for efficient cooling with water better use small pipes to achieve a turbulent flow (see Reynolds number).
german users find all neede info in this book: VDI Wärmeatlas
another aproach is to use heatpipes, perhaps also fanless.
Yes, you want turbulent flow for mixing near the wall. You want Reynolds number to be above 4000, at least close. But assuming all variables are held constant, larger pipe diameters produce larger Reynolds numbers. Re = (rho*V*Dh)/mu. However, if your larger pipe diameter reduces velocity then you will not see a Re increase.