劉錦濤,章少輝,許 迪,白美健,劉群昌
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灌溉輸配水系統(tǒng)明滿流的全隱式耦合模擬及驗(yàn)證
劉錦濤1,2,章少輝1※,許 迪1,白美健1,劉群昌1
(1. 中國(guó)水利水電科學(xué)研究院流域水循環(huán)模擬與調(diào)控國(guó)家重點(diǎn)實(shí)驗(yàn)室,北京 100038; 2. 中國(guó)農(nóng)業(yè)大學(xué)水利與土木工程學(xué)院,北京100083)
準(zhǔn)確合理地模擬具有自由表面的渠/管道明流和具有壓力的管道滿流運(yùn)動(dòng)過(guò)程,是設(shè)計(jì)、評(píng)價(jià)和管理灌區(qū)輸配水系統(tǒng)的基礎(chǔ)。為此,該文基于Preissmann窄縫法的概念,采用Saint-Venant方程組描述灌溉輸配水系統(tǒng)的明滿流過(guò)程,在交錯(cuò)空間離散單元格上,建立了基于全隱式標(biāo)量耗散有限體積法的明滿流耦合模擬模型。借助標(biāo)準(zhǔn)的室內(nèi)物理模型觀測(cè)數(shù)據(jù)和石家莊冶河灌區(qū)山尹村試驗(yàn)站野外原型觀測(cè)數(shù)據(jù),對(duì)模型的模擬效果進(jìn)行了驗(yàn)證。結(jié)果表明,與基于顯式向量耗散有限體積法建立的模型相比,該文建立的明滿流耦合模擬模型具有類似的模擬精度,但水量平衡誤差在室內(nèi)和野外試驗(yàn)條件下僅為前者的13%和1.2%,且計(jì)算效率提高了約5.2倍;與基于四點(diǎn)偏心有限差分法建立的模型相比,模擬精度顯著提高,水量平衡誤差在室內(nèi)和野外試驗(yàn)條件下僅為前者的7.6%和0.6%,且效率提高了1.3倍,故該文建立的模型有效克服了已有模型無(wú)法統(tǒng)一模擬精度和效率的缺陷,更適于實(shí)際工程問(wèn)題,為灌區(qū)輸配水系統(tǒng)的設(shè)計(jì)優(yōu)化和管理評(píng)價(jià)提供了數(shù)值模擬方法。
灌溉;輸水系統(tǒng); 有限差分法;明滿流;全隱式;耦合;模擬
灌溉輸配水系統(tǒng)由渠道和管道構(gòu)成,準(zhǔn)確合理地模擬該系統(tǒng)中的水流運(yùn)動(dòng)過(guò)程,是設(shè)計(jì)、評(píng)價(jià)和管理灌區(qū)輸配水過(guò)程的基礎(chǔ)[1-4]。在該系統(tǒng)中,渠道水流呈現(xiàn)出具有自由表面的無(wú)壓流,管道水流則同時(shí)呈現(xiàn)出具有自由表面的無(wú)壓流和滿管的有壓流,這種有壓-無(wú)壓交替變化的明滿流運(yùn)動(dòng)過(guò)程[5-7],導(dǎo)致數(shù)值模擬困難,是輸配水系統(tǒng)水動(dòng)力學(xué)模擬的難題[8-18]。
為了采用Saint-Venant方程組統(tǒng)一地描述輸配水系統(tǒng)中的明滿流過(guò)程,Preissmann提出了窄縫法(稱之為Preissmann窄縫法)[5-6]。在此基礎(chǔ)上,構(gòu)建了四點(diǎn)偏心有限差分法,在輸配水系統(tǒng)的明滿流過(guò)程模擬中獲得了極為廣泛的應(yīng)用[18-24]。然而,無(wú)壓流中的重力擴(kuò)散波和有壓流中的管道彈性波傳播速度巨大的差異,導(dǎo)致2種不同流態(tài)的穩(wěn)定性條件對(duì)時(shí)間步長(zhǎng)限制的顯著不同,致使明滿流的耦合模擬過(guò)程復(fù)雜且效率較低[5,13]。另外四點(diǎn)偏心有限差分法的質(zhì)量守恒性較差[25],而基于差分和插值概念的特征線法,在明滿流耦合模擬過(guò)程中表現(xiàn)出了類似的缺陷[13]。為此,學(xué)者們把在地表淺水流模擬中已被廣泛應(yīng)用的顯式向量耗散有限體積法推廣至明滿流的耦合模擬中[13],表現(xiàn)出了極高的精度,但其顯式和向量耗散特征導(dǎo)致效率極低[25-28],成為阻礙其推廣至工程實(shí)際應(yīng)用的主要障礙。
本文基于Preissmann窄縫法原理,采用Saint-Venant方程組描述灌溉輸配水系統(tǒng)的明滿流過(guò)程,在交錯(cuò)空間離散單元格上,建立基于全隱式標(biāo)量耗散有限體積法的明滿流耦合模擬模型,以克服四點(diǎn)偏心有限差分法精度較低和顯式向量耗散有限體積法效率低的缺陷,達(dá)到明滿流耦合模擬同時(shí)具備高精度和高效率的目的。借助標(biāo)準(zhǔn)的室內(nèi)物理模型觀測(cè)數(shù)據(jù)和石家莊冶河灌區(qū)山尹村試驗(yàn)站野外原型觀測(cè)數(shù)據(jù),通過(guò)與顯式向量耗散有限體積法及四點(diǎn)偏心有限差分法的模擬結(jié)果進(jìn)行對(duì)比,驗(yàn)證基于全隱式標(biāo)量耗散有限體積法的明滿流耦合模擬模型的模擬效果。
