---
author: Alvie Rahman
date: \today
title: MMME1048 // Fluid Dynamics
tags: [ uni, nottingham, mechanical, engineering, fluid_mechanics, mmme1048, fluid_dynamics ]
---

\newcommand\Rey{\mbox{\textit{Re}}}
\newcommand\textRey{$\Rey$}

# Introductory Concepts

These are ideas you need to know about to know what's going on, I guess?

## Control Volumes

A control volume is a volume with an imaginary boundry to make it easier to analyze the flow of a
fluid.
The boundry is drawn where the properties and conditions of the fluid is known, or where an
approximation can be made.
Properties which may be know include:

- Velocity
- Pressure
- Temperature
- Viscosity

The region in the control volume is analyed in terms of enery and mass flows entering and leaving
the control volumes.
You don't have to understand what's going on inside the control volume.

<details>
<summary>

### Example 1

The thrust of a jet engine on an aircraft at rest can be analysed in terms of the changes in
momentum or the air passing through the engine.

</summary>

![](./images/vimscrot-2021-11-03T21:51:51,497459693+00:00.png)

The control volume is drawn far enough in front of the engine that the air velocity entering can
be assumed to be at atmospheric pressurce and its velocity negligible.

At the exit of the engine the boundary is drawn close where the velocity is known and the air
pressure atmospheric.

The control volume cuts the material attaching the engine to the aircraft and there will be a force
transmitted across the control volume there to oppose the forces on the engine created by thrust
and gravity.

The details of the flows inside the control volume do not need to be known as the thrust can be 
determined in terms of forces and flows crossing the boundaries drawn.
However, to understand the flows inside the engine in more detail, a more detailed analysis would
be required.

</details>

## Ideal Fluid

The actual flow pattern in a fluid is usually complex and difficult to model but it can be
simplified by assuming the fluid is ideal.
The ideal fluid has the following properties:

- Zero viscosity
- Incompressible
- Zero surface tension
- Does not change phases

Gases and vapours are compressible so can only be analysed as ideal fluids when flow velocities are
low but they can often be treated as ideal (or perfect) gases, in which case the ideal gas equations
apply.

## Steady Flow

Steady flow is a flow which has *no changes in properties with respect to time*.
Properties may vary from place to place but in the same place the properties must not change in
the control volume to be steady flow.

Unsteady flow does change with respect to time.

## Uniform Flow

Uniform flow is when all properties are the same at all points at any given instant but can change
with respect to time, like the opposite of steady flow.

## One Dimensional Flow

In one dimensional (1D) flow it is assumed that all properties are uniform over any plane
perpedenciular to the direction of flow (e.g. all points along the cross section of a pipe have
identical properties).

This means properties can only flow in one direction---usually the direction of flow.

1D flow is never achieved exactly in practice as when a fluid flows along a pipe, the velocity at
the wall is 0, and maximum in the centre of the pipe.
Despite this, assuming flow is 1D simplifies the analysis and often is accurate enough.

## Flow Patterns

There are multiple ways to visualize flow patterns.

### Streamlines

A streamline is a line along which all the particle have, at a given instant, velocity vectors
which are tangential to the line.

Therefore there is no component of velocity of a streamline.

A particle can never cross a streamline and *streamlines never cross*.

They can be constructed mathematically and are often shown as output from CFD analysis.

For a steady flow there are no changes with respect to time so the streamline pattern does not.
The pattern does change when in unsteady flow.

Streamlines in uniform flow must be straight and parallel.
They must be parallel as if they are not, then different points will have different directions and
therefore different velocities.
Same reasoning with if they are not parallel.

### Pathlines

A pathline shows the route taken by a single particle during a given time interval.
It is equivalent to a high exposure photograph which traces the moevement of the particle marked.
You could track pathlines with a drop of injected dye or inserting a buoyant solid particle which
has the same density as the solid.

Pathlines may cross.

### Streaklines

A streakline joins, at any given time, all particles that have passed through a given point.
Examples of this are line dye or a smoke stream which is produced from a continuous supply.

## Viscous (Real) Fluids

### Viscosity

A fluid offers resisistance to motion due to its viscosity or internal friction.
The greater the resistance to flow, the greater the viscosity.

