3.24 A symmetrical sprinkler is shown as in Fig. 3-40. The rotating radius is R=200mm, ?=450, the nozzle diameter is d=8mm, the total flow rate is qv=0.563L/s, if the frictional moment is 0.2N?m, find the rotating speed n. And what is the magnitude of the moment needed to hold the sprinkler at rest while it is in operation?
¶Ô³ÆÈ÷Ë®Æ÷Èçͼ3-40Ëùʾ¡£Ðýת°ë¾¶ Fig. 3-40 Problem 3.24 R=200mm£¬?=450£¬Åç¿ÚÖ±¾¶d=8mm£¬×Ü
Á÷Á¿qv=0.563L/s£¬ÈôÒÑ֪Ħ²Á×èÁ¦¾ØÎª0.2N?m£¬ÇóתËÙn¡£ÈôÅçˮʱ²»ÈÃÆäÐýת£¬Ó¦Êܵ½¶à´óµÄÁ¦¾Ø£¿(103 r/min£¬0.441 N?m)
Problems
1.1 The velocity field of a rotational flow is given by
Find the average angular rotating velocity at point (2,2,2).
ÒÑÖªÓÐÐýÁ÷¶¯µÄËٶȳ¡Îª
ÇóÔڵ㣨2,2,2£©´¦Æ½¾ùÐýת½ÇËÙ¶È¡£(?x=0.5£¬?y=?2£¬?z=?0.5)
1.2 Determine whether the following flow field is rotational flow or irrotational flow.
È·¶¨ÏÂÁÐÁ÷³¡ÊÇÓÐÐýÔ˶¯»¹ÊÇÎÞÐýÔ˶¯£º
(1) (2)
( (1)ÓÐÐý£¬(2)ÎÞÐý)
4.3 The velocity distribution of a flow field is described by v=x2yi?xy2j Is the flow irrotational?
Á÷³¡µÄËÙ¶È·Ö²¼Îª
v=x2yi?xy2j
¸ÃÁ÷¶¯ÊÇ·ñÎÞÐý£¿(ÓÐÐý)
4.4 For a certain incompressible, two-dimensional flow field the velocity component in the y direction is given by vy=x2+2xy
Determine the velocity component in the x direction so that the continuity equation is satisfied.
ijһ²»¿ÉѹËõÆ½ÃæÁ÷³¡ÔÚy·½ÏòµÄËÙ¶È·ÖÁ¿Îª vy=x2+2xy
È·¶¨x·½ÏòµÄËÙ¶È·ÖÁ¿£¬ÒÔÂú×ãÁ¬ÐøÐÔ·½³Ì¡£(vx=?x2+C)
4.5 For a certain incompressible flow field it is suggested that the velocity components are given by the equations
vx=x2y vy=4y3z vz=2z Is this a physically possible flow field?
ijһ²»¿ÉѹËõÁ÷³¡Á÷³¡µÄËÙ¶È·ÖÁ¿ÓÉÏÂÁз½³Ì¸ø³ö vx=x2y vy=4y3z vz=2z ÊÔÎʸÃÁ÷³¡ÔÚÎïÀíÉÏÊÇ·ñ¿ÉÄÜ£¿(²»¿ÉÄÜ)
vvxxv
4.6 It is known that streamlines are concentric circles, and velocity distribution is ÒÑÖªÁ÷ÏßΪͬÐÄÔ²×壬ÆäËÙ¶È·Ö²¼Îª
Find the velocity circulation along circle x2+y2=R2, where the radiuses of the circle are
(1) R=3£»(2) R=5£»(3) R=10 respectively.
ÇóÑØÔ²ÖÜx2+y2=R2µÄËÙ¶È»·Á¿£¬ÆäÖÐÔ²µÄ°ë¾¶R·Ö±ðΪ (1) R=3£»(2) R=5£»(3) R=10¡£
((1) 18?/5£¬(2) 10?£¬(3) 10? )
4.7 Assume there is a vortex of ?=?0 locating at point (1£¬0), and another vortex of ?=-?0 at point (-1£¬0). Find the velocity circulation along the following routes: (1) x2+y2=4£» (2) (x-1)2+y2=1£»
(3) Square of x= ?2£¬y= ?2£» (4) Square of x= ?0.5£¬y= ?0.5.
