m_bubbles Module Reference

This module contains procedures shared by the ensemble-averaged and volume-averaged bubble models. More...

Functions/Subroutines
elemental real(wp) function	f_rddot (frho, fp, fr, fv, fr0, fpb, fpbdot, alf, fntait, fbtait, f_bub_adv_src, f_divu, fcson)
	Function that computes the bubble radial acceleration based on bubble models.
elemental real(wp) function	f_cpbw (fr0, fr, fv, fpb)
	Function that computes that bubble wall pressure for Gilmore bubbles.
elemental real(wp) function	f_h (fcpbw, fcpinf, fntait, fbtait)
	Function that computes the bubble enthalpy.
elemental real(wp) function	f_cgas (fcpinf, fntait, fbtait, fh)
	Function that computes the sound speed for the bubble.
elemental real(wp) function	f_cpinfdot (frho, fp, falf, fntait, fbtait, advsrc, divu)
	Function that computes the time derivative of the driving pressure.
elemental real(wp) function	f_hdot (fcpbw, fcpinf, fcpinf_dot, fntait, fbtait, fr, fv, fr0, fpbdot)
	Function that computes the time derivative of the enthalpy.
elemental real(wp) function	f_rddot_rp (fcp, frho, fr, fv, fcpbw)
	Function that computes the bubble radial acceleration for Rayleigh-Plesset bubbles.
elemental real(wp) function	f_rddot_g (fcpbw, fr, fv, fh, fhdot, fcgas, fntait, fbtait)
	Function that computes the bubble radial acceleration.
elemental real(wp) function	f_cpbw_km (fr0, fr, fv, fpb)
	Function that computes the bubble wall pressure for Keller–Miksis bubbles.
elemental real(wp) function	f_rddot_km (fpbdot, fcp, fcpbw, frho, fr, fv, fr0, fc)
	Function that computes the bubble radial acceleration for Keller–Miksis bubbles.
elemental subroutine	s_bwproperty (pb_in, ir0, chi_vw_out, k_mw_out, rho_mw_out)
	Subroutine that computes bubble wall properties for vapor bubbles.
elemental subroutine	s_vflux (fr, fv, fpb, fmass_v, ir0, vflux, fmass_n, fbeta_c, fr_m, fgamma_m)
	Function that computes the vapour flux.
elemental real(wp) function	f_bpres_dot (fvflux, fr, fv, fpb, fmass_v, ir0, fbeta_t, fr_m, fgamma_m)
	Function that computes the time derivative of the internal bubble pressure.
subroutine	s_advance_step (frho, fp, fr, fv, fr0, fpb, fpbdot, alf, fntait, fbtait, f_bub_adv_src, f_divu, bub_id, fmass_v, fmass_n, fbeta_c, fbeta_t, fcson, adap_dt_stop)
	Adaptive time stepping routine for subgrid bubbles (See Heirer, E. Hairer S.P.Nørsett G. Wanner, Solving Ordinary Differential Equations I, Chapter II.4)
subroutine	s_initial_substep_h (frho, fp, fr, fv, fr0, fpb, fpbdot, alf, fntait, fbtait, f_bub_adv_src, f_divu, fcson, h)
	Choose the initial time step size for the adaptive time stepping routine (See Heirer, E. Hairer S.P.Nørsett G. Wanner, Solving Ordinary Differential Equations I, Chapter II.4)
subroutine	s_advance_substep (err, frho, fp, fr, fv, fr0, fpb, fpbdot, alf, fntait, fbtait, f_bub_adv_src, f_divu, bub_id, fmass_v, fmass_n, fbeta_c, fbeta_t, fcson, h, myr_tmp, myv_tmp, mypb_tmp, mymv_tmp)
	Integrate bubble variables over the given time step size, h, using a third-order accurate embedded Runge–Kutta scheme.
elemental subroutine	s_advance_el (fr_tmp, fv_tmp, fpb_tmp, fmv_tmp, bub_id, fmass_n, fbeta_c, fbeta_t, fdpbdt_tmp, advance_el)
	Changes of pressure and vapor mass in the lagrange bubbles.

