2D Navier-Stokes Equations - Laminar Flow over Flat Plate
Location: hypar/Examples/2D/NavierStokes2D/FlatPlateLaminar_PETSc_Implicit (This directory contains all the input files needed to run this case. If there is a Run.m, run it in MATLAB to quickly set up, run, and visualize the example).
Governing equations: 2D Euler Equations (navierstokes2d.h)
Reference:
- Hirsch, "Numerical Computation of Internal & External Flows", Volume 1 (Fundamentals of Computational Fluid Dynamics), 2nd Edition, Elsevier, Section 12.3.2 (page 618-625).
Domain: \(-0.25 \le x \le 1\), \(0 \le y \le 0.25\)
Boundary conditions:
- Symmetry BC on \(y=0, -0.25 \le x < 0\) (imposed through "slip-wall" _SLIP_WALL_ with 0 wall velocity).
- No-slip wall BC on \(y=0, 0 \le x \le 1\) (_NOSLIP_WALL_ with 0 wall velocity).
- Subsonic outflow on \(y=0.25\) (_SUBSONIC_OUTFLOW_) with pressure \(p=1/\gamma\).
- Subsonic inflow on \(x=0\) (_SUBSONIC_INFLOW_) with density \(\rho=1\), and velocity \((u,v) = (0.3,0)\).
- Subsonic outflow on \(x=1\) (_SUBSONIC_OUTFLOW_) with pressure \(p=1/\gamma\).
Initial solution: Uniform flow with \(\rho=1, u=0.3, v=0, p=1/\gamma\).
Other parameters:
- \(\gamma = 1.4\) (NavierStokes2D::gamma)
- \(Re = \frac {\rho u L } {\mu} = 100,000\) (Reynolds number) (NavierStokes2D::Re), where \(L=1\) is the plate length .
- \(Pr = 0.72\) (Prandtl number) (NavierStokes2D::Pr)
- \(M_\infty = 0.3\) (freestream Mach number) (NavierStokes2D::Minf)
Note: Pressure is taken as \(1/\gamma\) in the above so that the freestream speed of sound is 1.
Numerical method:
- Spatial discretization (hyperbolic): 5th order WENO (Interp1PrimFifthOrderWENO())
- Spatial discretization (parabolic) : 4th order (FirstDerivativeFourthOrderCentral()) non-conservative 2-stage (ParabolicFunctionNC2Stage())
- Time integration: PETSc (SolvePETSc())
- Method Class: Additive Runge-Kutta method (TSARKIMEX - https://petsc.org/release/docs/manualpages/TS/TSARKIMEX.html) - Although the ARK methods are semi-implicit (IMEX), here they are used in the "fully implicit" mode, i.e., the implicit method is used to solve the complete equation (Note the flag -ts_arkimex_fully_implicit in .petscrc).
- Specific method: ARK2e (TSARKIMEX2E - https://petsc.org/release/docs/manualpages/TS/TSARKIMEX2E.html)
This is a steady state problem - the solution converges to a steady laminar flow over a flat plate, and the residuals decrease with time. The skin friction coefficient is given by
\begin{equation} c_f = \frac {0.664} {\sqrt{Re_x}}, Re_x = \frac {\rho u x} {\mu} \end{equation}
(Blasius solution).
Input files required:
.petscrc
# See PETSc documentation for more details (https://petsc.org/release/overview/).
# Note that if the following are specified in this file, the corresponding inputs in solver.inp are *ignored*.
# + "-ts_dt" (time step size): ignores "dt" in solver.inp
# + "-ts_max_steps" (maximum number of time iterations): ignores "n_iter" in solver.inp
# + "-ts_max_time" (final simulation time): ignores "n_iter" X "dt" in solver.inp
# If these are not specified here, then the values in solver.inp are used.
# Use PETSc time-integration
-use-petscts
# Time integration scheme type - ARK
-ts_type arkimex
-ts_arkimex_type 2e
-ts_arkimex_fully_implicit
# no time-step adaptivity
-ts_adapt_type none
# Nonlinear solver (SNES) type
-snes_type newtonls
# Set relative tolerance
-snes_rtol 1e-4
# Set absolute tolerance
-snes_atol 1e-4
# Set step size tolerance
-snes_stol 1e-16
# Linear solver (KSP) type
-ksp_type gmres
# Set relative tolerance
-ksp_rtol 1e-4
# Set absolute tolerance
-ksp_atol 1e-4
# use a preconditioner for solving the system
-with_pc
# preconditioner type - SOR
-pc_type sor
-pc_sor_omega 1.0
-pc_sor_its 1
# apply right preconditioner
-ksp_pc_side RIGHT
solver.inp
begin
ndims 2
nvars 4
size 226 502
iproc 2 3
ghost 3
n_iter 400
restart_iter 0
time_scheme rk
time_scheme_type 44
hyp_space_scheme weno5
hyp_interp_type components
par_space_type nonconservative-2stage
par_space_scheme 4
dt 0.05
screen_op_iter 1
file_op_iter 10
ip_file_type binary
op_file_format binary
op_overwrite yes
model navierstokes2d
end
boundary.inp
5
slip-wall 1 1 -0.25 0.00 0 0
0.0 0.0
noslip-wall 1 1 0.00 2.00 0 0
0.0 0.0
subsonic-outflow 1 -1 -0.25 2.00 0 0
0.7142857143
subsonic-inflow 0 1 0 0 0.00 0.25
1.0 0.3 0.0
subsonic-outflow 0 -1 0 0 0.00 0.25
0.7142857143
physics.inp
begin
gamma 1.4
upwinding roe
Pr 0.72
Minf 0.3
Re 100000
end
weno.inp (optional)
begin
mapped 0
borges 0
yc 0
no_limiting 1
epsilon 0.000001
p 2.0
rc 0.3
xi 0.001
end
To generate initial.inp (initial solution), compile and run the following code in the run directory.
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <string.h>
double GAMMA = 1.4;
int main()
{
int NI,NJ,ndims;
char ip_file_type[50];
strcpy(ip_file_type,"ascii");
FILE *in;
printf("Reading file \"solver.inp\"...\n");
in= fopen("solver.inp","r");
if (!in) printf("Error: Input file \"solver.inp\" not found. Default values will be used.\n");
else {
char word[500];
fscanf(in,"%s",word);
if (!strcmp(word, "begin")){
while (strcmp(word, "end")){
fscanf(in,"%s",word);
if (!strcmp(word, "ndims")) fscanf(in,"%d",&ndims);
else if (!strcmp(word, "size")) {
fscanf(in,"%d",&NI);
fscanf(in,"%d",&NJ);
} else if (!strcmp(word, "ip_file_type")) fscanf(in,"%s",ip_file_type);
}
} else printf("Error: Illegal format in solver.inp. Crash and burn!\n");
}
fclose(in);
if (ndims != 2) {
printf("ndims is not 2 in solver.inp. this code is to generate 2D solution\n");
return(0);
}
printf("Grid:\t\t\t%d x %d\n",NI,NJ);
double xmin = -0.25;
double xmax = 1.0;
double ymin = 0.0;
double ymax = 0.25;
double Lx = xmax - xmin;
double Ly = ymax - ymin;
int i,j;
double dx = Lx / ((double)(NI-1));
double dy = Ly / ((double)(NJ-1));
double *x, *y, *U;
FILE *out;
x = (double*) calloc (NI , sizeof(double));
y = (double*) calloc (NJ , sizeof(double));
U = (double*) calloc (4*NI*NJ, sizeof(double));
for (i = 0; i < NI; i++){
for (j = 0; j < NJ; j++){
x[i] = xmin + i*dx;
y[j] = ymin + j*dy;
int p = i + NI*j;
double rho, u, v, P;
rho = 1.0;
P = 1.0/1.4;
u = 0.3;
v = 0.0;
U[4*p+0] = rho;
U[4*p+1] = rho*u;
U[4*p+2] = rho*v;
U[4*p+3] = P/(GAMMA-1.0) + 0.5*rho*(u*u+v*v);
}
}
if (!strcmp(ip_file_type,"ascii")) {
printf("Writing ASCII initial solution file initial.inp\n");
out = fopen("initial.inp","w");
for (i = 0; i < NI; i++) fprintf(out,"%1.16E ",x[i]);
fprintf(out,"\n");
for (j = 0; j < NJ; j++) fprintf(out,"%1.16E ",y[j]);
fprintf(out,"\n");
for (j = 0; j < NJ; j++) {
for (i = 0; i < NI; i++) {
int p = i + NI*j;
fprintf(out,"%1.16E ",U[4*p+0]);
}
}
fprintf(out,"\n");
for (j = 0; j < NJ; j++) {
for (i = 0; i < NI; i++) {
int p = i + NI*j;
fprintf(out,"%1.16E ",U[4*p+1]);
}
}
fprintf(out,"\n");
for (j = 0; j < NJ; j++) {
for (i = 0; i < NI; i++) {
int p = i + NI*j;
fprintf(out,"%1.16E ",U[4*p+2]);
}
}
fprintf(out,"\n");
for (j = 0; j < NJ; j++) {
for (i = 0; i < NI; i++) {
int p = i + NI*j;
fprintf(out,"%1.16E ",U[4*p+3]);
}
}
fprintf(out,"\n");
fclose(out);
} else if ((!strcmp(ip_file_type,"binary")) || (!strcmp(ip_file_type,"bin"))) {
printf("Writing binary initial solution file initial.inp\n");
out = fopen("initial.inp","wb");
fwrite(x,sizeof(double),NI,out);
fwrite(y,sizeof(double),NJ,out);
fwrite(U,sizeof(double),4*NI*NJ,out);
fclose(out);
}
free(x);
free(y);
free(U);
return(0);
}
Output:
Note that iproc is set to
2 3
in solver.inp (i.e., 2 processor along x, and 3 processor along y). Thus, this example should be run with 6 MPI ranks (or change iproc).
