# 3.3 Test 6: Fluid Mechanics Problem

## 3.3.1 Description

Figure 3.9: Square cavity with free wall

The domain is a unit square with boundary conditions as follows: on the boundary

and on the remaining boundaries

We solve the Navier-Stokes problem with Stokes initialization using module NSQ2CA [59].

• Time-step: 0.15
• Number of iterations: 47
• Save solution every 20 iterations
• CPU time on Multics for 50 iterations with pressure calculation: 9 min 26 s.

Aim and limitations of module NSQ2CA:

• The aim of module NSQ2CA is to solve the two-dimensional Navier-Stokes equations for a viscous incompressible fluid.
• The problem data must be time-dependent.
• The volume forces are assumed zero.
• The following algorithm is used:
• The characteristic method is used for time discretization.
• A variational formulation with a zero divergence base is used in solving the linear problem resulting from this discretization.
• Discretization by Q2 straight finite elements for the velocity, and P1 completely discontinuous elements for the pressure.
• Remark: The stationary Navier-Stokes problem is the limit (i.e. ) of the evolutionary problem, and can be solved by this module. In this case, we begin iterating with a large time-step which we then refine towards the end.

The mesh consists of 64 regular quadrangles (see figure 3.10).

Figure 3.10: Suggested mesh

## 3.3.2 The continuous problem

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

is the velocity vector ,
is the pressure,
is the velocity on the boundary,
is the kinematic viscosity of the fluid,
where
is the Reynold's number, and
are the characteristic velocity and length,

is the initial velocity.

Set Re = 100

## 3.3.3 Execution of the test

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.

## 3.3.4 Convergence

The solution obtained at iterations 20, 40, and finally at 7.05 seconds, are:

 Relative Iteration N 20 Iteration N 40 Iteration N 47 Error Time = 3 sec Time = 6 sec Time = 7.05 sec L L L

The profile matrix occupies 933 words .

## 3.3.5 Results

The output of the 3 D.S. B, at iteration 47 is stored in library TESD under the following names:

S6VITE.D
output of the velocity
S6PRES.D
output of the P1 pressure
S6PRQ2.D
output of the Q2 pressure

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
```

## 3.3.6 Post-processing

The procedures for plotting the velocities, isobars, and streamlines, are described below:

1. Plot of the velocities

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)

2. Plot of the isobars

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)

3. Plot of the streamlines

The streamlines are calculated using the velocity values by solving the following system:

where

is the streamline function,
is the velocity, and
is the tangent.

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

Figure 3.13: Streamlines

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