渠道中的水流是具有自由表面的無(wú)壓流,常采用Saint-Venant方程組描述[5]。管道中的水流則呈現(xiàn)出無(wú)壓流和滿管的有壓流交替變化的狀態(tài),而有壓流需考慮管材及水流的可壓縮性[6]。為此,Preissmann提出了窄縫法(簡(jiǎn)稱Preissmann窄縫法)[7],以統(tǒng)一描述無(wú)壓和有壓流(圖1)。
注:B是管道頂端水面寬度,m。下同。
由圖1可直觀地看出,Preissmann窄縫法把有壓和無(wú)壓流看作一種具有特殊橫斷面的渠道水流運(yùn)動(dòng)過(guò)程。故可采用守恒型Saint-Venant方程組統(tǒng)一地描述明滿流運(yùn)動(dòng)過(guò)程,包括質(zhì)量守恒方程式(1)和動(dòng)量守恒方程式(2)[7]:
式中為時(shí)間坐標(biāo),s;為空間坐標(biāo),m;為過(guò)流斷面面積,m2;為通過(guò)任意斷面的過(guò)流量,m3/s;為通過(guò)任意斷面的流速,m/s;=z+為渠/管道內(nèi)的水位(自由水位或測(cè)壓管水頭)相對(duì)高程,且z為渠/管底相對(duì)高程,為渠/管內(nèi)的水深,m;為重力加速度,m/s2;為水力半徑,m;為曼寧糙率系數(shù),s/m1/3。
如圖1所示,管道水流處于滿管狀態(tài)時(shí),假設(shè)管道上端存在一個(gè)極窄的水面寬度,以此模擬有壓流動(dòng)。為模擬不同管徑與管材下的有壓流,圖1b中的按下式計(jì)算:
式中為管道彈性波速,m/s。
數(shù)值求解式(1)和式(2)之前,需首先給定渠/管道的橫斷面幾何參數(shù)。由于本文的驗(yàn)證部分僅涉及矩形和圓形斷面,故僅給出這2種橫斷面幾何參數(shù),其他諸如U型和梯形等橫斷面幾何參數(shù),可在相關(guān)著作中找到[29]。
圖2給出了矩形橫斷面示意圖及其參數(shù)表達(dá)。依據(jù)該圖,可獲得過(guò)流斷面面積、濕周和之間的關(guān)系式表達(dá)如下:
式中為濕周,m。
注:h為管道水深,m。下同。
圖3給出了圓形斷面的幾何參數(shù),依據(jù)該圖,可直接把、和之間的關(guān)系表達(dá)為[30]:
當(dāng)<時(shí),
當(dāng)<2時(shí),
基于上述矩形和圓形各水力關(guān)系式,可直接依據(jù)公式=/計(jì)算出水力半徑。
注:R為管道半徑m;a為水面線相對(duì)于管道橫斷面圓心的夾角,(°)。
在空間離散單元格中心處,定義過(guò)流斷面面積和與其相關(guān)的變量(、、);在空間離散單元格邊界上,定義流量和流速,此即交錯(cuò)空間離散單元格的概念。若涉及任意空間離散單元格中心處的流速,按照(u-1/2+u+1/2)/2計(jì)算,若涉及任意空間離散單元格邊界(+1/2)上的過(guò)流斷面面積A+1/2,則按照(A+A+1)/2計(jì)算,其他變量的獲取方式以此類推。
基于此,在空間離散單元格及其邊界上,采用標(biāo)量耗散有限體積法分別對(duì)式(1)和式(2)各項(xiàng)進(jìn)行空間離散,并利用全隱時(shí)間格式對(duì)控制方程式(1)和式(2)的空間離散表達(dá)式進(jìn)行時(shí)間離散,在實(shí)現(xiàn)無(wú)條件穩(wěn)定數(shù)值求解的基礎(chǔ)上,達(dá)到對(duì)輸配水系統(tǒng)中有壓和無(wú)壓流(明滿流)過(guò)程耦合模擬的目的。
借助流量與過(guò)流斷面面積之間的關(guān)系式=×,基于有限體積法的基本概念,在任意空間單元格和單元格邊界(+1/2)上,分別對(duì)式(1)和式(2)進(jìn)行空間積分平均如下[25]:
式中Dx=x+1/2-x-1/2,m;Dx+1/2=x+1-x,m。
式(15)中的第1項(xiàng)被空間離散如下:
式(15)中的第2項(xiàng)被空間離散如下:
采用標(biāo)量耗散有限體積法對(duì)式(18)中的(×)+1/2進(jìn)行空間離散如下:
式中c+1/2=(×A+1/2/B+1/2)1/2是單元格邊界處非滿管無(wú)壓流的重力波速或滿管有壓流的管道彈性波速,m/s;Fr+1/2=u+1/2/c+1/2是單元格邊界(+1/2)處的傅汝德數(shù)。
式(19)中的c+1/2和Fr+1/2被定義如下[31]:
式中和+1是傅汝德數(shù)分裂函數(shù),被定義如下:
通過(guò)把式(19)中的下標(biāo)(+1/2)變換成(-1/2),可獲得(×)-1/2的空間離散表達(dá)式。把這2個(gè)表達(dá)式及式(17)代入式(15),經(jīng)過(guò)合并同類項(xiàng)后,即可獲得質(zhì)量守恒方程式(1)的空間離散式:
