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**Lecture Objectives:**Simple algorithm Boundary conditions**Navier Stokes Equations**Continuity equation This velocities that constitute advection coefficients: F=rV Momentum x Momentum y Momentum z Pressure is in momentum equations which already has one unknown • In order to use linear equation solver we need to solve two problems: • find velocities that constitute in advection coefficients • 2) link pressure field with continuity equation**Pressure and velocities in NS equations**How to find velocities that constitute advection coefficients? For the first step use Initial guess And for next iterative steps use the values from previous iteration**Pressure and velocities in NS equations**How to link pressure field with continuity equation? SIMPLE (Semi-Implicit Method for Pressure-Linked Equations ) algorithm Dx Dx P E W Dx Ae Aw Aw=Ae=Aside We have two additional equations for y and x directions The momentum equations can be solved only when the pressure field is given or is somehow estimated. Use * for estimated pressure and the corresponding velocities**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 )**Surface boundarieswall 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 boundarieswall 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 modelin boundary layer**Wall shear stress Eddy viscosity V Wall function for e Wall function for k**Modeling of Turbulent Viscosityin 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