Add some fluid dynamics

mechanical/mmme1048_fluid_mechanics.md

@@ -5,7 +5,10 @@ title: MMME1048 // Fluid Mechanics
 tags: [ uni, nottingham, mechanical, engineering, fluid_mechanics, mmme1048 ]
 ---
 
-# Lecture 1 // Properties of Fluids (2021-10-06)
+\newcommand\Rey{\mbox{\textit{Re}}}  % Reynolds number
+\newcommand\textRey{$\Rey$}
+
+# Properties of Fluids (2021-10-06)
 
 ## What is a Fluid?
 
@@ -152,7 +155,9 @@ The -ve sign indicates that as $z$, height, increases, $p$, pressure, decreases.
 - It is usually better to use SI units
 - If in doubt, DA can be useful to check that your answer makes sense
 
-# Lecture 2 // Manometers (2021-10-13)
+# Fluid Statics
+
+## Manometers (2021-10-13)
 
 ![](./images/vimscrot-2021-10-13T09:09:32,037006075+01:00.png)
 
@@ -193,7 +198,7 @@ If $\rho_a << \rho_2$:
 
 $$\rho_{1,gauge} \approx \rho_2g\Delta z_2$$
 
-## Differential U-Tube Manometer
+### Differential U-Tube Manometer
 
 ![](./images/vimscrot-2021-10-13T09:37:02,070474894+01:00.png)
 
@@ -231,7 +236,7 @@ p_1 - p_2 &= \rho_wg(z_C-z_B-z_C+z_A) + \rho_mg\Delta z \\
 &= -\rho_wg\Delta z + \rho_mg\Delta z 
 \end{align*}
 
-## Angled Differential Manometer
+### Angled Differential Manometer
 
 ![](./images/vimscrot-2021-10-13T09:56:15,656796805+01:00.png)
 
@@ -247,7 +252,7 @@ p_1 - p_2 &= \rho_wg(z_C-z_B-z_C+z_A) + \rho_mg\Delta z \\
 <details>
 <summary>
 
-## Exercise Sheet 1
+### Exercise Sheet 1
 
 </summary>
 
@@ -333,13 +338,13 @@ p_1 - p_2 &= \rho_wg(z_C-z_B-z_C+z_A) + \rho_mg\Delta z \\
 
 </details>
 
-# Lecture 3 // Submerged Surfaces
+## Submerged Surfaces
 
-## Prepatory Maths
+### Prepatory Maths
 
-### Integration as Summation
+#### Integration as Summation
 
-### Centroids
+#### Centroids
 
 - For a 3D body, the centre of gravity is the point at which all the mass can be considered to act
 - For a 2D lamina (thin, flat plate) the centroid is the centre of area, the point about which the
@@ -374,13 +379,13 @@ Take the following lamina:
 <details>
 <summary>
 
-#### Example 1
+##### Example 1
 
 Determine the location of the centroid of a rectangular lamina.
 
 </summary>
 
-##### Determining Location in $y$ direction
+###### Determining Location in $y$ direction
 
 ![](./images/vimscrot-2021-10-20T10:14:17,688774145+01:00.png)
 
@@ -398,7 +403,7 @@ Determine the location of the centroid of a rectangular lamina.
 
    so $$y_c = \frac 1 {bd} \frac {bd} 2 = \frac d 2$$
 
-##### Determining Location in $x$ direction
+###### Determining Location in $x$ direction
 
 ![](./images/vimscrot-2021-10-20T10:24:48,372189101+01:00.png)
 
@@ -416,7 +421,7 @@ Determine the location of the centroid of a rectangular lamina.
 
 </details>
 
-## Horizontal Submereged Surfaces
+### Horizontal Submereged Surfaces
 
 ![](./images/vimscrot-2021-10-20T10:33:16,783724117+01:00.png)
 
@@ -428,7 +433,7 @@ Assumptions for horizontal lamina:
 
 ![](./images/vimscrot-2021-10-20T10:36:12,520683729+01:00.png)
 
-## Vertical Submerged Surfaces
+### Vertical Submerged Surfaces
 
 ![](./images/vimscrot-2021-10-20T11:05:33,235642932+01:00.png)
 
@@ -451,7 +456,7 @@ Therefore total force is
 
 $$F_p = \int_{area}\! \rho gh \,\mathrm{d}A = \int_{h_1}^{h_2}\! \rho ghw\,\mathrm{d}h$$
 
-### Finding Line of Action of the Force
+#### Finding Line of Action of the Force
 
 ![](./images/vimscrot-2021-10-20T11:15:51,200869760+01:00.png)
 
@@ -464,3 +469,382 @@ M_{OO} &= F_py_p = \int_{area}\! \rho gh^2 \,\mathrm{d}A \\
 \\
 y_p = \frac{M_{OO}}{F_p}
 \end{align*}
+
+# Fluid Dynamics
+
+## 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*}