BASICS(AERODYNAMICS)

Section 1:

        Fundamental dimensions and units
NOTE:-according to this blog we are not getting exponential terms so consider our error and plz understand  if it is m2=m square,m3=m cube.

1.1 The greek alphabets:- The Greek alphabet is used extensively in

Europe and the United States to denote

engineering quantities (see Table 1.1). Each

letter can have various meanings, depending on

the context in which it is used.

( Table is provided)

1.2 Conversions

Units often need to be converted. The least confusing way to do this is by expressing equality:

For example, to convert 600 lb thrust to kilograms (kg)

   Using 1 kg = 2.205 lb

Hence 600 lb = 272.1 kg

Setting out calculations in this way can help avoid confusion, particularly when they involve large numbers and/or several sequential stages

of conversion.

1.3 The Physical Properties of Air:-                             Air is a compressible fluid. It can therefore, flow or change its shape when subjected to very small outside

forces because there is little cohesion of the molecules. If there was no cohesion between the molecules

and therefore no internal friction then it would be an ‘ideal’ fluid, but unfortunately such is not the case.

1.3.1 Fluid Pressure

The pressure in a fluid at any point is the same in all directions. Any body, irrespective of shape or

position, when immersed in a stationary fluid is subject to the fluid pressure applied at right angles to the

surface of that body at that point.

1.3.2 Static Pressure

The pressure of a stationary column of air at a particular altitude is that which results from the mass of

air in the column above that altitude and acts in all directions at that point. The static pressure decreases

with increased altitude as shown in Table 1.3. The abbreviation for the static pressure at any altitude is P.

1.3.3 Dynamic Pressure

Air in motion has energy because it possesses density (mass per unit volume) that exerts pressure on any

object in its path. This is dynamic pressure, which is signified by the notation (q) and is proportional

to the air density and the square of the speed of the air. A body moving through the air has a similar

force exerted on it that is proportional to the rate of movement of the body. The energy due to this

movement is kinetic energy (KE), which is equal to half the product of the mass and the square of the speed.              

Bernoulli’s equation for incompressible airflow states that the kinetic energy of one cubic metre

of air travelling at a given speed can be calculated from the following formula:

KE = 1/2ρV2 joules

Where ρ is the air density in kg per m3 and V is the airspeed in metres per second.

Note: 1. A joule is the work done when the point of application of a force of one Newton is displaced

by one metre in the direction of the force.

Note: 2. A Newton is that force that when applied to a mass of one kg produces an acceleration of

1 metre per second per second.

If a volume of air is trapped and brought to rest in an open ended tube the total energy remains

constant. If such is the case then KE becomes pressure energy (PE), which for practical purposes is equal

to 1/2ρ(Vsquare) Newtons per (msqr). If the area of the tube is S square metres then:

S (dynamic + static pressure) = 1/2ρV2 Newtons

 1.4 Viscosity

Viscosity is a measure of the degree to which a fluid resists flow under an applied force. A fluid or liquid that is highly viscous flows less readily than a fluid or liquid that has low viscosity. The internal friction of the gas or liquid determines its ability to flow or its fluidity. Viscosity for air is the resistance of one layer of air to the movement over a neighbouring layer .Dynamic viscosity (μ) is measured in the SI system, in N s/m2 or pascal seconds (Pa s). Kinematic viscosity (v) is a function of dynamic viscosity.

Kinematic viscosity = dynamic viscosity/density.

1.5 The Equation of Continuity

Mass cannot be created or destroyed. Air mass flow is steady and continual. The equation related will be given later

1.6 Reynolds Number

Reynolds Number (Re) = Inertial Forces/Viscous Forces

ie.,Re= ρVL/μ

where ,

ρ is density in kg per m3;

 V is the velocity in metres per second; L is the chord length and

μ is the viscosity of the fluid.

A small Reynolds number is one in which the viscous force is predominant and indicates a steady

flow and smooth fluid motion(laminar flow). A large Reynolds number is one in which the inertial force is paramount

and indicates random eddies and turbulent flow.

