Two-dimensional Steady-state Solutions Research Articles

Linear global stability of liquid metal mixed convection in a horizontal bottom-heating duct under strong transverse magnetic field

Two-dimensional steady-state solutions of liquid metal mixed convection in a horizontal bottom-heating duct under a strong magnetic field are first computed numerically by the Newton iteration method along with the spatial discretization of the Taylor–Hood finite element. Two branches of steady solutions with symmetrical rolls and a pair of asymmetrical solutions with a single roll are identified and can be regarded as the base flow for linear global stability analysis. The symmetrical steady solution for the first branch has a nearly uniform distribution for the temperature field in the transverse direction, while the second branch occurs at much larger Grashof numbers and the temperature field becomes nonuniform transversely. The linear stability analysis is performed for a fixed Reynolds number and Prandtl number with Re = 5000 and Pr = 0.0321. For the symmetrical rolls of the first branch, with an increase in the Grashof number, two-dimensional stationary instabilities first occur at small Hartmann numbers, while three-dimensional oscillatory instabilities first appear at moderate or large Hartmann numbers. From the further study of the two-dimensional instabilities, it is revealed that the asymmetrical solution is actually bifurcated supercritically from the symmetrical solution at a two-dimensional critical Grashof number. In addition, the critical curve of the Grashof number with respect to the Hartmann number for the three-dimensional oscillatory mode shows that there exists a minimum critical Grashof number, which occurs at a moderate Hartmann number. The critical curves of the one-roll asymmetrical solution are also exhibited and determined by two three-dimensional oscillatory unstable modes. It is revealed that there exists a minimum Hartmann number below which the asymmetrical steady-state can always remain stable for all Grashof numbers (5 × 105–107). The energy analyses at the oscillatory critical thresholds with different Hartmann numbers are performed to exhibit that buoyancy is the dominant destabilizing term, and the magnetic force is always the main stabilization term for both symmetrical and asymmetrical solutions. In addition, both streamwise and cross-sectional shears of the basic flow are important for the determination of the linear stability boundary of the asymmetrical solution.

Stability analysis of a gravity-driven thermoviscous liquid film flowing over a heated flat substrate.

Two-dimensional steady-state solutions and their stability analysis are presented for a gravity-driven thin film of a thermoviscous liquid. The governing equations and boundary conditions are simplified using the lubrication approximation. The analytically obtained film thickness evolution equation consists of various dimensionless parameters such as the Marangoni number, Biot number and thermoviscosity number. The viscosity of the liquid is assumed as an exponential function of temperature. The viscosity decreases within the liquid film as the temperature increases. Due to localized heating interfacial temperature gradients generate surface tension gradient which results into thermocapillary or Marangoni stress. The Marangoni stress opposes the fluid flow at the leading edge of heater leading to an increase in the film thickness locally. This locally thick structure becomes unstable beyond critical values of the parameters that leads to formation of rivulets in the transverse direction. Using the linear stability analysis it is found that the Marangoni stress and the thermoviscous effect have a destabilizing effect on the thin-film flow. At much higher values of the thermoviscosity number another mode of instability appears which is known as thermocapillary instability which leads to oscillating film profiles. For streamwise perturbations, the destabilizing effect of the thermoviscosity number for localized and uniform heating remains consistent.

Aspect ratio effects on natural convection in a water-saturated porous cavity near its density maximum

In this study, two-dimensional steady-state solutions of buoyancy-driven convection in a water-saturated porous cavity is conducted numerically for a range of different aspect ratios. The left vertical wall is considered into a partially heating location. The Brinkman–Forchheimer extended Darcy model is used to investigate the average heat transfer rate. The governing equations are solved using a finite volume method. The results obtained for various values of parameters are presented graphically in the form of streamlines, isotherms and velocity at mid-plane of the cavity. In addition, numerical results for the average Nusselt number are presented for various parametric conditions.

