The actual Reynolds Number in the jacket will depend on the type of jacket and the jacket internals. The manufacturer may have empirical data on heat transfer used this information to answer your question. You may not be able to accurately calculate the Reynolds number. Wihtout any details about your set-up or guarantees of accuracy you could try the following.
This should yield a very conservative Re number.
Calculate the cross-sectional area of the jacket, A.
Flowrate/A = estimated velocity, v.
For the diameter a crude estimate would be 2 x (the distance from the inside of the jacket to the outside of the reactor). You then need the viscosity & density of the liquid in the jacket. Watch you're units.
This is very much a back of the napkin calculation and is the type of thing I normally keep to myself until I can do a more accurate calculation. If your answer looks totally off the wall then crumple up the napkin and toss it. One other note, if your chemical engineer is any good he should be able judge if the flow is or isn't turbulent if he has the reactor jacket specifications and flowrates in front of him.