Bounded flows of dense gases

Numerical solutions of the Enskog equation obtained employing a finite-difference lattice Boltzmann (FDLB) and a direct simulation Monte Carlo–like particle method (PM) are systematically compared to determine the range of applicability of the simplified Enskog collision operator implemented in the lattice Boltzmann framework using the half-range Gauss-Hermite quadratures. Three types of bounded flows of dense gases, namely the Fourier, the Couette, and the Poiseuille flows, are investigated for a wide range of input parameters. For low to moderate reduced density, the proposed FDLB model exhibits commendable accuracy for all bounded flows tested in this study, with substantially lower computational cost than the PM method.

Figure 1

Dense gas at rest near a reflective wall: Normalized density profiles $\eta /{\eta }_0$ for three values of the mean reduced density ${\eta }_0\in \{0.01,0.10,0.20\}$ and two values of the confinement ratios $R=10$ (a) and $R=4$ (b), corresponding to $\mathrm{Kn}\in \{1.1493,0.0904,0.0335\}$ and $\mathrm{Kn}\in \{2.8731,0.226,0.0838\}$, respectively.

Figure 2

Fourier flow: Heat transfer at wall temperature difference $\mathrm{\Delta }T=0.1$ and three values of the mean reduced density ${\eta }_0$: Transversal profiles of the normalized reduced density $\eta /{\eta }_0$ (first row) and of the normalized temperature $T/T_0$ (second row) for $R=10$ in the left column ($\mathrm{Kn}\in \{1.1493,0.0904,0.0335\}$) and $R=4$ in the right column ($\mathrm{Kn}\in \{2.8731,0.226,0.0838\}$).

Figure 3

Fourier flow: Heat transfer at wall temperature difference $\mathrm{\Delta }T=0.5$ and three values of the mean reduced density ${\eta }_0$: Transversal profiles of the normalized reduced density $\eta /{\eta }_0$ (first row) and of the normalized temperature $T/T_0$ (second row) for $R=10$ in the left column ($\mathrm{Kn}\in \{1.1493,0.0904,0.0335\}$) and $R=4$ in the right column ($\mathrm{Kn}\in \{2.8731,0.226,0.0838\}$).

Figure 4

Fourier flow: (a) Transversal heat flux values $q_x$ at temperature differences $\mathrm{\Delta }T\in \{0.1,0.5\}$ and confinement ratios of $R=\{4,10\}$. (b) Dependence of the ratio $q_x/n_0$ with respect to the Knudsen number Kn for three values of the reduced density $\eta \in \{0.01,0.1,0.2\}$.

Figure 5

Couette flow: Normalized reduced density profile $\eta /{\eta }_0$ at a wall velocity of $U_w=1$; three values of the mean reduced density ${\eta }_0=\{0.01,0.1,0.2\}$ and (a) $R=10$ ($\mathrm{Kn}\in \{1.1493,0.0904,0.0335\}$) and (b) $R=4$ ($\mathrm{Kn}\in \{2.8731,0.226,0.0838\}$).

Figure 6

Couette flow: Normalized velocity and temperature profiles at a wall velocity of $U_w=1$; three values of the mean reduced density ${\eta }_0=\{0.01,0.1,0.2\}$ and $R=10$ in the left column ($\mathrm{Kn}\in \{1.1493,0.0904,0.0335\}$) and $R=4$ in the right column ($\mathrm{Kn}\in \{2.8731,0.226,0.0838\}$).

Figure 7

Couette flow: Profiles of both the transversal ($q_x>0$) and the longitudinal ($q_y<0$) heat fluxes for the wall velocity $U_w=1$; three values of the mean reduced density ${\eta }_0=\{0.01,0.1,0.2\}$ and two values of the confinement ratio (a) $R=10$ ($\mathrm{Kn}\in \{1.1493,0.0904,0.0335\}$) and (b) $R=4$ ($\mathrm{Kn}\in \{2.8731,0.226,0.0838\}$).

Figure 8

Poiseuille flow (linear regime): Normalized density and velocity profiles generated using the external acceleration $a_y=0.001$ for three values of the mean reduced density ${\eta }_0=\{0.01,0.1,0.2\}$ and two values of the confinement, namely $R=10$ (left column) and $R=4$ (right column), corresponding to $\mathrm{Kn}\in \{1.1493,0.0904,0.0335\}$ and $\mathrm{Kn}\in \{2.8731,0.226,0.0838\}$, respectively.

Figure 9

Poiseuille flow (nonlinear regime): Normalized velocity and temperature profiles for the external acceleration $a_y=0.1$, three values of the mean reduced density ${\eta }_0=\{0.01,0.1,0.2\}, R=10$ in the left column ($\mathrm{Kn}\in \{1.1493,0.0904,0.0335\}$) and $R=4$ in the right column ($\mathrm{Kn}\in \{2.8731,0.226,0.0838\}$).

Figure 10

Poiseuille flow (nonlinear regime): Transversal and longitudinal heat flux, $q_x$ and $q_y$, at an external acceleration of $a_y=0.1$; three values of the mean reduced density ${\eta }_0=\{0.01,0.1,0.2\}$ and $R=10$ in the left column ($\mathrm{Kn}\in \{1.1493,0.0904,0.0335\}$) and $R=4$ in the right column ($\mathrm{Kn}\in \{2.8731,0.226,0.0838\}$).

Figure 11

Poiseuille flow (mass flow rate): (a) Normalized mass flow rate $\dot{m}/{\dot{m}}_0$ at an external acceleration of $a_y=0.001$ with respect to Knudsen number Kn; four values of the confinement ratio $R=\{2,5,10,20\}$. (b) Normalized mass flow rate $\dot{m}/{\dot{m}}_0$ with respect to the Knudsen number for four values of the quadrature order $Q_x\in \{8,16,32,64\}$. One can observe that as the Knudsen number increases one has to employ a higher-order quadrature in order to recover the mean flow rate, as is the case for dilute gases.

Figure 12

Comparison between simulation results performed with FDLB models using full- and half-range Gauss-Hermite quadratures: (a) Fourier flow, (b) Couette flow, and (c) Poiseuille flow at $R=10, {\eta }_0=0.01$ (resulting in $\mathrm{Kn}=1.1493$), and $\mathrm{\Delta }T=0.5, U_w=1$, and $a_y=0.001$, respectively. The full- and the half-range quadrature orders are denoted by $Q^H$ and $Q^h$, respectively. These plots indicate a convergence of the results obtained using full-range quadrature towards those obtained using half-range quadrature.

Figure 13

Heat flux: The dependence of the kinetic $q_x^{\mathrm{kin}}$ and the potential $q_x^{\mathrm{pot}}$ contributions to the heat flux on the Knudsen number. The kinetic component increases monotonically, while the potential part has two local maxima between $\mathrm{Kn}=1$ and 10, leading to a total heat flux that exceeds the free molecular limit.

Figure 14

Dependence of the $L_2^M$ error, Eq. (E1), with respect to the wall velocity $U_w$ for (a) $M=u_y$ and (b) $M=T$. The Knudsen number is $\mathrm{Kn}=2.8731$.