第六章 不可壓縮理想流體的無旋運(yùn)動(dòng)

1、 Mechanics of FluidChapter 6 1第六章 不可壓縮理想流體的無旋運(yùn)動(dòng)2Chapter 6 Irrotational flow of impressible ideal-fluid 61 Introduction 62 Dynamic analysis of micelle in liquid 6 3 Velocity potential of irrotational flow 64 Stream function of plane flow 65 Fundamental potential flow in plane 66 Combining theory of p

2、otential flow3第六章 不可壓縮理想流體的無旋運(yùn)動(dòng) 61 引言 62 流體微團(tuán)運(yùn)動(dòng)分析 6 3 無旋流動(dòng)的速度勢(shì) 64 平面流動(dòng)的流函數(shù) 65 基本平面勢(shì)流 66 勢(shì)流的疊加原理4Chapter 6 Irrotational flow of impressible ideal-fluid6-1 IntroductionIrrotational flow of impressible ideal-fluid In this chapter irrotational flow in plane of impressible ideal-fluid is studied, velocit

3、y field of irrotational flow can be obtained through three approach by calculating velocity potential, stream function and combining potential. 5第六章 不可壓縮理想流體的無旋運(yùn)動(dòng)6-1 引言不可壓縮理想流體的無旋運(yùn)動(dòng) 本章主要研究不可壓縮理想流體平面無旋運(yùn)動(dòng)，平面無旋運(yùn)動(dòng)的速度場(chǎng)可通過計(jì)算速度勢(shì)、流函數(shù)及復(fù)勢(shì)這三條途徑來確定。66-2 Dynamic analysis of liquid micelle Known from theory mecha

4、nics,in rigid body there are two motion forms：moving and rotation；but different in liquid motions.Due to different velocity of each point of liquid micelle in flow field,but need keep its continuity as well,so besides moving and rotation in liquid micelle,still has deform motions.In following three

5、motion forms of liquid micelle are analyzed. Figure 61 is plane motion of liquid micelle.Supposing liquid micelle is rectangle ABCD at t ,Through ,it changes into new position and deforms to ,again supposing at t velocity of angle point A is ,spreading due to Thaler progression,obtaining velocity of

6、 point B and CIrrotational flow of impressible ideal-fluid76-2 流體微團(tuán)運(yùn)動(dòng)分析 由理論力學(xué)可知，剛體有平移和旋轉(zhuǎn)兩種運(yùn)動(dòng)形式，而流體運(yùn)動(dòng)則不同。由于流體微團(tuán)在流場(chǎng)中各點(diǎn)的速度不同，但又要保持流體本身的連續(xù)性，因此流體微團(tuán)除有平移和旋轉(zhuǎn)運(yùn)動(dòng)外，還有變形運(yùn)動(dòng)。下面將分析流體微團(tuán)的三種運(yùn)動(dòng)形式。 不可壓縮理想流體的無旋運(yùn)動(dòng) 如圖61所示的平面運(yùn)動(dòng)中的流體微團(tuán)。設(shè)在 t 時(shí)刻流體微團(tuán)為矩形ABCD，經(jīng)過 時(shí)段后它移動(dòng)到新的位置并變形為 ，又設(shè) t 時(shí)刻角點(diǎn)A的速度為 ，根據(jù)泰勒級(jí)數(shù)展開，得B、C點(diǎn)的速度分別為8Figure 6-1 Ana

7、lysis on plane motion of liquid micelleIrrotational flow of impressible ideal-fluid9不可壓縮理想流體的無旋運(yùn)動(dòng)圖 61 分析流體微團(tuán)的平面運(yùn)動(dòng) 10 Velocity of each point concludes ,known from figure 61, is translating velocity. （1）Linear deformation Taking AB as example.Due to velocity of angle point B is faster than angle point

8、 A along direction x (or slower) , so through , protraction of side AB along x (or condense)is .Linear deformation of unit length in unit time called linear deformation velocity ,and written， then1、Translating motion2、Deformation motion（61）similarly Irrotational flow of impressible ideal-fluid11 各點(diǎn)的

