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1、四、計(jì)算題1 一流動(dòng)的速度場為：vx = 2xy+x，vy = x2-y2-y，試證明該流動(dòng)為有勢流動(dòng)，且存在流函數(shù)，并求速度勢及流函數(shù)。解：1 ， 那么 x= y= z = 0， 流動(dòng)為無旋流動(dòng)， 該流動(dòng)為有勢流動(dòng)。又 ，即流動(dòng)為不可壓縮流體的平面流動(dòng)，該流動(dòng)存在流函數(shù)。2 速度勢為： 流函數(shù)為：2 如下圖，兩圓筒內(nèi)裝的是水，用管子連接。第一個(gè)圓筒的直徑d1= 45 cm，其活塞上受力F1=320 N，密封氣體的計(jì)示壓強(qiáng)為981.0 Pa；第二個(gè)圓筒的直徑d2= 30 cm，其活塞上受力F2=490 N，開孔通大氣。假設(shè)不計(jì)活塞重量，求平衡狀態(tài)時(shí)兩活塞的高度差h。解： 3 ：一閘門如圖，h1
2、 = 2m，h2 =3m，h3 =5m，閘門寬B = 2m，1 =9806 N/m3，2 =12000 N/m3，3 =46000 N/m3。求作用在AB板上的合力，以及作用在B點(diǎn)的合力矩。解： 4 圖示為水自壓力容器定常出流，壓力表讀數(shù)為10atm，H=3.5m，管嘴直徑D1=0.06m，D2=0.12m，試求管嘴上螺釘群共受多少拉力？計(jì)算時(shí)管嘴內(nèi)液體本身重量不計(jì)，忽略一切損失。解：對容器液面和管嘴出口截面列伯努利方程：選管嘴外表和管嘴進(jìn)出口斷面所圍成的體積為控制體，列動(dòng)量方程：對管嘴的進(jìn)出口斷面列伯努利方程，得 5 如圖示，水流經(jīng)彎管流入大氣，d1=100mm，d2=75mm，v2=23m
3、/s，不計(jì)水頭損失，求彎管上所受的力。解：由連續(xù)方程： 得： 對彎管的進(jìn)、出口截面列伯努利方程：其中，P2 b= 0，z1 = z 2，代入得： 選彎管所圍成的體積為控制體，對控制體列動(dòng)量方程：求得：Fpnbx= - 710.6 (N) Fx= - Fpnbx= 710.6 (N) Fpnby= 1168.5 (N) Fy= - Fpnby= -1168.5 (N)6 油的密度=850 kg/m3，粘度=0.06 Pa.s，在圖示連接兩容器的光滑管中流動(dòng)，H=3 m。當(dāng)計(jì)及沿程和局部損失時(shí)，求：1管內(nèi)的流量為多少？2在管路中安一閥門，當(dāng)調(diào)整閥門使得管內(nèi)流量減小到原來的一半時(shí)，問閥門的局部損失系
4、數(shù)等于多少？水力光滑流動(dòng)時(shí)，= 0.3164/Re0.25。解：1對兩容器的液面列伯努利方程，得： 即： (1)設(shè)= 0.03，代入上式，得 v = 3.27 m/s，那么故，令=0.0291，代入1得：v=3.306m/s那么2那么求得：7為確定魚雷阻力，可在風(fēng)洞中進(jìn)行模擬試驗(yàn)。模型與實(shí)物的比例尺為1/3，實(shí)際情況下魚雷速度vp=6 km/h，海水密度p=1200 kg/m3，粘度p=1.14510-6 m2/s，空氣的密度m=1.29 kg/m3，粘度m=1.4510-5 m2/s，試求：1風(fēng)洞中的模擬速度應(yīng)為多大？2假設(shè)在風(fēng)洞中測得模型阻力為1000N，那么實(shí)際阻力為多少？解：（1） 由
5、Rep = Rem 得， k = kv kl， vm= kvvp= 386 =228 (km/h)2由kF= kkl2 kv2 得 FP = Fm/kF = 1000/0.1725 = 5798 (N)7 流體通過孔板流量計(jì)的流量qv與孔板前、后的壓差P、管道的內(nèi)徑d1、管內(nèi)流速v、孔板的孔徑d、流體密度和動(dòng)力粘度有關(guān)。試用定理導(dǎo)出流量qv的表達(dá)式。 dimP =ML-1T-2， dim=ML-1T-1。解：設(shè)qv= f (P, d1, v, d,)選d, v, 為根本變量上述方程的量綱方程為：由量綱一致性原那么，可求得：a1=0 a2=1 a3=0 a4=1b1=1 b2=2 b3=0 b4
6、=1c1=2 c2=0 c3=1 c4=1 8 如下圖，由上下兩個(gè)半球合成的圓球，直徑d=2m，球中充滿水。當(dāng)測壓管讀數(shù)H=3m時(shí)，不計(jì)球的自重，求以下兩種情況下螺栓群A-A所受的拉力。1上半球固定在支座上；2下半球固定在支座上。解：1上半球固定在支座上時(shí)2下半球固定在支座上時(shí)9. 新設(shè)計(jì)的汽車高1.5m，最大行駛速度為108km/h，擬在風(fēng)洞中進(jìn)行模型試驗(yàn)。風(fēng)洞試驗(yàn)段的最大風(fēng)速為45m/s，試求模型的高度。在該風(fēng)速下測得模型的風(fēng)阻力為1500N，試求原型在最大行駛速度時(shí)的風(fēng)阻。解： 根據(jù)粘性力相似準(zhǔn)那么， 又10. 連續(xù)管系中的90漸縮彎管放在水平面上，管徑d1=15 cm，d2=7.5 c
