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We present direct numerical simulation (DNS) and modelling of incompressible, turbulent, generalized Couette–Poiseuille flow. A particular example is specified by spherical coordinates 
$(Re,\theta,\phi )$, where 
$Re = 6000$ is a global Reynolds number, 
$\phi$ denotes the angle between the moving plate, velocity-difference vector and the volume-flow vector and 
$\tan \theta$ specifies the ratio of the mean volume-flow speed to the plate speed. The limits 
$\phi \to 0^\circ$ and 
$\phi \to 90^\circ$ give alignment and orthogonality, respectively, while 
$\theta \to 0^\circ,\ \theta \to 90^\circ$ correspond respectively to pure Couette flow in the 
$x$ direction and pure Poiseuille flow at angle 
$\phi$ to the 
$x$ axis. Competition between the Couette-flow shear and the forced volume flow produces a mean-velocity profile with directional twist between the confining walls. Resultant mean-speed profiles relative to each wall generally show a log-like region. An empirical flow model is constructed based on component log and log-wake velocity profiles relative to the two walls. This gives predictions of four independent components of shear stress and also mean-velocity profiles as functions of 
$(Re,\theta,\phi )$. The model captures DNS results including the mean-flow twist. Premultiplied energy spectra are obtained for symmetric flows with 
$\phi =90^\circ$. With increasing 
$\theta$, the energy peak gradually moves in the direction of increasing 
$k_x$ and decreasing 
$k_z$. Rotation of the energy spectrum produced by the faster moving velocity near the wall is also observed. Rapid weakening of a spike maxima in the Couette-type flow regime indicates attenuation of large-scale roll structures, which is also shown in the 
$Q$-criterion visualization of a three-dimensional time-averaged flow.
We discuss the modal, linear stability analysis of generalized Couette–Poiseuille (GCP) flow between two parallel plates moving with relative speed in the presence of an applied pressure gradient vector inclined at an angle 
$0\leqslant \phi \leqslant 90^\circ$ to the plate relative velocity vector. All possible GCP flows can be described by a global Reynolds number 
$Re$, 
$\phi$ and an angle 
$0\leqslant \theta \leqslant 90^\circ$, where 
$\cos \theta$ is a measure of the relative weighting of Couette flow to the composite GCP flow. This provides a novel and uncommon group of generally three-dimensional base velocity fields with wall-normal twist, for which Squire's theorem does not generally apply, requiring study of oblique perturbations with wavenumbers 
$(\alpha,\beta )$. With 
$(\theta,\phi )$ fixed, the neutral surface 
$f(\theta,\phi ;Re,\alpha,\beta )=0$ in 
$(Re, \alpha,\beta )$ space is discussed. A mapping from GCP to plane Couette–Poiseuille flow stability is found that suggests a scaling relation 
$Re^*\alpha /k = H(\theta ^*)$ that collapses all critical parameters, where 
${Re}^*= Re\,({\alpha _1}/{\alpha })\,({\sin \theta }/{\sin {\theta }^*})$ and 
$\tan \theta ^*=({\alpha _{1}}/\alpha )\tan \theta$, with 
$\alpha _1=\alpha \cos \phi +\beta \sin \phi$. This analysis does not, however, directly reveal global critical properties for GCP flow. The global 
$Re_{cr}(\theta,\phi )$ shows continuous variation, while 
$\alpha _{cr}(\theta,\phi )$ and 
$\beta _{cr}(\theta,\phi )$ show complex behaviour, including discontinuities owing to jumping of critical states across neighbouring local valleys (in 
$Re$) or lobes of the neutral surface. The discontinuity behaviour exists for all low 
$\phi$. For 
$\phi \gtrsim 21^\circ$, variations of 
$\alpha _{cr}(\theta )$ and 
$\beta _{cr}(\theta )$ are generally smooth and monotonic.
We present numerical simulation and mean-flow modelling of statistically stationary plane Couette–Poiseuille flow in a parameter space 
$(Re,\theta )$ with 
$Re=\sqrt {Re_c^2+Re_M^2}$ and 
$\theta =\arctan (Re_M/Re_c)$, where 
$Re_c,Re_M$ are independent Reynolds numbers based on the plate speed 
$U_c$ and the volume flow rate per unit span, respectively. The database comprises direct numerical simulations (DNS) at 
$Re=4000,6000$, wall-resolved large-eddy simulations at 
$Re = 10\,000, 20\,000$, and some wall-modelled large-eddy simulations (WMLES) up to 
$Re=10^{10}$. Attention is focused on the transition (from Couette-type to Poiseuille-type flow), defined as where the mean skin-friction Reynolds number on the bottom wall 
$Re_{\tau,b}$ changes sign at 
$\theta =\theta _c(Re)$. The mean flow in the 
$(Re,\theta )$ plane is modelled with combinations of patched classical log-wake profiles. Several model versions with different structures are constructed in both the Couette-type and Poiseuille-type flow regions. Model calculations of 
$Re_{\tau,b}(Re,\theta )$, 
$Re_{\tau,t}(Re,\theta )$ (the skin-friction Reynolds number on the top wall) and 
$\theta _c$ show general agreement with both DNS and large-eddy simulations. Both model and simulation indicate that, as 
$\theta$ is increased at fixed 
$Re$, 
$Re_{\tau,t}$ passes through a peak at approximately 
$\theta = 45^{\circ }$, while 
$Re_{\tau,b}$ increases monotonically. Near the bottom wall, the flow laminarizes as 
$\theta$ passes through 
$\theta _c$ and then re-transitions to turbulence. As 
$Re$ increases, 
$\theta _c$ increases monotonically. The transition from Couette-type to Poiseuille-type flow is accompanied by the rapid attenuation of streamwise rolls observed in pure Couette flow. A subclass of flows with 
$Re_{\tau,b}=0$ is investigated. Combined WMLES with modelling for these flows enables exploration of the 
$Re\to \infty$ limit, giving 
$\theta _c \to 45^\circ$ as 
$Re\to \infty$.
