GfsSourceViscosity

From Gerris

GfsSourceViscosity solves the diffusion equation for the velocity (a vector quantity obviously) while GfsSourceDiffusion solves the diffusion equation for a scalar quantity. In the case of a spatially uniform viscosity the two are strictly equivalent since — in the case of an incompressible fluid — the vector diffusion equation for the velocity can be split in independent scalar diffusion equations for each of the Cartesian components of the velocities.

To sum up, using GfsSourceViscosity is safe in all cases, regardless of whether the viscosity is uniform.

The syntax in parameter files is:

[ GfsSourceVelocity ] [ GfsDiffusion ]

where GfsDiffusion defines the dynamic viscosity of the fluid.

Examples

• Parallel simulation on four processors

  SourceViscosity 0.00078125

• Rayleigh-Taylor instability

  SourceViscosity {} 0.00313

• Collapse of a column of grains

    SourceViscosity {} {
	double Eta = MUF;
	if (P > 0. && D2 > 0.) {
	   double In = sqrt(2.)*D*D2/sqrt(P);
	   double muI = .32 + (.28)*In/(.4 + In);
	   double Etamin = sqrt(pow(D,3));
	   Eta = MAX((muI*P)/(sqrt(2.)*D2), Etamin);
	   Eta = MIN(Eta,10);
	}
	// Classic trick: use harmonic mean for the dynamic viscosity
	return 1./(T/Eta + (1. - T)/MUF);
    } {
	beta = 1 
	tolerance = 1e-4
    }

• Viscous folding of a fluid interface

    SourceViscosity (1/Re)

• Savart--Plateau--Rayleigh instability of a water column

    SourceViscosity 1e-2*MUR(T1)

• Atomisation of a pulsed liquid jet

    SourceViscosity 2.*radius/Re*rho(T)

• Forced isotropic turbulence in a triply-periodic box

  SourceViscosity MU

• Viscous flow past a sphere

    SourceViscosity 1./RE

• Boundary layer on a rotating disk

    SourceViscosity 0.2

• Poiseuille flow

    SourceViscosity 1. { beta = 1 }

• Bagnold flow of a granular material

    SourceViscosity {} {
	double In = sqrt(2.)*D*D2/sqrt(fabs(P));
	double muI = .38 + (.26)*In/(.3 + In);
	return MAX((muI*P)/(sqrt(2.)*D2), 0.);
    } { beta = 1 }

• Poiseuille flow with metric

    SourceViscosity 1. { beta = 1 }

• Creeping Couette flow of Generalised Newtonian fluids

  SourceViscosity {} {
    double mu, ty, N, mumax = 1000.;
    double m;

    switch (MODEL) {
    case 0: /* Newtonian */
      mu = 1.; ty = 0.; N = 1.; break;
    case 1: /* Power-law (shear-thinning) */
      mu = 0.08; ty = 0.; N = 0.5; break;
    case 2: /* Herschel-Bulkley */
      mu = 0.0672; ty = 0.12; N = 0.5; break;
    case 3: /* Bingham */
      mu = 1.; ty = 10.; N = 1.; break;
    }
    if (D2 > 0.)
      m = ty/(2.*D2) + mu*exp ((N - 1.)*log (D2));
    else {
      if (ty > 0. || N < 1.) m = mumax;
      else m = N == 1. ? mu : 0.;
    }
    return MIN (m, mumax);
  } {
    # Crank-Nicholson does not converge for these cases, we need backward Euler
    # (beta = 0.5 -> Crank-Nicholson, beta = 1 -> backward Euler)
    beta = 1
  }

• Momentum conservation for large density ratios

    SourceViscosity mu(T1)

• Hydrostatic balance with solid boundaries and viscosity

    SourceViscosity 1e-2

• Hydrostatic balance with quadratic pressure profile

    SourceViscosity 1e-2

• Coriolis formulation in 3-D

  SourceViscosity 10

• Wind-driven lake

    SourceViscosity 1./400.

• B\'enard--von K\'arm\'an vortex street behind a cylinder translating in a fluid at rest

  SourceViscosity 3.125e-5

• Circular droplet in equilibrium

  SourceViscosity MU

• Axisymmetric spherical droplet in equilibrium

  SourceViscosity MU

• Air-Water capillary wave

  SourceViscosity 0.0182571749236*MU(T1)

• Sessile drop

    SourceViscosity 0.2/(T + 100.*(1. - T))

• Creeping Couette flow between cylinders

    SourceViscosity 1

• Creeping Couette flow between eccentric cylinders

  SourceViscosity 1

• Flow between eccentric cylinders using bipolar coordinates

  SourceViscosity 1.

• Flow between eccentric cylinders on a stretched grid

  SourceViscosity 1

• Equilibrium of a droplet suspended in an electric field

    SourceViscosity Cmu