7、m，入口處水平均流速v1=2.5 m/s，靜壓p1e=6.86104 Pa計(jì)示壓強(qiáng)。如不計(jì)能量損失，試求支撐彎管在其位置所需的水平力。 解：由連續(xù)方程： 由能量方程： X方向動(dòng)量方程： Y方向動(dòng)量方程： 合力為:11. 小球在不可壓縮粘性流體中運(yùn)動(dòng)的阻力FD與小球的直徑D、等速運(yùn)動(dòng)的速度v、流體的密度、動(dòng)力粘度有關(guān)，試導(dǎo)出阻力的表達(dá)式。 dimF =MLT-2， dim=ML-1T-1。15分解：設(shè)FD = f (D, v, ,)選D、v、為根本變量上述方程的量綱方程為：由量綱一致性原那么，可求得：a1=1 a2=1 b1=2 b2=1 c1=2 c2=1 12. 如下圖，一封閉容器內(nèi)盛有油和
8、水，油層厚h1=40 cm，油的密度o=850 kg/m3，盛有水銀的U形測壓管的液面距水面的深度h2=60 cm，水銀柱的高度低于油面h=50 cm，水銀的密度hg= 13600 kg/m3，試求油面上的計(jì)示壓強(qiáng)15分。解： 13. 額定流量qm=35.69 kg/s的過熱蒸汽，壓強(qiáng)pe=981 N/cm2，蒸汽的比體積為v=0.03067 m3/kg，經(jīng)內(nèi)徑為227mm的主蒸汽管道鉛垂向下，再經(jīng)90彎管轉(zhuǎn)向水平方向流動(dòng)。如不計(jì)能量損失，試求蒸汽作用給彎管的水平力。解：由連續(xù)方程： 得： 選彎管所圍成的體積為控制體，對控制體列x方向動(dòng)量方程：14. 為測定90彎頭的局部阻力系數(shù)，在A、B兩斷
9、面接測壓管，流體由A流至B。管徑d=50 mm，AB段長度LAB = 0.8 m，流量qv = 15 m3/h，沿程阻力系數(shù)=0.0285，兩測壓管中的水柱高度差h = 20 mm，求彎頭的局部阻力系數(shù)。15分解： 對A、B列伯努利方程： 15. Pipe AB is of uniform diameter and h=10.5 m The pressure at A is 170 kPa and at B is 275 kPa. Find the direction of the flow, and what is the pipe friction head loss in meters
10、of the fluid if the liquid has a specific gravity of (a) 0.85; (b) 1.45? (10分)Solution: (a) the flow is from B to A (b) the flow is from A to B 16. A curved pipe is attached to a tank, and liquid flows out into atmosphere through the curved pipe. Determine the resultant force on the curved pipe. All
11、 the significant data are given in the figure. Assume an ideal liquid with =55 lb/ft3. Neglect the weight of the liquid. (10分)Solution: 17. An 180-mm-diameter pipeline (f=0.032) 150 m long discharges a 60-mm-diameter water jet into the atmosphere at a point that is 80 m below the water surface at in
12、take. The entrance to the pipe is reentrant, with ke=0.9, and the nozzle loss coefficient k2=kn=0.055. Find the flow rate and the pressure at B. (15分)Solution:(a) (b)18. What minimum value of b is necessary to keep the rectangular stone from sliding if it weighs 160 lb/ft3, a=14ft, c=16ft, and the c
13、oefficient of friction is 0.45? With this minimum b value, will it also be safe against overturning? Assume that water dose not get underneath the stone block. (10分)Solution1: to keep the rectangular stone from sliding, then Since it is also safe against overturning.19. A flow is defined by u=2(1+t)
14、, v=3(1+t), w=4(1+t). What is the velocity of flow at the point (3,2,4) at t=2? What is the acceleration at that point at t=2? (10分)Solution: At t=220. Water flows over the spillway of constant section. Given that y1 = 4.2m and y2 = 0.7m, determine the resultant horizontal force on the spillway per
