DOI

1

Turbulent 2.5-dimensional dynamos - Seshasayanan, K. and Alexakis, A.

JOURNAL OF FLUID MECHANICS 799, 246-264 (2016)

Abstract : We study the linear stage of the dynamo instability of a turbulent two-dimensional flow with three components (u(x, y, t), v(x, y, t), w(x, y, t)) that is sometimes referred to as a 2.5-dimensional (2.5-D) flow. The flow evolves based on the two-dimensional Navier-Stokes equations in the presence of a large-scale drag force that leads to the steady state of a turbulent inverse cascade. These flows provide an approximation to very fast rotating flows often observed in nature. The low dimensionality of the system allows for the realization of a large number of numerical simulations and thus the investigation of a wide range of fluid Reynolds numbers Re, magnetic Reynolds numbers Rm and forcing length scales. This allows for the examination of dynamo properties at different limits that cannot be achieved with three-dimensional simulations. We examine dynamos for both large and small magnetic Prandtl-number turbulent flows Pm = Rm/Re, close to and away from the dynamo onset, as well as dynamos in the presence of scale separation. In particular, we determine the properties of the dynamo onset as a function of Re and the asymptotic behaviour in the large Rm limit. We are thus able to give a complete description of the dynamo properties of these turbulent 2.5-D flows.

JOURNAL OF FLUID MECHANICS 799, 246-264 (2016)

LPS

Abstract : We study the linear stage of the dynamo instability of a turbulent two-dimensional flow with three components (u(x, y, t), v(x, y, t), w(x, y, t)) that is sometimes referred to as a 2.5-dimensional (2.5-D) flow. The flow evolves based on the two-dimensional Navier-Stokes equations in the presence of a large-scale drag force that leads to the steady state of a turbulent inverse cascade. These flows provide an approximation to very fast rotating flows often observed in nature. The low dimensionality of the system allows for the realization of a large number of numerical simulations and thus the investigation of a wide range of fluid Reynolds numbers Re, magnetic Reynolds numbers Rm and forcing length scales. This allows for the examination of dynamo properties at different limits that cannot be achieved with three-dimensional simulations. We examine dynamos for both large and small magnetic Prandtl-number turbulent flows Pm = Rm/Re, close to and away from the dynamo onset, as well as dynamos in the presence of scale separation. In particular, we determine the properties of the dynamo onset as a function of Re and the asymptotic behaviour in the large Rm limit. We are thus able to give a complete description of the dynamo properties of these turbulent 2.5-D flows.

DOI

2

Rotating Taylor-Green flow - Alexakis, A.

JOURNAL OF FLUID MECHANICS 769, 46-78 (2015)

Abstract : The steady state of a forced Taylor-Green flow is investigated in a rotating frame of reference. The investigation involves the results of 184 numerical simulations for different Reynolds numbers ReF and Rossby numbers RoF. The large number of examined runs allows a systematic study that enables the mapping of the different behaviours observed to the parameter space (ReF; RoF), and the examination of different limiting procedures for approaching the large ReF small RoF limit. Four distinctly different states were identified: laminar, intermittent bursts, quasi-twodimensional condensates and weakly rotating turbulence. These four different states are separated by power-law boundaries R-OF proportional to Re-F(-gamma) in the small RoF limit. In this limit, the predictions of asymptotic expansions can be directly compared with the results of the direct numerical simulations. While the first-order expansion is in good agreement with the results of the linear stability theory, it fails to reproduce the dynamical behaviour of the quasi-two-dimensional part of the flow in the nonlinear regime, indicating that higher-order terms in the expansion need to be taken into account. The large number of simulations allows also to investigate the scaling that relates the amplitude of the fluctuations with the energy dissipation rate and the control parameters of the system for the different states of the flow. Different scaling was observed for different states of the flow, that are discussed in detail. The present results clearly demonstrate that the limits of small Rossby and large Reynolds numbers do not commute and it is important to specify the order in which they are taken.

JOURNAL OF FLUID MECHANICS 769, 46-78 (2015)

LPS

Abstract : The steady state of a forced Taylor-Green flow is investigated in a rotating frame of reference. The investigation involves the results of 184 numerical simulations for different Reynolds numbers ReF and Rossby numbers RoF. The large number of examined runs allows a systematic study that enables the mapping of the different behaviours observed to the parameter space (ReF; RoF), and the examination of different limiting procedures for approaching the large ReF small RoF limit. Four distinctly different states were identified: laminar, intermittent bursts, quasi-twodimensional condensates and weakly rotating turbulence. These four different states are separated by power-law boundaries R-OF proportional to Re-F(-gamma) in the small RoF limit. In this limit, the predictions of asymptotic expansions can be directly compared with the results of the direct numerical simulations. While the first-order expansion is in good agreement with the results of the linear stability theory, it fails to reproduce the dynamical behaviour of the quasi-two-dimensional part of the flow in the nonlinear regime, indicating that higher-order terms in the expansion need to be taken into account. The large number of simulations allows also to investigate the scaling that relates the amplitude of the fluctuations with the energy dissipation rate and the control parameters of the system for the different states of the flow. Different scaling was observed for different states of the flow, that are discussed in detail. The present results clearly demonstrate that the limits of small Rossby and large Reynolds numbers do not commute and it is important to specify the order in which they are taken.

DOI

3

Bounding the scalar dissipation scale for mixing flows in the presence of sources - Alexakis, A. and Tzella, A.

JOURNAL OF FLUID MECHANICS 688, 443-460 (2011)

Abstract : We investigate the mixing properties of scalars stirred by spatially smooth, divergence-free flows and maintained by a steady source sink distribution. We focus on the spatial variation of the scalar field, described by the dissipation wavenumber, k(d), that we define as a function of the mean variance of the scalar and its gradient. We derive a set of upper bounds that for large Peclet number (Pe >> 1) yield four distinct regimes for the scaling behaviour of kd, one of which corresponds to the Batchelor regime. The transition between these regimes is controlled by the value of Pe and the ratio rho = l(u)/l(s), where l(e) and l(s) are, respectively, the characteristic length scales of the velocity and source fields. A fifth regime is revealed by homogenization theory. These regimes reflect the balance between different processes: scalar injection, molecular diffusion, stirring and bulk transport from the sources to the sinks. We verify the relevance of these bounds by numerical simulations for a two-dimensional, chaotically mixing example flow and discuss their relation to previous bounds. Finally, we note some implications for three-dimensional turbulent flows.

JOURNAL OF FLUID MECHANICS 688, 443-460 (2011)

LPS

Abstract : We investigate the mixing properties of scalars stirred by spatially smooth, divergence-free flows and maintained by a steady source sink distribution. We focus on the spatial variation of the scalar field, described by the dissipation wavenumber, k(d), that we define as a function of the mean variance of the scalar and its gradient. We derive a set of upper bounds that for large Peclet number (Pe >> 1) yield four distinct regimes for the scaling behaviour of kd, one of which corresponds to the Batchelor regime. The transition between these regimes is controlled by the value of Pe and the ratio rho = l(u)/l(s), where l(e) and l(s) are, respectively, the characteristic length scales of the velocity and source fields. A fifth regime is revealed by homogenization theory. These regimes reflect the balance between different processes: scalar injection, molecular diffusion, stirring and bulk transport from the sources to the sinks. We verify the relevance of these bounds by numerical simulations for a two-dimensional, chaotically mixing example flow and discuss their relation to previous bounds. Finally, we note some implications for three-dimensional turbulent flows.