1

9. Pre- and Post-Processing

● Pre-processing:

– creating geometry

– grid-generation

– governing equations

– boundary conditions

● Solving

● Post-processing

● Pre-processing

● Solving:

– discretisation

– solution of equations

● Post-processing

2

● Pre-processing

● Solving

● Post-processing:

– analysis

– visualisation

CFD Codes

● Commercial Codes:

– STAR-CCM+

– Fluent

– FLOW3D

– PowerFLOW

– COMSOL

● Open source:

– OpenFOAM

– CodeSaturne

● General portals:

– www.cfd-online.com

The Computational Mesh

3

Mesh Generation

● Function:

– to decompose the domain into control volumes

● Constraints:

– flow geometry

– capabilities of the solver

● Output:

– cell vertices (x, y, z)

– connectivity data

Mesh Arrangements

● Node locations (cell centres / vertices)

● Staggered / co-located

● Cell shapes (tetrahedra / hexahedra / arbitrary polyhedra)

● Structured (single- or multi-block) / unstructured

● Cartesian / curvilinear (orthogonal / non-orthogonal)

● …

Storage Locations

cell-centred cell-vertex

staggered

u p

v

4

Cell Shapes

2-d: triangles, quadrilaterals, …

3-d: tetrahedra, hexahedra, …

tetrahedron hexahedron

Areas and Volumes

● Vector areas are needed for fluxes:

● Volumes are needed for amounts, averages, gradients:

Auρ

Vρ

AΓ

mass flux:

diffusive flux:

amount in cell:

V

totalaverage:

Face Areas

2121 ssA

Triangles

Quadrilaterals

)()( 241321

241321 rrrrddA

● 4 points are not necessarily coplanar

● Vector area is independent of spanning surface

s1

s2

A

A

A

r3

4r

1r2r

5

Some Vector Calculus (Not Examinable)

),,()(gradzyx

z

f

y

f

x

f zyx

ffdivDivergence:

Gradient:

zyx fff

zyx

kji

ffcurlCurl:

),,(zyx

is both a vector and a differential operator

Integral Theorems

Gauss’ Divergence Theorem:

VV

V Aff dd

Stokes’ Theorem:

AA

sfAf dd

VV

dA

A

dsA

Volumes

V

V Ar d3

1

Integrate over an arbitrary closed volume V:

),,( zyxr

3

z

z

y

y

x

xr

VV

VV d3dr

VV

3d

Ar

Divergence:

Position vector:

Use the divergence theorem:

Volumes

V

V Ar d3

1Arbitrary volume:

General polyhedron: faces

ffV Ar3

1

Hexahedron

faces

ffV Ar31

)( 432141 rrrrr f

32161 sss V

Tetrahedron

s3

1s

2s

6

2-D Case

● Treat as cells of unit depth

● The “volume” of the cell is then the planar area

● Outward “face area” vectors derived from Cartesian projections:

x

y

Δ

ΔΔA

(cell boundary traversed anticlockwise)

V

s

A

y

x0

-x

y0

Cell-Averaged Derivatives

Vav

VxVx

d1

faces

fxf

av

AVx

1

faces

ffavV

A1

)(

V

VV

d),(1

VV

Ad1

Cartesian cell:

xx

we

Δ

Va

rea A

x

V

xAV

d1

(definition)

xA

ΑΑ we

Δ

)(

1wxwexe ΑΑ

V

Example (3D)

A tetrahedral cell has vertices at A(2, –1, 0), B(0, 1, 0), C(2, 1, 1) and D(0, –1, 1).

(a) Find the outward vector areas of all faces. Check that they sum to zero.

(b) Find the volume of the cell.

(c) If the values of at the centroids of the faces (indicated by their vertices) are

BCD = 5, ACD = 3, ABD = 4, ABC = 2,

find the volume-averaged derivatives

avavav zyx

,,

7

Example (2D)

In a 2-dimensional unstructured mesh, one cell has the

form of a pentagon. The coordinates of the vertices are

as shown in the figure, whilst the average values of a

scalar on edges a – e are:

a = –7, b = 8, c = –2, d = 5, e = 0

Find:

(a) the area of the pentagon;

(b) the cell-averaged derivatives avav yx

,

(2,-4)

(5,1)

(1,3)

(-3,0)

(-2,-3)

a

bc

d

e

Example

The figure shows the vertices of a single triangular cell in a 2-d unstructured finite-

volume mesh. The accompanying table shows the pressure p and velocity (u,v) on

the faces marked a,b,c at the end of a steady, incompressible flow simulation.

Find:

(a) the area of the triangle;

(b) the net pressure force (per unit depth) on the cell;

(c) the outward volume flow rate (per unit depth) for all faces;

(d) the missing velocity component uc;

(e) the cell-averaged velocity gradients u/x, u/y, v/x, v/y.

(f) Define, mathematically, the acceleration (material derivative) Du/Dt. If the

velocity at the centre of the cell is u = (16/3,–4), use this and the gradients from

part (e) to calculate the acceleration.

edge p u v

a 3 5 –1

b 5 7 –5

c 2 uc –6

(4,0)

(3,5)

(1,1)

bc

ax, u

y, v

Structured Grids

Control volumes indexed by (i, j, k)

Curvilinear Cartesian

8

Unstructured Grids

Fitting Complex Boundaries − Blocking Out Cells

● Implemented by a source-term modification:

● A lot of redundant matrix operations for cells inside block

)(,0 erlarge numbsb PP F

PPPFFPP sbaa

0

numberlargea

a

P

FF

P

Fitting Complex Boundaries − Volume-of-Fluid (VOF) Approach

● One way of handling free-surface calculations.

