Next: Part III: Installation
Up: 3 Batch tests
Prev: 3.2 Test 5: Elasticity Problem
Contents
Figure 3.9: Square cavity with free wall
The domain is a unit square with boundary conditions as follows: on the boundary
We solve the Navier-Stokes problem with Stokes initialization using module NSQ2CA [59].
Aim and limitations of module NSQ2CA:
The mesh consists of 64 regular quadrangles (see figure 3.10).
Let be a region in
(N = 2 in this case) with boundary
, then the flow of a
viscous fluid in
is governed by the Navier Stokes equations:
where
Set Re = 100
Step 1: Creation of Mesh
Execute program: | D6NOXX | |
Modules used: | APNOPO | ... 2D mesh generator [MODULEF User Guide - 3] |
QUACOO | ... mesh using quadrangles | |
ADPNOP | ... addition of nodes | |
GIBBS | ... renumbering of nodes | |
Libraries used: | NOP2, NOPO, UTSD, UTII | |
Input data file: | D6NOPO.D | |
Output data structure: | T6NOPO |
Step 2: Solution of problem
Execute program: | D6NSXX | |
Modules used: | NSQ2CA | ... solve Navier-Stokes equations |
PRP1Q2 | ... calculate the Q2 pressures | |
Libraries used: | FLUI RESD, UTSD, UTII | |
Output data structures: | T6VITE | ... contains 3 arrays corresponding to the velocity at iterations 20, 40 and at the solution, |
T6PRES | ... contains the P1 discontinuous pressure at iterations 20, 40 and at the final iteration, and | |
T6PRQ2 | ... contains the Q2 pressure (pressure values at the 9 nodes of the quadrangle). |
The user functions FX and FY represents the velocity as a function of x and y and of the reference numbers on the boundary of the domain. These functions are found at the end of program D6NSXX stored in file D6NSXX.F.
The solution obtained at iterations 20, 40, and finally at 7.05 seconds, are:
Relative | Iteration N![]() | Iteration N![]() | Iteration N![]() |
Error | Time = 3 sec | Time = 6 sec | Time = 7.05 sec |
L![]() | ![]() | ![]() | ![]() |
L![]() | ![]() | ![]() | ![]() |
L![]() | ![]() | ![]() | ![]() |
The profile matrix occupies 933 words .
The output of the 3 D.S. B, at iteration 47 is stored in library TESD under the following names:
The remainder of this section contains the partial or total printout of the various D.S. created in the different steps. These data structures can be viewed by calling preprocessor IMAGXX and specifying the names of the files containing the desired data structures.
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& PRINT OUT OF D.S. NOPO OF LEVEL 1 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& TITLE : BENCHMARK MODULEF NUMBER 6 :NAVIER STOKES DATE AND USER'S NAME : 14/09/90 dutoit DATA STRUCTURE TYPE : NOPO LEVEL AND STATE NUMBER : 1 0 NUMBER OF ASSOCIATED TABLES : 0 TABLE N O P 2 -------------- CHARACTERISTICS OF THE MESH : SPACE DIMENSION (NDIM ) : 2 MAXIMUM NUMBER OF REFERENCES (NDSR ) : 2 MAXIMUM NUMBER OF SUB-DOMAINS (NDSD ) : 1 NODES AND POINTS DO NOT COINCIDE (NCOPNP) : 0 NUMBER OF ELEMENTS (NE ) : 64 NUMBER OF QUADRANGLES (NQUA ) : 64 NUMBER OF BOUNDARY ELEMENTS (NEF ) : 28 NUMBER OF NODES (NOE ) : 289 NUMBER OF NODES BY SEGMENT (NO EXTREMITIES) : 1 NUMBER OF INTERNAL NODES : AT EACH QUADRANGLE (ISEQ ) : 1 NUMBER OF POINTS (NP ) : 81 TYPE OF COORDINATE VALUES (NTYCOO) : REEL1MOT MAX DIFFERENCE +1 BETWEEN 2 NODES OF AN ELEMENT : 55 NUMBER OF COARSE ELEMENTS (NBEGM ) : 0 NUMBER OF WORDS FOR TABLE NOP5 (LNOP5 ) : 1404 REFERENCE AXIS X,Y,Z (NTACOO) : 1 TABLE N O P 4 ---------------- COORDINATES OF POINTS ------------------------------------------------------------------------------- | POINT | X | Y | | POINT | X | Y | ------------------------------------------------------------------------------- | 1 | 0.000000 | 0.000000 | | 2 | 0.000000 | 0.250000 | | 3 | 0.000000 | 0.500000 | | 4 | 0.000000 | 0.750000 | | 5 | 0.000000 | 1.00000 | | 6 | 0.250000 | 0.000000 | | 7 | 0.250000 | 0.250000 | | 8 | 0.250000 | 0.500000 | | 9 | 0.250000 | 0.750000 | | 10 | 0.250000 | 1.00000 | --------------------------------------- --------------------------------------- TABLE N O P 5 ---------------- DESCRIPTION OF ELEMENTS | ELEMENT : 1 | QUADRANGLE | S-D : 1 | NUMBER OF STORED WORDS : 9 NUMBER OF NODES : 9 : 1 35 37 5 2 36 6 3 4 NUMBER OF POINTS : 4 : 1 26 66 29 EDGE REFERENCE : 1 0 0 1 VERTEX REFERENCE : 1 1 0 1 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& PRINT OUT OF D.S. B OF LEVEL 1 (T6VITE) &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& TITLE : DATE AND USER'S NAME : 14/09/90 dutoit DATA STRUCTURE TYPE : B LEVEL AND STATE NUMBER : 1 0 NUMBER OF ASSOCIATED TABLES : 0 TABLE B 2 ---------- TYPE OF THE TABLE (NTYT) : 2 NUMBER OF INDICES AND THEIR MAX VALUES (NIND..) : 2 1 578 OPTION (1: PAGES OF SAME SIZE , 0: IF NOT) (NCOD) : 1 NUMBER OF PAGES OF THE TABLE B4 (NBLOC) : 1 NUMBER OF NODES (NOE) : 289 NUMBER OF TABLES B4 IN THIS D.S. (NBBLOC) : 3 CONSTANT NUMBER OF D.O.F. AT EACH NODE OR 0 (ND) : 2 NUMBER OF D.O.F. OR LENGHT OF A B4 PAGE (NTDL) : 578 STORAGE OPTION FOR B4 (NCODSB) : -1 TABLE B 3 ---------- POINTER ON THE LAST ROW OF EACH PAGE OF B4 1 0 2 578 TABLE B4(NDSM,NTDL) NUMBER : 1 ----------------------------------- TABLE B4(NDSM,NTDL) PAGINATED IN 1 OR MORE PAGES ------------------------------------------------ PAGE 1 ---------- ------------------------------------------------------ | NODE | DOF | GLOBAL DOF| LOAD CASE(S) | ------------------------------------------------------ | 1 | 1 | 1 | 0.000000 | 2 | 2 | 0.000000 | 2 | 1 | 3 | 0.000000 | 2 | 4 | 0.000000 | 3 | 1 | 5 | 0.000000 | 2 | 6 | 0.000000 | 4 | 1 | 7 | -0.873772E-03 | 2 | 8 | 0.842329E-03 | 5 | 1 | 9 | 0.000000 | 2 | 10 | 0.000000 TABLE B4(NDSM,NTDL) NUMBER : 2 ----------------------------------- TABLE B4(NDSM,NTDL) PAGINATED IN 1 OR MORE PAGES ------------------------------------------------ PAGE 1 ---------- ------------------------------------------------------ | NODE | DOF | GLOBAL DOF| LOAD CASE(S) | ------------------------------------------------------ | 1 | 1 | 1 | 0.000000 | 2 | 2 | 0.000000 | 2 | 1 | 3 | 0.000000 | 2 | 4 | 0.000000 | 3 | 1 | 5 | 0.000000 | 2 | 6 | 0.000000 | 4 | 1 | 7 | -0.888998E-03 | 2 | 8 | 0.832968E-03 | 5 | 1 | 9 | 0.000000 | 2 | 10 | 0.000000 TABLE B4(NDSM,NTDL) NUMBER : 3 ----------------------------------- TABLE B4(NDSM,NTDL) PAGINATED IN 1 OR MORE PAGES ------------------------------------------------ PAGE 1 ---------- ------------------------------------------------------ | NODE | DOF | GLOBAL DOF| LOAD CASE(S) | ------------------------------------------------------ | 1 | 1 | 1 | 0.000000 | 2 | 2 | 0.000000 | 2 | 1 | 3 | 0.000000 | 2 | 4 | 0.000000 | 3 | 1 | 5 | 0.000000 | 2 | 6 | 0.000000 | 4 | 1 | 7 | -0.983558E-03 | 2 | 8 | 0.929105E-03 | 5 | 1 | 9 | 0.000000 | 2 | 10 | 0.000000 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& PRINT OUT OF D.S. B OF LEVEL 2 (T6PRES) &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& TITLE : DATE AND USER'S NAME : 14/09/90 dutoit DATA STRUCTURE TYPE : B LEVEL AND STATE NUMBER : 2 0 NUMBER OF ASSOCIATED TABLES : 0 TABLE B 2 ---------- TYPE OF THE TABLE (NTYT) : 2 NUMBER OF INDICES AND THEIR MAX VALUES (NIND..) : 2 1 192 OPTION (1: PAGES OF SAME SIZE , 0: IF NOT) (NCOD) : 1 NUMBER OF PAGES OF THE TABLE B4 (NBLOC) : 1 NUMBER OF NODES (NOE) : 64 NUMBER OF TABLES B4 IN THIS D.S. (NBBLOC) : 3 CONSTANT NUMBER OF D.O.F. AT EACH NODE OR 0 (ND) : 3 NUMBER OF D.O.F. OR LENGHT OF A B4 PAGE (NTDL) : 192 STORAGE OPTION FOR B4 (NCODSB) : -1 TABLE B 3 ---------- POINTER ON THE LAST ROW OF EACH PAGE OF B4 1 0 2 192 TABLE B4(NDSM,NTDL) NUMBER : 1 ----------------------------------- TABLE B4(NDSM,NTDL) PAGINATED IN 1 OR MORE PAGES ------------------------------------------------ PAGE 1 ---------- ------------------------------------------------------ | NODE | DOF | GLOBAL DOF| LOAD CASE(S) | ------------------------------------------------------ | 1 | 1 | 1 | -0.830884E-02 | 2 | 2 | 0.843354E-02 | 3 | 3 | 0.000000 | 2 | 1 | 4 | -0.364164E-02 | 2 | 5 | 0.131478E-01 | 3 | 6 | -0.115603E-02 | 3 | 1 | 7 | -0.170311E-01 | 2 | 8 | 0.475959E-02 | 3 | 9 | 0.124966E-02 | 4 | 1 | 10 | -0.344920E-02 TABLE B4(NDSM,NTDL) NUMBER : 2 ----------------------------------- TABLE B4(NDSM,NTDL) PAGINATED IN 1 OR MORE PAGES ------------------------------------------------ PAGE 1 ---------- ------------------------------------------------------ | NODE | DOF | GLOBAL DOF| LOAD CASE(S) | ------------------------------------------------------ | 1 | 1 | 1 | -0.781133E-02 | 2 | 2 | 0.778977E-02 | 3 | 3 | 0.000000 | 2 | 1 | 4 | -0.406640E-02 | 2 | 5 | 0.120007E-01 | 3 | 6 | -0.961857E-03 | 3 | 1 | 7 | -0.160561E-01 | 2 | 8 | 0.915468E-03 | 3 | 9 | 0.164742E-02 | 4 | 1 | 10 | -0.450665E-02 TABLE B4(NDSM,NTDL) NUMBER : 3 ----------------------------------- TABLE B4(NDSM,NTDL) PAGINATED IN 1 OR MORE PAGES ------------------------------------------------ PAGE 1 ---------- ------------------------------------------------------ | NODE | DOF | GLOBAL DOF| LOAD CASE(S) | ------------------------------------------------------ | 1 | 1 | 1 | -0.801085E-02 | 2 | 