武漢大學(xué)水力學(xué)習(xí)題解答水擊

1、第十章10-1 解：閥門突然開啟過程中，水擊波傳播四個(gè)階段示意圖如下：10-2 解：首先判斷水擊類型，然后計(jì)算各相末水擊壓強(qiáng)，從中找出最大水擊壓強(qiáng)。(1) 判斷水擊類型已知 m，m/s，則相長 s。而閥門關(guān)閉時(shí)間s，有，為間接水擊。(2) 計(jì)算首相末水擊壓強(qiáng)初始全開，有閥門開度；按線性啟閉關(guān)系，得首相末閥門開度；由于m/s，初始水頭m，則管道特性系數(shù) 。上述數(shù)據(jù)代入式(10-4-17) 得 整理 解此二次方程，得 或 m 或 m如果發(fā)生直接水擊 m而m，故不合理。則首相產(chǎn)生的水擊壓強(qiáng)增量為 m(水柱高) 或水擊壓強(qiáng)為 m(水柱高)(3) 計(jì)算第二相末水擊壓強(qiáng)(末相末水擊壓強(qiáng))將已知的，代入式(

2、10-4-17)得 解此方程 或 m(水柱高)則第二相(末相)產(chǎn)生得水擊壓強(qiáng)為 m(水柱高)如應(yīng)用式(10-4-20)計(jì)算末相水擊壓強(qiáng)，由于系數(shù)則有 可得水擊壓強(qiáng)增量 m(水柱高)或水擊壓強(qiáng)為 m(水柱高) 首相和末相水擊壓強(qiáng)比較，最大水擊壓強(qiáng)發(fā)生在末相，即為 m(水柱高)或?yàn)?m(水柱高)。10-3 解：首先判斷水擊類型，然后分別計(jì)算兩種情況下的水擊壓強(qiáng)。(1) 判斷水擊類型已知 m，cm，mm，查表10-2-1，由鋼管有，可計(jì)算水擊波速 m/s以及相長s。而閥門關(guān)閉時(shí)間s，有，為直接水擊。由于，閥門關(guān)閉結(jié)束時(shí)水擊壓強(qiáng)將保持到首相末，故本題所求閥門斷面水頭為首相末壓強(qiáng)水頭。(2) 初始開度等

3、，終止開度情況下閥門斷面水頭已知m，m/s，由于，以及管道特性系數(shù)代入式(10-4-17) 得 整理 解此二次方程，得 (不合理) ， ?？傻檬紫嗄╅y門斷面水頭 m(水柱高)。(3) 初始開度等，終止開度情況下首相末閥門斷面水頭此時(shí)，代入直接水擊壓強(qiáng)公式式(10-2-3)或例10-1給出的直接水擊壓強(qiáng)公式得 m(水柱高)10-4 解：根據(jù)題目要求，先對(duì)水擊類型進(jìn)行判別，后進(jìn)行水擊壓強(qiáng)的計(jì)算。(1) 判別水擊類型已知 m，m，mm，查表10-2-1，由鋼管有，可計(jì)算水擊波速 m/s以及相長s。當(dāng)閥門關(guān)閉時(shí)間s，有，發(fā)生直接水擊。當(dāng)關(guān)閉時(shí)間s，有，則發(fā)生間接水擊。在設(shè)置調(diào)壓室后，調(diào)壓室的作用類似于

4、水庫，水擊波將由調(diào)壓室而反射，這時(shí)管長m，相應(yīng)的相長s。當(dāng)閥門關(guān)閉時(shí)間s，有，則發(fā)生間接水擊。(2) 管長m時(shí)，計(jì)算水擊壓強(qiáng)增量已知流量m3/s，則m/s，水頭m，初始開度等，終止開度。當(dāng)閥門關(guān)閉時(shí)間s，發(fā)生直接水擊。則由直接水擊壓強(qiáng)公式式(10-2-3)，得水擊壓強(qiáng)增量 m(水柱高)當(dāng)閥門關(guān)閉時(shí)間s，發(fā)生間接水擊。由于，將計(jì)算首相末閥門斷面水擊壓強(qiáng)增量，和第二相末(即末相)末水擊壓強(qiáng)增量。為計(jì)算間接水擊壓強(qiáng)，可令閥門按線性規(guī)律啟閉，即閥門開度 。在首相末，s，閥門開度，以及管道特性系數(shù)代入式(10-4-17) 得 整理 解此二次方程，得 (不合理) ， 。可得首相末閥門斷面壓強(qiáng)增量 m(水柱

