Modeling Tools for Environmental Engineers and Scientists

These functions are valuable tools in groundwater studies, because theycan describe the path of a fluid particle, known as the streamline. Further,under steady flow conditions, the two functions, φ(x,y) and ψ(x,y), are linear.Hence, by taking advantage of the principle of superposition, functionsdescribing different simple flow situations can be added to derive potentialand stream functions, and hence, the streamlines for the combined flow field.The application of the stream and potential functions and the principle ofsuperposition can best be illustrated by considering a practical example. Thedevelopment of the flow field around a pumping well situated in a uniformflow field such as in a homogeneous aquifer is detailed in Worked Example6.3, starting from the functions describing them individually.Worked Example 6.3Develop the stream function and the potential function to construct theflow network for a production well located in a uniform flow field. Use theresulting flow field to delineate the capture zone of the well.SolutionConsider first, a uniform flow of velocity, U, at an angle, α, with the x-direction. The velocity components in the x- and y-directions are as follows:u = U cos and v = U sin Substituting these velocity components into the above definitions for thepotential and stream functions and integrating, the following expressions canbe derived: = 0 – U(x cos + y sin )ory = 0 – u sin – (cos )x = 0 + U(y cos – x sin )or0–y = u co s + (tan )xThe results indicate that the stream lines are parallel, straight lines at an angleof α with the x-direction, which is as expected. In the special case where the© 2002 by CRC Press LLC