2 | 0.795096E-02 | 3 | 3 | 0.000000 | 2 | 1 | 4 | -0.321434E-02 | 2 | 5 | 0.123081E-01 | 3 | 6 | -0.109873E-02 | 3 | 1 | 7 | -0.161097E-01 | 2 | 8 | -0.420842E-03 | 3 | 9 | 0.183181E-02 | 4 | 1 | 10 | -0.344618E-02 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& PRINT OUT OF D.S. B OF LEVEL 3 (T6PRQ2) &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& TITLE : DATE AND USER'S NAME : 14/09/90 dutoit DATA STRUCTURE TYPE : B LEVEL AND STATE NUMBER : 3 0 NUMBER OF ASSOCIATED TABLES : 0 TABLE B 2 ---------- TYPE OF THE TABLE (NTYT) : 2 NUMBER OF INDICES AND THEIR MAX VALUES (NIND..) : 2 1 289 OPTION (1: PAGES OF SAME SIZE , 0: IF NOT) (NCOD) : 1 NUMBER OF PAGES OF THE TABLE B4 (NBLOC) : 1 NUMBER OF NODES (NOE) : 289 NUMBER OF TABLES B4 IN THIS D.S. (NBBLOC) : 4 CONSTANT NUMBER OF D.O.F. AT EACH NODE OR 0 (ND) : 1 NUMBER OF D.O.F. OR LENGHT OF A B4 PAGE (NTDL) : 289 STORAGE OPTION FOR B4 (NCODSB) : -1 TABLE B 3 ---------- POINTER ON THE LAST ROW OF EACH PAGE OF B4 1 0 2 289 TABLE B4(NDSM,NTDL) NUMBER : 1 ----------------------------------- TABLE B4(NDSM,NTDL) PAGINATED IN 1 OR MORE PAGES ------------------------------------------------ PAGE 1 ---------- ------------------------------------------------------ | NODE | DOF | GLOBAL DOF| LOAD CASE(S) | ------------------------------------------------------ | 1 | 1 | 1 | 0.000000 | 2 | 1 | 2 | -0.519302E-03 | 3 | 1 | 3 | 0.527096E-03 | 4 | 1 | 4 | 0.779354E-05 | 5 | 1 | 5 | 0.144940E-02 | 6 | 1 | 6 | 0.657526E-03 | 7 | 1 | 7 | 0.214208E-02 | 8 | 1 | 8 | 0.107764E-02 | 9 | 1 | 9 | 0.288912E-02 | 10 | 1 | 10 | 0.145768E-02 TABLE B4(NDSM,NTDL) NUMBER : 2 ----------------------------------- TABLE B4(NDSM,NTDL) PAGINATED IN 1 OR MORE PAGES ------------------------------------------------ PAGE 1 ---------- ------------------------------------------------------ | NODE | DOF | GLOBAL DOF| LOAD CASE(S) | ------------------------------------------------------ | 1 | 1 | 1 | 0.000000 | 2 | 1 | 2 | -0.488208E-03 | 3 | 1 | 3 | 0.486861E-03 | 4 | 1 | 4 | -0.134727E-05 | 5 | 1 | 5 | 0.136779E-02 | 6 | 1 | 6 | 0.621930E-03 | 7 | 1 | 7 | 0.181907E-02 | 8 | 1 | 8 | 0.815563E-03 | 9 | 1 | 9 | 0.225265E-02 | 10 | 1 | 10 | 0.954031E-03 TABLE B4(NDSM,NTDL) NUMBER : 3 ----------------------------------- TABLE B4(NDSM,NTDL) PAGINATED IN 1 OR MORE PAGES ------------------------------------------------ PAGE 1 ---------- ------------------------------------------------------ | NODE | DOF | GLOBAL DOF| LOAD CASE(S) | ------------------------------------------------------ | 1 | 1 | 1 | 0.000000 | 2 | 1 | 2 | -0.500678E-03 | 3 | 1 | 3 | 0.496935E-03 | 4 | 1 | 4 | -0.374275E-05 | 5 | 1 | 5 | 0.138654E-02 | 6 | 1 | 6 | 0.632771E-03 | 7 | 1 | 7 | 0.175291E-02 | 8 | 1 | 8 | 0.746048E-03 | 9 | 1 | 9 | 0.206836E-02 | 10 | 1 | 10 | 0.794842E-03 TABLE B4(NDSM,NTDL) NUMBER : 4 ----------------------------------- TABLE B4(NDSM,NTDL) PAGINATED IN 1 OR MORE PAGES ------------------------------------------------ PAGE 1 ---------- ------------------------------------------------------ | NODE | DOF | GLOBAL DOF| LOAD CASE(S) | ------------------------------------------------------ | 1 | 1 | 1 | 0.000000 | 2 | 1 | 2 | -0.506843E-03 | 3 | 1 | 3 | 0.502850E-03 | 4 | 1 | 4 | -0.399328E-05 | 5 | 1 | 5 | 0.140208E-02 | 6 | 1 | 6 | 0.640137E-03 | 7 | 1 | 7 | 0.177119E-02 | 8 | 1 | 8 | 0.754146E-03 | 9 | 1 | 9 | 0.208460E-02 | 10 | 1 | 10 | 0.801812E-03
