Reynolds Number Simulator

What Is the Reynolds Number Simulator?

How to use the Reynolds number simulator

1. Choose the fluid. The four Fluid buttons, Water, Air, Glycerine and SAE 30 oil, each carry their kinematic viscosity in centistokes, and the line beneath them prints the density and dynamic viscosity the lab will use. Water at 20 °C is the default; the oil's values are for 40 °C.
2. Set the speed. Drag Mean velocity v between 0.010 m/s and 10.0 m/s. The slider is logarithmic, so each equal move multiplies the speed by the same factor, and the figure beside it always states the value the readouts are computed from.
3. Set the bore. Drag Pipe diameter D from 1.00 mm to 500 mm, also on a log scale. Re grows in proportion to both sliders, so doubling either one doubles the reading.
4. Read the number and the regime. Reynolds number is the main readout, and Flow regime beneath it prints Laminar, Transitional or Turbulent with the threshold sentence that applies. The dye streak in the picture changes shape with it: a straight filament, a wavering line, or a band of eddies filling the pipe.
5. Read what follows from it. Kinematic viscosity is the fluid's mu/rho; Darcy friction factor names the formula it used; Pressure drop per metre applies Darcy-Weisbach; Entrance length says how far from an inlet the profile takes to settle; and Velocity profile prints the centreline-to-mean ratio drawn as the gold curve on the right. In the transition band the last four of those print n/a; the kinematic viscosity never does.
6. Pause and start again. Pause flow freezes the dye where it is and Play flow resumes it; the readouts update either way. Reset restores water at 1.00 m/s in 25.0 mm with the flow running, and the six preset flows above are the quicker route to a specific real case.

Worked example: change one thing at a time

Begin from the Reset state and move one control per step. Each cell quotes what the running lab printed at those control positions (the fluid column is its button label), so the lab, not this table, is the authority if they ever differ.

Rows 1 to 3 leave the fluid on Water and the bore at 25.0 mm; only the velocity slider moves. Row 1 to row 2: slowing the flow twentyfold, from 1.00 m/s to 0.050 m/s, takes Re from 24,905 to 1,245 and the label from Turbulent to Laminar. The dye collapses into a single thread, the friction factor doubles from 0.02515 to 0.05139, yet the pressure drop falls from 502.2 Pa/m to 2.57 Pa/m, and the entrance length stretches from 0.594 m to 1.868 m because a laminar profile settles slowly.

Row 2 to row 3: doubling the speed to 0.100 m/s doubles the reading to 2,491 and lands it in the transition band. The label turns Transitional, the filament begins to snake down the pipe, and the friction factor, pressure drop, entrance length and profile ratio all print n/a, because the lab refuses to apply either formula where neither is reliable.

Row 3 to row 4: the sliders stay put and only the fluid changes. Glycerine is a quarter denser than water but about 1,400 times more viscous, so the same 0.100 m/s in the same bore reads 2.23, deep in laminar flow, with a friction factor of 28.67 and a pressure drop of 7.23 kPa/m. The entrance length shrinks to 3.3 mm: the profile is parabolic almost from the inlet.

Rows 5 and 6 are two turbulent flows from opposite ends of the lab. The air row, 5.00 m/s through a 100 mm bore, reads 32,986 with a friction factor of 0.02345, yet the pressure drop is a mere 3.53 Pa/m because there is so little mass in the moving gas. Water at 2.00 m/s in a 500 mm main reaches 996,208, where the friction factor readout switches from the Blasius fit to the Haaland equation and falls to 0.01159, while the entrance length grows to 21.99 m, some forty-four diameters of pipe.

Two more positions are worth finding. Slide water down to 0.092 m/s and the lab reads 2,291, still Laminar; one step more, 0.093 m/s, reads 2,316 and Transitional, so the 2,300 threshold sits between two adjacent slider steps in a 25.0 mm pipe. At the top of the band, 0.160 m/s reads 3,985 and Transitional while 0.161 m/s reads 4,010 and Turbulent, with the friction factor reappearing at 0.03971.

Formula and symbol reference

Every readout descends from the definition Re = ρ·v·D/μ, or Re = v·D/ν with ν = μ/ρ. Three regime-dependent relations do the rest: the friction factor for a hydraulically smooth wall (f = 64/Re when laminar, the Blasius fit f = 0.316·Re^(-0.25) as far as 100,000 and the Haaland equation beyond it), the Darcy-Weisbach gradient dp/L = f·ρ·v²/(2·D), and the entrance length. The ranges marked “in this lab” are the simulator's own displayed values at the control positions named.

