Three-Dimensional Incompressible Flow

1. Chapter 6 Three-Dimensional Incompressible Flow Yanjie Li Harbin Institute of Technology Shenzhen Graduate School

2. Outline • Lifting-surface theory • Vortex lattice numerical method • Three-dimensional source • Three-dimensional doublet • Flow over a sphere • General three-dimensional flows: panel techniques Chapter 5

3. Lifting-Surface Theory • Lifting-line theory is inappropriate for low-aspect-ratio straight wings, swept wings and delta wings.

4. Extend a simple lifting line model by placing a series of lifting lines on the plane of the wing. Line Surface

5. Downstream of the trailing edge has no spanwise vortex lines and only trailing vortices. The strength of this wake vortex is given by ,which depends only on

6. Consider point located at on the wing • The lifting surface and the wake vortex sheet both induce a normal component of velocity at point • Want the wing platform to be a steam surface of the flow, that is, the sum of the induced and the normal component of the freestream velocity to be zero at point • The central theme of lifting surface theory is to find and such that the flow- tangency condition is satisfied at all points

7. An expression for induced normal velocity in terms of and Consider a point given by the coordinates Spanwise vortex strength is The strength of filament of the spanwise vortex sheet of incremental length is Using Biot-Savart law, the incremental velocity induced at by a segment of this spanwise vortex filament of strength is Considering the direction and (5.78)

8. Similarly, the contribution of the elemental chordwise vortex of strength to the induced velocity at is The velocity induced at point by the complete wake vortex can be given by an equation analogous to the above equation Eq. (5.78) and Eq. (5.79) should be Integrated over the wing planform, Region S, Eq. (*) should be integrated over region W , Noting The normal velocity induced at P by both the lifting surface and the wake is (5.79) (*)

9. The central problem of lifting-surface theory is to solve the following equation for and (**) Numerical solution: 1. Dividing the wing platform into a number of panels and choosing control points on these panels, Eq. (**) results in simultaneous algebraic equations at these control points. Solving these equations, we can obtain the values of and 2. Vortex lattice method

10. Vortex lattice method Superimpose a finite number of horseshoe vortex of different strength on the wing surface At any control point , applying the Biot-Savart law and flow-tangency condition, we can obtain a system of simultaneous algebraic equations, which can be solved for the unknown

11. Chapter 6Three-Dimensional Incompressible Flow

12. Three-Dimensional Source Consider the velocity potential given by Satisfying Laplace’s equation (3.43) A physically possible incompressible, irrotational three-dimensional flow The gradient in spherical coordinates

13. Eq. (6.2) describes a flow with straight streamlines emanating from the origin. The velocity varies inversely as the square of the distance from the origin Such a flow is defined as a three-dimensional source or called simply a point source To calculate the constant C in Eq. (6.3a) Consider a sphere of radius and surface centered at the origin. Volume flow is defined as the strength of source. a point source is a point sink.

14. Three-Dimensional Doublet Consider a sink and source of equal but opposite strength located at point O and A From Eq. (6.7), the velocity potential at P is where . The flow field produced by Eq. (6.9) is a three-dimensional doublet.

15. From Eq. (2.18) and Eq. (6.9) The streamline of this velocity field are the same in all the planes. The flow induced by the three dimensional doublet is a series of stream surfaces generated by revolving the streamlines in this figure. The flow is independent of .Such a flow is defined as axisymmetric flow.

16. Flow over a Sphere Consider the superposition of a uniform flow and a three-dimensional doublet Spherical coordinates of the freestream Combining the flow of three-dimensional doublet

17. To find the stagnation points in the flow. Two stagnation points on Z axis, with coordinates

18. (flow-tangency condition) The impressible flow over a sphere of radius R On the surface of the sphere of radius R, the tangential velocity is From Eq. (6.16),

19. Three-dimensional relieving effect A general phenomenon for all types of three-dimensional flows Two examples: Maximum tangential velocity for three-D flow is Maximum tangential velocity for two-D flow is , The maximum surface velocity on a sphere is less than that for a cylinder The pressure distribution on the surface of the sphere is The pressure distribution on a cylinder is

20. Comments on Three-Dimensional Relieving Effect Physical reason for the three-dimensional relieving effect • First, visualize the two-dimensional flow over a circular cylinder. In order to move out of the way of the cylinder, the flow has only two ways to go: riding up-and-over and down-and-under the cylinder. • In contrast, visualize the three-dimensional flow over a sphere. In addition to moving up-and over and down-and-under the sphere, the flow can now move sideways, to the left and right over the sphere. • The sidewise movement relieves the previous constraint on the flow.

21. General Three-Dimensional FlowsPanel Techniques • Calculate the three-D flow by means of numerical panel techniques. • General idea behind all such panel programs: • Cover the three-dimensional body with panels • Unknown distributions are solved through a system of simultaneous linear algebraic equations generated by calculating the induced velocity and applying the flow-tangency condition • Consider the source and vortex panels • Geometric complexity of distributing panels • How to get the computer to see the precise shape? • How to distribute the panels over the body? • How many panels do you use? And so on……….

22. Thank you