A compatible Lagrangian hydrodynamic scheme for multicomponent flows with mixing

Abstract

We present a Lagrangian time integration scheme and compatible discretization for total energy conservation in multicomponent mixing simulations. Mixing behavior results from relative motion between species. Species velocities are determined by solving species momentum equations in a Lagrangian manner. Included in the species momentum equations are species artificial viscosity (since each species can undergo compression) and inter-species momentum exchange. Thermal energy for each species is also solved, including compression work and thermal dissipation caused by momentum exchange. The present procedure is applicable to mixing of an arbitrary number of species that may not be in pressure or temperature equilibrium. A traditional staggered stencil has been adopted to describe motion of each species. The computational mesh for the mixture is constructed in a Lagrangian manner using the mass-averaged mixture velocity. Species momentum equations are solved at the vertices of the mesh, and temporary species meshes are constructed and advanced in time using the resulting species velocities. Following the Lagrangian step, species quantities are advected (mapped) from the species meshes to the mixture mesh. Momentum exchange between species introduces work that must be included in an energy-conserving discretization scheme. This work has to be transformed to dissipation in order to effect a net change in species thermal energy. The dissipation between interacting species pairs is obtained by combining the momentum exchange work. The dissipation is then distributed to the species involved using a distribution factor based on species specific heats. The resulting compatible discretization scheme provides total energy conservation of the whole mixture. In addition, the numerical scheme includes conservative local energy exchange between species in mixture. Due to the relatively large species interaction coefficients, both the species momenta and energies are calculated implicitly. Sample calculations have yielded excellent results, including conservation of total energy in Lagrangian steps, symmetry preservation, and correct steady-state behavior.

Introduction

Lagrangian hydrodynamics applications have been applied primarily to problems with no relative motion between materials or species. All materials are initially placed in different regions and remain separated throughout the simulation. Interfaces between materials are usually located at zone boundaries; therefore, material interfaces are well maintained. The hydrodynamic velocity is determined at each mesh point in order to describe the motion of multiple materials. The material velocity at any location is simply identified as the local hydrodynamic velocity u.

When the mesh begins to tangle, ALE (arbitrary Lagrangian–Eulerian) methods are typically applied. After mesh relaxation, some computational zones will contain more than one material due to materials moving across zone boundaries via the ALE remapping step. There is no relative physical motion between materials in this case. Thus all materials present in a computational zone move at a single velocity, and material velocities can still be identified with the local hydrodynamic velocity. When ALE is active, materials may mix via numerical diffusion. But this is a numerical artifact and not a physical phenomenon. This numerical diffusion across zone boundaries is usually minimized by utilizing interface reconstruction techniques [1].

Of course, materials in contact can mix in certain circumstances. For example, gaseous materials will mix when they are in contact, and in this case of molecular mixing, sharp material interfaces cannot be defined. Furthermore, species have relative motion in this situation, and each species i moves with the species velocity u_i. Drift velocity relative to the average mixture motion is then u_i − u. Note that u in mixing situations represents the mass-weighted average velocity, defined as the total momentum divided by the total mass (the sum of species momenta divided by the sum of species masses).

Species motion in molecular mixing is described by the species momentum equation given by

$${\alpha }_i{\rho }_i\frac{D_i{\mathbf{u}}_i}{\mathit{Dt}}=-\nabla ({\alpha }_i{p}_i)+\sum _j{\mathbf{F}}_{\mathit{ij}}$$

where D_i/Dt = ∂/∂t + u_i · ∇, α_i is the volume fraction of species i in the context of Amagat’s law, and ρ_i and p_i are the density and pressure of pure material, respectively. (Partial density ${\tilde{\rho }}_i$ is α_iρ_i.) F_ij is the force due to the momentum exchange between species pair (i, j) by collision. (We will loosely refer to this as “friction”.) The mass-weighted average mixture velocity u is given by $\rho \mathbf{u}={\sum }_i{\alpha }_i{\rho }_i{\mathbf{u}}_i$, where $\rho ={\sum }_i{\alpha }_i{\rho }_i$ is the density of the mixture. These equations can be derived using kinetic theory by taking moments of the species Boltzmann equation [2].

Rather than solving species momentum equations, the species motion can be modeled by a simpler diffusion approach. Diffusion equations can be derived from Eq. (1) by ignoring inertial differences between species, thus implying large friction between species. The resulting, well-known, Stefan–Maxwell equations have been derived for various physical situations [3], [4], [5]. In these approaches, diffusion (drift) velocities u_i − u are obtained by solving the Stefan–Maxwell equations, leading to simulation of diffusional mixing. However, inertial differences between species play an essential role in hydrodynamic instabilities and mixing, and the selection of the species momentum equation (Eq. (1)) is preferred over simpler diffusion equations for these problems.