9、速度中均包含有 ，由圖61可見， 是平移速度。 （1）線變形 以AB為例。因?yàn)榻屈c(diǎn)B沿 x 方向的速度比角點(diǎn)A快（或慢） ，所以經(jīng)過 時(shí)段后，AB邊在 x 方向的伸長(zhǎng)（或縮短）量為 。單位時(shí)間單位長(zhǎng)度的線變形稱為線變形速度，并記為 ，則1、平移運(yùn)動(dòng)2、變形運(yùn)動(dòng)（61）同理不可壓縮理想流體的無旋運(yùn)動(dòng)12 Taking half angular deformation velocity in plane defined as shear deformation velocity of liquid micelle,written as known from figure 6-1,angle cha

10、nging of point A is According to definition of shear deformation velocity on liquid micelle,obtaining（62）and thatso（1）（2）（2）Shear deformationIrrotational flow of impressible ideal-fluid13 將平面上角變形速度之半定義為流體微團(tuán)的剪切變形速度，記為 由圖6-1可知，A點(diǎn)的角度變化為根據(jù)流體微團(tuán)剪切變形速度的定義得（62）而所以（1）（2）不可壓縮理想流體的無旋運(yùn)動(dòng)（2）剪切變形14 Taking average

11、value of rotational angular velocity on two straight-line in liquid micelle defined as rotational angular velocity of liquid micelle,written as , ,supposing straight-line rotational angular velocity along counter-clockwise is positive, then from formula(1)(2), rotational angular of side AB in unit t

12、ime is , rotational angular of side AC in unit time is ,according to definition of rotational angular velocity on liquid micelle, obtaining If all liquid micelle translating or deforming motion when liquid flowing, no rotating,viz. ,then called this flow is irrotational flow (potential flow). （63）3、

13、 Rotation Definition： If has rotation in liquid micelle,viz. at least one is not 0 of three, then called rotational motion(vortex motion).Irrotational flow of impressible ideal-fluid15 將流體微團(tuán)上兩條直線旋轉(zhuǎn)角速度的平均值定義為流體微團(tuán)的旋轉(zhuǎn)角速度，記為 ，假設(shè)直線逆時(shí)針方向旋轉(zhuǎn)的角速度為正，則由（1）（2）式可知，單位時(shí)間內(nèi)AB邊的旋轉(zhuǎn)角度為 ，單位時(shí)間內(nèi)AC邊的旋轉(zhuǎn)角度為 ，根據(jù)流體微團(tuán)旋轉(zhuǎn)角速度的定義得

14、如果流體流動(dòng)時(shí)所有流體微團(tuán)僅作平移和變形運(yùn)動(dòng)，沒有旋轉(zhuǎn)運(yùn)動(dòng)，即 ，則稱該流動(dòng)為無旋流動(dòng)（勢(shì)流）。 （63）不可壓縮理想流體的無旋運(yùn)動(dòng)3、旋轉(zhuǎn)運(yùn)動(dòng) 定義： 若流體微團(tuán)有旋轉(zhuǎn)運(yùn)動(dòng)，即 三者中至少有一個(gè)不 等于零，則稱為有旋流動(dòng)（有渦運(yùn)動(dòng)）。166-3 Velocity potential of irrotational flow Irrotational flow motion, in right-angle coordinate must have（64） formula (64) is sufficient and necessary condition of certain potentia

15、l function in complete differential ,where t is parameter variable, obtainingIlluminating no rotating musthave potential. And thatso （65）Irrotational flow of impressible ideal-fluid176-3 無旋流動(dòng)的速度勢(shì) 無旋運(yùn)動(dòng)， 在直角坐標(biāo)中必有（64） 式（64）是 為某一勢(shì)函數(shù) 的全微分的充分必需條件，其中 t 為參變量，必有又因說明無旋必有勢(shì)故 （65）不可壓縮理想流體的無旋運(yùn)動(dòng)18Column coordinate

16、（66）Spherical coordinate（67）Irrotational flow of impressible ideal-fluid19圓柱坐標(biāo)系（66）球坐標(biāo)系（67）不可壓縮理想流體的無旋運(yùn)動(dòng)20 Character of flow velocity potential function : where unit vector of this direction; angle between and grads;velocity component parallel to directionproving（68）1、direction derivative of random

