int main(int argc, char *argv[])
{
#include "setRootCase.H"
#include "createTime.H"
#include "createMesh.H"
#include "createFields.H"
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
Info<< "\nStarting time loop\n" << endl;
while (runTime.loop())
{
Info << "Time = " << runTime.timeName() << nl << endl;
// Update s
fvVectorMatrix sEqn
(
fvm::ddt(s)
- fvm::laplacian(nu,s)
+ corrDim*s
);
solve(sEqn == -fvc::grad(Theta));
// Update Theta
fvScalarMatrix ThetaEqn
(
(1/beta)*fvm::ddt(Theta)
);
solve(ThetaEqn == -fvc::div(s));
// Output
runTime.write();
Info<< "ExecutionTime = " << runTime.elapsedCpuTime() << " s"
<< " ClockTime = " << runTime.elapsedClockTime() << " s"
<< nl << endl;
}
Info<< "End\n" << endl;
return 0;
}
void Foam::kineticTheoryModel::solve()
{
if (!kineticTheory_)
{
return;
}
word scheme("div(phi,Theta)");
volScalarField alpha = alpha_;
alpha.max(1.0e-6);
const scalar sqrtPi = sqrt(mathematicalConstant::pi);
surfaceScalarField phi = 1.5*rhoa_*phia_*fvc::interpolate(alpha_);
volTensorField dU = fvc::grad(Ua_);
volTensorField dUT = dU.T();
volTensorField D = 0.5*(dU + dUT);
// NB, drag = K*alpha*beta,
// (the alpha and beta has been extracted from the drag function for
// numerical reasons)
volScalarField Ur = mag(Ua_ - Ub_);
volScalarField betaPrim = alpha_*(1.0 - alpha_)*draga_.K(Ur);
// Calculating the radial distribution function (solid volume fraction is
// limited close to the packing limit, but this needs improvements)
// The solution is higly unstable close to the packing limit.
gs0_ = radialModel_->g0(min(alpha, alphaMax_-1.0e-2), alphaMax_);
// particle pressure - coefficient in front of Theta (Eq. 3.22, p. 45)
volScalarField PsCoeff =
granularPressureModel_->granularPressureCoeff(alpha_,gs0_,rhoa_,e_ );
dimensionedScalar Tsmall
(
"small",
dimensionSet(0,2,-2,0,0,0,0),
1.0e-6
);
dimensionedScalar TsmallSqrt = sqrt(Tsmall);
volScalarField ThetaSqrt = sqrt(Theta_);
// 'thermal' conductivity (Table 3.3, p. 49)
kappa_ = conductivityModel_->kappa(alpha, Theta_, gs0_, rhoa_, da_, e_);
// particle viscosity (Table 3.2, p.47)
mua_ = viscosityModel_->mua(alpha, Theta_, gs0_, rhoa_, da_, e_);
// dissipation (Eq. 3.24, p.50)
volScalarField gammaCoeff =
12.0*(1.0 - e_*e_)*sqr(alpha)*rhoa_*gs0_*(1.0/da_)
*ThetaSqrt/sqrtPi;
// Eq. 3.25, p. 50 Js = J1 - J2
volScalarField J1 = 3.0*betaPrim;
volScalarField J2 =
0.25*sqr(betaPrim)*da_*sqr(Ur)
/(alpha*rhoa_*sqrtPi*(ThetaSqrt + TsmallSqrt));
// bulk viscosity p. 45 (Lun et al. 1984).
lambda_ = (4.0/3.0)*sqr(alpha_)*rhoa_*da_*gs0_*(1.0+e_)*ThetaSqrt/sqrtPi;
// stress tensor, Definitions, Table 3.1, p. 43
volTensorField tau = 2.0*mua_*D + (lambda_ - (2.0/3.0)*mua_)*tr(D)*I;
if (!equilibrium_)
{
// construct the granular temperature equation (Eq. 3.20, p. 44)
// NB. note that there are two typos in Eq. 3.20
// no grad infront of Ps
// wrong sign infront of laplacian
fvScalarMatrix ThetaEqn
(
fvm::ddt(1.5*alpha*rhoa_, Theta_)
+ fvm::div(phi, Theta_, scheme)
==
fvm::SuSp(-((PsCoeff*I) && dU), Theta_)
+ (tau && dU)
+ fvm::laplacian(kappa_, Theta_, "laplacian(kappa,Theta)")
+ fvm::Sp(-gammaCoeff, Theta_)
+ fvm::Sp(-J1, Theta_)
+ fvm::Sp(J2/(Theta_ + Tsmall), Theta_)
);
ThetaEqn.relax();
ThetaEqn.solve();
}
else
{
// equilibrium => dissipation == production
// Eq. 4.14, p.82
volScalarField K1 = 2.0*(1.0 + e_)*rhoa_*gs0_;
volScalarField K3 = 0.5*da_*rhoa_*
