I am doing an analysis of the motion of an object in air and so I'm first finding whether the flow is laminar or turbulent. From my research, I'm still a bit confused as to whether the drag equation in the link below applies to both turbulent and laminar flow or just turbulent.


Sorry, the wiki isn't that specific so could someone please clarify this for me?

  • $\begingroup$ It is for turbulent flow. $\endgroup$ – Deep Oct 28 '18 at 5:27
  • $\begingroup$ are you sure? en.wikipedia.org/wiki/Drag_(physics) the stuff under types of drag says that at low Re, it is proportional $\endgroup$ – txyriuc Oct 28 '18 at 5:37
  • $\begingroup$ @Deep i don't really get why though. how does cd being asymptotically proportional to Re^-1 make it linear $\endgroup$ – txyriuc Oct 28 '18 at 5:38
  • $\begingroup$ It has nothing to do with whether the flow is laminar or turbulent. It can even be used for flows with no viscosity at all (induced drag). Calling it the "drag equation" is also a bit misleading; it's essentially just the definition of the drag coefficient, which can be useful in some situations, but doesn't tell you anything about how to compute drag if you don't already have the coefficient. $\endgroup$ – D. Halsey Oct 28 '18 at 20:55
  • $\begingroup$ You can think of drag force as a function of velocity $v$ being expanded as a Taylor series in powers of $v$. Since drag is zero if $v=0$ the Taylor series doesn't have a constant term and begins with first power of $v$. For low enough velocity (more accurately Reynolds number $\ll1$) Taylor series upto the first power of $v$ will be sufficient, the rest of the terms adding small corrections. For higher velocities more terms in the Taylor series need be accounted for. Of course as Jan's answer says the dependence may not be any simple integer power of $v$. $\endgroup$ – Deep Oct 29 '18 at 4:14

The drag equation can be used to calculate drag force for both laminar and turbulent flow, if we allow $C$ to vary with velocity in a convenient way. This reinterpretation of the equation and meaning of $C$ is sometimes done in practice, especially if one is measuring the drag and expressing the results of measurements as function $C(v)$. This just a practical matter which is useful in the turbulent regime - physically there is no good reason to use single formula for all regimes.

If we don't allow $C$ to vary with velocity and fix it to a single number, the equation can give drag force accurately only in the turbulent regime, and only for limited range of velocities. When studying wide range of velocities, it is not that accurate. The drag force function of velocity is not exactly a single power. That is why function $C(v)$ is introduced, so that the drag equation can be accurate for a wide range of velocities.

For laminar flow, the expression with constant $C$ is completely wrong and should not be used. The accurate expression is closer to linear function:

$$ F = -kv. $$ (In fact the drag force depends also on the acceleration of the body, but this can be handled by redefining the mass of the body).

In common conditions in air, the drag force is linear function of velocity only if the Reynolds number Re is much less than 1. Stuff like microscopic water droplets in fog falling down. Things like falling rocks or flying planes in Earth's atmosphere have turbulent flow. The velocity and the body is just too big to allow for laminar flow.


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