Chapter 3. Hydrodynamics

What you will learn

You will understand the principles governing water dynamics within water supply systems.

Why

- An understanding of the physical laws that influence water supply systems is essential prerequisite in order to design efficient and long-lasting water supply systems.
  - You will therefore receive basic knowledge on water flow rate, law of conservation of energy

Duration of chapter 3

2 - 3 hours





This chapter will study the general notions of hydrodynamics, derived from the law of conservation of energy. Reynolds number is also explained; although it looks a bit abstract, it is a very important concept governing fluid mechanics.
Practical situations will be treated in the two next chapters for piped (closed) and open flows.

3.1. Flow *

According to the law of conservation of matter, for permanent conditions (i.e. not transient conditions such as water hammer), the mass flow rate is constant along a stream.
q: flow [kg/s]
ρ: density [kg/m3]
A: cross section area [m2]
v: mean velocity of the fluid
at a given i section

When the fluid is considered as incompressible, this equation can be simplified and a volumetric flow rate (m3/s) can be used. This is the case for water and is also valid for air under low pressure, such as in building ventilation systems.
Across a given cross-section, velocity usually varies. It may vary from zero at a peripheral contact with a "wall" to a maximum at the centre. The velocity that will be used in these chapters is taken as the mean velocity across a given section. In the majority of cases encountered in hydraulics, velocity is fairly homogenous, this point will be further developed with the Reynolds concept.

Q: flow [m3/s]
Ai: cross section area [m2]
vi: mean velocity of the fluid
at a given section, i.

As shown above, for a given flow, the smaller the section, the greater the velocity passing through it.

If there are junctions, the law of conservation of matter stipulates that the total flow going in must equal the total flow going out.
In this example the flow (Q1) entering the pipe must equal the sum of the flow leaving (Q2+Q3)

For transient conditions, like water hammer, the water is still entering the pipe but no water is exiting. In this case, water will accumulate for a certain period inside the pipe and the previous equations are not valid during this period. This case will be further studied later on.
As the international system of units standard for flow (m3/s) is quite large, more practical units are often encountered such as litres/second, litres/minute, m3/day, etc…

3.2. Law of Conservation of Energy (Bernoulli's Principle) *

In hydraulics, the energy of a flow at a given point is composed of three main elements: - the energy due to the pressure at this point or elastic energy, - the potential energy according to the difference of altitude in the system, - the kinetic energy due to the velocity of the fluid.
In the following expression, all of these elements are expressed in water column height [mWC] at a given point i :

Ei: energy
Pi: relative pressure
Hi: elevation
vi: velocity of the fluid

This equation is only valid for permanent, incompressible flow with negligible changes of internal energy (temperature) and without vortex.
NB: H is calculated from an arbitrary reference point (which is usually either the sea level or the lowest point occurring in calculations) but should stay coherent all along calculations.
According to the law of conservation of energy, the energy at point A equals the energy at point B minus any losses between these two points. The energy can change from one form to another, for instance from mainly potential energy to pressure or to kinetic, but the total will always respect the law of conservation of energy..

EA: total energy at position A
EB: total energy at position B
HA-B: energy losses between A and B

The evaluation of any losses is the most complicated problem, because it cannot be deduced from mathematical equations. Therefore it has to be calculated through experiments, from which empirical laws are developed. The most commonly encountered formulas are Chezy-Manning, Darcy-Weisbach and Hazen-Williams; their use will be reviewed in chapter 4.5. For the following examples, we will neglect losses in order to observe particular applications of Bernoulli's Principle.

Torricelli's theorem

For a large tank with a small orifice at the bottom, we can consider the energy at point A as being only potential, since the velocity is negligible and the water pressure is zero (EA).
Points between A and B gradually lose potential energy and increase pressure. At point B, all potential energy is lost, the velocity is still negligible but there is significant pressure (EB).
Points between B and C gradually increase in speed and lose pressure until the orifice (point C), where there is no more water pressure, no more potential energy and only velocity. Neglecting the losses, we can say that all the initial potential energy was ultimately transformed to kinetic energy (Ec).
Therefore, from A to B to C the energy type changes but the total energy is conserved.

Venturi effect

The Venturi effect can be used to measure the velocity of a fluid by comparing the pressure difference between two known sections of pipes.
Thanks to Eq. 2-5, Eq. 3-2 & Eq. 3-4, neglecting the losses the following speed can be calculated according to the difference of height (h).

