Turbulent Thermal Convection

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Turbulent thermal convection drives many natural and engineering flows. In such flows, the heat transport by thermal convection dominates those by thermal conduction and radiation.

In turbulent thermal convection, buoyancy injects energy into the flow at all scales, but most dominantly at large scales. Consequently, turbulent thermal convection has similar behavior as hydrodynamic turbulence, for example, it shows Kolmogorov-like energy spectrum and near isotropy in the inertial range. The large-scale quantities–the root mean square velocity and heat transport coefficient–exhibit nontrivial scaling due to the presence of the confining walls. Also, the viscous dissipation rate is suppressed in comparison to hydrodynamic turbulence. In turbulent convection, the walls also induce ordered structures–large scale circulation and turbulent superstructures–in the background of randomness.


Contents

Transition to turbulence

Thermal convection in realistic applications is quite involved due to complex configurations. Hence, to capture the essential features of turbulent thermal convection (TTC), a simple and generic model called Rayleigh-Bénard convection (RBC) is employed (Chandrasekhar 1961, Getling 1998). In RBC, a fluid is confined between a hot bottom plate and a cold top plate, which are separated by a distance d (see Fig. 1). We denote \( T_b - T_t = \Delta T \), where \( T_b \) and \( T_t \) are the respective temperatures of the bottom and top plates. The temperature varies sharply in the boundary layers, which are the regions near the plates.

Figure 1: (a) In RBC, a fluid is confined between hot and cold plates that are at separated by a distancd $d$. The temperature drops sharply in the boundary layers (BL), and it is approximately constant in the bulk. (b) Plots of the horizontally averaged temperature profile \( \bar{T}(z) \) vs \( z \), and conduction-state temperature profile \( T_c(z) \) vs \( z \). The thickness of the thermal boundary layer, \( \delta_T\), is computed using the intersection of the mean temperature, $((T_b+T_t)/2)$, with the linear temperature profile near the boundary. The viscous boundary layers are determined using the mean velocity profile.

The equations for RBC under Boussinesq approximation are

\[ \tag{1} \frac{\partial{\mathbf{u}}}{\partial{t}}+ (\mathbf{u}\cdot \nabla)\mathbf{u} = -\frac{1}{\rho} \nabla \sigma + \alpha g \theta \hat{z} + \nu \nabla^{2}\mathbf{u},\]

\[ \tag{2} \frac{\partial{\theta}}{\partial{t}}+(\mathbf{u}\cdot\nabla)\theta = \frac{\Delta T}{d} u_{z} + \kappa \nabla^{2}\theta, \]

\[ \tag{3} \nabla \cdot {\bf u} = 0, \]

where u is the velocity field; \( \theta = T-T_c(z) \) is the temperature fluctuation around the conduction profile \( T_c(z) \); $\sigma$ is the pressure field; \( \nu,\kappa,\alpha\) are respectively the kinematic viscosity, thermal difusivity, and thermal expansion coefficient of the fluid; \( g \) is the acceleration due to gravity; and \( \rho \) is the density of the fluid. Two imporant nondimensional parameters for RBC are Prandtl number, Pr \( = \nu / \kappa \), and Rayleigh number, Ra \( = \alpha g (\Delta T) d^3/(\nu \kappa) \), which is a measure of the strength of buoyancy.

For small Ra, the flow is stationary wherein the heat is transported by thermal conduction. At critical Rayleigh number, \( \mathrm{Ra}_c \), the flow becomes unstable and first convective rolls appear. On the further increase of Ra, the flow exhibits patterns and chaos via a set of secondary bifurcations (Getling 1998, Bhattacharjee 1987, Manneville 1990, and Transition to turbulence). For Pr ~ 1, the flow becomes turbulent when $\mathrm{Ra} \gtrapprox 10^6$. The transition to turbulence occurs at lower Ra for fluids with small Pr (for example, liquid metals). The number of interacting Fourier modes in turbulent thermal convection is many more than that in unstable and chaotic regimes.

