Murray Rudman

VOLUME-TRACKING METHODS FOR INTERFACIAL FLOW CALCULATIONS

SUMMARY A new algorithm for volume tracking which is based on the concept of flux-corrected transport (FCT) is introduced. It is applicable to incompressible 2D flow simulations on finite volume and difference meshes. The method requires no explicit interface reconstruction, is direction-split and can be extended to 3D and orthogonal curvilinear meshes in a straightforward manner. A comparison of the new scheme against well-known existing 2D finite volume techniques is undertaken. A series of progressively more difficult advection tests is used to test the accuracy of each scheme and it is seen that simple advection tests are inadequate indicators of the performance of volume-tracking methods. A straightforward methodology is presented that allows more rigorous estimates to be made of the error in volume advection and coupled volume and momentum advection in real flow situations. The volume advection schemes are put to a final test in the case of Rayleigh‐Taylor instability. 1997 by CSIRO. In the numerical computation of multifluid problems such as density currents or Rayleigh‐Taylor instability there is a need for an accurate representation of the interface separating two immiscible fluids. Free surface flows such as water waves and splashing droplets are an approximation to the multifluid problem in which one of the fluids (usually a gas) is neglected as having an insignificant influence on the dynamics of the system. In a general free surface flow problem, fluid coalescence and detachment may occur and deforming meshes cannot be used. In this case the need of an accurate and sharp interface is even greater than in true multifluid computations. Although a slightly diffuse interface may be acceptable in a problem where the continuity, momentum and energy equations are solved throughout the entire mesh, in a free surface simulation the location of the interface determines the size and shape of the computational domain and specifies where boundary conditions must be applied. In this case a diffuse interface cannot be tolerated. On finite volume (or difference) meshes, standard advection techniques can be used in multifluid problems to advect either the density or a material indicator function, however these methods are either diffusive (e.g. first order upwinding) or unstable (higher order schemes in which unphysical oscillations appear in the vicinity of the interface). Numerous techniques have been devised to limit the diffusiveness of low order schemes and to minimize the instability of high order schemes (see e.g.

A volume-tracking method for incompressible multifluid flows with large density variations

A numerical technique (FGVT) for solving the time-dependent incompressible Navier-Stokes equations in fluid flows with large density variations is presented for staggered grids. Mass conservation is based on a volume tracking method and incorporates a piecewise-linear interface reconstruction on a grid twice as fine as the velocity pressure grid. It also uses a special flux-corrected transport algorithm for momentum advection, a multigrid algorithm for solving a pressure-correction equation and a surface tension algorithm that is robust and stable. In principle, the method conserves both mass and momentum exactly, and maintains extremely sharp fluid interfaces. Applications of the numerical method to prediction of two-dimensional bubble rise in an inclined channel and a bubble bursting through an interface are presented

Analytical modeling and numerical simulation of forces in an endoluminal graft.

Purpose: To utilize mathematical analysis and computational fluid dynamics (CFD) to investigate the forces acting within the pressurized aorta and upon a stent-graft and how these forces may affect the ongoing performance of the stent-graft. Methods: Analytical force balance analysis and CFD simulations using the Fluent code were used to mimic blood flow through a bifurcated stent-graft in a person at rest. Steadystate blood flow was assumed in which the inlet pressure approximated the mean blood pressure (100 mm Hg) and the blood flow velocity was an approximation of the peak systolic flow rate (0.6 m/s). Two sizes of endoluminal grafts were analyzed: the larger graft had an inlet diameter of 3 cm and outlet diameters of 1 cm; the smaller graft diameters measured 2.4 cm proximally and 1.2 cm distally. The endografts were studied in 2 configurations: with the limbs straight and with one bent. Results: For the larger graft model, the normal peak blood flow induced a downward force of 7 to 9 N on the bifurcated grafts. Bending one of the limbs of the graft produced a sideways force of 1.3 N. For the smaller endograft, the downward force was in the range of 3.1 to 5.1 N and the sideways force on a curved limb was ∼1.5 N. The magnitude of the forces given by the analytical formulae and the CFD results agreed to within 2 significant figures. Conclusions: These results suggest that the downward force on a bifurcated stent-graft, which may exceed the force required to dislodge it when relying on radial attachment alone, is determined mostly by the proximal graft diameter. Curvature of the graft limbs creates an additional sideways force that works to displace the distal limbs of the graft from the iliac arteries.

VOLUME-OF-FLUID CALCULATION OF HEAT OR MASS TRANSFER ACROSS DEFORMING INTERFACES IN TWO-FLUID FLOW

A new volume-of-fluid (VOF)-based numerical method for calculating heat transfer or mass transfer of a species within and between fluids with deforming interfaces is described. The algorithm is tested first against an analytical solution for diffusion from a sphere, and good agreement between theory and calculation is shown. The method is then demonstrated by predicting (a) heat transfer from a rising bubble when the bubble forms a toroidal shape, and (b) mass transfer from a rising drop when the drop phase controls diffusion. The method is shown to be a viable approach for complex interfacial heat/mass transfer.

