We were discussing the basics of drag
force & lift force and drag
and lift coefficient in the subject of fluid mechanics, in our recent
posts. Before going in detail discussion about compressible flow, we must
have basic knowledge about various equations associated with the compressible
flow.
Above equation represent the velocity of sound wave in fluid.
Till now we were discussing the various concepts and
equations such as continuity
equation, Euler
equation, Bernoulli’s
equation and momentum
equation for incompressible fluid flow. In same way we have also
discussed above equations for compressible fluid flow.
We have already seen the derivation of continuity
equation, Bernoulli’s
equation and momentum equation for compressible
fluid flow in our previous posts. We will start here our discussion about the
compressible fluid flow with the derivation of expression for velocity of sound
wave in a fluid.
Expression for velocity of sound wave in a fluid
Compressible flow is basically defined as the flow
where fluid density could be changed during flow.
The disturbance in a solid, liquid or gas will be
transmitted from one point to other. Velocity with which the disturbance will
be transmitted will be dependent over the distance between the molecules of the
medium.
If we will consider the case of solid, molecules
will be very much closely packed and therefore the disturbance will be
transmitted instantaneously.
If we will consider the case of liquid, molecules
will be relatively apart and therefore disturbance will be transmitted from one
molecule to the next molecule. Velocity of disturbance will be less as compared
to the velocity of disturbance in case of solid.
If we will consider the case of gases, molecules
will be relatively apart and disturbance will be transmitted from one molecule
to the next molecule. There will be some distance between two adjacent molecules.
Each molecule will have to travel a certain distance before it can transmit the
disturbance.
Therefore, velocity of disturbance in fluids (liquid
and gas) will be less as compared to the velocity of disturbance in case of
solids.
This disturbance will develop the pressure waves in
fluids. These pressure waves will travel with velocity of sound waves in all the
directions. Let us consider here one dimensional case only.
Following figure, displayed here, indicates the
condition of one-dimensional propagation of the pressure waves. Let us consider
a cylinder of having uniform cross-sectional area attached with a piston as
displayed in following figure.
Let us assume that cylinder is filled with a
compressible fluid and compressible fluid is at rest initially.
If we apply a force through the piston in right
direction, force will develop a pressure as force will be applied uniformly. Due
to the application of force, piston will move by a certain distance let us say
x towards right direction as displayed here in following figure.
Due to the application of this small amount of
force, there will be generation of pressure waves inside the cylinder and
pressure will be applied over the fluid contained inside the cylinder.
In simple, we can say that disturbance will be
created inside the fluid due to the movement of piston by a distance x and this
disturbance will move in the form of pressure wave inside the cylinder with a
velocity of sound wave as discussed above.
Let us consider following terms from above figure as
mentioned here.
x = Distance of piston from initial position
L = Distance of sound wave from initial position
P = Pressure applied over the piston at initial
position
P + dP = Pressure inside the cylinder at final
position
ρ = Density of the fluid at initial position
ρ + d ρ = Density of the fluid at final
position
dt = Small amount of time taken by piston to travel
distance x
V = Velocity of piston
C = Velocity of pressure wave or sound wave
travelling in the fluid
Distance travelled by the piston in time dt from
initial position, x = v.dt
Distance travelled by pressure wave or sound wave in
time dt from initial position, L = C. dt
Let us recall the law of conservation of mass
Initial mass = final mass
As we know that mass will be equal to the product of
density and volume. We can write here the equation as mentioned here.
Mass = Density x Volume
Mass = Density x Area x Length
Mass at initial position, M1 = ρ A L = ρ
A C. dt
Mass at final position, M2 = (ρ + dρ) A (L-x)
= (ρ + dρ) A (C. dt- V. dt)
Mass at final position, M2 = (ρ + dρ) A. dt
(C - V)
Now, considering the conservation of mass, we will
have following equation as mentioned here.
Mass at initial position, M1 = Mass at final
position, M2
ρ A C. dt = (ρ + dρ) A. dt (C - V)
ρ C = (ρ + dρ) (C - V)
ρ C = ρ C - ρ V + C. dρ - V dρ
Here, the term dρ will be very small and velocity of
piston will also be very small and therefore product V dρ could be neglected.
ρ V = C. dρ
C
= ρ V/ dρ
-------------------------------- Eq 1
Let us determine the force at initial position of
piston and final position of piston as mentioned here
F1 = P.A
F2 = (P + dP). A
Change in force, ΔF
= (P + dP). A - P.A = dP. A
As we know that force could be written by recalling
the Newton’s second law of motion and we will have the following equation
Force = Mass x (Rate of change of velocity)
dP. A = (ρ A C. dt) [(V-u)/dt]
As we know that, V is the final velocity of the
piston and u is the initial velocity of the piston and initial velocity of
piston will be zero.
dP. A = (ρ A C. dt) V/dt
dP = ρ C V
C
= dP / (ρ V) --------------------------------
Eq 2
We will have following equation by multiplying the
equation 1 and equation 2
C2 = (ρ V/ dρ) x [dP / (ρ V)] = dP/
C2 = dP/
Further, we will go ahead to find out the expression for velocity of sound in terms of bulk modulus, in the subject of fluid mechanics, with the help of our
next post.
Do you have any suggestions? Please write in comment
box.
Reference:
Fluid mechanics, By R. K. Bansal
Image courtesy: Google
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