  A central obstacle to understanding the route to turbulence in wall-bounded
flows is that the flows are composed of complex, highly fluctuating, and
strongly nonlinear states. We address this challenge by deriving from the
Navier-Stokes equations a simplified model that describes transitional
turbulence in a planar setting. The Reynolds-averaged and
turbulent-kinetic-energy equations are projected onto a minimal set of
wall-normal modes and justified model closures are used for the Reynolds
stresses and turbulent dissipation and transport. The model reproduces the
oblique turbulent-laminar patterns ubiquitous in wall-bounded shear flows. It
also captures the pattern wavelengths and angles, and the large-scale flow
associated with both stationary patterns and growing turbulent spots. Patterns
are shown to arise with decreasing Reynolds number via a linear instability of
uniform turbulence. Linear analysis reveals implications for the critical angle
at onset.
