基于改進(jìn)4-方程摩擦模型的輸水管道水錘壓力計(jì)算

王振華，馬習(xí)賀，李文昊，鄭旭榮，張金珠

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基于改進(jìn)4-方程摩擦模型的輸水管道水錘壓力計(jì)算

王振華1,2※，馬習(xí)賀1,2，李文昊1,2，鄭旭榮1，張金珠1

（1. 石河子大學(xué)水利建筑工程學(xué)院，石河子 832000； 2. 現(xiàn)代節(jié)水灌溉兵團(tuán)重點(diǎn)試驗(yàn)室，石河子 832000）

摩擦耦合是流體與管壁之間相對(duì)運(yùn)動(dòng)產(chǎn)生粘性摩擦力而形成的邊界接觸耦合。在流體高頻運(yùn)動(dòng)的范圍內(nèi)，摩擦耦合的特性變得相對(duì)更加復(fù)雜，將直接影響管道系統(tǒng)的水錘演化。為了研究在實(shí)際管道中水錘的變化情況，本文基于Zielke模型對(duì)流固耦合作用（fluid-structure interaction，F(xiàn)SI）4-方程模型（four-equation model，4EM）建立的4-方程摩擦模型（four-equation friction model，4EFM）結(jié)合廣義不可逆熱力學(xué)理論（extended irreversible thermodynamics，EIT）進(jìn)行改進(jìn)，建立改進(jìn)4-方程摩擦模型。通過MATLAB軟件利用波速調(diào)整（wave-speed adjustment，WSA）插值方法的特征線法（method of characteristics，MOC），對(duì)新疆生產(chǎn)建設(shè)兵團(tuán)第十三師自壓輸水管道中的關(guān)閥水錘壓力進(jìn)行數(shù)值計(jì)算，結(jié)果表明改進(jìn)4-方程摩擦模型的計(jì)算結(jié)果相比4-方程摩擦模型以及其他計(jì)算模型與實(shí)測(cè)值具有更好的一致性，WSA相比其他線性插值方法可以減小插值誤差。該改進(jìn)模型可以應(yīng)用在計(jì)算機(jī)中進(jìn)行長(zhǎng)距離重力流輸水過程的水錘壓力計(jì)算。

水錘；摩擦；耦合；特征線法；數(shù)值計(jì)算

0 引 言

管道輸水是農(nóng)田灌溉中高效節(jié)約、清潔環(huán)保的灌溉輸水方式。但在管道系統(tǒng)充放水過程中，閥門的啟閉或水泵機(jī)組啟停等因素會(huì)誘發(fā)輸流管道系統(tǒng)產(chǎn)生水錘。水錘是一種流體的非恒定流動(dòng)，液體運(yùn)動(dòng)中所有空間點(diǎn)處的運(yùn)動(dòng)要素（流速、壓強(qiáng)、加速度、切應(yīng)力、密度等）都隨著空間位置和時(shí)間的變化而改變。水錘問題是輸流管道中常發(fā)生并且較難控制的問題，嚴(yán)重的水錘會(huì)導(dǎo)致管道部件的破裂以及爆管等事故。為保證管道系統(tǒng)的安全運(yùn)行，眾多專家學(xué)者結(jié)合實(shí)際工程對(duì)水錘的防護(hù)做了大量研究，比如防護(hù)設(shè)備的選用，閥門啟閉方式以及啟閉歷時(shí)控制等措施[1-4]。但在防護(hù)措施實(shí)施之前需要對(duì)管道水錘的大小以及防護(hù)位置進(jìn)行推算，目前大多數(shù)針對(duì)水錘計(jì)算的數(shù)值模型是基于恒定流摩阻項(xiàng)的一維水錘方程，忽略了管壁的切應(yīng)力和對(duì)流項(xiàng)，對(duì)水錘壓力波衰減過程和波形畸變難以做出準(zhǔn)確的計(jì)算。

