Dt Potential vorticity. The semi-discrete divergence equations. We now repeat the previous steps for deriving the semi-discrete potential vorticity equation, only this time, taking the discrete divergence of the semi-discrete Euler-Lagrange equations. Using the definition of div, after several calculations provided in A.
Dt Remark 6. Dt Intruiguingly, because the Lagrangian form of the continuum shallow water di- vergence equation is less visited by the literature, we find the need to emphasize the physical significance of the div2 U term, which has a very strong tendency to force the fluid back to a purely rotational state.
The discrete variant of this term also appears in the semi-discrete divergence equations. So like the semi-discrete potential vorticity, we are able to iden- tify analogous physically significant terms, but the extent to which these quantities are conserved in simulation is governed by the consistency of the choosen discrete operators with the approximation solution fields. Numerical Experiments This Section describes three numerical experiments using the VFL to simulate the rotating regularized shallow water equations on a periodic domain.
The pur- pose of the first experiment is to investigate the conservative properties of the VFL method for rotating 1D shallow water. In the second experiment, we con- sider the problem of geostrophic adjustment of rotating 1D shallow water and show that the VFL method produces results which are consistent with the the- ory of geostrophic adjustment established by Rossby It is precisely because this mechanism is consistent with the advection law of potential vorticity that geo- physical modelers use this experiment to assess the integrity of the discrete shallow water velocity and layer depth equations.
These two experiments are initialized by perturbing the Voronoi cell masses with a Gaussian perturbation and the velocity field is initially zero. In the third experiment, we simulate the motion of a vortex pair in purely ro- tational 2D shallow water flow and monitor the energy and potential vorticity conservative properties over a much longer period of 50 years.
Over the course of the simulation, the Voronoi diagram appears to retain the history of the flow field and deforms in the wake of the vortex pair. We also show that the relative energy and potential vorticity error does not exhibit secular drift.
Before describing each of the experiments in more detail, we mention a few per- tinent details common to all experiments. For the 1D simulations, the initial conditions were sufficiently smooth and the approximation sufficiently high in res- olution that the effect of regularization was marginal and not required to ensure stability.
Experiment 1: conservative properties of VFL. Unless otherwise stated, we use cells, where each cell is initially of the same size. The parameters of the simulations are provided in Table 1 below. Parameter Value Number of cells N Time step 0. This Table lists the simulation parameters for experi- ment 1, which investigates the conservative properties of the VFL method for 1D rotating shallow water.
The following Figures show the relative energy, total potential vorticity PV and total potential enstrophy PE sum of the square of the cellwise potential vorticity errors. Figure 2 shows the scaling of the energy, PV and total vorticity error with the size of the time step. All three quantities montonically decrease as the time step decreases.
This suggests that the VFL method introduces error in the discrete vorticity. Figure 3 shows from top to bottom the relative energy, potential vorticity and potential enstrophy error over time steps, with a time step of 0. This number of time steps is approximately equal to the number of time steps required to conduct a year climate change simulation. Each error does not drift which suggests that the VFL method is long-time stable and conservative.
This Figure shows the scaling laws of the mean energy error, potential vorticity and total vorticity with the size of the time step over time steps. Potential vorticity PV and total vorticity TV error do not scale with the order of the integrator suggesting that the order of the error of the PV and TV are not governed by the integrator but by the VFL method itself.
The simulation parameters are given in Table 1. Experiment 2: Geostrophic adjustment. The dynamics, governed by equations 31 , are observed to exhibit gravity waves which propagate energy and momentum away from the source leaving behind geostrophically balanced flow.
We verify in this numerical experiment that this mechanism only occurs if the scale of layer depth perturbation is smaller than the Rossby deformation radius by performing two simulations, one in which the scale of perturbation is larger and one is which it is smaller than the Rossby deformation radius. The parameters for this simulations are provided in Table 2. Figures 4 and 5 compare two different rotating shallow water regimes dis- tinguished by the scale of perturbation of the layer depth.