式中、和分別是與單元格(+1)、和(-1)相關(guān)的系數(shù),分別表達(dá)如下:
式(17)中的第1項(xiàng)被空間離散如下:
式(17)中的第2項(xiàng)被空間離散如下:
采用標(biāo)量耗散有限體積法對(duì)式(29)中的(×)+1進(jìn)行空間離散如下:
通過(guò)比較式(30)與式(19)可以看出,兩者的離散方式在形式上完全相同,差異僅在變量(和)及下標(biāo)。故通過(guò)變化式(20)~式(23)的下標(biāo),即可獲得式(30)中c+1和Fr+1的定義式。同理可獲得式(29)中(×)-1的空間離散式。
式(16)中的第3項(xiàng)被空間離散如下:
式中A是由初始水深假設(shè)(詳見“穩(wěn)定性條件和初始及邊界條件”)計(jì)算獲得初始過(guò)流斷面。
式(16)中的第4項(xiàng)被空間離散如下,
通過(guò)把以上各空間離散式代入式(16),即可獲得動(dòng)量守恒方程式(2)的空間離散式如下:
式中+1/2、+1/2、+1/2和+1/2分別是與單元格邊界(+3/2)、(+1/2)和(-1/2)相關(guān)的系數(shù),分別表達(dá)如下:
采用全隱時(shí)間格式,對(duì)質(zhì)量守恒方程和動(dòng)量守恒方程的空間離散式(25)及式(34)分別離散如下:
式中n為真實(shí)時(shí)間迭代步;D為真實(shí)時(shí)間離散步長(zhǎng),s。
Saint-Venant方程組的雙曲型數(shù)學(xué)屬性,使其時(shí)空離散式(38)和式(39)難以無(wú)條件收斂[31],為此,采用雙時(shí)間步法對(duì)這2個(gè)時(shí)空離散式進(jìn)行處理如下[32-33]:
式中為虛擬時(shí)間迭代步;D為虛擬時(shí)間離散步長(zhǎng),s。
從式(40)和式(41)可以看出,若虛擬時(shí)間迭代步收斂,這2個(gè)表達(dá)式左側(cè)第1項(xiàng)都將趨于0,式(40)和式(41)即是標(biāo)準(zhǔn)的皮卡迭代式。然而恰恰是這2個(gè)虛擬時(shí)間項(xiàng),使得式(40)和式(41)能以絕對(duì)收斂的形式達(dá)到無(wú)條件穩(wěn)定模擬的目的,這是皮卡迭代方法不具備的優(yōu)點(diǎn)。為明確這一點(diǎn),通過(guò)對(duì)式(40)和式(41)進(jìn)行合并同類項(xiàng)處理后,獲得如下2個(gè)待解式:
式中和+1/2分別是與單元格及其邊界(+1/2)相關(guān)的時(shí)空步長(zhǎng)比值,=D/Dx,+1/2=D/Dx+1/2;是真實(shí)時(shí)間步長(zhǎng)和虛擬時(shí)間步長(zhǎng)的比值D/Dt。
通過(guò)動(dòng)態(tài)判斷渠/管道內(nèi)的水流態(tài)并設(shè)置適當(dāng)?shù)腄t,獲得適宜的值,可使式(42)和式(43)始終保持對(duì)角占優(yōu)而達(dá)到絕對(duì)快速收斂,從而獲得無(wú)條件穩(wěn)定性模擬的目的,這即是皮卡迭代方法不具備的功能。由于式(42)和式(43)的系數(shù)矩陣在數(shù)值模擬中能始終保持對(duì)角占優(yōu),故采用高斯-賽德爾迭代法[34],即可高效解算該方程組,其收斂條件如下:
式中是預(yù)先設(shè)定的誤差值,下文取值10-5。
待解式(42)和式(43)形成的代數(shù)方程組的系數(shù)矩陣能始終保持對(duì)角占優(yōu)[11],故該方程組的求解過(guò)程可實(shí)現(xiàn)無(wú)條件穩(wěn)定收斂,這意味著時(shí)間離散步長(zhǎng)D可依據(jù)實(shí)際物理問(wèn)題而取任意值。
渠/管道內(nèi)初始無(wú)水的零水深,是Saint-Venant方程組的奇點(diǎn),故需假設(shè)初始水深值0=10-10m[31]。邊界條件包括入流、出流和無(wú)流3種條件,而空間交錯(cuò)單元格的另一個(gè)優(yōu)點(diǎn)是[35],僅需對(duì)質(zhì)量守恒方程時(shí)空離散式(42)在邊界處給定流量(入流量、出流量和零流量)條件,而無(wú)需再對(duì)動(dòng)量守恒方程時(shí)空離散式(43)設(shè)置邊界條件。這可有效保證渠/管道有壓與無(wú)壓流耦合模擬過(guò)程中的水量平衡性,提高模擬精度。
采用室內(nèi)物理模型試驗(yàn)和野外原型觀測(cè)試驗(yàn)數(shù)據(jù),并選取基于顯式向量耗散有限體積法和四點(diǎn)偏心有限差分法建立的明滿流耦合模擬模型作為對(duì)比模型,對(duì)上述建立的模型進(jìn)行模擬效果對(duì)比驗(yàn)證。在下文驗(yàn)證過(guò)程中,水量平衡誤差計(jì)算如下[31]:
