Fig.E-5 Starting flow in a pipe
supplied from constant
specific energy source.
The free motion case:
In this case the first step, spatial (path-) integration results in a differential equation - usually nonlinear - which contains derivatives with respect to time. Integrating it provides the
flow development in time as
the second step of solution, often quite demanding because of the nonlinearity. As a typical example, Fig. E-5 presents the task to compute
for t > 0 the time history of velocity in a pipe, which was closed until t = 0. The velocity is zero at
t = 0 and, because of fluid inertia, only gradually increases so that it reaches its asymptotic value theoretically at infinite time.
The asymptotic velocity: is evaluated by integrating, between and , the
steady form of the Bernoullis' Theorem equation. The pressure term is zero as well as the velocity
at , so that
. Here, of course, so that
The time development:
At this step we use the unsteady-flow equation containing the inertial term. Since the cross section between the starting point and the pipe entrance is very large, the corresponding acceleration may be neglected there. Integration of the inertial term therefore corresponds to the simple case of Fig.E-1. The integration result is
Moving the first and third term to the right-hand side:
... which, when compared with the above result for the asymptotic velocity, leads to the first-order differential equation containing, on the left-hand side, derivative with respect to time:
It is useful to introduce an auxiliary variable . Separating the variables leads to the easily integrable form:
The expression on the left-hand side is integrated by the method of partial fractions. This converts the left-hand side to the sum of elementary integrals:
The integrated form is:
It is now useful to introduce the inversion:
and from this, the resultant expression for the velocity in the pipe is as follows:
- this is simplified by the expression for hyperbolic functions as:
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This is page Nr. E03 from textbook Vaclav TESAR : "BASIC FLUID MECHANICS" Any comments and suggestions concerning this text may be mailed to the author
to his addresstesar@fsid.cvut.cz
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