Role of critical points of the skin friction field in formation of plumes in thermal convection

The dynamics in the thin boundary layers of temperature and velocity is the key to a deeper understanding of turbulent transport of heat and momentum in thermal convection. The velocity gradient at the hot and cold plates of a Rayleigh-Bénard convection cell forms the two-dimensional skin friction field and is related to the formation of thermal plumes in the respective boundary layers. Our analysis is based on a direct numerical simulation of Rayleigh-Bénard convection in a closed cylindrical cell of aspect ratio $\mathrm{\Gamma }=1$ and focused on the critical points of the skin friction field. We identify triplets of critical points, which are composed of two unstable nodes and a saddle between them, as the characteristic building block of the skin friction field. Isolated triplets as well as networks of triplets are detected. The majority of the ridges of linelike thermal plumes coincide with the unstable manifolds of the saddles. From a dynamical Lagrangian perspective, thermal plumes are formed together with an attractive hyperbolic Lagrangian coherent structure of the skin friction field. We also discuss the differences from the skin friction field in turbulent channel flows from the perspective of the Poincaré-Hopf index theorem for two-dimensional vector fields.

Figure 1

Critical points of a two-dimensional vector field. Phase portraits are displayed. (a) Saddle. (b) Unstable node. (c) Stable node. (d) Unstable focus. (e) Stable focus.

Figure 2

Simulation snapshot which displays the critical points of the skin friction field and thermal plumes. Top row: $z=0$; bottom row: $z=1$. Left: Magnitude (colored background) and streamlines of the skin friction field $\mathit{s}$. Unstable nodes and saddle point are indicated by arrows. Middle: Corresponding contour plot of the vertical temperature derivative, $\partial T/\partial z$. Linelike plumes are indicated again by arrows. Right: Contours of the divergence of the skin friction field together with the streamlines of $\mathit{s}$. Data are for $\mathrm{Ra}={10}^7,\mathrm{Pr}=0.7$, and $\mathrm{\Gamma }=1$.

Figure 3

Probability density function of the vertical temperature derivative $\partial T/\partial z$ which is taken at the bottom plate, $z=0$. Data are for $\mathrm{Ra}={10}^7,\mathrm{Pr}=0.7$, and $\mathrm{\Gamma }=1$. The dashed vertical line indicates the mean vertical temperature at the bottom plate which equals the Nusselt number of the convection flow.

Figure 4

(a) Probability density function of the two components of the skin friction field, $\partial u_x/\partial z$ and $\partial u_y/\partial z$ at $z=0$. The data set for this analysis is the same as in Fig. 3. Mean values are ${\overline{s}}_x=-3.12$ and ${\overline{s}}_y=-0.49$, respectively. (b) Lines of mean skin friction field averaged over 202 time snapshots which are separated by roughly one free-fall time unit $T_f=H/U_f$ of each other. The background contour stands for $\partial T/\partial z$.

Figure 5

Probability density function of the divergence of the skin friction field. We compare the unconditioned divergence with the divergence for which $\partial T/\partial z<{\langle \partial T/\partial z\rangle }_{A,t}$. The data set for this analysis is the same as that in Fig. 3.

Figure 6

Top: Number of saddles, nodes, and foci for the individual snapshots of the turbulent convection flow. The dashed lines with same color as the data correspond to the arithmetic means. Bottom: Sum of the indices of saddles and nodes. The dashed line at 1 indicates the Euler characteristic for the hot bottom or cold top plates $M$. Same data are taken as in top figure.

Figure 7

Network of critical points formed by N-S-N triplets of saddles (S) and nodes (N) at $z=0$. A snapshots of the skin friction field is shown. The solid lines between the nodes are a guide to the eye. Saddles are always found in between two nodes close to this straight connection. The contours represent the temperature derivative $\partial T/\partial z$.

Figure 8

Field lines of the surface vorticity field plotted together with the magnitude of the skin friction field, $\left|\mathit{s}\right|$, as background contour plot. The data are the same as in Fig. 7.

Figure 9

Comparison of Lagrangian analysis [(a)–(c)] and corresponding Eulerian analysis [(d)–(f)]. Three snapshots of the skin friction field dynamics are shown. The top row shows the attracting Lagrangian coherent structures as solid lines. The background shows the vertical temperature derivative. In the bottom row, field lines of the skin friction field and the node-saddle-node triplets are displayed.

Figure 10

Direct numerical approximation of the regions around unstable nodes of the skin friction field. Dominant eigenvectors of a stochastic matrix sampled from Lagrangian trajectories highlight the dynamically distinct regions, which compare well with the corresponding results for the Lagrangian coherent structures in Fig. 9.