Boundary Conditions

Thermal Wall

For Eulerian multiphase flows, it is assumed that the same local wall temperature $T_w$ is experienced by all phases in contact with the wall. This useful closure, together with an energy balance at the wall, allows the most common thermal boundary conditions to be specified for the wall, rather than for each individual phase.

The energy balance for the wall is:

where:

• ${{\dot{q}}^{″}}_w$ is the external heat flux from the environment to the wall
• ${{\dot{q}}^{″}}_{wi}$ is the heat flux from the wall to the phase i fluid next to the wall
• ${{\dot{q}}^{″}}_{w\left(ij\right)}$ is the heat flux from the wall to a phase interaction (ij) interface next to the wall

External Heat Flux

The four specifications available for modeling the external heat flux ${q″}_w$ are as follows:

• Adiabatic: ${{\dot{q}}^{"}}_w$ is zero and $T_w$ is computed.
• Heat Flux: ${{\dot{q}}^{"}}_w$ is specified and $T_w$ is computed.
• Temperature: $T_w$ is specified and ${{\dot{q}}^{"}}_w$ is computed.
• Convection: Ambient temperature $T_{\infty }$ and external heat transfer coefficient $h_{\infty }$ are specified, with both $q_w^{"}$ and $T_w$ computed.

  The external heat flux for the convection option is:

$${{\dot{q}}^{"}}_w=h_{\infty }\left(T_{\infty }-T_w\right)$$

(2564)

Phase Heat Flux

The standard model for internal heat flux from each phase i to the wall ${{\dot{q}}^{"}}_{wi}$ is based on single-phase wall treatments within each phase together with an assumption that each phase has a local wall contact area proportional to the volume fraction next to the wall.

For turbulent flow, these two models combine into:

$${{\dot{q}}^{"}}_{wi}={\alpha }_i\frac{{\rho }_i{c}_{pi}{u_i}^{*}}{t_i^{\text{+}}}\left(T_w-T_i\right)$$

(2565)

where:

• ${\alpha }_i$ is the volume fraction of phase i
• ${u_i}^{*}$ is the turbulent wall velocity scale of phase i
• $t_i^{\text{+}}$ is defined by the thermal wall law for phase i

For laminar flow, this relationship is:

$${{\dot{q}}^{"}}_{wi}=\frac{-{{\dot{q}}^{"}}_{wi}\cdot \mathbf{\text{a}}}{\left|\mathbf{\text{a}}\right|}=\frac{{\alpha }_i{k}_i\nabla T_i\cdot \mathbf{\text{a}}}{\left|\mathbf{\text{a}}\right|}$$

(2566)

where $\mathbf{\text{a}}$ is the outward facing wall face area vector.

You can also override the standard relationship for phase heat flux, by specifying three coefficients $A_i$ , $B_i$ and $C_i$ :

$$-{{\dot{q}}^{"}}_{wi}=A_i+B_i{T}_i+C_i{T}_w$$

(2567)

This allows various opportunities for alternative heat flux relationships, wall contact models, or linearization methods.

Note

The sign convention, which is based on the discretization of fluxes using outward facing areas. Since either side of this formula is defined to be positive when heat flows from the fluid to the wall, the coefficient of the fluid temperature, $B_i$ , is always positive or zero. The coefficient of the wall temperature, $C_i$ , is always negative or zero.

The internal coefficients from single-phase wall treatments are available as field functions for constructing multiphase coefficients. For example, the standard turbulent or laminar wall contact model is equivalent to specifying the coefficients:

$$\begin{array}{c}{A}_i={\alpha }_i{A}_{INTERNAL,i}\\ B_i={\alpha }_i{B}_{INTERNAL,i}\\ C_i={\alpha }_i{C}_{INTERNAL,i}\end{array}$$

(2568)

Phase Interaction Heat Flux

This term is absent from the wall heat balance, unless the phase interaction (ij) includes phase change, in which case this heat flux drives the rate of mass transfer at the wall by:

$${{\dot{q}}^{"}}_{w\left(ij\right)}=-{{\dot{m}}^{"}}_{w\left(ij\right)}{h}_{\left(ij\right)}$$

(2569)

where:

• ${{\dot{m}}^{"}}_{w\left(ij\right)}$ is the rate of interphase mass transfer from phase j to phase i per unit wall area
• $h_{\left(ij\right)}$ is the enthalpy of phase change from phase i to phase j.

