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

${\displaystyle v=v_c-<!--\; -\; -->\left(\frac{\mathrm{\gamma <!--\; ?\; -->}{h}_L}{4\mathrm{\mu <!--\; \mu \; -->}L}\right)r^2}$

 

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

${\displaystyle v=v_c{\left(y/r_o\right)}^n}$

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

${\displaystyle v=v_{\ast <!--\; *\; -->}[5.5+5.75\log <!--\; ?\; -->(y{v}_{\ast <!--\; *\; -->}/v)]}$

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

${\displaystyle (v_c-<!--\; -\; -->v)=-<!--\; -\; -->2.5\sqrt{v_o/\mathrm{\rho <!--\; ?\; -->}}\ln <!--\; ?\; -->(y/r_o)=-<!--\; -\; -->2.5v_{\ast <!--\; *\; -->}\ln <!--\; ?\; -->(y/r_o)}$

In terms of average velocity V, Vennard suggests that V/v_c may be written

 

Eq. 6

${\displaystyle \frac{V}{v_c}=\frac{1}{1+4.07\sqrt{f/8}}}$

(d) For rough pipes,

 

Eq. 7

${\displaystyle v=v_{\ast <!--\; *\; -->}[8.5+5.75\log <!--\; ?\; -->(y/\mathrm{ϵ<!--\; ?\; -->})]}$

where ε is the absolute roughness of the boundary.

 

(e) For rough or smooth boundaries,

 

Eq. 8

${\displaystyle \frac{u-<!--\; -\; -->V}{V\sqrt{f}}=2\log <!--\; ?\; -->\frac{y}{r_o}+1.32}$

also

 

Eq. 9

${\displaystyle v_c/V=1.43\sqrt{f}+1.32}$

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 ft²/sec or m²/s
v_c = Center velocity of fluid
v_* = Shear velocity
r = radius, ft ot m
r_o = radius of pipe
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/ft² or N-s/m²

Schaum's Outline of Fluid Mechanics and Hydraulics

 

Related

 

• Convergent Nozzle Flow Velocity and Area Equation and Calculator
• Convergent-Divergent Nozzle Flow Velocity and Area Equation and Calculator
• Viscosity of Fluid with Falling Sphere at Terminal Velocity Equation and Calculator Measurement of Viscosity in a Vertical Falling Ball Viscometer Calculator
• Pitot-Static Tube Velocity of Fluid Flow Measurement Calculator and Equation
• Velocity of Water Due To Head in Feet Table
• Velocity of Escaping Compressed Air Equation and Calculator
• Total Head Pressure Velocity Equations and Calculator
• Pipe Whip If a pipe were to rupture, the reaction force created by the high velocity fluid jet could cause the piping to displace and cause extensive damage
• Fluid Flow Open Channel Calculator Chezy and Manning's equation to determine the wetted perimeter, hydraulic radius, flow area, Chezy coefficient and flow velocity
• Velocity and Flow Capacity of Oil Standard Tubes Velocity and Flow Capacity of Oil Standard Tubes
• Velocity and Flow Capacity of Oil Schedule 40 Pipe Velocity and Flow Capacity of Oil Schedule 40 Pipe
• Velocity and Flow Capacity of Oil Schedule 80 Pipe Velocity and Flow Capacity of Oil Schedule 80 Pipe
• Velocity and Flow Capacity of Oil Schedule 160 Pipe Velocity and Flow Capacity of Oil Schedule 160 Pipe
• Flow Velocity Profiles Not all fluid particles travel at the same velocity within a pipe.
• Average (Bulk) Velocity Ia single average velocity to represent the velocity of all fluid at that point in the pipe.
• Liquid in Horizontal and Vertical Motion Equations and Calculator
• Compressibility of a Fluid Equations and Calculator
• Compressibility of Liquids Equations and Calculators
• Flow Through a Sluice Gate Formula and Calculator
• Flow over Channel Rise Hump Formula
• Fluid Flow Discharge Time under Falling Head Formula and Calculator
• Fluid Kinetic Force in Conduits Calculator and Equation