Fluid Model of Plasma and Computational Methods for Solution E

WDS'06 Proceedings of Contributed Papers, Part III, 180–186, 2006. ISBN 80-86732-86-X © MATFYZPRESS Fluid Model of Plasma and Computational Methods for Solution E. Havl´ıˇcková Charles University, Faculty of Mathematics and Physics, V Holeˇsoviˇckách 2, 180 00 Prague, Czech Republic. Abstract. This contribution gives basic insight into the fluid modelling in plasma physics. The main aim of the paper is to offer brief conspectus of equations that form the fluid model of plasma and to point out the differences between the theoretical model in general form as derived from the Boltzmann equation and the fluid model which is commonly used in practical simulations. Second part of the contribution summarizes computational methods which are often used for solution of the described model, as they were the most widely mentioned in literature. Introduction The simulation of plasma processes can be based on various approaches. The microscopic description of plasma is for many practical purposes too detailed or computationally too complicated and in some applications we need not to know the information about the behaviour of individual particles. Fluid model describes macroscopic plasma phenomena and reveals how the statistical plasma parameters evolve in time and space. Fluid model of plasma Theoretical description of plasma Fluid model of plasma is based on partial differential equations (PDEs) which describe the macroscopic quantities such as density, flux, average velocity, pressure, temperature or heat flux. Governing PDEs can be derived from the Boltzmann equation (BE) by taking velocity moments. Zero moment of ∞ + 3 the Boltzmann equation ( −∞ (BE) d v) yields continuity equation for the particle density R ∂n Qρ + ∇ · (n~u) = (1) ∂t m ∞ ∞ + 3 1 + 3 Density and average velocity are defined as n = −∞ f d v and ~u = n −∞ ~vf d v. The source term on the right side of the equation (1) correspondsR to the collision term ofR the Boltzmann equation and ∞ ρ + 3 describes mass production and annihilation. It is defined as Q = m −∞ (∂f/∂t)c d v. Similarly, the momentum transport equation (equation of motion) (2, 3) and heat transporR t equation (energy equation) ∞ ∞ + 3 1 + 2 3 (4, 5) can be found as first moment (m −∞ ~v (BE) d v) and second moment ( 2 m −∞ v (BE) d v) of the Boltzmann equation. R R ∂(n~u) ↔ m + ∇ P + m∇(n~u~u) − nq(E~ + ~u × B~ ) = ~uQρ + Q~p (2) ∂t ∂~u ↔ mn + mn(~u · ∇)~u + ∇ P − nq(E~ + ~u × B~ ) = Q~p (3) ∂t ∂ 1 1 1 ↔ 1 1 p + mnu2 + ∇ · p~u + ~u P + mnu2~u + L~ − nqE~ · ~u = u2Qρ + ~u · Q~p + QE (4) ∂t µγ − 1 2 ¶ µγ − 1 2 ¶ 2 1 ∂p ↔ + ∇ · (p~u) + (P ·∇)~u + ∇ · L~ = QE (5) γ − 1 µ ∂t ¶ ∞ ~ 1 + − − 2 3 Mean quantities are defined by the following equations. L = 2 m −∞ (~v ~u)(~v ~u) f d v defines heat ↔ ∞ ↔ ~ + 3 R flux L, P = m −∞ (~v − ~u)(~v − ~u)f d v defines pressure tensor P and Pij = p δij defines scalar pressure p. γ in (4) andR (5) is the ratio of specific heats. Source terms in (2) and (4) describe transport of 180 HAVLÍCKOVˇ A:´ FLUID MODELLING IN PLASMA PHYSICS ∞ ~p + − 3 momentum and heat transfer due to collisions and they are defined as Q = m −∞ (~v ~u)(∂f/∂t)c d v, ∞ E 1 + − 2 3 R Q = 2 m −∞ (~v ~u) (∂f/∂t)c d v. We canR continue to derive the moments for high order terms, however the equation chain must be truncated somewhere. In many practical problems this is made in the first order by substituting the energy equation by an equation of the state, or in the second order by using algebraic expression for the heat flux as closure approximation. Therefore the equations (1), (2) and (4) establish the typical set of fluid equations which are used in many simulations. Moreover, to complete the system of governing equations and to obtain the self-consistent description of plasma, we must consider the Maxwell’s equations describing the electromagnetic behaviour of plasma. The meaning of individual terms in the fluid equations and their derivation is explained in Golant et al. [1980], Chen [1984] or Braginskii [1965] in more detail. Specific application of the fluid theory leads to various forms of the model. Generally we can use one set of fluid equations for each plasma species and considering a simple two component (ion and electron) plasma one obtains the so-called two-fluid model. Magnetohydrodynamics, the most widely known plasma theory, is another modification - plasma is considered as a single fluid in the center of mass frame [Chen, 1984; Vold et al., 1991]. A choice of the model depends on the character of the simulated plasma. Classical fluid formulation Practical use of the fluid model of plasma is not based on the general form of the equations (1), (2) and (4) as they were derived from the Boltzmann equation, but it is always connected with various approximations. Therefore let’s introduce the classical fluid modelling approach used most commonly in simulations [Chen