Consolidation Week

Over the course of this term we have covered many areas of fluid dynamics. In this consolidation blog I aim to revisit some of these, pinpointing particularly important areas. I shall include some more example questions I have completed, a few definitions and comment on areas I may still need to work on.

Pathlines, Streamlines and Streaklines:

For this topic I think it is vital to remember the key definitions:

Pathline: The resulting trajectory when a marked particle is thrown into a flow.

Streamline: A curve tangential to the fluid velocity.

Streakline: The injection of dye or bubbles into a fluid at a fixed point at different time intervals. The streakline itself is the path connecting all the points formed in time order.

When I first looked at this topic at the start of term I used my notes to help me in my calculations. Now I am able to answer both simple and more complex questions on this topic independently. Completing Example sheet 0 and various questions on the internet has enabled me to feel quite confident with this topic.

I have attached my workings to Question 3 to demonstrate this:

 

Flow Kinematics and Visualisation:

Flow visualisation around a car

I didn’t write a blog entry specifically on this material but flow visualisation is an area that is important in most of the material we have covered. I have practised numerous questions which involve the sketching of Pathlines, streamlines and isopotential lines but in some cases I do still struggle. For me this is an area I need to practise again and again. However, I am quite comfortable with working out the kinematics of the flow.

I have included my workings to Example Sheet 1, Question 2:

  

Stream Function and the Principle of Superposition:

I enjoyed learning about different flows in this topic such as a source, a sink, a vortex and a doublet. I found some of the questions on this topic a little more difficult but my class notes were really helpful in improving my understanding. It is important to know both what the principle of superposition is and how to use it. I have added the definition below:

The Principle of Superposition: If the velocity fields u₁ and u₂ have associated stream functions ψ₁ and ψ₂ then the superposed velocity field u₁ + u₂ has stream function: ψ₁ + ψ₂. Put more simply, we get another flow by adding two stream functions together.

Dimensional Analysis and Buckingham 𝜋 Theorems:

Once I memorised the dimension table for this topic I found it quite straight forward. The flash cards I made to help me learn the dimensions were extremely helpful and I still go through them every so often. I enjoyed answering the Buckingham 𝜋 questions and found plenty of examples online to work through as well as the example sheet questions.

Similarity in Fluids

What I liked most about this topic was that it is very useful in many areas. When designing boats, cars or aeroplanes models that are geometrically similar to the prototype are always made. Because this area is so relevant I found many examples and feel confident with these calculations.

Euler’s Equation

I shall start by defining Euler’s equation again:

This is a relationship between velocity, density and pressure for an inviscid flow. We did a nice example in class of the force on a lock gate and I found it intriguing to discover the mathematics behind this.

Bernoulli’s Equation

Bernoulli’s equation plays a vital part in many of the topics we covered and it is very important to remember what it is:

Because we have looked at it quite a few times I feel that I understand how to use Bernoulli’s equation and the assumptions related to it (inviscid, incompressible and steady flow with negligible height change).

Fluid Flow around a Cylinder

Fluid flow around a cylinder

This was one of the topics where I did quite a lot of further research as to why we were studying a cylinder. I found many interesting uses such as the NASA article I mentioned in my blog entry. As for the calculations themselves I was relatively content carrying them out, although I didn’t get as many questions as I would have liked done as it often took me a while to simplify the acceleration vectors for the given flow! This is a topic I would like to work on more thoroughly over the Christmas period.

Velocity potential and the Stream Function

I found the velocity potential topic relatively straight forward and there wasn’t too much to remember! The involvement of the Cauchy-Riemann equation wasn’t a problem as I was familiar with these from my previous Vector Calculus module. The only issue was interpreting my answer in order to sketch the streamlines and isopots.

Water Waves

I found that there was quite a lot to learn in this topic. Recalling how to calculate the potential, dispersion relation, phase speed and group velocity for both Infinite and finite depth is quite challenging and as I mentioned in the related entry it is definitely an area I need to look at over the holidays.

Open Channel flows

A river, an example of an open channel flow

The final topic covered this term was Open Channel Flows. It was a nice one to end on and I feel as though I understood all of the taught material and how to apply what I know to questions. This was my favourite topic as it is easy to visualise an open channel flow and I found that drawing sketches really helped me carry out examples.

