RAYLEIGH-BÉNARD-, MARANGONI-, RAYLEIGH-BENARD DOUBLE DIFFUSIVE-, CONVECTION

BENARD’S Convection EXPERIMENT

Heat transfer by diffusion Solar panels T instability —> 1 Heat transport by convection

Forced Convection (pump, fan, suction device) T0 or Natural convection (buoyancy force)

Buildings & Environment

industry mantel of the Earth, Atmospheres 3 mantel of the Sun... HEAT CONVECTION

onset Optical deformation due to heat gradient using index refraction

wavelength

Cell formation stability onset t >> tc

turbulence Ra >> Rac Cell formation

RAYLEIGH BENARD CONVECTION HEAT CONVECTION

T1 < T0 h Driven by density gradients due to gradients in concentration, temperature or composition T ρ

STATIC STABILITY u=0 ΔT 0 ρ(z) = ρ (1+ βz) β = α T0 0 h EQUATIONS With u=(u,v,w) and g=(0,0,-g) STEADY SOLUTION ρ(z) ρ(z) ⌅u Momentum ⇥ + u. u = p ⇥gk + µ 2u ⌅t ⇥ ⇥ ⇥ U=0 € ⇥ ⇤T Diffusion of heat + u. T = 2T ⇤t Hydrostatic balance dp Continuity .u =0 = (z)g Stable Statically unstable dz

⇥(T ) ⇥0 ⇥(z) ⇥0 (T0 T1) Density = (T ) = (T T0) = z ⇥0 ⇥0 h parameters g ν κ β Dimensions LT-2 L2T-1 L2T- L-1 Boussinesq approximation ∆ρ ≈ 1% , (ρ - ρ0)/ρ0 =α(T0 - T)<< 1 only density variation in z considered

€ * * 2 -1 * 2 2 * X=x H, t=t H /κ, u=u*H κ, p=p ρ0κ /Η T=T ΔΤ Note: Boussinesq approximation thus gives for density perturbations ρ = ρ +ρ’ 0 Substitution gives for the linearized equations: and related pressure perturbation p=p0 +p’ with hydrostatic equilibrium :

1 ⌅p0 + p 1 ⌅p0 + p g = g u 2 + ⌅z (1 + / ) ⌅z = p + Ra P r T k + Pr u 0 0 0 t 1 ⌅p + p 1 ⌅p 1 ⌅p ⇥ ⇥ 0 0 + ... g = + g ⇥ ⌅z ⌅z ⌅z T 2 0 0 0 ⇥ 0 0 w = T t ⇥ Linearized perturbation equations around the basic state: .u =0 u = (u’ ,v’,w’) ; T = T0 + Τ’ and with ρ’/ρ0 = α T’ into the equations and keep linear terms in the perturbation

2 2 2 ⇤u 1 ⇤p 2 2 ⇤ ⇤ ⇤ 3 4 = + u and = 2 + 2 + 2 ⇥ diusivity of momentum gTh g⇥h ⇤t ⇥0 ⇤x ⇥ ⇥ ⇤x ⇤y ⇤z Pr = = Ra = = diusivity of heat ⌅⇤ ⌅⇤ ⇤v 1 ⇤p 2 ⇥u ⇥v ⇥w = + v .u = + + ⇤t ⇥0 ⇤y ⇥ ⇥x ⇥y ⇥z

⌅w 1 ⌅p 2 ⇥T T0 T1 2 = + gT + ⇥ w w = T Pr = 7 for heat in water and ~O(100) for salt in water ⌅t ⇤ ⌅z ⇥ ⇥t h ⇥ 0 0.7 in air and has very small values in liquid metals