Particle-size and -density segregation in granular free-surface flows

When a mixture of particles, which differ in both their size and their density, avalanches downslope, the grains can either segregate into layers or remain mixed, dependent on the balance between particle-size and particle-density segregation. In this paper, binary mixture theory is used to generalize models for particle-size segregation to include density differences between the grains. This adds considerable complexity to the theory, since the bulk velocity is compressible and does not uncouple from the evolving concentration fields. For prescribed lateral velocities, a parabolic equation for the segregation is derived which automatically accounts for bulk compressibility. It is similar to theories for particle-size segregation, but has modified segregation and diffusion rates. For zero diffusion, the theory reduces to a quasilinear first-order hyperbolic equation that admits solutions with discontinuous shocks, expansion fans and one-sided semi-shocks. The distance for complete segregation is investigated for different inflow concentrations, particle-size segregation rates and particle-density ratios. There is a significant region of parameter space where the grains do not separate completely, but remain partially mixed at the critical concentration at which size and density segregation are in exact balance. Within this region, a particle may rise or fall dependent on the overall composition. Outside this region of parameter space, either size segregation or density segregation dominates and particles rise or fall dependent on which physical mechanism has the upper hand. Two-dimensional steady-state solutions that include particle diffusion are computed numerically using a standard Galerkin solver. These simulations show that it is possible to define a Péclet number for segregation that accounts for both size and density differences between the grains. When this Péclet number exceeds 10 the simple hyperbolic solutions provide a very useful approximation for the segregation distance and the height of rapid concentration changes in the full diffusive solution. Exact one-dimensional solutions with diffusion are derived for the steady-state far-field concentration.

Bifurcation phenomena in viscoelastic flows through a symmetric 1:4 expansion

In this work we present an investigation of viscoelastic flow in a planar sudden expansion with expansion ratio D/ d = 4. We apply the modified FENE–CR constitutive model based on the non-linear finite extensibility dumbbells (FENE) model. The governing equations were solved using a finite volume method with the high-resolution CUBISTA scheme utilised for the discretisation of the convective terms in the stress and momentum equations. Our interest here is to investigate two-dimensional steady-state solutions where, above a critical Reynolds number, stable asymmetric flow states are known to occur. We report a systematic parametric investigation, clarifying the roles of Reynolds number (0.01 < Re < 100), Weissenberg number (0 < We < 100) and the solvent viscosity ratio (0.3 < β < 1). For most simulations the extensibility parameter of the FENE model was kept constant, at a value L 2 = 100, but some exploration of its effect in the range 100–500 shows a rather minor influence. The results given comprise flow patterns, streamlines and vortex sizes and intensities, and pressure and velocity distributions along the centreline (i.e. y = 0). For the Newtonian case, in agreement with previous studies, a bifurcation to asymmetric flow was observed for Reynolds numbers greater than about 36. In contrast viscoelasticity was found to stabilise the flow; setting β = 0.5 and We = 2 as typical values, resulted in symmetric flow up to a Reynolds number of about 46. We analyse these two cases in particular detail.

Analytic solutions to the shallow water equations

Analytic, two-dimensional steady-state solutions to the rotating shallow water equations over variable topography are derived, by exploiting a drastic simplification of the equilibrium problem that occurs for nondivergent flows. For such flows, the equilibrium system decouples, and the cross-stream component of the momentum equation formally reduces to the barotropic vorticity equation. Thus, any solution to the barotropic vorticity equation can be used, in principle, to construct exact equilibrium solutions to the shallow water equations.

Symmetry-breaking instability and strongly peaked periodic clustering states in a driven granular gas.

An ensemble of inelastically colliding grains driven by a horizontally vibrating wall in two dimensions exhibits clustering. Working in the limit of nearly elastic collisions and employing granular hydrodynamics, we predict, by a marginal stability analysis, a spontaneous symmetry breaking of the laterally uniform clustering state. Two-dimensional steady-state solutions found numerically describe laterally periodic clustering states. Well within the instability region the density of these states is strongly peaked, with most of the granulate located in "density islands." Time-dependent granular hydrodynamic simulations show that strongly peaked states can develop from small-amplitude single-mode density perturbations.

Limits of Existence of a Two-Dimensional Steady-State Structure of a Liquid Film in Combustion-Wave Propagation

A two-dimensional steady-state structure of a film flow of a combustible liquid on a heat-conducting substrate in the case of combustion-wave propagation is considered in the hydrodynamic formulation. The physical mechanism of formation of this structure is analyzed. It is shown that the thermocapillary effect plays an important role. The conclusion is justified that the existence of a two-dimensional regime is possible only for rather low values of the temperature gradient on the film surface. A critical condition is obtained, which determines the transition to a three-dimensional regime. This condition implies that the flow velocity is equal to the velocity induced by the thermocapillary force. If the temperature gradient is higher than a certain critical value, a zone with a reverse flow should appear, in accordance with the two-dimensional model. It is assumed that such a regime cannot exist due to its instability to three-dimensional perturbations. Experiments with a liquid film flowing down due to gravity in the presence of an immovable heat source (without the combustion wave) support the conclusion of the transition to a three-dimensional regular flow structure if the temperature gradient is rather high. The first part of the paper deals with simulation of the film structure for the critical condition satisfied. The second part deals with generalization of the problem to the case of a movable heat source moving with a constant velocity. This formulation of the problem includes the situation with combustion-wave propagation. The mathematical formulation of this problem allows us to assume that the existence of a two-dimensional steady-state regime in this case is limited by the same critical condition. If the temperature gradient on the film surface is higher than the critical value, the two-dimensional steady-state solution does not exist. This concept justified in the present work offers a generic explanation of phenomena observed in liquid films in the presence of local heat sources of various natures.

