04_streamline - 1 Equation of Motion in Streamline...

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Unformatted text preview: 1 Equation of Motion in Streamline Coordinates Ain A. Sonin, MIT 2.25 Advanced Fluid Mechanics Euler’s equation expresses the relationship between the velocity and the pressure fields in inviscid flow. Written in terms of streamline coordinates, this equation gives information about not only about the pressure-velocity relationship along a streamline (Bernoulli’s equation), but also about how these quantities are related as one moves in the direction transverse to the streamlines. The transverse relationship is often overlooked in textbooks, but is every bit as important for understanding many important flow phenomena, a good example being how lift is generated on wings. A streamline is a line drawn at a given instant in time so that its tangent is at every point in the direction of the local fluid velocity (Fig. 1). Streamlines indicate local flow direction, not speed, which usually varies along a streamline. In steady flow the streamline pattern remains fixed with time; in unsteady flow the streamline pattern may change from instant to instant. Fig. 1: Streamline coordinates In what follows, we simplify the exposition by considering only steady, inviscid flows with a conservative body forces (of which gravity is an example). A conservative force per unit mass G is one that may be expressed as the gradient of a time-invariant scalar function, G = U ( r ), (1) 2 and the steady-state Euler equation reduces to 1 V V = p U ( r ). (2) A uniform gravitational force per unit mass g pointing in the negative z direction is represented by the potential U = gz . (3) A streamline coordinate system is not chosen arbitrarily, but follows from the velocity field (which, we note, is not known à priori). Associated uniquely with any point r and time t in a flow field are (Fig. 2): the streamline that passes through the point (streamlines cannot cross), the streamline’s local radius of curvature R and center of curvature, and the following triad of orthogonal unit vectors:...
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04_streamline - 1 Equation of Motion in Streamline...

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