Heat and momentum transport in turbulent horizontal convection at low Prandtl numbers
aa r X i v : . [ phy s i c s . f l u - dyn ] M a y Heat and momentum transport in turbulent horizontal convection atlow Prandtl numbers
Pierre-Yves Passaggia ∗ Alberto Scotti † Brian L. White ‡ September 18, 2018
Abstract
The transition to a new turbulent regime in horizontal convection in the case of low Prandtlnumbers is analyzed using the Shishkina, Grossmann & Lohse (SGL) theory. The flow driven bythe horizontal gradient along a horizontal surface, perpendicular to the acceleration of gravity isshown to transition to turbulence in the plume and the core. This transition to turbulence setsa sequence of heat transfer and momentum transport scalings which are found to follow the SGLprediction for the scaling factors and the prediction of Hughes, Griffith & Mullarney (HGM) forlarger forcing amplitudes. These results embed the HGM model in the SGL theory, agreed andextends the known regime diagram of horizontal convection, and provide the first evidence of bothregimes at low and intermediate Prandtl numbers and sheds new insights on the role of HC in theearth’s inner core dynamics.
Turbulent Horizontal Convection (HC) in highly thermally conductive liquids plays a major rolein geophysical flows. For example, it drives the Earth’s outer core dynamics via either heat transportfrom the inner to the outer core [2] or Joule effect due to Earth’s magnetic field [20]. Although a lot ofattention has been devoted to Rayleigh-B´enard Convection (RBC) for the outer core’s dynamics, it isonly very recently that HC has attracted the attention of planetary scientists [1]. Lateral motions inthe outer core are responsible for driving large scale horizontal flows in the mantle and have long beenthought to be at the origin of striping and faulting of tectonic plates and strike-slip earthquakes [7].At the edge of Earth’s inner core, horizontal regions of thermally stable (crystallizing) and unstable(melting) stratified layers explain the East-West asymmetry of the inner core[1]. However, only verylittle is known about the properties of the turbulent horizontal flows generated in these regions. In thisletter, we report Direct Numerical Simulation (DNS) results on how the Reynolds number (Re), theturbulent kinetic energy dissipation ( ǫ u ) and the Nusselt number (Nu) depend on the Rayleigh number(Ra) and the Prandlt number (Pr) in turbulent HC at low Pr for different Prandtl numbers charac-teristic of convection in gases where 0 . < Pr < O (10 − ) (seeref.[20]). The results are in agreement with the scaling power laws recently derived by Shishkina et al. [16] based on the original work of Grossmann & Lohse [4] (GL) and numerical simulations of Takehiro[20]. It also provides a connection between the GL theory and the plume driven dynamics derivedby Hughes et al. [6]. Our simulations cover the laminar Rossby regime I l (see ref.[12]), the high-Prlaminar regime I ∗ l recently reported by Shishkina & Wagner [17] and a new low-Pr turbulent regimesnamed II l (see ref.[19, 4] for theoretical predictions of HC and RBC), which is the first turbulentlimiting regime reported in HC. We also observe the plume dominated flow regime of Hughes et al. ∗ Department of Marine Sciences, University of North Carolina, Chapel Hill, NC 27599, USA. Email: [email protected] . † Department of Marine Sciences, University of North Carolina, Chapel Hill, NC 27599, USA. Email: [email protected] . ‡ Department of Marine Sciences, University of North Carolina, Chapel Hill, NC 27599, USA. Email: [email protected] . / /
160 1 z x y z x y (a) (b) b b (c) − − − − k − / Ψ ( k ) kL/ π Figure 1: Snapshot of the iso-contours of Λ = − Pr − / b (background)at (a) Ra = 6 . × , P r = 0 .
01 showing the new regime ( II l ) and (b) Ra = 6 . × , P r = 1corresponding to the Hughes’ regime [6] ( II u ).(c) Turbulent spectra Ψ( k ) at Ra = 6 . × atPr = 0 .
