Of the visualization lines described previously, the streamlines have the most value as a flow-visualization device throughout this book. This is because for the most part, we concentrate on steady-flow applications. Because streamlines are such a useful tool, we now must learn how to describe them mathematically.
Suppose that we have carried out an analysis for a flow problem that results in a solution for the velocity vector as a function of position in an appropriate coordinate system. That is, we found the velocity vector throughout the domain in Eulerian form:
V = V(r, t),
where r is the position vector for any point in the domain. This might be a complicated algebraic function that is difficult to interpret physically. Mathematical flow visualization is necessary to bring out the details. To illustrate the method, consider a case in which V is described in terms of a Cartesian coordinate system in which a point in the flow is identified by its position vector, r = xi + yj + zk, where i, j, and k are the unit vectors along the three coordinate axes. Then, in component form,
V = u(x, y,z, t)i + v(x, y,z, t)j + w(x, y,z, t)k,
where u, v, and w are the velocity components as a function of position in the field. How can we find the streamline for this velocity vector that passes through a given point? By definition, the streamline must be parallel to the velocity vector. Therefore, the slope of the streamline projected on each of the three planes normal to the coordinate directions must be equal to the slope of the projection of the velocity vector. Thus, we must have:
dy = v dz = w dz = w dx u dx u ’ dy v
as the condition of tangency at any given time. Because these can be written in the form:
dy = dx dz = dx dz = dy v u ’ w u ’ w v
it is clear that this information can be conveyed best in the simpler form:
dx = dy = dz u v w ‘
To find the equation of the streamline at an instant of time, it is necessary to integrate these equations to define, for example, z = z(x, y), which is an equation for any streamline. To find a particular streamline, which can be plotted for flow visualization, we simply evaluate the constants of integration by inserting values of x, y, and z corresponding to a point through which that streamline passes.
This is a simple process for two-dimensional problems because we then must integrate only the simple equation:
dy = v dx u
to find y = y(x), which describes the required streamline.
The following example illustrates the technique used in finding a mathematical description of streamlines in a given flow. Notice that the presence of a function of time is of no concern because only the instantaneous streamline is meaningful.
EXAMPLE 2.5 Required: Find the equation for streamlines in a flow field described by the unsteady velocity vector:
V = x(1 + at2)i + yebtj.
Find the specific streamline passing through the point (3, -2) at time t = 0. Approach: Integrate Eq. 2.26.
Solution: Substituting the velocity components into Eq. 2.26 yields:
dy = v = ye
dx u x(l + at2)’
The equation is separable so, for a given time, we can write:
which immediately can be integrated to give:
where C is the constant of integration. Notice that the streamline is time – dependent. Because we want the streamline passing through point (3, -2) at time t = 0, C is readily found to be C = -2/3, and the required streamline equation is:
which is obviously a straight line at t = 0. The shape of the streamline changes as time progresses unless constants a and b are both zero.