(
(sqrtPi/(3.0*(3.0-e_)))
*(1.0 + 0.4*(1.0 + e_)*(3.0*e_ - 1.0)*alpha*gs0_)
+ 1.6*alpha*gs0_*(1.0 + e_)/sqrtPi
);
volScalarField K2 =
4.0*da_*rhoa_*(1.0 + e_)*alpha*gs0_/(3.0*sqrtPi) - 2.0*K3/3.0;
volScalarField K4 = 12.0*(1.0 - e_*e_)*rhoa_*gs0_/(da_*sqrtPi);
volScalarField trD = tr(D);
volTensorField D2 = D & D;
volScalarField tr2D = trD*trD;
volScalarField trD2 = tr(D2);
volScalarField t1 = K1*alpha + rhoa_;
volScalarField l1 = -t1*trD;
volScalarField l2 = sqr(t1)*tr2D;
volScalarField l3 = 4.0*K4*alpha*(2.0*K3*trD2 + K2*tr2D);
Theta_ = sqr((l1 + sqrt(l2 + l3))/(2.0*(alpha + 1.0e-4)*K4));
}
Theta_.max(1.0e-15);
Theta_.min(1.0e+3);
volScalarField pf =
frictionalStressModel_->frictionalPressure
(
alpha_,
alphaMinFriction_,
alphaMax_,
Fr_,
eta_,
p_
);
PsCoeff += pf/(Theta_+Tsmall);
PsCoeff.min(1.0e+10);
PsCoeff.max(-1.0e+10);
// update particle pressure
pa_ = PsCoeff*Theta_;
// frictional shear stress, Eq. 3.30, p. 52
volScalarField muf = frictionalStressModel_->muf
(
alpha_,
alphaMax_,
pf,
D,
phi_
);
// add frictional stress for alpha > alphaMinFriction
mua_ = viscosityModel_->mua(alpha, Theta_, gs0_, rhoa_, da_, e_) + muf;
mua_.min(1.0e+2);
mua_.max(0.0);
lambda_ = (4.0/3.0)*sqr(alpha_)*rhoa_*da_*gs0_*(1.0 + e_)*ThetaSqrt/sqrtPi;
Info<< "kinTheory: max(Theta) = " << max(Theta_).value() << endl;
volScalarField ktn = mua_/rhoa_;
Info<< "kinTheory: min(nua) = " << min(ktn).value()
<< ", max(nua) = " << max(ktn).value() << endl;
Info<< "kinTheory: min(pa) = " << min(pa_).value()
<< ", max(pa) = " << max(pa_).value() << endl;
}
void Foam::kineticTheoryModel::solve(const volTensorField& gradU1t)
{
if (!kineticTheory_)
{
return;
}
const scalar sqrtPi = sqrt(constant::mathematical::pi);
volScalarField da_(phase1_.d());
surfaceScalarField phi(1.5*phi1_*fvc::interpolate(rho1_*alpha1_));
volTensorField dU(gradU1t.T());
volSymmTensorField D(symm(dU));
// NB, drag = K*alpha1*alpha2,
// (the alpha1 and alpha2 has been extracted from the drag function for
// numerical reasons)
volScalarField Ur(mag(U1_ - U2_));
volScalarField alpha2Prim(alpha1_*(1.0 - alpha1_)*draga_.K(Ur));
// Calculating the radial distribution function (solid volume fraction is
// limited close to the packing limit, but this needs improvements)
// The solution is higly unstable close to the packing limit.
gs0_ = radialModel_->g0
(
min(max(alpha1_, scalar(1e-6)), alphaMax_ - 0.01),
alphaMax_
);
// particle pressure - coefficient in front of Theta (Eq. 3.22, p. 45)
volScalarField PsCoeff
(
granularPressureModel_->granularPressureCoeff
(
alpha1_,
gs0_,
rho1_,
e_
)
);
// 'thermal' conductivity (Table 3.3, p. 49)
kappa_ = conductivityModel_->kappa(alpha1_, Theta_, gs0_, rho1_, da_, e_);
// particle viscosity (Table 3.2, p.47)
mu1_ = viscosityModel_->mu1(alpha1_, Theta_, gs0_, rho1_, da_, e_);
dimensionedScalar Tsmall
(
"small",
dimensionSet(0 , 2 ,-2 ,0 , 0, 0, 0),
1.0e-6
);
dimensionedScalar TsmallSqrt = sqrt(Tsmall);
volScalarField ThetaSqrt(sqrt(Theta_));
// dissipation (Eq. 3.24, p.50)
volScalarField gammaCoeff
(
12.0*(1.0 - sqr(e_))*sqr(alpha1_)*rho1_*gs0_*(1.0/da_)*ThetaSqrt/sqrtPi
);
// Eq. 3.25, p. 50 Js = J1 - J2
volScalarField J1(3.0*alpha2Prim);
volScalarField J2
(
0.25*sqr(alpha2Prim)*da_*sqr(Ur)
/(max(alpha1_, scalar(1e-6))*rho1_*sqrtPi*(ThetaSqrt + TsmallSqrt))
);
// bulk viscosity p. 45 (Lun et al. 1984).
