PHOENICS SEGREGATED SOLVER

• To get a converged numerical

solution the solver visits each

transport equation separately.

• Consider for example a 2D (XY plane)

flow case with energy equation and

laminar regime.

• The solver does a loop visiting the

momentum equations, energy equation

and ,at last, the mass (pressure)

equation and return.

• A SWEEP is completed after the Solver

has worked on each transport

equation.

Initial data &

Boundary Conditions

Solver U1

Solver V1

Solver TEM1

Solver P1

(mass conserved field)

Compare residuals

against a reference

End

Sw

eep

Solver Sweeping Direction(1)

• Each transport equation is visited by the solver individually;

• The computational domain is not tackled at once by the solver. It slices the

domain in slabs (XY plane) along the Z direction.

• The sweep starts from low Z to high Z, or IZ = 1 to IZ = last.

• The solver sweeping direction has a direct influence on the computer time

and some times on the convergency rate.

• For 3D simulation align the Z direction along the main flow direction. This

practice propagates faster the upstream information to the downstream

direction

Solver’ sweeping direction: two dimensional cases (2)

• For 2D cases employ the XY plane; the solution is faster because the solver

has only one slab to sweep. Furthermore, it solves at once the whole slab .

X

Y

Z X

Y

Z

4 s

wee

ps

• Workshop #1: compare the CPU time for the

cylinder in cross flow using the XY and YZ

planes. One can download the q1XY and

q1YZ.

CO

ME

NT

S C

ase 2

D • Both simulations have the same inlet values, fluid properties, grid

size and reference residuals.

•The XY plane requires 5200 sweeps to fall below the threshold level

and stop simulation. The YZ plane doesn’t reach threshold even

after the 8000 sweeps

• XY CPU time = 22sec (5200 sweeps) YZ CPU time = 222s (8000

sweeps); it is a 10 times faster if one uses the XY plane.

• One concludes that using XY plane reduces the CPU time and also

speeds convergence!

• The Z direction is the Phoenics natural choice to be the axis

parallel to the coordinate system axial direction.

• For 3D it is highly recommended to choose the main flow

direction coincident with the Z axis.

• For 3D BFC it is mandatory that the main flow direction be

coincident with the Z axis. BFC two dimensional cases one must

choose the XY plane only.

Cilin

dri

ca

l-P

ola

r, 3

D

Cas

es a

nd

BF

C

•The next sections

introduces Phoenics’ tools

to access and control

convergence.

• The type and the use of

these new tools are closely

related to the solver’s

segregated structure.

Initial data &

Boundary Conditions

Solver U1

Solver V1

Solver TEM1

Solver P1

(mass conserved field)

Compare residuals

against a reference

End

Sw

eep

Getting a Numerical Solution is a Difficult Task

1. Grid independent - if you increase the number of nodes the

solution does not change significantly.

2. Convergence– the solution achieves a constant value, it does

not oscillates neither changes with further interactions;

3. Negligible Residuals – if the mass, momentum, energy (and

other variables) residuals are negligible one can say the

numerical solution satisfies GLOBALLY the conservation

equations.

• Usually if you get convergence but not get negligible residuals

your solution is converged to a non-physical (wrong) solution!

– A number of devices are provided in PHOENICS to monitor and

control the convergence of the solution. They are fully described

at ´Convergence Monitoring´ link and are briefly presented now.

1. STUDYING RESIDUALS

THE RESIDUALS & CONVERGENCE MONITORING

• The residuals are the quantities used in PHOENICS to monitor the

convergence procedure. These are the imbalances in the FVE, defined as:

• During the computation, the residuals are printed by Earth to the VDU, or

are displayed graphically.

• The frequency of display update is controlled by the PIL variable TSTSWP,

which is set in the menu from the Monitoring controls/Residuals display

frequency panel.

• The default value of 1 prints to the screen, which is suitable for batch

operation. A setting TSTSWP = -1 activates the graphical display.

THE SPOT VALUE (GROUP 22)

• In addition to the display of residuals to the screen, the values

of chosen variables for a single cell can also be displayed.

• A monitoring location (or cell) is selected in the Q1 file, or

from the menu by setting the variables IXMON, IYMON, IZMON

to the values of IX, IY and IZ of the cell to be monitored.

• The monotoring spot can be set on VR/Output box.

MONITORING SPOT VALUES AND RESIDUALS ON SCREEN

SPOT VALUES RESIDUALS

• The quantities printed out are, in fact:

where the sum is extended to the current slab (in PARABolic cases) or to the

whole field (in elliptic cases).

