2.1 Joint scalar PDF (SPDF)
The approach shown here has firstly been published in [
15] and it is briefly revised. The joint scalar PDF model includes enthalpy and mass fractions variables into its sample space. The one-time one-point fine-grained Eulerian probability density function (PDF) is therefore defined as:
$$ f^{\prime}\left( \eta,\boldmath{Z};{x},t\right) = \delta \left( h\left( \boldmath{x},t\right) - \eta\right) \times \prod\limits_{\alpha=1}^{N_{s}} \delta (Y_{\alpha}\left( \boldmath{x},t\right) - Z_{\alpha}) $$
(1)
where
η,
Zα are the sample enthalpy and mass fractions,
h(
x,
t) and
Yα(
x,
t) are the real enthalpy and mass fractions, respectively. A non-closed filtered transport equation for LES-PDF can be obtained by deriving a transport equation for
\(f^{\prime }\) and applying a spatial filter to it [
8]:
$$ \begin{array}{@{}rcl@{}} \frac{\partial \overline{\rho}\widetilde{f}}{\partial t} + \frac{\partial \overline{\rho}\widetilde{u}_{i}\widetilde{f}}{\partial x_{i}} &=& \frac{\partial} {\partial x_{i}} \left( {\Gamma} \frac{\partial \widetilde{f}}{\partial x_{i}} \right) - \frac{\partial} {\partial Z_{\alpha}} \left( \overline{\rho} \widetilde{ S_{\alpha} \mid \boldmath{\Psi}} \widetilde{f} \right)\\ &&+ \frac{\partial} {\partial x_{i}} \left( \overline{\rho}\widetilde{u}_{i}\widetilde{f} - \overline{\rho} \widetilde{u_{i} \mid \boldmath{\Psi}} \widetilde{f} \right)- \frac{\partial} {\partial \eta}\left( \overline{\rho} {\widetilde{\frac{1}{\rho}\frac{D p}{D t}} \mid \boldmath{\Psi}} \widetilde{f} + \overline{\rho} \widetilde{ \frac{\tau_{ij}}{\rho} \frac{\partial u_{i}}{\partial x_{j}}} \mid \boldmath{\Psi}\widetilde{f} \right)\\ &&- \frac{\partial^{2}} {\partial \psi_{\alpha} \partial \psi_{\beta}} \left( \overline{\rho} \widetilde{{\Gamma} \frac{\partial \phi_{\alpha}}{\partial x_{i}} \frac{\partial \phi_{\beta}}{\partial x_{i}}} \mid \boldmath{\Psi} \widetilde{f} \right) \end{array} $$
(2)
where
\(\left (\ \widetilde {\cdot } \ \right )\) is the filtering operator and
\(\widetilde {f}\) is the filtered PDF. The proposed LES-PDF equation relies on the assumption that the reactive source term is a function of sample variables and the filtered pressure. In this way, the reactive source term is partially modelled as
\(\widetilde {S}_{\alpha } \approx S_{\alpha }\left (\overline {p},\boldmath {\Psi }\right )\), where Ψ = [
η,
Zα], in a similar fashion as [
11,
13], although it is still accurate closed to regions of small compressibility or low-Mach number flows. The Smagorinsky model [
17] is used to close the convective terms. The several remaining unclosed terms are modelled with the IEM micro-mixing model [
18]. The joint scalar LES-PDF equation obtained is:
$$ \begin{array}{@{}rcl@{}} \frac{\partial \overline{\rho}\widetilde{f}}{\partial t} + \frac{\partial \overline{\rho}\widetilde{u}_{i} \widetilde{f}} {\partial x_{i}} &= & \frac{\partial}{\partial x_{i}}\left( {{\varGamma}}^{\prime} \frac{\partial \widetilde{f}}{\partial x_{i}} \right) - \frac{\partial} {\partial Z_{\alpha}}\left( \overline{\rho} S_{\alpha}\left( \overline{p},\boldmath{\Psi}\right) \widetilde{f} - \frac{1}{2} \frac{C_{Y_{\alpha}}}{\tau_{sgs}} \overline{\rho} \left( Z_{\alpha} - \widetilde{Y}_{\alpha} \right) \widetilde{f} \right)\\ && - \frac{\partial}{\partial \eta}\left( \frac{D\overline{p}}{Dt}\widetilde{f} + \widetilde{\tau}_{ij} \frac{\partial \widetilde{u}_{i}}{\partial x_{j}}\widetilde{f} - \frac{1}{2} \frac{C_{H}}{\tau_{sgs}} \overline{\rho} \left( \eta - \widetilde{h} \right) \widetilde{f} \right) \end{array} $$
