In this study, the entire rubble mound of the breakwater is modeled with DEM particles. Since the caisson block, larger than the soil particles, is not permeable to fluids, a resolved coupled model is applied to treat it as a rigid body and a moving wall boundary for the fluid. On the other hand, the fluid problem inside the porous media should be governed by the Navier-Stokes equations in a microscopic sense. This requires a high computational cost if a coupled resolving model is used, which resolves the details of the inside of the porous media. Therefore, this study applies an unresolved coupling model to the interaction between soil particles and fluid. To reduce the computational cost, a macroscopic fluid analysis is carried out to obtain the Darcy flow velocity, which is the volume-averaged flow velocity. Note that in the unresolved coupled model, the drag forces acting on the particles are calculated from a semi-empirical model. This drag force model for simple shape structures is already established, but the application to the porous media flow composed of fine particles is still discussing. The drag force model for the porous media should be intensely dependent on the porosity and its micro-structures. The drag force models in the geomechanics are nicely summarized [39]. The selection of the drag force models will be discussed again in the next section.
After confirming the performance of the conventional drag model for heaving and boiling behavior during piping destruction, some modifications will be made based on the traditional critical hydraulic gradient criterion proposed by Terzaghi [37].
An unified governing equation for Navier-Stokes flow and permeable porous media flows
Before we discuss the drag force model in the unresolved CFD-DEM coupled problem, we summarize the governing equations for a unified equation for Navier-Stokes flow and permeable porous media flows governed by the extended Darcy law. In the seepage-induced failure simulation of the rubble mound, water modeled as seepage flow goes through the rubble mound and becomes general surface flows outside the mound. The Darcy-Brinkman type equations are applied in this research to unify these equations. According to Akbari et al. [40, 41], a set of unified governing equations modeling fluid flows in the saturated porous domain \(\Omega _\textrm{m}\) and outside porous domain \(\Omega _\textrm{f}\) are derived as:
$$\begin{aligned}{} & {} \dfrac{C_r(\varepsilon _f)}{\varepsilon _f} \dfrac{D {\varvec{v}}_D}{D t} = - \dfrac{1}{\rho _f} \nabla p + {\varvec{g}} + \nu _E(\varepsilon _f) \, \nabla ^{2} {\varvec{v}}_D + {\varvec{F}}_{r}({\varvec{v}}_D, \varepsilon _f) \quad \textrm{in} \,\, \Omega _ \textrm{f} \, \cup \, \Omega _\textrm{m}, \end{aligned}$$
(1)
$$\begin{aligned}{} & {} \dfrac{ D \hat{\rho _f} }{D t} + \hat{\rho _f} \nabla \cdot \left( \dfrac{ {\varvec{v}}_D }{\varepsilon _f} \right) = 0 \quad \textrm{in} \,\, \Omega _\textrm{f} \, \cup \, \Omega _\textrm{m}, \end{aligned}$$
(2)
where \(\rho _f\) and \({\varvec{g}}\) represent the fluid density and the gravitational acceleration vector, respectively(\(\rho _f = 1.0 \, \mathrm {g/cm}^3\), \(|{\varvec{g}}| = 981 \, \mathrm{cm/s}^2\)). p is the fluid pressure, and \(\varepsilon _f\) is the porosity. \({\varvec{v}}_D\) is the Darcy velocity, defined as a locally averaged velocity, defined as the intrinsic velocity \({\varvec{v}}_f\) multiplied by the porosity i.e., \({\varvec{v}}_D = \varepsilon _f {\varvec{v}}_f\). Here, \(\hat{\rho }\) denotes the apparent density, which is given by \(\hat{\rho _f}=\varepsilon _f \rho _f\). This relation regarding the apparent density must be employed to guarantee the volume conservation of fluid inside the porous domain with the SPH. In this study, the fluid is assumed to be incompressible, and the porosity is also assumed to be unchanged at each time step. Therefore, the equation of continuity becomes a divergence-free condition of the Darcy velocity, as follows:
$$\begin{aligned} \nabla \cdot {\varvec{v}}_D = 0 \quad \textrm{in} \,\, \Omega _\textrm{f} \, \cup \, \Omega _\textrm{m}. \end{aligned}$$
(3)
The inertial coefficient \(C_r(\varepsilon _f)\) to add resistance force related to the virtual mass and the effective viscosity \(\nu _E(\varepsilon _f)\) including the kinematic viscosity \(\nu _f\) and an eddy viscosity \(\nu _T\) modeled by the Smagorinsky turbulent model are given as:
