Ph.D Thesis
Ph.D Student Ishay Mark Incompressible Flow in Conical Tubes and around Conical Bodies with Elliptical Cross-Section Department of Mechanical Engineering Professor Emeritus David Degani Research Professor E Daniel Weihs

Abstract

An analytical solution of incompressible potential and boundary layer flow-fields about cone-shaped forebodies and inside conical tubes with elliptical cross-section at zero angle of attack is obtained.

We reduce the three-dimensional potential flow equations to generalized axisymmetric equations in an elliptic-conical coordinate system. This further allows the use of a Mangler transformation’s generalization to obtain expressions for the boundary layer velocity field, for a flow about an elliptic cone. The thickness of the boundary layer inside a conical tube is obtained by using an integral method, guessing the flow profile in the boundary layer.

The solutions are functions of the cone spatial vertex angle and the cross-section ellipticity ratio. The friction and drag forces on such cones are obtained. We show that for a flow about an elliptic cone there exists a range of ellipticity ratios for which the drag is less than the drag on circular cones of equal surface area. Due to the unique coordinate system, the solutions are actually symmetric about both vertical and horizontal axes, and therefore the velocities, pressure, and derived properties from them are given for each quadrant separately.

Using this property we present several applications for symmetric and asymmetric elliptical cones, while other lifting bodies can be designed by using halves of elliptic cones, either as the upper or lower surfaces. As examples of such bodies, which mostly fly at supersonic and hypersonic speeds, we show several waveriders configurations. Lift and drag forces on such lifting bodies are calculated, showing that except for engineering constraints, a friction force will influence the choice of the lifting body ellipticity ratio.

In addition, we present another interesting application of the flow about an elliptic cone, which is the train nose modeled by using a half elliptic cone as the upper surface of the train bow. Due to lack of analytical solutions on the train, we extended our model to the whole generic train, showing that despite of simplicity of the model train geometry, a good correlation is obtained with experimental data from the literature.