Equivalence of three thermal boundary conditions in compressible turbulent channel flows

In this paper, direct numerical simulations have been performed to explore the equivalence of different thermal boundary conditions in compressible turbulent channel flows at fixed $\mathrm{Re}=6000,\mathrm{Ma}=1.5$. Three canonical types of thermal boundary conditions will be investigated at almost equivalent setups, including the first boundary condition with fixed wall temperature $T_w$ (constant Dirichlet boundary condition), the second boundary condition with fixed wall heat-flux $q_w$ (constant Neumann boundary condition), and the third boundary condition (Robin boundary condition). The turbulent statistics of the temperature and velocity fields, including mean profiles, root-mean-square values, second-order statistics, and normalized probability density functions, temperature stripes near the wall and the budget of internal energy have been analyzed in detail to clarify the differences caused by the different thermal boundary conditions. The results show that the three thermal boundary conditions have almost negligible effect on the velocity field, whereas some discernible deviations can be observed for the temperature field in the near-wall region with $y^{+}\lesssim 30$. Furthermore, the statistics from the second and third thermal boundary conditions are very close, enabling the usage of the second boundary condition to mimic the more complex third boundary conditions.

Figure 1

Schematic of a compressible turbulent channel flow. On the top wall, the isothermal boundary condition is applied. On the bottom wall, three different thermal boundary conditions, i.e., (1)–(3), are applied in three cases at almost equivalent setups.

Figure 2

Profiles of (a) mean streamwise velocity (normalized by $u_m$), (b) mean temperature (normalized by $T_{\mathrm{ref}}$), and (c) mean density (normalized by ${\rho }_m$) in global coordinate (normalized by $h$). The inset in (c) shows the corresponding near-wall behavior.

Figure 3

Profiles of turbulence intensities from the three cases in (a) the bottom half and (b) the top half. Here, the rms values are normalized by $u_{\tau }$.

Figure 4

Profiles of the rms values of the temperature fluctuations (normalized by $T_{\mathrm{ref}}$) in (a) the bottom half and (b) the top half.

Figure 5

Profiles of (a) $\langle u^{\prime }{v}^{\prime }\rangle$ (normalized by $u_m{u}_m$), (b) $\langle u^{\prime }{T}^{\prime }\rangle$ (normalized by $u_m{T}_{\mathrm{ref}}$) and (c) $\langle v^{\prime }{T}^{\prime }\rangle$ (normalized by $u_m{T}_{\mathrm{ref}}$) in the bottom half of the channel. Please note that they are zero for all cases at the wall. The near-wall asymptotic behaviors of $\langle u^{\prime }{T}^{\prime }\rangle$ and $\langle v^{\prime }{T}^{\prime }\rangle$ are also shown in the insets in (b) and (c).

Figure 6

Normalized p.d.f.s of the temperature fluctuations from different cases at different wall-normal locations. (a) $y^{+}\approx 5$, (b) $y^{+}\approx 15$, (c) $y/h=0.3$ from the bottom wall, and (d) $y^{+}\approx 5$ and $y/h=0.3$ from the top wall.

Figure 7

Normalized p.d.f.s of the streamwise velocity fluctuations (a) and (b) and wall-normal velocity fluctuations (c) and (d) at different wall-normal locations. Here, (a) and (c) are in the bottom half and (b) and (d) are in the top half.

Figure 8

The instantaneous temperature contours $T^{\prime }/T_{\mathrm{ref}}$ at the bottom wall from the cases of (a) M15T21 and (b) M15T31.

Figure 9

The instantaneous temperature contours $T^{\prime }/T_{\mathrm{ref}}$ at $y^{+}\approx 15$ away from the bottom wall from the cases of (a) M15T11, (b) M15T21, and (c) M15T31.

Figure 10

(a) Spanwise and (b) streamwise two-point correlations for the temperature at $y^{+}\approx 15$ away from the bottom wall for the three cases.

Figure 11

(a) Spanwise and (b) streamwise two-point correlations for the streamwise velocity component at $y^{+}\approx 15$ away from the bottom wall for the three cases.

Figure 12

Budget of the mean internal equation for case of M15T11 in the bottom half of the channel. $C_k\approx 0$ is not shown. The terms are normalized by ${\rho }_m{u}_m^3/h$.

Figure 13

Budget terms for all three cases in the bottom half of the channel: (a) $D_{em1}$ and $D_{em2}$; (b) ${\varepsilon }_{em}$, ${\varepsilon }_k$, and $P_{em}$. The terms are normalized by ${\rho }_m{u}_m^3/h$.

Figure 14

Profiles of the budget terms in M15T11(normalized by ${\rho }_m{u}_m{T}_{\mathrm{ref}}^2/h$).

Figure 15

Near-wall behavior of the dominated terms: (a) $P_{TV}$, (b) $C_{TV2}$, (c) $D_{TV2}$, and (d) ${\varepsilon }_{TV}$ for all three cases.