# Fibers Effects on Contract Turbulence Using a Coupling Euler Model
## 4 Results and Discussion
### Effect of Fibers Concentrations on Fibers Orientation  
The orientation alignment index \(a_{1111}\) is simulated for the suspension of Table 1 by using the coupling RSM model. Six groups of volume concentration parameters including dilute phase (\(nL^{3}<<1\)) and semi dilute phase (\(nL^{3}>>1\) and \(nL^{2}d<1\)) were used. The impact of fibers concentration on fibers orientation is illustrated as Figures 3 and 4 below. Figure 2: Comparison of consequences at the center streamline by [PERSON]’s experiments, one-way and two-way simulation. It is showed from the Figure 3 that the fiber orientation aligns along the streamline more and more quickly with the increasing fiber concentration though almost all of the fibers at different concentration have a higher aligning orientation at the outlet. One of the more interesting findings is that the fibers orientations of \(nL^{3}=0.3\) and \(nL^{3}=1.2\) are not only slightly lower than that of the semi dilute phase \(nL^{3}=4.8\) and \(nL^{3}=12\) but also slightly lower than that of the dilute phase \(nL^{3}=0.0053\) and \(nL^{3}=0.075\) near the center streamline. However, the fibers orientations are all close to the even distribution near the wall in Figure 3. It may be inferred that whether the orientation of the suspended fibers is not uniform, the fiber orientation is not uniform. Figure 4: _Probability distribution f(\(\varphi\)) of fibers orientation angley varying with \(nL^{3}\) in the contraction where (**a**) \(nL^{3}=0.0053\), (**b**) \(nL^{3}=0.075\), (**c**) \(nL^{3}=0.3\), (**d**) \(nL^{3}=1.2\), (**e**) \(nL^{3}=4.8\), (**f**) \(nL^{3}=12\). Figure 3: \(a_{1111}\) distribution in the contract flow at different \(nL^{3}\). fibers is uniform or aligned depends largely on the local flow structure. Moreover, the probability distribution of fiber orientation in a dilute phase is more dependent on the velocity gradient structure of the flow field than that in a semi-dilute phase. With the increase of the contraction ratio, its distribution in the semi-dilute phase shows a certain degree of viscous hysteresis, whose pattern is obviously different from that of velocity field. From Figure 4, there is little difference between the probability density distribution_f(\(\varphi\))_ of the orientation angle \(\varphi=0^{\circ}\)and \(\varphi=90^{\circ}\)near the inlet no matter the fibers concentration. However, with the increase of contract ratio, the higher the concentration is, the more obviously \(\varphi=0^{\circ}\)is dominant. At the same time, the probability _f(\(\varphi\))_ of fibers orientation \(\varphi=60^{\circ}\) and \(\varphi=90^{\circ}\) decreases sharply in the semi dilute phase, which indicates that the fibers tend to be oriented along the flow direction due to the tensile effect of local flow in the field with large contract ratio. Based on the near wall velocity, the boundary layer thickness of the case is estimated to be about 1.38 mm. Then the boundary layer grid is divided accordingly. The rest area is a turbulence central region. In the turbulent boundary layer, the fiber orientation alignment index \(a_{1111}\) is shown in the Figure 5.  
It is generally believed that the viscous bottom layer is located at the wall distance \(0<y^{+}<40\); The corresponding region \(y^{+}<5\) is the linear bottom layer, where its velocity presents a linear distribution along the normal direction of the wall. It can be seen from the Figure 5 that the semi-dilute phase is the dividing line. When \(nL^{3}>1\), it is more difficult for the fibers to align with the increase of phase concentration in \(y^{+}<5\). However, when \(nL^{3}<1\) fibers will approach the homogeneous orientation of \(1/\pi\) more easily with the decrease of phase concentration. When \(40<y^{+}<300\), the flow is in the logarithmic layer where its velocity presents a logarithmic distribution along the normal direction of the wall. The average viscous shear  
Figure 5: \(a_{1111}\) varying with \(Y^{+}\) from the wall to the boundary layer at different \(nL^{3}\).