2D Navier-Stokes Equations - Laminar Flow over Flat Plate
Location: hypar/Examples/2D/NavierStokes2D/FlatPlateLaminar/UniformGrid (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: SSPRK3 (TimeRK(), _RK_SSP3_)
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:
solver.inp
begin
ndims 2
nvars 4
size 226 502
iproc 2 4
ghost 3
n_iter 50000
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 0.0005
screen_op_iter 100
file_op_iter 1000
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 4
in solver.inp (i.e., 2 processors along x, and 4 processor along y). Thus, this example should be run with 8 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:
Expected screen output:
HyPar - Parallel (MPI) version with 8 processes
Reading solver inputs from file "solver.inp".
No. of dimensions : 2
No. of variables : 4
Domain size : 226 502
Processes along each dimension : 2 4
No. of ghosts pts : 3
No. of iter. : 50000
Restart iteration : 0
Time integration scheme : rk (44)
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-04
Check for conservation : no
Screen output iterations : 100
File output iterations : 1000
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: 3.1451541361720559E-01
1: 9.4354624085189354E-02
2: 0.0000000000000000E+00
3: 5.7578786078635780E-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 time integration.
Solving in time (from 0 to 50000 iterations)
Writing solution file op.bin.
Iteration: 100 Time: 5.000E-02 Max CFL: 1.011E+00 Max Diff. No.: -1.000E+00 Norm: 4.7750E-05
Iteration: 200 Time: 1.000E-01 Max CFL: 1.013E+00 Max Diff. No.: -1.000E+00 Norm: 3.2736E-05
Iteration: 300 Time: 1.500E-01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.7768E-05
Iteration: 400 Time: 2.000E-01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.4610E-05
Iteration: 500 Time: 2.500E-01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.0734E-05
Iteration: 600 Time: 3.000E-01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 1.7873E-05
Iteration: 700 Time: 3.500E-01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 1.5475E-05
Iteration: 800 Time: 4.000E-01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 1.3750E-05
Iteration: 900 Time: 4.500E-01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 1.2439E-05
Iteration: 1000 Time: 5.000E-01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 1.2392E-05
Writing solution file op.bin.
Iteration: 1100 Time: 5.500E-01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 1.0491E-05
Iteration: 1200 Time: 6.000E-01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 9.7967E-06
Iteration: 1300 Time: 6.500E-01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 9.1987E-06
Iteration: 1400 Time: 7.000E-01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 8.7849E-06
Iteration: 1500 Time: 7.500E-01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 8.3762E-06
Iteration: 1600 Time: 8.000E-01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 7.8794E-06
Iteration: 1700 Time: 8.500E-01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 7.5538E-06
Iteration: 1800 Time: 9.000E-01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 7.1740E-06
Iteration: 1900 Time: 9.500E-01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 6.8866E-06
Iteration: 2000 Time: 1.000E+00 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 5.9095E-06
Writing solution file op.bin.
Iteration: 2100 Time: 1.050E+00 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 6.4236E-06
Iteration: 2200 Time: 1.100E+00 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 6.2284E-06
Iteration: 2300 Time: 1.150E+00 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 6.1467E-06
Iteration: 2400 Time: 1.200E+00 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 6.1082E-06
Iteration: 2500 Time: 1.250E+00 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 5.9863E-06
Iteration: 2600 Time: 1.300E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 5.6731E-06
Iteration: 2700 Time: 1.350E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 5.5233E-06
Iteration: 2800 Time: 1.400E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 5.2993E-06
Iteration: 2900 Time: 1.450E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 5.0664E-06
Iteration: 3000 Time: 1.500E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 5.2579E-06
Writing solution file op.bin.
Iteration: 3100 Time: 1.550E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 4.8691E-06
Iteration: 3200 Time: 1.600E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 4.8381E-06
Iteration: 3300 Time: 1.650E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 4.9002E-06
Iteration: 3400 Time: 1.700E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 4.8980E-06
Iteration: 3500 Time: 1.750E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 4.8452E-06
Iteration: 3600 Time: 1.800E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 4.6270E-06
Iteration: 3700 Time: 1.850E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 4.5194E-06
Iteration: 3800 Time: 1.900E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 4.3036E-06
Iteration: 3900 Time: 1.950E+00 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 4.1520E-06
Iteration: 4000 Time: 2.000E+00 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 3.8470E-06
Writing solution file op.bin.
