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  • In-Form.htm
    and if a General description As has already been stated the general form of an In Form statement in a Q1 file consists of four items enclosed between brackets keyword what where when indicators formula optional conditions This will now be explained under the headings keywords pre formula conditions i e what where when indicators the formula containing the expression to be evaluated by the solver post formula options i e optional conditions It should be noted that The opening and closing brackets of the statement must always be present The opening bracket must start in the first or second column Lines must not extend beyond the 68th column but a dollar sign in or before the 68th column will be taken as an append next line instruction Lines can contain symbols specifically any of the following symbols 0123456789 ABCDEFGHIJKLMNOPQRSTUVWXYZ but no others Characters may be upper or lower case without consequences Blank spaces separate the items several successive spaces having the same effect as one See also Appendix 2 for a succinct summary of In Form statement features in which it will be seen from the appearance of brackets which items may be dispensed with b Keywords The keywords used in In Form statements are PROPERTY This has already been encountered in section 2 1 It will be explained in connexion with all material properties used by PHOENICS in section 4 STORED This keyword is used for the creation of auxiliary variables which can have distinct values for each cell in the domain Such variables are of two main kinds namely those which are used only for post processing purposes as when exact solutions are computed for comparison with the numerically derived values and those which require to be updated continually during the computations because their values influence how the computation proceeds These will be discussed in section 5 MAKE This keyword is also concerned with the creation of auxiliary variables but of lesser dimensionality It is also discussed in section 5 STORE1 This keyword concerns what is to be done with single auxiliary variables introduced by MAKE It is also discussed in section 5 INITIAL This is concerned with the setting of initial values of solved for or auxiliary values Initial values can be set for the whole domain for limited volumes defined by patches i e those of which the bounding boxes are aligned with the computational grid for limited volumes defined by facetted objects i e those of which the bounding boxes are not wholly aligned with the computational grid for limited volumes defined by In Form objects the shapes and positions of which are defined by formulae This keyword is discussed in section 6 INFOB As its name suggests this keyword indicates that the position and shape of an In Form object are to be defined It also is discussed in section 6 SOURCE This extremely important keyword the subject of section 7 is used for introducing formulae defining the sources of mass momentum energy and other conserved properties In accordance with the usual PHOENICS practices boundary conditions are also introduced by its means Sources may like initial values be applied to the whole domain or to variously defined limited spaces MOVOB This keyword represents In Form s contribution to MOFOR the moving frame of reference feature of PHOENICS As originally introduced the motion of objects had to be specified by way of a bvh later mof file Whereas this was necessary and convenient for complex motions of articulated objects such as the dancing man it was over complicated for others for which formulae suffice The MOVOB keyword discussed in section 8 1 makes this possible indeed easy TGRID XGRID YGRID ZGRID These four keywords concerned with the specification of time intervals for transient calculations and of longitudinal distance interval and lateral grid expansion or contraction in steady parabolic flows are discussed together in section 3 PRINT and LONGNAME In Form has greatly extended the ability of the user to print useful and easily understandable information whether to the RESULT file or to a special one called INFOROUT The relevant keywords are explained in section 9 REALREAD INTREAD LOGREAD These keywords provide the opportunity to convey to EARTH more information than SPEDAT can It is described in section 10 c Pre formula conditions var at var the what indicator In the statements for density in section 2 1a the words between the keyword and the formula were RHO1 is Here RHO1 is of the what where when kind and specifically what it says this formula applies to the variable with name RHO1 and by implication to no other It can thus be regarded as a pre formula condition As it happens the words of RHO1 is or var RHO1 is would have had the same effect but since the of and var add little to the understandability of the statement they are best omitted The word is is a signal that the formula follows It must not be omitted at the where when indicator applied to a patch The applicability of formulae can be limited in space and time by use of the word at followed by the name of a patch or object the names of which must appear in the Q1 file An example of a Q1 file which defines density differently for each of four different patches SW NW NE SW now follows 249 case 249 square cavity with moving lid has now been loaded STORE RHO1 in order that PHOTON will be able to display it now declare the patches to which at can refer PATCH SW CELL 1 NX 2 1 NY 2 1 1 1 1 south west quadrant property RHO1 at SW is 2 0 PATCH NW CELL 1 NX 2 NY 2 1 NY 1 1 1 1 north west quadrant property RHO1 at NW is 3 0 PATCH NE CELL NX 2 1 NX NY 2 1 NY 1 1 1 1 north east quadrant property RHO1 at NE is 4 0 PATCH SE CELL NX 2 1 NX 1 NY 2 1 1 1 1 south east quadrant property RHO1 at SE is 5 0 Here the where is specified by the IXF IXL IYF etc arguments of the patch commands and the when by the final 1 1 s which since the case is a steady state one signify at all times The result is as shown with the smearing engendered by the very coarse grid in the following PHOTON plot Readers interested in the transmission details may like to know that the relevant EARDAT contains the lines PROPERTY RHO1 SW C 2 0 PROPERTY RHO1 NW C 3 0 PROPERTY RHO1 NE C 4 0 PROPERTY RHO1 SE C 5 0 wherein it will be noted that the exclamation mark replaces at and the corresponding lines in the Q1EAR and RESULT files are SPEDAT SET PROPERTY RHO1 SW C 2 0 SPEDAT SET PROPERTY RHO1 NW C 3 0 SPEDAT SET PROPERTY RHO1 NE C 4 0 SPEDAT SET PROPERTY RHO1 SE C 5 0 The same with more complex formulae It might be argued that the same effect could have been arrived at without the use of In Form this is true for RHO1 could have been set by means of four INIVAL type patches However INIVAL patches allow only uniform or linear in x or y or z via LINVLX LINVLY and LINVLZ values to be set whereas In Form can set any non uniform ones which can be described by formulae for example property rho1 at sw is 100 xg yg property rho1 at nw is 1 0 1 e 1 u1 v1 property rho1 at ne is 10 0 0 0001 p1 h1 property rho1 at se is 5 0 0 1 xg 2 yg 2 from which it can be deduced that the formulae may involve other 3D stored variables u1 v1 p1 h1 and also geometrical quantities xg yg It should be noted however that the ability to set density freely in this manner does not signify that the setting makes physical sense or that a converged solution will be obtained A when example Core library case 756 simulates rather crudely the motion of a paddle in a chemical reactor It does so by shifting the PRPS distribution be reference to different patches at different times In association with In Form SOURCE settings which depend upon the PRPS value to be described below the fluid is set in motion as seen here at the where when indicator applied to a faceted object The name of an object may be substituted for the name of a patch with the difference that the formula is then operative only in that part of the bounding box of the object which lies within the facet described outer surfaces of the object Input Library case 764 shows how the viscosity can be set at different values according to whether