fluid_mechanics

4.2 The Acceleration Field 163
V4.13 Streamline
coordinates
a n
a
a s
V
4.2.4 Streamline Coordinates
In many flow situations it is convenient to use a coordinate system defined in terms of the streamlines
of the flow. An example for steady, two-dimensional flows is illustrated in Fig. 4.8. Such flows
can be described either in terms of the usual x, y Cartesian coordinate system 1or some other system
such as the r, u polar coordinate system2 or the streamline coordinate system. In the streamline
coordinate system the flow is described in terms of one coordinate along the streamlines, denoted
s, and the second coordinate normal to the streamlines, denoted n. Unit vectors in these two
directions are denoted by ŝ and nˆ , as shown in the figure. Care is needed not to confuse the coordinate
distance s 1a scalar2 with the unit vector along the streamline direction, ŝ.
The flow plane is therefore covered by an orthogonal curved net of coordinate lines. At any
point the s and n directions are perpendicular, but the lines of constant s or constant n are not
necessarily straight. Without knowing the actual velocity field 1hence, the streamlines2 it is not
possible to construct this flow net. In many situations appropriate simplifying assumptions can be
made so that this lack of information does not present an insurmountable difficulty. One of the major
advantages of using the streamline coordinate system is that the velocity is always tangent to the s
direction. That is,
This allows simplifications in describing the fluid particle acceleration and in solving the equations
governing the flow.
For steady, two-dimensional flow we can determine the acceleration as
a DV
Dt
V V ŝ
a s ŝ a n nˆ
where a s and a n are the streamline and normal components of acceleration, respectively, as indicated
by the figure in the margin. We use the material derivative because by definition the acceleration
is the time rate of change of the velocity of a given particle as it moves about. If the streamlines
y
s = s 1
s = s 2
n = n 2
n = n 1
n = 0
s = 0
Streamlines
^
n
^
s
V
s
x
F I G U R E 4.8
Streamline coordinate system
for two-dimensional flow.