A Staggered Pressure Correction Numerical Scheme to Compute a Travelling Reactive Interface in a Partially Premixed Mixture

Abstract

We address a turbulent deflagration model with a flow governed by the compositional Euler equations and the flame propagation represented by the transport of the characteristic function. The numerical scheme works on staggered, unstructured meshes with a time-marching algorithm solving first the chemical species mass balances and then the mass, momentum and energy balances. A pressure correction technique is used for this latter step, which solves a balance equation for the sensible enthalpy with corrective terms to ensure consistency of the total energy. The approximate solutions respect the physical bounds and satisfy a conservative weakly-consistent discrete total energy balance equation. Numerical evidence shows that they converge to the solution of the infinitely fast chemistry continuous problem when the chemical time scale tends to zero with the space and time steps.

Appendices

8 Appendix A: The MUSCL Scheme

The MUSCL discretization of the convection operators of the chemical species balance and G-equation closely follows the technique proposed in [15]. To present this discretization, we consider the following system of equations:

$$ \begin{array}{l} \displaystyle \partial _t\rho + \mathrm{div}(\rho {\boldsymbol{u}}) = 0, \\ \displaystyle \partial _t(\rho {\boldsymbol{u}}) + \mathrm{div}(\rho {\boldsymbol{u}}y) = 0. \end{array} $$

We suppose for short that this system is complemented by impermeability boundary conditions, i.e. that the normal velocity, both at the continuous and the discrete level, vanishes on the boundary of the computational domain.

The discretization of the above system reads:

$$ \begin{array}{ll} \forall K\in {\mathcal M}, \quad &{} \displaystyle \frac{\rho _K^{n+1}-\rho _K^n}{\delta t} + \frac{1}{|K|} \sum _{\sigma \in {\mathcal E}(K)} F_{K,\sigma }^{n+1} = 0, \\ &{} \displaystyle \frac{\rho _K^{n+1} y_K^{n+1} - \rho _K^n y_K^n}{\delta t} + \frac{1}{|K|} \sum _{\sigma \in {\mathcal E}(K)} F_{K,\sigma }^{n+1} y_\sigma ^n = 0. \end{array} $$

For any $\sigma \in {\mathcal E}$, the procedure consists in three steps:

• calculate a tentative value for $y_\sigma$ as a linear interpolate of nearby values,

• calculate an interval for $y_\sigma$ which guarantees the stability of the scheme,

• project the tentative value $y_\sigma$ on this stability interval.

For the tentative value of $y_\sigma$, let us choose some real coefficients $(\alpha _K^\sigma )_{K\in {\mathcal M}}$ such that

$$ {\boldsymbol{x}}_\sigma = \sum _{K\in {\mathcal M}} \alpha _K^\sigma {\boldsymbol{x}}_K, \qquad \qquad \sum _{K\in {\mathcal M}} \alpha _K^\sigma = 1. $$

The coefficients used in this interpolation are chosen in such a way that as few as possible cells, to be picked up in the closest cells to $\sigma$, take part. For example, for $\sigma =K|L$ and if ${\boldsymbol{x}}_K,\ {\boldsymbol{x}}_\sigma ,\ {\boldsymbol{x}}_L$ are aligned, only two non-zero coefficients exist in the family $(\alpha _K^\sigma )_{K\in {\mathcal M}}$, namely $\alpha _K^\sigma$ and $\alpha _L^\sigma$. Then, these coefficients are used to calculate the tentative value of $y_\sigma$ by

$$ y_\sigma = \sum _{K\in {\mathcal M}} \alpha _K^\sigma y_K. $$

The construction of the stability interval must be such that the following property holds:

$$\begin{aligned} \begin{array}{c} \forall K\in {\mathcal M},\ \forall \sigma \in {\mathcal E}(K)\cap {\mathcal E}_{\mathrm{int}},\ \exists \beta _K^\sigma \in [0,1] \text { and } M_K^\sigma \in {\mathcal M}\text { such that} \\ y_\sigma ^n - y_K^n = \left| \begin{array}{l} \beta _K^\sigma \, (y_K^n - y_{M_K^\sigma }^n), \text { if } F_{K,\sigma }^{n+1} \ge 0,\\ \beta _K^\sigma \, (y_{M_K^\sigma }^n - y_K^n), \text { otherwise.} \end{array} \right. \end{array} \end{aligned}$$

(33)

Indeed, under this latter hypothesis and a CFL condition, the scheme preserves the initial bounds of y.

Remark 4

Note that, in Assumption (33), only internal faces are considered, since the fluxes through external faces are supposed to vanish. However, the present discussion may easily be generalized to cope with convection fluxes entering the domain.