• When the sum of errors divided by RESREF falls below 1.0, solution for

that variable stops. When this happens for all variables, sweeping stops.

• EARTH can calculate values for RESREF, based on the net sum of fluxes

for a variable, if the PIL variable SELREF=T.

•The supplementary variable RESFAC acts as a tolerance - a setting of 0.01

means that sweeping stops when the errors are 1% of the reference fluxes.

NORMALIZING THE RESIDUALS

RESREF , RESFAC & SELREF (GROUP 15)

• RESREF, the reference-residual for a solved-for variable

• It is evaluated by RESREF = RESFAC*(Typical flow rate of the variable)

• The "typical flow rate" is the sum of the absolute values, for all cell

volumes and cell faces, of the convection fluxes, diffusion fluxes,

sources and transient terms.

• If SELREF = T Earth evaluates internally RESREF

• If SELREF = F RESREF is user set.

• Users should be aware that the employment of SELREF=T, with a value

of RESFAC which is the same for all variables, is a convenient but crude

method of controlling the cut-off point of the solution procedure. To set

SELREF = F, and set the individual RESREFs for themselves may be

safer, when obtaining a well-converged solution is especially important.

VR SETTINGS:

SWEEPS

RESFAC

SELREF

1.1 WORKSHOP#2:

ANALYSING RESIDUALS

WORKSHOP NOTES – CONVERGENCE AND CONTROL #2

• Load core library case 240. This case deals with the entry flow in a 2-D

parallel walls channel.

• Now make the changes:

– y size from 1 m to 0.5 m;

–NY = 10 and NZ = 15;

–set the domain to material 67 (water);

–LSWEEP = 200 ;

– Turn off the Energy Equation

–and finally make WIN = 0.01 m/s instead of 5 m/s

WORKSHOP NOTES – CONVERGENCE AND CONTROL #2

• Run EARTH.

• Inspect the RESULT file to check the assigned RESREF values.

variable resref (res sum)/resref (res sum)

P1 5.018E-04 4.528E-04 2.272E-07

V1 1.213E-08 8.297E+03 1.006E-04

W1 4.693E-06 7.178E+01 3.369E-04

• Can you give a physical meaning to them?

DOWNLOAD WKSH Q1

WORKSHOP #2 NOTE (1) Whole-field residuals before solution with RESREF values determined by EARTH & RESFAC = 1.00E-03 & TIME/(VARIABLES*CELLS*TSTEPS*SWEEPS*ITS = 9E-05 sec.

variable resref (res sum)/resref (res sum)

P1 5.018E-04 4.528E-04 2.272E-07

V1 1.213E-08 8.297E+03 1.006E-04

W1 4.693E-06 7.178E+01 3.369E-04 1- the 4th column means that the sum of residuals of pressure, for example, is nearly 2.3E-7 (kg/s). Similarly to the other variables.

2- The meaning of the 2nd column is a reference flux. Thus for pressure (mass conservation) it is 5.0E-04 kg/s, and for V1 and W1 it is the mass flux times a reference velocity, 1.0E-08 kgm/s2 (N) and 4.7E-06 kgm/s2 (N).

3- How big or small these fluxes are? One have to compare against some characteristic value of the problem.

4- Inlet mass flux = RHO1*WIN*H = 5 kg/s; W1 momentum flux = 5E-02 N, V1 momentum flux (very small, boundary layer)

5- The mass imbalance (sum of residuals) is 2.3E-07 kg/s or 5-10 % of the mass inlet! If the run had stopped at (res sum)/resref = 1, then the global mass imbalance would correspond to 0.01% of the mass inlet. The same reasoning applied to W1 and V1 one sees that the residuals are far from satisfy the convergence criterion.

WORKSHOP NOTES – CONVERGENCE AND CONTROL #2

• Open q1 file and edit the following lines:

–SELREF=F

–RESREF(V1) = 5.0E-06 kg.m/s2 [N],

–RESREF(W1) = 5.0E-06 kg.m/s2 [N] and

–RESREF(P1) = 5.0E-04 kg/s.

• These changes can not be made using VR, you have to edit q1 file directly.

