Vortical and thermal interfacial layers in wall-bounded turbulent flows under transcritical conditions

The outer region of fully developed turbulent boundary layers can be viewed as a collection of uniform momentum zones separated by thin (but finite-thickness) shear layers referred to as momentum internal interface layers (MIILs). We first show the existence of such interfacial layers under transcritical thermodynamic conditions and introduce their thermal counterpart, named uniform thermal zones (UTZs), based on temperature. The UTZs, and the corresponding thermal internal interfacial layers (TIILs), are studied for a database of turbulent channel flow at transcritical thermal conditions [K. Kim et al., J. Fluid Mech. 871, 52 (2019)]. The thermal and vortical interfaces are identified using a recently proposed clustering approach by Fan et al. [D. Fan et al., J. Fluid Mech. 872, 198 (2019)]. It is shown that the MIILs and TIILs are correlated, but not collocated; their location is related to the underlying turbulent structures in the flow. On average, the TIILs are located about halfway between the wall and the MIIL, and the relationship between these layers is studied from the perspective of the attached eddy model. Under high near-wall thermal gradients the pseudoboiling line and the outer MIIL are collocated, which is explained using a shear stress balance analysis. Ultimately, the study of the thermal layering permits a simplification of the wall scaling in the presence of complex transcritical thermodynamics.

Figure 1

Shown on the left is an illustration of the numerical setup of a turbulent channel flow with differential wall heating. Shown on the right are the density (solid line) and Prandtl number (dashed line) of R-134a versus temperature at $p=1.1p_{cr}$. Closed circles represent pseudoboiling conditions.

Figure 2

Reynolds-averaged velocity and thermodynamic profiles of the differentially heated simulations for $\mathrm{\Delta }\mathrm{T}=5\mathrm{K}$ (solid line), 10 K (dashed line), and 20 K (dotted line). The symbol corresponds to the pseudoboiling point based on the averaged density and temperature.

Figure 3

Instantaneous slice of the $\mathrm{\Delta }T=20\mathrm{K}$ channel flow showing the MIILs (solid lines) separating the UMZs at the top and bottom walls.

Figure 4

Instantaneous slice of the $\mathrm{\Delta }T=20\mathrm{K}$ channel flow showing the TIILs (dotted lines) separating the UTZs at top and bottom walls.

Figure 5

Shown on the left are instantaneous MIILs (solid lines) and TIILS (dashed lines) overlaid on vorticity magnitude (top) and temperature gradient magnitude (bottom) contours. Shown on the right are instantaneous (dotted lines) and mean (solid lines) profiles overlaid with local location of MIILs (solid lines) and TIILs (dashed lines). Data are taken from the $\mathrm{\Delta }T=20\mathrm{K}$ transcritical channel flow.

Figure 6

Conditional sampling of the magnitude of the temperature gradient across the outer TIIL and the conditional sampling of the vorticity across the outer MIIL of the bottom wall for the $\mathrm{\Delta }T=20\mathrm{K}$ channel flow. The conditionally sampled vorticity is computed from the curl of the instantaneous velocity field (not the fluctuating component).

Figure 7

Histogram of (a) the inner MIIL bottom wall, (b) the inner MIIL top wall, (c) the outer MIIL bottom wall, and (d) the outer MIIL top wall.

Figure 8

Histogram of (a) the inner TIIL bottom wall, (b) the inner TIIL top wall, (c) the outer TIIL bottom wall, and (d) the outer TIIL top wall.

Figure 9

Structure of the thermal and vortical interface 2D contour plot for the bottom wall of the $\mathrm{\Delta }T=20\mathrm{K}$ channel flow. (a) The inner layer is much smoother than (b) the outer layer. $Y_{\mathrm{TIIL}}$ is the height of the TIIL and $Y_{\mathrm{MIIL}}$ is the height of the MIIL.

Figure 10

Structure of the thermal and vortical interface 2D contour plot for the top wall of the $\mathrm{\Delta }T=20\mathrm{K}$ channel flow. (a) The outer layer is much rougher than (b) the inner layer. $Y_{\mathrm{TIIL}}$ is the height of the TIIL and $Y_{\mathrm{MIIL}}$ is the height of the MIIL.

Figure 11

Location of the local pseudoboiling point (symbols) for the $\mathrm{\Delta }T=5$, 10, and 20 K channel flows. The MIIL (solid line) and TIIL (dashed line) are also shown.

Figure 12

Slice of the bottom wall for the $\mathrm{\Delta }T=20\mathrm{K}$ channel flow showing the MIILs (solid lines) and TIILs (dashed lines).

Figure 13

Velocity vectors of the $\mathrm{\Delta }T=20\mathrm{K}$ channel flow and the corresponding MIILs in a frame of reference convecting at $U_c=0.9U_{\infty }\approx 36\mathrm{m}/\mathrm{s}$.

Figure 14

Three-dimensional visualization of the outer MIIL (left) and TIIL (right) for a section of the bottom wall. Structures are identified using the $Q$ criterion.

Figure 15

Two-dimensional slices of the MIILs (solid blue lines) and TIILs (dashed red lines) at two different spanwise locations overlaid on the contour plot of the $Q$ criterion.