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Reynolds number is the ratio of the fluid speed to some characteristic speed, $v_R = \frac{\mu}{\rho L}$ where $L$ is some characteristic length scale of the situation.

Is there an intuitive interpretation or some kind of physical picture of what this speed $v_R$ means? Is there any interpretation beyond "the speed above which turbulence arises"?

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    $\begingroup$ I don't think the concept of "Reynolds speed" is particularly useful. You could also introduce a "Reynolds length scale" $l_{Re} = \frac{\mu}{\rho U}$ or a "Reynolds viscosity" $\mu_{Re} = \rho U L$ or a "Reynolds density" $\rho_{Re} = \frac{\mu}{U L}$ and try to find a meaning for them. $\endgroup$ – 2b-t Nov 10 '19 at 13:34
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Dimensionless parameters such as the Reynolds number are of uttermost importance in fluid dynamics as they appear in the dimensionless conservation equations. The precise material parameters are not important as long as the relevant dimensionless numbers have a similar magnitude (law of similarity). This allows us to conduct experiments with properly designed smaller systems. For the Reynolds number for instance this means that only rescaling a model and thus the characteristic length is not sufficient to have similar turbulence behaviour such as similar turbulent structures. Instead you have to adapt either characteristic velocity or viscosity or both as well. If you additionally demand other dimensionless numbers to be similar as well, such as the Mach number, you soon have very limited options for conducting model experiments. Most of these dimensionless numbers are a ratio of certain opposed properties and allow to estimate which are the dominating effects in a particular situation.

The Reynolds number

$$ Re := \frac{ U L }{ \nu } = \frac{\rho \, U \, L}{\mu} \label{1}\tag{1}$$

appears in the conservation equations several times

$$\frac{\partial \rho^*}{\partial t^*} + \sum\limits_{j \in \mathcal{D}} \frac{\partial (\rho^* u_j^* )}{\partial x_j^* }=0 \label{2}\tag{2}$$

$$\rho^* \frac{\partial u_i^*}{\partial t^*} + \rho^* \sum\limits_{j \in \mathcal{D}} u_j^* \frac{\partial u_i^*}{\partial x_j^*} = - \frac{\partial p^*}{ \partial x_i^* } + \frac{1}{Re} \sum\limits_{j \in \mathcal{D}} \frac{\partial \tau_{ij}^*}{\partial x_j^* } + \frac{1}{Fr^2} g_i^* \label{3}\tag{3}$$

$$\rho^* \frac{\partial T^*}{\partial t^*} + \rho^* \sum\limits_{j \in \mathcal{D}} u_j^* \frac{\partial T^*}{\partial x_j^*} = Ec \left( \frac{\partial p^*}{\partial t^*} + \sum\limits_{j \in \mathcal{D}} u_j^* \frac{\partial p^*}{\partial x_j^*} \right) + \frac{1}{Pr Re} \sum\limits_{j \in \mathcal{D}} \frac{\partial}{\partial x_j^*} \left( \frac{\partial T^*}{\partial x_j^*} \right) + \frac{Ec}{Re} \sum\limits_{i, j \in \mathcal{D}} \tau _{ij}^* \frac{\partial u_i^*}{\partial x_j^*} \label{4}\tag{4}$$

It appears in front of the diffusive terms in the momentum \ref{3} and energy equation \ref{4}. Diffusive/dissipative terms act similar to dampers, they dissipate energy: they take macroscopic energy and convert it to thermal energy and thus smooth out gradients, they equalise. The corresponding damping coefficient is given by the dynamic viscosity $\mu$ with appears in the denominator of equation \ref{1}. For large Reynolds number the dissipative effects can be neglected and we end up with the Euler's equations for inviscid flow.

Additionally you can see that the numerator takes the form of some sort of momentum $\propto \rho U$. So our basic idea is to cast this into a form of the two opposed properties of inertia and dissipation.