I have attatched my solution to Sheet 5 Question 1 to display my understanding:

Overall Summary:

All in all I think I have understood and learnt how to apply a great deal of the material covered this term. Looking back over my blog entries and carrying out questions from the Example Sheets and online has allowed me to progress so that I can now carry out questions confidently and independently.

I have highlighted 3 main areas I feel that I’m weakest at in order to tailor my revision appropriately:

• Sketching graphs
• Water Waves
• Fluid flow around a cylinder

In the lead up to next term I hope not only to concentrate on the material covered this term, but also to look ahead at what we will be covering up until Easter.

References:

http://wallpho.com/36721-beautiful-river-forest-id-23838.htm (Open Channel Flow image)

http://www.cs.swan.ac.uk/~csbob/research/application/image/bmw.jpg (Flow visualisation car image)

http://www.scienceclarified.com/photos/bernoulli-s-principle-2803.jpg (Fluid flow image)

http://upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Potential_cylinder.svg/2000px-Potential_cylinder.svg.png (Fluid flow around a cylinder image)

Open Channel Flows

An Open Channel Flow is the flow of a fluid with a free surface open to the atmosphere. Examples include streams and rivers. Open channel flow assumes that the pressure at the surface is constant.

The volume flow rate is the volume of fluid that flows past a given cross sectional area per second. It is defined:

We also have an equation for the volume/ unit width, Q:

Where:

We can derive the specific energy, E, from the Bernoulli equation along a free surface as follows:

Specific Energy Function:

We can rearrange Q to get:

and substituting this into our equation for specific energy gives:

We are able to calculate the first and second derivatives of E to find out turning points for when the equation is plotted.

We calculate a value, Emin, using the critical value of h and rearranging E. This gives us:

Using our curve E and our value Emin we are able to produce three different graphs which help us to classify the flows:

Froude Number:

The Froude number, Fr, is a dimensionless value that describes flows in an open channel. It is a ratio of inertial and gravitational forces.

It is useful to calculate the Froude Number for our given flow. It is defined:

If…

• Froude number <1 : subcritical, implies deep and slow flow.
• Froude number =1 : critical
• Froude number >1 : supercritical, implies shallow and fast flow.

I have included an example from the notes which clearly demonstrates how to carry out the above calculations and classify flows:

    

From this calculation it is easy to see how we use the volume flow rate, volume / unit width and specific energy to classify flows and how it is also useful to find the Froude number as this checks our results.

Reflection from last week:

As I had anticipated I didn’t get much of a chance, due to by interviews, to do much practise of last weeks water waves material. However, as we have now finished lectures for the term I plan to look at this area in great detail over the next few weeks. I did do another question on this weeks topic though, Example Sheet 5, Question 2 and have included my workings:

I enjoyed completing the question and would even say it is my favourite topic from this term! I think perhaps this is because it is quite easy to visualise an open channel flow (such as a river) and the material is more interpretable when there are strong real life applications.

What to work on this week:

As we have consolidation week next week followed by three weeks off I would like to look over all my previous blog entries. I shall carry out a range of questions on a variety of topics (and hopefully include some examples in next weeks consolidation blog).

References:

http://faculty.wwu.edu/vawter/physicsnet/Topics/Pressure/VolumeFlowRate.html

http://www.fsl.orst.edu/geowater/FX3/help/8_Hydraulic_Reference/Open_Channel_Flow.htm

Water Waves: Infinite and Finite depth

Recently in lectures we have been exploring water waves and how we can model them. In this blog entry I shall clarify related definitions and discuss how we would carry out calculations for waves of both infinite and finite depth.

Here is a graphical representation:

In order to model these water waves we must first take note of the following assumptions:

We use Bernoulli’s equation to find boundary conditions which gives us the general solution:

We can then calculate the following for both infinite and finite depths:

Potential:

As discussed in my previous blog entry, velocity potential 𝜙 is useful in analysing fluid flow when the flow is assumed irrotational.

