Immersed Boundary Method Assignment

Transcription

1 Immersed Boundary Method Assignment This assignment is modification of the immersed boundary code that I downloaded from the course website. I constructed new IB objects with changing ibinit.m and afibdriver.m. The code runs with following time steps: 1 N=64, h =,dt =0.2 h, Nt =400, n where N is N N grid, h is mesh space, dt is time step, and Nt is number of time steps to be taken. The result of the time steps ran stable and IB objects moves around the forces. Specifically, I modified two different IB objects, which named demo number1 and demo number2. For demo1, I constructed large ellipse that fits in the domain and put a small circle inside the ellipse but not at the center of the ellipse. The ellipse computed as x =a cos 2 π θ 0.5,, where a=0.48, b=0.2. y =b sin 2 π θ 0.5 Inside the circle computed as 2 π r, where r =0.1,center=0.58. The circle moving around the ellipse along with the forces. I also defined a spatially periodic swirling initial flow and applied an extra background force density to the continue to drive the swirling flow using given matlab code. The speed of the forces increase when maximum speed A is increased. However, the IB is unstable when maximum speed A is up to 4. The process illustrated in following images and parameters listed in each tables. Figure 1. Immerse Boundary solution for periodic swirling flow A = 1 from Demo1.

2 Figure 2. Immerse Boundary solution for periodic swirling flow A = 1 from Demo1. Figure 3. Immerse Boundary solution for periodic swirling flow A = 1 from Demo1.

3 Figure 4. Immerse Boundary solution for periodic swirling flow A = 1 from Demo1. Figure 5. Immerse Boundary solution for periodic swirling flow A = 1 from Demo1.

4 Figure 1 through 5. Comparison of the IB solution to the problem at a various different times for periodic swirling flow A = 1. The parameters listed in Table 1. Table 1. Parameters used for the solution shown in Figure 1 through 5. Parameter Value Method Immersed Boundary Method Interval 0 x 1, 0 y 1 ellipse x =a cos 2 π θ 0.5,, where a=0.48, b=0.2. y =b sin 2 π θ 0.5 circle 2 π r, where r =0.1,center=0.58. N ( N N grid) 64 h(mesh space) 1 N dt(time step) 0.2 h Nt(number of time steps to be taken) 400 Following five images tested by IB solution to the problem at a various different times for periodic swirling flow A = 3. Figure 6. Immerse Boundary solution for periodic swirling flow A = 1 from Demo1.

5 Figure 7. Immerse Boundary solution for periodic swirling flow A = 1 from Demo1. Figure 8. Immerse Boundary solution for periodic swirling flow A = 1 from Demo1.

6 Figure 9. Immerse Boundary solution for periodic swirling flow A = 1 from Demo1. Figure 10. Immerse Boundary solution for periodic swirling flow A = 1 from Demo1.

7 Figure 6 through 10. Comparison of the IB solution to the problem at a various different times for periodic swirling flow A = 3. The parameters listed in Table 2. Table 2. Parameters used for the solution shown in Figure 6 through 10. Parameter Value Method Immersed Boundary Method Interval 0 x 1, 0 y 1 ellipse x =a cos 2 π θ 0.5,, where a=0.48, b=0.2. y =b sin 2 π θ 0.5 circle 2 π r, where r =0.1,center=0.58. N ( N N grid) 64 h(mesh space) 1 N dt(time step) 0.2 h Nt(number of time steps to be taken) 400

8 For demo2, I also constructed large ellipse that fits in the domain and put three circles, which one of the small circle is inside the circle inside the ellipse. The new IB objects modified from ibinit.m and afibdriver.m in original matlab code. The code runs with same time steps as demo 1. After execute the program, the IB methods ran stable with various time steps and also IB objects moves around the forces. In specific, the large ellipse computed as x =a cos 2 π θ 0.5,, where a=0.48, b=0.2. y =b sin 2 π θ 0.5 Inside the circle1 computed as 2 π r, where r =0.1, center =0.57. Circle2 computed as r =0.05,center =0.4. Circle3 computed as r =0.05,center =0.55. Circle3 is located in inside of circle1. Inside two circles moving around the ellipse along with the forces. However, compare to original given demo4, inside two circles pushed away with each other so the external forces to each IB point moving opposite direction with each other as given amplitude is The speed of the forces also increase when maximum speed A is increased but the IB is unstable when maximum speed A is up to 4. The process illustrated in following images and parameters listed in each tables. Figure 11. Immersed Boundary solution for initial periodic swirling flow.

9 Figure 12. Immersed Boundary solution for periodic swirling flow A = 1 Demo 2. Figure 13. Immersed Boundary solution for periodic swirling flow A = 1 Demo 2.

10 Figure 11 through 13. Comparison of the IB solution to the problem at a various different times for periodic swirling flow A = 1. The parameters listed in Table 3. Table 3. Parameters used for the solution shown in Figure 11 through 13. Parameter Value Method Immersed Boundary Method Interval 0 x 1, 0 y 1 ellipse x =a cos 2 π θ 0.5,, where a=0.48, b=0.2. y =b sin 2 π θ 0.5 circle1 2 π r, where r =0.1,center =0.57. circle2 2 π r, where r =0.05,center =0.4. circle3 2 π r, where r =0.05,center =0.55. N ( N N grid) 64 h(mesh space) 1 N dt(time step) 0.2 h Nt(number of time steps to be taken) 400 Following six images tested by IB solution to the problem at a various different times for periodic swirling flow A = 3. Figure 14. Immersed Boundary solution for periodic swirling flow A = 3 Demo 2.

11 Figure 15. Immersed Boundary solution for periodic swirling flow A = 3 Demo 2. Figure 16. Immersed Boundary solution for periodic swirling flow A = 3 Demo 2.

12 Figure 17. Immersed Boundary solution for periodic swirling flow A = 3 Demo 2. Figure 18. Immersed Boundary solution for periodic swirling flow A = 3 Demo 2.

13 Figure 19. Immersed Boundary solution for periodic swirling flow A = 3 Demo 2. Figure 14 through 19. Comparison of the IB solution to the problem at a various different times for periodic swirling flow A = 3. The parameters listed in Table 4. Table 4. Parameters used for the solution shown in Figure 14 through 19. Parameter Value Method Immersed Boundary Method Interval 0 x 1, 0 y 1 ellipse x =a cos 2 π θ 0.5,, where a=0.48, b=0.2. y =b sin 2 π θ 0.5 circle1 2 π r, where r =0.1,center =0.57. circle2 2 π r, where r =0.05,center =0.4. circle3 2 π r, where r =0.05,center =0.55. N ( N N grid) 64 h(mesh space) 1/N dt(time step) Nt(number of time steps to be taken) 0.2 h 400