Computational Investigation of Weakly Turbulent Flame Kernel Growths in Iso-Octane Droplet Clouds in CVC Conditions

Abstract

Numerical simulations of turbulent flame kernel growths in monodisperse clouds of iso-octane liquid droplets are conducted in conditions relevant to constant volume combustors. The simulations make use of a low-Mach number Navier-Stokes solver and a thermodynamic pressure evolution model has been implemented to reproduce the pressure variation that may be issued from either experiments or from a standard (i.e., analytical) compression law. Chemistry is described with a representative skeletal mechanism featuring 29 species and 48 elementary reaction steps. The computational results clearly confirm the enhancement of flame propagation in constant volume combustion conditions. The impact of the droplet diameter on the turbulent flame development is scrutinized for two distinct values of the Stokes number St equal to 0.1 and 1.0. Significant influence on the flame dynamics is put into evidence. This is a direct outcome of the equivalence ratio and temperature heterogeneities, which are themselves very sensitive to the choice of the Stokes number value. Then, small-scale turbulence-scalar interactions (TSI) are studied by analyzing the fields of the scalar gradients and strain-rate. Their dynamics is investigated for both non-reactive and reactive two-phase flows conditions. The TSI analysis is performed on the basis of time evolution equations written for quantities that characterize the couplings between the velocity gradient tensor and scalar gradients vectors. Special emphasis is placed on the possible influence of mass exchange terms between the liquid and gaseous phases.

Appendices

Appendix A: Thermodynamic Pressure Variation with Artificial Mass Sources and Chemical Heat Release

The thermodynamic pressure evolution is deduced from the analysis of the normalized EoS together with a non-dimensional form of the NSE featuring an artificial mass source term $\dot {m}_{p}$, see Eq. 2d in Section 2. Since Asphodele resolves the non-dimensional form of NSE, the thermodynamic pressure evolution is considered within the corresponding non-dimensional framework:

$$ \begin{array}{@{}rcl@{}} \frac{\partial \rho^{+}}{\partial t^{+}} + \frac{\partial (\rho^{+} u_{j}^{+})}{\partial x_{j}^{+}} &=& \dot{m}_{p}^{+} + \dot{d}_{\rho}^{+} \qquad\qquad\qquad\qquad\qquad\quad\left( \rho_{\infty} u_{ref}x_{ref}^{-1} \right) \end{array} $$

(26a)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \rho^{+} u_{i}^{+}}{\partial t^{+}} + \frac{\partial \left( \rho^{+} u_{i}^{+} u_{j}^{+}\right)}{\partial x_{j}^{+}} &=& - \frac{\partial p_{1}^{+}}{\partial x_{i}^{+}} +\frac{\partial \tau^{+}_{ij}}{\partial x_{j}^{+}}+\dot{m}_{p}^{+} u_{i}^{+}+\dot{d}_{\rho u_{i}}^{+} \qquad\left( {\rho_{\infty} u_{ref}^{2}}{x_{ref}^{-1}} \right) \end{array} $$

(26b)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \rho^{+} h_{s}^{+}}{\partial t^{+}}+\frac{\partial \left( \rho^{+} h_{s}^{+} u_{j}^{+}\right)}{\partial x_{j}^{+}}&=& \frac{\gamma_{\infty}-1}{\gamma_{\infty}}\frac{d p_{0}^{+}}{d t^{+}}+\dot{\omega}_{T}^{+} -\frac{\partial q_{j}^{+}}{\partial x_{j}^{+}}+\text{Ma}_{\infty}^{2}\left( \gamma_{\infty}-1 \right)\tau_{ij}^{+} \frac{\partial u^{+}_{i}}{\partial x_{j}^{+}} \\ &&+\dot{d}_{\rho h_{s}}^{+}+\dot{m}_{p}^{+} h_{s}^{+} \quad\qquad\qquad\left( {\rho_{\infty} C_{p\infty} T_{\infty} u_{ref}}{x_{ref}^{-1}} \right) \end{array} $$