9

Five equations, five unknown --> eliminate the pressure with continuity to obtain a single equation by taking the curl of the velocity ¯ = u¯ and using .u¯ =0 ⇥ ⇥ ⇥¯ 2 2 ¯ = u¯ = Pr ¯ + Ra P r T k¯ (A) 2 6 4 2 ⇤ ⇥ ⇥t ⇤ ⇤ 2 w + Pr w (1 + Pr) w = Ra P r hw ⇥ ⇥T t ⇥ ⇥ t⇥ ⇥ .u¯ =0 2u¯ = Ra P r 2T k¯ + Pr 4u¯ ⇤ ⇥ ⇥t⇤ ⇤ ⇤⇥z ⇤ ⇥ Normal modes w(x, y, z)=Re [W (z) exp i[(kxx + kyy)+st]] 2 2 2 2 4 w = Ra P r hT + Pr w and h = + (B) 2 2 2 2 1 2 2 2 t x2 y2 ( k s)( k Pr s)( k )W + k RaW =0 z z z T 2 (C) w = T 2 2 2 t ⇥ k = kx + ky Eliminate w or T from B, C to obtain a single equation in T or w (same eq.) 2 2w + Pr 6w (1 + Pr) 4w = Ra P r 2 w t2 ⇥ ⇥ t⇥ ⇥h Boundary conditions --->

Note that from (A) we can solve the z-component NOTE: FOR TAYLOR COUETTE FLOW, THE RELATION IS ≈ THE SAME ⇥v ⇥u ⇥ = z = Pr 2 z ⇥x ⇥y ⇥t ⇥ z T0 1 Stress free, perfectly conducting boundaries ( Rayleigh 1916) 2 2 2 2 1 2 2 2 T(z) ρ(z) ( k s)( k Pr s)( k )W + k RaW =0 z z z Marginal stability curve s=0 (Neutral stability curve) Boundary conditions 0 ρ0 2 2 3 2 [( @z k ) + Ra k ] W =0 In all cases W=0 and Tz=0 at the boundaries z=0 and h (0 and 1 scaled) the temperature is maintained at a constant at z=0 and h 2 2 2 Boundaries: W=∂z W=(∂z - k ) W=0 (z=0,1) A) Bounding surfaces are rigid: no slip (zero tangential velocity) most realistic Solution W=sin nπz with (n2π2+k2)3=Ra k2 u=0, v=0 at z=0, 1. Since this is valid for all x and y, this is also valid for 3 4 ux=0 and vy=0 so that ⇤w αgΔTh βgh .u =0 = 0 and idem for vorticity =0 First unstable mode for n=1 Ra = = ⇥h ⇤z z κν κν

n=3 n=2 B) Bounding surfaces are free surfaces: free slip (zero tangential stress) (easiest!) Ra unstable n=1 w1(z) ⇧u ⇧v 2 2 3 2 = =0 = 0 and =0 Neutral curve Ra = (k +π ) /k xz yz ⇥ ⇧z ⇧z €

⇧ ⇧2w w2(z) .u¯ =0 = 0 at z =0, 1 2 4 ⇥ ⇧z ⇤ ⇥ ⇧z Ra Rac=27π /4 =658 ⇧ ⇧u ⇧v ⇧⇥ c stable = z = 0 at z =0, 1 (k=π/21/2, n=1) ⇧z ⇧y ⇧x ⇧z k ⇥ k 0 z 0.5 ⇧4w c with equations = 0 at z =0, 1 Racrit does not depend on Pr but the growth rate does! ⇧z4

NOTES Perfectly conducting boundaries - critical value O(103) depends on boundary conditions Free and one rigid boundary (Benard problem) Rac≈ 1101 at kc= 2.68 Realistic temp differences result in Ra>>Rac

Ra increases rapidly with h: deeper layers produce higher Ra numbers ! Two rigid boundaries Rac≈ 1708 at kc=3.12

3 4 at 200 αair≈ 0.003K-1 αgΔTh βgh Ra = = αwater≈0.0002 K-1 κν κν

- Lowest mode n=1 has the least vertical structure. Vertical velocity has With fixed heat fluxes Tz=constant at boundaries (different T-curve) only one sign but changes sign for higher modes. € Ra (correlation between vertical motion and buoyant fluid results from the momentum generated by the buoyancy forces.)

- linear analyses determines only the horizontal scale and not the structure ! Rac Two stress free boundaries Rac= 120 k Two rigid boundaries Rac= 720 0 Flows related to Rayleigh Benard ! How do we measure the transition to convection ?

