Finite Volume Discretization

MMVN05ROBERT SZASZ

Goal & Outline

• 1D Cartesian

– Diffusion

– Source

– Convection

– Time dependent

• 2D Cartesian

• 2D Unstructured

Partial Differential

Equation(s)

Set of Algebraic

Equations

The Finite Volume Method

• Generic transport equation

• Integrate over a control volume

𝜕𝜌𝜙

𝜕𝑡+ 𝑑𝑖𝑣 𝜌𝑢 𝜙 = 𝑑𝑖𝑣 Γ𝑔𝑟𝑎𝑑 𝜙 + 𝑆

Time

evolutionConvection Diffusion Source

term

𝑉

𝜕𝜌𝜙

𝜕𝑡𝑑𝑉 +

𝑉

𝑑𝑖𝑣 𝜌𝑢 𝜙 𝑑𝑉 = 𝑉

𝑑𝑖𝑣 Γ𝑔𝑟𝑎𝑑 𝜙 𝑑𝑉 + 𝑉

𝑆 𝑑𝑉

Discretization in 1D

Control volume boundaries

Control volume

P EWew

Dx

dxPe

dxPEdxWP

Diffusion problems in 1D

𝑑𝑖𝑣 Γ𝑔𝑟𝑎𝑑 𝜙 + 𝑆 = 0

𝑉

𝑑𝑖𝑣 Γ𝑔𝑟𝑎𝑑 𝜙 𝑑𝑉 + 𝑉

𝑆 𝑑𝑉 = 0

P EWew

DxdxPe

dxPEdxWP

𝐴

𝑛. Γ𝑔𝑟𝑎𝑑 𝜙 𝑑𝐴 + 𝑉

𝑆 𝑑𝑉 = 0

Γ𝐴𝜕𝜙

𝜕𝑥𝑒

− Γ𝐴𝜕𝜙

𝜕𝑥𝑤

+ 𝑆𝑉 = 0

Γ𝑒𝐴𝑒𝜕𝜙

𝜕𝑥𝑒

− Γ𝑤𝐴𝑤𝜕𝜙

𝜕𝑥𝑤

+ Su + Sp𝜙P = 0

Linear approximation

Diffusion problems in 1D

P EWew

DxdxPe

dxPEdxWP

Γ𝑒𝐴𝑒𝜕𝜙

𝜕𝑥𝑒

− Γ𝑤𝐴𝑤𝜕𝜙

𝜕𝑥𝑤

+ Su + Sp𝜙P = 0

𝜙𝑃Γ𝑒𝐴𝑒𝛿𝑥𝑃𝐸

+Γ𝑤𝐴𝑤𝛿𝑥𝑊𝑃

− 𝑆𝑃 = 𝜙𝑊Γ𝑤𝐴𝑤𝛿𝑥𝑊𝑃

+ 𝜙𝐸Γ𝑒𝐴𝑒𝛿𝑥𝑃𝐸

+ 𝑆𝑢

Γ𝑒 =Γ𝑊+Γ𝑃

2, Γ𝑤 =

Γ𝐸+Γ𝑃

2, 𝜕𝜙

𝜕𝑥 𝑒=𝜙𝐸−𝜙𝑃

𝛿𝑥𝑃𝐸, 𝜕𝜙

𝜕𝑥 𝑤=𝜙𝑃−𝜙𝑊

𝛿𝑥𝑊𝑃

These are usually stored

𝜙𝑃𝑎𝑃 = 𝜙𝑊𝑎𝑊 + 𝜙𝐸𝑎𝐸 + 𝑆𝑢

Boundaries

