**Related Resources: fluid flow**

### Fluid Velocity Distribution within Pipes

Fluid Velocity Distribution within Pipes Equations

Velocity distribution at a cross section will follow a parabolic law of variation for laminar flow. Maximum velocity is at the center and is twice the average velocity. The following application are discussed:

- turbulent flows,
- smooth pipes,
- rough pipes,
- rough or smooth boundaries

The equation of the velocity profile for laminar flow can be expressed as

Eq. 1

$v={v}_{c}-\left(\frac{\gamma {h}_{L}}{4\mu L}\right){r}^{2}$

For turbulent flows, more uniform velocity distribution results. From experiments of Nikuradse and others, equations of velocity profiles in terms of center velocity vc or shear velocity µ_{c} follow.

(a) An empirical formula is

Eq. 2

$v={v}_{c}{\left(y/{r}_{o}\right)}^{n}$

where:

n = 1/7 for smooth tubes, up to Re = 100,000

n = 1/8 | for smooth tubes for Re from 100,000 to 400,000

(b) For smooth pipes,

Eq. 3

$v={v}_{\ast}[5.5+5.75\mathrm{log}(y{v}_{\ast}/v)]$

For the yv_{*}/v term,

Eq. 4

v_{*} = (τ/ρ)^{0.5}

(c) For smooth pipes (for 5000 < Re < 3,000,000) and for pipes in the wholly rough zone,

Eq. 5

$({v}_{c}-v)=-2.5\sqrt{{v}_{o}/\rho}\text{}\mathrm{ln}(y/{r}_{o})=-2.5{v}_{\ast}\text{}\mathrm{ln}(y/{r}_{o})$

In terms of average velocity V, Vennard suggests that V/v_{c} may be written

Eq. 6

$\frac{V}{{v}_{c}}=\frac{1}{1+4.07\sqrt{f/8}}$

(d) For rough pipes,

Eq. 7

$v={v}_{\ast}[8.5+5.75\text{}\mathrm{log}(y/\u03f5)]$

where ε is the absolute roughness of the boundary.

(e) For rough or smooth boundaries,

Eq. 8

$\frac{u-V}{V\sqrt{f}}=2\text{}\mathrm{log}\frac{y}{{r}_{o}}+1.32$

also

Eq. 9

${v}_{c}/V=1.43\sqrt{f}+1.32$

Where:

γ = specific (or unit) weight of fluid

h_{L} = lost head

µ = absolute viscosity

L = Length of pipe

*v* = kinematic viscosity of the fluid in ft^{2}/sec or m^{2}/s

*v _{c}* = Center velocity of fluid

v

_{*}= Shear velocity

*r*= radius, ft ot m

*r*= radius of pipe

_{o}y = depth distance

V = mean (average) velocity in ft/sec or m/s

*f*= friction factor (Darcy) for pipe flow

µ = absolute viscosity in lb-sec/ft

^{2}or N-s/m

^{2}

Schaum's Outline of Fluid Mechanics and Hydraulics

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