Analysis of uncertainties and convergence of the statistical quantities in turbulent wall-bounded flows by means of a physically based criterion

Abstract

The present paper provides an analysis of the statistical uncertainties associated with direct numerical simulation (DNS) results and experimental data for turbulent channel and pipe flows, showing a new physically based quantification of these errors, to improve the determination of the statistical deviations between DNSs and experiments. The analysis is carried out using a recently proposed criterion by Thompson et al. [“A methodology to evaluate statistical errors in DNS data of plane channel flows,” Comput. Fluids 130, 1–7 (2016)] for fully turbulent plane channel flows, where the mean velocity error is estimated by considering the Reynolds stress tensor, and using the balance of the mean force equation. It also presents how the residual error evolves in time for a DNS of a plane channel flow, and the influence of the Reynolds number on its convergence rate. The root mean square of the residual error is shown in order to capture a single quantitative value of the error associated with the dimensionless averaging time. The evolution in time of the error norm is compared with the final error provided by DNS data of similar Reynolds numbers available in the literature. A direct consequence of this approach is that it was possible to compare different numerical results and experimental data, providing an improved understanding of the convergence of the statistical quantities in turbulent wall-bounded flows.

Letter: Convergence rate of individual and global quantities in direct numerical simulations

“…Recently, Thompson et al 2 proposed and applied a methodology to evaluate the statistical error plaguing the averaged velocity and the Reynolds stress in plane channel flows. This methodology was used by Andrade et al 3 . Their Figure 6 suggested a residual norm decreasing as 1 T , with T the averaging time.…”

“…The following is tailored to the time scheme used by Andrade et al 3 . It could be simply adapted to fit another time scheme.…”

“…The definition of E i R , as given by equation 10, uses an integration constant to enforce E i R (y + = 0) = 0. In the present study, we weakly enforce this condition at y + = 0 by using an integration constant derived from the theoretical friction velocity, as in Andrade et al 3 . This choice, combined with the symmetry of the case at hand, strongly enforces E i R (y + = Re τ ) = 0.…”

Análise spectral da cascata de energia turbulenta e desenvolvimento de um novo modelo submalha não linear para simulação das grandes escalas.

“…Information about the numerical results is given by Table A.2 for Re τ ≈ 180, and by Table A. 3 for Re τ ≈ 544 − 685. AKC [1]; BOP [13]; AJ [36]; EUWWAFN [45]; FK [47]; ISK [65]; KSNFBJ [74]; LM [88]; MKM [117]; TGM [146]; VK-FD2 [162]; VK [163]; VK-S2B3 [161]; VK-S4B3 [161]; WBA [171]; WM [172].…”

“…Actually, it occurs in both directions, the so-called backscatter and forward scatter [124]. Conflicting situations with the Richardson cascade have already been observed, such as, for example, the net transfer of turbulent kinetic energy from the smallest to the largest scales, the so-called inverse energy cascade phenomenon, regularly evinced in laboratory and numerical experiments [3].…”