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9.4 Summary of Dimensionless Numbers

 This section summarizes all the major dimensionless parameters which are commonly used in the fluid mechanics field.

Common Dimensionless Parameters of Thermo–Fluid Field

Name Symbol Equation Interpretation Application
Archimedes Number \(Ar\) \(\dfrac {g\, {\ell}^3 \rho_f (\rho - \rho_f)}{\mu^2} \) \(\dfrac{\text{buoyancy forces}}{\text{viscous forces}}\) in nature and force convection
Atwood Number \(A\) \(\dfrac {(\rho_a - \rho_b)}{\rho_a + \rho_b}\) \(\dfrac{\text{buoyancy forces}}{\text{``penetration'' force}}\)

in stability of liquid layer \(a\) over \(b\) Rayleigh–Taylor instability etc

Bond Number \(Bo\) \(\dfrac{\rho\,g\,\ell^2}{\sigma}\) \(\dfrac{\text{gravity forces}}{\text{surface tension force}}\) in open channel flow, thin film flow
Brinkman Number \(Br\) \(\dfrac {\mu U^2}{k\,\Delta T}\) \(\dfrac{\text{heat forces}}{\text{heat force}}\) in open channel flow, thin film flow
Capillary Number \(Ca\) \(\dfrac {\mu U}{\sigma}\) \(\dfrac{\text{viscous forces}}{\text{surface tension force}}\) For small \(Re\) and surface tension involve problem
Cauchy Number \(Cau\) \(\dfrac{\rho\,U^2}{E}\) \(\dfrac{\text{inertia forces}}{\text{elastic forces}}\) For large \(Re\) and surface tension involve problem
Cavitation Number \(\sigma\) \(\dfrac{P_l - P_v}{\dfrac{1}{2} \rho U^2}\) \(\dfrac{\text{pressure difference}}{\text{inertia energy}}\) pressure difference to vapor pressure to the potential of phase change (mostly to gas)
Courant Number \(Co\) \(\dfrac{\Delta t\, U}{\Delta x}\) \(\dfrac{\text{wave distance  }}{\text{typical distance}}\) A requirement in numerical schematic to achieve stability
Dean Number \(D\) \(\dfrac{Re}{\sqrt{R / h}}\) \(\dfrac{\text{inertia forces}}{\text{viscous deviation forces}}\) related to radius of channel with width \(h\) stability
Deborah Number \(De\) \(\dfrac{t_c}{t_p}\) \(\dfrac{\text{stress relaxation time}}{\text{observation time}}\) the ratio of the fluidity of material primary used in rheology

Drag Coefficient

\(C_D\) \(\dfrac{D}{\dfrac{1}{2}\,\rho\,U^2\,A }\) \(\dfrac{\text{drag force}}{\text{inertia effects }}\) Aerodynamics, hydrodynamics, note this coefficient has many definitions

Eckert Number

\(Ec\) \(\dfrac{U^2}{C_p\,\Delta T}\) \(\dfrac{\text{inertia effects}}{\text{thermal effects }}\) during dissipation processes
Ekman Number \(Ek\) \(\dfrac{\nu}{2\ell^2\,\omega}\) \(\dfrac{\text{viscous forces}}{\text{Coriolis forces }}\) geophysical flow like atmospheric flow
Euler Number \(Eu\) \(\dfrac{P_0-P_{\infty}}{\dfrac{1}{2}\,\rho\,U^2} \) \(\dfrac{\text{pressure effects}}{\text{inertia effects }}\) potential of resistance problems
Froude Number \(Fr\) \( \dfrac{U}{\sqrt{g\,\ell}} \) \(\dfrac{\text{inertia effects}}{\text{gravitational effects }}\) open channel flow and two phase flow
Galileo Number \(Ga\) \(\dfrac{\rho\, g\,\ell^3}{\mu^2} \) \(\dfrac{\text{gravitational effects}}{\text{viscous effects }}\) open channel flow and Stokes flow
Grashof Number \(Gr\) \(\dfrac{\beta\,\Delta T \,g\,\ell^3\,\rho^2}{\mu^2 } \) \(\dfrac{\text{buoyancy effects}}{\text{viscous effects }}\) natural convection

