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Mechanical Engineering · 400 Level
This is the complete lecture note for MEE404 Applied fluid mechanics, prepared for Mechanical Engineering students at 400 Level. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.
Course code: MEE 404 Course title: Applied Fluid Mechanics Course credit load: 2 Units Series 1: Unsteady Flow, Surge Tanks, and Water Hammer • Part 1.1: Fundamentals of Unsteady Flow o Sub Part 1.1.1: Concept of Unsteady Flow o Sub Part 1.1.2: Causes of Unsteady Flow in Pipe Systems o Sub Part 1.1.3: Governing Equations of Unsteady Flow • Part 1.2: Oscillation in a U-Tube o Sub Part 1.2.1: Description of U-Tube Oscillation o Sub Part 1.2.2: Equation of Motion for U-Tube Oscillation o Sub Part 1.2.3: Period and Damping of U-Tube Oscillation • Part 1.3: Surge Tanks o Sub Part 1.3.1: Purpose and Function of a Surge Tank o Sub Part 1.3.2: Surge Tank Oscillation Analysis o Sub Part 1.3.3: Types of Surge Tanks • Part 1.4: Water Hammer o Sub Part 1.4.1: Mechanism of Water Hammer o Sub Part 1.4.2: Pressure Rise Due to Water Hammer o Sub Part 1.4.3: Water Hammer Mitigation Measures Introduction Much of fluid mechanics as commonly taught assumes that flow conditions at any point remain constant over time, a convenient simplification that nonetheless breaks down whenever a valve is closed suddenly, a pump trips offline, or a hydropower plant's demand changes abruptly, and it is
precisely these transient, time-varying conditions, together with the specific phenomena of U-tube oscillation, surge tank behaviour, and water hammer that arise from them, that this opening Series of the course sets out to examine. The aim of this Series is to equip you with an understanding of unsteady flow and its causes, the oscillation of liquid within a U-tube, the purpose and behaviour of surge tanks, and the mechanism, pressure effects, and mitigation of water hammer in pipeline systems. We shall commence this Series with the fundamentals of unsteady flow, before turning to oscillation in a U-tube. We will then examine surge tanks, and conclude this Series by considering water hammer. Series Learning Outcomes Upon completing this Series, you should be able to: • SLO 1.1: define unsteady flow and distinguish it from steady flow, • SLO 1.2: explain the causes of unsteady flow in pipe systems, • SLO 1.3: describe the oscillation of liquid in a U-tube, • SLO 1.4: derive the equation of motion for U-tube oscillation, • SLO 1.5: explain the purpose and function of a surge tank, • SLO 1.6: analyse surge tank oscillation, • SLO 1.7: explain the mechanism of water hammer, • SLO 1.8: calculate the pressure rise due to water hammer, 1.1 Fundamentals of Unsteady Flow 1.1.1 Concept of Unsteady Flow Unsteady flow is fluid flow in which velocity, pressure, or other flow properties at a given point within the fluid change with time, in direct contrast to steady flow, where these same properties remain constant at every point regardless of how much time has elapsed, and this time-dependence means that a complete description of unsteady flow requires tracking how conditions evolve rather than simply recording a single, unchanging value for each property at each location.
