What's the meaning of the Reynolds number
 

Hi,

I've heard a lot, especially in cfd context, that the Reynolds number is the ratio between the inertial to viscous forces. Now what I understand from inertial force is (mass times acceleration) so if this is correct then for pipe flow where the velocity profile doesn't change there must be zero acceleration for fluid particles and yet we define a Reynolds number. What am I missing?

Nick

no, "inertial" refers to the lagrangian (convective) term, not to the eulerian (time derivative) term! is the ratio between convective and diffusive fluxes

by the convective term you mean (u.del) u where u is the velocity right? well this is also zero for pipe flow for instance where velocity doesn't change on a streamline

I mean the ratio of the fluxes, the convective flux is rho * u^2

rho * u^2 has the dimension of pressure) may be just rho * u ? ;)

no, not at all, the ratio is
[convective flux]/[diffusive flux] = rho*u^2/ (mu * u/L) = rho*u*L/mu = Re

You can define it as Re=Dh*V/visc

where:

Dh=Hydraulic diameter of the transversal area
V= Velocity of the fluid
visc = Cinematic viscosity of the fluid

I guess it is term misunderstanding (math vs phys). The Reynolds number it is ratio of forces (inertial vs viscous). Term momentum flux has force (per volume unit) dimention for momentum equation.

the meaning of the Re number is even more general ... for example, it can stand for the ratio between the convective to the diffusive time scales...

I guess you don't see my point. Take a control volume in a pipe where velocity profile is not changing with distance. Now if we compute the convective fluxes over the ENTIRE control volume, it is ZERO because the right and left surfaces cancel each other out and there's no flux thru the top and bottom surfaces. On each side (say the right surface of the control volume) the momentum convective flux is Integral of (u ro u.dA )/A , this is non-zero. So at each cross-section the rate of momentum is that Integral (u ro u.dA) which in dimensional analysis is ro u^2 but I don't like the name "the inertial force" it doesn't make much sense to me. Since inertial suggests acceleration. Unless someone has a better physical interpretation for me. Also, for the same example, the lagrangian convective term is actually zero since the velocity of a fluid particle don't change as they convect downstream.

u^2/L is actually an acceleration .... as well as mu*u/rho/L^2 ... the ratio is again the Re number... Note that in channel the surface integral of the mass flux (rho*u) is zero over the volume, not the mass flux alon ... the same comment is valid for the integral of the convective flux, not for the single flux ....
So I do not understand your doubt

so again..here's the problem ..saying the reynolds number is the ratio of inertial FORCES to viscous forces is misleading ...here's why I think this is so.... look...from what inertial force means in dynamics : it is the mass times acceleration right? what else could it mean? so when fluid particles DO NOT ACCELERATE (like in a pipe where velocity profile is constance over distance) there is NO inertial forces...but we have a Reynolds number ..so saying that Re is the ratio of inertial to viscous forces is not accurate...hope that makes sense.

The classical physics of a rigid body expresses the Newton law as m*a = F, this is a Lagrangian framework wherin a is the total acceleration. Here you see the inertial term...
In fluid dynamics the total acceleration is decomposed in
a = du/dt + u*grad u

du/dt is the Eulerian acceleration
u*grad u is the "lagrangian" acceleration

u*grad u can be written as div * (u u) using the mass equation.

Therefore, we define the Reynolds number as ratio of "partial" acceleration, without involving the eulerian acceleration du/dt. The non-dimensional number that takes into account for this acceleration is the Strohual number.

Yeah u.grad u is zero if.velocity doesn't change with distance.like a pipe situation yet we have Reynolds number

but you don't use u *grad u ... you take a characteristic velocity of the problem, for example the averaged velocity U, a characteristic lenght of the problem, that means a lenght in the direction where the velocity changes (in channel where u=u(y) you take L = height).

Then [rho*U^2/H] / [mu* U /L^2] = Re

I know how.to.calculate the Reynolds number ..my problem was that the term inertial force is not accurate because inertia is zero in that example. I've said this a few times now inertial force is zero if velocity doesn't change with distance.. saying that Reynolds is the ratio between inertial and viscous forces is not accurate

we simply do not agree on that definition of inertial, no problem ;)

Note that, rigorously, in channel flow the motion is driven by the pressure gradient Delta P/L, therefore:

P/L = O(rho*U^2/L) that balance the viscous term. Maybe using the "pressure force" is more appealing for you

Just a quick question for my own edification - if the velocity is not changing with distance, then aren't the viscous forces also zero? Wouldn't the whole flow system become trivial, rendering the Reynolds number moot?

In channel flow the velocity changes with distance y alone if the flow is fully devoloped ... however, channel flow can be also the case in which the inviscid velocity inlet (for example a plug profile) develops into a boundary layer... then u = u(x,y). In all cases the Re definition can be suitably introduced

How do you define inertial force? You.previously said that it is the convective term in the navier stokes equations..that term is zero for.the.described channel flow.

Viscous forces are not zero because there s a velocity gradient in the normal to wall direction but they are balanced by pressure difference forces .the.velocity for fully developed flow doesn't change in the parallel to wall direction .that means fluid particles don't accelerate because of the balance of forces

Your observation about the null lagrangian acceleration u * grad u is correct provided that you have a fully developed steady flow, this is only an academic framework. In the case you are citing it is the equilibrium of the isotropic to deviatoric part of the stress tensor that balance each other, since the isotropic part (the pressure) is of order rho*u^2 that leads to some misleading in the Re.
Again, I don't see the problem in the definition of Re as ratio between convective and diffusive fluxes (or convective to diffusive time) ... both do not vanish...
But the real flow in channel is far to be the Poiseulle solution, it is turbulent, with unsteady behaviour and inertial term always present, therefore I am used to define the Re in channel as rho*u_tau*H/mu wherein u_tau is the stress velocity that is defined by means of the pressure gradient

I think we can agree on the definition of Re based on the momentum comvective flux thru one.cross section divided by viscous flux ... I'm not convinced of the presence.of any inertial forces ..the flow.could be laminar too .anyways thanks for.your.input and.time