Higher viscosity also reduces the rate of shear deformation between layers for a given shear stress.

Viscosity comes from two effects:

- In liquids, the inter-molecular forces act as drag between layers of fluid moving at different
  velocities
- In gases, the mixing of faster and slower moving fluid causes friction due to momentum transfer.
  The slower layers tend to slow down the faster ones

### Newton's Law of Viscosity

Viscosity can be defined in terms of rate of shear or velocity gradient.

![](./images/vimscrot-2021-11-17T14:14:05,079195275+00:00.png)

Consider the flow in the pipe above.
Fluid in contact with the surface has a velocity of 0 because the surface irregularities trap the
fluid particles.
A short distance away from the surface the velocity is low but in the middle of the pipe the
velocity is $v_F$.

Let the velocity at a distance $y$ be $v$ and at a distance $y + \delta y$ be $v + \delta v$.

The ratio $\frac{\delta v}{\delta y}$ is the average velocity gradient over the distance
$\delta y$.

But as $\delta y$ tends to zero, $\frac{\delta v}{\delta y} \rightarrow$ the value of the
differential $\frac{\mathrm{d}v}{\mathrm{d}y}$ at a point such as point A.

For most fluids in engineering it is found that the shear stress, $\tau$, is directly proportional
to the velocity gradient when straight and parallel flow is involved:

$$\tau = \mu\frac{\mathrm{d}v}{\mathrm{d}y}$$

Where $\mu$ is the constant of proportinality and known as the dynamic viscosity, or simply the
viscosity of the fluid.

This is Newton's Law of Viscosity and fluids that ovey it are known as Newtonian fluids.

### Viscosity and Lubrication

Where a fluid is a thin film (such as in lubricating flows), the velocity gradient can be
approximated to be linear and an estimate of shear stress obtained:

$$\tau = \mu \frac{\delta v}{\delta y} \approx \mu \frac{v}{y}$$

From the shear stress we can calculate the force exerted by a film by the relationship:

$$\tau = \frac F A$$

# Fluid Flow

## Types of flow

There are essentially two types of flow:

- Smooth (laminar) flow

  At low flow rates, particles of fluid are moving in straight lines and can be considered to be
  moving in layers or laminae.

- Rough (turbulent) flow

  At higher flow rates, the paths of the individual fluid particles are not straight but disorderly
  resulting in mixing taking place

Between fully laminar and fully turbulent flows is a transition region.

## The Reynolds Number

### Development of the Reynolds Number

In laminar flow the most influentialfactor is the magnitude of the viscous forces:

$$viscous\, forces \propto \mu\frac v l l^2 = \mu vl$$

where $v$ is a characteristic velocit and $l$ is a characteristic length.

In turbulent flow viscous effects are not significant but inertia effects (mixing, momentum
exchange, acceleration of fluid mass) are.
Interial forces can be represented by $F = ma$

\begin{align*}
m &\propto \rho l^3 \\
a &= \frac{dv}{dt} \\
&\therefore a \propto \frac v t \text{ and } t = \frac l v \\
&\therefore a \propto \frac {v^2} l \\
&\therefore \text{Interial forces} \propto \rho l^2\frac{v^2} l = \rho l^2v^2
\end{align*}

The ratio of internalforces to viscous forces is called the Reynolds number and is abbreviated to
Re:

$$\Rey = \frac{\text{interial forces}}{\text{viscous forces}} = \frac {\rho l^2v^2}{\mu vl} = \frac {\rho vl} \mu$$

where $\rho$ and $\mu$ are fluid properties and $v$ and $l$ are characteristic velocity and length.

- During laminar flow, $\Rey$ is small as viscous forces dominate.
- During turbulent flow, $\Rey$ is large as intertial forces dominate.

\textRey is a non dimensional group.
It has no units because the units cancel out.

Non dimensional groups are very important in fluid mechancics and need to be considered when scaling
experiments.

If \textRey is the same in two different pipes, the flow will be the same regardless of actual
diameters, densities, or other properties.