Dispersion Relation:

Is the relationship between angular frequency (ω) and wavenumber (k) . Two different forces; gravity and surface tension give rise to the dispersion relation. (Note: we assume no surface tension in our model).

Phase Speed:

The speed at which the phase of a wave is propagated.

Group velocity:

The speed at which wave “packets” travel (i.e. The speed at which wave energy travels).

Infinite Depth:

Finite Depth:

Particle Paths:

The particle paths are calculated using the equations for pathlines:

We can then calculate the paths for infinite and finite depths.

Circular particle motion (infinite depth)

Motions:

• Infinite depth:
• -particles have circular motions
• -amplitude decreases with depth
• -maximum at surface with radius; kA/ω

• 

  (A) Circular particle motion (infinite depth) compared to (B) elliptical particle motion (finite depth)

  Finite depth:

• -particles have elliptical motions
• -amplitude decrease with depth
• -maximum is at the surface

• Shallow water:
• -particles have side to side motions
• -kh0 is a lot smaller than 1
• -amplitude is independent of z0,x0

Reflection from last week:

Last week I stated that I wanted to do some more velocity potential questions. I found the the calculations to be similar to ODE’s last year so the questions didn’t take too long. I have included my answer to the second question of Example sheet 6:

What to work on next week:

I have a busy week next week with interviews, so i’m hoping to do some Example questions when I get a chance!

References:

https://www.haverford.edu/physics/…/211-7DispersionRelation.doc

www.thefreedictionary.com/phase+speed

http://faculty.gvsu.edu/videticp/waves.htm (circular motion image)

http://www.newworldencyclopedia.org/entry/Wave (circular and elliptical motion image)

Velocity Potential and Stream Function

This week we have been looking at Velocity Potential and the Stream Function. Flow in which the velocity of flow is the gradient of a scalar function is known as the velocity potential. Velocity potential is a powerful tool in the analysis of irrotational flows. The flow must also be incompressible.

Here is the stream function, which we have come across previously:

This is our velocity potential:

If we combine the streamfunction and the velocity potentials, this implies the Cauchy Riemann equations:

Using these equations we are then able to calculate the velocity potential and analyse various flows.

We can find and sketch both the streamlines and the isopotential lines (Isopots) for the flow. It is important to note that in all cases the streamlines and isopots intersect at right angles.

I shall use some examples from class to demonstrate how we perform these calculations and the relevant sketches for a uniform flow, line source and line vortex:

Uniform Flow:

 

Line Source:

 Line Vortex:

I found that recapping these examples was a useful way of revising this topic. I found the calculations themselves rather straight forward, my main difficulty is interpreting the data so I can sketch the graphs.

Reflection from last week:

Last week I stated that I would like to do some extra reading and discover some interesting applications for fluid flow around a cylinder. I came across a useful article on the NASA website and have included the link: http://www.grc.nasa.gov/WWW/k-12/airplane/cyl.html The dynamics were slightly different to what we have been covering but I found the “Historical Note” section at the end particularly interesting. It stated that in the early 1920’s the force from a rotating cylinder was used to power a sailing ship as a replacement for sails. The power was less than that of a standard propeller but it did work!

What to work on this week:

This week I would quite like to do some more example questions on the Velocity Potential and Stream Function topic. I feel this is not too difficult an area and to lose marks would be a shame, hence the extra practise!

References:

 http://www-mdp.eng.cam.ac.uk/web/library/enginfo/aerothermal_dvd_only/aero/fprops/poten/node22.html

http://www.grc.nasa.gov/WWW/k-12/airplane/cyl.html

Fluid Flow around a Cylinder

This week we have been looking at the fluid flow around a cylinder and how Bernoulli’s equation can be applied to this.

For this model to be valid the fluid flow must be tangential to the cylinder. When an ideal fluid flows around a cylinder, the stream lines and velocity potentials can be represented as a doublet and vortex placed in a constant, horizontal uniform flow. It follows that we may use a doublet and vortex to study the flow pattern around a cylinder. The resulting expression is:

Potential Flow:

Potential flow is a frictionless, irrotational flow. All real fluids have some amount of viscosity but if it is small enough then the frictional effects may be negligible. Therefore potential flow can be useful for finding flows over solid surfaces such as our cylinder.

