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Solver Settings
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Outlineu Using the Solver
l Setting Solver Parameters
l Convergence
n Definition
n Monitoringn Stability
n Accelerating Convergence
l Accuracy
n Grid Independence
n Adaption
u Appendix: Background
l Finite Volume Method
l Explicit vs. Implicit
l Segregated vs. Coupled
l Transient Solutions
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Modify solution
parameters or grid
NoYes
No
Set the solution parameters
Initialize the solution
Enable the solution monitors of interest
Calculate a solution
Check for convergence
Check for accuracy
Stop
Yes
Solution Procedure Overviewu Solution Parameters
l Choosing the Solver
l Discretization Schemes
u
Initializationu Convergence
l Monitoring Convergence
l Stability
n Setting Under-relaxation
n Setting Courant number
l Accelerating Convergence
u Accuracy
l Grid Independence
l Adaption
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Choosing a Solveru Choices are Coupled-Implicit, Coupled-Explicit, or Segregated (implicit)
u The Coupled solversare recommended if a strong inter-dependence exists
between density, energy, momentum, and/or species.
l e.g., high speed compressible flow or finite-rate reaction modeled flows.
l In general, the Coupled-Implicitsolver is recommended over the coupled-explicit
solver.
n Time required: Implicit solver runs roughly twice as fast.
n Memory required: Implicit solver requires roughly twice as much memory as coupled-
explicit orsegregated-implicit solvers! (Performance varies.)
l The Coupled-Explicit solver should only be used for unsteady flows when the
characteristic time scale of problem is on same order as that of the acoustics.
n e.g., tracking transient shock wave
u The Segregated (implicit) solveris preferred in all other cases.
l Lower memory requirements than coupled-implicit solver.
l Segregated approach provides flexibility in solution procedure.
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Discretization (Interpolation Methods)u Field variables (stored at cell centers) must be interpolated to the faces of
the control volumes in the FVM:
u FLUENToffers a number of interpolation schemes:
l First-Order Upwind Scheme
n easiest to converge, only first order accurate.
l Power Law Scheme
n more accurate than first-order for flows when Recell< 5 (typ. low Re flows).
l Second-Order Upwind Scheme
n uses larger stencil for 2nd order accuracy, essential with tri/tet mesh or
when flow is not aligned with grid; slower convergence
l Quadratic Upwind Interpolation (QUICK)
n applies to quad/hex mesh, useful for rotating/swirling flows, 3rd order
accurate on uniform mesh.
VSAAVVt
f
faces
fff
faces
fff
ttt
+=+
+
,)()()(
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Interpolation Methods for Pressureu Additional interpolation options are available for calculating face pressure when
using the segregated solver.
u FLUENTinterpolation schemes for Face Pressure:
l Standard
n default scheme; reduced accuracy for flows exhibiting large surface-normal pressuregradients near boundaries.
l Linear
n useful only when other options result in convergence difficulties or unphysical
behavior.
l Second-Order
n
use for compressible flows or when PRESTO! cannot be applied.l Body Force Weighted
n use when body forces are large, e.g., high Ra natural convection or highly swirling
flows.
l PRESTO!
n applies to quad/hex cells; use on highly swirling flows, flows involving porous
media, or strongly curved domains.
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Pressure-Velocity Couplingu Pressure-Velocity Coupling refers to the way mass continuity is
accounted for when using the segregated solver.
u Three methods available:
l SIMPLE
n default scheme, robust
l SIMPLEC
n Allows faster convergence for simple problems (e.g., laminar flows with
no physical models employed).
l PISO
n useful for unsteady flow problems or for meshes containing cells with
higher than average skew.
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Initializationu Iterative procedure requires that all solution variables be initialized
before calculating a solution.
SolveInitializeInitialize...
l Realistic guesses improves solution stability and accelerates convergence.
l In some cases, correctinitial guess is required:
n Example: high temperature region to initiate chemical reaction.
u Patch values for individual
variables in certain regions.