After running the code, there should be one output file op.bin, since HyPar::op_overwrite is set to yes in solver.inp.
HyPar::op_file_format is set to binary in solver.inp, and thus, all the files are written out in the binary format, see WriteBinary(). The binary file contains the conserved variables \(\left(\rho, \rho u, \rho v, e\right)\). The following two codes are available to convert the binary output file:
- hypar/Extras/BinaryToTecplot.c - convert binary output file to Tecplot file (works only for 2D and 3D).
- hypar/Extras/BinaryToText.c - convert binary output file to an ASCII text file (to visualize in, for example, MATLAB).
The following plot was obtained by converting the binary file to the Tecplot format, and using VisIt to plot it (it shows the density and velocity):
The following plot shows a magnified view of the boundary layer:
The following file computes the skin friction as a function of the Reynolds number and writes to to a text file SkinFriction.dat with 4 columns: Reynolds number, computed skin friction coefficient, exact skin friction coefficient ( \(0.664/\sqrt{Re_x}\)), and normal velocity gradient on the plate surface ( \(\left.\partial u/\partial y\right|_{y=0}\)). Compile and run it in the run directory.
/*
This code calculates the skin friction
as a function of Reynolds number for
the flow over a flat plate.
*/
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
int main()
{
FILE *out, *in, *inputs;
char filename[50], op_file_format[50];
double Re_inf, M_inf;
inputs = fopen("solver.inp","r");
if (!inputs) {
fprintf(stderr,"Error: File \"solver.inp\" not found.\n");
return(1);
} else {
char word[100];
fscanf(inputs,"%s",word);
if (!strcmp(word, "begin")){
while (strcmp(word, "end")){
fscanf(inputs,"%s",word);
if (!strcmp(word, "op_file_format")) fscanf(inputs,"%s" ,op_file_format);
}
}
fclose(inputs);
}
if (strcmp(op_file_format,"binary") && strcmp(op_file_format,"bin")) {
printf("Error: solution output needs to be in binary files.\n");
return(0);
}
inputs = fopen("physics.inp","r");
if (!inputs) {
fprintf(stderr,"Error: File \"physics.inp\" not found.\n");
return(1);
} else {
char word[100];
fscanf(inputs,"%s",word);
if (!strcmp(word, "begin")){
while (strcmp(word, "end")){
fscanf(inputs,"%s",word);
if (!strcmp(word, "Re")) fscanf(inputs,"%lf",&Re_inf);
else if (!strcmp(word, "Minf")) fscanf(inputs,"%lf",&M_inf);
}
}
fclose(inputs);
}
strcpy(filename,"op.bin");
in = fopen(filename,"rb");
out = fopen("SkinFriction.dat","w");
if (!in) {
printf("File op.bin found. Exiting.\n");
return(0);
} else {
printf("Reading file %s.\n",filename);
int ndims, nvars, dims[3];
double *U,*x,*y,*z;
fread(&ndims,sizeof(int),1,in);
fread(&nvars,sizeof(int),1,in);
if (ndims != 2) {
printf("Error: ndims in %s not equal to 2!\n",filename);
return(0);
}
if (nvars != 4) {
printf("Error: nvars in %s not equal to 4!\n",filename);
return(0);
}
fread(dims,sizeof(int),ndims,in);
printf("Dimensions: %d x %d\n",dims[0],dims[1]);
int NI = dims[0];
int NJ = dims[1];
int sizeu = NI*NJ*nvars;
x = (double*) calloc (NI ,sizeof(double));
y = (double*) calloc (NJ ,sizeof(double));
U = (double*) calloc (sizeu,sizeof(double));
fread(x,sizeof(double),NI ,in);
fread(y,sizeof(double),NJ ,in);
fread(U,sizeof(double),sizeu,in);
int i, j, q;
j = 0;
for (i = 1; i < NI-1; i++) {
int q1,q2,q3;
