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Boundary condition: intermediate velocity for fractional time step method 

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June 19, 2015, 18:37 
Boundary condition: intermediate velocity for fractional time step method

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Rime
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Hi!
How to calculate u*=u_n+1 for the intermediate velocity u* 

June 20, 2015, 03:47 

#2 
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Filippo Maria Denaro
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are you talking about the intermediate BC.s in the fractional time step method? The literature is full of paper about this issue, starting from the hystorical paper of Kim & Moin on JCP you can find many many studies and proposals. However, u_n+1 is the physical BC you have to know, a zeroorder approximation simply set it as u*. 

June 20, 2015, 07:25 

#3 
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Thank u for the reply.
yes for fractional time step method 

June 20, 2015, 14:24 

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I use FSM with free pressure
boundary : 1)n.u*=n.u_n+1 u*=u_n+1 => u*=u_n+dt*u_t =u_n+dt*(u*grad_u+(1/rho)*(mu*laplace(u)+grad_pression+F) is correct?? (depent to density rho) 2) t.u*=t.(u_n+1+(dt*grad_Phi) 3) for pressure: laplace(phi)=(rho/dt)*div u^* boundary n.grad_phi=0 4) and to calculate u_n+1 u_n+1=u*(dt/rho)*grad_phi No boundary conditions, ie I compute the velocity for interior and boundary points. PS: I am beginner in CFD I need your help Thank you 

June 20, 2015, 18:27 

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Filippo Maria Denaro
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I think your process need a careful revision...
1) Are you using a second order implicit time integration in the momentum equation? 2) The pressure equation is nothing else but the continuity equation in which the Hodge decomposition is substituded. 3) You need to take care of the product Div Grad, I suggest not to write it as Laplacian for a correct setting of the Neuman BC.s 4) What books/papers are you studying for the FTS method? 

June 20, 2015, 18:48 

#6 
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for time discretisation I use Adams Bashforth and Crank Nicolson scheme.
[1] Application of a FractionalStep Method to Incompressible NavierStokes Equations J. KIM AND P. MOIN [2] David L. Brown, Ricardo Cortez,y and Michael L. Minionz Accurate Projection Methods for the Incompressible Navier–Stokes Equations [3]J.L. Guermond, P. Minev c, Jie Shen An overview of projection methods for incompressible flows and other paper for spatial discretisation (Radial basis function). 

June 21, 2015, 03:21 

#7  
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Filippo Maria Denaro
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Quote:
Ok, therefore you solve the implicit scheme for v* using the BC.s v* = v_n+1 + dt Grad phi_n, right? Then you solve the elliptic equation Dic Grad phi_n+1 = Div q, wherein q is the source term. What's wrong with your results? 

June 21, 2015, 07:01 

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Thank you for the reply.
my result diverge and when I use steady boundary I get a false solution. 

June 21, 2015, 07:10 

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June 21, 2015, 07:16 

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I get div u around 0.04


June 21, 2015, 07:29 

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June 21, 2015, 07:42 

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June 21, 2015, 08:31 

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June 21, 2015, 08:36 

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June 21, 2015, 08:43 

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June 21, 2015, 09:19 

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Quote:
n.u^* = n.u_n+1 + n.grad_phi so n.grad_phi= n.(u^*  u_n+1^*) I look that in my code I consider that the velocity in the boundary equal to zero (therefore that is false) 

June 21, 2015, 11:58 

#17  
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Quote:
As you see, if you set in n.u^* = n.u_n+1 + n.grad_phi the BC.s : n.grad_phi=0 , you MUST set also n.u^* = n.u_n+1 therfore in computing the source term Div u^* you have to use the physical BC.s n.u_n+1. One you have fulfilled such constraints, you have the compatibility condition satisfied and the Poisson equation will converge. Finally, if you use a staggered secondo order discretization you should have the continuity equation satisfied up to machine precision 

June 21, 2015, 12:34 

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Quote:
n.u^* = n.u_n+1 <=> n.u^* = n.u_bc (bc=boundary condition) because I can't see the difference between u_n+1 and steady u_bc 

June 21, 2015, 12:37 

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June 21, 2015, 12:44 

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