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The Lagrangian and Eulerian descriptions of motion

In general, a given thermal or kinematic quantity in the ocean depends not only upon position but upon time as well. For example, the fields of temperature and salinity vary with time. The only stationary field which will be rpesumed is that of the earth's gravity.gif The magnetic field of the earth is known to vary with time.

Consider any scalar quantity S which is a continuous function of the four independent variables x, y, z and t, with t being time, for which the space and time derivatives exists.

The total rate of change of S with time is in general given by the expression

where the differential displacements dx, dy, dz are specified for the elapsed time dt. If we let

be the components of fluid velocity in the x, y, z directions at the point x, y, z, then the expression (22) takes on an important enough meaning to attach a special symbol to the operation therein indicated. We shall define the operator D/Dt such that

This will be referred to hereafter as the material time rate of change of S. The operator can also be written in the form

where

is the fluid velocity vector. The term represents the local time rate of change of the quantity S at a fixed point. The term is a scalar representing the advectional or field changes in the fluid associated with the motion of the fluid. The latter quantity can vanish in one of three ways:

If one of the above conditions is true then there is no advection of the property S by virtue of the fluid velocity and

However, in general the material rate of change differs from the local rate of change of S.

We can attach a physical significance to the term DS / Dt if we select a particular small particle of fluid. Let the symbol A identify this particular parcel. Then it can be shown that

where denote the value of S of the parcel of fluid at any time t, and the subscript on the right ahnd term of (28) denotes the value of the material derivative when each term of the expression is evaluated at the point occupied by the parcel A at the instant t. More specifically, if represent the position of the parcel A at time t, then

where it is understood that are functions of time which describe the path of the particular parcel A. The term implies that each term of the expression

is to be evaluated at the position at the instant t, after carrying out the partial differentiations indicated.

It is clear that physically represents the time rate of change of S in the fluid contained in the small parcel A as the latter is carried along with the fluid velocity. For this reason we might refer to or as the individual time rate of change of S for the parcel A. Consequently, we can intepret DS/Dt in general as the individual time rate of change of that parcel which happens to occupy the position x, y, z at the instant t.

If we are interested in the time rate of change of a given property for a particular parcel of fluid, it is uually somewhat easier to compute DS/Dt and then evaluate this at the position of the parcel, rather than evaluate and take its time derivative. This is particularly true if the field of is specified. This leads into the two methods by which the field of motion can be described.

We can specify either the field of or we can specify the paths (trajectories) of the particles of fluid. The former is referred to as the Eulerian description of motion while the latter is endowed with the title of Lagrangian description of motion. (It is believed however that Euler is actually responsible for both approaches.)

Specifically, in the Eulerian system we specify implying that

are known.

In the Lagrangian system we specify

for each particle of fluid.

The connection between these two approaches leads to the differential equation for the trajectories, namely:

The above equations determine the path of the particle A if the position is specified at some initial instant in its path history.

We will first illustrate the connection between the Eulerian and Lagrangian descriptions of motion with a simple one-dimensional problem.

Suppose

where a and b are constants. Then the equation for the trajectory of A is

This is a linear equation of the type discussed by Sokolnikoff and Sokolnikoff (1941) (Sec. 85). The solution of (31) can be shown to be

where

it being required to specify the initial position for the particle A. By substituting the solution for in Eq. (31) we find that the velocity of particle A at any time t is

which approaches the constant value b/a with time, if .

The Eulerian and Lagrangian modes of depicting the fluid motion for this example are compared graphically on the diagram of Fig. 1.03-1.

The diagram shows isolines of u (dashed lines) and also the path histories of several different particles which start at different points equally spaced along the x axis at the initial instant. For simplicity, the values of a and b were taken as unity in the illustration. It is evident from this picture that, because of the crowding together of the particles as time progresses, the fluid density must be increasing with time in the neighborhood of a given particle as we follow along with the motion of that particle. Obviously this type of motion, if it existed at all, could only occur in a compressible fluid.

Now suppose the salinity of the fluid varies with time and distance according to

where , and m are constants and S is now the salinity of the fluid. It will be presumed that this relationship holds in the presence of the foregoing velocity distribution. The time rate of change of the salinity following along with parcel A can be computed in two ways.

From Eqs. (32) and (35) we see that the salinity of parcel A is

Therefore

but in view of Eq. (34) this can be written

This same result can be obtained without even knowing the form of the trajectory. The material derivative S for this one-dimensional case is simply

Inserting S in terms of t and x from (35) yields

When this is evaluted for the parcel A the result on the right is identical with that of Eq. (38). Thus we have at least one specific check on the validity of the general statement Eq. (28).

