• Research paper •

### Experimental research on complex eddy viscosity modeling of multi-scale coherent structures in wall turbulence

Aidong Guo Nan Jiang

1. Department of Mechanics, Tianjin University, Tianjin 300072, China Department of Mechanics, Tianjin University, Tianjin 300072, China
• Received:2008-09-17 Revised:2009-01-19 Online:2010-03-18 Published:2010-03-18

Abstract: In the dynamics equation of coherent structures with periodical phasic average, Reynolds stress term \tilde {r}_{ij} = - (\langle {u}'_i {u}'_j \rangle - \overline {{u}'_i {u}'_j } ) that noncoherent structures contributes to coherent structures, is taken into the model, in which the velocity strain rate of coherent structures is multiplied by a complex eddy viscosity coefficient, as follows: \tilde {r}_{ij} = \nu _T \Big[ \dfrac{\partial \tilde {u}_i }{\partial x_j } + \dfrac{\partial \tilde {u}_j }{\partial x_i } \Big] + \dfrac{1}{3}\delta _{ij} ( \tilde {r}_{kk} ) , where \nu _T is a complex eddy viscosity coefficient. Eddy viscosity coefficient is defined to be a complex, supposing that there exist phase difference between velocity strain rate of coherent structures and the Reynolds stress. The phase relation has been experimentally measured between velocity distortion \dfrac{\partial \tilde {u}}{\partial y} of multi-scales coherent structures in turbulent boundary layer over smooth plate and the Reynolds stress, \tilde {r}_{12} and analyzed in a close-circuit low-speed wind tunnel based on the expression \tilde {r}_{12} = \nu _T \Big[ \dfrac{\partial \tilde {u}}{\partial y} + \dfrac{\partial \tilde {v}}{\partial x} \Big]. Through the phase difference analysis between the Reynolds stress and the velocity distortion of coherent structures in wall turbulence during the bursting process of coherent structures, conclusions can be drawn as following: Relaxation time in dynamic interaction between the turbulence of coherent structures with macroscopical scale and movement distortion of coherent structures can't be ignored. Hence, there exist phase difference between Reynolds stress component that noncoherent structures contribute to coherent structures and the velocity distortion of coherent structures. It is an important factor that needs to be taken into account for eddy viscosity modeling theory for non-steady turbulence macrostructure. The above phase difference depends on not only the scale of coherent structures, but also the detail of physical burst process of coherent structures, such as eject and sweep. The phase of the Reynolds stress is posterior than that of velocity distortion of coherent structures in ejecting process, while anterior in sweeping process. In addition, phase difference between them does not vary with boundary normal positions in logarithmic sublayer.