As I understand it, drag force of an object in a fluid is given by $F_D = \frac{1}{2}\rho v^2C_DA$, where $\rho$ is density of the fluid, and $v$ is the relative velocity of the flow with respect to the object. We can then solve for drag coefficient $C_D = \frac{2F_D}{\rho v^2A}$.

First question: Does a stationary object in a flow of velocity $v$ result in equal drag force to the same object moving at a velocity $v$ in a stationary fluid?

Second question: It makes sense to me that drag force is related to velocity, and a higher velocity means a higher drag force. However, the drag coefficient, I'm not so sure about. Given an object, does its $C_D$ vary with relative flow velocity? Or is that already factored in because $F_D$ changes as well, and the $C_D$ is only dependent on the geometry of the object?

So basically, if I dragged a submerged square plate through the water, once as fast as possible and once as slow as possible, would the calculated drag coefficients be theoretically the same?

  • $\begingroup$ Suggestion to the post (v2): Replace the word velocity with the word speed. $\endgroup$
    – Qmechanic
    Nov 10, 2022 at 21:06
  • $\begingroup$ You need to include viscosity effects to get the correct result. Your formula will have ranges of applicability for speed. Also for shape and size of the test object. Just as an example, when you get near the speed of sound in air, drag does weird things. $\endgroup$
    – Boba Fit
    Nov 10, 2022 at 22:57
  • $\begingroup$ One of my pet peeves (speaking as an Aero engineer) is that so many students are simply taught that first equation as if it told them anything about the drag, when in fact Cd can vary by several orders of magnitude (as shown by @basics excellent answer). I prefer to think of it as simply the definition of Cd. $\endgroup$
    – D. Halsey
    Nov 11, 2022 at 0:02
  • $\begingroup$ @Qmechanic More often than not, Aero engineers will use the word "velocity" when they mean "speed" $\endgroup$
    – D. Halsey
    Nov 11, 2022 at 0:03

2 Answers 2

  1. yes, only relative velocity matters;

  2. drag coefficient $C_D$ and other adimensional coefficients, like the lift coefficient of an airfoil or a wing, can be functions of adimensional parameters of the flow, like the Reynolds' number

    $Re = \dfrac{\rho U L}{\mu} = \dfrac{\text{inertia}}{\text{viscosity}}$

    or the Mach's number

    $M = \dfrac{U}{c} = \dfrac{\text{inertia}}{\text{compressibility}}$

    of the flow.

    Thus, aerodynamic coefficients can be function of the velocity, through these adimensional numbers

    $C_x(\alpha, \beta, Re, M, \dots)$,

    being $\alpha$, $\beta$ the angles that describe the relative velocity of the flow w.r.t. the orientation of the object.

    In order to have an example of how the drag coefficient can vary as a function of the velocity, through the Reynolds' number, we can take a look at the plot of the $C_D(Re)$ of a sphere.

enter image description here


The drag forces on a object may vary with its geometric form, the angle-of-attack and indeed its velocity normally indicated as a dimensionless metric called Reynolds number. For certain objects it is numerically and practically impossible to measure the drag for low Reynolds number. A typical case is from a sphere in a flow. Scientists and researchers have been trying in vain to calculate a distribution function for the drag coefficient for this particular case.

With the found analytic and general solution of the system of the Navier-Stokes differential equations I found (for the first time ever) the following distribution function for the Drag per meter span divided by the free-stream velocity what presumedly equivalent is with the general used, but transient version, of the "drag coefficient". an animation of the multi-dimensional contour-plot of the transient "drag coefficient" is as follows:enter image description here

Vertically is the angle-of-attack (in radians) offset against the Reynoldsnumber The N-soliton solution to the Navier-Stokes can be found in the open-access published R. Meulens; A note on N-soliton solutions for the viscid incompressible Navier–Stokes differential equation. AIP Advances 1 January 2022; 12 (1): 015308. https://doi.org/10.1063/5.0074083 and the application formulae to calculate the drag and lift forces on an arbitrary or a hypothetical airfoil may be found in Rensley Meulens; Exact distributions for the solutions of the compressible viscous Navier Stokes differential equations: An application in the aeronautical industry. AIP Conf. Proc. 28 September 2023; 2872 (1): 120085. https://doi.org/10.1063/5.0163230

The used formulae for the Drag forces per meter span can also be found in a documentary Understanding Aerodynamic Drag from curiositystream.com and the university aerospace engineering lectures of prof. Hester Bijl of Leiden university. The typical cusp near Reynolds number = 550 000 is identified as the neon green and yellow colors arcs in above graph. Above animation is copyrighted 2024 and is planned to be used in my Phd-dissertation


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