Reynolds Number:

It is the ratio of inertia forces to viscous forces in the velocity boundary layer. It is used in forced convection and approximated as:

Re=$\frac{(Inertia forces)}{(Viscous forces)}$=$\frac{(ρu_∞ L_c)}{μ}$=$\frac{(u_∞ L_c)}{v}$

Where u is the upstream velocity (equivalent to the free-stream velocity $u_∞$ for a flat plate), Lcis the characteristic length of the geometry, and v=$\frac{μ}{ρ}$ is the kinematic viscosity of the fluid. For a flat plate, the characteristic length is the distance x from the leading edge. Note that kinematic viscosity has the unit m2/s, which is identical to the unit of thermal diffusivity, and can be viewed as viscous diffusivity or diffusivity for momentum.

The transition from laminar to turbulent flow depends on the surface geometry, surface roughness, free-stream velocity, surface temperature, and type of fluid, among other things. After exhaustive experiments in the 1880s, OsbornReynolds discovered that the flow regime depends mainly on the ratio of the inertia forces to viscous forces in the fluid. This ratio is called the Reynoldsnumber, which is a dimensionless quantity, and is expressed for external flow. The Reynolds number at which the flow becomes turbulent is called the critical Reynolds number. The value of the critical Reynolds number is different for different geometries. For flow over a flat plate, the generally accepted value of the critical Reynolds number is $Re_cr$=$\frac{(Vx_cr)}{(v}$=$\frac{(u_∞ x_cr)}{v}$=5×$10^5$ ) where $x_cr$ is the distance from the leading edge of the plate at which transition from laminar to turbulent flow occurs. The value of $Re_cr$ may change substantially, however, depending on the level of turbulence in the free stream.

Prandtl Number: The relative thickness of the velocity and the thermal boundary layers is best described by the dimensionlessparameter Prandtl number, defined as

Pr=$\frac{(Molecular diffusivity of momentum)}{(Molecular diffusivity of heat)}$=$\frac{v}{α}$=$\frac{(μC_P)}{k}$

It is named after Ludwig Prandtl, who introduced the concept of boundary layer in 1904 and made significant contributions to boundary layer theory. The Prandtl numbers of fluids range from less than 0.01 for liquid metals to more than 100,000 for heavy oils. Note that the Prandtl number is in the order of 10 for water.

The Prandtl numbers of gases are about 1, which indicates that both momentum and heat dissipate through the fluid at about the same rate. Heat diffuses very quickly in liquid metals (Pr<<1) and very slowly in oils (Pr>>1) relative to momentum. Consequently the thermal boundary layer is much thicker for liquid metals and much thinner for oils relative to the velocity boundary layer.