Reynolds Number — From Laminar Flow to Turbulence

Reynolds Number — From Laminar Flow to Turbulence

Why does water flow smoothly through a narrow straw but chaotically through a garden hose?
Why is blood flow laminar in capillaries but turbulent in the aorta?
The answer lies in a single dimensionless number: the Reynolds number.

1. Osborne Reynolds and the 1883 Experiment

In 1883, Irish-English physicist Osborne Reynolds performed a beautifully simple experiment.
He injected dye through a thin needle into a pipe carrying water at various flow speeds.

  • At low speed: the dye formed a thin, straight thread — laminar flow
  • At high speed: the dye dispersed chaotically — turbulent flow

Reynolds discovered that the transition between these regimes is governed by a simple combination of
physical parameters — today called the Reynolds number.

“The nature of the motion of fluid in a pipe depends upon the ratio of the inertial to the viscous forces acting on the fluid.”

— Osborne Reynolds, 1883

2. Definition and Formula

The Reynolds number is a dimensionless quantity defined as:

$$\boxed{\text{Re} = \frac{\rho\, v\, L}{\mu} = \frac{v\, L}{\nu}}$$

Parameters:
• ρ = fluid density [kg/m³] — water ≈ 1000, air ≈ 1.2
• v = characteristic velocity [m/s] — usually mean flow speed
• L = characteristic length [m] — pipe diameter, body diameter, etc.
• μ = dynamic viscosity [Pa·s] — resistance to shearing
• ν = μ/ρ = kinematic viscosity [m²/s] — water (20°C) ≈ 1×10⁻⁶, air ≈ 1.5×10⁻⁵

2.1 Characteristic Length by Geometry

Geometry Characteristic Length L Definition
Circular pipe D = inner diameter D = 2r
Rectangular channel Dh = hydraulic diameter 4A/P
Sphere or cylinder D = body diameter —
Flat plate x = distance from leading edge variable
Airfoil c = chord length —

3. Physical Meaning — Ratio of Forces

Re is the ratio of inertial forces (which drive fluid motion) to viscous forces (which resist it):

$$\text{Re} = \frac{F_{\text{inertia}}}{F_{\text{viscous}}} \approx \frac{\rho v^2 L^2}{\mu v L} = \frac{\rho v L}{\mu}$$

Order-of-magnitude derivation:
• Inertial force (Newton’s 2nd law): \(F_i \sim \rho a \cdot \text{Vol} \sim \rho \frac{v^2}{L} \cdot L^3 = \rho v^2 L^2\)
• Viscous force (Newton’s viscosity law): \(F_v \sim \mu \frac{dv}{dy} \cdot A \sim \mu \frac{v}{L} \cdot L^2 = \mu v L\)
• Ratio: \(\dfrac{F_i}{F_v} = \dfrac{\rho v^2 L^2}{\mu v L} = \dfrac{\rho v L}{\mu} = \text{Re}\)
Re ≪ 1: Viscous forces dominate → smooth, laminar, predictable flow
Re ≫ 1: Inertial forces dominate → instability, eddies, turbulence

4. Dimensional Analysis

Reynolds number arises naturally from Buckingham’s π theorem. With four governing parameters (ρ, v, L, μ)
and three fundamental dimensions (M, L, T), there is exactly one independent dimensionless group:

$$\Pi = \rho^a \, v^b \, L^c \, \mu^d$$

Solving the dimensional equations gives a = 1, b = 1, c = 1, d = −1 → exactly Re.
This proves Re is the unique dimensionless group for viscous flow with these parameters.

5. Flow Regimes in Pipes

Regime Re Range Characteristics
Laminar Re < 2300 Parallel streamlines, parabolic profile, low friction losses, predictable
Transition 2300 – 4000 Unstable, alternates between laminar and turbulent bursts
Turbulent Re > 4000 Random eddies, intense mixing, flatter profile, higher losses

5.1 Velocity Profiles

Laminar flow in a circular pipe (Hagen-Poiseuille):

$$v(r) = v_{\max}\left(1 – \frac{r^2}{R^2}\right), \qquad v_{\max} = 2\bar{v}$$

Turbulent flow follows the power law:

$$\frac{v}{v_{\max}} \approx \left(\frac{R-r}{R}\right)^{1/n}, \quad n \approx 7 \text{ at Re} \approx 10^5$$

⚠️ Warning: Recr = 2300 applies to a circular pipe with a quiet inlet. Under carefully controlled laboratory conditions, laminar flow has been maintained up to Re = 100,000!