(there is a hint on editing in note 2). Compare with the actual ´run performance´

with the one from last case.

variable resref (res sum)/resref (res sum)

P1 5.018E-04 4.528E-04 2.272E-07

V1 1.213E-08 8.297E+03 1.006E-04

W1 4.693E-06 7.178E+01 3.369E-04

WORKSHOP #2 NOTE (2)

• Edit GROUP 15 in q1 file using the notepad:

************************************************************

Group 15. Terminate Sweeps

LSWEEP = 200

RESREF(P1 ) = 5.000000E-04 ;RESREF(V1 ) = 5.000000E-06

RESREF(W1 ) = 5.000000E-06; SELREF = F

•Save and close notepad

•DO NOT FORGET: RELOAD WORKING FILE

•Whole-field residuals with resref values set by the user

variable resref (res sum)/resref (res sum)

P1 5.000E-04 5.016E-04 2.508E-07

V1 5.000E-06 3.409E+01 1.704E-04

W1 5.000E-06 1.758E+02 8.788E-04

•The criteria to stop V1 was eased. As a consequence the solution for W1 got worse probably because the V1 residuals were too coarse.

• One may use the SELREF =T or decide by himself the reference residuals, this is the most important convergence criterion.

2. PROMOTING & ACHIEVING

CONVERGENCE

• What if your numerical solution blows up?

• What if the residuals of your numerical solution does not

fall bellow a specified threshold value ?

• Phoenics has several commands to control the numerical

convergency rate which are represented by :

1. Control of the number of iteration per variable and

the termination criteria;

2. The prescription for a variable of the range of values

it can take;

3. The use of relaxation;

ASSESING CONVERGENCE

• There are several guidelines to follow and observe when trying to assess

whether a run has converged or not. These are:

• Have the values at the monitor point stopped changing?

• Have the residuals reached the cut-off point, or reduced by several orders

of magnitude in any case?

• Do the sums of sources printed in the RESULT file balance?

• Are plots taken from PHI files dumped, say 200, sweeps apart essentially

identical?

• If the answers the first three questions are all yes, then the run has almost

definitely converged. The fourth test can be used just to make sure, if you

are not convinced. Intermediate PHI files can be dumped from the EARTH

interrupt, or by appropriate settings in the Q1.

2.1 - MATRIX SOLVER CONTROLER:

NUMBER OF ITERATIONS PER VARIABLE

AND

TERMINATION CRITERIA

MATRIX SOLVER CONTROLER: NUMBER OF ITERATIONS PER VARIABLE AND TERMINATION CRITERIA

Each variable is visited by the solver LITER times

before the solver moves to the next one.

LITER(phi)....maximum number of iterations which

are to be performed by the linear-equation solver

for the variable phi, when such a solver is being

employed for it.

ENDIT----- Real array; default= 1.E-3; Satellite Help -

When greater than zero, ENDIT influences the

termination of iterations in the built-in linear-

equation solver by requiring that, before

termination can occur, the average change of value

is less than ENDIT times the average change at the

first iteration (unless LITER iterations have been

performed).

Initial data &

Boundary Conditions

Solver U1

Solver V1

Solver TEM1

Solver P1

(mass conserved field)

End

Sw

eep

Compare residuals

against a reference

LITER

ENDIT

MATRIX SOLVER CONTROLER: NUMBER OF ITERATIONS PER VARIABLE AND TERMINATION CRITERIA

The convergence process is interactive. The solver visits one variable at a

time.

Therefore is not necessary to get a error free solution at the beginning of the

process since the other variables are not fully converged. This means:

• LITER large will demand large machine time to get a solution;

• LITER small probably will not guarantee a converged solution since the

intermediate solutions are not well resolved;

• ENDIT small will use all the sweeps specified in the LITER;

• ENDIT big will cause the solver leave the variable without a ´reasonable´

solution;

VR SETTINGS UNDER NUMERICS ENTRY

2.2 - MATRIX SOLVER CONTROLER:

LIMITING THE RANGE OF VALUES

VARIABLE CONTROL LIMITING THE RANGE OF VALUES (GROUP 18)

• A variable can be prevented from overshooting, during the iteration

procedure, the range of physically-realistic values by using the PIL

commands VARMIN (variable) and VARMAX(variable) to prescribe that

range.

• Then, PHOENICS will use those values as lower and upper bounds for the

field of the variable. It is usefull for solved variables such as Turbulent

Kinetic Energy and Dissipation which have always positive values!

•The use of VARMIN and VARMAX is not limited to solved for variables; it

can also be used with any STOREd one.

• For example, after STOREing the density or the turbulent viscosity,

VARMIN can be used to avoid negative values during the iteration process.