The bottom profile of each of these graphs in this Figure also shows that a geostrophically balanced region is recovered in the region of the source. Figure 6 show the corresponding energy and potential vorticity errors over 2 rotation units i. We firstly observe that the profiles exhibit high fre- quency oscillations arising from gravity waves but do not exhibit drift.
There are also jumps in the profiles with an approximate period of 0. These jumps occur when the gravity waves collide recall that the domain is periodic. Figure 7 shows how the discrete approximation of the layer depth, over the central region of the domain, converges to the steady state after 0. The outer regions of the domain exhibit gravity waves which have propagated away from the source. These graphs show from top to bottom the relative energy, potential vorticity and potential enstrophy error over time steps.
The simulation pa- rameters are given in Table 1. The top two graphs show the layer depth perturbation and layer depth respectively. The third and fourth graphs show the horizontal and meridional velocities. The bottom graph shows the difference of the meridional velocity with the layer depth gradient.
Note that the source region is not restored to a geostrophically balanced state because geostrophic adjustment does not occur. The top graph show the layer depth perturbation and layer depth respectively.
This difference is zero when the flow is geostrophically balanced. Note that the source region does restore to a geostrophically balanced state because gravity waves prop- agate away energy and momentum to restore the balance. Both profiles exhibit high frequency oscillations arising from gravity waves but do not exhibit drift. These jumps occur when the gravity waves collide recall that the domain is pe- riodic. This graph shows the layer depth at 0. In the central region of the domain, the layer depth has reached a steady state geostrophic balance.
The graph shows how the discrete approximation of the layer depth converges to the steady state. This outer region is not in geostrophic balance after 0. Experiment 3: Vortex pair in 2D rotating shallow water flow. This experiment shows the motion of a vortex pair in a purely rotational shallow water flow over a doubly periodic domain without bottom topography. The simulation parameters are given in Table 3. A contour plot of the regularized layer thickness is shown in Figure 8 in which each snapshot is taken at increments of 40 days and is viewed from left to right, starting at the top of the page.
The vortex pair is observed to undergo pure rotation without significant shape deformation or a change in strength of the poles. The Voronoi diagram corresponding to these layer thickness snapshots is shown in Figure 9. The diagram retains history of the flow field and largely deforms in the wake of the vortex pair to exhibit some intriguiging rotational structure with loss of the radial symmetry. Over the course of the simulation, an increasing number of cells deform from their initial state and the resulting effect is a more homogeneously deformed diagram.
Experiment 3: the simulation parameters for the rotat- ing shallow water equations over a doubly-periodic domain initial- ized by perturbing a geostrophically balanced vortex pair. This graph shows the layer depth at increments of 40 days during simulation of a vortex pair in rotating shallow water over a doubly periodic domain.
The simulation parameters for this experiment are given in Table 3. This graph shows the Voronoi diagram at increments of 40 days during simulation of a vortex pair in rotating shallow water over a doubly periodic domain. Conclusion The development of new numerical methods for climate modeling is challenged by the simultaneous need to tractably simulate long-time dynamics and exhibit consis- tent potential vorticity dynamics.
This paper considers the conservative properties of a variational formulation of the free-Lagrange method for the regularized ro- tating shallow water equations. RINGLER method, a Lagrangian method which remedies the diffusive transport computa- tions of Eulerian methods, by building the transport into the evolution of the cell positions.
Our implementation of the method has the distinguishing feature of us- ing both a Voronoi diagram to formulate a local mass conservation law and the dual Delaunay triangulation to approximate a dispersive regularization of the layer thickness. The semi-discrete regularized shallow water equations preserve symplec- tic structure and, when integrated with a symplectic time stepper, conserve energy over long-time simulations to the order of the symplectic integrator.
The discrete variational principle provides a unifying point of conferred mathe- matical understanding from which to systematically derive new geometric methods. In the absence of this property, we turn to the diagnostic semi-discrete potential vorticity dynamics and choose a discrete curl op- erator which has the property that it operates on the layer thickness gradient to give a zero vector field. This property guarantees the existence of a semi-discrete potential vorticity law.
The semi-discrete potential vorticity and the semi-discrete divergence equations kinematically equate to the respective solenoidal and irrota- tional components of the velocity field. Together, they augment the description of how the momentum evolves in discrete space.
