This paper extends our recent theoretical work concerning the feasibility of stable and accurate computation of turbulence using a large eddy simulation [Phys. Rev. E 68, 036705 (2003)]]. In our previous paper, it was shown, based on a simple assumption regarding the instantaneous streamwise velocity, that the application of the Gaussian filter to the incompressible Navier-Stokes equations can result in the appearance of a numerically unstable term that can be decomposed into positive and negative viscosities. That result raises the question as to whether an accurate solution can be achieved by a numerically stable subgrid-scale model. In the present paper, based on assumptions regarding the statistically averaged velocity, we present similar theoretical investigations to show that in several situations, the shears appearing in the statistically averaged velocity field numerically destabilize the fluctuation components because of the derivation of a numerically unstable term that represents negative diffusion in a fixed direction. This finding can explain the problematic numerical instability that has been encountered in large eddy simulations of wall-bounded flows. The present result suggests that this numerical problem is universal in large eddy simulations, and that if there is no failure in modeling, the resulting subgrid-scale model can still have unstable characteristics; that is, the known instability problems of several existing subgrid-scale models are not something that one may remove simply by an artificial technique, but must be taken seriously so as to treat them accurately.