Variables
real(wp)	chi_vw
	Bubble wall properties (Ando 2010)
real(wp)	k_mw
	Bubble wall properties (Ando 2010)
real(wp)	rho_mw
	Bubble wall properties (Ando 2010)

Detailed Description

This module contains procedures shared by the ensemble-averaged and volume-averaged bubble models.

Function/Subroutine Documentation

f_bpres_dot()

elemental real(wp) function m_bubbles::f_bpres_dot	(	real(wp), intent(in)	fvflux,
		real(wp), intent(in)	fr,
		real(wp), intent(in)	fv,
		real(wp), intent(in)	fpb,
		real(wp), intent(in)	fmass_v,
		integer, intent(in)	ir0,
		real(wp), intent(in), optional	fbeta_t,
		real(wp), intent(in), optional	fr_m,
		real(wp), intent(in), optional	fgamma_m )

Function that computes the time derivative of the internal bubble pressure.

Parameters

fvflux	Vapour flux
fR	Current bubble radius
fV	Current bubble velocity
fpb	Current internal bubble pressure
fmass_v	Current mass of vapour
iR0	Bubble size index (EE) or bubble identifier (EL)
fbeta_t	Mass transfer coefficient (EL)
fR_m	Mixture gas constant (EL)
fgamma_m	Mixture gamma (EL)

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f_cgas()

elemental real(wp) function m_bubbles::f_cgas	(	real(wp), intent(in)	fcpinf,
		real(wp), intent(in)	fntait,
		real(wp), intent(in)	fbtait,
		real(wp), intent(in)	fh )

Function that computes the sound speed for the bubble.

Parameters

fCpinf	Driving bubble pressure
fntait	Tait EOS parameter
fBtait	Tait EOS parameter
fH	Bubble enthalpy

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f_cpbw()

elemental real(wp) function m_bubbles::f_cpbw	(	real(wp), intent(in)	fr0,
		real(wp), intent(in)	fr,
		real(wp), intent(in)	fv,
		real(wp), intent(in)	fpb )

Function that computes that bubble wall pressure for Gilmore bubbles.

Parameters

fR0	Equilibrium bubble radius
fR	Current bubble radius
fV	Current bubble velocity
fpb	Internal bubble pressure

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f_cpbw_km()

elemental real(wp) function m_bubbles::f_cpbw_km	(	real(wp), intent(in)	fr0,
		real(wp), intent(in)	fr,
		real(wp), intent(in)	fv,
		real(wp), intent(in)	fpb )

Function that computes the bubble wall pressure for Keller–Miksis bubbles.

Parameters

fR0	Equilibrium bubble radius
fR	Current bubble radius
fV	Current bubble velocity
fpb	Internal bubble pressure

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f_cpinfdot()

elemental real(wp) function m_bubbles::f_cpinfdot	(	real(wp), intent(in)	frho,
		real(wp), intent(in)	fp,
		real(wp), intent(in)	falf,
		real(wp), intent(in)	fntait,
		real(wp), intent(in)	fbtait,
		real(wp), intent(in)	advsrc,
		real(wp), intent(in)	divu )

Function that computes the time derivative of the driving pressure.

Parameters

fRho	Local liquid density
fP	Local pressure
falf	Local void fraction
fntait	Tait EOS parameter
fBtait	Tait EOS parameter
advsrc	Advection equation source term
divu	Divergence of velocity

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f_h()

elemental real(wp) function m_bubbles::f_h	(	real(wp), intent(in)	fcpbw,
		real(wp), intent(in)	fcpinf,
		real(wp), intent(in)	fntait,
		real(wp), intent(in)	fbtait )

Function that computes the bubble enthalpy.

Parameters

fCpbw	Bubble wall pressure
fCpinf	Driving bubble pressure
fntait	Tait EOS parameter
fBtait	Tait EOS parameter

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f_hdot()

elemental real(wp) function m_bubbles::f_hdot	(	real(wp), intent(in)	fcpbw,
		real(wp), intent(in)	fcpinf,
		real(wp), intent(in)	fcpinf_dot,
		real(wp), intent(in)	fntait,
		real(wp), intent(in)	fbtait,
		real(wp), intent(in)	fr,
		real(wp), intent(in)	fv,
		real(wp), intent(in)	fr0,
		real(wp), intent(in)	fpbdot )

Function that computes the time derivative of the enthalpy.

