摘要
在明渠水利工程中，涉及許多重要評估一維漸變流水面線的問題。渠道斷面除了一般常見的矩形與梯形之外，圓形的渠道在都市下水道方面也是常見的。傳統以積分方式求解一維漸變流水面線，往往需要使用變流函數表（Flow-Varied Function Table）查表推估積分值。近年Jan and Chen (2012)提出使用高斯超幾何函數(Gaussian hypergeometric function, GHF）求解漸變流方程式，免除繁複查表的問題，但是他們的研究僅限於寬廣矩形渠道，尚未處理圓形渠道問題。因此本研究將利用高斯超幾何函數(GHF)求解圓形渠道之漸變流水面線。
本研究以Jan and Chen (2012)求解圓形渠道之漸變流水面線，有兩個重點：1.仿照Hager(1991)利用簡單解(固定水力指數M=N=4)並建立軸向距離修正係數修正β。 2.利用Vatankhah and Easa (2013)半解析解尋找合適之緩坡與陡坡之圓形渠道水力指數，而本文求得之圓形渠道水力指數定義為 及 值。最後本研究以案例驗證利用高斯超幾何函數(GHF)求解圓形渠道漸變流水面線之適宜性，並與Standard Fourth Order Runge-Kutta (SFORK)作差異量與計算時間之比較。
關鍵字：高斯超幾何函數(GHF)、圓形渠道、漸變流水面線、水力指數

Abstract
Many hydraulic engineering works involve the computation of surface profiles of one-dimensional gradually-varied flow (GVF) that is a steady non-uniform flow in an open channel with gradual changes in its water surface elevation. Rectangular channels and trapezoidal channels are general in open channel, but circular channels in urban drainage systems are common. Such GVF equation can be analytically integrated by the use of the Gaussian hypergeometric function (GHF) without recourse to the so-called varied-flow function, as done by Jan and Chen (2012). However, their solution is treated with the GVF profile problems of wide rectangular channels, but does not involve the GVF profiles problems of circular channels yet. The characteristics of the analytical solutions of GHF-based GVF profiles in circular channels are discussed in this paper.
This paper uses Jan and Chen’s (2012) method to slove circular channels GVF profiles.There are two main focus in this study：1. Based on the simplified, empirical GVF equation (hydraulic exponents are constant M=N=4) presented by Hager (1991), we improve the transformation parameter β (fix the axial distance). 2. Then we compare with Vatankhah and Easa (2013) semi-analytical solution and find the suitable hydraulic exponents (denoted as and ). The results reveal the GHF can appropriately slove circular channels GVF profiles via the validation of case study. Furthermore, compare our method with the Standard Fourth Order Runge-Kutta (SFORK) to discuss their discrepancy and the calculation of time.
Key words：Gaussian hypergeometric function, circular channels, gradually-varied flow water profiles, hydraulic exponents

參考文獻
1. Allen, J., Enever, K.J. (1968), “Water surface profiles in gradually varied open-channel flow. Proc. Inst.” Civ. Eng. ASCE 41 (December), 783–811.
2. Bakhmeteff, B.A. (1932), “Hydraulics of Open Channels.” McGraw-Hill, New York, NY.
3. Bresse, J.A.Ch. (1860), “Cours de mecanique applique.” 2e partie, hydraulique. Mallet-Bachelier, Paris.
4. Chow, V.T. (1955), “Integrating the equation of gradually varied flow.” Proceedings Paper No. 838, ASCE, 81, 1–32.
5. Chow, V.T. (1957), “Closure of discussions on ‘‘Integrating the equation of gradually varied flow.” J. Hydraul. Div., ASCE 83(1), 9–22 (paper 1177).
6. Chow, V.T. (1959), “Open-Channel Hydraulics.” McGraw-Hill, New York, NY.
7. Chow, V.T. (1981), “Hydraulic exponents.” J. Hydraul. Div., ASCE 107 (HY11), 1489–1499.
8. Gill, M.A. (1976), “Exact solution of gradually varied flow.” J. Hydraul. Div., ASCE 102 (HY9), 1353–1364.
9. Hager, W.H. (1991), “Backwater curves in circular channels.” J. Irrig. Drain Eng., 117(2), 173–183.
10. Hager, W.H. (1989), “Disussion to Nincircular sewer flow by P. K. Swamee.” R. Bhargava and A. K. Sharma. J. Envir. Eng., ASCE, 115(1), 274-276.
11. Hager, W.H. (2010), “Wastewater hydraulics, Theory and practice.” Springer, New York, N. Y.
12. Jan, C.D., and Chen, C.L. (2012), “Use of the Gaussian hypergeometric function to solve the equation of gradually-varied flow.” Journal of Hydrology, 456-457, 139-145.
13. Jan, C.D., and Chen, C.L. (2013), “Gradually varied open-channel flow profiles normalized by critical depth and analytically solved by using Gaussian hypergeometric functions.” Hydrol. Earth Syst. Sci., 17, 973-987..
14. Kumar, A. (1978), “Integral solutions of the gradually varied equation for rectangular and triangular channels.” Proc. Inst. Civ. Eng. ASCE 65 (2), 509–515.
15. Sauerbrey, M. (1969), “Abfluss in Entwasserungsleitungen.” Wasser und Abwasser in Forschung ynd Praxis, 1, Inst. Wasserverversorgung, Abwasserbeseitigung und Stadtbauwesen, TH Darmstadt, E. Schmidt Verlag (in German).
16. Subramanya, K. (2009), “Flow in Open Channels, third ed.” McGraw-Hill, Singapore.
17. Vatankhah A.R. and Easa S.M. (2013 in press), “Accurate gradually varied flow model for water surface profile in circular channels.” Ain Shams Engineering Journal.
18. Zaghloul N.A. & Shahin M.M. (1993), “Computer simulation of gradually varied profiles in circular sections,” Advances in Engineering Software 16.37-46.