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Computational Fluid Dynamics –Lecture 8

Prof. Jiyuan Tu

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Solution Errors--Causes Solution error depends on:

Discretion error -- usually the dominant contribution Equation solver error  Choice of computational domain Implementation of boundary and initial conditions

Discretization error depends on: Grid size (overall refinement Grid quality (aspect ratio! ortho"onality Grid density (local refinement 

Discretisation formula (lo#$hi"h order

Equation solver is: (usually a minor source of solution error  can be source of instability (or poor iterative convergence )

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Sources of errors in CFD

（ I） Discretization error (DE

Computer round-off error (%&E

Errors due to physical modelin" (E'

  ()urbulence modelin" *uman errors + ine,perience 

Wrong Boundary Condition

 Bad numerical scheme

Wrong computational domain

 Bad computational model 

Garbage in!

Garbage out!

 Mesh

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Sources of errors in CFD

（ II） DE-)runcation error 

∂

∂∆+

∆

−=

∂

∂   +-

-

.

 x

 xo

 x x

ii

i

φ φ φ φ 

∂

∂∆+

∆

−=

∂

∂   +

.

 x

T  xo

t t 

n

i

n

i

i

φ φ φ 

)runcation error /irst order 

0ocal error  Space

)imeGlobal error 

The local and global discretization errors

of finite difference method at the third

time step at a specified nodal point 

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Sources of errors in CFD

（ II） %&E++Di"its1 di"its++Sin"le precision

.2 di"its++Double precision

 SP 

:

4444.6667 4444.666666 

4444.6666 

 A!" A!"

≠Example:A simple arithmetic

operation performed

with a computerin a single precision

using seven significant

digits

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Sources of errors in CFD

（ III）

 3o4 of Computations

5ccumulated %&E   ↑

↑

As the mesh or time step size decreases,

the discretization error decreases !

but the round-off error increase!

a6or error source in C/D

E'++0aminar /lo#

  ++)urbulence /lo#

odelin"

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  Solution Integrity 

7hy is predictive reliability important 8 

 Is the computer  (#uman$ #ard%are& infallible8  7hat should #e e,pect:

solutions are accurate

 ' can be validated against reliable eperiments

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Testing Solution

Integrity Set up physical e,periment and measure 9ey data  )pensive$ time*consuming 

Compare #ith personal e,perience

+e ,no% %#at to epect (most of t#e time&

Compare #ith standard cases;  )quivalent to -alidation/ 

%ely on theoretical foundation  )quivalent to -erification/  

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Computational Solution

C/D is implemented by t#o-sta"e process:

 iscretisation ‑ !onversion of t#e governing partial

differential equations into a system of algebraic equations

 

 )quation Solver  ‑ iterative solution of t#e algebraic

equations to provide t#e approimate solutions

 

Overview of the omputational olution "rocess

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(Grid Conver"ence 

C/D produces an appro,imate solution solution error 5 eact solution ** approimate solution

(Grid Conver"ence  epect solution error 5 $ as $ t 5 refine grid until t#e solution no longer c#anges

 Consistency@Stability AB (Grid Conver"ence

 

#terative convergence

 

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Comments -- Conver"ence

C&3SIS)E3C @ S)5I0I) AB C&3

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Consistency Definition:  As $ y$ 2$ t 55 $ t#e system of algebraic

equations s#ould recover t#e governing partial differential

equation at eac# grid point 

Comments: 0est by epanding all nodal values of t#edependent variables about t#e control volume centre

E,ample: ;ass conservation equation 

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  Taylor Series Expansion about

point    )qn (ran,el sc#eme

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  /inite Grid Solutions (. Comments: Grid refinement may be restricted by memory si2e or !P? time

@btain t#e most accurate solution %it# fied 9$ 9B$ 9C 

 Some grids can increase accuracy but increase t#e number of iterations to

convergence of t#e algebraic equation solution  )pect solution error to follo% truncation error 

0ypical truncation error:

  (    = 36  &D ∂ E( ρu&3 ∂  E F (   y= 36  &D ∂ E( ρv&3  ∂ yE F

0#erefore refine grid  %#ere solution gradients large:  boundary layers$ up%ind stagnatn points$ for%ard*facing corners

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  /inite Grid Solutions (

Is the "rid fine enou"h8 refine grid until important parameter no longer variant  eg force against a %all 

Parameter

Value

Number of elements

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E!uation Structure ost industrial fluid flo#s involve si"nificant motion

omentum equations describe three ma6or interactions

(convective& transport****************** motion of fluid 

diffusion********************* (turbulent& eddy diffusivity source terms********production of turbulent ,inetic energy

Is solution accuracy sensitive to discretisation of specific

terms 8  (YES

 

$%&!

xxx

u

t ' ' ' 

'  

∂

φ

Γ

∂

∂

 

∂

φ

 

∂

ρφ

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"ig#er $rder Interpolation%I&

 !omments:

  So far #ave interpolated i.e. depends on local values

 9o% interpolateassuming -u/ is positive

( ) E  pe

  !"" #  "   =

( )   ( ) E W WW W  E  pW e  !"" !" #  "and  !"" !" #  "   ==

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"ig#er $rder Interpolation%II&

(eneral three point interpolation:

  and equivalent formula for and  

q  iscretisation Sc#eme @rder (0.).& 

< !entered difference =  E*pt up%ind =

  E34 ?1!H (4 pt& =

  =3E 4*pt up%ind E

[ ]   ( )( )[ ]   ( )   ($% &x &x '  &x"&& &x")($

 &x &x '  &x" &x") "

 pW  pW  pW  p

 E  p p E  E  pe

+−+−+

++=

W "  '&' "(" x"'  W e −=∂∂

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'ounded "ig#er $rderSc#eme 3umerical dispersion may appear as #i""les ounded second order scheme:  1n (

