Reynolds stress ($R_{ij}=\rho \overline{u_i^{\prime }{u}_j^{\prime }}$)

Velocity-fluctuation covariances transport mean momentum. For positive wall shear, $-\overline{u^{\prime }{w}^{\prime }}$ is the downward momentum-transport magnitude and contributes to the total shear stress. Some stress conventions include the minus sign in the definition of the tensor itself; the momentum equation must use the same convention.