FLUENT: THEORY GUIDE

CU 1: PHNG PHP EULER 1.7.1 Euler Equations For inviscid flows, ANSYS FLUENT solves the Euler equations. The mass conservation equation is the same as for a laminar flow, but the momentum and energy conservation equations are reduced due to the absence of molecular diffusion.

In this section, the conservation equations for inviscid flow in an inertial (non-rotating) reference frame are presented. The equations that are applicable to non-inertial reference frames are described in Chapter 2. The conservation equations relevant for species transport and other models will be discussed in the chapters where those models are described.

The Mass Conservation Equation

The equation for conservation of mass, or continuity equation, can be written as follows:

(1.71)

Equation 1.7-1 is the general form of the mass conservation equation and is valid for

incompressible as well as compressible flows. The source is the mass added to the continuous phase from the dispersed second phase (e.g., due to vaporization of liquid droplets) and any user-defined sources.

For 2D axisymmetric geometries, the continuity equation is given by

(1.72)

where is the axial coordinate, is the radial coordinate, is the axial velocity, and is the radial velocity.

Momentum Conservation Equations

Conservation of momentum is described by

(1.73)

where is the static pressure and and are the gravitational body force and external body forces (e.g., forces that arise from interaction with the dispersed phase), respectively. also contains other model-dependent source terms such as porous-media and user-defined sources.

For 2D axisymmetric geometries, the axial and radial momentum conservation equations are given by

(1.74)

and

(1.75)

where

(1.76)

Energy Conservation Equation

Conservation of energy is described by

(1.77)

CU 2: PHNG PHP RUNGE-KUTTA 18.5.4 Steady-State Flow Solution Methods The coupled set of governing equations (Equation 18.5-6) in ANSYS FLUENT is discretized in time for both steady and unsteady calculations. In the steady case, it is assumed that time marching proceeds until a steady-state solution is reached. Temporal discretization of the coupled equations is accomplished by either an implicit or an explicit time-marching algorithm. These two algorithms are described below.

Explicit Formulation In the explicit scheme a multi-stage, time-stepping algorithm [ 146] is used to discretize the time

derivative in Equation 18.5-6. The solution is advanced from iteration to iteration with an -stage Runge-Kutta scheme, given by

where and is the stage counter for the -stage scheme.

is the multi-stage coefficient for the stage. The residual is computed from the

intermediate solution and, for Equation 18.5-6, is given by

(18.5-13)

The time step is computed from the CFL (Courant-Friedrichs-Lewy) condition

(18.5-14)

where is the cell volume, is the face area, and is the maximum of the local eigenvalues defined by Equation 18.5-9.

For steady-state solutions, convergence acceleration of the explicit formulation can be achieved with the use of local time stepping, residual smoothing, and full-approximation storage multigrid.

Local time stepping is a method by which the solution at each control volume is advanced in time with respect to the cell time step, defined by the local stability limit of the time-stepping scheme.

Residual smoothing, on the other hand, increases the bound of stability limits of the time-stepping scheme and hence allows for the use of a larger CFL value to achieve fast convergence (Section 18.5.4).

The convergence rate of the explicit scheme can be accelerated through use of the full-approximation storage (FAS) multigrid method described in Section 18.6.4.

By default, ANSYS FLUENT uses a 3-stage Runge-Kutta scheme based on the work by Lynn [ 211] for steady-state flows that use the density-based explicit solver.

Implicit Formulation In the implicit scheme, an Euler implicit discretization in time of the governing equations (Equation 18.5-6) is combined with a Newton-type linearization of the fluxes to produce the following linearized system in delta form [ 370]:

18.5.5 Unsteady Flows Solution Methods For time-accurate calculations, explicit and implicit time-stepping schemes are available. (The time-implicit approach is also referred to as "dual time stepping''.)

Explicit Time Stepping The explicit time stepping approach, is available only for the explicit scheme described above. The time step is determined by the CFL condition. To maintain time accuracy of the solution the explicit time stepping employs the same time step in each cell of the domain (this is also known as global-time step), and with preconditioning disabled. By default, ANSYS FLUENT uses a 4-stage Runge-Kutta scheme for unsteady flows.

Implicit Time Stepping (Dual-Time formulation)

The implicit-time stepping method (also known as dual-time formulation) is available in the density-based explicit and implicit formulation.

When performing unsteady simulations with implicit-time stepping (dual-time stepping), ANSYS FLUENT uses a low Mach number time-derivative unsteady preconditioner to provide accurate results both for pure convective processes (e.g., simulating unsteady turbulence) and for acoustic processes (e.g., simulating wave propagation) [ 350, 263].

Table 18.5.1: Summary of the Density-Based Solver

Solution Method Density-Based Solver - Explicit Formulation

Density-Based Solver - Implicit Formulation

Steady-State

- 3-stages Runge-Kutta - local time step - time-derivative preconditioning - FAS

- local time step - time-derivative preconditioning

Unsteady - Explicit Time Stepping

- 4-stages Runge-Kutta - global time step - no time-derivative reconditioning - No FAS

N/A

Unsteady - Implicit Time Stepping (dual-time formulation) First Order

- dual-time formulation - Physical time: first order Euler backward - preconditioned pseudo-time derivative - inner iteration: explicit pseudo-time marching, 3-stage Runge-Kutta

- dual-time formulation - Physical time: first order Euler backward - preconditioned pseudo-time derivative - inner iteration: implicit pseudo-time marching

Unsteady - Implicit Time Stepping (dual-time formulation) Second Order

- dual-time formulation - Physical time: second order Euler backward - preconditioned pseudo-time derivative - inner iteration: explicit pseudo-time marching, 3-stage Runge-Kutta

- dual-time formulation - Physical time: second order Euler backward - preconditioned pseudo-time derivative - inner iteration: implicit pseudo-time marching

CU 3: PHNG PHP MULTIGRID 18.6 Multigrid Method The ANSYS FLUENT solver contains two forms of multigrid: algebraic (AMG) and full-approximation storage (FAS). AMG is an essential component of both the pressure-based and density-based implicit solvers, while FAS is an important, but optional, component of the density-based explicit solver. (Note that when the density-based explicit solver is used, AMG will also be used, since the scalar equations (e.g., turbulence) are solved using the approach described in Section 18.2.)