Heat Flux : The (⌘ = µ) Heat transfer rate F=λ(∆T)/H, λ is heat transfer coefficient FH Nu = T

total heat flux Nu = conducted heat flux

In unsteady flows the Nusselt number is averaged in time

no flow Nu=1 flow Nu>1 Guyon, Hulin et Petit, 1982

Boundary layer structure

= Ra/Rac

increasingly turbulent (mixed) in Ra/Rac≤1 the interior for higher Ra-numbers

T – T0 |∆T| onset of Rayleigh Benard convection For larger Ra the boundary layer thickness decreases Interior ≈ homogenous Turbulence convection

laminar boundary layer δ laminar boundary layer H T Turbulent interior H T Turbulent interior laminar boundary layer δ laminar boundary layer δ u=w=0

the boundary layer thickness grows with δ~(κt)1/2 until it becomes heat flux in each boundary layer δ is Fδ = λ Tδ / δ unstable and releases its heat (see*) 3 F ↵g(T/2) the Nusselt number for the boundary layer is then Nu = for the boundary layer δ Ra = = Raδ crit.=1708. T/H ⌫ ✓ ◆ T H Nu = 2Ra 1/3 T ⟹ c = crit H Ra ✓ H ◆ From observations Tδ ≈ ∆T/2 so that Nu ≈ H/2δ the scaling of the Nusselt number was Nu ≈ H/2δ:

H 1 1 Ra 1/3 Nu = = 0.077Ra1/3 2 2 2 Ra ⇡ ✓ c ◆

(*note: there are other scalings, see e.g. Lohse & Xia Ann. Rev Fluid Mech22 2009, Ahlers et al (2009))

Turbulence convection *Plumes released from the boundary laminar boundary layer T=T0+∆T/2 h T turbulent interior h Turbulent interior laminar boundary layer thermal boundary layer δ(t) T=T0+∆T kT HEAT FLUX => F = Nu Howard 1966 H h in the interior Nu ~ Ra1/3 The thermal BL grows by thermal diffusion Interior is turbulent and uniform : FH independent of h, T z Then we have for a fixed Prandtl number P T = T0 + 1 + erfc = p⇤⇥t 1/3 2 2pt 3 ✓ ◆ 1/ 3 ' αgΔTh * Nu = f (Ra,P) ∝ Ra = ) , this goes on until it is unstable and breaks down at tc ( κν + (i.e. at Ra_c) and δ breaks down to zero: and for the turbulent heat flux: gT ⇥3 2⇤⌅ 1/3 3 1/3 c = Ra ⇥ = Ra1/3 ↵gk 4/3 4/3 2⇤⌅ c ) c gT c F = c(P ) T = C T ✓ ◆ H ⌫ € ✓ ◆ the time cycle of the plume formation can be calculated as Also called 4/3 law. (Howard, Appl Mech.1964) h2 Ra 2/3 t = c 24 ⇥ Ra ✓ ◆ Turbulence convection Castaing et al, J. Fluid Mech. 1989: boundary layer relation with turbulent interior general comments

These laws are also known as Malkus, Kraichnan, Howard, Ginsburg-Landau (GL) model. The general law is written as Nu~Ra2/7 Nu=C Raβ Prγ Nu~Ra1/3 allows to represent all models for infinite horizontal dimensions. For a finite horizontal size, D, the constant C=C(Γ)depends on the aspect ratio Γ=D/H.

Malkus (1954) proposed β= 1/3 and γ= 0 supposing a heat flux that is independent with height.

possible cartoons for the flow pattern in the cell centre. 3 regions: BL, mixing zone, turbulent interior Kraichnan’s law, β= 1/2, and boundary layers can become unstable. recent research: - Geometry effects: geometry modifies temperature and vertical velocity fluctuations at the cell The GL theory proposes a phenomenological law with a linear combination center and their Rayleigh-number dependence; interior turbulent fluctuations are nonuniversal in of expressions like above and some adjustable parameters that easily allow to fit the data. low-aspect ratio convection (Daya &Ecke 2001 Phys Rev Lett; for tilt of container Chilla et al In the fully turbulent regime, for 106 < Ra <1011 , all experiments agree within an error of 10% 2004). - A possible reason are different thicknesses of the kinetic boundary layers at the sidewalls and at the top/bottom plates (Lohse & Xia Ann. Rev Fluid Mech 2009, Ahlers et al (2009), Rev Mod For Ra > 1011 there is a clear divergence up to a even a factor 2 in Nusselt number Phys 81:503) ... 26

