  High-frequency very-long-baseline interferometry (VLBI) observations can now
resolve the horizon-scale emission from sources in the immediate vicinity of
nearby supermassive black holes. Future space-VLBI observations will access
highly lensed features of black hole images -- photon rings -- that will
provide particularly sharp probes of strong-field gravity. Focusing on the
particular case of the supermassive black hole M87*, our goal is to explore a
wide variety of accretion flows onto a Kerr black hole and to understand their
corresponding images and visibilities. We are particularly interested in the
visibility on baselines to space, which encodes the photon ring shape and whose
measurement could provide a stringent test of the Kerr hypothesis. We develop a
fully analytical model of stationary, axisymmetric accretion flows with a
variable disk thickness and a matter four-velocity that can smoothly
interpolate between purely azimuthal rotation and purely radial infall. We then
determine the observational appearance of such flows, taking care to include
the effects of thermal synchrotron emission and absorption. Our images
generically display a "wedding cake" structure composed of discrete, narrow
photon rings (n=1,2,...) stacked on top of broader primary emission that
surrounds a central brightness depression of model-dependent size. We find that
the "black hole shadow" is a model-dependent phenomenon -- even for diffuse,
optically thin sources -- and should not be regarded as a generic prediction of
general relativity. At 230 GHz, the n=1 ring is always visible, but the n=2
ring is sometimes suppressed due to absorption. At 345 GHz, the medium is
optically thinner and the n=2 ring displays clear signatures in both the image
and visibility domains, identifying this frequency as more promising for future
space-VLBI measurements of the photon ring shape.
