DESCRIPTION OF FLUID MOTION

The fluid being studied here is modeled as a continuum and infinitesimally small regions of the fluild (with a fixed mass) are called fluid elements or fluid

Подпись: FIGURE 1.1 Particle trajectory lines in a steady-state flow over an airfoil as viewed from a body-fixed coordinate system.

particles. The motion of the fluid can be described by two different methods. One adopts the particle point of view and follows the motion of the individual particles. The other adopts the field point of view and provides the flow variables as functions of position in space and time.

The particle point of view, which uses the approach of classical mechanics, is called the Lagrangian method. To trace the motion of each fluid particle, it is convenient to introduce a cartesian coordinate system with the coordinates де, у, and z. The position of any fluid particle P (see Fig. 1.1) is then given by

x = xP(x0, y0, Zo, t)

У =Уг(хо, Уо, Zo> 0 (1.1)

z = Zp(x0, y0, Zo, t)

where (jc0, y0, Zq) is the position of P at some initial time t = 0. (Note that the quantity (jc0, уо, Zo) represents the vector with components jc0, y0, and z0.) The components of the velocity of this particle are then given by

Эх

dy

dz

(1.2)

u~ dt

v = — at

w = — at

cPx

cPy

&z

(1.3)

‘ at2

Яу’ at2

*г at2

and the acceleration by

The Lagrangian formulation requires the evaluation of the motion of each fluid particle. For most practical applications this abundance of informa­tion is neither necessary nor useful and the analysis is cumbersome.

The field point of view, called the Eulerian method, provides the spatial

distribution of flow variables at each instant during the motion. For example, if a cartesian coordinate system is used, the components of the fluid velocity are given by

и = n(x, y, z, t)

v = v(x, y, z, t) (1.4)

w = w(x, y, z, t)

The Eulerian approach provides information about the fluid variables that is consistent with the information supplied by most experimental techniques and is in a form that is appropriate for most practical applications. For these reasons the Eulerian description of fluid motion is the most widely used.