Characterization of anisotropic turbulence behavior in pulsatile blood flow

Abstract

Turbulent-like hemodynamics with prominent cycle-to-cycle flow variations have received increased attention as a potential stimulus for cardiovascular diseases. These turbulent conditions are typically evaluated in a statistical sense from single scalars extracted from ensemble-averaged tensors (such as the Reynolds stress tensor), limiting the amount of information that can be used for physical interpretations and quality assessments of numerical models. In this study, barycentric anisotropy invariant mapping was used to demonstrate an efficient and comprehensive approach to characterize turbulence-related tensor fields in patient-specific cardiovascular flows, obtained from scale-resolving large eddy simulations. These techniques were also used to analyze some common modeling compromises as well as MRI turbulence measurements through an idealized constriction. The proposed method found explicit sites of elevated turbulence anisotropy, including a broad but time-varying spectrum of characteristics over the flow deceleration phase, which was different for both the steady inflow and Reynolds-averaged Navier-Stokes modeling assumptions. Qualitatively, the MRI results showed overall expected post-stenotic turbulence characteristics, however, also with apparent regions of unrealizable or conceivably physically unrealistic conditions, including the highest turbulence intensity ranges. These findings suggest that more detailed studies of MRI-measured turbulence fields are needed, which hopefully can be assisted by more comprehensive evaluation tools such as the once described herein.

Keywords: Barycentric anisotropy invariant map; MRI turbulence measurements; Patient-specific scale-resolved computational hemodynamics; Reynolds stress and dissipation tensor; Verification and validation.

Fig. 1

Turbulence states characterized by the anisotropy tensor in principal invariant coordinates. All physically realizable states of turbulence are constrained to a limited region, called the anisotropy invariant map (AIM) or Lumley triangle, and describe the relative size of the eigenvalues along the orthogonal principal axes (see glyph examples), i.e., the turbulence componentiality. The AIM corners bound three primary states: one-component (1C), two-component axisymmetric (2C), and three-component isotropic (3C) turbulence. These states are joint by different boundaries: axisymmetric expansion (rod-like turbulence), axisymmetric contraction (disk-like turbulence), and the two-component limit (pancake-like turbulence). Along the plain-strain region, one anisotropy eigenvalue is zero, whereby turbulence only commutes in planes. Reprinted from Andersson et al. (2019), with permission from Elsevier

Fig. 2

Barycentric colormap of turbulence anisotropy. With barycentric mapping all limiting states are connected by lines, in contrast to the nonlinear Lumley triangle (Fig. 1), forming a equilateral triangle. The local states are governed by the combined weights $\{C_{1C},C_{2C},C_{3C}\}$, i.e., the anisotropy tensor coordinates, which individually scales from 1 to 0, from the limiting state to the opposite corner, respectively. By associating color triplets to these weights, e.g., red, green, and blue, each realizable state in the barycentric map can be represented by a specific color. Further details are given in Fig. 1

Fig. 3

Patient-specific Reynolds stress characteristics. (a and b) Rows represent: a snapshot during the flow deceleration phase (top), time-averaged over the early (middle) and late (bottom) flow deceleration phase, denoted EFD and LFD, respectively. (a) Axial planes through the turbulent region (vessel inset in b, black region), colored by the turbulence kinetic energy (k), anisotropy index ($A{I}_b$), and barycentric map. Cross-sectional planes were added normal to the centerline at 0.5D and 1.5D downstream the smallest stenotic diameter (D). For reference, the left (L), anterior (A), and posterior (P) sides of the aorta were included. (b) Barycentric maps with 50k points extracted from the turbulence region (vessel inset, black region), colored by the wall-normal offset distance (left column), and k (right column). The seed points were randomly selected with an even spatial distribution. At EFD, the dashed lines demonstrate suggested borders of the, respectively, turbulent state, for reference also shown at LFD

Fig. 4

Patient-specific turbulence dissipation characteristics. (a) Axial planes colored by the dissipation rate of turbulence kinetic energy ($ϵ$), anisotropy index ($A{I}_d$), and barycentric map with addition of cross-sectional planes at 0.5D and 1D. (b) Barycentric maps with 50k points, colored by the wall normal offset distance (left column) and $ϵ$ (right column). Additional details are given in Fig. 3

Fig. 5

Steady versus pulsatile inflow condition effects on the Reynolds stress characteristics. (a) Axial planes colored by the turbulence kinetic energy (k), anisotropy index ($A{I}_b$), and barycentric AIM. (b) Deviation maps, showing the root–mean–square ($C_{\mathit{rms}}$, Eq. 16) and individual differences ($C_{\mathit{ic}}^s-C_{\mathit{ic}}^p$, superscripts: $s=$ steady, $p=$ pulsatile), of the AIM weights ranging from [0, 1]. (c) Barycentric maps with 50k points colored by k. Additional details are given in Fig. 3

Fig. 6

RANS-based Reynolds stress characteristics. (a) Axial planes colored by the turbulence kinetic energy (k), anisotropy index ($A{I}_b$) , and barycentric AIM with addition of cross-sectional planes at 0.5D and 1.5D. Note, $A{I}_b$ upper limit is set to 0.5. (b) Barycentric maps with 50k points. Additional details are given in Fig. 3

Fig. 7

MRI-measured Reynolds stress characteristics in an idealized constriction. Axial symmetry planes (2D upstream to 6D downstream) at two different Reynolds numbers (${\text{Re}}_{\text{D}}$) based on the large diameter (D) of the pipe. Planes are colored by the velocity magnitude (Velocity), turbulence kinetic energy (TKE, k), anisotropy index (Anisotropy, $A{I}_b$), and barycentric AIM (States). Additional planes were added to show voxels that fall outside the AIM (black voxels), i.e., nonphysical turbulence states; considering all voxels (Unrealizable) and only the upper $25\%$ ($>Q_3$) of the post-orifice TKE values. The latter data range was also projected into the barycentric maps (bottom), which show the unrealizable voxels outside the triangular domain for both flow cases