SolveInitializePatch...
l Free jet flows(patch high velocity for jet)
l Combustion problems
(patch high temperature
for ignition)
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Convergence Preliminaries: Residualsu Transport equation for can be presented in simple form:
l Coefficients ap, anbtypically depend upon the solution.
l Coefficients updated each iteration.
u At the start of each iteration, the above equality will not hold.
l The imbalance is called the residual,Rp, where:
l Rpshould become negligibleas iterations increase.
l The residuals that you monitor are summed over all cells:
n By default, the monitored residuals are scaled.
n You can also normalize the residuals.
u Residuals monitored for the coupled solver are based on the rmsvalue of
the time rate of change of the conserved variable.
l Only for coupled equations; additional scalar equations use segregated
definition.
p
nb
nbnbpp baa =+
p
nb
nbnbppp baaR +=
||=cells
pRR
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Convergenceu At convergence:
l All discrete conservation equations (momentum, energy, etc.) are
obeyed in all cells to a specified tolerance.
l Solution no longer changes with more iterations.
l Overall mass, momentum, energy, and scalar balances are obtained.
u Monitoring convergence with residuals:
l Generally, a decrease in residuals by 3 orders of magnitude indicates at
least qualitative convergence.
n Major flow features established.
l Scaled energy residual must decrease to 10-6for segregated solver.
l Scaled species residual may need to decrease to 10-5to achieve speciesbalance.
u Monitoring quantitative convergence:
l Monitor other variables for changes.
l Ensure that property conservation is satisfied.
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Convergence Monitors: Residualsu Residual plots show when the residual values have reached the
specified tolerance.
SolveMonitors Residual...
All equations converged.
10-3
10-6
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Convergence Monitors: Forces/Surfacesu In addition to residuals, you can also monitor:
l Lift, drag, or moment
SolveMonitorsForce...
l Variables or functions (e.g., surface integrals)
at a boundary or any defined surface:SolveMonitorsSurface...
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Checking for Property Conservation
u In addition to monitoring residual and variable histories, you should
also check for overall heat and mass balances.
l Net imbalance should be less than 0.1% of net flux through domain.
Report
Fluxes...
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Decreasing the Convergence Toleranceu If your monitors indicate that the solution is converged, but the
solution is still changing or has a large mass/heat imbalance:
l Reduce Convergence Criterion
or disable Check Convergence.
l Then calculate until solution
converges to the new tolerance.
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Convergence Difficultiesu Numerical instabilities can arise with an ill-posed problem, poor
quality mesh, and/or inappropriate solver settings.
l Exhibited as increasing (diverging) or stuck residuals.
l Diverging residuals imply increasing imbalance in conservation equations.
l Unconverged results can be misleading!
u Troubleshooting:
l Ensure problem is well posed.
l Compute an initial solution with
a first-order discretization scheme.
l
Decrease under-relaxation forequations having convergence
trouble (segregated).
l Reduce Courant number (coupled).
l Re-mesh or refine grid with high
aspect ratio or highly skewed cells.
Continuity equation convergence
trouble affects convergence of
all equations.
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Modifying Under-relaxation Factorsu Under-relaxation factor, , is
included to stabilize the iterative
process for the segregated solver.
u Use default under-relaxation factors
to start a calculation.
SolveControlsSolution...
u Decreasing under-relaxation for
momentumoften aids convergence.
l Default settings are aggressive but
suitable for wide range of problems.l Appropriate settings best learned
from experience.
poldpp += ,
u For coupled solvers, under-relaxation factors for equations outsidecoupled
set are modified as in segregated solver.
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Modifying the Courant Numberu Courant number defines a time
step size for steady-state problems.
l A transient term is included in the
coupled solver even for steady state
problems.
u For coupled-explicit solver:
l Stability constraints impose a
maximum limit on Courant number.
n Cannot be greater than 2.
s Default value is 1.
n Reduce Courant number when
having difficulty converging.
u
xt
=
)CFL(u For coupled-implicit solver:
l Courant number is not limited by stability constraints.
n Default is set to 5.
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Accelerating Convergence
u Convergence can be accelerated by:
l Supplying good initial conditions
n Starting from a previous solution.
l
Increasing under-relaxation factors or Courant numbern Excessively high values can lead to instabilities.
n Recommend saving case and data files before continuing iterations.
l Controlling multigrid solver settings.
n Default settings define robust Multigrid solver and typically do not need
to be changed.
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Starting from a Previous Solutionu Previous solution can be used as an initial condition when changes are
made to problem definition.
l Once initialized, additional iterations uses current data setas starting point.