6. Critical Reynolds Number by Geometry

System Recr Notes
Circular pipe 2300 Transition from parabolic to turbulent profile
Flat plate (BL) 5×10⁵ Boundary-layer transition
Sphere ~1 Onset of flow separation
Sphere (drag crisis) ~2×10⁵ Sudden CD drop — turbulent BL attachment
Parallel plates ~1000 Channel flow between infinite plates
Cylinder in crossflow ~40 Onset of von Kármán vortex street

7. Worked Examples

Example 1: Blood in the Aorta

Data: D = 2.5 cm, v = 0.25 m/s, ρ = 1060 kg/m³, μ = 4×10⁻³ Pa·s

$$\text{Re} = \frac{1060 \times 0.25 \times 0.025}{4 \times 10^{-3}} \approx 1656 \quad \text{(laminar ✓ at rest)}$$
During intense exercise (v ≈ 1 m/s): Re ≈ 6625 → turbulent.
The heart is most efficient when aortic flow is laminar.

Example 2: Air in Funnel Stem (IYPT P7)

Data: rs = 1 mm → D = 2 mm, vavg ≈ 2.7 m/s, ρ = 1.2 kg/m³, μ = 1.8×10⁻⁵ Pa·s

$$\text{Re} = \frac{1.2 \times 2.7 \times 0.002}{1.8 \times 10^{-5}} \approx 360 \quad \text{(laminar ✓ — Hagen-Poiseuille valid)}$$
For rs = 3 mm, vavg ≈ 15 m/s → Re ≈ 6000 → turbulent — H-P breaks down.

Example 3: Tennis Ball in Air

Data: D = 6.7 cm, v = 50 m/s, ν = 1.5×10⁻⁵ m²/s

$$\text{Re} = \frac{50 \times 0.067}{1.5 \times 10^{-5}} \approx 2.2 \times 10^5$$
Near the drag crisis! At this Re, the boundary layer transitions from laminar to turbulent,
CD drops sharply — this is why tennis balls feel “heavy” in slow serves.

Example 4: Bacteria Swimming

Data: L = 1 μm, v = 30 μm/s, ν = 1×10⁻⁶ m²/s (water)

$$\text{Re} = \frac{30 \times 10^{-6} \times 10^{-6}}{10^{-6}} = 3 \times 10^{-5}$$
Bacteria live in a world where inertia is essentially zero. If a bacterium stops swimming,
it stops within a distance of ~0.1 Å — less than an atomic diameter. Purcell called this
“Life at Low Reynolds Number” (1977).

8. Navier-Stokes Equations and Re

Re appears naturally when we non-dimensionalize the Navier-Stokes equations:

$$\rho\left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v}\cdot\nabla\mathbf{v}\right) = -\nabla p + \mu\,\nabla^2 \mathbf{v}$$

Using \(\mathbf{v}^* = \mathbf{v}/v_0\), \(\mathbf{x}^* = \mathbf{x}/L\), \(p^* = p/(\rho v_0^2)\):

$$\frac{\partial \mathbf{v}^*}{\partial t^*} + \mathbf{v}^*\cdot\nabla^*\mathbf{v}^* = -\nabla^* p^* + \frac{1}{\text{Re}}\,\nabla^{*2} \mathbf{v}^*$$

The viscous term has coefficient exactly 1/Re. Large Re → tiny viscous term → flow approaches inviscid (Euler) behavior → instability and turbulence.

9. Dynamic Similarity

Two flows with the same Re have geometrically similar streamline patterns — even at vastly different scales.
This is the foundation of wind-tunnel testing and ship model experiments.