Zielke模型是1968年由Zielke針對(duì)層流問題提出的非穩(wěn)態(tài)摩擦模型[5]。該模型考慮了與摩擦損失成正比的瞬時(shí)管壁剪切應(yīng)力，將非恒定摩阻項(xiàng)同加權(quán)函數(shù)和歷史加速度聯(lián)系起來(lái)[6]。Adamkowsky和Lewandowsky驗(yàn)證了Zielke模型和其他一些非穩(wěn)態(tài)摩擦模型，但沒有考慮固定閥門的連接耦合問題[7]。4-方程模型（four-equation model, 4EM）是一組形式簡(jiǎn)單的線性偏微分方程組，描述了流固耦合作用（fluid-structure interaction, FSI）管道的軸向振動(dòng)問題[8]。Lee[9]提出了描述輸流管道非線性流固耦合運(yùn)動(dòng)的4-方程模型，建立了描述管道和液體的控制方程，但所得的偏微分方程不完全耦合，流固互動(dòng)機(jī)制的描述不完整，并且忽略了輸送高脈動(dòng)頻率流體時(shí)壓力波對(duì)動(dòng)力響應(yīng)的影響。張立翔等[10-11]通過Hamilton變分原理和變形體內(nèi)流體運(yùn)動(dòng)的微分方程建立管道-流體系統(tǒng)，得到了一個(gè)能反映管道內(nèi)部流流固耦合、彎曲運(yùn)動(dòng)等因素對(duì)水錘特性影響的改進(jìn)4-方程模型。楊超等[12-14]以經(jīng)典4-方程為基礎(chǔ)，依據(jù)Timoshenko梁理論，從對(duì)充液直管耦合振動(dòng)的建模入手，得到了非恒定流充液管道的考慮摩擦耦合和泊松耦合的非線性的軸向振動(dòng)4-方程模型。陳婷[15]等基于FSI處理得到用于計(jì)算耦合水錘的改進(jìn)基本連續(xù)性方程，與簡(jiǎn)化后的流體動(dòng)量方程、管道運(yùn)動(dòng)方程及物理方程構(gòu)成了改進(jìn)的軸向4-方程模型，將連續(xù)性方程中的水錘波速與流速的關(guān)系定義為能夠反映耦合水錘特性的管道與流體在縱橫兩個(gè)方向均耦合的耦合波速，解決了4-方程模型在應(yīng)用特征線法求解時(shí)，基本方程的建立和特征線定義不一致的情況。但Zielke模型和4-方程模型均難以提供水錘問題的精確求解方案，Ghodhbani A等[16]結(jié)合Zielke模型和4-方程模型在考慮摩擦耦合和連接耦合下提出了可以模擬簡(jiǎn)單管道中水錘變化的4-方程摩擦模型（four-equation friction model，4EFM），但該模型未對(duì)管壁剪切應(yīng)力做出定義，未在工程實(shí)際中進(jìn)行驗(yàn)證。本文結(jié)合Axworth等[17]從廣義不可逆熱力學(xué)理論（extended irreversible thermodynamics，EIT）推導(dǎo)出的一維非恒定管流的管壁切應(yīng)力替換4-方程摩擦模型原有的管壁切應(yīng)力計(jì)算方程，建立改進(jìn)4-方程摩擦模型。該改進(jìn)4-方程摩擦模型可以準(zhǔn)確的反映非恒定摩阻項(xiàng)對(duì)于水錘波的影響。通過對(duì)新疆生產(chǎn)建設(shè)兵團(tuán)第十三師自壓輸水管道進(jìn)行末端關(guān)閥試驗(yàn)，對(duì)比分析了Zielke模型、4-方程模型、4-方程摩擦模型以及改進(jìn)4-方程摩擦模型對(duì)管道水錘模擬的準(zhǔn)確性以及波速調(diào)整（wave-speed adjustment，WSA）插值方法對(duì)水錘壓力模擬精度的影響。

1 改進(jìn)4-方程摩擦模型及求解

卷積關(guān)系展開式為：

加權(quán)函數(shù)定義為：

在充液滿流管道系統(tǒng)中，所含的液體具有均勻性、各向同性和牛頓力學(xué)特性[20]，用Timoshenko理論和胡克定律對(duì)流體管道系統(tǒng)進(jìn)行建模[21-23]，改進(jìn)4EFM如下：

