Lecture 16: Convection and Diffusion (Cont’d)

Last Time … We l Looked at CDS/UDS schemes to unstructured meshes l Look at accuracy of CDS and UDS schemes l Look at false diffusion in UDS using model equation

This Time… l We will use model equation to look at behavior of CDS scheme l Look at some first-order schemes based on exact solutions to the convection-diffusion equation »Exponential scheme »Hybrid scheme »Power-law scheme

CDS Model Equations l Pure convection equation: l Apply CDS: l Expand in Taylor series Do same type of expansion in y direction

Model Equation (Cont’d) l Subtract to obtain: l Do same in y direction: l Substitute into discrete equation Dispersion Term

Discussion l Model equation for CDS has extra third-derivative (dispersive) term l This type of odd-derivative term tends to cause spatial wiggles l Note that truncation error for CDS is O(  x 2 ) l Thus, UDS is dissipative and CDS is dispersive

First-Order Schemes Based on Exact Solutions l 1D Convection-diffusion equation  x Pe -Pe Pe=0 What are the limits of this equation for different Pe?

Exponential Scheme l Use 1-D exact solution as profile assumption in doing discretization l Consider convection-diffusion equation: l Integrate over control volume:

Exponential Scheme (Cont’d) l Area vectors l Flux*Area: l Use exact solution to write convection and diffusion terms

Exponential Scheme: Discrete Equations l Both convection and diffusion terms estimated from exact solution l If S=0, we would get the exact solution in 1D problems l But obviously not exact for non-zero S, multi- dimensional problems… l Discretization has boundedness, diagonal dominance l Only first-order accurate

Approximations to Exponential Scheme l Exponentials are expensive to compute l Approximations to the exponential profile assumption have been used to offset the cost. »Hybrid difference scheme »Power-law scheme l Both these approximations are also only first-order accurate

Hybrid Difference Scheme l Consider the a E coefficient in exponential scheme l Limits with respect to Pe:

Hybrid Difference Scheme (Cont’d) Instead of using the exact curve for a E /D e, use three tangents Similar manipulation for other coefficients

Hybrid Difference Scheme (Cont’d) l Guaranteed bounded solutions l Satisfies Scarborough criterion l O(  x) accurate

Power-Law Scheme l Employs fifth-order polynomial approximation to l Similar approach to other coefficients l Scheme is bounded and satisfies the Scarborough criterion l Is O(  x) accurate

Multi-Dimensional Schemes l Exact solutions have been used as profile assumptions in multi- dimensional situations l Control volume-based finite element method of Baliga and Patankar (1983) l This form is the solution to the 2D convection-diffusion equation X Y U

Multi-Dimensional Schemes l Finite analytic scheme (Chen and Li, 1979) l Write 2D convection diffusion equation with source term for “element”: l Fix coefficient using (i,j) values l Find analytical solution using separation of variables l Use exact solution for profiles assumptions (i,j) (i-1,j) (i+1,j) (i,j+1) (i, j-1)

Closure In this lecture, we l Looked at the model equation for CDS »Shown dispersive nature of model equation l Looked at differencing schemes based on exact solution to 1D convection-diffusion equation l Looked at schemes which are approximations to the exponential scheme l Looked at multidimensional schemes based on exact solutions