riaology
Unit 03 · Lesson 07 of 09

How do flow regimes differ?

Reynolds number compares inertia with viscosity.

Explore the model. Derive the result. Check its limits.

What you’ll be able to do
  • Form a dimensionless comparison.
  • Use regimes as qualified guides.
  • Laminar flow has an exact profile.
Before you touch the controls

Which changes favor turbulence: higher speed, larger diameter, or higher viscosity?

An experiment you can change

Change Reynolds number and the velocity profile

Predict the direction of change, move one control, then explain the response using the equation.

Pipe flow, Reynolds number 300. Profile is shown in the graph.Orderly laminar pathsChange Reynolds number and the velocity profile. Radius / pipe radius versus Speed / mean speed. Selected result 300 . Parabolic fully developed laminar flow has maximum speed twice the mean.0010.521Speed / mean speedRadius / pipe radius
Re=0.015(0.02)10−6=300\mathrm{Re}=\frac{0.015(0.02)}{10^{-6}}=300

Parabolic fully developed laminar flow has maximum speed twice the mean.

Model assumptions and units

Water ν = 10⁻⁶ m²/s in a circular pipe. Re < 2300 is typically laminar; 2300–4000 transitional; > 4000 typically turbulent. Thresholds depend on disturbances and entrance conditions.

Re=ρVDμ=VDν,u(r)V=2[1−(rR)2](laminar)\mathrm{Re}=\frac{\rho VD}{\mu}=\frac{VD}{\nu},\quad \frac{u(r)}V=2\left[1-\left(\frac rR\right)^2\right]\quad\text{(laminar)}
01

Form a dimensionless comparison.

For circular-pipe flow, Reynolds number uses mean speed and pipe diameter. Dynamic viscosity and density can be combined into kinematic viscosity. Consistent SI units make Reynolds number dimensionless.

Re=ρVDμ=VDν\mathrm{Re}=\frac{\rho VD}{\mu}=\frac{VD}{\nu}
02

Use regimes as qualified guides.

Below roughly 2300, flow is typically laminar. Above roughly 4000, it is typically turbulent. Between them, disturbances and entrance conditions matter. Reynolds number alone cannot supply a deterministic answer in transition.

03

Laminar flow has an exact profile.

For steady fully developed Newtonian flow in a straight circular pipe, the axial pressure gradient balances radial viscous stress. A finite centerline velocity and no-slip wall condition give a parabola.

u(r)=Δp4μL(R2−r2),umax=2Vu(r)=\frac{\Delta p}{4\mu L}(R^2-r^2),\quad u_{max}=2V
04

Turbulence changes the mean and fluctuations.

Turbulent flow has instantaneous fluctuations and a fuller mean profile. Statistically steady turbulence can have a time-independent mean while individual velocities vary. The power-law profile shown is illustrative, not a universal exact solution.

A worked example

A slow water flow.

  1. For V=0.015 m/sV=0.015\,\mathrm{m/s}, D=0.020 mD=0.020\,\mathrm m, and ν=10−6 m2/s\nu=10^{-6}\,\mathrm{m^2/s}, Re=300\mathrm{Re}=300.
  2. The fully developed flow is laminar under the stated conditions.
  3. Its maximum centerline speed is 2V=0.030 m/s2V=0.030\,\mathrm{m/s}. Raising speed tenfold puts Re at 3000, where the regime is transitional.
Make the idea your own

Calculate. Compare. Test the assumptions.

Work out your answer, check it, then explore the visual solution. The answer checker uses a 1.5% rounding tolerance; for model-limit questions it checks the displayed model result.

1 · Calculate

A laminar regime check

For water with ν = 10⁻⁶ m²/s, use mean speed 0.015 m/s and diameter 0.020 m. Find Reynolds number.

Hint

Use VD/ν with speed and diameter in SI units.

Explore the visual solution
Pipe flow, Reynolds number 300. Profile is shown in the graph.Orderly laminar pathsChange Reynolds number and the velocity profile. Radius / pipe radius versus Speed / mean speed. Selected result 300 . Parabolic fully developed laminar flow has maximum speed twice the mean.0010.521Speed / mean speedRadius / pipe radius
Re=0.015(0.02)10−6=300\mathrm{Re}=\frac{0.015(0.02)}{10^{-6}}=300

Parabolic fully developed laminar flow has maximum speed twice the mean.

Model assumptions and units

Water ν = 10⁻⁶ m²/s in a circular pipe. Re < 2300 is typically laminar; 2300–4000 transitional; > 4000 typically turbulent. Thresholds depend on disturbances and entrance conditions.

Re=ρVDμ=VDν,u(r)V=2[1−(rR)2](laminar)\mathrm{Re}=\frac{\rho VD}{\mu}=\frac{VD}{\nu},\quad \frac{u(r)}V=2\left[1-\left(\frac rR\right)^2\right]\quad\text{(laminar)}

Controls explore variations. The steps below retain the original question’s values.

