# 英文论文代写 Thermal Counterflow Past A Cylinder

## Analyse the motion of vortex – antivortex pairs of four vortices downstream of the cylinder

The next task of the project was to see how the vortices move when there are two vortex – antivortex pairs this time, that means two vortices at the upper half plane and two vortices at the lower half -plane as shown in figure 5.1 below:

Figure 5.1: Point vortices in the potential flow around the disk, two vortex – antivortex pairs downstream (at rear) of the disk.

At this case z11 and z12 vortices have negative circulation (clockwise) and z21 and z22 vortices have positive circulation (anticlockwise) and now the equations are eight in total (two for each vortex). The stationary points are the same as in the case in section 3 and the m – files in Matlab were made with the same idea as before and can be seen in Appendix 1. One example of the graphs when λ = 1 is shown below in figure 5.2

Figure 5.2: 1: Trajectories of the vortex points for vortices z11, z12, z21 and z22. 2: Trajectories of z11 and z12. 3: Trajectories of z21 and z22.

Again at this case the from the time – depended figures that are not shown here but will be shown in the final report it was observed that for vortices initially located at the rear of the cylinder, the period of time during the vortex points remain close to their initial locations increases with the non – dimensional circulation λ as in the case with two vortices. Another observation from figures 5.2 is that the trajectories now are not just a line and moving but are making a spiral motion, rotating and moving simultaneously.

## Analyse the motion of vortex – antivortex pairs of four vortices downstream of the cylinder when there is no other flow

This objective was to see the motion of vortices when other flows are removed and leave just one pair of vortices of the same polarity. This can be done by removing some terms from the Lagrangian Motion Equations. The first attempt was to remove the term from equation 4.2 for example. To do this this term was removed from the Strem2down.m for all the equations and so one example of the graphs for λ = 1 that was made is shown below in figure 6.1.

Figure 6.1: NO FLOW case 1. Trajectories of the vortex points for vortices z11, z12, z21 and z22. 2: Trajectories of z11 and z12. 3: Trajectories of z21 and z22.

In this case it is expected to see the trajectories of the vortices to propagate to a straight line because the vortices in the upper half – pane are negative and the vortices in the lower half – plane are positive and they do not oppose each other. In figure 6.1 this thing is observed but not at the beginning. To see the behaviour of the vortices when there is no flow correct another term was removed from the Lagrangian equations, this term was of the equation 4.2 for example. The Stream2down.m was modified and both terms were removed and renamed to Stream2downnf2.m (Appendix 1). Motion m – file and stationarypoints m – file are the same as before with small changes. Below an example graph is presented about the no flow case for λ = 1 in figure 6.2.

Figure 6.1: NO FLOW case 2. Trajectories of the vortex points for vortices z11, z12, z21 and z22. 2: Trajectories of z11 and z12. 3: Trajectories of z21 and z22.

Now from figure 6.1 it can be seen clearly what was expected. The vortices motion propagates to straight line (spiral motion on a straight line) because the upper vortices are negative and the lower vortices are positive and do not oppose each other. Since the vortices on the upper half – plane have the same signs (negative) they rotate around each other and on the lower half – plane the vortices also have the same signs (positive) they rotate around each other.