The procedures for plotting the velocities, isobars, and streamlines, are described below:
The Q2 velocity values are stored in file T6VITE. The plot of the velocity distribution is generated by the preprocessor TRMCXX [MODULEF User Guide - 6] (see figure 3.11). Before execution of TRMCXX it is necessary to create the data structures MAIL and COOR corresponding to the NOPO data structure for the Q2 mesh.
Step 1:
Call preprocessor: | COMAXX | |
Specify input data file: | D6COMA.D | ... in library TESD |
Output data structures: | T6MAIL | |
T6COOR |
Step 2:
Call preprocessor: | TRMCXX |
Specify input data structures: | T6MAIL |
T6COOR | |
T6VITE |
Figure 3.11: Velocity distribution ( TRMCXX)
The values of the Q2 pressure are stored in file T6PRQ2. A plot of the isobars is generated by preprocessor TRMCXX [MODULEF User Guide - 6] (see figure 3.12). As before, data structures MAIL and COOR are necessary as input in preprocessor TRMCXX.
Call preprocessor: | TRMCXX |
Specify input data structures: | T6MAIL |
T6COOR | |
T6PRQ2 |
Figure 3.12: Isobars ( TRMCXX)
The streamlines are calculated using the velocity values by solving the following system:
where
Remark: For the case where on a section
of the boundary, we may impose that
on
and the preceding Neumann condition on the remainder of the boundary.
A better conditioned system is obtained admitting one unique solution. The system is solved by the same preprocessors as was used in the solution of the thermal problem with a QUAD 2Q2D element:
Step 1: Interpolation [13]
Call preprocessor: | COMAXX | |
Element type: | QUAD 2Q2D | |
Specify input data file: | D6COMA.D | ... in library TESD |
Output data structures: | T6MAIL | |
T6COOR |
Step 2: Description of data [14]
Call preprocessor: | FOMIXX | |
Specify input data file: | D6FOMI.D | ... in library TESD |
Output data structures: | T6FORC | |
T6MILI |
Step 3: Calculation of the element arrays [14]
Execute program: | D6CAXX | ... in library TEST |
Specify input data structures: | T6MAIL | |
T6COOR | ||
T6MILI | ||
T6FORC | ||
T6VITE | ||
Output data structure: | T6TAE |
Step 4: Description of boundary conditions [18]
Call preprocessor: | COBDXX | |
Specify input data file: | D6BDCL.D | ... in library TESD |
Output data structure: | T6BDCL |
Step 5: Solution [MODULEF User Guide - 5]
Call preprocessor: | CHOLXX | |
Specify input data file: | D6B.D | ... in library TESD |
Output data structure: | T6B |
A plot of the streamlines is generated by preprocessor TRMCXX as follows:
Call preprocessor: | TRMCXX |
Specify input data structures: | T6MAIL |
T6COOR | |
T6B |