Inverse-cubic scaling of normalized static-pressure spatial variations in low-Reynolds-number air-film flows over suction-type floating stages

This study investigates characteristics of flow kinetic energy and static pressure within a floating stage with suction holes using direct numerical simulation (DNS). The analysis targets situations where a thin film is floated and stabilized at a height of $30\text{–}80\mu m$ above the floating stage, reflecting practical applications. On this stage, jet inlets and suction holes are arranged in a grid pattern. A characteristic fundamental flow element is extracted and analyzed in this study. The flow is confirmed to be an incompressible flow governed by the continuum approximation, as evidenced by the Mach number and Knudsen number. Here, maximum Mach number: $0.17\text{–}0.18$ and Knudsen number range (for the levitation gaps of $30\text{–}80\mu m$): $0.0009\text{–}0.002$, respectively. Meanwhile, the bulk Reynolds number indicates that the flow is within the applicability range of the Reynolds lubrication equation, allowing the use of governing equations that exclude inertial terms. Here, the bulk Reynolds numbers examined: 200, 50, and 12.5. Three key parameters are defined to characterize this flow: the ratio of the inlet radius to the suction hole radius $R_{\mathrm{in}}/R_{\mathrm{out}}$, the non-dimensionalized floating height $h$, and the horizontal domain width of the fundamental flow element $L_x$. The DNS is performed using a validated computational code previously applied to direct numerical simulations of turbulent channel flows. The results are verified in terms of grid convergence and zero viscous divergence required by mass conservation. Subsequent analysis is then applied to elucidate the flow kinetic energy and static pressure characteristics. The results demonstrate that the steady-state values of the flow kinetic energy and the spatial root mean square (RMS) of the static pressure variation, normalized by their respective values at $h=1$, are independent of $R_{\mathrm{in}}/R_{\mathrm{out}}$ and $L_x$. These quantities are inversely proportional to $h^2$ and $h^3$, respectively.