Multiple states in turbulence

Abstract

Turbulence is ubiquitous in nature, and is known as one of the unresolved problems in classical physics. The classical theory of turbulence assumed that turbulence is ergodic, which means that when turbulence is in a stationary state, even though the instantaneous properties are sensitive to initial conditions, the statistical averages of the instantaneous properties, such as the mean profiles and the skin friction in wall-bounded turbulence, are unique at any fixed set of parameters. This classical ergodic theory is the foundation of turbulence theory and modeling, and makes it possible to extrapolate data from lower Reynolds number turbulence to higher ones. However, in recent years, a few experimental and numerical evidences showed that multiple states exist in several flow problems. That is, the turbulent statistics and flow structures are not the same even at the same control parameters. In this paper, several published flow problems with multiple states are reviewed, including Rayleigh-Benard convection (RBC), von Karman flow (VKF), Taylor-Couette flow (TCF), spherical Couette flow (SCF), rotating homogeneous turbulence with Taylor-Green forcing (TGF), and the spanwise rotating plane Couette flows (RPCF). Two different categories of multiple states can be observed from these flow problems. One is in RBC, where several research groups found that the system will have different flow states and it will switch between different states as time evolves. The other can be generally seen from other five flows, where initial conditions or hysteresis effect can be observed. For example, hysteresis loops were reported in the experiments of TCF for global torque and local velocities at very high Reynolds numbers, while in RPCF, different flow statistics and flow structures were obtained with different initial flow fields based on the same code, the same computational domain and the same grid resolution at the same control parameters. Whether the second type of multiple states will finally turn into the first one in a long enough time is an open question which deserves further investigations. The underlying mechanism of multiple states is not clear at the present moment, which also demands continued effort and studies. In the six flow problems mentioned above, large-scale flow structures persist, which may highlight the importance of the coherent structures and their selectability in turbulence, as concluded by Huisman et al. Another possible guess from myself is the multiple competing flow mechanisms in the flow problems with multiple states. For example, in rotating homogeneous turbulence with Taylor-Green forcing (TGF), there are two competing mechanisms, one is the viscosity dissipation which causes the forward cascade, the other is the system rotation which results in the inverse cascade. The two competing mechanisms will make the system has more equilibrium points and multiple states. I believe that there will be more and more flow problems with multiple states reported in the future, together with a better understanding of the underling mechanism.