![Direct Scheme
• Discretize u, v momentum equations and
continuity at all cells
• Assemble a big matrix: Ax = b
• Solution vector is:
x = [u1,u2,----uN,v1,v2,---vN,p1,p2,---pN]T
• No need to identify which unknown gets solved
using which PDE
• Unfortunately too expensive for practical use as
of this writing](https://image.slidesharecdn.com/primitivevariable-250410093128-711d5b91/85/primitive-variable-for-claculation-of-flow-field-6-320.jpg)
primitive variable for claculation of flow field
- 1. Primitive Variable Formulation MEL 807 Computational Heat Transfer (2-0-4) Dr. Prabal Talukdar Assistant Professor Department of Mechanical Engineering IIT Delhi
- 2. N-S Equation 0 y v x u y p y v x v y v v x v u t v x p y u x u y u v x u u t u 2 2 2 2 2 2 2 2 = ∂ ∂ + ∂ ∂ ∂ ∂ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ μ = ∂ ∂ ρ + ∂ ∂ ρ + ∂ ∂ ρ ∂ ∂ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ μ = ∂ ∂ ρ + ∂ ∂ ρ + ∂ ∂ ρ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ζ ∂ + ∂ ζ ∂ μ = ∂ ζ ∂ ρ + ∂ ζ ∂ ρ + ∂ ζ ∂ ρ 2 2 2 2 y x y v x u t 2-D, Navier-Stokes equation, We discussed—Stream-function-vorticity approach ζ − = ∂ ψ ∂ + ∂ ψ ∂ 2 2 2 2 y x
- 3. Sequential Solution • In a sequential approach, we solve set of coupled PDEs in a sequential manner • We identify • u-momentum equation for u velocity component • v-momentum for v velocity component • Continuity as PDE for pressure • Possible solution loop • Discretize and solve u over domain, assuming v, p known from current iterates • Similarly solve v from discrete v-mom equations • Similarly solve p from discrete continuity • Continue until convergence
- 4. Pressure into Continuity • Look into the problems of introducing pressure into continuity equation for incompressible flows • We have 3 equations (u, v momentum for 2D and continuity) and 3 unknowns (u, v and p) • But how can pressure be linked to continuity? • For compressible flow- not a problem • But for incompressible flow density is constant – No obvious way to introduce pressure into continuity. RT / P 0 ) .( t = ρ = ρ ∇ + ∂ ρ ∂ V
- 5. Sequential vs. Direct Schemes • Important to recognize that the problem of pressure computation for incompressible flows is an artifact of sequential solution procedure • Only for sequential procedures it is necessary to identify a PDE for each unknown variable – U-mom for u velocity; v-mom for v velocity, continuity equation for pressure; energy equation for temperature etc. • For direct solution procedures, no such one-on-one identification is necessary
- 6. Direct Scheme • Discretize u, v momentum equations and continuity at all cells • Assemble a big matrix: Ax = b • Solution vector is: x = [u1,u2,----uN,v1,v2,---vN,p1,p2,---pN]T • No need to identify which unknown gets solved using which PDE • Unfortunately too expensive for practical use as of this writing
- 7. Pressure Based Methods • Sequential methods are preferred for incompressible flows – Low storage and reasonable convergence rate • For incompressible flows, how to obtain pressure from continuity? – Density unrelated to pressure for incompressible flows • Pressure based methods are techniques to introduce pressure into continuity equation
- 8. Other Methods • Density based methods – Popular in compressible flow community – (u,v,w,ρ,T) used as unknowns • Artificial compressibility/preconditioning schemes – Effectively add some artificial compressibility to enable use of density-based schemes for incompressible flows – We will focus on pressure based methods as they are widely used
- 9. Representation of the pressure gradient N-S equation ) ( y 3 2 y u x y v y f S ) ( x 3 2 x v y x u x f S S j p ) v ( ) v ( S i p ) u ( ) u ( v v u u v u V V V V ⋅ ∇ μ ∂ ∂ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ μ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ μ ∂ ∂ + = ⋅ ∇ μ ∂ ∂ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ μ ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ μ ∂ ∂ + = + ⋅ ∇ − ∇ μ ⋅ ∇ = ρ ⋅ ∇ + ⋅ ∇ − ∇ μ ⋅ ∇ = ρ ⋅ ∇ Body force Source term
- 10. Representation of the pressure gradient (cont’d) u S i p ) u ( ) u ( + ⋅ ∇ − ∇ μ ⋅ ∇ = ρ ⋅ ∇ V dx dp − In 1D, Integrating this term over a control volume, we get 2 p p 2 p p 2 p p p p dx dx dp E W E P P W e w − = + − + = − = − ∫ w is assumed midway between P and W, But not a restriction
- 11. Implications 2 p p dx dp E W − = − Implies that the momentum equation will contain a term of pressure difference of alternate grid points instead of adjacent grid points Is that a problem?? Implication: The pressure is calculated from a coarser grid than the one actually employed -reduces accuracy But there is another serious problem-follows!!!
- 12. Checkerboard pressure field • Serious Problem: – See the zig-zag field which is unrealistic. The proposed scheme will give PW-PE zero at every grid point – Although it represents a wavy pressure field, proposed scheme will result a uniform pressure field in the momentum equation – Patterns are even more unrealistic in 2D and 3D 0 0
- 13. Representation of the Continuity Equation • Continuity equation • 1D steady, incompressible flow • Integrating over a CV • Using a piecewise linear profile for u 0 u u 0 2 u u 2 u u 0 u u 0 dx du 0 ) V .( W E P W E P w e = − = + − + = − = = ρ ∇
- 14. Checkerboard Velocity Field • So, the discretized continuity equation demands the equality of velocities at alternate grid points and not at adjacent ones • Although the following wavy velocity field satisfies the continuity equation, it is unrealistic • Conclusions: first order derivative always creates problem 0 0 U =
- 15. Remedy • The staggered grid – Store pressure at main cell centroid – Store velocities at staggered control volumes
- 16. Staggered grid: Momentum Eq. v u S j p ) v ( ) v ( S i p ) u ( ) u ( + ⋅ ∇ − ∇ μ ⋅ ∇ = ρ ⋅ ∇ + ⋅ ∇ − ∇ μ ⋅ ∇ = ρ ⋅ ∇ V V u-momentum v-momentum x ) p p ( j p y ) p p ( i p N P E P Δ − = ⋅ ∇ − Δ − = ⋅ −∇ Advantages: • No need to Interpolate pressure • Pressure Checker boarding avoided
- 17. Discretization of Momentum Eq. • Discretization of other terms are similar to the scalar transport equation e E P nb nb nb e e b y ) p p ( u a u a + Δ − + = ∑ n N P nb nb nb n n b x ) p p ( v a v a + Δ − + = ∑
- 18. Discretization of Continuity Eq. • Discretize continuity on main control volume 0 x ) v ( x ) v ( y ) u ( y ) u ( s n w e = Δ ρ − Δ ρ + Δ ρ − Δ ρ • No need to interpolate velocities -No checkerboarding