Rayleigh–Bénard configuration is one of the most well-known natural convection problems in enclosed spaces where buoyancy-driven fluid motion takes place between differentially heated horizontal walls with the heated bottom wall. This configuration has been widely analysed because of its conceptual simplicity and relevance to several applications ranging from astrophysics, geophysics and meteorology to process industries. Interested readers can be referred to Bodenschatz et al.1 for an extensive review in Rayleigh–Bénard convection. Recently, immense heat transport enhancement (e.g. 500 %) was reported for Rayleigh–Bénard convection applications, by using water-heavy liquid (hydrofluoroether) mixture2 and vibration-induced boundary-layer destabilization3.

The flow becomes turbulent for high values of Rayleigh number (Ra=rho ^2c_{p}gbeta Delta TL^3/mu k) where (rho ,c_{p},beta ,mu) and k are density, specific heat, volume expansion coefficient, viscosity and thermal conductivity, respectively and g, (Delta T) and L are the acceleration due to gravity, temperature difference between hot and cold walls and the enclosure height, respectively in the Rayleigh–Bénard convection. In the aforementioned applications, turbulent Rayleigh–Bénard convection is obtained for fluids with different Prandtl numbers (Pr=mu ,c_{p}/,k) (e.g. (Prapprox 1) is relevant to weather predictions, whereas (Prgg 1) is relevant to geophysics and process industries).

The relative thicknesses of hydrodynamic and thermal boundary layers is dependent on Pr, which is known to affect the scalar spectrum and it is possible to obtain roll-off of the scalar spectrum in the inertial range and an inertial-diffusive range is obtained for (Pr<1)4. By contrast, the roll-off of the scalar spectrum is obtained for the length scales smaller than the Kolmogorov length scale and a viscous-diffusive range is observed for the scalar spectrum for (Prgg 1)4. As the temperature distribution in turbulent flows is affected by Pr, it can be expected that the velocity distribution in natural convection will also be affected by Pr because the flow is induced by the temperature difference.

The Prandtl number has indeed been shown to affect the turbulent kinetic energy spectrum in a recent analysis by the present authors5. However, the Prandtl number of the fluid does not only affect the distribution of turbulent kinetic energy in Rayleigh–Bénard convection but also has the potential to alter the distribution of flow topologies, as they can be categorised in terms of the invariants of the velocity gradient (partial {u_i}/partial {x_j}) tensor (i.e. PQ and R) where (u_i) is the ith component of the velocity vector6,7. Depending on the values of the invariants of the velocity gradient (partial u_i/partial x_j) tensor, 8 different topologies (i.e. S1–S8 topologies) can be identified in the three-dimensional PQR phase space. The velocity-gradient tensor can be split into symmetric and skew-symmetric parts: (A_{ij}=partial u_i/partial x_j=S_{ij}+W_{ij}), where (S_{ij}=0.5(A_{ij}+A_{ji})) and (W_{ij}=0.5left( A_{ij}-A_{ji}right)) are the symmetric and skew-symmetric components, which are referred to as strain and rotation rates, respectively. Three eigenvalues, (lambda _1), (lambda _2) and (lambda _3), of (A_{ij}) can be obtained from solutions of the characteristic equation (lambda ^3+Plambda ^2+Qlambda +R=0) where PQR are the invariants of (A_{ij})6,7:

$$begin{aligned} P=-left( lambda _1+lambda _2+lambda _3right) ; ,Q=0.5left( (P^2-S_{ij}S_{ij})+{(W}_{ij}W_{ij})right) =Q_s+Q_w; ,R=(-P^3+3PQ-S_{ij}S_{jk}S_{ki}-3W_{ij}W_{jk}S_{ki})/3 end{aligned}$$


The discriminant, (D=left[ 27R^2+left( 4P^3-18PQright) R+4Q^3-P^2Q^2right] /108), of the characteristic equation divides the (P-Q-R) phase-space into two regions depending on the sign of the discriminant. For (D>0, (D<0)), a focal (nodal) topology is obtained6,7 and the velocity gradient tensor exhibits one real eigenvalue and two complex conjugate eigenvalues for focal topologies, whereas three real eigenvalues are obtained for nodal topologies. The solutions of (D=0) are given by two surfaces in the (P-Q-R) phase space6,7: ( r_{1a}=Pleft( Q-2P^2/9right) /3-2left( -3Q+P^2right) ^{{3}/{2}}/27) and (r_{1b}=Pleft( Q-2P^2/9right) /3+2left( -3Q+P^2right) ^{{3}/{2}}/27). For a positive discriminant (i.e. (D>0)), the (A_{ij}) tensor has purely imaginary eigenvalues on the surface (r_2), which is given by (R=PQ). The surfaces (r_{1a}), (r_{1b}) and (r_2), divide the (P-Q-R) phase space into eight flow topologies. The first invariant (P=-partial u_{i}/partial x_{i}) of the velocity gradient tensor vanishes for incompressible fluids, and therefore only topologies S1–S4 are observed for (P=0), as shown in Fig. 1. Therefore, in Rayleigh–Bénard convection of incompressible fluids the flow topologies are determined by the behaviours of the second and third invariants (i.e. Q and R) of the velocity gradient tensor6,7 and only S1–S4 topologies can be seen.

