thermal.h File Reference

#include "poisson.h"
#include "compressible/two-phase.h"

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Data Structures
struct	Thermal
	We then define the corresponding relaxation and residual functions which will be used by the multigrid solver. More...

Macros
#define	kappa(f)   (clamp(f,0.,1.)*(kappa1 - kappa2) + kappa2)
#define	poisson(...)   poisson_thermal(__VA_ARGS__)
	Finally, we overload the poisson() function called by [two-phase.h]() with our new function.

Functions
double	average_temperature (Point point, double p)
	These functions are provided by the Equation Of State.
double	thermal_expansion (Point point)
void	event_defaults (void)
	If the thermal conductivities are not-zero, we need to allocate a new field.
static void	relax_thermal (scalar *al, scalar *bl, int l, void *data)
static double	residual_thermal (scalar *al, scalar *bl, scalar *resl, void *data)
mgstats	poisson_thermal (scalar p, scalar rhs, const vector alpha, const scalar lambda, double tolerance=0., int nrelax=4, int minlevel=0, scalar *res=NULL)
	This is the interface for the coupled solver.
void	event_acceleration (void)
	Event: acceleration (i++)
void	event_end_timestep (void)
	Event: end_timestep (i++)

Variables
scalar	T []
double	kappa1 = 0.
	... and the thermal conductivities and specific heat capacities.
double	kappa2 = 0.
double	cp1 = 0.
double	cp2 = 0.
const vector	kappa0 [] = {0,0,0}
const vector	kappa = kappa0
scalar	Ts []
scalar	lambda1 []
scalar	lambda2 []
scalar	lambda4 []

Macro Definition Documentation

kappa

#define kappa

(

f

)

(clamp(f,0.,1.)*(kappa1 - kappa2) + kappa2)

Definition at line 85 of file thermal.h.

poisson

#define poisson

(

...

)

poisson_thermal(__VA_ARGS__)

Finally, we overload the poisson() function called by [two-phase.h]() with our new function.

Definition at line 243 of file thermal.h.

Function Documentation

average_temperature()

double average_temperature ( Point  point, double  p  )

extern

These functions are provided by the Equation Of State.

See for example the Noble-Able Stiffened-Gas Equation Of State.

Average temperature

Definition at line 99 of file NASG.h.

References b1, b2, clamp(), cp1, cp2, cv1, cv2, f, frho1, frho2, p, PI1, PI2, and x.

Referenced by event_acceleration().

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event_acceleration()

void event_acceleration

(

void

)

Event: acceleration (i++)

Computation of the provisional temperature and Poisson–Helmholtz parameters

This is done before the projection, which will call the coupled Poisson–Helmholtz solver.

We compute \(\lambda_1\), \(\lambda_2\), \(\lambda_4\) and r.h.s. for the temperature which will be used by the coupled Poisson–Helmholtz solver. The \(\lambda\) parameter and r.h.s. for the pressure are computed by the two-phase compressible solver.

We also compute the face-averaged thermal conductivity.

Definition at line 254 of file thermal.h.

References _i, average_temperature(), beta, cm, cp1, cp2, dt, f, fm, frho1, frho2, kappa, kappa1, kappa2, lambda1, lambda2, lambda4, point, ps, thermal_expansion(), Ts, vector::x, and x.

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event_defaults()

void event_defaults

(

void

)

If the thermal conductivities are not-zero, we need to allocate a new field.

Event: defaults (i = 0)

The default density field is set to unity (times the metric).

We reset the multigrid parameters to their default values.

Definition at line 93 of file thermal.h.

References kappa, kappa1, and kappa2.

event_end_timestep()

void event_end_timestep

(

void

)

Event: end_timestep (i++)

Heat flux contribution to the total energy

This adds the \(-\nabla\cdot\mathbf{q}_i\) term of the energy equation.

Definition at line 292 of file thermal.h.

References _i, clamp(), cm, dimension, dt, energy(), f, face_gradient_x, fE1, fE2, kappa, kappa1, kappa2, T, vector::x, and x.

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poisson_thermal()

mgstats poisson_thermal	(	scalar	p,
		scalar	rhs,
		const vector	alpha,
		const scalar	lambda,
		double	tolerance = 0.,
		int	nrelax = 4,
		int	minlevel = 0,
		scalar *	res = NULL
	)

This is the interface for the coupled solver.

Definition at line 215 of file thermal.h.

References alpha, kappa, lambda, lambda1, lambda2, lambda4, max, mg_solve(), Thermal::minlevel, Thermal::nrelax, p, relax_thermal(), Thermal::res, residual_thermal(), restriction, s, T, Thermal::tolerance, TOLERANCE, tp, Ts, and x.

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relax_thermal()

static void relax_thermal ( scalar *  al, scalar *  bl, int  l, void *  data  )

static

Definition at line 134 of file thermal.h.

References _i, alpha, data, dimension, kappa, l, lambda, lambda1, lambda2, lambda4, p, sq(), T, tp, vector::x, and x.

Referenced by poisson_thermal().

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residual_thermal()

static double residual_thermal ( scalar *  al, scalar *  bl, scalar *  resl, void *  data  )

static

Definition at line 169 of file thermal.h.

References _i, alpha, data, dimension, face_gradient_x, kappa, lambda, lambda1, lambda2, lambda4, max, p, T, tp, vector::x, and x.

Referenced by poisson_thermal().

