Sensitivity dependence of the Navier-Stokes turbulence of a two-dimensional Rayleigh-Bénard convection on time-step

Abstract

A two-dimensional Rayleigh-Bénard convection governed by the Navier-Stokes (NS) equations is solved by traditional direct numerical simulation (DNS) using double precision and various time-steps. It is found that there are two kinds of final flow types, one is vortical flow, the other is zonal flow, and their statistics are completely different. Especially, the two flow types frequently alternate as the time-step decreases to a very small value, suggesting that the time-step corresponding to each type of turbulent flows should be densely distributed. Thus, stochastic numerical noise has huge influences on final flow type and statistics of numerical simulation of the NS turbulence (i.e. turbulence governed by NS equations), since time-step has a close relationship with numerical noise. However, the NS equations as turbulence model have such an assumption that all small stochastic disturbances for $t>0$ are negligible. This leads to a logic paradox in theory. Obviously, more investigations are necessary to reveal the essential differences between the NS turbulence, its numerical simulation, and real turbulence.

Figures (4)

• Figure 1: Schematic of 2D turbulent RBC tilted at an angle $\beta$. The fluid layer, between two parallel plates separated by a distance $H$, obtains heat from the bottom (red) plate and is cooled by the top (blue) plate due to a constant temperature difference $\Delta T>0$. Here $L$ is the width of computational domain and the downward direction of gravity acceleration $g$ is as indicated.
• Figure 2: $\theta$ (temperature departure from the linear variation background) fields at $t=500$ of the turbulent RBC tilted at the angles (a)-(b) $\beta=10^\circ$, (c)-(d) $\beta=15^\circ$, (e)-(f) $\beta=18^\circ$, and (g)-(h) $\beta=20^\circ$, given by DNS with the time steps $\triangle t=7\times10^{-4}$ (left, vortical flow) and $\triangle t=8\times10^{-4}$ (right, zonal flow).
• Figure 3: Final flow type of the tilted turbulent RBC versus time step $\triangle t$ of DNS: either vortical/roll-like flow (blue circle) or zonal flow (red square). Tilt angle: (a) $\beta=10^\circ$, (b) $\beta=15^\circ$, (c) $\beta=18^\circ$, and (d) $\beta=20^\circ$.
• Figure 4: Final flow type of the turbulent RBC tilted at the angle $\beta=18^\circ$ versus time step $\triangle t$ of DNS: either vortical/roll-like flow (blue circle) or zonal flow (red square). (a) $\triangle t \in [0.0001, 0.002]$, (b) $\triangle t \in [0.00172, 0.00198]$, and (c) $\triangle t \in [0.001742, 0.001778]$.