  The detection of gravitational waves (GWs) provides a direct way to measure
the luminosity distance, which enables us to probe cosmology. In this paper, we
continue to expand the application of GW standard sirens in cosmology, and
propose that the spatial curvature can be estimated in a model-independent way
by comparing the distances from future GW sources and current
cosmic-chronometer observations. We expect an electromagnetic counterpart of
the GW event to give the source redshift, and simulate hundreds of GW data from
the coalescence of double neutron stars and black hole--neutron star binaries
using the Einstein Telescope as reference. Our simulations show that, from 100
simulated GW events and 31 current cosmic-chronometer measurements, the error
of the curvature parameter $\Omega_{K}$ is expected to be constrained at the
level of $\sim0.125$. If 1000 GW events are observed, the uncertainty of
$\Omega_{K}$ would be further reduced to $\sim0.040$. We also find that adding
50 mock $H(z)$ data (consisting of 81 cosmic-chronometer data and 1000
simulated GW events) could result in much tighter constraint on the zero cosmic
curvature, for which, $\Omega_{K}=-0.002\pm0.028$. Compared to some actual
model-independent curvature tests involving the distances from other cosmic
probes, this method with GW data achieves constraints with much higher
precision.