(26c)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \rho^{+} Y_{\alpha}}{\partial t^{+}} + \frac{\partial \left( \rho^{+} Y_{k} u^{+}_{j}\right) }{\partial x_{j}^{+}}&=& - \frac{\partial {J_{j}^{\alpha +}}}{\partial x_{j}^{+}} +\dot{d}_{\rho Y_{\alpha}}^{+} +\dot{\omega}_{\alpha}^{+}+ \dot{m}_{p}^{+}Y_{\alpha}\quad\quad\left( {\rho_{\infty} u_{ref}}{x_{ref}^{-1}} \right) \end{array} $$

(26d)

In the above equations, variables with superscript (⁺) are normalized quantities, and the subscript (_ref) and $(_{\infty })$ denote either normalization (reference) parameters or values of physical quantities taken at infinity. The quantities that are reported at the end of each line (in brackets) correspond to values that are used to normalize the corresponding equations. A standard normalization procedure for low-Mach conditions is retained:

$$ \begin{array}{@{}rcl@{}} x_{i}&=&x_{ref}x_{i}^{+};\quad u_{i}=u_{ref}u_{i}^{+};\quad t=({x_{ref}}/{u_{ref}}) \hspace*{1.7pt} t^{+}; \end{array} $$

(27a)

$$ \begin{array}{@{}rcl@{}} \rho&=&\rho_{\infty}\rho^{+};\quad \dot{m}_{p}=\rho_{\infty}({u_{ref}}/{x_{ref}}) \hspace*{1.7pt} \dot{m}_{p}^{+};\quad T=T_{\infty}T^{+}; \end{array} $$

(27b)

$$ \begin{array}{@{}rcl@{}} p&=&p_{\infty}p^{+}=\rho_{\infty}({\mathcal{R}T_{\infty}}/{\mathcal{M}_{\infty}}) \hspace*{1.7pt} p^{+};\hspace*{1.7pt}\hspace*{1.7pt} \mathcal{M}=\mathcal{M}_{\infty}\mathcal{M}^{+};\quad h_{s}=C_{p\infty}T_{\infty} h_{s}^{+}; \end{array} $$

(27c)

$$ \begin{array}{@{}rcl@{}} \gamma_{\infty}&=&\frac{C_{p\infty}}{C_{p\infty}-\mathcal{R}/\mathcal{M}_{\infty}};\quad \text{Ma}_{\infty} = \frac{u_{ref}}{\sqrt{\gamma_{\infty}T_{\infty}\mathcal{R}/\mathcal{M}_{\infty}}}; \end{array} $$

(27d)

$$ \begin{array}{@{}rcl@{}} \dot{\omega}_{T}&=&\rho_{\infty} C_{p\infty}T_{\infty} ({u_{ref}}/{x_{ref}}) \hspace*{1.7pt} \dot{\omega}_{T}^{+};\quad \dot{\omega}_{k}=\rho_{\infty} ({u_{ref}}/{x_{ref}}) \hspace*{1.7pt} \dot{\omega}_{k}^{+}; \end{array} $$

(27e)

$$ \begin{array}{@{}rcl@{}} \tau_{ij}&=&\rho_{\infty}C_{p\infty}T_{\infty}\tau_{ij}^{+};\quad q_{j}=\rho_{\infty}C_{p\infty}T_{\infty}u_{ref} q_{j}^{+};\quad J_{j}^{\alpha}=\rho_{\infty}u_{ref}J_{j}^{\alpha\ +}; \end{array} $$

(27f)

$$ \begin{array}{@{}rcl@{}} \dot{d}_{\rho}&=&\rho_{\infty} ({u_{ref}}/{x_{ref}}) \hspace*{1.7pt} \dot{d}_{\rho}^{+};\quad \dot{d}_{\rho u_{i}}=\rho_{\infty} u_{ref} ({u_{ref}}/{x_{ref}}) \hspace*{1.7pt} \dot{d}_{\rho u_{i}}^{+}; \end{array} $$

