Dimensionless Numbers

A. Salih

Dept. of Aerospace Engineering
IIST, Thiruvananthapuram

The nondimensionalization of the governing equations of fluid flow is important for both theoretical and computational reasons. Nondimensional scaling provides a method for developing dimensionless groups that can provide physical insight into the importance of various terms in the system of governing equations. Computationally, dimensionless forms have the added benefit of providing numerical scaling of the system discrete equations, thus providing a physically linked technique for improving the ill-conditioning of the system of equations. Moreover, dimensionless forms also allow us to present the solution in a compact way. Some of the important dimensionless numbers used in fluid mechanics and heat transfer are given below.


      Nomenclature

  • Archimedes Number:
  • Ar  =   Re2
    Fr
      =   gL3ρ(ρsρ)
    μ2
  • Atwood Number:
  • A  =   (ρ1ρ2)
    (ρ1 + ρ2)
    Note: Used in the study of density stratified flows.

  • Biot Number:
  • Bi  =   hL
    Ks
      =   conductive resistance in solid
    convective resistance in thermal boundary layer

  • Bond Number:
  • Bo  =   We
    Fr
      =   ρgL2
    σ

  • Brinkman Number:
  • Br  =   μU2
    K(Tw − To)
    Note: Brinkman number is related to heat conduction from a wall to a flowing viscous fluid. It is commonly used in polymer processing.

  • Capillary Number:
  • Ca  =   We
    Re
      =  
    σ

  • Cauchy Number:
  • Ca  =  M2 =   ρU2
    K

  • Centrifuge Number:
  • Ce  =   We
    Ro2
      =   ρΩ2L3
    σ

  • Dean Number:
  • De  =   Re
    (R ⁄ h)1/2
    Note: Dean number deals with the stability of two-dimensional flows in a curved channel with mean radius R and width 2h.

  • Deborah Number:
  • De  =   τ
    tp
      =   relaxation time
    characteristic time scale
    Note: Commonly used in rheology to characterize how "fluid" a material is. The smaller the De, the more the fluid the material appears.

  • Eckert Number:
  • Ec  =   U2
    cpΔT
    Note: Eckert number represents the kinetic energy of the flow relative to the boundary layer enthalpy difference. Ec plays an important role in high speed flows for which viscous dissipation is significant.

  • Ekman Number:
  • Ek  =   μ
    ρσL2
      =   viscous force
    Coriolis force
  • Eötvös Number:
  • Eo  =   We
    Fr
      =   ΔρgL2
    σ

  • Euler Number:
  • Eu  =   Δp
    ρU2
      =   pressure force
    inertial force

  • Fourier Number:
  • Fo  =   αt
    L2
      =   rate of heat conduction
    rate of thermal energy stored
    Note: Fourier number represents the dimensionless time. It may be interpreted as the ratio of current time to time to reach steady-state.

  • Froude Number:
  • Fr  =   U2
    gL
      =   inertial force
    gravitational force

  • Galileo Number:
  • Ga  =   Re2
    Fr
      =   ρ2gL3
    μ2

  • Graetz Number:
  • Gz  =   di Pe
    L
      =   Udi
    ν

  • Grashof Number:
  • Gr  =   (Thot − Tref)L3
    ν2
      =   buoyancy force
    viscous force

  • Hagen Number:
  • Hg  = − dp
    dx
      ρL3
    μ2
    Note: It is the forced flow equivalent of Grashof number.

  • Jakob Number:
  • Ja  =   cp(Tw − Tsat)
    hfg
    Note: Jakob number represents the ratio of sensible heat to latent heat absorbed (or released) during the phase change process.

  • Knudsen Number:
  • Kn  =   λ
    L
      =   length of mean free path
    characteristic dimension

  • Laplace Number:
  • La  =   Re2
    We
      =   ρσL
    μ2

  • Lewis Number:
  • Le  =   α
    DAB
      =   thermal diffusivity
    mass diffusivity

  • Mach Number:
  • M  =   U
    a
      =   inertial force
    elastic (compressibility) force

  • Marangoni Number:
  • Ma  = −
    dT
      LΔT
    μα
    Note: Marangoni number is the ratio of thermal surface tension force to the viscous force.

  • Morton Number:
  • Mo  =   We3
    Fr Re4
      =   4
    Δρσ3

  • Nusselt Number:
  • Nu  =   hL
    Kf
    Note: Nusselt number represents the dimensionless temperature gradient at the solid surface.