5、高)。在第二相末，將已知的，代入式(10-4-17)得 解此方程 或 m(水柱高)最大水擊壓強(qiáng)增量發(fā)生在首相末。(3) 管長m時(shí)，計(jì)算水擊壓強(qiáng)增量相應(yīng)的相長s，當(dāng)閥門關(guān)閉時(shí)間s，發(fā)生間接水擊。由于，將計(jì)算首相末、第二相末、第三相末、第四相末閥門斷面水擊壓強(qiáng)增量，從中找出最大水擊壓強(qiáng)增量。采用例10-3所給的程序進(jìn)行計(jì)算，計(jì)算結(jié)果如下tttnhnh0.9855 0.6715 0.4495 40.4581.9710 0.3430 0.4714 42.4272.9565 1.4487 0.4639 41.7513.9421 0.0000-4.2282-38.054從計(jì)算結(jié)果可見，第二相末水擊壓強(qiáng)增量

6、最大。10-5 解：已知m，mm，可計(jì)算水擊波速 m/s以及相長s。由于最大水擊壓強(qiáng)值產(chǎn)生在第一相末，又考慮閥門按線性關(guān)閉，則此時(shí)閥門開度為。根據(jù)題意，可令，其中K為待求系數(shù)。由此第一相末的閥門開度為。又根據(jù)題給條件，m/s，m，初始開度，可求管道特性系數(shù)并代入式(10-4-17) 得 由于，和題給，則有，代入上式得1解此不等式，得3.64，即閥門關(guān)閉時(shí)間至少應(yīng)為相長的3.64倍。10-6 解：本題可使用特征線法進(jìn)行計(jì)算，可采用例10-3所給的程序進(jìn)行計(jì)算，計(jì)算結(jié)果如下(1) 關(guān)閉指數(shù)t0L0.2L0.4L0.6L0.8LL開度.000090.089.555789.111488.667188.

7、222887.77851.00000.259290.089.555789.111488.667188.222891.6467.956800.518490.089.555789.111488.667192.075994.7822.913600.777690.089.555789.111492.505295.199698.0647.8704001.036890.089.555792.934595.617198.4702101.4760.8272001.296090.093.364096.034898.8758101.8695105.0463.7840001.555290.096.452599.28

8、15102.2631105.4277108.7569.7408001.814490.095.9082102.6569105.8092109.1261112.6391.6976002.073690.096.2053102.4251109.4955112.9961116.6735.6544002.332890.096.5149103.0426109.6000117.0184120.8927.6112002.592090.096.8355103.6860110.5624117.4837125.2766.5680002.851290.097.1693104.3516111.5643118.815712

9、4.0071.5248003.110490.097.5144105.0442112.6000118.0863123.7999.4816003.369690.097.8733105.7597111.5662117.5868123.3786.4384003.628890.098.2438104.3966110.7502116.8610122.7122.3952003.888090.096.5256103.2387109.6947115.8786121.7814.3520004.147290.094.9996101.8275108.3707114.6187120.5528.3088004.40649

10、0.095.3035100.1355106.7554113.0488118.9996.2656004.665690.095.1358100.2330104.8174111.1403117.0849.2224004.924890.094.929999.8180104.6198108.8574114.7726.1792005.184090.094.682599.3173103.8587108.2543112.0208.1360005.443290.094.387798.7237102.9527107.0231112.7560.0928005.702490.094.041598.0236101.88

11、89107.4548112.8002.0496005.961690.093.636097.2070102.5257107.6659112.9935.0064006.220890.093.165698.1381102.9839108.0645108.7584.0000006.480090.094.502098.9424103.6768104.0767103.9403.0000006.739290.095.7762100.0402100.035899.553999.3957.0000006.998490.095.537796.870395.919695.356195.1682.0000007.25