We might next consider a slightly more complicated form of velocity field and investigate the particle trajectories which it generates.

Suppose

where , k and C are constants. This might represent the velocity field associated with a longitudinal acoustic wave in a tube. In this case C is the sound velocity in the fluid, is the wavelength, kC is the frequency, and is the amplitude of the oscillatory particle velocity. The form of the field of u indicates that energy is being propagated down the tube at a velocity C. The maximum particle velocity in an acoustical wave is extremely small compared with C even for very high acoustical energy levels (loud noises).

In general, the trajectory of a particle A is governed by

However, if as in the case of an ordinary acoustic wave, it can be shown that the variation of with time has very little effect on the right hand side of the equation, and can be taken therefore as the mean value, which happens to be the value at in this example.

Thus

for , and hence

where is the frequency of the sound. It we taken then Eq. (43) reduces to

Thus the fluid particle A moves back and forth with a frequency and with an amplitude equal to about the mean position . It will be noted that the condition is equivalent to , i.e., the amplitude of vibration of the particles must be much less than , where is the wavelength of the acoustic wave.

If we have a situation of a velocity field represented by Eq. (41) but is not small compared with C, then the particles do not simply vibrate about a fixed position but rather also have a net forward motion. In order to solve Eq. (42) in this case, it is convenient to make the substitution

Then Eq. (42) takes the form

which can be solved for t as

The integral can be found in standard tables. Upon evaluation of the integral and subsequent eliminatin of through Eq. (44), the desired solution is

whcih holds as long as is not equal to or greater than C. The new terms introduced in Eq. (47) are

where is the value of at , as before. When is very small then Eq. (47) reduces to the special case given by Eq. (43), however, the algebra is somewhat tedious. It is more readily shown from Eq. (46) that when is very small then the integral can be approximated by

which leads to

which can be verifed as being essentially equal to Eq. (43).

In the limiting case where then Eq. (47) is no longer applicable and we must return to Eq. (46). The solution is readily found to be of the form

In this case the particle which starts at the origin, i.e. , simply travels with the constant speed , since the second term in Eq. (49) vanishes in this case. Particle starting at other positions at do not travel with speed C at first but as time increases the second term in Eq. (49) approaches zero and eventually all of the particles that started between and lump together and travel with speed C. This is best illustrated graphically in the diagram (see Fig. 1.03-2). The dashed lines in Fig. 1.03-2 are isolines of equal particle velocity, while the full lines are trajectories of different particles which start out at positions equally spaced along the x axis at . The effect is somewhat analogous to that of Case A, except that only those particles contained initially in an interval lump together, there being formed many such compressional regions at intervals of one wavelength.

For the trajectory given by Eq. (47) has an oscillatory component as well as a net motion. A graph of the trajectory for a particle starting at when is given in Fig. 1.03.3 (curve 2). The curve (1) corresponds to the case of very much smaller than C, and curve (3) corresponds to the extreme case of , each curve applying to that particle which starts at the origin.

The net displacement of a particle which starts at the origin for an elapsed time of one period is plotted in Fig. 1.03-4 as a function of . It is apparent that the magnitude of net displacement is a small fraction of in one period as long as u is less than 2/3 C and is negligible if u is less than about C/10.

As a third illustration of the interrelationship between the Eulerian and the Lagrangian representations of fluid motion, we will consider the following two-dimensional case:

In this case there are two equations which determine the trajectory of a particle A; these are

Suppose it is desired to find the trajectory of that particle which starts at the origin at , i.e.

Equations (51 a,b) are to be satisfied simultaneously. Consequently, if we differentiate (51a) and eliminate using (51b) we obtain the second order equation

the general solution of which is

From Eq. (51a) we find therefore that

But Eqs. (52a,b) require that

Consequently, the final solutions are

for that particle which starts at the origin at . These are the parametric equations of a cycloid with a cusp at the origin and other cusps spaced along the y axis at intervals of (see Fig. 1.03-5, curve 1). Values of t/T where are plotted along the trajectory to indicate the time taken for the particle to traverse the distance from the origin to the point in question along the indicated path in the x-y plane.

In order to compare the trajectory with the field of velocity with which it is associated, it is convenient to construct the instantaneous streamlines at a given time. The streamlines are defined as those lines which are everywhere tangent to the velocity vectors at a given instant. Thus the streamlines indicate merely the direction of the flow at a given time, i.e. they give the synoptic flow pattern. Like trajectories there is no number associated with the lines, and the spacing is quite arbitrary. Additional isolines are required in general to give the magnitude of the velocities (such a set of lines will be referred to as isovels).

