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<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 3.2 Final//FR"><!-- Converted with LaTeX2HTML 95.1 (Fri Jan 20 1995) --><!-- by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds --><!-- Modified Simulog 03/97 --><HTML><HEAD><TITLE>3.3 Test 6: Fluid Mechanics Problem</TITLE><LINK REL=STYLESHEET TYPE="text/css" HREF="./Modulef.css" TITLE="Modulef CSS"><meta name="description" value="3.3 Test 6: Fluid Mechanics Problem"><meta name="keywords" value="Guide1"><meta name="resource-type" value="document"><meta name="distribution" value="global"></HEAD><BODY BGCOLOR="#FFFFFF"><P> <IMG SRC="../icons/smallmod.gif" WIDTH=211 HEIGHT=50 ALIGN=BOTTOM ALT="Modulef"><A NAME=tex2html479 HREF="node23.html"><IMG BORDER=0 ALIGN=BOTTOM SRC="../icons/previous_motif.gif" ALT="previous"></A><A NAME=tex2html483 HREF="node21.html"><IMG BORDER=0 ALIGN=BOTTOM SRC="../icons/up_motif.gif" ALT="up"></A><A NAME=tex2html485 HREF="node25.html"><IMG BORDER=0 ALIGN=BOTTOM SRC="../icons/next_motif.gif" ALT="next"></A><A NAME=tex2html487 HREF="node2.html"><IMG BORDER=0 ALIGN=BOTTOM SRC="../icons/contents_motif.gif" ALT="contents"></A><A HREF="../Guide1-18/node24.html"><IMG BORDER=0 SRC="../icons/zoom18.gif" ALIGN=BOTTOM ALT="[BIG]"></A><A HREF="../Guide1-14/node24.html"><IMG BORDER=0 SRC="../icons/zoom14.gif" ALIGN=BOTTOM ALT="[Normal]"></A><A HREF="../Guide1-10/node24.html"><IMG BORDER=0 SRC="../icons/zoom10.gif" ALIGN=BOTTOM ALT="[small]"></A><BR><B> Next: </B> <A NAME=tex2html486 HREF="node25.html">Part III: Installation</A><B>Up: </B> <A NAME=tex2html484 HREF="node21.html">3 Batch tests</A><B> Prev: </B> <A NAME=tex2html480 HREF="node23.html">3.2 Test 5: Elasticity Problem</A><B><A HREF="node2.html" >Contents</A></B><HR SIZE=3 WIDTH="75%"><H1><A NAME=SECTION04330000000000000000>3.3 Test 6: Fluid Mechanics Problem</A></H1><P><P><P><H2><A NAME=SECTION04331000000000000000>3.3.1 Description</A></H2><P><P><P><P><A NAME=3064> </A><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img234.gif"><BR><STRONG>Figure 3.9:</STRONG> <i> Square cavity with free wall</i><A NAME=figtes6> </A><BR><P><P>The domain <IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img11.gif"> is a unit square with boundary conditions as follows: on the boundary<P><P><P><DIV ALIGN=center><IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img235.gif"></DIV> and on the remaining boundaries<DIV ALIGN=center><IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img236.gif"></DIV><P>We solve the Navier-Stokes problem with Stokes initialization using module <b> NSQ2CA</b> [<A HREF="node65.html#mod_59">59</A>].<UL><LI> Time-step: 0.15 <LI> Number of iterations: 47 <LI> Save solution every 20 iterations <LI> CPU time on Multics for 50 iterations with pressure calculation: 9 min 26 s.</UL><P>Aim and limitations of module <b> NSQ2CA</b>:<UL><LI> The aim of module <b> NSQ2CA</b> is to solve the two-dimensional Navier-Stokes equations for a viscous incompressible fluid. <LI> The problem data must be time-dependent. <LI> The volume forces are assumed zero. <LI> The following algorithm is used: <UL><LI> The characteristic method is used for time discretization. <LI> A variational formulation with a zero divergence base is used in solving the linear problem resulting from this discretization. <LI> Discretization by Q2 straight finite elements for the velocity, and P1 completely discontinuous elements for the pressure. </UL> <LI> <b> Remark:</b> The stationary Navier-Stokes problem is the limit (i.e. <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img237.gif">) of the evolutionary problem, and can be solved by this module. In this case, we begin iterating with a largetime-step which we then refine towards the end.</UL><P>The mesh consists of 64 regular quadrangles (see figure <A HREF="node24.html#fig20">3.10</A>).<P><P><A NAME=3073> </A><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img238.gif"><BR><STRONG>Figure 3.10:</STRONG> <i> Suggested mesh</i><A NAME=fig20> </A><BR><P><P><P><P><H2><A NAME=SECTION04332000000000000000>3.3.2 The continuous problem</A></H2><P><P><P>Let <IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img11.gif"> be a region in <IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img239.gif"> (<b>N = 2</b> in this case) with boundary <IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img240.gif">, then the flow of a viscous fluid in <IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img11.gif"> is governed by the Navier Stokes equations:<P><DIV ALIGN=center><IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img241.gif"></DIV><P>where<DL COMPACT><DT><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img242.gif"><DD> is the velocity vector <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img243.gif">, <DT><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img244.gif"><DD> is the pressure, <DT><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img245.gif"><DD> is the velocity on the boundary, <DT><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img246.gif"><DD> is the kinematic viscosity of the fluid, <DIV ALIGN=center><IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img247.gif"></DIV> where <DL COMPACT><DT><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img248.gif"><DD> is the Reynold's number, and <DT><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img249.gif"><DD> are the characteristic velocity and length,<P> </DL> <DT><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img250.gif"><DD> is the initial velocity.<P> </DL><P>Set <b>Re = 100</b><P><P><P><H2><A NAME=SECTION04333000000000000000>3.3.3 Execution of the test</A></H2><P><P><P><b> Step 1:</b> Creation of Mesh<P><TABLE COLS=3 RULES=GROUPS><COL ALIGN=LEFT><COL ALIGN=LEFT><COL ALIGN=LEFT><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP>Execute program: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> D6NOXX</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Modules used: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> APNOPO</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> ... 