Nonlinear oscillatory rarefied gas flow inside a rectangular cavity

The nonlinear oscillation of rarefied gas flow inside a two-dimensional rectangular cavity is investigated on the basis of the Shakhov kinetic equation. The gas dynamics, heat transfer, and damping force are studied numerically via the discrete unified gas-kinetic scheme for a wide range of parameters, including gas rarefaction, cavity aspect ratio, and oscillation frequency. Contrary to the linear oscillation where the velocity, temperature, and heat flux are symmetrical and oscillate with the same frequency as the oscillating lid, flow properties in nonlinear oscillatory cases turn out to be asymmetrical, and second-harmonic oscillation of the temperature field is observed. As a consequence, the amplitude of the shear stress near the top-right corner of the cavity could be several times larger than that at the top-left corner, while the temperature at the top-right corner could be significantly higher than the wall temperature in nearly the whole oscillation period. For the linear oscillation with the frequency over a critical value, and for the nonlinear oscillation, the heat transfer from the hot to cold region dominates inside the cavity, which is contrary to the anti-Fourier heat transfer in a low-speed rarefied lid-driven cavity flow. The damping force exerted on the oscillating lid is studied in detail, and the scaling laws are developed to describe the dependency of the resonance and antiresonance frequencies (corresponding to the damping force at a local maximum and minimum, respectively) on the reciprocal aspect ratio from the near hydrodynamic to highly rarefied regimes. These findings could be useful in the design of the micro-electro-mechanical devices operating in the nonlinear-flow regime.

Figure 1

Schematic diagram of the oscillatory flow in a rectangular cavity. The coordinate origin is located at the bottom-left corner of the cavity.

Figure 2

The horizontal velocity on the oscillating lid when $\omega t/2\pi =0, \text{St}=2, A=1$, and (a) $\text{Ma}=0.01$ and (b) $\text{Ma}=1.2$. Along the direction of the arrow, the Knudsen number is respectively 10, 1, 0.1, 0.01, and 0.001 for each line. The velocity has been normalized by the velocity amplitude $U_0$ of the oscillating lid.

Figure 3

The evolution of the temperature at $(x,y)=(0.95L,0.95H)$ for (a) $\text{Kn}=0.1, \text{Ma}=0.01$, (b) $\text{Kn}=0.1$, Ma = 1.2, (c) Kn = 0.01, $\text{Ma}=0.01$, and (d) $\text{Kn}=0.01, \text{Ma}=1.2$, with $\text{St}=2$ and $A=1$. The solid lines are obtained by fitting the numerical results (circle symbol) with Eq. (29). Note that time $t$ has been normalized by the oscillation period $2\pi /\omega$.

Figure 4

Temperature contours and streamlines of heat flux inside the cavity at different times, when $\text{Kn}=0.1, \text{St}=2, A=1$, and the Mach numbers are $\text{Ma}=0.01$ (top row) and 1.2 (bottom row).

Figure 5

Temperature contours overlaid by the heat flux streamlines inside the cavity for $\text{St}=0$, 0.5 and 1 (from let to right column), and $\text{Ma}=0.01$ (top row) and 1.2 (bottom row), with $\text{Kn}=0.1$ and $A=1$, when the horizontal velocity of the lid is maximum along the positive direction ($\omega t/2\pi =0$).

Figure 6

The magnitude of shear stress on the oscillating lid when $t=0, \text{St}=2$, and (a) $\text{Ma}=0.01$ and (b) $\text{Ma}=1.2$. Along the direction of the arrow, Kn is respectively 10, 1, 0.1, 0.01, and 0.001 for each line. Note that the shear stress has been normalized by ${\rho }_{0}R{T}_w{U}_0/v_m$.

Figure 7

Variation of the average amplitudes of shear stress on the oscillating lid with St, for $A=0.5$, 1 and 2, and $\text{Kn}=10$, 1, 0.1, and 0.01 (from top to bottom), at $\text{Ma}=0.01$ (left column) and 1.2 (right column).

Figure 8

The variation of the amplitude of the average shear stress on the oscillating lid with the Mach number, when $\text{Kn}=0.1$, 1 and $\text{St}=2$, 10. Note that the shear stress has been normalized by ${\rho }_{0}R{T}_w{U}_0/v_m$.

Figure 9

The antiresonance Strouhal number ${\text{St}}_a$, at which the amplitude of the average shear stress at the lid is minimum, as a linear function of the inverse aspect ratio $1/A$ for (a) $\text{Kn}=0.1$ and (b) $\text{Kn}=1$. The results of the linearized Boltzmann equation solved by the FSM [17] are also included for comparison.

Figure 10

The resonance (antiresonance) Strouhal numbers, at which the amplitude of the average shear stress at the lid is a local maximum (minimum), as a linear function of the inverse aspect ratio $1/A$ when $\text{Kn}=0.01$, in both the linear ($\text{Ma}=0.01$) and nonlinear ($\text{Ma}=1.2$) oscillations.