a normal shock moving in a tube. At any instant of time ahead of the moving shock fluid (air) is stationary and behind the shock fluid is moving with a constant velocity Ub. The Mach number of the stock with to the sound speed ahead of the shock is 5.0. Undistribution pressure and temperature of air ahead of the shock is 100Pa and 3000K respectively. Find Ub and the Mach number of the flow behind the moving shock. Assume specific heat ratio of air is 1.4.Q5. Water flows at a uniform speed out of a pipe vertically downward on to a flat plate a distance L=10cm below, as shown in the figure below. The volumetric flow rate is Q=100cm3/s, and the pipe area is A=1.0 cm2. The water spreads out smoothly out over

the plate so that at a distance r from the flow axis the radial velocity Vr {r} is uniform, that is independent of z. The flow is axisymmetric so that thickness of the water layer h{r} depend only on r. Derive expression for Vr{r} and h{r}, assuming that h << L, and calculate their values at r=10cm.

Q6. Two thin rectangular plates of dimension L by W, are connected at one end by a closed- end flexible bellows arrangement, as sketched in fig below. The parallel plates are separated by a distance h{t} that is increasing at a constant rate w, ie dh/dt= -w.The fluid between the plates is inviscid and in compressible having a density ρ.The plate width w is much grater than the length L so that we ca assume that the floe is quasiunidirectional and that the velocity V{x,t} is uniform at any crossection x.The pressure at outlet x=L is atmospheric pressure,Pa. {a} Derive an expression for the fluid velocity V{x,t} in terms of the parameters x,h and w {b} Express P{x,t} as a function of ρ,x,L and h{t} {c} Using the momentum theorem, derive an expression for the force F required to keep the plates from moving horizontally.

Q7. A layer of oil of thickness of a and viscosity μo floats on top of a layer of water of thickness b and viscosity μw both layer are contained between two large flat plates, the lower of which is stationary and the upper of which moves at a speed U in the x direction {see figure below}.Derive expressions for (a) The speed Vi of the water-oil interface and (b)The volumetric flow rates of oil and water, Qo/w and Qw/W, per unit distance normal to the direction of the flow .Express your answers in terms of the known parameters a,b,μw,μo and U.

Q8. A plane, inviscid, constant-density flow consists of circular streamlines about an axis normal to the plane of the flow. The re is no radial component of the velocity (Vr=0) and the tangential component V is proportional to the radius r when 0<r<R and inversely proportional to r when

V=Vm(r/R) if 0<r<R; V=Vm(R/r) if .

Where Vm is a constant.

(a) Is the flow for an irrotational flow? (Prove your answer) (b) Is the flow for 0<r <R an irrotational flow? (Prove your answer) (c) At the pressure is . Derive an expression for the pressure pr at r=R in terms of the density ρ, Vm and . (d)Derive an expression for the pressure at r=0 in terms of the density ρ, Vm and

Q1. (a) Define the following (i)Vorticity (ii) Circulation (iii) isentropic flow and (IV) Irrotational flow

(b) Explain why: In a source pressure at radius r (beyond the critical radius ) will always be less Than the pressure at infinity. (c) A circular cylinder of diameter 10 cm rotating at 1000 rpm in an air stream flowing in a direction perpendicular to the axis of the cylinder at 10 cm/s . If air is an ideal fluid. What will be the drag on the cylinder? (d) At the inlet of a diverging channel flow is supersonic. The chance of flow separation is high in such flows.

Q2. A tornado may be modeled as a circulating flow with radial velocity. Vr= Axial

velocity, Vz=0 and azimuthal velocity,

V=ωr for r≤R =ωR2/r for r≥R Where R is the Core radius, ω is constant. (a) Determine whether the flow pattern is irrotational in both the inner (r<R) and the outer (r>R) region.

(b) Using the radial momentum equation, determine the pressure distribution P(r) in the

tornado, assuming as r→ ∞.Find out the location of minimum pressure.

Q3. (a) Consider laminar boundary layer flow of a fluid over flat plate alignment with the X- direction that is the direction of free stream velocity V∞. Assume incompressible flow .Using moment integral theory show that the shears stress τw=ρu2

∞ d/dx. Where ρ is the density of the fluid and is the momentum thickness.

( b) If the velocity profile is assumed to be u/u∞=2 y/ - (y/)2 : 0≤ y ≤ Show that the

skin friction coefficient Cf = 0.73/ Rex ½

Where y is the direction perpendicular to the plane, is the boundary layer t thickness, Re x is the Reynolds number based on x and U ∞.

Q4.

A viscous liquid of constant density ρ and viscosity μ falls due to gravity between two plates a distance 2h apart as shown. The flow is fully developed with the single velocity component U=U (x). There are no applied pressure gradients. Flow is only due to gravity.

(a) Write the governing equations and boundary condition s for finding U(x). (b) Show that the y- component of velocity is zero. ( c ) Find the velocity profile. (d) Sketch qualitatively the velocity and Vorticity profile.

