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Fundamental Concepts of Fluid Mechanics

8.1. Trajectories, streamlines and streaklines

The understanding of the organization of a flow cannot be based solely on knowledge of its local properties, velocity, pressure, density, temperature, etc. at each point in space (x, y, z) and every time t. We often need a more “dynamic” representation by using the motion of fluid particles, or at least the direction of the displacement. Such a visualization of the flow is a valuable aid in the design of vehicles or systems usinsg fluids. It plays a vital role in the study of separated three-dimensional flows whose understanding is based on the topological analysis of the velocity and skin friction fields. To assist this representation, we use the concepts of trajectory and streamlines, and to a lesser extent, streak lines.

8.1.1. Trajectories in a moving fluid

We consider an unsteady flow in the three-dimensional space whose velocity field (x, y, z, t) is known at any point (x, y, z) and at any time t. Given a fluid element (E1) passing through the point P0 at time t0 (see Figure 8.1a), at a later time t = t0 + Δt, that same element can be found at a point P1. The path followed by (E1) on its displacement from P0 to P1 is called a trajectory.

We consider another fluid element (E2) passing through the same point P0 at a later time t1. At time t = t0 + Δt, (E2) has no reason to be in ...

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