SPRING 2005

Part 1. Fluid-Flow Principles

1. Introduction

1.1 Definitions

1.2 Notation and fluid properties

1.3 Hydrostatics

1.4 Fluid dynamics

1.5 Control volumes

1.6 Visualising fluid flow

1.7 Real and ideal fluids

1.8 Laminar and turbulent flow

2. Continuity (mass conservation)

2.1 Flow rate

2.2 The steady continuity equation

2.3 Unsteady continuity equation

3. The Equation of Motion

3.1 Forms of the equation of motion

3.2 Fluid acceleration

3.3 Bernoulli’s equation

3.4 Application to flow measurement

3.5 Other applications (flow through an orifice; tank-emptying)

4. The Momentum Principle

4.1 Steady-flow momentum principle

4.2 Applications (pipe contractions; pipe bends; jets)

5. Energy

5.1 Derivation of Bernoulli’s equation from an energy principle 5.2 Fluid head

5.3 Departures from ideal flow (discharge coefficients; loss coefficients; momentum & energy coefficients)

Part 2. Applications (Separate Notes)

1. Hydraulic Jump

2. Pipe Flow (Dr Lane-Serff)

Recommended Reading

Hamill, 2001, Understanding Hydraulics, 2nd Edition, Palgrave, ISBN 0-333-77906-1 Chadwick, Morfett and Borthwick, 2004, Hydraulics in Civil and Environmental Engineering, 4th Edition, Spon Press, ISBN 0-415-30609-4

Massey, 1998, Mechanics of Fluids, 7th Edition, (Revised by Ward-Smith, J.), Stanley Thornes, ISBN 0-748-74043-0

White, 2003, Fluid Mechanics, 5th Edition, McGraw-Hill, ISBN 0-07-240217-2

Hydraulics 1

1

David Apsley

1. Introduction and Basic Principles

1.1 Definitions

A fluid is a body of matter that can flow; i.e. continues to deform under a shearing force. Fluids may be liquids (definite volume; free surface) or gases (expand to fill any container). Fluids obey the usual laws of Newtonian mechanics, but as a continuum. Unlike rigid bodies, fluid particles may move relative to each other, interacting via internal forces. These are usually expressed in terms of stresses (stress = force / area), the principal ones being: pressure, p – normal stress: pushing or pressing;

shear stress, – tangential stress: frictional drag; opposing relative motion.

An ideal fluid has no viscosity. This is never exactly true, but is often a useful approximation. Fluids may be regarded as incompressible (density or volume not changed by the flow) at speeds much less than that of sound. This is usually the case in hydraulics. Fluid flow may be described as laminar (adjacent layers “slide” smoothly past each other) or turbulent (irregular, with constant intermingling of adjacent layers).

1.2 Notation and Fluid Properties

The main flow variables are:

p

pressure (force/area)

u

velocity

These are field variables because they are functions of position and time t; e.g. u = u(x,t) Vector quantities like position and velocity are often decomposed into components: x ≡ ( x, y , z )

u ≡ (u , v, w)

U (or occasionally V) will be used for the magnitude of velocity. Important fluid properties are:

density = mass / volume

¡

dynamic viscosity; defined by Newton’s viscosity law:

du

=

dy

= / is called the kinematic viscosity

¢

¢

¡

¢

£

surface tension = force / length (or surface energy/area)

F= l

¤

¥

K

bulk modulus = pressure change / volumetric strain

−V

p = K(

)

V

¦

¦

Hydraulics 1

2

David Apsley

Most gases at normal temperatures and pressures satisfy the ideal gas law, which in fluid mechanics is usually written as

p = RT

The gas constant R depends on the particular gas (R = R*/m = universal gas constant / molar mass). It has a value of 287 J kg–1 K–1 for dry air.

1.3 Hydrostatics

Hydrostatics concerns the balance of forces in a fluid at rest.

p is the normal force per unit area

≡ g is the specific weight (weight per unit volume)

is the force per unit length

¡

¡

The principal forces are:

pressure;

weight;

surface tension;

¤

In the interior of a stationary fluid, pressure forces...