Session FJ: Convection and Buoyancy-Driven Flows IV

Simulation of Flows and Heat Transfer in a Supercritical Carbon Dioxide Filled Enclosure

Supercritical fluids are characterized by high compressibilities and large densities. In addition to conduction and convection, thermomechanical conversion of acoustic energy to heat is significant in fluids near their critical point. Supercritical fluids also exhibit a number of specific interesting properties such as non-zero bulk viscosity, low viscosity and low thermal diffusivity - which make them quite different from gases and liquids. In this paper, thermally generated wave induced convection in a supercritical carbon dioxide filled square enclosure is investigated. The top and bottom walls of the enclosure are thermally insulated and the left and right walls are heated (either rapidly or gradually) differentially. Rapid heating causes stronger acoustic waves within the enclosure that enhance mixing and homogenization. The role of the thermally induced acoustic waves in enhancing mixing, and heat transfer in supercritical fluids is examined. The time-dependent flow and temperature fields within the enclosure are obtained by solving a fully compressible form of the Navier-Stokes equations. A flux-corrected transport algorithm is used to discretize the convective terms while the central difference scheme is used to discretize the viscous and the conduction terms. The state relation p = f($\rho $,T), the internal energy i = f($\rho $,T) and the speed of sound c = f($\rho $,T) in supercritical carbon dioxide are obtained from the NIST Standard Reference Database 12. Acknowledgment: NASA grants: NNC04AA22A and NNC04IA09I.   

Pattern Crystallization due to Spatially Periodic Forcing

We present experimental results on Rayleigh-Benard convection subjected to spatially periodic forcing. We identify a new type of coherent structure - a localized orientational defect - arising out of spatial entrainment of the convection pattern to the forcing. Localized, ``crystalline'' domains with robust orientational order are a characteristic feature of chaotic states containing these defects. Local analysis reveals that the structure of these domains is a consequence of three elements: underlying reflection symmetry, entrainment, and an annulus of preferred periodicities. We are grateful for support from the National Science Foundation, under grant no. DMR- 0305151, and from the Max Planck Society.   

Feedback Control of Pattern Formation

Pattern formation in Rayleigh-Benard and Marangoni convection is often modeled by the real Swift-Hohenberg (SH) equation. Global feedback control of these spatially-regular patterns described by the SH equation is studied. Two cases are considered: (i) the effect of control on the competition between roll and hexagonal patterns; (ii) the suppression of sub-critical instability by feedback control. In case (i), it is shown that control can change the stability boundaries of hexagons and rolls as well as stabilize mixed-mode (non-equilateral hexagon) patterns. Transitions between up- and down-hexagons are also observed. In case (ii), the feedback control can suppress the unbounded solutions of the sub-critical SH equation and lead to the formation of spatially-localized patterns.   

Transient buoyancy driven front dynamics in near horizontal tilted tubes.

The front velocity $V_f$ for interpenetrating light and heavy fluids in a long tube tilted at a small angle $\alpha$ from horizontal is studied as a function of the time $t$ and of $\alpha$ for different Atwood numbers $At$. Above a critical angle $\alpha_c$ which decreases as $At$ increases ($\alpha_c = 5^o$ for $At = 4 \times 10^{-3}$), the front velocity is always controlled by inertia and constant with time with $ V_f = C (At gd)^{0.5}$ ($C \simeq 0.7$). At lower angles ($\alpha \le \alpha_c$), the front dynamics is initially inertial and the constant $C$ decreases to $0.5$ as $\alpha \rightarrow 0$. Then, after a distance $x_c$ diverging to infinity as $\alpha \rightarrow \alpha_c$, the front velocity becomes controlled by viscosity and decreases with distance towards a value $V_{f\infty}$ decreasing with $\alpha$. For an horizontal tube, in the viscous regime, the front velocity decreases to zero as $t^{-0.5}$ ($V_{f\infty} = 0$); the mean concentration profile $C(x,t)$ (reflecting the height of the interface) is self-similar and a function of the reduced variable $x/t^{0.5}$. The variations of $V_f$ with time are well described by a simple equation including the height of the front as a control parameter.   