the cell in question lies inside or outside a sphere defined by way of facets The sphere object is of course that defined by the line OBJ NAME SPHERE near the bottom of the Q1 file The resulting viscosity distribution is here shown in a VR Viewer contour plot Further scrutiny of the Q1 file for case 764 reveals the at condition in use also for statements with the keywords PROPERTY INITIAL STORED and SOURCE in each case showing that the formula is to be applied to the sphere Evidently it is very versatile A question of order The alert reader may have noticed that for the square cavity case the In Form statement containing at SW was placed below the PATCH SW declaration and that the same order PATCH first at later was adopted for the three further statements This appeared natural and conducive to orderly thought but was it necessary In case 764 the lines with at sphere all appear above the line which defines the sphere object This suggests that the patch first at later principle may not a necessity and the suggestion can be proved to be correct by running the following Q1 249 case 249 square cavity with moving lid has now been loaded STORE RHO1 in order that PHOTON will be able to display it now declare the patches to which at can refer property RHO1 at SW is 2 0 property RHO1 at NW is 3 0 property RHO1 at NE is 4 0 property RHO1 at SE is 5 0 PATCH SW CELL 1 NX 2 1 NY 2 1 1 1 1 south west quadrant PATCH NW CELL 1 NX 2 NY 2 1 NY 1 1 1 1 north west quadrant PATCH NE CELL NX 2 1 NX NY 2 1 NY 1 1 1 1 north east quadrant PATCH SE CELL NX 2 1 NX 1 NY 2 1 1 1 1 south east quadrant which will be found to give the same results as before The patch first at later prescription is a good rule to follow but it is not required by In Form d The formula Suppose that one wished to specify the density as proportional to the inverse square of the distance from the origin of the coordinate system Then any algebraically literate person would recognise this as being expressed symbolically with x y and z as the cartesian coordinates by density 1 x 2 y 2 z 2 0 5 with signifying exponentiation This is almost exactly how it is expressed in an In Form statement which might be property RHO1 is 1 XG 2 YG 2 ZG 2 0 5 wherein xg yg and zg denote the cartesian coordinates of the cell centres and replaces as the exponentiation sign If the distances were to be measured not from the origin but from some point with coordinates x0 y0 z0 these values would need to be declared in the Q1 file and given some values for example by REAL X0 Y0 Z0 X0 1 2 Y0 2 2 Z0 3 3 Then the formula could be expressed as 1 XG X0 2 YG Y0 2 ZG Z0 2 0 5 This is still easy to read but if the pre formula and post formula options were at all extensive the In Form statement would have more than one line then ease of reading would be diminished It is therefore useful to recognise that PIL is flexible enough to allow the formula to be built up in stages each one being brief enough be easily understood For example one might declare and define the character variables XDSQ YDSQ and ZDSQ thus CHAR XDSG YDSQ ZDSQ XDSQ XG X0 2 YDSQ YG Y0 2 ZDSQ ZG Z0 2 Then the formula could be abbreviated to 1 XDSQ YDSQ ZDSQ 0 5 wherein the colons signify that because XDSQ etc are character variables they must be evaluated as soon as read by SATELLITE Finally one might decide to declare and define the character variable FORM thus CHAR FORM FORM 1 XDSQ YDSQ ZDSQ 0 5 whereupon the whole In Form statement would be reduced to property RHO1 is FORM The equivalence of all these statements can be tested by running PHOENICS from the following Q1 file which contains all four of the above specifications talk t run 1 1 store rho1 nx 50 ny 50 nz 50 xulast 2 4 yvlast 4 4 zwlast 6 6 unigrid inform9begin REAL X0 Y0 Z0 X0 1 2 Y0 2 2 Z0 3 3 Case 1 property of RHO1 is 1 XG X0 2 YG Y0 2 ZG Z0 2 0 5 Case 2 CHAR XDSQ YDSQ ZDSQ XDSQ XG X0 2 YDSQ YG Y0 2 ZDSQ ZG Z0 2 property of RHO1 is 1 XDSQ YDSQ ZDSQ 0 50 Case 3 CHAR FORM FORM 1 XDSQ YDSQ ZDSQ 0 5 colons are needed here too FORM 1 XDSQ YDSQ ZDSQ 0 5 This would be incorrect property of RHO1 is FORM inform9end If Group 19 of the RESULT file is examined it will be seen that all four of the formulae have been converted into identical SPEDAT lines and the VR Viewer plot shown here indicates qualitatively that the resulting density field is as expected A comprehensive discussion of the many types of formulae allowed by In Form is given in section 2 3 below The present preliminary discussion now concluded has been inserted so as to make clear that even the most complex formulae of which several will be shown below can be broken into more easily digested fragments e Post formula options with if and infob There are numerous post formula options but many of them apply only to particular keywords as is shown in Appendix 2 Keyword specific options will be dealt with keyword by keyword in section 4 However there are three which apply to many keywords being characterised by IMAT IF and INFOB These will be dealt with in order here Limitation to a particular material by use of with IMAT etc It is possible to limit the action of a formula to those locations at which the material property index IMAT alias PRPS has a specific value Core library case 756 for example uses this technique so as to apply a momentum source only to those cells within the space occupied by the associated patch in which the material designating property PRPS alias IMAT is in excess of a prescribed value namely 100 The flow is transient so this is a when as well as a where condition with IMAT 100 0 is what appears in the In Form statement in question The complete set of with IMAT conditions allowable in In Form statements is with IMAT value means for PRPS greater than value with IMAT value means for PRPS less than value with IMAT value means for PRPS greater than or equal to value with IMAT value means for PRPS less than or equal to value with IMAT value means for PRPS equal to value with IMAT value means for PRPS not equal to value EARDAT manifestations the use of Inspection of the corresponding entries written into EARDAT when the SATELLITE has finished reading the Q1 of case 756 reveals that the counterparts of the In Form statements just inspected namely SOURCE of U1 at I is 1 E5 VEL YIC YG U1 with IMAT 100 LINE SOURCE of V1 at I is 1 E5 VEL XG XIC V1 with IMAT 100 LINE are with and blanks removed for ease of reading SOURCE U1 I C 1 E5 7 8540E 01 10 YG U1 IMAT 100 LINE SOURCE V1 I C 1 E5 7 8540E 01 XG 10 V1 IMAT 100 LINE Evidently the with has been replaced by an exclamation mark Consider now the LINE which follows 100 in both the In Form statements and their EARDAT counterparts The following needs to be noted LINE is a further post formula option applicable only to the source keyword its meaning is linearize the source i e express it as constant1 constant2 the as yet unknown value of the dependent variable in the cell in question despite the apparent equivalence of with and substitution of the former for the latter in this In Form statement is not allowed it can be concluded that although several post formula options can be applied simultaneously only the first can be preceded by a with and the following ones must be separated by exclamation marks Other variables than PRPS can be used in a similar way Thus it is possible to define a new whole field stored MARK variable use In Form to ascribe values for it over the whole field ascribe to density one value where and when the marker exceeds a prescribed value and another when it falls below the prescribed value IF Another generally applicable post formula option is the IF condition construct wherein condition can be a Fortran like expression as exemplified in library cases 776 in connection with the ZGRID keyword 777 in connection with the YGRID keyword 778 in connection with the STORE1 keyword 779 in connection with the SOURCE keyword 781 in connection with the STORED keyword 783 in connection with the several keywords 785 in conjunction with INFOB and 786 in connection with the SOURCE keyword It should be noted that either with or must always precede the IF INFOB N In Form objects created by the INFOB keyword will