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:

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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:

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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:

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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

Flow Visualisation

For now, we are moving back to inviscid flows. I will remind myself of what a fluid actually is.

What is a fluid?

A fluid is a substance that continues to deform in the presence of sheer stress.

We can model a fluid with a continumm model, where the fluid properties are defined by:

  • velocity u(x,t) [vector]
  • temperature T(x,t) [scalar]
  • density p(x,t) [scalar]
  • viscosity mu(x,t) [scalar]
  • and many more..

We can also visualise fluids. One way we can do this is by using pathlines and streamlines.

Pathlines are defined as trajectories of marked fluid parcels. These can be written mathematically:

pathlines

This can also be adapted for polar coordinates:

pathlines in polar coordinates

Streamlines are defined as a curve which is always tangential to the fluid velocity. These can be written mathematically:

Streamlines

We can now use these equations to calculate fluid flows and we can show the graphically.

We have also seen that pathlines and streamlines are identical if they do not depend on time.

Please find below examples on pathlines and streamlines.

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This week I need to work on a few more of these examples and try to find some useful links which would help me understand more about how to display the pathlines and steamlines on a graph.

To start of with, I was really struggling to understand pathlines and streamlines, but since we have been working through the example sheets it feels like I’m starting to understand it more and have a go at some of the harder examples.