Riemannian geometry of gravity waves turbulent black hole analogs

The gravity water wave black (GWBH) hole analog discovered by Schutzhold and Unruh (SU) is extended to allow for the presence of turbulent shear flow. The Riemannian geometry of turbulent black holes (BH) analogs in water waves is computed in the case of laminar tirbulent shear flow. The Riemann curvature is constant and the geodesic deviation equation shows that the curvature acts locally as a diverging lens and the stream lines on opposite sides of the analog black hole flow apart from each other. In this case it is shown that the curvature quantities can be expressed in terms of the Newtonian gravitational constant in the ergoregion. The dispersion relation is obtained for the case of constant flow injection.