● For moving surfaces, solve a transport equation for fluid fraction f.

● Related technique: level-set method.

f = 0

f = 1

f = 0

0 < f < 1

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Fitting Complex Boundaries − Curvilinear (Body-Fitted) Grids

Curvilinear Grids

sourcediffusionadvectionamountt faces

)()(d

d

Diffusion:

● also requires derivatives parallel to cell faces

Advection:

● all velocity components contribute to mass flux

AAn PE

PE

ΔΓΓ

P

E = const.

=const.

u

v

Fitting Complex Boundaries − Multi-Block Grids

● Multiple structured blocks

● Grid lines may or may not match at block boundaries

● Arbitrary interfaces allow non-coincident grid vertices

● Sliding grids used for rotating machinery (pumps; turbines)

2 3 4

51

2 3 4

587

61

6 85321

47

10

Fitting Complex Boundaries − Overset (Chimera) Grids

Disposition of Grid Cells

● Local refinement needed where gradients are large:

– solid boundaries

– shear layers

– separation, reattachment and impingement points

– discontinuities (shocks, hydraulic jumps)

● Grid-dependence tests necessary

● Turbulent calculations impose constraints:

– low-Re calculations: y+ < 1 (ideally)

– wall-function calculations: 30 < y + < 150 (ideally)

– many codes now adopt a y+-dependent blending approach

Multiple Levels of Grid

● Used to confirm grid independence

● Exploited by multi-grid methods

● Permit estimation of error (and solution improvement)

by Richardson extrapolation

11

Richardson Extrapolation

nΔerror

12

2* Δ2Δ

n

n

n

n

C

C

)Δ2(*

Δ*

Δ2

Δ

Order n

Improved solution by weighted

average from two meshes

12Δ ΔΔ2

n

nC Estimate of error from difference

between solutions on two meshes

Example

A numerical scheme known to be second-order accurate is used to

calculate a steady-state solution on two regular Cartesian meshes A

and B, where the finer mesh A has half the grid spacing of mesh B.

The values of the solution at a particular point are found to be

0.74 using mesh A and 0.78 using mesh B. Use Richardson

extrapolation to:

(a) estimate an improved value of the solution at this point;

(b) estimate the error at this point using the mesh-A solution.

Lewis Fry Richardson Father of modern weather forecasting

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Summary of Grids

● Dictated by:

– flow geometry

– solver capabilities

● Grid generator provides vertex and connectivity data

● Vector geometry to find areas, volumes and cell-averages

● Structured/unstructured meshes

● Complex geometries via:

– blocked-out cells

– volume-of-fluid methods

– curvilinear (body-fitted) meshes

– multiblock grids

– overset meshes

● Grid refinement in rapidly-varying regions

Boundary Conditions

Boundary Conditions

● Inlet:

– velocity inlet

– stagnation / reservoir inlet

● Outlet:

– standard outlet

– pressure boundary

– radiation boundary

● Wall:

– non-slip (rough/smooth; moving/stationary; adiabatic/heat-transfer)

– slip (inviscid case)

● Symmetry plane

● Periodic boundary

● Free surface

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Flow Visualisation

Uses of Flow Visualisation

● Understanding flow behaviour

● Locating important regions

● Summarising data

● Optimising design

● Finding reasons for non-convergence

● Publicity

Types of Plot

● x-y line graphs

● Contour plots

● Vector plots

● Streamline plots

● Mesh plots

● Composite plots

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x-y Graphs

x-y Graphs − Assessment

● Simple

● Widely-available software

● Precise and quantitative

● Direct comparison with experimental data

● Linear or logarithmic scales

● Limited view of flow field

Line or Shaded Contour Plots

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Contour Plots − Assessment

● Isoline (2D) or isosurface (3D)

● Optional smooth or discrete colour shading

● Global view of the flow

● Geometric spacing of lines indicates gradient

● Not as quantitative as line graphs

● Miss detail in small, but important, flow regions

Vector Plots

Vector Plots − Assessment

● Direction and magnitude

● Used to plot vector quantities (velocity and stress)

● May be coloured to indicate magnitude

● Excellent first indication of flow behaviour

● Interpolation often necessary for non-uniform grids

● Not good in flows with wide range of magnitudes

● Miss important detail in small regions

● Orientation effects deceptive in 3d

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Streamline Plots

Calculating Streamlines

3D: integrate a particle path ux

td

d

xv

yu

ψ,

ψ

fluxvolumexvyu ddψd

2D: contour the stream function

1

2

volume flux

1

2

dy

dx-v

u

(a) Two adjacent cells in a 2-dimensional Cartesian mesh are shown below, along

with the cell dimensions and some of the velocity components (in m s–1) normal

to cell faces. The value of the stream function at the bottom left corner is

A = 0. Find the value of the stream function at the other vertices B to F. (You

may use either sign convention for the stream function.)

(b) Sketch the pattern of streamlines across the two cells in part (a).

Example

A

D F

C

0.3 m 0.2 m

0.1 m

E

B

5

2

12 3

17

Mesh Plots

Composite Plots

Composite Plots

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Composite Plots

Composite Plots