Corotational and Compressibility Aspects Leading to a Modification of the Vortex-Identification Q-Criterion

Abstract

Nomenclature Ma= Mach number Q = vortex-identification criterion Q D = measure based on the deviatoric part of ∇u Q M = modification of the vortex-identification Q-criterion Re = Reynolds number S = strain-rate tensor, symmetric part of ∇u S D = deviatoric part of the strain-rate tensor S u, u i = velocity vector, components of u u i;j = spatial partial derivatives of components of u II = second invariant of an arbitrary second-order tensor A is defined by II A equals trA 2 − trA 2 =2 II S = second invariant of (the strain-rate tensor) S II SD = second invariant of (the deviatoric part of the strain-rate tensor) S D II ∇u = second invariant (see the definition of II above) of ∇u λ 2 = second-largest eigenvalue of S 2 Ω 2 , vortexidentification criterion σ = principal strain-rate difference vector,= deviatoric principal strain rates Ω = vorticity tensor, antisymmetric part of ∇u ω, ω 1 , ω 2 , ω 3 = vorticity vector, vorticity vector components ∇u = velocity-gradient tensor ∇ × u = curl of velocity vector

Vortex and the Balance between Vorticity and Strain Rate

“…Although applicable to compressible flows, Q D lacks a clear kinematic interpretation similarly as Q. This drawback led us in Kolář and Šístek [4] to a further modification of Q based on both corotational and compressibility arguments (strictly said, derived from comparing the magnitudes of the vorticity vector and the principal strain-rate difference vector)…”

“…where II SD is the second invariant of the (deviatoric) strain-rate tensor employed in the original expression for Q M in [4]. In fact, Q M is nothing but the overall measure of vorticity and strain-rate balance associated with three orthogonal strain-rate principal planes obtained by summing the three corresponding values of q D .…”

“…Nevertheless, it still lacks a clear kinematic interpretation. In a later attempt to add such property to the Q-criterion, we have presented the Q M -criterion in [4]. In this method, we have substituted the strain-rate tensor with the principal strain-rate difference vector, which is independent of compressibility but responsible for the shape deformation.…”

Computational aeroacoustics beneath high speed transitional and turbulent boundary layers

“…Two flow cases are considered here, with the first one being a hypersonic flow over a flat plate subjected to von Kármán atmospheric spectrum at the inlet. The flow transitions to fully turbulent at a downstream location, as shown in Figure 1 by isosurfaces of compressible Q-criterion [29,30] and contour plots of density and temperature. The results presented here are from simulations performed at Mach 6 and turbulence intensity of the free-stream velocity T u = 3%.…”

“…The structure of turbulence at the compression/expansion ramp is clearly visualized by iso-surfaces of the compressible Q-criterion [29,30] for all meshes in Figure 4. The iso-surfaces are colored with the streamwise velocity showing the flow separation at the corner.…”