Parameters

fCpbw	Bubble wall pressure
fCpinf	Driving bubble pressure
fCpinf_dot	Time derivative of the driving pressure
fntait	Tait EOS parameter
fBtait	Tait EOS parameter
fR	Current bubble radius
fV	Current bubble velocity
fR0	Equilibrium bubble radius
fpbdot	Time derivative of the internal bubble pressure

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f_rddot()

elemental real(wp) function m_bubbles::f_rddot	(	real(wp), intent(in)	frho,
		real(wp), intent(in)	fp,
		real(wp), intent(in)	fr,
		real(wp), intent(in)	fv,
		real(wp), intent(in)	fr0,
		real(wp), intent(in)	fpb,
		real(wp), intent(in)	fpbdot,
		real(wp), intent(in)	alf,
		real(wp), intent(in)	fntait,
		real(wp), intent(in)	fbtait,
		real(wp), intent(in)	f_bub_adv_src,
		real(wp), intent(in)	f_divu,
		real(wp), intent(in)	fcson )

Function that computes the bubble radial acceleration based on bubble models.

Parameters

fRho	Current density
fP	Current driving pressure
fR	Current bubble radius
fV	Current bubble velocity
fR0	Equilibrium bubble radius
fpb	Internal bubble pressure
fpbdot	Time-derivative of internal bubble pressure
alf	bubble volume fraction
fntait	Tait EOS parameter
fBtait	Tait EOS parameter
f_bub_adv_src	Source for bubble volume fraction
f_divu	Divergence of velocity
fCson	Speed of sound from fP (EL)

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f_rddot_g()

elemental real(wp) function m_bubbles::f_rddot_g	(	real(wp), intent(in)	fcpbw,
		real(wp), intent(in)	fr,
		real(wp), intent(in)	fv,
		real(wp), intent(in)	fh,
		real(wp), intent(in)	fhdot,
		real(wp), intent(in)	fcgas,
		real(wp), intent(in)	fntait,
		real(wp), intent(in)	fbtait )

Function that computes the bubble radial acceleration.

Parameters

fCpbw	Bubble wall pressure
fR	Current bubble radius
fV	Current bubble velocity
fH	Current enthalpy
fHdot	Current time derivative of the enthalpy
fcgas	Current gas sound speed
fntait	Tait EOS parameter
fBtait	Tait EOS parameter

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f_rddot_km()

elemental real(wp) function m_bubbles::f_rddot_km	(	real(wp), intent(in)	fpbdot,
		real(wp), intent(in)	fcp,
		real(wp), intent(in)	fcpbw,
		real(wp), intent(in)	frho,
		real(wp), intent(in)	fr,
		real(wp), intent(in)	fv,
		real(wp), intent(in)	fr0,
		real(wp), intent(in)	fc )

Function that computes the bubble radial acceleration for Keller–Miksis bubbles.

Parameters

fpbdot	Time-derivative of internal bubble pressure
fCp	Driving pressure
fCpbw	Bubble wall pressure
fRho	Current density
fR	Current bubble radius
fV	Current bubble velocity
fR0	Equilibrium bubble radius
fC	Current sound speed

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f_rddot_rp()

elemental real(wp) function m_bubbles::f_rddot_rp	(	real(wp), intent(in)	fcp,
		real(wp), intent(in)	frho,
		real(wp), intent(in)	fr,
		real(wp), intent(in)	fv,
		real(wp), intent(in)	fcpbw )

Function that computes the bubble radial acceleration for Rayleigh-Plesset bubbles.

Parameters

fCp	Driving pressure
fRho	Current density
fR	Current bubble radius
fV	Current bubble velocity
fR0	Equilibrium bubble radius
fCpbw	Boundary wall pressure

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s_advance_el()

elemental subroutine m_bubbles::s_advance_el	(	real(wp), intent(in)	fr_tmp,
		real(wp), intent(in)	fv_tmp,
		real(wp), intent(in)	fpb_tmp,
		real(wp), intent(in)	fmv_tmp,
		integer, intent(in)	bub_id,
		real(wp), intent(in)	fmass_n,
		real(wp), intent(in)	fbeta_c,
		real(wp), intent(in)	fbeta_t,
		real(wp), intent(inout)	fdpbdt_tmp,
		real(wp), intent(out)	advance_el )

Changes of pressure and vapor mass in the lagrange bubbles.

