Waves: Particle Paths

Continuing on from last week, we looked at water waves, and I will start with the example that we did last week.

Image

We have to make assumptions with this:

  • v=0; and the partial derivative with respect to y=0.
  • It is inviscid.
  • It is incompressible.
  • It is irrotational. If it’s incompressible and irrotatinal, this implies that nabla^2*Phi=0.
  • g is the only body ‘force’.
  • no surface tension.
  • the displacement and the velocities are small (ignore products).
  • water at the free surface stays there.
  • Starts from rest.

We use these equations to find the solution for both finite depth and infinite depth:

waterequations This is equation 4.1

waterequations2 This is equation 4.6

waterequations3 This is equation 4.5

waterequations4 This is equation 4.3

We can work out the solution by following these steps:

Image

We can now imply the boundary conditions:

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Implying that B=0, we get the solution to be:

solutiontoinfinite

Dispersion Relation

If we use equation 4.6 we get:

dispersionrelation

this implies that w^2=gk and this is equation 4.9.

Surface Displacement

If we use equation 4.3 we get:

surfacedisplacement

Potential

The potential is:

potential

Phase Speed

The phase speed is the speed at which the phase of a wave is propagated, the product of the frequency times the wavelength.

In our solution, this is given by: c=w/k.

The following is the example of finite depth:

Image

 

 

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This example is to do with infinite depth:

Image

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This week, going through the examples I’m not sure I fully understand it, so I aim to go back over the examples till I am 100% sure I know where we get the solution from and the stages at each step.

Sources:

Phase speed definition: http://www.thefreedictionary.com/phase+speed

 

 

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Potential Flow

This week we looked more into potential flow as well as doing some examples.

We looked into a Doublet, which was 2-dimensional and steady. The streamline that was given is:

CodeCogsEqn(8)

To find the isopots of this, we have to find:

CodeCogsEqn(8)

and:

CodeCogsEqn(8)

with some manipulation we can get this in the form of x^2+y^2=r^2

CodeCogsEqn(8)

As we can see from this, the circle is going to be centered around CodeCogsEqn(8) and with radius CodeCogsEqn(8).

If we plot the streamlines and isopots on the same graph, we get:

Image

We can see these cross at right angles, this seems to be the case for every stream function.

Now we looked at Bernouilli’s equation in potential flow and how to derive it.

This is here:

Image

We also looked at another example with water waves.

This week I enjoyed more since we were deriving new equations. I think I will need to go over this week thoroughly so that I understand fully the derivation.

 

Flow around a cylinder

This week we will be going through some examples where we can apply Bernouilli’s equation.

We were first looking at a flow around a cylinder. Before we could do anything, we have to make observations of this. We need to see if:

  • Flow tangential to cylinder.
  • Streamlines = Solid surfaces in inviscid flow.
  • Neglect boundary layer effects due to viscosity.

We are given the stream function:

CodeCogsEqn (16)

If we play around with this equation, getting terms for Ur and Utheta when r=a:

We can now substitute this into Bernouilli’s equation which is:

CodeCogsEqn (15)

We get:

CodeCogsEqn(8)

Ps is the stagnation pressure at the point where u=0.

We have also looked at Potential flow this week. This describes the velocity field as the gradient of a scalar function; the velocity potential.

Sources:

Picture and definition. http://en.wikipedia.org/wiki/Potential_flow

Example sheet 4, question 1:

IMG_0092[1] IMG_0094[1]

Bernouilli’s Equation

This week we looked more into Bernouilli’s Equation. But to start off with, we had a quick ‘clicker’ quiz to recap on what we have learnt so far, and this highlighted some of the area I need to work on more.

Euler’s equation can be simplified when we integrate along streamlines, or when the flow is both inviscid and irrotational. In either case, we get a form of Bernouilli’s equation:

IMG_0033[2]

We also touched on Bernouilli’s principle. This is when the body forces are negligible. i.e. Phi=0.

Therefore Bernouilli’s equation becomes:

23

Where C is a constant, P is the pressure, Rho is the density and u is the speed.

Thus an increase in speed gives a decrease in pressure and vice versa.

We used this equation to go through some examples on pilot tubes within an oil refinery and looking at aircrafts to judge their speed.

I’m starting to understand Bernouilli’s equation a little more, but I feel I need to do some examples on this to help further my understanding.

Example sheet 3, question 5:

IMG_0091[1]

Euler’s Equation

This week we have been looking at Euler’s equation.

We can look at the dynamics of inviscid flows through Euler’s equation. This equation, together with the equation of continuity, specifies the dynamic of inviscid flows. Euler’s equation can be simplified under certain circumstances to give Bernoulli’s equation.

If we use S to denote 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 thus Rho*V, where Rho is the density of the fluid.