$$\begin{aligned}{} & {} C_r(\varepsilon _f)=1+0.34\dfrac{1-\varepsilon _f}{\varepsilon _f} \, , \end{aligned}$$
(4)
$$\begin{aligned}{} & {} \nu _E(\varepsilon _f) = \dfrac{\nu _f + \nu _T }{\varepsilon _f} \,. \end{aligned}$$
(5)
When the porosity \({\varepsilon _f = 1.0}\), \({C_r = 1.0}\) and \({{\varvec{F}}_r = {\varvec{0}}}\) , the unified Eq. (1) is consistent with the original incompressible Navier-Stokes equation. Then, the last term in Eq. (1), resistance force \({\varvec{F}}_{r}({\varvec{v}}_D, \varepsilon _f)\) for the ’fixed’ porous with the porosity, is assumed by:
$$\begin{aligned}{} & {} {\varvec{F}}_r({\varvec{v}}_D, \varepsilon _f) = -a(\varepsilon _f) \, {\varvec{v}}_D -b(\varepsilon _f) \, |{\varvec{v}}_D|{\varvec{v}}_D, \end{aligned}$$
(6)
$$\begin{aligned}{} & {} \quad a (\varepsilon _f) = \alpha \dfrac{\nu _f \, (1-\varepsilon _f)^{2}}{\varepsilon _f^{3}\, D_{50}^{2}} \,, \end{aligned}$$
(7)
$$\begin{aligned}{} & {} \quad b (\varepsilon _f) = \beta \dfrac{1-\varepsilon _f}{\varepsilon _f^{3} \, D_{50}} \,, \end{aligned}$$
(8)
where \(a (\varepsilon _f)\) and \(b (\varepsilon _f)\) are the linear and non-linear coefficients in the extended Darcy law, respectively. These two coefficients are determined from the average diameter \(D_{50}\) of rubble mound particle and the empirically derived parameters \(\alpha \) and \(\beta \).
The above equations are assumed that the rubble mound domain is fixed with an initial porosity distribution. Most of them can be applied to the deformable and movable rubble mound situation. However, the extended Darcy law is not applicable in such situations, and the resistance model must be modified. In this research, deformation and motion in the rubble mound are represented by DEM spherical particles with a constant diameter equal to the average diameter \(D_{50}\) of the rubble mound particle. The resistance force should be determined by considering the porosity change referring to the DEM particle motion and the relative velocity of fluid flow and DEM particle velocity. We selected one of the most fundamental and the most cited drag force models proposed by Wen and Yu [42] in the particulate multi-phase flow simulation as the modified resistance force. Wen and Yu proposed a drag force model with two different low and high porosity states. In this study, the resistance force is assumed by Eq. (6) in the low porosity seepage state, and the drag force in the high porosity suspended state is given as the equation proposed by Wen and Yu. The resistance force \({\varvec{F}}_r({\varvec{v}}_D, \varepsilon _f)\) in the Eq. (1), defined by Eq. (6), is replaced by this modified resistance force \({\varvec{F}}_r^*({\varvec{v}}_r, \varepsilon _f)\):
$$\begin{aligned}{} & {} {\varvec{F}}_r^*({\varvec{v}}_r, \varepsilon _f)= {\left\{ \begin{array}{ll} -a(\varepsilon _f) \, \varepsilon _f \, {\varvec{v}}_r -b(\varepsilon _f)\, \varepsilon _f^{2}\, |{\varvec{v}}_r| \, {\varvec{v}}_r &{} (\varepsilon _f < 0.80) \\ -c(\varepsilon _f) \, |{\varvec{v}}_r| \, {\varvec{v}}_r &{} (\varepsilon _f \ge 0.80) \end{array}\right. } \,, \end{aligned}$$
(9)
$$\begin{aligned}{} & {} \quad c (\varepsilon _f) = - \dfrac{3}{4} \, C_d \, \dfrac{1-\varepsilon _f}{\varepsilon _f^{2.7} \, D_{50} }\,, \end{aligned}$$
(10)
$$\begin{aligned}{} & {} \quad C_d = {\left\{ \begin{array}{ll} \dfrac{24 \, (1+0.15 \, Re^{0.687})}{Re} &{} (Re \le 1000) \\ 0.44 &{}(Re > 1000) \end{array}\right. } \,, \end{aligned}$$
(11)
$$\begin{aligned}{} & {} \quad Re = \dfrac{\hat{\rho }_f \, d_s \, \varepsilon _f \, |{\varvec{v}}_r|}{\mu _f} \,, \end{aligned}$$
(12)
where \({C_d}\) represents the drag coefficient depending on the Reynolds number Re. \({\mu _f}\) represents the fluid viscosity. Note here that the first equation in Eq. (9) has the same function shape as Eq. (6), but the variable was changed from the Darcy velocity \({\varvec{v}}_D\) to a relative velocity \({\varvec{v}}_r := {\varvec{v}}_f - {\varvec{v}}_s\). The modified resistance force can treat a more comprehensive condition ranging from flow through the mound as a seepage flow to flow with gravels suspended in the mound.
