University:
Massachusetts Institute of Technology
Course:
2.29 | Numerical Marine Hydrodynamics (13.024)
Academic year:

2022
Views:

281

Pages:

17
Author:

Alexa Livingston

Z y X n object U Sketch of an Object in a Uniform Stream Boundary Condition on Perturbation Potential ӘФ θη = 0 = Uin Three Dimensional Flows (x, y, z) outside S 0 - 2nd(x, y, z) (x,y,z) on S -4πφ(г, у, г) (x, y, z) inside S G 一 G(x, y, z, ξ. η. ζ) = 1 √(x)2 + (yn)² + (z-C)2 where: r = √(x - 5)2 + (y - 1)² + (2-6)2 1 G can be taken as: G = +H(x, y, z, ξ, η, ζ) where H is an analytic function (VH0 in both (x, y, z) and in (ξ,η, 5) coordinates. It is used sometimes when a particular H makes the integrand zero on flow boundaries external to an object, thereby removing the necessity of integrating over them. Numerical Form of the Two Dimensional Integral equation The 2D curve is divided into panels which become segments of a curve: L4 レッ V 4 L3 C3 V3 L2 2 L1 GA 5 V CN LN U Panelization of a Two Dimensional Object • Each panel(actually a line) is defined by two vertices on the object curve. Panel 1 extends from vertex 1 (labeled v₁ in the figure) to vertex 2, panel 2 extends from vertex 2 to vertex 3, etc. • The centroids of each panel, are at the mid-points of the arc length of each panel. For flat panels, straight lines are drawn between adjacent vertices and the centroids, called are moved to the midpoints of each straight line panel. • The unit normal vector, n, on panel i, is a constant along the length of the panel. The panels are defined by the (x,y) values of the two vertices at the panel ends. For the simplest implementation the potential, is approximated as being a constant on each panel. With these approximations and definitions, the numerical form of the integral equation is: N Git dl; + πΦ, δις = N j-1 Ui-ns, Gudl nj Analogous to the 3D case, we define: and Lan N j=1 Un Gijdly = Bi = Aj Again, the final set of equations for the N unknown values of & is: N j=1 Calculation of A and B₁ For a field point i located at the midpoint of the i'th panel we need to G determine for each panel j the value of A by integrating an over the j'th panel. We also need to determine B, by summing up the terms in the equation for B. General purpose MATLAB m-functions for doing these integrals exist and will be provided to students. Situations With the Generation of Lift • If the object in a streaminf flow has a sharp trailing edge, it will usually generate lift. • The lift is related to circulation around the object so that jumps acorss the trailing edge. This results in a "wake" across which the potential jumps by the same amount as at the trailing edge. • To obtain a flow domain in which the velocity potential is analytic (sat- isfies V20) the wake must be excluded from the domain. The line of the wake can be treated as part of the object upon which there is a uniform strength dipole sheet which makes the velocity potential jump when crossing it. • Here we will use an approximation in which the wake is presumed to follow the free stream direction. Theoretically, the wake is infinitely long. For practical purposes, we can model the wake as being about two airfoil chord lengths long since dipoles further than this from the object. (airfoil) will have negligible effect on the flow on the object. N+1 N N 1 2 U 1 Ny Expanded View of Wake n = ny An = -ny A Two-Dimensional Lifting Airfoil With a Wake

Week 10 Notes: Panel Methods

Related Documents

Recommended Documents

Description

Related Documents

Almost There!

Free up your schedule!

Take 5 seconds to unlock