lambda_ = (4.0/3.0)*sqr(alpha1_)*rho1_*da_*gs0_*(1.0+e_)*ThetaSqrt/sqrtPi;
// stress tensor, Definitions, Table 3.1, p. 43
volSymmTensorField tau(2.0*mu1_*D + (lambda_ - (2.0/3.0)*mu1_)*tr(D)*I);
if (!equilibrium_)
{
// construct the granular temperature equation (Eq. 3.20, p. 44)
// NB. note that there are two typos in Eq. 3.20
// no grad infront of Ps
// wrong sign infront of laplacian
fvScalarMatrix ThetaEqn
(
fvm::ddt(1.5*alpha1_*rho1_, Theta_)
+ fvm::div(phi, Theta_, "div(phi,Theta)")
==
fvm::SuSp(-((PsCoeff*I) && dU), Theta_)
+ (tau && dU)
+ fvm::laplacian(kappa_, Theta_, "laplacian(kappa,Theta)")
+ fvm::Sp(-gammaCoeff, Theta_)
+ fvm::Sp(-J1, Theta_)
+ fvm::Sp(J2/(Theta_ + Tsmall), Theta_)
);
ThetaEqn.relax();
ThetaEqn.solve();
}
else
{
// equilibrium => dissipation == production
// Eq. 4.14, p.82
volScalarField K1(2.0*(1.0 + e_)*rho1_*gs0_);
volScalarField K3
(
0.5*da_*rho1_*
(
(sqrtPi/(3.0*(3.0-e_)))
*(1.0 + 0.4*(1.0 + e_)*(3.0*e_ - 1.0)*alpha1_*gs0_)
+1.6*alpha1_*gs0_*(1.0 + e_)/sqrtPi
)
);
volScalarField K2
(
4.0*da_*rho1_*(1.0 + e_)*alpha1_*gs0_/(3.0*sqrtPi) - 2.0*K3/3.0
);
volScalarField K4(12.0*(1.0 - sqr(e_))*rho1_*gs0_/(da_*sqrtPi));
volScalarField trD(tr(D));
volScalarField tr2D(sqr(trD));
volScalarField trD2(tr(D & D));
volScalarField t1(K1*alpha1_ + rho1_);
volScalarField l1(-t1*trD);
volScalarField l2(sqr(t1)*tr2D);
volScalarField l3
(
4.0
*K4
*max(alpha1_, scalar(1e-6))
*(2.0*K3*trD2 + K2*tr2D)
);
Theta_ = sqr((l1 + sqrt(l2 + l3))/(2.0*(alpha1_ + 1.0e-4)*K4));
}
Theta_.max(1.0e-15);
Theta_.min(1.0e+3);
volScalarField pf
(
frictionalStressModel_->frictionalPressure
(
alpha1_,
alphaMinFriction_,
alphaMax_,
Fr_,
eta_,
p_
)
);
PsCoeff += pf/(Theta_+Tsmall);
PsCoeff.min(1.0e+10);
PsCoeff.max(-1.0e+10);
// update particle pressure
pa_ = PsCoeff*Theta_;
// frictional shear stress, Eq. 3.30, p. 52
volScalarField muf
(
frictionalStressModel_->muf
(
alpha1_,
alphaMax_,
pf,
D,
phi_
)
);
// add frictional stress
mu1_ += muf;
mu1_.min(1.0e+2);
mu1_.max(0.0);
Info<< "kinTheory: max(Theta) = " << max(Theta_).value() << endl;
volScalarField ktn(mu1_/rho1_);
Info<< "kinTheory: min(nu1) = " << min(ktn).value()
<< ", max(nu1) = " << max(ktn).value() << endl;
Info<< "kinTheory: min(pa) = " << min(pa_).value()
<< ", max(pa) = " << max(pa_).value() << endl;
}
void Foam::kineticTheoryModel::solve(const volTensorField& gradUat)
{
if(kineticTheory_)
{
//if (!kineticTheory_)
//{
// return;
//}
//const scalar sqrtPi = sqrt(mathematicalConstant::pi);
if(Berzi_)
{
Info << "Berzi Model is used" << endl;
}
else{
const scalar sqrtPi = sqrt(constant::mathematical::pi);
surfaceScalarField phi = 1.5*rhoa_*phia_*fvc::interpolate(alpha_);
volTensorField dU = gradUat.T();//fvc::grad(Ua_);
volSymmTensorField D = symm(dU);
// NB, drag = K*alpha*beta,
// (the alpha and beta has been extracted from the drag function for
// numerical reasons)
volScalarField Ur = mag(Ua_ - Ub_);
volScalarField betaPrim = alpha_*(1.0 - alpha_)*draga_.K(Ur);
// Calculating the radial distribution function (solid volume fraction is
// limited close to the packing limit, but this needs improvements)
// The solution is higly unstable close to the packing limit.
gs0_ = radialModel_->g0
(
min(max(alpha_, 1e-6), alphaMax_ - 0.01),
alphaMax_
);
// particle pressure - coefficient in front of Theta (Eq. 3.22, p. 45)
volScalarField PsCoeff = granularPressureModel_->granularPressureCoeff
(
alpha_,
gs0_,
rhoa_,
e_
);
// 'thermal' conductivity (Table 3.3, p. 49)
kappa_ = conductivityModel_->kappa(alpha_, Theta_, gs0_, rhoa_, da_, e_);
// particle viscosity (Table 3.2, p.47)
mua_ = viscosityModel_->mua(alpha_, Theta_, gs0_, rhoa_, da_, e_);
dimensionedScalar Tsmall
(
"small",
dimensionSet(0 , 2 ,-2 ,0 , 0, 0, 0),
1.0e-6
);
dimensionedScalar TsmallSqrt = sqrt(Tsmall);
volScalarField ThetaSqrt = sqrt(Theta_);
// dissipation (Eq. 3.24, p.50)
volScalarField gammaCoeff =
12.0*(1.0 - sqr(e_))*sqr(alpha_)*rhoa_*gs0_*(1.0/da_)*ThetaSqrt/sqrtPi;
// Eq. 3.25, p. 50 Js = J1 - J2
volScalarField J1 = 3.0*betaPrim;
volScalarField J2 =
0.25*sqr(betaPrim)*da_*sqr(Ur)
/(max(alpha_, 1e-6)*rhoa_*sqrtPi*(ThetaSqrt + TsmallSqrt));
// bulk viscosity p. 45 (Lun et al. 1984).