• Note, however, that if any values are left at VARMIN or VARMAX in the

converged flow-field, the flow is NOT converged. The limits must be eased,

and the run continued.

VR SETTINGS UNDER NUMERICS ENTRY & q1 COMMAND LINES

2.3 - MATRIX SOLVER CONTROLER:

RELAXATION (very important)

RELAXATION • Relaxation is a commonly-used device that

promotes convergence by "slowing down"

("relaxing") the (sometimes excessive) changes

made to the values of the variable during the

solution procedure.

• Relaxation involves, therefore, changing the values

obtained from the solution algorithm, so that they

are closer to the pre-existing ones.

• How the values are changed gives rise to two kinds

of relaxation:

1.Linear relaxation

2.False time-step (or inertial) relaxation

• Both options are available in PHOENICS (PIL group

17 or the menu), and their activation is discussed

next.

• However, before proceeding with the discussion, it

must be stressed that relaxation does NOT change

the converged solution, but only the RATE at which

the solution is achieved.

Initial data &

Boundary Conditions

Solver U1

Solver V1

Solver TEM1

Solver P1

(mass conserved field)

Compare residuals

against a reference

End

Sw

eep

relax

1. Linear Relaxation

When linear relaxation is applied to a variable f, the new value fnew for the

variable at each cell is taken as:

fnew = (1 - a) fold + a f*

where: fold is the current in-store value, resulting from the previous

iteration; and f* is the value which results from the current iteration; and

a is the relaxation factor.

Obviously:

if a = 0, then fnew = fold (i.e., no change at all)

if a = 1, then fnew = f* (i.e., no relaxation at all)

The PHOENICS command to activate linear relaxation for a variable is:

RELAX( f , LINRLX , a )

2. False-Time-Step Relaxation

False-time-step relaxation results from adding to the RHS of the

FVE an extra term:

where: fP is the cell-value of to be computed, f is the cell-value

of resulting from the previous iteration, V is the cell volume and

dtf is a user-set false-time-step

• Large values of dtf, the added extra term is negligible, and

the solution of the equation is the same as the unrelaxed one.

• Small values of dtf, the solution is (i.e., no

change at all).

•The name false-time-step arouse because the extra added

term has the same form as the one resulting from the

transient term even if the problem is steady state.

False-Time-Step Relaxation - cont

• dtf can be set finding a characteristic time scale of the phenomena:

Convective time scale: dtf ~ L/U

Difusive time scale: dtf ~ L2 /n

• The PHOENICS command to activate linear relaxation for a variable is:

RELAX ( f , FALSDT , time-step ) where time-step = dtf.

VR SETTINGS UNDER NUMERICS ENTRY & q1 COMMAND LINES

Expert Systems to Achieve Fast Convergency

• EXPERT: A set of 'auto-pilot' devices have been introduced which make

'in- flight' adjustments to the numerical parameters (such as relaxation

factors) in order to speed up the convergence of the solution procedure.

Look after the Encyclopedia to activate Expert.

• SARAH (Self-Adjusting Relaxation AlgoritHm): Much in the same way

that EARTH can calculate normalising factors for residuals when

SELREF=T based on sums of fluxes, it can also calculate average time-

scales for each variable. Look after the Encyclopedia to activate Expert.

• CONWIZ (Convergency Wizard): is a feature of PHOENICS 3.6 which

is designed to ensure that PHOENICS solutions will converge,

whenever the input data are sufficient and self-consistent. See entry at

Encyclopedia

2.4 – WORKSHOP ON CONTROLING CONVERGENCY

WORKSHOP #3 – CONVERGENCE AND CONTROL

• Reload 2D rectangular channel flow .

•Set the Auto Converg Control to OFF

• Set FALSDT to W1 and V1 to 1

• Set SWEEPS to 600

•--------------------------------------------

DOWNLOAD WKSH Q1

•Comments:

• The residuals are lower than previous case.

• Return to the Auto-Convergence mode.

• Set the reference velocity and scale to: 0.01m/s & 0.5m and

run again the case.

WORKSHOP #4 – CONVERGENCE AND CONTROL

• Load core library case 252. This case

deals with natural convection in an

annular cavity.

• At the OUTPUT box activate the graphics

monitor convergence; TSTSWP = -1

• At the NUMERICS box change the sweeps

from 150 to 600; LSWEEP = 600

• After you completed these changes run

EARTH. The solution must converge

within the 300 sweeps.