流體動(dòng)量方程：

流體連續(xù)方程：

管道動(dòng)力方程：

管道物理方程：

改進(jìn)4EFM為雙曲型偏微分方程，將式（5）～（8）耦合方程寫成矩陣形式如下：

利用特征線法（method of characteristics，MOC）得到其特征方程為：

求解出相應(yīng)的特征值：

利用MOC進(jìn)行瞬態(tài)數(shù)值計(jì)算[24-25]，其計(jì)算網(wǎng)格見圖1，其中A1、A2、A3、A4為4條特征線，線上點(diǎn)為計(jì)算結(jié)點(diǎn)，p點(diǎn)為4條特征線交點(diǎn)。

為了進(jìn)一步分析和驗(yàn)證改進(jìn)4-方程摩擦模型對(duì)管道運(yùn)動(dòng)特性的影響，通過MOC推導(dǎo)以下相容性方程:

由于管道系統(tǒng)存在多種波，在使用特征線法計(jì)算時(shí)，需要通過插值方法來(lái)獲取沒有特征線通過的計(jì)算網(wǎng)格節(jié)點(diǎn)上的參數(shù)值，從而會(huì)產(chǎn)生插值誤差。目前常用的有空間線性插值（space-line interpolation，SLI），時(shí)間線性插值（time-line interpolation，TLI）和波速調(diào)整（wave-speed adjustment，WSA）插值方法，Tijsseling[26]認(rèn)為在計(jì)算水錘壓力時(shí)WSA比線性插值更準(zhǔn)確。為了便于在Matlab軟件中計(jì)算，本文根據(jù)文獻(xiàn)[27]提出以下矩陣公式：

2 初始條件及邊界條件

2.1 初始條件

初始條件根據(jù)穩(wěn)定流計(jì)算，在瞬態(tài)流產(chǎn)生之前，假設(shè)輸流管道系統(tǒng)處于平衡狀態(tài)，根據(jù)式（5）～（8）推導(dǎo)的方程為：

2.2 邊界條件

構(gòu)建如圖2所示的水箱-管道-閥門簡(jiǎn)單管道系統(tǒng)，管道兩端有水箱和閥門2個(gè)固定端。

圖2 水箱-管道-閥門簡(jiǎn)單管道系統(tǒng)

水箱壓力恒定

固端約束

閥門處流體和管道同時(shí)運(yùn)動(dòng)

管道系統(tǒng)動(dòng)態(tài)過程

3 算例分析

從圖3可以看出各模型的水錘壓力模擬計(jì)算值與實(shí)測(cè)值的初始?jí)毫猩仙厔?shì)，水錘初始?jí)毫焖偕仙脑蚺c管道邊界的變化方式有關(guān)[29]。Zielke模型計(jì)算出的初始?jí)毫ι咧岛蛯?shí)測(cè)值的第1個(gè)周期第1個(gè)壓力上升值具有一致性（2=0.89），并且數(shù)據(jù)相近，相比4EM（2=0.79）、4EFM（2=0.83）和改進(jìn)4EFM（2=0.85）可以較好的模擬出水錘壓力的初期變化。但由于管道水錘壓力波速的實(shí)測(cè)值與計(jì)算值之間的差異，Zielke模型和4EM在預(yù)測(cè)水錘變化趨勢(shì)上與實(shí)測(cè)值具有較大誤差，2模型分別在2 s和2.5 s時(shí)出現(xiàn)水錘壓力衰減變化，相比實(shí)測(cè)值衰減提前1 s和0.5 s，從此出現(xiàn)了相位偏移，且隨著時(shí)間的增大相位偏移量增大。Zielke模型在第4個(gè)周期之后出現(xiàn)與實(shí)測(cè)值相反的水錘壓力波衰減變化情況，并且衰減速度加快，4EM計(jì)算的最大和最小水錘壓力都與實(shí)測(cè)值相差較大，水錘壓力振蕩幅度大。4EFM（2=0.87）和改進(jìn)4EFM（2=0.91）相比Zielke模型（2=0.79）和4EM（2=0.76）可以更加精確的模擬水錘發(fā)生瞬間的最大水錘壓力升壓值和降壓值，并且能準(zhǔn)確的描述第1個(gè)周期之后水錘壓力波的畸變和衰減過程，同時(shí)改進(jìn)4EFM相比4EFM的數(shù)值計(jì)算結(jié)果與實(shí)測(cè)值更具有良好的一致性，并且模擬相位偏移量不會(huì)隨時(shí)間增加而逐漸增大，由此可以說(shuō)明改進(jìn)4EFM具有更佳的水錘壓力計(jì)算效果。

