Vorticity and the Entropy Jump Across a Shock Wave

This work studies the relationship between vorticity and entropy in three-dimensional, compressible, inviscid flow fields. The Crocco equation is applied to analyze flows with shock waves by expressing the entropy gradient in terms of the cross product of the velocity and vorticity. Shock waves are modeled as two-dimensional curved surfaces embedded in three-dimensional flow fields. The relationship between the entropy jump condition and the kinematic jump conditions across a shock surface is derived using singular generalized functions cast as generalized derivatives concentrated on moving surfaces. For the perfect gas case, the entropy jump condition across a shock wave, which satisfies both the Crocco equation and the second law of thermodynamics, is shown to be a nonlinear generalized jump function of pressure and specific volume. It is also shown that the vorticity jump conditions do not change the functional form of the entropy jump function across a shock wave; this results because the vorticity jump conditions are purely kinematic relationships that depend only on the curl of velocity and the conservation of mass and momentum.