6. Dimensional analysis Flashcards
What is the purpose of dimensional analysis?
- To determine what scaling relationships govern a particular engineering problem/system.
- We can then use these relationships to appropriately size a small or larger system and to infer results between a small and full - scale system
- And for checking your answers for dimensional homogenitiy.
What is Buckingham π method?
A method of dimensional analysis for large and small scale systems.
* States that for n quantities (variables) which contain m fundamental dimensions the quantities can be arranged into (n-m) indepentend dimensionless groups. So for 5 fundamental vars, and 3 fundamental dimensions we get 2 dimensionless groups for scaling.
* Using laws of indicies and by repeating groups of variables we can form the dimensionless groups used for scaling purposes.
* Still need to determine any constants through experiment, etc.
What are the steps for Buckingham π method?
- List all the variables and their dimensions
- Work out how many groups we will have: (for n quantities (variables) which contain m fundamental dimensions the quantities can be arranged into (n - m) independent dimensionless groups)
- We will assign one variable to each group that will only appear in that group. Such a variable is Fixed.
- The remaining variables are repeating and appear in all groups.
- All the variables in the group must make a dimensionless group so write 0 = (variables go here)
- Raise each repeating variable to a power a, b, c, etc.
- Replace variables with their fundamental dimensions (M, L, T etc.)
- Solve for each power a = , b = etc.
- Replace dimensions with variables and calculated power. Form equations for each dimensionless group.
- You can now manipulate the groups to simplify them taking care to keep them dimensionless. (more on this later on…)
Choosing repeating variables
* When combined, these repeating variables variable must contain all of the dimensions in the problem
* A combination of the repeating variables must not form a dimensionless group.
* The repeating variables do not have to appear in all π groups.
* The repeating variables should be chosen to be measurable by experiment and should be of most interest to the problem. For example, fluid depth (dimension L) is more useful and measurable than roughness height (also dimension L).
* Fixed variables should be exotic and unlikely to appear in all groups. Experience helps!!!!!
Use Buckingham pi’s method to solve this problem.
These pictures show answers, that can be reformatted into Fn and Rn.
Full solution : https://southampton.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=55fb4075-9627-4f04-957c-ac5500ae4fd9
True or false:
We can manipulate dimensionless groups
True:
- So long as you preserve the dimensionless nature of a group you can multiply/divide by other groups, invert etc. essentially anything so long as it remains dimensionless.
- Why? To give us more useful, simple or established (named) groups.
How can we use Froude and Reynolds in real world applications?
- We can write that the model and the prototypes froude or reynolds number must be equal or constant. i.e. model froude or reynolds number divided by prototypes froude or reynolds number must equal 1.
- We can then rearrange and get rid of factors that will be constant in both, i.e. gravity or maybe even dynamic viscosity if the water is the same. Thus we get a simplified equation that relates velocities to one or two variables we can then find velocity for models and prototypes scaling appropriately so we have similar working system.
Can we do simultaneous scaling of both Froude and Reynolds number?
Not really. They both are different, often to high magnitudes, and so (without doing complex maths) we just have to see where which number needs to be fixed and one we can get close to simulate a similar flow or effect.
Often Froude number should be kept close, and if we get Reynolds number in the same category (such as turbulent or laminar) then it should behave similarly in the prototype or model.