Actual Problem Initial Condition
flow with heat transfer isothermal solution
natural convection lowerRa solution
combustion cold flow solution
turbulent flow Euler solution
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Multigridu The Multigrid solver accelerates convergence by using solution on
coarse mesh as starting point for solution on finer mesh.
l Influence of boundaries and far-away points are more easily transmitted to
interior of coarse mesh than on fine mesh.
l Coarse mesh defined from original mesh.n Multiple coarse mesh levels can be created.
s AMG- coarse mesh emulated algebraically.
s FAS- cell coalescing defines new grid.
a coupled-explicit solver option
n Final solution is for original mesh.
l Multigrid operates automatically in the background.
u Accelerates convergence for problems with:
l Large number of cells
l Large cell aspect ratios, e.g., x/y > 20
l Large differences in thermal conductivity
fine (original) mesh
coarse mesh
solution
transfer
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Accuracyu A converged solution is not necessarily an accurate one.
l Solve using 2nd order discretization.
l Ensure that solution is grid-independent.
n Use adaption to modify grid.
u If flow features do not seem reasonable:
l Reconsider physical models and boundary conditions.
l Examine grid and re-mesh.
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Mesh Quality and Solution Accuracyu Numerical errors are associated with calculation of cell gradients and
cell face interpolations.
u These errors can be contained:
l Use higher order discretization schemes.
l Attempt to align grid with flow.l Refine the mesh.
n Sufficient mesh density is necessary to resolve salient features of flow.
s Interpolation errors decrease with decreasing cell size.
n Minimize variations in cell size.
s Truncation error is minimized in a uniform mesh.
s Fluent provides capability to adapt mesh based on cell size variation.
n Minimize cell skewness and aspect ratio.
s In general, avoid aspect ratios higher than 5:1.
s Optimal quad/hex cells have bounded angles of 90 degrees
s Optimal tri/tet cells are equilateral.
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Determining Grid Independenceu When solution no longer changes with further grid refinement, you
have a grid-independent solution.
u Procedure:
l Obtain new grid:
n
Adapts Save original mesh before adapting.
If you know where large gradients are expected, concentrate the
original grid in that region, e.g., boundary layer.
s Adapt grid.
Data from original grid is automatically interpolated to finer grid.
n filereread-grid
andFile
Interpolate...
s Import new mesh and initialize with old solution.
l Continue calculation to convergence.
l Compare results obtained w/different grids.
l Repeat adaption/calculation procedure if necessary.
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Unsteady Flow Problemsu Transient solutions are possible with both segregated and coupled solvers.
l Solver iterates to convergence at each time level, then advances automatically.
l Solution Initialization provides initial condition, must be realistic.
u For segregated solver:
l Time step size, t, is input in Iterate panel.n t should be small enough to resolve
time dependent features and to ensure
convergence within 20 iterations.
n May need to start solution with small t.
l Number of time steps, N, is also required.
n N*t =total simulated time.l Use TUI command it # to iterate without advancing time step.
u For Coupled Solver, Courant number defines in practice:
l global time step size for coupled explicit solver.
l pseudo-time step size for coupled implicit solver.
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Summary
u Solution procedure for the segregated and coupled solvers is the same:
l Calculate until you get a converged solution.
l Obtain second-order solution (recommended).
l
Refine grid and recalculate until grid-independent solution is obtained.u All solvers provide tools for judging and improving convergence and
ensuring stability.
u All solvers provide tools for checking and improving accuracy.
u Solution accuracy will depend on the appropriateness of the physical
models that you choose and the boundary conditions that you specify.
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Appendixu Background
l Finite Volume Method
l Explicit vs. Implicit
l Segregated vs. Coupled
l Transient Solutions
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Background: Finite Volume Method - 1u FLUENTsolvers are based on the finite volume method.
l Domain is discretized into a finite set of control volumes or cells.
u General transport equation for mass, momentum, energy, etc. is
applied to eachcell and discretized. For cellp,
+=+
dVSdddV
tAAV
AAV
unsteady convection diffusion generation
Eqn.
continuity 1x-mom. u
y-mom. v
energy h
Fluid region of pipe flow
discretized into finite set of
control volumes (mesh).
control
volume
u Allequations are solved to render flow field.