Energy spectrum and flux

Kolmogorov’s theory for homogeneous and isotropic hydrodynamic turbulence (Turbulence, Cascade and scaling) often acts as a starting point for modeling more complex turbulent flows, including turbulent thermal convection (TTC). In Kolmogorov's theory, a pure hydrodynamic flow is forced at large scales. Kinetic energy thus fed at the large-scales cascades to intermediate scales and then to small scales. This multiscale energy flux, \( \Pi_u \), is constant in the intermediate scales called inertial range, and \( \Pi_u \) equals the total viscous dissipation rate, \( \epsilon_u \). Using these observations and several other assumptions, it has been shown that in the inertial range of hydrodynamic turbulence (HDT) (Turbulence, Cascade and scaling),

\[ \tag{4} E_u(k) = K_{Ko} \Pi_u^{2/3} k^{-5/3} , \]

where \( k \) is the wavenumber, and \( K_{Ko} \) is Kolmogorov’s constant.

In thermal convection, however, buoyancy is expected to act at all scales, and consequently one anticipates that Kolmogorov's theory will fail. However, the following equation for the variable energy flux in the inertial range, where the viscous dissipation is weak, provides valuable insights:

\[ \tag{5} \frac{d }{dk} \Pi_u(k) = \mathcal{F}_u(k), \]

where \( \mathcal{F}_u(k) \) is the energy injection rate by the external force, here buoyancy. Physically, under steady state, the rate of change of energy flux of an inertial-range wavenumber shell \( (k, k+dk)\) is the difference between the energy injection rate by buoyancy (see Fig. 2).

Figure 2: (a) The net energy flux of an 'inertial-range wavenumber shell $(k,k+dk)$ is the difference between the energy injection rate by buoyancy (\( \mathcal{F}_u(k) \), denoted by rotating wheels). (b) In TTC, the inerital-range \( \Pi_u(k) \) is approximately constant because \( \mathcal{F}_u(k) \approx 0 \) here.

Since buoyancy drives thermal convection, \( \mathcal{F}_u(k) > 0 \). In addition, it has been shown that for Pr ~ 1, \( \mathcal{F}_u(k) \) drops sharply as \( k^{-5/3} \), and even steeper for other Prandtl numbers (Verma et al. 2017, Verma 2018). Hence, in the inertial range,

\[ \tag{6} \mathcal{F}_u(k) \approx 0 \implies \Pi_u(k) \approx \mathrm{const}, \]

similar to that in Kolmogorov’s theory for HDT where the forcing is at large scales. Therefore, TTC exhibits \( k^{-5/3} \) spectrum as in Eq. (4). Note that \( \mathcal{F}_u(k) \equiv 0 \) in the inertial range of HDT. There are, however, several minor differences between TTC and HDT. Buoyancy at small scales does not contribute to the energy flux, but the energy feed at these scales is dissipated by viscosity. Consequently, the inertial-range \( \Pi_u(k) \) of TTC is around 30-50 % smaller than the total viscous dissipation rate \( \epsilon_u \). Also, as will be discussed in the next section, \( \epsilon_u \ne U^3/d \), where U is the root mean square velocity.

In TTC, \( \mathcal{F}_u(k) > 0 \), hence its inertial-range \( \Pi_u(k) \) is a nondecreasing function of \( k\). Consequently, the phenomenology of Bolgiano (1959) and Obukhov (1959) (\(\Pi_u(k) \sim k^{-4/5}\); \(E(k) \sim k^{-11/5} \)) is ruled out for TTC. This is not surprising because Bolgiano-Obukhov scaling is applicable to stably stratified turbulence where the kinetic energy is transferred to the potential energy (\( \mathcal{F}_u(k) < 0 \)), opposite to that in TTC.

The temperature spectrum in TTC is very different from those of passive scalar turbulence and Bolgiano-Obukhov phenomenology; it exhibits a bi-spectrum with \( k^{-2} \) as an upper branch and a fluctuating lower branch. This feature is due to the presence of walls (Verma et al. 2017, Verma 2018). It is important to note that the flows in the bulk and in the transition region between the bulk and boundary layers are turbulent, while the flows near the walls are essentially laminar. Therefore, the subdominant fluctuations in the boundary layers contribute to \( E_u(k) \) in the dissipation range, not to the inertial-range \( E_u(k) \).

