Flow Boundary Conditions

Free Stream

For a free surface simulation, the shape of the free surface is not known in advance and is calculated as part of the solution using an Arbitrary Lagrangian-Eulerian (ALE) approach.

Kinematic Condition

The free stream kinematic condition is:

$$\mathbf{v}\cdot \mathbf{n}=0$$

(1050)

where $\mathbf{v}$ is the velocity vector and $\mathbf{n}$ is the unit normal to the boundary.

Dynamic Condition

The dynamic boundary condition at a free surface is defined as:

$$-{\mathbf{n}}_{\mathrm{\Gamma }}\cdot ({\mathbf{T}}_1-{\mathbf{T}}_2)=2H\sigma {\mathbf{n}}_{\mathrm{\Gamma }}+{\nabla }_s\sigma$$

(1051)

where:

• ${\mathbf{T}}_1$ and ${\mathbf{T}}_2$ are the stress tensors of the fluids in regions 1 and 2 on either side of the free surface or interface.
• $\sigma$ is the surface tension.
• $H$ is the mean curvature related to the unit normal according to $2H\gamma =-{\nabla }_s\cdot {\mathbf{n}}_{\mathrm{\Gamma }}$
• ${\mathbf{n}}_{\mathrm{\Gamma }}$ is the unit normal pointing from region 1 to region 2 on either side of the free surface or interface.
• ${\nabla }_s$ is the surface gradient defined as ${\nabla }_s=(\mathbf{I}-{\mathbf{n}}_{\mathrm{\Gamma }}{\mathbf{n}}_{\mathrm{\Gamma }})\cdot \nabla$

Since this term concerns only the boundary term in the weak form, only the last term, the boundary term, of Eqn. (1024) is equal to:

$$\begin{array}{cc}-\underset{\mathrm{\Gamma }}{\oint }{\tilde{\mathbf{v}}}_h\cdot \left[({\mathbf{T}}_1-{\mathbf{T}}_2)\cdot {\mathbf{n}}_{\mathrm{\Gamma }}\right]d\mathrm{\Gamma }& =\underset{\mathrm{\Gamma }}{\oint }{\tilde{\mathbf{v}}}_h\cdot (2H\sigma {\mathbf{n}}_{\mathrm{\Gamma }}+{\nabla }_s\sigma )d\mathrm{\Gamma }\hfill \\ & =\underset{\mathrm{\Gamma }}{\oint }2H\sigma ({\tilde{\mathbf{v}}}_h\cdot {\mathbf{n}}_{\mathrm{\Gamma }})d\mathrm{\Gamma }+\underset{\mathrm{\Gamma }}{\oint }{\tilde{\mathbf{v}}}_h\cdot {\nabla }_s\sigma d\mathrm{\Gamma },\forall \tilde{\mathbf{v}}\in {\left(H_1^0\right)}^3\hfill \end{array}$$

(1052)

Following the approach proposed by Ruschak [281], the above equation transforms such that the explicit calculation of the surface tension gradient and the curvature are circumvented, giving:

$$-\underset{\mathrm{\Gamma }}{\oint }{\tilde{\mathbf{v}}}_h\cdot \left[({\mathbf{T}}_1-{\mathbf{T}}_2)\cdot {\mathbf{n}}_{\mathrm{\Gamma }}\right]d\mathrm{\Gamma }=\underset{C}{\oint }\sigma ({\tilde{\mathbf{v}}}_h\cdot \mathbf{m})dC-\underset{\mathrm{\Gamma }}{\oint }\sigma {\nabla }_s\cdot {\tilde{\mathbf{v}}}_{h}d\mathrm{\Gamma },\forall \tilde{\mathbf{v}}\in {\left(H_1^0\right)}^3$$

(1053)

where:

• $\mathbf{m}$ is an outward-pointing bi-normal unit vector, tangent to the surface and normal to the boundary curve along the edge of the surface: $\mathbf{m}=\mathbf{t}\times {\mathbf{n}}_{\mathrm{\Gamma }}$ .
• $\mathbf{t}$ is the unit tangent vector.
• $C$ is the contour enclosing the surface $\mathrm{\Gamma }$ .

Note that both surface tension gradients (Marangoni stress) and capillary pressure show up implicitly from the application of this equation.

Traction

The traction boundary concerns only the weak form of the boundary term. See the last term of Eqn. (1024). The traction boundary conditions are:

No Traction

The boundary term is zero and therefore nothing is done regarding the computation of the boundary term.