not be discussed systematically until section 6 Here it suffices to say that their existence and location may but need not be indicated by the value of a marker variable usually MARK in the cells which they occupy This value is usually 1 2 3 or other integer Then the post formula option with INFOB BALL say signifies that the formula is to be applied to cells where MARK has the value 2 strictly 2 0 because real values are in question Library case 768 illustrates its use Specifically with INFOB 1 will be seen to modify INITIAL of MARK and SOURCE of TEM1 while with LINE INFOB 1 will be seen to modify SOURCEs of U1 V1 and W1 It should be noted that it is no longer necessary as it was when case 768 was created that INFOB should be followed by a numeral Now any string of characters can be used as in BALL above It may be asked why since this condition is of the what where when variety it is not among the pre formula options The answer is that at has already been used for a different purpose for an In Form object itself requires to be localised which is why at PATCH1 appears in each of the In Form statements in question 2 3 Allowable formulae Contents General description Operators Functions Operands Further examples a General description The formulae employed by In Form whether for setting properties initial values sources or anything else are arrangements of operators functions and operands which conform to rules which are similar to those of algebra and or Fortran No significance attaches to whether upper or lower case characters are used b Operators The operators which may be used are all of which have their usual significances and which represents exponentiation thereby replacing the of Fortran c Functions The syntax of all In Form functions corresponds to Fortran At first the function name is specified Further in brackets the its arguments follow separated from each other by commas or ampersands Note When written in the Q1EAR EARDAT and RESULT files only ampersands appear but commas are preferable in Q1 files because they are easier to read In general case arguments of many functions can be the formulas containing constants and names of solved stored and user defined variables or any variables from list in Appendix 4 In Form operands Except for some functions of which will be commented specially below The functions listed in alphabetical order in Appendix 1 are as follows conventional mathematical functions ABS ACOS ASIN ATAN COS COSH EXP MAX MIN SIN SINH SQRT TAN TANH which have their Fortran significances except that it should be noted that MAX and MIN operate on real values thereby replacing the AMAX1 and AMIN1 of Fortran LOGE which stands for the natural logarithm with base e LOG10 which stands for the Napierian logarithm with base 10 0 Special mathematical function ARR for calculation of Arrhenius value from next formulae EXP arg1 R T where R is the universal gas constant 8314 31 T is the absolute temperature and arg1 is a formula The mathematical functions can be used in any In Form statements In general case their arguments can be the formulas containing constants and names of any variables The examples of use are specified in Appendix 1 In Form functions formula name functions POL2 POL3 POL4 POL5 POL6 which signify that a polynomial of the appropriate order is to be used These functions are used usually for calculation stored or solved variables in STORED or PROPERTY statements Syntax of POLx POLx is followed by a bracket then the name of the variable in question then a comma or ampersand i e or followed by the requisite number separated by further commas or ampersands Thus POL2 x a0 a1 a2 signifies a0 x a1 x a2 POL3 x a0 a1 a2 a3 signifies a0 x a1 x a2 x a3 POL4 x a0 a1 a2 a3 a4 signifies a0 x a1 x a2 x a3 x a4 POL5 x a0 a1 a2 a3 a4 a5 signifies a0 x a1 x a2 x a3 x a4 x a5 POL6 x a0 a1 a2 a3 a4 a5 a6 signifies a0 x a1 x a2 x a3 x a4 x a5 x a6 where x may be a constant or a solved stored variable but a0 a1 a2 etc must be constants Examples of use of polynomial functions are to be found in library cases 701 for POL3 345 for POL4 089 for POL6 and POL5 whereby it may be remarked that the practice of case 345 in which the coefficients are declared and allocated before being placed as symbols in the In Form statement is preferable from the view point of clarity PWL3 which signifies a piece wise linear function with three parts Syntax of PWL3 The syntax of PWL3 is similarly constructed Thus PWL3 is followed by a bracket then the name of the variable in question followed by a comma or an ampersand Thereafter follow pairs of numbers of which the first is the abscissa and the second is the corresponding ordinate thus PWL3 x x0 y0 x1 y1 x2 y2 x3 y3 represents y as a function of x consisting of straight lines which pass through x0 y0 x1 y1 x2 y3 x3 y3 Here x may be a constant or a solved stored variable and x0 y0 x1 y1 x2 y2 x3 y3 are constants Library case 763 illustrates the use of this function It enables different formulae for setting each of four fluid properties to be compared It calculates the four relevant properties of the fluid ethylene glycol in four different ways namely by way of the polynomial formulae in macro 089 three part patchwise linear formulae five point cubic spline formulae and multi part piece wise linear formulae SPL5 which signifies a cubic interpolation spline function passing through five points Syntax of SPL5 The syntax of SPL5 is similarly constructed Thus SPL5 is followed by a bracket then the name of the variable in question followed by a comma or an ampersand Thereafter follow pairs of numbers of which the first is the abscissa and the second is the corresponding ordinate thus SPL5 x x0 y0 x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 represents y as a spline function of x which passes through x0 y0 x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 As in the previous function x may be a constant or a solved stored variable and x0 y0 x1 y1 etc are constants Library case 763 illustrates the use of this function also PWLF which signifies a piece wise linear function of which the defining points are specified in a file Syntax of PWLF PWLF is followed by an opening bracket the name of the file with full path if it is not in the working directory a comma the name of the independent variable and a closing bracket Examples of the use of PWLF can be found in library case 763 in which the property values at a range of values are read from the files denprp for density enuprp for kinematic viscosity cpprp for specific heat and cndprp for thermal conductivity neighbour location functions EAST WEST NORTH SOUTH HIGH LOW and OLD which have meanings which are conventional in PHOENICS They calculate of variable value at neighbouring cell beside appropriate face and at previous time step Syntax of neighbour location functions is conventional in PHOENICS The function name is followed by a bracket then the name of the variable in question Examples of the use of neighbour location functions may be found for EAST in 367 and 368 cases for WEST in 367 368 710 722 and 735 cases for HIGH in 710 722 and 735 cases for LOW in 366 710 722 and 735 cases for OLD in 360 365 368 and 786 cases special functions ALL BOX SPHERE LINE and POINT which can be used in INFOB statements with infob n flag for making geometrical shapes more numerous than the names suggest They will be discussed in Object related functions ALL function sets BOX In Form object which fills the whole boundary box limited by PATCH command connected with current INFOB statement Unlike other this function has not arguments and therefore brackets See more BOX x0 y0 z0 xsize ysize zsize alpha beta theta function sets BOX In Form object which lies in the boundary box limited by PATCH command connected with current INFOB statement The position and size of the box are defined in its nine parameters The syntax of BOX in more details can be found in section 6 4 b SPHERE xc yc zc radius function sets SPHERE In Form object which lies in the boundary box limited by appropriate PATCH The position and size of the sphere are defined in its four parameters The syntax of SPHERE in more details is described