Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Complete Matched Asymptotic Expansions for Velocity Statistics in Turbulent Channels

Peter A. Monkewitz

Abstract

Complete high fidelity matched asymptotic expansions (abbreviated MAE) are developed for the first and second order turbulence statistics in channel flow from 11 direct numerical simulations (DNS). To put the crucial identification of overlaps on a solid footing, a simple a priori test is devised, which only requires a DNS or experimental profile and the presumed overlap of the MAE for the quantity in question. This test fully supports the form c0 - c1 Y^1/4 of the overlaps for the stream-wise and cross-stream normal stresses <uu> and <ww>, which has been advocated by Chen and Sreenivasan (2022, 2023, 2025) and Monkewitz (2022, 2023). The first MAE analysis of the wall-normal stress <vv> then reveals an overlap of the form c0 - c1 Y^5/4 , which is extensively documented. Finally, the logarithmic indicator function Xi = y dU/dy for the mean velocity overlap is reanalyzed, with focus on its spatial oscillations. The latter are compared in the concluding section to the spatial oscillations of <uu>, together with further observations.

Complete Matched Asymptotic Expansions for Velocity Statistics in Turbulent Channels

Abstract

Complete high fidelity matched asymptotic expansions (abbreviated MAE) are developed for the first and second order turbulence statistics in channel flow from 11 direct numerical simulations (DNS). To put the crucial identification of overlaps on a solid footing, a simple a priori test is devised, which only requires a DNS or experimental profile and the presumed overlap of the MAE for the quantity in question. This test fully supports the form c0 - c1 Y^1/4 of the overlaps for the stream-wise and cross-stream normal stresses <uu> and <ww>, which has been advocated by Chen and Sreenivasan (2022, 2023, 2025) and Monkewitz (2022, 2023). The first MAE analysis of the wall-normal stress <vv> then reveals an overlap of the form c0 - c1 Y^5/4 , which is extensively documented. Finally, the logarithmic indicator function Xi = y dU/dy for the mean velocity overlap is reanalyzed, with focus on its spatial oscillations. The latter are compared in the concluding section to the spatial oscillations of <uu>, together with further observations.
Paper Structure (12 sections, 34 equations, 20 figures, 1 table)

This paper contains 12 sections, 34 equations, 20 figures, 1 table.

Table of Contents

  1. Introduction
  2. A brief review of MAE and a simple test to identify overlaps in DNS profiles
  3. A simple test to identify overlaps
  4. The stream-wise normal stress $\langle uu\rangle$
  5. The cross-stream normal stress $\langle ww\rangle$
  6. The wall-normal stress $\langle vv\rangle$
  7. The log indicator function for the mean velocity
  8. Conclusions and some speculations
  9. Comments on normal stresses
  10. Speculation on a possible connection between the oscillations of $\langle uu\rangle$ and of the MVP indicator function
  11. Speculation on the flow dependence of the outer MVP
  12. Appendix: Brief comparison to the $\langle uu\rangle$ overlaps in pipes

Figures (20)

  • Figure 1: Streamwise normal stress $\langle uu\rangle$ versus $Y$. (a) Figure 1 of MMHS13 replotted versus $Y$; --- (orange: Melbourne windtunnel; green: LCC; violet: Superpipe; SLTEST omitted). $- \cdot - \cdot -$ (black), logarithmic fit (\ref{['uuAEasymp']}); $\bullet\bullet\bullet$ (red), "bounded dissipation" overlap (\ref{['uuOLBD']}). (b) Analogue for the channel data of table \ref{['TableDNS']} with addition of full outer $\langle uu\rangle_{out}$ (equ. \ref{['uuout']} of section \ref{['sec3']}) (wake indicated by yellow $\bullet\bullet\bullet$ beyond $Y \approxeq 0.75$).
  • Figure 2: Difference between the $\langle uu\rangle_{DNS}$ profiles #3, 5, 7 and 8 of table \ref{['TableDNS']} and the "CS" overlap (\ref{['uuOLBD']}) (color $- - -$), as well as the logarithmic law (\ref{['uuAEasymp']}) (grey $- \cdot - \cdot -$). Vertical red line: start of the "CS" overlap at $y \approxeq 800$; vertical colored lines: approximate end of the "CS" overlap at $Y\approxeq 0.75$. Horizontal arrows: extent of overlap for the four $\mathrm{Re}_{\tau}$.
  • Figure 3: Indicator functions $\Xi_{\mathrm{CS}}= 4\,Y^{3/4} \mathrm{d} \langle uu\rangle/\mathrm{d} Y$ in panel (a) and $\Xi_{\mathrm{AE}}= Y \mathrm{d} \langle uu\rangle/\mathrm{d} Y$ in panel (b) for the experimental channel data of SchultzFlack2013, with $\mathrm{Re}_{\tau} = 1010, 1960, 4040, 5900$ (increasingly dark blue $-\bullet-$). Grey lines: reproduction, for comparison, of the four channel indicator functions for profiles #2, 3 7 and 10 of table \ref{['TableDNS']}.
  • Figure 4: Order $\mathcal{O}(\mathrm{Re}_{\tau}^{-1/4})$ of stream-wise normal stress $\langle uu\rangle$ extracted from DNS in fig. 1b of monkewitz22Monkarxiv23. $\cdot \cdot \cdot$ (black), 1st term $-0.405\,y^2$ of Taylor series; --- (grey), outer overlap $-10\,y^{1/4}$; $\bullet\bullet\bullet$ (red), new complete fit (\ref{['uu025']}) of the $\mathcal{O}(\mathrm{Re}_{\tau}^{-1/4})$ contribution to $\langle uu\rangle$.
  • Figure 5: Order $\mathcal{O}(1)$ of stream-wise normal stress $\langle uu\rangle_{in, 0}$ (\ref{['uu01']}-\ref{['uu03']}) plus wake (\ref{['Wuu']}) in panel (a) and without wake in (b). $\cdot \cdot \cdot$ (black), 1st term $(1/4)\,y^2$ of Taylor series; --- (grey), small-$Y$ limit $10.6$ of overlap; $- \cdot - \cdot -$ (green), exponential (\ref{['uu01']}); $- \cdot \cdot - \cdot \cdot -$ (green), exponential (\ref{['uu02']}); $\bullet \bullet \bullet$ (light yellow), complete inner fit (\ref{['uu01']}) - (\ref{['uu03']}) of the $\mathcal{O}(1)$ contribution to $\langle uu\rangle$.
  • ...and 15 more figures