10. Buoyancy-driven flow

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Geraldine Miller
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1 10. Buoyancy-driven flow For such flows to occur, need: Gravity field Variation of density (note: not the same as variable density!) Simplest case: Viscous flow, incompressible fluid, density-variation effects only present in body force term Archimedes, c. 87 BCE c. 1 BCE

2 Natural convection Movement is due to buoyancy Important forces: Forced convection Viscous Buoyancy Movement is due to other forces Important forces: Density variation mostly relevant for body-force term only Viscous Buoyancy Other Density - unknown Need energy equation and equations of state to close the system!

3 10.1. Boussinesq approximation Incompressible Navier-Stokes with gravity as body force u=0 u ρ +ρ ( u ) u= p+μ u ρ g e z t Hydrostatics (u = 0, p = p0, r = r0): 0= p0 ρ0 g e z Buoyancy-driven convective motion: u = u*, p = p0+p*, r = r0 + r* Joseph Valentin Boussinesq ( )

4 Plug expressions for p, r, u into momentum equation: * u * * * ( ρ0 +ρ ) t +(ρ0 +ρ ) (u ) u = * * * ( p + p0 )+μ u ( ρ0 +ρ ) g e z * Throw out hydrostatic terms Linearize, assuming * * ρ0 ρ, ρ=ρ0+ρ ρ0 Drop * and 0, replace r* = Dr u ρ +ρ ( u ) u= p+μ u Δ ρ g e z t Boussinesq approximation to momentum equation

5 10.. Thermal convection Fluid density change due to small temperature variations: ρ=ρ0 (1 β ( T T 0 )) Thermal expansion coefficient (valid for r r0, T T0) For ideal gas, b =1/T0, so Boussinesq equation becomes u ρ +ρ ( u ) u= p+μ u ρ g β ( T T 0) e z t r is not variable, but still need energy equation!

6 Note: Boussinesq approximation assumes incompressible fluid of constant density in momentum and continuity equations, but body force is due to density variation! (self-contradictory assumption, but works quite well for finite variations of density with temperature!)

7 10.3. Boundary-layer approximation x Away from surface: T = T0 U=0 d T0 g y dt Ts On the surface: T = Ts (or other condition) U=0

8 BL thicknesses: thermal dt and velocity d Three possible cases (both for natural and forced convection): dt << d: velocity in thermal BL small, heat transfer dominated by conduction, energy equation uncouples dt >> d: temperature gradient in velocity BL small, can assume it plays negligible role, energy equation uncouples dt ~ d: have to solve energy equation with momentum and continuity to get profiles for natural convection (Prandtl number ~ 1!)

9 Consider dt ~ d Full energy equation (1.14) ( ) u k e e ρ +ρ u k = p + k +Φ t xk xk x j x j ( ) ( ) u k u i u j u j Φ=λ +μ + xk x j xi xi Dissipation function Rewrite in terms of enthalpy h = e + p/r (r = const)

10 ( ) ( ) p p ρ h ρ +ρ uk h ρ = t xk uk p + k +Φ xk x j x j ( ) Open brackets... h p h p ρ +ρu k u k = t t xk xk u k p + k +Φ xk x j x j ( ) For incompressible fluid, this term is zero! (continuity)

11 ( ) h h p p ρ +ρu k = +u k + k +Φ t xk t xk x j x j Steady flow, D, k = const ( ) h h p p T T ρu +ρ v =u +v +k + +Φ Let h = cpt ρc p ( ( ) ) p p T T u +v =u +v +k + +Φ finite small Terms of same order Toss this term thin BL, no yvariation of pressure This << this

12 Equations we need to solve... u p T u +v = +κ +Φ y ρcp x Thermal diffusivity k κ= ρcp For F, assume it's negligibly small (works for low viscosity, high Re - and all boundary layer flows are high-re flows!) u u 1dp u u +v = ρ +ν +g β(t T 0 ) dx Body-force term (Boussinesq u v BL equations from approx.) + =0 Ch. 9 y

13 10.4. Vertical isothermal surface x Away from surface: T = T0 U=0 p = const d T0 g y dt Ts On the surface: T = Ts u=0

14 Thermal BL equations (zero pressure gradient) T u +v =κ u u u u +v =ν +g β(t T 0) u v + =0 y u y=0, y =0 Note no condition on v at edge v y=0=0 of BL (that would overdefine the system!) T y=0=t s T y =T 0

15 Solution approaches No length scale in x-direction similarity solution Can be simplified with polynomials (nd order for T, 3rd for u) - Pohlhausen Dimensionless parameters Nusselt number Convection coefficient hl Nu= k Length scale Grashof number 3 Gr= g l ( T s T 0) ν T0 Ra=Pr Gr Rayleigh number Prandtl number ν Pr= κ

16 Physical meaning of dimensionless characteristics of thermal convection Nu how much more efficient convection (hl) is when compared to conduction (k); sometimes referred to as dimensionless convection coefficient Gr non-dimensional temperature differential driving the convection Pr measure of respective importance of mechanical (n) to thermal (k) dissipation Ra dimensionless buoyancy force

17 Pohlhausen's result for thermal convection in air (Pr ~ 0.7) 1 /4 ( ) δ =5 Gr x x 4, δ x Nu Gr 1/ 4 1 /4 Thermal BL stability Ra x,c = g β ( T s T 0 ) x νκ 3 10 Above Rax,c transition to turbulence! 9

18 10.7. Stability of horizontal layers y T = T r = r y=h Buoyancy x T = T1 r = r1 T1 < T (hot above cold) stable (and stably stratified) horizontal thermal layer, heat transfer by conduction only T1 > T (cold above hot) for some DT, unstable stratification leads to convection?

19 Governing parameter g β( T 1 T ) h Ra= νκ 3 Low Ra no movement, heat transfer by conduction only (viscous forces >> buoyancy) Consider small perturbation (velocity and temperature, 3D, time-dependent), look for Ra when any such perturbation can start growing Thermal convection should start at Ra > Rac =1708 Great agreement with experiment!

6.2 Governing Equations for Natural Convection

6.2 Governing Equations for Natural Convection 6. Governing Equations for Natural Convection 6..1 Generalized Governing Equations The governing equations for natural convection are special cases of the generalized governing equations that were discussed