Accelerated evolution of convective simulations

High-Peclet-number, turbulent convection is a classic system with a large timescale separation between flow speeds and the thermal relaxation time. In this paper, we present a method of fast-forwarding through the long thermal relaxation of convective simulations, and we test the validity of this method. This accelerated evolution (AE) method involves measuring the dynamics of convection early in a simulation and using its characteristics to adjust the mean thermodynamic profile within the domain toward its evolved state. We study Rayleigh-Bénard convection as a test case for AE. Evolved flow properties of AE solutions are measured to be within a few percent of solutions which are reached through standard evolution (SE) over a full thermal diffusion timescale. At the highest values of the Rayleigh number at which we compare SE and AE, we find that AE solutions require roughly an order of magnitude fewer computing hours to evolve than SE solutions.

Figure 1

(a) Kinetic energy (black) and $〈T〉-T_{\text{top}}$ (blue) vs time are shown for a SE run at $S={10}^5$. The mean evolved values of kinetic energy and mean temperature, averaged over the time shaded in green, are denoted by the horizontal dashed lines. (b) The time- and horizontally averaged flux profiles are shown for the times highlighted in orange in panel (a). (c) The same quantities as in panel (a) are shown, but for AE at the same parameters. The axes are scaled identically in panels (a) and (c), and the AE method is used three times, marked by the numbered arrows. The fluxes averaged over the green shaded region of panel (c) are shown in panel (d). The difference between the fluxes in the AE and SE solutions is shown in panel (e).

Figure 2

The time evolution of Nu (black) and $〈T〉-T_{\text{top}}$ (blue) are shown for simulations at $S={10}^5$ in SE (a) and AE (b). A moving average of Nu, using a centered boxcar with a span of 50 freefall time units, is overplotted as a gray line. We see that the mean value of Nu reaches its relaxed value quickly compared to $〈T〉$. Regardless, fluctuations of Nu about the mean value are much smaller in the thermally relaxed state. The times at which AE solves occur are denoted by arrows in panel (b), and the gray boxed region is examined in more detail in Fig. 3.

Figure 3

The time variation of the Nusselt number is shown for two AE cases at $S={10}^5$ (top) and $S={10}^7$ (bottom). On the left, the instantaneous value of Nu is shown as a function of time. On the right, temperature snapshots are shown for Nu maxima (Ia and IIa) and minima (Ib and IIb). The suppressed value of Nu at the minma arises from entrainment of fluid elements whose temperature perturbations are wrongly signed (e.g., hot material going downward and cold material going upward in Ib and IIb). The color scaling of panels Ia and Ib are the same, as are the scalings of panels IIa and IIb. The minimum temperature is chosen by the fixed-temperature boundary condition at the top, $T_{\text{top}}=-0.5$. The decreased range of the color scaling for IIa and IIb was chosen to better display the convective dynamics.

Figure 4

Volume- and time-averaged measurements of the Nusselt number (Nu), the RMS Peclet number ($\langle \mathrm{Pe}\rangle$), and the mean temperature ($〈T〉$) for AE runs are shown in panels (a)–(c). Symbols are located at the mean value of each measurement and denote 2D (purple circles) and 3D (red stars). The run at $S={10}^5$ marked as a black square is examined in more detail in Figs. 1, 2, 3, 5, and 6. Vertical lines represent the standard deviation of the measurement, and quantify natural variation over the averaging window. (a) Nu scales as ${\mathrm{Ra}}^{1/5}$; at high $S$ in 2D the value of Nu fluctuates over time (see Fig. 3). (b) Pe, which measures turbulence in the solution, scales as ${\mathrm{Ra}}^{0.45}$. (c) The difference between $〈T〉$ and the value of $T$ at the fixed-temperature top boundary is shown; this quantity scales as ${\mathrm{Ra}}^{-1/5}$, the inverse of Nu. Relative error for measurements of (d) Nu, (e) Pe, and (f) $〈T〉-T_{\text{top}}$ between AE solutions and SE solutions are shown. The grayed area of the plots indicates the region in which only AE runs were carried out due to computational expense.

Figure 5

Comparisons of the evolved thermodynamic states of an AE and SE run at $S={10}^5$ are shown. (a) Evolved horizontally and time-averaged temperature profiles, as a function of height. (b) Probability distribution functions (PDFs, $P\left(T\right)$) and their integrated cumulative distribution functions (CDFs, $F\left(T\right)$) of point-by-point measurements of the temperature field. (c) The percentage difference between the mean temperature profiles as a function of height. The difference between the mean profiles is very small, $O$(0.5%). (d) $D_{\text{KS}}\left(T\right)$, as defined in Eq. (15), is shown. The small difference in the mean interior temperature between AE and SE results in a large difference between the two temperature distributions near the values of the temperature maxima. The spread of temperature around the maxima, which includes the fluctuations that drive convection, are nearly identical between the two runs.

Figure 6

Probability distribution functions (PDFs, $P\left(q\right)$) of (a) the vertical velocity ($q=w$), (b) the horizontal velocity ($q=u$), and (c) nonlinear convective transport ($q=w(T_1-{\langle T_1\rangle }_{x,y})$) are shown for 2D runs achieved through SE (blue) and AE (red) at $S={10}^5$. The cumulative distribution function (CDF, $F\left(q\right)$) is overplotted for each PDF. [(d)–(f)] The $D_{\text{KS}}$ profiles, as defined in Eq. (15), are shown for the related distributions; solid lines indicate positive values while dashed lines are negative values. Unlike the temperature distributions in Fig. 5, these distributions show very good agreement and small values of the $D_{\text{KS}}$ statistic.