# Reynolds numbers and inertial force

1. Nov 13, 2012

### forks11

Reynolds numbers and "inertial force"

I am an undergrad physics major taking an engineering course that just introduced the concept of reynolds numbers. When I try to get an idea of how the Reynolds number is physically derived, I keep running into the definition that it is the "ratio of inertial forces to viscous forces acting on a fluid." Where the "inertial force" is defined as F=mass*acceleration.

As I understand newtons 2nd law, it means that for an object with mass M to accelerate with acceleration A, the object must be acted on by a net force ∑F=M*A. But F = M*A is not an actual force, but rather a description that says that "acceleration is directly proportional to force and inversely proportional to mass."

The only other time I have seen this concept of "inertial force" is with respect to non-inertial reference frames where objects experience a pseudo-force due to the acceleration of the frame, but that doesn't seem to apply here.

Can anybody explain to me what is meant by "inertial force," or otherwise give me a intuitive or straightforward description of what the Reynolds number describes??

2. Nov 13, 2012

### Noumenon

Re: Reynolds numbers and "inertial force"

"The inertial forces, which characterize how much a particular fluid resists any change in motion are not to be confused with inertial forces defined in the classical way" - Wiki

A high Reynolds number indicates high turbulence, chaotic flow, while a low number indicates laminar flow, smooth flow. Wiki gives a good explanation of the specific ratio that is the Reynolds number (of the scales of turbulence wrt energy dissipation).

3. Nov 13, 2012

Re: Reynolds numbers and "inertial force"

The "inertial forces" in this case do effectively refer to the fluid's mass and therefore tendency to resist movement by an external force. The term in the numerator is essentially a mass flux term multiplied by a length scale.

The Reynolds number basically represents the importance of viscosity to the flow field relative to effects of the overall fluid motion. For very high Reynolds numbers, the flow can often be approximated reasonably accurately as inviscid, for example.

This is incredibly false. There is no universal correlation between Reynolds number and the onset of turbulence. In general, as the Reynolds number gets higher, the flow is more likely to transition, but it is not a measure of turbulence or an absolute indicator of turbulence.

4. Nov 13, 2012

### Staff: Mentor

Re: Reynolds numbers and "inertial force"

The Reynolds number can be manipulated mathematically as follows:

Re = ρvL/μ = (ρv2)/(μv/L)

where L is the characteristic length scale for the system. The quantity μv/L is like the viscous shear stress, and is loosely referred to as the "viscous force." The quantity ρv2 is like the kinetic energy per unit volume, and is loosely referred to as the "inertial force." Saying that the Re is the ratio of inertial forces to viscous forces is no more precise than this, so don't take it too seriously. It is more of an intuitive notion.

5. Nov 16, 2012

### The Jericho

Re: Reynolds numbers and "inertial force"

The Reynold's number is a way of quantifying how much force is needed to change the direction of the fluid flow whether comp. or incomp. so it's not a measure of turbulence, it's a dimensionless value to depict exactly what you said in your question..."ratio of inertial forces to viscous forces acting on a fluid". It is used to essentially unify an understanding of the effects of viscosity and the momentum of each particle. The penny will drop so to speak when you perform any wind tunnel tests and you'll see why it's relevant.

Sincerely, The Jericho.

6. Nov 16, 2012

### Staff: Mentor

Re: Reynolds numbers and "inertial force"

Based on decades of experience if fluid mechanics, I can tell you that I have never heard anyone describe the Reynolds number in this way. The force needed to change the direction of a fluid flow is primarily related to a change in momentum (inertial), and has very little to do with viscous effects. In addition, for a given flow (e.g., flow in a pipe), the Reynolds number is used in practice to determine the transition from laminar flow to turbulent flow (for pipe flow, the transition is 2100), and also correlates quantitatively with the rms magnitude of the turbulent velocity fluctuations. In fluid mechanics, the fluid is treated as a continuum, and there are no individual particles involved. Certainly, viscosity is a continuum concept.

How is the Reynolds number used in practice? In both fluid mechanics and heat transfer, it is used to determine the forces and heat fluxes at interfaces. In turbulent pipe flow, for example, the shear stress at the wall can be predicted if the Reynolds number is known. Knowledge of the shear stress at the wall is necessary to predict the pressure drop. In turbulent heat transfer, the heat transfer coefficient at the wall is a function of two dimensionless groups, the Reynolds number and the Prantdl number. Knowledge of the heat transfer coefficient at the wall is necessary to predict the temperature change of a fluid passing through a heated pipe.

7. Nov 16, 2012

Re: Reynolds numbers and "inertial force"

In the end it doesn't REALLY predict heat transfer or shear stress either in the normal sense of "predict." It is a similarity variable that therefore correlates well with quite a few parameters on interest in fluid mechanics.

Take for example heat transfer. Yes the Nusselt number depends on Reynolds number and Prandtl number, but that relationship changes for different ranges of both and for laminer or turbulent flow. The Reynolds number itself doesn't have any real predictive capability, therefore. It just works nicely with many correlations and models due to its nature as a similarity variable.

8. Nov 16, 2012

### Staff: Mentor

Re: Reynolds numbers and "inertial force"

Yes. I totally agree. The wording I employed was that it is used to predict the drag and the heat transfer. I didn't mean to imply that, without the experimental correlations, it can have predictive power on its own. But, of course, for laminar flow, the experimental correlations aren't really necessary, since the behavior can be predicted analytically.

9. Nov 16, 2012