An Experimental and Numerical Study of the Isothermal Flowfield Behind a Bluff Body Flameholder

An experimental and numerical investigation was conducted to study the turbulent velocities and stresses behind a two-dimensional bluff body. Simultaneous three-component laser-Doppler velocimeter (LDV) measurements were made in the isothermal incompressible turbulent flowfield downstream of a bluff body placed at midstream in a rectangular test section. Mean velocities and Reynolds stresses were measured at various axial positions. Spanwise velocity measurements indicated that the flow is three dimensional in the recirculation zone of the bluff body. Confidence in the accuracy of the data was gained by calculating the mass fluxes at each axial station. These were found to agree with each other to within ±3 percent. A parallel Computational Fluid Dynamics (CFD) study was initiated to gage the predictive accuracy of currently available CFD techniques. Three solutions were computed: a two-dimensional steady-state solution using the standard k-ε model, a two-dimensional time-accurate solution using the standard k-ε model, and a two-dimensional time-accurate solution using a Renormalized-Group (RNG) k-ε turbulence model. The steady-state solution matched poorly with the data, severely underpredicting the Reynolds stresses in the recirculation zone. The time-accurate solutions captured the unsteady vortex shedding from the base of the bluff body, providing a source for the higher Reynolds stresses. The RNG k-ε solution provided the best match to the data.

Convective heat transfer inhibition in an annular porous layer rotating at weak angular velocity

A study is made of natural convection in a horizontal annular porous layer in rotation about its axis. Isothermal boundary conditions are applied on both inner and outer boundaries, with the outer boundary being warmer. For such conditions, no symmetry with respect to a vertical diameter can be expected for the flow and temperature fields and the whole annular region must be considered. Two-dimensional steady state solutions are obtained numerically. Results indicate that the rotation generates a net circulating flow around the annulus relative to the porous matrix and drastically reduces the overall heat transfer. These effects occur at relatively low angular velocities for which the centrifugal forces are negligible compared to gravity, and inertial effects may be neglected.

Computational vortex dynamics in two and three dimensions

We present a discussion of some solutions and outstanding problems of dissipationless and nearly dissipationless flows in two and three dimensions. In particular, we review two-dimensional steady-state solutions, linear stability and nonlinear evolutions including axisymmetrization of isolated vortex regions and merger or condensation of like-signed vorticity regions. These results are obtained from pseudo-spectral codes, contour dynamical (CD) codes and a new Hamiltonian moment-model of the Euler equations.

Free Convection in a Two-Dimensional Porous Loop

A study is made of the natural convection in an annular porous layer having an isothermal inner boundary and its outer boundary subjected to a thermal stratification arbitrarily oriented with respect to gravity. For such conditions, no symmetry can be expected for the flow and temperature fields with respect to the vertical diameter and the whole circular region must be considered. Two-dimensional steady-state solutions are sought by perturbation and numerical approaches. Results obtained indicate that the circulating flow around the annulus attains its maximum strength when the stratification is horizontal (heating from the side). This circulating flow is responsible for an important heat exchange between the porous layer and its external surroundings. The flow field is also characterized by the presence of two convective cells near the inner boundary, giving rise to flow reversal on this surface. When the maximum temperature on the outer boundary is at the bottom of the cavity, the convective motion becomes potentially unstable; for a Rayleigh number below 80, there exists a steady-state solution symmetrical with respect to both vertical and horizontal axes; for a Rayleigh number above 80, an unsteady periodic situation develops with the circulating flow alternating its direction around the annulus.

Propagation of sound waves in a moving medium

The propagation of sound waves radiated by a two-dimensional source in a fluid moving with subsonic velocity between two perfectly reflecting parallel walls is considered. The steady state problem appears to have a non-unique solution, for which Sommerfeld's radiation condition does not apply. Two methods are used for obtaining the unique solution. First the corresponding problem for a fluid with non-zero bulk viscosity is solved, which has a unique solution and then the limit for zero bulk viscosity is taken. Secondly, the initial value problem for a source being switched on at timet=0 is solved and it is shown that its solution tends to the same steady state solution in the limit fort → ∞. In the last section the results for the corresponding axisymmetric case are given. In the appendix some properties of the twodimensional steady state solution are explained qualitatively.