Quadratic drag
= Quadratic drag
Quadratic drag has magnitude proportional to the square of the relative speed and points opposite to the relative velocity.
For an <isotropic> model of a body of fixed shape, let $\mathbf U=\mathbf v_{\rm body}-\mathbf u_\infty$ denote its <relative velocity> through the distant <fluid>, with $U=|\mathbf U|$. A constant positive resistance $k$ gives
$$
\mathbf F_D=-k|\mathbf U|\mathbf U,\qquad F_D=kU^2,\qquad k=\tfrac12\rho AC_D.
$$
Here <density> $\rho$, reference <area> $A$ and <drag coefficient> $C_D$ are held fixed over the modeled range of relative <speeds>. The vector formula makes the direction explicit: the <drag force> on the body opposes its motion relative to the <fluid>, and vanishes at $U=0$. For a general body, orientation and other flow conditions can affect the <drag coefficient>; treating it as a constant requires a regime in which that is an adequate approximation. https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/drag-coefficient/[NASA Glenn Research Center, Drag Coefficient, Air Viscosity and Compressibility Effects]{ref}
Compare two distinct models at a chosen relative <speed> $U_0>0$. A <linear drag> model has $F_D(U)/F_D(U_0)=U/U_0$, while a constant-coefficient <quadratic drag> model has $F_D(U)/F_D(U_0)=(U/U_0)^2$. At $U=2U_0$ their <drag force> ratios are two and four respectively.
\Image[/linear-versus-quadratic-drag-scaling.png]
{title=Normalized linear and quadratic drag models: doubling speed gives force ratios two and four}
{description=Each model is divided by its own <drag force> at $U_0$. The <linear drag> curve applies, for example, to a rigid <sphere> under the <Stokes drag law> assumptions while its <particle Reynolds number> remains small. The <quadratic drag> curve assumes constant <density>, reference <area> and <drag coefficient> over its modeled range. The comparison specifies neither an exact transition nor a single set of physical parameters that realizes both laws throughout the plotted range.}
This distinction matters when estimating <terminal velocity> or trajectories such as <descent from rest under quadratic drag>: doubling relative <speed> increases the resistance much more strongly in the <quadratic drag> model. An approximately constant <drag coefficient> can occur in another <Reynolds number> regime; its use must be justified there. Conversely, substituting the creeping-flow <sphere> coefficient $C_D=24/\mathrm{Re}_d$ into $F_D=\tfrac12\rho U^2AC_D$ recovers <linear drag>, so writing this general formula does not make the resistance quadratic. Neither approximation should be extended across all <Reynolds numbers>.
The distinction already appears in the original investigation underlying the <Stokes drag law>, read on 9 December 1850: article 42 identifies resistance proportional to relative <speed> and distinguishes it from a contribution proportional to the square of relative <speed>. https://mural.uv.es/daroig/documentos/stokes1850.pdf[George Gabriel Stokes, On the Effect of the Internal Friction of Fluids on the Motion of Pendulums, article 42, collected-paper reprint]{ref}