lambda_ = (4.0/3.0)*sqr(alpha_)*rhoa_*da_*gs0_*(1.0+e_)*ThetaSqrt/sqrtPi;
// stress tensor, Definitions, Table 3.1, p. 43
volSymmTensorField tau = 2.0*mua_*D + (lambda_ - (2.0/3.0)*mua_)*tr(D)*I;
if (!equilibrium_)
{
// construct the granular temperature equation (Eq. 3.20, p. 44)
// NB. note that there are two typos in Eq. 3.20
// no grad infront of Ps
// wrong sign infront of laplacian
fvScalarMatrix ThetaEqn
(
fvm::ddt(1.5*alpha_*rhoa_, Theta_)
+ fvm::div(phi, Theta_, "div(phi,Theta)")
==
fvm::SuSp(-((PsCoeff*I) && dU), Theta_)
+ (tau && dU)
+ fvm::laplacian(kappa_, Theta_, "laplacian(kappa,Theta)")
+ fvm::Sp(-gammaCoeff, Theta_)
+ fvm::Sp(-J1, Theta_)
+ fvm::Sp(J2/(Theta_ + Tsmall), Theta_)
);
ThetaEqn.relax();
ThetaEqn.solve();
}
else
{
// equilibrium => dissipation == production
// Eq. 4.14, p.82
volScalarField K1 = 2.0*(1.0 + e_)*rhoa_*gs0_;
volScalarField K3 = 0.5*da_*rhoa_*
(
(sqrtPi/(3.0*(3.0-e_)))
*(1.0 + 0.4*(1.0 + e_)*(3.0*e_ - 1.0)*alpha_*gs0_)
+1.6*alpha_*gs0_*(1.0 + e_)/sqrtPi
);
volScalarField K2 =
4.0*da_*rhoa_*(1.0 + e_)*alpha_*gs0_/(3.0*sqrtPi) - 2.0*K3/3.0;
volScalarField K4 = 12.0*(1.0 - sqr(e_))*rhoa_*gs0_/(da_*sqrtPi);
volScalarField trD = tr(D);
volScalarField tr2D = sqr(trD);
volScalarField trD2 = tr(D & D);
volScalarField t1 = K1*alpha_ + rhoa_;
volScalarField l1 = -t1*trD;
volScalarField l2 = sqr(t1)*tr2D;
volScalarField l3 = 4.0*K4*max(alpha_, 1e-6)*(2.0*K3*trD2 + K2*tr2D);
Theta_ = sqr((l1 + sqrt(l2 + l3))/(2.0*(alpha_ + 1.0e-4)*K4));
}
Theta_.max(1.0e-15);
Theta_.min(1.0e+3);
volScalarField pf = frictionalStressModel_->frictionalPressure
(
alpha_,
alphaMinFriction_,
alphaMax_,
Fr_,
eta_,
p_
);
PsCoeff += pf/(Theta_+Tsmall);
PsCoeff.min(1.0e+10);
PsCoeff.max(-1.0e+10);
// update particle pressure
pa_ = PsCoeff*Theta_;
// frictional shear stress, Eq. 3.30, p. 52
volScalarField muf = frictionalStressModel_->muf
(
alpha_,
alphaMax_,
pf,
D,
phi_
);
// add frictional stress
mua_ += muf;
//-AO Inconsistency of equations
const scalar constSMALL = 0.001; //1.e-06;
mua_ /= (fvc::average(alpha_) + scalar(constSMALL));
lambda_ /= (fvc::average(alpha_) + scalar(constSMALL));
//-AO
mua_.min(1.0e+2);
mua_.max(0.0);
Info<< "kinTheory: max(Theta) = " << max(Theta_).value() << endl;
volScalarField ktn = mua_/rhoa_;
Info<< "kinTheory: min(nua) = " << min(ktn).value()
<< ", max(nua) = " << max(ktn).value() << endl;
Info<< "kinTheory: min(pa) = " << min(pa_).value()
<< ", max(pa) = " << max(pa_).value() << endl;
//}
/*
volScalarField& Foam::kineticTheoryModel::ppMagf(const volScalarField& alphaUpdate)
{
volScalarField alpha = alphaUpdate;
gs0_ = radialModel_->g0(min(alpha, alphaMinFriction_), alphaMax_);
gs0Prime_ = radialModel_->g0prime(min(alpha, alphaMinFriction_), alphaMax_);
// Computing ppMagf
ppMagf_ = Theta_*granularPressureModel_->granularPressureCoeffPrime
(
alpha,
gs0_,
gs0Prime_,
rhoa_,
e_
);
volScalarField ppMagfFriction = frictionalStressModel_->frictionalPressurePrime
(
alpha,
alphaMinFriction_,
alphaMax_,
Fr_,
eta_,
p_
);
// NOTE: this might not be appropriate if J&J model is used (verify)
forAll(alpha, cellI)
{
if(alpha[cellI] >= alphaMinFriction_.value())
{
ppMagf_[cellI] = ppMagfFriction[cellI];
}
}
ppMagf_.correctBoundaryConditions();
return ppMagf_;
}
*/}
}
else if(mofidiedKineticTheoryPU_)
{
//if (!mofidiedKineticTheoryPU_)
//{
// return;
//}
Info << " " << endl;
Info << "Modified kinetic theory model - Chialvo-Sundaresan " << endl;
bool testMKTimp(false);
if(kineticTheoryProperties_.found("testMKTimp"))
{
testMKTimp = true;
Info << "Modified kinetic theory model - testing implementation (chi=1,eEff=e, ksi=1) " << endl;
}
bool diluteCorrection(false);
if(kineticTheoryProperties_.found("diluteCorrection"))
{
testMKTimp = false;
diluteCorrection = true;
Info << "Modified kinetic theory model - Only dilute correction " << endl;
}
bool denseCorrection(false);
if(kineticTheoryProperties_.found("denseCorrection"))
{
testMKTimp = false;
diluteCorrection = false;
denseCorrection = true;
Info << "Modified kinetic theory model - Only dense correction " << endl;
}
bool frictionBlending(false);
if(kineticTheoryProperties_.found("frictionBlending"))
{
frictionBlending = true;
Info << "Modified kinetic theory model - Include Friction Blneding " << endl;
}
if(decomposePp_) Info << "Decompose Pp into Pp - PpStar " << endl;
bool verboseMKT(false);
if(kineticTheoryProperties_.found("verboseMKT")) verboseMKT = true;
const scalar Pi = constant::mathematical::pi;
const scalar sqrtPi = sqrt(constant::mathematical::pi);
const scalar constSMALL = 1.e-06; //1.e-06; 1.e-03;
// Read from dictionary
muFric_ = readScalar(kineticTheoryProperties_.lookup("muFriction"));
eEff_ = e_ - 3.0 / 2.0 * muFric_ * exp(-3.0 * muFric_);
// If only test MKT implementation
if(testMKTimp) eEff_ = e_;
alphaf_ = readScalar(kineticTheoryProperties_.lookup("alphaDiluteInertialUpperLimit"));
alphac_ = readScalar(kineticTheoryProperties_.lookup("alphaCritical"));
alphad_ = readScalar(kineticTheoryProperties_.lookup("alphaDelta"));
upsilons_ = readScalar(kineticTheoryProperties_.lookup("yieldStressRatio"));
// Model parameters
dimensionedScalar I0(0.2); // Table 2, p.15
dimensionedScalar const_alpha(0.36); // Table 2, p.15
dimensionedScalar const_alpha1(0.06); // Table 2, p.15
// Calculating the radial distribution function (solid volume fraction is
// limited close to the packing limit, but this needs improvements)
// The solution is higly unstable close to the packing limit.
