Mechanics of Fluids

12.3.4 The method of characteristics
Since the velocity and pressure in a pipe subject to pressure transients are
continuous functions of position and time, they are described essentially
by partial differential equations. The method of characteristics is a general
technique in which such partial differential equations are converted into particular
simultaneous total differential equations which, after being expressed
in finite-difference form, can be solved by computer. The effects of friction
and difference of elevation can be retained in the equations and complex
pipe system can be dealt with. This method is not the only one possible for
solving the equations but it is increasingly used for the study of problems
involving pressure transients, so we shall here derive the finite-difference
equations and outline the method to the point where a computer program
can be written.
Pressure transients 575
Figure 12.9 shows a short length δx of the pipe through which a pressure The equation of motion
wave travels upstream. The velocity of the fluid is assumed uniform over
each cross-section and so the fluid in the length δx may be regarded as a
single particle to which Newton’s Second Law can be directly applied. Then
the force in the direction of flow
�
pA − p + ∂p
∂x δx
��
A + ∂A
∂x δx
�
−τ0Pδx − mg sin α
= Mass × Acceleration = m
�
+ p + 1
2
�
u ∂u
�
∂u
+
∂x ∂t
∂p
∂x δx
�
∂A
∂x δx
(12.14)
Here the third term on the left-hand side consists of the average pressure at
the sides of the element multiplied by the projected area of the sides perpendicular
to the flow direction. The term therefore represents the component in
the flow direction of the force exerted by the pipe walls on the fluid particle.
Fig. 12.9