圖3 不同模型之間數(shù)值結(jié)果與實(shí)測(cè)值對(duì)比圖

圖4 不同插值方法對(duì)改進(jìn)4EFM水錘壓力計(jì)算的對(duì)比圖

從圖4計(jì)算得出，WSA、SLI和TLI的2值依次為0.83、0.79、0.76，WSA方法相比SLI和TLI對(duì)改進(jìn)4EFM計(jì)算水錘壓力的精度有所提升。在7.5 s時(shí)SLI和TLI插值計(jì)算的水錘壓力值出現(xiàn)衰減，產(chǎn)生了相位偏移，并且相位偏移量隨時(shí)間的延長(zhǎng)而增大，TLI相比SLI所得到的水錘壓力值具有更大的相位偏移以及數(shù)值振蕩幅度。SLI所得水錘壓力值振蕩幅度較小，但其衰減速度快，水錘壓力升高值遠(yuǎn)低于實(shí)測(cè)值。WSA插值計(jì)算的水錘壓力和實(shí)測(cè)值具有較小的阻尼誤差，出現(xiàn)相位偏移時(shí)間滯后，數(shù)值振蕩較小，提高了改進(jìn)4EFM計(jì)算水錘壓力的穩(wěn)定性和收斂性。

各計(jì)算模型與實(shí)測(cè)值以及使用不同插值方法的改進(jìn)4EFM對(duì)水錘壓力計(jì)算的定量分析如表1、表2所示。

表1 不同模型計(jì)算的水錘壓力峰值相對(duì)誤差分析

注：表1中1～8分別代表水錘壓力波動(dòng)第1～8個(gè)周期，水錘壓力波動(dòng)周期為前一個(gè)壓力峰值到后一個(gè)壓力峰值所經(jīng)歷的時(shí)間。下同。 Note：In Table 1~8 represents water hammer pressure fluctuation first to eighth cycle, respectively. The fluctuation period of water hammer pressure is the time of peak value of the previous pressure peak to the next pressure peak. The same bellow.

表2 改進(jìn)4EFM在不同插值方法下水錘壓力峰值相對(duì)誤差分析

由表1可知：Zielke模型和4EM對(duì)于管道閥門處水錘壓力峰值的捕捉效果不理想，Zielke模型在第1個(gè)周期相對(duì)誤差高達(dá)15.63%，水錘壓力峰值誤差隨著時(shí)間增加具有整體下降趨勢(shì)，最低誤差在第7個(gè)周期為1.39%。4EM在第1個(gè)周期相對(duì)誤差達(dá)到8.04%，且誤差隨著時(shí)間增加而增大。4EFM和改進(jìn)4EFM的閥門處水錘壓力峰值捕捉誤差都相對(duì)較小，4EFM在第3個(gè)周期水錘壓力峰值誤差較大，最高達(dá)到7.07%，誤差多數(shù)大于3%，改進(jìn)4EFM的最大誤差出現(xiàn)在第1個(gè)周期為5.13%，并隨著時(shí)間的增加誤差振幅較小并無(wú)增大趨勢(shì)，多數(shù)誤差控制在1.5%以內(nèi)，數(shù)值計(jì)算趨于穩(wěn)定。

