|
Incompressible Fluid Mechanics - Glossary (Under Construction) |
Bernoulli Equation
where p is pressure, rho is density and V is flow velocity
This equation defines the relation between pressure and velocity, and is applicable for an inviscid, incompressible fluid, with no body forces.
For rotational flow, this equation applies only along a streamline , but for irrotational flows the equation is applicable between any two points in the flow.
The physical significance of the equation is that when velocity increases
pressure drops, and vice-versa.
See also the
Bernoulli Effect.
Circulation in a clockwise direction is classed as positive.
The formula above defines circulation. The word is used in a different way than the 'normal'
English definition - the fluid particles are definitely not 'circulating around'.
In the form
These equations hold, in general, for three-dimensional flows, steady or unsteady flows, viscous or inviscid flows,
and compressible or incompressible flows.
However, for incompressible flow, the latter form reduces to
For a simplified flow across an area of fixed size, the equation can take the form
which is valid for both
compressible
and
incompressible
flows.
However for a
compressible
flow it will obviously reduce to just
This equation relates the change in velocity along a streamline to the change in pressure
along the same streamline, and applies to
inviscid
flows, with
no body forces.
When the flow is
incompressible,
it can be integrated to give
Bernoulli's Equation.
Irrotational flows allow a potential function expressed by
Combining these two equations gives
In cartesian coordinates, Laplace's Equation is expressed as
Laplace's Equation is linear, therefore a sum of any particular solutions of the equation is itself a solution of
the equation. This property allows us to build up a realistic flow from the superposition of different flows which are
not, of themselves, realistic flows.
Different flows are obtained from the same governing equation by applying different boundary conditions.
By reference to
Bernouilli's Equation, we can state the following equation from the conditions at
points A and H
Using a bit of algebra, we can then work out the velocity, using the pressure difference (given here by a manometer).
The advantage it offers is that the flow field can now be described using one variable (the potential), instead of three (the
velocities in three directions).
The lines of potential and streamlines will be perpendicular to each other.
Like the stream function, the potential can be used to derive the flow velocities. Difference are
The potential satisfies Laplace's Equation.
where
For incompressible flows
Note that for
incompressible flows,
the coefficient will equal 1 at most. For
compressible flows
the coefficient can be greater than 1.
where c is speed of sound, k is a constant and T is absolute temperature
Since temperature varies within the troposphere, the speed of sound will vary with altitude
also. At sea level, at 15 degrees, it is 1224 km/hr (about 340 m/sec) and at 9100 meters
altitude it is 1090,8 km/hr. The
Earth's troposphere ends near 11000 meters, and since the temperature in the stratosphere
is fairly constant, the speed of sound is also approximately a constant there - about 1060 km/hr.
Strictly speaking, the speed of sound is to do with compressible flow.
and for compressible flow in polar coordinates
For incompressible flow, in cartesian coordinates (where the psi is a 'different' psi from above)
The stream function cab be used for both rotational and irrotational flows, but can only be used
for two dimensions.
For
irrotational flows,
psi satisfies the Laplace Equation.
By analogy, in general, a streamline pattern is like a single frame in a moving picture - each frame will be different.
A pathline, on the other hand, is like a time-exposure picture of a particular fluid element.
Pathlines and streamlines will, in general, be different. However for
steady flow, the pathlines and streamlines will coincide, and the streamline pattern will
remain unaltered from one 'frame' to the next.
From the equation
the differential equation for a streamline is
psi = constant will be the equation of a streamline.
The continuity equation expressed using the substantial derivative is
and the Momentum Equation for non-steady, inviscid flow with no body forces is
Physically, the substantial derivative gives the time rate of change following a moving fluid element
as oppposed to
which is the time rate of change at a fixed point.
It is physically the time rate of change due to the movement of the fluid element from one location to another
in the flow field where the flow properties are spatially different.
For a symmetric tube, we can use the symmetry to
reduce the equations to just "quasi-one dimension" form. All we need to enter into the equations
are the cross-sectional area and the (for our diagram, horizontal) velocity.
The flow will speed up towards the neck in the middle, but drop in pressure. It will slow
down and increase in pressure as it approaches the exit.
Carburettors and wind tunnels operate on the Venturi principle.
Giovanni Battista Venturi lived from 1746-1822, and worked at the University of Pavia,
on hydraulics and acoustics.
If this formula does not equal zero at every point in a flow, the flow is called rotational, implying that
the fluid elements have a finite angular velocity.
If the formula equals zero at every point, the flow is irrotational and any motion is pure translation.
In component form, the vorticity is given by
It is related to angular velocity, so
Circulation
Continuity Equation (Conservation of Mass)
The full continuity (or principle of mass conservation) equation is, in integral form
Doublet Flow
Energy Equation
Equation of State
Euler's Equation
Ideal (or Perfect) Fluid
An inviscid, incompressible fluid.
Kutta-Joukowski Theorem
Laplace's Equation
Incompressibility gives, via the principle of
continuity
Magnus Effect
Perfect (or Ideal) Fluid
An inviscid, incompressible fluid.
Pitot Tube
A Pitot tube can be used to measure the velocity of a fluid.
Potential
For a irrotational flow, a potential can be defined. From this potential
the flow velocities can be derived thus
Pressure Coefficient
Sound, Speed of
The speed of sound increases proportionally to the absolute temperature.
Source Flow
Steady Flow
Stream Function
For compressible flow in cartesian coordinates
Streamlines
Streamlines are curves whose tangent at any point is in the direction of the velocity vector at that point.
Substantial Derivative (or Total Derivative)
The Substantial Derivative is defined in the following manner
Total Derivative
See Substantial Derivative
Uniform Flow
any time derivative vanishes.
Venturi Tube
The flow of an inviscid, incompressible flow thru a Venturi Tube acts in
accordance with
Bernoulli's Equation, taking into account the
Continuity Equation.
Vorticity
The vorticity is given by
