Boundary Conditions in Phoenics

8/12/2019 Boundary Conditions in Phoenics

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UNDARY CONDITIONS IN PHOENICS

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Encyclopaedia Index

BOUNDARY CONDITIONS IN PHOENICS

Contents

Title And Purpose

The main conceptsBoundary And Internal Conditions And SourcesTreatment Of Boundary Conditions In PHOENICSThe Algebraic EquationsThe Source Term

Boundary conditionsPIL Commands For Boundary ConditionsExamples Of Boundary Condition SettingsFixed Value Boundary ConditionFixed Flux / Fixed Source Boundary ConditionLinear Boundary ConditionNon-Linear Boundary Condition

Wall boundary conditionsLaminar Wall ConditionsTurbulent Wall Conditions

Inlets, outlets and aperturesInflow Boundary ConditionFixed Pressure Boundary Condition

SourcesGeneral Source TermsExamples Of Non-Linear SourcesExamples Of Source LinearizationNon-Linear Sources From Q1

Stagnation Pressure Boundary ConditionQuadratic SourcesPower-Law SourcesRadiative Heat Loss To The Surroundings

Summary Of Main Points

Title And Purpose

his lecture explains how boundary conditions and sources are specified in PHOENICS.

he basis of the implementation is given, and some simple examples taken from library cases are given.

ull details of Boundary Conditions and Sources are available through the Satellite help-facility, the PHOENICSncyclopaedia, and in TR/200. Many examples can be found in the Library.

NOTE: By default all domain edges are impervious to flow, frictionless and adiabatic. They represent symmetry plr axes.

Boundary And Internal Conditions And Sources

Differential equations need to be supplemented by boundary conditions before they can be solved.

he boundary conditions which define a fluid- or heat-flow problem usually convey the necessary information abo
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ow much fluid enters the domain, where it can leave, what is its temperature on entry, what are the temperatures ohe walls, etc.

Of course, fluid may be caused to enter or leave at points within the domain, not only at its external limits; andemperatures of structural elements within the domain may also be externally imposed, and exert an influence uponow.

HOENICS makes no distinction between "boundary" and "internal" conditions, or between these and "source" termo the former term will be used here for all of them.

Treatment Of Boundary Conditions In PHOENICS

n PHOENICS, boundary conditions and sources appear on the r.h.s. of the differential equation for a variable f. Th

where:

frepresents conventionally recognised source terms, such as pressure gradients or viscous heating terms. These ar

built-in' to EARTH.

Gfrepresents the diffusive exchange coefficient for f.

bc1etc. represent various boundary conditions. These may be present only in certain regions of the domain. More

erms of this kind may be also be present in other regions of the domain, and these regions may overlap.

The Algebraic Equations

he differential equation is integrated over a control volume to yield the finite-volume equation actually solved. Thntegral of the boundary source is represented in linearized form:

he finite-volume discretization of the differential equation thus yields, for each cell P in the domain, the followinglgebraic equation:

where:

fis the 'true' source

C is the coefficient

V is the value

is the type, a geometrical multiplier

The Source Term

As a consequence of the integration procedure, the source is required per cell. The units of the source are (f kg/s). Type is used to convert the source from any given set of units.


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hus, if the source is defined as 'per unit volume', type supplies the cell volumes. If the source is 'per unit area', typn appropriate cell face area.

Note that TCV is added to the numerator, and TC to the denominator. As will be shown, this allows for easymanipulation of the solution.

IL Commands For Boundary Conditions

As set out in preceding panels, the specification of boundary conditions requires two kinds of information:

1. Where (and when) the boundary is2. The values of T, C and V

1) and the first part of (2) is conveyed by a PATCH command:

ATCH ( name, t ype, I XF, I XL, I YF, I YL, I ZF, I ZL, I TF, I TL)

where:

ame is a unique patch name for future reference;

ype is T

XF,IXL are the first and last IX in the patch; and similarly for y, z and t (time)

NOTE that any of the PATCH limits can be specified in terms of REGION number by using #IXF,#IXL etc.

he remainder of (2) is specified by a COVAL command, the format of which is:

OVAL ( name, var i abl e, coef f i ci ent , val ue) ,

where:

ame is the patch name to which the command refers

ariable is a SOLVEd-for variable

oefficient is C

alue is V

How these are used can be seen by inspecting Library cases, or Menu-generated Q1 files. Some simple examples arow given.

urther information can be found in the Encyclopaedia, under the headings: Boundary Conditions, Sources, PATCCOVAL

Examples Of Boundary Condition Settings

he types of boundary conditions which have to be provided for are:

Fixed valueFixed flux / fixed source


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Linear boundary conditionNon-linear boundary conditionWall conditionsInflows and outflowsGeneral sources

ach of these will now be explained and illustrated with examples.

ixed Value Boundary Condition

ractical example: we wish to fix the temperature in one corner of a cube to 0.0, and to 1.0 in the diagonally opposorner.