1.7 Perfect gas

A perfect (or ‘ideal’) gas is one which follows Boyle’s/Charles’ law

 pv = RT

where:

p = pressure of the gas

v = specific volume

T = absolute temperature

R = the universal gas constant

1.8 Compressibility

The extent to which a fluid can be compressed in

volume is expressed using the compressibility

coefficient B =dv/v  =1

                        dp      k

where dv = change in volume

v = initial volume

dp = change in pressure

K = bulk modulus

SECTION 2

Aeronautical definitions

2.1 Forces and moments

Forces and moments play an important part in

the science of aeronautics. The basic definitions are:

Weight force (W):

Weight of aircraft acting vertically downwards.

Aerodynamic force:

         Force exerted (on an aircraft) by virtue of

the diversion of an airstream from its original

path. It is divided into three components:

lift, drag and lateral.

Lift force (L)

Force component perpendicularly ‘upwards’ to the flight direction.

Drag force (D)

Force component in the opposite direction to flight. Total drag  is subdivided into pressure drag and surface friction drag.

2.2 Aerofoil Profile

The definitions used with reference to an aerofoil’s shape are shown in Figure  and are as follows:

a)Camber. The curvature of the profile view of an aerofoil is its camber. The amount of camber and

its distribution along the chord length is dependent on the performance requirements of the aerofoil.

Generally, low-speed aerofoils require a greater amount of camber than high-speed aerofoils. See Figure

b) Chordline. The straight line joining the centre of curvature of the leading-edge radius and the trailing

edge of an aerofoil is the chordline. See Figure 2.1(a).

c)Chord. The distance between the leading edge and the trailing edge of an aerofoil measured along

the chordline is the chord.

d) Fineness Ratio. The fineness ratio is the ratio of the length of a streamlined body to its maximum

width or diameter. A low fineness ratio has a short and fat shape, whereas a high fineness ratio describes a long thin object.

e)Leading Edge Radius. The radius of a circle, centred on a line tangential to the curve of the leading

edge of an aerofoil, and joining the curvatures of the upper and lower surfaces of the aerofoil is the leading-edge radius or nose radius of an aerofoil.

f) Maximum Thickness. The maximum depth between the upper and lower surfaces of an aerofoil is its maximum thickness.

g) Mean Aerodynamic Chord. The chordline that passes through the geometric centre of the plan area (the centroid) of an aerofoil is the Mean Aerodynamic Chord (MAC) and for any planform is equal to the chord of a rectangular wing having the same wing span and the same pitching moment and lift.

It is usually located at approximately 33% of the semi-span from the wing root. The MAC is often used to reference the location of the centre of gravity particularly with American-built aeroplanes. It is the primary reference for longitudinal stability.

h) Mean Camber Line. The line joining points that are equidistant from the upper and lower surfaces of an aerofoil is the mean camber line. Maximum camber occurs at the point where there is the greatest difference between the mean camber line and the chordline. When the mean camber line is

above the chordline the aerofoil has positive camber.

i)Mean Geometric Chord. This is the average chord length of an aerofoil is the mean geometric chord or alternatively it is defined as the wing area divided by the wingspan.

j) Quarter Chordline. A line joining the quarter chord points along the length of a wing is the quarter chordline, i.e. a line joining all of the points 25% of the chord measured from the leading edge of the aerofoil.

k) Root Chord. The line joining the leading edge and trailing edge of a wing at the centreline of the wing is the root chord

l)Thickness/Chord Ratio. The ratio of the maximum thickness of an aerofoil to the chord length expressed as a percentage is the thickness/chord ratio. This is usually between 10% and 12%.

m) Washout. A reduction in the angle of incidence of an aerofoil from the wing root to the wing tip is the washout. This is also known as the geometric twist of an aerofoil.