Some RB Convection setups Ilmenau, Germany (Eberhard Bodensatz) Present state of the art of heat transfer power law for very high Ra number ENS Lyon (Francesca Chilla) Chilla & Schumacher 2012 Inst Neel Grenoble (Philippe Neel) Cryogenic Eindhoven Pays Bas (rotating convection, Rudie Kunnen) NLNA US, (Bob Ecke) Santa Barabara, (US Günter Ahlers) HongKong China (Ke-Qing Xia) …. Expériences de laboratoire LEGI CONVECTION T4 PATTERN T T1 2 FORMATION

T3

LEGI Grenoble

Dependence on Prandtl number Ra Rayleigh-Benard Convection (h2 + 2)(Ra Ra ) n =1 s = c high aspect ratio low aspect ratio (1 + Pr 1)Ra c Ra heat diffusion c pattern control opposes momentum diffusion Route to chaos: instability opposes instability k without defects with defects k Rac is the critical Rayleigh number, Ra > Rac c high Pr low Pr low Pr high Pr period doubling Quasi periodicity The Prandtl number does affect the growth rate and convection type instability rolls rolls rolls rolls with imperfection (even at large Ra and large Pr) intermittency skew -varicose cross-rolls dislocation For large Pr-number ν/κ>>1 (e.g. salt and momentum diffusion) (larger rolls) movement Faster momentum diffusion!! oscillating oscilating localised large Ra number (∆T large): rolls rectangles oscillations Large (broad) convection cells temperature anomalies in narrow regions loss of periodicity spatial structure; temporal randomness loss of spatial structure For small Pr-number ν/κ<1 (liquid metals etc.) 4 Ra>10 turbulent convection (earlier for small Pr<1) Ra Narrow convection cells (slower momentum diffusion, faster heat diffusion) temperature anomalies broaden into large regions. 31 illustrations of convection patterns homogeneous time dependent convection spots

turbulence

steady rolls

33

Cell patterns

Disturbances are characterized by a particular wave number, but the convection 2 pattern is unspecified, because the wave vector k can be resolved into kx and ky in infinitely many ways.

One has to consider a certain symmetry in advance in θ (x,y,z,t) or w(x,y,z,t) For instance for

2 2 1/2 rolls: θ= θ(z) cos k . x k= (kx + ky )

Rentangular & square cells: θ= θ(z) cos kxx cos kyy

Hexagonal cells: θ= θ(z) [2cos k x/√3 cos k y /√3 + cos 2/3k y ] 2 2 1/2 k= (kx + ky ) - 2π/L and L the size of the hexagon e.g Chandrasekhar 1961 p 43-52. Flow structures with increasing Ra-number

oscillatory « natural » Convection No structure In the forcing

Osillatory instability 3 - 10 ×

Ra Ra MARANGONI 3 - Pr

10 × OR Ra Ra 3

- RAYLEIGH -BENARD k 10 ×

Ra Ra k CONVECTION ? Ra × 10-3 k k

Busse 1981 Springer ed. Swinney & Gollub 1981 MARANGONI EFFECT z Near a wall dγ dT > 0 < 0 Some solubles (soap) or temperature gradients create surface tension. x a dx dx h(x) a Soap or heat vx(z)dz =0 € € vx(z) Z0 F F See Guyon, Hulin & Petit p 250 γ= γ0 vz(0) = 0 γ <γ0 γ =γ0 L

for the pressure we have: p=patm + ρg(a-z) dx Since the pression does not depend on x and the fluid is at rest, NS gives near the wall: This results in a tangential stress and deformation of the liquid layer 2 ⇤ vx dh for small temperature gradients, γ(Τ)= γ(Τ0)(1−b(T-T0)) = ⇥g ⇤z2 dx dγ dγ dT $ dT ' surface tension = = −bγ(T0 )& ) dx dT dx % dx ( with the balance above and boundary conditions we obtain then:

g dF dγ % dT ( at the free surface : σ xyz = = = −bγ(T0 )' * Ldx dx & dx ) 2 € ⇤g dh z az dh 3 b(T0) dT @vx viscous stress: ⌫ = µ σ γ + σ ν = 0 vx(z)= = xyz xyz xyz ⇥ dx 2 3 dx 2 ⇤ga dx @yz  €