(27g)

$$ \begin{array}{@{}rcl@{}} \dot{d}_{\rho h_{s}}&=&\rho_{\infty} C_{p\infty}T_{\infty} ({u_{ref}}/{x_{ref}}) \hspace*{1.7pt} \dot{d}_{\rho h_{s}}^{+};\quad \dot{d}_{\rho Y_{\alpha}}=\rho_{\infty} ({u_{ref}}/{x_{ref}}) \hspace*{1.7pt} \dot{d}_{\rho Y_{\alpha}}^{+}\ . \end{array} $$

(27h)

and the normalized EoS can be written as follows

$$ \begin{array}{@{}rcl@{}} p^{+}_{0}&=& \frac{\rho^{+}T^{+}}{\mathcal{M}^{+}}\ ; \end{array} $$

(28a)

$$ \begin{array}{@{}rcl@{}} h_{s}^{+}&=& \frac{1}{C_{p\infty}T_{\infty}}\sum\limits_{\alpha=1}^{N_{sp}}\left( Y_{\alpha} {\int}_{T_{0}}^{T}C_{p,\alpha} (\theta) d\theta \right)\ . \end{array} $$

(28b)

On the one hand, the derivative analysis of Eq. 28b gives

$$ \begin{array}{@{}rcl@{}} dp^{+}_{0}&=&\frac{\rho^{+}}{\mathcal{M}^{+}}dT^{+}+ \rho^{+}T^{+}d \left( \frac{1}{\mathcal{M}^{+}} \right)+ \frac{T^{+}}{\mathcal{M}^{+}}d\rho^{+}\\ &=&\frac{\gamma_{\infty}}{\gamma_{\infty}-1}\left[ \rho^{+}r^{+}dT^{+}+\rho^{+}T^{+}\sum\limits_{\alpha=1}^{N_{sp}}\left( r_{\alpha}^{+} dY_{\alpha} \right) +r^{+}T^{+}d\rho^{+}\right], \end{array} $$

(29)

where

$$ r_{\alpha}^{+}\equiv \frac{1}{C_{p\infty}}\frac{\mathcal{R}}{\mathcal{M}_{\alpha}},\quad r^{+}\equiv \frac{1}{C_{p\infty}}\sum\limits_{\alpha=1}^{N_{sp}}\frac{\mathcal{R}Y_{\alpha}}{\mathcal{M}_{\alpha}}\ . $$

On the other hand, the derivative of the sensible enthalpy definition (i.e., Eq. 28b) reads

$$ {dh_{s}^{+} = C_{p}^{+}dT^{+}+\sum\limits_{\alpha=1}^{N_{sp}}h_{s,\alpha}^{+}dY_{\alpha}}\ , $$

(30)

Using Eq. 30 and considering ${r^{+}}/{C_{p}^{+}}={(\gamma -1)}/{\gamma }$, Eq. 29 becomes

$$ \frac{\gamma_{\infty}-1}{\gamma_{\infty}}\gamma dp^{+}_{0}=\left( \gamma-1 \right)\rho^{+}dh_{s}^{+} +\sum\limits_{\alpha=1}^{N_{sp}}\left[\gamma r_{\alpha}^{+}T^{+}-\left( \gamma-1 \right)h_{s,\alpha}^{+} \right]\rho^{+}dY_{\alpha} +\gamma r^{+}T^{+}d\rho^{+} $$

(31)

from which one may obtain

$$ \frac{\gamma_{\infty}-1}{\gamma_{\infty}}\gamma \frac{dp^{+}_{0}}{dt^{+}}=\left( \gamma-1 \right)\rho^{+}\frac{dh_{s}^{+}}{dt^{+}} +\sum\limits_{\alpha=1}^{N_{sp}}\left[\gamma r_{\alpha}^{+}T^{+}-\left( \gamma-1 \right)h_{s,\alpha}^{+} \right]\rho^{+}\frac{dY_{\alpha}}{dt^{+}}+\gamma r^{+}T^{+}\frac{d\rho^{+}}{dt^{+}}\ , $$