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Background: Finite Volume Method - 2u Each transport equation is discretized into algebraic form. For cell p,
face f
adjacent cells, nb
cellp
u Discretized equations require information at cell centers andfaces.
l Field data (material properties, velocities, etc.) are stored at cell centers.
l Face values can be expressed in terms of local andadjacent cell values.
l Discretization accuracy depends upon stencil size.
u
The discretized equation can be expressed simply as:
l Equation is written out for every control volume in domain resulting in an
equation set.
p
nb
nbnbppbaa =+
VSAAVVt
f
faces
fff
faces
fff
t
p
tt
p +=+
+
,)()()(
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u Equation sets are solved iteratively.
l Coefficients apand anbare typically functions
of solution variables(nonlinear and coupled).
l Coefficients are written to use values of solution variables from previous
iteration.
n Linearization: removing coefficients dependencies on .
n De-coupling: removing coefficients dependencies on other solution
variables.
l Coefficients are updated with each iteration.
n For a given iteration, coefficients are constant.
s pcan either be solved explicitly or implicitly.
Background: Linearization
p
nb
nbnbpp baa =+
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u Assumptions are made about the knowledge of nb:
l Explicit linearization- unknown value in each cell computed from relations
that include only existingvalues (nb assumed known from previous
iteration).
n psolved explicitly using Runge-Kutta scheme.
l Implicit linearization- p andnb are assumed unknown and are solved
using linear equation techniques.
n Equations that are implicitly linearized tend to have less restrictive stability
requirements.
n
The equation set is solved simultaneously using a second iterative loop (e.g.,point Gauss-Seidel).
Background: Explicit vs. Implicit
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Background: Coupled vs. Segregated
u Segregated Solver
l If the only unknowns in a given equation are assumed to be for asingle
variable, then the equation set can be solved without regard for the
solution of other variables.
n coefficients apand anbare scalars.
u Coupled Solver
l If more than one variableis unknown in each equation, and each
variable is defined by its own transport equation, then the equation set is
coupledtogether.
n coefficients apand anbareNeqx Neqmatricesn is a vector of the dependent variables, {p, u, v, w, T, Y}T
p
nb
nbnbpp baa =+
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Background: Segregated Solver
u In the segregated solver, each equation is
solved separately.
u The continuity equation takes the form
of a pressure correction equation as partof SIMPLE algorithm.
u Under-relaxation factors are included in
the discretized equations.
l Included to improve stability of iterative
process.
l Under-relaxation factor, , in effect,limits change in variable from one
iteration to next:
Update properties.
Solve momentum equations (u, v, w velocity).
Solve pressure-correction (continuity) equation.
Update pressure, face mass flow rate.
Solve energy, species, turbulence, and other
scalar equations.
Converged?
Stop
No Yes
poldpp += ,
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Background: Coupled Solveru Continuity, momentum, energy, and
species are solved simultaneously in the
coupled solver.
u Equations are modified to resolve
compressible andincompressible flow.u Transient term is always included.
l Steady-state solution is formed as time
increases and transients tend to zero.
u For steady-state problem, time step is
defined by Courant number.l Stability issues limit maximum time step
size for explicit solver but not for
implicit solver.
Solve continuity, momentum, energy,
and species equations simultaneously.
Stop
No Yes
Solve turbulence and other scalar equations.
Update properties.
Converged?
u
xt
=
)CFL( CFL= Courant-Friedrichs-Lewy-numberwhere u= appropriate velocity scale
x= grid spacing
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Background: Segregated/Transientu Transient solutions are possible with both segregated and coupled solvers.
l 1st- and 2nd-order time implicit discretizations (Euler) available for coupled
and segregated solvers.
n Procedure: Iterate to convergence at each time level, then advance in time.
l
2nd order time-explicit discretization also available for coupled-explicit solver.u For segregated solver:
l Time step size, t, is input in Iterate panel.
n t should be small enough to resolvetime dependent features.
l Number of time steps, N, is also required.
n N*t equals total simulated time.
l Generally, use t small enough to ensureconvergence within 20 iterations.
l Note: Use TUI command it # to iterate
further without advancing time step.
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Background: Coupled/Transientl If implicit scheme is selected, twotransient terms are included in discretization.
n Physical-time transient
s Physical-time derivative term is discretized implicitly (1st or 2nd order).
s Time step size, t, defined as with segregated solver.
n Pseudo-time transient
s At each physical-time level, a pseudo-time transient is driven to zero through a
series of inner iterations (dual time stepping).
s Pseudo-time derivative term is discretized:
explicitly in coupled-explicit solver.
implicitly in coupled-implicit solver.
s Courant number defines pseudo-time step size, .
l For explicit time stepping, physical-time derivative isdiscretized explicitly.
n Option only available with coupled-explicit solver
n Physical-time step size is defined by Courant number.
s Same time step size is used throughout domain (global time stepping).
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