gs0_ = radialModel_->g0jamming
(
Ua_.mesh(),
//max(alpha, scalar(constSMALL)),
min(max(alpha_, scalar(constSMALL)),alphaMax_ - 0.01), //changed by YG
alphaMax_,
alphad_, ///changed by YG
alphac_
);
// particle pressure - coefficient in front of T (Eq. 1, p. 3)
volScalarField PsCoeff // -> rho_p * H
(
granularPressureModel_->granularPressureCoeff
(
alpha_,
gs0_,
rhoa_,
e_
)
);
PsCoeff.max(1.0e-15);
// PsCoeff.min(1.0e+10);
// PsCoeff.max(-1.0e+10);
// Solid kinetic+collisional viscosity mua_ = nu_k^star + nu_c^star, Eq. 8,9, p.4
// If Garzo-Dufty viscosity is used (viscosity is dimensionless), there is issue with dimension of mu1
mua_ = viscosityModel_->mua(alpha_, Theta_, gs0_, rhoa_, da_, e_);
// Solid bulk viscosity mua_ = nu_k^star + nu_c^star, Eq. 10, p.4
// If Garzo-Dufty viscosity is used (viscosity is dimensionless), there is issue with dimension of mu1
// Create dimensionedScalar
dimensionedScalar viscDim("zero", dimensionSet(1, -1, -1, 0, 0), 1.0);
lambda_ = viscDim * 384.0 / ( 25.0 * Pi ) * ( 1.0 + e_ ) * alpha_ * alpha_ * gs0_ ;
//lambda_ = (4.0/3.0)*sqr(alpha_)*rhoa_*da_*gs0_*(1.0+e_)*sqrt(Theta_)/sqrtPi;
volScalarField ratioBulkShearVisc(lambda_/(mua_+lambda_));
// J Eq.5, p3
volScalarField J_( 5.0 * sqrtPi / 96.0 * ( mua_ + lambda_ ) / viscDim ); // Dimension issue
// K Eq.6, p3
volScalarField K_(12.0/sqrtPi*alpha_*alpha_*gs0_*(1.0-e_*e_));
// K' Eq.26, p8 modified dissipation due to friction
volScalarField Kmod_(K_*(1.0 - eEff_*eEff_)/(1.0 - e_*e_));
// M Eq.30 p.9
volScalarField M_( max( J_ / max( Kmod_, constSMALL) , const_alpha1 / sqrt( max(alphac_ - alpha_, constSMALL) ) ) );
// Shear stress rate tensor
volTensorField dU(gradUat.T());
volSymmTensorField D(symm(dU));
// Shear stress rate (gammaDot)
volScalarField gammaDot(sqrt(2.*magSqr(D)));
dimensionedScalar gammaDotSmall("gammaDotSmall",dimensionSet(0 , 0 , -1 , 0 , 0, 0, 0), constSMALL);
// Dilute inertia temperature Eq.24, p8
volScalarField ThetaDil_ = ( J_ / max ( Kmod_ , 1e-1 ) ) * ( gammaDot * da_ ) * ( gammaDot * da_ );
// Dense inertia temperature Eq.27, p8
// volScalarField ThetaDense_ = const_alpha1 * ( gammaDot * da_ ) * ( gammaDot * da_ )
// / sqrt( max(alphac_ - alpha_, constSMALL) );
volScalarField ThetaDense_ = const_alpha1 * ( gammaDot * da_ ) * ( gammaDot * da_ )
/ sqrt( max(alphac_ - alpha_, alphad_) )
+ max(alpha_ - (alphac_ - alphad_),0.0) * 0.5 *const_alpha1*( gammaDot * da_ ) * ( gammaDot * da_)*pow(alphad_,-1.5);
// Theta
Theta_ = max(ThetaDil_,ThetaDense_) ;
if(testMKTimp || diluteCorrection) Theta_ = ThetaDil_;
if(denseCorrection) Theta_ = ThetaDense_;
// Limit granular temperature
Theta_.max(1.0e-15);
Theta_.min(1.0e+3);
// Particle pressure
pa_ = PsCoeff * Theta_;
if(frictionBlending)
{
/* volScalarField pf = frictionalStressModel_->frictionalPressure
(
alpha_,
alphaMinFriction_-0.001,
alphaMax_,
Fr_,
eta_,
p_
);
pa_ = pa_ + pf;
*/
// pa_ =pa_ + dimensionedScalar("1e24", dimensionSet(1, -1, -2, 0, 0), Fr_.value())*pow(max(alpha_ - (alphaMinFriction_), scalar(0)), 2/3);
pa_ =pa_ + dimensionedScalar("5810", dimensionSet(1, 0, -2, 0, 0), 581.0)/da_*pow(max(alpha_ - (alphaMinFriction_-0.0), scalar(0)), 2.0/3.0);
// pa_ =pa_ + dimensionedScalar("4.7e9", dimensionSet(1, -1, -2, 0, 0), 4.7e9)*pow(max(alpha_ - (alphaMinFriction_-0.0), scalar(0)), 1.56);
// forAll(alpha_, cellI)
// {
// if(alpha_[cellI] >= (alphaMinFriction_.value()-0.00001))
// {
// pa_[cellI] = pa_[cellI] + 581.0/da_.value()*pow(alpha_[cellI] - (alphaMinFriction_.value()-0.00001), 2.0/3.0);
// }
// }
}
// Psi Eq.32, p.12
dimensionedScalar psi(1.0 + 3.0/10.0*pow((1.0-e_*e_),-1.5)*(1.0-exp(-8.0*muFric_)));