P Eew

DxdxPe

dxPE

Case 1: is known𝜙𝑤 = 𝐹𝑤

Γ𝑒𝐴𝑒𝜕𝜙

𝜕𝑥𝑒

− Γ𝑤𝐴𝑤𝜕𝜙

𝜕𝑥𝑤

+ Su + Sp𝜙P = 0

𝜕𝜙

𝜕𝑥𝑤

=𝜙𝑃 − 𝐹𝑤𝛿𝑥𝑤𝑃

Case 2: is known

Eastern boundary similarly

𝜕𝜙

𝜕𝑥𝑤

= 𝐺𝑤

𝜙𝑃𝑎𝑃 = 𝜙𝐸𝑎𝐸 + 𝑆𝑢

Building the system of equations

1 2 N

𝜙𝑃𝑎𝑃 = 𝜙𝑊𝑎𝑊 + 𝜙𝐸𝑎𝐸 + 𝑆𝑢

𝜙1𝑎1,1 = 𝜙2𝑎2,1 + 𝑆1

3

Building the system of equations

1 2 N

𝜙𝑃𝑎𝑃 = 𝜙𝑊𝑎𝑊 + 𝜙𝐸𝑎𝐸 + 𝑆𝑢

𝜙1𝑎1,1 = 𝜙2𝑎2,1 + 𝑆1

𝜙2𝑎2,2 = 𝜙1𝑎1,2 + 𝜙3𝑎3,2 + 𝑆2

3

Building the system of equations

1 2 N

𝜙𝑃𝑎𝑃 = 𝜙𝑊𝑎𝑊 + 𝜙𝐸𝑎𝐸 + 𝑆𝑢

𝜙1𝑎1,1 = 𝜙2𝑎2,1 + 𝑆1

𝜙2𝑎2,2 = 𝜙1𝑎1,2 + 𝜙3𝑎3,2 + 𝑆2

3

𝜙3𝑎3,3 = 𝜙2𝑎2,3 + 𝜙4𝑎4,3 + 𝑆3

Building the system of equations

1 2 N

𝜙𝑃𝑎𝑃 = 𝜙𝑊𝑎𝑊 + 𝜙𝐸𝑎𝐸 + 𝑆𝑢

𝜙1𝑎1,1 = 𝜙2𝑎2,1 + 𝑆1

𝜙2𝑎2,2 = 𝜙1𝑎1,2 + 𝜙3𝑎3,2 + 𝑆2

3

𝜙3𝑎3,3 = 𝜙2𝑎2,3 + 𝜙4𝑎4,3 + 𝑆3

𝜙𝑁𝑎𝑁,𝑁 = 𝜙𝑁−1𝑎𝑁−1,𝑁 + 𝑆𝑁

Convection-Diffusion problems in 1D

𝑑𝑖𝑣 𝜌𝑢 𝜙 = 𝑑𝑖𝑣 Γ𝑔𝑟𝑎𝑑 𝜙

𝑉

𝑑𝑖𝑣 𝜌𝑢 𝜙 𝑑𝑉 = 𝑉

𝑑𝑖𝑣 Γ𝑔𝑟𝑎𝑑 𝜙 𝑑𝑉

𝐴

𝑛. 𝜌𝜙𝑢 𝑑𝐴 = 𝐴

𝑛. Γ𝑔𝑟𝑎𝑑 𝜙 𝑑𝐴

P EWew

DxdxPe

dxPEdxWP

𝑢𝑒𝑢𝑤

(source terms not considered for simplicity)

𝜌𝑢𝐴𝜙 𝑒 − 𝜌𝑢𝐴𝜙 𝑤 = Γ𝐴𝜕𝜙

𝜕𝑥𝑒

− Γ𝐴𝜕𝜙

𝜕𝑥𝑤

Assume Ae=Aw=A and denote fluxes as:

𝐹𝑤 = 𝜌𝑢 𝑤 𝐹𝑒 = 𝜌𝑢 𝑒 𝐷𝑤 =Γ𝑤𝛿𝑥𝑊𝑃

𝐷𝑒=Γ𝑒𝛿𝑥𝑃𝐸

𝐹𝑒𝜙𝑒 − 𝐹𝑤𝜙𝑤 = 𝐷𝑒 𝜙𝐸 − 𝜙𝑃 − 𝐷𝑤(𝜙𝑃 − 𝜙𝑊)

