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10. Buoyancy-driven flow ● For such flows to occur, need: ● field ● Variation of (note: not the same as variable density!) ● Simplest case: ● Viscous flow, incompressible , density-variation effects only present in

body term , c. 287 BCE – c. 212 BCE ● Natural ● Forced convection ● Movement is due to ● Movement is due to buoyancy other ● Important forces: ● Important forces: – Viscous – Viscous – Buoyancy – Buoyancy – Other ● Density variation ● Density - unknown mostly relevant for body-force term only ● Need energy equation and equations of state to close the system!

10.1. Boussinesq approximation

● Incompressible Navier-Stokes with gravity as body force ∇⋅u=0 ∂u ρ +ρ(u⋅∇ )u=−∇ p+μ ∇ 2u−ρ g e ∂t z Joseph Valentin ● (u = 0, p = p , r = r ): Boussinesq 0 0 (1842-1929) =−∇ −ρ 0 p0 0 g ez ● Buoyancy-driven convective motion:

u = u*, p = p0+p*, r = r0 + r* Plug expressions for p, r, u into equation: ∂ * * u * * * (ρ +ρ ) +(ρ +ρ )(u ⋅∇ )u = 0 ∂ t 0 −∇ ( *+ )+μ ∇ 2 *−(ρ +ρ*) p p0 u 0 g ez Throw out hydrostatic terms Linearize, assuming ρ ≫ρ* ρ=ρ +ρ*≈ρ 0 , 0 0 Drop * and , replace r* = Dr ∂u 0 ρ +ρ(u⋅∇ )u=−∇ p+μ ∇ 2 u−Δρ g e ∂t z

Boussinesq approximation to momentum equation 10.2. Thermal convection Fluid density change due to small variations: ρ=ρ ( −β( − )) 0 1 T T 0

Thermal expansion coefficient

(valid for r  r0, T  T0)

For ideal , b =1/T0 , so Boussinesq equation becomes ∂u ρ +ρ(u⋅∇ )u=−∇ p+μ ∇ 2 u−ρ gβ(T −T )e ∂t 0 z

r is not variable, but still need energy equation! 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!)

10.3. Boundary-layer approximation

x

Away from surface: d T = T 0 T U = 0 s T0 dT On the surface:

T = Ts (or other condition) U = 0

g

y

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!) Consider dT ~ d Full energy equation (1.14) ∂ ∂ ∂u ∂ ∂ ρ e+ρ e =− k + ( T )+Φ ∂ uk ∂ p ∂ ∂ k ∂ t xk xk x j x j

∂u 2 ∂u ∂u ∂ u Φ=λ( k ) +μ( i + j ) j ∂ ∂ ∂ ∂ xk x j xi xi Dissipation function

Rewrite in terms of enthalpy h = e + p/r (r = const)

∂ ∂ ρ ( − p)+ρ ( − p)= ∂ h ρ uk ∂ h ρ t xk ∂ ∂ ∂ − uk + T +Φ p ∂ ∂ (k ∂ ) xk x j x j Open brackets... ∂ ∂ ∂ ∂ ρ h− p +ρ h − p = ∂ ∂ uk ∂ uk ∂ t t xk xk ∂ ∂ ∂ − uk + T +Φ p ∂ ∂ (k ∂ ) xk x j x j

For incompressible fluid, this term is zero! (continuity)

∂ ∂ ∂ ∂ ∂ ρ h+ρ h = p+ p + ∂ ( T )+Φ ∂ uk ∂ ∂ uk ∂ ∂ k ∂ t xk t xk x j x j Steady flow, 2D, k = const ∂ ∂ ∂ ∂ ∂2 ∂2 ρ h+ρ h = p+ p+ T + T +Φ u ∂ v ∂ u ∂ v ∂ k( ) x y x y ∂ x2 ∂ y2

Let h = cpT ∂ ∂ ∂ ∂ ∂2 ∂2 ρ ( T + T )= p + p+ ( T + T )+Φ c p u ∂ v ∂ u ∂ v ∂ k x y x y ∂ x2 ∂ y2

finite small Toss this term – This << this Terms of same order thin BL, no y- variation of Equations we need to solve... ∂T ∂T u ∂ p ∂2T u +v = +κ +Φ ∂ ∂ ρ ∂ ∂ 2 x y c p x y κ= k Thermal diffusivity ρ c p For F, assume it's negligibly small (works for low , high Re - and all boundary layer flows are high-Re flows!) ∂ ∂ ∂2 u+ u =−ρ1 d p+ν u + β( − ) u ∂ v ∂ g T T 0 x y d x ∂ y2 ∂ ∂ Body-force term BL equations from u v (Boussinesq + =0 approx.) Ch. 9 ∂ x ∂ y 10.4. Vertical isothermal surface

x

Away from surface: d

T = T0 Ts U = 0 T 0 d On the surface: p = const T T = Ts u = 0

g

y

Thermal BL equations (zero ) ∂ ∂ ∂2 T + T =κ T u ∂ v ∂ x y ∂ y2 ∂ ∂ ∂2 u+ u =ν u + β( − ) u ∂ v ∂ g T T 0 x y ∂ y2 ∂u ∂ v + =0 ∂ x ∂ y ∣ = u y=0, y →∞ 0 Note – no condition on v at edge ∣ = of BL (that would overdefine the v y=0 0 system!) ∣ = T y=0 T s ∣ →∞= T y T 0 approaches No length scale in x-direction – similarity solution Can be simplified with polynomials (2nd order for T, 3rd for u) - Pohlhausen Dimensionless parameters Prandtl Nusselt number Length number scale Convection coefficient 3 − hl g l (T T ) ν Nu= Gr= s 0 Pr= κ k ν2 T 0 = ⋅ Ra Pr Gr 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

Pohlhausen's result for thermal convection in air (Pr ~ 0.7) −1/4 δ Gr / =5( x ) , δ∼x1 4 x 4 / Nu≈0.359 Gr1 4 Thermal BL stability gβ(T −T )x3 = s 0 ∼ 9 Ra x ,c ν κ 10

Above Rax,c – transition to turbulence!

10.7. Stability of horizontal layers y

T = T2 r = r2 y = h

Buoyancy x

r = r T = T1 1

T1 < T2 (hot above cold) – stable (and stably stratified) horizontal thermal layer, – by conduction only

T1 > T2 (cold above hot) – for some DT, unstable leads to convection? Governing parameter β( − ) 3 = g T1 T 2 h Ra νκ

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!