Definition 1

The so-called CFL number reads for any $0 \le n \le N$:

$$ \mathrm{CFL}^{n+1} = \max _{K\in {\mathcal M}}\Big \{ \frac{\delta t}{\rho _K^{n+1}\ |K|} \sum _{\sigma \in {\mathcal E}(K)} \big |F_{K,\sigma }^{n+1}\big | \Big \}. $$

Lemma 3

Let us suppose that $\mathrm{CFL}^{n+1}\le 1$. For $K\in {\mathcal M}$, let us note by $\mathcal V(K)$ the union of the set of cells $M_K^\sigma ,\ \sigma \in {\mathcal E}(K) \cap {\mathcal E}_{\mathrm{int}}$ such that (33) holds. Then $\forall K\in {\mathcal M}$, the value of $y_K^{n+1}$ is a convex combination of $\{y_K^n, (y_M^n)_{M\in \mathcal V(K)} \}$.

Proof  The discrete mass balance equation yields:

$$ \rho _K^n = \rho _K^{n+1} + \frac{\delta t}{|K|} \sum _{\sigma \in {\mathcal E}(K)} F_{K,\sigma }^{n+1}. $$

Replacing this expression of $\rho _K^n$ in the discrete balance equation of y and using the relations provided by (33), we obtain:

$$ \begin{array}{ll} \rho _K^{n+1} y_K^{n+1} &{} \displaystyle = \rho _K^n y_K^n - \frac{\delta t}{|K|} \sum _{\sigma \in {\mathcal E}(K)} F_{K,\sigma }^{n+1} y_\sigma ^n \\ &{} \displaystyle = \rho _K^{n+1} y_K^n - \frac{\delta t}{|K|} \sum _{\sigma \in {\mathcal E}(K)} F_{K,\sigma }^{n+1} (y_\sigma ^n - y_K^n) \\ &{} \displaystyle = \rho _K^{n+1} y_K^n - \frac{\delta t}{|K|} \sum _{\sigma \in {\mathcal E}(K)} \big (F_{K,\sigma }^{n+1}\big )^+ (y_\sigma ^n - y_K^n) + \frac{\delta t}{|K|} \sum _{\sigma \in {\mathcal E}(K)} \big (F_{K,\sigma }^{n+1}\big )^- (y_\sigma ^n - y_K^n) \\ &{} \displaystyle = \rho _K^{n+1} y_K^n - \frac{\delta t}{|K|} \sum _{\sigma \in {\mathcal E}(K)} \big (F_{K,\sigma }^{n+1}\big )^+ \beta _K^\sigma (y_K^n - y_{M_K^\sigma }^n) + \frac{\delta t}{|K|} \sum _{\sigma \in {\mathcal E}(K)} \big (F_{K,\sigma }^{n+1}\big )^- \beta _K^\sigma (y_{M_K^\sigma }^n - y_K^n). \end{array} $$

This relation yields

$$ y_K^{n+1} = y_K^n \Big ( 1 - \frac{\delta t}{\rho _K^{n+1}\ |K|} \sum _{\sigma \in {\mathcal E}(K)} \beta _K^\sigma \, \big |F_{K,\sigma }^{n+1}\big | \Big ) + \frac{\delta t}{|K|} \sum _{\sigma \in {\mathcal E}(K)} y_{M_K^\sigma }^n \beta _K^\sigma \, \big |F_{K,\sigma }^{n+1}\big |, $$

which concludes the proof under the hypothesis that $\mathrm{CFL}\le 1$.

We now need to reformulate (33) in order to construct the stability interval. Let $\sigma \in {\mathcal E}$, let us denote by $V^-$ and $V^+$ the upstream and downstream cell separated by $\sigma$, and by $\mathcal V_\sigma (V^-)$ and $\mathcal V_\sigma (V^+)$ two sets of neighbouring cells of $V^-$ and $V^+$ respectively, and let us suppose:

$$ \begin{array}{l} \text {(H1)} -\exists M\in \mathcal V_\sigma (V^+) \text { s.t. } u_\sigma ^n\in |[u_M^n, u_M^n + \dfrac{\zeta ^+}{2}(u_{V^+}^n - u_M^n) ]|,\\ \text {(H2)} -\exists M\in \mathcal V_\sigma (V^-) \text { s.t. } u_\sigma ^n\in |[u_{V^-}^n, u_{V^-}^n + \dfrac{\zeta ^-}{2}(u_{V^-}^n - u_M^n) ]|, \end{array} $$

where for $a,\,b\in \mathbb R$, we denote by |[a, b]| the interval $\{\alpha a + (1-\alpha )b,\ \alpha \in [0,1] \}$, and $\zeta ^+$ and $\zeta ^-$ are two numerical parameters lying in the interval [0, 2].

Conditions (H1)-(H2) and (33) are linked in the following way: let $K\in {\mathcal M}$ and $\sigma \in {\mathcal E}(K)$. If $F_{K,\sigma }^n\le 0$, i.e. K is the downstream cell for $\sigma$, denoted above by $V^+$, since $\zeta ^+\in [0,2]$, condition (H1) yields that there exists $M\in {\mathcal M}$ such that $u_\sigma ^n\in |[u_K^n,u_M^n]|$, which is (33). Otherwise, i.e. if $F_{K,\sigma }^n\ge 0$ and K is the upstream cell for $\sigma$, denoted above by $V^-$, condition (H2) yields that there exists $M\in {\mathcal M}$ such that $y_\sigma ^n\in |[y_K^n,2y_K^n-y_M^n]|$, so $y_\sigma ^n-y_K^n\in |[0,y_K^n-y_M^n]|$, which is once again (33).

