# Fibers Effects on Contract Turbulence Using a Coupling Euler Model
## 2 Coupling Models and Methods
### Coupling Equations of Fiber Suspension Flow  
According to the volume concentration \(nL^{3}\) or volume fraction \(\Phi_{c}\) of the additive, the suspension flow may be divided into a dilute phase, a semi-dilute phase and a dense phase. It is generally believed that the distance \(D\) between fibers is much larger than the fiber length \(L\) to meet the condition of \(nL^{3}<<1\) or \(\Phi_{c}<<a_{r}{}^{-2}\) and there is almost no contact between the fibers in the dilute phase. In a semi-dilute phase, the distance \(D\) between the fibers is less than \(L\) but is much larger than fiber diameter \(d\), that is, \(D\leq L\) & \(D>>d\). Although there are little physical contacts among fibers to meet \(nL^{3}>>1\) and \(nL^{2}d<1\) or \(a_{r}{}^{-2}<<\Phi_{c}<<a_{r}{}^{-1}\), the fibers are mainly affected by the hydraulic action within a range of fiber length. In the case of a dense phase, the distance \(D\) between fibers is a multiple of the fiber diameter \(d\) to meet \(nL^{2}d>1\) or \(\Phi_{c}>>a_{r}{}^{-1}\). Here \(a_{r}=L/d\), is the aspect ratio of fibers. \(n\) is the number density of fibers. A coupling Reynolds stress model is needed to reflect the interaction mechanism between the fluid-solid phase of dilute as Equation (14) [19]. Thus compared with Newtonian fluid, the Reynolds average series equation for the suspension will produce the fiber additional stress terms. The concrete forms of the terms are given in detail by [PERSON] etc. [18] and can be found in the Equations (15)-(25). Then the effects of fiber concentration on the carrier fluid were investigated by solving these equations. \[\tau_{f}=\mu_{f}(<pppp>-\frac{I}{3}<pp>):E \tag{14}\]  
\[\frac{\partial\overline{V_{i}}}{\partial t}+\overline{V_{i}}\frac{\partial \overline{V_{i}}}{\partial x_{j}}=(\cdots)+Q_{i}^{V} \tag{15}\]  
\[Q_{i}^{V}=\frac{\mu_{f}}{\rho}\frac{\partial}{\partial x_{j}}\bigg{(}(\overline {a_{ijmn}}-\frac{I_{ij}}{3}\overline{a_{mn}})\frac{\partial\overline{V_{m}}}{ \partial x_{n}}\bigg{)} \tag{16}\]  
\[\frac{D(\overline{V^{\prime}_{i}V^{\prime}_{j}})}{Dt}=(\cdots)+Q_{ij}^{R}+S_{ ij}^{R} \tag{17}\]  
\[Q_{ij}^{R}=\frac{\mu_{f}}{2\mu}\Bigg{\{}\frac{\partial}{\partial x_{k}}\bigg{(}( \overline{a_{ijmn}}-\frac{I_{jk}}{3}\overline{a_{mn}})v\frac{\partial\overline{ V^{\prime}_{i}V^{\prime}_{m}}}{\partial x_{n}}\bigg{)}+\frac{\partial}{ \partial x_{k}}\bigg{(}(\overline{a_{ikmn}}-\frac{I_{ik}}{3}\overline{a_{mn}}) v\frac{\partial\overline{V^{\prime}_{j}V^{\prime}_{m}}}{\partial x_{n}}\bigg{)} \Bigg{\}} \tag{18}\]  
\[S_{ij}^{R}=-\frac{\mu_{f}}{\mu}\bigg{(}\sum_{k}(\overline{a_{ikmn}}-\frac{I_{ ik}}{3}\overline{a_{mn}})\frac{2\epsilon}{3}\delta_{im}\delta_{nk}\delta_{ij} \bigg{)} \tag{19}\]  
\[\frac{\partial k}{\partial t}+\overline{V}_{k}\frac{\partial k}{\partial x_{k}} =(\cdots)+Q^{k}+S^{k} \tag{20}\]  
\[Q^{k}=\frac{\mu_{f}}{2\mu}\frac{\partial}{\partial x_{k}}\bigg{(}(\overline{a_{ ikmn}}-\frac{I_{ik}}{3}\overline{a_{mn}})v\frac{\partial\overline{V^{\prime}_{i}V^{ \prime}_{m}}}{\partial x_{n}}\bigg{)} \tag{21}\]\[S^{k}=-\frac{\mu_{f}}{\mu}\left(\sum_{i,k}\left(\overline{a_{ikmn}}-\frac{I_{ik}}{3 }\overline{a_{mn}}\right)\frac{\varepsilon}{3}\delta_{im}\delta_{nk}\right) \tag{22}\]  
\[\frac{D\varepsilon}{Dt}=\left(\cdots\right)+\frac{\mu_{f}}{\mu}\left(Q^{S}+S^{S }\right) \tag{23}\]  
\[\frac{\partial\overline{V_{i}}}{\partial t}+\overline{V_{i}}\frac{\partial \overline{V_{i}}}{\partial x_{j}}=(\cdots)+Q_{i}^{V} \tag{15}\]0  
\[\frac{\partial\overline{V_{i}}}{\partial t}+\overline{V_{i}}\frac{\partial \overline{V_{i}}}{\partial x_{j}}=(\cdots)+Q_{i}^{V} \tag{15}\]1  
\[\frac{\partial\overline{V_{i}}}{\partial t}+\overline{V_{i}}\frac{\partial \overline{V_{i}}}{\partial x_{j}}=(\cdots)+Q_{i}^{V} \tag{15}\]2  
where: _<pppp_> and _<pp_> are the fourth order and second order tensor of averaged fibers orientation respectively \(\langle pp\rangle=\int ppf(p,t)dp\), \(\langle pppp\rangle=\int ppppf(p,t)dp\); \(a_{ijmn}\) and \(a_{ij}\) are the components of _<pppp_> and _<pp_> respectively.