12、7690.091.096591.420892.192891.534991.3171.0000007.516890.085.889186.423687.038888.154887.9022.0000007.776090.085.330481.511682.388383.407284.9928.0000008.035290.085.622681.297177.882979.227778.9129.0000008.294490.085.966681.993878.137873.390473.4639.0000008.553690.086.371082.806877.501472.374767.869

13、0.0000008.812890.086.840281.878677.043971.980071.2856.0000009.072090.085.507681.077476.357475.954576.0904.0000009.331290.084.237879.986979.987480.466580.6227.0000009.590490.084.479783.147184.093884.654384.8419.0000009.849690.088.907088.582987.811788.468188.6853.00000010.108890.094.097393.567092.9546

14、91.841892.0938.00000010.368090.094.656898.464597.594396.579194.9980.00000010.627290.094.367198.6821102.0861100.7491101.0638.00000010.886490.094.025497.9888101.8357106.5690106.4991.00000011.145690.093.621997.1794102.4717107.5848112.0731.00000011.404890.093.154098.1047102.9285107.9757108.6705.00000011

15、.664090.094.482798.9029103.6086104.0144103.8790.00000011.923290.095.748399.986299.989499.513299.3590.00000012.182490.095.503096.835695.892995.335495.1480.00000012.441690.091.089691.413592.183791.528991.3124.00000012.700890.085.916386.442387.052088.161687.9102.00000012.960090.085.355981.559382.422983

16、.434585.0112.00000013.219290.085.643281.338677.944879.273978.9594.00000013.478490.085.982582.028478.190873.471473.5378.00000013.737690.086.385182.834477.555272.455567.9846.00000013.996890.086.851881.911977.099172.068371.3731.00000014.256090.085.526881.116776.425276.016676.1515.00000014.515290.084.26

17、5580.040680.033580.507180.6592.00000014.774490.084.514283.181684.120384.674984.8620.00000015.033690.088.913888.590287.820888.474188.6900.00000015.292890.094.070293.548592.941591.835092.0858.00000015.552090.094.631598.417097.559996.552094.9796.00000015.811290.094.346698.6408102.0246100.7031101.0175.0

18、0000016.070490.094.009697.9544101.7829106.4883106.4255.000000(2) 關(guān)閉指數(shù)t0L0.2L0.4L0.6L0.8LL開度.000090.089.555789.111488.667188.222887.77851.00000.259290.089.555789.111488.667188.222894.6442.915466.518490.089.555789.111488.667195.0621100.8436.834665.777690.089.555789.111495.4801101.2392107.2383.7575961.

19、036890.089.555795.8983101.6349107.6123113.7543.6842601.296090.096.3165102.0308107.9865114.1077120.3959.6146561.555290.0102.4269108.3610114.4614120.7300127.0777.5487851.814490.0102.0251114.8154121.0644127.3939133.7995.4866462.073690.0102.3886114.7098127.7104134.0992140.4664.4282392.332890.0102.683011

20、5.2824127.7281140.7514147.0756.3735652.592090.0102.8927115.6992128.3221140.6913153.5259.3226242.851290.0103.0156115.9313128.6608141.0961148.0676.2754153.110490.0103.0384115.9769128.7048136.0432143.0135.2319393.369690.0102.9613115.8120123.3640130.6324136.9661.1921953.628890.0102.7737110.3508117.74541

21、24.2954129.9431.1561833.888090.097.3898104.7083111.2856117.0629122.0056.1239044.147290.091.938898.3268104.0275109.0004113.2181.0953574.406490.090.940591.263696.0444100.1856103.6744.0705434.665690.089.326588.663187.427290.721393.4900.0494624.924890.087.724785.493583.344980.735582.8038.0321135.184090.