15、meter of spillway width (perpendicular to the spillway section). Assume ideal flow. (10分)Solution: 將y1 = 4.2m, y2 = 0.7m, p1 = p2 , =9810 N/m2 , 代入得 解得：v1 =1.4 , v2 =8.4 To the right21. Water and oil in an open storage tank. (a) Find the total forces exerted by the fluids on a tank wall, and (b) the
16、 location of the center of pressure. (10分)Solution:(a) (b) 22. A diverging nozzle that discharges an 8-in-diameter water jet into the air is on the right end of a horizontal 6-in-diameter pipe. If the velocity in the pipe is 12fps, find the magnitude and direction of the resultant axial force the wa
17、ter exerts on the nozzle. Neglect fluid friction. (10分)Solution: to the right23. A horizontal 100-mm-diameter pipe (f=0.027) projects into a body of water (ke=0.8) 1m below the surface. Considering all losses, find the pressure at a point 5 m from the end of the pipe if the velocity is 4 m/s and the
18、 flow is (a) into the body of water; (b) out of the body of water. (10分)Solution:(a) (b) 24. A rectangular plate 5 ft by 4 ft is at an angle of 30with the horizontal, and the 5-ft side is horizontal. Find the magnitude of the force on one side of the plate and the depth of its center of pressure whe
19、n the top edge is (a) at the water surface; (b) 1 ft below the water surface. (10分)Solution:(a) (b)1. 25. Water in a reservoir is discharged from a vertical pipe. If a=25 ft, b= 60 ft, c=40 ft, d= 2ft. All the losses of energy are to be ignored when the stream discharging into the air at E has a dia
20、meter of 4 in. What are pressure heads at B, C and D, if the diameter of the vertical pipe is 5 in? (10分) 26. A nozzle that discharges a 60-mm-diamater water jet into the air is on the right end of a horizontal 120-mm-diameter pipe. In the pipe the water has a velocity of 4 m/s and a gage pressure o
21、f 400 kPa. Find the magnitude and direction of the resultant axial force the water exerts on the nozzle, and the head loss in the nozzle. (10分)Solution: to the right27. A 450-ft-long pipeline runs between two reservoirs, both ends being under water, and the intake end is square-edged and nonprojecti
22、ng. The difference between the water surface levels of the two reservoirs is 150ft. (a) What is the discharge(流量) if the pipe diameter is 12 in and f = 0.028? (b) When this same pipe is old, assume that the growth of tubercles has reduced the diameter to 11.25 in and that f=0.06. What will the rate of discharge be then? (10分)Solution:(a) (b)
【力學(xué)課件】流體力學(xué)習(xí)題庫