€

y BENARD-MARANGONI CONVECTION Equations are given by

T1+θ y=a T1

Vy(x) y=0 x

T2 Boundary conditions Motion driven by surface tension, heat diffuses By continuity hot fluid aliments the process bottom top viscosity and diffusion oppose the convective motion as in Rayleigh Benard convection bTa 1 @ heat transfer at the boundaries (with Bi the , Bit indicating the heat Marangoni number Ma = and b = µ @T transfer at the top (t) or bottom (b) of the boundary layer, λ is the thermal conductivity) Its derivation is similar to that of the Rayleigh number Ra 2 ga2 The ratio is Ra αρga δρ (same δΤ for δρ and δγ) = = For discussions of Bi see Bragard & Velarde JFM 1998 48 Ma dγ dT δγ

€ Linear stability analyses: Bit=0, Bib ––> ∞ Pearson JFM 1958 , perturbation Bit=1, Bib ––> ∞

Bit=0, Bib=0.01

The Marangoni number is Amplitude equations etc. to consider the structures above the critical Ma number (see Bragard & Velarde 1998) see Bragard & Velarde JFM 1998 for further details

49 50

Effect was first observed by James Thompson 1855 in a glass of wine

alcohol < water different alcohol concentrations

liquid flows away from regions with lower alcohol concentration (=higher tension)

What happens ? Explain

51 52 Flows related to Rayleigh Benard !

(⌘ = µ)

Skotheim, J.M, and Bush, J.W.M., 2000. Evaporatively-driven convection in a draining soap film, Gallery of Fluid Motion, Physics of Fluids , 12 (9)

Hosoi, A.E. and Bush, J.W.M., 2000. Evaporative instabilities in climbing films, J. Fluid Mech. , 442, 217-229

example in paints

53 Guyon, Hulin et Petit, 1982

DOUBLE DIFFUSIVE CONVECTION

Stable stratification and unstable… Two or more stratifying agents compose a stable density stratification DOUBLE but diffuse at different rates. DIFFUSIVE Opposing effects: heat diffusion and dissipation due to viscous effects CONVECTION hot salty

ρ ρT ρs

cold fresh

−2 κs/κΤ=10 for salt and heat (1/3 for sugar and salt) Consider a blobs displacement (up and down) and see whether it is stable OSCILLATORY DIFFUSIVE CONVECTION

Τ+ cold fresh τa

Τ+ Τ+ ρT ρs ρ etc.

hot salty Τ_

Ta Consider a blobs displacement and see whether it is stable….

Over-stable oscillation = oscillation but unstable ! i.e. w~ eiωt+ikx with ω=ζ+iξ ζ ≠0 and ξ≠0 !

Consider different cases of diffusion of salt &heat ⇤(S, T )=⇤ (1 T + ⇥S) m (A) cold salty (B) hot fresh

ρT ρs ρT ρs

ρ(T,s)

hot fresh cold salty

(C) hot salty (D) cold fresh

ρT ρs ρT ρs C et D deja vue

cold fresh hot salty 60 Layer formation ... EXTENDED RAYLEIGH-BENARD PROBLEM More precisely (following Baines and Ghil 1969) Linear stability analyses:

Boussinesq perturbation equations are After linearization the equations for salt and heat are

∂S ∂ψ κ ∂T ∂ψ + = τ∇ 2S τ = s + = ∇ 2T ∂t ∂x κT ∂t ∂x and the vorticity equation is

−1 ∂T ∂S 4 P ∇ψt = −RT +€R s + ∇ ψ € ∂x ∂x gSh3 αgΔTh 3 ν Rs = RT = P = s⌫ κTν κT linear stability anal. Schmitt Ann. Rev Fluid Mech 1994; 62 € Baines & Gill 1969 J. Fluid Mech

€ €

Boundary conditions at z=0,1 are:

Solutions are of the form

The dispersion relation for p=k2q is then (with k2=π2(α2+n2) )

From this relation the different instability regimes as a function of R’ and R’s are obtained (see Baines & Ghil 1969).

63 64 D: hot salty Examples 1: The ocean water below

Marginal A: Warm fresh Stability water below line

C: Cold fresh B : Cold salty water below water below

Examples 2: Coffee

heated bottom below a salt gradient Remark Inclined isolated slopes SUMMARY

-BENARD MARANGONI CONVECTION

-RAYLEIGH BENARD CONVECTION

-HEAT TRANSPORT

-DOUBLE DIFFUSIVE CONVECTION

Ficks law at the wall is: ∂ρ/∂n=0