標(biāo)準(zhǔn)馬蹄形斷面正常水深的直接近似計(jì)算公式

李風(fēng)玲,文 輝

(惠州學(xué)院建筑與土木工程系,廣東惠州 516007)

標(biāo)準(zhǔn)馬蹄形斷面正常水深的直接近似計(jì)算公式

李風(fēng)玲,文 輝

(惠州學(xué)院建筑與土木工程系,廣東惠州 516007)

針對(duì)目前標(biāo)準(zhǔn)馬蹄形斷面正常水深計(jì)算過(guò)程煩瑣、公式復(fù)雜的缺陷,對(duì)標(biāo)準(zhǔn)馬蹄形斷面均勻流基本方程進(jìn)行數(shù)學(xué)變換,根據(jù)水工隧洞設(shè)計(jì)規(guī)范的要求和工程實(shí)際應(yīng)用情況確定公式的適用范圍,應(yīng)用擬合優(yōu)化原理得到標(biāo)準(zhǔn)馬蹄形斷面正常水深的簡(jiǎn)捷、實(shí)用的計(jì)算公式。計(jì)算結(jié)果表明:在工程常用范圍內(nèi)計(jì)算的正常水深最大相對(duì)誤差為0.585%,整個(gè)區(qū)間內(nèi)95%以上的計(jì)算點(diǎn)相對(duì)誤差小于0.20%,精度較高,能夠滿(mǎn)足工程實(shí)踐的需要。

標(biāo)準(zhǔn)馬蹄形斷面;正常水深;直接計(jì)算公式;無(wú)量綱水深;水力計(jì)算

目前國(guó)內(nèi)外學(xué)者對(duì)輸水隧洞工程的圓形、蛋形、城門(mén)洞形等斷面形式的水力計(jì)算問(wèn)題進(jìn)行了較為深入的分析和研究,也得到了許多簡(jiǎn)捷計(jì)算公式,解決了不少常規(guī)算法存在的工程實(shí)際問(wèn)題[1-11]。馬蹄形斷面隧洞力學(xué)條件好、水力學(xué)條件優(yōu)越,特別適合于圍巖壓力大的地質(zhì)情況,是水利水電工程最常用的斷面形式之一。為了滿(mǎn)足工程建設(shè)的需要,自20世紀(jì)80年代就開(kāi)始了標(biāo)準(zhǔn)馬蹄形斷面的水力計(jì)算研究,較好地解決了相關(guān)工程問(wèn)題[12-18],但也存在著以下缺陷:①公式應(yīng)用范圍被無(wú)限地?cái)U(kuò)大,最大無(wú)量綱水深竟達(dá)1.85,遠(yuǎn)超出規(guī)范的上限要求,無(wú)法滿(mǎn)足無(wú)壓明流的條件;最小無(wú)量綱水深太小,也嚴(yán)重脫離工程實(shí)際需要。②公式大多為對(duì)應(yīng)3種水深工況下的分段函數(shù),使用時(shí)首先需要進(jìn)行判別,然后選擇對(duì)應(yīng)的公式,最后才能得到結(jié)果,計(jì)算過(guò)程煩瑣。③過(guò)分追求計(jì)算精度也導(dǎo)致公式形式較復(fù)雜。鄭博等[19]較好地解決了標(biāo)準(zhǔn)馬蹄形斷面的臨界水深計(jì)算,但關(guān)于正常水深的直接計(jì)算未見(jiàn)報(bào)道。本文從馬蹄形斷面均勻流基本方程出發(fā),依據(jù)給水排水工程規(guī)范及水利工程規(guī)范等要求,考慮工程實(shí)際情況,合理地確定公式的應(yīng)用范圍,對(duì)無(wú)量綱水深和無(wú)量綱參數(shù)之間的關(guān)系進(jìn)行研究分析,應(yīng)用優(yōu)化擬合方法,得到馬蹄形斷面正常水深的直接近似計(jì)算公式,為標(biāo)準(zhǔn)馬蹄形斷面的工程設(shè)計(jì)和運(yùn)用提供參考。

