Toy models for Rayleigh- Taylor instability:

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Noel Lawrence
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1 Toy models for Rayleigh- Taylor istability: Stuart Dalziel Departmet of Applied Mathematics ad Theoretical Physics iversity of Cambridge Iteratioal Workshop o the Physics of Compressible Turbulet Mixig 4 December 00 (9:0 9:30) with thaks to Joae Holford (DAMTP) David Yougs (AWE) IWPCTM 8 December 00

2 The growth questio: h αagt, where But what is α? A 0.0, 0.07, 0.06, 0.03, 0.0? ρ ρ ρ ρ Timescale: T H Ag If δ h/h, ad τ t/t, the δ α τ IWPCTM 8 December 00

3 500 mm Experimets Top view 00 mm Ed view 400 mm IWPCTM 8 3 December 00

4 Appropriate modellig (?) IWPCTM 8 4 December 00

5 Growth Dimesioal aalysis/similarity theory Sigle mode Layzer (955) For ζ ( x y) if h α Ag t. πx, a0 cos λ dh w, dt dw w CD, dt λ the ( E) Ag( E) where 6πh E exp. λ Experimetally C D ~ 0 Does this make sese? Early time liear theory d h dt Late time costat velocity w πag λ h Agλ C D h w (t t 0 ) IWPCTM 8 5 December 00

6 Structure Ofte described as bubbles but more like thermals i miscible fluids IWPCTM 8 6 December 00

7 Thermals Self-similar r θ z V γ r 3. Buoyacy coserved g V g γ r 3 g 0 V 0. Costat Froude umber F Itegratig w dz/dt F γ ( g V ) 0 θ w g r 0 z t Experimetal results F.. IWPCTM 8 7 December 00

8 Rayleigh-Taylor as thermals Froude umber ~. (aspect ratio 0.7) C Thermal.3. Rayleigh-Taylor bubbles a little like thermals C D.3 But i Rayleigh-Taylor eviromet Desity field ot hydrostatic i ambiet Hydrostatic i mea desity halve buoyacy force C D.6 Flow aroud bubble affected by bubble movig i opposite directio Drag due to twice rise speed of bubble C D 0.4 I agreemet with sigle mode experimets BT atural R-T has more tha oe mode IWPCTM 8 8 December 00

9 Multi-mode What happes if λ grows with h? Let λ ψ h Late times approximatio: Ag dh dt Ag ψ C D ( E) h h ( E) ψ ( t t ) αag( t t ) C D For C D 0 ad ψ, α [Full Layzer growth with ψ gives α 0.03.] 0 0 Growth rate maximised with ψ ~ 0 givig α ~ h/dt h/h β ψ IWPCTM 8 9 December 00

10 Where do the modes begi? How do they iteract? Noliear iteractio? Iitial perturbatio? If modes idepedet ad equal amplitude: δ h/h Key δ α τ with α 0.06 λ/h 0.00 λ/h λ/h λ/h 0.06 λ/h 0.03 λ/h λ/h 0.8 λ/h 0.56 λ/h 0.5 λ/h τ (Ag/H) / t Istataeous oliear mode halvig iteractio whe h λ: δ h/h Key δ α τ with α λ/h 0.00 λ/h λ/h λ/h 0.06 λ/h 0.03 λ/h λ/h 0.8 λ/h 0.56 λ/h Which is it? τ (Ag/H) / t IWPCTM 8 0 December 00

11 Mixig See talk by Joae Holford Eergy budget Ca decompose PE ito Backgroud PE ad Available PE. PE back is the miimum eergy state that is achieved by adiabatic rearragemet of fluid parcels. Mixig icreases PE back it caot decrease it! PE avail is the compoet of PE that ca be coverted ito KE, heat (through dissipatio) ad, if mixig occurs, ito PE back. PE PE Back PE Avail I the absece of exteral work: D D KE KE KE E avail PE avail PE avail PE avail PE PE back PE back PE back time IWPCTM 8 December 00

12 Mixig efficiecy η Itegral E E back avail PEBack PEBack ε dt PE Back ( KE PE ) Avail Overall mixig efficiecy η Joae Holford Agle of tak α o.00 η istataeous δpe δe Back Avail δpe Back δke δpe Avail 0.80 Joae Holford η istataeous Time t/τ IWPCTM 8 December 00

13 Thermal Etraimet ito a thermal dv βwa dt β 0.8. Eergetics of a thermal Mixig efficiecy ot well defied: depeds o size of domai! Rayleigh-Taylor ρ λ h H ρ IWPCTM 8 3 December 00

14 h αagt δ ατ w αagt ω ατ V L h Total potetial eergy PE Total PE PE Total Backgroud potetial eergy 0 4 4α τ ρ ρ a h H ρ b ρ Chages due to etraimet betwee couter-flowig streams. Ivoke etraimet hypothesis: u e βw Area of etraimet idepedet of h depth of etraimet comparable with λ etraiig area ϕ pla area. PE Back 4 ( ϕβα τ ) IWPCTM 8 4 December 00

Available potetial eergy PE Avail PETot PEBack 4 ( 4 ϕβ ) α τ.00 τ α 4 ( ϕβ ) 4 Potetial eergy PE(t)/PE(0) 0.50 0.00-0.

