Ye Zhou Lawrence Livermore National Laboratory, Livermore, California

1. Ye Zhou Lawrence Livermore National Laboratory, Livermore, California UCRL-PRES-205961 A perspective on turbulent flows: Cascade dynamics in rotating flows

2. Turbulent flows subject to strong rotation occur in many important applications • Rotating turbomachinery; geophysical problems involving the rotation of the Earth • Example: rotation suppresses the growth of turbulent mixing layers induced by Rayleigh-Taylor instability RT mixing layer without rotation RT mixing layer with strong rotation Carnevale, Orlandi, Zhou, and Kloosterziel,JFM

3. Rotation has subtle, yet profound effects on the fundamental properties of the energy cascade • But rotation does not even enter the kinetic energy equation • The physics of rotating turbulent flows must be better understood: • energy transfer process • dissipation • anisotropy • This improved understanding should be incorporated into models: • subgrid models for large-eddy simulations (LES) • Reynolds averaged Navier-Stokes (RANS) models

4. The first step in our approach is to consider how strong rotation modifies the energy spectrum • A strong similarity between magnetohydrodynamic turbulence and isotropic turbulence subject to solid body rotation has been noted (Zhou, Phys. Fluids) • For MHD turbulence, Kraichnan (Phys. Fluids) argued that the propagation of Alfvénic fluctuations disrupts phase relations, thereby decreasing energy transfer • Uniform rotation causes plane waves to propagate with phase speed 2Ωkz/k • This argument is also analogous to that given by Herring (Meteorol. Atmos. Phys., vol. 38, 106, 1988) for stratified flows

5. A rotation modified energy spectrum E(k) can be obtained from a phenomenological argument • Kolmogorov’s theory cannot be directly applied to complex problems with imposed time scales • The time scale for decay of triple correlations, 3, is crucial to turbulent spectral transfer • We assume that 3 results from local interactions determined by the rotation time scale, Ω=1/Ω, not by the nonlinear time scale nl • Because energy is conserved by nonlinear interactions and a local cascade has been assumed, dimensional analysis leads to where k is the wavenumber and εisthe energy dissipation rate

6. The rotation modified energy spectrum is supported by both experiments and simulations A direct substitution of 3 =1/Ω results in the energy spectrum for turbulence subject to strong rotation (Zhou, Phys. Fluids): DNS with forcing at small k and strong rotation Experiment with Ro=0.06 and Reλ=360 Yeung and Zhou, Phys. Fluids Baroud et al., Phys. Rev. Lett., 88, 114501 (2002)

7. The phenomenology can be made more precise by appealing to DIA and other closure theories • For a constant energy flux steady state, the DIA inertial range energy balance (Kraichnan, 1971) is: where the integration operators are defined by and  = t-s denotes time difference. The time integrals will take the form With the ansatz (Rubinstein and Zhou) The leading order solution of the DIA energy balance equation reduces to

8. A generalized time scale leads to a spectrum intermediate between the “-5/3” and “-2” spectra Schematic of generalized spectral law (Zhou, Phys. Fluids DNS with forcingatlarge k (Smith and Waleffe, Phys. Fluids rotation No rotation

9. Scaling analysis shows that a -2 spectrum leads to an eddy viscosity with 1/Ω dependence • The eddy viscosity ~ 1/Ω • Speziale pointed out in turbulent flows with a mean velocity gradient in a rotating frame, the effect of rotation appears in conjunction with the mean vorticity (Gatski and Speziale, 1993) • The rotation rate, Ω, is replaced by where (Zhou; Mahalov and Zhou)

10. Generalized eddy viscosity for both strong and weak rotation can be constructed in Padé form • Closure provides perturbation expansions of both the time scale and energy spectrum: • Strong rotation limit: 2. Weak rotation limit (Shimomura and Yoshizawa): 3. Generalized eddy viscosity form which reduces to the correct limits for both strong and weak rotation Note: K – kinetic energy (Rubinstein and Zhou)