This section describes the mathematical basis of the multigrid approach. Common aspects of AMG and FAS are presented first, followed by separate sections that provide details unique to each method.

18.6.1 Approach 18.6.2 Multigrid Cycles 18.6.3 Algebraic Multigrid (AMG) 18.6.4 Full-Approximation Storage (FAS) Multigrid

18.6.1 Approach ANSYS FLUENT uses a multigrid scheme to accelerate the convergence of the solver by computing corrections on a series of coarse grid levels. The use of this multigrid scheme can greatly reduce the number of iterations and the CPU time required to obtain a converged solution, particularly when your model contains a large number of control volumes.

The Need for Multigrid Implicit solution of the linearized equations on unstructured meshes is complicated by the fact that there is no equivalent of the line-iterative methods that are commonly used on structured meshes. Since direct matrix inversion is out of the question for realistic problems and "whole-field'' solvers that rely on conjugate-gradient (CG) methods have robustness problems associated with them, the methods of choice are point implicit solvers like Gauss-Seidel and ILU . Although the Gauss-Seidel and ILU schemes rapidly remove local (high-frequency) errors in the solution, global (low-frequency) errors are reduced at a rate inversely related to the mesh size. Thus, for a large number of nodes, the solver "stalls'' and the residual reduction rate becomes prohibitively low.

The multi-stage scheme used in the density-based explicit solver can efficiently remove local (high-frequency) errors as well. That is, the effect of the solution in one cell is communicated to

adjacent cells relatively quickly. However, the scheme is less effective at reducing global (low-frequency) errors-errors which exist over a large number of control volumes. Thus, global corrections to the solution across a large number of control volumes occur slowly, over many iterations. This implies that performance of the multi-stage scheme will deteriorate as the number of control volumes increases.

Multigrid techniques allow global error to be addressed by using a sequence of successively coarser meshes. This method is based upon the principle that global (low-frequency) error existing on a fine mesh can be represented on a coarse mesh where it again becomes accessible as local (high-frequency) error: because there are fewer coarse cells overall, the global corrections can be communicated more quickly between adjacent cells. Since computations can be performed at an exponentially decaying expense in both CPU time and memory storage on coarser meshes, there is the potential for very efficient elimination of global error. The fine-grid relaxation scheme or "smoother'', in this case either the point-implicit linear solvers (Section 18.6.3) or the explicit multi-stage scheme, is not required to be particularly effective at reducing global error and can be tuned for efficient reduction of local error.

The Basic Concept in Multigrid Consider the set of discretized linear (or linearized) equations given by

(18.6-1)

where is the exact solution. Before the solution has converged there will be a defect

associated with the approximate solution :

(18.6-2)

We seek a correction to such that the exact solution is given by

(18.6-3)

Substituting Equation 18.6-3 into Equation 18.6-1 gives

(18.6-4)

(18.6-5)

Now using Equations 18.6-2 and 18.6-5 we obtain

(18.6-6)

which is an equation for the correction in terms of the original fine level operator and the defect . Assuming the local (high-frequency) errors have been sufficiently damped by the

relaxation scheme on the fine level, the correction will be smooth and therefore more effectively solved on the next coarser level. Restriction and Prolongation Solving for corrections on the coarse level requires transferring the defect down from the fine level (restriction), computing corrections, and then transferring the corrections back up from the

coarse level (prolongation). We can write the equations for coarse level corrections as

(18.6-7)

where is the coarse level operator and the restriction operator responsible for transferring the fine level defect down to the coarse level. Solution of Equation 18.6-7 is followed by an update of the fine level solution given by

(18.6-8)

where is the prolongation operator used to transfer the coarse level corrections up to the fine level. Unstructured MultigridThe primary difficulty with using multigrid on unstructured meshes is the creation and use of the coarse grid hierarchy. On a structured mesh, the coarse meshes can be formed simply by removing every other mesh line from the fine meshes and the prolongation and restriction operators are simple to formulate (e.g., injection and bilinear interpolation).

The difficulties of applying multigrid on unstructured meshes are overcome in a separate fashion by each of the two multigrid methods used in ANSYS FLUENT. While the basic principles discussed so far and the cycling strategy described in Section 18.6.2 are the same, the techniques for construction of restriction, prolongation, and coarse mesh operators are different, as discussed in Section 18.6.3 and Section 18.6.4 for the AMG and FAS methods, respectively.

CU 4: DNG DI M, BNG M, TRN M 1.6.1 When to Use the Compressible Flow Model Compressible flows can be characterized by the value of the Mach number:

(1.6-1)

Here, is the speed of sound in the gas:

(1.6-2)

and is the ratio of specific heats .

When the Mach number is less than 1.0, the flow is termed subsonic. At Mach numbers much

less than 1.0 ( or so), compressibility effects are negligible and the variation of the gas density with pressure can safely be ignored in your flow modeling. As the Mach number approaches 1.0 (which is referred to as the transonic flow regime), compressibility effects become important. When the Mach number exceeds 1.0, the flow is termed supersonic, and may contain shocks and expansion fans which can impact the flow pattern significantly. ANSYS FLUENT provides a wide range of compressible flow modeling capabilities for subsonic, transonic, and supersonic flows.