(32)

where ${dh_{s}^{+}}/{dt^{+}}$ must be further expanded using the energy conservation (26d), because it includes a pressure derivative in its RHS. Then, we define $\rho ^{+} {d\hbar _{s}^{+}}/{dt^{+}} \equiv \dot {\omega }_{T}^{+}-{\partial q^{+}_{j}}/{\partial x_{j}^{+}}+ \text {Ma}_{\infty }^{2}\left (\gamma _{\infty }-1 \right )\tau _{ij}^{+} {\partial u^{+}_{i}}/{\partial x_{j}^{+}} +\dot {d}_{\rho h_{s}}^{+}-\dot {d}_{\rho }^{+} h_{s}^{+}$, where it is noteworthy that the second term, which is associated to the dissipation function, can be neglected within the low Mach number framework.

The governing equation of pressure can be expressed as

$$ \frac{\gamma_{\infty}-1}{\gamma_{\infty}}\frac{dp_{0}}{dt}\hspace*{1.7pt}=\hspace*{1.7pt}\left( \gamma\hspace*{1.7pt}-\hspace*{1.7pt}1\right)\rho \frac{d\hbar_{s}}{dt} \hspace*{1.7pt}+\hspace*{1.7pt}\sum\limits_{\alpha=1}^{N_{sp}} \left[\gamma r_{\alpha} T\hspace*{1.7pt}-\hspace*{1.7pt}\left( \gamma\hspace*{1.7pt}-\hspace*{1.7pt}1 \right) h_{s,\alpha} \right] \rho \frac{dY_{\alpha}}{dt} \hspace*{1.7pt}+\hspace*{1.7pt}\gamma r T \frac{d\rho}{dt} \hspace*{1.7pt} . $$

(33)

For the sake of simplicity, the super/subscript for normalized (⁺) and leading order (⁰) are no longer considered in the above expression. It is worth noting that, this equation is valid everywhere in the calculation domain. However, due to the spatial uniformity of the thermodynamic pressure, an averaged form of this equation over a volume $\mathcal {V}$ is preferred so as to avoid numerical difficulties. Finally, the thermodynamic pressure evolution will be determined from

$$ \frac{dp_{0}}{dt}\hspace*{1.7pt}=\hspace*{1.7pt}\frac{\gamma_{\infty}}{\gamma_{\infty}-1}\frac{1}{\mathcal{V}} {\int}_{\mathcal{V}}\left( \underbrace{\left( \gamma\hspace*{1.7pt}-\hspace*{1.7pt}1 \right)\rho \frac{d\hbar_{s}}{dt} }_{\text{(I)}}\hspace*{1.7pt}+\hspace*{1.7pt}\underbrace{\sum\limits_{\alpha=1}^{N_{sp}}\left[\gamma r_{\alpha} T\hspace*{1.7pt}-\hspace*{1.7pt}\left( \gamma\hspace*{1.7pt}-\hspace*{1.7pt}1 \right)h_{s,\alpha} \right]\rho\frac{dY_{\alpha}}{dt}}_{\text{(II)}} \hspace*{1.7pt}+\hspace*{1.7pt}\underbrace{\gamma rT\left( \dot{m}_{p}\hspace*{1.7pt}+\hspace*{1.7pt}\dot{d}_{\rho}\hspace*{1.7pt}-\hspace*{1.7pt}\rho \frac{\partial u_{j}}{\partial x_{j}} \right) }_{\text{(III)}} \right) d \mathcal{V}\ , $$

(34)

with $\rho {d\hbar _{s}} \equiv \dot {\omega }_{T}-{\partial q_{j}}/{\partial x_{j}}+\dot {d}_{\rho h_{s}}-\dot {d}_{\rho } h_{s}$.

