Different scaling behaviors, such as Kolmogorov (K41) scaling and Bolgiano and Obukhov (BO) scaling, have been reported in various shell models proposed for turbulent thermal convection. However, two coexistent subranges with K41 and BO scaling are not set up with Bolgiano scale interlaying between the largest scale and the dissipation scale. In this paper, we summarize fixed-point solution study of the Brandenburg model with small perturbation theory by introducing a small disturbance term as the impact of buoyancy. Three groups of fixed-point solutions with different locations of the so-called buoyancy scale, above/below which buoyancy is significant/insignifant. Both theoretical and numerical results show that a modified K41 scaling, instead of K41 and BO coexistent scaling, is set up even though buoyancy may be significant over the scaling range, which suggests that the buoyancy scale is not related exactly to the Bolgiano scale. Thus, a K41 and BO coexistent scaling behavior is not setup for the Brandenburg model.
It is necessary to build turbulence model to study the response of aircraft to atmospheric turbulence for high resolution earth observation. The conventional method is on the basis of Dryden’s model with the assumption that individual patches are Gaussian. In this paper,based on Kraichnan’s refined similarity idea,a new 1D atmospheric turbulence model is set up by introducing the energy transfer rate as an intermittency disturbance to a Gaussian process. Our results show that the turbulent fields generated by our new method exhibit an anomalous scaling described by the She-Leveque (SL) formula,which is now well accepted for homogenous and isotropic turbulence.
The attractive fixed-point solution of a nonlinear cascade model is studied for the homogeneous isotropic turbulence containing a parameter C, introduced by Desnyansky and Novikov. With a traditional constant positive external force added on the first shell equation, it can be found that the attractive fixed-point solution of the model depends on both the parameter C and the external force. Thus, an explicit force is introduced to remove the effects of the external force on the attractive fixed-point solution. Furthermore, two groups of attractive fixed-point solutions are derived theoretically and studied numerically. One of the groups has the same scaling behavior of the velocity in the whole inertial range and agrees well with those observed by Bell and Nelkin for the nonnegative parameters. The other is found to have different scaling behaviors of the velocity at the odd and even number shells for the negative parameters. This special characteristic may be used to study the anomalous scaling behavior of the turbulence.