1 標(biāo)準(zhǔn)馬蹄形斷面的構(gòu)成及其水力要素

1.1 斷面構(gòu)成

馬蹄形斷面是由1個(gè)底弧、2個(gè)圓弧側(cè)拱和1個(gè)圓弧頂拱構(gòu)成,如圖1所示。圖1中h為過(guò)水?dāng)嗝鎸?duì)應(yīng)的水深,m;r為頂拱半徑,m;R為兩側(cè)拱半徑和底拱半徑(標(biāo)準(zhǔn)Ⅰ型馬蹄形斷面R=3r、標(biāo)準(zhǔn)Ⅱ型馬蹄形斷面R=2r),m;θ為底拱的弦對(duì)應(yīng)的圓心半角或兩側(cè)拱對(duì)應(yīng)的圓心角,rad;β為過(guò)水?dāng)嗝娴坠皩?duì)應(yīng)的圓心半角,rad;γ為過(guò)水?dāng)嗝鎮(zhèn)裙皩?duì)應(yīng)的圓心角,rad;φ為過(guò)水?dāng)嗝骓敼皩?duì)應(yīng)的圓心角,rad; e為標(biāo)準(zhǔn)馬蹄形斷面的底拱高度,m。

圖1 標(biāo)準(zhǔn)馬蹄形過(guò)水?dāng)嗝?/p>

1.2 水力要素

過(guò)水?dāng)嗝婷娣eA、濕周χ及水深h分別為

1.3 無(wú)量綱水深表達(dá)式

設(shè)無(wú)量綱水深X為水深與斷面頂拱半徑之比:

3種不同水深條件下的未知圓心角就可用無(wú)量綱水深來(lái)表示:

式中Xe為標(biāo)準(zhǔn)馬蹄形斷面的無(wú)量綱底拱高度。

1.4 計(jì)算公式適用范圍

從理論上講,無(wú)量綱正常水深的取值范圍應(yīng)是[0,2]。根據(jù)DL/T 5195—2004《水工隧洞設(shè)計(jì)規(guī)范》[20],為了保證無(wú)壓隧洞能在水位變化或波浪起伏的情況下仍能保持無(wú)壓流狀態(tài),要求通過(guò)最大流量時(shí)洞內(nèi)水面以上的空間面積不小于隧洞橫斷面面積的15%,凈空高度大于0.40m,即

將φ=0代入式(1),得馬蹄形全斷面面積A0:

計(jì)算通過(guò)最大流量時(shí)洞內(nèi)過(guò)水?dāng)嗝婷娣eA*,有

式中φ*為通過(guò)最大流量時(shí)洞內(nèi)過(guò)水?dāng)嗝骓敼皩?duì)應(yīng)的圓心角,rad。

將式(7)和式(8)代入式(6),可得三角超越方程:

方程(9)無(wú)法直接求解,經(jīng)過(guò)試算:當(dāng)標(biāo)準(zhǔn)Ⅰ型馬蹄形斷面t=3時(shí),φ*=1.9468,代入式(2)得無(wú)量綱水深的上限值X*=1.563;當(dāng)標(biāo)準(zhǔn)Ⅱ型馬蹄形斷面t=2時(shí),φ*=1.9310,代入式(2)得無(wú)量綱水深的上限值X*=1.569。

考慮到適當(dāng)外延,馬蹄形斷面無(wú)量綱水深的上限值統(tǒng)一取1.60;根據(jù)絕大多數(shù)工程實(shí)際應(yīng)用情況,過(guò)小水深是沒(méi)有工程實(shí)際意義的,無(wú)量綱水深的下限值統(tǒng)一取0.20,故確定的無(wú)量綱水深計(jì)算公式的工程適用范圍為[0.20,1.60]。

2 標(biāo)準(zhǔn)馬蹄形斷面的簡(jiǎn)化計(jì)算公式

2.1 正常水深的基本方程

根據(jù)DL/T5195—2004《水工隧洞設(shè)計(jì)規(guī)范》和工程實(shí)際要求,馬蹄形過(guò)水?dāng)嗝嫠矶幢仨毷菬o(wú)壓流動(dòng),應(yīng)按恒定均勻流計(jì)算,采用以曼寧公式表示的明渠均勻流方程計(jì)算其正常水深。

式中:Q為流量,m3/s;i為隧洞底坡比降;n為糙率系數(shù)。

2.2 正常水深的隱函數(shù)表達(dá)式

將式(5)代入式(1)～(3),再代入式(10)并整理,可得標(biāo)準(zhǔn)馬蹄形斷面無(wú)量綱正常水深的隱函數(shù)表達(dá)式:

