We study the two-dimensional, time-dependent hydrodynamics of radiatio
n-driven winds from luminous accretion discs in which the radiation fo
rce is mediated primarily by spectral lines. We assume that the disc i
s flat, Keplerian, geometrically thin and optically thick, radiating a
s an ensemble of blackbodies according to the alpha-disc prescription.
The effect of a radiant central star is included both in modifying th
e radial temperature profile of the disc and in providing a contributi
on to the driving radiation field. Angle-adaptive integration techniqu
es are needed to achieve an accurate representation of the driving for
ce near the surface of the disc. Our hydrodynamic calculations use non
-uniform grids to resolve both the subsonic acceleration zone near the
disc and the large-scale global structure of the supersonic wind. We
find that line-driven disc winds are produced only when the effective
luminosity of the disc (i.e. the luminosity of the disc times the maxi
mum value of the force multiplier associated with the line-driving for
ce) exceeds the Eddington limit. If the dominant contribution to the t
otal radiation field comes from the disc, then we find that the outflo
w is intrinsically unsteady and characterized by large-amplitude veloc
ity and density fluctuations. Both infall and outflow can occur in dif
ferent regions of the wind at the same time. The cause of this behavio
ur is the difference in the variation with height of the vertical comp
onents of gravity and radiation force: the former increases while the
latter is nearly constant. On the other hand, if the total luminosity
of the system is dominated by the central star, then the outflow is st
eady. In either case, we find that the two-dimensional structure of th
e wind consists of a dense, slow outflow, typically confined to angles
within similar to 45 degrees of the equatorial plane, that is bounded
on the polar side by a high-velocity, less dense stream. The flow geo
metry is controlled largely by the geometry of the radiation held - a
brighter disc/star produces a more polar/equatorial wind. Global prope
rties such as the total mass loss rate and terminal velocity depend ma
inly on the system luminosity and are insensitive to geometry. The mas
s-loss rate is a strong function of the effective Eddington luminosity
; at values of less than 1 there is virtually no wind at all, whereas
above 1 the mass-loss rate in the wind scales with the effective Eddin
gton luminosity as a power law with index 1.5. Matter is fed into the
fast wind from within a few stellar radii of the central star. Our sol
utions agree qualitatively with the kinematics of outflows in cataclys
mic variable (CV) systems inferred from spectroscopic observations. We
predict that low luminosity systems may display unsteady behaviour in
wind-formed spectral lines. Our study also has application to winds f
rom active galactic nuclei and from high mass young stellar objects (Y
SOs).