Incompressible Flow around a Sphere
The three-dimensional point-source flow described herein can be extended to the representation of a three-dimensional doublet by superposing a source-sink pair similarly to the analysis detailed for two-dimensional, planar, incompressible flow in Chapter 4. Superposing uniform flow and a doublet at the origin with its axis parallel to the freestream leads to the solution for the flow around a sphere (see Anderson, 1991 and Karamcheti, 1967). The radius of the sphere is given by:
where о is the doublet strength.
In Chapter 4, a two-dimensional, planar, flow field generated by the superposition of a doublet and a uniform flow yielded the flow over a right-circular cylinder placed normal to the flow. The superposition of a three-dimensional doublet and a uniform flow gives the flow around a sphere. Both the radius of the cylinder and the radius of the sphere depend on the doublet strength. However, the radius of the cylinder varies as the doublet strength to the one-half power, whereas the radius of the sphere varies as the doublet strength to the one-third power.
According to the inviscid-flow model, the flow field about the right cylinder with its axis placed normal to the flow and also about the sphere is symmetrical fore and aft, meaning that there is a stagnation point on both the upstream and downstream surfaces of the bodies along the flow axis of symmetry. Boundary-layer separation profoundly influences the flow field on the downstream side of both the cylinder and the sphere in the case of a real (i. e., viscous) flow situation.
The maximum velocity on the surface of both of these bodies occurs 90° in polar angle away from the two stagnation points (i. e., at the top and bottom). The results are, for the same radius:
cylinder: Vmax = 2.0V». sphere: Vmax = 1.5V».
Thus, the freestream does not accelerate to as high a velocity (or to as low a static pressure) around the surface of the sphere as it does around the surface of the right-circular cylinder of the same radius. This is evidence of a three-dimensional relief effect. The two-dimensional flow approaching a right-circular cylinder placed normal to the flow must split and pass either above or below the body. In the threedimensional case, the flow passes around the sphere. Thus, the sphere causes less of a disturbance in a flow than the cylinder if both bodies are of the same radius. This same relief effect is observed when comparing supersonic flow around a twodimensional wedge and an axisymmetric cone as developed in tests on compressible gas dynamics.