Fluid Mechanics 3(2+1)

LESSON 18. DYNAMICS OF FLUID FLOW

Continuity Equation

Consider a fluid that is flowing through a pipe. The pipe has a cross sectional area that is not constant. Let the area on the left end of the pipe be A1 and the area on the right end be A2. Let the velocity of the fluid entering the pipe from the left be labeled V1 and the velocity of the fluid leaving the pipe from the right be V2.

If the fluid is incompressible,

Volume entering = Volume leaving

A1V1 = A2V2

If A2 is smaller than A1, then V2 must be larger than V1 so the amount of water coming out equals the amount going in.

Bernouli's Equation

5.1 Frictionless Flow Along Streamlines

• Application of the second Newton’s law of motion along streamlines of fluid flow leads to a very famous equation in Fluid Mechanics, i.e. the Bernoulli equation.

• There are four assumptions used to derive the equation and these four assumptions must always be remembered to ensure that it is used correctly, i.e.

1. The flow is inviscid or frictionless, i.e. viscous effects are negligible which is valid for low viscosity fluids such as water and air,

2. The flow is steady, i.e. the flow pattern is fully developed and does not change with time,

3. The flow is incompressible, which is valid for all fluids and low speed gas of Mach 0.3 or below since the change in gas density is less than 5%,

4. The flow considered is along the same streamline, as the variation of properties for fluid molecules travelling in the same path can be simulated more accurately through conservation laws of physics.

• In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.

• Bernoulli's principle is named after the Swiss scientist Daniel Bernoulli who published his principle in his book Hydrodynamica in 1738.

• Bernoulli's principle can be applied to various types of fluid flow, resulting in what is loosely denoted as Bernoulli's equation. In fact, there are different forms of the Bernoulli equation for different types of flow.

• The simple form of Bernoulli's principle is valid for incompressible flows (e.g. most liquid flows) and also for compressible flows (e.g. gases) moving at low Mach numbers. More advanced forms may in some cases be applied to compressible flows at higher Mach numbers (see the derivations of the Bernoulli equation).

• Bernoulli's principle can be derived from the principle of conservation of energy. This states that, in a steady flow, the sum of all forms of mechanical energy in a fluid along a streamline is the same at all points on that streamline. This requires that the sum of kinetic energy and potential energy remain constant. Thus an increase in the speed of the fluid occurs proportionately with an increase in both its dynamic pressure and kinetic energy, and a decrease in its static pressure and potential energy. If the fluid is flowing out of a reservoir, the sum of all forms of energy is the same on all streamlines because in a reservoir the energy per unit volume (the sum of pressure and gravitational potential ρgh) is the same everywhere.

In most flows of liquids, and of gases at low Mach number, the density of a fluid parcel can be considered to be constant, regardless of pressure variations in the flow. Therefore, the fluid can be considered to be incompressible and these flows are called incompressible flow. Bernoulli performed his experiments on liquids, so his equation in its original form is valid only for incompressible flow. A common form of Bernoulli's equation, valid at any arbitrary point along a streamline, is:

Where :

v is the fluid flow speed at a point on a streamline,

g is the acceleration due to gravity,

z is the elevation of the point above a reference plane, with the positive z-direction pointing upward – so in the direction opposite to the gravitational acceleration,

Ρ is the pressure at the chosen point, and

ρ is the density of the fluid at all points in the fluid.

For conservative force fields, Bernoulli's equation can be generalized as:^[8]

where Ψ is the force potential at the point considered on the streamline. E.g. for the Earth's gravity Ψ = gz.

The following two assumptions must be met for this Bernoulli equation to apply:

• the flow must be incompressible – even though pressure varies, the density must remain constant along a streamline;
• friction by viscous forces has to be negligible.

By multiplying with the fluid density , equation (A) can be rewritten as:

or:

where:

is dynamic pressure,

is the piezometric head or hydraulic head (the sum of the elevation z and the pressure head) and

is the total pressure (the sum of the static pressure p and dynamic pressure q).

The constant in the Bernoulli equation can be normalized. A common approach is in terms of total head or energy head H:

The above equations suggest there is a flow speed at which pressure is zero, and at even higher speeds the pressure is negative. Most often, gases and liquids are not capable of negative absolute pressure, or even zero pressure, so clearly Bernoulli's equation ceases to be valid before zero pressure is reached. In liquids – when the pressure becomes too low – cavitation occurs. The above equations use a linear relationship between flow speed squared and pressure. At higher flow speeds in gases, or for sound waves in liquid, the changes in mass density become significant so that the assumption of constant density is invalid.

Final Ideas:

• Bernouli's equation states that if one moves around in the fluid, points of fast velocity are points of low pressure, and points of lower speed have higher pressure. This does make "sense", since to obtain a large velocity places of larger pressure somewhere else are needed to "push" the fluid to these higher speeds where the pressure is lower.

• If the fluid is at rest, velocity is zero everywhere, and Bernouli's equation reduces to the equation for a fluid at rest: P + ρgz = constant.

Energy and Hydraulic Grade Lines

• Static pressure p – representing the actual or thermodynamic pressure at a particular point in the streamline.

• Dynamic pressure ½ρV² – representing the kinetic energy for fluid molecules passing at the same point.

• Hydrostatic pressure ρgz – representing the potential energy for fluid molecules at the same point which changes with elevation.

• If the fluid has a certain velocity V travelling along one streamline with small elevation, the hydrostatic pressure is usually small and insignificant compared to the static pressure and the dynamic pressure.  The combination of the static pressure and the dynamic pressure forms the stagnation pressure p₀, or

                         p + ½ρV ² =  p₀