Incompressible Fluid Mechanics - Glossary (Under Construction)
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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'.
Continuity Equation (Conservation of Mass)The full continuity (or principle of mass conservation) equation is, in integral form
in this form it is applicable in a finite region of space.
In the form
the equation is applicable only to a point in the flow.
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
where rho is density, A is the cross-sectional area and V is flow velocity
which is valid for both compressible and incompressible flows.
However for a compressible flow it will obviously reduce to just
Equation of State
where p is pressure, rho is density and V is flow velocity
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.
Ideal (or Perfect) FluidAn inviscid, incompressible fluid.
Laplace's EquationIncompressibility gives, via the principle of continuity
Irrotational flows allow a potential function expressed by
Combining these two equations gives
usually expressed as which is Laplace's Equation
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.
Perfect (or Ideal) FluidAn inviscid, incompressible fluid.
Pitot TubeA Pitot tube can be used to measure the velocity of a fluid.
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).
PotentialFor a irrotational flow, a potential can be defined. From this potential the flow velocities can be derived thus
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 can be used in three dimensions, the stream function only in two
- the potential can only be used for irrotational flows, the stream function can be used for both irrotational and rotational flows.
The potential satisfies Laplace's Equation.
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.
Sound, Speed ofThe speed of sound increases proportionally to the absolute temperature.
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.
Stream FunctionFor compressible flow in cartesian coordinates
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.
StreamlinesStreamlines are curves whose tangent at any point is in the direction of the velocity vector at that point.
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.
Substantial Derivative (or Total Derivative)The Substantial Derivative is defined in the following manner
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.
is called the convective derivative.
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.
Total DerivativeSee Substantial Derivative
Uniform Flowany time derivative vanishes.
Venturi TubeThe flow of an inviscid, incompressible flow thru a Venturi Tube acts in accordance with Bernoulli's Equation, taking into account the Continuity Equation.
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.
VorticityThe vorticity is given by
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