The flow velocity of a fluid describes the motion of a fluid, here are the flow velocities for our flow around a cylinder:

We set r=a assuming the radius of the cylinder is r, therefore cancels out and we are just left with:

Since our fluid is invisicid and irrotational, Bernoulli’s equation allows us to calculate the pressure:

Where Ps is stagnation pressure.

Stagnation pressure:

The stagnation pressure is the pressure when the fluid comes to rest (ie. The speed = 0). The pressure at any point in the flow never exceeds this pressure.

Two of the forces acting upon our cylinder are lift and drag:

Note: In the context of potential flow theory, drag is always 0.

Reflection from last week:

I tried a few more complex questions but didn’t get as many done as I would have liked due to numerous coursework deadlines. I did however do another example from Example Sheet 3 now that we have covered more on Bernoulli, here is my solution to Question 2:

  

I wasn’t sure how to do part (c) of the question: Sketch the streamline pattern for the case m = 1,a = 0. So I shall have another go at this in the future when I have a better understanding of the topic.

What to work on this week:

I would like to have a look at why we study flow around a cylinder and see some practical applications of this as it is an area that interests me. I will also try and do some more example questions.

References:

www.freestudy.co.uk/fluid%20mechanics/t5203.pdf

https://www.princeton.edu/~asmits/Bicycle_web/Bernoulli.html

faculty.poly.edu/~rlevicky/Handout14_6333.pdf

http://en.wikipedia.org/wiki/Potential_flow_around_a_circular_cylinder (flow image)

The Bernoulli equation

A special form of the Euler’s equation derived along a fluid flow streamline is often called the Bernoulli Equation named after its discoverer, the Swiss scientist Daniel Bernoulli. It describes the relationship between pressure and speed:

 

Where:

• u is speed
• p is pressure
• ρ is density
• 𝛟 represents forces

Increase in speed results from a decrease in pressure and a decrease in speed is due to an increase in pressure.

Bernoulli’s equation relates a moving fluid’s pressure, density, speed, and height from Point 1 to Point 2 in this way:

Where:

• u is speed
• p is pressure
• ρ is density
• g is gravity
• z is height of fluid

For the Bernoulli equation to apply, we assume that the flow must be:

• inviscid
• incompressible
• steady
• negligible height change

For Bernoulli’s equation to be valid not only along a streamline but everywhere else in the flow, it must also be irrotational.

Another useful application of the Bernoulli equation is in the derivation of Torricelli’s law for flow out of a sharp edged hole in a reservoir.

Torricelli’s Law:

Also known also as Torricelli’s theorem relates the speed of fluid flow from a tank opening to the height of fluid above the opening. The maximum possible drainage rate for a tank with a tap at the bottom can be calculated directly from Bernoulli’s equation, and is found to be proportional to the square root of the height of the fluid in the tank.

In class this week we looked at how Bernoulli’s equation can be used in practical situations, I have included an example of flow from a reservoir:

 

Adapting Bernoulli’s Equation:

The Bernoulli equation can be adapted to use on flows that are unsteady and compressible but the flow must still remain inviscid. Compressibility effects depend on speed of the flow relative to the speed of sound in the fluid. This is determined by the Mach number.

What to work on:

I have enjoyed studying Bernoulli’s equation this week as I like to see the practical uses of what we are covering. I have carried out my own calculations on Bernoulli’s equation when applied to a hose pipe:

I shall continue to try out some more questions using Bernoulli’s equation but as I feel pretty comfortable with this topic I shall try out some more complex examples on topics covered this term so far.

