Numerical Investigation of High-Speed Turbulent Boundary Layers of Dense Gases

Abstract

High-speed turbulent boundary layers of a dense gas (PP11) and a perfect gas (air) over flat plates are investigated by means of direct numerical simulations and large eddy simulations. The thermodynamic conditions of the incoming flow are chosen to highlight dense gas effects, and laminar-to-turbulent transition is triggered by suction and blowing. In the paper, the behavior of the fully developed turbulent flow region is investigated. Due to the low characteristic Eckert number of dense gas flows ($\hbox {Ec}=U_\infty ^2/c_{p,\infty }T_\infty$), the mean velocity profiles are largely insensitive to the Mach number and very close to the incompressible case even at high speeds. Second-order velocity statistics are also weakly affected by the flow Mach number and the velocity spectra are characterized by a secondary peak in the outer region of the boundary layer because of the higher local friction Reynolds number. Despite the incompressible-like velocity and Reynolds-stress profiles, the strongly non-ideal thermodynamic and transport-property behavior of the dense gas results in unconventional distributions of the fluctuating thermo-physical quantities. Specifically, density and viscosity fluctuations reach a peak close to the wall, instead of vanishing as in perfect gas flows. Additionally, dense gas boundary layers exhibit higher values of the fluctuating Mach number and velocity divergence and a larger dilatational-to-solenoidal dissipation ratio in the near-wall region, which represents a major deviation from high-Mach-number perfect gas boundary layers. Other significant deviations are represented by the more symmetric probability distributions of fluctuating quantities such as the density and velocity divergence, due to the more balanced occurrence of strong expansion and compression events.

Appendix: Validations

In this appendix, we present validations of our numerical solver against well-established literature results for perfect-gas (air) high-speed boundary layers at $M=2.25$ and 6. The simulation at $M=2.25$ is compared to the DNS data of Pirozzoli and Bernardini (2011). This study focused on the fully turbulent flow behavior and adopted a rescaling/recycling strategy to shorten the computational domain required to achieve fully-developed turbulence. Sutherland’s law was used to model the viscosity, along with a constant Prandtl number hypothesis. Figure 16 shows wall-normal profiles of selected flow statistics for the present PG ILES and the DNS of Pirozzoli and Bernardini (2011). An excellent agreement is observed.

Fig. 16

Wall-normal profiles of Reynolds stresses (a), normalized wall pressure (b) and Van Driest-scaled streamwise velocity profile (c), for current ILES of Air at $M=2.25$ (lines) and Pirozzoli and Bernardini (2011) (symbols), extracted at $\hbox {Re}_\tau =580$

The calculation at $M=6$ has been performed in the same conditions and with the same thermodynamic and transport-property models as the DNS study of Franko and Lele (2013), except that our computational domain is much longer to achieve a fully turbulent state. These authors focused their analysis on the transition mechanisms, so that a comparison is possible only in the transitional regime. Another difference in our numerical setup is that the inlet of the domain corresponds to the leading edge of the flat plate, whereas Franko and Lele (2013) started with a finite laminar boundary layer thickness, such that the Reynolds number based on the inflow displacement thickness, $\delta ^*_{\text{in}},{\text{F}} \& {\text{L}}$, is 3000. For the comparisons in Fig. 17, we use this displacement thickness as reference ($\delta ^*_{\text{in}},{\text{F}} \& {\text{L}}=\delta ^*_\text {ref}$). Panel a shows that the distribution of the skin friction coefficient for the present and the reference calculation are in excellent agreement. The present simulation finally reaches a fully developed turbulent state where $\hbox {C}_f$ follows the trend of classical skin friction correlations (see Sect. 3). Selected velocity profiles at various stations in the laminar, transitional, and nearly turbulent flow regimes are reported on panel b of the same figure. Once again, the present results match remarkably well the reference data, thus confirming the quality of the present simulations.

Fig. 17

Skin friction coefficient a blue line present DNS; circle DNS of Franko and Lele; dotted line White’s turbulent correlation. Mean streamwise velocity profiles b present DNS (solid lines) and Franko and Lele (2013) (symbols) at locations $x/\delta ^*_\text {ref}$ = 400 (orange line, orange triangle), 650 (green line, green diamond), 800 (blue line, blue inverted triangle), and 950 (red line, red circle)

Finally, in Fig. 18, we compare temperature profiles from the present PG simulations at $M=2.25$ and $M=6$ at $\hbox {Re}_\theta =4000$ with the classical temperature law of Walz (1969). The numerical results are found to match very well the analytical model.

Fig. 18

Normalized temperature profiles for PG runs (lines) and prediction from Walz’s law (Walz 1969) (symbols) at $\hbox {Re}_\theta =4000$. Red dashed line Air $M=2.25$, blue line Air $M=6$