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What I'm trying to understand is the relationship between the friction force (on an object moving through a fluid) and the velocity of the object. A few places I see "friction force is linear with velocity for laminar flow and goes as velocity squared for turbulent flow." I'm trying to undestand this on a physical level.

I'm more familiar with fluid flow through pipes. In this case, the problems are usually solved by using a friction factor (f) multipied by velocity squared: f*(v^2)/2g. The factor f is found from the Moody chart, which shows f as a function of reynolds number. At low Re, f is proportional to 1/Re; at fully turbulent flow (high Re), f is constant. Since Re is proportional to v, this gives us the described behavior: when Re is low the losses go as v^2/v (ie, linear in v) and when Re is high the losses go as v^2.

OK, so now my questions: 1) why is the loss =f*v^2? and (2) why is the shape of f vs Re as shown on the Moody chart? I'm looking for the physics behind this. Anyone able to give me a clue?

Thanks