double rho1, u1, v1, e1, p1;
double rho2, u2, v2, e2, p2;
double rho3, u3, v3, e3, p3;
/* calculate du_dy at the wall */
q1 = i + NI*j;
rho1 = U[nvars*q1+0];
u1 = U[nvars*q1+1] / rho1;
v1 = U[nvars*q1+2] / rho1;
e1 = U[nvars*q1+3];
p1 = 0.4 * (e1 - 0.5*rho1*(u1*u1+v1*v1));
q2 = i + NI*(j+1);
rho2 = U[nvars*q2+0];
u2 = U[nvars*q2+1] / rho2;
v2 = U[nvars*q2+2] / rho2;
e2 = U[nvars*q2+3];
p2 = 0.4 * (e2 - 0.5*rho2*(u2*u2+v2*v2));
q3 = i + NI*(j+2);
rho3 = U[nvars*q3+0];
u3 = U[nvars*q3+1] / rho3;
v3 = U[nvars*q3+2] / rho3;
e3 = U[nvars*q3+3];
p3 = 0.4 * (e3 - 0.5*rho3*(u3*u3+v3*v3));
double u0 = -u1;
double u_wall = 0.5*u0 + 0.5*u1;
double u_flow1 = 0.5*u1 + 0.5*u2;
double u_flow2 = 0.5*u2 + 0.5*u3;
double dy = y[j+1] - y[j];
double du_dy = (-3.0*u_wall + 4.0*u_flow1 - u_flow2) / (2.0*dy);
/* Calculate Re_x */
double Re_x = Re_inf * x[i];
/* Calculate exact solution */
double exact_cf = 0.664 / sqrt(Re_x);
/* Calculate the computed cf */
double cf = (M_inf/Re_inf) * (2.0/(M_inf*M_inf)) * du_dy;
if (x[i] > 0) fprintf(out,"%1.16E %1.16E %1.16E %1.16E\n",Re_x,cf,exact_cf,du_dy);
}
free(U);
free(x);
free(y);
fclose(in);
fclose(out);
}
return(0);
}
The following figure showing the exact and computed skin friction coefficients was obtained by plotting SkinFriction.dat:
The file function_counts.dat reports the computational expense (in terms of the number of function counts):
400
22324
22324
22324
2331
2331
1131
19993
The numbers are, respectively,
- Time iterations
- Number of times the hyperbolic term was evaluated
- Number of times the parabolic term was evaluated
- Number of times the source term was evaluated
- Number of calls to the explicit right-hand-side function (PetscRHSFunctionIMEX() or PetscRHSFunctionExpl())
- Number of calls to the implicit right-hand-side function (PetscIFunctionIMEX() or PetscRHSFunctionImpl())
- Number of calls to the Jacobian (PetscIJacobianIMEX() or PetscIJacobian())
- Number of calls to the matrix-free Jacobian function (PetscJacobianFunctionIMEX_Linear(), PetscJacobianFunctionIMEX_JFNK(), PetscJacobianFunction_JFNK(), or PetscJacobianFunction_Linear()).
Expected screen output:
HyPar - Parallel (MPI) version with 6 processes
Compiled with PETSc time integration.
Reading solver inputs from file "solver.inp".
No. of dimensions : 2
No. of variables : 4
Domain size : 226 502
Processes along each dimension : 2 3
No. of ghosts pts : 3
No. of iter. : 400
Restart iteration : 0
Time integration scheme : PETSc
Spatial discretization scheme (hyperbolic) : weno5
Split hyperbolic flux term? : no
Interpolation type for hyperbolic term : components
Spatial discretization type (parabolic ) : nonconservative-2stage
Spatial discretization scheme (parabolic ) : 4
Time Step : 5.000000E-02
Check for conservation : no
Screen output iterations : 1
File output iterations : 10
Initial solution file type : binary
Initial solution read mode : serial
Solution file write mode : serial
Solution file format : binary
Overwrite solution file : yes
Physical model : navierstokes2d
Partitioning domain.
Allocating data arrays.
Reading array from binary file initial.inp (Serial mode).
Volume integral of the initial solution:
0: 3.1451541361714314E-01
1: 9.4354624085191963E-02
2: 0.0000000000000000E+00
3: 5.7578786078649591E-01
Reading boundary conditions from "boundary.inp".
