From 373cb801112cfcb1c4ec7374b4594ea6f875b038 Mon Sep 17 00:00:00 2001 From: Alvie Rahman Date: Wed, 17 Nov 2021 14:34:49 +0000 Subject: [PATCH] split fluid dynamics into separate file --- mechanical/mmme1048_fluid_dynamics.md | 386 ++++++++++++++++++++++++ mechanical/mmme1048_fluid_mechanics.md | 387 +------------------------ 2 files changed, 390 insertions(+), 383 deletions(-) create mode 100755 mechanical/mmme1048_fluid_dynamics.md diff --git a/mechanical/mmme1048_fluid_dynamics.md b/mechanical/mmme1048_fluid_dynamics.md new file mode 100755 index 0000000..d174d13 --- /dev/null +++ b/mechanical/mmme1048_fluid_dynamics.md @@ -0,0 +1,386 @@ +--- +author: Alvie Rahman +date: \today +title: MMME1048 // Fluid Dynamics +tags: [ uni, nottingham, mechanical, engineering, fluid_mechanics, mmme1048, fluid_dynamics ] +--- + +\newcommand\Rey{\mbox{\textit{Re}}} % Reynolds number +\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. + +
+ + +### 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. + + + +![](./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. + +
+ +## 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*} diff --git a/mechanical/mmme1048_fluid_mechanics.md b/mechanical/mmme1048_fluid_mechanics.md index ff5d42d..9dbcea3 100755 --- a/mechanical/mmme1048_fluid_mechanics.md +++ b/mechanical/mmme1048_fluid_mechanics.md @@ -1,14 +1,14 @@ --- author: Alvie Rahman date: \today -title: MMME1048 // Fluid Mechanics -tags: [ uni, nottingham, mechanical, engineering, fluid_mechanics, mmme1048 ] +title: MMME1048 // Fluid Mechanics Intro and Statics +tags: [ uni, nottingham, mechanical, engineering, fluid_mechanics, mmme1048, fluid_statics ] --- \newcommand\Rey{\mbox{\textit{Re}}} % Reynolds number \newcommand\textRey{$\Rey$} -# Properties of Fluids (2021-10-06) +# Properties of Fluids ## What is a Fluid? @@ -157,7 +157,7 @@ The -ve sign indicates that as $z$, height, increases, $p$, pressure, decreases. # Fluid Statics -## Manometers (2021-10-13) +## Manometers ![](./images/vimscrot-2021-10-13T09:09:32,037006075+01:00.png) @@ -469,382 +469,3 @@ 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. - -
- - -#### 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. - - - -![](./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. - -
- -### 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*}