We perform a numerical study of the heat transfer and flow structure of Rayleigh-Benard (RB) convection in (in most cases regular) porous media, which are comprised of circular, solid obstacles located on a square lattice. This study is focused on the role of porosity in the flow properties during the transition process from the traditional RB convection with (so no obstacles included) to Darcy-type porous-media convection with approaching 0. Simulations are carried out in a cell with unity aspect ratio, for Rayleigh number from to and varying porosities , at a fixed Prandtl number , and we restrict ourselves to the two-dimensional case. For fixed , the Nusselt number is found to vary non-monotonically as a function of ; namely, with decreasing , it first increases, before it decreases for approaching 0. The non-monotonic behaviour of originates from two competing effects of the porous structure on the heat transfer. On the one hand, the flow coherence is enhanced in the porous media, which is beneficial for the heat transfer. On the other hand, the convection is slowed down by the enhanced resistance due to the porous structure, leading to heat transfer reduction. For fixed , depending on , two different heat transfer regimes are identified, with different effective power-law behaviours of versus , namely a steep one for low when viscosity dominates, and the standard classical one for large . The scaling crossover occurs when the thermal boundary layer thickness and the pore scale are comparable. The influences of the porous structure on the temperature and velocity fluctuations, convective heat flux and energy dissipation rates are analysed, further demonstrating the competing effects of the porous structure to enhance or reduce the heat transfer.

We report on a three-dimensional direct numerical simulation study of flow structure and heat transport in the annular centrifugal Rayleigh-Bdnard convection (ACRBC) system, with cold inner and hot outer cylinders corotating axially, for the Rayleigh number range Ra is an element of [10(6), 10(8)] and radius ratio range eta = R-i/R-o is an element of [0.3, 0.9] (R-i and R-o are the radius of the inner and outer cylinders, respectively). This study focuses on the dependence of flow dynamics, heat transport and asymmetric mean temperature fields on the radius ratio eta. For the inverse Rossby number Ro(-1) = 1, as the Coriolis force balances inertial force, the flow is in the inertial regime. The mechanisms of zonal flow revolving in the prograde direction in this regime are attributed to the asymmetric movements of plumes and the different curvatures of the cylinders. The number of roll pairs is smaller than the circular roll hypothesis as the convection rolls are probably elongated by zonal flow. The physical mechanism of zonal flow is verified by the dependence of the drift frequency of the large-scale circulation (LSC) rolls and the space- and time-averaged azimuthal velocity on eta. The larger eta is, the weaker the zonal flow becomes. We show that the heat transport efficiency increases with eta. It is also found that the bulk temperature deviates from the arithmetic mean temperature and the deviation increases as eta decreases. This effect can be explained by a simple model that accounts for the curvature effects and the radially dependent centrifugal force in ACRBC.

Flow reversals in two-dimensional Rayleigh-Benard convection led by non-Oberbeck-Boussinesq (NOB) effects due to large temperature differences arc studied by direct numerical simulation. Perfect gas is chosen as the working fluid and the Prandtl number is 0.71 for the reference state. 11 NOB effects are included, the flow pattern P-11 with only one dominant roll often becomes unstable by the growth of the cold corner roll, which sometimes results in cession-led flow reversals. By exploiting the vorticity transport equation, it is found that the asymmetries of buoyancy and viscous forces are responsible for the growth of the cold corner roll because both such asymmetries cause an imbalance between the corner rolls and the large-scale circulation (I.SC). The buoyancy force near the cold wall increases and decreases near the hot wall originating from the temperature-dependent isobaric thermal expansion coefficient alpha = 1/T if NOB effects are included; Moreover, the decreased dissipation due to lower viscosity is favourable for the growth of the cold corner roll, while the increased viscosity further suppresses the growth of the hot corner roll. Finally, it is found that the boundary layer near the cold wall plays an important role in the mass transport from LSC to corner rolls subject to mass conservation.

For a fixed geometric configuration, hydrodynamic instabilities and bifurcation processes of laminar isothermal planar opposed jet flows with symmetric and slightly asymmetric inlet boundary conditions are investigated numerically using a high-resolution approach based on spectral element method. In current configuration, when inlet boundary conditions are symmetric, in the range of the Reynolds number considered (Re <= 200), multiple new symmetry-breaking bifurcations are observed and four new flow patterns are identified. Their hydrodynamic characteristics are analyzed, in particular their symmetries. In addition, the case that inlet boundary conditions are slightly asymmetric is investigated. It is found that bifurcation processes are extremely sensitive to this small symmetry-breaking imperfection and much different from those in the symmetric case. Furthermore, model equations are constructed by symmetry consideration to explain the numerical results based on hydrodynamic equations.