WORKSHOP NOTES – CONVERGENCE AND CONTROL #1

•THE DANGER OF OVER-RELAXATION

•Now change the relaxation settings of U1, V1 and H1 to FALSDT

= 1.0E+04 and watch the monitor plots (see NOTE 2 for help if

you need).

• In many cases even moderate amounts of relaxation can speed

up convergence dramatically. In other cases, it is the only

difference between convergence and divergence!

WORKSHOP NOTES – CONVERGENCE AND CONTROL

THE DANGERS OF UNDER RELAXATION

• Run the case again. This time put in extremely heavy

relaxation (FALSDT = 1.0E-8) and see how the solution

behaves. Can you explain this behavior? See note 3 if you

need help.

• Answer: you have introduced so much relaxation that the momentum

equations can no longer change themselves. Pressure correction is still

active, and will change the velocities so as to satisfy continuity. The P1

residual will therefore be very low, but the velocity residuals will be high

and stuck. Setting FALSDT to 1.0E-8 is an extremely heavy under

relaxation which will probably freeze the variables.

ADDTIONAL REMARKS

• When divergence occurs it is necessary to establish the cause. It

may be found in the strength of the linkages between two or more

sets of equations, which are being solved. in turn rather than

simultaneously.

• For example in Free Convection heat transfer is a strongly coupled

with the energy and the velocity field. They influence each other and

may be a common source of divergence.

• Whether it is or not THE source in a particular case can be

established by ´freezing´ the temperature field before the divergence

has progressed too far and seeing if the divergence persists.

• If it DOES NOT, the velocity-temperature link can be regarded as the

source of divergence; otherwise the cause must be sought in some

other linkage.

PLAYING WITH RELAXATION

• Explore how different choices of relaxation factors affect the

required number of SWEEPS to get convergence.

• Fill in the Table below

P1 V1 U1 H1 SWEEPS

-1 1 1 1

AUTO AUTO AUTO AUTO

-1 2 2 10

-1 200 200 100

-1 0.1 0.1 0.1

CHANGING THE RESFAC

• Objective: evaluate the number of sweeps necessary to get

CONVERGENCY by changing the RESFAC

• Fill in the Table below

RESFAC N. SWEEPS

1%

0.1%

0.01%

0.001% 0

100

200

300

400

0.001 0.01 0.1 1

SW

EE

PS

RESFAC (%)

PLAYING WITH LITER

• Explore how different choices of LITER affect the required

number of SWEEPS and the CPU time to get convergence.

• Use RESFAC = 0.1% and ENDIT = 1E-3

• Fill in the Table below

P1 V1 U1 H1 SWEEPS CPU Time/Var

20 10 10 20

30 20 20 30

10 5 5 10

2 1 1 2

LAST REMARK ON ‘CONVERGENCE AND

NEGLIGIBLE RESIDUALS’

• After applying the available tools to access convergence

and negligible residuals with no success consider REVIEW

YOUR GRID.

Try to work with coarse grids at start and then

progressively refine them.

Be alert to flow regions with steep gradients, typical of

Boundary Layer. Try to refine the grid on these regions to

capture the steep gradients.

Avoid working with grids exhibiting large aspect ratios, i.e.

larger than 20:1

END

PLAYING WITH RELAXATION

• Explore how different choices of relaxation factors affect the

required number of SWEEPS to get convergence.

• Fill in the Table below

P1 V1 U1 H1 SWEEPS

-1 1 1 1 240

AUTO AUTO AUTO AUTO 569

-1 2 2 10 54

-1 200 200 100 no conv

-1 0.1 0.1 0.1 too slow

CHANGING THE RESFAC

• Objective: evaluate the number of sweeps necessary to get

CONVERGENCY by changing the RESFAC

• Fill in the Table below

RESFAC N. SWEEPS

1%

0.1%

0.01%

0.001% 0

100

200

300

400

0.001 0.01 0.1 1

SW

EE

PS

RESFAC (%)

PLAYING WITH LITER

• Explore how different choices of LITER affect the required

number of SWEEPS and the CPU time to get convergence.

• Use RESFAC = 0.1% and ENDIT = 1E-3

• Fill in the Table below

P1 V1 U1 H1 SWEEPS CPU Time/Var

20 10 10 20 240 2 2.08x10-5

30 20 20 30 238 2 2.10x10-5

10 5 5 10 245 2 2.04x10-5

2 1 1 2 260 5 4.80x10-5