if(testMKTimp) psi = 1.0;
// Shear stress ratio in dilute regime, Eq.33, p.12
dimensionedScalar paSmall("paSmall",dimensionSet(1, -1, -2, 0, 0), constSMALL);
volScalarField inertiaNumber( gammaDot * da_ / sqrt( (pa_ + paSmall) / rhoa_ ) );
// Modified inertia number Eq.35, p.13
volScalarField modInertiaNumber( inertiaNumber / max( alpha_, constSMALL ) );
// Model parameters
volScalarField chi( 1.0 / ( pow( I0 / max( modInertiaNumber,constSMALL ) , 1.5 ) + 1.0 ));
if(testMKTimp || diluteCorrection) chi = max( modInertiaNumber,constSMALL ) / max( modInertiaNumber,constSMALL ) ;
if(denseCorrection) chi= modInertiaNumber - modInertiaNumber;
// Beta + Sigma_tau Eq.49 p.14
volScalarField beta(alpha_ * psi * J_ * sqrt( K_ /( max ( (Kmod_ * ( PsCoeff / rhoa_)), constSMALL ) ) ) );
volScalarField sigmaTau( const_alpha / max( beta, constSMALL ) + ( 1 - const_alpha / max( beta, constSMALL ) ) * chi);
// Sigma_gamma Eq.51 p.14
volScalarField sigmaGamma( beta * sqrt( PsCoeff/rhoa_ ) / max( ( Kmod_ * M_ ), constSMALL ) * sigmaTau);
// dissipation
volScalarField gammaCoeff
(
// van Wachem (Eq. 3.24, p.50) 12.0*(1.0 - sqr(e_))*sqr(alpha_)*rhoa_*gs0_*(1.0/da_)*ThetaSqrt/sqrtPi
// Chialvo & Sundaresan Eq.50 p.14
//rhoa_ / da_ * Kmod_ * Theta_ * sqrt(Theta_) * sigmaGamma
rhoa_ / da_ * Kmod_ * sqrt(Theta_) * sigmaGamma
);
// Blending function
volScalarField func_B( const_alpha + ( beta-const_alpha ) * chi );
// Shear stress ratio
upsilon_ = upsilons_ * (1 - chi) + func_B * modInertiaNumber;
// Shear stress
volSymmTensorField S( D - 1./3.*tr(D)*I );
volSymmTensorField hatS( 2. * S / max( gammaDot, gammaDotSmall ) );
// Shear stress based on pressure and ratio
tau_ = pa_ * upsilon_ * hatS;
// Viscosity
mua_ = ( pa_ * upsilon_ ) / (max( gammaDot, gammaDotSmall )) ;
// Divide by alpha (to be consistent with OpenFOAM implementation)
/* mua_ /= (fvc::average(alpha_) + scalar(0.001));
tau_ /= (fvc::average(alpha_) + scalar(0.001));
lambda_ /= (fvc::average(alpha_) + scalar(0.001)); */
mua_ /= max(alpha_, scalar(constSMALL));
// Limit mua
mua_.min(3e+02);
mua_.max(0.0);
// Limit lambda
lambda_ = mua_ * ratioBulkShearVisc;
// Limit shear stress
tau_ = mua_ * gammaDot * hatS;
// tau_ /= max(alpha_, scalar(constSMALL));
// lambda_ /= max(alpha_, scalar(constSMALL));
//mua_ /= max(alpha_, scalar(constSMALL));
//tau_ /= max(alpha_, scalar(constSMALL));
//lambda_ /= max(alpha_, scalar(constSMALL));
if(verboseMKT)
{
#include "verboseMKT.H"
}
//-AO, YG - Decompose particle pressure, Sundar's idea
if(decomposePp_)
{
pa_ /= (fvc::average(alpha_) + scalar(0.001));
//pa_.correctBoundaryConditions();
pa_.max(1.0e-15);
}
}
else
{
return;
}
}
int main(int argc, char *argv[])
{
#include "setRootCase.H"
#include "createTime.H"
#include "createMesh.H"
#include "createFields.H"
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
const double epsilon = 0.000001;
double s_residual = 2*epsilon,
Theta_residual = 2*epsilon,
residual_max = 2*epsilon,
div_s_MAX = 2*epsilon;
Info<< "\nStarting time loop\n" << endl;
while (runTime.loop() && epsilon < residual_max)
{
Info << "Time = " << runTime.timeName() << nl << endl;
fvVectorMatrix sEqn
(
fvm::ddt(s)
- fvm::laplacian(nu, s)
);
// save previous s-field
volVectorField s_old = s;
// Update s
solve(sEqn == -fvc::grad(Theta));
// calculate residual
s_residual = max(mag(s_old-s)).value();
volScalarField div_s = fvc::div(s);
div_s_MAX = max(mag(div_s)).value();
// Info << "div_s.internalField()[1]: " << div_s.internalField()[1] << endl;
fvScalarMatrix ThetaEqn
(
(1/beta)*fvm::ddt(Theta)
);
