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We know the non-dimensionalized distance of the viscous sub-layer is 5 wall units for a turbulent boundary layer. It is common to see this in Fluid Mechanics books but seems somewhat arbitrary.

I want to know if this true for all conditions or does it depend on the Reynolds number (high viscosity, high speed, etc).

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As far as I know, there are no exceptions of this law when increasing the Reynolds number. When decreasing the Reynolds number, at some point your boundary layer will not be turbulent, and thus be different.

I am not sure if you can derive the value $y^+=5$ without experimental evidence. However, it is common to, in plus units, write the expression for the viscous sublayer as

$$u^+=y^+$$

However, if you write this out, it is nothing more then writing

$$u=\frac{du}{dy}y,$$ which, close enough to the boundary, is always true as a linearization. Of course this close enough highly depends on the viscosity and velocity, but that is exactly the reason to write it in non-dimensional for. The definition of this close enough is actually the thickness of the viscous sublayer (and called viscous because that is the dominant force, leaving it linear).

The tricky thing here is, that you cannot really estimate the friction velocity without knowing something about the velocity. Therefore, in the dimensional form, the Reynolds number IS important.

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