Numerical practice: the value of phi can be fixed in any cell by setting C to a large number, and V to the requiredalue.

he equation then becomes:

he PIL variable FIXVAL is provided for this purpose. A typical PATCH/COVAL would be:

ATCH ( FI XED, CELL, I XF, I XL, I YF, I YL, I ZF, I ZL, I TF, I TL)OVAL ( FI XED, phi , FI XVAL, r equi r ed_val ue)

Example From The Library

Case 100 models a solid cube of material, in which one corner is held at a low temperature, and the diagonallypposite corner is held at a high temperature:

ROUP 13. Boundary condi t i ons and speci al sour ces*Cor ner at I X=I Y=I Z=1ATCH( COLD, CELL, 1, 1, 1, 1, 1, 1, 1, 1)*Fi x t emper at ur e t o zer oOVAL( COLD, TEMP, 1. E2, 0. 0)*Cor ner at I X=NX, I Y=NY, I Z=NZATCH( HOT, CELL, NX, NX, NY, NY, NZ, NZ, 1, 1)*Fi x t emper at ur e to 1. 0OVAL( HOT, TEMP, 1. E2, 1. 0)

Note that the coefficient has been set to 100. This in effect sets a finite conductivity between the external value and

entre of the cell. FIXVAL would lock the cell-centre value to the external value.ixed Flux / Fixed Source Boundary Condition

ractical example: heat is being generated at a constant (fixed) rate.

Numerical practice: a fixed source can be put into the equation by setting C to a small number, so that the denomins not changed, and by setting V to (source/C). T then ensures that the final source is per cell.

he equation then becomes:


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he PIL variable FIXFLU is provided for this purpose. A typical PATCH/COVAL setting would be:

ATCH ( SOURCE, ar ea, I XF, I XL, I YF, I YL, I ZF, I ZL, I TF, I TL)OVAL ( SOURCE, phi , FI XFLU, sour ce_per_uni t _ar ea)

ibrary case 921 concerns the prediction of the flow and temperature fields in a closed cavity with one moving walnd a heated block. The heated block appears as shown below:

Heat source in block - 10 MW/m**3

ATCH( HEATEDBL, VOLUME, NX/ 4+1, 3*NX/ 4, NY/ 4+1, 3*NY/ 4, 1, 1, 1, 1)OVAL( HEATEDBL, TEM1, FI XFLU, 1. 0E7)

he patch type VOLUME converts the source from W/m3to W per cell, by multiplying by the appropriate cellolumes.

Linear Boundary Condition

ractical example: One of the domain boundaries is losing heat to the surroundings. The external heat transfer

oefficient, H (W/m2/K), and the external temperature, Text (K), are both known and constant.

Numerical practice: The heat source for a cell with area A is:

Q = A H (Text- Tp)

his is obviously in TC(V-phi) form if T=A, C=H and V=Text. A typical PATCH/COVAL would be:

ATCH ( HEATLOSS, ar ea, I XF, I XL, I YF, I YL, I ZF, I ZL, I TF, I TL)OVAL ( HEATLOSS, TEM1, heat _t r ansf er_coef f i ci ent , ext ernal _t emperat ur e)

Non-Linear Boundary Condition

n the previous example, either or both the heat transfer coefficient and external temperature are likely to be a funcf some solved-for variable or other suitable expression.

he non-linear source can always be linearized into TC(V-phi) form by arranging to update C and/or V during theourse of the calculation.

he updating of C and V is signalled to EARTH by setting C or V to one of the 'ground flags' - GRND, GRND1,GRND2, ..., GRND10.

hus the following COVALs may be seen:

OVAL ( HEATLOSS, TEM1, GRNDn, ext ernal _t emper at ure)

r


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OVAL ( HEATLOSS, TEM1, heat _t r ansf er_coef f i ci ent , GRNDn)

r

OVAL ( HEATLOSS, TEM1, GRNDn, GRNDn)

Many commonly-found examples have already been coded into GREX by CHAM.

Laminar Wall Boundary Condition

aminar shear stress at stationary wall is expressed by:

where area is the cell face area, and dely is the distance from the cell face to the cell centre.

his can be put into form if .

he problem with this approach is that the density and laminar viscosity may be varying, whilst the distance to the

wall will change as the grid is refined, and indeed may change from cell to cell in a BFC grid. This simple method herefore not recommended, as all the quantities causing problems to the user are known to EARTH.

