LIFT COEFFICIENT

Variation of Lift at least depends on

1.       Free-stream velocity (V∞)

2.       Free stream density(r)

3.       Size of an aerodynamic surface. We will use the wing area S to indicate the size

4.       Angle of attack

5.       The shape of the airfoil

6.       Viscosity coefficient µ_∞(because the aerodynamic forces are generated in part from skin friction distribution)

7.       Compressibility of the airflow, compressibility effects are governed by the value of the free-stream number M_∞=V_∞/a_∞ , hence L, D&M depend on the speed of flow also the speed of sound

....1

 Using dimensional analysis

...2

...3

Consider mass m. The exponent of m on the left side is 1. Thus, the exponents of m on the right must add to 1, hence  

...4

Similarly, for time t

...5

And for length l,

...6

Solving equation 4, 5&6 for a,b and d in terms of e and f

...7

...8

...9

Substituting equation 7 and 9

Rearranging equation 10

We know that

 Equation 11 becomes

We will now define a new quantity called as coefficient c_l as

Dynamic pressure is

..15

Equation number 15 is a very important the result, as it tells us  that lift, is directly proportional to dynamic pressure, hence the square root of velocity, it is also directly proportional to wing area and to the lift coefficient  

The lift coefficient is a function of M_∞, Re as reflected in equation 12

Since, M_∞, Re is dimensionless and we assumed initially Z as dimensionless from equation 12 c_l is dimensionless.

MACH NUMBER

Velocity, it tells us at what magnitude and direction a moving object, similarly we need a term which we define us the speed of your aircraft, rocket, missile and all kind of air vehicle, its MACH NUMBER(M). As Mach number is directly related to the speed of sound

SPEED OF SOUND

The speed at which sound waves is travels through the air is called speed of sound. In aerodynamics speed of sound plays a very pivotal role

Speed of sound depends on

1.       Pressure

2.       Temperature

3.       Density

Let us assume that we are in a room with motionless air, density (r), Temperature (T), Pressure (P). Now if some sound is produced let as a firecracker sound. If u are standing in the middle of the room, the sound wave sweeps by you at velocity (a) m/s, ft. /s.

The sound wave is a thin region of disturbance in the air across temperature, pressure, density slightly changes, and this change in pressure activates your eardrums and helps us to hear. 

As the wave is passing in the room, there is a slight change in temperature, density, pressure, airspeed by the amount dT,dr,dp and da respectively. Thus the air behind the way is moving a+da

Let’s us apply the fundamental equation of gas flow, to get an equation for a

From the continuity equation  

A1 and A2 are an area of steam tube and there is a geometric reason for a change in an area

So A1=A2=A

Since drda is very small, hence we can ignore them

Form Euler’s equation

substitute equation 3 into 4 

On a physical basis, the flow through a sound wave involves no heat addition on the effect is negligible. Hence sound wave is isentropic

From isentropic flow

Equation 7 says that the ratio 

is the same constant value at every point

We can write

Substituting the value of c in equation 9 from equation 8

Substitute equation 10 into 6

For a perfect gas p=ρRT

Substituting this result into equation 11

Equation 12 is a very important result in aerodynamics it also tells us, the speed of sound in a perfect gas depends only on the temperature of the gas

The speed of sound leads to another, vital definition of Mach number

With the help of Mach number, we will define 5 different of aerodynamic flow

1.       Subsonic flow is a flow in which M<1

2.       Sonic flow is a flow in which M=1

3.       Supersonic flow is a flow in which M>1

4.       Transonic flow is a flow in which 0.8≤M≤1.2

5.       Hypersonic flow is a flow in which M>5

ANATOMY OF THE AIRPLAN

1 Fuselage: Centre body of the aircraft, it cares the major payloads & weight, commonly used shape of the fuselage is similar to cylindrically shape. It is a semi-monocoque structure, the fuselage is a French word meaning ‘’ spindle’’ shape.

2 Wings: they are the major lift producing components both left and right wings are identical to each other. The volume of the wings is used for storage of main landing gear and fuel tank. There are two controlling surfaces on the wing i.e. flaps and aileron

      Flaps: they are used at the time of climbing and landing (decent), at the time of climbing the flaps are used to increase in the lift.

       Aileron: they are the controlling surfaces of an aircraft, they are used the longitudinal direction(Y-axis) it is also called a rolling moment both the aileron work simultaneously. When we left aileron deflects downwards then simultaneously the right will go up and then aircraft will rotate anticlockwise, and vice and versa.

