Hi, everyone. I'm Japanese. I have a simple question on definition of friction velocity.

Def of the friction velocity is U_tau = sqrt(tau_wall/rho) & In 2D, tau_wall = mu*du/dy

So, in 3D? tau_wall = mu*(vorticity magnitude) is correct?

Best regards. kei-tee

For 3D flows you use your 2D formula with

tau_wall = mu*sqrt( (du/dy)**2 + (dw/dy)**2 )

i.e. it's the magnitude of the surface stress vector

(assuming the wall is flat; y=0). This is effectively the magnitude of the vorticity. In a more general flow (curved wall) it would not be exactly equal to the vorticity. Although for an attached boundary layer the difference should be small - of the order of 1/sqrt(R) where R is the Reynolds number.

Thanks, Tom.

But still I wonder why you think so. Though, your definition, tau_wall = mu*sqrt( (du/dy)**2 + (dw/dy)**2 ) is ofcourse approximation of the vorticity magnitude. in terms of computatinal cost, cost save of this approximation is negligible, I think. For what?

Please give me your opinion, if possible.

Best regards. kei-tee.

Think it this general way, then it does not matter if it's 2D or 3D or which components of U, V, W are parallel to wall.

tau_wall = mu * d(velocity_parallel_to_wall)/dy

where one can easily work out velocity component parallel to wall using velocity vector and wall area vector.

Thanks, John.

Actually, if the wall is flat (y=0) it's not an approximation (the no-slip condition forces the missing terms to vanish). For an attached boundary-layer the missing terms are O( 1/sqrt(R) ) as the Reynolds number R -> infinity.

In the general case of a curved wall with unit normal n and tangent vectors s,t in the pane of the body then the surface stress has the two components

tau_s = mu.(Dn).s , tau_t = mu.(Dn).t

where D is the symmetric part of the velocity gradient (deformation). tau_wall is the magnitude of this vector. If y=0 is the wall then n=j, t=i, s=j and you obtain my original formula on using no-slip.

Hope this helps,

Tom.

Hi all,

I guess you are dealing with laminar boundary layers, or not??

Regards,

Gorka

Ref.Mr Gorka's query

I feel that the friction velocity is associated with turbulent flow.

By the way, some one can kindly tell about the physical meaning for it.Though this may be Fluid Mechanics question rather than a CFD query,I feel that this site is frequented by good Fluid mech. experts also.

The friction velocity is nothing more than a conventient way of defining a velocity scale close to the wall and can be defined for both a laminar and turbulent flow (but it's not much use in a laminar flow!). To see where it comes from you observe that very close to the wall (within the laminar sublayer of a turbulent flow) u can be approximated by it's Taylor series expansion

u ~ tau.y/nu

where tau = nu.du/dy evaluated at y=0 (flat wall) is the wall stress and nu is the kinematic viscosity. If you now nondimensionalize this with some velocity scale U you obtain

u/U ~ (tau/U).y/nu.

Now since tau has units of velocity squared we may take U = sqrt(tau) to obtain

u/U ~ U.y/nu (= y+)

This scaling argument is the sole purpose for the introduction of the friction velocity.

Thanks Mr.Tom for your nice insight explanation.Courtesy to this forum.

One observation please.In the Newton's law of viscosity "mew" Ns/sq m should come in place of "nu".Regards

Or, another way of looking at the friction velocity is that it is the velocity whose momentum flux/unit area equals the shear force per unit area (remember that, whether we're talking about laminar or turbulent flow, the friction velocity is defined within the laminar sublayer - which exists even in turbulent flow). As one of the other respondents pointed out, the physical meaning is a bit nebulous, but it gives a usable length and velocity scale for turbulent flow analysis.