Convection-Diffusion problems in 1D

𝜕(𝜌𝑢)

𝜕𝑥= 0

The continuity equation must be also fullfilled:

P EWew

DxdxPe

dxPEdxWP

𝑢𝑒𝑢𝑤

𝜌𝑢𝐴 𝑒 − 𝜌𝑢𝐴 𝑤 = 0

𝐹𝑒 − 𝐹𝑤 = 0

Convection-Diffusion problems in 1D

P EWew

DxdxPe

dxPEdxWP

𝑢𝑒𝑢𝑤

𝐹𝑒𝜙𝑒 − 𝐹𝑤𝜙𝑤= 𝐷𝑒 𝜙𝐸 − 𝜙𝑃 − 𝐷𝑤(𝜙𝑃 − 𝜙𝑊)

How to estimate fe, fw?

Central difference scheme:𝜙𝑒 =

𝜙𝑃 + 𝜙𝐸2

𝜙𝑤 =𝜙𝑊 + 𝜙𝑃

2

𝐷𝑤 +𝐹𝑤2+ 𝐷𝑒 −

𝐹𝑒2𝜙𝑃 = 𝐷𝑤 +

𝐹𝑤2𝜙𝑊 + 𝐷𝑒 −

𝐹𝑒2𝜙𝐸

𝑎𝑃𝜙𝑃 = 𝑎𝑊𝜙𝑊 + 𝑎𝐸𝜙𝐸

Fig. 5.4. N=5, u=0.1 m/s, F=0.1, D=0.5 Fig.5.5. N=5, u=2.5 m/s, F=2.5, D=0.5

Fig.5.6. N=20, u=2.5 m/s, F=2.5, D=2.0

WHY?

Properties of discretization schemes

• Conservativeness

– Estimate the fluxes in a consistent manner!

Fig.5.7

Fig.5.8

Properties of discretization schemes

• Boundedness

– In the absence of sources, the value of a property

should be bounded by its boundary values

– Requirements:

» All coefficients the same sign

»∑ 𝑎𝑛𝑏

𝑎𝑝′≤ 1 𝑎𝑡 𝑎𝑙𝑙 𝑛𝑜𝑑𝑒𝑠, < 1 𝑎𝑡 𝑜𝑛𝑒 𝑛𝑜𝑑𝑒 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡

Properties of discretization schemes

• Transportiveness

– Where is the information transported?

Fig.5.9

Assessment of the central differencing

scheme

• Conservativeness: OK

• Boundedness:

ae<0 for Pee=Fe/De>2 !!!

• Transportiveness: Not OK!

• Accuracy: 2nd order

𝐷𝑤 +𝐹𝑤2+ 𝐷𝑒 −

𝐹𝑒2𝜙𝑃 = 𝐷𝑤 +

𝐹𝑤2𝜙𝑊 + 𝐷𝑒 −

𝐹𝑒2𝜙𝐸

Upwind differencing scheme

• Compute the convective term depending on the flow

direction:

P EWew

DxdxPe

dxPEdxWP

𝑢𝑒𝑢𝑤

𝜙𝑒 = 𝜙𝑃 𝜙𝑤= 𝜙𝑊

P EWew

DxdxPe

dxPEdxWP

𝑢𝑒𝑢𝑤

𝜙𝑒 = 𝜙𝐸 𝜙𝑤= 𝜙𝑃

Upwind differencing scheme

• Assessment:

– Conservativeness: OK

– Boundedness: OK

– Transportiveness: OK

– Accuracy: 1st order

Fig.5.15

Hybrid differencing scheme

• Combine schemes:

– Central for Pe=F/D < 2

– Upwind for Pe > 2

• Assessment:

