Dimensional Analysis and Units

• Dimensional analysis combines dimensional variables, nondimensional variables, and dimensional constants into nondimensional parameters to reduce the number of independent parameters in a problem.
• The method of repeating variables reduces the number of independent parameters in a problem using a step-by-step method based solely on the dimensions of the variables and constants.

Dimensions and Units

• A dimension is a measure of a physical quantity without numerical values.
• A unit is a way to assign a number to a dimension.
• Length is a dimension measurable in units like microns (μm), feet (ft), centimeters (cm), meters (m), or kilometers (km).
• There exist seven primary, fundamental, or basic dimensions: mass, length, time, temperature, electric current, amount of light, and amount of matter.
• Nonprimary dimensions consist of combinations of the seven primary dimensions.
• Force has the same dimensions as mass times acceleration, giving the dimensions {mL/t²}.

Dimensional Homogeneity

• Every additive term in an equation must have the same dimensions.
• The change in total energy of a simple compressible closed system equation is ΔE = ΔU + ΔKE + ΔPE, with each term having dimensions of energy.
• Calculations are valid when the units are homogeneous in each additive term.
• It’s advisable to write out all units when performing mathematical calculations to avoid errors.

Nondimensionalization of Equations

• Dividing each term in an equation by a collection of variables and constants with the same dimensions renders the equation nondimensional.
• A normalized equation is one where the nondimensional terms are of order unity.
• Normalization is more restrictive than nondimensionalization.
• Nondimensional parameters such as the Reynolds number and the Froude number often appear when nondimensionalizing an equation of motion.

Dimensional Analysis and Similarity

• Dimensional analysis is useful for generating nondimensional parameters for designing experiments and obtaining scaling laws for predicting prototype performance from model performance.
• Complete similarity between a model and a prototype requires geometric, kinematic, and dynamic similarity.
• Geometric similarity refers to the model having the same shape of the prototype
• Kinematic similarity refers to the velocity being proportional
• Dynamic similarity refers to a constant scale factor
• Complete similarity exists only when there is geometric, kinematic, and dynamic similarity.
• Complete similarity between the model and prototype is attainable by ensuring that all independent groups match between the model and prototype.

Method of Repeating Variables and Buckingham Pi Theorem

• The method of repeating variables involves six steps to obtain nondimensional parameters:
• List all parameters (dimensional variables, nondimensional variables, and dimensional constants) and count them (n).
• List the primary dimensions for each of the n parameters.
• Guess the reduction j and calculate k
• Choose j repeating parameters with the potential to appear in each II.
• Generate the II's one at a time
• Check that all the II’s are indeed dimensionless.

Experimental Testing and Incomplete Similarity

• Nondimensionalization reduces the number of parameters in a problem
• A full factorial test matrix is a complete set of experiments conducted by testing every possible combination of several levels of each of the independent parameters
• It’s not always possible to match all II’s of a model to the corresponding II’s of the prototype, even with geometric similarity, leading to incomplete similarity.
• Many wind tunnel tests rely on Reynolds number independence above a threshold value