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:
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 m 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:
- n (=3) repeating variables (from the second theorem).
- 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).
- A combination of the repeating variables must not form a dimensionless group.
- The repeating variables do not have to appear in all Pi groups.
- 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.