Like the continuum divergence equation expressed in the Lagrangian frame , the semi-discrete divergence equa- tion exhibits a div2 U term which indicates that the flow has a very strong tendency to reach a purely rotational state and therefore has a marked effect upon potential vorticity evolution.
Simulation of a geostrophic adjustment mechanism provides further evidence that the VFL method exhibits consistent potential vorticity dynamics. References Augenbaum, J. Bonet, J. Buneman, O. Dixon, M. Harlen, O. Fluid Mech. Lewis, D. Margolin, L. Petera, J. Reich, S. Ringler, T. Weather Rev. Ripa, P. J West 76, — Rossby, C. Salmon, R. Tabor and Y. Treve 88, — Serrano, M. Whitham, G.
Appendix A. Calculation of the semi-discrete vorticity equations. Dt Remark A. Calculation of the semi-discrete divergence equations. E-mail address: mfdixon ucdavis. EE; dashed line: EL. Figure 13 shows field of local volume fraction of solid particles for the EE computation. Local droplet accumula- tion is also observed upstream of the stagnation point within the central jet. Figure Radial profiles of RMS axial particle velocities at Figure Instantaneous volume fraction in the central plane 7 stations along z axis.
Symbols: experiment; solid line: EE; from Euler-Euler simulation. This can be quantified by plotting mean velocities along the axis for the gas Fig. On this axis, both AVBP results match but are slightly off the experimental results. This also demonstrates the importance of injecting not only the proper mean profile for the gas velocity but also fluctuations with a reasonably well-defined turbulent spectrum. Additional tests also reveal that the injection of white noise on the particle velocities has a very limited effect on the results.
Figure Radial profiles of mean radial particle velocities Figures 16 and 17 display axial profiles of RMS velocities at 7 stations along z axis. Symbols: experiment; solid line: for the gas and the particles.
These plots confirm that the EE; dashed line: EL. Symbols: experiment; solid line: EE; dashed line: EL. Task parallelization in which some processors compute the gas flow and others compute the droplets flow. Domain partitioning in which droplets are computed to- gether with the gas flow on geometrical subdomains mapped on parallel processors. Droplets must then be exchanged between processors when leaving a subdo- main to enter an adjacent domain.
For LES, it is easy to show that only domain partitioning is efficient on large grids because task parallelization would require the communication of very large three-dimensional data sets at each iteration between all processors. How- ever, codes based on domain partitioning are difficult to Figure Axial profiles of mean particle velocities.
Sym- optimize on massively parallel architectures when droplets bols: experiment; solid line: EE; dashed line: EL. Moreover, the distribution of droplets may change during the computation: for a gas turbine reignition sequence, for example, the chamber is filled with droplets when the ignition begins thus ensuring an almost uniform droplet distribution; these droplets then evaporate rapidly during the computation, leaving droplets only in the near injector regions.
This leads to a poor speedup on a parallel machine if the domain is decomposed in the same way for the entire computation. As a result, dynamic load balancing strategies are required to redecompose the domain during the computation itself to preserve a high parallel efficiency Ham et al. Sym- means of two basic parameters used to measure the efficiency bols: experiment; solid line: EE; dashed line: EL. The former is defined as the ratio between the CPU time of a simulation with 1 processor Analysis of code scalability and the CPU time of a simulation with a given number of processors, Nprocs : In terms of code implementation EE techniques are naturally parallel because the flow and the droplets are solved using Trun 1 the same solver Kaufmann The EE formulation additional cost is of the order of 80 percent for this test case since the computational cost does not depend on the number of particles.
Table and Figs. Nprocs 1 2 4 8 16 32 64 Ideal scaling 1 2 4 8 16 32 64 Single-phase 1 2. On the other hand, Fig. This points out the need of dynamic load balancing for two-phase flow simulations with a Lagrangian approach, for example, by using multi-constraint partitioning algorithms which take into account particle loading on each processor Ham et al. Figure Speedup of the single-phase and the two-phase EL simulation.