For the seepage flow problem on the porous structure, the coefficients of the resistance model are often determined as \({\alpha =150}\) and \(\beta = 1.75\) based on the Ergun model [43], for example, by Larese et al. [44] and Peng et al. [45]. However, These empirical coefficients \({\alpha }\) and \({\beta }\) have varieties in previous studies, and the determination should be carefully considered, as discussed by Losada et al. [39]. In this study, a preliminary study to determine this coefficient is carried out in the dam-break simulation through the porous media in “Preliminary validation test for the unified ISPH through a fixed porous media” section.
Spatial discretization of fluid using the SPH based on the Lagrangian description
The SPH is based on the Lagrangian description and divides a continuum into a set of discrete particles. These particles have a spatial distance, known as the smoothing length, over which a kernel function smoothes their properties. In a general SPH analysis, the physical quantity of the target particle i can be obtained by summing all the neighboring particles j, which exist within the range of the kernel area. Then, the contribution of each neighboring particle is weighted according to its distance from the target particle using a smoothing function. Finally, the motion of the particles can be described by the interpolated physical quantity based on a simple algorithm. Here, the basic concept of the SPH method is described by referring to Asai et al. [36].
A spatial discretization using scattered particles is summarized. First, The scalar function \(\phi ({\varvec{x}}_i,t)\) of the field represented by the domain \(\Omega \) can be expressed as a volume integration using the Dirac delta function \(\delta \) at any point \( {\varvec{x}} \) in the domain \(\Omega \). The Dirac delta functions \(\delta \) is a sharp function that is \(\infty \) at the origin and 0 elsewhere. It has the property of becoming 1 when integrated over a domain \(\Omega \) with \(\phi ({\varvec{x}})=1\) as the function to be integrated.
$$\begin{aligned}{} & {} \phi ({\varvec{x}}_i,t) = \int _\Omega \phi ({\varvec{x}}_j,t) \, \delta ( {\varvec{x}}_j - {\varvec{x}}_i) d {\varvec{x}} \, , \end{aligned}$$
(13)
$$\begin{aligned}{} & {} \delta ( {\varvec{x}}) = {\left\{ \begin{array}{ll} \infty &{} ({\varvec{x}} = {\varvec{0}}) \\ 0 &{} ({\varvec{x}} \ne {\varvec{0}}) \end{array}\right. } \,. \end{aligned}$$
(14)
Instead of this sharp delta function \(\delta \), the SPH method deals with approximations using a smooth kernel function:
$$\begin{aligned} \phi ({\varvec{x}}_i,t) \approx \int _\Omega \phi ({\varvec{x}}_j,t) \, W(r_{ij},h) d {\varvec{x}} \, , \end{aligned}$$
(15)
where W is a weight function called the smoothing kernel function. In the smoothing kernel function, \(r_{ij}\) and h are the particle distance between neighbor particles and the smoothing length, respectively. The distance \(r_{ij}\) is simply the length of the relative coordinate vector \({\varvec{r}}_{ij} = {\varvec{x}}_j - {\varvec{x}}_i\) (the relative distance \(r_{ij} = | {\varvec{r}}_{ij} |\)). In SPH discretization, the differential operators (e.g., the gradient \(\nabla \phi \), the divergence \(\nabla \cdot {\varvec{\phi }}\) and the Laplacian \(\nabla ^2 \phi \)) acting on a target particle i are evaluated using the neighboring particles j within an effective radius \(r_e\). Let \(\mathbb {S}_i\) be defined as the set of neighbor particles j of the target particle i, as follows:
$$\begin{aligned} \mathbb {S}_i \equiv \{ j = 1,...,N^\textrm{SPH} \mid r_e > r_{ij} \, \wedge \, j \ne i \, \wedge \, {\varvec{x}} _j \in \Omega \} \, , \end{aligned}$$
(16)
where \(N^\textrm{SPH}\) is the number of SPH particles (serial ID number). For an SPH simulation, the volume integration in Eq. (15), \(\nabla \cdot {\varvec{\phi }}\), \(\nabla \phi \) and \(\nabla ^2 \phi \) can be approximated as:
$$\begin{aligned} \phi ({\varvec{x}}_i, t) \approx \langle \phi \rangle _i= & {} \sum _{j \in \mathbb {S}_i} \frac{m_j}{\rho _j} \phi _j W(r_{ij},h) \,, \end{aligned}$$
(17)
$$\begin{aligned} \nabla \cdot {\varvec{\phi }} ({\varvec{x}}_i, t) \approx \langle \nabla \cdot {\varvec{\phi }} \rangle _i= & {} \frac{1}{\rho _i} \sum _{j \in \mathbb {S}_i} m_j \left( {\varvec{\phi }}_j- {\varvec{\phi }}_i \right) \cdot \nabla W(r_{ij},h) \,, \end{aligned}$$
(18)
$$\begin{aligned} \nabla \phi ({\varvec{x}}_i, t) \approx \langle \nabla \phi \rangle _i= & {} \frac{1}{\rho _i} \sum _{j \in \mathbb {S}_i} m_j \left( \phi _j- \phi _i \right) \nabla W(r_{ij},h) \,, \end{aligned}$$
(19)
$$\begin{aligned}= & {} {\rho _i} \sum _{j \in \mathbb {S}_i} m_j \left( \frac{\phi _j}{\rho _j^2} + \frac{\phi _i}{\rho _i^2} \right) \nabla W(r_{ij},h)\,, \end{aligned}$$
(20)
$$\begin{aligned} \nabla ^2 \phi ({\varvec{x}}_i, t) \approx \langle \nabla ^2 \phi \rangle _i= & {} \sum _{j \in \mathbb {S}_i} m_j \left( \frac{\rho _i + \rho _j}{\rho _i \rho _j} \frac{{\varvec{r}}_{ij} \cdot \nabla W(r_{ij},h)}{ r_{ij}^2 + \eta ^2} \right) \left( \phi _j - \phi _i \right) \,. \end{aligned}$$
(21)
The subscripts i and j indicate the positions of a labeled particle; for example, \(m_j\) and \(\rho _j\) mean the representative mass and density of a neighbor particle j, respectively. Note that the triangle bracket \(\langle \cdot \rangle \) indicates the SPH approximation of a particular function. Furthermore, note that the two sets of expressions of the gradient models have various properties that can be converted to each other analytically. \(\eta \) in Eq. (21) is the parameter to avoid division by zero and is defined by the following expression, i.e., \(\eta ^{2}=0.0001(h/2)^{2}\). In this study, the kernel function adopts the cubic spline curve with the smoothing length \(h=1.2r_0\) (\(r_0\) being the initial distance between SPH particles) and an effective radius \(r_e=2h\).
Discretization of the unified governing equations with the ISPH method
In the ISPH: Incompressible SPH method, the unified governing equations, Eqs. (1) and (2), are first discretized in time by following the projection method based on the predictor and corrector scheme. Then, the spatial Discretization is implemented for the time-discretized equations.