式中in、out和domain分別是在模擬時(shí)段內(nèi)流入、流出和駐留在渠/管道內(nèi)的水量,m3。
本實(shí)例是一個(gè)國(guó)際標(biāo)準(zhǔn)算例[13],試驗(yàn)裝置見圖4。該試驗(yàn)裝置的主體管道是一長(zhǎng)度為10 m的封閉管道,橫斷面為矩形,高度為0.148 m,寬為0.51 m。曼寧糙率= 0.012 s/m1/3。初始流量為0= 0,初始水頭為0= 0.128 m。圖5是實(shí)測(cè)獲得的上下游邊界處水深隨時(shí)間的變化過(guò)程。在距離上游閘門3.5 m處設(shè)置測(cè)點(diǎn),用于觀測(cè)該點(diǎn)的水深-壓力變化過(guò)程。
圖4 室內(nèi)物理模型試驗(yàn)裝置示意圖
圖5 室內(nèi)物理模型試驗(yàn)上下游水深演變過(guò)程實(shí)測(cè)值
在數(shù)值模擬過(guò)程中,3種解法(本文解法、顯式向量耗散有限體積法和四點(diǎn)偏心有限差分法)的空間離散步長(zhǎng)均取0.01 m,全隱式標(biāo)量耗散有限體積法的時(shí)間步長(zhǎng)取0.01 s(由于該解法無(wú)條件穩(wěn)定,即沒有穩(wěn)定性條件限制,故還可取其他值,但必須能分辨出滿管條件下水擊波動(dòng)的運(yùn)動(dòng)過(guò)程),顯式向量耗散有限體積法和四點(diǎn)偏心有限差分法的時(shí)間步長(zhǎng),則需滿足CFL穩(wěn)定性條件(以Courant、Friedrichs和Lewy 3人的名字命名的時(shí)間格式穩(wěn)定性必要條件[25]),即CFL數(shù)小于1(明滿流條件下滿足該穩(wěn)定性的時(shí)間步長(zhǎng)值遠(yuǎn)小于0.01 s)。另外,全隱式標(biāo)量耗散有限體積法和顯式向量耗散有限體積法的管道彈性波速取值60 m/s[13],四點(diǎn)偏心有限差分法的管道彈性波速取值10 m/s[13]。
依據(jù)參數(shù)及初始與邊界條件,圖6給出了模擬值與實(shí)測(cè)值之間的對(duì)比。可以直觀地看出,基于全隱式標(biāo)量耗散有限體積法和顯式向量耗散有限體積法建立的模型的模擬結(jié)果,均與實(shí)測(cè)數(shù)據(jù)之間具有良好的擬合度,兩者的水量平衡誤差分別為0.16%和1.2%,所耗用的計(jì)算時(shí)間分別為3.2和19.8 s(CPU i7-6700,WIN7),全隱式標(biāo)量耗散有限體積法的水量平衡誤差僅為顯式向量耗散有限體積法的13%,計(jì)算效率提高了約5.2倍。四點(diǎn)偏心有限差分法的水量平衡誤差為2.1%,隱式標(biāo)量耗散有限體積法水量平衡誤差值在野外試驗(yàn)條件下僅為四點(diǎn)偏心有限差分法的7.6%。四點(diǎn)偏心有限差分法模擬結(jié)果與實(shí)測(cè)值之間的差異則較大,這與該解法無(wú)法再取更大的管道彈性波速值(否則模擬結(jié)果即失穩(wěn))密切相關(guān)[15]。與此同時(shí),四點(diǎn)偏心有限差分法的計(jì)算耗時(shí)為7.4 s,本文提出的模型的計(jì)算效率比之提高了1.3倍。故在室內(nèi)物理模型條件下,本文建立的模型達(dá)到了同時(shí)提高模擬精度和效率的目的。
該算例位于河北省石家莊冶河灌區(qū)山尹村試驗(yàn)站。觀測(cè)試驗(yàn)于2013年4月5日進(jìn)行。觀測(cè)渠/管段長(zhǎng)為混凝土材質(zhì),水平向長(zhǎng)度為1018 m,垂向剖面參數(shù)見圖7。其中矩形斷面為方形,寬和高都為1 m,圓形斷面的直徑為1 m。在距離下游200 m的位置處,設(shè)置觀測(cè)點(diǎn),用于觀測(cè)該點(diǎn)處的水深/水壓變化過(guò)程。從上游開始進(jìn)水計(jì)時(shí),上游入流量為0.08 m3/s。在計(jì)時(shí)開始后的30 min時(shí)打開下游閥門,40 min時(shí)關(guān)閉下游閥門,打開和關(guān)閉閥門的時(shí)間為13和15 s,按照線性方式打開和關(guān)閉閥門。
注:e為水量平衡誤差。下同。
圖7 野外原型試驗(yàn)裝置示意圖
在數(shù)值模擬過(guò)程中,空間離散步長(zhǎng)均取1 m,全隱式標(biāo)量耗散有限體積法的時(shí)間步長(zhǎng)取1 s,顯式向量耗散有限體積法和四點(diǎn)偏心有限差分法的時(shí)間步長(zhǎng),則需滿足CFL穩(wěn)定性條件,即CFL數(shù)小于1(該條件下的時(shí)間步長(zhǎng)值遠(yuǎn)小于1 s)。另外,全隱式標(biāo)量耗散有限體積法和標(biāo)量耗散有限體積法的管道彈性波速取值60 m/s[13],四點(diǎn)偏心有限差分法的管道彈性波速取值10 m/s[13]。
