North East Aircraft Museum

Incompressible Fluid Mechanics - Glossary (Under Construction)

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Bernoulli Equation

Bernouilli 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

Circulation Formula

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

Continuity Equation
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

Continuity Equation

Doublet Flow

Energy Equation

Equation of State

Equation of State

Euler's Equation

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) Fluid

An inviscid, incompressible fluid.

Kutta-Joukowski Theorem

Laplace's Equation

Incompressibility 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.

Magnus Effect

Perfect (or Ideal) Fluid

An inviscid, incompressible fluid.

Pitot Tube

Pitot Tube A 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

Bernouilli Equation applied to a Pitot Tube

Using a bit of algebra, we can then work out the velocity, using the pressure difference (given here by a manometer).

Potential

For a irrotational flow, a potential can be defined. From this potential the flow velocities can be derived thus

Potential Function        Potential Function        Potential Function

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.

Pressure Coefficient

Pressure Coefficient

where

Dynamic Pressure

For incompressible flows

Pressure Coefficient for Incompressible Flow

Note that for incompressible flows, the coefficient will equal 1 at most. For compressible flows the coefficient can be greater than 1.

Sound, Speed of

The speed of sound increases proportionally to the absolute temperature.

Relation between Speed of Sound and 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.

Source Flow

Steady Flow

Steady State Condition

Stream Function

For compressible flow in cartesian coordinates
Stream Function        Stream Function

and for compressible flow in polar coordinates

Stream Function        Stream Function

For incompressible flow, in cartesian coordinates (where the psi is a 'different' psi from above)

Stream Function        Stream Function

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.

Streamlines

Streamlines 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

No rotation about the z axis

the differential equation for a streamline is

Differential Equations for Streamlines

psi = constant will be the equation of a streamline.

Substantial Derivative (or Total Derivative)

The Substantial Derivative is defined in the following manner

Substantial Derivative

The continuity equation expressed using the substantial derivative is

Continuity Equation expressed using the Substantial derivative

and the Momentum Equation for non-steady, inviscid flow with no body forces is

Momentum Equation expressed using the Substantial derivative

Physically, the substantial derivative gives the time rate of change following a moving fluid element as oppposed to

Partial Derivative

which is the time rate of change at a fixed point.

Convective Derivative
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 Derivative

See Substantial Derivative

Uniform Flow

any time derivative vanishes.

Venturi Tube

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.

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.

Vorticity

The vorticity is given by

Vorticity Formula

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

Vorticity Formula

It is related to angular velocity, so

Relation between Vorticity and Angular Velocity


Brian Daugherty