  Numerical simulations of wall-turbulence using the restricted nonlinear (RNL)
model generate realistic mean velocity profiles in plane Couette and channel
flow at low Reynolds numbers. The results are less accurate at higher Re, and
while a logarithmic region is observed, its von-K\'arm\'an constant is not
consistent with the standard logarithmic law. In half-channel flow we show that
limiting the streamwise-varying wavenumber support of RNL turbulence to one or
few empirically determined modes improves its predictions considerably. In
particular, the mean velocity profiles obtained with the band-limited RNL model
follow standard logarithmic behavior for the higher Reynolds numbers in this
study.