Many real fluid systems that are ordinarily treated as steady for simplicity actually pass through genuinely unsteady conditions during start-up, shutdown, or any sudden change in operating demand, and while a steady-flow assumption remains entirely adequate for most everyday design calculations, the transient behaviour a system exhibits during these changing conditions can produce effects, elevated pressures chief among them, that a steady-flow analysis alone would never reveal or predict. Worked example: A pipeline carrying a constant flow rate under normal operation behaves as steady flow, but the same pipeline experiences genuinely unsteady flow for the brief period immediately after a downstream valve is suddenly closed. Application: Recognising when a system's operating condition has shifted from steady to unsteady is the essential first step before any of the transient phenomena examined later in this Series can be properly analysed. Fill in the blank: Flow in which velocity or pressure at a given point changes with time is called ________ flow. 1.1.2 Causes of Unsteady Flow in Pipe Systems Unsteady flow in a pipe system most commonly arises from a deliberate or accidental change at one of the system's boundaries, such as a valve being opened or closed, a pump being started, stopped, or tripping offline unexpectedly, or a downstream demand changing suddenly, each of these boundary changes disturbing the previously steady flow condition and propagating that disturbance through the fluid as a transient wave that takes a finite, calculable time to travel along the pipeline's length. The severity of the resulting unsteady flow depends heavily on how rapidly the boundary change actually occurs, with a very sudden change, such as an almost instantaneous valve closure, generally producing far more severe transient effects than the same overall change carried out gradually over a longer period, a relationship that underlies much of the practical guidance given later in this Series regarding water hammer mitigation. Worked example: Closing a valve at the end of a long pipeline very rapidly produces a considerably more severe pressure transient than closing that same valve gradually over several seconds, even though both actions ultimately stop the same flow. Application: Recognising that the rate of a boundary change, not merely its overall magnitude, governs the severity of the resulting unsteady flow is essential background for the water hammer mitigation measures examined later in this Series. Fill in the blank: A very sudden boundary change, such as an almost instantaneous valve closure, generally produces ________ severe transient effects than the same change made gradually. 1.1.3 Governing Equations of Unsteady Flow
Unsteady flow in a pipeline is governed by two coupled partial differential equations, a continuity equation expressing conservation of mass and a momentum equation expressing Newton's second law applied to the moving fluid, both of which retain an explicit time derivative term that a steady-flow analysis would simply set to zero, and it is precisely this retained time-dependence that allows these equations to capture how pressure and velocity evolve together as a disturbance propagates along the pipe. Solving these governing equations analytically in closed form is possible only for a limited number of simplified cases, so most practical unsteady flow problems of real engineering interest, including the water hammer analysis introduced later in this Series, are instead solved using numerical methods, most notably the method of characteristics, which converts the governing partial differential equations into a form well suited to computation along a pipeline discretised into a finite number of calculation points. Worked example: Setting the time derivative terms in the unsteady flow governing equations to zero recovers the familiar steady-flow continuity and momentum equations used throughout ordinary pipe flow calculations, confirming that steady flow is simply a special case of the more general unsteady formulation. Application: Recognising steady flow as a special case of the more general unsteady governing equations, rather than as an entirely separate set of equations, clarifies why the same underlying physics connects both types of analysis. Fill in the blank: Unsteady pipe flow is governed by a continuity equation and a ________ equation, both retaining an explicit time derivative term. Figure 1.1: Velocity variation with time during unsteady flow following a valve closure.
Pilot questions 1.1 1. Define unsteady flow and distinguish it from steady flow. (Linked to SLO 1.1) 2. Give an example of a pipeline system passing from steady to unsteady flow. (Linked to SLO 1.1) 3. Explain the common causes of unsteady flow in pipe systems. (Linked to SLO 1.2) 4. Explain why the rate of a boundary change affects the severity of unsteady flow. (Linked to SLO 1.2) 5. Describe the two governing equations used to analyse unsteady flow. (Linked to SLO 1.2) 1.2 Oscillation in a U-Tube 1.2.1 Description of U-Tube Oscillation U-tube oscillation occurs when a liquid column within a U-shaped tube, displaced from its equilibrium position by an initial disturbance, subsequently swings back and forth about that equilibrium position under the restoring influence of gravity, the liquid's own inertia carrying it past the equilibrium point on each swing before gravity pulls it back again, producing an oscillation broadly analogous to a pendulum swinging under gravity, though governed by its own distinct equation of motion. This oscillation provides one of the simplest and most instructive physical models for understanding the more complex transient phenomena examined later in this Series, since a surge tank connected to a pipeline behaves, in its essential physics, remarkably similarly to a U-tube, with the tank's liquid level oscillating about its own equilibrium position in direct response to a disturbance in the connected pipeline's flow, exactly as the U-tube's liquid column oscillates in response to its own initial displacement. Worked example: Displacing the liquid in one limb of a U-tube upward and then releasing it produces an oscillation in which the liquid level swings symmetrically above and below its original equilibrium position, gradually diminishing in amplitude as friction dissipates the system's energy over successive swings. Application: Understanding U-tube oscillation as a simple physical model prepares the ground directly for analysing the more practically important surge tank oscillation examined later in
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