//ThetaEqn.setReference(ThetaRefCell, ThetaRefValue);
// save previous Theta-field
volScalarField Theta_old = Theta;
// Update Theta
solve(ThetaEqn == -fvc::div(s));
// calculate residual
Theta_residual = max(mag(Theta_old-Theta)).value();
Info << "Residual in s: " << s_residual << endl <<
"Residual in Theta: " << Theta_residual << endl <<
"div_s.internalField_MAX: " << div_s_MAX << endl;
residual_max = max(Theta_residual, max(s_residual, div_s_MAX));
runTime.write();
Info<< "ExecutionTime = " << runTime.elapsedCpuTime() << " s"
<< " ClockTime = " << runTime.elapsedClockTime() << " s"
<< nl << endl;
}
Info<< "End\n" << endl;
return 0;
}
void Foam::kineticTheoryModel::solve(const volTensorField& gradUat, const volScalarField& kb,const volScalarField& epsilonb,const volScalarField& nutf,const dimensionedScalar& B, const dimensionedScalar& tt)
{
if (!kineticTheory_)
{
return;
}
const scalar sqrtPi = sqrt(constant::mathematical::pi);
// surfaceScalarField phi = 1.5*rhoa_*phia_*fvc::interpolate(max(alpha_,DiluteCut_));
surfaceScalarField phi = 1.5*rhoa_*phia_*fvc::interpolate((alpha_+1e-8));
volTensorField dUU = gradUat.T(); //that is fvc::grad(Ua_);
volSymmTensorField D = symm(dUU); //0.5*(dU + dU.T)
volTensorField dU = dUU;//dev(dUU);
// NB, drag = K*alpha*beta,
// (the alpha and beta has been extracted from the drag function for
// numerical reasons)
// this is inconsistent with momentum equation, since the following form missed
// the drift velocity
volScalarField Ur_ = mag(Ua_ - Ub_);
// volScalarField Ur = mag(Ua_ - Ub_ + nuft/(max(alpha_,1e-9)*(1.0-alpha_))*fvc::grad(alpha_));
// volScalarField Ur = mag(Ua_ - Ub_ + nuft/(max(alpha_,1e-9))*fvc::grad(alpha_));
volScalarField betaPrim = (1.0 - alpha_)*draga_.K(Ur_);
// Calculating the radial distribution function (solid volume fraction is
// limited close to the packing limit, but this needs improvements)
// The solution is higly unstable close to the packing limit.
gs0_ = radialModel_->g0
(
min(alpha_, alphaMax_ - 1e-9),
alphaMax_
);
// particle pressure - coefficient in front of Theta (Eq. 3.22, p. 45)
volScalarField PsCoeff = granularPressureModel_->granularPressureCoeff
(
alpha_,
// (alpha_+1e-12),
gs0_,
rhoa_,
e_
);
// 'thermal' conductivity (Table 3.3, p. 49)
kappa_ = conductivityModel_->kappa(alpha_, Theta_, gs0_, rhoa_, da_, e_);
// particle viscosity (Table 3.2, p.47)
// mua_ = viscosityModel_->mua(alpha_, Theta_, gs0_, rhoa_, da_, e_);
dimensionedScalar Tsmall
(
"small",
dimensionSet(0 , 2 ,-2 ,0 , 0, 0, 0),
scalar(1.0e-40)
);
dimensionedScalar TsmallEqn
(
"small",
dimensionSet(0 , 2 ,-2 ,0 , 0, 0, 0),
scalar(1.0e-30)
);
dimensionedScalar TsmallSqrt = sqrt(Tsmall);
volScalarField ThetaSqrt = sqrt(Theta_);
volScalarField muaa = rhoa_*da_*
(
(4.0/5.0)*sqr(alpha_)*gs0_*(1.0 + e_)/sqrtPi
+ (1.0/15.0)*sqrtPi*gs0_*(1.0 + e_)*sqr(alpha_)
+ (1.0/6.0)*sqrtPi*alpha_
+ (10.0/96.0)*sqrtPi/((1.0 + e_)*gs0_)
)/(ThetaSqrt+TsmallSqrt);
// dissipation (Eq. 3.24, p.50)
volScalarField gammaCoeff =
3.0*(1.0 - sqr(e_))*sqr(alpha_)*rhoa_*gs0_*((4.0/da_)*ThetaSqrt/sqrtPi-tr(D));
// 3.0*(1.0 - sqr(e_))*sqr(alpha_)*rhoa_*gs0_*((4.0/da_)*ThetaSqrt/sqrtPi);
// Eq. 3.25, p. 50 Js = J1 - J2
// The following is to calculate parameter tmf_ in u_f*u_s correlation
dimensionedScalar Tpsmall_
(
"Tpsmall_",
dimensionSet(1, -1, -3, 0, 0, 0, 0),
scalar(1e-30)
);
volScalarField tmf_ = Foam::exp(-B*rhoa_*scalar(6.0)*epsilonb/(max(kb*betaPrim,Tpsmall_)));
volScalarField J1 = 3.0*alpha_*betaPrim;
volScalarField J2 =
0.25*alpha_*sqr(betaPrim)*da_*sqr(Ur_)
/(rhoa_*sqrtPi*(max(ThetaSqrt,TsmallSqrt)));
// bulk viscosity p. 45 (Lun et al. 1984).