Reflection from last week:

Last week I spent time consolidating what we have covered so far, but I only looked at the more basic questions so that I had time to try out exercises from a range of topics. For this reason I have decided to try more in depth questions this week. I have attached my solution to Example Sheet 3 Question 1 as this is an area i’ve been working on over the last few days:

References:

https://www.boundless.com/physics/textbooks/boundless-physics-textbook/fluid-dynamics-and-its-applications-11/bernoulli-s-equation-99/application-of-bernoulli-s-equation-pressure-and-speed-357-4588/

http://cavity.ce.utexas.edu/kinnas/COURSES/ce319/ebook/stube/stube.html

http://www.peaseofgarforth.co.uk/products/2422-30m-hose-reel-with-15m-hose-pipe (hose pipe image)

https://www.teachengineering.org/view_lesson.php?url=collection/cub_/lessons/cub_bernoulli/cub_bernoulli_lesson01.xml (Bernoulli pipe image)

Euler’s Equation

This week we have been introduced to Euler’s equation which is stated below:

We are able to derive a relationship between velocity and pressure for an inviscid flow using Newton’s second law of motion. The resulting equation, known as Euler’s Equation, was first derived by Leonhard Euler in the 1750’s.

Euler’s equation describes how the velocity, pressure and density of a flow are related. It can be applied to compressible and incompressible flows.

We can use it to carry out calculations eg. The force on a lock gate:

 

It is however, a simplified version of the Navier-Stokes equations of fluid dynamics. The Euler equation doesn’t account for the viscosity of the fluid which is included in the Navier-Stokes equations. A solution of the Euler equation is therefore only an approximation to a real fluids problem.

It is noted that Euler’s equation is useful in deriving other results, for example the Vorticity Equation for inviscid flows, Kelvins Theorem and Bernoulli’s Equation.

Vorticity Equation: The Vorticity Equation of fluid dynamics describes the change in the spinning motion of a particle as it moves in a flow.

Kelvin’s Theorem: This states that the circulation around a closed loop in an inviscid fluid is independent of time.

Bernoulli’s Equation: The significance of the Bernoulli equation is that when velocity increases in a fluid stream, pressure decreases and when velocity decreases, pressure increases. Bernoulli’s principle is valid for incompressible flows and also for compressible flow and can be applied, for example, to calculate lift force of an aerofoil or the rate at which a tank drains.

What I shall work on this week:

As it is currently consolidation week I feel it is important to recap all the material covered so far this term. Primarily I shall look over my previous blogs as I believe this is a useful revision tool. I intend to continue working through examples and hopefully look ahead to future topics in order to have a firm basis for the rest of this term’s work.

Reflection from last week:

I have made myself the dimensional analysis flash cards that I mentioned last week. They have helped a great deal. Every day since I made them I have gone through them a few times and now I have memorised all the dimensions! I have attached a photo below:

References:

https://www.grc.nasa.gov/WWW/K-12/airplane/eulereqs.html

http://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)

http://nptel.ac.in/courses/112104118/lecture-12/12-3_euler_eqn_motion.htm

http://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node59.html

Similarity in Fluid Dynamics

The material I shall cover in todays blog shows how we can draw conclusions about a prototype from a model with similar features. I shall build upon the previous Buckingham π groups blog entry, define relevant terms and use worked examples to display similarity in fluid dynamics.

Model: A lot of the time the structure itself is too complicated to be analysed mathematically and so models must be made. The actual structure is known as the prototype and the model is usually built to be geometrically similar. Measurements and analysis can be carried out on the model and used to calculate values for the prototype.

Geometric Similarity: Geometric similarity exists between a model and a prototype if the ratio between all dimensions in the model and prototype are the same:

where λ_L is the scale factor for length.

Kinematic Similarity: Kinematic similarity is the similarity of time between the model and prototype, ie) the streamlines are the same:

Dynamic Similarity: Dynamic similarity exists between both the geometric and kinematic and also if the forces in the model and prototype are the same:

I shall demonstrate the use of similarity of the drag resistance on a submarine:

 

Free-surface models: When trying to models fluids which have free surfaces we must take into account the effect of gravity and the major governing non-dimensional number becomes the Froude (Fn) number:

To conclude I have added a numerical dynamic similarity example for water-borne missiles:

Reflection from last week: Although I worked on dimensional analysis and carried out questions using the dimensional analysis table, I am still slightly struggling to memorise the dimensions.

What to work on this week: I have decided to make some flash cards over the next week in order to help me memorise these dimensions.

References:

http://www.efm.leeds.ac.uk/CIVE/FluidsLevel1/Unit04/T4.html

http://blisty.cz/art/45218.html (missile picture)

Dimensional Analysis and Buckinghams π groups

This week we began to look at dimensional analysis and Buckinghams π groups.

