本論文對黏彈性流體在湍流、過渡和層流狀態下的管道流動進行了直接數值模擬，並使用FENE-P模型求解高分子聚點物的構象方程式。這項研究有兩個目的；第一個是找到合適的方法來實現構象方程式的數值邊界條件。結果表明，構象方程的邊界計算使用線性外推法，可為二階精度並提供最小的誤差。使用相同的方法模擬不同Weissenberg值對平板間紊流的影響，當Weissenberg值增加則減阻增加，當Weissenberg值增加到200時，接近最大減阻 (MDR) 的極限。
第二個目標是研究黏彈性流體在方管拉板流的自持紊流臨界雷諾數。結果發現，層流和紊流中的平均速度對摩擦速度關係式與壓力驅動方管流有著相同比例，表明了方管流場的相似性。在層流轉變到紊流的臨界雷諾數較大於牛頓流體，並隨著魏森伯格數的增加而增加。對於雷諾數相對較低的紊流，黏彈性流體也可以被觀察到在方管流中存在兩種流場狀態的交替。然而，隨著Weissenberg值的增加則二次流的強度變弱。在雷諾數為85和Weissenberg值為11時，可獲得的紊流統計數據與高減阻條件下的實驗數據一致。

This dissertation presents direct numerical simulation for the duct flow of a drag-reducing viscoelastic fluid in both turbulent, transitional and laminar regimes. The FENE-P dumbbell model was used for solving the conformation tensor. There were two aims in this study; the first one was to find the appropriate approach of implementing numerical boundary conditions for the conformation tensor and the second objective was to investigate the critical Reynolds number for self-sustaining turbulence in viscoelastic fluid of Couette duct flow. The results showed that, boundary treatments for calculation of the conformation equation based on the linear extrapolation scheme could provide more accurate results of error norms for second-order accurate numerical method. By applying the same treatment in simulation of different Weissenberg numbers in turbulent channel flow, it allowed observation of drag-reducing phenomenon. When the Weissenberg number increased to $We_\tau\approx 200$, the drag reduction would approach the limit of maximum drag reduction (MDR).


In further application on the square duct Couette flow, the relationship between the bulk velocity and friction velocity, both in laminar and turbulent flow, were found to maintain the same scaling as those in pressure-driven duct flow; thus, indicating a degree of universality in the two classes of wall-bounded flows. An investigation of the laminar-turbulent transition indicated that there was a delay, where the critical Reynolds number increased with higher Weissenberg number and was greater than that of the Newtonian case. For turbulent flows of relatively low Reynolds numbers, the alternation between two states normally observed in Newtonian duct flow persisted in the viscoelastic fluid. The two states however became increasingly similar as the Weissenberg number was increased and the secondary motion was weaker in magnitude. The statistics of turbulence obtained at a shear Reynolds number of $85$ and Weissenberg number of $11$ were consistent with the experimental data in the high drag reduction regime.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .i
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .viii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1.1 Introductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Drag reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 MDR Asymptote in mean velocity profiles . . . . . . . . . . . . . 3
1.2.3 Critical Reynolds number of turbulent duct flow . . . . . . . . . . 5
1.3 The numerical simulation of viscoelastic flow . . . . . . . . . . . . . . . . 6
1.3.1 The simplified model of long-chain polymer . . . . . . . . . . . . 7
1.3.2 The conformation tensor of continuous polymer model . . . . . . 10
1.4 Two difficulties of Numerical simulation for viscoelastic flow . . . . . . . 12
1.5 Case analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16
2.1 Flow velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Constitutive equations of FENE-P model . . . . . . . . . . . . . . . . . . 18
2.3 Conformation equations of FENE-P model . . . . . . . . . . . . . . . . . 19
2.4 The detailed derivation of dimensionalized for the equation . . . . . . . . 21
3 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
3.1 Grid Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Discretization of the Momentum Equation . . . . . . . . . . . . . . . . . 26
3.2.1 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Discretization of conformation equation . . . . . . . . . . . . . . . . . . . 28
3.3.1 MINMOD scheme using Flux-limiter method . . . . . . . . . . . 28
3.4 Temporal discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.5 Flowchart of solving procedure . . . . . . . . . . . . . . . . . . . . . . . 32
3.6 Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Preliminary tests of prediction procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37
4.1 Laminar channel flow with polymer additives using FENE-P model . . . 37
4.2 DNS of Newtonian turbulent Poiseuille channel flow . . . . . . . . . . . . 43
4.2.1 Statistics of turbulent flow field . . . . . . . . . . . . . . . . . . . 44
4.3 The drag reduction of turbulent channel flow . . . . . . . . . . . . . . . . 49
4.4 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5 DNS of Turbulent Couette duct flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 Validation for Couette duct flow . . . . . . . . . . . . . . . . . . . . . . . 61
5.2.1 Laminar Couette duct flow in Newtonian fluid . . . . . . . . . . . 62
5.2.2 Turbulent Couette duct flow in Newtonian fluid . . . . . . . . . . 63
5.3 Case description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3.1 Couette duct flow at low Reynolds number . . . . . . . . . . . . . 69
5.4 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2 The suggestions of the future work . . . . . . . . . . . . . . . . . . . . . 87
7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88

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