Full text loading...

^{1}and Matteo Bernardini

^{1}

### Abstract

We study the high-Reynolds-number behavior of a turbulent boundary layer in the low supersonic regime through very-large-scale direct numerical simulation (DNS). For the first time a Reynolds number is attained in DNS ( , where δ is the boundary layer thickness and δ v is the viscous length scale) at which theoretical predictions and experiments suggest the occurrence of phenomena pertaining to the asymptotic Reynolds number regime. From comparison with previous DNS data at lower Reynolds number we find evidence of a continuing trend toward a stronger imprint of the outer-layer structures onto the near-wall region. This effect is clearly manifested both in flow visualizations, and in energy spectra. More than a decade of nearly-logarithmic variation is observed in the mean velocity profiles, with log-law constants k ≈ 0.394, C ≈ 4.84, and a trend similar to experiments. We find some supporting evidence for the debated existence of a k −1 region in the power spectrum of streamwise velocity fluctuations, which extends up to y + ≈ 150, and of a k −5/3 spectral range in the outer layer.

We acknowledge that the results in this paper have been achieved using the PRACE Research Infrastructure resource JUGENE based at the Forschungszentrum Jülich (FZJ) in Jülich, Germany.

### Key Topics

- Reynolds stress modeling
- 24.0
- Friction
- 9.0
- Eddies
- 8.0
- Turbulent flows
- 7.0
- Boundary layer turbulence
- 5.0

## Figures

Mean velocity distributions. The thin red line in panel (a) denotes the standard law of the wall, compounding u + = y + with u + = y +/k + C, k = 0.394, C = 4.84. The circles indicate experimental data by Smith 22 at . Panel (b) shows the diagnostic function Ξ (as defined in Eq. (1) ), the horizontal line denoting the reference 1/k log-law value. See Table I for line legend.

Mean velocity distributions. The thin red line in panel (a) denotes the standard law of the wall, compounding u + = y + with u + = y +/k + C, k = 0.394, C = 4.84. The circles indicate experimental data by Smith 22 at . Panel (b) shows the diagnostic function Ξ (as defined in Eq. (1) ), the horizontal line denoting the reference 1/k log-law value. See Table I for line legend.

Distribution of density-scaled Reynolds stress components, . The circles indicate experimental data by Smith 22 at . See Table I for line legend.

Distribution of transformed skin friction (C f i, as defined in Eq. (2) ), and peak streamwise velocity variance. Circles denote data from the TBL1-4 DNS datasets. Incompressible DNS data are shown from Schlatter and Örlü 13 (squares), Simens et al. 14 (diamonds), and Sillero et al. 15 (triangles). Low-speed experimental data from Smith 22 are shown with gradient symbols. In panel (a), the dashed line indicates Eq. (3) , and the solid line Eq. (4) . In panel (b), the solid line indicates Eq. (5) , and the dashed line Eq. (6) .

Distribution of transformed skin friction (C f i, as defined in Eq. (2) ), and peak streamwise velocity variance. Circles denote data from the TBL1-4 DNS datasets. Incompressible DNS data are shown from Schlatter and Örlü 13 (squares), Simens et al. 14 (diamonds), and Sillero et al. 15 (triangles). Low-speed experimental data from Smith 22 are shown with gradient symbols. In panel (a), the dashed line indicates Eq. (3) , and the solid line Eq. (4) . In panel (b), the solid line indicates Eq. (5) , and the dashed line Eq. (6) .

Instantaneous streamwise velocity field in x − z plane at y + = 15 for TBL4 dataset. Contour levels are shown for −0.25 ⩽ u ′/u ∞ ⩽ 0.25, from dark to light shades. The figure inset shows a zoom of a small rectangular region, to highlight the turbulence fine scales.

Instantaneous streamwise velocity field in x − z plane at y + = 15 for TBL4 dataset. Contour levels are shown for −0.25 ⩽ u ′/u ∞ ⩽ 0.25, from dark to light shades. The figure inset shows a zoom of a small rectangular region, to highlight the turbulence fine scales.

Pre-multiplied spanwise spectral densities of u ′ in inner scaling. In panel (a) data are shown at y + = 15 for TBL1-4 (see Table I for line legend). In panel (b) spectra are shown for TBL4 at y + = 15 (solid), 44 (dashes), 70 (dash-dots), 97 (dots), 125 (long dashes), 156 (dash-dot-dot). The horizontal line in (a) is indicative of behavior.

Pre-multiplied spanwise spectral densities of u ′ in inner scaling. In panel (a) data are shown at y + = 15 for TBL1-4 (see Table I for line legend). In panel (b) spectra are shown for TBL4 at y + = 15 (solid), 44 (dashes), 70 (dash-dots), 97 (dots), 125 (long dashes), 156 (dash-dot-dot). The horizontal line in (a) is indicative of behavior.

Compensated spanwise spectral densities of u ′ in outer scaling. In panel (a) data are shown at y/δ = 0.3 or TBL1-4 (see Table I for line legend). In panel (b) spectra are shown for TBL4 at y/δ = 0.15 (dots), 0.2 (dashes), 0.3 (solid), 0.5 (dash-dots), 0.7 (long dashes), and 0.9 (dash-dot-dot). The horizontal line in (a) is indicative of behavior.

Compensated spanwise spectral densities of u ′ in outer scaling. In panel (a) data are shown at y/δ = 0.3 or TBL1-4 (see Table I for line legend). In panel (b) spectra are shown for TBL4 at y/δ = 0.15 (dots), 0.2 (dashes), 0.3 (solid), 0.5 (dash-dots), 0.7 (long dashes), and 0.9 (dash-dot-dot). The horizontal line in (a) is indicative of behavior.

## Tables

Summary of parameters for DNS study. L x and L z are the streamwise and spanwise domain lengths, respectively. is the “incompressible” momentum thickness Reynolds number, defined in Eq. (2) . Grid spacings are given in wall units taken at for TBL1, for TBL2, for TBL3, and for TBL4.

Summary of parameters for DNS study. L x and L z are the streamwise and spanwise domain lengths, respectively. is the “incompressible” momentum thickness Reynolds number, defined in Eq. (2) . Grid spacings are given in wall units taken at for TBL1, for TBL2, for TBL3, and for TBL4.

Article metrics loading...

Commenting has been disabled for this content