/// <summary>
/// The non-linear equation F=0 to be solved
/// </summary>
/// <param name="density"></param>
/// <param name="r"></param>
/// <param name="eos"></param>
/// <param name="c1"></param>
/// <returns></returns>
private static double F(double density, double r, CovolumeGas eos, double c1)
{
double gamma = eos.HeatCapacityRatio;
double covolume = eos.Covolume;
return((gamma - covolume * density) / (gamma - 1.0) / density *
Math.Pow(density / (1.0 - covolume * density), gamma)
- c1 + 0.5 * Math.Exp(1.0 - r * r));
}
/// <summary>
/// Computes the density from a non-linear equation using a simple
/// bisection method
/// </summary>
/// <param name="r"></param>
/// <param name="eos"></param>
/// <param name="c1"></param>
/// <returns></returns>
private static double GetDensity(double r, CovolumeGas eos, double c1)
{
double gamma = eos.HeatCapacityRatio;
double covolume = eos.Covolume;
double densityLeft = 0.0 + 1e-14;
double densityRight = 1.0 / covolume - 1e-14;
double fOld = double.PositiveInfinity;
int maxIterations = 1000;
double density = double.NaN;
double f = double.NaN;
int n = 0;
while (true)
{
if (n >= maxIterations)
{
throw new Exception("No convergence");
}
density = 0.5 * (densityRight + densityLeft);
f = F(density, r, eos, c1);
if (Math.Abs(fOld - f) < 1e-13)
{
break;
}
else if (f > 0.0)
{
densityRight = density;
}
else
{
densityLeft = density;
}
fOld = f;
n++;
}
return(density);
}
/// <summary>
/// Computes the solution at (x, t).
/// </summary>
/// <param name="x"></param>
/// <param name="y"></param>
/// <param name="u"></param>
/// <param name="t"></param>
/// <param name="c"></param>
/// <param name="c1"></param>
/// <returns></returns>
public static StateVector GetSolution(double x, double y, double u, double t, CNSControl c, double c1)
{
double xBar = x - u * t;
double r = Math.Sqrt(xBar * xBar + y * y);
double theta = Math.Atan2(y, xBar);
CovolumeGas eos = c.EquationOfState as CovolumeGas;
double gamma = eos.HeatCapacityRatio;
double covolume = eos.Covolume;
double rho = GetDensity(r, eos, c1);
double p = Math.Pow(rho / (1 - covolume * rho), gamma);
double normU = r * Math.Exp(0.5 * (1.0 - r * r));
Vector3D U = new Vector3D(
u - Math.Sin(theta) * normU,
Math.Cos(theta) * normU,
0.0);
return(new StateVector(
new Material(c),
rho,
rho * U,
p * (1.0 - covolume * rho) / (gamma - 1.0) + 0.5 * rho * U * U));
}
/// <summary>
/// Test case using <see cref="CovolumeGas"/> in combination with the
/// <see cref="RusanovFlux"/>.
/// </summary>
/// <returns></returns>
public static VortexControl IsentropicVortexCovolumeGasRusanov()
{
VortexControl c = Template(1, 2);
CovolumeGas eos = new CovolumeGas(1.4, 0.1);
double integrationConstant = 3.6;
c.EquationOfState = eos;
c.ConvectiveFluxType = ConvectiveFluxTypes.Rusanov;
Func <double[], double, StateVector> solution = (X, t) =>
CovolumeVortexExactSolution.GetSolution(X[0], X[1], c.VortexSpeed, t, c, integrationConstant);
c.InitialValues_Evaluators.Add(Variables.Density, X => solution(X, 0.0).Density);
c.InitialValues_Evaluators.Add(Variables.Velocity.xComponent, X => solution(X, 0.0).Velocity[0]);
c.InitialValues_Evaluators.Add(Variables.Velocity.yComponent, X => solution(X, 0.0).Velocity[1]);
c.InitialValues_Evaluators.Add(Variables.Pressure, X => solution(X, 0.0).Pressure);
c.Queries.Add("L2ErrorDensity", QueryLibrary.L2Error(Variables.Density, (X, t) => solution(X, t).Density, 10));
c.Queries.Add("L2ErrorPressure", QueryLibrary.L2Error(Variables.Pressure, (X, t) => solution(X, t).Pressure, 10));
c.Queries.Add("L2ErrorEntropy", QueryLibrary.L2Error(Variables.Entropy, (X, t) => 1.0, 10));
return(c);
}