05 (red) and Ψ( k ) / k − / turbulence cascade whereas the green vertical line shows the wavenumber of the forcing.(see ref.[6]), that we name II u according to the SGL theory. These results also agree and extend theregime diagram of horizontal convection proposed in Hughes & Griffith (see ref.[5]).Similarly to Shishkina & Wagner [17], we exploit the idea that in turbulent thermal convection,the time- and volume-averaged thermal and viscous dissipation rates are determined to leading orderby their bulk or Boundary Layer (BL) contributions. For the ease of comparison, we follow the sameway of presenting as Shishkina & Wagner [17]. We consider here the problem of convection in theBoussinesq limit, where the density difference ∆ ρ = ρ max − ρ min across the horizontal surface is asmall deviation from the reference density ρ min . In this limit, the equations of fluid motion are D u Dt = −∇ p + b e z + (cid:18) PrRa (cid:19) / ∇ u (1a) ∇ · u = 0 (1b) DbDt = (Pr Ra) − / ∇ b, (1c)where D/Dt denotes the material derivative, u = ( u, v, w ) T is the velocity vector, b = − g ( ρ − ρ min ) /ρ min is the buoyancy, g is the acceleration of gravity along the vertical unit vector e z and p is the hydrodynamic pressure. The Prandtl number is given by Pr = ν/κ where ν and κ are2 − − − I ∗ l ( , I l ( , ) II l ( , ) ( , ) II u ( , ) P r Ra Figure 2: (a) Sketch of the phase diagram in the (Ra , Pr) plane for the laminar regimes I l and I ∗ l together with the turbulent scalings II l with the conducted DNS. The yellow stripes shows thetransition from I ∗ l to I l , and I l to II l , with a slope P r ≈ Ra / . The transition from II l to II u with a slope P r ≈ Ra − . Symbols reflect the computational meshes in ( x, y, z ), used in the DNS:512 × ×
256 (circle), 1024 × ×
128 (squares), and 2048 × ×
256 (squares). The values ( α, β )in each region provide the exponents Nu ∼ Ra α Pr β measured in the DNS and derived in the theory.the viscous and stratifying agent’s diffusion coefficients. The Navier-Stokes equations are solved ona Cartesian grid, stretched near the upper boundary using second-order finite volumes while thepressure is solved using a standard projection method. Laminar and turbulent flow solutions wereintegrated in time using an explicit second order Adams-Bashforth type-scheme. The Rayleigh num-ber is defined such that Ra = ∆ L / ( νκ ) where L is the horizontal length scale of the domain and∆ = − g ( ρ max − ρ min ) /ρ min . The computational domain is a parallelepiped of aspect ratio Γ = 4with dimensions [ L, W, H ] = [1 , / , /
4] where W is the width of the computational domain [13].A buoyancy profile is imposed at the surface z = H where H is the height of the domain using abuoyancy profile such that b ( x ) | z = H = (1 + tanh(9 . x )) / x = ± L/ y . This is in contrast with Shishkina & Wagner[17] wherethey used no-slip boundary conditions and end walls in the transverse direction. Our approach avoidsthe numerical difficulties involved with resolving the no-slip BL and a finite domain in the transversedirection. Instead we privilege numerical efficiency and report results for Ra up to 6 . × for awide range of Prandtl numbers.The turbulent scalings for momentum and buoyancy transport are computed using Direct Numer-ical Simulations (DNS) in the range Ra = [6 . × , . × ] and 0 . ≤ Pr ≤
2. For Ra < and 0 . ≤ Pr ≤
2, the HC flows are steady [17, 10]. With increasing Ra and/or decreasing val-ues of Pr, HC flows become increasingly unsteady, leading to turbulence (as shown in figure 1(c))and the mesh size is increased in order to resolve the Kolmogorov length scale (see ref.[14] for de-tails about turbulent HC). Mesh sizes are reported in Fig.2(b) in the (Ra , Pr) plane along with thedifferent regimes reported later in this manuscript. Note that turbulence in HC for moderate val-ues of Pr is confined to a narrow region located under the cooling/heavy boundary consisting of theplume and the BL where the fluid is statically unstable (cf. Fig.1) [3, 14]. Decreasing values of Pr3 . Nu ∼ Ra / Nu ∼ Ra / Nu ∼ Ra / Nu ∼ Ra / N u R a − / Ra − − − Nu ∼ P r / Nu ∼ P r / Nu ∼ P r / Nu ∼ P r / N u P r − / P r (a) (b) − Re ∼ Ra / Re ∼ Ra / Re ∼ Ra / Re ∼ Ra / Re ∼ Ra / R e R a − / Ra − − − Re ∼ P r − / Re ∼ P r − / Re ∼ P r − Re ∼ P r − / R e P r P r (c) (d)Figure 3: (a),(c) Ra dependencies and (b),(d) Pr dependencies of (a),(b) the Nusselt number and(c),(d) the Reynolds number, as obtained in the DNS for (a),(c) Pr = 1 (squares), Pr = 0 . .