22、086.168982.409878.805375.430171.7832.0184965.443290.084.686679.483374.497669.854570.6554.0086125.702490.083.315376.775970.533769.723469.5359.0024605.961690.082.089774.366172.001870.215169.6431.0000416.220890.081.051177.315574.047271.921571.1368.0000006.480090.085.224880.730377.234374.968574.2038.000

23、0006.739290.089.674585.140481.649679.515578.7997.0000006.998490.089.912690.588487.417985.478884.8263.0000007.257690.090.913092.185994.411792.725692.1564.0000007.516890.092.271794.733897.4890101.0850100.6227.0000007.776090.093.818997.5719101.4040105.3833110.0107.0000008.035290.095.2987100.4867105.463

24、7110.3282110.1431.0000008.294490.096.6668103.1889109.4097110.2231110.6457.0000008.553690.097.8899105.5894107.9482109.7272110.3030.0000008.812890.098.9224102.6492105.9072108.0283108.8087.0000009.072090.094.760399.2420102.7302104.9892105.7537.0000009.331290.090.324494.844498.3259100.4566101.1701.00000

25、09.590490.090.087189.413792.574694.508895.1606.0000009.849690.089.090287.821385.602487.281687.8491.00000010.108890.087.735985.281582.532978.947079.4048.00000010.368090.086.193182.450478.629174.659070.0477.00000010.627290.084.716079.543274.578869.731369.9139.00000010.886490.083.351176.846070.646669.8

26、34269.4149.00000011.145690.082.130374.454872.101570.330269.7545.00000011.404890.081.104077.385874.138172.021771.2454.00000011.664090.085.254480.785577.305175.052974.2887.00000011.923290.089.676885.170681.698379.571178.8599.00000012.182490.089.913290.584387.432985.503484.8525.00000012.441690.090.9066

27、92.171494.383592.711392.1453.00000012.700890.092.256694.703397.4452101.0212100.5678.00000012.960090.093.794997.5275101.3380105.2990109.8943.00000013.219290.095.2694100.4271105.3788110.2095110.0294.00000013.478490.096.6312103.1192109.2975110.1088110.5248.00000013.737690.097.8495105.5013107.8491109.61

28、28110.1883.00000013.996890.098.8698102.5794105.8168107.9286108.7008.00000014.256090.094.730999.1871102.6598104.9053105.6692.00000014.515290.090.322194.814598.2775100.4014101.1102.00000014.774490.090.086589.417892.559794.484495.1345.00000015.033690.089.096687.835885.630487.295987.8601.00000015.292890

29、.087.750985.311882.576579.010479.4594.00000015.552090.086.217082.494578.694874.742870.1634.00000015.811290.084.745179.602474.663269.849370.0270.00000016.070490.083.386476.915370.758169.947869.5351.00000010-7 解：在同一水位情況下，有流量公式，其中 恒定流流量 非恒定流流量 由于 此處，為附加水面坡度。對(duì)于恒定流有，；對(duì)于非恒定流，漲水時(shí)一般總是上游先漲水，則有，；落水時(shí)一般總是上游先退水，

30、則有，。這樣由于附加水面坡度的影響，對(duì)于同一水位，有。整個(gè)洪水期的水位流量關(guān)系曲線將呈現(xiàn)繩套形曲線?？蓞⒁娊滩膱D10-5-2(a)。10-8 解：由明槽非恒定流運(yùn)動(dòng)方程式(10-6-11)，有將代入上式，得在上式中，有 其中由連續(xù)性方程(無旁側(cè)入流)可得，又由棱柱體渠道則 將上述各項(xiàng)代入運(yùn)動(dòng)方程，得由此得證。10-9 解：由于 將上式對(duì)求偏導(dǎo)，有又對(duì)水深與水位的關(guān)系式或求偏導(dǎo)，并考慮°，可得式中為底坡。另外，對(duì)于同一斷面有，以及，則得證畢。10-10解：首先從數(shù)學(xué)上說明最大值出現(xiàn)的性質(zhì)，再討論各種水力要素最大值的出現(xiàn)順序。1 從數(shù)學(xué)上說明某函數(shù)A出現(xiàn)最大值的性質(zhì)(1) 從時(shí)間方面分析

