Morphing continuum analysis of energy transfer in compressible turbulence

A shock-preserving finite volume solver with the generalized Lax-Friedrichs splitting flux for morphing continuum theory (MCT) is presented and verified. The numerical MCT solver is showcased in a supersonic turbulent flow with Mach 2.93 over an $8^{\circ }$ compression ramp. The simulation results validated MCT with experiments as an alternative for modeling compressible turbulence. The required size of the smallest mesh cell for the MCT simulation is shown to be almost an order larger than that in a similar direct numerical simulation study. The comparison shows MCT is a much more computationally friendly theory than the classical Navier-Stokes equations. The dynamics of energy cascade at the length scale of individual eddies is illuminated through the subscale rotation introduced by MCT. In this regard, MCT provides a statistical averaging procedure for capturing energy transfer in compressible turbulence, not found in classical fluid theories. Analysis of the MCT results show the existence of a statistical coupling of the internal and translational kinetic energy fluctuations with the corresponding eddy rotational energy fluctuations, indicating a multiscale transfer of energy. In conclusion, MCT gives a new characterization of the energy cascade within compressible turbulence without the use of excessive computational resources.

Figure 1

Boundary conditions for a 2D Couette flow.

Figure 2

Velocity and gyration profile for the MCT Couette flow.

Figure 3

Size of the compression ramp computational domain.

Figure 4

Velocity profile of the incoming flow implemented in both MCT cases versus the experimental data (KAA 1987: [46, 47]).

Figure 5

Turbulent intensity at the inlet for MCT and experimental results (KAA 1987: [46, 47]).

Figure 6

Van Driest transformed streamwise velocity profile at the inlet.

Figure 7

Mean wall pressure distribution from MCT and experimental results (KAA 1987: [46, 47]; OLSBA 2007: [19]; AMS 2013: [49]).

Figure 8

Velocity profiles at (a) $3\delta$, (b) $4.2\delta$, and (c) $5.4\delta$ downstream of the ramp edge from the MCT solution and the experiments.

Figure 9

Turbulent Mach number at different location at (a) $1.8\delta$, (b) $3\delta$, (c) $4.2\delta$, and (d) $5.4\delta$ along the ramp.

Figure 10

Rotational kinetic energy component of the mean total energy density, mean $(——)$ and fluctations $\left(\text{−}\text{−}\text{−}\text{−}\right)$, at (a) $1.8\delta$, (b) $3\delta$, (c) $4.2\delta$, and (d) $5.4\delta$ along the ramp.

Figure 11

Translation kinetic energy component of the mean total energy density, mean $(——)$ and fluctations $\left(\text{−}\text{−}\text{−}\text{−}\right)$, at (a) $1.8\delta$, (b) $3\delta$, (c) $4.2\delta$, and (d) $5.4\delta$ along the ramp.

Figure 12

Internal energy component of the mean total energy, mean $(——)$ and fluctations $\left(\text{−}\text{−}\text{−}\text{−}\right)$, at (a) $1.8\delta$, (b) $3\delta$, (c) $4.2\delta$, and (d) $5.4\delta$ along the ramp.

Figure 13

Instantaneous total energy density profiles, translational component $(——)$, internal component $\left(\text{−}\text{−}\text{−}\text{−}\right)$ and rotational component $(.........)$, at (a) $1.8\delta$, (b) $3\delta$, (c) $4.2\delta$, and (d) $5.4\delta$ along the ramp.