The drop of performances shown in Fig 18 is not related to large communications costs between processors as it might be thought at first sight but merely to the parallel load imbalance generated by the partitioning algorithm Garcia et al.
This effect can be observed by plotting the number of nodes, cells and particles presented in each processor. Figure Number of cells and nodes per processor for a Figure 20 reports the number of nodes and cells presented partition by using a recursive inertial bisection RIB par- per processor for a partition simulation by using a titioning algorithm. On the prediction of gas-solid flows with two-way coupling using large eddy sim- ulation.
Physics of Fluids, Volume 12, Number 8, — The effect of mass loading and inter-particle collisions on the development of the poly- dispersed two-phase flow downstream of a confined bluff body. Modeling of liquid a partition by using a recursive inertial bisection RIB fuel injection, evaporation and mixing in a gas turbine burner partitioning algorithm. Cambridge Mathematical Library Edition. Cambridge University Press digital reprint For the present test case mass loading of 22 percent , the total number of particles present in the domain for the La- grange codes is of the order of , For such a small Colin, O.
A thickened number of particles, the computing power required by the flame model for large eddy simulations of turbulent premixed Lagrangian solvers compared to the power required for the combustion. Physics of Fluids, Volume 12, Number 7, gas flow remains low: the additional cost due to the parti- — cles is small even with the load balancing problem observed when increasing the number of parallel processors.
The EE Colin, O. Development of high-order formulation additional cost of the order of 80 percent is in- Taylor-Galerkin schemes for unsteady calculations.
Journal dependent of the mass loading, so that, for such a dilute case, of Computational Physics, Volume , Number 2, — the EL formulations proved to be faster up to 64 processors. On the decay rate of plemented into the AVBP solver are very close showing that isotropic turbulence laden with microparticles. Physics of both formulations lead to equivalent results in this situation. Fluids, Volume 11, Number 3, — An important factor controlling the quality of the results is the introduction of turbulence on the gas flow in the injec- Fede, P.
Numerical Study of Subgrid Fluid tion duct: without these turbulent fluctuations, the results are Turbulence Effects on the Statistics of Heavy Colliding Parti- not as good on the axis in terms of positions of the recircu- cles. Physics of Fluids, Volume 18, 17 pages lation zones. Future developments of the La- Solid Flows. Partitioning of particle improve scalability of the EL simulations. Large- eddy simulation of swirling particle-laden flows in a coaxial- Gatignol, R. Direct numerical sim- ulation of turbulence modulation by particles in isotropic tur- Ham, F.
Journal of Fluid Mechanics, Volume , — stantinescu, G. Unstructured LES of reacting multiphase flows in realistic gas turbine combustors. Center for Turbulence Research, Roux, S. Combustion and Flame, Volume , 40 — 54 of a two phase flow data based obtained on the flow loop Her- cule. A drag coefficient correlation.
Merseburg, M. Sommerfeld Editor , 3 — 18 V. Zeitung, Volume 77, — Kaufmann, A. Large- lie, J. Dynamics and dispersion in 3D unsteady simulations eddy simulation and experimental study of heat transfer, ni- of two phase flows.
In Supercomputing in Nuclear Applica- tric oxide emissions and combustion instability in a swirled tions. Paris: CEA. Selle, L. Compressible large-eddy simulation of turbulent combustion Kaufmann, A.
Compar- in complex geometry on unstructured meshes. Proceedings of the Estonian Academy of Sciences. Simonin, O.
Equation of motion for a small rigid bulence, Volume 3, sphere in a nonuniform flow. Numeri- cal study and modelling of turbulence modulation in a parti- Moin, P.
A dynamic cle laden slab flow. Journal of Turbulence, Volume 4, Num- subgrid-scale model for compressible turbulence and scalar ber , 1 — 39 transport. Subgrid modeling in — les of compressible flow. Applications Scientific Research, Moreau, M. Large eddy simulation of particle- euler-lagrange simulations of gas-particle turbulent flow. In laden turbulent channel flow. Large-eddy simulation of turbulent gas-particle flow Colin, O. High-order methods for dns and in a vertical channel: effect of considering inter-particle col- les of compressible multicomponent reacting flows on fixed lisions.
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