Wind-tunnel test: Full-size airplane wing (chord c = 3 m, v = 250 m/s, air) → Re = 5×10⁷.
Scale model 1/10 (c = 0.3 m) in same air at same v: Re = 5×10⁶ — wrong! Must either compress air
(higher ρ, lower ν) or use a different fluid to match Re.

10. Drag Coefficient vs. Re

$$F_D = \frac{1}{2}\,C_D(\text{Re})\,\rho\,v^2\,A$$

Regime Re CD (sphere) Relation
Stokes (creeping) Re ≪ 1 24/Re CD ∝ 1/Re
Transitional 1 – 1000 0.4 – 24/Re complex
Turbulent (Newton) 10³ – 2×10⁵ ≈ 0.44 roughly constant
Drag crisis ~2×10⁵ drops to ~0.1 BL turbulent attachment

11. Re in Biological Systems

System Re Regime
E. coli bacterium ~3×10⁻⁵ Stokes — inertia-free world
Blood in capillaries ~0.008 Deep laminar — diffusion-based O₂ exchange
Blood in aorta (rest) ~1600 Laminar
Blood in aorta (exercise) ~6000 Turbulent
Hummingbird wings ~2000 Near-critical
Tuna swimming ~10⁷ Fully turbulent — exploits eddies

12. Key Takeaways

  1. Re = ρvL/μ = vL/ν — dimensionless ratio of inertial to viscous forces
  2. Low Re → laminar (smooth, predictable, parabolic profile)
  3. High Re → turbulent (chaotic, mixing, flatter profile)
  4. Critical value depends on geometry (pipe: ~2300)
  5. Dynamic similarity: same Re → same flow pattern at any scale
  6. In dimensionless N-S equations, Re appears as the coefficient of the viscous term (1/Re)
  7. Biological flows span 10⁻⁵ (bacteria) to 10⁷ (large fish)

Interactive Reynolds Number Simulator

Adjust parameters and click Simulate to generate plots.

Fluid presets:












References

  1. Reynolds, O. (1883). "An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels." Philosophical Transactions of the Royal Society of London, 174, 935–982.
  2. White, F.M. (2016). Fluid Mechanics, 8th ed. McGraw-Hill. Chapter 6: Viscous Flow in Ducts.
  3. Munson, B.R., Okiishi, T.H., Huebsch, W.W., & Rothmayer, A.P. (2013). Fundamentals of Fluid Mechanics, 7th ed. Wiley. Chapter 8.
  4. Landau, L.D. & Lifshitz, E.M. (1987). Fluid Mechanics, 2nd ed. Pergamon Press. §27: Turbulent Flow.
  5. Tritton, D.J. (1988). Physical Fluid Dynamics, 2nd ed. Oxford University Press.
  6. Purcell, E.M. (1977). "Life at low Reynolds number." American Journal of Physics, 45(1), 3–11.
  7. Cengel, Y.A. & Cimbala, J.M. (2018). Fluid Mechanics: Fundamentals and Applications, 4th ed. McGraw-Hill.
  8. Denn, M.M. (1980). Process Fluid Mechanics. Prentice-Hall.
physicsme.ir · Fluid Mechanics · Reynolds Number
Prepared for advanced high-school students, first-year university physics, and IYPT teams.

Have a question? 🤔

If something isn't clear or you have a question, ask it here. The answer will be published on this page.

💬 جواب بهتری داری؟ یا یه سؤال جدید؟

اگه به سؤالای بالا پاسخی داری که فکر می‌کنی روشن‌تر یا کامل‌تر از مال منه، یا یه سؤال جدید برای دانش‌آموزای دیگه داری — تو بخش نظرات پایین صفحه ارسال کن. هر پیامی رو می‌خونم، تأیید می‌کنم و منتشر می‌شه. این‌جوری همه از تجربه‌ی همدیگه استفاده می‌کنیم. 🌱

Tagged , , , , , ,
در حال آپلود فایل...
لطفاً صبر کنید — صفحه را نبندید
۰٪