The physics: inertia against viscosity

The number compares the stress that carries a disturbance along, of order ρ·v², with the viscous stress that smooths it out, of order μ·v/D; the derivation, and the 1883 experiment the picture re-enacts, are told in full in the article linked below, and this page keeps to what the sliders and readouts show. At small Re the dye stays a thread because every wobble is damped before it has travelled its own length; at large Re the wobbles are stretched by the mean flow faster than viscosity can drain them, and the streak becomes a band of eddies.

Because only the ratio matters, the individual values do not. Water at 2.00 m/s in 25.0 mm and water at 1.00 m/s in 50.0 mm are different pipes with different speeds, yet the lab reads 49,810 for both and draws the same profile with the same 0.02115 friction factor. That is dynamic similarity, and it is the whole basis of testing a model in a wind or water tunnel: match Re and you have matched the flow pattern.

The lab's two thresholds are conventions, and the physics underneath them is softer than the labels suggest. Turbulent puffs in a circular pipe stop persisting somewhere near 2,000, whatever disturbed the flow; the 2,300 the lab switches at is the design value most textbooks adopt, with a margin above that point, and in the gap a disturbed flow alternates between laminar stretches and bursts of turbulence. Above about 4,000 practically any pipe is turbulent throughout. The lab marks its band by wavering the dye and declining to quote a friction factor; verify the exact figures against the textbook you are working from.

Its readouts bracket the band cleanly. With water in the 25.0 mm bore, 0.092 m/s reads 2,291 and Laminar, 0.093 m/s reads 2,316 and Transitional, 0.160 m/s reads 3,985 and Transitional, and 0.161 m/s reads 4,010 and Turbulent, so each threshold falls between two neighbouring slider steps.

The regime also fixes the shape of the velocity profile, the gold curve on the right of the picture. In laminar flow viscosity alone sets the profile and the answer is Poiseuille's parabola, whose peak on the centreline is exactly twice the mean, which the lab prints as 2.00. In turbulent flow the eddies carry fast fluid outwards and slow fluid inwards, flattening the middle and squeezing the whole velocity change into a thin layer at the wall; the classic 1/7-power fit puts the centreline at 1.22 times the mean.

The steep gradient in that wall layer is what the friction factor readout is measuring. In laminar flow f = 64/Re, so the pressure drop is proportional to the speed: water at 0.025 m/s reads 1.28 Pa/m and 0.050 m/s reads 2.57 Pa/m. In turbulent flow the Blasius fit gives a drop that grows roughly as the speed to the power 1.75: 1.00 m/s reads 502.2 Pa/m and 2.00 m/s reads 1.69 kPa/m, more than three times as much for twice the speed.

Above 100,000 the Blasius fit drifts and the lab switches to the Haaland smooth-pipe equation, which you can watch happen with water at 1.00 m/s: a 100 mm bore reads 99,621 with the friction factor labelled Blasius, and a 500 mm bore reads 498,104 with it labelled Haaland. The two fits agree to about 0.3 % at the hand-over, so the readout does not jump.

Finally, all of this describes fully developed flow, and a real pipe needs a run of length before the profile has formed. The lab estimates it as 0.06·Re·D for laminar flow and 4.4·Re^(1/6)·D for turbulent flow. Laminar flow is the slow settler: glycerine at 5.00 m/s in the 500 mm bore is still laminar at 2,233 and needs 66.98 m of straight pipe, while the turbulent water main at 996,208 needs 21.99 m and the default flow only 0.594 m.

Where the simple pipe model breaks down

The lab shows an idealised case: a straight, smooth, circular pipe, a Newtonian fluid at a fixed temperature, and flow that has already settled. Move away from that picture and the readouts stop meaning quite what they say; here is where, and what you would notice.

Rough walls

The lab has no roughness control, so its friction factor always describes glass-smooth walls. In a real turbulent line of steel, cast iron or concrete the wall texture pushes f above the Blasius or Haaland figure, by more the higher Re climbs. Carry the Re readout into the friction factor calculator, where the Colebrook equation adds the roughness the sim leaves out.

Symptom: a measured pressure drop in an old pipe that sits well above the lab's Pa/m figure at the same Re.

Ducts that are not circular

The D slider means a round bore. A rectangular duct has no single diameter, and the usual stand-in, four times the cross-section over the wetted perimeter, shifts the thresholds by an amount the lab cannot show; a result that lands near either threshold should be checked against measured duct data rather than the lab's label.

The transition band depends on the inlet

The lab's Transitional label begins at 2,300 whatever you do, because the sim has no inlet to adjust; a real apparatus does. With a still tank and a rounded entry Reynolds himself watched the filament survive to roughly 13,000 (verify before use), while a vibrating bench or a sharp bend upstream can break it up a little below 2,300, down towards the 2,000 where puffs stop persisting. Read the band the lab draws as a warning zone rather than a prediction.