2.3 正常水深的簡(jiǎn)化計(jì)算公式

為了方便研究,將式(11)左側(cè)的已知量設(shè)為無(wú)量綱參數(shù),即

本文以?xún)绾瘮?shù)構(gòu)造計(jì)算公式形式,通過(guò)進(jìn)行大量試算,并采用編程逐步優(yōu)化擬合的方法,得出標(biāo)準(zhǔn)無(wú)量綱正常水深的近似計(jì)算公式,定義Ⅰ型、Ⅱ型馬蹄形斷面無(wú)量綱正常水深分別為XnⅠ、XnⅡ。

標(biāo)準(zhǔn)Ⅰ型馬蹄形斷面無(wú)量綱正常水深的簡(jiǎn)化計(jì)算公式為

標(biāo)準(zhǔn)Ⅱ型馬蹄形斷面無(wú)量綱正常水深的簡(jiǎn)化計(jì)算公式為

3 公式評(píng)價(jià)

給出范圍為[0.20,1.60]的無(wú)量綱正常水深X,由式(11)和式(12)求得無(wú)量綱參數(shù)M,由式(13)或

公式(13)和式(14)均為簡(jiǎn)單的冪函數(shù)形式,不是分段函數(shù)表示,不需通過(guò)判別適用范圍來(lái)選擇公式,因此簡(jiǎn)捷實(shí)用。通過(guò)對(duì)工程常用范圍(0.2≤h/r≤1.6)的10000個(gè)計(jì)算點(diǎn)進(jìn)行觀(guān)察得出:求解正常水深的最大相對(duì)誤差為0.585%,其中95%以上的計(jì)算點(diǎn)相對(duì)誤差小于0.20%,精度較高,完全滿(mǎn)足工程實(shí)踐的需要。

4 實(shí)例應(yīng)用

某輸水隧洞擬采用標(biāo)準(zhǔn)Ⅰ型馬蹄形斷面,Q= 32m3/s,r=1.5m,i=0.006,n=0.014。由式(12)求得參數(shù)M=1.961 669;由式(13)求得無(wú)量綱正常水深XnⅠ=1.513 644;由式(4)求得正常水深hnⅠ= 2.270m(試算的精確值為2.271m),相對(duì)誤差為-0.04%。

圖2 計(jì)算誤差分布

5 結(jié) 語(yǔ)

目前對(duì)于標(biāo)準(zhǔn)馬蹄形斷面的正常水深計(jì)算仍是十分復(fù)雜而煩瑣的問(wèn)題。本文對(duì)正常水深均勻流基本方程進(jìn)行變換,用冪函數(shù)構(gòu)造直接計(jì)算公式形式,運(yùn)用擬合原理,提出了標(biāo)準(zhǔn)馬蹄形斷面正常水深的直接計(jì)算公式,所得公式不是分段函數(shù),不需通過(guò)條件判別來(lái)選取計(jì)算公式,直接計(jì)算即可得到結(jié)果,簡(jiǎn)捷、方便、實(shí)用。在工程適用范圍(0.20≤X≤1.60)內(nèi),求解正常水深的最大相對(duì)誤差為0.585%,其中超過(guò)95%的計(jì)算點(diǎn)其相對(duì)誤差絕對(duì)值小于0.20%,精度較高,完全滿(mǎn)足工程實(shí)際需要。

[1]李風(fēng)玲,文輝,彭波.圓形過(guò)水?dāng)嗝媾R界流的近似水力計(jì)算[J].人民長(zhǎng)江,2008,39(11):77-78.(LIFengling, WEN Hui,PENG Bo,et al.Approximate hydraulic calculation for critical flow in circular cross-section[J]. Yangtze River,2008,39(11):77-78.(in Chinese))

[2]張寬地,呂宏興,趙延風(fēng).明流條件下圓形隧洞正常水深與臨界水深的直接計(jì)算[J].農(nóng)業(yè)工程學(xué)報(bào),2009,25 (3):1-5.(ZHANG Kuandi,LüHongxing,ZHAO Yanfeng.Direct calculation for normal depth and critical depth of circular section tunnel under free flow[J]. Transactions of the Chinese Society of Agricultural Engineering:Transactions of the CSAE,2009,25(3):1-5. (in Chinese))

[3]文輝,李風(fēng)玲.再論圓管明渠均勻流正常水深的直接計(jì)算公式[J].水利水電科技進(jìn)展,2012,32(6):15-17. (WEN Hui,LIFengling.Further study on explicit formula for normal water depth of uniform flows in circular pipes [J].Advances in Science and Technology of Water Resources,2012,32(6):15-17.(in Chinese)).

