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Chapter:
1. Find expression for pressure and friction drag. What do you understand by a streamlined body? Give some examples of streamlined body. [2073-Chaitra ][5]
2. Given: Military aircraft with M=8000 kg, lands on 350 km/hr and is slowed by a parachute with `A=10 m^2` area. Find
- Estimate time needed to slow to 200 km/hr.
- Maximum deceleration rate.
Model chute as an open hemisphere. [2070-Chaitra ][8]
3. Derive an expression for forces acting on a submerged body.
A body wholly immersed in a real fluid may be subjected to the following two kinds of forces due to the relative motion between the body and fluid. These are:
Drag Force
Lift Force
Drag Force:
The component of force in the direction of flow (free stream) acting on a submerged body which tends to resist the motion is called drag force.Drag force is equal to Thrust force at steady condition of aircraft(When aircraft travelling with constant speed).
Lift Force:
The component of force at right angles to the direction of flow is called lift force.Lift is the force on the aircraft which is equal to weight force of the aircraft in steady condition. Lift is in vertically upward direction of the aircraft. This force keeps aircraft in the air.
Differences between drag and lift force:
S.NO | Drag Force | Lift Force |
1 | The component of force in the direction of flow (free stream) acting on a submerged body which tends to resist the motion is called drag force | The component of force at right angles to the direction of flow is called lift force. |
2 | It is possible to create drag without lift. | It is impossible to create drag without lift. |
3 | The drag force always exists. | The production of lift force requires assymmetry of flow. |
4 | Drag force is equal to Thrust force at steady condition of aircraft(When aircraft travelling with constant speed). | Lift force on the aircraft is equal to weight force of the aircraft in steady condition. |
Consider an arbitrary shaped solid body placed in a real fluid, which is flowing with a uniform velocity `U` in the horizontal direction as shown in figure. Let us consider a small elemental area `dA` on the surface of the body. Now, the forces acting on the small surface `dA` are:
Pressure force equal to `P*dA` acting perpendicular to the surface.
Shear force equal to `tau_0*dA` acting along the tangential direction to the surface.
Let `theta` be the inclination of the tangent to the element with the direction of flow.
Drag Force:
The component of force due to P and `tau_x` along the direction of motion is known as drag force.
Drag force on the elemental area= force due to pressure in the direction of flow + flow due to shear stress in teh direction of fluid motion.
`dF_d=P*dA*sintheta + tau_0dA*costheta`
Total drag force is,
`F_D=intP*dA*sintheta+inttau_0*dA*costheta`
Lift Force:
The component of force perpendicular to the direction of flow is called lift force.
Lift force on the elemental area= force due to pressure in the direction perpendicular to the direction of motion + flow due to shear stress in the direction perpendicular to the direction of motion
`dF_L=tau_0*dA*sintheta - P*dA*costheta`
Now, the total lift is,
`F_L=inttau_0*dA*sintheta-intP*dA*costheta`
The drag and lift for a body moving in a fluid of density `rho` at uniform velocity `U` are calculated as:
`F_D=C_D*(rho*U^2)/2*A`
and `F_L=C_L*(rho*U^2)/2*A`
Where, `C_D=` coefficient of drag
`C_L=` coefficient of life
`A=` characteristic area
For calculating drag force `F_D`, the area A is taken as area projected on the plane perpendicular to the relative motion of fluid.
For calculating the lift force `F_L`, the area A is taken as area projected on a plane at right angle to the direction of lift force.
In case of air foil, the projection is conventionally taken on the plane of the chord i.e, the area of wings itself, independent of its inclination to the direction of the flow.
area= span `l`* mean chord `c `
4. Aerofoil
Aerofoil (Airfoil):
An airfoil or aerofoil is a streamlined body or a a structure with curved surfaces designed to give the most favourable ratio of lift to drag in flight, used as the basic form of the wings, fins, and tailplanes of most aircraft.
Some terms related to airfoil are:
Aerofoil Section
Chord Line
Camber line
Angle of attack
Stall
Aspect ratio
Aerofoil Section:
The shape of a wing section is called aerofoil section.
Chord Line:
The line joining the leading and trailing edge of an airfoil is called chord and its length is called chord length.
Camber line:
The line connecting the mid-points of the upper and lower faces of the aerofoil section is called Camber line.
Angle of Attack:
It is the angle between the chord line and direction of the fluid stream.
