Physical units in fluid dynamics

Classical Mechanics Level 1

In fluid dynamics, the drag equation is a formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid

$F_D=\frac{1}{2} C_D ρv^2 A$ where

$F_D$ is the drag force, which is by definition the force component in the direction of the flow velocity,
$ρ$ is the density of the fluid,
$v$ is the velocity of the object relative to the fluid,
$A$ is the reference area, and
$C_D$ is the drag coefficient.

What units can the quantity $(C_D )^2$ have?

kg^2 m^{-6}

Dimensionless (no units)

m^{-2}

kg^2 m^{-6} s^{-2}

1 solution

Wee Xian Bin
Jun 30, 2016

Let units of $C_D$ be $[C_D]$ .

Force expressed in SI units: $kg m s^{-2}$

Density expressed in SI units: $kg m^{-3}$

Velocity^2 expressed in SI units: $m^2 s^{-2}$

Area expressed in SI units: $m^2$

$kg m s^{-2} = [C_D] (kg m^{-3}) (m^2 s^{-2}) (m^2)$

Therefore $C_D$ is unit-less and hence $(C_D)^2$ is unit-less.

@François Pineau That is not correct. For the purposes of this problem (and we can safely extend the presumption to other problems on Brilliant.org) the terms density and volumetric mass density (VMD) are sufficiently interchangeable. The dimensionless quantity you are referring to is "specific gravity" aka. "SG" or "relative density" which compares the VMD of a body with respect to the VMD of a specified reference body at a specified reference state. These two terms are, however, definitely strictly not interchangeable with the term "density" as used for interpretation in common nomenclature.

Wee Xian Bin - 4 years, 11 months ago

Density is without dimension as it compares two volumic masses. Youre talking about volumic mass for ro....

François Pineau - 4 years, 11 months ago

Physical units in fluid dynamics

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