  • Ohnesorge Number:
  • Oh  =   We1/2
    Re
      =   μ
    (ρσL)1/2

  • Peclet Number:
  • Pe  =   UL
    α
      =   inertia (convection)
    Diffusion

  • Prandtl Number:
  • Pr  =   ν
    α
      =   momentum diffusivity
    thermal diffusivity

  • Rayleigh Number:
  • Ra  = Gr Pr =   (Thot − Tref)L3
    να
      =   buoyancy
    viscous × rate of heat diffusion

  • Reynolds Number:
  • Re  =   ρUL
    μ
      =   inertial force
    viscous force

  • Richardson Number:
  • Ri  =   Gr
    Re2
      =   (Thot − Tref)L
    U2
      =   buoyancy force
    inertial force

  • Rossby Number:
  • Ro  =   U
    ΩL
      =   inertial force
    Coriolis force

  • Rotating Froude Number:
  • FrR  =   Fr
    Ro2
      =   Ω2L
    g

  • Schmidt Number:
  • Sc  =   Le Pr   =   ν
    DAB
      =   momentum diffusivity
    mass diffusivity

  • Sherwood Number:
  • Sh  =   hmL
    DAB
    Note: Sherwood number represents the dimensionless concentration gradient at the solid surface.

  • Stanton Number:
  • St  =   Nu
    Re Pr
      =   h
    ρUcp
    Note: Stanton number is the modified Nusselt number. It is used in analogy between heat transfer and viscous transport in boundary layers.

  • Stefan Number:
  • St  =   cpdT
    Lm
      =   specific heat
    latent heat
    Note: Stefan number is useful in the study of heat transfer during phase change.

  • Stokes Number:
  • Stk  =   τUo
    dc
      =   stopping distance of a particle
    characteristic dimension of the obstacle
    Note: Commonly used in particles suspended in fluid.
    For Stk << 1, the particle negotiates the obstacle.
    For Stk >> 1, the particle travels in straightline and eventually collides with obstacle.


  • Strouhal Number (for oscillatory flow):
  • St  =   L
    Utref
      =   inertia (local)
    inertia (convection)
    Note: If tref is taken as the reciprocal of the circular frequency ω of the system, then
    St  =   ωL
    U
  • Taylor Number:
  • Ta  =   ρ2Ωi2L4
    μ2
    where L = [ri(rori)3]1/4

  • Weber Number:
  • We  =   ρU2L
    σ
      =   inertial force
    surface tension force

  • Womersley Number:
  • α  =   (π Re St)1/2 =   L  (ρω)1/2
    μ1/2
    Note: Womersley number is used in biofluid mechanics. It is a dimensionless expression of the pulsatile flow frequency in relation to the viscous effects.


    Nomenclature:
    aspeed of sound
    cpspecific heat at constant pressure
    DABmass diffusivity coefficient
    dTtemperature difference between phases
    dccharacteristic dimension of the obstacle
    dihydraulic diameter of the duct
    ggravitational acceleration
    hheat transfer coefficient
    hwidth of the channel
    hfglatent heat of condensation
    hmmass transfer coefficient
    Kbulk modulus of elasticity
    Kthermal conductivity of fluid
    Kfthermal conductivity of fluid
    Ksthermal conductivity of solid
    Lcharacteristic length scale
    Lmlatent heat of melting
    Rradius of the channel
    riradius of the inner cylinder
    roradius of the outer cylinder
    Thottemperature of the hot wall
    Trefreference temperature
    Tobulk fluid temperature
    Tsatsaturation temperature
    Twwall temperature
    Tquiescent temperature of the fluid
    ttime
    trefreference time
    tpcharacteristic time scale
    Ucharacteristic velocity scale
    Uofluid velocity far away from the object
    dp/dxpressure gradient
    dσ/dTrate of change of surface tension with temperature
    αthermal diffusivity of fluid
    βvolumetric thermal expansion coefficient
    Δpcharacteristic pressure difference of flow
    ΔTcharacteristic temperature difference
    Δρdifference in density of the two phases
    λlength of mean free path
    μviscosity of fluid
    νkinematic viscosity of fluid
    ρdensity of fluid
    ρ1density of heavier fluid
    ρ2density of lighter fluid
    ρldensity of solid
    σsurface tension
    τrelaxation time
    Ωangular velocity
    ωcircular frequency
    ωiangular velocity of inner cylinder