Download presentation
Presentation is loading. Please wait.
1
Lecture Objectives: Newton-Raphson method
Discuss Airpak and other CFD software Boundary Conditions
2
Newton-Raphson method (example of Jacobean solver)
Faster convergence Used in many professional tools (MathCAD, EES, MatLab, Mathematica, etc) More complex for programming Requires linear solver Based on Taylor-Series Expansion You need first derivative for each function to create the Jacobean matrix Equations in the form where all side are on one side of equality sign Our simple example: X-Y/2= → X-Y/2+1=0 X2-Y= → X2-Y+3=0
3
Newton-Raphson method
Section 6.11 of handouts Our simple example: f1 = X-Y/2+1=0 f2 = X2-Y+3=0 Steps: 0) Find derivatives d(f1)/dX = , d(f1)/dY =-1/2 d(f2)/dX =2X , d(f2)/dY =-1 1) Initial guess: Y(0)=2, X(0)=2 2) Find f1(Y(0),X(0))=2-2/2+1=2 f2(Y(0),X(0))=22-2+3=5 3) Using derivatives and guess values find the Jacobean matrix 4) Solve the matrix using linear solver and find DX and DY 5) Find Y(1)=Y(0)+ DY, X(1)=X(0)+ DX, Repeat step (2) with Y(1) and X(1) … Follow the procedure till convergence Unknowns (correction Dxi) Jacobean matrix Function values for guessed variables
4
Surface boundaries wall functions
Wall surface Introduce velocity temperature and concentration Use wall functions to model the micro-flow in the vicinity of surface Using relatively large mesh (cell) size.
5
Surface boundaries wall functions
Course mesh distribution in the vicinity of surface Y Wall surface Velocity in the first cell will depend on the distance y.
6
Surface boundary conditions and log-wall functions
E is the integration constant and y* is a length scale Friction velocity u+=V/Vt y*=(n/Vt) y+=y/y* k- von Karman's constant The assumption of ‘constant shear stress’ is used here. Constants k = 0.41 and E = 8.43 fit well to a range of boundary layer flows. Surface cells Turbulent profile Laminar sub-layer
7
K-e turbulence model in boundary layer
Wall shear stress Eddy viscosity V Wall function for e Wall function for k
8
Modeling of Turbulent Viscosity in boundary layer
forced convection natural convection
9
Temperature and concentration gradient in boundary layer
Depend on velocity field Temperature q=h(Ts-Tair) Concentration F=hc(Cs-Cair/m) m=Dair/Ds m- segregation coefficient h = f(V) = f(k,e) Tair Ts Into source term of energy equation hC = f(V, material prop.) Cair Cs
Similar presentations
© 2025 SlidePlayer.com Inc.
All rights reserved.