The free (open) boundary condition (FBC) in viscoelastic flow simulations

Abstract

The Free (or Open) Boundary Condition (FBC, OBC) was proposed by Papanastasiou et al. (A New Outflow Boundary Condition, Int. J. Numer. Meth. Fluids, 1992; 14:587–608) to handle truncated domains with synthetic boundaries where the outflow conditions are unknown. In the present work, implementation of the FBC has been extended to viscoelastic fluids governed by explicit differential constitutive equations. As such we consider here the Criminale-Ericksen-Filbey (CEF) model, which also reduces to the Second-Order Fluid (SOF) for constant material parameters. The Finite Element Method (FEM) is used to provide numerical results in simple Poiseuille flow where analytical solutions exist for checking purposes. Then previous numerical results are checked against Newtonian highly non-isothermal flows in a 4:1 contraction. Finally, the FBC is used with the CEF fluid with data corresponding to a Boger fluid of constant material properties. Particular emphasis is based on a non-zero second normal-stress difference, which seems responsible for earlier loss of convergence. The results with the FBC are in excellent agreement with those obtained from long domains, due to the highly convective nature of viscoelastic flows, for which the FBC seems most appropriate. The FBC formulation for fixed-point (Picard-type) iterations is given in some detail, and the differences with the Newton–Raphson formulation are highlighted regarding some computational aspects.

Appendix

Our recent work [13] contains detailed derivations of the FEM formulation based on the “stiffness” matrix and “load” vector approach advocated by Huebner and Thornton [22]. Here we concentrate on the appropriate modifications to incorporate the FBC for differential viscoelastic models.

Mass and momentum discrete equations

Combining the discrete forms of the conservation equations of mass and momentum (including compressibility) into one matrix equation leads (in two-dimensional axisymmetric domains, r-z-θ corresponding to 1-2-3) to the following system of an element (stiffness) matrix [S], a vector of unknowns {x}, and a RHS (load) vector {F} for each element:

$$ \left[ {\begin{array}{*{20}{c}} {{S_{{11}}}} & {{S_{{12}}}} & {{S_{{13}}}} \\ {{S_{{21}}}} & {{S_{{22}}}} & {{S_{{23}}}} \\ {{S_{{31}}}} & {{S_{{32}}}} & {{S_{{33}}}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}{c}} {\bar{U}} \\ {\bar{V}} \\ {\bar{P}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}{c}} {{{\bar{F}}_1}} \\ {{{\bar{F}}_2}} \\ 0 \\ \end{array} } \right] $$

(A.1)

The entries for each term in the above system are given in detail in [13].

Contribution from the FBC

With the FBC, the extra terms along the outflow boundary are:

$$ \bar{F} = \underbrace{{\int\limits_{{{\Gamma_{{FBC}}}}} {\left( {\bar{n} \cdot \left( { - p\overline{\overline I} + \overline{\overline \tau } } \right)} \right)\bar{\varphi }d\Gamma = } }}_{\text{free boundary condition}}\underbrace{{\int\limits_{{{\Gamma_{{FBC}}}}} {\left( {_{{{n_r}{\tau_{{rz}}} + {n_z}\left( { - p + {\tau_{{zz}}}} \right)}}^{{{n_r}\left( { - p + {\tau_{{rr}}}} \right) + {n_z}{\tau_{{rz}}}}}} \right){\varphi^i}d\Gamma, } }}_{\text{free boundary condition}}\quad i = 1,3 $$

(A.2)

$$ {\bar{F}_q} = \underbrace{{\int\limits_{{{\Gamma_{{FBC}}}}} {\left( {\bar{n} \cdot k\nabla T} \right)\bar{\varphi }d\Gamma } = }}_{\text{free boundary condition}}\underbrace{{\int\limits_{{{\Gamma_{{FBC}}}}} {k \left( {{n_r}\frac{{\partial T}}{{\partial r}} + {n_z}\frac{{\partial T}}{{\partial z}}} \right){\varphi^i}d\Gamma } }}_{\text{free boundary condition}},\quad {\text{i}} = {1},{3} $$

(A.3)