01 (triangles) and for (b),(d) Ra = 6 . × (diamonds) and Ra = 1 . × (pentagones).Pr dependence of ¯ ǫ u with Re (d). The DNS results support the scaling in the regime I l (solid lines)[Eqs. (7a) and (7b)], transition to II l (dotted lines) [Eqs. (9a),(9b) and 11),(11)], transition to II u (dotted lines) [Eqs. (14a) and (14b)].increases the volume of fluid subject to turbulence (see Fig.1) and extends the depth of the circulation.The dependences of Nu and Re with respect to Ra and Pr are summarized in Fig. 3(a-d). For allvalues of Pr and Ra, the Nusselt number Nu ∼ Ra α [see Fig. 3(a)] transitions from: • the enhanced laminar scaling α = 1 / • the classical laminar scaling α = 1 / • A new α = 1 / • the entrainment-type regime α = 1 / u · u ) / L /ν where the overbar denotes thespatio-temporal average over the computational domain. We observe the laminar scalings Re ∼ Ra γ with γ = 1 / γ = 2 / γ = 1 / γ = 2 / ∼ Pr β with: • β = 1 / • β = 1 /
10 for Ra < see (see ref.[12]), • the new exponent β = 1 / • β = 1 / > × see (see ref.[6]).The Reynolds number dependence Re ∼ Pr δ with δ = − / δ = − − . Pr . . δ = − / − − ∼ Ra / − / ∼ Ra / − / N u R e − / − − − − − − ∼ P r / ∼ P r ∼ P r / (a) (b) − L ν − ¯ ǫ u R a − Ra − − − − ∼ P r − P r (c) (d) − − − − − ∼ P r ¯ ǫ u R e − P r (e)Figure 4: (a),(c) Ra dependences and (b),(d) Pr dependences of (a),(b) NuRe − / and (c),(d) L ν − ¯ ǫ u Ra − , as obtained in the DNS for (a),(c) Pr = 1 (squares), Pr = 0 . . Ra = 10 (diamonds) and Ra = 2 × (pentagones). The upper figuressupport (3) and (11b), while the lower figures illustrate (6). The correction in (e) supports the scaling(10).HGM scaling δ = − / ub x + vb z = κb zz , (2)and reduces to U ∆ /L = κ ∆ /λ b where λ b is the thickness of the thermal BL, which scales as λ b ∼ Nu − .Combining the above reduces to Nu = Re / Pr / , (3)and provides a relation tying Nu, Re and Pr. This result is supported by our DNS of laminar toturbulent HC [Fig. 4(a)], for small Pr. A small correction was found for Pr = 1 and can also beobserved in the laminar DNS described in ref.[17]. Also note that the Prandtl number dependence on(3) is modified from ∼ Pr / to ∼ Pr for 10 − . Pr . . h ∂b/∂z i z = H = 0where hi z = H denotes the surface and time average at z = H . Combining the time average of eq. (1c)with the PY constraint (i.e. h wb i z = κ h ∂b/∂z i z ) (see ref.[9]), and integrating over z leads to wb ≤ κ ( h b i z = H − h b i z =0 ) /H = B (Γ / κ ∆ /L, (4)where 1 < B < ǫ u = wb ≤ B (Γ / ν L − RaPr − , (5)which is supported for all regimes. SGL recast this argument in a spatio-temporal volume-averagedkinetic energy dissipation rate ǫ u ≡ ν P i,j ( ∂u j /∂x i ) which in the case of a BL-dominated regime,matches the dissipation in the boundary layer ǫ u ∼ ( νU ) / ( λ u L ) where λ u is the thickness of theviscous BL. Together with the scaling for the BL thickness such that λ u ∼ Re − / , the scaling for themean dissipation in the particular case of laminar BL[8] is ǫ u ∼ ν L − Re / . (6)Combining (3), (5) and (6), one recovers the laminar scaling [12, 3, 16]Re ∼ Ra / Pr − / , (7a)Nu ∼ Ra / Pr / . (7b)By analogy to the notation in the GL theory for RBC [4, 16], this scaling regime is denoted as I l ,where the subscript l stands for low-Pr fluids. With decreasing Pr and/or increasing Ra, the bulkdynamics is driven by the large-scale overturning flow whose horizontal length scale is L . In this case,it is the large-scale velocity U which drives the dissipation of kinetic energy and the latter is given by ǫ u ∼ ν L − Re . (8)From (3), (5) and (8), it follows that low-Pr HC exhibits dependences of the formRe ∼ Ra / Pr − / , (9a)Nu ∼ Ra / Pr / , (9b)where this scaling regime is denoted as II l [see Fig. 2(b) and ref.