In today’s lesson, we will be discussing the following topics:
1. Nondimensionalization and scaling of the Continuity Equation
2 Nondimensionalization and scaling of the Navier-Stokes Equation
video coming soon
If you haven’t read the background on nondimensionalization, check out this previous post here. We are going to be converting the Navier-Stokes and Continuity equations into a dimensionless form (we will nondimensionalize them). This brings out the base parameters that each equation relies on and reduces the equation to its natural form, therefore simplifying the equation.
There are some common scaling parameters that are discussed in this earlier post. We will choose some of these to help us scale these equations.
We need to nondimensionalize each variable in the original fluids equations by multiplying or dividing it by another variable with the same units. Doing so creates dimensionless variables, which will be used to replace these variables in the original equation. These are the scaling parameters we will choose. They are denoted with an asterisk.
The reference velocity (U) and reference length (L) are two common units used to nondimensionalize variables.
Nondimensionalization and scaling of the Continuity Equation
We can use the same scaling parameters and dimensionless groups as above to nondimensionalize the continuity equation.
Here is the original continuity equation:
After substituting in the dimensionless variables it becomes:
Nondimensionalization and scaling of the Navier-Stokes Equation
Given the original Navier-Stokes equation, substituting in the dimensionless variables and dividing through by density gives the final nondimensionalized form of the equation:
Some of the terms can also be replaced with known groups of variables (the image below shows common ones – these are called dimensionless groups), such as the Reynolds number or the Mach Number. This reduces the variables in the equation, for example instead of writing U/a we can write Ma for the Mach Number, and so on (eg. you will see the last term with viscosity and density above is just the inverse of the Reynolds Number, a dimensionless group in the table below. So that whole group of variables could be replaced with 1/Re). These dimensionless groups are derived from the non-dimensionalization of the continuity, Navier-Stokes energy equations and more. Although we did not do the nondimensionalization of the energy equation to derive these dimensionless groups, all the common ones are listed here:
