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I read here that the equation of resistive drag $F = -bv$ can be used, assuming the particle is moving

through a fluid at relatively slow speeds where there is no turbulence (i.e. low Reynolds number, $R_{e}<1$)

In order to decide whether I can use $F = -bv$, I need to identify the Reynold's number of the rigid object I am working with.

In my case, I have a solid plate of width $w$, length $l$ and height $h$ which is moving perpendicular to the direction of the fluid at a velocity $v$. The fluid has a dynamic viscosity $μ$.

With the information mentioned above how can I calculate the Reynold's number of this rigid object?

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With the Reynolds Number:

$$\text{Re}=\frac{\rho v L}{\mu}$$

the characteristic length $L$ that can be used here is (e.g.):

$$L=\frac{l+h}{2}$$

assuming that $L \gg w$

Why is the formula for the characteristic length $L$ what you have written it to be and why does $L$ need to be much bigger than $w$?

In many cases, the choice of a 'characteristic length' for the determination of $\text{Re}$ can be somewhat arbitrary and there's no 'hard and fast rule' to guide us. The case of a rectangular plate is such a case.

However, the point of transition from fully laminar to fully turbulent flow in terms of $\text{Re}$ number is not sharp. Here's Wikipedia's note on this:

For flow in a pipe of diameter $D$, experimental observations show that for "fully developed" flow,[n 2] laminar flow occurs when $\text{Re}_D < 2300$ and turbulent flow occurs when $\text{Re}_D > 2900$. At the lower end of this range, a continuous turbulent-flow will form, but only at a very long distance from the inlet of the pipe. The flow in between will begin to transition from laminar to turbulent and then back to laminar at irregular intervals, called intermittent flow. This is due to the different speeds and conditions of the fluid in different areas of the pipe's cross-section, depending on other factors such as pipe roughness and flow uniformity. Laminar flow tends to dominate in the fast-moving center of the pipe while slower-moving turbulent flow dominates near the wall. As the Reynolds number increases, the continuous turbulent-flow moves closer to the inlet and the intermittency in between increases, until the flow becomes fully turbulent at $\text{Re}_D > 2900$. This result is generalized to non-circular channels using the hydraulic diameter, allowing a transition Reynolds number to be calculated for other shapes of channel.

Because of this uncertainty on $\text{Re}$, some uncertainty on $L$ is not problematic. Without direct observation, whether flow will be laminar of turbulent will always be a bit of and 'educated guess'.

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  • $\begingroup$ Why is the formula for the characteristic length $L$ what you have written it to be and why does $L$ need to be much bigger than $w$? $\endgroup$ – user10764803 Jul 30 at 13:45
  • $\begingroup$ I'll answer that as an edit to my answer, ta. $\endgroup$ – Gert Jul 30 at 13:50
  • $\begingroup$ Another question: The rigid object in my question is falling down at a constant velocity $V$ through air in a room (the air is the fluid). The air in the room has no velocity. Can I assume that the $v$ in the Reynold's number equation is the velocity of the rigid object? $\endgroup$ – user10764803 Jul 30 at 14:18
  • $\begingroup$ Absolutely: velocity is relative. Calculation of $\text{Re}$ is done for boats, ships, planes etc using these objects' velocity (or the relative velocity wrt the sea or air) $\endgroup$ – Gert Jul 30 at 14:32
  • $\begingroup$ What about the growth of the laminar boundary layer along the plate and its collapse to a turbulent boundary layer at a Reynolds number based on the distance from the leading edge? $\endgroup$ – Chet Miller Jul 30 at 22:01

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