圖8給出了全隱式標(biāo)量耗散有限體積法、顯式向量耗散有限體積法和四點(diǎn)偏心有限差分法的模擬結(jié)果與實(shí)測(cè)值結(jié)果之間的對(duì)比情況。從中可以看出,全隱式標(biāo)量耗散有限體積法和顯式向量耗散有限體積法的模擬精度顯著高于四點(diǎn)偏心有限差分法。四點(diǎn)偏心有限差分法模擬精度較低的原因,是其取的管道彈性波速值較低密切相關(guān),但該解法無(wú)法再取更大的管道彈性波速值,否則模擬結(jié)果將失穩(wěn)。另外,全隱式標(biāo)量耗散有限體積法、顯式向量耗散有限體積法和四點(diǎn)偏心有限差分法的水量平衡誤差分別為0.016%、1.35%和2.68%,計(jì)算耗用時(shí)間分別為16.6、103.8和38.2 s(CPU i7-6700,WIN7),即與顯式向量耗散有限體積法和四點(diǎn)偏心有限差分法相比,全隱式標(biāo)量耗散有限體積法的計(jì)算效率分別提高了約5.2倍和1.3倍。全隱式標(biāo)量耗散有限體積法水量平衡誤差值在野外試驗(yàn)條件下僅為四點(diǎn)偏心有限差分法的0.6%。這表明,在野外原型觀測(cè)試驗(yàn)條件下,本文建立的模型達(dá)到了同時(shí)提高模擬精度和效率的目的。
圖8 3種方法模擬實(shí)例2的計(jì)算結(jié)果與實(shí)測(cè)值之間的比較
本文基于Preissmann窄縫法的概念,采用Saint- Venant方程組描述灌溉輸配水系統(tǒng)的明滿流過(guò)程,在交錯(cuò)空間離散單元格上,構(gòu)造了該方程組描述明滿流時(shí)的全隱式標(biāo)量耗散有限體積法時(shí)空離散式,借助高斯-賽德爾迭代法解算該時(shí)空離散式,建立了高效和高精度模擬灌溉輸配水系統(tǒng)明滿流的數(shù)值模擬模型。
借助標(biāo)準(zhǔn)的室內(nèi)物理模型觀測(cè)數(shù)據(jù)和石家莊冶河灌區(qū)山尹村試驗(yàn)站野外原型觀測(cè)數(shù)據(jù),以基于顯式標(biāo)量耗散有限體積法和四點(diǎn)偏心有限差分法建立的模型作為對(duì)比模型,對(duì)本文建立的模型的模擬效果進(jìn)行了驗(yàn)證。結(jié)果表明,與基于顯式向量耗散有限體積法相比,本文建立的明滿流耦合模擬模型具有類似的模擬精度,但在室內(nèi)和野外試驗(yàn)條件下僅為前者的13%和1.2%,且計(jì)算效率提高了約5.2倍;與四點(diǎn)偏心有限差分法相比,模擬精度顯著提高,水量平衡誤差值在室內(nèi)和野外試驗(yàn)條件下僅為前者的7.6%和0.6%,故本文建立的模型有效克服了無(wú)法統(tǒng)一模擬精度和效率的缺陷,更適于實(shí)際工程問(wèn)題,為灌區(qū)輸配水系統(tǒng)的設(shè)計(jì)優(yōu)化和管理評(píng)價(jià)提供了數(shù)值模擬方法。
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劉錦濤,章少輝,許 迪,白美健,劉群昌. 灌溉輸配水系統(tǒng)明滿流的全隱式耦合模擬及驗(yàn)證[J]. 農(nóng)業(yè)工程學(xué)報(bào),2017,33(19):124-130. doi:10.11975/j.issn.1002-6819.2017.19.016 http://www.tcsae.org
Liu Jintao, Zhang Shaohui, Xu Di, Bai Meijian, Liu Qunchang. Coupled simulation and validation with fully implicit time scheme for free-surface-pressurized water flow in pipe/channel[J]. Transactions of the Chinese Society of Agricultural Engineering (Transactions of the CSAE), 2017, 33(19): 124-130. (in Chinese with English abstract) doi:10.11975/j.issn.1002-6819.2017.19.016 http://www.tcsae.org
Coupled simulation and validation with fully implicit time scheme for free-surface-pressurized water flow in pipe/channel
