model-equation - Derivation of Model Equation for Explicit...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Derivation of Model Equation for Explicit UDS Following (Tannehill et al. , 1997), we can derive the model equation for the explicit UDS scheme in the following way. The explicit UDS scheme may be written as: 0 φP φP u 0 φ 0 φW (1) ∆t ∆x P 0 0 0 Expanding φ P in a Taylor series about φ P in the time term and φW about φP in the spatial term, we get: ∆t u∆x φtt φxx O ∆t 2 O ∆x2 2 2 We wish to convert the φtt term into one involving φ xx . To do this, differentiate Eq. 3 with respect to time to obtain: Also, differentiate Eq. 3 with respect to x and multiply by u to obtain: u2 ∆x φxxx 2 Add Eqns. 4 and 5 to obtain: Substituting for φtt in Eq. 3, we obtain where ν is the Courant number, u∆t ∆x . References Tannehill, J.C., Anderson, D., & Pletcher, R.H. 1997. Computational Fluid Mechanics and Heat Transfer. 2nd edn. Series in Computational Methods and Physical Processes in Mechanics and Thermal Sciences. Taylor and Francis. 1 £ φt uφx £ ¢ ¡ ¢ ¡   u∆x 1 2 ν φxx O ∆t 2 O ∆x2 φtt u2 φxx ∆t O ∆t u∆x ©  ¡ ¡ ©  ¡  ¡ φttt 2 uφttx 2 uφxxt 2 u2 φxxx 2 £ ¤ ¤ ¡ uφtx u2 φxx £ ¢¡¢¡ u∆t φ 2 ttx O ∆t 2 O ∆x2 £ φtt uφxt £ ¢¡¢¡   ¡ ∆t φ 2 ttt u∆x φ 2 xxt O ∆t 2 O ∆x2 £ φt uφx £ ¢¡¢¡ Re-arranging, we obtain 0 φP φt ∆t φtt φttt 0 φP φx ∆x φxx φxxx O ∆x © ¥ © §¥¨¥¡ ¡ ¡ ¡¥ © ¨¥§¥¦¡    ¡ ¡ ¡ ¡ ¡  1 ∆t ∆t 2 2 ∆t 3 6 u φ0 ∆x P £ ¢ ¤ ¤ ¡ ¡¤ ∆x2 2 ∆3 6 0 (2) (3) (4) (5) (6) (7) ...
View Full Document

This note was uploaded on 12/29/2011 for the course ME 608 taught by Professor Na during the Fall '10 term at Purdue University-West Lafayette.

Ask a homework question - tutors are online