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Dimensionless Numbers Dimensionless Numbers https://www.iist.ac.in/sites/default/files/people/numbers.html 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: 3 Re 2 gL ρ(ρs − ρ) Ar = = Fr µ2 Atwood Number: (ρ1 − ρ2) A = (ρ1 + ρ2) Note: Used in the study of density stratified flows. Biot Number: hL conductive resistance in solid Bi = = Ks convective resistance in thermal boundary layer Bond Number: We ρgL 2 Bo = = Fr σ Brinkman Number: µU2 Br = K(Tw − T o) 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: We Uµ Ca = = Re σ Cauchy Number: ρU2 Ca = M2 = K 1 06.10.2017, 09:49 Dimensionless Numbers https://www.iist.ac.in/sites/default/files/people/numbers.html Centrifuge Number: We ρΩ 2L3 Ce = = Ro 2 σ Dean Number: Re De = (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: τ relaxation time De = = tp 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: U2 Ec = cp∆T 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: µ viscous force Ek = = ρσ L2 Coriolis force E�tv �s Number: We ∆ρ gL 2 Eo = = Fr σ Euler Number: ∆p pressure force Eu = = ρU2 inertial force Fourier Number: αt rate of heat conduction Fo = = L2 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: U2 inertial force Fr = = gL gravitational force Galileo Number: Re 2 ρ2gL 3 Ga = = Fr µ2 Graetz Number: Gz = di Pe = Ud i 2 06.10.2017, 09:49 Dimensionless Numbers https://www.iist.ac.in/sites/default/files/people/numbers.html L ν Grashof Number: gβ(T − T )L 3 buoyancy force Gr = hot ref = ν2 viscous force Hagen Number: dp ρL3 Hg = − dx µ2 Note: It is the forced flow equivalent of Grashof number. Jakob Number: cp(Tw − T sat ) Ja = hfg Note: Jakob number represents the ratio of sensible heat to latent heat absorbed (or released) during the phase change process. Knudsen Number: λ length of mean free path Kn = = L characteristic dimension Laplace Number: Re 2 ρσ L La = = We µ2 Lewis Number: α thermal diffusivity Le = = DAB mass diffusivity Mach Number: U inertial force M = = a elastic (compressibility) force Marangoni Number: dσ L∆T Ma = − dT µα Note: Marangoni number is the ratio of thermal surface tension force to the viscous force. Morton Number: We 3 gµ4 Mo = = Fr Re 4 ∆ρσ 3 Nusselt Number: hL Nu = Kf Note: Nusselt number represents the dimensionless temperature gradient at the solid surface. Ohnesorge Number: We 1/2 µ Oh = = Re (ρσ L)1/2 3 06.10.2017, 09:49 Dimensionless Numbers https://www.iist.ac.in/sites/default/files/people/numbers.html Peclet Number: UL inertia (convection) Pe = = α Diffusion Prandtl Number: ν momentum diffusivity Pr = = α thermal diffusivity Rayleigh Number: gβ(T − T )L 3 buoyancy Ra = Gr Pr = hot ref = να viscous × rate of heat diffusion Reynolds Number: ρUL inertial force Re = = µ viscous force Richardson Number: Gr gβ(Thot − T ref )L buoyancy force Ri = = = Re 2 U2 inertial force Rossby Number: U inertial force Ro = = ΩL Coriolis force Rotating Froude Number: Fr Ω2L Fr R = = Ro 2 g Schmidt Number: ν momentum diffusivity Sc = Le Pr = = DAB mass diffusivity Sherwood Number: hmL Sh = DAB Note: Sherwood number represents the dimensionless concentration gradient at the solid surface. Stanton Number: Nu h St = = Re Pr ρUc p Note: Stanton number is the modified Nusselt number. It is used in analogy between heat transfer and viscous transport in boundary layers. Stefan Number: cpdT specific heat St = = Lm latent heat Note: Stefan number is useful in the study of heat transfer during phase change. Stokes Number: 4 06.10.2017, 09:49 Dimensionless Numbers https://www.iist.ac.in/sites/default/files/people/numbers.html τUo stopping distance of a particle Stk = = dc 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): L inertia (local) St = = Ut ref inertia (convection) Note: If tref is taken as the reciprocal of the circular frequency ω of the system, then ωL St = U Taylor Number: ρ2Ω 2L4 Ta = i µ2 3 1/4 where L = [r i(ro − ri) ] Weber Number: ρU2L inertial force We = = σ surface tension force Womersley Number: (ρω )1/2 α = ( π Re St )1/2 = L 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: a → speed of sound cp → specific heat at constant pressure DAB → mass diffusivity coefficient dT → temperature difference between phases dc → characteristic dimension of the obstacle di → hydraulic diameter of the duct g → gravitational acceleration h → heat transfer coefficient h → width of the channel hfg → latent heat of condensation hm → mass transfer coefficient K → bulk modulus of elasticity K → thermal conductivity of fluid Kf → thermal conductivity of fluid Ks → thermal conductivity of solid L → characteristic length scale Lm → latent heat of melting R → radius of the channel ri → radius of the inner cylinder 5 06.10.2017, 09:49 Dimensionless Numbers https://www.iist.ac.in/sites/default/files/people/numbers.html ro → radius of the outer cylinder Thot → temperature of the hot wall Tref → reference temperature To → bulk fluid temperature Tsat → saturation temperature Tw → wall temperature T∞ → quiescent temperature of the fluid t → time tref → reference time tp → characteristic time scale U → characteristic velocity scale Uo → fluid velocity far away from the object dp/dx → pressure gradient dσ/dT → rate of change of surface tension with temperature α → thermal diffusivity of fluid β → volumetric thermal expansion coefficient ∆p → characteristic pressure difference of flow ∆T → characteristic temperature difference ∆ρ → difference in density of the two phases λ → length of mean free path µ → viscosity of fluid ν → kinematic viscosity of fluid ρ → density of fluid ρ1 → density of heavier fluid ρ2 → density of lighter fluid ρl → density of solid σ → surface tension τ → relaxation time Ω → angular velocity ω → circular frequency ωi → angular velocity of inner cylinder 6 06.10.2017, 09:49.
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