{
  "id": "reynolds-number",
  "version": "308d5e040d43",
  "status": "published",
  "name": "Reynolds Number Calculator",
  "question": "What Reynolds number is my flow?",
  "summary": "Reynolds number calculator for flow in a pipe: Re = ρvD ÷ η from the fluid, its speed and the pipe diameter, with laminar, transitional or turbulent flow and the kinematic viscosity.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/reynolds-number-calculator",
  "markdown": "https://www.acalculator.org/physics/reynolds-number-calculator.md",
  "kind": "function",
  "method": "Re = ρ × v × D ÷ η; kinematic viscosity ν = η ÷ ρ; laminar up to v = 2,000 × η ÷ (ρ × D).",
  "assumptions": [
    "Flow in a full round pipe, with D the inside diameter and v the average speed. OpenStax writes the same number as 2ρvr ÷ η with r the radius.",
    "Laminar below 2,000, transitional from 2,000 to 3,000, turbulent above 3,000 (OpenStax gives these as approximate limits)."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "f": {
        "title": "Fluid",
        "description": "The fluid, which sets its density and viscosity (OpenStax Tables 14.1 and 14.4).",
        "type": "string",
        "enum": [
          "water20",
          "water0",
          "water100",
          "air0",
          "custom"
        ]
      },
      "v": {
        "title": "Flow speed (v)",
        "description": "The average speed of the fluid through the pipe.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "speed",
        "minimum": 0.000001,
        "maximum": 10000
      },
      "D": {
        "title": "Pipe diameter (D)",
        "description": "The inside diameter of the pipe (the characteristic length).",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0.000001,
        "maximum": 1000
      },
      "rho": {
        "title": "Density (ρ)",
        "description": "The density of the fluid.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "density",
        "minimum": 0.001,
        "maximum": 30000
      },
      "eta": {
        "title": "Viscosity (η)",
        "description": "The dynamic viscosity of the fluid, in the unit chosen below.",
        "type": "number",
        "minimum": 1e-9,
        "maximum": 1000000000
      },
      "eu": {
        "title": "Viscosity unit",
        "description": "The unit of the viscosity you type.",
        "type": "string",
        "enum": [
          "Pa-s",
          "mPa-s",
          "P",
          "lb-fts"
        ]
      }
    }
  },
  "outputs": {
    "re": {
      "label": "Reynolds number",
      "description": "Re = ρ × v × D ÷ η, a pure number.",
      "format": "number"
    },
    "regime": {
      "label": "Flow",
      "description": "Laminar below 2,000, transitional from 2,000 to 3,000, turbulent above 3,000.",
      "format": "text"
    },
    "nu": {
      "label": "Kinematic viscosity (m²/s)",
      "description": "η ÷ ρ, in square metres per second.",
      "format": "number"
    },
    "rho": {
      "label": "Density used (kg/m³)",
      "description": "The density of the fluid in kilograms per cubic metre.",
      "format": "number"
    },
    "eta": {
      "label": "Viscosity used (Pa·s)",
      "description": "The dynamic viscosity in pascal seconds.",
      "format": "number"
    },
    "laminarSpeed": {
      "label": "Fastest laminar speed (m/s)",
      "description": "The speed at which Re reaches 2,000 in this pipe: 2,000 × η ÷ (ρ × D).",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "f": "water20",
      "v": "0.5 m/s",
      "D": "2 cm",
      "rho": 1000,
      "eta": 1,
      "eu": "mPa-s"
    },
    "outputs": {
      "re": 9962.075848303393,
      "regime": "Turbulent",
      "nu": 0.0000010038068523342016,
      "rho": 998.2,
      "eta": 0.001002,
      "laminarSpeed": 0.10038068523342016
    },
    "text": "The Reynolds number is 9,962.08: Turbulent flow."
