Identification of Strong, Near-Wall Quasi-Streamwise Vortices and Their Behavior

Abstract

It is widely accepted that the near-wall region of wall turbulence is dominated by an organized structure the most important ingredients of which are quasi-streamwise vortices. In order to identify these vortices, several indices have been proposed. Among them, Q, which is the second invariant of velocity gradient tensor $A = A_{ij} = \partial v_{i}/\partial x_{j}$ and which is defined by $Q = (1/2)\left \{ \left [ tr(A)^{2} \right ] -trA^{2} \right \} = -(1/2)(\Omega _{ij} \Omega _{ji} + S_{ij} S{ji})$ (Hunt et al., 1988) where $\Omega _{ij} = \left\{ {{{\partial \upsilon _i } \mathord{\left/ {\vphantom {{\partial \upsilon _i } {\partial x_j - \partial \upsilon _i }}} \right. \kern-\nulldelimiterspace} {\partial x_j - \partial \upsilon _i }}} \right\}/2,S_{ij} = \left( {{{\partial \upsilon _i } \mathord{\left/ {\vphantom {{\partial \upsilon _i } {\partial x_j + \partial \upsilon _i }}} \right. \kern-\nulldelimiterspace} {\partial x_j + \partial \upsilon _i }}} \right)/2{\rm{ and }}\lambda _2$ and λ₂(Jeong et al., 1995, Jeong et al., 1997), which is the second largest eigenvalue of a symmetric part of second order tensor of Hessian ${{\partial ^2 p} \mathord{\left/ {\vphantom {{\partial ^2 p} {\partial x_i }}} \right. \kern-\nulldelimiterspace} {\partial x_i }}\partial x_j$ where p is pressure, are most widely used. In the near-wall layer of y ⁺ =} 10 ~ 50, where $y^{+} (= yu_{\tau}/\nu)$ is the non-dimensional distance from the wall, Q and −λ₂ do not differ greatly in their ability to identify quasi-streamwise vortices, at least statistically. For example, the volume fractions of Q > 0 and of $\lambda _{2} < 0$ are both about 40% of the whole volume in this layer, though that of — λ₂ is slightly smaller than that of Q, particularly in the buffer layer. Furthermore the wall normal distribution of both their mean values $Q_{mean}$ and $\lambda _{2},_{mean}$, and the r.m.s. of their fluctuations $Q{rms}$ and - $\lambda _{2},_{rms}$ are also nearly identical.