15 Available potetial eergy PE Avail PETot PEBack 4 ( 4 ϕβ ) α τ.00 τ α 4 ( ϕβ ) 4 Potetial eergy PE(t)/PE(0) Key Experimet: PE Tot Experimet: PE Back Model: PE Back Model: PE Tot Joae Holford s experimets Time t/τ IWPCTM 8 5 December 00

16 Kietic eergy KE 3 4 6σα τ Kietic eergy KE(t)/PE(0) Key Experimets Model: σ Model: σ Joae Holford s experimets Time t/τ Available eergy chagig de Avail dke dpe dτ dτ dτ 4 τ Avail 3 ( 4 ϕβ 6σα ) α Hece, eergy is lost wheever α < ¼ (for β 0, σ ). IWPCTM 8 6 December 00

17 Istataeous mixig efficiecy η Ist dpeback dτ dpe Avail dke dτ dτ ϕβ 4 ϕβ 6σα So for ϕ 6, β 0.8, σ, ad α 0.06, the η Ist η istataeous Thermal predictio Joae Holford s experimets Time t/τ IWPCTM 8 7 December 00

18 Itegral mixig efficiecy If there o mixig after reachig the bottom η Itegral ( bot ) ( 0) PE PE Back PE ( 0) Avail Back η Itegral ϕβ 8 For ϕ 6 ad β 0.8, the η Itegral IWPCTM 8 8 December 00

19 If there is mixig after reachig the bottom If E E ( ) bot Avail ( After bot ) ( bot ) Back η η stab Itegral E Avail 4σα ϕβ 4, the For η stab 0., the η Itegral 0.4. ϕβ ηstab 4σα ϕβ Overall mixig efficiecy η Joae Holford s experimets Agle of tak α o IWPCTM 8 9 December 00

20 Extesios ρ λ h H c ρ Let c be the fractioal displacemet of the cetroid of the bubble from z 0. η Ist 4 ϕβ 4 δpeback δpe δke Avail ϕβ ( 4 ϕβ ) 6σα c Pyramid ( c /4): η Ist 0.6. Parabolic ( c /6): η Ist (gives liear mea cocetratio) IWPCTM 8 0 December 00

21 IWPCTM 8 December 00 How ca we avoid havig to specify C D? Shell model GOY model (Gledzer Ohkitai Yamada): ( ) F k ck bk ak dt d ν with k β k 0, a, b ε ad c ε. I Rayleigh-Taylor istability, eergy iput at all scales. ( ) ( ) F k k k k dt d ν ε ε Recall Layzer model: ( ) ( ) λ w C E Ag dt dw E D Hece E E g A F, where h E λ 6π exp ad A A h /h. The mode peetratios h ad total peetratio h are obtaied from dt dh ad h max h.

22 YF; F E-03XX; Erms.499E-0 YF; F-8.640E E-03X E-03XX; Erms.7973E-03 Peetratio (h) Time (t) Approximate quadratic growth Coefficiet depeds o iitial spectrum Possible to replicate α ~ 0.06 IWPCTM 8 December 00

23 Coclusios Geeral Iitial coditios are importat for gross features Iteral details relatively isesitive to iitial coditios Appropriate modellig of iitial coditios gives close agreemet Thermals model Sigle-mode growth rate cosistet with isolated thermal Simple model for trasfer betwee modes replicates t growth Mixig efficiecy cosistet with thermal etraimet Shell model Barocliic iput at all scales Very simple model replicates t growth Growth rate sesitive to iitial spectrum A explaatio? No, but it helps. IWPCTM 8 3 December 00

Fluid Physics 8.292J/12.330J % (1)

Fluid Physics 8.292J/12.330J % (1) Fluid Physics 89J/133J Problem Set 5 Solutios 1 Cosider the flow of a Euler fluid i the x directio give by for y > d U = U y 1 d for y d U + y 1 d for y < This flow does ot vary i x or i z Determie the