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Explanation of constant mean angular momentum in high-Reynolds-number Taylor--Couette turbulence in terms of history effects

Kazuhiro Inagaki, Yasufumi Horimoto

Abstract

This study discusses the mechanism of the emergence of nearly constant mean angular momentum profiles, which are widely observed in curved turbulent flows including the bulk region of Taylor--Couette (TC) flows. For high-Reynolds-number TC flows where the inner and outer cylinders are weakly counter-rotating and co-rotating, both the bulk and boundary layers become turbulent without Taylor rolls, referred to as the featureless ultimate regime (UR). Thus, we utilize the Reynolds-averaged Navier--Stokes (RANS) equations to explain the mechanism of the nearly constant mean angular momentum. High-Reynolds-number experiments of TC turbulence are performed for reference, where the angular velocity ratio $a = -ω_\mathrm{out}/ω_\mathrm{in}$ is in the range $-0.5 \le a \le 0.1$. Verification of the RANS based on the conventional algebraic Reynolds stress model suggests that convection of the Reynolds stress is essential for predicting the angular momentum profile. This indicates that the physical origin of the nearly constant angular momentum is the history effect of the Reynolds stress. To rigorously incorporate the convection effect into the Reynolds stress, we employ the Jaumann derivative as a covariant time derivative. The model that takes into account the history effect involving the normal stress difference successfully predicts the nearly constant mean angular momentum in the co-rotating cases. This study suggests the significance of the history effects for understanding curved or rotating turbulent flows in terms of the statistical analysis.

Explanation of constant mean angular momentum in high-Reynolds-number Taylor--Couette turbulence in terms of history effects

Abstract

This study discusses the mechanism of the emergence of nearly constant mean angular momentum profiles, which are widely observed in curved turbulent flows including the bulk region of Taylor--Couette (TC) flows. For high-Reynolds-number TC flows where the inner and outer cylinders are weakly counter-rotating and co-rotating, both the bulk and boundary layers become turbulent without Taylor rolls, referred to as the featureless ultimate regime (UR). Thus, we utilize the Reynolds-averaged Navier--Stokes (RANS) equations to explain the mechanism of the nearly constant mean angular momentum. High-Reynolds-number experiments of TC turbulence are performed for reference, where the angular velocity ratio is in the range . Verification of the RANS based on the conventional algebraic Reynolds stress model suggests that convection of the Reynolds stress is essential for predicting the angular momentum profile. This indicates that the physical origin of the nearly constant angular momentum is the history effect of the Reynolds stress. To rigorously incorporate the convection effect into the Reynolds stress, we employ the Jaumann derivative as a covariant time derivative. The model that takes into account the history effect involving the normal stress difference successfully predicts the nearly constant mean angular momentum in the co-rotating cases. This study suggests the significance of the history effects for understanding curved or rotating turbulent flows in terms of the statistical analysis.
Paper Structure (30 sections, 96 equations, 9 figures, 1 table)

This paper contains 30 sections, 96 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Basic equations and covariance
  3. Governing equations
  4. Transformation rules
  5. Velocity field and Reynolds stress
  6. Velocity gradient
  7. Time derivative of tensors
  8. Constant mean angular momentum and zero mean absolute vorticity
  9. Basic equations for TC flow in featureless UR
  10. Reynolds stress transport equations for TC turbulence
  11. Implicit algebraic model incorporating convection effects
  12. Verification of RANS model compared with experiments
  13. Experimental setup
  14. Numerical setup
  15. Results
  16. ...and 15 more sections

Figures (9)

  • Figure 1: Mean velocity profiles of the experiments at $\hbox{$\mathrm{Re}$}_\mathrm{in}=8.5\times 10^4$ for various angular velocity ratios $a$.
  • Figure 2: Mean angular momentum profiles of the experiments at $\hbox{$\mathrm{Re}$}_\mathrm{in}=8.5\times 10^4$ for various $a$, normalized by (a) the angular momentum of the inner cylinder and (b) the angular momentum difference between the inner and outer cylinders.
  • Figure 3: Mean angular momentum profiles of AKN and ccARSM compared with those of experiments at $\hbox{$\mathrm{Re}$}_\mathrm{in}=8.5\times 10^4$ for (a) $a=0$ and (b) $a=-0.33$.
  • Figure 4: Mean angular momentum profiles for several sets of model parameters of ccARSM at $\hbox{$\mathrm{Re}$}_\mathrm{in}=8.5\times 10^4$ for $a=-0.33$.
  • Figure 5: Profiles of dissipation rate normalized by (a) Eq. (\ref{['eq:exactproductionscaling']}) and (b) Eq. (\ref{['eq:standardkepsilonscaling']}) with $y = r-r_\mathrm{in}$ for the AKN model.
  • ...and 4 more figures