Dimensional Homogeneity: This means that physical equations are only valid if both sides of the equation have the same dimensions. This is a useful characteristic as it allows us to carry out dimensional analysis.

Fundamental Units:

length = L (metre in SI units)

mass = M (kilogramme in SI)

time = T (second in SI)

temperature = Θ (Kelvin in SI)

electric current = A (amp in SI)

luminous intensity = I (candela in SI)

amount of substance = N (mole in SI)

Below are some dimensions for the common physical properties:

Buckingham’s π Theorems: The Buckingham π theorems are key in dimensional analysis. It is a way of finding dimensionless groups from the variables in the problem. The first theorem states m variables can be expressed as a relationship between m-n non dimensional groups of variables (these variables are the π groups and n is the number of fundamental dimensions required to express the variables). The second theorem tells us each dimensionless group is a function of n governing or repeating variables plus one of the remaining variables.

Common π groups: The common π groups, listed below are groups that are found in many different dimensional analysis problems. They are important for us to note.

What to work on this week: This week so far I have been finding and working through some examples requiring dimensional analysis to find Buckingham π groups. Here is an example of my revision:

 

For the remainder of the week I shall recap the different dimensions from the analysis table as I feel it is very important for this topic to know this.

Reflection from last week: Last week I stated that I would like to work on my graphical representations. I have tried my best to do this by finding examples of streamlines for different flows online to see how they sketched the graph from their results. I also wanted to do some questions on the Stream Function and the Principle of Superposition which I have done. To demonstrate this I have attatched my solution to Example Sheet 2, question 1 below:

Once I understood what to do I enjoyed solving questions on this topic.

References:

Fluid dynamics class notes, D. Graham (table of dimensions)

 

Stream Function and the Principle of Superposition

In this weeks blog entry I shall be reflecting on some of the material covered in lectures and clarifying related definitions. Notably I will be looking over the stream function and the principle of superposition.

The stream function tells us the flow speed and can be defined for both 2 and 3 dimensional flows. The stream function gives us streamlines (lines parallel to velocity, see previous blog for more details). The stream function is defined:

We can check incompressibility:

In class this week we looked at two examples where the streamfunction is used in order to give us streamlines. We are able to graphically display our streamline solutions to ψ=constant. It is visible from these graphs that streamlines are equally spaced for equally spaced ψ values and get closer together for uniformly spaced ψ values.

Some useful definitions:

Source: A source is a point from which a flow is issued at the same rate in all directions. (3d definition)

Sink: A sink is a point into which fluid flows at the same rate from all directions. (3d definition)

Vortex: An area in a fluid where the flow is rotating.

Image displaying a vortex and a doublet and their directions of flow.

The principle of superposition: If the velocity fields u₁ and u₂ have associated stream functions ψ₁ and ψ₂ then the superposed velocity field u₁ + u₂ has stream function: ψ₁ + ψ₂. Put more simply, we get another flow by adding two stream functions together.

Doublet: A doublet is the flow which results when the distance between the source and the sink are decreased and approach zero.

The image on the near right is a graphical representation of a source and sink and the figure on the far right shows the streamlines of this combined flow.

Clicker Quiz and what to work on: This week one of the clicker quiz questions was to identify the shape of the streamlines for the flow: u=(x,-y). The answer was hyberbolae which I did not know and this has highlighted to me that in this weeks revision I should target graphical representations. I would also like to do some questions on this weeks topic of the Stream Function and Principle of Superposition.

Reflection from last week: Last week I thought it would be beneficial to do some questions on Pathlines and Streamlines and over the course of the week I worked on questions from Example Sheet 0. I have attatched my solutions to question 1 and 2 below:

I found the calculations themselves relatively straight forward. To start with I referred to my notes but as I went on I became more confident in answering the questions. I wasn’t totally sure how to sketch the streamlines for 1a which again hints that I must work on my graphical representation.

References:

http://www.ecourses.ou.edu/cgi-bin/ebook.cgi?doc&topic=fl&chap_sec=07.4&page=theory (doublet/ vortex image)