A special PATCH type is provided which automatically sets:

he coefficient is then a further multiplier, which is usually set to 1.0.

hese PATCH types are NWALL, SWALL, EWALL, WWALL, HWALL and LWALL.

o further simplify matters, a WALL command generates all the PATCH and COVAL statements required.

A typical WALL command would be:

WALL ( SI DE, ar ea, I XF, I XL, I YF, I YL, I ZF, I ZL, I TF, I TL)

Note that WALL uses the ordinary area types, not those above.


he stationary and moving walls in case 921 are specified as shown:

Movi ng wal l at Sout h si de of domai n at 1 degWALL ( MOVI NG, SOUTH, 1, NX, 1, 1, 1, 1, 1, 1)OVAL( MOVI NG, U1, 1. 0, - WALLVEL) ; COVAL( MOVI NG, TEM1, 1. 0, 1. 0)

St at i onar y wal l at Nor t h si de at 0 degWALL ( NORTHW, NORTH, 1, NX, NY, NY, 1, 1, 1, 1)OVAL( NORTHW, U1, 1. 0, 0. 0) ; COVAL( NORTHW, TEM1, 1. 0, 0. 0)

St at i onar y wal l at West si de at 0 degWALL (WESTW, WEST, 1, 1, 1, NY, 1, 1, 1, 1)OVAL( WESTW, V1, 1. 0, 0. 0) ; COVAL( WESTW, TEM1, 1. 0, 0. 0)

St at i onar y wal l at East si de at 0 degWALL ( EASTW, EAST, NX, NX, 1, NY, 1, 1, 1, 1)OVAL( EASTW, V1, 1. 0, 0. 0) ; COVAL( EASTW, TEM1, 1. 0, 0. 0)


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Note that the coefficients have all been set to 1.0, and the values to the wall surface values. This suffices for laminawall conditions.

Turbulent Wall Boundary Condition

n a turbulent flow, the near-wall grid node normally has to be in the fully-turbulent region, otherwise the assumptin the turbulence model are invalid. The wall shear stress and heat transfer can no longer be obtained from the simpnear laminar relationships.

Unless a low-Reynolds number extension of the turbulence model is used, the normal practice is to bridge the laminub-layer with wall functions. These use empirical formulae for the shear stress and heat transfer coefficients.

hree types of wall function are available, selected by the COVAL settings:

Coefficient GRND1 for Blasius power lawCoefficient GRND2 for equilibrium Logarithmic wall functionCoefficient GRND3 for Generalised (non-equilibrium) wall functionCoefficient GRND5 for Fully-rough equilibrium Logarithmic wall function


ibrary case 172 concerns the prediction of developing flow in a duct. The k-epsilon turbulence model is used. Theuct surface at the north side of the duct is represented as:

GROUP 13. Boundary condi t i ons and speci al sour ces* Nort h- Wal l Boundary

WALL ( WFUN, NORTH, 1, 1, NY, NY, 1, NZ, 1, 1)OVAL( WFUN, TEMP, GRND2, TWALL)

Note the use of the WALL command to locate the wall and activate wall friction effects.

he surface temperature is supplied via the COVAL for TEM1. The GRND2 in the coefficient slot activatesogarithmic wall functions.

nflow Boundary Condition

All mass flow boundary conditions are introduced as linearized sources in the continuity equation, with pressure (Ps the variable. A mass source is thus:

where Cm and Vm are coefficient and value for P1.

At an inflow boundary, the mass flow is fixed irrespective of the internal pressure. This effect is achieved by settingCm to FIXFLU, and Vm to the required mass flow.

he sign convention is that inflows are +ve, outflows are -ve. A fixed outflow rate can thus be fixed by setting aegative mass flow.


ibrary case 274 concerns the flow over a simplified van geometry. The inflow boundary at the low end of the soluomain is represented as:

GROUP 13. Boundary condi t i ons and speci al sour ces


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* Upst r eam boundaryNLET( UPSTR, LOW, #1, #NREGX, #1, #NREGY, #1, #1, 1, 1)ALUE(UPSTR, P1, 14. 0) ; VALUE(UPSTR, W1, 14. )

Note that for an INLET, the VALUE command for P1 sets the mass flux. This is often set as RHOIN*VELIN, thenlet density) * (inlet velocity).

he mass flux is fixed, and the in-cell pressure is allowed to float.

he VALUE command for W1 sets the velocity of the inflowing stream. In this case all other variables are taken to

.0 at the inlet. If they are not, then VALUE commands would have to be added.

ixed Pressure Boundary Condition

his is the case of a mass flow boundary where the pressure is fixed irrespective of the mass flow.

