GOVERNING EQUATIONS
2.1 FLOW LINES
Written in this form, it is clear that these three equations may be expressed in the following, more compact form:
dx u ¼dy
v ¼dz w
Integration of these equations for ¢xedtwill yield, for that instant in time, an equation of the formz¼zðx;yÞ,which is the required streamline. The easiest way of carrying out the required integration is to try to obtain the parametric equations of the curvez¼zðx;yÞin the formx¼xðsÞ,y¼yðsÞ, andz¼zðsÞ.
Elimination of the parametersamong these equations will then yield the equation of the streamline in the formz¼zðx;yÞ.
Thus a parametersis introduced whose value is zero at some reference point in space and whose value increases along the streamline. In terms of this parameter the equations of the streamline become
dx u ¼dy
v ¼dz w ¼ds
These three equations may be combined in tensor notation to give dxi
ds ¼uiðxi;tÞ t fixed ð2:1Þ in which it is noted that if the velocity components depend upon time, the instantaneous streamline for any ¢xed value of t is considered. If the streamline that passes through the pointðx0;y0;z0Þis required, Eqs. (2.1) are integrated and the initial conditions that when s¼0, x¼x0, y¼y0, and z¼z0are applied. This will result in a set of equations of the form
xi¼xiðx0;y0;z0;t;sÞ
which, asstakes on all real values, traces out the required streamline.
As an illustration of the determination of streamline patterns for a given £ow ¢eld, consider the two-dimensional £ow ¢eld de¢ned by
u¼xð1þ2tÞ v¼y
w¼0
From Eqs. (2.1), the equations to be satis¢ed by the streamlines in thexy plane are
dx
ds ¼xð1þ2tÞ dy
ds¼y Integration of these equations yields
x¼C1eð1þ2tÞs y¼C2es
which are the parametric equations of the streamlines in the xyplane. In particular, suppose the streamlines passing through the point (1;1) are required. Using the initial conditions that whens¼0,x¼1 andy¼1 shows thatC1 ¼C2¼1. Then the parametric equations of the streamlines passing through the point (1, 1) are
x¼eð1þ2tÞs y¼es
The fact that the streamlines change with time is evident from the preceding equations. Suppose the streamline passing through the point (1, 1) at time t¼0 is required; then
x¼es y¼es Hence the equation of the streamline is
x¼y
This streamline is shown in Fig. 2.1 together with other £ow lines which are discussed below.
Pathlines
A pathline is a line traced out in time by a given £uid particle as it £ows. Since the particle under consideration is moving with the £uid at its local velocity, pathlines must satisfy the equations
dxi
dt ¼uiðxi;tÞ ð2:2Þ The equation of the pathline that passes through the pointðx0;y0;z0Þat time t¼0 will then be the solution to Eq. (2.2),which satis¢es the initial condition
that whent¼0,x¼x0,y¼y0, andz¼z0. The solution will therefore yield a set of equations of the form
xi¼xiðx0;y0;z0;tÞ
which, asttakes 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 £ow ¢eld de¢ned by
u¼xð1þ2tÞ v¼y
w¼0
From Eqs. (2.2), the di¡erential equations to be satis¢ed by the pathlines are dx
dt ¼xð1þ2tÞ dy
dt ¼y
FIGURE2.1 Comparison of the streamline through the point (1, 1) att¼0 with the pathline of a particle that passed through the point (1, 1) att¼0 and the streakline through the point (1, 1) att¼0 for the flow fieldu¼xð1þ2tÞ;v¼y;w¼0.
Integration of these equations gives
x¼C1etð1þtÞ y¼C2et
These are the parametric equations of all the pathlines in thexyplane for this particular £ow ¢eld. In particular, if the pathline of the particle that passed through the point (1, 1) at t¼0 is required, these parametric equations become
x¼etð1þtÞ y¼et
Eliminatingtfrom these equations shows that the equation of the required pathline is
x¼y1þlogy
This pathline is shown in Fig. 2.1, from which it will be seen that the streamline that passes through (1, 1) at t¼0 does not coincide with the pathline for the particle that passed through (1, 1) att ¼0.
Streaklines
A streakline is a line traced out by a neutrally buoyant marker £uid that is continuously injected into a £ow ¢eld at a ¢xed point in space. The marker
£uid may be smoke (if the main £ow involves air or some other gas) or a dye (if the main £ow involves water or some other liquid).
A particle of the marker £uid that is at the locationðx;y;zÞat timet must have passed through the injection pointðx0;y0;z0Þat some earlier time t¼t. Then the time history of this particle may be obtained by solving the equations for the pathline [Eqs. (2.2)] subject to the initial conditions that x¼x0,y¼y0, andz¼z0whent ¼t. Then asttakes on all possible values in the range1 tt, all £uid particles on the streakline will be obtained.
That is, the equation of the streakline through the pointðx0;y0;z0Þis obtained by solving Eqs. (2.2) subject to the initial conditions that whent¼t,x¼x0, y¼y0, andz¼z0. This will yield an expression of the form
xi¼xiðx0;y0;z0;t;tÞ
Then asttakes on the valuestt, these equations will de¢ne the instanta- neous location of that streakline.
As an illustrative example, consider the £ow ¢eld that was used to illustrate the streamline and the pathline.Then the equations to be solved for the streakline are
dx
dt ¼xð1þ2tÞ dy
dt ¼y which integrate to give
x¼C1etð1þtÞ y¼C2et
Using the initial conditions that x¼y¼1 when t¼t, these equations become
x¼etð1þtÞtð1þtÞ
y¼ett
These are the parametric equations of the streakline that passes through the point (1, 1), and they are valid for all timest. In particular, att¼0 these equations become
x¼etð1þtÞ y¼et
Eliminatingtfrom these parametric equations shows that the equation of the streakline that passes through the point (1, 1) is, at timet¼0,
x¼y1logy
This streakline is shown in Fig. 2.1 along with the streamline and the pathline that were obtained for the same £ow ¢eld. It will be noticed that none of the three £ow lines coincide.