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CFD

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Published in: Mechanical
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Finite Volume Method

Uday K / Hyderabad

1 year of teaching experience

Qualification: M.Tech (JNTUH COLLEGE OF ENGINEERING HYDERABAD (JCEH), Hyderabad - 2018)

Teaches: English, EVS, Mathematics, Science, All Subjects, Reading Skills, Story Telling, Writing Skills

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  1. Lecture 16: Convection and Diffusion (Cont'd)
  2. Last Time o Looked at CDS/UDS schemes to unstructured meshes o Look at accuracy of CDS and UDS schemes o Look at false diffusion in UDS using model equation
  3. This Time, o We will use model equation to look at behavior of CDS scheme o Look at some first-order schemes based on exact solutions to the convection-diffusion equation » Exponential scheme Hybrid scheme » Power-law scheme
  4. CDS Model Equations o Pure convection equation: Apply CDS: 2Ar 2Ay Expand in Taylor series c)q5 (Ax)2 029b qbw = — Ar— + Ox 2! öx2 (Ar)2 D 29b E = -f- AX -F Ox 2! c)x2 (Ar)3 (Ar)3 (p 110) -F (p O Do same type of expansion in y direction
  5. Model Equation (Cont'd) o Subtract to obtain: 2Ar o Do same in y direction: 2Ay o Substitute into discrete equation Pil— + pv— — Ox p 11Ax2 3! c)qb (Ar)2 c)3qb Ox (Ay)2 03 q) + O ((Ar)3) Oy,3 Dispersion Term
  6. Discussion Model equation for CDS has extra third-derivative (dispersive) term This type of odd-derivative term tends to cause spatial wiggles Note that truncation error for CDS is 0( Ax2 ) Thus, UDS is dissipative and CDS is dispersive
  7. First-Order Schemes Based on Exact Solutions o ID Convection-diffusion equation (pug)) — — Ox Ox atx 0 qbL atx L exp((Pe)x/L) — I exp(Pe) I puL x What are the limits of this equation for different Pe?
  8. Exponential Scheme o Use I-D exact solution as profile assumption in doing discretization o Consider convection-diffusion equation: — (pug)) — — Ox Ox o Integrate over control volume: Je.Ae +Jw.Aw (SC + spop)
  9. Exponential Scheme (Cont'd) o Area vectors Flux*Area: CIO — (puo)e-re dx Jw.Aw = -(pué)w+rw dx o Use exact solution to write convection and diffusion terms exp(Pee) —
  10. Exponential Scheme: Discrete Equations b exp(Fe/De) 1 Fwexp(Fw/Dw) exp(Fw/Dw) 1 CIE + SpAUp (F scAVp Both convection and diffusion terms estimated from exact solution If S=O, we would get the exact solution in ID problems But obviously not exact for non-zero S, multi- dimensional problems.. Discretization has boundedness, diagonal dominance Only first-order accurate
  11. Approximations to Exponential Scheme o Exponentials are expensive to compute o Approximations to the exponential profile assumption have been used to offset the cost. Hybrid difference scheme » Power-law scheme o Both these approximations are also only first-order accurate
  12. Hybrid Difference Scheme o Consider the aE coefficient in exponential scheme De exp(Pee) — o Limits with respect to Pe: aE/D =-Pe 1- -+ 0 for Pee -+ —Pee for Pee -+ Pee at Pe L Exa
  13. Hybrid Difference Scheme (Cont'd) Instead of using the exact curve for adD tangents 0 for Pee > 2 Pee for — 2 > Pee < 2 —Pee for Pee < —2 Similar manipulation for other aE/D = -Pe , use three L Exa
  14. Hybrid Difference Scheme (Cont'd) CIE b Guaranteed bounded solutions Satisfies Scarborough criterion O(Ax) accurate Max[Fw, Dw + CIE -F + (F scAUp
  15. Power-Law Scheme Employs fifth-order polynomial approximation to o. llFel = Mar[O, 1 + Max[O, -Fel De Similar approach to other coefficients Scheme is bounded and satisfies the Scarborough criterion Is O(Ax) accurate
  16. o Multi-Dimensional Schemes Exact solutions have been used as profile assumptions in multi- dimensional situations Control volume-based finite element method of Baliga and Patankar (1983) - Aexp(pUX + BY + C This form is the solution to the 2D convection-diffusion equation x
  17. Multi-Dimensional Schemes Finite analytic scheme (Chen and Li, 1979) Write 2D convection diffusion equation with source term for "element": Dc D2x (i- l,j) < x < and x J -1 < y < YJ+I' Fix coefficient using (i,j) values Find analytical solution using separation of variables Use exact solution for profiles assumptions (i,j-l)
  18. Closure In this lecture, we o Looked at the model equation for CDS Shown dispersive nature of model equation o Looked at differencing schemes based on exact solution to 1 D convection-diffusion equation o Looked at schemes which are approximations to the exponential scheme o Looked at multidimensional schemes based on exact solutions

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