Is the coefficient of drag the same on two similar objects?

if I had two objects, scaled perfectly to each other so that one is $$5$$ times the size of another while keeping the shape the same, would their coefficients of drag, $$C_d$$, be the same in the formula $$D = \frac12 \rho C_d S V^2$$?

If we take the fluid dynamical system under consideration to be only a function of fluid density $$\rho$$, some characteristic speed $$V$$, fluid viscosity $$\mu$$, and some characteristic length $$R$$, then it has to be the case from dimensional analysis that the drag coefficient is only a function of the Reynolds number: $$C_d = f(Re)$$ Since altering the characteristic length of the object changes the Reynolds number, you'd expect the drag coefficient to change as the size of the object changes.