Nondimensionalisation

Nondimensionalisation, also known as normalization or scaling, is a mathematical technique used to simplify the analysis of physical phenomena by reducing the number of parameters involved. It involves transforming a set of governing equations into a dimensionless form, where all variables are expressed in terms of dimensionless quantities. This process is particularly useful in fluid dynamics, heat transfer, and other fields where the governing equations are partial differential equations.

The primary goal of nondimensionalisation is to identify the key parameters that influence a physical system and to separate them from the less important ones. This is achieved by expressing the variables in terms of a characteristic length, velocity, time, and other relevant scales. For example, in fluid dynamics, the Reynolds number is a dimensionless quantity that characterizes the flow regime, combining the effects of viscosity, density, and velocity.

The process of nondimensionalisation typically involves the following steps:

1. Identify the relevant physical parameters and their dimensions.

2. Choose a set of characteristic scales for each dimension.

3. Express each variable in terms of its characteristic scale.

4. Substitute these expressions into the governing equations and simplify.

The resulting dimensionless equations are often easier to analyze and solve, as they reveal the underlying physics of the problem. Moreover, they can be used to identify the key parameters that control the behavior of the system and to predict the scaling laws that govern its response to changes in these parameters.

Nondimensionalisation is a powerful tool for simplifying the analysis of complex physical systems and for gaining insights into their underlying mechanisms. It is widely used in engineering, physics, and other fields where the governing equations are partial differential equations.