Appendix B: Evaluation of the Premixedness Index

The premixedness index is evaluated from the generalized definition introduced in reference [66]:

(35)

where denotes the molecular diffusion velocity of species α. The above expression can also be written as follows: $\xi _{p} = \left (1+ \boldsymbol {n}_{F} \cdot \boldsymbol {n}_{O} \right )/2$ and, since Fickian diffusion has been assumed, the unit vector n_k, which gives the diffusion fluxes direction, may be expressed from the species mass fraction isolines: n_α = ∇Y_α/|∇Y_α|. This allows to relate the premixedness index ξ_p to the Takeno index G_FO = ∇Y_F ⋅∇Y_O [67]:

$$ \xi_{p} = \frac{1}{2}\left( 1+ \frac{G_{FO}}{\vert \boldsymbol{\nabla}Y_{F} \vert \cdot \vert \boldsymbol{\nabla}Y_{O} \vert} \right) $$

(36)

In one-dimensional flame structures described with single-step chemistry, the above index is zero in non-premixed conditions while it is unity in premixed conditions [67]. In contrast to the standard definition of reference [67], which was based on single-step chemistry, Y_F and Y_O do not presently correspond to the mass fractions of fuel and oxidizer. The definition has been indeed generalized herein by considering the mass fractions of unburnt carbon and oxygen atoms

$$ Y_{F}\equiv \sum\limits_{\alpha}^{\alpha \neq \text{CO}_{2},\mathrm{H}_{2}\mathrm{O}}\frac{\mathcal{M}_{\mathrm{C}}}{\mathcal{M}_{\alpha}}Y_{\alpha}\ ,\ Y_{O}\equiv \sum\limits_{\alpha}^{\alpha \neq \text{CO}_{2},\mathrm{H}_{2}\mathrm{O}}\frac{\mathcal{M}_{\mathrm{O}}}{\mathcal{M}_{\alpha}}Y_{\alpha} $$

(37)

Appendix C: Derivation of Transport Equations of Scalar Alignments

For a scalar field ξ(x, t) the transport equation of which can be written as

$$ \frac{\partial \xi}{\partial t} + \boldsymbol{u}\cdot \boldsymbol{\nabla} \xi = \frac{1}{\rho}\boldsymbol{\nabla} \cdot (\rho D \boldsymbol{\nabla} \xi) + \dot{\omega}_{\xi} $$

(38)

with D the diffusion coefficient of the scalar and $\dot {\omega }_{\xi }$ the source term, the transport equation for its gradient g ≡∇ξ can be obtained

$$ \frac{D \boldsymbol{g}}{Dt}=-\boldsymbol{S} \boldsymbol{g}-\frac{1}{2}\boldsymbol{g}\times\boldsymbol{\omega}+\boldsymbol{\nabla}(\text{RHS}_{\xi}) $$

(39)

where S is the strain tensor, ω is the vorticity and $\text {RHS}_{\xi } \equiv \boldsymbol {\nabla }\cdot (\rho D \boldsymbol {\nabla } \xi )/\rho + \dot {\omega }_{\xi }$ denotes the right hand side (RHS) of the transport equation of the scalar. Then, the transport equation for its norm direction vector $\boldsymbol {n}\equiv {\boldsymbol {g}}/{\left | \boldsymbol {g} \right |}$ can be deduced from Eq. 39

$$ \frac{D \boldsymbol{n}}{Dt}=\frac{1}{\left| \boldsymbol{g} \right|}\boldsymbol{\mathcal{I}}\boldsymbol{\mathcal{N}}\frac{D \boldsymbol{g}}{Dt}=-\boldsymbol{\mathcal{I} \mathcal{N}S}\boldsymbol{n}-\frac{1}{2}\boldsymbol{\mathcal{I}}\boldsymbol{\mathcal{N}}\left( \boldsymbol{n}\times\boldsymbol{\omega} \right)+ \frac{\boldsymbol{\mathcal{I} \mathcal{N}}}{\left| \boldsymbol{g} \right|}\boldsymbol{\nabla}(\text{RHS}_{\xi}) $$

(40)

where the matrix $\boldsymbol {\mathcal {I}}\boldsymbol {\mathcal {N}}$ is defined as follows

$$ \boldsymbol{\mathcal{I}} \boldsymbol{\mathcal{N}} \equiv \boldsymbol{I}-\boldsymbol{n}\boldsymbol{n}^{T} $$

(41)

with I the identity matrix.