The flow structures associated with S1–S4 topologies are schematically shown in Fig. 1c. One aspect of this work focuses on flow topologies close to the active walls and therefore only the upper part (above the horizontal plane crossing the origin of the coordinate system) of the velocity field is shown in Fig. 1, representative of a situation close to the lower wall, but in principle it can be mirrored at the horizontal plane (see e.g. Ref.8). It has been demonstrated by Perry and Chong6 and Soria et al.9 that S4 topologies are obtained predominantly for positive values of Q, whereas Blackburn et al.10 demonstrated that the topologies S2 and S4 are predominantly obtained in the regions away from the wall in boundary layer flows. The ‘teardrop’ structure in the joint probability density function (PDF) between Q and R has been demonstrated by Chong et al.7 and Chacin and Cantwell11. The analysis by Ooi et al.12 and experimental evidences11,13 suggested that the same qualitative behaviour is observed in a range of different incompressible turbulent flows indicating some degree of universality in the joint PDFs between Q and R. The theoretical justifications of the ‘teardrop’ shape of the (Q-R) joint PDF for incompressible flows have been provided by Elsinga and Marusic14 and the loss of ‘teardrop’ structure was shown to be a mark of intermittency in some previous analyses15. Tsinober16 postulated that the enstrophy production is large in S4 topology whereas the strain rate production is concentrated in regions of S1 topology. The flow topology distributions in Rayleigh–Bénard convection, where temperature and velocity fields are intrinsically coupled, are yet to be analysed in detail17,18,19,20 in comparison to the vast body of literature (e.g. Refs.5,6,7,8,9,10,11,12,13,14,15,16) on other wall-bounded flows.

The analyses by Dabbagh et al.17,18,19 revealed the existence of the teardrop shape in the bulk region away from the walls in Rayleigh–Bénard convection but the small scale structures in the vicinity of the hot and cold walls have not been discussed there in terms of Q and R. Xi et al.20 reported a transition of flow topologies from a quadruple structure to a dipole structure based on Rayleigh number in turbulent Rayleigh–Bénard convection, which has implications on the Nusselt number (or heat transfer rate). A recent analysis revealed that large-scale circulation in Rayleigh–Bénard convection is affected by Prandtl number21. However, the effects of Prandtl number on the flow topology are yet to be analysed and the present work addresses this gap in the existing literature. In this respect, the main objectives of the present analysis are: (a) to demonstrate and explain the effects of Prandtl number on the statistical behaviours of Q and R and their joint PDFs and (b) to indicate the implications of the above findings on flow topology distribution for Rayleigh–Bénard convection of Newtonian fluids. According to Buckingham’s pi theorem22, the Nusselt number (Nu={h,L}/{k}), where h represents the convective heat transfer coefficient, can be taken to be a function of Ra and Pr (i.e. (Nu=f(Ra,Pr))) for Rayleigh–Bénard convection in a cubic enclosure. Therefore, three-dimensional Direct Numerical Simulations (DNS) of Rayleigh–Bénard convection in a cubic enclosure for different values of Ra (i.e. (Ra={10}^7-{10}^9) ) and Prandtl number ((Pr=1) and 320) have been carried out in order to meet these objectives. It is worth noting that (Pr=320) corresponds for example to silicone oil at ({20}^{o}C) which exhibits Newtonian rheological behaviour23.

The conservation equations for mass, momentum and energy for incompressible Newtonian fluids under transient conditions take the following form:

$$begin{aligned}&partial u_{i}/partial x_{i}=0; end{aligned}$$


$$begin{aligned}&{[}partial u_{i}/partial t + u_{j} , (partial u_{i}/partial x_{j})]= -1/rho , (partial P/partial x_{i}) + nu , [partial ^2 u_{i}/(partial x_{j},partial x_{j})] + g[beta (T-T_{ref})], delta _{i2} ; end{aligned}$$


$$begin{aligned}&{[}partial T/partial t + u_{j} , (partial T/partial x_{j})]= alpha , [partial ^2 T/(partial x_{j},partial x_{j})]. end{aligned}$$


The last term on the right-hand side of Eq. (3) originates due to Boussinesq’s approximation and the temperature difference between horizontal walls is considered to be small enough so that this approximation remains valid. Also, in Eq. (3), (nu) is the kinematic viscosity and the Kronecker delta ((delta _{i2})) indicates that buoyancy forces affect the flow only in the vertical direction (i.e. (x_{2}) direction). The reference temperature ((T_{ref})) is taken to be the cold wall temperature (i.e. (T_{ref}=T_{C})). In Eq. (4), α=k/(ρ cp) is the thermal diffusivity of the fluid.

Equations (24) are solved in a coupled manner in conjunction with the following boundary conditions. The simulation configuration is schematically shown in Fig. 1a which demonstrates that the differentially heated horizontal walls are subjected to constant wall temperature boundary conditions (i.e. (T=T_{H}) at (x_{2}=0) and (T=T_{C}) at (x_{2}=L) where (T_{H}>T_{C})). All the other walls are considered to be adiabatic (i.e. (partial T/partial x_{1,3}=0) at (x_{1,3}=0,,L)). Finally, no-slip and impermeability conditions are specified for all walls (i.e. (u_{1,2,3}=0) at (x_{1,2,3}=0,,L)).

Figure 1

(a) Schematic diagram of the simulation domain. (b) Classification of topologies S1–S4. (c) Graphical representation of topologies S1–S4. Symbols correspond to UF unstable focus; UN unstable node; SN stable node; SF stable focus; C compressing; S saddle; ST stretching. The blue circles indicate the origin of the blue streamlines.

Table 1 Summary of the mean Nusselt number (i.e. ({overline{Nu}})) for different values of Ra and Pr of Rayleigh–Bénard convection of Newtonian fluids in a cubic enclosure.

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