[4]李風(fēng)玲,文輝,涂寧宇蛋形斷面管道正常水深近似算法[J].人民長(zhǎng)江,2008,39(18):77-78.(LI fengling, WEN Hui,TU Ningyu.Approximate calculation approach of normalwater depth in an egg-shape pipe[J].Yangtze River,2008,39(18):77-78.(in Chinese))

[5]文輝,李風(fēng)玲,李霞,等.蛋形斷面管道臨界水深的近似算法[J].人民黃河,2008,30(12):111-112.(WEN Hui,LI fengling,LIXia,et al.Approximate calculation of critical depth of oval section pipe[J].Yellow River, 2008,30(12):111-112.(in Chinese))

[6]馬吉明,謝省宗,梁元博.城門(mén)洞形及馬蹄形輸入隧洞內(nèi)的水躍[J].水利學(xué)報(bào),2000,31(7):20-24.(MA Jiming,XIE Shengzong,LIANG Yuanbo.Hydraulic jumps in rectangular conduit with circular upper wall and horseshoe tunnel[J].Journal of Hydraulic Engineering, 2000,31(7):20-24.(in Chinese))

[7]王正中,陳濤,張新民,等.城門(mén)洞形斷面隧洞臨界水深度的近似算法[J].清華大學(xué)學(xué)報(bào):自然科學(xué)版,2004, 44(6):812-814.(WANG Zhengzhong,CHEN Tao, ZHANG Xinming,et al.Approximate solution for the critical depth of a arched turnel[J].Journal of Tsinghua University:Science and Technology,2004,44(6):812-814.(in Chinese))

[8]文輝,李風(fēng)玲,歐軍利,等.城門(mén)洞形斷面隧洞正常水深的近似算法[J].給水排水,2007,33(7):18-21.(WEN Hui,LIFengling,OU Junli,et al.Approximate solution of the normal depth inside tunnel with arch cross-section [J].Water and Wastewater Engineering,2007,33(7): 18-21.(in Chinese))

[9]文輝,李風(fēng)玲.再論城門(mén)洞形斷面隧洞正常水深的近似計(jì)算[J].給水排水,2008,34(11):42-43.(WEN Hui, LI Fengling.Discussion again of the approximate calculation for the normal water depth of the city-opening Section Tunnel[J].Water and Wastewater Engineering, 2008,34(11):42-43.(in Chinese))

[10]文輝,李風(fēng)玲再論城門(mén)洞形斷面隧洞臨界水深的近似計(jì)算[J].人民長(zhǎng)江,2009,40(11):78-79.(WEN Hui, LI Fengling.Discussion on approximate calculation of critical water depth in U-shape cross-section tunnel[J]. angtze River,2009,40(11):78-79.(in Chinese))

[11]趙延風(fēng),劉軍,梅淑霞,等.普通城門(mén)洞形斷面正常水深的近似計(jì)算方法[J].武漢大學(xué)學(xué)報(bào):工學(xué)版,2009 42 (6)773-775.(ZHAO Yanfeng,LIU Jun,MEI Shuxia,et al.Approximatemethod for calculating normalwater depth in common city-opening shaped cross-section[J]. Engineering Journal of Wuhan University,2009,42(6): 773-775.(in Chinese))[12]王正中,陳濤,蘆琴,等.馬蹄形過(guò)水?dāng)嗝媾R界水深的直接計(jì)算[J].水力發(fā)電學(xué)報(bào),2005,24(5):95-98. (WANG Zhengzhong,CHEN Tao,LU Qin,et al.The direct solution on critical depth of horseshoe section tunnel [J].Journal of Hydroelectric Engineering,2005,24(5): 95-98.(in Chinese))

[13]張寬地,呂宏興,陳俊英.馬蹄形過(guò)水?dāng)嗝媾R界水深的直接計(jì)算法[J].農(nóng)業(yè)工程學(xué)報(bào),2009,25(4):15-18. (ZHANG Kuandi,LüHongxing,CHEN Junying.Direct calculation of critical depth of horseshoe section tunnel [J].Transactions of the Chinese Society of Agricultural Engineering,2009,25(4):15-18.(in Chinese))

