Law of the Wall

Explore how pipe resistance, wall shear and near-wall velocity scaling describe the same flow state.

Physical inputs

Mean velocity, V

Pipe diameter, D

Water temperature

Flow regime presets

Moody chart

Numerical summaryInputs, pipe resistance and wall scaling

Physical state

Mean velocity, Vinput

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Diameter, Dinput

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Temperature, Tinput

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Kinematic viscosity, νwater property ν(T)

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Roughness, k_sinput

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Pipe resistance

Reynolds numberRe = VD/ν

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Relative roughnessk_s/D

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Darcy friction factorf(Re, k_s/D)

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Flow regimeclassified by Re

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Wall scaling

Shear velocity, u*u* = V√(f/8)

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Velocity scaleu⁺ = u/u*

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Distance scaley⁺ = yu*/ν

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Roughness in wall unitsk_s⁺ = k_su*/ν

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Turbulent roughness regimeclassified by k_s⁺

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About this model

This model links the velocity distribution close to the wall with the resistance of the whole pipe. The physical profile, wall-coordinate chart and Moody chart always represent the same flow state.

From the wall to the whole pipe

The physical view follows the velocity from the wall, where u = 0, to the pipe centreline. Laminar flow has a parabolic profile, while turbulent mixing makes the central part much flatter.

In turbulent flow, most of the rapid velocity change is concentrated in a thin layer next to the wall. The linked wall chart expands this region so the viscous, transitional and logarithmic parts of the profile can be seen together.

The Moody-chart connection

For fully developed pipe flow, the Darcy–Weisbach equation relates the head loss to the mean velocity:

\[h_f=f\frac{L}{D}\frac{V^2}{2g}.\]

Here f is the Darcy friction factor. The Moody chart shows how it depends on Reynolds number and relative roughness:

\[Re=\frac{VD}{\nu},\qquad\frac{k_s}{D}.\]

Changing velocity, diameter or roughness moves the Moody point and reshapes the linked velocity profile at the same time.

Why wall coordinates help

The friction factor determines the shear velocity u_*, which rescales velocity and wall distance:

\[u_*=V\sqrt{\frac{f}{8}},\qquad u^+=\frac{u}{u_*},\qquad y^+=\frac{yu_*}{\nu}.\]

The colored bands identify the viscous sublayer, buffer region, logarithmic region and outer flow. The log-law color blends gradually into the outer region, because that transition is not sharp. The fractional-logarithmic control changes only how these distances are spaced on screen; the morph curves show where each level lies in the physical pipe.

How roughness becomes hydraulic

The hatched band shows the same physical roughness height k_s in both velocity views. In wall coordinates it becomes

\[k_s^+=\frac{k_su_*}{\nu}.\]

A surface can therefore behave as hydraulically smooth at low velocity and as rough at higher velocity. The guide uses k_s⁺ < 5 for smooth turbulent flow, 5 ≤ k_s⁺ < 70 for transitionally rough flow, and k_s⁺ ≥ 70 for fully rough flow.