Iteration: 4100 Time: 2.050E+00 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 4.1073E-06
Iteration: 4200 Time: 2.100E+00 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 4.0925E-06
Iteration: 4300 Time: 2.150E+00 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 4.2089E-06
Iteration: 4400 Time: 2.200E+00 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 4.1999E-06
Iteration: 4500 Time: 2.250E+00 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 4.1013E-06
Iteration: 4600 Time: 2.300E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 3.9459E-06
Iteration: 4700 Time: 2.350E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 3.8661E-06
Iteration: 4800 Time: 2.400E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 3.6527E-06
Iteration: 4900 Time: 2.450E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 3.5941E-06
Iteration: 5000 Time: 2.500E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 3.7403E-06
Writing solution file op.bin.
Iteration: 5100 Time: 2.550E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 3.6347E-06
Iteration: 5200 Time: 2.600E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 3.6373E-06
Iteration: 5300 Time: 2.650E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 3.7370E-06
Iteration: 5400 Time: 2.700E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 3.7658E-06
Iteration: 5500 Time: 2.750E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 3.7053E-06
Iteration: 5600 Time: 2.800E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 3.5679E-06
Iteration: 5700 Time: 2.850E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 3.5446E-06
Iteration: 5800 Time: 2.900E+00 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 3.3854E-06
Iteration: 5900 Time: 2.950E+00 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 3.2592E-06
Iteration: 6000 Time: 3.000E+00 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 3.2110E-06
Writing solution file op.bin.
Iteration: 6100 Time: 3.050E+00 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 3.3176E-06
Iteration: 6200 Time: 3.100E+00 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 3.3200E-06
Iteration: 6300 Time: 3.150E+00 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 3.3329E-06
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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.
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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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Iteration: 40600 Time: 2.030E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1911E-06
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Iteration: 40800 Time: 2.040E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.2340E-06
Iteration: 40900 Time: 2.045E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.2018E-06
Iteration: 41000 Time: 2.050E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.1861E-06
Writing solution file op.bin.
Iteration: 41100 Time: 2.055E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.3085E-06
Iteration: 41200 Time: 2.060E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.3074E-06
Iteration: 41300 Time: 2.065E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.1671E-06
Iteration: 41400 Time: 2.070E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.2999E-06
Iteration: 41500 Time: 2.075E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.2630E-06
Iteration: 41600 Time: 2.080E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1781E-06
Iteration: 41700 Time: 2.085E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.2326E-06
Iteration: 41800 Time: 2.090E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.3194E-06
Iteration: 41900 Time: 2.095E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.2268E-06
Iteration: 42000 Time: 2.100E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1624E-06
Writing solution file op.bin.
Iteration: 42100 Time: 2.105E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1134E-06
Iteration: 42200 Time: 2.110E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.0722E-06
Iteration: 42300 Time: 2.115E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.0565E-06
Iteration: 42400 Time: 2.120E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1552E-06
Iteration: 42500 Time: 2.125E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.2222E-06
Iteration: 42600 Time: 2.130E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.2178E-06
Iteration: 42700 Time: 2.135E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.2022E-06
Iteration: 42800 Time: 2.140E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1955E-06
Iteration: 42900 Time: 2.145E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1539E-06
Iteration: 43000 Time: 2.150E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1148E-06
Writing solution file op.bin.
Iteration: 43100 Time: 2.155E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1138E-06
Iteration: 43200 Time: 2.160E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.0393E-06
Iteration: 43300 Time: 2.165E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.0894E-06
Iteration: 43400 Time: 2.170E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.1770E-06
Iteration: 43500 Time: 2.175E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.1833E-06
Iteration: 43600 Time: 2.180E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.1327E-06
Iteration: 43700 Time: 2.185E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.2290E-06
Iteration: 43800 Time: 2.190E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1961E-06
Iteration: 43900 Time: 2.195E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1697E-06
Iteration: 44000 Time: 2.200E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.0958E-06
Writing solution file op.bin.