[14]劉計(jì)良,王正中,趙延風(fēng).馬蹄形隧洞斷面收縮水深的迭代法計(jì)算[J].農(nóng)業(yè)工程學(xué)報(bào),2010,26(增刊2): 167-171.(LIU Jiliang,WANG Zhengzhong,ZHAO Yanfeng.Iterative calculation for contracted depth of horseshoe tunnel cross-section[J].Transactions of the Chinese Society of Agricultural Engineering,2010,26 (Sup2):167-171.(in Chinese))

[15]文輝,李風(fēng)玲,李霞.標(biāo)準(zhǔn)Ⅰ型馬蹄形斷面正常水深的近似算法[J].人民黃河,2008,30(7):89-90.(WEN Hui,LIFengling,LIXia.Approximate solution on normal depth of standard I type horseshoe section tunnel[J]. Yellow River,2008,30(7):89-90.(in Chinese))

[16]趙延風(fēng),王正中,蘆琴.馬蹄形斷面正常水深的直接計(jì)算公式[J].水力發(fā)電學(xué)報(bào),2012,31(1):173-178. (ZHAO Yanfeng,WANG Zhengzhong,LU Qin.Direct calculation formulae for normal depth of horseshoe section [J].Journal of Hydroelectric Engineering,2012,31(1): 173-178.(in Chinese))

[17]文輝,李風(fēng)玲.平底馬蹄形斷面的水力計(jì)算[J].農(nóng)業(yè)工程學(xué)報(bào),2013,29(10):130-135.(WEN Hui,LI Fengling.Hydraulic calculation of horseshoe cross-section with flat-bottom[J].Transactions of the Chinese Society of Agricultural Engineering,2013,29(10):130-135.(in Chinese))

[18]文輝,李風(fēng)玲.平底Ⅰ型馬蹄形斷面臨界水深的直接求解[J].長(zhǎng)江科學(xué)院院報(bào),2013,30(4):40-43.(WEN Hui,LI fengling,Direct solution to the critical depth of I-type horseshoe cross-section with flat bottom[J].Journal of Yangtze River Scientific Research Institute,2013,30 (4):40-43.(in Chinese))

[19]鄭博,滕凱.馬蹄形隧洞過(guò)水?dāng)嗝媾R界水深的簡(jiǎn)化計(jì)算方法[J].中國(guó)水能及電氣化,2012,90(8):7-11. (ZHENG Bo,TENG Kai.Simplified calculation method of criticalwater depth on the U-shaped tunnel water passing cross-section[J].China Water Power&Electrification, 2012,90(8):7-11.(in Chinese))

[20]DL/T 5195—2004 水工隧洞設(shè)計(jì)規(guī)范[S].

The exp licit formula for normal water depth in the standard horseshoe cross-section

LI Fengling,WEN Hui (Department of Architecture and Civil Engineering,Huizhou University,Huizhou 516007,China)

The calculation formula of normalwater depth in the standard type horseshoe cross-section presents shortcomings such as complicated formula and cumbersome process.To overcome this,herewemathematically transform the fundamental equations of uniform flow in the standard type horseshoe cross-section tunnels.By determining the range of the formula according to the hydraulic tunnel design specification and engineering application,we obtained a simple and feasible calculation formula for normal water depth in the standard type horseshoe cross-section tunnels based on the principle of optimization.Results showed that themaximum relative error of normal water depth was 0.585%in the utility range,and the relative error over 95%of the whole range was less than 0.20%.The proposed formula satisfies the needs of engineering practice due to its high accuracy and simple form.

standard horseshoe cross-section,normal water depth,direct calculation formula,dimensionless depth; hydraulic calculation

TV131.4

A

1006-7647(2015)02-0043-04

10.3880/j.issn.1006 7647.2015.02.009

2013-12-02 編輯:周紅梅)

惠州學(xué)院引進(jìn)教授、博士科研啟動(dòng)基金(C510.0211);惠州學(xué)院重點(diǎn)培育學(xué)科項(xiàng)目(ZDPYXK1404)

李風(fēng)玲(1964—),女,重慶梁平人,副教授,主要從事水力學(xué)研究。E-mail:mmlflmm@163.com