Vertical Stabilizer: Vertical stabilizer or tail they are used for providing the necessary stability to the aircraft. It has only one controlling surface that is the rudder.

     The rudder is one of the controlling surfaces of vertical stabilizer it is used for changing directional direction (z-axis) it is called a yawing moment. It turns the nose of the aeroplane according to the rudder motion.

Horizontal stabilizer: horizontal stabilizer or tail they are used for providing necessary stability to the aircraft. It has only one controlling surface that is the elevator.

     The elevator is one of the controlling surfaces which is used for changing lateral direction (x-axis) it is called a pitching moment. pitch up means aircraft nose is up and it is going in upwards direction and vice versa for pitch down                   

INTRODUCTION TO AERODYNAMIC

What do u think what is aerodynamics …

Aerodynamics (aero mean something involved to air or atmosphere, dynamic means characterized by constant change) 

As for aerodynamics means something involved with air with  the constant change in its properties so here in the introduction to aerodynamics we look at the different types of flow, how does pressure, density and speed of the air changes the property of the flow, we will also drive the following equations Continuity, Momentum (Euler’s, Bernoulli’s ), Energy equations.

DIFFERENT TYPES OF FLOW

    TYPES OF FLOW ON THE BASES OF PRESSURE

                   I.  COMPRESSIBLE FLOW:  compressible flow is a flow in which we don’t assume density (density₁≠density₂) of the flow to be constant, such type of flow is particularly important at high speed, such as for high-performance subsonic aircraft, all supersonic aircraft vehicles and rocket engine.

                 II.  INCOMPRESSIBLE FLOW:  incompressible flow is the flow in which the density of the flow is assumed to be constant(r₁=r₂), it is only a theoretical type of flow we assume density to be constant to simplify the calculations it is done in the flow in which the actual variation of r is negligibly small, otherwise, such type of doesn’t exist in reality  

TYPES OF FLOW ON THE BASES OF FRICTION

       I.        VISCOUS FLOW: it is a type of flow with friction. Types of viscous flow.

           a)   LAMINAR FLOW: in this type of flow Reynolds number is less than 2100. The streamline lines are smooth and regular and a fluid element moves smoothly along a streamline.

           b)  TURBULENT FLOW: in this type of flow Reynolds number is greater than 4000. The streamlines break up and fluid element moving in a random, irregular, and tortuous fashion

  II. INVISCID FLOW : in this flow velocity of the fluid is 0  in this flow we neglected viscosity

CONTINUITY EQUATION 

We will apply the basic physics principle in flowing gases to get the laws of aerodynamics.

Physics principle: mass can be neither be created nor be destroyed.

To apply this principle on flowing gases we will consider an imaginary circle drawn perpendicular to the flow direction as shown in the figure 1

    figure 1

The cross-section area of the stream tube (streamlines that go through the circumference of the circle, these streamlines form stream tube) may change with from point 1 to 2 as shown in figure 1

As the flow is steady the mass flow will remain constant throughout steam tube i.e. mass at point 1 =mass at point 2 because of the mass flowing through the stream tube is confined by the streamlines of the boundary much as the flow of water through a flexible garden hose is confined by the wall  

Consider all the fluid elements that are momentarily in the plane of A₁. After a lapse of time dt, these same fluid elements all moving a distance , as shown in figure. The elements have a swept out volume A₁ V₁ dt downstream at point 1.

The mass of the gas dm in this volume is equal to the density times of the volume

                                

         

This is the mass of the gas that as a swept through area A1 during time interval dt.        (Definition: the mass flow through an area A is the mass crossing A per unit time)

From equation 1, for area A_{1
Mass flow is} 

                                     

This is the continuity equation for steady fluid flow.

It gives us

1.       the algebraic equation that relates the value of r& V& area at one section of the stream tube to be the same quantities at any other point.

2.       There is a caveat in the equation

      I.   V₁&V₂ are assumed to be uniform over the entire area A₁&A₂ respectively.

      II.   Density1&dnsity₂ are assumed to be uniform over the entire area A₁&A₂ respectively.

     III.   In real life, it is an approximation, in real life V&r vary across cross-section area.

3.       The continuity equation is a workhorse in the calculation of the flow through all type of ducts and tube, such as wind tunnel and rocket moto 

   

figure 2

4.       Consider the streamline of flow over an airfoil, as shown in the figure. The space between two adjacent streamlines, such as the shaded space is a stream tube in figure 2  applies to the stream tube in the figure, is a stream tube. Equation 2 applies to the stream tube in where r₁&V1 are appropriate mean values over A1&r1 are appropriate mean value over A₁&r₂&V₂ are appropriate value over A₂.