Liu Jintao1,2, Zhang Shaohui1※, Xu Di1, Bai Meijian1, Liu Qunchang1
(1., 100038;2.100083,)
In the irrigation water distribution system including pipe and cannel as well as control valve/gate, water flow presents both free surface and pressurized flows. Saint-Venant equations are often applied to discribe the free-surface-pressurized water flow in pipe/channel by means of Preissmann slot approach and then four-point implicit finite difference and vector-dissipation finite-volume approaches with explicit-time scheme are applied to simulation of unsteady flow in pipe/channel. However, it is very different for gravity diffusion wave in free-surface water flow and pipe elastic wave in pressure water flow, which induces different constraint on time step size, low computational efficiency and large water balance error in the modes based on these 2 numerical solutions. To solve these problems, Saint-Venant equations was applied to describe the free surface and pressure water flows in irrigation water distribution system, conjunctive with the Preissmann slot approach. Then a scalar-dissipation finite-volume scheme was developed to spatially discretize all terms of the governing equations. This scheme exhibited more simple expression and was more suitable to written computational code than the four-point implicit finite difference approach and vector-dissipation finite-volume approaches. On the basis of the spatial scheme, a fully implicit time scheme was implemented to temporally discretize all terms of the governing equations to result in a nonlinear algebraic equation system. To efficiently solve this nonlinear algebraic equation system, a dual time approach was introduced, which included real- and pseudo-time steps, to make a linearization. The advantage of the dual time approach was the existence of a ratio between real- and pseudo-time steps. The value of the ratio could be automatically adjusted according to the known pipe water flow conditions and then the coefficient matrix of the algebraic equation system could maintain diagonally dominant all the time. In such case, the absolute convergence could be achieved whether