  This paper extends a new class of positivity-preserving, entropy stable
spectral collocation schemes developed for the one-dimensional compressible
Navier-Stokes equations in [1,2] to three spatial dimensions. The new
high-order schemes are provably L2 stable, design-order accurate for smooth
solutions, and guarantee the pointwise positivity of thermodynamic variables
for 3-D compressible viscous flows. Similar to the 1-D counterpart, the
proposed schemes for the 3-D Navier-Stokes equations are constructed by using a
flux-limiting technique that combines a positivity-violating entropy stable
method of arbitrary order of accuracy and a novel first-order
positivity-preserving entropy stable finite volume-type scheme discretized on
the same Legendre-Gauss-Lobatto grid points used for constructing the
high-order discrete operators. The positivity preservation and excellent
discontinuity-capturing properties are achieved by adding an artificial
dissipation in the form of the low- and high-order Brenner-Navier-Stokes
diffusion operators. To our knowledge, this is the first family of
positivity-preserving, entropy stable schemes of arbitrary order of accuracy
for the 3-D compressible Navier-Stokes equations.