As with any other variable, the pressure is fixed by putting a large number for Cm, and the required pressure for Vm

or numerical reasons, FIXVAL tends to be too big. A Cm of about 1E3 usually suffices. The Encyclopaedia givesurther guidance.

he direction of flow is then determined for each cell in the PATCH by whether Pp>Vm, or Pp< Vm. The first

roduces local outflow, the second local inflow.


he exit boundary in case 274 is a fixed pressure boundary, set as:

Downst r eam boundar yATCH( DWSTR, HI GH, #1, #NREGX, #1, #NREGY, #NREGZ, #NREGZ, 1, 1)OVAL( DWSTR, P1, FI XP, 0. )OVAL( DWSTR, U1, ONLYMS, 0. 0) ; COVAL( DWSTR, V1, ONLYMS, 0. 0)OVAL( DWSTR, W1, ONLYMS, 0. 0)

n this case, the in-cell pressure is fixed by the COVAL for P1, and the mass flux is adjusted to satisfy continuity. T

irection of flow is determined by whether the in-cell pressure is > or < the fixed value.he COVALs for U1, V1 and W1 are supplied in case part of the boundary should be an inflow - they specifyelocities to be brought in. They are not used in cells where in-cell pressure > external.

General Source Terms

HOENICS makes no distinction between boundary conditions and source terms. Both are represented in the by noamiliar TC(V - phi) form.

n the lecture so far, we have made a distinction between the 'true' source, Sf, and the boundary source, Sbc.

he true source represents the fundamental parts of an equation which are not covered by the convective, diffusive ransient terms. These are coded in EARTH, and hence are called the built-in sources.

Built-in Sources

xamples of these are:

1. The pressure gradient sources in the momentum equations2. The centrifugal and Coriolis sources in the momentum equations in cylindrical-polar co-ordinates3. The negative of the substantial derivative of pressure in the enthalpy equation;4. The viscous dissipation of heat in the enthalpy equation.


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he built-in sources can be switched off for individual variables, by setting N as the first argument of TERMS. Forxample:

ERMS ( W1, N, P, P, P, P, P)

will deactivate the pressure gradient terms in the W1 equation.

User-set Sources

All other sources are treated as external user-set sources. Examples are:1. Inlets, outlets, walls, etc. - all those covered in the first part of this lecture;2. Sources in 'model' equations, such as those forming the turbulence, or combustion models.

Many commonly used examples of the second type have already been programmed by CHAM. The source code fof these can be found in the exemplary GROUND routine, GREX, and several other files whose names start with G

All of these can be found in phoenics/d_earth/d_core/..., and can be viewed through POLIS. If required, they can bopied and used as templates for further modification.

mplementation

oundary conditions (or sources) cannot always be specified through a constant coefficient and a constant value.

n many instances, the coefficient and/or the value are functions of one or several solved-for variables, the value ofwhich cannot be foreseen at the time of data input.

As we have already seen, PHOENICS copes with these complex relationships through the insertion of FORTRANoding in the GROUND module.

o do so, special flags (GRND, GRND1, ... GRND10) can be specified as coefficients and/or values in the COVALommand. These instruct EARTH to visit special sections of GROUND, where coefficients and/or values can beomputed and set back into EARTH.

Nearly all the conditions specified in GREX are of this type.

Non-linearity

All user-defined sources are introduced in the linear form. PATCH and COVAL commands are used toefine the location, duration and T, C and V.

inear sources can be inserted directly from Q1 or the Menu.

xample: A chemical species, C1, is being created throughout the whole domain at a constant rate of 10 kg/m3/s. T

IL commands are:ATCH ( CREATEC1, VOLUME, 1, NX, 1, NY, 1, NZ, 1, LSTEP)OVAL ( CREATEC1, C1, FI XFLU, 10)

However, many common CFD sources cannot be expressed directly in linear form, because:

They are not linear; and/orThey interconnect several variables, the values of which are unknown before the calculation is completed.

Examples Of Non-Linear Sources


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xample 1: The pressure drop through a porous medium can frequently be written in the form:

his can be turned into a momentum source (force):

where vol is the cell volume.

he source is proportional to velocity squared.

xample 2: The sink of a chemical species (C1) undergoing chemical reaction can frequently be expressed by theArrhenius law:

where a and b are empirical constants, C2 is the mass fraction of another species, and T is the temperature.

Although the expression is linear in C1, the local values of C2 and T cannot be predetermined.

Note that the Arrhenius expression is available as an option in the standard PHOENICS combustion model.

