| Author | Message | |||
     Renato Pantuliani (cheng) New member Username: cheng Post Number: 1 Registered: 11-2007 |
I'm dealing with a simple laminar flow in a duct. I would like to evaluate the average temperature at each position along the z-axis. It is defined as Tb=r_Integral(2*r*vz*T)/r1^2*vz0 in which vz0 is the average velocity (a constant in my case). I tried to introduce a new variable in my problem, and the related equation is dr(Tb)-2/(r1^2*vz0)*r*vz*T=0. However, I get very large oscillation in the value of Tb. Is there a simple way to solve the problem? Here is the whole program: ============= TITLE 'Heat transfer Flow in a Tube' COORDINATES ycylinder('r','z') VARIABLES vr vz p T Tb DEFINITIONS L=1.3e-2 r1=5.0e-3 vz0=1e-4 visc=1e-3 dens=1e3 cond=0.6 Cp=4100 rcp=dens*Cp T0=300 Tw=350 natp=0 EQUATIONS vr: dr(p)- visc*[ 1/r*dr(r*dr(vr))- vr/r^2+ dzz(vr)]=0 vz: dz(p)- visc*[ 1/r*dr(r*dr(vz))+ dzz(vz)]=0 p: 1/r*dr( r*dr(p))+ dzz(p)- 1e4*visc/L^2* div_v=0 T: -cond*[ 1/r*dr(r*dr(T))+ dzz(T)]+ rcp*[ vr*dr( T)+ vz*dz( T)]=0 Tb: dr(Tb)-2/(r1^2*vz0)*r*vz*T=0 BOUNDARIES region 'domain' start 'outer' (0,-L) natural(vr)=0 value(vz)=vz0 value(T)=T0 natural(p)=natp value(Tb)=0 line to (r1,-L) { In } value(vr)=0 value(vz)=0 value(T)=Tw natural(p)=natp natural(Tb)=0 line to (r1,L) { Wall } value(vr)=0 natural(vz)=0 natural(T)=0 value(p)=0 natural(Tb)=0 line to (0,L) { Out } value(vr)=0 natural(vz)=0 natural(T)=0 natural(p)=0 value(Tb)=0 line to close { Axis } PLOTS elevation(Tb) from (r1,-L) to (r1,L) export file='Tb.tbl' END ================ | |||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 982 Registered: 06-2003 |
1. Your boundary conditions on TB are wrong. TB is not zero along the bottom boundary. This introduces a condition that cannot be met, so the trial solution oscillates strongly. 2. Introducing the equation dr(Tb) = Q*v*T does not simply define Tb, it imposes the constraint that v*T = dr(Tb)/Q. Nonlinear problems require a reasonable starting value to guarantee convergence, and Tb=0 is not a good starting estimate of Tb. There are several ways to deal with this issue: 1. Simply create two scripts. Exclude Tb from the first script, and TRANSFER the results to a file. Copy the first script to the second. Instead of solving for Vr,Vz,P and T in the second script, simply TRANSFER them in. Use only the Tb equation to solve for Tb. 2. Alternatively, you can use STAGES to eliminate the coupling to Tb from the first stage. Include it in the second stage, after the primary variables have established their proper values. The attached script shows this method. The parameter "switch" turns off the nonlinear coupling to Tb in the first stage. I have also added a small z-diffusion term in the Tb equation. Finite element equations are based on integrals over the volume surrounding each mesh node. If there is no coupling in one of the spatial dimensions, the integrals can support oscillatory solutions in this dimension. Since your Tb equation has no z-derivative terms, I have added a small one to couple the z dimension. Use a coefficient equal to the square of the distance over which oscillations should be smoothed. (See Tech Note "Smoothing Operators in PDE's") There are still some oscillations in the resulting Tb plot. This is due to insufficient mesh density to model the shape exactly. If you are using the Student version of FlexPDE, there is not much you can do about this. If you are using the professional version, try using RESOLVE TB.
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