in section 6 4 c POINT x y z diam function creates the point sub grid object which lies in the boundary box limited by appropriate PATCH The position and size of the point object are defined in its four parameters The syntax of POINT in more details can be found in section 6 4 a LINE xf yf zf xl yl zl diam function creates the line sub grid object which lies in the boundary box limited by appropriate PATCH The position and size of the line are defined in its seven parameters The syntax of LINE in more details is described in section 6 4 a It is exemplified in core library for BOX in 375 383 754 769 770 783 and 784 cases for SPHERE in 360 765 766 767 768 and 772 cases SUM and SSUM which perform special summation operations The SUM and SSUM functions are used for summation of results of the formula calculation on all cells inside the area of which can be limited by Patch command connected with current In Form statement The result of summation is a real number Therefore as a rule the summation result is assigned to real user defined variable connected with current In Form statement and previously to described by make In form statement The SUM function is operation of the global summation at all IZ slabs inside Patch region Before each fulfillment of sum operation the sum is zeroized and accumulates results of summation on all iz slabs The SSUM function is operation of the slab summation at one current IZ slab lying inside Patch region At each iz slab before performance of SSUM operation the sum is zeroized and accumulates result of summation at current iz slabs only Syntax of SUM SUM is followed by formula with condition where the formula is a expression containing the constants or any variables For example in 786 case the SUM function is used for setting the bulk room temperature where TEM1 is temperature VOL is cell volume and YVLAST XULAST are y x sizes of domain The summation is executed at finish of each iz slab calculation on all domain area and is assigned to user defined variable TBUL MAKE of TBUL is 0 STORE1 TBUL is SUM TEM1 VOL YVLAST XULAST with ZSLFIN In 785 case the SUM function is used for definition the sum of factors on which inlet mass fluxes should be multiplied Here AEAST is east face area of calculation grid and PI RAD1 2 is real inlet area from task PATCH PATCH1 INIVAL 1 1 1 NY 1 NZ 1 1 Primary inlet mass flux coefficient MAKE PRIMF is 1 STORE1 PRIMF at PATCH1 is SUM AEAST PI RAD1 2 w ith INFOB 1 IF ISWEEP LE 2 Secondary inlet mass flux coefficient MAKE SECNF is 1 STORE1 SECNF at PATCH1 is SUM AEAST PI RAD2 2 RAD1 2 w ith INFOB 2 IF ISWEEP LE 2 Here the Patch command with PATCH1 name contained any IZ slabs If the Patch command contained one IZ slab then it was possible to use for these purposes other function SSUM as will be shown below In 781 case the SUM function provide the re calculation of reference residuals for G solved variable The summation is executed on all domain area and is assigned to the PIL real array RESREF STORE1 of RESREF G is SUM VOL GENG 2 RHO1 EPKE G NY NZ In 345 case the SUM function calculate the surface heat flux where TEM1 and TCOLD are temperatures at cell and cold wall accordingly WALL COLD EAST NX NX 1 NY 1 NZ 1 LSTEP STORED of QWAL at COLD is SUM PART TEM1 TCOLD NZ NY with IF I SWEEP EQ LSWEEP The result of summation is assigned the QWAL stored variable which is used hereinafter calculation the bulk heat transfer coefficient However will more economically use for these purposes single user defined variable In 362 case the SUM function calculate the average temperature at each IZ slab where ASUM and TSUM user defined variables are sums at one IZ slab of high cell face areas and the cell temperatures accordingly MAKE TSUM is 0 MAKE ASUM is 0 DO II 1 NZ Start of IZ cycle PATCH PATCH II CELL 1 NX 1 NY II II 1 1 One PATCH per slab Summation of HIGH area for each IZ slab STORE1 ASUM at PATCH II is SUM AHIGH Summation of temperature multiplied by HIGH area for each IZ slab STORE1 TSUM at PATCH II is SUM AHIGH TEM1 Determination of average temperature for each IZ slab and storage inside 3D variable TAVE STORED TAVE at PATCH II is TSUM ASUM ENDDO End of DO loop The calculations are executed within the limits of one IZ slab of which set by Patch command with PATCH II name The DO command of cycle is used for fulfilment of calculation in all domain area Syntax of SSUM The syntax of SSUM is similarly constructed Thus SSUM is followed by formula with condition where the formula is a expression containing the constants or any variables Example of SSUM use for calculation the average temperature at each IZ slab may be found in 363 case which is identical to 362 PATCH PATCH1 CELL 1 NX 1 NY 1 NZ 1 LSTEP One PATCH per whole domain Summation of HIGH area for each IZ slab STORE1 ASUM at PATCH1 is SSUM AHIGH Summation of temperature multiplied by HIGH area for each IZ slab STORE1 TSUM at PATCH1 is SSUM AHIGH TEM1 Determination of average temperature for each IZ slab and storage inside 3D variable TSLB STORED TAVE at PATCH1 is TSUM ASUM The use of SSUM function permits to avoid the creation iz slab loop as in 362 case Here the summation operations of are executed consistently for each iz slab for calculation of the total area total temperature multiplied by high area and average temperature AECO AWCO ANCO ASCO AHCO ALCO APCO GAMM RESI SORC and CORR which can used for extraction of some terms in the finite volume equations according to function name AECO AWCO ANCO ASCO AHCO ALCO APCO functions get coefficients of the finite volume equation See more RESI function stores residuals See more CORR function extracts corrections See more GAMM function gets exchange coefficients See more SORC function gets sources values These functions have one argument Namely name of solved variable They are used in the STORED statements section 9 3 in more detail Examples of the use may be found for AECO AWCO ANCO ASCO APCO in 703 and 788 cases for AHCO ALCO in 788 case for RESI in 249 768 and 788 cases for CORR in 064 249 and 768 cases for GAMM in 703 and 788 cases COVAL which can used for linearized source setting in the SOURCE statement The functions has two arguments which in general case can be the formulas for calculation coefficient and value Examples of the use may be found in 737 745 746 747 774 and 789 core library cases In more details it will be discussed and exemplified in section 7 1 and section 7 6 POS OFFSET which can used for mofor purposes in the MOVOB In Form statement POS Xpos Ypos Zpos Xang Yang Zang function describes the coordinates of position of moving object Its six parameters can be formulas If they are functions of the TIM current time then the object will be moved during time steps Examples of the use may be found in 360 365 369 370 371 372 373 376 378 379 380 381 and 382 core library cases More details see section 8 OFFSET Xorigin Yorigin Zorigin function declares the frames of reference of object related coordinate systems In general case its three parameters can be formulas also Examples of the use may be found in 381 and 382 core library cases See more d Operands See also Appendix 4 The foregoing operators and functions can be associated with the following operands Names of stored and solved variables The NAME of any SOLVEd or STOREd variable e g P1 H2 KE ENUL or RH01 may be an operand If no square or curly brackets follow the name of the variable the value prevailing at the current cell is intended However neighbour cell or more distant cell values may be referred to either

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  • MOFOR: MOving Frames Of Reference