Iteration: 44100 Time: 2.205E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.0213E-06
Iteration: 44200 Time: 2.210E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 1.9354E-06
Iteration: 44300 Time: 2.215E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.0575E-06
Iteration: 44400 Time: 2.220E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.0893E-06
Iteration: 44500 Time: 2.225E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1691E-06
Iteration: 44600 Time: 2.230E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1351E-06
Iteration: 44700 Time: 2.235E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1437E-06
Iteration: 44800 Time: 2.240E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1762E-06
Iteration: 44900 Time: 2.245E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1602E-06
Iteration: 45000 Time: 2.250E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1251E-06
Writing solution file op.bin.
Iteration: 45100 Time: 2.255E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.0295E-06
Iteration: 45200 Time: 2.260E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.0001E-06
Iteration: 45300 Time: 2.265E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 1.9198E-06
Iteration: 45400 Time: 2.270E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 1.9230E-06
Iteration: 45500 Time: 2.275E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.0499E-06
Iteration: 45600 Time: 2.280E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.1371E-06
Iteration: 45700 Time: 2.285E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1341E-06
Iteration: 45800 Time: 2.290E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1897E-06
Iteration: 45900 Time: 2.295E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.0504E-06
Iteration: 46000 Time: 2.300E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 1.9263E-06
Writing solution file op.bin.
Iteration: 46100 Time: 2.305E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 1.9280E-06
Iteration: 46200 Time: 2.310E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.0235E-06
Iteration: 46300 Time: 2.315E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.0344E-06
Iteration: 46400 Time: 2.320E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.0221E-06
Iteration: 46500 Time: 2.325E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.0169E-06
Iteration: 46600 Time: 2.330E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 1.9558E-06
Iteration: 46700 Time: 2.335E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 1.9743E-06
Iteration: 46800 Time: 2.340E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.0630E-06
Iteration: 46900 Time: 2.345E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1651E-06
Iteration: 47000 Time: 2.350E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.2773E-06
Writing solution file op.bin.
Iteration: 47100 Time: 2.355E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.2293E-06
Iteration: 47200 Time: 2.360E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.0926E-06
Iteration: 47300 Time: 2.365E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 1.9563E-06
Iteration: 47400 Time: 2.370E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 1.8237E-06
Iteration: 47500 Time: 2.375E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 1.8648E-06
Iteration: 47600 Time: 2.380E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 1.9746E-06
Iteration: 47700 Time: 2.385E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.1151E-06
Iteration: 47800 Time: 2.390E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.3198E-06
Iteration: 47900 Time: 2.395E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.3022E-06
Iteration: 48000 Time: 2.400E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.3245E-06
Writing solution file op.bin.
Iteration: 48100 Time: 2.405E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.2523E-06
Iteration: 48200 Time: 2.410E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.0435E-06
Iteration: 48300 Time: 2.415E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 1.8089E-06
Iteration: 48400 Time: 2.420E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 1.7796E-06
Iteration: 48500 Time: 2.425E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 1.8753E-06
Iteration: 48600 Time: 2.430E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 1.9723E-06
Iteration: 48700 Time: 2.435E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1904E-06
Iteration: 48800 Time: 2.440E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.3146E-06
Iteration: 48900 Time: 2.445E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.2946E-06
Iteration: 49000 Time: 2.450E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.2081E-06
Writing solution file op.bin.
Iteration: 49100 Time: 2.455E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.1098E-06
Iteration: 49200 Time: 2.460E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.0178E-06
Iteration: 49300 Time: 2.465E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 1.8594E-06
Iteration: 49400 Time: 2.470E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 1.7898E-06
Iteration: 49500 Time: 2.475E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 1.9376E-06
Iteration: 49600 Time: 2.480E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.0604E-06
Iteration: 49700 Time: 2.485E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.2076E-06
Iteration: 49800 Time: 2.490E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.3387E-06
Iteration: 49900 Time: 2.495E+01 Max CFL: 1.014E+00 Max Diff. No.: -1.000E+00 Norm: 2.3651E-06
Iteration: 50000 Time: 2.500E+01 Max CFL: 1.015E+00 Max Diff. No.: -1.000E+00 Norm: 2.2600E-06
Writing solution file op.bin.
Completed time integration (Final time: 25.000000).
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): 8.8939867780000004E+03
Total runtime (in seconds): 8.8940228050000005E+03
Deallocating arrays.
Finished.