PROJECTS SUMMARYPROJECTS SUMMARY

PENGTAO SUNPENGTAO SUN

6/25/20036/25/2003

MATHEMATICS DEPT.MATHEMATICS DEPT.

PENN STATEPENN STATE

BLACK HOLE SIMULATIONBLACK HOLE SIMULATIONParticipators: Pablo Laguna, Jinchao Xu, Pengtao SunParticipators: Pablo Laguna, Jinchao Xu, Pengtao Sun

Numerical evolutions of black holes have been improved slowly but steadily Numerical evolutions of black holes have been improved slowly but steadily over the last few years and now first attempts are being made to extract over the last few years and now first attempts are being made to extract physical information from these evolutions. Most notably one wants to predict physical information from these evolutions. Most notably one wants to predict the gravitational radiation emitted during black hole coalescence the gravitational radiation emitted during black hole coalescence

Initial data are the starting point for any numerical simulation. In the case of Initial data are the starting point for any numerical simulation. In the case of numerical relativity, Einstein's equations constrain our choices of these initial numerical relativity, Einstein's equations constrain our choices of these initial data.data.

The quality of the initial data will be crucial to the success of the predictions The quality of the initial data will be crucial to the success of the predictions of the gravitational wave forms. Unphysical gravitational radiation present in of the gravitational wave forms. Unphysical gravitational radiation present in the initial data will contribute to the gravitational waves computed in an the initial data will contribute to the gravitational waves computed in an evolution and might overwhelm the true gravitational wave signature of the evolution and might overwhelm the true gravitational wave signature of the physical process under consideration. Therefore an important question is how physical process under consideration. Therefore an important question is how to control the gravitational wave content of initial-data sets, and how to to control the gravitational wave content of initial-data sets, and how to specify astrophysically relevant initial data with the appropriate gravitational specify astrophysically relevant initial data with the appropriate gravitational wave content, for e.g. two black holes orbiting each other. wave content, for e.g. two black holes orbiting each other.

BLACK HOLE IN 3DBLACK HOLE IN 3D

BLACK HOLE IN 2DBLACK HOLE IN 2D Extreme mass ratios binary systems, binaries

involving compact objects such as stellar mass black holes or neutron stars orbiting super-massive black holes, are considered to be a primary source of gravitational radiation to be detected by the space-based interferometer LISA.

The numerical modeling of these binary systems is extremely challenging because the scales involved expand orders of magnitude. One needs to handle large wavelength scales comparable to the super-massive black hole and, at the same time, to resolve the scales in the vicinity of the small compact object where radiation reaction effects play a crucial role.

Finite element methods are a natural choice to achieve this high level of adaptivity. To demonstrate this, we present results of a toy problem consisting of a point-like source orbiting a black hole in scalar gravitation.

510

510

TWO PHASE FLOWSTWO PHASE FLOWSExample 1— Thermo-induced Marangoni effectsExample 1— Thermo-induced Marangoni effects

One of two-phase fluids flow example, is two-fluid Marangoni One of two-phase fluids flow example, is two-fluid Marangoni convection in which the heated boundary is embedded in the convection in which the heated boundary is embedded in the free surface between two liquid with different densities (with free surface between two liquid with different densities (with the lighter one on the top), the induced temperature gradient the lighter one on the top), the induced temperature gradient on the surface drives convective motion, and induces vorticity on the surface drives convective motion, and induces vorticity in the bulk fluid. This motion deforms the free surface and in the bulk fluid. This motion deforms the free surface and lead to a complicated flow pattern. Preliminary numerical lead to a complicated flow pattern. Preliminary numerical experiments for a two dimensional model have already been experiments for a two dimensional model have already been performed using grid adaptation techniquesperformed using grid adaptation techniques

The experimental counter part has been constructed in the The experimental counter part has been constructed in the Pritchard Lab by Belmonte.Pritchard Lab by Belmonte.

It is well known that certain surfactants can form long cylindrical micelles in aqueous It is well known that certain surfactants can form long cylindrical micelles in aqueous solution at low concentration, if they are mixed with certain large and semi-hydrophobic solution at low concentration, if they are mixed with certain large and semi-hydrophobic organic counterions such as sodium salicylate. Such solutions, known as wormlike organic counterions such as sodium salicylate. Such solutions, known as wormlike micellar fluids, are similar in some ways to polymer solutions. micellar fluids, are similar in some ways to polymer solutions.