To begin with the time discretization, \({\varvec{v}}_D\) at \(n+1\) step is written as:
$$\begin{aligned} {\varvec{v}}_D^{n+1} = {\varvec{v}}_D ^{*} + \Delta {\varvec{v}}_D ^{*} \, , \end{aligned}$$
(22)
where \({\varvec{v}}_D^{*}\) is the predictor term calculated explicitly from the physical quantities at n step, while \(\Delta {\varvec{v}}_D ^{*}\) is the corrector term which is implicitly given from the physical quantities at \(n+1\) step to correct the predictor term. Based on the projection method, Eq. (22) can be separated as:
$$\begin{aligned} {\varvec{v}}_D ^{*}= & {} {\varvec{v}}_D ^{n} + \dfrac{\varepsilon _f^n \Delta t}{C_r(\varepsilon _f^n)} \left\{ {\varvec{g}} + \nu _E(\varepsilon _f^n) \nabla ^{2} {\varvec{v}}_D ^{n} + {\varvec{F}}_r(\varepsilon _f^n) \right\} \, , \end{aligned}$$
(23)
$$\begin{aligned} \Delta {\varvec{v}}_D ^{*}= & {} - \dfrac{1}{\rho _f} \dfrac{\varepsilon _f \Delta t}{C_r(\varepsilon _f)} \, \nabla p^{n+1} \, . \end{aligned}$$
(24)
The pressure \(p^{n+1}\) in Eq. (24) is determined by solving the simultaneous linear equation called the Pressure Poisson Equation (PPE):
$$\begin{aligned} \nabla ^2 p^{n+1} = \dfrac{C_r(\varepsilon _f)\, \rho _f}{\varepsilon _f \Delta t} \, \nabla \cdot {\varvec{v}}_D ^{*} \, . \end{aligned}$$
(25)
Next, the PPE is discretized into the particle quantities using Eqs. (18) and (21) as:
$$\begin{aligned} \langle \nabla \cdot {\varvec{v}}_{D} \rangle _i= & {} \dfrac{1}{\hat{\rho }_{fi}} \sum _{j \in \mathbb {S}_i} m_j \left( {\varvec{v}}_{Dj} - {\varvec{v}}_{Di} \right) \cdot \nabla W(r_{ij},h) \, , \end{aligned}$$
(26)
$$\begin{aligned} \langle \nabla ^2 p \rangle _i= & {} \sum _{j \in \mathbb {S}_i} m_j \left( \frac{ \hat{\rho }_{fi} + \hat{\rho }_{fj} }{\hat{\rho }_{fi} \hat{\rho }_{fj}} \frac{{\varvec{r}}_{ij} \cdot \nabla W( r_{ij},h)}{ r_{ij}^2 + \eta ^2} \right) \left( p_j - p_i \right) \, . \end{aligned}$$
(27)
Here, \(\hat{\rho }_f\) denotes the apparent fluid density that is defined by the following equation, i.e., \(\hat{\rho }_f=\varepsilon _f \rho _f\). This concept must be employed to guarantee the volume conservation of fluid inside the rubble mound. The representative volume changes as the fluid density changes depending on the porosity. Consequently, in an entire analysis domain, including outside and inside of the rubble mound, the total amount of the fluid volume can be theoretically conserved.
Using the above SPH approximate operators, the PPE in Eq. (25) is described as:
$$\begin{aligned} \langle \nabla ^2 p \rangle _i^{n+1} = \dfrac{C_r(\varepsilon _{fi}) \, \rho _{fi}}{\varepsilon _{fi} \, \Delta t} \langle \nabla \cdot {\varvec{v}}_{D}^{*} \rangle _i \, . \end{aligned}$$
(28)
This derivation of the PPE is well-known as the formulation under the divergence-free condition. Next, the velocity at \(n+1\) step \({\varvec{v}}_D^{n+1}\) is determined from the obtained pressure calculated through Eq. (28) by using Eq. (22) and Eq. (24). The pressure gradient term needs to be determined to implement this updating procedure. By referring to Eq. (20), it can be written as:
$$\begin{aligned} \langle \nabla p \rangle _i = \hat{\rho }_{fi} \sum _{j \in \mathbb {S}_i} m_j \left( \dfrac{p_j}{\hat{\rho }_{fj}^2}+\dfrac{p_i}{\hat{\rho }_{fi}^2} \right) \nabla W(r_{ij},h) \, . \end{aligned}$$
(29)
Finally, the position of a particle is updated from the updated velocity:
$$\begin{aligned} {\varvec{r}}_{i}^{n+1}={\varvec{r}}_{i}^{n}+ \dfrac{{\varvec{v}}_{Di}^{n+1}}{\varepsilon _{fi}}\Delta t \, . \end{aligned}$$
(30)
A particle’s intrinsic Lagrange velocity \({\varvec{v}}_f\) is the Darcy velocity \({\varvec{v}}_D\) divided by the porosity \(\varepsilon _f\). In the next step, this intrinsic velocity \({\varvec{v}}_f\) is used to calculate the particle position \({\varvec{r}}_{i}^{n+1}\).