(3)
where
\({{\varGamma }}^{\prime } = \mu /\sigma + \mu _{sgs}/\sigma _{sgs}\) is the total diffusion coefficient. A sub-grid mixing timescale is defined as in [
19]:
$$ \tau_{sgs} = \left( \frac{\mu+\mu_{sgs}} {\overline{\rho}{\Delta}^{2}} \right){~}^{-1}\left( 1 - \exp(-\mathscr{R}^{2}) \right) $$
(4)
where
\(\mathscr{R}\) is the ratio
μsgs/
μ (akin to a sub-grid Reynolds number). The time-scale reverts to a molecular one when the local flow is laminar. The model constants
\(C_{Y_{\alpha }}\) and
CH are equal to 2 following [
20]. Equal diffusivity is assumed (unity Lewis number), where the Schmidt number
σ and Prandtl number are equal to unity, while their equivalent sub-grid number,
σsgs are equal to 0.7. The conventional Smagorinsky model is employed to obtain the sub-grid viscosity,
μsgs. The equations for the n
th-set of Eulerian stochastic fields for the mass fractions and enthalpy can be obtained following [
10]:
$$ \begin{array}{@{}rcl@{}} \frac{ \partial \overline{\rho} \mathscr{Y}^{n}_{\alpha}}{\partial t} +\frac{\partial \overline{\rho}\widetilde{u}_{i} \mathscr{Y}^{n}_{\alpha}}{\partial x_{i}} &=& \frac{\partial} {\partial x_{i}} \left( {\varGamma}^{\prime} \frac{\partial \mathscr{Y}^{n}_{\alpha}}{\partial x_{i}} \right) + \overline{\rho} S^{n}_{\alpha}\left( \overline{p},\boldmath{\Psi}\right)- \frac{1}{2} \frac{C_{Y_{\alpha}}}{\tau_{sgs}} \overline{\rho} \left( \mathscr{Y}^{n}_{\alpha} - \widetilde{Y}_{\alpha} \right)\\ &&+ \left( 2 \overline{\rho} {\varGamma}^{\prime} \right){~}^{1/2} \frac{\partial \mathscr{Y}^{n}_{\alpha}}{\partial x_{i}}\frac{\text{d} {W^{n}_{i}}}{\text{d} t} \end{array} $$
(5)
$$ \begin{array}{@{}rcl@{}} \frac{\partial \overline{\rho}\mathscr{H}^{n}}{\partial t} +\frac{\partial \overline{\rho}\widetilde{u}_{i} \mathscr{H}^{n}}{\partial x_{i}} &=& \frac{\partial} {\partial x_{i}} \left( {\varGamma}^{\prime} \frac{\partial \mathscr{H}^{n}}{\partial x_{i}} \right) + \frac{D\overline{p}}{Dt} + \widetilde{\tau}_{ij} \frac{\partial \widetilde{u}_{i}}{\partial x_{j}}- \frac{1}{2} \frac{C_{H}}{\tau_{sgs}} \overline{\rho} \left( \mathscr{H}^{n} - \widetilde{h} \right)\\ &&+ \left( 2 \overline{\rho} {\varGamma}^{\prime} \right){~}^{1/2}\frac{\partial \mathscr{H}^{n}}{\partial x_{i}}\frac{\text{d} {W^{n}_{i}}}{\text{d} t} \end{array} $$
(6)
where
\(\mathscr{Y}^{n}_{\alpha }\) and
\(\mathscr{H}^{n}\) are the stochastic variables for mass fractions of the chemical specie
α and enthalpy, respectively, of the stochastic field
n. The Wiener process
\(d{W^{n}_{i}}\) is approximated by
dt1/2γ, where
γ = {− 1,1} is a dichotomic vector, ensuring
\(\left \langle dW_{i} \right \rangle = 0\). Filtered quantities are calculated from the average of the stochastic fields as
\( \widetilde {Q} \approx \left \langle Q \right \rangle \).