17、direction is equal to component velocity of this direction, viz.Irrotational flow of impressible ideal-fluid21證 流速勢(shì)函數(shù) 的性質(zhì)： （68）其中 該方向的單位矢量； 與梯度 的夾角；速度在 方向的分量。不可壓縮理想流體的無旋運(yùn)動(dòng)1、 對(duì)于任意方向 的方向?qū)?shù)等于該方向的分速，即22 Curved area with constant potential function of flow velocity is equipotential area.In its area line

18、in it called equipotential line. So wherevelocity vector；infinitesimal arc vector in equipotential surface.2、Equipotential lines perpendicular with streamlines definition：explaining：velocity u and ds is perpendicular.Equipotential line is cross flow section line . one family streamline and equipoten

19、tial line constitute perpendicular flow net.Irrotational flow of impressible ideal-fluid23 流速勢(shì)函數(shù)等于常數(shù)的曲面積為等勢(shì)面。在其面上位于等勢(shì)面上的線稱為等勢(shì)線。 所以式中速度向量；等勢(shì)面上微元弧向量。不可壓縮理想流體的無旋運(yùn)動(dòng)2、等勢(shì)線與流線正交 定義：說明：速度u與ds正交。等勢(shì)線既是過流斷面線。 一族流線與等勢(shì)線構(gòu)成相互正交的流網(wǎng)。243、Potential function of flow velocity increasing along streamlines direction soFro

20、m character1,obtaining velocity parallel to streamline direction Velocity parallel to direction streamline ，so , showing that direction of value increasing parallel to direction s .Irrotational flow of impressible ideal-fluid253、流速勢(shì)函數(shù)沿流線 s 方向增大。 從而得不可壓縮理想流體的無旋運(yùn)動(dòng)由性質(zhì)1得沿流線方向的速度為 沿流線方向速度 ，所以 ，即說明 值增大的方向

21、與 s 方向相同。264、Potential function of flow velocity is harmonic function Substituting continuous equation of imcompressible liquid into, obtainingsoor（6 9）（610) above formula showing that potential functions of flow velocity satisfy Laplace equation ,in math function that satisfy Laplace equation calle

22、d harmonic function, so potential function of flow velocity is harmonic function.Potential flow in paneIrrotational flow of impressible ideal-fluid274、流速勢(shì)函數(shù)是調(diào)和函數(shù) 代入不可壓縮流體的連續(xù)方程中得從而得或者（6 9）（610) 上式說明流速勢(shì)函數(shù) 滿足拉普拉斯 方程式，在數(shù)學(xué)上稱滿足拉普拉斯方程式的函數(shù)為調(diào)和函數(shù)，所以流速勢(shì)函數(shù) 是調(diào)和函數(shù)。不可壓縮理想流體的無旋運(yùn)動(dòng)平面勢(shì)流中286-4 Stream function of plane

23、flow一、Definition and confirmation of stream function （611）Viz. it make become sufficient and necessary condition of certain function ,thenso（612）For imcompressible liquid of plane flow,its continuous equation isIrrotational flow of impressible ideal-fluid296-4 平面流動(dòng)的流函數(shù)一、流函數(shù)的定義及其確定 （611）即 它是使 成為某一函數(shù)

24、的全微分的充要條件，則有故（612）不可壓縮理想流體的無旋運(yùn)動(dòng)對(duì)于不可壓縮流體的平面流動(dòng)，其連續(xù)方程式為30 called stream function of imcompressible liquid of plane flow.Proving analogously,in polar coordinate（613） Due to condition of stream function existing is that flow satisfy continuous function of imcompressible liquid, and that to any flow ,cont

25、inuous function is must be satisfied, so in any plane flow must have stream function .Irrotational flow of impressible ideal-fluid31 就稱為不可壓縮流體平面流動(dòng)的流函數(shù)。類似地可證，在極坐標(biāo)中（613） 因?yàn)榱骱瘮?shù)存在的條件是要求流動(dòng)滿足不可壓縮流體的連續(xù)方程式，而連續(xù)方程式是任何流動(dòng)都必須滿足的，所以說任何平面流動(dòng)中一定存在著一個(gè)流函數(shù) 。不可壓縮理想流體的無旋運(yùn)動(dòng)32二、Basic character of stream function proving：se

