Chapter V: Direct Numerical Simulation of Sandia flame B: grid resolu-
5.5 Convergence of statistics
Figure 5.4: Time series over three time periods of centerlineUvelocity (top) and the OH mass fraction atr/d=1(bottom), forx/d=1(black),x/d=10(blue), andx/d=30(red), from case BL. Left and center: every time step is shown; the vertical dotted lines represent every data file. Right: entire simulation time; the vertical dotted lines represent every tenth data file.
A convective flow through time may be defined as τ≡
Z L x
0
dx
hU(x)i ≈79.1 d Ujet
!
=0.0313s, (5.8) where hU(x)i is the mean centerline velocity, and Lx is the longitudinal domain length. The averaging time of each simulation inτunits is reported in Table5.2.
In the analysis carried out in Ch. 3, it was shown that for small chemical mod- els (tens of species and hundreds of reactions) and three dimensional DNS, the computational cost associated with mixture-averaged transport properties is only marginally greater than using constant Lewis numbers. That is why for all four cases considered in this chapter, mixture-averaged transport properties are consid- ered.
The instantaneous and averaged temperature field for case BL is shown in Figs.5.3a and5.3b, respectively. All averages considered in this chapter are carried out both in time and in the azimuthal direction, which is statistically homogeneous. In the next sections, comparisons between case BL and the refined cases are carried out along the centerline, and the cross-stream planes shown in Fig.5.3b.
5.5. Convergence of statistics 86
Figure 5.5: Solid lines: two-time auto-correlation for the centerline U forx/d=1(black), x/d=10(blue) andx/d=30(red). Dashed lines: osculating parabolas.
5.5 Convergence of statistics
The high cost of DNS often leads to relatively short running times, and as a result, sampling errors for the statistics may be of the same order as or even larger than discretization errors [164]. In this section, the estimation of sampling errors for sample means is discussed. Figure5.4shows time series of the centerline U and the OH mass fraction atr/d=1, which is close to the radial location of its peak mass fraction, for several downstream stations. As can be seen, for the same downstream locations, the two quantities present significantly different fluctuation levels, and the oscillations occur with a much different frequency. Further, the behavior of each quantity changes qualitatively and quantitatively moving downstream. Thus, for different quantities at different locations, achieving the same sampling error will require different simulation running times.
5.5.1 Randomness and time-correlation
When assessing sampling errors, it is desirable to have independent samples, and a direct way to estimate the sample decorrelation distance is through the two-time autocorrelation function [147]
ρcorr(lag)≡
X0(t)X0(t+lag)
hX0(t)2i , (5.9)
where X is some quantity of interest, and X0=X − hZi. Figure5.5shows ρcorr for U at three downstream locations along the centerline.
5.5. Convergence of statistics 87
Figure 5.6:τlagfor U (centerline) and YOH(r/d=1) for several downstream stations up to x/d=45.
From Eq. (5.9), an integral length scale may be computed as τint≡
Z ∞
0
ρcorr(lag)dlag. (5.10)
Typically, samples are considered decorrelated if the separation time is at least τint [173]. However, directly computing ρcorr through its definition tends to be noisy [174] due to the short running time of the simulations, which may lead to large errors when evaluating Eq. (5.10). As an alternative approach, one can extract a time scale from the curvature of ρcorrfor zero lag, since for small lag ρcorr can be considered to be converged. More precisely, the time scale may be computed as the x-axis intercept of the osculating parabola
ρ(lag)≈ 1− lag2
λ2lag. (5.11)
Osculating parabolas toρcorrare shown in Fig.5.5, and Fig.5.6showsλlagfor both U (at the centerline) and OH (atr/d=1), for x/d<45. As expected, λlag for both quantities is found to increase with axial distance. Figures5.5and5.6also show the data file saving rate for the baseline and refined cases. As can be seen, the chosen data file saving rate for the baseline case is larger than the largest λlag for both U and YOH, suggesting that the data files with that separation may be considered independent up tox/d≈45.
As an alternative approach to investigate sample independence for a quantity of interest X, one may consider the convergence of the sample standard deviation
5.5. Convergence of statistics 88
(a)Mean centerline streamwise velocity. (b)Mean OH atr/d=1.
Figure 5.7: Average of σN/hXiN with the normalized sample separation time, for the centerline U (left) and the OH mass fraction atr/d=1(right). Four downstream locations are considered: x/d=1, black;x/d=15, blue; x/d=30, red; x/d=45, green. Solid lines and full symbols: N=9; dashed lines and empty symbols: N=18. The vertical dashed line represents the separation time between data files.
relative to the sample average σN
hXiN ≡ 1 hXiN
vu t 1
N−1
N
X
i=1
(Xi− hXi)2, (5.12) where
hXiN ≡ 1 N
N
X
i=1
Xi, (5.13)
for increasing sample separation distances. In Eq. (5.12), the sum is carried out over N samples generated by the simulation. Here, two sample sizes are considered, including N=9 and N=18. Equation (5.12) is then computed for every set of 9 and 18 samples obtained with the given time separation distance, and then an average is plotted. This is shown in Fig. 5.7, for U and OH. For both quantities, the average of the ratioσN/hXiN initially grows with the sample separation time, and then plateaus when the separation time is greater than the decorrelation time.
As expected, for large sample separation times, the standard deviation becomes independent of the sample size and the separation distance. More specifically, for both the centerline U and OH at r/d=1, data files stored at the baseline saving rate may be considered decorrelated for all downstream stations considered. In Fig. 5.7b, the averageσN/hXiN values for x/d=1 and x/d=10 are much lower than the other stations. That occurs because for these stations, the flame is likely to be embedded within a laminar mixing region, and the rms of YOH is much lower than further downstream (see Fig.5.4).
5.6. Grid independence 89