Immersed Boundary Formulation
 

Hi, I am writing my own code and I want to use immersed boundary method for simulating flow around moving bodies. I have a question regarding the formulation.

According to formulation of Mohd.Yusof we have

u(n+1)(i,j)-u(n)(i,j)/dt=Rhs(n)(i,j)+f(n)(i,j)

where n is the time advanceme and (i,j) are the x and y co-ordinates

If the boudnary is conciding with a grid point and has velocity V we have

-->f(n)=V-u(n)/dt-Rhs(n). --> u(n+1)=u(n)+Rhs(n)*dt+f(n)*dt --> u(n+1)=u(n)+Rhs(n)+((V-u(n)/dt)-Rhs(n))*dt --> u(n+1)=V

If stationary and no -slip is used V=0 --> u(n+1)=0

am I correct to this point??

If the immersed point is not coinciding with the grid point use linear interpolation and instead of V we have velocity at a particular point for example at u(n)(i,j) to be u(n)(i,j+1)*fac (where the fac is the factor used to multiply that can be computed using linear interpolation so that zero falls in between them)i.e.

u(l)(i,j)=fac*u(n)(i,j+1)

using same procedure as above we have

--> u(n+1)(i,j)=fac*u(n)(i,j+1)

am I correct here ???

Is my formulation ok?? Thanks in advance