Treatment Of Non-Linear Sources

Non-linear and interconnected sources will normally allow several representations in the linear form.

ractice 1: The source can be introduced as a fixed flux (FIXFLU) source, with the value of the source computed frn-store values.

his practice is not recommended, as it tends to be numerically unstable. However, in some cases it is the only way

ractice 2: If at all possible, the source should be linearized.

requently, there will be more than one way to linearize a source. Only experience will show which is the most sta

Examples Of Source Linearization

he quadratic momentum source from above can be written as:

where v* is the current in-store velocity. In the converged solution, v = v*, and the source is the required one.

his is in form if:


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he PIL implementation for, say, the W1 equation is:

ATCH ( PDROP, PHASEM, 1, NX, 1, NY, 1, NZ, 1, LSTEP)OVAL ( PDROP, W1, GRND, 0. 0)

Note that the type PHASEM sets T to r*Vol .

he GROUND code would set coefficient = K W1*.

he Arrhenius expression in panel 30 can be written as:

his is in C(V-phi) form if:

he PIL implementation is (assuming that the source is 'per unit volume'):

ATCH (REACT, VOLUME. 1, NX, 1, NY, 1, NZ, 1, LSTEP)OVAL ( REACT, C1, GRND, 0. 0)

he GROUND code would set:

Non-Linear Sources From Q1

o far, it has been stressed that linear sources can be specified directly from Q1, but that non-linear sources need

dditional coding in GROUND. Many such sources have already been provided by CHAM.

n addition to the GROUND code already supplied in GREX, a wide range of non-linear sources CAN be introduceirectly from Q1 without the need for any GROUND coding. A range of these will now be described. A full list caound in the Encyclopaedia, under the PATCH entry.

hese sources are introduced by 'unconventional' coefficient settings, and/or by special PATCH names.

Normally, the coefficient in the source must be positive, otherwise numerical instability may result. In vief this, negative coefficients in the Q1 are used to flag special sources.

he following non-linear Q1-set sources will be exemplified:

Stagnation Pressure ConditionQuadratic SourcePower Law SourceRadiative Heat Loss to the Surroundings

hese sources can also all be set in VR, using the 'User-Defined' object type.

tagnation Pressure Boundary Condition


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or incompressible flows, the stagnation pressure is:

his can be manipulated to give the mass flux:

or P1 (and also P2), a negative coefficient flags a mass source of the form:

where Cmand Vmare the coefficient and value for mass (P1).

he desired stagnation source can thus be set by putting:

hese settings must be supplemented by a COVAL for the velocity component normal to the boundary, withoefficient set to ONLYMS, and value set to SAME.

IL example:

ATCH ( PSTAG, LOW, 1, N, 1, NY, 1, 1, 1, LSTEP)OVAL ( PSTAG, P1, - 2*RHO1, PSTAG)OVAL ( PSTAG, W1, ONLYMS, SAME)

Note that for compressible flows, it is still possible to set stagnation conditions, though not in such an easy fashion

Quadratic Sources

f a negative coefficient is supplied for any other variable, the source introduced is:

IL example: The quadratic pressure drop shown on panel 29 can thus be implemented without any GROUND cods shown below:

ATCH ( PDROP, PHASEM, 1, NX, 1, NY, 1, NZ, 1, LSTEP)OVAL ( PDROP, W1, - K, 0. 0)

Note that the negative coefficient options only apply to constant coefficients. They will not apply to GROUND-setoefficients.

ower Law Sources

ources of the form:

an be introduced from Q1 by using 'special' PATCH names.


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wo PIL commands convey the information required: a PATCH command which sets "where", "when" and T, andCOVAL command which sets the values of C and V.

or a variable f, the main kinds of boundary conditions are:

Fixed-value boundary condition (coefficient = FIXVAL)Fixed-flux boundary condition (coefficient = FIXFLU)Wall-type boundary condition (patch type = *WALL)Linear boundary condition (coefficient = proportionality constant)

oundary conditions for mass and pressure are both treated as linearized sources in the continuity equation, with

ressure as the variable in the linear source: . FIXFLU is used as coefficient for the specification ofmass fluxes, and FIXVAL or FIXP for fixing the pressure.

oundary conditions must be supplied for all the variables when there is an inflow mass into the domain. This is doy using ONLYMS as coefficient in the COVAL command for the property, and the inflowing value as value.

inear sources, and certain non-linear sources can be introduced directly from the Q1 or Menu, by appropriate settif coefficient and/or value.

Non-linear sources have to be linearized, and the non-constant part programmed in GROUND. This is flagged bysing GRND, GRND1 ... GRND10 for coefficient or value. Many such sources are already provided in GREX.

Boundary Conditions in Phoenics

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