Lecture Objectives: Simple algorithm Boundary conditions

Navier Stokes Equations In order to use linear equation solver we need to solve two problems: 1)find velocities that constitute in advection coefficients 2) link pressure field with continuity equation This velocities that constitute advection coefficients: F=  V Pressure is in momentum equations which already has one unknown Continuity equation Momentum x Momentum y Momentum z

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 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 P E W xx xx xx Ae Aw Aw=Ae=A side We have two additional equations for y and x directions

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: Solution is: x: y: z: ……………….. P = P* + P’ 2) The pressure and velocity correction P’ – pressure correction V = V* + f(P’) For all nodes E,W,N,S,… V’ – velocity correction Substitute P=P* + P’ into momentum equations (simplify equation) and obtain 3) Substitute V = V* + f(P’) into continuity equation solve P’ and then V V’=f(P’) V = V* + V’ 4) Solve T, k,  equations

SIMPLE algorithm 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 Guess p* start end Converged (residual check) yes no p=p*

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

Surface boundaries wall functions Wall surface Use wall functions to model the micro-flow in the vicinity of surface Using relatively large mesh (cell) size. Introduce velocity temperature and concentration

Surface boundaries wall functions Wall surface Velocity in the first cell will depend on the distance y. Course mesh distribution in the vicinity of surface Y

Surface boundary conditions and log-wall functions E is the integration constant and y * is a length scale 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. Laminar sub-layer Turbulent profile Surface cells y*=( / V t )  - von Karman's constant Friction velocity u + =V / V t y + =y/y*

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

Modeling of Turbulent Viscosity in boundary layer natural convection forced convection

Temperature and concentration gradient in boundary layer Depend on velocity field Temperature q=h(T s -T air ) Concentration F=h c (C s -C air /m) m=D air /D s m- segregation coefficient Tair Ts Into source term of energy equation h = f(V) = f(k,  ) Cair Cs h C = f(V, material prop.)