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Comments 5bove bounded schemes available in /0FE3) ounded schemes more accurate but less robust than po#er

la# scheme /or fast iterative conver"ence #ith hi"her accuracy! start from

conver"ed po#er la# solution oundin" is effectively introducin" very localised numerical

dissipation

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  (nstructured )ridDiscreti*ation

'o#er-la# (segregated eqns. only >ace value obtained from solution toace values obtained t#roug# multi*dimensional reconstruction

FIC scheme( for quad.3#e. cells and segIated eqns.  Jig#er*order construction of face values from S@? and

interpolation in mes# direction  ;ore accurate for structured mes#es t#at are mainly

  flo% aligned 

7c

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  Second $rder (p+ind %S$(&

0inear reconstruction  provides =nd order accuracy on unstructured grids up%ind values obtained from linear$ piece%ise discontinuous

s#ape functions limiting  is used to -suppress %iggles/ 

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 Linear ,econstruction 0inear reconstruction provides:

better accuracy t#an stencil*based sc#emes compatibility %it# arbitrary cell s#apes (tetra#edrals$

triangles&

improved accuracy on s,e%ed grids

Comments: uses more

information t#an

stencil*based 

 sc#eme eample:

diffusion terms

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 Structured s (nstructured

5ccuracy: bot# can ac#ieve =nd rder accuracy for t#e convective terms structured grids rely on truncation error reduction unstructured grids rely on linear reconstruction

Economy: structured grids lead to fe%er operations in t#e discretised equations unstructured grids can cover a domain %it# fe%er cells

%obustness:

reliable algorit#ms available for bot# types solution adaption on unstructured grids is less li,ely to affect

robustness limiters can be introduced for bot# to avoid -%iggles/

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'roblem definition  turbulent or laminar flo%K steady or transient L  is t#e p#ysical model$ eg granular multip#ase$ inaccurate L

Geometry and "rid  is t#e imported !A file correct L

oundary Conditions  is t#e upstream boundary too close to t#e body L

Solution method  is a #ig#er*order sc#eme required L

Systematic rocedure forSolution Integrity -- $erie+

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Define clearly #hat the problem is +#at do you %ant to find outL

+#at are t#e important parameters you need to inputL

+#at %ill be t#e defining c#aracteristics of t#e flo%(eg turbulent #eat transfer L&

0oo9 for computational efficiencies !an you ma,e any simplificationsL

 Jo% muc# of t#e real domain do you need to modelL

!an you run any simple cases first to test your modelL

)uidelines – roblem

De.nition

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5ny possibility of import8  A!1S 3 1G)S 

5ny simplifications8  SymmetryL

 Periodic "oundariesL

Fse >top do#n? approach to "eometry creation

Consider dividin" the domain up into smaller sections formore control over the "rid

a9e use of 6ournal files K

 Parametric modelling   )asy transport of geometry specification files

)uidelines – )eometry

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)rid /uality Grid aspect ratio: 

A) * + x Comments:

 9eed to c#oose y small if rapidsolution c#ange in t#e y direction 

#f A) ./0 or A) 1 2! possible reduction in accurac+ ma+be poor iterative convergence &or divergence$

Grid distortion: @rt#ogonality (   5 M deg& desirable Comments:

!#oose grid so t#at 4N deg O O

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  Sudden C#anges in )rid Si*e

 Comments:   !ould occur at bloc, boundaries in multibloc, procedure

 !ould occur at duct inlet to a plenum c#amber 

 E,ample:  Mass conser1ation e2uation

 Comments: 0.). contains diffusion terms (=nd derivs&**destabilising %#en r     <  ;a,e sure grid c#anges slo%ly and smoot#ly

 iscretisation of =nd derivatives requires very smoot# grid c#anges

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Does your selection of boundary conditions match the real

#orld conditions 8 eg ,$ epsilon c#ange rapidly ust do%nstream

  of inlet value specification

Is it possible to limit the domain size by specifyin" the

 boundary condition in more detail 8 eg reduce upstream pipe lengt#$ if specify inlet profile

Fse the >patch? command to fill areas after initialization4

)his is particularly useful for free surface problems4 5re the boundaries in the correct locations8

eg are far*field boundaries far enoug# a%ayL

)uidelines – 'oundary

Conditions

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If the residuals are diver"in":  isplay t#e contours after initiali2ation. Are t#e initial conditions

correctL

!#ec, t#e models. ;aybe start as laminar and s%itc# to turbulent

later in t#e solution$ for eample. If the residuals initially reduce = then are oscillatory:

 1f flo% is assumed steady$ rerun as a transient problem

!ould a different type of boundary condition be more stableL (i.e.

outflo% instead of pressure boundaryL&

!#ec, for %#ic# equation residual is largest 

GuidelinesLSolution Conver"ence

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*i"her order differencin" schemes are required foraccuracy  Qun t#e solution first %it# default sc#emes$ t#en s%itc# to #ig#er

order once converged 

Is the problem #ell-posed 8

  o t#e boundary conditions suit t#e problem L

 1ncorrect specification of nearby boundary conditions

5dequate "rid resolution 8  istorted volumes solution adaption or revise t#e grid 

 Jig# gradients ' coarse grid solution adaption

Guidelines L Solution 5ccuracy

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Conclusions

5ssess C/D solution inte"rity p#ysical eperiments personal eperience t#eoretical foundation

E,pect computational solution to conver"e to the e,actsolution as ∆,! ∆y! ∆z! ∆t AAB M

(see, Rgrid * independentR solution&

/inite "rid solutions Avoid  *** sudden c#anges in grid si2e

  *** large c.v. aspect ratios*** grid distortion

  *** large c.v. area variation over domain