Standard relationships for phase interaction heat fluxes are described in the Wall Boiling section.

Override a standard relationship for phase interaction heat flux, by specifying three coefficients $A_{\left(ij\right)}$ , $B_{\left(ij\right)}$ , and $C_{\left(ij\right)}$ :

$$-{{\dot{q}}^{"}}_{w\left(ij\right)}=A_{\left(ij\right)}+B_{\left(ij\right)}{T}_{\left(ij\right)}+C_{\left(ij\right)}{T}_w$$

(2570)

where:

• $T_{\left(ij\right)}$ is the temperature of the interface between phases i and j, which is typically specified as the saturation temperature

For example, an evaporation heat flux that depends on some power of the wall superheat, such as:

$${{\dot{q}}^{"}}_{w\left(ij\right)}=K\text{ max}{\left(T_w-T_{\left(ij\right)},0\right)}^n$$

(2571)

Can be linearized by differentiation with respect to each temperature:

$${{\dot{q}}^{"}}_{w\left(ij\right)}(T_{\left(ij\right)},T_w)={{\dot{q}}^{"}}_{w\left(ij\right)}({T_{\left(ij\right)}}^{*},{T_w}^{*})+\left(T_{\left(ij\right)}-{T_{\left(ij\right)}}^{*}\right){\left[\frac{\partial }{\partial T_{\left(ij\right)}}{{\dot{q}}^{"}}_{w\left(ij\right)}\right]}^{*}+\left(T_w-{T_w}^{*}\right){\left[\frac{\partial }{\partial T_w}{{\dot{q}}^{"}}_{w\left(ij\right)}\right]}^{*}$$

(2572)

where an asterisk signifies a value at the previous iteration.

Collecting terms in $T_{\left(ij\right)}$ and $T_w$ leads to the implementation:

$$\begin{array}{c}{A}_{ij}=\left(n-1\right)K\text{ max}{({T_w}^{*}-{T_{\left(ij\right)}}^{*},0)}^n\\ B_{ij}=-nK\text{ max}{({T_w}^{*}-{T_{\left(ij\right)}}^{*},0)}^{n-1}\\ C_{ij}={-B}_{\left(ij\right)}\end{array}$$

(2573)

Permeable Wall

A wall boundary that has been set to phase permeable for a particular phase allows a phase mass flux approaching the boundary to leave the system at this boundary.

The mass flux for phase i is given by:

where:

• $m_i$ is the mass flux leaving the system
• ${\rho }_i$ is the cell-centered density
• $v_i$ is the cell-centred velocity
• ${\alpha }_i$ is the cell-centered volume fraction
• $a$ is the cell wall area vector (normal to the wall in the outward facing direction)
• ${\gamma }_i$ is the wall permeability factor

The wall permeability takes a value from 0 through 1 to control the amount of the flux incident on the surface that is removed from the flow. Any component of the wall velocity parallel to the wall is the same as for a standard wall (zero in this case for a no-slip wall). When a permeable wall phase condition is activated, the mass flux that is calculated at the wall cell is used in all the other transport equations.

Mass Flow Inlet

The mass flow inlet can be specified by a known mass flow rate. This boundary condition is designed primarily for specifying mass flow inward across the boundary, but also works for specifying mass flowing outward. For details on specifying mass flow inlet variables see Mass Flow Inlet. In addition, for multiphase simulations you specify the volume fraction for each phase at the mass flow inlet boundary.

The inlet velocity for each phase i is then defined as:

where:

• $m$ is the mass flow rate
• ${\rho }_i$ is the density of phase i
• ${\alpha }_i$ is the phase volume fraction of phase i
• $A$ is the inlet cross-sectional area

Outlet

The outlet flow can be specified by flow split ratio, mass fluxes, or by volumetric fluxes.

Specified Volumetric Fluxes

For specified volumetric fluxes, the mass flow rate is scaled by the density to give a volumetric flux:

$$\dot{V}=\sum _f\frac{\dot{m}\left|a\right|}{{\rho }_m}$$

(2576)

The mixture density is given by:

$${\rho }_m=\sum _k^{phases}{\alpha }_k{\rho }_k$$

(2577)

So the total volumetric flux is calculated by adding a summation over all phases:

$$\dot{V}=\sum _k^{phases}\sum _f\frac{1}{{\rho }_m}{\dot{m}}_k\left|a\right|$$

(2578)