Stabilized ISPH method for the unified equation
The heterogeneity of the particle distribution directly affects the accuracy of the SPH discretization. Therefore, the stabilized ISPH method proposed by Asai et al. [36] is arranged for the unified equation. In the incompressible Navier-Stokes equation, numerical density \( \langle {\rho }_f \rangle _i\) should be preserved with the true fluid density value \(\rho _f\). On the other hand, the numerical density in the pressure Poisson equation using the Darcy-Brinkman type unified equation must be \( \hat{\rho }_{fi} = \varepsilon _{fi}^n \rho _f\), consistent with the apparent density following the surrounding porosity distribution. The discrete PPE (28) is slightly modified by adding a stabilization term, which tries to maintain the reasonable density as:
$$\begin{aligned} \langle \nabla ^2 p \rangle _i^{n+1} \simeq \dfrac{C_r(\varepsilon _{fi})}{\varepsilon _{fi}} \left( \dfrac{\rho _{fi}}{\Delta t} \langle \nabla \cdot {\varvec{v}}_{D}^{*} \rangle _i + \gamma \dfrac{ \hat{\rho }_{fi} - \langle \hat{\rho }_f \rangle _i^{n} }{\Delta t^2} \right) \, , \end{aligned}$$
(31)
where \(\gamma \left( 0\le \gamma \le 1\right) \) is called the relaxation coefficient and is generally set to be much less than 1.
Porosity and apparent density/volume
In this study, the ISPH and DEM are coupled through drag forces in an unresolved coupling scheme. Therefore, the porosity \(\varepsilon _f\), which is the volume fraction of voids in a unit fluid volume, is necessary to calculate the space-averaged fluid density \(\hat{\rho }_f\) and Darcy velocity \({\varvec{v}}_D\). Based on the concept of interpolation approximation of the SPH method, the porosity is obtained from the total volume of wall particles and DEM particles contained within the influence zone of the target particles as follows:
$$\begin{aligned} \langle \varepsilon _{f} \rangle _i= & {} 1-\dfrac{ \displaystyle \sum _{j \in \mathbb {D}_i } V_{sj} W(r_{ij},h) }{1- {\displaystyle \sum _{j \in \mathbb {S}_i^{\textrm{c}} } \dfrac{m_{j}}{\rho _{fj}} W(r_{ij},h)} } \, , \end{aligned}$$
(32)
$$\begin{aligned} \mathbb {S}_i^{\textrm{c}} \equiv \{ j= & {} 1,...,N^\textrm{SPH} \mid r_e > r_{ij} \, \wedge \, j \ne i \, \wedge \, {\varvec{x}} _j \in \Omega _\textrm{c} \} \, , \end{aligned}$$
(33)
$$\begin{aligned} \mathbb {D}_i \equiv \{ j= & {} 1,...,N^\textrm{DEM} \mid r_e > r_{ij} \, \wedge \, j \ne i \, \wedge \, {\varvec{x}} _j \in \Omega _\textrm{m} \cup \Gamma _\textrm{c} \} \, , \end{aligned}$$
(34)
where \(V_s\) represents the volume of DEM particle. The sets \(\mathbb {S}_i^{\textrm{c}}\) and \(\mathbb {D}_i\), which are subject to the sum of the numerator-denominator, are the sets of all DEM particles and the sidewall/caisson particles of the SPH contained within the influence area of the target particle i, respectively. Figure 3 shows that the fluid deforms to avoid solid particles in the gap. On the other hand, the unresolved coupling allows for the overlap of solids and fluid, so the apparent increase in fluid volume and an apparent decrease in fluid density must be considered. Therefore, under the constant mass of fluid particles, spatial averaging is performed using porosity \(\varepsilon _f\), and the apparent density \(\hat{\rho }_f\), volume \(\hat{V}_f\), and Darcy flow velocity \({\varvec{v}}_D\) are defined as follows:
$$\begin{aligned} \hat{\rho }_{fi}= & {} \langle \varepsilon _f \rangle _i \rho _{fi}\, , \end{aligned}$$
(35)
$$\begin{aligned} \hat{V}_{fi}= & {} \dfrac{m_{i}}{ \hat{\rho }_{fi}} \, , \end{aligned}$$
(36)
$$\begin{aligned} {\varvec{v}}_{Di}= & {} \langle \varepsilon _f\rangle _i {\varvec{v}}_{fi}\, . \end{aligned}$$
(37)