Non-Newtonian fluids

Water, air, glycerine and the oil are all Newtonian, so one viscosity figure describes each of them at its temperature. Ketchup, drilling mud, wet paint or blood squeezing through a capillary thins or thickens with the rate of shear, which leaves no single μ to put in the denominator; the engineer's workaround, an apparent viscosity at the working shear rate, is not something the buttons here can offer.

Temperature

The four buttons pin the temperature: 20 °C for water, air and glycerine, 40 °C for the oil, and nothing in the lab warms or cools them. Yet viscosity moves with temperature faster than any other input, warm water thinning and cold oil stiffening, so a real line at another temperature has a different Re from the one shown; the calculator takes any viscosity you have.

Fast gases

The air button treats the gas as incompressible, which is fine at the 10.0 m/s top of the slider but not for a gas approaching a substantial fraction of the speed of sound. Beyond roughly a third of Mach 1 the gas compresses along the pipe, its density stops being the single fixed figure on the fluid line, and Re on its own no longer describes the flow.

The lab's own limits

The drawn pipe is only about one diameter long, so the turbulent dye reaches the walls sooner in the picture than the few diameters a real Reynolds apparatus needs; the streak is an illustration of the regime, not a scale drawing. The sliders move in thousandths of a decade and both readings are rounded to three significant figures, which is exactly the value the physics uses, so the printed setting and the readouts always agree. Both the transitional profile and the dye's waver are drawn schematically; the lab computes no stability theory.

Where the Reynolds number is actually used

Pipeline and pump sizing

A pump is sized against the pressure the pipe will eat per metre, and the lab prints that figure the moment Re has chosen the friction law. The whole chain is on screen: the Domestic water pipe preset, 1.00 m/s in 15.0 mm, reads 14,943 and a pressure drop of 951.0 Pa/m, while the 500 mm Water main at 2.00 m/s loses only 46.29 Pa/m despite moving vastly more water.

Ventilation and air-conditioning ducts

Air's density is tiny, but so is its viscosity, and the kinematic viscosity the lab prints, 1.516e-5 m²/s against water's 1.004e-6 m²/s, is only about fifteen to one, so duct flows are turbulent almost by default. Press Air and leave the sliders at their defaults, 1.00 m/s in 25.0 mm, and the lab already reads 1,649; the Ventilation duct preset, four times the bore and five times the speed, reads 32,986. The practical question in a duct is therefore rarely whether the flow is turbulent, only how much pressure the fan must supply, and the Pa/m readout answers it directly.

Lubrication and oil lines

Press the SAE 30 oil button and the lab rarely leaves the laminar regime; the Oil lubrication line preset reads 263, with a pressure drop of 532.5 Pa/m for 0.500 m/s through a 50.0 mm bore, the kind of figure a hydraulic line is sized against. Because the friction factor there is 64/Re, the drop scales with the speed itself rather than its square: nudge the velocity slider and watch the Pa/m readout move in step with it.

Blood flow

The aorta is roughly the size of the lab's default 25.0 mm bore, and at the peak of each heartbeat the blood in it runs at a Reynolds number in the low thousands, near the band the lab marks Transitional; where a diseased valve narrows the passage the local flow goes turbulent, and that turbulence is what a stethoscope hears as a murmur. There is no blood button, and blood's kinematic viscosity is roughly three to four times water's, so use the lab for the shape of the argument and verify physiological figures against a medical source before relying on them.

Aircraft and model testing

An airliner wing's chord Reynolds number at cruise is of the order of ten million, a figure to verify for any particular aircraft; the lab tops out at 4,981,038 with water at 10.0 m/s in 500 mm, so its largest pipe flow and a wing share the same thin-wall-layer physics. A tunnel model answers for the full-size wing only if its Re matches, and the similarity demonstration above, 49,810 at two different speeds and bores, is that requirement in its simplest form.

Microfluidics

Shrink the bore far enough and inertia stops mattering altogether. The lab's narrowest tube is 1.00 mm, and glycerine creeping through it at 0.010 m/s reads 0.00893 with a filament that could not wobble if it tried; a lab-on-a-chip channel a few tens of micrometres wide sits in the same regime with plain water, which is why two liquids fed into such a chip flow along it side by side and blend only by molecular diffusion.

Where to go next

When you have your own fluid rather than one of four buttons, the Reynolds number calculator accepts any density and viscosity, prints the substitution with units, and can be run backwards to find the speed or bore that lands on a chosen Re. The article Reynolds Number: Formula and Meaning tells the story behind the dye, derives the ratio and works seven problems, while Bernoulli's principle covers the loss-free relation between pressure and speed that pipe friction eats into. Carry the result into the friction factor calculator for a rough pipe, watch the property in the denominator behave on its own in the viscosity simulator, or browse the whole library of physics simulations.