11. Strong rotation leads to reduced energy transfer with external energy input to large-scales • The skewness factor (Batchelor) DNS with forcing at small k and strong rotation Kinetic energy Ro=0.0195 dissipation Ro=0.0039 Yeung and Zhou, Phys. Fluids

12. The response of isotropic turbulence under strong rotation is revealed through spectral dynamics The evolution equation for the energy spectrum tensor is Ro=∞ Ro=0.0195 Yeung and Zhou, Phys. Fluids

13. Anisotropy in the energy transfer is clearly seen Inverse cascade Transfer function component perpendicular to Ω Transfer function component parallel to Ω Yeung and Zhou, Phys. Fluids

14. RANS models must satisfy two major consistency conditions in rotating flows • In rapidly rotating flows, based on the previous discussion: • Cascade is disturbed by suppressing phase coherence • Two-dimensionalization and its often associated laminarization Therefore, two conditions must be met at Ω→ 1. The dissipation rate ε→0 • The eddy viscosity  K2/ε→0 (Speziale, Younis, Rubinstein, and Zhou, Phys. Fluids)

15. Analytical theory of the destruction term in the dissipation rate equations can be derived • Many RANS models do not satisfy these two conditions. For example, Launder, Reece and Rodi model: ε is not affected by rotation, so that the first condition is not met Launder, Priddin, and Sharma model: the eddy viscosity is decreased by having ε increased • Once the inertial range theory is given, rotation effects arise naturally in the dissipation rate equation • The -2 energy spectrum • The energy flux function as given by closure

16. The procedure of Schiestel offers an alternative dissipation rate equation for rotating flows • The large-scale energy and the inertial range energy are linked through a self-similarity assumption • Energy becomes trappedin the largest scales of motion, where it undergoes purely viscous decay • The large-scale and inertial range energies evolve independently • Thus, the kinetic energy and transfer in the inertial range vanish, but the kinetic energy and the integral scale both approach constants in the absence of viscosity Rubinstein and Zhou, Comput. Math. Appl., 46, 633, 2003

17. In the long time limit, K is a constant and ε=0, indicating that the energy transfer to small scales has stopped • The solution for decaying turbulence is • In the long time limit, K is a constant and ε=0, indicating that the energy transfer to small scales has stopped • This rotation dependence is in agreement with the Bardina model (JFM, 1985)

18. Two distinctive dissipation range spectra may exist for rotating flow • Matching the appropriate Kolmogorov and wave frequency • The ordering parameter is • This parameter is related to Ro (micro) • Normal ordering, If Od,Ω>> 1: Kolmogorov scaling is recovered at small k • Inverted ordering, If Od,Ω << 1: Rotation effects are important in the dissipation range

19. DIA provides a starting point for studying the dissipation range energy spectrum The DIA energy balance equation in the dissipation range is • Under the inverted ordering • The dissipation range dynamics is dominated by interactions q<< p ≈k (Kraichnan, JFM, 1959) • The resonance condition gives • For q near zero Note that is << Rubinstein and Zhou, 1999

20. Energy moves toward the horizontal plane inwavenumber space, where dissipation is possible • The -2 spectrum is established at moderate Ro or large Ro Re1/2when Kolmogorov scaling is recovered at small inertial range, and an isotropic dissipation range exists • At asymptotically low Ro or Ro Re1/2small, the argument leading to a -2 scaling breaks down. These conditions can exist in decaying turbulence • The dissipation range dynamics depends on Ω; it isnot exactly two-dimensional.

21. Summary and conclusion • We have investigated the effects of strong rotation on cascade dynamics energy transfer process • Rotation-modified energy transfer provides the starting point for our model development • An eddy viscosity that depends on rotation rate is derived • Two constraints on RANS models under strong rotation are obtained: • Dissipation rate must vanish • Eddy viscosity must vanish • Applications to engineering flows, based on the energy spectrum have shown promise (Thangam, Wang, and Zhou, Phys. Fluids) • We are interested in the stratified turbulence and stratified rotating turbulent flows