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Инженерия / Гидравлика

Reynolds number

Reynolds number indicates the ratio of inertial to viscous forces in a fluid and is used to determine flow regime.

Опубликовано: Обновлено:

Гидравлика

Формула

$$Re = \frac{\rho v D_h}{\mu} = \frac{v D_h}{\nu}$$
chart Reynolds regime map

Dimensionless Reynolds value versus flow conditions, showing laminar and turbulent bands.

A practical divider used for friction model selection.

Обозначения

$Re$
Reynolds number (dimensionless), -
$\rho$
Fluid density, kg/m^3
$v$
Average velocity, m/s
$D_h$
Hydraulic diameter, m
$\mu$
Dynamic viscosity, Pa·s
$\nu$
Kinematic viscosity, m^2/s

Условия применения

  • Single-phase flow in a duct or equivalent hydraulic conduit.
  • Characteristic length is chosen consistently (hydraulic diameter for non-circular ducts).
  • Material properties are valid for operating temperature.

Ограничения

  • Local transitional effects near entrances are not captured by global Re alone.
  • Variable viscosity and density with temperature can require local evaluation.
  • Non-Newtonian fluids need generalized Reynolds definitions.

Подробное объяснение

This nondimensional ratio directly compares inertial transport against viscous diffusion; high values indicate turbulence.

Как пользоваться формулой

  1. Compute characteristic velocity and size.
  2. Use dynamic or kinematic viscosity form consistently.
  3. Compare Re with regime thresholds (typically 2000 and 4000).

Историческая справка

Introduced by Osborne Reynolds as a similarity criterion for flow transition.

Пример

With v=2 m/s, D_h=0.1 m, and ν=1.0e-6 m^2/s: Re = 2e5, indicating turbulent flow.

Частая ошибка

Using pipe diameter for non-circular sections instead of hydraulic diameter can misclassify flow regime.

Практика

Задачи с решением

Determine flow regime

Условие. v = 0.8 m/s, D_h = 0.2 m, ν = 1.2e-6 m^2/s.

Решение. Re = v D_h / ν = 0.8·0.2/1.2e-6 = 133333.

Ответ. Re = 1.33×10^5 (turbulent).

Find hydraulic diameter for target Re

Условие. Re target = 2000, v = 1.5 m/s, ν = 1.0e-6 m^2/s.

Решение. D_h = Re·ν/v = 2000·1.0e-6/1.5 = 1.33e-3 m.

Ответ. D_h ≈ 1.33 mm.

Дополнительные источники

  • White, F. M. (2011). Fluid Mechanics, 7th ed.
  • Fox, R. W., Pritchard, P. J., & McDonald, A. T. (2011). Introduction to Fluid Mechanics, 8th ed.

Связанные формулы

Инженерия

Hydraulic diameter

$D_h = \frac{4A}{P_w}$

Hydraulic diameter extends circular-pipe relations to non-circular ducts using equivalent diameter based on wetted area and perimeter.

Инженерия

Darcy–Weisbach head loss

$h_f = f\frac{L}{D_h}\frac{v^2}{2g}$

The Darcy–Weisbach equation estimates major head loss in fully developed pipe flow using friction factor and geometric ratio.

Инженерия

Velocity from flow rate

$v = \frac{Q}{A}$

If flow rate and flow area are known, this equation yields the average velocity in the section.