The projection of n in the eigenframe of S can be characterized with the vector

$$ \widehat{\boldsymbol{n}} \equiv \left[ \begin{array}{l} \hat{{n}}_{1}\\ \hat{{n}}_{2}\\ \hat{{n}}_{3} \end{array} \right] = \left[ \begin{array}{l} \boldsymbol{{n}} \cdot \boldsymbol{e}_{1}\\ \boldsymbol{{n}} \cdot \boldsymbol{e}_{2}\\ \boldsymbol{{n}} \cdot \boldsymbol{e}_{3} \end{array} \right] = \boldsymbol{R}^{T} \boldsymbol{{n}} $$

(42)

with R is the eigenmatrix of S

$$ \boldsymbol{R} \equiv [\boldsymbol{e}_{1} |\boldsymbol{e}_{2} | \boldsymbol{e}_{3}] $$

(43)

where e_i are the eigenvectors of S corresponding to eigenvalues λ_i with a descending order, i.e., λ₁ > λ₂ > λ₃. Finally, the transport equation for $\boldsymbol {\widehat {n}}$ can be achieved

$$ \begin{array}{@{}rcl@{}} \frac{D\widehat{\boldsymbol{n}} }{Dt} &=& \frac{D (\boldsymbol{R}^{T}\boldsymbol{n})}{Dt}=R^{T}\frac{D\boldsymbol{n}}{Dt}+\frac{D R^{T}}{Dt}\boldsymbol{n} \\ &=&-\boldsymbol{R}^{T}\boldsymbol{\mathcal{I} \mathcal{N}S}\boldsymbol{n}-\frac{1}{2}\boldsymbol{R}^{T}\boldsymbol{\mathcal{I}}\boldsymbol{\mathcal{N}}\left( \boldsymbol{n}\times\boldsymbol{\omega} \right)+ \frac{\boldsymbol{R}^{T}\boldsymbol{\mathcal{I}}\boldsymbol{\mathcal{N}}}{\left| \boldsymbol{g} \right|}\boldsymbol{\nabla}(\text{RHS}_{\xi}) +\frac{D \boldsymbol{R}^{T}}{Dt}\boldsymbol{n} \\ &= &\left[\left( \widehat{\boldsymbol{n}}^{T}\boldsymbol{{\Lambda} }\widehat{\boldsymbol{{n}}} \right)\boldsymbol{I} - \boldsymbol{{\Lambda} } \right]\widehat{\boldsymbol{{n}}} -\frac{\boldsymbol{R}^{T}}{2}\left( \boldsymbol{{n}}\times{\boldsymbol{\omega}} \right) + \frac{\left( \boldsymbol{I}- \widehat{\boldsymbol{n}}\widehat{\boldsymbol{n}}^{T}\right)}{\left| \boldsymbol{g} \right|}\boldsymbol{R}^{T}\boldsymbol{\nabla}(\text{RHS}_{\xi}) +\frac{D \boldsymbol{R}^{T}}{Dt}\boldsymbol{n} \end{array} $$

(44)

where Λ is the diagonal matrix based on the eigenvalues of S,

$$ \boldsymbol{\Lambda} \equiv \left[ \begin{array}{lll} \lambda_{1}&&\\ &\lambda_{2}&\\ &&\lambda_{3} \end{array} \right]\ . $$

(45)

Since we have

$$ \boldsymbol{n}=\sum\limits_{j=1}^{3}{\hat{n}}_{j} \boldsymbol{e}_{j}\ , $$

(46)

the last term in Eq. 44 can be expanded as follows

$$ \frac{D \boldsymbol{R}^{T}}{Dt}\boldsymbol{n} = \boldsymbol{\mathcal{W}} \hspace*{1.7pt} \widehat{\boldsymbol{n}} $$

(47)

where the anti-symmetric tensor $\boldsymbol {\mathcal {W}}$ denotes the rate of rotation of the principal axes

$$ \boldsymbol{\mathcal{W}}_{ij}=\frac{D \boldsymbol{e}_{\boldsymbol{i}}}{Dt}\cdot \boldsymbol{e}_{\boldsymbol{j}}\ . $$

