Lecture 17:
Convection and Diffusion
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Last Time…
In the last lecture, we

Developed the least-squares method for finding cellbased gradients on unstructured meshes

Looked at influence of secondary gradients in
destroying boundedness

Considered implementation issues, including the use
of face- and cell-based data structures
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This Time …
We will

Start looking at adding convection terms to our
transport equation

Look at two different schemes for discretizing the
convection terms
» Central difference scheme
» Upwind difference scheme
Consider the properties of these schemes vis-à-vis
» Boundedness
» Stability
» Accuracy

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Steady 2D Convection-Diffusion Equation

Governing equation:
Assume flow field
known
Assume Cartesian
structured mesh
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Discretization

As usual, integrate over control volume

Apply divergence theorem, linearize source term:
Nothing different
so far !
5
Discretization (Cont’d)

Area vectors:

Flux on east face:
Units?

Flow rate on east face:
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Diffusion Term

Write as usual assuming linear profile between (E,P)
etc:
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Grid Peclet Number

Face flux*area written in terms of two coefficients
Multiplies
face gradient
Multiplies face
value e: How to
evaluate?

Define grid Peclet number
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Central Difference Scheme (CDS)

Find face value of  using cell average:
Assuming
uniform mesh

Convection through face e:
Same sign! Trouble
ahead…
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CDS: Discrete Equation

Note possibility of
negative
coefficients

Note extra flow
rate term in aP
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CDS: Discussion

Consider V= u i+ v j with u>0, v>0. When Fe > 2De ,
i.e., if Peclet number Pee >2 , aE <0.

Similarly aN becomes negative if Fn > 2Dn or if Pen >2

For other configurations of the velocity vector, the
other coefficients can also become negative

This implies that if neighbor values go up, value at
point P can go down!


This is true even though aP  anb for S=0 and extra
mass flow rate term =0
What about Scarborough criterion ?
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CDS: Discussion

Scarborough criterion not satisfied:

In fact aP =0 is possible for zero diffusion and uniform
flow – how would you do Gauss-Seidel?
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CDS: Discussion

Notice extra flow rate term in aP:
This is the net mass flow rate out of the control volume

If the assumed flow field is continuity-satisfying, this term
would be zero. If not, it can cause loss of diagonal
dominance

Summary:
» Spatial wiggles are possible because of negative
coefficients – for uniform mesh, keep Pe<2
» Scarborough criterion not satisfied – can’t use iterative
schemes
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Upwind Difference Scheme (UDS)

Write face value as:

“Upwind” in the direction of the mass flow

Note asymmetric nature of the discretization!
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UDS: Discrete Equation

Here

What are the signs
on the neighbor
coefficients?

Note extra flow
rate term in aP
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UDS: Discussion

Notice that all coefficients are positive

Since aP  anb for S=0 and extra mass flow rate term
=0
» Solution is bounded

Scarborough criterion satisfied in the equality for S=0
and extra mass flow rate term=0

Notice extra flow rate term in aP:
Zero for
continuitysatisfying
velocity field
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Summary

Central difference scheme (CDS)
» Makes linear profile assumptions between grid
points to get face value
» Can lead to spatial wiggles in convectiondominated flows
» Can lose diagonal dominance – difficult to use
iterative solvers
» Can show that it is O(x2) accurate
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Summary (Cont’d)

Upwind difference scheme (UDS)
» Makes linear profile assumption for diffusion term,
but upwinds convective term
» Bounded solutions guaranteed for continuitysatisfying fields regardless of grid Peclet number
» Satisfies Scarborough criterion – iterative solutions
possible

Can show that UDS is O(x) accurate

Neither very satisfactory for practical use
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Closure
In this lecture:

We considered the steady 2D convection-diffusion
equation

Convection term requires evaluation of  on the faces

Considered two different scheme (CDS and UDS) and
examined the properties of the discretizations
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