I. Effect of Neutral Fluid Turbulence in the Gas Blanket Region of the Divertor

It is well known that the projected heat loads onto the divertor plate in ITER-like tokamaks pose a severe problem. There are two somewhat divergent ideas in controlling this energy flux which comes into the scrape-off-layer (SOL) from the bulk plasma: (a) impurity radiation, and (b) formation of a neutral gas blanket between the plasma flame front and the divertor plate. Both ideas need to be considered in detail since the use of impurity radiation could lead to large impurity influx into the plasma core, while the gas blanket regime requires high neutral densities (on the order of 10**22 - 10**23 m**-3 for a pure hydrogen plasma) near the divertor plate.

The basic idea behind the gas blanket is that the heat flux onto the divertor plate target can be reduced if there exists a sufficiently dense neutral gas layer so that the plasma will detach from the target. Similar to the plasma detachment seen in current experiments, this cold neutral layer will act as a catalyst for recombination plasma detachment which will result in reduced heat loads to the divertor plate and might be considered as an option for divertor design for reactor scale machines. A simple poloidal one dimensional (1D) heat conduction model of Krasheninnikov et. al., for the gas blanket estimates its thickness as only
Lff = 3.5 cm [hydrogen plasma]
Lff = 2.2 cm [D-T plasma]
In this laminar model, typical reactor relevant conditions are assumed: a SOL heat flux is around 50 MW/m2 with 1 MW/m2 reaching the target.

Excitation of convective/turbulent structures in the neutral gas layer should increase the effective heat conduction coefficient of the neutral gas and, hence, the distance between the target and the flame front.

Of course, strong impurity seeding and neutral gas blanket regimes are two extreme approaches to the divertor flux problem - and each have their advantages and disadvantages. It seems attractive to actually try to combine these approaches in such a way as to minimize their respective weaknesses - (a) the hazard of bulk plasma contamination by impurities on the one hand, and (b) high neutral gas pressure in the divertor which can result in unacceptably high upstream plasma density.

Recent results from 2D divertor plasma modeling, where a fluid plasma code [UEDGE] is coupled to a laminar (2D) Navier-Stokes neutral package, show that for relevant reactor upstream plasma densities ( < 10**20 m**-3) and energy fluxes, a neutral gas layer (with densities ª10**21 m**-3) already forms near the target under relatively low impurity (Carbon) concentration of around 3%. The effect of convective energy fluxes in the gas layer region is significant. Moreover the plasma density in front of the target and the plasma flux onto the target drastically reduced due to plasma recombination. The toroidal component of the neutral gas velocity in these regimes approaches the local sound speed and the Reynolds number of the neutral flow is on the order of 1000.

While it has been customary to think of neutral turbulence as being induced in channel flow at Reynolds numbers Re > 2000, where Re = U L / n (U is a mean velocity, L is the full width of the channel and n the molecular viscosity), there is experimental evidence that turbulence can be triggered at Reynolds numbers as low as Re ª 650 by using eddy promoters. This is of great significance to the divertor regime since recent laminar studies have indicated that Re > 1000 are readily found within the gas blanket. At such Reynolds numbers, turbulent effects can be expected to be important and will significantly effect the parameters of the divertor plasma as well as the gas layer.

We are considering how 3D turbulence can significantly increase the gas blanket region and yield extensive spreading of the energy onto the toroidal sidewalls and thereby decrease the heat flux to the target. Both of these effects lead to an increased Lff. In determining the turbulent enhancement of heat flux to the toroidal walls, we simplify the problem to 2D mean flow over toroidal cavities. The cavities provide both a source for the turbulence (since vorticity is shed from the walls) as well as a larger surface area for heat deposition. It is critical to note that even though we are restricting ourselves to 2D mean flows and geometries, one must treat the neutral turbulence in 3D - otherwise one will obtain the wrong energetics. Indeed, in 2D turbulence ( as well as 2D laminar flows) energy is cascaded to large spatial scales, while in 3D the energy cascades to small spatial scales. In neutral turbulence, the turbulence is 3D. Thus if one restricted the analysis to 2D (whether laminar or turbulent), large scale convection patterns would form purely as a result of the imposed 2D geometry and thereby give incorrect physics.

A typical plot of the turbulent pressure contours is shown below using the compressible K-e NASA code ISAAC which solves for the evolution of the mean density, mean flow, mean total energy, the turbulent kinetic energy K and the turbulent dissipation rate e. One can notice the shock structures coming off the back edge of the cavity. The upper horizontal plane is a symmetry plane.

The corresponding heat flux to the toroidal walls shows significant heat reductions to the target due to the turbulence. Indeed, the enhanced heat flux to the walls is shown in the next Figure. Before the cavity region, the turbulent heat flux is a factor of 2 - 3 higher than that for laminar flow. However, within and after the cavity region there is an order of magnitude increase.

Turbulent flows are difficult to compute since they involve a wide range of length and time scales that must be adequately resolved. In particular, turbulent flows of practical importance are even more difficult to solve because of the presence of nontrivial boundaries. Direct numerical simulation (DNS) of complex flows are just not feasible on any foreseeable supercomputer architecture, while large eddy simulations (LES) have to contend with various closure approximations and subgrid scale modeling. A problem of considerable importance is turbulent separated flows - and the canonical backward facing step as well as cavity flows in possible divertor designs for ITER-like conditions. DNS solutions on the backward facing step are computationally extremely expensive and can only serve as a database on which to test simpler turbulence models. Of course, once validated, these simpler models can handle non-trivial geometries.

Another area of great importance is compressible turbulent flows. There are some excellent DNS simulations that have produced a wealth of statistical information on simple compressible shear flow turbulence. The idea behind these DNS calculations is to understand energy transfer mechanisms as well as properties of the rate of strain tensor.

We are addressing these questions by utilizing the recent lattice Boltzmann methods (LBE), and their extension (TLBE) to compressible flows. The idea behind LBE is to replace the nonlinear macroscopic equations by a linearized Boltzmann equation in which the Krook-like collision operator is so chosen that by performing Chapman-Enskog expansions one recovers the original macroscopic system. Moreover, by solving this LBE on a suitably chosen spatial lattice, the underlying discrete symmetry in the kinetic model is exactly smoothed out when the appropriate moments are taken to obtain the continuum equations. The beauty of TLBE is that it is ideal for multi-parallel processors and results in very simple codes. For a periodic system, there are only 2 basic operations: (a) free streaming from one lattice node to another, and (b) relaxation at each lattice site. These are purely local operations and ideal for parallelization. In fact, on a dedicated CRAY C90-time (on the DoE machine at Livermore), our 2D TLBE code runs at over 600 Mflops/processor and over a concurrency of 15.5 CPU/wallclock time [of a 16 CPU machine]. The average vector length per operation is 127.87 [of a 128 vector length machine]. We show the evolution of vorticity contours in 2D shear turbulence when there is a strong initial temperature gradient. Parallelization (using MPI) is also being done on the T3E (512 processor machine). We show plot , plot the effect of the Reynolds number on the evolution of vorticity contours in the 2D decay of shear turbulence in the presence of a sharp temperature gradient.

Another very important advantage of TLBE methods is the ease of incorporating boundary conditions in realistic geometries.