Parameters

bub_id	Bubble identifier
fmass_n	Current mass of gas
fbeta_c	Mass transfer coefficient
fbeta_t	Heat transfer coefficient
fR_tmp	Bubble radius
fV_tmp	Bubble radial velocity
fPb_tmp	Internal bubble pressure
fMv_tmp	Mass of vapor in the bubble
fdPbdt_tmp	Rate of change of the internal bubble pressure
fdMvdt_tmp	Rate of change of the mass of vapor in the bubble

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s_advance_step()

subroutine m_bubbles::s_advance_step	(	real(wp), intent(in)	frho,
		real(wp), intent(in)	fp,
		real(wp), intent(inout)	fr,
		real(wp), intent(inout)	fv,
		real(wp), intent(in)	fr0,
		real(wp), intent(inout)	fpb,
		real(wp), intent(in)	fpbdot,
		real(wp), intent(in)	alf,
		real(wp), intent(in)	fntait,
		real(wp), intent(in)	fbtait,
		real(wp), intent(in)	f_bub_adv_src,
		real(wp), intent(in)	f_divu,
		integer, intent(in)	bub_id,
		real(wp), intent(inout)	fmass_v,
		real(wp), intent(in)	fmass_n,
		real(wp), intent(in)	fbeta_c,
		real(wp), intent(in)	fbeta_t,
		real(wp), intent(in)	fcson,
		integer, intent(inout)	adap_dt_stop )

Adaptive time stepping routine for subgrid bubbles (See Heirer, E. Hairer S.P.Nørsett G. Wanner, Solving Ordinary Differential Equations I, Chapter II.4)

Parameters

fRho	Current density
fP	Current driving pressure
fR	Current bubble radius
fV	Current bubble radial velocity
fR0	Equilibrium bubble radius
fpb	Internal bubble pressure
fpbdot	Time-derivative of internal bubble pressure
alf	bubble volume fraction
fntait	Tait EOS parameter
fBtait	Tait EOS parameter
f_bub_adv_src	Source for bubble volume fraction
f_divu	Divergence of velocity
bub_id	Bubble identifier (EL)
fmass_v	Current mass of vapour (EL)
fmass_n	Current mass of gas (EL)
fbeta_c	Mass transfer coefficient (EL)
fbeta_t	Heat transfer coefficient (EL)
fCson	Speed of sound (EL)
adap_dt_stop	Fail-safe exit if max iteration count reached

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s_advance_substep()

subroutine m_bubbles::s_advance_substep	(	real(wp), intent(out)	err,
		real(wp), intent(in)	frho,
		real(wp), intent(in)	fp,
		real(wp), intent(in)	fr,
		real(wp), intent(in)	fv,
		real(wp), intent(in)	fr0,
		real(wp), intent(in)	fpb,
		real(wp), intent(in)	fpbdot,
		real(wp), intent(in)	alf,
		real(wp), intent(in)	fntait,
		real(wp), intent(in)	fbtait,
		real(wp), intent(in)	f_bub_adv_src,
		real(wp), intent(in)	f_divu,
		integer, intent(in)	bub_id,
		real(wp), intent(in)	fmass_v,
		real(wp), intent(in)	fmass_n,
		real(wp), intent(in)	fbeta_c,
		real(wp), intent(in)	fbeta_t,
		real(wp), intent(in)	fcson,
		real(wp), intent(in)	h,
		real(wp), dimension(4), intent(out)	myr_tmp,
		real(wp), dimension(4), intent(out)	myv_tmp,
		real(wp), dimension(4), intent(out)	mypb_tmp,
		real(wp), dimension(4), intent(out)	mymv_tmp )

Integrate bubble variables over the given time step size, h, using a third-order accurate embedded Runge–Kutta scheme.