Scaling of dissipation rates and Nusselt number

The heat transport in TTC is often quantified using Nusselt number, which is the ratio of the total heat transport and the conductive heat transport:

\[ \tag{7} \mathrm{Nu} = \frac{ H_\mathrm{cond} + H_\mathrm{conv}}{ H_\mathrm{cond}} = 1+ \frac{\langle u_z \theta \rangle} {\kappa (\Delta T) /d}. \]

Theoretical arguments predict that for TTC, \( \mathrm{Nu} \sim \mathrm{Ra}^{\beta} \) with \( \beta \) as 1/4, 2/7, 1/3, 1/2 (Ahlers et al. 2009, Lohse and Xia 2012, Chilla and Schumacher 2012). The exponent 1/3 is derived by assuming that the heat transport in TTC is independent of the distance between the thermal plates (Priestley 1954, Malkus 1954). Considerations of the role of boundary layers on heat transport yields \(\beta \) = 2/7 (Castaing et al. 1989). Experimental and numerical observations, however, indicate that \( \beta \approx 0.3 \) up to \( \mathrm{Ra} \sim 10^{14} \) or so (Niemela et al. 2000, Grossmann and Lohse 2000, Ahlers et al. 2009). Disentangling these diverging values of \( \beta \) remains a challenge till date. Exact relations of Shraiman and Siggia (1990) and the scaling of dissipation rates provide further insights into the Nu scaling.

In Eq. (1), under steady state, the average energy injection rate by buoyancy is balanced by the average viscous dissipation rate, \( \epsilon_u \). This equality yields the following exact relation (Shraiman and Siggia 1990):

\[ \tag{8} \epsilon_u = \frac{\nu^3}{d^4} \frac{(\mathrm{Nu}-1) \mathrm{Ra}}{\mathrm{Pr}^2}. \]

Similar calculation using Eq. (2) yields an exact relation on the thermal dissipation rate (dissipation of \( \theta^2/2 \)):

\[ \tag{9} \epsilon_\theta = \frac{\kappa ( \Delta T)^2}{d^2} (\mathrm{Nu}-1). \]

Interestingly, numerical solutions reveal that up to Ra \( \sim 10^{11} \), and possibly beyond,

\[ \tag{10} \epsilon_u \sim \frac{U^3}{d} \mathrm{Ra}^{-0.18}, \] \[ \tag{11} \epsilon_\theta \sim \frac{U(\Delta T)^2}{d} \mathrm{Ra}^{-0.22} . \]

Thus, the viscous and temperature dissipation rates in TTC differ significantly from those in homogeneous and isotropic passive scalar turbulence. The factors \( \mathrm{Ra}^{-0.18} \) and \( \mathrm{Ra}^{-0.22} \) in the above equations arise due to the walls (Verma 2018, Scheel and Schumacher 2017).

Experiments and numerical simulations reveal that \( Ud/\kappa \sim \mathrm{Ra}^{1/2} \), substitution of which in Eqs. (10) and (11) yields \( \epsilon_u \sim \mathrm{Ra}^{1.32} \) and \( \epsilon_\theta \sim \mathrm{Ra}^{0.28} \). Substiution of these dissipation rates in Eqs. (8) and (9) yields

\[ \tag{12} \mathrm{Nu} \sim \mathrm{Ra}^{1/2-0.2} \sim \mathrm{Ra}^{0.30}, \]

consistent with those observed in numerical simulations and experiments up to Ra \( \sim 10^{14} \) or so.

Kraichnan (1962) argued that for a fully-developed turbulent thermal convection (called ultimate regime),

\[ \tag{13} \mathrm{Nu} \sim \langle u_z \theta \rangle \sim U \Delta T \sim \mathrm{Ra}^{1/2}. \]

Derivation of Eq. (13) assumes a perfect correlation between \( u_z\) and \(\theta \), and that \( \theta_\mathrm{rms} \approx \Delta T \). Both these assumptions do not hold, at least for Ra up to 1011; for this range of Ra, normalised correlation between \( u_z \) and \( \theta \) varies as \( \sim \mathrm{Ra}^{-0.05} \), and \( \theta_\mathrm{rms} \sim (\Delta T) \mathrm{Ra}^{-0.15} \) (Niemela et al. 2000, Verma et al. 2017, Verma 2018). These corrections take the Nusselt number exponent from Kraichnan's 1/2 to \( \approx 0.3 \), consistent with the numerical and experimental observations. A cautionary remark–the above scaling relations on dissipation rates and Nusselt number are for Pr ~ 1, and they do not include the Pr dependence. Flows with small or very large Pr have more complex behaviour (Grossmann and Lohse 2000, Scheel and Schumacher 2017, Verma 2018).

