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

In the dynamics equation of coherent structures withperiodical phasic average, Reynolds stress term \tilde r_ij = -(\langle u'_i u'_j \rangle - \overline u'_i u'_j ) thatnoncoherent structures contributes to coherent structures, is taken into themodel, in which the velocity strain rate of coherent structures ismultiplied 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 +\dfrac13\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 thereexist phase difference between velocity strain rate of coherent structuresand the Reynolds stress.The phase relation has been experimentally measured between velocitydistortion \dfrac\partial \tilde u\partial y of multi-scales coherentstructures in turbulent boundary layer over smooth plate and the Reynoldsstress, \tilde r_12 and analyzed in a close-circuit low-speed windtunnel based on the expression \tilde r_12 = \nu _T \Big\dfrac\partial \tilde u\partial y + \dfrac\partial \tildev\partial x \Big. Through the phase difference analysis betweenthe Reynolds stress and the velocity distortion of coherent structures inwall turbulence during the bursting process of coherent structures,conclusions can be drawn as following:Relaxation time in dynamic interaction between the turbulence of coherentstructures with macroscopical scale and movement distortion of coherentstructures can't be ignored. Hence, there exist phase difference betweenReynolds stress component that noncoherent structures contribute to coherentstructures and the velocity distortion of coherent structures. It is animportant factor that needs to be taken into account for eddy viscositymodeling theory for non-steady turbulence macrostructure.The above phase difference depends on not only the scale of coherentstructures, but also the detail of physical burst process of coherentstructures, such as eject and sweep. The phase of the Reynolds stress isposterior than that of velocity distortion of coherent structures inejecting process, while anterior in sweeping process. In addition, phasedifference between them does not vary with boundary normal positions inlogarithmic sublayer.