free surface or pressurized flow was in pipe according to numerical analysis theory. As a result, a fully coupled model of free-surface-pressure flow for irrigation water distribution system was proposed. A standard physical test, which strictly controlled the initial and boundary conditions under the indoor condition, was firstly applied to validate the performance of the proposed model. The validated results showed that the proposed model could well simulate the free surface and pressurized water flow processes, which was similar to vector-dissipation finite-volume approach and better than four-point implicit finite difference approach. Meanwhile, the water balance error of the proposed model was only 0.16%. By contrast, the error values of the models based on four-point implicit finite difference and vector-dissipation finite-volume approaches were 2.1% and 1.2%, respectively. The computational efficiency of the proposed model was 1.3 and 5.2 times higher than the existing 2 models. Furthermore, a field experiment was performed in Hebei Yehe irrigation area, April 5, 2013. On the basis of the field observed data, the proposed model still exhibited better performance than the 2 existing models. In details, the water balance error of the proposed model was only 0.016%, by contrast, 2.68% and 1.35% for the 2 existing models. The efficiency of the proposed model was still 1.3 and 5.2 times higher than the existing 2 models. Consequently, the proposed model overcomes the disadvantages of the existing models and is suitable to practical engineering, and provides a useful method for tool design, evaluation and management of water distribution system in irrigation area.
irrigation; water piping systems; finite difference method; free-surface-pressurized water flow; fully implicit time scheme; coupling; simulation
10.11975/j.issn.1002-6819.2017.19.016
S275; S11+.1
A
1002-6819(2017)-19-0124-07
2017-05-02
2017-08-10
國(guó)家科技支撐計(jì)劃課題(2015BAD24B01)
劉錦濤,山西沁源人,博士生,從事灌溉管網(wǎng)水動(dòng)力學(xué)模擬與試驗(yàn)分析研究。Email:ljtpenny@163.com
※通信作者:章少輝,河北石家莊人,高級(jí)工程師,博士,從事灌區(qū)水循環(huán)動(dòng)力學(xué)耦合模擬與調(diào)控研究。Email:zhangsh@iwhr.com