英文论文代写 Thermal Counterflow Past A Cylinder
Finding the Stationary Positions of Point Vortices
One of the project’s objectives which are related to the experiment was to find if there exist stationary locations of the vortex – antivortex pair, both downstream and upstream of the disk. The stationary points can only be complex conjugates on a complex plane, in the upper, and in the lower half – plane. The stationary points are shown in the figure 3.1 below:
Figure 3.1: Point vortices around the disk. At the left are is the vortex – antivortex pair downstream and at the right is the vortex – antivortex upstream of the disk.
To find the stationary point positions the following cubic equation was used:
λ: is the non – dimensional circulation
This equation was solved with 3 ways. The first method was Cardano’s method using excel, also was solved using Matlab with root command and finally with fsolve command. The two m – files are shown in Appendix 1. The solution of this equation has given the values of and then these values where used in the following equation to find the values: (3.2) where
The range of λs that was used was from 0 to 5 with 0.1 intervals. So by the solution of the cubic equation the following graph was produced:
Figure 3.2: Coordinates red dashed lines and blue solid line in the upper half – plane as functions of the non – dimensional circulation λ.
这是solved方程与3种。cardano’ method was the first method was also Excel的使用，solved using MATLAB命令与fsolve与根和最后的命令。两个are shown the M在附录1。the solution of this has given the values of方程，然后这些值在used in the values to find the following方程式（3.2）：在
λs that was used of the范围是从0到5 intervals和0.1。我知道by the solution of the following the三次方程为：图的制
Analyse the motion of vortex – antivortex pairs of two vortices downstream of the cylinder
所有的M文件被称为第一stream1down。M（附录1）建立常微分方程如何涡动。另一个M文件名为motion1down。M（附录1）并制定所需的初始条件求解常微分方程。在motion1down固定点。我是从stationary1down。M（附录1）和λ值（Q m文件）所需的stream1down。M和stationary1down。我是从lambdainput M（附录1）。这是一例图所示，这三个图显示在上半平面X1与Y1的涡和时间依赖的轨迹X1涡点的轨迹（T）和Y1（t）为λ= 1，1.5，1.8，2和3
从时间依赖图4.1.2和4.1.3，据观察，最初位于气缸的旋涡涡点后，在一段时间保持接近其初始位置，随着非流通λ维。例如λ= 1的涡点保持接近其初始位置约5非时间的单位和λ= 2约35。当λ= 3点涡仍接近其初始位置超过70没有时间单
The next objective of the project was to see how the vortices move when particular stationary points chosen, the stationary points are depending on the non – dimensional circulation λ from the range that was mentioned before in section 3. To do this the Lagrangian Equations of Motion of Point Vortices in the Inviscid Flow around the Disk were used. The equations are shown below:
The equation 4.2 is for the upper half – plane vortex z1 and the equation 4.3 is for the lower half – plane vortex z2. Each of the vortices has two equations, one for real and one for imaginary solution, so the total equations are four. To analyse the motion, these four equations were solved with some initial conditions using Matlab.
First of all an m – file was made called Stream1down.m (Appendix 1) to set the ode equations to find how the vortices are move. Another m – file called Motion1down.m (Appendix 1) was made to set the required initial conditions to solve the ode equations. The stationary points in Motion1down.m are taken from the stationary1down.m (Appendix 1) and the λ value (q in m – files) that is required for Stream1down.m and in stationary1down.m is taken from the lambdainput.m (Appendix 1). One example of the graphs that were made is shown below, these three graphs are shown the trajectories for the vortex points of the vortex in the upper half-plane x1 vs y1 and the time – dependent trajectories x1(t) and y1(t) for λ = 1, 1.5, 1.8, 2 and 3.
Figure4.1: Trajectories of the vortex points x1 vs y1, time – dependent coordinates x1(t) and y1(t).
From the time – depended figures 4.1.2 and 4.1.3, it is 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. For example for λ = 1 the vortex point stays close to its initial location for approximately 5 non – dimensional units of time and for λ = 2 for approximately 35. When λ = 3 the vortex point remain close to its initial location for more than 70 no – dimensional units of time.