FLOW PAST A CYLINDER:

I looked at four blockage ratio's using femlab:

B25	25% blockage ratio.  This was taken from femlab's demo
B50	50% blockage
B70	70% blockage
B90	90% blockage

I have simulations of flow at Reynolds numbers near Hoph bif pts at
all these blockage ratios.  I also have some similar runs for the heated
cylinder problem.

_____________________
I started with the B25 case, first trying to replicate the results
of the demo.  The demo was dimensional, and I gradually changed
parameters.

Demo values for  
Flow past a cylinder

I'll include the weak boundary info in ()'s
and time dependent values in []'s

Draw 
	cylinder	0 < x < 2.2 
 				0 < y < .41
	circle   dia = 0.1, center (0.2,0.2)

Constants	rho0 = 1 	
				eta0 = 1e-3
				Umax = 1.5

Boundary		Inlet	u = Umax*4*s*(1-s) 
						v = 0 	  % s is parameterization of the edge
				Outlet p = 0 
				No slip (u,v) = 0 on rest
				(weak boundary active only on cylinder:
												  lmx = u lmy = v)

Equation		[rho u'] - div(eta (grad(u)+grad(u)^T) + ...
                         div(rho u grad(u)) + grad(p) = F; div(u) = 0;
Subdomain	rho = rho0
				eta = eta0
				F = 0

Mesh			Max edge = 0.03
				Growth = 1.2
				Curvature = 0.1

Solve			Stationary nonlinear 
				[Time dependent  
					Time stepping
						output times = [0:0.2:3.5 3.52:0.02:5]
						Algorithm = fldaspk
						Tolerances
							rel = 0.01
 							abs = u 1e-4 v 1e-4 p Inf ...
                                  (lmx Inf lmy Inf) 
				]

Questions:
   Dimensional.  
		Easy to non-dimensionalize?
   Blockage ratio.  
		What effects does higher blockage ratio have?
   Weak Boundary conditions.  
		Are these necessary?
	The mesh/domain isn't symmetric. 
		Can I force it to be symmetric?
	Boundary conditions.  
		Introduce asymmetry here instead of domain?
	Mesh  
		What effect does mesh have on soln?
	Solver
		How to choose optimal solver?
		How to set tolerances?
		How to set appropriate time scale? 

___________________________________________________

Modifications to demo.

Non-dimensional.
	Easy to non-dimensionalize?
			Let rho = Reynolds #;
			eta = 1;   (or 1-gamma0*T in heated cylinder)
			
Blockage ratio
What effect does blockage ratio have?
			in B25 case, both velocity gradient and pressure gradient
			are about order 1.  In B90 case, velocity gradient ~ 1e+3
			pressure gradient ~ 1e+5.
			This makes interplay between rel tol and absolute tol more
			important. I always used rel tol = 1e-2 and had to modify
			abs tol for the velocity field.  Solving the the steady
			state can give an idea for reasonable values to use for abs
         tol.   
																						
Weak boundary conditions.  
	Could weak boundary conditions be eliminated?

			When I first removed the conditions, the periodic soln
			failed to appear.  Later I found that removing boundary
			conditions resets the solver tolerances.  When I re-ran
			the experiment with original relative tolerances, vortex
			street was produced.

Perturbations
	What types of perturbation can be used?

	Domain  
			I first tried to use a asymmetric domain to initialize a
			asymmetric flow, then revert to symmetric domain.  This
			had drawbacks (at least in GUI).  The mesh had to be 
         reinitialized, mesh parameters had to be reset.  I then 
         started to think about perturbing inflow, outflow and 
         boundary conditions.

         I figured out how control domain with scripts - and it
         seems to work fairly well.   

	Inflow	
			I tried time dependent inflow perturbation (inflow +
		  	gaussian in time perturbation). A large perturbation is 
			needed, and the time scale is hard to establish. 

   Boundary 
			I added a boundary component just behind the cylinder, and 
			added flow field (u,v) perturbations and pressure (p)
			perturbations. 
         Flow perturbations were hard to control, but more dramatic 
			then inlet perturbations.  Time scale was also hard to 
			establish.  I needed to be careful with rel tol when using
			flow control.  Restarting flow perturbations usually
			worked, however.
			Pressure perturbations were easier to control from GUI.  I would
			solve steady state, look at pressure field, and choose a
			reasonable constant perturbation to solve another steady
			state problem.  In practice, I could often set p=0.
			I'd then restart the steady state as time dependent.
			Perturbing pressure at outlet also work, and was easier 
			to get to converge in experiments with large pressure
			gradients. I'd used p = eps*s on the outlet boundary, and
			play around with the values of eps, which was easy.

Mesh 	    
	How important are mesh parameters?
			Mesh needed to capture enough features to be
			reliable.  The femlab mesh is asymmetric.  The edge nodes
			can be forced to be symmetric (or anti-symmetric) in 
			pairs, but I don't think this makes the entire grid 
			symmetric.  Refining the mesh was enough in several cases
			to induce periodic solutions.  Coarsening mesh eliminated	
			periodic solutions in several cases. I think that the
			mesh should be visualized before each solve.  
 
Solver.  
	How to choose optimal solver?
			I generally used fldaspk.  For new problems, I suggest
			short time solves comparisons.  There is a listing in 
			the manual for the solvers.  Running as scripts through
         matlab lets you use tic/toc to time various solves. 

	How to set tolerances?
			These were sometimes hard to set.  For NS, both rel and
			abs tolerances are needed. The relative tolerance is set
			for all variables, while abs tol is set per variable. The
			time dependent variables (u,v) needed a much higher 
			tolerance than pressure (p).  I still don't know what 
			the rel tol really measures. I used a estimate for max
         flow to set abs tol in my experiments, and this seems to work.
				
	How to set appropriate time scale?
			The non-dim problem lets me figure the time scale easily.
			I did this in the add/modify constants section. Also, the
         scipts allow me to restart a computation easily, and so this
         wasn't terribly important.