Dimensionless Numbers

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Marilynn Gardner
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1 , 09:49 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: Atwood Number: Ar Re2 gl3 ρ(ρ s ρ) A (ρ 1 ρ 2 ) (ρ 1 + ρ 2 ) Note: Used in the study of density stratified flows. Biot Number: Bi hl K s conductive resistance in solid convective resistance in thermal boundary layer Bond Number: Bo We ρgl2 Brinkman Number: Br µu 2 K(T w 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: Ca We Re Uµ Cauchy Number: Ca M 2 ρu2 K

2 Centrifuge Number: Ce We Ro 2 ρω2 L 3 Dean Number: De Re (R h) 1/2 Note: Dean number deals with the stability of twodimensional flows in a curved channel with mean radius R and width 2h. Deborah Number: De τ t p 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 U 2 c p 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: Ek µ viscous force ρl2 Coriolis force E tv s Number: Eo We ρgl2 Euler Number: Eu p pressure force ρu2 Fourier Number: Fo αt L 2 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. oude Number: U2 gl gravitational force Galileo Number: Ga Re2 ρ2 gl 3 Graetz Number: Gz d i Pe Ud i , 09:49

3 , 09:49 L ν Grashof Number: Gr gβ(t hot T ref )L 3 ν 2 buoyancy force viscous force Hagen Number: Hg dp ρl 3 dx Note: It is the forced flow equivalent of Grashof number. Jakob Number: Ja c p(t w T sat ) h fg 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 ρl Lewis Number: Le α thermal diffusivity mass diffusivity Mach Number: M U a elastic (compressibility) force Marangoni Number: Ma d L T dt µα Note: Marangoni number is the ratio of thermal surface tension force to the viscous force. Morton Number: Mo We 3 Re 4 gµ 4 ρ 3 Nusselt Number: Nu hl K f Note: Nusselt number represents the dimensionless temperature gradient at the solid surface. Ohnesorge Number: Oh We1/2 Re µ (ρl) 1/2

4 , 09:49 Peclet Number: Pe UL α inertia (convection) Diffusion Prandtl Number: Pr ν α momentum diffusivity thermal diffusivity Rayleigh Number: Ra Gr Pr gβ(t hot T ref )L 3 να buoyancy viscous rate of heat diffusion Reynolds Number: Re ρul µ viscous force Richardson Number: Ri Gr Re 2 gβ(t hot T ref )L U 2 buoyancy force Rossby Number: Ro U ΩL Coriolis force Rotating oude Number: R Ro 2 Ω2 L g Schmidt Number: Sc Le Pr ν momentum diffusivity mass diffusivity Sherwood Number: Sh h ml Note: Sherwood number represents the dimensionless concentration gradient at the solid surface. Stanton Number: St Nu Re Pr h 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: St c pdt L m specific heat latent heat Note: Stefan number is useful in the study of heat transfer during phase change. Stokes Number:

5 , 09:49 Stk τu o d c 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 Ut ref inertia (local) inertia (convection) Note: If t ref is taken as the reciprocal of the circular frequency ω of the system, then Taylor Number: where L [r i (r o r i ) 3 ] 1/4 St ωl U Ta ρ2 Ω i 2 L 4 Weber Number: We ρu2l surface tension force Womersley Number: α (π Re St) 1/2 L (ρω)1/2 µ 1/2 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 c p dt d c d i g h h h fg h m K K K f K s L L m R r i speed of sound specific heat at constant pressure mass diffusivity coefficient temperature difference between phases characteristic dimension of the obstacle hydraulic diameter of the duct gravitational acceleration heat transfer coefficient width of the channel latent heat of condensation mass transfer coefficient bulk modulus of elasticity thermal conductivity of fluid thermal conductivity of fluid thermal conductivity of solid characteristic length scale latent heat of melting radius of the channel radius of the inner cylinder

6 , 09:49 r o radius of the outer cylinder T hot temperature of the hot wall T ref reference temperature T o bulk fluid temperature T sat saturation temperature T w wall temperature T quiescent temperature of the fluid t time t ref reference time t p characteristic time scale U characteristic velocity scale U o fluid velocity far away from the object dp/dx pressure gradient d/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 ρ 1 density of heavier fluid ρ 2 density of lighter fluid ρ l density of solid surface tension τ relaxation time Ω angular velocity ω circular frequency ω i angular velocity of inner cylinder

Summary of Dimensionless Numbers of Fluid Mechanics and Heat Transfer

1. Nusselt number Summary of Dimensionless Numbers of Fluid Mechanics and Heat Transfer Average Nusselt number: convective heat transfer Nu L = conductive heat transfer = hl where L is the characteristic