Red giant branch(RGB) and asymptotic giant branch(AGB) stars are the luminous and red giants in late periods of stellar evolution from the main sequence stage of low to intermediate-mass stars. Their effective temperatures are rather low, however, their luminosities are rather high, which leads to the fact that their outer envelope is convective and the convective envelopes are enormous in mass or radius. Based on a new second-order turbulent convection model(TCM), we have studied the characteristics of turbulent convection in RGB and AGB stars, and made comparison with the results of the classical mixing length theory(MLT). Furthermore, we have utilized the TCM to research the effect of chemical mixing caused by the turbulent convective motion on stellar structure and evolution. At the same time, the TCM and some free parameters have been tested and confined according to the observational data on the Cepheid instability strip and the number ratio of blue to red evolved stars. It is found that, the convective motions become stronger and stronger when RGB and AGB stellar models are located up along the Hayashi line. The TCM usually gives larger convective heat fluxes and larger turbulent velocity than the MLT does, however, when the turbulent velocity is adjusted to close to that of the MLT by decreasing the value of the mixing length parameter in the TCM, the superadiabatic convection zone will be much more extended inward and a lower effective temperature of the stellar model will be obtained. Besides, the e-folding lengths of the turbulent velocity in both the top and bottom overshooting regions decrease as the stellar model is located up along the Hayashi line, but both the extents of the decrease are not obvious. Moreover, in the overshooting regions there is a nearly linear relationship between the turbulent kinetic energy and the pressure. And the slope of the relationship is mainly determined by the TCM, however, the constant term in it is mainly determined by the stellar model. After taking the effect of the chemical mixing(disposed as a diffusive process) in the bottom overshooting region of the outer convective envelpe into account, it is found that in the stage of the RGB the convective envelope of the stellar model will be enlarged inward, and the effective temperature and the luminosity of the stellar model at the RGB tip will be increased and decreased a little, respectively. When the stellar model evolves from the RGB tip down to its bottom again, the convective envelope will be suppressed to a rather small extent, which leads to a trend that the stellar model will evolve blueward to increase the length of blue loop. And the blue loop will become longer and longer as both the parameter CX in the diffusive mixing model and aTCM in the TCM increase, furthermore, the time interval of the stellar model staying around the blue end of the blue loop will also be increased as both the CX and aTCM increase. The obtained results are in agreement with the observations of the Cepheid instability strip and the number ratio of blue to red evolved stars. And both the two observations respectively confine the CX and aTCM in a same range: 0.5
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