Entropy-Based Analysis for Application to Highly Compressible Flows

Matrix normalizations are a critical component of mathematically rigorous aerodynamics analysis, especially where kinematic and thermodynamic behaviors are of interest. Here, a matrix normalization based around the entropy of a perturbation is derived according to the principles of mathematical entropy analysis and using a general definition of entropy amendable to physical phenomena such as thermal nonequilibrium and caloric and thermal imperfection. This normalization is shown to be closely related to the contemporary Chu energy normalization, expanding the range of validity of that normalization and clarifying the details of its interpretation. This relationship provides a basis for deriving other normalizations. Entropy analysis has also been applied to develop an entropy transport equation with which the entropy generation in a flow can be calculated.

The entropy norm is applied to predict the growth of instabilities in compressible boundary layers and was found to predict amplification consistent with the Chu norm. The norms were found to diverge in their predictions above Mach 5. The entropy norm was found to be more responsive to temperature gradients in the flow, which is a key factor in amplifying second-mode instabilities. This represents an advantage of the entropy norm over the Chu norm.

The norm has also been applied in POD analysis of two test cases. The first of these cases is a jet in supersonic crossflow, a well-characterized canonical case which is useful in evaluating the behavior of the entropy norm in cases with large temperature gradients. The entropy norm places more importance on these temperature fluctuations than the Chu norm. The entropy norm has the advantage in reduced-order modeling in that such models will converge to an entropy-respecting solution.

Entropy production in highly compressible turbulence as it varies with critical flow parameters has been analyzed. Thermal conduction and nonequilibrium losses are found to be critical mechanisms of entropy production. Entropy is also demonstrated to behave monotonically even when turbulent kinetic energy does not describe the turbulent cascade. These findings provide a potential avenue for the development of robust turbulence models and analysis tools.

Updated: added some references and made grammar revisions.