The filtered pressure field is therefore calculated, when applying the ideal gas law, without neglecting its sub-grid fluctuations terms:
$$ \overline{p} = \overline{\rho} \widetilde{RT} \approx \overline{\rho} R_{u} \left[ \frac{1}{N_{F}} \sum\limits_{n=1}^{N_{F}} \left( \sum\limits_{\alpha=1}^{N_{s}} \frac{\mathscr{Y}^{n}_{\alpha}}{MW_{\alpha}} \right) T^{n} \right] $$
(7)
where
Ru is the gas universal constant,
MWα is the molecular weight of the chemical specie
α and
Tn is the temperature of the
nth field. This approach has not been tried previously with the stochastic fields approach, only in Lagrangian formulations [
13]. Accounting for sub-grid pressure fluctuations may improved accuracy in hydrocarbon fuels-pure oxygen systems [
21].
2.2 Joint velocity-scalar PDF (VSPDF)
This new formulation, also published in [
15], is a more complete approach. Here, the velocity components, along with density, mass fractions and total energy are included into the PDF sample space. This allows the exact closure of the convective and chemical source terms in the LES framework. The one-time one-point joint velocity-scalar fine-grained Eulerian PDF is defined as follows:
$$ \begin{array}{@{}rcl@{}} f^{\prime}(d,\boldmath{v},\zeta, \boldmath{Z};\boldmath{x},t)& = & \delta (\rho(\boldmath{x},t) - d) \prod\limits_{i=1}^{3} \delta (u_{i}(\boldmath{x},t) - v_{i})\\ &&\times\delta (e_{t}(\boldmath{x},t) - \zeta) \prod\limits_{\alpha=1}^{N_{s}} \delta (Y_{\alpha}(\boldmath{x},t) - Z_{\alpha}) \end{array} $$
(8)
where
d,
vi,
ζ and
Zα are the sample density, velocity, total energy and mass fraction, while
ρ(
x,
t),
ui(
x,
t),
et(
x,
t) and
Yα(
x,
t) are the respective real fields. Following the same methods for the SPDF, it is possible to derive the following closed transport equation:
$$ \begin{array}{@{}rcl@{}} \frac{\partial \overline{\rho} \widetilde{f}}{\partial t} + \frac{\partial \overline{\rho} v_{i} \widetilde{f}}{\partial x_{i}} &=& - \frac{\partial}{\partial d}\left( - \overline{\rho}^{2} \frac{\partial \widetilde{u}_{i}}{\partial x_{i}} \widetilde{f} \right)\\ &&-\frac{\partial}{\partial Z_{\alpha}} \left( \frac{\partial \widetilde{J}_{\alpha,i}} {\partial x_{i}} \widetilde{f} + \overline{\rho} S_{\alpha}\left( \boldmath{\Phi}\right) \widetilde{f} - \frac{1}{2} C_{Y_{\alpha}} \frac{\epsilon}{k} \overline{\rho} \left( Z_{\alpha} - \widetilde{Y}_{\alpha} \right) \widetilde{f} \right)\\ &&- \frac{\partial} {\partial v_{i}} \left( -\frac{\partial \overline{p}}{\partial x_{i}} \widetilde{f} + \frac{\partial \widetilde{\tau}_{ij}}{\partial x_{i}} \widetilde{f} + \overline{\rho} G_{ij} \left( v_{j} - \widetilde{u}_{j} \right) \widetilde{f} \right)\\ &&+ \frac{\partial^{2}}{\partial v_{i} \partial v_{i}} \left( \frac{1}{2} C_{0} \epsilon_{sgs} \widetilde{f} \right)- \frac{\partial}{\partial \zeta}\left( \frac{\partial \widetilde{q}_{i}}{\partial x_{i}} \widetilde{f} - \frac{\partial \overline{p} \widetilde{u}_{i}}{\partial x_{i}}\widetilde{f} + \frac{\partial \widetilde{\tau}_{ij}\widetilde{u}_{j}}{\partial x_{i}} \widetilde{f}\right.\\ &&\left.- \frac{1}{2} C_{e_{t}} \frac{\epsilon_{sgs}}{k_{sgs}} \overline{\rho} \left( \zeta - \widetilde{e_{t}} \right) \widetilde{f}\right) \end{array} $$