et al., 2004; Bukowski et al., 1996]. Approximative form of the equations is connected especially with a simplification of the source terms which are very complicated functions of velocity. The mass balance collision term gives the rates of creation and loss of species. The productivity of the reactions is defined by the reaction rate coefficients kr corresponding to the collision of type r. The coefficients kr are the input parameters of the simulation. Assuming an electron collision with neutral, the source term for the formation of ρ species k can be evaluated as Qk = mk r lrkkr(Te)nenn, where lrk is the number of particles of species k created or lost per collision type r [BukowskiP et al., 1996]. The useful approximation of the collision term in the momentum transport equation (2) and the energy equation (4) is the Krook’s approximation p mkml E 3 2mkml ~ − − − 2 − Qk = l mk+ml nk ( ~uk ~ul) νkl and Qk = l 2 k nk (mk+ml) (Tk Tn) νkl, where νkl is the mean collisionP frequency [Golant, 1980; Chen et al., 2004].P Fluid equations as used in simulations are often simplified by neglecting some terms: - neglecting viscosity effect in the transfer equation characterizing the anisotropic part of the pressure, which yields the reduction of pressure tensor to the scalar pressure [Golant et al., 1980]. - neglecting kinetic energy contribution compared to the thermal one, which leads to the significant simplification of the energy equation [Bukowski et al., 1996; Chen et al., 2004]. ~ 5 ~ − 5 ∇ - neglecting the first term in the thermal flux closure for electrons Le = 2 ΓekTe 2 kDene Te, which is known as Fourier’s approximation [Chen et al., 2004]. The expressions for thermal flux of various species can be found in Golant et al. [1980]. The most widely mentioned approximation in the classical fluid formulation is the so-called drift- diffusion approximation [Herrebout et al., 2001; Herrebout et al., 2002; Chen et al., 2004; Boeuf, 1987; Donkó, 2001; Passchier et al., 1993]. This approximation reduces the number of partial differential equations included in model by the use of the algebraic expression for particle flux Γ~k (7) instead of full equation of motion (2). The equation (7) describes the transport of species due to density gradient and the transport of charged species under the influence of the electric field. ∂n Qρ k + ∇ · Γ~ = k (6) ∂t k m Γ~k = ± µknkE~ − ∇(Dknk) (7) Coefficients of mobility µ and diffusion D are input parameters of the simulation. The expression (7) is equivalent to the equation of motion (3) after neglecting unsteady (first term) and inertial (second term) contributions. Conditions under which the drift-diffusion approximation and the approximations 181 HAVLÍCKOVˇ A:´ FLUID MODELLING IN PLASMA PHYSICS mentioned above are fulfilled can be found in Golant et al. [1980] and Chen et al. [2004]. These publications also include more detailed characteristics of the individual terms in the equations and the definition of all quantities. Computational methods of solution The physical aspects of any fluid flow are governed by three fundamental principles - mass conservation, Newton’s second law and energy conservation. These principles can be expressed in terms of mathematical equations, which are usually partial differential equations. This section briefly resumes the methods of computational fluid dynamics (CFD) that give the solution of PDEs describing the plasma as a fluid. The basic computational techniques are based on the construction of a discrete grid and the replacement of individual differentiated terms in PDEs by algebraic expressions connecting nodal values on a grid (Fig. 1). GOVERNING SYSTEMOF EQUATION APPROXIMATE PARTIAL DISCRETIZATION ALGEBRAIC SOLVER SOLUTION DIFF.EQS. EQUATIONS Figure 1. Overview of the computational solution procedure. The most common choices for converting the PDE to the algebraic one are finite difference, finite element and finite volume methods. In practice, time derivations in the time-dependent equation are discretized almost exclusively using the finite difference method and spatial derivatives are discretized by either the finite difference, finite element or finite volume method, typically. Finite difference method The method of finite differences (FDM) is widely used in CFD [Meeks et al., 1993; Montierth et al., 1992; Baboolal, 2002]. Finite difference representation of derivatives is based on Taylor’s series expansions. The discretization process introduces an error dependent on the order of terms in the Taylor’s series which are truncated. Derivation of elementary finite differences of various order accuracy is described in Wendt [1992]. To represent the derivatives by the differences, a number of choices is available, especially when the dependent variable appearing in the governing equation is a function of both coordinates and time.

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