Newton Linearization
 

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!

Re: Newton Linearization
 

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.

Re: Newton Linearization
 

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?

Re: Newton Linearization
 

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)

Re: Newton Linearization
 

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!

Re: Newton Linearization
 

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) ).