The bifurcations of penetrative Rayleigh-Benard convection in cylindrical containers are studied by the linear stability analysis (LSA) combined with the direct numerical simulation (DNS) method. The working fluid is cold water near 4 degrees C, where the Prandtl number Pr is 11.57, and the aspect ratio (radius/height) of the cylinder ranges from 0.66 to 2. It is found that the critical Rayleigh number increases with the increase in the density inversion parameter (m). The relationship between the normalized critical Rayleigh number (Ra-c((m))/Ra-c(0)) and (m) is formulated, which is in good agreement with the stability results within a large range of (m). The aspect ratio has a minor effect on Ra-c((m))/Ra-c(0). The bifurcation processes based on the axisymmetric solutions are also investigated. The results show that the onset of axisymmetric convection occurs through a trans-critical bifurcation due to the top-bottom symmetry breaking of the present system. Moreover, two kinds of qualitatively different steady axisymmetric solutions are identified.

A key difficulty in the analysis and numerical approximation of the shallow water equations is the nonconservative product of measures due to the gravitational force acting on a sloped bottom. Solutions may be nonunique, and numerical schemes are not only consistent discretizations of the shallow water equations, but they also determine how to model the physics. Our derivation is based on a subcell reconstruction using infinitesimal singular layers at the cell boundaries, as inspired by S. Noelle, Y. Xing, and C.-W. Shu [J. Comput. Phys., 226 (2007), pp. 29-58]. One key step is to separate the singular measures. Another aspect is the reconstruction of the solution variables in the singular layers. We study three reconstructions. The first leads to the well-known scheme of Audusse et al., [SIAM J. Sci. Comput., 25 (2004), pp. 2050-2065], which introduces the hydrostatic reconstruction. The second is a modification proposed in [T. Morales de Luna, M. J. Castro D´ıaz, and C. Par´es, Appl. Math. Comput., 219 (2013), pp. 9012-9032], which analyzes whether a wave has enough energy to overcome a step. The third is our new scheme, which borrows its structure from the wet-dry front. For a number of cases discussed in recent years, where water runs down a hill, Audusse’s scheme converges slowly or fails. Morales’ scheme gives a visible improvement. Both schemes are clearly outperformed by our new scheme.

In this paper, we are concerned with the motion of electrically conducting fluid governed by the two-dimensional non-isentropic viscous compressible MHD system on the half plane with no-slip condition on the velocity field, perfectly conducting wall condition on the magnetic field and Dirichlet boundary condition on the temperature on the boundary. When the viscosity, heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate, there is a boundary layer which is described by a Prandtl-type system. Under the non-degeneracy condition on the tangential magnetic field instead of monotonicity of velocity, by applying a coordinate transformation in terms of the stream function of magnetic field as motivated by the recent work [27], we obtain the local-in-time well-posedness of the boundary layer system in weighted Sobolev spaces.

In this paper, we prove the global existence of solutions with analytic regularity to the 2D magnetohydrodynamic (MHD) boundary layer equations in the mixed Prandtl and Hartmann regime derived by formal multiscale expansion in [D. Gerard-Varet and M. Prestipino, <i>Z. Angew. Math. Phys.</i>, 68 (2017), 76]. The analysis shows that the combined effect of the magnetic diffusivity and transverse magnetic field on the boundary leads to a linear damping on the tangential velocity field near the boundary. And this damping effect yields the global-in-time analytic norm estimate in the tangential space variable on the perturbation of the classical steady Hartmann profile.

The paper aims to estimate the thickness of the boundary layer for the planar MHD system with vanishing shear viscosity . Under some conditions on the initial and boundary data, we show that the thickness is of the order | In |. Note that this estimate holds also for the Navier-Stokes system so that it extends the previous works even without the magnetic effect.

In this paper, we will survey some recent results on the study of the viscous and invisid compressible flow with vacuum. It is wellknown that the study on vacuum has significance in the investigation on some important physical phenomena. However, most of the important questions about vacuum are still open due to the singularities caused by vacuum which need new mathematical tools and techniques to handle.