Symmetry analysis of the turbulent dissipation rate

A core attribute of any turbulent flow is the rate at which it dissipates energy $\varepsilon$. In his classic study from 1935, Taylor invoked rotational symmetry to transform the original cumbersome expression for $\varepsilon$ into a remarkably simple formula but for which it would be practically impossible to compute $\varepsilon$ in most experiments. Taylor's analysis, though ingenious, leaves it unclear if the formula truly conforms with rotational symmetry. We use the rigorous approach of Lie groups and show that Taylor's formula indeed holds for rotational symmetry. Further, we find that the formula is surprisingly robust—it holds, as is, for a distinctly different symmetry: reflectional symmetry. Additionally, we highlight that the widely used tests for identifying flow symmetries can yield misleading results. With rigor, precision, and clarity, the machinery of Lie groups delineates the underlying symmetries that dictate turbulent flows.