## Characteristic Numbers

In fluid dynamics and heat transfer, **characteristic numbers** are dimensionless numbers used to describe a character of the flow or to describe a character of heat transfer. **Characteristic numbers** can be used to compare a **real situation** (e.g. air flow around an airfoil and water flow in a pipe) with a **small-scale model**. It is necessary to keep the important characteristic numbers the same. Names of these numbers were standardized in **ISO 31-12**, which gives name, symbol and definition for 25 selected characteristic numbers used for the description of transport phenomena.

## Reynolds Number

**The Reynolds number** is the ratio of **inertial forces **to **viscous forces** and is a convenient parameter for predicting if a flow condition will be **laminar or turbulent**. It can be interpreted that when the **viscous forces** are dominant (slow flow, low **Re**) they are sufficient enough to keep all the fluid particles in line, then the flow is laminar. Even very low **Re** indicates viscous creeping motion, where inertia effects are negligible. When the **inertial forces dominate** over the viscous forces (when the fluid is flowing faster and Re is larger) then the flow is turbulent.

**It is a dimensionless number** comprised of the physical characteristics of the flow. An increasing Reynolds number indicates an increasing turbulence of flow.

where:

V is the flow velocity,

D is a** characteristic linear dimension**, (travelled length of the fluid; hydraulic diameter etc.)

ρ fluid density (kg/m^{3}),

μ dynamic viscosity (Pa.s),

ν kinematic viscosity (m^{2}/s); ν = μ / ρ.

## Prandtl Number

The **Prandtl number** is a dimensionless number, named after its inventor, a German engineer **Ludwig Prandtl**, who also identified the boundary layer. The** Prandtl number** is defined as the** ratio** of **momentum diffusivity** to **thermal diffusivity**. The **momentum diffusivity**, or as it is normally called, kinematic viscosity, tells us the material’s resistance to shear-flows (different layers of the flow travel with different velocities due to e.g. different speeds of adjacent walls) in relation to density. That is, the **Prandtl number** is given as:

where:

**ν** is **momentum diffusivity** (kinematic viscosity) [m^{2}/s]

**α** is **thermal diffusivity** [m^{2}/s]

**μ** is **dynamic viscosity** [N.s/m^{2}]

**k** is **thermal conductivity** [W/m.K]

**c _{p }**is

**specific heat**[J/kg.K]

**ρ** is **density** [kg/m^{3}]

Small values of the **Prandtl number**, **Pr << 1**, means the thermal diffusivity dominates. Whereas with large values, **Pr >> 1**, the momentum diffusivity dominates the behavior. For example, the typical value for liquid mercury, which is about 0.025, indicates that the **heat conduction** is more significant compared to **convection**, so thermal diffusivity is dominant. When Pr is small, it means that the heat diffuses quickly compared to the velocity.

In comparison to Reynolds number, the **Prandtl number** is not dependent on geometry of an object involved in the problem, but is dependent solely on the fluid and the fluid state. As such, the **Prandtl number** is often found in property tables alongside other properties such as viscosity and thermal conductivity.

See also: Prandtl Number

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