Biological nanopores such as aquaporins combine the opposing functions of high water permeation and total ion exclusion in part by the virtue of their hourglass shape. Here, we perform molecular dynamics simulations to examine water and ion conduction through hourglass shaped nanopores created from carbon nanotubes (CNTs) of chirality (6,6), (8,8), and (10,10) in combination with carbon nanocones of half cone angles 41.8°, 30.0°, 19.45°, 9.6° and 0.0°. We observe large variations in flow through the nanopores with change in half cone angles and tube diameters. By computing the pore-water interactions we find a correlated change between the flux and the density profiles of water inside the nanopores. Further, from the orientation, and the hydrogen bonding characteristics of water, we uncover some unexplored facets of flow through hourglass shaped nanopores. The results are insightful for devising novel separation membranes based on nanopores that mimic the shape of biological nanochannels.