Three-Dimensional Incompressible Flow

Yanjie Li

Harbin Institute of Technology

Shenzhen Graduate School

Chapter 6

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

Lifting-Surface Theory

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

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

Line Surface

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

wy

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

P ),( yx

P

),( yxw

P

),( yx

The central theme of lifting surface theory is to

find and such that the flow-

tangency condition is satisfied at all points

),( yx

An expression for induced normal velocity in terms of and ),( yxw ,, w

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

),( ),(

d d

P dd

Considering the direction and rx /)(sin

(5.78)

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

P

P

(5.79)

3

)(

4)(

r

ddydw w

w

(*)

The central problem of lifting-surface theory is to solve the following equation for

and

0freestream theofcomponent normal),( yxw

),( yx ),( yx

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

(**)

Numerical solution:

2. Vortex lattice method

Vortex lattice method

Superimpose a finite number of horseshoe vortex of different strength on the wing surface

n

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

P

n

Chapter 6

Three-Dimensional Incompressible Flow

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

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. r S

Volume flow is defined as the strength of source.

a point source is a point sink.

0

Three-Dimensional DoubletConsider 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 .l

The flow field produced by Eq. (6.9) is a three-dimensional doublet.

From Eq. (2.18) and Eq. (6.9)

The streamline of this velocity field are the same in all the planes. zr

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.

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

To find the stagnation points in the flow.

Two stagnation points on Z axis, with coordinates),( r

The impressible flow over a sphere of radius R

(flow-tangency condition)

On the surface of the sphere of radius R, the tangential velocity is

From Eq. (6.16),

Maximum tangential velocity for three-D flow is

Maximum tangential velocity for two-D flow is , V2

V2

3The maximum surface velocity on a sphere is less than that for a cylinder

Three-dimensional relieving effect

A general phenomenon for all types of three-dimensional flows

Two examples:

The pressure distribution on the surface of the sphere is

The pressure distribution on a cylinder is 2sin41

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.

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……….

Thank you