Calculating the Grashof and Reynolds number in tangential flow
 

Hello, I have a project for studying the case of a concentric annulus with a moving core (2 concentric cylinders), through which the fluid flows. There is a gravitational field being exerted perpendicular to the axis of the cylinders. The inner cylinder is spinning at constant speed and there is a temperature difference between the two cylinders (the inner cylinder has a higher temperature.) I want to study the effects of mixed convection on temperature and velocity contours on various lengths in the tube.

Now my question is how to make sure that the natural convection in the tube is dominant? I know the solution is to calculate the Richardson number but the Reynolds number parameter present in the Richardson number formula needs a hydraulic diameter and as my natural convection is occurring in the tangential direction, I don't know how to give the hydraulic diameter a value corresponding to the direction of the flow in natural convection. The problem is that there is flow in mainly two directions: axial and tangential. As I want the tangential flow (natural convection) to be dominant, should I use the same formula for axial hydraulic diameter for the tangential one? How do you suggest I make the natural convection dominant?