After the appropriate manipulations, the following matrix system is obtained:

$$ \left[ \begin{gathered} \begin{array}{*{20}{c}} {{S_{{O11}}}} & {{S_{{O12}}}} & {{S_{{O13}}}} \\ \end{array} \hfill \\ \begin{array}{*{20}{c}} {{S_{{O21}}}} & {{S_{{O22}}}} & {{S_{{O23}}}} \\ \end{array} \hfill \\ \end{gathered} \right]\left[ {\begin{array}{*{20}{c}} {\bar{U}} \\ {\bar{V}} \\ {\bar{P}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}{c}} {{{\bar{F}}_{{O1}}}} \\ {{{\bar{F}}_{{O2}}}} \\ \end{array} } \right] $$

(A.4)

where the components of the element (stiffness) matrix [S _O] of Eq. A.1 are:

$$ {S_{{O11}}} = \int\limits_{\Gamma } {\eta \frac{{\partial \bar{\varphi }}}{{\partial z}}{\varphi^i}d\Gamma }, \quad {\text{i}} = {1},{3} $$

(A.5)

$$ {S_{{O12}}} = \int\limits_{\Gamma } {\eta \frac{{\partial \bar{\varphi }}}{{\partial r}}{\varphi^i}d\Gamma }, \quad {\text{i}} = {1},{3} $$

(A.6)

$$ {S_{{O13}}} = 0 $$

(A.7)

$$ {S_{{O21}}} = \int\limits_{\Gamma } {\left( {{\Psi_1} + {\Psi_2}} \right){{\dot{\gamma }}_{{rz}}}\frac{{\partial \bar{\varphi }}}{{\partial z}}{\varphi^i}d\Gamma - } \int\limits_{\Gamma } {\frac{{2n}}{3}\left( {\frac{{\partial \bar{\varphi }}}{{\partial r}} + \frac{{\bar{\varphi }}}{r}} \right){\varphi^i}d\Gamma, } \quad {\text{i}} = {1},{3}\quad \left( {{{2}^{\text{nd}}}{\text{term}} = 0{\text{ for incompressible fluids}}} \right) $$

(A.8)

$$ {S_{{O22}}} = \int\limits_{\Gamma } {2\eta \frac{{\partial \bar{\varphi }}}{{\partial z}}{\varphi^i}d\Gamma + \int\limits_{\Gamma } {\left( {{\Psi_1} + {\Psi_2}} \right){{\dot{\gamma }}_{{rz}}}\frac{{\partial \bar{\varphi }}}{{\partial r}}{\varphi^i}d\Gamma } - \int\limits_{\Gamma } {\frac{{2\eta }}{3}\frac{{\partial \bar{\varphi }}}{{\partial z}}{\varphi^i}d\Gamma, } } \quad {\text{i}} = {1},{3}\quad \left( {{{3}^{\text{rd}}}{\text{term}} = 0{\text{ for incomp}}.{\text{ fluids}}} \right) $$

(A.9)

$$ {S_{{O23}}} = - \int\limits_{\Gamma } {\bar{\psi }{\varphi^i}d\Gamma } - \int\limits_{\Gamma } {\frac{{\partial \rho }}{{\partial p}}\bar{\psi }{\varphi^i}d\Gamma, } \quad {\text{i}} = {1},{3}\quad \left( {{{2}^{\text{nd}}}{\text{term}} = 0{\text{ for incomp}}.{\text{ fluids}}} \right) $$

(A.10)

$$ {\bar{F}_{{O1}}} = {\bar{F}_{{O2}}} = 0 $$

(A.11)

The above contributions of [S _O] and [F _O] must be added to the corresponding terms of Eq. A.1 for the elements having the FBC on one side.

For the contribution to the energy equation from the FBC (Eq. A.3), the matrix Eq. A.4 has a 4th unknown, $\bar{T}$, and the stiffness element corresponding to this is simply:

$$ {S_{{O44}}} = {K_{{O44}}} = \int\limits_{\Gamma } {k\left( {{n_r}\frac{{\partial \bar{\varphi }}}{{\partial r}} + {n_z}\frac{{\partial \bar{\varphi }}}{{\partial z}}} \right){\varphi^i}d\Gamma }, \quad {\text{i}} = {1},{3} $$

(A.12)

It should be noted that when using the N-R iteration, Eqs. A.2 and A.3, as such, simply constitute the residuals, $\{ \bar{R}\}$, from which the Jacobian $[J] = [\partial \bar{R}/\partial \bar{x}]$ is derived, and the system is solved for the vector of unknowns $\{ \Delta \bar{x}\}$, according to $[J]\{ \Delta \bar{x}\} = - \{ \bar{R}\}$. Thus, it is not necessary to derive “stiffness” matrices and “load” vectors, as in the above.