(48)

In fact, this tensor can be evaluated from the transport equation of the strain-rate tensor in eigenframe (D(R^TSR)/Dt), it reads

$$ \boldsymbol{\mathcal{W}}_{ij}\hspace*{1.7pt}=\hspace*{1.7pt}\frac{D \boldsymbol{e}_{\boldsymbol{i}}}{Dt}\cdot \boldsymbol{e}_{\boldsymbol{j}}\hspace*{1.7pt}=\hspace*{1.7pt}\frac{1-\delta_{ij}}{\lambda_{i}-\lambda_{j}}\left( \!-\frac{1}{4}\hat{\omega}_{i}\hat{\omega}_{j}\hspace*{1.7pt}+\hspace*{1.7pt}\frac{1}{2}\left[\boldsymbol{R}^{T} \left( \boldsymbol{\nabla} \left( \frac{D \boldsymbol{u}}{D t} \right)\hspace*{1.7pt}+\hspace*{1.7pt}\left( \!\boldsymbol{\nabla} \!\left( \frac{D \boldsymbol{u}}{D t}\right) \right)^{T} \right) \boldsymbol{R} \right]_{ji} \right) $$

(49)

where δ_ij is the Kronecker delta tensor and the substantial derivative of the velocity is given by

$$ \frac{D\boldsymbol{u}}{Dt} = \frac{1}{\rho}\left( -\boldsymbol{\nabla} p +\boldsymbol{\nabla} \cdot \tau+ \boldsymbol{f}\right)\ . $$

(50)

In summary, the transport equation for the projection of n in the eigenframe of the strain-tensor reads:

$$ \begin{array}{@{}rcl@{}} \frac{D\hat{n}_{1}}{Dt} &=& \left( \sum\limits_{j=1}^{3}\lambda_{j} \hat{n}_{j}^{2}-\lambda_{1}\right)\hat{n}_{1}+{\mathbf{T}\mathbf{W}}_{1} +\tilde{n}_{2}\boldsymbol{\mathcal{W}}_{12} + \hat{n}_{3}\boldsymbol{\mathcal{W}}_{13}+({\mathbf{G}\mathbf{R}})_{1} \end{array} $$

(51a)

$$ \begin{array}{@{}rcl@{}} \frac{D\hat{n}_{2}}{Dt} &=& \left( \sum\limits_{j=1}^{3}\lambda_{j} \hat{n}_{j}^{2}-\lambda_{2}\right)\hat{n}_{2}+{\mathbf{T}\mathbf{W}}_{2} +\hat{n}_{1}\boldsymbol{\mathcal{W}}_{21} + \hat{n}_{3}\boldsymbol{\mathcal{W}}_{23}+({\mathbf{G}\mathbf{R}})_{2} \end{array} $$

(51b)

$$ \begin{array}{@{}rcl@{}} \frac{D\hat{n}_{3}}{Dt} &=& \left( \sum\limits_{j=1}^{3}\lambda_{j} \hat{n}_{j}^{2}-\lambda_{3}\right)\hat{n}_{3}+{\mathbf{T}\mathbf{W}}_{3} +\hat{n}_{1}\boldsymbol{\mathcal{W}}_{31} + \hat{n}_{2}\boldsymbol{\mathcal{W}}_{32}+({\mathbf{G}\mathbf{R}})_{3} \end{array} $$

(51c)

where the vector TW and GR are defined as

$$ {\mathbf{T}\mathbf{W}}\equiv -\frac{\boldsymbol{R}^{T}}{2}\left( \boldsymbol{n}\times\boldsymbol{\omega} \right)\ , \quad {\mathbf{G}\mathbf{R}}\equiv \frac{\left( \boldsymbol{I}- \widehat{\boldsymbol{{n}}}\widehat{\boldsymbol{{n}}}^{T}\right)}{\left| \boldsymbol{g} \right|}\boldsymbol{R}^{T}\boldsymbol{\nabla}(\text{RHS}_{\xi})\ . $$

(52)