[16]]. Note that these scalings areonly observed for Pr . − where the large-scale flow spans the full depth of the domain [see Fig.1(a)]. For intermediate Pr, the size of the turbulent overturning flow decreases with increasing Pr,from a deep to a shallow region [see Fig. 1(a-b)]. Similarly, the dissipation increases linearly withincreasing Pr and eq. (3) together with eq. (8) can be modified to account for that transitional regimesuch that ǫ u ∼ ν L − Re Pr , (10a)Nu ∼ Re / Pr (10b)which is verified empirically in our DNS [see Fig. 3(e) and Fig. 4(b)]. Combining eq. (5), and eq.(10a-b)) provides a correction for this Pr transition in the II l regimeRe ∼ Ra / Pr − , (11a)Nu ∼ Ra / Pr / , (11b)found in the range 10 − . Pr . . λ u saturates and we observe the laminar scaling reported by Shishkina & Wagner [17]Re ∼ Ra / Pr − , (12a)Nu ∼ Ra / Pr , (12b)denoted as I ∗ l [see Fig. 2(b)]. Increasing Ra and at high Pr, the dynamics are driven by the plume,detached from the bottom [see Fig. 1(b)]. This particular case was theorized by Hughes et al. [6] with6 plume model inside a filling box. Here we recast their model according to the SGL theory (i.e. seethe plume model definition eq. (2.15)-(2.20) in ref.[6]) and the dissipation in the boundary layer writes ǫ u,P l ∼ ν L − Re / Pr − / , (13)where the dissipation now scales with the thickness of the thermal layer, not the kinetic BL and isgiven by ǫ u,P l ∼ νU / ( λ b L ). Combining (3), (5) and (13)Re ∼ Ra / Pr − / , (14a)Nu ∼ Ra / Pr / , (14b)which is denoted as II u [see Fig. 2(b)].The slope of the transition regions in the (Ra , Pr) plane, between the laminar regimes I l , II l , II u and I ∞ l , is determined by matching the Reynolds numbers in these neighboring regimes [4, 16]. Thus,from eqs. (7) and (12), we obtain the slope of the transition region between the regimes I l and II l ,which is Pr ∼ Ra / and Pr ∼ Ra − between II l and II u [17]. Each transition region is highlighted bya yellow line in the (Ra , Pr) plane [see Fig. 2(b)] and is estimated from the DNS data, by consideringthe changes in the Nu(Ra , Pr) and Re(Ra , Pr) dependencies. Note that the transition is not smoothand suggests that a bifurcation takes place when transitioning from the I l to II u regime, and can beaffected by the geometry of particular HC setups (as previously suggested in ref.[17]).In conclusion, we report evidences of a new regime in turbulent simulation of horizontal convec-tion based on scaling arguments at low Pr. The transition occurs from Re ∼ Ra / Pr − / , Nu ∼ Ra / Pr / to Nu ∼ Ra / Pr / , Re ∼ Ra / Pr − / at low Pr and sufficiently large Ra and thenRe ∼ Ra / Pr − / , Nu ∼ Ra / Pr / . Our results, integrate previous evidence from Shishkina & Wag-ner [17] and the model of Hughes et al. (see ref. [6]) in the SGL theory of HC (see ref. [4, 16]). Thetransition to the turbulent limiting regime denoted as IV u or the ultimate regime IV l have yet to beobserved (see ref. [16]). It is therefore of particular interest to attain Rayleigh number of Ra ≈ to determine whether core driven turbulent HC regimes can be reached, as suggested in ref.[16].The authors acknowledge the support of the National Science Foundation Grant Number OCE–1155558 and OCE–1736989. References [1] T. Alboussiere, R. Deguen, and M. Melzani. Melting-induced stratification above the earth’sinner core due to convective translation.
