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Classical dimensional analysis has two limitations: (i) the computeddimensionless groups are not unique, and (ii) the analysis does not measurerelative importance of the dimensionless groups. We propose two algorithms forestimating unique and relevant dimensionless groups assuming the experimentercan control the system's independent variables and evaluate the correspondingdependent variable; e.g., computer experiments provide such a setting. Thefirst algorithm is based on a response surface constructed from a set ofexperiments. The second algorithm uses many experiments to estimate finitedifferences over a range of the independent variables. Both algorithms aresemi-empirical because they use experimental data to complement the dimensionalanalysis. We derive the algorithms by combining classical semi-empiricalmodeling with active subspaces, which---given a probability density on theindependent variables---yield unique and relevant dimensionless groups. Theconnection between active subspaces and dimensional analysis also reveals thatall empirical models are ridge functions, which are functions that are constantalong low-dimensional subspaces in its domain. We demonstrate the proposedalgorithms on the well-studied example of viscous pipe flow---both turbulentand laminar cases. The results include a new set of two dimensionless groupsfor turbulent pipe flow that are ordered by relevance to the system; theprecise notion of relevance is closely tied to the derivative based globalsensitivity metric from Sobol' and Kucherenko.

Data-driven dimensional analysis: algorithms for unique and relevant dimensionless groups