  A high order morphing continuum theory (MCT) is introduced to model highly
compressible turbulence. The theory is formulated under the rigorous framework
of rational continuum mechanics. A set of linear constitutive equations and
balance laws are deduced and presented from the Coleman-Noll procedure and
Onsager's reciprocal relations. The governing equations are then arranged in
conservation form and solved through the finite volume method with a second
order Lax-Friedrichs scheme for shock preservation. A numerical example of
transonic flow over a three-dimensional bump is presented using MCT and the
finite volume method. The comparison shows that MCT-based DNS provides a better
prediction than NS-based DNS with less than 10% of the mesh number when
compared with experiments. A MCT-based and frame-indifferent Q-criterion is
also derived to show the coherent eddy structure of the downstream turbulence
in the numerical example. It should be emphasized that unlike the NS-based
G-criterion, the MCT-based Q-criterion is objective without the limitation of
Galilean-invariance.