This is a very cheap and efficient way to measure flow rate. Transparent hoses or glass pipes are necessary to measure the difference of level and need to be regularly cleaned.
When the section A2 is small enough, air can be sucked into the pipe by the passing water, this principle is used in several practical applications, such as in vacuum pumps or chlorine injectors. When air is sucked in the water, air bubbles will be taken in to the flow and remain there unless absorbed. In this case, we have a compressible fluid with a mixture of liquid and gaseous phases and the equations seen previously are no longer valid.

Cavitation while passing a gate valve or an obstruction

When a section in a pipe is very small, such as at a gate valve that is almost closed, the pressure can decrease until it reaches vapour pressure (as the velocity increases). At this stage, the water will start to cavitate and a typical noise (like small stones hitting the pipe) can be heard.

Cavitation produces small vapour bubbles (not air bubbles) that might grow a bit if the velocity further increases. After the restriction, when the velocity decreases again, the pressure rises and the bubbles implode suddenly, progressively eroding the pipes downstream of the restriction. It is these implosions that make the typical cavitation noise. Similar phenomena can occur in open flow in dams' spillways, and the effect can easily destroy concrete structures.
When cavitation occurs, the equations seen previously are not applicable as we have two phases present: compressible gas and water; the pressure will be kept at its vapour value. In most simulation software (like Epanet) this situation cannot be taken into consideration and a warning message is given but the equations are still solved with negative values for the pressure.

Representation of energy (or piezo) line

As all components of energy are expressed in meters of water column, they can easily be represented as elevations on a diagram. In this way, it is easy to see and understand the transformation of energy from potential to kinetic and pressure and to see the losses.
The example represented in the above figure shows a simple piped scheme running from an elevated tank to a house. At the top of the tank all of the energy is potential. When entering the pipe, it takes on speed and so part of the available energy is kinetic. Along the pipe, the kinetic energy will be constant if the pipe diameter is constant, and the pressure will rise as potential energy decreases, and a part of the available energy will be lost in friction losses. At the delivery point, all of the initial potential energy has been transformed into lost, kinetic and pressure energies.

3.3. Reynolds number

In fluid mechanics, the Reynolds number (Re) is a dimensionless number that indicates the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions.
It is used to characterize different flow regimes, laminar or turbulent flow. Laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion. While turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities.

Reynolds number depends only on three values:
- the viscosity, fixed for a given fluid at a certain temperature
- the velocity
- the specific dimension (the internal diameter for a pipe, the thickness for a wing, etc).

Re: Reynolds number
D: specific dimension [m]
v: velocity of the fluid [m/s]
ν: kinematic viscosity


Laminar flow around a body: in this situation each flow line returns to its initial position after the obstacle. There is no mixing in the fluid and velocities are changed smoothly to pass the obstacle and find their original values after it.

Turbulent flow around a body: in this situation the flow is disturbed over a long distance, the flow lines are mixed in vortices formed after the obstacle. The velocities are changing quickly when approaching the obstacle.

Transient conditions: Flow can only be laminar or turbulent; there is no intermediate flow regime between these conditions. Flow is considered to jump from one state to the other. The exact Re values of this jump might vary with some parameters like roughness, or speed of change. It might also not be the same value when passing from laminar to turbulent and from turbulent to laminar. Therefore the transition Re is given over a wide range (23'000 to 40'000 for pipes).

Basic exercises:

1. What is the flow in a pipe of 150mm of diameter with a 1m/s speed?
2. What is the speed in the same pipe after a reduction in diameter to 100mm and to 75mm?
3. What is the flow in the same pipe after a Tee with a 25mm pipe with a 1 m/s speed?
4. What is the speed of water going out of the base of a tank with 2.5m, 5m and 10m height?
5. What are the speed and the flow in a pipe of 150mm of diameter showing a difference of 20cm height in a Venturi section of 100mm diameter?

Intermediary exercises:

6. What are the Reynolds numbers for the flows in the exercises 1,2 & 5?
7. For a pipe of 150mm of diameter, with 2 bar pressure and a a velocity of 1 m/s, what should be the diameter reduction to cause cavitation at 20°C (neglect head losses)?

Advanced exercises:

8. With equations Eq. 2-5, Eq. 3-2 & Eq. 3-4, demonstrate Eq. 3-6.
9. For a pipe of 150mm of diameter, with 1 bar pressure and a velocity of 1 m/s, what should be the diameter reduction to cause air suction?
10. A tank 2m high and 1m diameter has a 75mm valve at the bottom. When the tank is full and the valve is quickly opened, how long does it take to empty the tank (the losses and contraction factor are neglected)? (Difficult exercise to be solved with integral calculations).