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thermal_expansion()

double thermal_expansion ( Point  point )

extern

Thermal expansion coefficient

Definition at line 111 of file NASG.h.

References b2, clamp(), cv2, f, gamma2, PI2, ps, Ts, and x.

Referenced by event_acceleration().

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Variable Documentation

cp1

double cp1 = 0.

Definition at line 66 of file thermal.h.

Referenced by average_temperature(), and event_acceleration().

cp2

double cp2 = 0.

Definition at line 66 of file thermal.h.

Referenced by average_temperature(), and event_acceleration().

kappa

const vector kappa = kappa0

Definition at line 82 of file thermal.h.

kappa0

const vector kappa0[] = {0,0,0}

Phase-averaged thermal conductivity

By default the arithmetic average of the two phase conductivities is used to compute the face-centered thermal conductivity.

Definition at line 81 of file thermal.h.

kappa1

double kappa1 = 0.

... and the thermal conductivities and specific heat capacities.

Definition at line 65 of file thermal.h.

Referenced by event_acceleration(), event_defaults(), and event_end_timestep().

kappa2

double kappa2 = 0.

Definition at line 65 of file thermal.h.

Referenced by event_acceleration(), event_defaults(), and event_end_timestep().

lambda1

scalar lambda1[]

Definition at line 117 of file thermal.h.

Referenced by event_acceleration(), poisson_thermal(), relax_thermal(), and residual_thermal().

lambda2

scalar lambda2[]

Definition at line 117 of file thermal.h.

Referenced by event_acceleration(), poisson_thermal(), relax_thermal(), and residual_thermal().

lambda4

scalar lambda4[]

Definition at line 117 of file thermal.h.

Referenced by event_acceleration(), poisson_thermal(), relax_thermal(), and residual_thermal().

T

scalar T[]

Thermal effects extension for the compressible two-phase solver

This extension of the two-phase compressible solver is described in detail in Saade et al, 2023.

This solves the two-phase compressible Navier-Stokes equations including the total energy equation.

\[ \frac{\partial (f \rho_i)}{\partial t } + \nabla \cdot (f \rho_i \mathbf{u}) = 0 \]

\[ \frac{\partial (\rho_i \mathbf{u})}{\partial t } + \nabla \cdot ( \rho_i \mathbf{u} \mathbf{u}) = -\nabla p + \nabla \cdot \tau_i' \]

\[ \frac{\partial [\rho_i (e_i + \mathbf{u}^2/2)]}{\partial t } + \nabla \cdot [ \rho_i \mathbf{u} (e_i + \mathbf{u}^2/2)] = -\nabla \cdot (\mathbf{u} p_i) + \nabla \cdot \left( \tau'_i \mathbf{u} \right) {\color{blue} - \nabla \cdot \mathbf{q}_i} \]

an advection equation for the volume fraction \(f\)

\[ \frac{\partial f}{\partial t} + \mathbf{u} \cdot \nabla f = 0 \]

The term in blue in the energy equation is the heat flux which is added by this extension. The other terms are treated by the two-phase compressible solver. The heat flux is given by Fourier's relation as

\[ \nabla \cdot \mathbf{q}_i = - \kappa_i \nabla T_i \]

where \(\kappa_i\) is the thermal conductivity and \(T\) is the temperature field.

The temperature and pressure fields are then coupled through the two equations (8 and 10 in Saade et al, 2023)

\[ \rho_ic_{p,i}\frac{DT_i}{Dt} = \beta_iT_i\frac{Dp_i}{Dt} - \nabla\cdot\mathbf{q}_i \]

with \(c_p\) the specific heat capacity and \(\beta\) the thermal expansion coefficient.

\[ \left(\frac{\gamma_i}{\rho_ic_i^2} - \frac{\beta_i^2T_i}{\rho_ic_{p,i}}\right)\frac{Dp_i}{Dt} = -\frac{\beta_i}{\rho_ic_{p,i}}\nabla\cdot\mathbf{q}_i - \nabla\cdot\mathbf{u}_i \]

with \(\gamma\) the ratio of specific heats.

The system is closed by specifying an Equation Of State (EOS) relating \(p\), \(\rho\) and \(T\).

In addition to the primary variables of the compressible two-phase solver we have the temperature field...

Definition at line 60 of file thermal.h.

Referenced by diagonalization_2D(), energy(), equilibrium_tide_constituents(), event_end_timestep(), poisson_thermal(), relax_thermal(), residual_thermal(), and RPE().

Ts

scalar Ts[]

Coupled Poisson-Helmholtz equations for temperature and pressure

The robustness of the solver relies on solving equations (8) and (10) in a strongly-coupled manner with the multigrid solver. After temporal discretisation this leads to the following coupled Poisson–Helmholtz system for the temperature \(T^{n+1}\) and pressure \(p^{n+1}\) (see eqs. 25 and 26 in Saade et al, 2023)

\[ \nabla\cdot\left(\kappa\nabla T^{n+1}\right) + \lambda_1 T^{n+1} + \lambda_2 p^{n+1} = \lambda_T \]

\[ \nabla\cdot\left(\alpha\nabla p^{n+1}\right) + \lambda p^{n+1} + \lambda_4\nabla\cdot\left(\kappa\nabla T^{n+1}\right) = \lambda_p \]

We allocate fields to store the provisionary temperature \(T^n\) and the coefficients.

Definition at line 117 of file thermal.h.

Referenced by event_acceleration(), poisson_thermal(), and thermal_expansion().