Lecture Objectives: Simple Algorithm vs. Direct Solvers Discuss HW 3 Boundary Conditions

SIMPLE algorithm Guess pressure field: P*W, P*P, P*E, P*N , P*S, P*H, P*L 1) For this pressure field solve system of equations: x: y: ……………….. ……………….. z: Solution is: 2) The pressure and velocity correction P = P* + P’ P’ – pressure correction For all nodes E,W,N,S,… V = V* + V’ V’ – velocity correction Substitute P=P* + P’ into momentum equations (simplify equation) and obtain V’=f(P’) V = V* + f(P’) 3) Substitute V = V* + f(P’) into continuity equation solve P’ and then V 4) Solve T , k , e equations

SIMPLE algorithm start Guess p* p=p* Step1: solve V* from momentum equations Step2: introduce correction P’ and express V = V* + f(P’) Step3: substitute V into continuity equation solve P’ and then V Step4: Solve T , k , e equations no Converged (residual check) yes end

Other methods SIMPLER SIMPLEC variation of SIMPLE PISO COUPLED - use Jacobeans of nonlinear velocity functions to form linear matrix ( and avoid iteration )

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=-1 → X-Y/2+1=0 X2-Y=-3 → X2-Y+3=0

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 =1 , 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

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.

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.

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

K-e turbulence model in boundary layer Wall shear stress Eddy viscosity V Wall function for e Wall function for k

Modeling of Turbulent Viscosity in boundary layer forced convection natural convection

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