Water ν = 10⁻⁶ m²/s in a circular pipe. Re < 2300 is typically laminar; 2300–4000 transitional; > 4000 typically turbulent. Thresholds depend on disturbances and entrance conditions.

Re=ρVDμ=VDν,u(r)V=2[1−(rR)2](laminar)\mathrm{Re}=\frac{\rho VD}{\mu}=\frac{VD}{\nu},\quad \frac{u(r)}V=2\left[1-\left(\frac rR\right)^2\right]\quad\text{(laminar)}
2 · Compare

Into transition

At diameter 0.020 m, increase speed to 0.15 m/s. Find Reynolds number and decide whether a unique regime follows.

Hint

The usual transition interval is approximately 2300 to 4000.

Explore the visual solution
Pipe flow, Reynolds number 3000. Profile is shown in the graph.Transition: no unique predictionChange Reynolds number and the velocity profile. Radius / pipe radius versus Speed / mean speed. Selected result 3,000 . Transition: regime cannot be decided by Reynolds number alone. The displayed turbulent-like profile is only an illustrative comparison.000.6120.51.2241Speed / mean speedRadius / pipe radius
Re=0.15(0.02)10−6=3000\mathrm{Re}=\frac{0.15(0.02)}{10^{-6}}=3000

Transition: regime cannot be decided by Reynolds number alone. The displayed turbulent-like profile is only an illustrative comparison.

Model assumptions and units

Water ν = 10⁻⁶ m²/s in a circular pipe. Re < 2300 is typically laminar; 2300–4000 transitional; > 4000 typically turbulent. Thresholds depend on disturbances and entrance conditions.

Re=ρVDμ=VDν,u(r)V=2[1−(rR)2](laminar)\mathrm{Re}=\frac{\rho VD}{\mu}=\frac{VD}{\nu},\quad \frac{u(r)}V=2\left[1-\left(\frac rR\right)^2\right]\quad\text{(laminar)}

Controls explore variations. The steps below retain the original question’s values.

Water ν = 10⁻⁶ m²/s in a circular pipe. Re < 2300 is typically laminar; 2300–4000 transitional; > 4000 typically turbulent. Thresholds depend on disturbances and entrance conditions.

Re=ρVDμ=VDν,u(r)V=2[1−(rR)2](laminar)\mathrm{Re}=\frac{\rho VD}{\mu}=\frac{VD}{\nu},\quad \frac{u(r)}V=2\left[1-\left(\frac rR\right)^2\right]\quad\text{(laminar)}
3 · Challenge the model

A turbulent mean

At speed 0.20 m/s and diameter 0.050 m, find Reynolds number. Explain whether a steady mean excludes instantaneous fluctuations.

Hint

Distinguish a time-averaged velocity field from instantaneous velocity.

Explore the visual solution
Pipe flow, Reynolds number 10000.000000000002. Profile is shown in the graph.Fluctuating turbulent paths (illustration)Change Reynolds number and the velocity profile. Radius / pipe radius versus Speed / mean speed. Selected result 10,000 . Displayed normalized one-seventh-power turbulent profile is illustrative; real profiles depend on Reynolds number and roughness.000.6120.51.2241Speed / mean speedRadius / pipe radius
Re=0.2(0.05)10−6=10000\mathrm{Re}=\frac{0.2(0.05)}{10^{-6}}=10000

Displayed normalized one-seventh-power turbulent profile is illustrative; real profiles depend on Reynolds number and roughness.

Model assumptions and units

Water ν = 10⁻⁶ m²/s in a circular pipe. Re < 2300 is typically laminar; 2300–4000 transitional; > 4000 typically turbulent. Thresholds depend on disturbances and entrance conditions.

Re=ρVDμ=VDν,u(r)V=2[1−(rR)2](laminar)\mathrm{Re}=\frac{\rho VD}{\mu}=\frac{VD}{\nu},\quad \frac{u(r)}V=2\left[1-\left(\frac rR\right)^2\right]\quad\text{(laminar)}

Controls explore variations. The steps below retain the original question’s values.

Water ν = 10⁻⁶ m²/s in a circular pipe. Re < 2300 is typically laminar; 2300–4000 transitional; > 4000 typically turbulent. Thresholds depend on disturbances and entrance conditions.

Re=ρVDμ=VDν,u(r)V=2[1−(rR)2](laminar)\mathrm{Re}=\frac{\rho VD}{\mu}=\frac{VD}{\nu},\quad \frac{u(r)}V=2\left[1-\left(\frac rR\right)^2\right]\quad\text{(laminar)}
Your learning path

Ready to carry this forward?

Mark this lesson complete when you can explain the main idea and work through its practice. Completion is your choice.

PreviousNext lesson

Completion is saved on this browser.