Morphing continuum theory for turbulence: Theory, computation, and visualization

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 direct numerical simulation (DNS) provides a better prediction than Navier-Stokes (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 $Q$ criterion, the MCT-based $Q$ criterion is objective without the limitation of Galilean invariance.

Figure 1

Pressure coefficient comparison among experiment, DNS and MCT along the centerline on the bump.

Figure 2

Hairpin eddy structure identified by the objective $Q$ criterion with MCT.