從表2可知：改進(jìn)4EFM利用SLI方法計(jì)算的水錘壓力峰值相對(duì)誤差較大，多數(shù)在5%以上，并且具有繼續(xù)增大的趨勢(shì)，最高可達(dá)23.24%。TLI方法計(jì)算所得的水錘壓力峰值在水錘壓力波動(dòng)前3周期相對(duì)誤差較低，均小于3%，最低誤差為0.24%，但從第4周期以后水錘壓力峰值相對(duì)誤差增大，并且基本高于WSA方法所計(jì)算的水錘壓力峰值，WSA方法計(jì)算得的水錘壓力峰值相對(duì)誤差在第1周期以后都相對(duì)較小，因此WSA可以提高改進(jìn)4EFM的水錘壓力計(jì)算精度。

在水錘壓力波傳播過程中，近壁區(qū)在水錘壓力波傳到的同時(shí)先產(chǎn)生反向流動(dòng)，而核心區(qū)由于流體慣性仍保持正向流動(dòng)，由此在近壁區(qū)產(chǎn)生很大的瞬時(shí)流速梯度，進(jìn)而產(chǎn)生較大的附加切應(yīng)力和能量耗散，從而使水錘壓力波的衰減速度加快[30]。Zielke模型和4EM中無(wú)法通過提高對(duì)摩阻項(xiàng)積分計(jì)算精度的方式妥善解決，4EFM考慮了附加摩阻項(xiàng)的影響，模擬精度將更加精確，而改進(jìn)4EFM能夠充分體現(xiàn)附加切應(yīng)力的變化，因此能更加真實(shí)地反映管道中的水錘壓力瞬變過程。

4 結(jié) 論

本文建立了改進(jìn)4-方程摩擦模型（4EFM），通過MATLAB軟件，使用波速調(diào)整（WSA）插值方法的特征線法（MOC）計(jì)算在關(guān)閥情況下簡(jiǎn)單管道中水錘的變化，并與實(shí)測(cè)值、Zielke模型、4EM和4EFM進(jìn)行比較。結(jié)果表明：

1）在摩擦耦合以及連接耦合發(fā)生的情況下，Zielke模型和4EM都不適合計(jì)算管道水錘的變化，并且均具有嚴(yán)重的數(shù)值振蕩和誤差，2個(gè)模型相比實(shí)測(cè)值提前出現(xiàn)壓力衰減變化，并出現(xiàn)相位偏移。改進(jìn)4EFM和4EFM可以準(zhǔn)確描述1個(gè)周期之后壓力波的畸變和衰減過程，且模擬相位偏移量不會(huì)隨時(shí)間增加而逐漸增大，契合度高。

2）改進(jìn)4EFM相比4EFM在模擬效果上具有較大改善，各周期水錘壓力峰值相對(duì)誤差基本上小于1.5%，優(yōu)于4EFM。在實(shí)際工程中，除客觀試驗(yàn)條件影響外，改進(jìn)4EFM計(jì)算的水錘壓力值與試驗(yàn)數(shù)據(jù)具有較高的吻合度以及水錘波形、時(shí)間的契合度，能準(zhǔn)確的捕捉長(zhǎng)距離輸水管道水錘壓力的初期變化，水錘波衰減以及相位偏移等特征，可以進(jìn)行長(zhǎng)距離輸水關(guān)閥水錘的模擬計(jì)算。

WSA方法相比SLI和TLI方法可以提高改進(jìn)4EFM的水錘壓力計(jì)算精度，提高水錘壓力的預(yù)測(cè)能力。目前該改進(jìn)4EFM在粘彈性管、多相流等環(huán)境下的水錘計(jì)算效果仍需要進(jìn)一步的驗(yàn)證。

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Calculation of water hammer pressure of flow pipeline based on modified four-equation friction model

Wang Zhenhua1,2※, Ma Xihe1,2, Li Wenhao1,2, Zheng Xurong1, Zhang Jinzhu1

(1.832000,2.832000,)

