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)
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
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 2nd law
of motion
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
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
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