  },
  "examples": [
    {
      "given": {
        "f": "water20",
        "v": 0.5,
        "D": 0.02
      },
      "expect": {
        "re": 9962.075848303393,
        "regime": "Turbulent",
        "nu": 0.0000010038068523342016
      },
      "source": "OpenStax, University Physics Volume 1, §14.7 Viscosity and Turbulence (N_R = 2ρvr ÷ η; laminar below about 2,000, turbulent above about 3,000; Table 14.4), https://openstax.org/books/university-physics-volume-1/pages/14-7-viscosity-and-turbulence (retrieved 2026-10-05); OpenStax, University Physics Volume 1, §14.1 Fluids, Density, and Pressure (Table 14.1), https://openstax.org/books/university-physics-volume-1/pages/14-1-fluids-density-and-pressure (retrieved 2026-10-05); hand calculation in content.mdx: 998.2 × 0.5 × 0.02 ÷ 0.001002 = 9,962.08",
      "tolerance": 1e-12
    },
    {
      "given": {
        "f": "custom",
        "rho": 1000,
        "eta": 1,
        "eu": "Pa-s",
        "v": 0.1,
        "D": 0.01
      },
      "expect": {
        "re": 1,
        "regime": "Laminar"
      },
      "source": "OpenStax, University Physics Volume 1, §14.7 Viscosity and Turbulence (N_R = 2ρvr ÷ η; laminar below about 2,000, turbulent above about 3,000; Table 14.4), https://openstax.org/books/university-physics-volume-1/pages/14-7-viscosity-and-turbulence (retrieved 2026-10-05); hand calculation in content.mdx: 1,000 × 0.1 × 0.01 ÷ 1 = 1",
      "tolerance": 1e-12
    },
    {
      "given": {
        "f": "custom",
        "rho": 1000,
        "eta": 1,
        "eu": "mPa-s",
        "v": 0.2,
        "D": 0.01
      },
      "expect": {
        "re": 2000,
        "regime": "Transitional (unstable)",
        "laminarSpeed": 0.2
      },
      "source": "OpenStax, University Physics Volume 1, §14.7 Viscosity and Turbulence (N_R = 2ρvr ÷ η; laminar below about 2,000, turbulent above about 3,000; Table 14.4), https://openstax.org/books/university-physics-volume-1/pages/14-7-viscosity-and-turbulence (retrieved 2026-10-05); hand calculation in content.mdx: 1,000 × 0.2 × 0.01 ÷ 0.001 = 2,000",
      "tolerance": 1e-12
    },
    {
      "given": {
        "f": "air0",
        "v": 10,
        "D": 0.1
      },
      "expect": {
        "re": 75438.59649122808,
        "regime": "Turbulent"
      },
      "source": "OpenStax, University Physics Volume 1, §14.7 Viscosity and Turbulence (N_R = 2ρvr ÷ η; laminar below about 2,000, turbulent above about 3,000; Table 14.4), https://openstax.org/books/university-physics-volume-1/pages/14-7-viscosity-and-turbulence (retrieved 2026-10-05); OpenStax, University Physics Volume 1, §14.1 Fluids, Density, and Pressure (Table 14.1), https://openstax.org/books/university-physics-volume-1/pages/14-1-fluids-density-and-pressure (retrieved 2026-10-05); hand calculation in content.mdx: 1.29 × 10 × 0.1 ÷ 0.0000171 = 75,438.6",
      "tolerance": 1e-12
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 1, §14.7 Viscosity and Turbulence: N_R = 2ρvr ÷ η, limits of about 2,000 and 3,000, Table 14.4 viscosities, retrieved 2026-10-05. https://openstax.org/books/university-physics-volume-1/pages/14-7-viscosity-and-turbulence",
    "OpenStax, University Physics Volume 1, §14.1 Fluids, Density, and Pressure: Table 14.1 densities of water and air, retrieved 2026-10-05. https://openstax.org/books/university-physics-volume-1/pages/14-1-fluids-density-and-pressure",
    "NIST Special Publication 811 (2008), Appendix B.8: 1 ft = 0.3048 m, 1 in = 0.0254 m, 1 lb = 0.45359237 kg, 1 mph = 0.44704 m/s, retrieved 2026-10-05. https://www.nist.gov/pml/special-publication-811"
  ],
  "related": [
    "flow-rate",
    "density",
    "pressure"
  ],
  "changelog": []
}