An investigation of the flow regimes resulting from splashing drops

Numerical simulations have been used to investigate the flow regimes resulting from the impact of a 2.9 mm water drop on a deep water pool at velocities in the range 0.8–2.5 m/s. The results were used to identify the conditions leading to the formation of vortex rings, entrapment of a bubble during cavity collapse and the formation of vertical Rayleigh jets. Bubble entrapment and the associated growth of a thin high speed jet were shown to be the result of a capillary wave that propagates down the walls of the crater resulting from drop impact. Although the existence of a capillary waves is a necessary condition for bubble entrapment, bubbles will only occur when the wave speed and maximum crater size is such that the wave reaches the bottom of the crater before collapse has resulted in the formation of a thick Rayleigh jet. Simulations also clarified the conditions for which drop impact leads to axi-symmetric vortex rings. Results not reported previously, include the observation that a single drop can pr...

Simulation of the near field of a jet in a cross flow

Abstract A direct numerical simulation technique is used to predict the time-dependent behavior of a weakly compressible axisymmetric jet exiting normally into a cross flow. Time-averaged velocity fields from the unsteady simulation are in qualitative agreement with previously published experimental and numerical results. The bound vortex system that is observed experimentally is also in evidence in the results of the unsteady simulation, as is the Kelvin -Helmholtz roll-up of the jet shear layer. The horseshoe vortex system often observed experimentally is not seen in the numerical results due to insufficient resolution. The formation of wake vortices in a numerical simulation is observed for the first time. The wake vortex system is found to have a Strouhal frequency of 0.098. The results show that cross-flow boundary layer fluid is lifted up from the boundary layer into these vortices near the rear of the jet exit. This observation is consistent with experimental observations that suggest that the vorticity in the wake system originates in the cross-flow boundary layer upstream of the jet exit.

Topological mixing study of non-Newtonian duct flows

Tracer advection of non-Newtonian fluids in reoriented duct flows is investigated in terms of coherent structures in the web of tracer paths that determine transport properties geometrically. Reoriented duct flows are an idealization of in-line mixers, encompassing many micro and industrial continuous mixers. The topology of the tracer dynamics of reoriented duct flows is Hamiltonian. As the stretching per reorientation increases from zero, we show that the qualitative route from the integrable state to global chaos and good mixing does not depend on fluid rheology. This is due to a universal symmetry of reoriented duct flows, which we derive, controlling the topology of the tracer web. Symmetry determines where in parameter space global chaos first occurs, while increasing non-Newtonian effects delays the quantitative value of onset. Theory is demonstrated computationally for a representative duct flow, the rotated arc mixing flow.

Assessment of higher-order upwind schemes incorporating FCT for convection-dominated problems

Abstract A new generalized formulation is suggested for four-point discretization schemes on nonuniform grids. The central difference scheme, the QUICK scheme, and the second-order upwind scheme fall into this formulation. A second-order hybrid scheme is also presented on nonuniform grids. The unbounded behavior of the generalized formulation is examined. A flux-corrected transport algorithm is then applied to the above four schemes on a uniform grid. Four two-dimensional convection-dominated problems are used to test the schemes. Incorporation of flux-corrected transport (FCT) into the high-order schemes improves the solution accuracy greatly. The unmodified multidimensional FCT limiter is found to be unable to completely suppress the small-scale oscillation of a velocity component which has discontinuities in a direction normal to the advection.

Two-phase natural convection: implications for crystal settling in magma chambers

Abstract Crystal settling is no longer thought to be the most likely mechanism operating in the formation of layered igneous intrusions. The main objection raised against its feasibility is the prediction that crystal settling velocities are likely to be orders of magnitude smaller than convective velocities, and that vigorous convection would therefore keep crystals well mixed within the interior of a magma chamber. In previous theoretical and numerical investigations of crystal settling, no account has been taken of compositional buoyancy gradients created by the interpenetration of crystals and melt. This paper includes these effects. The results of two-dimensional numerical simulations show that parameter regimes exist in which effective crystal settling occurs in infinite Prandtl number, thermally convecting, two-phase fluids even for small ratios of settling to convective velocity.

Computed oscillations of a confined submerged liquid jet

Abstract The jet oscillation observed in thin slab continuous casting is studied numerically by modelling the flow of liquid injected through a submerged entry nozzle and into a cavity. The oscillation relies on the exchange of fluid between recirculation cells on each side of the jet via a cross-flow through the gap between the nozzle shaft and the broad face of the cavity wall. Features of the oscillating jet are investigated by varying the resistance to cross-flow. This resistance occurs naturally since the nozzle obstructs cross-flow. The predicted oscillation can be manipulated by altering the cross-flow (through the use of an effective resistance force in the model) or stopped altogether to form a static asymmetrical flow pattern. Flow calculations are performed using a transient, two-dimensional, turbulent, fluid flow model.