We found a thick gel-like micellar phase at the interface between an aqueous surfactant We found a thick gel-like micellar phase at the interface between an aqueous surfactant solution and an aqueous organic salt solution. When mixed homogeneously, these two solution and an aqueous organic salt solution. When mixed homogeneously, these two solutions are known to form a highly elastic fluid. This observation are made during the solutions are known to form a highly elastic fluid. This observation are made during the slow injection of one fluid into the other by a tube or pipette.slow injection of one fluid into the other by a tube or pipette.

TWO PHASE FLOWSTWO PHASE FLOWSExample 2 — Gel fingering phenomenonExample 2 — Gel fingering phenomenon

Fingering of Gel Fingering of Gel

Numerical experiment is ongoing, the primary results are as follows.Numerical experiment is ongoing, the primary results are as follows.

TWO PHASE FLOWSTWO PHASE FLOWSExample 3 — Surface water wavesExample 3 — Surface water waves

Another even more challenging example is in the area of Another even more challenging example is in the area of surface water waves. The Pritchard Lab contains a wave basin surface water waves. The Pritchard Lab contains a wave basin with a segmented, programmable wave-maker system that is with a segmented, programmable wave-maker system that is capable of generating both 2D and 3D water waves. These capable of generating both 2D and 3D water waves. These wave motions are typically modeled by the (inviscid) Euler wave motions are typically modeled by the (inviscid) Euler equations assuming the flow to be irrotational. Yet, both equations assuming the flow to be irrotational. Yet, both viscous and rotational effects have been observed in many viscous and rotational effects have been observed in many experiments. In particular, a remarkably stable 2D vortex has experiments. In particular, a remarkably stable 2D vortex has been observed in 2D and weakly 3D experiments carried out by been observed in 2D and weakly 3D experiments carried out by J. Hammack and D. Henderson. The vortex forms near the J. Hammack and D. Henderson. The vortex forms near the center of the basin, spanning its width,and then propagates center of the basin, spanning its width,and then propagates slowly to the wave-maker where it is extinguished.slowly to the wave-maker where it is extinguished.

We will use our grid adaptation and multigrid techniques to We will use our grid adaptation and multigrid techniques to develop a \numerical wave basin" based on the Navier-Stokes develop a \numerical wave basin" based on the Navier-Stokes equations that will be used to predict wave motions as well as equations that will be used to predict wave motions as well as resulting vortical motions.resulting vortical motions.

FUEL CELL MODELFUEL CELL MODEL

We propose a model of liquid and heat flux, ignoring the gas dynamics. More We propose a model of liquid and heat flux, ignoring the gas dynamics. More specifically we assume that the pressure and vapour pressure are constant and specifically we assume that the pressure and vapour pressure are constant and solve for the water volume fraction and the temperature as functions of space solve for the water volume fraction and the temperature as functions of space and time. The water motion is driven by capillary pressure, and a heat flux is and time. The water motion is driven by capillary pressure, and a heat flux is generated by boundary conditions. The two equations are coupled by generated by boundary conditions. The two equations are coupled by condensation, which exchanges heat for liquid, generating a liquid flux condensation, which exchanges heat for liquid, generating a liquid flux opposite that of the heat flux.opposite that of the heat flux.

Numerical simulation is done with the adaptive finite element methodNumerical simulation is done with the adaptive finite element method

CONVECTION-DIFFUSION CONVECTION-DIFFUSION PROBLEMPROBLEM

multilevel discretization and grid adaptationmultilevel discretization and grid adaptation

ANISOTROPIC ADAPTIVITYANISOTROPIC ADAPTIVITY

All edges are equal under some metric which depends on p norm, All edges are equal under some metric which depends on p norm, which give us a criteria to construct a nonlinear functional which which give us a criteria to construct a nonlinear functional which satisfies the equidistribution and isotropy simultaneously.satisfies the equidistribution and isotropy simultaneously.