31、。對(duì)某斷面，設(shè)t時(shí)刻某函數(shù)值為，時(shí)刻某函數(shù)值為，則有函數(shù)對(duì)時(shí)間的偏導(dǎo)數(shù)為。還未出現(xiàn)最大值時(shí)，最高點(diǎn)未到，函數(shù)隨時(shí)間增加，有，故；當(dāng)已出現(xiàn)最大值時(shí)，最高點(diǎn)已過，函數(shù)隨時(shí)間減少，有，故；正處在最大值時(shí)，函數(shù)隨時(shí)間不變化，有，故 。(2) 從位置方面分析。某函數(shù)A最大值的出現(xiàn)是從上游到下游依次出現(xiàn)的。假定最大值發(fā)生在斷面處，在以某斷面為中心的領(lǐng)域內(nèi)，即在范圍內(nèi)，必有 。下面分兩種情況進(jìn)行說明。在說明之前對(duì)此偏微分作下列變形還未出現(xiàn)函數(shù)最大值的地方，在此最大值處的下游，函數(shù)曲線的斜率小于零，并且隨時(shí)間的增加，函數(shù)的最大值點(diǎn)處將向下游移動(dòng)，還未出現(xiàn)最大值處的函數(shù)值將逼近最大值點(diǎn)，函數(shù)曲線的斜率的絕對(duì)值將

32、趨近于零，即有，可得 ；已出現(xiàn)過函數(shù)最大值的地方，在此最大值處的上游，函數(shù)曲線的斜率大于零，并且隨時(shí)間的增加，函數(shù)的最大值點(diǎn)處將向下游移動(dòng)，出現(xiàn)過最大值處的函數(shù)值將遠(yuǎn)離最大值點(diǎn)，函數(shù)曲線的斜率將由趨近于零逐漸增加，即有，可得 。2 分析各種水力要素最大值的出現(xiàn)順序。關(guān)于與。由于，而。如果水位最大值未出現(xiàn)，按照前述數(shù)學(xué)推論，有，這時(shí)，說明的最大值已出現(xiàn)過。也就是說，出現(xiàn)在之前。關(guān)于與、。由z、v表示的運(yùn)動(dòng)方程，令，即忽略慣性力影響，可得 ，并微分得當(dāng)時(shí)，同時(shí)有，也就是水位值正出現(xiàn)最大值時(shí)，由前知水面坡度已出現(xiàn)，即，由上式可得，也就是說出現(xiàn)在之前。另外，當(dāng)時(shí)，即正在出現(xiàn)時(shí)，還未出現(xiàn)，也未出現(xiàn)，即有

33、，由上式可知，應(yīng)有，也就是說在說之前出現(xiàn)。關(guān)于與、。由于，對(duì)其求偏導(dǎo)，有。當(dāng)已知水位正出現(xiàn)時(shí)，面積A也應(yīng)處于最大值，即 。此時(shí)如果流速已出現(xiàn)，亦即，則可得，也就是說出現(xiàn)在之前。另外，當(dāng)時(shí)，即正在出現(xiàn)時(shí)，如果有，也就是還未出現(xiàn)，可有，也就是還未出現(xiàn)，即在說之前出現(xiàn)。由上述討論可知，所討論的這些水力要素最大值出現(xiàn)的先后順序是：、 。10-11 解：可使用例10-5所給出的計(jì)算程序進(jìn)行計(jì)算，計(jì)算時(shí)需將水深h折算成水位z進(jìn)行計(jì)算，上下游邊界條件需改寫。下表給出了用特征線法計(jì)算的部分成果th1Q1h2Q2h3Q3h4Q401.26015.00001.26015.00001.26015.00001.260

34、15.0000301.34016.78771.33116.57991.32216.37751.31316.1823601.42018.58721.41118.36941.40118.15251.39217.9369901.53021.28001.51820.97431.50520.67131.49320.37251201.64024.05181.62823.73761.61623.42541.60523.11681501.73526.51611.72526.24051.71525.96731.70625.69881801.83029.14641.82128.86921.81228.59681.

35、80428.33322101.91531.56021.90731.31161.90031.06941.89430.83842402.00034.10651.99333.85821.98733.61911.98233.39522702.07536.38582.06936.16932.06535.96402.06235.77683002.15038.78142.14538.56722.14138.36742.14038.19003302.18539.71642.18439.63262.18539.56372.18839.51813602.22040.85782.22040.78022.22140.