    object to which property values and sources should be applied and which cells if the PARSOL method is to be used are partially filled by object material and which by outside object material It is necessary to retain in use both methods because objects such as human beings can be described only by way of facets as shown here objects supplied to PHOENICS by CAD packages even though they truly have simple geometrical shapes are often supplied via STL format files which are facet like in structure Nevertheless it must be recognised that where it is usable the words and numbers method is superior Thus it takes hundreds of facets to describe adequately even a simple cylinder and each must requires three Cartesian coordinates to be supplied for each of its four vertices The words and numbers method is much more compact 10 Describing their position and orientation the various frames of reference next back or contents Whichever method is used for describing the shape and size of an object a single method is used for describing its position and orientation relative to the reference frame i e grid which is being used to describe the CFD cells Note In BVH terminology frame is used in the cinematographic sense as a shot or instantaneous state That is not the sense used here This single method has the following elements which will be described in terms of the bounding box concept The facet wise description of an object lists the x y and z coordinates of each of the vertices of each facet in the Cartesian grid having one corner of the box defined as the origin and the edges which meet there as its own coordinate axes The words and numbers method equally must define its shape and size relative to a Cartesian coordinate system of its own Therefore the position of the object in the CFD grid space begins by stating the three numbers which define the position of the object s origin relative to the origin of the CFD grid Those three numbers concern the imagined translation of the object from the CFD grid origin to its actual position but how it is there oriented has to be defined by way of three further numbers which signify the rotations which it has undergone These rotations are sometimes called the Euler angles 11 Articulated objects the MOF format next back or contents Were all objects to move independently of one another motion could be described by listing the values of the three translations and three rotations of the object s reference frame at successive instants of time However many objects are connected together so that the movement of one enforces some movement of the another Such objects are here called articulated A mechanical example is the crank connecting rod piston trio A human being whose limbs are connected at joints elbow knee etc is also articulated It is this connectedness that the MOF format expresses by way of a hierarchy whereby the torso say is regarded as the root element the upper arms are subservient to it but have certain degrees of freedom called channels for an unclear reason relative to it the lower arm is subservient to the upper but can move independently so far as the elbow allows the hand is subservient to the lower arm because it must remain joined at the wrist the fingers and so to the fingertips The MOF data file which describes a set of connected objects therefore has to convey information about the hierarchy what is connected to what and about the offset of each dependent element also called child from its next superior in the hierarchy also called parent It also indicates in what way each object can move Finally it states for a succession of times what have been the movements in each of the degrees of freedom 12 Describing their motion next back or contents At this point it is best to inspect a simple example namely that pertaining to a particular crank conn rod piston motion The information about the motion is conveyed in this MOF file which contains explanatory comments after A further explanation of the file is supplied here This file if it exists is read by EARTH and the information is placed in the appropriate locations in memory at the start of the run Alternatively the information is supplied to the same locations by way of In Form formulae expressed via SPEDATs and conveyed via EARDAT It should be clearly understood that the time steps used in the CFD calculation are not necessarily the same as those in the MOF description They may be larger or smaller The necessary interpolation is performed inside EARTH 13 Effects on the fluid velocity next back or contents It is necessary to convey to the solver module that the velocity at velocity grid nodes lying within the moving object is that prescribed through the MOF file or the In Form formulae for that location This is performed by introducing appropriate momentum sources at the relevant nodes It is of course necessary to determine for each time step which cells are inside which body because more than one moving body may be present as was illustrated by case 360 The necessary coding for this exists inside EARTH It requires no user intervention If the computational grid which is of course itself a frame of reference is in motion relative to a surrounding fluid and if its boundaries are open as when a grid which is fixed relative to a tennis ball travels with it through the air then its motion must be expressed by suitable in flow and out flow sources at the grid boundaries 14 Effects on the fluid acceleration next back or contents If the computational grid is in accelerating motion as when it is constrained to follow an accelerating rocket so as to simulate the launch aerodynamics closely the effect must be simulated by supplying corresponding body forces at all velocity cells which lie within the fluid The necessary coding is provided in EARTH The values of the velocity and acceleration pertaining to each time instant and each time interval are obtained by differentiation of the position and rotation data contained in the MOF data store in EARTH 15 Other effects next back or contents When scalar variables such as temperature or concentration are being solved it is necessary to introduce further sources and sinks into the relevant finite volume equations What needs to be done is readily recognised by consideration of the rectilinear motion of a heated but non conducting solid which pushes colder non conducting fluid ahead of it and is followed by a plug of the same fluid Obviously the temperature profiles at two successive times ought to be as shown here direction earlier later of movement Profiles approximating to these will be produced by PHOENICS if the built in temperature solution procedure is activated but smearing will occur by reason of false diffusion and when specific heats are not uniform conservation of energy is not assured To achieve what is required appropriate corrections to the finite volume equations must be made in order to enforce correctness 16 Implementation in PHOENICS The Input File library next back or contents Click here to learn about the MOFOR cases in the Input File Library 17 A typical Q1 file next back or contents Inspection of library case v112 htm as an example will show the conventional statement about the objects comprising the block and its tip each with the essential MOFOR attribute of domain material MATERIAL 1 the useful but not obligatory storage of OBID the object identifier the spedat line which tells the solver which MOF file to use the conventional statement about the objects comprising the block and its tip and finally a non obligatory but sometimes useful constant VELCON which controls the stiffness of the connexion between body motion and fluid motion There is thus very little which needs to be placed in the Q1 file to initiate a MOFOR calculation 18 The associated MOF file next back or contents The MOF file associated with the just examined Q1 is to be seen here Its connexion with the Q1 is made by the presence in it of the words BLOCK and TIP The MOTION table at the bottom of the file shows only two positions for the beginning and the end it is the value of LSTEP set by PHOENICS which breaks the time into 20 intervals It may be observed that both the degrees of freedom i e channels are rotations 19 The description of faceted objects next back or contents The facetdat file associated with the above example is shown here Inspection shows that it corresponds to the initial position of the objects Its use for mofor has therefore involved no enlargement or complication 20 The associated results next back or contents An animated PHOTON display of the results of the computation can be seen here Contours of OBID the object identifier are used for showing where the two moving objects are at each instant