#### \textRey for a Circular Section Pipe

The characteristic length for pipe flow is the diameter $d$ and the characteristic velocity is
mean flow in the pipe, $v$, so \textRey of a circular pipe section is given by:

$$\Rey = \frac{\rho vd} \mu$$

For flow in a smooth circular pipe under normal engineering conditions the following can be assumed:

- $\Rey < 2000$ --- laminar flow
- $2000 < \Rey < 4000$ --- transition
- $\Rey > 4000$ --- fully turbulent flow

These figures can be significantly affected by surface roughness so flow may be turbulent below
$\Rey = 4000$.

# Euler's Equation

In a static fluid, pressure only depends on density and elevation.
In a moving fluid the pressure is also related to acceleration, viscosity, and shaft work done on or
by the fluid.

$$\frac 1 \rho \frac{\delta p}{\delta s} + g\frac{\delta z}{\delta s} + v\frac{\delta v}{\delta s} = 0$$

## Assumptions / Conditions

The Euler euqation applies where the following can be assumed:

- Steady flow
- The fluid is inviscid
- No shaft work
- Flow along a streamline

# Bernoulli's Equation

Euler's equation comes in differential form, which is difficult to apply.
We can integrate it to make it easier

\begin{align*}
\frac 1 \rho \frac{\delta p}{\delta s} + g\frac{\delta z}{\delta s} + v\frac{\delta v}{\delta s} &= 0
 & \text{(Euler's equation)} \\
\int\left\{\frac{\mathrm{d}p} \rho + g\mathrm{d}z + v\mathrm{d}v \right\} &= \int 0 \,\mathrm{d}s \\
\therefore \int \frac 1 \rho \,\mathrm{d}p + g\int \mathrm{d}z + \int v \,\mathrm{d}v &= \int 0 \,\mathrm{d}s \\
\therefore \int \frac 1 \rho \,\mathrm{d}p + gz + \frac{v^2}{2} &= \text{constant}_1
\end{align*}

The first term of the equation can only be integrated if $\rho$ is constant as then:

$$\int \frac 1 \rho \,\mathrm{d}p = \frac 1 \rho \int \mathrm{d}p = \frac p \rho$$

So, if density is constant:

$$\frac p \rho + gz + \frac{v^2}{2} = \text{constant}_2$$

## Assumptions / Conditions

All the assumptions from Euler's equation apply:

- Steady flow
- The fluid is inviscid
- No shaft work
- Flow along a streamline

But also one more:

- Incompressible flow

## Forms of Bernoulli's Equation

### Energy Form

This form of Bernoulli's Equation is known as the energy form as each component has the units
energy/unit mass:

$$\frac p \rho + gz + \frac{v^2}{2} = \text{constant}_2$$

It is split into 3 parts:

- Pressure energy ($\frac p \rho$) --- energy needed to move the flow against the pressure
  (flow work)
- Potential energy ($gz$) --- elevation
- Kinetic energy ($\frac{v^2}{2}$) --- kinetic energy

### Elevation / Head Form

Divide the energy form by $g$:

$$\frac\rho{\rho g} + z + \frac{v^2}{2g} = H_T$$

where $H_T$ is constant and:

- $\frac{p}{\rho g}$ --- static/pressure haed
- $z$ --- elevation head
- $\frac{v_2}{2g}$ --- dynamic/velocity head
- $H_T$ --- total head

- Each term now has units of elevations
- In fluids the elevation is sometimes called head
- This form of the equation is also useful in some applications

### Pressure Form

Multiply the energy form by $\rho$ to give the pressure form:

$$p + \rho gz + \frac 1 2 \rho v^2 = \text{constant}$$

where:

- $p$ --- static pressure (often written as $p_s$)
- $\rho gz$ --- elevation pressure
- $\frac 1 2 \rho v^2$ --- dynamic pressure

- Density is constant
- Each term now has the units of pressure
- This form is useful is we are interested in pressures

### Comparing two forms of the Bernoulli Equation (Piezometric)

$$\text{piezometric} = \text{static} + \text{elevation}$$

Pressure form:

\begin{align*}
p_s + \rho gz + \frac 1 2 \rho v^2 &= \text{total pressure} \\
p_s + \rho gz  &= \text{piezometric pressure}
\end{align*}

Head form:

\begin{align*}
\frac{p_s}{\rho g} + z + \frac{v^2}{2g} &= \text{total head} \\
\frac{p_s}{\rho g} + z &= \text{piezometric head}
\end{align*}