// lambda_ = (4.0/3.0)*sqr(alpha_)*rhoa_*da_*gs0_*(1.0+e_)*ThetaSqrt/sqrtPi;
volScalarField lambdaa = (4.0/3.0)*sqr(alpha_)*rhoa_*da_*gs0_*(1.0+e_)/(sqrtPi*(ThetaSqrt+TsmallSqrt));
// stress tensor, Definitions, Table 3.1, p. 43
// volSymmTensorField tau = 2.0*mua_*D + (lambda_ - (2.0/3.0)*mua_)*tr(D)*I;
volSymmTensorField tau = scalar(2.0)*muaa*D + (lambdaa - (scalar(2.0)/scalar(3.0))*muaa)*tr(D)*I;
dimensionedScalar alphasmall
(
"alphasmall",
dimensionSet(0, 0, 0, 0, 0, 0, 0),
scalar(1e-9)
);
if (!equilibrium_)
{
// construct the granular temperature equation (Eq. 3.20, p. 44)
// NB. note that there are two typos in Eq. 3.20
// no grad infront of Ps
// wrong sign infront of laplacian
fvScalarMatrix ThetaEqn
(
fvm::ddt(1.5*(alpha_+1e-8)*rhoa_, Theta_)
+ fvm::div(phi, Theta_, "div(phi,Theta)")
==
//Ps term.
fvm::SuSp(-((PsCoeff*I) && dU), Theta_)
//production due to shear.
+ fvm::SuSp((tau && dU),Theta_)
// + (tau && dU)
// granular temperature conduction.
+ fvm::laplacian(kappa_, Theta_, "laplacian(kappa,Theta)")
//energy disipation due to inelastic collision.
+ fvm::SuSp(-gammaCoeff, Theta_)
//
+ fvm::SuSp(-J1, Theta_)
+ (scalar(2.0)/scalar(3.0))*J1*tmf_*kb
);
ThetaEqn.relax();
ThetaEqn.solve();
}
else
{
// equilibrium => dissipation == production
// Eq. 4.14, p.82
volScalarField K1 = 2.0*(1.0 + e_)*rhoa_*gs0_;
volScalarField K3 = 0.5*da_*rhoa_*
(
(sqrtPi/(3.0*(3.0-e_)))
*(1.0 + 0.4*(1.0 + e_)*(3.0*e_ - 1.0)*alpha_*gs0_)
+1.6*alpha_*gs0_*(1.0 + e_)/sqrtPi
);
volScalarField K2 =
4.0*da_*rhoa_*(1.0 + e_)*alpha_*gs0_/(3.0*sqrtPi) - 2.0*K3/3.0;
volScalarField K4 = 12.0*(1.0 - sqr(e_))*rhoa_*gs0_/(da_*sqrtPi);
volScalarField trD = tr(D);
volScalarField tr2D = sqr(trD);
volScalarField trD2 = tr(D & D);
volScalarField t1 = K1*alpha_;
volScalarField l1 = -t1*trD;
volScalarField l2 = sqr(t1)*tr2D;
volScalarField l3 = 4.0*K4*alpha_*(2.0*K3*trD2 + K2*tr2D);
Theta_ = sqr((l1 + sqrt(l2 + l3))/(2.0*(alpha_ + 1.0e-4)*K4));
}
forAll(alpha_, cellk)
{
// for initial stability
if(tt.value() <= ttzero.value() && alpha_[cellk]>= 0)
{
Theta_[cellk] = 0.0*Theta_[cellk];
}
if(tt.value() <= ttone.value() && alpha_[cellk] >= 0.5)
{
Theta_[cellk] = 0.0*Theta_[cellk];
}
// to cut off in dilute limit, set DiluteCut < 0 to turn this off
if(tt.value()>=ttone.value() && alpha_[cellk] <= DiluteCut_.value())
{
Theta_[cellk] = 0.0*Theta_[cellk];
}
}
Theta_.max(0.0e-40);
Theta_.min(MaxTheta);
// need to update after solving Theta Equation.
PsCoeff = granularPressureModel_->granularPressureCoeff
(
alpha_,
gs0_,
rhoa_,
e_
);
// update bulk viscosity and shear viscosity
// bulk viscosity p. 45 (Lun et al. 1984).
lambda_ = (4.0/3.0)*sqr(alpha_)*rhoa_*da_*gs0_*(1.0+e_)*sqrt(Theta_)/sqrtPi;
// particle viscosity (Table 3.2, p.47)
mua_ = viscosityModel_->mua(alpha_, Theta_, gs0_, rhoa_, da_, e_);
pf = frictionalStressModel_->frictionalPressure
(
alpha_,
alphaMinFriction_,
alphaMax_,
Fr_,
eta_,
p_
);
// yes, after solving Theta, because frictional stress is not a part of kinetic theoty
// PsCoeff = PsCoeff + pf/(max(Theta_,Tsmall));
PsCoeff.min(1e+6);
PsCoeff.max(-1e+6);
// update particle pressure, just from the kinetic part
pa_ = PsCoeff*Theta_;
// frictional shear stress, Eq. 3.30, p. 52
volScalarField muf = frictionalStressModel_->muf
(
alpha_,
Theta_,
alphaMinFriction_,
alphaMax_,
pf,
D,
phi_
);
// add frictional stress
mua_ = mua_+muf;
mua_.max(0.0);
Info<< "kinTheory: max(Theta) = " << max(Theta_).value() <<" min(Theta) = "<<min(Theta_).value()<< endl;
volScalarField ktn = mua_/rhoa_;
Info<< "kinTheory: min(nua) = " << min(ktn).value()
<< ", max(nua) = " << max(ktn).value() << endl;
Info<< "kinTheory: min(pa) = " << min(pa_).value()
<< ", max(pa) = " << max(pa_).value() << endl;
}