– Conservativeness: OK

– Boundedness: OK

– Transportiveness: OK

– Accuracy: 1st order

Power-law scheme

• More accurate than hybrid

• Diffusion set to 0 for Pe>10

• For Pe < 10 polynomial expression is used to evaluate

the fluxes

The QUICK scheme

• Quadratic Upstream Interpolation for Convective Kinetics

• Higher order & Upwind

• Face values of f obtained from quadratic functions

• Diffusion terms can be evaluated from the gradient

of the parabola

• For uw>0

• For ue>0

𝜙𝑤 =6

8𝜙𝑊 +

3

8𝜙𝑃 −

1

8𝜙𝑊𝑊

𝜙𝑒 =6

8𝜙𝑃 +

3

8𝜙𝐸 −

1

8𝜙𝑊

Fig.5.17

The QUICK scheme

• Generic form:

• Issues at boundaries: no second neighbours

– Create virtual ‘mirror nodes’ by linear extrapolation

𝑎𝑃𝜙𝑃 = 𝑎𝑊𝜙𝑊 + 𝑎𝐸𝜙𝐸 + 𝑎𝑊𝑊𝜙𝑊𝑊 + 𝑎𝐸𝐸𝜙𝐸𝐸

Fig.5.18

The QUICK scheme

• Assessment

– Conservativeness: OK

– Boundedness:

» Only conditionnaly stable!

– Reformulated versions

to improve stability

– Transportiveness: OK

– Accuracy: better formal

accuracy than upwind

– Possible over/undershoots

Fig.5.20

General upwind-biased schemes

• ‘Pure’ upwind:

• Linear upwind differencing:

• QUICK:

• Central differencing:

• General:

𝜙𝑒 = 𝜙𝑃

𝜙𝑒 = 𝜙𝑃 +1

2(𝜙𝑃 − 𝜙𝑊)

𝜙𝑒 = 𝜙𝑃 +1

8(3𝜙𝐸 − 2𝜙𝑃 − 𝜙𝑊)

𝜙𝑒 = 𝜙𝑃 +1

2(𝜙𝐸 − 𝜙𝑃)

𝜙𝑒 = 𝜙𝑃 +1

2𝜓(𝜙𝐸 − 𝜙𝑃)

General upwind-biased schemes

• UD

• CD

• LUD

• QUICK𝜙𝑒 = 𝜙𝑃 +

1

2𝜓(𝜙𝐸 − 𝜙𝑃)

𝜓 = 𝜓(𝑟)

𝑟 =𝜙𝑃 − 𝜙𝑊𝜙𝐸 − 𝜙𝑃

𝜓 𝑟 = 0

𝜓 𝑟 = 1

𝜓 𝑟 = 𝑟

𝜓 𝑟 = (3 + 𝑟)/4

Fig.5.21

TVD schemes

• Total Variation Diminishing

• Criteria:

– Upper limit for TVD:

f1

f2

f3

f4

f5

𝑇𝑉 𝜙= 𝜙2 − 𝜙1 + 𝜙3 − 𝜙2+ 𝜙4 − 𝜙3 + |𝜙5 − 𝜙4|

Fig.5.23

𝜓 𝑟 ≤ 2𝑟 𝑟 ≤ 1

𝜓 𝑟 ≤ 2 𝑟 > 1

TVD

• For second order:

– Must go through (1,1)

– Limited by CD and LUD

• To treat forward and

backward differencing

consistently: symmetry

property:

• Flux limiters

𝜓 𝑟

r= 𝜓(1/𝑟)