Stall:
An airfoil is said to be in stall condition when the angle of attack of an airfoil is greater than the angle of attack at the minimum lift.
At stall, the air separates from the airfoil or wings and eddies are formed as a consequence of which there is a considerable increase in the drag coefficient.
Aspect Ratio:
The ratio of span of wing to its mean chord is called the aspect ratio.
I.e, A.R=`L/c`
Where, `L=`span of the wing
`C=` mean chord.
Since, `A=L*C`
`C=A/L`
Thus, A.R`=L^2/A`
We know the lift force is,
`F_l=rho*L*U*gamma`
Where, `gamma=` circulation developed on airfoil
`gamma=pi c*U*sinalpha`
where,`c=` chord length
`alpha=`angle of attack
`U=` free stream velocity of an air foil.
Thus, `F_l=rhoL*U*picUsinalpha`
`F_l=pi rho cLU^2sinalpha`...(i)
The lift force is also given by,
`F_l=C_L*(rho*U^2)/2*A`
`F_l=C_L*(rho*U^2)/2*C*L`..(ii)
From (i) and (ii),
`C_L=2pisinalpha`
NOTE: when a flying object such as aeroplane is in steady state, then
Weight of aeroplane = lift force,
`W=C_l*(rho*U^2)/2*A`
Thrust developed by the engine = the drag force
5. Water flows over 0.3 m long nad 0.1 m wide flat plate at 15 m/s parallel to it. Calculate
- drag force on that portion of plate over which boundary layer is laminar.
- total drag force on both sides of plate.
Take `rho=998` kg/`m^3` and `mu= 10^(-6)` `m^2`/s. [2074-Chaitra ][8]
6. Define aerofoil with accepted terminology with neat sketch. A kite which may be assumed to be a flat plate and mass 1 kg soars at an angle to horizontal. The tension in string holding kite is 60N when wind velocity is 50 km/hr horizontally and angle of string to horizontal direction is `35^0`. The density of air is 1.2 kg/`m^3`. Calculate drag coefficient for kite in given position if lift coefficient in same position is 0.45. Both coefficients have been used in full area of kite. [ 2068-Shrawan ][8]
7. Explain drag of a Sphere.
Let us consider a case when real fluid flows past a sphere. Let D be the diameter of the sphere, v be the velocity of flow pf fluid of mass density `rho` and viscosity `mu`.
We have,
`R_e=(rho*U*D)/mu`
When velocity of flow is very small or the fluid is very viscous such that the Reynolds number is very small (`R_e<=0.2`), then the viscous forces are more predominant than the inertial force. Stokes analysed theoretically the flow around sphere under `R_e<0.2` and found that the total drag force is,
`F_D=3pimuDU`....(i)
He also found that out of total drag, two third is contributed by skin friction and one third by pressure difference.
Thus,
Skin friction drag`=2/3F_D=2pimuDU`
Pressure drag `=1/2F_d=pimuDU`
Again, the total drag is,
`F_D=C_D*(rho*U^2)/2*A`
Where `A` is the projected area of sphere.
`A=(pi*D^2)/4`
Thus,
`C_D*(rho*U^2)/2*A=3pimuDU`
`C_D=(24mu)/(rhoUD)=24/R_e`
For `R_e` between 0.2 and 5
`C_D=24/R_e(1+3/(16R_e))` this is called onseen formula.
For `R_e` between 5 and 1000, `C_D=0.4`
For `R_e` between 1000 and 100,000, `C_D=0.5`.
For `R_e>10^5`, `C_D=0.2`.
8. A 2000 kg aircraft is designed to carry a payload of 5000N when cruising at 300 km/hr. The effective wing area is `25 m^2`. Assuming a conventional airfoil, calculate takeoff speed if angle of attack of `10^0` is desired and stall speed when landing. [2071-Chaitra ][8]
9. A 3mm diameter sphere made of steel (sp.wt 75 KN/`m^3`) falls in glycerine (sp.wt 12.5 KN/`m^3`) of viscosity 0.893 NS/`m^2` at a terminal velocity. Determine terminal velocity and drag force on sphere. [2075-Chaitra ][4]
10. A kite has an effective area of `0.6 m^2` and mass 0.4 kg. It experience a drag of 15 N in a wind speed of 40 km/hr. Determine
- tension in chord if it makes an angle of `45^0` with horizontal.
- lift coefficient for kite
Consider density of air as 1.2 kg/`m^3`. [2070-Ashad ][8]
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