Re should represent the fluid flow state. I use to define it in front of obstacle. After obstacle that are not "aerodynamic" , flow may be turbulent and unstable according to fluid inlet speed, and distance.

To recover a stable flow, a  distance of 3 to 5 times the equivalent length may be necessary.

This length can be evaluated by hydrolic diameter Dh defined as 4 times the ratio between surface and wetted perimeter.

For an external flow with a step (Blue rectangle). You can use the higth of step to calculate the Re number. 

Reynolds number should represent the flow condition. For pipe flow, the characteristic length is taken as D (diameter of pipe). For the free surface flow like channel flow, it is taken as hydraulic depth (R = A/P; A being flow depth and P being wetted perimeter). Thus, for free surface flow in a circular pipe, it is D/4 and y (water depth) for wide rectangular channel. It is not clear from the diagram whether cross-section is circular or rectangular. It seems the flow is channel flow and I guess it is having rectangular cross-section. Thus, you can calculate R for this which will be function of flow depth over the obstacle and obstacle width. 

It really depends on the dimensions of the duct and inlet velocity profiles. If the Duct is too large, or flow is not fully developed within the duct, then characteristic length typically used to calculate Re will be the distance from the leading edge of the object to the point of interest.  If the duct is too small or flow is fully developed, Re can be calculated based on Hydraulic diameter at that section normal to the flow where the point of interest is located.

If it is a ship, it should be the waterline length. If it is a cylinder, then the diameter. It is easily understood from the flow directions with respect to the body and its characteristic/geometric dimension like length, diameter, width, etc.

For flow of fluid over a flat plate, the boundary layer grows along the length of the plate which is the characteristic dimension.  For flow in a tube, the boundary grows in the radial direction which is known as the developing length.  In the fully developed region, the boundary layer thickness is equal to the diameter of the tube.  Hence, the diameter of the tube is the characteristic dimension for flow of fluid in the tube.  

In Fig.1, the direction in which the boundary layer grows in the fluid flow direction is the characteristic dimension.  The second figure is not clear.  The basic concept of characteristic dimension may be extended to the second figure also.   

The average velocity of a flowing fluid, for a given mass flow rate, is a strong function of diameter(ex flow through pipe, avg velocity is directly proportional to square of diameter ) in general the area of cross section of the flow channel.  For a non-circular sections the equivalent diameter is used in the calculations of Re. Effect of length on velocity is not significant.   

Generally characteristic length for internal flow is taken as ratio of 4 times area to the perimeter (4A/P)

Whatever is the Case use the dimension of the inlet and inlet velocity for Defining Re and mention that. even if the cross section changes near the obstacle the velocity will increase and from your design it looks like it will not put any effect on that.

However if you are trying to study the obstacle and flow around it then I suggest you to use bulk Reynolds number.

Hi dear,

I agree with the most proposed answers. The best way to avoid the influence of boundary layer, we must consider the following ratio: (Length / Diameter) this ratio muste be more than 10.

With my best regards!

Drr. Adel OUESLATI