Fig.5.24

Fig.5.25

Unsteady flows

𝜕𝜌𝜙

𝜕𝑡+ 𝑑𝑖𝑣 𝜌𝑢 𝜙 = 𝑑𝑖𝑣 Γ𝑔𝑟𝑎𝑑 𝜙 + 𝑆

𝑉

𝜕𝜌𝜙

𝜕𝑡𝑑𝑉 +

𝑉

𝑑𝑖𝑣 𝜌𝑢 𝜙 𝑑𝑉 = 𝑉

𝑑𝑖𝑣 Γ𝑔𝑟𝑎𝑑 𝜙 𝑑𝑉 + 𝑉

𝑆 𝑑𝑉

𝑡

𝑡+Δ𝑡

𝑉

𝜕𝜌𝜙

𝜕𝑡𝑑𝑉𝑑𝑡 +

𝑡

𝑡+Δ𝑡

𝑉

𝑑𝑖𝑣 𝜌𝑢 𝜙 𝑑𝑉 𝑑𝑡

= 𝑡

𝑡+Δ𝑡

𝑉

𝑑𝑖𝑣 Γ𝑔𝑟𝑎𝑑 𝜙 𝑑𝑉 𝑑𝑡 + 𝑡

𝑡+Δ𝑡

𝑉

𝑆 𝑑𝑉𝑑𝑡

Unsteady 1D heat conduction

• c – specific heat [J/(kg K)]

𝜌𝑐𝜕𝑇

𝜕𝑡=𝜕

𝜕𝑥𝑘𝜕𝑇

𝜕𝑥+ 𝑆

P EWew

DxdxPe

dxPEdxWP

𝑡

𝑡+Δ𝑡

𝑉

𝜌𝑐𝜕𝑇

𝜕𝑡𝑑𝑉𝑑𝑡 =

𝑡

𝑡+Δ𝑡

𝑉

𝜕

𝜕𝑥𝑘𝜕𝑇

𝜕𝑥𝑑𝑉𝑑𝑡 +

𝑡

𝑡+Δ𝑡

𝑉

𝑆𝑑𝑉𝑑𝑡

𝑉

𝑡

𝑡+Δ𝑡

𝜌𝑐𝜕𝑇

𝜕𝑡𝑑𝑡𝑑𝑉 =

𝑡

𝑡+Δ𝑡

𝑘𝐴𝜕𝑇

𝜕𝑥𝑒

− 𝑘𝐴𝜕𝑇

𝜕𝑥𝑤

𝑑𝑡 + 𝑡

𝑡+Δ𝑡

𝑆Δ𝑉𝑑𝑡

Unsteady 1D heat conduction

𝑉

𝑡

𝑡+Δ𝑡

𝜌𝑐𝜕𝑇

𝜕𝑡𝑑𝑡𝑑𝑉 =

𝑡

𝑡+Δ𝑡

𝑘𝐴𝜕𝑇

𝜕𝑥𝑒

− 𝑘𝐴𝜕𝑇

𝜕𝑥𝑤

𝑑𝑡 + 𝑡

𝑡+Δ𝑡

𝑆Δ𝑉𝑑𝑡

𝜌𝑐 𝑇𝑃 − 𝑇𝑃0 ΔV =

𝑡

𝑡+Δ𝑡

𝑘𝐴𝑇𝐸 − 𝑇𝑃𝛿𝑥𝑃𝐸

− 𝑘𝐴𝑇𝑃 − 𝑇𝑊𝛿𝑥𝑊𝑃

𝑑𝑡 + 𝑡

𝑡+Δ𝑡

𝑆Δ𝑉𝑑𝑡

Assume uniform T

In control volume

and use backward

differencing

Use central

differencing

How do TE,TW and TP vary in time? Assume

a linear function:@ t+Dt @ t

𝐼𝑇 = 𝑡

𝑡+Δ𝑡

𝑇𝑑𝑡 = 𝜃𝑇 + 1 − 𝜃 𝑇0 Δ𝑡

Unsteady 1D heat conduction

𝜌𝑐 𝑇𝑃 − 𝑇𝑃0 ΔV =

𝑡

𝑡+Δ𝑡

𝑘𝐴𝑇𝐸 − 𝑇𝑃𝛿𝑥𝑃𝐸

− 𝑘𝐴𝑇𝑃 − 𝑇𝑊𝛿𝑥𝑊𝑃

𝑑𝑡 + 𝑡

𝑡+Δ𝑡

𝑆Δ𝑉𝑑𝑡

Divide by A and Dt:

𝜌𝑐 𝑇𝑃 − 𝑇𝑃0Δ𝑥

Δ𝑡

= 𝜃 𝑘𝑇𝐸 − 𝑇𝑃𝛿𝑥𝑃𝐸

− 𝑘𝑇𝑃 − 𝑇𝑊𝛿𝑥𝑊𝑃

+ (1 − 𝜃) 𝑘𝑇𝐸0 − 𝑇𝑃

0

𝛿𝑥𝑃𝐸− 𝑘

𝑇𝑃0 − 𝑇𝑊

0

𝛿𝑥𝑊𝑃+ 𝑆Δ𝑥

𝑎𝑃𝑇𝑃= 𝑎𝑊 𝜃𝑇𝑊 + 1 − 𝜃 𝑇𝑊

0 + 𝑎𝐸 𝜃𝑇𝐸 + 1 − 𝜃 𝑇𝐸0

+ 𝑎𝑃0 − 1 − 𝜃 𝑎𝑊 − 1 − 𝜃 𝑎𝐸 𝑇𝑃

0 + 𝑏

Explicit scheme

𝜃 = 0

𝑎𝑃𝑇𝑃 = 𝑎𝑊𝑇𝑊0 + 𝑎𝐸𝑇𝐸

0 + 𝑎𝑃0 − 𝑎𝑊 − 𝑎𝐸 + 𝑆𝑃 𝑇𝑃

0 + 𝑆𝑢

TP can be directly computed

First order

Can be negative! To assure stability:

Δ𝑡 < 𝜌𝑐Δ𝑥 2

2𝑘

Very small timesteps when the grid is refined!

Implicit scheme

𝜃 = 1

𝑎𝑃𝑇𝑃 = 𝑎𝑊𝑇𝑊 + 𝑎𝐸𝑇𝐸 + 𝑎𝑃0𝑇𝑃0 + 𝑆𝑢

Several unknowns -> System of equations

Unconditionally stable

First order

Crank-Nicolson scheme

𝜃 =1

2

𝑎𝑃𝑇𝑃 = 𝑎𝑊𝑇𝑊 + 𝑇𝑊

0

2+ 𝑎𝐸

𝑇𝐸 + 𝑇𝐸0

2+ 𝑎𝑃

0 −𝑎𝑊2−𝑎𝐸2+𝑆𝑃2𝑇𝑃0 + 𝑆𝑢

System of equations

To assure stability:

Δ𝑡 < 𝜌𝑐Δ𝑥 2

𝑘

Very small timesteps when the grid is refined!

Second order

Unsteady 1D convection-diffusion

• Similarly to unsteady diffusion

• Additional stability limits due to convection

𝜕𝜌𝜙

𝜕𝑡+𝜕𝜌𝑢𝜙

𝜕𝑥=𝜕

𝜕𝑥Γ𝜕𝜙

𝜕𝑥+ 𝑆

Finite volume method for 2D Cartesian

grids

• E.g. diffusion problem:

Fig.4.12

𝜕

𝜕𝑥Γ𝜕𝜙

𝜕𝑥+𝜕

𝜕𝑦Γ𝜕𝜙

𝜕𝑦+ 𝑆 = 0

Δ𝑉

𝜕

𝜕𝑥Γ𝜕𝜙

𝜕𝑥𝑑𝑉 +

Δ𝑉

𝜕

𝜕𝑦Γ𝜕𝜙

𝜕𝑦𝑑𝑉

+ Δ𝑉

𝑆𝑑𝑉 = 0

Γ𝑒𝐴𝑒𝜙𝐸 − 𝜙𝑃𝛿𝑥𝑃𝐸

− Γ𝑤𝐴𝑤𝜙𝑃 − 𝜙𝑊𝛿𝑥𝑊𝑃

+ Γ𝑛𝐴𝑛𝜙𝑁 − 𝜙𝑃𝛿𝑦𝑃𝑁

− Γ𝑠𝐴𝑠𝜙𝑃 − 𝜙𝑆𝛿𝑦𝑆𝑃

+ 𝑆Δ𝑉 = 0

𝑎𝑃𝜙𝑃 = 𝑎𝑊𝜙𝑊 + 𝑎𝐸𝜙𝐸 + 𝑎𝑆𝜙𝑆 + 𝑎𝑁𝜙𝑁 + 𝑆𝑢 = 0

Finite Volume Method on unstructured

grids

P

ni

DA

𝑉

𝜕𝜌𝜙

𝜕𝑡𝑑𝑉 +

𝑉

𝑑𝑖𝑣 𝜌𝑢 𝜙 𝑑𝑉

= 𝑉

𝑑𝑖𝑣 Γ𝑔𝑟𝑎𝑑 𝜙 𝑑𝑉 + 𝑉

𝑆 𝑑𝑉

𝜕

𝜕𝑡

𝑉

𝜌𝜙 𝑑𝑉 + 𝐴

𝑛. 𝜌𝑢 𝜙 𝑑𝐴 = 𝐴

𝑛. Γ𝑔𝑟𝑎𝑑 𝜙 𝑑𝐴 + 𝑉

𝑆 𝑑𝑉

𝜕

𝜕𝑡

𝑉

𝜌𝜙𝑑𝑉 + ∑ Δ𝐴𝑖

𝑛𝑖 . 𝜌𝑢 𝜙 𝑑𝐴 = ∑ Δ𝐴𝑖

𝑛𝑖 . Γ𝑔𝑟𝑎𝑑 𝜙 𝑑𝐴 + 𝑉

𝑆 𝑑𝑉

• ni can be computed based on grid topology

Diffusion term

• Central differencing

• Only true if the grid is

orthogonal

Δ𝐴𝑖

𝑛𝑖 . Γ𝑔𝑟𝑎𝑑 𝜙 𝑑𝐴 ≈

𝑛𝑖 . Γ𝑔𝑟𝑎𝑑 𝜙 Δ𝐴𝑖 ≈

Γ𝜙𝐴 − 𝜙𝑃Δ𝜉

Δ𝐴𝑖

Fig.11.15

Non-orthogonal grids

• Can be computed as:

n. grad𝜙Δ𝐴𝑖

=𝑛. 𝑛Δ𝐴𝑖𝑛. 𝑒𝜉

𝜙𝐴 − 𝜙𝑃Δ𝜉

−𝑒𝜉 . 𝑒𝜂 Δ𝐴𝑖

𝑛. 𝑒𝜉

𝜙𝑏 − 𝜙𝑎Δ𝜂

Fig.11.17

Non-orthogonal grids

• Compare to:

n. grad𝜙 Δ𝐴𝑖 =𝑛. 𝑛Δ𝐴𝑖𝑛. 𝑒𝜉

𝜙𝐴 − 𝜙𝑃Δ𝜉

−𝑒𝜉 . 𝑒𝜂 Δ𝐴𝑖

𝑛. 𝑒𝜉

𝜙𝑏 − 𝜙𝑎Δ𝜂

Direct gradient Cross-diffusion

Known Interpolated

Can be (pre)computed (and stored) based on grid topology.

n. grad𝜙 Δ𝐴𝑖 =𝜙𝐴 − 𝜙𝑃Δ𝜉

Δ𝐴𝑖

Convective term

• Approximate as:

Δ𝐴𝑖

𝑛𝑖 . 𝜌𝜙𝑢 𝑑𝐴 ≈

𝜙𝑖 Δ𝐴𝑖

𝑛𝑖 . 𝜌𝑢 𝑑𝐴 ≈ 𝜙𝑖𝐹𝑖

Fi = Convective flux

normal to the surface

element.

Usually stored.Fig.11.15

fi

• Upwind differencing

– Fi>0 fi=fP

– Fi<0 fi=fA

• Linear upwind differencing

• QUICK

• TVD

• …

𝜙𝑖 = 𝜙𝑃 + 𝛻𝜙𝑃 . Δ𝑟

Needs to be computed!