If we call deltaS the surface element of Sn is the unit outward-pointing normal to deltaS and p is the fluid pressure on deltaS. The force on the surface element is:  CodeCogsEqn(8)

We can now say that the total surface force acting over S is:  CodeCogsEqn(8)

The total body force is:  CodeCogsEqn(8)

If we use Newton’s Second Law, the equation of motion of the fluid volume V is:  CodeCogsEqn(8)

Now, if we allow the volume V to tend to zero, we get:  CodeCogsEqn(8)

And lastly, if we divide through by Rho*V  and taking the limit as V->0 we get:  CodeCogsEqn(8) which is Euler’s Equation.

This week, I feel like I’m starting to grasp more on fluids, and luckily we went over dimensional analysis so I could refresh my memory as we are about to settle down to learn a new topic!

Example sheet 3, question 1:

IMG_0090[2]

Dimensional Analysis

This week we have been asked to do some reading due to it being consolidation week on Dimensional Analysis.

Dimensional analysis provides a strategy for choosing relevant data and effective ways of presenting them. This is a useful technique in all experimentally based areas of physics or engineering. IF we can identify the factors involved in a physical situation, then we can use dimensional analysis can form a relationship between them. The expressions we get from this may not appear rigorous but these qualitative results converted to quantitative forms can be used to obtain any unknown factors from experimental analysis.

In any physical situation can be described by certain familiar properties:

  • length = L (metre in SI units)
  • mass = M (kilogram 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)

Note: SI is the international system of units. More information here. http://en.wikipedia.org/wiki/International_System_of_Units

The following table lists dimensions of some common physical quantities:

Untitled

One of the key principles of dimensional analysis is dimensional homogeneity. This is any equation describing a physical situation that will only be valid of both dies have the same dimensions.

This property can be useful for:

  • checking units of equations
  • converting between two sets of units
  • defining dimensionless relationships

We can obtain dimensionless groups by using Buckingham’s Pi theorems which are stated below:

First theorem: A relationship between variables can be expressed as a relationship between m-n non-dimensional groups of variables (called Pi groups), where n is the number of fundamental dimensions required to express the variables.

Second theorem: Each dimensionless group is a function of n governing or repeating variables plus one of the remaining variables.

Repeating variables are those which we think will appear in all or most of the Pi groups and are an influence in the problem. There are rules to follow when finding these variables:

  1. n (=3) repeating variables (from the second theorem).
  2. When combined, these repeating variable/s must contain all of the dimensions (M,L,T). (That is not to say that they each must contain M, L and T).
  3. A combination of the repeating variables must not form a dimensionless group.
  4. The repeating variables do not have to appear in all Pi groups.
  5. The repeating variables should be chosen to be measurable in an experimental investigation. They should be of major interest to the designer.

Within fluid dynamics, we often take Rho, u, and d as the three repeating variables (although, sometimes we use the viscosity instead of density).

During dimensional analysis, we will be using Reynolds, Froude, and Mach numbers. Some non dimensional numbers are listed below.

  • Reynolds number: Re=(Rho*u*d)/Mu (inertial, viscous force ratio)
  • Euler number: En=p/(Rho*(u^2)) (pressure, inertial force ratio)
  • Froude number: Fn=(u^2)/gd (inertial, gravitational force ratio)

Geometric similarity exists between model and prototype if the ratio of all corresponding dimensions in the model and prototype are equal.

Kinematic similarity is the similarity of time as well as geometry. It exists between model and prototype if:

  • the paths of moving particles are geometrically similar.
  • the ratios of the velocities of particles are similar.

This has the consequence that streamline patterns look the same.

Dynamic similarity exists between geometrically and kinematically similar systems if the 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 model and prototype.

Stream Functions, Incompressibility and Fluid accelerations

This week we are looking further into flow visualisation and looking at Stream Functions.

We are now concentrating on incompressible 2D flows (which are flows in which the velocity is in the form (u,v,0). The continuity equation in these flows is:

1

Suppose that we can find a scalar function si such that:

2

This scalar function is called the stream function for the flow. Once we know the velocity field, we can find the stream function and also the other way round!

If we say that the streamline is parametrised by writing x=x(s) and y=y(s) for some parameter s, then dx/ds is the tangent vector to the streamline and is therefore parallel to the fluid velocity at x. Therefore the vector cross product between u and dx/ds is zero. I.e:

3

We can simplify this down to:

4

Therefore this becomes dsi/dsk=0. Therefore dsi/ds=0 along a streamline and hence si is constant along a streamline. We can then solve the equation of the stream function, and this gives us another method for producing streamlines!

We have also seen the Principle of Superposition which stated if we add two different velocity fields together it has the effect of combining the two relative stream functions together.

Please find below a few examples from example sheet 1:

IMG_0027IMG_0028