# Newton Linearization

 Register Blogs Members List Search Today's Posts Mark Forums Read

 September 26, 2002, 10:48 Newton Linearization #1 seabank Guest   Posts: n/a Sponsored Links Hello, everyone: I am solving the boundary layer equations with Newton linearization method. But I meet one problem, i.e. how to set the guessed values. The following is the specific question. u(i-1,j),u(i,j) and u(i+1,j)are the unknown U-velocity I want to get. But in x-moment difference equation, there are five terms, i.e. ug(i-2,j),ug(i-1,j),ug(i,j),ug(i+1,j), and ug(i+2,j), which mean the guessed U-velocity value and I should set value to them. How to set value to these five terms? Thanks for your help very much!

 September 27, 2002, 05:16 Re: Newton Linearization #2 Tom Guest   Posts: n/a I'm mot too sure I can see your problem. You are solving uu_x + vu_y = -p'(x) + u_yy, u_x + v_y =0 subject to u=v =0 on y=0 and u-> us(x) as y->infinity ( with ' = d/dx and p = -.5*(us)^2 ) with some compatable initial condition at x=0; i.e u = ui(y), v=vi(y) at x=0 in finite difference form we have momentum (j=1,N) 0.5*( u(i,j)+u(i-1,j) )*( u(i,j)-u(i-1,j)/h +0.125*( v(i,j)+v(i-1,j) )*( u(i,j+1)-u(i,j-1) +u(i-1,j+1)-u(i-1,j-1) )/l = -p(i) + 0.5*( u(i,j+1)-2*u(i,j)+u(i,j-1) +u(i-1,j+1)-2*u(i-1,j)+u(i-1,j-1) )/l^2 continuity 0.25*( u(i,j)-u(i-1,j)+u(i,j-1)-u(i-1,j-1) )/h +0.5*( v(i,j)-v(i,j-1)+v(i-1,j)-v(i-1,j-1) )/l =0 where h is the x spacing and l the y spacing and the y boundary condtions go in j=0 and j=N+1. These are 2N equations in 2N unknowns ( u(i,j),v(i,j) j=1,..,N) - the u(i-1,j) and v(i-1,j) are known from the previous x step and ultimately from the initial condition at x=0. Now since u(i-1,j) and v(i-1,j) are known you can set u(i,j) = u(i-1,j) & v(i,j)=v(i-1,j) as your initial guess for the Newton iteration. (You should be able to get away with setting it to be almost anything if you're using Newton iterations!) Tom.

 September 27, 2002, 13:24 Re: Newton Linearization #3 Seabank Guest   Posts: n/a Tom, thank you for your help very much! I will try the method that you told me. But I have another question about your x-momentum difference equation. That is the terms underlined with "~~~~~". 0.5*( u(i,j)+u(i-1,j) )*( u(i,j)-u(i-1,j)/h ~~~~~~ ~~~~~~~ From the above form, we know uu_x term is not linearized. What will you do? So I try to linearize it with Newton method, lik that F(u,v)=F(ug,vg)+(dF/du)(u-ug)+(dF/dv)(v-vg). After I simplify x-moment equation, I meet my problmes in my previous post. How to set the guessed values becomes the biggest problme. What is the good way?

 September 28, 2002, 09:46 Re: Newton Linearization #4 Tom Guest   Posts: n/a Basically what I'd do for this problem is full Newton iteration - since the equations are only quadratic you can calculate the Jacobian explicitly (it's a five diagonal matrix = tridiagonal of 2x2 matrices). Now because the boundary-layer equations are parabolic in the downstream direction I must supply values of u and v at some initial position - so I know u and v at i=0 say. If I now set u,v at i=1 to these values I've initialized the iteration scheme (similarly at a later station I can initialize using u,v at i-1). One important point is your initial condition must be consistent with the equations. For example if you are calculating flow past an aerofoil the equations should be initiated at the stagnation point using the Falkner-Skan stagnation point solution (which you also need to find numerically!) hope this helps Tom. P.S. It's probably worth looking at H.B. Kellers review paper in the Ann. Rev. Fluid Mech. (early 1970's)

 September 28, 2002, 11:34 Re: Newton Linearization #5 Seabank Guest   Posts: n/a Hi, Tom, thank you very much for your help. I also find that initial condition is very important for my current problem. It is really a good idea to get the initial value from Falkner-Skan stagnation point solution. Best wishes to you!

 September 30, 2002, 04:46 Re: Newton Linearization #6 Tom Guest   Posts: n/a I'd like to take credit for this but using Falkner-Skan similarity solutions to initialize the boundary-layer equations is how I was taught to do it. The reason you need to do it this way is the following:- near the stagnation point, which we'll place at x=0, the slip velocity has the Taylor expansion Us = ax + ... where a is a positive constant. This gives the pressure gradient as -p' = a^2 x + ... This then suggests looking for a solution of the form u = xu_1 + ..., v = v_1 + ... where u_1 and v_1 are functions of y and must be the Falkner-Skan stagnation point solution; i.e. you are not permitted to arbitrarily pick your initial condition. The reason for this behaviour stems from the fact that the steady state Navier-Stokes equations are elliptic while the boundary-layer equations are parabolic; the NS equations say that there is upstream influence and the bl equations say there is no upstream influence - this is the reason for the separation singularity. Tom. P.S. for other types slip-velocity you can use one of the other Falkner-Skan solutions; i.e. if you have a sharp leading edge use the Blasius solution and solve the equations in Blasius coordinates! (x and eta=y/sqrt(x) ).

 Thread Tools Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is OffTrackbacks are On Pingbacks are On Refbacks are On Forum Rules

 Similar Threads Thread Thread Starter Forum Replies Last Post kus CFX 9 April 21, 2013 01:54 lost.identity Main CFD Forum 7 May 8, 2010 11:07 Lixian CFX 3 November 27, 2008 08:57 Mehmood Main CFD Forum 1 May 22, 2008 07:17 Rebecca FLUENT 2 June 20, 2007 14:23