Q5. (a) Sketch the streamline pattern and surface pressure distribution for flow over a diamond shape aerofoil as shown.

(b) Consider one-dimensional flow in a constant area duct with rough wall. The entry flow is supersonic. What happened to the flow Mach number as the flow proceeds from entry to exit. (c) Consider supersonic flow over a bluff body shown in the picture; sketch the streamline pattern. Is the flow Irrotational everywhere? Explain.

Q6. A supersonic air flow at M1=3.2 and p1=50kpa undergoes a compression shock followed by an isentropic expansion turn. The flow deflection is 300 for each turn. Compute Mach number, M2 and pressure, p2 after the expansion wave, if the shock is followed by the expansion.

Q7. Air flows isentropically from a reservoir , where p=300kpa and T=5000K .to section 1 in a duct , where area A1=0.2 m2 and velocity , V1=550m/s. Compute (a) Mach number ,M1,(b) Temperature,T1,(c) pressure ,p1, (d) mass flow rate , m and (e)and area ,A*.Is the flow chocked ?

Q8. Derive the fundamental equation of thin airfoil theory. For flows over a flat plate show that the lift coefficient is proportional to angle of attack.

I. Comment on the following statements and verify its correctness in not more than five sentences with a plot/sketch.

a) A flow dominated by viscous effects is called irrotational flow.b) A cylinder kept in a free stream is subjected to Magnus effect.c) When the Mach number upstream of a normal shock is infinity, the

Mach number downstream of the shock is finite.d) The propagation of sound is an isothermal process.e) In subsonic flow through a converging passage the chances of

boundary layer separation are high.II. (a) Define stream function and circulation.

(b) Show that the combination of a source and sink of equal but opposite strengths situated at a distance 2C apart has the stream function given by

Where m is the strengths of source and sink.

Show that this is the equation of a family of circles of radius given by

and center given by .

III. A pattern in Z plane (z=x+iy) is transformed to a different pattern in ζ plane (ζ = ξ + ¡ y ) by the transformation ζ =1/Z . If the flow pattern of Z plane is that due to uniform parallel flow parallel to OX axis show that the transformed pattern is a family of circles. Also find out the radius and centre of the circle. Sketch both the flow patterns.

IV. A thin aerofoil has a camberline defined by the relation

Where x and y are its X and Y coordinates and c is the chord of the aerofoil.

The origin at the leading edge and the maximum camber is 2% of the chord. determine the coefficient of lift and pitching moment coefficient at 30 incidence assuming it to be two dimensional.

V. (1) Explain what is an expansion fan and derive the expression for Prandtl – Meyer function.

(2 ) Show that the maximum possible flow deflection angle around a centred expansion corner is 130.50 .

VI. (1) Explain what is meant by hodograph and shock polar. Also explain how shock polar can be represented in non- dimensional form and highlight its characteristics and advantages.

(2) Derive the Area- Mach number relation and explain why convergent- divergent nozzle is needed for achieving supersonic flow.

VII. Write a brief note on a) Isentropic relationsb) Supersonic flow through constant area ductsc) Detached shockd) Method of characteristics

VIII. A supersonic inlet is to be designed to handle air at Mach 2.4 with static pressure and temperature of 0.5*105 N/m2 and 280 K, as shown in figure.

(a) Determine the diffuser inlet area Ai if the device is to handle 20 Kg/s of air.(b) The diffuser has to further decelerate the flow behind the normal shock so that the velocity entering the compressor is not to exceed 30 m/s. Assuming isentropic flow behind the normal shock, determine the area Ae required and the static pressure Pe there.

IX. (a) Explain what is meant by displacement thickness and momentum thickness. (b) Discuss briefly various method of boundary layer control. (c) Write a short note on laminar flow airfoils.

X. Explain the following statements in not more than ten sentences plus a plot/sketcha. Which of the two methods of boundary layer

control is more effective and why? BLOWING / SUCTION.b. Which of two flows is Reynolds Number

dependent and why? LAMINAR / TURBULENTc. Which of two designated Mach numbers is higher and why? CRITICAL MACH NUMBER / DRAG DIVERGENCE MACH

NUMBER.d. Which of the two Vortices produces lift force on an airfoil and why? STARTING VORTEX / BOUND VORTEXe. Which of the two methods of describing flow is usually adopted and why? STREAM LINES / EQUIPOTENTIAL LINES

XI. a. Show that in a velocity filed is equal to the vorticity. Hence define an irrotational flow filed.

b. Show that Navier Stokes equations describe a viscous flow. c. A velocity field is described by

u = 2y - y / (x2 + y2) 1/2

u = -2x - x / (x2 + y2) 1/2

Does the flow exist? If so, find the stream function and vorticity. Sketch the stream lines.

XII. a. Obtain Bernoulli equation from Euler equation for a rotational flow field.

b. An incompressible, inviscid flow field is described by where

is constant

i. Sketch streamlines and equipotential lines ii. Comment on the type of flow field.

