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1、412Flow KinematicsThis chapter explores some of the results that may be deduced about the nature of a flowing continuum without reference to the dynamics of the continuum.The first topic,flow lines,introduces the notions of streamlines,pathlines,and streaklines.These concepts not only are useful for
2、 flow-visualization experiments but also supply the means by which solutions to the governing equations may be interpreted physically.The concepts of circulation and vorticity are then introduced.Although these quantities are treated only in a kinematic sense at this stage,their full usefulness will
3、 become apparent in the later chapters when they are used in the dynamic equations of motion.The concept of the streamline leads to the concept of a stream tube or a stream filament.Likewise,the introduction of the vorticity vector permits the topic of vortex tubes and vortex filaments to be discuss
4、ed.Finally,this chapter ends with a discussion of the kinematics of vortex filaments or vortex lines.In this treatment,a useful analogy with the flow of an incom-pressible fluid is used.The results of this study form part of the so-called Helmholtz equations,with the remaining parts being taken up i
5、n the next chapter,which deals with,among other things,the dynamics of vorticity.2.1 Flow LinesThree types of flow lines are used frequently for flow-visualization pur-poses.These flow lines are called streamlines,pathlines,and streaklines,and in a general flow field,they are all different.The defin
6、itions and equations of these various flow lines will be obtained separately below.2.1.1 StreamlinesStreamlines are lines whose tangents are everywhere parallel to the veloc-ity vector.Since,in unsteady flow,the velocity vector at a given point will change both its magnitude and its direction with t
7、ime,it is meaningful to consider only the instantaneous streamlines in the case of unsteady flows.42Fundamental Mechanics of FluidsIn order to establish the equations of the streamlines in a given flow field,consider first a two-dimensional flow field in which the velocity vector u has components u
8、and v in the x and y directions,respectively.Then,by virtue of the definition of a streamline,its slope in the xy plane,namely,dy/dx,must be equal to that of the velocity vector,namely,v/u.That is,the equation of the streamline in the xy plane is ddyxvu=where,in general,both u and v will be function
9、s of x and y.Integration of this equation with respect to x and y,holding t fixed,will then yield the equation of the streamline in the xy plane at that instant in time.In the case of a three-dimensional flow field,the foregoing analysis is valid for the projection of the velocity vector on the xy p
10、lane.By similarly treating the projections on the xz plane and on the yz plane,the slopes of the stream-lines are found to be ddzxwu=ddzywv=on the xz and yz planes,respectively.These three equations defining the streamline may be written in the form ddddddyvxuzwxuzwyv=.Written in this form,it is cle
11、ar that these three equations may be expressed in the following more compact form:dddxuyvzw=.Integration of these equations for fixed t will yield,for that instant in time,an equation of the form z=z(x,y),which is the required streamline.The easiest way of carrying out the required integration is to
12、 try to obtain the parametric equations of the curve z=z(x,y)in the form x=x(s),y=y(s),and z=z(s).Elimination of the parameter s among these equations will then yield the equation of the streamline in the form z=z(x,y).43Flow KinematicsThus,a parameter s is introduced whose value is zero at some ref
13、erence point in space and whose value increases along the streamline.In terms of this parameter,the equations of the streamline become ddddxuyvzws=.These three equations may be combined in tensor notation to give ddfixedxsu x ttiii=(,)(2.1)in which it is noted that if the velocity components depend
14、upon time,the instantaneous streamline for any fixed value of t is considered.If the stream-line that passes through the point(x0,y0,z0)is required,Equation 2.1 is inte-grated and the initial conditions that when s=0,x=x0,y=y0,and z=z0 are applied.This will result in a set of equations of the form x
15、i=xi(x0,y0,z0,t,s)which,as s takes on all real values,traces out the required streamline.As an illustration of the determination of streamline patterns for a given flow field,consider the two-dimensional flow field defined by u=x(1+2t)v=y w=0.From Equation 2.1,the equations to be satisfied by the st
16、reamlines in the xy plane are ddxsxt=+()12 ddysy=.Integration of these equations yields x=C1e(1+2t)s y=C2es 44Fundamental Mechanics of Fluidswhich are the parametric equations of the streamlines in the xy plane.In par-ticular,suppose the streamlines passing through the point(1,1)are required.Using t