A second animation in which the VR Viewer is the displaying medium is shown here This shows the boundaries of the box more correctly Since the macro using feature of Viewer is rather new it is worth mentioning that it uses precisely the same use file commands as does PHOTON 21 The tutorials next back or contents In order to enable PHOENICS users quickly to learn how to use MOFOR a set of tutorials has been provided They can be accessed by clicking here and here 22 Next steps Thermal effects next back or contents As mentioned above further steps must be taken to ensure energy conservation in MOFOR No difficulties of principle are foreseen but there exist many combinations of circumstance to be provided for Careful planning is therefore needed 23 Acceleration effects next back or contents The situation is similar to but perhaps easier than that for thermal effects no difficulties of principle but careful planning and testing are needed Many exemplifications are needed not just to test the method but also to convey to PHOENICS users what a large number of applications can be made 24 Influence of fluid forces on body motion next back or contents In order to be able to couple the motion of the body with the forces upon it the latter must first be accurately calculated This is not difficult in principle because PHOENICS always does calculate the momentum balances on each cell The task is therefore one of using correctly information which is already available rather than that of starting anew A brief indication of what must be done is hinted at below Use of a fixed value patch to set a velocity to what is required is a blunt instrument which achieves it ends but does not report what force was needed Use of a linearised source patch to set the velocity nearly to what is required enables the force necessary to be computed Sensitive use of the linearised source is therefore all that is required 25 Merging with PARSOL next back or contents Until now the development work on PARSOL which has involved almost complete re writing has been kept separate from that on MOFOR Now however both have advanced sufficiently for merging to be desirable and possible It is envisaged that for handling the fluid dynamical and thermal effects of moving bodies more accurately use will be made of the volume and area porosities which have been available from the earliest days of PHOENICS but which were disregarded by PARSOL The merged coding for the faceted bodies will therefore compute these quantities as is indeed already done for In Form described bodies 26 Concluding remarks next back or contents CHAM regards the work done so far as having confirmed the feasibility and economy of the MOFOR approach to the simulation of moving bodies It is certainly much simpler in concept and easier to implement than the approaches of other CFD code vendors who modify the grid at every time step so as to accommodate the moving body Of course simplicity and economy are not the only criteria realism of the predictions is another It will therefore be a major objective of CHAM s further development program to test and where necessary improve the quality of the predictions made by MOFOR Appendix domain movement back or contents A 1 Introduction Attention has been focussed in the above lecture on the motions induced by objects moving through the domain However there is another important class of phenomena in which fluids are caused to move by motions imparted to the domain itself A familiar example is the sloshing of a layer of liquid in a tank which is jolted tilted or caused to oscillate Such motions can also be handled by MOFOR as illustrated by the following file fragments namely part of a q1 which specifies dom mof as the MOF file which defines the movement of the tank the part of dom mof which defines the motion of the joint called DOMAIN as a rotation about the z axis which varies with time A 2 Domain motion without use of a mof file for an infinite fluid at rest The situation considered Let it first be supposed that there exists a large fluid volume at rest and that within it there is present a computational grid which is in motion Let this motion be in the first instance rectilinear and such that the coordinates in absolute space of the origin of the grid are expressible as known functions of time thus xO t yO y zO t A question How must this motion be represented in PHOENICS if the solution is everywhere to accord with the exact solution u1 xO t v1 yO t w1 zO t p1 constant Here the denotes differentiation with respect to time The answer At the boundaries of the grid the velocities must of course be given the prescribed values of xO yO and zO and within the volume the fluid must be subjected to accelerating forces of rho1 xO rho1 yO and rho1 zO per unit volume where stands for the second derivative with respect to time to that the xO vO and zO are the accelerations in the three coordinate directions Since PHOENICS employs finite time steps Dt say the question arises of how xO and xO are to be evaluated and the answer is not obvious because for example the velocity to be used in the boundary conditions could be that at the end of the time interval or the average velocity during the time interval and the same choice exists in relation to the acceleration The correct answer is the choices must be such as produce the exact solution described above which is to say that the must accord with the choices adopted in the finite volume formulations within PHOENICS Trial and error may well be the best mode of investigation When a solid body is present within the grid An empty moving grid which if properly represented has no effect on the fluid is of little interest Suppose however that a solid body exists within the grid and indeed that the grid is moving solely so as to keep pace with the movement of that solid Then the pressure within the grid will not remain constant in reality and if PHOENICS is properly solving the time dependent equations of motion the pressure variations which it predicts will be in accordance with reality Consider for example the case of a sphere of heavy material which initially at rest relative to surrounding air is suddenly released Until its velocity increases sufficiently for the pressure and frictional forces exerted on it to be large compared with the force of gravity the sphere will fall at a velocity w given by w g t where g is the acceleration due to gravity and t is time Around the sphere as time proceeds a region of non uniform pressure and velocity will be predicted and the extent of this region will increase with time However until it extends to the outer boundaries of the grid the above mentioned velocity boundary conditions and within grid body forces will suffice In the wake of the sphere at the outlet boundary of the grid the physically realistic velocity profile is certainly non uniform It would therefore be unrealistic to enforce the yO t boundary condition there It is also unnecessary a uniform pressure boundary condition should suffice The built in PARSOL procedures will represent the shape of the sphere adequately and because the grid and the object move together the cut cell calculation has to be performed only once The built in forces on solid calculation procedure can be expected to function appropriately as all times The magnitude of the calculated forces will increase in proportion to the third power of time for it is proportional velocity squared and velocity is proportional to time if the acceleration is constant Accounting for varying acceleration But of course the acceleration will not remain constant it will be g F sphere mass and with the force F increasing at first in proportion to the t 3 it will soon fall to zero Therefore the acceleration must be calculated at each time step then the velocity at the next time step can be calculated from it Implementation in PHOENICS The prescription of inlet velocity values and of acceleration related body forces is easily accomplished by way of In Form However at the present moment the last interval forces on bodies are not accessible via In Form Once this deficiency is remedied implementation should be quite straightforward It should be remarked that for the falling sphere problem only one quarter of the whole domain needs to be simulated because of the prevailing symmetries A 3 Examples Three cases