   INCOMPRESSIBLE AND COMPRESSIBLE FLOW

1.        COMPRESSIBLE FOW:  compressible flow is a flow in which we don’t assume density (density₁≠density₂) of the flow to be constant, such type of flow is particularly important at high speed, such as for high-performance subsonic aircraft, all supersonic aircraft vehicles and rocket engine.

2.         INCOMPRESSIBLE FLOW:  incompressible flow is the flow in which the density of the flow is assumed to be constant(density₁=density₂), it is only a theoretical type of flow we assume density to be constant to simplify the calculations it is done in the flow in which the actual variation of r is negligibly small, otherwise, such type of doesn’t exist in reality  

                   A₁ V₁ = A₂ V₂

                      V₂= V1 (A₁/A₂)

This explains why all common garden hose nozzle is convergent shape.

This same principle is used in the nozzle for a subsonic wind tunnel. Built for aerodynamics testing.

MOMENTUM EQUATION

The continuity equation says nothing about pressure in the flow

From intuition, that the pressure is an important flow variable, the difference in pressure creates a force that acts on the fluid element and causes them to move.so the relation between pressure and velocity is given by  Euler’s, Bernoulli’s equation

EULER’S EQUATION

We will apply the basic physics law in flowing gases to get the laws of aerodynamics.

We will use newton’s second law of motion

\       Force= mass ×acceleration

                                 F=ma

To apply this physic law inflowing gas consider an infinitesimally small fluid element moving along a streamline with velocity V, as shown in the figure3

figure 3 

At some given instant, the element is located at point P. the element is moving in the x-direction, where the x-axis is oriented parallel to the streamline at point P, Y&Z axis are mutually perpendicular to X-axis

Force on the fluid element is given by

     1.  The pressure acting in a normal direction on all the six faces

     2.  Frictional shear is acting tangentially on all the 6 faces of the element

     3.  Gravity is acting on the mass inside the element  

Here we will ignore the presence of friction forces, the gravity force is generally a small contribution to the total force.

We will assume the only force acting on the fluid element is pressure.

How to calculate the pressure acting on a fluid element

      a)  Dimensions of the fluid element are dx dy dz  

      b )  Consider the faces which are perpendicular to x-axis (in x-direction because of the pressure on  the faces parallel to the streamline doesn’t affect the motion of the element along the streamline.)

     c)   Pressure on the left face is P

The area on the left face is dy dz

       Force is p(dydz)                                          it is in a positive X direction

     d)   We know that pressure varies from point to point in the flow, \ there is a change in pressure per          unit length, symbolized by the derivative dp/dx

     e)   If we move away from the left face by a distance dx along the x-axis , the change in pressure is           (dp/dx)dx

     f)      Pressure on the right face is p+(dp/dx)d
             The area on the right face is  dy d
            Force on the right face is [p+(dp/dx)dx](dydz)                    is in negative x direction

From newton’s 2^nd law of motion

      F=ma
combine equation 3,4&5

 The above equation is Euler’s equation

Point to be taken into consideration when dealing with Euler’s equation

1.  It is a rate of change of momentum   

2.  We neglected friction and gravity

3.  It can also be said as a momentum equation of inviscid flow.

4.  Assumed to be steady, invariant w.r.t to time

5.  The equation is valid for both compressible and incompressible flow

6.   It gives us change in pressure dp to change in velocity

7.   A differential equation, it describes the phenomena in an infinitesimally small neighbourhood around the given point

To obtain Bernoulli’s equation

We will take 2 points, far removed from each other in the flow but on the same streamline p&v at point 1&2 as shown in figure 4

figure 4

We will consider the case of incompressible flow.

These are the following points about Bernoulli’s equation

1. This equation hold for inviscid & incompressible flow

2.  Properties between different points along a streamline

ISENTROPIC FLOW

 There is neither heat exchange nor any defect due to friction (adiabatic process, reversible)

By using thermodynamics law’s and the appropriate calculation we get

Derivation of this equation will be there in a future blog