流体力学流体力学 (2).pdf
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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
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