~Fluids – Lecture 8 Notes 1. Streamlines 2. Pathlines 3. Streaklines Reading: Anderson 2.11 Three types of fluid element trajectories are defined: Streamlines, Pathlines, and Streaklines. They are all equivalent for steady flows, but differ conceptually for unsteady flows. Streamlines Streamline equations A streamline is defined as a line which is everywhere parallel to the local velocity vector ~ˆV (x, y, z, t) = u ˆı + v ˆ+ w k. Define d~s = dx ˆı + dy ˆ + dz ˆk as an infinitesimal arc-length vector along the streamline. Since this is parallel to ~V , we must have d~s × V = 0 + (v dx − u dy)ˆ(w dy − v dz)ˆı + (u dz − w dx) ˆ k = 0 Separately setting each component to zero gives three differential equations which define the streamline. The three velocity components u, v, w, must be given a s functions of x, y, z before these equations can be integrated. To set the constants of integration, it is sufficient to specify some point xo, yo, zo through which the streamline passes, yy V ds Vds xo o ozy x z 3−D streamline 2−D streamline In 2-D we have dz = 0 and w = 0, and only the ˆk component of the equation above is non-trivial. It can be written as an Ordinary Differential Equation for the streamline shape y(x). dy v = dx u x u v dx dy xo oy Again, u(x, y) and v(x, y) must be given to allow integration, and xo, yo must be given to set the integration constants. In a numerical integration, xo, yo would serve as the initial values. 1� � � � � � � � � Streamtubes Consider a set of xo, yo, zo points arranged in a closed loop. The streamlines passing through all these points form the surface of a streamtube. Because there is no flow across the surface, each cross-section of the streamtube carries the same mass flow. So the streamtube is equivalent to a channel flow embedded in the rest of the flowfield. x o o ozy x y z 3−D streamtube y x o oy x 2−D streamtube In 2-D, a streamtube is defined by two streamlines passing through two specified xo, yo points. The flow between these two streamlines carries the same mass flow/span at each cross-section, and can be considered as a 2-D channel flow embedded in the rest of the flowfield. Pathlines The pathline of a fluid element A is simply the path it takes through space as a function of time. An example of a pathline is the trajectory taken by one puff of smoke which is carried by the steady or unsteady wind. This path is fully described by the three position functions xA(t), yA(t), zA(t), which can be computed by integrating the three velocity-field components u(x, y, z, t), v(x, y, z, t), w(x, y, z, t) along the path. The integration is started at time to, f r om the element’s initial position xo, yo, zo (e.g. the smoke release point), and proceeds to some later time t. t xA(t) = xo + u xA(τ), yA(τ), zA(τ), τ dτ to t yA(t) = yo + v xA(τ), yA(τ), zA(τ), τ dτ to t zA(t) = zo + w xA(τ), yA(τ), zA(τ), τ dτ to The dummy var ia ble of integration τ runs from to to t. Streaklines A streakline is associated with a particular point P in space which has the fluid moving past it. All points which pass through this point are said to form the streakline of point P . An example of a streakline is the continuous line of smoke emitted by a chimney at point P , which will have some curved shape if the wind has a time-varying direction. 2Unlike a pathline, which involves the motion of only o ne fluid element A in time, a streakline involves the motion of all the fluid elements along its length. Hence, the trajectory equations for a pathline are applied t o all the fluid elements defining the streakline. The figure below illustrates streamlines, pathlines, and streaklines for the case of a smoke being continuously emitted by a chimney at point P , in the presence of a shifting wind. One particular smoke puff A is also identified. The figure corresponds to a snapshot when the wind everywhere is along one particular direction. V instantaneousstreamlines instantaneous wind velocity A streakline from point P (smoke line) pathline of fluid element A (smoke puff) streakline at successive times P In a steady flow, streamlines, pathlines, and streaklines all cooincide. In this example they would a ll be marked by the smoke line.
View Full Document
Unlocking...