ÉèÔÚµã(1£¬0)´¦ÖÃÓÐ?=?0µÄÐýÎУ¬ÔÚµã(-1£¬0)´¦ÖÃÓÐ?=-?0µÄÐýÎС£ÊÔÇóÏÂÁзÏßµÄËÙ¶È»·Á¿£º
(1) x2+y2=4£» (2) (x-1)2+y2=1£»
(3) x= ?2£¬y= ?2µÄ·½Ðοò£»
(4) x= ?0.5£¬y= ?0.5µÄ·½Ðοò¡£( (1) 0£¬(2) ?0£¬(3) 0£¬(4) 0 )
4.8 For incompressible fluid, determine if there exist stream functions in the following flow fields, where K is a constant.
¶ÔÓÚ²»¿ÉѹËõÁ÷Ì壬ÊÔÈ·¶¨ÏÂÁÐÁ÷³¡ÊÇ·ñ´æÔÚÁ÷º¯Êý£¿Ê½ÖÐKΪ³£Êý¡£ (1) vx=Ksin(xy)£¬vy=-K sin(xy)
(2) vx=Kln(xy)£¬vy=-Ky/x ( (1) ²»´æÔÚ£¬(2) ´æÔÚ)
4.9 Demonstrate the following planar flow of an incompressible fluid
?satisfies continuity equation, and is a potential flow, then find the potential function. ÊÔÖ¤Ã÷ÒÔϲ»¿ÉѹËõÁ÷ÌåÆ½ÃæÁ÷¶¯
v
Âú×ãÁ¬ÐøÐÔ·½³Ì£¬ÊÇÓÐÊÆÁ÷¶¯£¬²¢ÇóÊÆº¯Êý¡£( ?=x2/2+x2y?y2/2?y3/3)
4.10 A velocity field is given by vx=x2y+y2£¬vy=x2?xy2£¬vz=0£¬questions£º (1) If there exist stream function and potential function?
(2) Find the expressions of stream function and potential function if they exist. ¸ø¶¨Ëٶȳ¡vx=x2y+y2£¬vy=x2?xy2£¬vz=0£¬ÎÊ£º (1) ÊÇ·ñ´æÔÚÁ÷º¯ÊýºÍÊÆº¯Êý£¿ (2) Èç¹û´æÔÚ£¬ÇóÆä¾ßÌå±í´ïʽ¡£
((1)´æÔÚÁ÷º¯Êý£¬²»´æÔÚÊÆº¯Êý£»(2) ?=x2y2/2+y2/3?x2/3) 4.11 The velocity potential in a certain flow filed is ?=4xy
Determine the corresponding stream function. ijÁ÷³¡µÄËÙ¶ÈÊÆÎª
?=4xy ÇóÏàÓ¦µÄÁ÷º¯Êý¡£( ?=2x2?2y2)
4.12 The velocity potential for an incompressible, planar flow is ?=x2?y2+x Find its stream function.
²»¿ÉѹËõÁ÷ÌåÆ½ÃæÁ÷¶¯µÄÊÆº¯ÊýΪ ?=x2?y2+x ÊÔÇóÁ÷º¯Êý¡£( ?=2xy+y) 4.13 The stream function for an incompressible, planar flow is
?=xy+2x?3y
Find the potential function.
²»¿ÉѹËõÁ÷ÌåÆ½ÃæÁ÷¶¯µÄÁ÷º¯ÊýΪ ?=xy+2x?3y ÊÔÇóÊÆº¯Êý¡£( ?=(x2?y2)/2?3x?2y)
4.14 Demonstrate the following two flow fields are identical:
(1) the potential function is ?=x2+x ?y2 (2) the stream function is ?=2xy+y Ö¤Ã÷ÏÂÁÐÁ½¸öÁ÷³¡ÊÇÏàͬµÄ¡£ (1) ÊÆº¯Êý ?=x2+x ?y2 (2) Á÷º¯Êý ?=2xy+y
4.15
Given the velocity distribution of a flow field as
vx=Ax+By
vy=Cx+Dy
If the flow is incompressible and irrotational, find
vx