2D mesh generator [<A HREF="node65.html#mod_104"><A NAME=tex2html37 HREF="../Guide3/welcome.html">MODULEF User Guide - 3</A></A>] </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> QUACOO</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> ... mesh using quadrangles </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> ADPNOP</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> ... addition of nodes </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> GIBBS</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> ... renumbering of nodes </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Libraries used: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP COLSPAN=2> <b> NOP2</b>, <b> NOPO</b>, <b> UTSD</b>, <b> UTII</b></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Input data file: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> D6NOPO.D</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Output data structure: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> T6NOPO</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD></TR></TABLE><P><P><P><b> Step 2:</b> Solution of problem<P><TABLE COLS=3 RULES=GROUPS><COL ALIGN=LEFT><COL ALIGN=LEFT><COL ALIGN=JUSTIFY WIDTH="0.33"><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP>Execute program: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> D6NSXX</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Modules used: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> NSQ2CA</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"> ... solve Navier-Stokes equations </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> PRP1Q2</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"> ... calculate the Q2 pressures </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Libraries used: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP COLSPAN=2> <b> FLUI</b> <b> RESD</b>, <b> UTSD</b>, <b> UTII</b></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Output data structures: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> T6VITE</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"> ... contains 3 arrays corresponding to the velocity at iterations 20, 40 and at the solution, </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> T6PRES</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"> ... contains the P1 discontinuous pressure at iterations 20, 40 and at the final iteration, and </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> T6PRQ2</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"> ... contains the Q2 pressure (pressure values at the 9 nodes of the quadrangle). </TD></TR></TABLE><P>The user functions <b> FX</b> and <b> FY</b> represents the velocity as a function of <b>x</b> and <b>y</b> and of the reference numberson the boundary of the domain. These functions are found at the end of program <b> D6NSXX</b> stored in file<b> D6NSXX.F</b>.<P><P><P><H2><A NAME=SECTION04334000000000000000>3.3.4 Convergence</A></H2><P><P><P>The solution obtained at iterations 20, 40, and finally at 7.05 seconds, are:<P><P><P><TABLE COLS=4 BORDER FRAME=BOX RULES=GROUPS><COLGROUP><COL ALIGN=CENTER><COLGROUP><COL ALIGN=RIGHT><COLGROUP><COL ALIGN=RIGHT><COLGROUP><COL ALIGN=RIGHT><TBODY><TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>Relative </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> Iteration N<IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img251.gif"> 20 </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> Iteration N<IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img251.gif"> 40 </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> Iteration N<IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img251.gif"> 47 </TD></TR><TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> Error </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> Time = 3 sec </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> Time = 6 sec</TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> Time = 7.05 sec </TD></TR></TBODY><TBODY><TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP></TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> </TD></TR><TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> L<IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img252.gif"> </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img253.gif"> </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img254.gif"> </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img255.gif"> </TD></TR><TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> L<IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img256.gif"> </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img257.gif"> </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img258.gif"> </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img259.gif"> </TD></TR><TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> L<IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img260.gif"> </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img261.gif"> </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img262.gif"> </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img263.gif"> </TD></TR><TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> </TD></TR></TBODY></TABLE><P><P><P>The profile matrix occupies 933 words .<P><P><P><H2><A NAME=SECTION04335000000000000000>3.3.5 Results</A></H2><P><P><P>The output of the 3 D.S. <b> B</b>, at iteration 47 is stored in library <b> TESD</b> under the following names:<DL COMPACT><DT>S6VITE.D<DD> output of the velocity <DT>S6PRES.D<DD> output of the P1 pressure <DT>S6PRQ2.D<DD> output of the Q2 pressure<P> </DL><P>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 <b> IMAGXX</b> andspecifying the names of the files containing the desired data structures.<P><P><P><PRE> &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& 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 | --------------------------------------- ---------------------------------------⌨️ 快捷键说明
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