High Accuracy Numerical Simulation of Non-Isothermal Viscoelastic Polymer Fluid Past a Cylinder
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摘要:
采用同位网格有限体积(coupled and linked equations algorithm revised，CLEAR)方法求解黏性和XPP (eXtended Pom-Pom)黏弹性流动的控制方程，基于延时修正方法构造了动量和本构方程对流项的高精度AVLsmart格式。首先，为了验证该文方法的有效性，对不同Reynolds数下不可压黏性流体圆柱绕流问题进行了模拟。随后，对等温及非等温不可压XPP黏弹性流体圆柱绕流问题进行了有效模拟，给出了速度矢量、应力分量、拉升量以及温度的分布规律，分析了We数对水平速度、法向应力及拉升量的影响。该文研究成果能为精确预测复杂型腔纤维增强黏弹性聚合物熔体动态充填过程提供理论基础。
Abstract:The collocated grid finite volume CLEAR (coupled and linked equations algorithm revised) method was applied to solve the governing equations for viscous and XPP (eXtended Pom-Pom) viscoelastic fluids. The high accuracy AVLsmart schemes for the convection terms of momentum and constitutive equations were constructed based on the deferred correction method. Firstly, the incompressible viscous fluids past a cylinder at different Reynolds numbers were simulated to verify the validity of the developed numerical method. Then, the isothermal and non-isothermal XPP viscoelastic fluids past a cylinder were studied numerically, with the distribution patterns of velocity vectors, stress components, stretches and temperatures obtained. Especially, the effects of We on horizontal velocities, normal stresses and stretches were analyzed. The results provide a theoretical foundation for accurate prediction of fiber reinforced viscoelastic polymer dynamic filling process in complex cavities.
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Key words:
- non-isothermal /
- XPP fluid /
- flow past a cylinder /
- high accuracy /
- numerical simulation
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图 2 不同Re下，圆柱附近的压力(上)和流线分布(下)
Figure 2. Distributions of pressure (top) and streamlines (bottom) near the cylinder for different Re numbers
图 3 不同Re下，圆柱绕流的速度u(上)和速度v(下)分布
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同。
Figure 3. Distributions of velocity u (top) and velocity v (bottom) near the cylinder for different Re numbers
图 4 t=100时，不同 Reynolds数下圆柱绕流的流线图：(a) Re=60；(b) Re=80；(c) Re=100；(d) Re=200
Figure 4. Streamlines of the flow near the cylinder at t=100 for different Re numbers: (a) Re=60; (b) Re=80; (c) Re=100; (d) Re=200
图 5 Re=100时，半个周期内圆柱附近流线图：(a) t=T/12；(b) t=2T/12；(c) t=3T/12；(d) t=4T/12；(e) t=5T/12；(f) t=6T/12
Figure 5. Streamlines of the flow near the cylinder in a half cycle for Re=100: (a) t=T/12; (b) t=2T/12; (c) t=3T/12; (d) t=4T/12; (e) t=5T/12; (f) t=6T/12
图 7 We=1，Re=1时，速度、应力分量及拉升量的分布：(a) u；(b) v；(c) τxx；(d) τyy；(e) τxy；(f) λ
Figure 7. Distributions of velocity vectors, stress components and stretches at We=1, Re=1: (a) u; (b) v; (c) τxx; (d) τyy; (e) τxy; (f) λ
图 8 当y=2时，We对水平速度、拉升量及法向应力的影响：(a) u；(b) λ；(c) τxx；(d) τyy
Figure 8. Effects of We on horizontal velocities, stretches and normal stresses at y=2: (a) u; (b) λ; (c) τxx; (d) τyy
图 9 不同Pe下的温度分布：(a) Pe=100；(b) Pe=1 000；(c) Pe=10 000；(d) Pe=100 000
Figure 9. Distributions of temperatures at different Pe numbers: (a) Pe=100; (b) Pe=1 000; (c) Pe=10 000; (d) Pe=100 000
表 1 方程(18)中函数和常数的表达式
Table 1. Definition of functions and constants in eq. (18)
equation $ \varPhi $ $ \theta $ $ \delta $ $ {S_\varPhi } $ continuity 1 1 0 0 u-momentum $ u $ $ Re $ 1 $ - \dfrac{{\partial p}}{{\partial x}} + \left( {\dfrac{{\partial {\tau _{xx}}}}{{\partial x}} + \dfrac{{\partial {\tau _{xy}}}}{{\partial y}}} \right) + (\beta - 1)\left( {\dfrac{{{\partial ^2}u}}{{\partial {x^2}}} + \dfrac{{{\partial ^2}u}}{{\partial {y^2}}}} \right) $ v-momentum $ v $ $ Re $ 1 $ - \dfrac{{\partial p}}{{\partial y}} + \left( {\dfrac{{\partial {\tau _{xy}}}}{{\partial x}} + \dfrac{{\partial {\tau _{yy}}}}{{\partial y}}} \right) + (\beta - 1)\left( {\dfrac{{{\partial ^2}v}}{{\partial {x^2}}} + \dfrac{{{\partial ^2}v}}{{\partial {y^2}}}} \right) $ $ {\tau _{xx}} $ normal stress $ {\tau _{xx}} $ $ We $ 0 $ {S_{{\tau _{xx}}}} $ $ {\tau _{xy}} $ shear stress $ {\tau _{xy}} $ $ We $ 0 $ {S_{{\tau _{xy}}}} $ $ {\tau _{yy}} $ normal stress $ {\tau _{yy}} $ $ We $ 0 $ {S_{{\tau _{yy}}}} $ energy $ T $ $ Pe $ 1 ${S_{{T} } }$ 
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表 3 LDPE熔体在
$ {T_{\text{r}}}{\text{ = 443}} $ 时的XPP和Arrhenius流变参数[9]Table 3. Rheology parameters for LDPE melt for the XPP and Arrhenius models at
$ {T_{\text{r}}}{\text{ = 443}} $ [9]parameters ${\lambda _{ {\text{0b} } } }({T_{\text{r} } })/{\rm{s }}$ $\lambda {}_{ {\text{0s} } }({T_{\text{r} } })/{\rm{s}}$ $ q $ $ \alpha $ ${\eta _{\text{s} } }/({\rm{Pa \cdot s}})$ ${\eta _0}/({\rm{Pa \cdot s} })$ $ {E_0}/({\text{J}} \cdot {\text{mo}}{{\text{l}}^{ - 1}}) $ $ R{\kern 1pt} /({\text{J}} \cdot {\text{mo}}{{\text{l}}^{ - 1}} \cdot {{\text{K}}^{ - 1}}{\text{)}} $ value 1.741 5 0.580 5 2 0.15 200 4 600 48 200 8.314
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