17、he initial conditions that when s=0,x=1 and y=1 shows that C1=C2=1.Then,the parametric equations of the streamlines passing through the point(1,1)are x=e(1+2t)s y=es.The fact that the streamlines change with time is evident from the preced-ing equations.Suppose the streamline passing through the poi
18、nt(1,1)at time t=0 is required;then,x=es y=es.Hence,the equation of the streamline is x=y.This streamline is shown in Figure 2.1 together with other flow lines,which are discussed below.0123456012345678910PathlineStreamlineStreaklineyxFIGURE 2.1Comparison of the streamline through the point(1,1)at t
19、=0 with the pathline of a particle that passed through the point(1,1)at t=0 and the streakline through the point(1,1)at t=0 for the flow field u=x(1+2t),v=y,w=0.45Flow Kinematics2.1.2 PathlinesA pathline is a line traced out in time by a given fluid particle as it flows.Since the particle under cons
20、ideration is moving with the fluid at its local velocity,pathlines must satisfy the equation ddxtu x tiii=(,).(2.2)The equation of the pathline that passes through the point(x0,y0,z0)at time t=0 will then be the solution to Equation 2.2,which satisfies the initial condi-tion that when t=0,x=x0,y=y0,
21、and z=z0.The solution will therefore yield a set of equations of the form xi=xi(x0,y0,z0,t)which,as t takes on all values greater than zero,will trace out the required pathline.As an illustration of the manner in which the equation of a pathline is obtained,consider again the flow field defined by u
22、=x(1+2t)v=y w=0.From Equation 2.2,the differential equations to be satisfied by the path-lines are ddxtxt=+()12 ddyty=.Integration of these equations gives x=C1et(1+t)y=C2et.These are the parametric equations of all the pathlines in the xy plane for this particular flow field.In particular,if the pa
23、thline of the particle that 46Fundamental Mechanics of Fluidspassed through the point(1,1)at t=0 is required,these parametric equations become x=et(1+t)y=et.Eliminating t from these equations shows that the equation of the required pathline is x=y1+log y.This pathline is shown in Figure 2.1,from whi
24、ch it will be seen that the streamline that passes through(1,1)at t=0 does not coincide with the path-line for the particle that passed through(1,1)at t=0.2.1.3 StreaklinesA streakline is a line traced out by a neutrally buoyant marker fluid that is continuously injected into a flow field at a fixed
25、 point in space.The marker fluid may be smoke(if the main flow involves air or some other gas)or a dye(if the main flow involves water or some other liquid).A particle of the marker fluid that is at the location(x,y,z)at time t must have passed through the injection point(x0,y0,z0)at some earlier ti
26、me t=.Then,the time history of this particle may be obtained by solving the equa-tion for the pathline(Equation 2.2)subject to the initial conditions that x=x0,y=y0,and z=z0 when t=.Then,as takes on all possible values in the range t,all fluid particles on the streakline will be obtained.That is,the
27、 equation of the streakline through the point(x0,y0,z0)is obtained by solving Equation 2.2 subject to the initial conditions that when t=,x=x0,y=y0,and z=z0.This will yield an expression of the form xi=xi(x0,y0,z0,t,).Then,as takes on the values t,these equations will define the instan-taneous locat
28、ion of that streakline.As an illustrative example,consider the flow field that was used to illus-trate the streamline and the pathline.Then,the equations to be solved for the streakline are ddxtxt=+()12 ddyty=47Flow Kinematicswhich integrate to give x=C1et(1+t)y=C2et.Using the initial conditions tha
29、t x=y=1 when t=,these equations become x=et(1+t)(1+)y=et.These are the parametric equations of the streakline that pass through the point(1,1),and they are valid for all times t.In particular,at t=0,these equa-tions become x=e(1+)y=e.Eliminating from these parametric equations shows that the equatio
30、n of the streakline that passes through the point(1,1)is,at time t=0,x=y1log y.This streakline is shown in Figure 2.1 along with the streamline and the pathline that were obtained for the same flow field.It will be noticed that none of the three flow lines coincide.2.2 Circulation and VorticityThe c
31、irculation contained within a closed contour in a body of fluid is defined as the integral around the contour of the component of the velocity vector that is locally tangent to the contour.That is,the circulation is defined as=u d1(2.3)where d1 represents an element of the contour.The integration is taken counterclockwise around the contour,and the circulation is positive if this integral is positive.