Boundary slip-wall: Along dimension 1 and face +1
Boundary noslip-wall: Along dimension 1 and face +1
Boundary subsonic-outflow: Along dimension 1 and face -1
Boundary subsonic-inflow: Along dimension 0 and face +1
Boundary subsonic-outflow: Along dimension 0 and face -1
5 boundary condition(s) read.
Initializing solvers.
Reading WENO parameters from weno.inp.
Initializing physics. Model = "navierstokes2d"
Reading physical model inputs from file "physics.inp".
Setting up PETSc time integration...
Implicit-Explicit time-integration:-
Hyperbolic term: Implicit
Parabolic term: Implicit
Source term: Implicit
SolvePETSc(): Problem type is nonlinear.
** Starting PETSc time integration **
Writing solution file op.bin.
Iteration: 1 Time: 5.000E-02 Max CFL: 1.011E+02 Max Diff. No.: -1.000E+00
Iteration: 2 Time: 1.000E-01 Max CFL: 1.013E+02 Max Diff. No.: -1.000E+00
Iteration: 3 Time: 1.500E-01 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 4 Time: 2.000E-01 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 5 Time: 2.500E-01 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 6 Time: 3.000E-01 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 7 Time: 3.500E-01 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 8 Time: 4.000E-01 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 9 Time: 4.500E-01 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 10 Time: 5.000E-01 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Writing solution file op.bin.
Iteration: 11 Time: 5.500E-01 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 12 Time: 6.000E-01 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 13 Time: 6.500E-01 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 14 Time: 7.000E-01 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 15 Time: 7.500E-01 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 16 Time: 8.000E-01 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 17 Time: 8.500E-01 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 18 Time: 9.000E-01 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 19 Time: 9.500E-01 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 20 Time: 1.000E+00 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Writing solution file op.bin.
Iteration: 21 Time: 1.050E+00 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 22 Time: 1.100E+00 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 23 Time: 1.150E+00 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 24 Time: 1.200E+00 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 25 Time: 1.250E+00 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 26 Time: 1.300E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 27 Time: 1.350E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 28 Time: 1.400E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 29 Time: 1.450E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 30 Time: 1.500E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Writing solution file op.bin.
Iteration: 31 Time: 1.550E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 32 Time: 1.600E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 33 Time: 1.650E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 34 Time: 1.700E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 35 Time: 1.750E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 36 Time: 1.800E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 37 Time: 1.850E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 38 Time: 1.900E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 39 Time: 1.950E+00 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 40 Time: 2.000E+00 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Writing solution file op.bin.
Iteration: 41 Time: 2.050E+00 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 42 Time: 2.100E+00 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 43 Time: 2.150E+00 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 44 Time: 2.200E+00 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 45 Time: 2.250E+00 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Iteration: 46 Time: 2.300E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 47 Time: 2.350E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 48 Time: 2.400E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 49 Time: 2.450E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 50 Time: 2.500E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Writing solution file op.bin.
Iteration: 51 Time: 2.550E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 52 Time: 2.600E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 53 Time: 2.650E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 54 Time: 2.700E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 55 Time: 2.750E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 56 Time: 2.800E+00 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
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Iteration: 60 Time: 3.000E+00 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
Iteration: 291 Time: 1.455E+01 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
Iteration: 321 Time: 1.605E+01 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
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Writing solution file op.bin.
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Writing solution file op.bin.
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Writing solution file op.bin.
Iteration: 351 Time: 1.755E+01 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
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Writing solution file op.bin.
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Writing solution file op.bin.
Iteration: 371 Time: 1.855E+01 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
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Writing solution file op.bin.
Iteration: 381 Time: 1.905E+01 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
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Writing solution file op.bin.
Iteration: 391 Time: 1.955E+01 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Iteration: 392 Time: 1.960E+01 Max CFL: 1.014E+02 Max Diff. No.: -1.000E+00
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Iteration: 398 Time: 1.990E+01 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
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Iteration: 400 Time: 2.000E+01 Max CFL: 1.015E+02 Max Diff. No.: -1.000E+00
Writing solution file op.bin.
** Completed PETSc time integration **
Computed errors:
L1 Error : 0.0000000000000000E+00
L2 Error : 0.0000000000000000E+00
Linfinity Error : 0.0000000000000000E+00
Conservation Errors:
0.0000000000000000E+00
0.0000000000000000E+00
0.0000000000000000E+00
0.0000000000000000E+00
Solver runtime (in seconds): 1.9543866630000000E+03
Total runtime (in seconds): 1.9544529339999999E+03
Deallocating arrays.
Finished.