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### Abstract

The relevance of Orr's inviscid mechanism to the transient amplification of disturbances in shear flows is explored in the context of bursting in the logarithmic layer of wall-bounded turbulence. The linearized problem for the wall normal velocity is first solved in the limit of small viscosity for a uniform shear and for a channel with turbulent-like profile, and compared with the quasiperiodic bursting of fully turbulent simulations in boxes designed to be minimal for the logarithmic layer. Many properties, such as time and length scales, energy fluxes between components, and inclination angles, agree well between the two systems. However, once advection by the mean flow is subtracted, the directly computed linear component of the turbulent acceleration is found to be a small part of the total. The temporal correlations of the different quantities in turbulent bursts imply that the classical model, in which the wall-normal velocities are generated by the breakdown of the streamwise-velocity streaks, is a better explanation than the purely autonomous growth of linearized bursts. It is argued that the best way to reconcile both lines of evidence is that the disturbances produced by the streak breakdown are amplified by an Orr-like transient process drawing energy directly from the mean shear, rather than from the velocity gradients of the nonlinear streak. This, for example, obviates the problem of why the cross-stream velocities do not decay once the streak has broken down.

This work was supported by the European Research Council Grant No. ERC-2010.AdG-20100224. The careful critique by Dong Siwei of a previous version of this paper is gratefully acknowledged.

I. INTRODUCTION II. THE LINEARIZED PROBLEM A. The equations of motion B. Inviscid homogeneous shear flow C. The linear Orr mechanism in channels D. The wall parallel velocities III. BURSTING IN MINIMAL BOXES IV. THE EFFECT OF NONLINEARITY A. How linear is wall-bounded turbulence? B. A nonlinear Orr mechanism V. DISCUSSION AND CONCLUSIONS

### Key Topics

- Turbulent flows
- 36.0
- Viscosity
- 28.0
- Rotating flows
- 21.0
- Shear flows
- 15.0
- Vortex dynamics
- 14.0

_{τ}= 2003,” Phys. Fluids 18, 011702 (2006).

_{0})|, and equating the energies of the two configurations in which the rows are either aligned along the centerline, z

_{0}= π/4, or maximally separated at z

_{0}= iH.

_{τ}≈ 550,” Phys. Fluids 24, 105101 (2012).

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### Abstract

The relevance of Orr's inviscid mechanism to the transient amplification of disturbances in shear flows is explored in the context of bursting in the logarithmic layer of wall-bounded turbulence. The linearized problem for the wall normal velocity is first solved in the limit of small viscosity for a uniform shear and for a channel with turbulent-like profile, and compared with the quasiperiodic bursting of fully turbulent simulations in boxes designed to be minimal for the logarithmic layer. Many properties, such as time and length scales, energy fluxes between components, and inclination angles, agree well between the two systems. However, once advection by the mean flow is subtracted, the directly computed linear component of the turbulent acceleration is found to be a small part of the total. The temporal correlations of the different quantities in turbulent bursts imply that the classical model, in which the wall-normal velocities are generated by the breakdown of the streamwise-velocity streaks, is a better explanation than the purely autonomous growth of linearized bursts. It is argued that the best way to reconcile both lines of evidence is that the disturbances produced by the streak breakdown are amplified by an Orr-like transient process drawing energy directly from the mean shear, rather than from the velocity gradients of the nonlinear streak. This, for example, obviates the problem of why the cross-stream velocities do not decay once the streak has broken down.

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