(9)
The simplified Langevin model [
2] is used to close the unknown terms for the velocity and density part. The tensor
Gij is then defined as:
$$ G_{ij} = -\frac{\epsilon_{sgs}}{k_{sgs}}\left( \frac{1}{2} + \frac{3}{4}C_{0} \right)\delta_{ij} $$
(10)
where
ksgs and
𝜖sgs are the sub-grid kinetic energy and its dissipation respectively. The IEM micro-mixing model [
18] is also chosen to close the remaining unknown terms on the total energy and mass fractions part.
Two further approximations are necessary to derive the stochastic field equations. The first is to substitute the pressure terms by the stochastic pressure field, \(\mathscr{P}(\boldmath {x},t)\), instead of the filtered pressure, \(\overline {p}\), neglecting sub-grid pressure fluctuations. This results in a different momentum equation from the Burgers’ equation, preventing the occurrence of numerical shocks. The stochastic pressure can be directly obtained from stochastic variables \(\mathscr{P} \equiv \mathscr{P}(\varrho ,\mathscr{U}_{i},\mathscr{E}_{t},\mathscr{Y}_{\alpha })\) and present the same properties of the real pressure field.
The other approximation used is to neglect the stochastic difference on the right hand side of the continuity equation if written on conservative form, so it can ensure mass conservation for all set of stochastic fields. This approximation does not affect the first-moments and is exact if the density is constant. Although this approximation can be avoided in a Lagrangian framework, the SPDEs are aimed to be coupled with a Eulerian solver. This numerical approximation act as an additional force to prevent the fields from diverge striongly by adding or removing stochastic density. The proposed set of Eulerian SPDEs is therefore:
$$ \frac{\partial \varrho^{n}}{\partial t} + \frac{\partial \varrho^{n} \mathscr{U}^{n}_{i}} {\partial x_{i}} = 0 $$
(11)
$$ \frac{\partial \varrho^{n} \mathscr{U}^{n}_{i}}{\partial t} + \frac{\partial \varrho^{n} \mathscr{U}^{n}_{j} \mathscr{U}^{n}_{i}}{\partial x_{j}} = -\frac{\partial \mathscr{P}^{n}}{\partial x_{i}} + \frac{\varrho^{n}}{\overline{\rho}} \frac{\partial \widetilde{\tau}_{ij}}{\partial x_{i}} + \varrho^{n} G_{ij} \left( \mathscr{U}^{n}_{j} - \widetilde{u}_{j} \right) + \varrho^{n} \sqrt{C_{0} \frac{\epsilon_{sgs}}{\overline{\rho}}} \frac{\text{d} {W^{n}_{i}}}{\text{d} t} $$
(12)