36、72162.22540.69163902.21540.47312.21940.52632.22440.59772.23240.69734202.21040.29392.21440.35092.22040.42862.22940.53804502.16538.61242.17338.79432.18238.99492.19439.22444802.12037.16442.12837.34492.13737.54532.14937.77555102.08536.11532.09236.26082.10036.42552.11036.61875402.05035.03522.05635.17632.

37、06435.33422.07435.51835701.97532.58271.98532.83871.99633.10812.00933.39856001.90030.38761.91030.63851.92130.90281.93431.18756301.83528.59361.84428.80831.85429.03501.86529.28006601.77026.81371.77827.02101.78827.23811.79827.46996901.68524.49161.69524.74801.70625.01091.71825.28427201.60022.33781.61122.

38、58761.62122.84321.63323.10797501.57521.92301.57922.00551.58322.09601.58922.19857801.55021.32171.55321.39801.55721.47841.56221.56528101.51020.31501.51520.42951.52020.54611.52620.66658401.47019.37431.47519.48781.48019.60371.48619.72358701.38517.32201.39517.54911.40517.77511.41618.00019001.30015.49461.

39、31115.71881.32115.94351.33216.1690th5Q5h6Q6h7Q7h8Q8 01.26015.00001.26015.00001.26015.00001.26015.0000301.30515.99601.29715.82051.29015.65671.28315.5053601.38217.72371.37317.51421.36517.31091.35817.1183901.48220.08001.47119.79751.46219.53081.45519.28981201.59422.81501.58422.52521.57722.25651.57322.02

40、461501.69825.43921.69225.19581.68824.98131.69024.81651801.79728.08401.79327.85951.79327.67581.80027.55782101.89030.62611.88930.44461.89330.31341.90630.26082401.97933.19601.98133.03591.98832.93722.00532.93102702.06235.61892.06735.50632.07835.46412.10035.52513002.14138.04772.14837.95942.16337.95112.19

41、038.05863302.19539.50942.20739.55692.22839.68882.26139.93903602.23440.70522.24840.78312.27140.95402.30841.25293902.24540.84012.26341.04662.29141.34562.33241.77114202.24240.69412.26140.91832.28941.23862.33141.68954502.21139.49702.23339.83312.26440.26032.30740.81154802.16538.04932.18738.38672.21738.81

42、422.25939.36445102.12536.85382.14437.14992.17237.53262.21138.03415402.08835.74082.10636.01982.13236.38002.16936.85325702.02533.72062.04634.09092.07334.53232.11035.07486001.94931.50301.96931.86371.99532.29122.02932.81296301.87829.55221.89629.86531.91830.23791.94830.69606601.81127.72431.82628.01281.84

43、628.35221.87328.76616901.73225.57441.74825.89071.76726.24731.79226.66487201.64623.38661.66123.68731.67924.02131.70124.40557501.59522.31791.60422.46131.61522.63891.63022.86577801.56721.66221.57421.77481.58221.91221.59522.08758101.53220.79351.53920.93161.54921.08851.56121.27568401.49219.84961.49919.98

44、591.50820.13891.51920.31818701.42618.22551.43618.45291.44818.68651.46118.93269001.34316.39631.35416.62621.36516.86071.37717.1017th9Q9h10Q10h11Q11 01.26015.00001.26015.00001.26015.0000301.27715.36531.27215.23691.26915.1201601.35316.94491.35216.80591.35916.7286901.45219.09111.45818.96311.47718.9487120

45、1.57621.85541.59121.78831.62621.87581501.70224.73311.72724.77631.77624.99821801.81827.54191.85427.67461.91528.00192101.93130.32601.97630.55482.04830.98572402.03733.05812.09033.36172.17333.87072702.13935.73102.20036.12062.29236.71423002.23538.32122.30338.77452.40439.42813302.31340.34782.38840.94592.49541.74003602.36341.71672.44342.37422.55343.22603902.39142.36032.47443.14022.58744.11484202.39042.30662.47443.11712.58744.1238450