Let's take Newton's first law, and rewrite it using characteristic measures

$$ F_{in} = m a \approx \underbrace{ \rho A L }_{m = \rho V} \frac{L}{T^2} = \rho A U^2 \label{5}\tag{5}$$

on the other hand someone could define a viscous force in a similar manner:

$$ F_{\mu} \propto \frac{\mu A \Delta U}{\Delta L} \approx \frac{\mu A U}{L} \label{6}\tag{6}$$

as the shear-rate $S_{ij}$ is connected to viscous stresses $\tau_{ij}$

$$ \frac{F_{\mu}}{A} = \tau_{ij} = 2 \mu S_{ij} - \frac{2}{3} \mu \sum\limits_{k \in \mathcal{D}} S_{kk} \delta_{ij} \label{7}\tag{7}$$

where

$$S_{ij} := \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right). \label{8}\tag{8}$$

We can see that dividing these two forces - $F_{in}$ and $F_{\mu}$ - results in the Reynolds number and thus

$$ Re \propto \frac{\text{inertial forces}}{\text{viscous damping forces}}. \label{9}\tag{9}$$

It basically is a measure wherever our flow will be dominated by inertia, become turbulent with small whirls and large local gradients or smooth out quickly and be laminar as for low Reynolds number flow (in particular Stokes' flow $Re \lesssim \mathcal{O}(1)$).

In my opinion thinking in terms of a "Reynolds speed" is not particularly useful: You could also introduce a "Reynolds length scale" $l_{Re} = \frac{\mu}{\rho U}$ or a "Reynolds viscosity" $\mu_{Re} = \rho U L$ or a "Reynolds density" $\rho_{Re} = \frac{\mu}{U L}$ and try to find a meaning for them. Instead always think for dimensionless numbers in term of ratios of forces and diffusivities. In the end we are only trying to estimate orders of magnitude.

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Well, a small Reynolds number indicates that the flow is dominated by viscous forces, and tends to be laminar, while a large Reynolds number indicates that the flow is dominated by inertial forces and tends to be turbulent and chaotic.

Therefore, you could interpret this “Reynolds speed” to be the velocity scale above which flow instabilities (vortices, eddies, general turbulence) start to arise.

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  • $\begingroup$ Thanks, though I guess I meant an interpretation beyond that one. Put another, why is there a special speed that controls turbulence? What's special about that special speed and not others? $\endgroup$ – Luke Pritchett Nov 9 '19 at 22:09
  • $\begingroup$ @LukePritchett I'm not sure what kind of interpretation you're looking for. Consider a smooth flow and then introduce a small perturbation. When the flow speed is small, then the viscous forces are sufficient to damp out the perturbation, so the flow is stable. At high speeds, the inertia of the fluid parcels is too high to be tamed by the viscous forces and the perturbations tend to grow. The cutoff between these two behaviors depends on the viscosity, the mass density (i.e. the inertia), and the length scale over which the existing flow changes - $\frac{\mu}{\rho L}$. $\endgroup$ – J. Murray Nov 10 '19 at 0:36
  • $\begingroup$ The existence of a velocity scale above which the motion is unstable is not unique to fluids - think about why driving at 150 mph is dangerous. The maximum safe speed to drive at depends on how much friction you have between your car and the road, how massive your car is, and on how tight the curves in the road are. $\endgroup$ – J. Murray Nov 10 '19 at 0:39
  • $\begingroup$ That's a helpful analogy, thank you. Helps me understand why a velocity should be the cutoff for turbulent behavior. $\endgroup$ – Luke Pritchett Nov 10 '19 at 1:22
  • $\begingroup$ Don't think of it as a strict cutoff. Dimensionless numbers in fluid dynamics are ratios that are used to estimate orders of magnitudes of certain effects. It's like defining the ratio between the money you ear and the money you spend. If that is close to unity then you are making a living, if you earn much more than you spend then that's an indicator that you are rich, if you spend much more it is likely you are broke. This concept is only vague: you might spend much more in a month than you ear when you buy a car but if you average you might obtain a good measure of your financial status. $\endgroup$ – 2b-t Nov 10 '19 at 2:50

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