A burning question in the community is whether the Nu exponent will reach 1/2. If yes, at what Ra? In this regime, the aforementioned Ra dependence is expected to disappear. Some argue that the ultimate regime may not be realizable in the presence of walls. However, some others claim that the nature of the boundary layers changes at large Ra, which may be construed as a signature for a transition towards the emergence of ultimate regime.

Using the exact relations of Shraiman and Siggia (1992), bulk dissipation rates as \( \epsilon_u \sim U^3/d \) and \( \epsilon_\theta \sim U (\Delta T)^2/d \), and formulas for the dissipation rates in the boundary layers, Grossmann and Lohse (2000) derived formulas for the Reynolds number Re and Nu as functions of Ra and Pr. Their predictions on Re and Nu are in good agreement with the experimental and numerical results. In an alternative formalism, an expression for Re has been derived using Eq. (1) (Verma et al. 2017).

Large scale circulation and anisotropy

Buoyancy is expected to induce strong anisotropy in TTC. Numerical simulations, however, reveal that TTC is nearly isotropic. For example, for Pr = 1, \( \langle |\mathbf{u}_\perp|^2 \rangle /[2 \langle u^2_\parallel \rangle] \approx 0.73 \), where \( \mathbf{u}_\perp, u_\parallel \) are respectively the components of the velocity field perpendicular and parallel to the buoyancy direction (Verma 2018). This isotropy is due to the similarity between TTC and HDT, as discussed in the section Energy spectrum and flux.

The velocity field in homogeneous and isotropic HDT is random. But, TTC exhibits ordered structures, referred to as large scale circulation (LSC) and turbulent superstructures, embedded in a sea of turbulence (Niemela et al. 2000, Pandey et al. 2018). The confining walls of TTC play a key role in the formation of these structures. These structures exhibit many interesting properties, for example, large spatial and temporal correlations, random reversals of the velocity field (Ahlers and Brown 2006, Xi et al. 2006), etc. These reversals have a strong resemblance to the magnetic field reversals in the geodynamo and solar dynamo.

In summary, turbulent thermal convection is more complex than hydrodynamic turbulence with many open issues. The complexity is increased further in more complex convective flows, such as horizontal convection, rotating convection, magneto-convection, forced convection, moist convection, convection with rough walls, convection with open surfaces (as in lakes and oceans), etc. These topics continue to be investigated rigorously by researchers.

References


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Internal references

  • John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
  • Guenther Ruediger (2008) Solar dynamo. Scholarpedia, 3(1):3444.

Further reading

  • S. Bhattacharya, A. Pandey, A. Kumar, and M. K. Verma, Complexity of viscous dissipation in turbulent thermal convection, Phys. Fluids, 30, 031702, 2018.
  • E. Bodenschatz, W. Pesch, and G. Ahlers, Recent developments in Rayleigh-Bénard convection, Annu. Rev. Fluid Mech. 32, 709, 2000.
  • G. Falkovich and K. R. Sreenivasan, Lessons from hydrodynamic turbulence, Phys. Today, 59, 43, 2006.
  • X. He, D. Funfschilling, H. Nobach, E. Bodenschatz, and G. Ahlers, Transition to the ultimate state of turbulent Rayleigh-Bénard Convection, Phys. Rev. Lett., 108, 024502, 2012.
  • M. Lappa, Thermal convection: Patterns, Evolution and Stability, John Wiley & Sons, Chichester, 2010.
  • V. S. L’vov and G. Falkovich, Conservation laws and two-flux spectra of hydro-dynamic convective turbulence, Physica D, 57, 85, 1992.
  • H. K. Pharasi, D. Kumar, K. Kumar, and J. K. Bhattacharjee, Spectra and probability distributions of thermal flux in turbulent Rayleigh-Bénard convection, Phys. Fluids, 28, 055103, 2016.
  • E. D. Siggia, High Rayleigh number convection, Annu. Rev. Fluid Mech., 26, 137, 1994.
  • K. R. Sreenivasan, A. Bershadskii, and J. J. Niemela, Mean wind and its reversal in thermal convection, Phys. Rev. E, 65, 056306, 2002.
  • D. J. Tritton, Physical Fluid Dynamics, Clarendon Press, Oxford, 1998.
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