$$ \frac{\partial \varrho^{n} \mathscr{Y}^{n}_{\alpha}}{\partial t} +\frac{\partial \varrho^{n} \mathscr{U}^{n}_{i} \mathscr{Y}^{n}_{\alpha}}{\partial x_{i}} = \frac{\varrho^{n}}{\overline{\rho}}\frac{\partial \widetilde{J}_{\alpha,i}} {\partial x_{i}} + \varrho^{n} S_{\alpha}\left( \boldmath{\Psi}\right) - \frac{1}{2} C_{Y_{\alpha}} \frac{\epsilon_{sgs}}{k_{sgs}} \varrho^{n} \left( \mathscr{Y}^{n}_{\alpha} - \widetilde{Y}_{\alpha} \right) $$
(13)
$$ \frac{\partial \varrho^{n} \mathscr{E}^{n}_{t}}{\partial t} + \frac{\partial \varrho^{n} \mathscr{U}^{n}_{i} \mathscr{E}^{n}_{t}}{\partial x_{i}} = \frac{\varrho^{n}}{\overline{\rho}}\frac{\partial \widetilde{q}_{i}}{\partial x_{i}} - \frac{\varrho^{n}}{\overline{\rho}}\frac{\partial \overline{p} \widetilde{u}_{i}}{\partial x_{i}} + \frac{\varrho^{n}}{\overline{\rho}}\frac{\partial \widetilde{\tau}_{ij}\widetilde{u}_{j}}{\partial x_{i}} - \frac{1}{2} C_{e_{t}} \frac{\epsilon_{sgs}}{k_{sgs}} \varrho^{n} \left( \mathscr{E}^{n}_{t} - \widetilde{e_{t}} \right) $$
(14)
The employed closure relation for the dissipation of the sub-grid kinetic energy,
𝜖sgs, is the following:
$$ \epsilon_{sgs} = C_{\epsilon} k_{sgs}^{3/2}/{\Delta} $$
(15)
where the constant
C𝜖 is equal to 1.05. The micro-mixing constants
\(C_{Y_{\alpha }}\) and
\(C_{e_{t}}\) are set to 2 and the Langevin constant is set to 2.1, following LES-PDF convention [
8]. The sub-grid kinetic energy,
ksgs, can be directly obtained from the stochastic fields information:
$$ k_{sgs}= \frac{1}{2} \left( \frac{1}{N_{f}} \sum\limits_{n=1}^{N_{f}} (\mathscr{U}^{n}_{i}-\widetilde{u}_{i})^{2} \right) $$
(16)
At last, the filtered variables can be obtained from the average of the Eulerian stochastic fields. For a variable
Q(
x,
t), it is possible to recover the filtered and the Favre-filtered values:
$$ \overline{Q} = \frac{1}{N_{f}}\sum\limits_{n=1}^{N_{f}} Q^{n} \ \ \widetilde{Q} = \frac{{\sum}_{n=1}^{N_{f}} \varrho^{n} Q^{n}}{{\sum}_{n=1}^{N_{f}} \varrho^{n}} $$
(17)
The developed Eulerian stochastic differential equations are equivalent to Eq.
9 with mild assumptions. The continuity and momentum equations resemble those of Azarnykh et al. [
22]. This set of equations also does not generate the numerical shocks predicted in [
23] if a conventional discretisation scheme is employed.
The novelty of the present VSPDF formulation stems from both the selection of independent variables and its solution method, which has advantages in high speed flows. The VSPDF approach includes a set of variables similar to Kollmann [
5] but without a dilatation variable and with density instead of pressure, unlike the formulation of Nouri et al. [
24]. This has benefits when combining the method with a density-based solver, which are typically used in high-speed flows. The use of
Eulerian stochastic fields method to solve the VSPDF has not been used before for fully compressible and reactive flows. Moreover, the addition of an stochastic pressure, unlike other VPDF methods [
6,
23] , allowed sub-grid pressure fluctuations as well as increase the stability of the SPDE system (
11)-(
14).