Water hammer problem is a problem that often occurs in the pipeline and is difficult to control. Friction coupling is a boundary contact coupling formed by the relative motion between the fluid and the wall of the tube to produce viscous friction. In the range of high frequency motion of the fluid, the characteristics of the friction coupling become more complex, which will directly affect the water hammer evolution of the pipeline system. At present, most numerical models of water hammer calculation are based on the constant flow friction equation of one-dimensional water hammer, ignoring the wall shear stress and convection item, and the attenuation and waveform distortion processes of water hammer pressure wave are difficult to make an accurate calculation. There are few studies on the friction coupling and connection coupling and convection term as to the pipe fluid water hammer calculation. Unsteady friction models are only validated with uncoupled formulation. Additionally, coupled models such as four-equation model (4EM), provide more accurate prediction of water hammer since fluid-structure interaction (FSI) is taken into account, but they are limited to steady-state friction formulation. In this paper, the four-equation friction model (4EFM) based on Zielke model and FSI 4EM is modified according to the one-dimensional unsteady flow shear stress on tube wall, which was derived from the extended irreversible thermodynamics (EIT) by Axworth and others, and the modified 4EFM can accurately reflect the influence of the unsteady friction term on the water hammer wave. In the self-pressure pipeline water delivery project on Hongxing Farm of Thirteenth Division, Xinjiang Production and Construction Corps, the closed valve water hammer test was performed, this model was applied to quasi rigid pipeline with axial movement valve, and the numerical calculation of closed valve water hammer in the pipeline was carried out by using the method of characteristics (MOC) with the MATLAB software. Then the accuracy of Zielke model, 4EM, 4EFM and modified 4EFM in the simulation of water hammer of the pipeline was compared and analyzed. The results show that under the friction coupling and junction coupling condition, Zielke model and 4EM are not suitable for calculating the change of pipeline water hammer, and numerical oscillation and the error are serious, in the third s two model pressure decay process, compared with measured data of early 0.5 s, phase deviation. The modified 4EFM and 4EFM can accurately describe the distortion and attenuation process of pressure wave after a cycle, the simulation deviation will not increase with time, the phase deviation is small, and the fit degree is high. The modified 4EFM has better consistency with the measured value through the comparison. The 4EFM and other calculation models have better consistency. The peak error of water hammer pressure in each cycle for the modified 4EFM is basically less than 1.5%, which is better than 4EFM with the pressure peak error of more than 3%. In practical engineering, in addition to the impact of objective conditions, water hammer pressure calculated by the modified 4EFM has high fitting degree with the tested data, as well as water hammer wave form and time, and this method can accurately capture the characteristics of the early change of pressure in long distance pipeline, the water hammer wave attenuation and the phase shift, so the modified model can be applied in water hammer calculation during long distance water transfer process of gravity flow with the computer. And the reduction of the time step of the operation can improve the simulation precision of the modified 4EFM. At present, the effect of the modified 4EFM on the calculation of water hammer in the environment of viscoelastic tube and multiphase flow still needs further verification.

water hammer; friction; coupling; method of characteristics; numerical simulation

10.11975/j.issn.1002-6819.2018.07.015

O353.1

A

1002-6819(2018)-07-0114-07

王振華，馬習(xí)賀，李文昊，鄭旭榮，張金珠. 基于改進(jìn)4-方程摩擦模型的輸水管道水錘壓力計(jì)算[J]. 農(nóng)業(yè)工程學(xué)報(bào)，2018，34(7)：114－120. doi：10.11975/j.issn.1002-6819.2018.07.015 http://www.tcsae.org

Wang Zhenhua, Ma Xihe, Li Wenhao, Zheng Xurong, Zhang Jinzhu. Calculation of water hammer pressure of flow pipeline based on modified four-equation friction model [J]. Transactions of the Chinese Society of Agricultural Engineering (Transactions of the CSAE), 2018, 34(7): 114－120. (in Chinese with English abstract) doi：10.11975/j.issn.1002-6819.2018.07.015 http://www.tcsae.org

2017-12-05

2018-03-04

國(guó)家重點(diǎn)研發(fā)計(jì)劃“自流灌區(qū)用水調(diào)控技術(shù)集成與應(yīng)用示范”（2017YFC0403205）

王振華，男，河南扶溝人，教授，博士，博士生導(dǎo)師，主要從事干旱區(qū)節(jié)水灌溉理論與技術(shù)研究。Email：wzh2002027@163.com

中國(guó)農(nóng)業(yè)工程學(xué)會(huì)會(huì)員：王振華（E041200608S）