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    turbulent provided that the cell is cut by the interface into no more than two sub cells one containing fluid and the other solid Conjugate heat transfer is also correctly computed in such circumstances It is currently a requirement that the information about the geometry of the solid fluid interface should be conveyed to EARTH by means of a FACETDAT file of the kind which can be produced by the

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    1 wfree DZ YVLAST This increases YVLAST in proportion to the step size DZ and also to a user chosen factor yrfac times the solution generated expression 1 w1 ny 1 wfree in which the quantity is the velocity in the cell closest to the north boundary When this velocity aproaches the free stream velocity the enlargement of the width of the grid is much reduced as can be seen from the following table which shows the total enlargement of the grid when this formula is used for core library case 190 yrfac w ratio YVLAST YINLET 0 1 9 419406E 01 1 132596E 00 1 0 9 897953E 01 1 450421E 00 10 0 9 985036E 01 1 944726E 00 100 0 9 998467E 01 2 314851E 00 It is here seen that a thousand fold variation in the user selected parameter yrfac leads to only a two fold variation of total enlargement factor The first of the next two contour diagrams corresponding to yrfac 0 1 reveals that indeed that factor is too small for the contours intersect the upper edge The second for yrfac 100 0 shows that the enlargement is rather excessive for a significant region of the upper region has uniform velocity nevertheless even these extremely different grids produce not very different solutions in the important region close to the wall Allowance for changes in main stream velocity are best accomplished by including an additional multiplier in the YRAT or XRAT formula namely wfree last wfree current This will be illustrated below 4 4 The pressure gradient for IPARAB EQ 1 Gradients of pressure and of free stream velocity This heading serves as a reminder that the two there mentioned quantities are linked by the Bernoulli equation p free 0 5 w free 2 constant Therefore changes in the pressure gradient which provides sources in the w momentum equation must correspond exactly to changes in the free sream velocity which manifests itself as a boundary condition of the w equation The former may be represented by way of In Form thus source of w1 at prgrd is dpdz whereof the in Q1 corresponding lines might be MAKE DPDZ is 0 0 STORE1 DPDZ is rho1 dwfz wfre patch prgrd volume 1 nx 1 ny 1 nz 1 1 source of w1 at prgrd is dpdz What are dwfz and wfre The following in Q1 lines answer MAKE wfre is wfree STORE1 wfre is WFREE dwfz ZZW wherein dwfz must have been declared as a PIL variable to represent the uniform increase of w1 with z As mentioned in section 4 3 the width of the grid should vary with the mainstream velocity Specifically the width velocity product should remain constant The following yrat formula effects this YGRID of YRAT is 1 0 dwfz dz wfre Human error being always to be expected it is wise to check the compatibility of the dpdz and wfre settings The following image provides such a check by supplying a uniform upstream profile and de activating the solid wall boundary condition The free stream velocity is caused by the widening of the grid without diffusional effects to further enlarge the boundary layer to fall linearly from 33 to 27 m s The fact that the contour lines are precisely vertical shows that the wfre formula which controls the upper edge is entirely consistent with the dpdz formula which controls the values below it This satisfactory check lends credibility to the quite different contours which are generated when the non uniform upstream profile and frictional wall boundary condition are reactivated It is shown below and it reveals that the adverse pressure gradient has the effect of causing what is called boundary layer separation i e a zero velocity region close to the wall In the case in question which was derived from input library case 190 the grid expansion factor accounted for both diffusional and velocity reduction effects and was YGRID of YRAT is 1 0 dwfz dz wfre yrfac 1 w1 ny 1 wfree DZ YVLAST The imposed deceleration value dwfz was 15 sec 1 and the arbitrary factor yrfac was 0 5 The vertical inclinations of the contours near the upper boundary show that this suffices to spread the grid into the inviscid region but not too far for accuracy The Parelliptic problem An important practical use of the parabolic solution procedure is to refine an elliptic flow solution of the region outside the boundary layer In this case it is from the elliptic flow solution that the pressure must be extracted pressures for the nearest to surface cells of the elliptic grid are transferred to all the cells in the parabolic grid at the same z location In general because the cells of the parabolic and elliptic grids are likely to be of different sizes as shown below interpolation is needed In such cases the values of dpdz and dwfz extracted by way of interpolation will vary with the distance z along the surface and will appear as tables of numbers In Form allows these to be input into the calculation by way of the PWL piece wise linear functions but care is necessary in formulating them to ensure that the two functions are wholly consistent The above sketch containers a reminder that a two way interchange of information may take place between the elliptic and parabolic calculations Thus the elliptic calculation may take place at first with the assumption that friction at the solid surfaces is absent Its predicted pressure distribution is for that reason not quite correct However the ensuing parabolic calculation takes detailed account of friction and can report the so called displacement thickness of the boundary layer If this is transmitted back to the elliptic solver that can repeat its calculation on the assumption that the effective size ot the solid object is larger than it first supposed Its second flow prediction will be correspondingly more accurate Iterative exchanges between the two solvers can

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  • about_pc.htm
    is created by phoenics d pc htms english pc pages standard htm the new user panel is created by phoenics d pc htms english pc pages newuser htm the about PHOENICS panel is created by phoenics d pc htms english guidance learn htm and so on Extending the Commander therefore implies adding content to existing html files and or creating new html files Warning These files can be safely edited only by those who are conversant with the writing of HTML files and it is to those that the remainder of this document is addressed in the form of Instructions and warnings to creators of new PHOENICS Commander panels as follows You are free to provide as much material as you wish written in valid html including links to gifs except animated ones and htms However the Commander is not as tolerant of departures from the rules as are some browsers Moreover back slashes should not appear anywhere and it is unsafe to comment out lines by use of the conventional HTML tags and because these have special significance as explained below Buttons and tabs are created by the introduction of an unordered list by means of the ul tag and inclusion of a colon in the first space following the li Then coding in phoenics d pc tcls newstyle tcl will create either a corresponding left hand margin button or a bottom margin tab Buttons are created if the colon is followed by icon name gif Tabs are created if the colon is followed by addtab Format of lines with colons A line having a colon dictates the appearance and function of the buttons or tabs which appear on the screen as follows The characters to the left of the colon appear as button or tab names Other features are defined between the html comment delimiters and they indicate the icon in the case of a button the action which will ensue after pressing the button the file to be displayed in the case of a tab The first line of words following appears as hover help for a button The type of action viz exhibit file in browser or Word display another panel edit a file or run an executable is conveyed by the choice of icon for buttons There are some special icon names which produce other effects The Commander is very clever at finding files which are somewhere in the PHOENICS system It is therefore possible to insert a file name without its full path as for example in addtab howtotut which finds the file howto htm wherever it is But what if there are two files of the same name within the PHOENICS system Then one cannot be sure as to which one the Commander will find and present Example to be found in phoenics d pc htms english quickst quickst htm li Sphere execsev gif input file library case 805 will create a button labelled with the name Sphere and the icon When the

