Mechanics of Fluids

and velocities are known, then for a time interval �t later, the conditions at
points such as P and Q in the next row of intersections of characteristics can
be calculated from the initial values respectively at A and B and at B and C.
(The value of �t must be �x/2c, as shown by the geometry of Fig. 12.10.)
Values for point R can then be established from those at P and Q, and
similarly for the whole x–t map.
If c is constant and large compared with u, the slopes of the characteristics
become 1/c and −1/c, respectively, and their intersections occur at regular
intervals of x and t over the whole map. Strictly, the velocity u changes with
both x and t (and c may change too if the liquid contains air or other gases
because K may then change with p). The characteristics are then curved
rather than straight and interpolation becomes necessary to obtain results
for the intersections.
At the upstream end of the pipe (x = 0) no positive-slope characteristic is
available for calculating conditions at a point such as N (Fig. 12.11). Therefore
the appropriate upstream boundary condition has to be used instead.
This may be, for example, a constant value of either u or p, or a specified
variation of one of them as a function of t, or an algebraically specified connection
between u and p as for flow through a valve. Similarly, the boundary
condition at the downstream end of the pipe (x = l) must be used there in
place of a negative-slope characteristic. (If u is not negligible compared with
c or if c is not constant, the characteristics with not necessarily intersect on
the boundary lines x = 0 and x = l. Interpolation then becomes necessary.)
At a junction of two or more pipes the equation of continuity must be
satisfied at every instant and (if any additional losses are disregarded) the
pressures where each pipe meets the junction must at every instant be the
Pressure transients 579
Fig. 12.11