Week 12 – Vorticity and Final Reflection

This week is consoladation week, or as many students come to think of it as, deadline week. As it suggests all mathematics students have several deadlines this week. But within the chaos of this final week I have managed to squeeze in some time for Fluid Dynamics and this blog.

I have been working on vorticity and circulation this week. We briefly touched on this in lectures so I have decided to self-study to help with next terms understanding of the topic.

Vorticity is the swirling motion of a fluid. An obvious example of vorticity is a tornado. The Photograph below shows a tornado that hit South Dakota. This tornado was classed as an F3 tornado with winds reaching up to 206mph. The diagram shows how the winds spiral to form a tornado.

Photo: Tornado on the South Dakota prairie

I personally am very excited to study this, the natural phenomenon of tornadoes, in more detail at the start of next term.

Final Reflection

In my final reflection I will look back at the blog as a whole. I will share with you how I have found writing it, what i have learned, and what I can take into working life.

Before that though I must reflect upon this blog. As I mentioned at the start this week has been very busy, so as a result I have not spent the same amount of time researching and completing as many exercises as in previous weeks. I believe I have a good understanding but more practice may be needed in order to master the topic for the exam.

Now my reflection for the entire blog. At first, if I am honest with myself, I was nervious about writing my own blog as I have never done anything similar before. As the first weeks progress I believe I have improved the quality and relevance of each week. As a result I have enjoyed the challenge and grown into this work.

This coursework assessment has been a good way of going over the material of each weeks lectures. As a result I believe I am in a better position for the final year exams than in other modules.

Writing this blog has improved how I articulate my work. It has improved my research abilities and my time management. All of these skills I can transfer into the working world after I finish my degree.

Overall this has been an enjoyable task that has helped me learn much of the first term material, and also helped me develop key skills for the future.

References

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Week 11 – Water Waves

This week we have looked into water waves. But before we can study them we need to make several assumptions;

  • Channel is very wide so that v=0, as well as d/dy=0.
  • inviscid flow.
  • incompressible flow.
  • gravity is the only body force.
  • flow is irrotational.
  • surface tension affects are negligible.
  • all displacements and velocities are small. Also assumed that wave amplitude<<wavelength.
  • water initially at the free surface remains there.

The assumptions of incompressibility and irrotationality lead to \frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial z^2}=0.  The momentum equations can be reduced to Bernoulli’s equation. The Bernoulli’s equation is once again key to this weeks work.

 The motion of water waves results from the transfer of energy from the atmosphere to water.  Most water waves are generated by wind moving over the water’s surface.  The size of the resulting waves is dependent upon the wind velocity, wind duration, and the fetch (the area and distance over which the wind travels).  Deep water waves have a sinuous pattern as illustrated below.  The highest point on the wave is called the wave crest, and the low point on the water surface between crests is called the wave trough.  The distance from crest to trough is the wave height, and the distance from one crest to another is called the wavelength. 

 
 

 Although deep water waves may visually appear to be moving water over horizontal distances, most of the water is actually moving in a circular pattern.  The water particles return to approximately the same position from which they started. Thus, most water molecules remain in the same general area, and the same can be said for an object floating on the surface in deep water.  As shown in the animation below, the diameters of the circles making up the wave decrease with depth until the wave base is reached at one half the wavelength.

Reflection

I have enjoyed writing this weeks entry to my blog as we have studied water waves, which we can all picture in our minds. This has made it easier to understand what is happening in a water wave and in particular the cyclic circular movement that each molecule travels. Although the worked examples are challenging I believe with more practice I can master this topic.

References

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Week 10 – Potential Flows

I briefly touched upon Potential flows in my last entry, but as we have studied them in more detail in this weeks lectures I will also go into more detail.

In order to define the velocity potential, ϕ, we must assume the curl of the velocity is zero and that the divergence of the velocity is also zero.. In other words the fluid flow is irrotational   and incompressible incom. This implies that it is an Ideal flow in which the velocity potential can be defined as;

velpot

This means that the ideal flow satisfies Laplace’s equation;

laplace

We then find the Cauchy-Riemann equations for the Stream function and the Velocity Potential in cartesian form;

streamfunction3              velocitypot

Below is the equivalent equations but in polar co-ordinates.

streamfunctionpolar        velpotentialpolar

Worked Example

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Reflection

I feel like I have grasped the initial idea of Potential flows but I do not feel like I have perfect understanding yet. I will therefore do more worked examples given in the notes and the portal to fully understand the subject.

References

  • MATH3402 Inviscid Flow Lecture Notes , David Graham
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Week 9 – Flow around a cylinder

This week we spent time studying examples of Bernoulli’s equation. In particular we looked at an example of fluid flow around a cylinder.

This image shows fluid travelling from left to right slows over and around a cylinder. We can see that as the fluid flows around the cylinder, the gap between the streamlines get smaller. Hence the velocity of the fluid increases as it travels around the cylinder.
As we know in practical real life situations the flow cannot be this perfect. There are several other situations that can occur. As seen below, we have two similar situations but the flow of air eddies as in pass the cylinder.

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But as we are studying inviscid flow we can make some observations;

Boundary Conditions

The flow velocity must be tangential to the surface of the cylinder, since no fluid can penetrate through the solid surface of the cylinder. This is the only surface condition for an inviscid flow. This means that for any inviscid flow, streamlines can be thought of as solid surfaces.

