It is of broad interest to understand how the evolution of non-equilibrium systems can be triggered and the role played by external perturbations. A famous example is the origin of randomness in the laminar-turbulence transition, which is raised in the pipe flow experiment by Reynolds as a century old unresolved problem. Although there exist different hypotheses, it is widely believed that the randomness is “intrinsic", which, however, remains as an open question to be verified. Simulating the modeled Rayleigh-Bénard convection system by means of the so-called clean numerical simulation (CNS) with negligible numerical noises that are smaller even than thermal fluctuation, we verify that turbulence can be self-excited from the inherent thermal fluctuation without any external disturbances, i.e. out of nothing. This reveals a relationship between microscopic physical uncertainty and macroscopic randomness. It is found that in physics the system nonlinearity functions as a channel for randomness information, and energy as well, to transport microscopic uncertainty toward large scales. Such scenario can generally be helpful to understand the various relevant phenomena. In methodology, compared with direct numerical simulation (DNS), CNS opens a new direction to investigate turbulent flows with largely improved accuracy and reliability.
This work was partly supported by the National Natural Science Foundation of China (Grant Nos. 11272209, and 11432009). The parallel algorithms were performed on TH-1A at National Supercomputer Centre in Tianjin, China. The authors thank Jing Li for using the open resource DEDALUS to gain the DNS results.
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Figure 1
(Color online) Schematic representation of the Rayleigh-Bénard convective flow. The two-dimensional incompressible fluid between two parallel free surfaces separated by $H$ obtains heat from the bottom boundary because of the prescribed constant temperature difference $\Delta~T~>~0$, where $g$ is the gravity acceleration.
Figure 2
(Color online) Verification of CNS algorithm. The Nusselt number $Nu$ is calculated by CNS (in symbol) with the double Fourier expansion modes $M~=~N~=~31$ at different Rayleigh numbers above the critical value $Ra_c~=~27\pi^4/4$. From the curve fitting (line) the estimated critical Rayleigh number agrees well with the theoretical value.
Figure 3
(Color online) Reliability check of the CNS results. The results are for the case A (of the initial thermal fluctuation) with the Rayleigh number $Ra~=~10^{7}$ and the double Fourier expansion modes $M~=~N~=~127$ at three probe points: (a) ($3L/4,~H/2$), (b) ($3L/4,~2H/5$) and (c) ($3L/4,~H/10$). The curves denote the dimensionless deviations of $\Delta^\theta_{10}=~|\theta_{P=12}-\theta_{P=10}|/\theta_{RMS}$ (left) and $\Delta^V_{10}=~|V_{P=12}-V_{P=10}|/\sqrt{E_{RMS}}$ (right), which are much less than the micro-level thermal fluctuation. Here $P$ is the order of the Taylor series in time; $\theta_{RMS}$ is the spatial root mean square of $\theta$ (the temperature departure from a linear variation background); $E_{RMS}$ is the spatial root mean square of the kinetic energy $(u^2+w^2)/2$, respectively.
Figure 4
(Color online) Evolution of the $\theta$ (temperature departure from a linear variation background) field. The results are for (a) $t=0$; (b) $t=2$; (c) $t=8$; (d) $t=28$ and (e) $t=31$, with the Rayleigh number $Ra~=~10^{7}$ and the double Fourier expansion modes $M~=~N~=~127$. Case A and case B have different initial micro-level randomness, generated by the same variance of temperature $\sigma_T=10^{-10}$ and velocity $\sigma_u=10^{-9}$.
Figure 5
(Color online) Evolution of the kinetic and thermal energy at different scales. The results are for case A with Rayleigh number $Ra~=~10^{7}$ and the double Fourier expansion modes $M~=~N~=~127$, where $E$ and $E_\theta$ denote the total kinetic and thermal energy, $k$ is the wave number, respectively. Lines in black: kinetic energy; Lines in red: thermal energy. The points A (black dot) and B (red dot) represent transition when the nonlinear interaction (with the large scale components) is strong enough to dominate to evolution process.
Figure 6
(Color online) Evolution of the normalized kinetic energy spectrum. The results are for case A with Rayleigh number $Ra~=~10^{7}$ and the double Fourier expansion modes $M~=~N~=~127$. (a) Initially energy shifts from small scales to larger scales, at which the spatial structure remains stable in most of the evolution process. (b) When turbulence transition occurs the large scale disintegrates and energy shifts inversely from large scales to small ones.
Figure 7
(Color online) Correlation between the $\theta$ field and $w$ field. Here $\theta$ denotes the temperature departure from a linear variation background and $w$ is the velocity component along the opposite gravity direction, respectively. The results are for (a) $t=0$; (b) $t=2$; (c) $t=8$; (d) $t=28$ and (e) $t=31$, with the Rayleigh number $Ra~=~10^{7}$ and the double Fourier expansion modes $M~=~N~=~127$.
Figure 8
(Color online) Evolution of the correlation coefficient and its PDF. (a) Evolution of the correlation coefficient $C(t)$ between $w$ and $\theta$ for the case A with $Ra~=~10^{7}$ and the double Fourier expansion modes $M~=~N~=~127$. From the initial state $C(t)$ increases rapidly (for $t<1$) from $C\sim~0$ because of the initial random independence to $C\sim~1$, corresponding to a strong correlation, which then remains till the transition to turbulence. (b) Change of the PDF of the normalized kinetic energy source (for the case A with $Ra~=~10^{7}$). The PDF is initially close to be symmetric and evolve rapidly to be positively skewed. At the turbulence state, the PDF returns to be symmetric, but broadens largely.
Figure 9
(Color online) Evolution of the $\theta$ (temperature departure from a linear variation background) field. Here the Rayleigh number is $Ra~=~2000$ and the double Fourier expansion modes $M~=~N~=~31$. The results are for (a) $t=0$; (b) $t=0.5$; (c) $t=1$; (d) $t=5$; (e) $t=50$; (f) $t=400$. Case 1 and case 2 have different initial micro-level randomness due to thermal fluctuation, generated by the same variance of temperature $\sigma_T=10^{-10}$ and velocity $\sigma_u=10^{-9}$.
Figure 10
(Color online) Comparison of evolution of the $\theta$ (temperature departure from a linear variation background) given by the CNS and DNS. Here, Rayleigh number is $Ra~=~10^{7}$. Left: CNS results, obtained using the same parameters in Figure
Figure 11
(Color online) Reliability check of the numerical results given by the CNS and DNS. 1) for DNS (dash line)$\Delta^\theta~=~|\theta_{cfl=0.2}-\theta_{cfl=0.1}|/\theta_{RMS}$ and $\Delta^V~=~|V_{cfl=0.2}-V_{cfl=0.1}|/\sqrt{E_{RMS}}$; 2) for CNS (solid line) $\Delta^\theta~=~|\theta_{P=12}-\theta_{P=10}|/\theta_{RMS}$ and $\Delta^V~=~|V_{P=12}-V_{P=10}|/\sqrt{E_{RMS}}$. $\theta_{RMS}$ and $E_{RMS}$ are the root mean squares of the temperature $\theta$ and the kinetic energy $E=(u^2+w^2)/2$ in the domain $x\in[0,L],~z\in[0,H]$. (a) At probe point ($3L/4,~H/10$);(b) at probe point ($3L/4,~2H/5$); (c) at probe point ($3L/4,~H/2$).
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