Nature , 466:744–747, 2010.[2] J. Bloxham and A. Jackson. Fluid flow near the surface of earth’s outer core.
Rev. Geophys. ,29(1):97–120, 1991.[3] B. Gayen, R. W. Griffiths, and G. O. Hughes. Stability transitions and turbulence in horizontalconvection.
J. Fluid Mech. , 751:698–724, 7 2014.[4] S. Grossmann and D. Lohse. Scaling in thermal convection: a unifying theory.
J. Fluid Mech. ,407:27–56, 2000.[5] G. O. Hughes and R. W. Griffiths. Horizontal convection.
Annu. Rev. Fluid Mech. , 40:185–208,2008. 76] G. O. Hughes, R. W. Griffiths, J. C. Mullarney, and W. H. Peterson. A theoretical model forhorizontal convection at high rayleigh number.
J. Fluid Mech. , 581:251–276, 2007.[7] L. Knopoff. Horizontal convection in the earth’s mantle: A mechanism for strike-slip faulting.
Science , 140(3565):383–383, 1963.[8] L. D. Landau and E. M. Lifschitz.
Statistische Physik . Akademie-Verlag, 1987.[9] F. Paparella and W. R. Young. Horizontal convection is non-turbulent.
J. Fluid Mech. , 466:205–214, 2002.[10] P.-Y. Passaggia, A. Scotti, and B. L. White. Transition and turbulence in horizontal convection:linear stability analysis.
J. Fluid Mech. , 821:31–58, 2017.[11] P.-E. Roche, B. Castaing, B. Chabaud, and B. H´ebral. Prandtl and rayleigh numbers dependencesin rayleigh-b´enard convection.
Europhys. Lett.) , 58(5):693, 2002.[12] H. T. Rossby. On thermal convection driven by non-uniform heating from below: an experimentalstudy.
Deep-Sea Res. , 12:9–16, 2 1965.[13] A. Scotti. A numerical study of the frontal region of gravity currents propagating on a free-slipboundary.
Theo. Comput. Fluid Dyn. , 22(5):383, 2008.[14] A. Scotti and B. L. White. Is Horizontal convection really ”non turbulent”?
Geophys. Res. Lett. ,38:L21609, 2011.[15] G. J. Sheard and M. P. King. Horizontal convection: effect of aspect ratio on rayleigh numberscaling and stability.
App. Math Model. , 35(4):1647–1655, 2011.[16] O. Shishkina, S. Grossman, and D. Lohse. Heat and momentum transport scalings in horizontalconvection.
Geophys. Res. Lett. , 43(3):1219–1225, 2016.[17] O. Shishkina and S. Wagner. Prandtl-number dependence of heat transport in laminar horizontalconvection.
Phys. Rev. Lett. , 116(2):024302, 2016.[18] O. Shishkina, S. Wagner, and S. Horn. Influence of the angle between the wind and the isother-mal surfaces on the boundary layer structures in turbulent thermal convection.
Phys. Rev. E ,89(3):033014, 2014.[19] Olga Shishkina, Mohammad S Emran, Siegfried Grossmann, and Detlef Lohse. Scaling relationsin large-prandtl-number natural thermal convection.
Physical review fluids , 2(10):103502, 2017.[20] S.-I. Takehiro. Fluid motions induced by horizontally heterogeneous joule heating in the earth’sinner core.
Phys. Earth Planet. Inter. , 184(3):134–142, 2011.[21] M. F. Taylor, K. E. Bauer, and D. M. McEligot. Internal forced convection to low-prandtl-numbergas mixtures.
Int. J. Heat Mass Trans. , 31(1):13 – 25, 1988.8, 31(1):13 – 25, 1988.8