Remark 5

For $\sigma \in {\mathcal E}$, if $V^-\in \mathcal V_\sigma (V^+)$, the upstream choice $y_\sigma ^n=y_{V^-}^n$ always satisfies the conditions (H1)-(H2), and is the only one to satisfy them if we choose $\zeta ^-=\zeta ^+=0$.

Fig. 10

Conditions (H1) and (H2) in 1D.

Remark 6

(1D case) Let us take the example of an interface $\sigma$ separating $K_i$ and $K_{i+1}$ in a 1D case (see Fig. 10 for the notations), with a uniform meshing and a positive advection velocity, so that $V^-=K_i$ and $V^+=K_{i+1}$. In 1D, a natural choice is $\mathcal V_\sigma (K_i)=\{K_{i-1}\}$ and $\mathcal V_\sigma (K_{i+1})=\{K_i \}$. On Fig. 10, we sketch: on the left, the admissible interval given by (H1) with $\zeta ^+=1$ (green) and $\zeta ^+=2$ (orange); on the right, the admissible interval given by (H2) with $\zeta ^-=1$ (green) and $\zeta ^-=2$ (orange). The parameters $\zeta ^-$ and $\zeta ^+$ may be seen as limiting the admissible slope between $({\boldsymbol{x}}_i,y^n_i)$ and $({\boldsymbol{x}}_\sigma , y^n_\sigma )$ (with ${\boldsymbol{x}}_i$ the abscissa of the mass centre of $K_i$ and ${\boldsymbol{x}}_\sigma$ the abscissa of $\sigma$), with respect to a left and right slope, respectively. For $\zeta ^-=\zeta ^+=1$, one recognizes the usual minmod limiter (e.g. [7, Chapter III]). Note that, since, on the example depicted on Fig. 10, the discrete function $y^n$ has an extremum in $K_i$, the combination of the conditions (H1) and (H2) imposes that, as usual, the only admissible value for $y^n_\sigma$ is the upwind one.

Finally, we need to specify the choice of the sets $\mathcal V_\sigma (V^-)$ and $\mathcal V_\sigma (V^+)$. Here, we just set $\mathcal V_\sigma (V^+) = \{V^-\}$; such a choice guarantees that at least the upstream choice is in the intersection of the intervals defined by (H1) and (H2), as explained in Remark 6. The set $\mathcal V_\sigma (V^-)$ may be defined in two different ways (cf. Fig. 11):

Fig. 11

Notations for the definition of the limitation process. In orange, control volumes of the set $\mathcal V_\sigma (V^-)$ for $\sigma =V^-|V^+$, with a constant advection field $\mathbf{F}$: upwind cells (a) or opposite cells (b).

• as the “upstream cells” to $V^-$, i.e.

  $$\begin{aligned} \mathcal V_\sigma (V^-) = \{ L\in {\mathcal M},\ L \text { shares a face }\sigma \text { with }V^- \text { and } F_{V^-,\sigma } \le 0 \}, \end{aligned}$$

• when this makes sense (i.e. with a mesh obtained by $Q_1$ mappings from the $(0,1)^d$ reference element), the opposite cells to $\sigma$ in $V^-$ are chosen. Note that for a structured mesh, this choice allows to recover the usual minmod limiter.

9 Appendix B: An Anti-diffusive Scheme

The scheme proposed in [5] by of B. Després and F. Lagoutière for the constant velocity advection problem presents some interesting properties in one space dimension (and may be extended to structured multi-dimensional meshes using alternate directions techniques); in particular, it notably limits the numerical diffusion. We extend here this scheme to work with unstructured meshes for which the “opposite cell to a face” (in the sense introduced in the previous section) may be defined and with a variable density. With the same notations as in the previous section, for $\sigma \in {\mathcal E}_{\mathrm{int}}$, $\sigma =K|L$ with $F_{K,\sigma }^{n+1} \ge 0$,

• the tentative value for $y_\sigma$ is chosen as the downwind value, i.e. $y_\sigma ^n=y_L^n$,

• Then we project $y_\sigma ^n$ on the interval

  $$ I_\sigma = \bigl [ y_K^n,\,y_K^n+\frac{1-\nu }{\nu }(y_K-y_M) \bigr ],\quad \nu =\frac{|F_{K,\sigma }^{n+1}|\,\delta t}{\rho _K^{n+1}\ |K|}, $$

  where $M\in {\mathcal M}$ is the control volume which stands at the opposite side of K with respect to L.

The original scheme presented in [5] is recovered by this formulation for the one-dimensional constant velocity convection equation. Note however, that if the space dimension is greater than one, the above limitation may be not sufficient to reserve the maximum principle.