Traction Velocity

A Dirichlet boundary condition for the velocity vector is imposed in all three directions, with the specified value given in the normal direction to the boundary, and with zero for the tangential directions to the boundary. Therefore the boundary term is identically zero (as $\tilde{\mathbf{v}}$ is zero by definition at boundaries that have a Dirichlet condition, as above).

Aligned Exits

Imposes zero traction in the normal direction of extrusion. Imposes two zero-Dirichlet conditions for the tangential directions to the boundary.

Normal Velocity

Similar to the traction velocity except for the tangential directions, which have zero traction instead of zero Dirichlet.

Open Boundary

This option is available only when both free surface and viscoelastic models are active. The integral on the boundary in the above is exactly calculated based on the values of the variables in the domain, such that the presence of the boundary leads to no change from the domain values.

Mass Flow Inlet

For the mass flow inlet boundary, the tangential component of the velocity vector must be zero:

The value of the pressure at the mass flow inlet is:

where:

• $R$ is the flow resistance which is a constant.
• $\dot{m}$ is the mass flow rate provided at the boundary.
• $\rho$ is the density of the fluid.
• $A$ is the area of the inlet boundary.

Pressure Outlet

For the pressure outlet boundary, the tangential component of the velocity vector must be zero:

The value of the pressure must be equal to the specified value given for the outlet boundary:

Symmetry Plane

For the symmetry plane boundary, the normal component of the velocity vector must be zero:

Velocity Inlet

For the velocity inlet boundary, the tangential component of the velocity vector must be zero:

The value of the pressure at the inlet is:

where $v_{\text{avg}}$ is the specified average value of the velocity at the boundary.

Wall

No-Slip Wall

At the no-slip wall, all components (both normal and tangential components) of the velocity vector must be zero:

$$\begin{array}{c}\mathbf{v}\cdot \mathbf{n}=0\\ \mathbf{v}-(\mathbf{v}\cdot \mathbf{n})\mathbf{n}=0\end{array}$$

(1061)

Slip Wall

When simulating viscous flow using the Viscous Flow model, the slip wall boundary conditions are computed according to a power-law slip relation. The shear force per unit surface ${\mathbf{f}}_s$ is related to the slip velocity ${\mathbf{v}}_s$ by [178][185]:

$${\mathbf{f}}_s=-k_s{\left|{\mathbf{v}}_s\right|}^{n_s-1}{\mathbf{v}}_s$$

(1062)

where:

• $k_s$ is the slip coefficient.
• $n_s$ is the slip exponent.

The shear force and the slip velocity are defined as:

$${\mathbf{f}}_s=\mathbf{T}\cdot {\mathbf{n}}_w-\left[(\mathbf{T}\cdot {\mathbf{n}}_w)\cdot {\mathbf{n}}_w\right]{\mathbf{n}}_w$$

$${\mathbf{v}}_s=(\mathbf{v}-{\mathbf{v}}_w)-\left[(\mathbf{v}-{\mathbf{v}}_w)\cdot {\mathbf{n}}_w\right]{\mathbf{n}}_w$$

where:

• ${\mathbf{n}}_w$ is the unit normal pointing out from the wall.
• ${\mathbf{v}}_w$ is the wall velocity.
• $\mathbf{T}$ is the total hydrodynamic stress tensor.

Traction

The traction boundary concerns only the weak form of the boundary term. See the last term of Eqn. (1024). The traction boundary conditions are:

No Traction

The boundary term is zero and therefore nothing is done regarding the computation of the boundary term.

Traction Velocity

A Dirichlet boundary condition for the velocity vector is imposed in all three directions, with the specified value given in the normal direction to the boundary, and with zero for the tangential directions to the boundary. Therefore the boundary term is identically zero (as $\tilde{\mathbf{v}}$ is zero by definition at boundaries that have a Dirichlet condition, as above).

Aligned Exits

Imposes zero traction in the normal direction of extrusion. Imposes two zero-Dirichlet conditions for the tangential directions to the boundary.

Normal Velocity

Similar to the traction velocity except for the tangential directions, which have zero traction instead of zero Dirichlet.

Open Boundary

This option is available only when both free surface and viscoelastic models are active. The integral on the boundary in the above is exactly calculated based on the values of the variables in the domain, such that the presence of the boundary leads to no change from the domain values.