2.3 Aerofoil Attitude

The definitions used with reference to an aerofoil’s attitude shown in Figure 2.2 are as follows:

a. Angle of Attack. The angle subtended between the chordline of an aerofoil and the oncoming airflow is the angle of attack.

b. Angle of Incidence. The angle subtended between the chordline and the longitudinal axis is the angle of incidence. It is usual for the chordline of a commercial transport aeroplane to be at an angle of incidence of 4◦ when compared with the longitudinal axis because in level flight, at relatively high speeds, it produces the greatest amount of lift for the smallest drag penalty. If such is the case, the angle of attack and the pitch angle will be 4◦ apart, no matter what the attitude of the aeroplane.

c. Critical Angle of Attack. The angle of attack at which the maximum lift is produced is the critical angle of attack often referred to as the stalling angle of attack. A fixed-wing aeroplane is stalled at or above the critical angle of attack.

d. Climb or Descent Angle. The angle subtended between the flight path of an aeroplane and the horizontal plane is the climb angle or descent angle.

e. Longitudinal Dihedral. The difference between the angle of incidence of the mainplane and the tailplane is the longitudinal dihedral. See Figure 2.2(b).

f. Pitch Angle. The angle subtended between the longitudinal axis and the horizontal plane is the pitch angle.

g. Pitching Moment. Certain combinations of the forces acting on an aeroplane cause it to change its pitch angle. The length of the arm multiplied by the pitching force is referred to as a pitching moment and is counteracted by the use of the tailplane. A pitching moment is positive if it moves the aircraft nose-upward in flight and negative if it moves it downward.

2.4 Wing Shape

The definitions used with reference to the shape of an aeroplane’s wing are shown in Figure 2.3 and are as follows:

a. Aspect Ratio. The ratio of the span of an aerofoil to the mean geometric chord is the aspect ratio,

which is sometimes expressed as the square of the span divided by the wing area.

Aspect Ratio = span

                          chord

           = span²    

             wing area    

b. Mean Aerodynamic Chord. The chordline that passes through the geometric centre of the plan area (the centroid) of an aerofoil is the Mean Aerodynamic Chord (MAC).

c. Quarter Chordline. The line joining all points at 25% of the chord length from the leading edge of an aerofoil is the quarter chordline.

d. Root Chord. The straight line joining the centres of curvature of the leading and trailing edges of an aerofoil at the root is the root chord.

e. Sweep Angle. The angle subtended between the leading edge of an aerofoil and the lateral axis of an aeroplane is the sweep angle.

f. Taper Ratio. The ratio of the length of the tip chord expressed as a percentage of the length of the root chord is the taper ratio.

g. Tapered Wing. Any wing on which the root chord is longer than the tip chord is a tapered wing.

h. Tip Chord. The length of the wing chord at the wing tip is the tip chord.

i. Wing Area. The surface area of the planform of a wing is the wing area.

j. Swept Wing. Any wing of which the quarter chordline is not parallel to the lateral axis is a swept wing.

k. Wing Centroid. The geometric centre on a wing plan area is the wing centroid.

l. Mean Geometric Chord. The average chord length of an aerofoil is the mean geometric chord, or alternatively it is defined as the wing area divided by the wingspan.

m. Wingspan. The shortest distance measured between the wing tips of an aeroplane is its wingspan.

2.5 WING LOADING

The angle of inclination subtended between the front view of a wing 25% chordline (the wing plane) and the lateral horizontal axis is referred to as:

a. dihedral if the inclination is upward.

b. anhedral if the inclination is downward.

2.5.1 Wing Loading

The forces acting on a wing are defined below:

a. Centre of Pressure (CP). The point on the chordline through which the resultant of all of the aerodynamic forces acting on the wing is considered to act is the centre of pressure.

b. Coefficient. A numerical measure of a physical property that is constant for a system under specified conditions is a coefficient, e.g. Cl is the coefficient of lift.

c. Load Factor (n). The total lift of an aerofoil divided by the total mass is the load factor. i.e.

     n = total lift

              mass

d. Wing Loading. The mass per unit area of a wing is the wing loading. i.e.

Wing Loading = mass

                          wing area

             = M  Kg

                S    m²

2.6 Types of flow;-

   

1)Steady and unsteady flow:

We will consider the flow around a body such as a sphere, a cylinder, or an aerofoil. In a steady flow, there are no fluctuations and the properties in a point fixed with respect to the body do not change with time. when the flow properties

in a point moving along with the body do change in time, we have an unsteady flow.