A Simulation of a Convective flow to give a Key of defining the Shape of the Human Brain

Since the principal rules for a self-organizing neural network are not sufficient for determining the shape and anatomical features of the brain, these must be governed by other rules. Ontogeny of the global shape of the brain is guided by radial glial fibers. The rules defining the growth pattern of radial glial fibers, therefore, should be the rules for a self- organization for the shape of the brain. In the vortex theory of the human brain (T. Nakada, 2003), it is shown that radial glial fibers grow along a pattern of a flow similar to thermal convection. That is, a convective flow determines the overall shape and structural anatomical detail of the human brain. In this paper, to establish the vortex theory of the human brain, a simulation of a convective flow is carried out. In the computation, the incompressible Navier-Stokes equations based on Boussinesq approximation are solved by multi-directional finite difference method. The computational domain of sphere shape is employed. The computed flow is visualized suitably, and investigated qualitatively.   

Flow instabilities in open buoyant-thermocapillary pools

The flow structure in open buoyant-thermocapillary pools heated or cooled from above is investigated numerically. For pools of cylindrical shape and axisymmetric heat loads with a parabolic profile we carry out a systematic variation of the governing parameters such as the heating rate, Bond number, and the aspect ratio of the pool. The flow instability with respect to three-dimensional perturbations is is studied by a detailed analysis of the neutral modes.   

Near-wall SGS modeling for LES of an Isothermal Wall using ODT

A novel coupling approach is developed for One-Dimensional Turbulence (ODT) as near-wall subgrid scale model for Large-Eddy Simulation (LES). In this approach, all near-wall molecular processes are completely resolved using a 1D embedded grid that lies normal to the wall surface which partially overlaps with the filtered field. The SGS stresses are constructed using the vector and variable density ODT model and are used to evolve the LES equations. Within the ODT domain the effects of unresolved 3D turbulent mixing processes associated with the wall surface tangential directions is modeled using a sequence of triplet mapping stirring events. A modification to eddy selection is proposed in this description to take into account the physical process of air engulfment from long wavelength instability modes by adding a buoyancy generation production term. The near-wall vortex structures generated in the coupled LES-ODT simulation of a turbulent boundary layer along an isothermal wall indicates that the near-wall small scales have a profound affect on the large scale structures. The results show a significant improvement in the prediction of near-wall quantities using the LES-ODT approach over standard LES approaches at only a fraction of the cost associated with DNS.   

Self-similar thermals in Stokes flow

Similarity solutions are obtained for the rise of a buoyant thermal in Stokes flow (i.e. infinite Prandtl number), in which all lengths scale like $t^{1/2}$ and velocities like $t^{-1/2}$. The dimensionless problem depends only on the Rayleigh number $Ra=B/(\nu\kappa)$, where $B$ is the (conserved) total buoyancy. For small $Ra$ there are only slight deformations to a spherically symmetric Gaussian temperature distribution. For large $Ra$ the temperature distribution is greatly elongated in the vertical direction, with a long `wake', which contains most of the buoyancy and dominates the flow, and a small `head', which is asymptotically unimportant. This structure contrasts with the usual view of mantle plumes and laboratory experiments. The large-$Ra$ behaviour is explained using a simple analytic model based on slender-body theory. The width of the thermal increases like $(\kappa t)^{1/2}$ while the wake length and rise height both increase like $(Ra\ln Ra)^{1/2}(\kappa t)^{1/2}$.   

Intrusive gravity currents in two-layer stratified media

Intrusive gravity currents, in which a well-defined fluid volume of intermediate density propagates horizontally along a stable interface, arise in a variety of natural settings. Unless the intrusion density is the depth-weighted mean density of the upper and lower layers, a propagating interfacial wave will be generated ahead of the intrusive gravity current. We describe the results of a new study, which combines a Benjamin (1968)-type analysis with shallow-water theory in developing a coupled intrusion-wave model. Favorable agreement is observed with existing experimental and numerical data.