Parameters

err	Estimated error
fRho	Current density
fP	Current driving pressure
fR	Current bubble radius
fV	Current bubble velocity
fR0	Equilibrium bubble radius
fpb	Internal bubble pressure
fpbdot	Time-derivative of internal bubble pressure
alf	bubble volume fraction
fntait	Tait EOS parameter
fBtait	Tait EOS parameter
f_bub_adv_src	Source for bubble volume fraction
f_divu	Divergence of velocity
bub_id	Bubble identifier (EL)
fmass_v	Current mass of vapour (EL)
fmass_n	Current mass of gas (EL)
fbeta_c	Mass transfer coefficient (EL)
fbeta_t	Heat transfer coefficient (EL)
fCson	Speed of sound (EL)
h	Time step size
myR_tmp	Bubble radius at each stage
myV_tmp	Bubble radial velocity at each stage
myPb_tmp	Internal bubble pressure at each stage (EL)
myMv_tmp	Mass of vapor in the bubble at each stage (EL)

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s_bwproperty()

elemental subroutine m_bubbles::s_bwproperty	(	real(wp), intent(in)	pb_in,
		integer, intent(in)	ir0,
		real(wp), intent(out)	chi_vw_out,
		real(wp), intent(out)	k_mw_out,
		real(wp), intent(out)	rho_mw_out )

Subroutine that computes bubble wall properties for vapor bubbles.

Parameters

	pb	Internal bubble pressure
	iR0	Current bubble size index
[out]	chi_vw_out	Bubble wall properties (Ando 2010)
[out]	k_mw_out	Bubble wall properties (Ando 2010)
[out]	rho_mw_out	Bubble wall properties (Ando 2010)

s_initial_substep_h()

subroutine m_bubbles::s_initial_substep_h	(	real(wp), intent(in)	frho,
		real(wp), intent(in)	fp,
		real(wp), intent(in)	fr,
		real(wp), intent(in)	fv,
		real(wp), intent(in)	fr0,
		real(wp), intent(in)	fpb,
		real(wp), intent(in)	fpbdot,
		real(wp), intent(in)	alf,
		real(wp), intent(in)	fntait,
		real(wp), intent(in)	fbtait,
		real(wp), intent(in)	f_bub_adv_src,
		real(wp), intent(in)	f_divu,
		real(wp), intent(in)	fcson,
		real(wp), intent(out)	h )

Choose the initial time step size for the adaptive time stepping routine (See Heirer, E. Hairer S.P.Nørsett G. Wanner, Solving Ordinary Differential Equations I, Chapter II.4)

Parameters

fRho	Current density
fP	Current driving pressure
fR	Current bubble radius
fV	Current bubble velocity
fR0	Equilibrium bubble radius
fpb	Internal bubble pressure
fpbdot	Time-derivative of internal bubble pressure
alf	bubble volume fraction
fntait	Tait EOS parameter
fBtait	Tait EOS parameter
f_bub_adv_src	Source for bubble volume fraction
f_divu	Divergence of velocity
fCson	Speed of sound (EL)
h	Time step size

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s_vflux()

elemental subroutine m_bubbles::s_vflux	(	real(wp), intent(in)	fr,
		real(wp), intent(in)	fv,
		real(wp), intent(in)	fpb,
		real(wp), intent(in)	fmass_v,
		integer, intent(in)	ir0,
		real(wp), intent(out)	vflux,
		real(wp), intent(in), optional	fmass_n,
		real(wp), intent(in), optional	fbeta_c,
		real(wp), intent(out), optional	fr_m,
		real(wp), intent(out), optional	fgamma_m )

Function that computes the vapour flux.

Parameters

fR	Current bubble radius
fV	Current bubble velocity
fpb
fmass_v	Current mass of vapour
iR0	Bubble size index (EE) or bubble identifier (EL)
fmass_n	Current gas mass (EL)
fbeta_c	Mass transfer coefficient (EL)
fR_m	Mixture gas constant (EL)
fgamma_m	Mixture gamma (EL)

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Variable Documentation

chi_vw

real(wp) m_bubbles::chi_vw

Bubble wall properties (Ando 2010)

k_mw

real(wp) m_bubbles::k_mw

Bubble wall properties (Ando 2010)

rho_mw

real(wp) m_bubbles::rho_mw

Bubble wall properties (Ando 2010)