In general, the streamlines and the trajectories differ. Only in those cases where the field of velocity is steady is the pattern of streamlines and trajectories identical. In the case at hand the velocity field is changing with time and we must therefore expect to find a difference between the streamlines and trajectories.

The evaluation of the shape of the streamlines at a given instant ismost readily accomplished by setting time dependent terms in the equations for the velocity field equal to a constant and then evaluate the trajectories. That is, we find those trajectories which would exist if the velocity field were frozen, which is the special case where the streamlines are identical with the trajectories.

Thus if we let and represent the positions along a particular streamline at a given time then

where (a constant) is the instant for which the streamline pattern is to be evaluated. These equations lead to the homogeneous second order equation

for which the solutions are of the form

and

These equations can be written in the more convenient form

(where r and are constants) which are the parametric equations of a circle of radius r, centered at the point , ; i.e.

where the radius r is arbitrary.

Thus the streamlines for Case C at any instant are concentric circles, the center of which is on the y axis at a position equal to . This streamline pattern progresses along the y axis consequently at a speed . The streamlines at are shown in Fig. 1.03-5.

The magnitude of the velocity V at any instant is given by

or

If we let x, y be the positions along a particular streamline at instant then we see from Eq. (62) that

or

for this particular example. Thus the velocity on a particular streamline is constant and its magnitude varies from one streamline to the next directly proportional to the radius of the streamline. These results represent a very special type of motion corresponding to a traveling deformationlessgif vortex. The closest thing to this in nature is the horizontal circulation in th eye of a hurricane.

The streamlines and isovels could also be obtained for this example by use of polar coordinates. This is convenient when a radially symmetical pattern of this sort is being dealt with. In order to convert u and v to the polar velocity components we must refer to Figure 1.03-6. Consider a point x, y also described by r and . The components of along the radius vector and

perpendicular to it (in the sense of positive ) are and , respectively. It will be noted from the diagram that

and also that

Now in this example

Inserting these last expression in Eqs. (66) and simplifying gives

However, if we refer the velocities to a relative coordinate system whose origin moves with speed along the y axis, then the relative polar coordinates r and are given by

and the polar velocity components and relative to the moving coordinates are

Thus has the same form as , the velocity along the streamline at any instant.

Returning now to Eqs. (54) and (55) we may investigate the shape of other trajectories. One case of interest is the particle which starts at the point , at . For this trajectory we find from Eq. (54) and from Eq. (55). Thus, the parametric equations of the trajectory are

Thus the particle which starts at and at simply moves along a straight line parallel to the y axis and travels at constant speed . This is to be expected since at the initial position of this particle, the fluid velocity is given by

and since the fluid velocity pattern is traveling at the rate , the particle always remains at the same relative position on the streamlines and maintains the velocity . The trajectory is shown in Fig. 1.03-5 (curve 2).

As a third example trajectory, consider that a particle which starts at the position , at . In this case we find from Eqs. (54) and (55) that

Thus

These are the parametric equations of a degenerate troichoid which has loops at intervals of along the y axis. This trajectory corresponds to curve (3) of Fig. 1.03-5.

One more example (not shown on figure) is that of a particle starting at , at . The equations of this trajectory are

The latter trajectory has a minimum value of x of and a maximum value of . The curve is a troichoid with rounded peaks rather than cusps (at the minimum x values). This curve would lie intermediate between curves (1) and (2).

Those particles which start at a position between and would have a path similar to that just described except that the range of variation of x would be smaller and approach zero for as seen from curve (2). Those paths which lie very close to curve (2) would have very nearly a sinusoidal shape.

In general, it can be stated that the trjectories of all particles which are subject to the moving vortex under consideration are troichoids of some type, with curves 1, 2 and 3 being special cases of this general class of curves. It may be remarked in passing that the field of velocity discussed does not lead to any compressional effects whatever. That is, the separation distance between particles remains constant. This will be proven for this example in a later section (Art. 1.07). For the time being, it is sufficient to note that although the trajectories converge and cross each other, the particles do not actually pile up because in no case to two particles reach the crossing point simultaneously.




next up previous contents
Next: Methods of representation of Up: Kinematics of Fluids Previous: Curl of a vector

Steve Baum
Mon Dec 1 08:50:29 CST 1997