XIII. A flat plate with 20% chord flap is set at α = 5o and = 5 o.. Determine the chord is

reduced to 10% , calculate change in CL.

XIV. a. An aircraft flies at a pressure altitude of 10500m. A Pitot measures a pressure 38000

N/m2. Determine the Mach number. If the outside temperature is -10o C standard.

Determine TAS and EAS.

b. Define Boundary layer velocity thickness, displacement thickness, momentum

thickness and energy thickness. How can you relate displacements thickness and

momentum thickness.

XV. Cp = 1- 4 Sin2 describe pressure distribution over a stationary circular cylinder usual

notations. Show that lift generated in this case is zero. Hence stat and prove d’

Alembert’s paradox.

b. Transform a circular cylinder in to a symmetric airfoil by an appropriate technique.

Explain the salient features of this procedure along with.

XVI. a. Show that flow after a Normal shock is always subsonic.

b. Consider Mach 2.5 airflow at a pressure of 1 atm. Which is desired to be slowed

sub-sonic speed with minimum loss in total pressure. Use following options,

a. A simple Normal Shock.

b. A oblique shock with = 20o, followed by a normal shock.

c. A n oblique shock with = 20o, followed a second oblique shock with =

followed by a normal shock.

Present your work out with comments.

XVII. a. Describe flow between two parallel plates, one of which is moving and the other is

rest an exact solution of Navier Stokes Equations.

b. Water flows between two parallel plates one of which is fixed and the other moves

velocity U. If the volumetric flow Q per unit width is zero, find the relation between

dp/dx in terms of dynamic viscosity μ, and h the spacing between the plates. Calculate

pressure gradient for U = 0.15 m / sec and h = 10mm.

XVIII. a. Show that Hagen Poiseulle flow is an exact solution of Navier-Stokes Equations.

b. Sketch turbulent boundary layer on a flat plat and explain its flow field.

XIX. a. Explain the following statements in not more than five sentences plus a plot / sketch.

a. Reynolds Number is a important similarity parameter in Aerodynamics.

b. Pressure measurements over an airfoil / wing are presented in no-dimensional

parameter Cp.

c. A favorable pressure gradient delays Boundary layer transition whereas a adverse

pressure gradient promotes it.

d. In iso-energic flows, rotation depends upon rate of change of entropy.

e. The wing of space shuttle has well rounded leading edge.

XX. a. Prove that Curl Grad = 0. What does it mean?

b. Show that Euler equation describes an inviscid flow. Is it the same as potential

flow?

c. A velocity field is described by

u = x +y+z2

v= x-y+z

Does the flow exist? If so, determine and Curl V.

XXI. a. Obtain Bernoulli equation from Euler equation. Hence state restrictions in its us if

any.

b. A incompressible, inviscid flow field is described by , Where

A=1m/s ,x and y are in meters,

i. Sketch streamlines for

ii. Indicate the direction of resultant velocity vector at (0, 0) on the above plot.

iii. Determine the magnitude of flow rate between the streamlines passing

through (2, 2) and (4, 1).

XXII. a. Write a note on Kutta-Joukowaski transformation.

b. A long right circular cylinder of dia a.m is placed in a steady horizontal stream of

velocity u m/sec. and rotates at w rad/sec. Establish an expression in terms of and u

for the ration where q is the free stream dynamic head. Explain the

behaviour of stagnation lines as builds up from zero onwards.

XXIII. A 2-D laminar boundary layer flow along a flat plate is assumed to be

Evaluate the expression for

a. rate of growth of as a function of x.

b. as a function of x.

c. total frictional force on the plate of length L and width b.

XXIV. a. Explain the aerodynamics of the trailing edge flap in effecting Ct on an airfoil.

b. A flat plate of chord 2 units is placed at an angle α in a stream of velocity V.

Determine vorticity distribution over the plate, assuming that the Kutta-Joukowaski

condition of circulation is satisfied.

XXV. a. Show that flow after an oblique an oblique shock is almost always supersonic.

b. Consider a Mach 4 Airflow at a pressure of 1 atm, which is desired to be slowed

down sub-sonic speed with minimum loss in total pressure. Use the following options,

a. A simple Normal shock.

b. An oblique shock with followed by a Normal shock.

c. An Oblique shock with , followed a second oblique shock

with followed by a Normal shock.

Present your work cut with comments.

XXVI. a. Find velocity distribution u(r) in fully developed laminar flow through circular pipe.

b. The velocity along the center line of Hagen-Poiseulle flow in a 0.1m dia pipe is 2

m/sec. If the viscosity of the flow is 0.07 kg/ m-s, and its specific gravity is 0.92,

calculate the volumetric flow in m3 / sec, the shearing stress of the fluid at the pipe

wall and the load friction coefficient Cf.

XXVII. a. Explain the formation of detached shock waves in a super-sonic flow.

b. Write a note on shock wave-Boundary layer interaction.

c. Verify that and of a 2D vortex flow, satisfy the Laplace’s equation.

d. Write a note on turbulent boundary layer on a flat plate.