 Knudsen Number

\(Kn\) \(\dfrac{\lambda}{\ell} \) \(\dfrac{\text{LMFP }}{\text{characteristic length }}\) length of mean free path, LMFP, to characteristic length
Laplace Constant \(La\) \(\sqrt{\dfrac{2\,\sigma}{g(\rho_1-\rho_2)}}\) \(\dfrac{\text{surface force }}{\text{gravity effects }}\) liquid raise,  surface tension problem, (also ref Capillary constant)
Lift Coefficient \(C_L\) \( \dfrac{L}{\dfrac{1}{2}\,\rho\,U^2\,A } \) \(\dfrac{\text{lift force }}{\text{inertia effects }}\) Aerodynamics, hydrodynamics, note this coefficient has many definitions
Mach Number \(M\) \( \dfrac{U}{c} \) \(\dfrac{\text{velocity }}{\text{sound speed }}\) Compressibility and propagation of disturbance
Marangoni Number \(Ma\) \(-{\dfrac{d\sigma}{dT}}\dfrac{\ell \, \Delta T}{\nu \alpha} \) \(\dfrac{\text{ `thermal' s. tension }}{\text{viscous force }}\) Compressibility and propagation of disturbance
Morton Number \(Mo\) \(\dfrac{g \mu_c^4 \, \Delta \rho}{\rho_c^2 \sigma^3} \) \(\dfrac{\text{ viscous force }}{\text{surface tension }}\) bubble and drop flow
Ozer Number \(Oz\) \(\dfrac{\dfrac
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{\rho} }{\left(\dfrac{Q_{max}}{A}\right)^2 } \)
\(\dfrac{\text{ `maximum' supply }}{\text{`maximum' demand }}\)  supply and demand analysis such pump and pipe system, economy
Prandtl Number \(Pr\) \(\dfrac{\nu}{\alpha}\) \(\dfrac{\text{ viscous diffusion}}{\text{thermal diffusion }}\) Prandtl number is fluid property important in flow due to thermal forces
Reynolds Number \(Re\) \(\dfrac{\rho\,U\,\ell}{\mu}\) \(\dfrac{\text{ inertia force}}{\text{viscous force }}\) in most fluid mechanics issues
Rossby Number \(Ro\) \(\dfrac{U}{\omega\,ll_{0}}\) \(\dfrac{\text{ inertia force}}{\text{Coriolis force }}\) in rotating fluids
Shear Number \(Sn\) \(\dfrac{\tau_c\,ll_c}{\mu_c\,U_c}\) \(\dfrac{\text{actual shear}}{\text{`potential' shear }}\) Shear flow
Stokes Number \(Stk\) \(\dfrac{\rho\, g\,\ell^3}{\mu^2}\) \(\dfrac{\text{particle relaxation time }}{\text{Kolmogorov time }}\) in aerosol flow dealing with penetration of particles
Strouhal Number \(St\) \(\dfrac{\omega\,\ell}{U}\) \(\dfrac{\text{`unsteady' effects }}{\text{inertia effect }}\) The effects of natural or forced frequency in all the field that is how much the `unsteadiness' of the flow is
Taylor Number \(Ta\) \(\dfrac{\rho^2\,{\omega_i}^2\,\ell^4}{\mu^4}\) \(\dfrac{\text{centrifugal force}}{\text{viscous force }}\) Stability of rotating cylinders Notice \(\ell\) has special definition
Weber Number \(We\) \(\dfrac{\rho\,U^2\,\ell}{\sigma}) \(\dfrac{\text{inertia force}}{\text{viscous force }}\) For large \(Re\) and surface tension involve problem

The dimensional parameters that were used in the construction of the dimensionless parameters in Table 9.8 are the characteristics of the system. Therefore there are several definition of Reynolds number. In fact, in the study of the physical situations often people refers to local \(Re\) number and the global \(Re\) number. Keeping this point in mind, there several typical dimensions which need to be mentioned. The typical body force is the gravity \(g\) which has a direction to center of Earth. The typical length is denoted as \(\ell\) and in many cases it is referred to as the diameter or the radius. The density, \(\rho\) is referred to the characteristic density or density at infinity. The area, \(A\) in drag and lift coefficients is referred normally to projected area.

Fig. 9.4 Oscillating Von Karman Vortex Street.

The frequency \(\omega\) or \(f\) is referred to as the "unsteadiness'' of the system. Generally, the periodic effect is enforced by the boundary conditions or the initial conditions. In other situations, the physics itself instores or forces periodic instability. For example, flow around cylinder at first looks like symmetrical situation. And indeed in a low Reynolds number it is a steady state. However after a certain value of Reynolds number, vortexes are created in an infinite parade and this phenomenon is called Von Karman vortex street (see Figure 9.4) These vortexes are created in a non–symmetrical way and hence create an unsteady situation. When Reynolds number increases, these vortexes are mixed and the flow becomes turbulent which, can be considered a steady state. The pressure \(P\) is the pressure at infinity or when the velocity is at rest. \(c\) is the speed of sound of the fluid at rest or characteristic value. The value of the viscosity, \(\mu\) is typically some kind averaged value. The inability to define a fix value leads also to new dimensionless numbers which represent the deviations of these properties. 

Contributors

  • Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.