Stream Function

Given the stream function;

CodeCogsEqn (16)

We can derive the velocities for the cylinder at r=a, using this equation, to give;

CodeCogsEqn (18)

CodeCogsEqn (19)

We can now substitute the velocities into Bernoulli’s equation to find the pressure;

CodeCogsEqn (20)

upon simplifying this equation we find an expression for P and once plotted we see that there is a relative low-level of pressure on top and a relative high pressure on the bottom of the cylinder. Therefore we can see that the cylinder will be pushed up as a consequence of the flow, in a similar way to an aeroplane wing. Using the definition of the force;

CodeCogsEqn (21)

We can find the lift and drag from this as;

  • Lift = \int_{0}^{2\pi} -p sin(\theta)a d\theta = Vortex strength
  • Drag =  \int_{0}^{2\pi} -p cos(\theta)a d\theta = 0 , due to our assumptions.

Potential Flow

Potential flow describes the velocity field as the gradient of the velocity potential. The Ideal flow and its two components are taken to define this. It is incompressible as we take the divergence and find it to equal zero. We then find the it to be irrotational as the curl is also equal to zero.

Therefore we can assume the existence of a velocity potential, also note that in the case of an incompressible flow we also see the velocity potential satisfies Laplace’s equation. Now using the equations for the stream functions and matching with the velocity potential we can derive;

CodeCogsEqn (23)

These are the Cauchy-Reimann equations which can be used to show that a complex function is differentiable.

Reflection

We worked through various in class examples. I feel going over these examples after the lecture has helped me with the understanding of this weeks work.

References

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Week 8 – Bernoulli Equation and Application

This week we have learned the Bernoulli equation and its Applications. The Bernoullis were a family of merchants and scholars who developed many theories in Mathematics. In particular this week, we have been looking at an equation that Daniel Bernoulli developed.

Derivation of Bernoulli Equation

Streamline Form

To derive Bernoulli’s Equation we must begin with Euler’s equation. We can define force as a potential. Then substituting this into Euler’s equation we get;

CodeCogsEqn (13)

If we now write the convective derivative in its explicit form, and we make the assumption that we have a steady flow, we get;

CodeCogsEqn (14)

Now if we define a unit vector s along a streamline this is parallel to the velocity u and dot product this with both sides we can set our equation equal to zero. This is due to the fact if we dot a vector which is parallel to u we get zero. Now for the left hand side of the above equation, we would then have a directional derivative which is set equal to zero therefore by definition the part in the bracket must be equal to a constant. Therefore we have Bernoulli’s Equation;

CodeCogsEqn (15)

Potential Flow Form

If the flow is irrotational, the curl of the velocity vector is zero. This means we can express the velocity u as the gradient of a velocity potential. Below shows the derivation;

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Worked Example

Below is a worked example of Bernoulli’s equation.

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Reflection

This week has been very productive. I have learned more about Euler’s equation by studying Bernoull’s equation. I have learned how to use Bernoulli’s equation in modelling a steady fluid flow.

References:
Bernoulli equation derivation http://www.princeton.edu/~asmits/Bicycle_web/Bernoulli.html

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Week 7 – Euler’s Equation

Leonhard Euler was a Swiss mathematician and physicist. He made many important discoveries in calculus and graph theory. He pioneered much of the modern mathematical terminology and notation and in particular for mathematical analysis. He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy. As we have found out in our studies, his work was diverse and appears in many of our modules. Euler is considered to be one of the greatest mathematicians to have ever lived.

Derivation of Euler’s Equation

I will first show how we derive Euler’s equation. If S is the surface of a small volume V of a moving fluid, the volume moving with the fluid with velocity u. The mass of the volume of fluid is therefore ρV were ρ is the density of the fluid.

Above is a picture of my notes showing the derivation of Euler’s equation.

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Example of Euler’s Equation

Below is an example question of Euler’s equation. We cannot find the full answer as we have not yet studied Bernoullis equations.

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Reflection
We have been given information about Euler’s equation this week and by working through examples I believe I have a good understanding of the topic. Things I can improve upon, is memorising the derivation to expand my knowledge of Fluid Dynamics.

References

  • Lecture Notes Chapter 2 The Euler’s and Bernoulli Equations, David Graham
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Week 6 – Dimensional analysis

This week is consolidation week so we have been assigned to read up about Units, dimensional analysis, flow similarity and fluid flow experiments.

Dimensional analysis is used to understand the properties of physical quantities independent of the units used to measure them. It is the analysis of the relationships between different physical quantities by identifying their dimensions. A physical quantity can be expressed as a product of physical dimensions. The difference between a dimension and a unit is a dimension is more abstract than a unit e.g. mass is a dimension, while kilograms are a scale unit in the mass dimension. Here are the dimensions and their corresponding symbols the SI standard recommends:

Mass =M (kg in SI units)
Length=L (m)
Time=T (s)
Electric Current= I(amps/A)
Thermodynamic Temperature=Θ(k)
Velocity is measured as Length/Time or L/T or LT−1 

Similarity

There are three types of Similarity. Geometric, Kinematics and Dynamic. Geometric Similarity exists between model and prototype if the ratio of all corresponding dimensions in the model are equal. For example the length of the model is proportional to the length of the prototype and the same for the area. Kinematic Similarity is the similarity of time as well as geometry, this exists if the paths of moving particles are geometrically similar and the ratios of velocities of particles are similar. Dynamic similarity exists between geometrically and kinematically similar system if ratios of all forces in the model and prototype are the same. This occurs when the controlling dimensionless group on the right hand side of the defining equation is the same for the model and prototype.

Reflection

 Dimensional Analysis is a topic I studied last year in Vector Calculus Last year I found them particularly confusing but now I feel confident as one years practice would suggest.

References

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