26、eing about discharge of any curve AB(direction z is unit length). (figure 62)to discharge through infinitesimal vectorThen discharge through any join line between point A and B.（614）Figure 62 relation of stream function and dischargeDue toViz. is streamline equation. 1、Uniform flow function line is

27、streamline2、Discharge through two streamline is equal with difference of flow function.Irrotational flow of impressible ideal-fluid33二、流函數(shù)的基本性質(zhì)因?yàn)榧?為流線方程。 證：考察通過任意一條曲線AB（ z 方向?yàn)閱挝婚L(zhǎng)度）的流量。（圖62）對(duì)于通過微元矢量 的流量則通過AB兩點(diǎn)的任意連線AB的流量（614）圖62流函數(shù)與流量的關(guān)系不可壓縮理想流體的無旋運(yùn)動(dòng) 1、等流函數(shù)線為流線2、兩條流線間通過的流量等于兩條流線的流函數(shù)之差。343、Uniform stre

28、am function line(streamline) perpendicular with equipotential line showing gradient of stream function correctitude join with grads of velocity potential (viz. velocity), so uniform flow function line (streamline)which normal to them perpendicular with equipotential line.example61 Stream function of

29、 imcompressible flow field （1）flowing is rotational or irrotational ？ （2）if is irrotational, finding velocity potential of flowing solution (1)Due toSo is irrotational flowbecauseIrrotational flow of impressible ideal-fluid353、等流函數(shù)線（流線）與等勢(shì)線正交 說明流函數(shù)的梯度與速度勢(shì)的梯度（即速度）正交，故分別與它們垂直的等流函數(shù)線（即流線）與等勢(shì)線正交。 例題61不可壓

30、縮流體流場(chǎng)的流函數(shù) （1）流動(dòng)是無旋還是有旋？ （2）若無旋，確定流動(dòng)的速度勢(shì)。 解 （1）因故是無旋流。不可壓縮理想流體的無旋運(yùn)動(dòng) 這是因?yàn)?64、 In plane irrotational flowStream function satisfied Laplace equation, is harmonic equation.Proving：plane irrotational flow need satisfythenbecauseSubstitute into above formula，Relation between plane irrotational flow functio

31、n and potential function of flow velocity ：Irrotational flow of impressible ideal-fluid37不可壓縮理想流體的無旋運(yùn)動(dòng)4、在平面無旋流動(dòng)中流函數(shù)滿足拉普拉斯方程，是調(diào)和函數(shù)。證：平面無旋流動(dòng)需滿足則因?yàn)榇肷鲜?，平面無旋流動(dòng)的流函數(shù)和流速勢(shì)函數(shù)之間的關(guān)系式為：38 in mathematic analysis,this relation called Cauchy-liman condition, two harmonic function that satisfy this condition called

32、 conjugated harmonic function, known one of them can obtain the other.Irrotational flow of impressible ideal-fluid39不可壓縮理想流體的無旋運(yùn)動(dòng) 在數(shù)學(xué)分析中，這個(gè)關(guān)系式稱為柯西黎曼條件，滿足這個(gè)條件的兩個(gè)調(diào)和函數(shù)稱共軛調(diào)和函數(shù)，已知其中一個(gè)函數(shù)就可以求出另一個(gè)函數(shù)。40（2）soIntegrating And thenthenIrrotational flow of impressible ideal-fluid41故（2）積分于是則不可壓縮理想流體的無旋運(yùn)動(dòng)426-5 Fund

33、amental potential flow in plane一、parallel uniform velocity flowstreamline equationorAfter integrating（615）They are parallel straight-lines with slope , shown as figure 6-3figure 63 parallel uniform velocity flow Supposing liquid flowing with uniform velocity at parallel line, magnitude and direction

