Navier-Stokes Method

Computational Fluid Dynamics have been well developed to numerically solve the Navier-Stokes equations during the past few decades. Unstructured grid methods are characterized by their ease in handling completely unstructured meshes and have become widely used in computational fluid dynamics. The relative increase of computer memory and CPU time with unstructured grid methods is not trivial, but can be offset by using parallel techniques in which many processors are put together to work on the same problem.

Traditionally, numerical methods developed for compressible flow simulations use an unsteady form of the Navier-Stokes or Euler equations. In general, either density or pressure (density-based or pressure-based) is chosen as one of the primary variables when building up the discretized governing equations. However, the density-based method in cases of incompressible or low Mach number flows is questionable, since in low compressibility limit, and the pressure-density coupling becomes very weak. For flows for all speed regimes, there are some important researches reported by Hirt et al. [1989], Karki and Patankar [1989]

and Chen [1989]. The Navier-Stokes solvers, called HYB3D [Koomullil et al., 1996a,

Koomullil et al., 1996b, Koomullil et al., 1996c, Koomullil and Soni, 1999], are used as flow solver of continuum flow region in the first phase of developing a coupled method. Then, the UNIC-UNS code [Chen,1989, Shang et al.,1995a, Shang et al., 1995b, Shang et al., 1995c, Shang et al., 1997 and Zhang et al., 2001] developed by Chen has been used as the continuum domain solver in the current couple method instead of HYB3D code because of its powerful capabilities in handling flows of interest involving low-speed (or incompressible) region, slip boundary condition and chemical reactions.

2.2.1 General Description of Navier-Stokes Method

The continuum method, employed to solve the Navier-Stokes (NS) or Euler equation for continuum flows, is computationally efficient in simulating a wide variety of flow problems.

In general, numerical methods developed for compressible flow simulations use an unsteady form of the Navier-Stokes or Euler equations. The general form of mass conservation, Navier-Stokes equation, energy conservation and other transport equations can be written in Cartesian tensor form:

where µ_φ is an effective diffusion coefficient,S denotes the source term, ρ is the fluid _φ

density and φ =( l, u, v, w, h, k, ε) stands for the variables for the mass, momentum, total

energy and turbulence equations, respectively. Detailed expressions for the k-ε turbulence models and wall functions can be found in [Launder and Spalding, 1974]. For spatial discretization in cell-centered scheme, the transport equations can also be written in integral form as

direction and ^Fis the flux function of the variables φ and ρ. The flux integral formulation in finite volume scheme can be evaluated by the summation of the flux vectors over each face,

j

In the present work, two cell-centered unstructured finite volume methods HYB3D [Koomullil et al., 1996a, Koomullil et al., 1996b, Koomullil et al., 1996c, Koomullil and Soni, 1999] and UNIC-UNS [Chen,1989, Shang et al.,1995a, Shang et al., 1995b, Shang et al., 1995c, Shang et al., 1997 and Zhang et al., 2001] are used as the continuum solver in the proposed coupled DSMC-NS method and the general features of both Navier-Stokes solver will be described in the following sections. General features of the HYB3D and UNIC-UNS

codes will be introduced briefly and interested readers are referred to Koomullil [Koomullil et

al., 1996a, Koomullil et al., 1996b, Koomullil et al., 1996c, Koomullil and Soni, 1999] and

Chen [Chen,1989, Shang et al.,1995a, Shang et al., 1995b, Shang et al., 1995c, Shang et al., 1997 and Zhang et al., 2001] for the details.

2.2.2 Navier-Stokes Solvers: HYB3D and UNIC-UNS Code

HYB3D

The HYB3D is a Navier-Stokes solver using a generalized- or an unstructured- grid topology and has the following important features: 1) Cell-centered finite-volume upwind scheme for the numerical integration of governing equations, 2) Roe’s approximate Riemann solver for convective flux evaluation, 3) Parallel computing using message passing interface (MPI), 4) Laminar or turbulent flow simulation capability with various turbulence models, and 5) Application of overset grid topology for flow simulation over moving or complex bodies. Implicit time integration is used with local time stepping, where the maximum allowable time step in each cell is determined by the CFL condition constrained by advection and viscous stability criteria. A second order spatial accuracy is achieved using Taylor’s series expansion and the gradients of the flow properties are computed using a least-square method. The creation of local extrema during the higher order linear reconstruction is eliminated by the application of Venkatakrishnan's type [Venkatakrishnan, 1995] limiter.

Parallel computing of the HYB3D also incorporates the graph-partition tool, METIS [Karypis and Kumar, 1998], which is the same as that in the PDSC. The dynamic domain decomposition in the current HYB3D computation is circumvented by not adaptively refining in this phase of computation. Details of the algorithms and numerical methods used in HYB3D are omitted here for brevity and interested readers are referred to Koomullil [Koomullil et al., 1996a, Koomullil et al., 1996b, Koomullil et al., 1996c, Koomullil and Soni, 1999] for the details.

UNIC-UNS

In the thesis, UNIC-UNS is the Navier-Stokes solver using 3D hybrid unstructured- grid topology for the application of the thesis. Its importance features, which are similar to the HYB3D code, are list as follows: 1) Cell-centered upwind finite-volume scheme, 2) Roe’s approximate Riemann solver for convective flux evaluation, 3) Parallel computing using message passing interface (MPI), 4) Capability of Laminar or turbulent flow simulation, and 5) the local extreme limiter of flux with data reconstruction proposed by Barth [1993] is employed in the UNIC-UNS. Implicit time integration is used. For general applications, a dual-time sub-iteration method is now used in the UNIC-UNS code for time accurate time-marching computations. A multi-dimensional linear reconstruction approach, proposed by Barth and Jespersen [1989], is used in the UNIC-UNS code with modifications for high order accurate estimation of flux at the cell faces. The most important specialties of

UNIC-UNS, different form HYB3D, are listed as follows: 1) Pressure-velocity-density coupling. Unlike density-based method, SIMPLE [Karki and Patankar, 1989 and Chen, 1989]

family pressure correction equation, formulated using the perturbed equation of state, momentum and continuity equations, can be applied for all-speed flow, even for low Mach number or incompressible flow. 2) Automatics Slip boundary condition near the solid wall in the flow region 0.01<Kn<0.1. 3) Capability of chemical reaction modeling. A general chemical reacting flow module is included in the UNIC-UNS code for simulations such as CVD reactor simulations or plume related study of flying object at high altitude issued from combustion chamber. 4) Mesh adaptive refinement with handing nodes. Mesh adaptation with refinement and coursing modules [Shang et al., 1997 and Zhang et al., 2001] in UNIC-UNS code can maintain a smooth grid density variation required for solver to guarantee the stability and accuracy. Details of the algorithms and numerical methods used in UNIC-UNS are omitted here for brevity and interested readers are referred to Chen [Chen,1989, Shang et

al.,1995a, Shang et al., 1995b, Shang et al., 1995c, Shang et al., 1997 and Zhang et al., 2001]

for the details.