6 Colleen M. Kaul, Venkat Raman, Guillaume Balarac and Heinz Pitsch
10 20
30 40
50 60
70 80
0.5 1
1.5 2
2.5 3
3.5 4
4.5 x 10
−4
Lag
γ
a
10 20
30 40
50 60
70 80
0.2 0.4
0.6 0.8
1 1.2
1.4 1.6
1.8 2
x 10
−4
Lag
γ
b
10 20
30 40
50 60
70 80
0.2 0.4
0.6 0.8
1 1.2
1.4 x 10
−3
Lag
γ
c
10 20
30 40
50 60
70 80
1 2
3 4
5 6
x 10
−4
Lag
γ
d Fig. 2.
Semivariograms of the subfilter variance field computed with exact DNS variance black and modeled variance from filtered DNS scalar gray and LES
scalars evolved using spectral gray dots, CD-2 dashed, CD-4 black dots, and P-6 dash-dot schemes at t = 0.7τ . a ∆=8η, Eq. 1 b ∆=8η, Eq. 2 c ∆=16η,
Eq. 1 d ∆=16η, Eq. 2
3.2 Evolution of subfilter variance distributions Quantile-quantile q-q plots provide a means of comparing two distributions
without assuming a parametric form while avoiding ad hoc bin assignments necessary for constructing histograms. The p quantile ζ
p
indicates the value of a random variable ζ at which its cumulative distribution function, F , equals p,
i.e. it satisfies F ζ
p
= p. In Fig. 3, quantiles of the modeled subfilter variances are plotted against quantiles of the exact DNS subfilter variance. A match
between the distributions is indicated when the q-q plot forms a 45 degree line. Departures from that line can be used to diagnose the differences between the
distribution. Provided the distributions are of the same shape, a linear plot is still formed. A difference in the location parameter for example, the mean of
the normal distribution shifts the intercept of the line. Inequality of the scale parameter e.g, the variance of the normal distribution modifies the slope.
More general differences in the shape of the distribution are manifested by a nonlinear q-q plot [4]. It appears that the differences in the subfilter variance
A Posteriori Analysis of Numerical Errors in Computing Scalar Variance
7
distributions can be largely explained by changes in the scale parameter, which accords with the results of Section 3.1.
In Figs. 3 and 4, symbols indicate the position of specific quantile esti- mates. These are the 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.925, 0.95,
and 0.975 quantiles. The distribution of subfilter variance values is strongly left-skewed. Results are shown for ∆ = 8η simulations only, as the ∆ = 16η
results are qualitatively very similar.
Initially, the differences between the modeled Z
′2
quantiles are small for all but the highest quantiles examined and agree fairly well with the distribution
of exact subfilter variance from the DNS. As the scalar fields evolve, the disparities between the distributions of modeled subfilter variance increase
Fig. 3.
Before pursuing the causes and implications of these findings, it should be recalled that the plots show estimated quantiles, based on a limited sample.
Confidence intervals for the data can be obtained through bootstrap tech- niques [4, 3] if we continue to treat the data as independent. Fig. 4 plots
differences of the estimated p quantiles Z
′2 model
p
− Z
′2 exact
p
and their 95 per- cent confidence intervals. These confidence intervals are obtained by forming
1000 random resamples of the original data and repeating the quantile dif- ference calculations for each sample. The results are then sorted in ascending
order, with the 25th and 975th values forming the lower and upper bounds. The behavior of the quantile differences is consistent with the hypothesis of
a scale change between the modeled and exact differences, with some more complex changes perhaps occurring in the extreme right tails. The confidence
intervals of the differences overlap at their edges in several areas, but do sup- port the claim that the second order scheme results in a different subfilter
variance distribution than the fourth or sixth order schemes. Fig. 5 provides another indication of the uncertainty of the quantile estimates. In addition to
the data used in Fig. 3a, alternative samples of the CD-2 and exact subfil- ter variance fields, formed by selecting points at the same interval but with
a different starting point, are plotted as dashed lines. Intersample variabil- ity is clearly evident, but does not meaningfully alter our assessment of the
scheme’s performance. Also shown is the result when all points in the field are used for the quantile calculations. As expected, it does not differ greatly
from the other estimates for this kind of statistic with this kind of data.
It is found that the model coefficient calculated by a dynamic procedure [1] increases with decreasing order of the numerical scheme. Qualitatively,
the same relationship is found in a priori analysis [5]. However, the differ- ences increase in time in the a posteriori case Fig. 6, a feature that was
not strongly apparent for a priori results at different times. Additionally, the coefficient predicted by second and fourth order accurate finite difference
methods strongly depends on the form of the diffusion term in the filtered scalar equation.
8 Colleen M. Kaul, Venkat Raman, Guillaume Balarac and Heinz Pitsch
In a priori tests, lower order schemes produce larger coefficient values by underestimating the gradient-based term M of the closure. In the a posteriori
case, evolution using lower order schemes shifts the bulk of the distribution of the Leonard term L to higher values, as shown in Fig. 7a. A higher value
implies greater filtered scalar energy near the cutoff lengthscale, a feature that can be readily inferred from the scalar spectrum Fig. 1. This result is a
direct consequence of the increased errors in finite difference schemes at high wavenumbers, as quantified by the modified wavenumber of the discretization.
Although higher values of L are associated with higher exact values of M, finite difference errors cause underestimation of M that roughly balances the
error of the scalar evolution Fig. 7b.
0.01 0.02
0.03 0.04
0.05 0.06
0.07 0.08
0.01 0.02
0.03 0.04
0.05 0.06
0.07 0.08
Model Quantiles DNS Quantiles
a
0.01 0.02
0.03 0.04
0.05 0.06
0.01 0.02
0.03 0.04
0.05 0.06
Model Quantiles DNS Quantiles
b Fig. 3.
Quantiles of modeled subfilter variance from ∆=8η filtered DNS circle and LES using spectral square, CD-2 triangle, CD-4 diamond, and P-6 star
schemes plotted against quantiles of exact DNS variance at t=0.7τ . a Filtered scalar evolved with Eq. 1 b Filtered scalar evolved with Eq. 2
4 Conclusions
A posteriori tests of a dynamic subfilter variance model show that errors
in the numerical solution of the filtered scalar field can have significant ef- fects on model predictions. Lower order schemes produce higher values of the
model coefficient C, as observed in a priori tests of finite difference model implementation. While in the DNS-based tests this effect resulted solely from
underestimation of the gradient term M , the most important factor in the current LES-based tests is the general increase in the Leonard term L. The
average rise in values of L can be attributed in part to poor representation of the diffusion operator at high wavenumbers by finite difference methods. This
problem can be ameliorated by adopting the modified although analytically equivalent form of the filtered scalar transport equation given in Eq. 2.
A Posteriori Analysis of Numerical Errors in Computing Scalar Variance
9
0.1 0.2
0.3 0.4
0.5 −0.01
−0.008 −0.006
−0.004 −0.002
Probability Quantile differences
a
0.5 0.6
0.7 0.8
0.9 1
−0.05 −0.04
−0.03 −0.02
−0.01 0.01
Probability Quantile differences
b Fig. 4.
Estimated quantile differences solid lines and 95 bootstrap confidence intervals dashed lines for subfilter variance of ∆=16η, Eq. 2 case at t=0.7τ . Exact
DNS quantiles are subtracted from quantiles of modeled variance from filtered DNS circle and LES using spectral square, CD-2 triangle, CD-4 diamond, and P-6
star methods. a 0.05 to 0.5 quantiles b 0.5 to 0.975 quantiles
0.01 0.02
0.03 0.04
0.05 0.06