34、 of velocity of any point in flow field,that is both are fixed value.Irrotational flow of impressible ideal-fluid436-5 基本平面勢(shì)流一、平行等速流 流線方程或積分后得（615）它是一組斜率為 平等直線，如圖63所示 圖 63 平 行 等 速 流不可壓縮理想流體的無旋運(yùn)動(dòng) 設(shè)液體作平行直線等速流動(dòng)，流場(chǎng)中各點(diǎn)速度的大小和方向均相同，即 均為定值。44And then ,velocity potentialAnd that stream function Due tobecause

35、（616）（617）Irrotational flow of impressible ideal-fluid45而流函數(shù)為由于于是，速度勢(shì)為又（616）（617）不可壓縮理想流體的無旋運(yùn)動(dòng)46Liquid outflows from certain radial called source,as shown in figure6-4(a).Liquid inflows from certain radial called converge,as shown in figure64(b).Supposing thickness of water layer is 1 along radius r

36、,discharge of source (converge) is Q，thentherefore（ a ）（ b ）figure 6 - 4 source and converge二、Source and convergedefinition：Irrotational flow of impressible ideal-fluid47 流體從某一點(diǎn)徑向流出稱為源，如圖64（a）所示。 流體從某一點(diǎn)徑向流入稱為匯，如圖64（b）所示。 設(shè)半徑 r 方向水層的厚度為1，源（匯）的流量為Q，則由此（ a ）（ b ）圖 6 4 源 與 匯不可壓縮理想流體的無旋運(yùn)動(dòng)二、源流匯 定義：48 Due

37、to source and converge only flow along radial, so component velocity along circumference .In polar coordinate, from formula（66）Integrating and obtaining（618）From formula（613）Integrating and obtaining where is integrating constant about ,from above two should be equal, thenIrrotational flow of impres

38、sible ideal-fluid49 由于源匯只有徑向流動(dòng)，所以圓周方向的速度分量 。在極坐標(biāo)中，由式（66）積分得（618）由式（613） 積分得不可壓縮理想流體的無旋運(yùn)動(dòng) 式中 分別是關(guān)于 的積分常數(shù)，根據(jù)上面兩個(gè)應(yīng)該相等，得50 where is integrating constant about ,from two should be equal, obtaining （619）so Assuming discharge of outflow is positive,then source get“ ” ,converge get “-”.Equipotential line of

39、 source and converge are group of concentric circles.Irrotational flow of impressible ideal-fluid51不可壓縮理想流體的無旋運(yùn)動(dòng) 式中 分別是關(guān)于 的積分常數(shù)，由兩個(gè) 應(yīng)該相等，得（619）故 假定流出流量為正，則源流取“ ”號(hào)，匯流取“-”號(hào)。源匯流的等勢(shì)線為一組同心圓。52thereforeFrom formula (6-6)（620）Integrating and obtaining Now we study circumference motion of liquid,that is onl

40、y have velocity along circumference , and that velocity at radius direction . shown as figure 65, and define linear integrating of velocity in tangential of circumference as velocity loop quantity ,that is三、Potential vortex flowIrrotational flow of impressible ideal-fluid53 現(xiàn)在我們來研究流體的圓周運(yùn)動(dòng)，即只有圓周方向速度

41、，而徑向速度 。如圖65所示，并且定義速度 在圓周切線上的線積分為速度環(huán)量 ，即不可壓縮理想流體的無旋運(yùn)動(dòng)三、勢(shì)渦流所以由式（66）（620）積分得54thereforeIntegrating and obtaining （621）Equipotential lines are one family radialsFigure 64（a）potential vortex flowIrrotational flow of impressible ideal-fluid55不可壓縮理想流體的無旋運(yùn)動(dòng)由此得積分得（621）等勢(shì)線是一族射線。圖64（a）勢(shì)渦流56 If taking source t

42、hat in point and intensity is Q combined with converge that in point B and with uniform intension.(figure 6-5）making are velocity potential and stream function of source and converge,then after combining,velocity potential of certain point（622）stream function（623）四、Combining of source and convergefigure 65 source and converge Irrotational flow of impressible ideal-fluid57 若將位于 點(diǎn)，強(qiáng)度為Q的源與位于B 點(diǎn)等強(qiáng)度的匯疊加（圖65）令 分別為源與匯的速度勢(shì)和流函數(shù)，則疊加后某點(diǎn) 的速度勢(shì)（622）流函數(shù)