# Fibers Effects on Contract Turbulence Using a Coupling Euler Model
## 1 Introduction  
Fiber suspensions are widely used in many fields, such as wood pulp used in the modern papermaking industry. Main components of the mixture are water and the fibers. Previous studies have indicated that the addition of fibers can change the transport properties of the flow field, and in turn influence the fibers orientation and distribution, which will ultimately affect the quality of finished products. Related studies focus on the pipe and channel flow more than the contraction.  
Contract flow is mainly used to accelerate the flow rate, such as jet nozzle and the headbox. The headbox is the key part of the modern paper machine, which can produce high intensity micro-turbulence to effectively disperse the fibers and determine their distribution along the horizontal and longitudinal direction of paper. Thus, it can prevent the fiber from settling and flocculating to improve the strength of the paper sheet. The internal flow field in the headbox is similar to the turbulent contraction. The contraction ratio \(C_{x}\) is an important parameter of the contract field, which is the ratio of the local average velocity to the inlet average velocity. In view of the complexity of the fluid-solid coupling mechanism between the carrier fluid and the fibers, some simulations have been made in many researches. [PERSON] et al. [1], [PERSON] et al. [2, 3] and [PERSON] [4] studied the fiber orientation distribution in the dilute turbulent contraction. It is found that the anisotropy of fibers orientation is mainly determined by the contraction ratio rather than the Reynolds number. Increasing contraction ratio would increase fiber alignment significantly and vice versa. Because the turbulent effect is not taken into account in their studies, the relationship between the orientation anisotropy of the fibers and the turbulence pulsation of the flow field is not known clearly.  
[PERSON] et al. [5] thought that fibers orientation depends on the combined effect of fibers motion and turbulence. Ignoring the influence of the latter is the reason why the theoretical prediction of fiber orientation alignment is higher than the experimental measurements. Then [PERSON] et al. [6] used the Euler model to predict the fiber orientation distribution. When the contraction ratio grew from 5 to 50, the fiber alignment increases obviously. The fibers orientation distribution changed slightly with inlet velocity. [PERSON] et al. [7; 8] analyzed the effects of four kinds of wall shapes on the anisotropy of fiber orientation by ignoring the interaction between fibers and fluids. It was shown that the anisotropy of fibers orientation varied with the contract shape and was dominated by the rotating Peclet number. When the contract ratio and Peclet number are both large, the turbulence effect can be ignored. Otherwise, turbulence has a great influence on fiber orientation. [PERSON] et al. [9] used slender body theory to simulate fiber orientation distribution. Then [PERSON] et al. [10] obtained the orientation angle of a single fiber by the analytical and numerical method and concluded that contraction ratio had an important effect on fiber orientation distribution. [PERSON] [11] gave the optimal contract wall for desired fibers orientation in dilute suspension by the one-way coupling RSM method.  
The existing methods mainly consider the effect of the carrier fluid on the fibers (e.g., [PERSON] [5; 6; 12], [PERSON] [13] and [PERSON] et. [14]). In fact, the effect of additive on the turbulence varies with its concentration and scale (e.g., [PERSON] et. [10; 15; 16; 17] and [PERSON] etc. [18]). Therefore, the coupling numerical simulation will contribute to a further research about the orientation characteristics of the fiber under the different additive concentration. It'll also achieve a better control over flow behaviors during the fiber-reinforced materials production.