2)compressible flow:

In a flow around any orbitary body the property density of the fluid is considered and is no longer a constant is called a compressible flow 

    ρ is not constant

3)incompressible flow:

In a flow around any body if density of the fluid is constant the flow is said to be incompressible flow.

     d ρ=0 , ρ=constant

4)Inviscid flow

Some theoretical approaches of aerodynamics ignore the viscosity of air, which gives an inviscid or ideal flow. In such a flow model there is no internal friction and inviscid flow is also characterized as frictionless flow.

5)Streamlines

We will examine the two-dimensional flow around an aircraft component, in this case an aerofoil section, that is, a cross section in a plane parallel

to the flow direction (Figure 3.1). The path of an air element in the flow is called a streamline.1 The flow element has a velocity V at the point A far upstream, where the other flow properties are p∞, ρ∞ and T∞.When the element approaches the aerofoil these properties change so that, for instance, at point B the velocity is greater than V and has a different direction. At each point of the element’s path, the velocity vector is tangential to the streamline and it follows that through every point in the flow field there passes one and only one – streamline. There is one exceptional case: streamlines may be intersecting at a point in the flow where the local velocity is zero.

  Such a point is called a stagnation point.

                                     

             2.6 f :Streamlines over an aerofoil

2.7  Equations for steady flow

2.7.1 Continuity equation

Since there is no mass transport through the wall of a stream tube, the air mass per unit of time entering a stream tube or a stream filament is equal to the air mass exiting in the same period of time, provided the flow is steady.

This continuity law is based on the principle of mass conservation.Figure 2.7(a)shows a stream filament with cross sectional areas A₁ and A₂ normal to the local direction of flow. Through A₁ a volume A₁  v₁ dt enters each unit of time dt ,

with a mass equal to dm = ρ₁(A₁ v₁ dt) = (ρ₂A₂v₂ )dt

or  ρAv = constant.

This (algebraic) continuity equation only holds for steady flows and makes up a simple relation connecting the local density and velocity to the cross section of a stream filament. For incompressible flows, the density is constant and the equation simplifies to

A₁ v₁ = A₂ v₂ or Av = constant.

2.7.2 Euler’s equation

In an incompressible flow The resultant of all external forces on a body is equal to the product of the body mass and its acceleration.

             dp + ρvdv =0 or

dp/dv       = −ρv .

This differential equation holds for both compressible and incompressible inviscid flows and is known as                                              

 Euler’s equation

2.7.3 Bernoulli’s equation

P₁+ ½ ρv₁²+ ρgh₁ = p₂ + ½ ρv₂²+ ρgh2.

Because P₁ and P₂ were chosen arbitrarily we can say that along a streamline

                p + ½ ρv² + ρgh is constant.

 This theorem is known as Bernoulli’s equation,When

we write this equation per unit of mass,

p/ρ+ ½ v²+ gh = constant.

In the flow around an aircraft the vertical displacement of elements is mostly small enough to neglect the change in potential energy relative to the other terms. In this case, Bernoulli’s equation simplifies to

p + ½ ρv² = constant. (as.,h₁ =h₂)

The term ½ ρv² is called the dynamic pressure q. This is equivalent to the kinetic energy imparted per second to a unit volume of air of density ρ when it is accelerated to velocity v. In an inviscid incompressible flow, the sum of

the static and dynamic pressure is a constant referred to as the total pressure,

p_t=p + q = p + ½ ρv².

A special case is a stagnation point, where the velocity and the dynamic pressure are zero. The stagnation pressure ps is therefore equal to the total pressure of the undisturbed flow,

P_s = P_t = P∞ + ½ ρv².

Momentum equation

Newton’s second law of motion for a solid body with a given mass can be

written as

F = m (dV)

            dt            = d(mV )

                                 dt

where mV is the body’s momentum.

Fig1:Element in streamtube  fig2:Control  volume over an aerofoil

ref:-aeronautical engineers data book. by Clifford Matthews BSc, CEng, MBA, FIMechE.
      FAA pilots handbook,by FAA usa.