Some materials will interpenetrate and mix in some situations, although they are not miscible at the molecular level. For example, water and air frequently coexist and can flow at different velocities, resulting in mixing and/or separation. For example, rainfall can be viewed as large scale water and air interpenetration. (We will refer to this case as “multiphase”, while referring to the molecular mixing case as “multicomponent”. “Multimaterial” or “multifluid” will be used to refer to both cases.) Multiphase flow models have been used primarily in modeling of two-phase flows in various devices such as nuclear power reactors. Multiphase flow models involve terms that represent forces exerted by other phases, resulting in a species momentum equation in a different form than Eq. (1). In particular, the pressure forces exerted on a material (phase) by another phase are included [6], [7], [8], resulting in a more complicated discretization of expansion/compression work. Development of an adequate numerical scheme for multiphase mixtures is currently being undertaken and details will be reported in due course, but this paper and the present scheme are restricted to multicomponent mixtures.

Multifluid flows can be modeled by constructing and tracking Lagrangian meshes for each species involved. In this design, the mixture region would have multiple overlapping meshes, each representing a single species. Local mixture information can be constructed by combining information from the species meshes. Although this approach may be conceptually simple, considerable difficulties are expected in practice, e.g., separately applying ALE to each species. We thus choose an approach in which the whole mixture is represented with a single common mesh. Species velocities are obtained on this common mesh by Eq. (1) and used to construct temporary species meshes. Species quantities are then advected (mapped) from each species mesh to the common mesh, resulting in changes of composition on the common mesh.

Construction of the common mesh is not restricted to any particular approach. For example, the common mesh can be fixed in space, representing an Eulerian calculation, or movement of the common mesh can be governed by some equation of motion. In any case, a model for mesh motion is obviously required. In this paper, we use the mass-averaged mixture velocity u, as this is a logical, simple approach for prescribing mesh motion. In this case, u and u_i need to be described in a numerically consistent manner (the Eulerian approach is not used in the mixing regions because continuity of mesh motion is desired throughout the domain, which may include single-fluid regions modeled in a Lagrangian manner).

Once u is chosen for the construction of the common mesh, our scheme reduces to a Lagrangian scheme from a general moving-mesh paradigm. Using the mixture velocity u implies that the net mass flux across a zone surface (sum of drift species mass fluxes) is zero, thereby keeping the computational zone mass constant. The energy-conserving scheme is developed based on the Lagrangian motion of each species using a temporary species mesh whose vertex velocity is u_i. The advection of species quantities to the common mesh introduces a non-Lagrangian aspect in the present scheme. However, this procedure only changes the species concentration, not zone mass, and is congruent with the spirit of Lagrangian simulation. We thus choose to use the word “Lagrangian” in our paper.

Numerical schemes describing multicomponent flows include features common to single-fluid flow modeling as well as features unique to multifluid modeling. A species can experience the same degree of shock compression as the whole mixture. Therefore, artificial viscosities need to be applied to each species motion, similar to the single-fluid modeling approach. On the other hand, the presence of other species affects species motion via frictional forces, and this frictional momentum exchange results in friction work. Also, energy changes caused by artificial viscosity need to be handled properly. Finally, as mentioned above, advection of species quantities to the common mesh is required.

To address these issues, we have reformulated and generalized the compatible discretization scheme commonly used in Lagrangian hydrodynamic simulations [9], [10], [11] to multicomponent fluid flows. In particular, species thermal energy change caused by frictional interaction between species is included in the discretization along with compression work. Similar to the previous developments [9], [10], [11], species momentum conservation is used in the derivation of kinetic energy, naturally leading to species friction work. This friction work does not represent species thermal energy change, but must be transformed to frictional dissipation to correctly represent the physical phenomena of thermal energy change caused by frictional interaction (dissipative heating). The resulting discretization scheme provides total energy conservation in the Lagrangian calculations to machine precision.

We first present the implemented governing equations for simulation of nonequilibrium multifluid flows in Section 2. The time integration scheme is described in Section 3, and the compatible discretization developed for multifluid flow simulations is presented in Section 4. Sections 3 Time integration scheme, 4 Discretization for total energy conservation are the main focus of this paper. Numerical results are presented in Section 5.