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  • PIL: The PHOENICS Input Language
    change to a polar grid It is asked only if the grid is cartesian ie not polar already The signs appear so as to indicate that what follows is to be read by SATELLITE even though it does not start in the first or second column This enables indentation to be used so as to clarify the nested logical structure IF ANS EQ Y THEN CARTES F GOTO DISPLAY ENDIF ENDIF If the answer is Y the PIL variable CARTES is set FALSE then the data settings are displayed again Otherwise control proceeds to the next question IF NOT BLOCK THEN MESG introduce blockage Y n READVDU ANS CHAR Y IF ANS EQ Y THEN BLOCK T GOTO DISPLAY ENDIF ENDIF This concerns evidently whether a blockage is to be placed inside the box If it is the logical variable BLOCK is set to TRUE MESG change dimensions of box Y n READVDU ANS CHAR Y IF ANS EQ Y THEN The user is now given the opportunity to change the enclosure dimensions MESG x dimension is xulast New value MESG in metres if cartesian otherwise radians READVDU XULAST REAL XULAST Taking this opportunity enables the user to set new values of XULAST in metres or radians according to the value of CARTES MESG y dimension is yvlast New value READVDU YVLAST REAL YVLAST and then YVLAST MESG z dimension is zwlast New value READVDU ZWLAST REAL ZWLAST GOTO DISPLAY and then ZWLAST IF NOT CARTES THEN MESG Inner radius rinner New value READVDU RINNER REAL RINNER ENDIF and finally if CARTES TRUE RINNER ENDIF ENDIF LABEL PROCEED Here is the PROCEED label after which no further questions are posed IF BLOCK THEN STORE PRPS PATCH BLOCK INIVAL NX 4 1 3 NX 4 NY 4 1 3 NY 4 NZ 4 1 3 NZ 4 1 1 COVAL BLOCK PRPS 0 0 100 0 ENDIF However consequences of the menu made settings are of course to be found Here it is seen how if and only if BLOCK is TRUE the PRPS value is stored and then given a solid value in a central portion of the enclosure It should be remarked that no questions are asked about the size or location of the blockage which are therefore fixed by the PATCH arguments NX 4 1 3 NX 4 NY 4 1 3 NY 4 NZ 4 1 3 NZ 4 Of course the menu options could easily be extended by declaring integer variables indicating what cells of the grid the solid was to occupy and providing appropriate question and answer sequences Whether a user prefers to make this extension or simply to edit the default values is a matter of choice to be made by reference to the supposed skills of those for whom the Q1 is being provided and the frequency with which changes of data settings are likely to be desired b Calling in complete sets of data The setting of individual values may become rather

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  • PRELUDE, the Relational-Input Module for PHOENICS
    can if they wish modify the data on its way to the solver by way of the VR Editor 2 What PRELUDE does 2 1 A list of main features PRELUDE processes input data provided by one or more of the following a single instance Q1 file a multiple instance Q3 file a script containing human editable commands input especially of formulae expressing data inter relationships provided interactively by the user dat files defining shapes ac files created by AC3D stl files pob files This list will be extended to include all file formats required by users PRELUDE allows relationships to be typed into boxes whereafter they are acted upon instantly This is probably the most important of all the features of PRELUDE PRELUDE can record the results of an interactive input session in the form of a Q3 file which because it preserves the algebraically expressed relationships between the data elements can be used for the creation of an unlimited number of individual instance simulations PRELUDE has undo capabilities of the recall previous box contents kind PRELUDE is capable of summoning not hitherto existing shapes by passing parameters to Shapemaker which generates them as required It will thus enable the contents of the OBJECTS folders of PHOENICS to be greatly reduced being indeed limited to those shapes e g man or horse which Shapemaker cannot yet make PRELUDE connects its objects in a tree with parent child relationships PRELUDE has visual display features which are not yet possessed by the VR Editor or Viewer Notable among these are the clipper which enables which parts of the scene to be made invisible the texturing feature control of eye position and field of view via script and attribute boxes as well as by mouse movement PRELUDE can launch a series of

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    0 008 meters Further it has approximately twice this value near the convex corners and it becomes zero in the concave corners Contours of auxiliary quantities used in the fluid flow calculation The flow field was calculated by means of the LVEL turbulence model which makes use of the wall distance field which is also derived from the LTLS distribution The contours of WDIS are displayed in Fig 16 which exhibits the expected maximum of 0 004 between the parallel horizontal walls and a somewhat greater value near the cavity where the true distance from the wall depends on the direction in which it is measured LVEL like IMMERSOL is a heuristic model by which is meant that it is incapable of rigorous justification but is nonetheless useful WDIS is calculated once for all at the start of the computation From it and from the developing velocity distribution the evolving distribution of ENUT the effective turbulent viscosity is derived The resulting contours of ENUT are shown in Fig 17 Since the laminar viscosity is of the order of 1 e 5 m 2 s it is evident that turbulence raises the effective value far from the walls by an order of magnitude 3 The mathematics of the method a Similarities between the equations for displacement and velocity The similarities referred to in section 1 d are here decribed for only one cartesian direction but they prevail for all three directions The x direction displacement U obeys the equation del 2 U d dx D C1 Te C3 Fx C2 0 where Te local temperature measured above that of the un stressed solid in the zero displacement condition multiplied by the thermal expansion coefficient D d dx U d dy V d dz W which is called the dilatation Fx external force per unit volume in x direction V and W displacements in y and z directions C1 C2 and C3 are functions of Young s modulus and Poisson s ratio When the viscosity is uniform and the Reynolds number is low so that convection effects are negligible the x direction velocity u obeys the equation del 2 u d dx p c1 fx c2 0 where p pressure fx external force per unit volume in x direction c1 c2 the reciprocal of the viscosity Notes The two equations are here set one below the other so that they can be easily compared del 2 U d dx D C1 Te C3 Fx C2 0 del 2 u d dx p c1 fx c2 0 The equations can thus be seen to become identical if p c1 D C1 Te C3 which implies D p c1 Te C3 C1 and fx c2 Fx C2 The expressions for C1 C2 and C3 are C1 1 1 2 PR C2 2 1 PR YM where PR Poisson s Ratio and YM Young s Modulus and C3 2 1 PR 1 2 PR A solution procedure designed for computing velocities will therefore in fact

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