2. turbulence ; turbulence modellingusers.abo.fi/rzevenho/icfd19-rz2.pdf · 2019-10-10 ·...

Introduction to Computational Fluid Dynamics 424512 E #2 - rz

Introduction to Computational Fluid Dynamics(iCFD) 424512.0 E, 5 sp

2. Turbulence ; Turbulence modelling(lecture 2 of 4)

Ron ZevenhovenÅbo Akademi University

Process and Systems EngineeringThermal and Flow Engineering Laboratory

tel. 3223 ; [email protected]


oktober 2019 Åbo Akademi Univ -Thermal and Flow Engineering -Piispankatu 8, 20500 Turku

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2.1 Turbulence as a phenomenon

See also HKTJ07Section 7.1.,7.2



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Turbulence Most flows in nature and engineering are

turbulent:

Turbulence as seen by Leonardo da Vinci (B92)

See for example the tidal flows atSaltstraumen, Norway



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Laminar vs. turbulent pipe flow

HKTJ07



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Turbulent flows: eddy formation

vD82,HKTJ07



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Large-scale structure in turbulent mixing layer

N2 at 1000 cm/s over He/Ar mixture (same density)at 380 cm/s, pressure p = 4 bar.

Same as above, but at higher pressure, i.e.double Re number.More small-scale structure, unalteredlarge-scale structure

vD82

N2

He/Ar

N2

He/Ar



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Crude oil leaking from a grounded tankship (1976) at ~ 45° angle to the current (Re ~107)

vD82



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2.2 Turbulence modelling

See also HKTJ07Section 7.2, 7.3



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Equation of motion for iso-thermal, incompressible Newtonian flow: Navier-Stokes equations

Fupuut

uFF

2

L*,uρ

p*p,

L

tu*t,

u

u*u

F

0

2*

Re

******

*

*

u

Fupuu

t

u

FF

velocity vector u,fluid density F,

time t, static pressure p,

dynamic viscosity F,external force(s) F

With dimensionless variables:

FF

FF ν

Lu

η

LuρRe

(H75,W93,BSL60)

See lecture 1



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Turbulence characteristics Round tube flow : Re < ~ 2100 – 2300 laminar

~ 2200 < Re < ~ 6000 turbulent (fine scale)

Re > 6000 turbulent (large scale)

Velocity fluctuations, v', superimposed on average velocity. Reynolds decomposition: v(t) = vavg + v'(t)

Energy dissipation (W/m³ or W/kg) Fluid dynamic, “kinematic” viscosity

= dynamic viscosity/density = / (m²/s)

(H75,W93,BSL60)



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1. Turbulence is not unorganised: hence, random distributions for describing turbulent fluctuations are wrong.

2. Turbulent flows are often characterised by vortices.3. Turbulence is 3-dimensional !!!4. Turbulence may be isotropic: all gradients the same in all

directions5. Turbulence may be homogeneous: independent of position6. Turbulence is dissipative

e.g. turbulent kinetic energy k = ½ (ux´² + uy´² + uz´²)(unit: J/kg) energy dissipation = - dk/dt (unit: W/kg)

k- model (isotropic)

Turbulence characteristics /2

(B92,W93)


Turbulent flow and time averaging


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S10



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Turbulent scales: length, time, velocity Turbulent macroscale : length l0 (also called integral length scale),

~ 0.1 × system size scaletime scale = l0 /v'.

Turbulent microscale : Kolmogorov length scale lK = (³/)¼

Kolmogorov time scale K = ( /)½

Kolmogorov velocity scale vK = ( )¼

Turbulent Re-number : Rel = v' l0 / = (l0/lK)4/3

Average scaling : Taylor length scale, lT, and lT/l0 = 10/Rel

lT ~ average size of dissipating eddies.

Turbulent Taylor Re- number : ReT² = 10·Rel.

(H75,W93,BSL60)

See also slide 45



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Turbulent energy spectrum, showing energy cascade (left to right)

lK = Kolmogorovlength scale

lT = Taylor length scale

l0 = integral length scale (~ 0.1 × system scale)

units: e(k) = W/m³k 1/m

note : k = wavenumber,is not k of k- model !

ENERGY

1/k


Turbulent energy spectrum


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S10



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Turbulence solutions: DNS, RANS, LES

HKTJ07



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Turbulence solutions: DNS, RANS, LES

HKTJ07

Chemistry,if taking place,would occur

here



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2.3 Turbulence – statistical description

Material from: HKTJ07Section 7.3

Comment:

RZ doesn’t favour usingthe material derivative



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Solutions of the N-S equations



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Turbulent fluctuations

Measured velocity and temperature fluctuations at a point in a turbulent flow



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Turbulent pipe flow

Flow velocity distribution or profile across a pipe cross-sectionleft: time realisations right: profiles for time-averagedturbulent flow and laminar flow

Largergradients

near the wallresult in

higher pressuredrop, and

more intensewall

heat transfer



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Time averaging

A stationary system vs. a non-stationary system with fluctuations

pic: T06



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Reynolds averaging /1

7.1

7.2

7.3

7.4



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Reynolds averaging /2

7.7

7.6

7.5



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Reynolds decomposition - rules

For exampleΦ = a velocity termΨ = density

7.8!!



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Correlated / non-correlated fluctuations

7.7



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Reynolds averaged conservation equations /1

γm = body forceγa = surface force

7.10

7.9



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Reynolds averaged conservation equations /2

7.10c

7.13



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Reynolds averaged conservationequations /3 - incompressible flow

7.13

7.12

7.11



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Reynolds averaged Navier-Stokes (RANS) equations /1

(Navier-Stokes)

”Reynolds stresses”

See section 1.4b

7.14



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Reynolds averaged Navier-Stokes (RANS) equations /2 - energy, mass

7.16

7.15



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Turbulence modelling: averaging Reynolds averaging:

signal = average signal + fluctuation

Taking into account density fluctuations density weighted average / Favre averaging

0' where),(')(),( utxuxutxu

tt

tt

dttxut

xu ),(1

)(0

lim

0" with ),("),(~),( utxutxutxu

'')')('( and /~ uuuuuuu

ρ

'uρk i

example for

~



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2.4 Turbulence – features




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Turbulence features /1



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Turbulent energy cascade



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Turbulent structure isotropy



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Turbulence mechanisms



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2.5 Turbulence – characterisation and scales of turbulence




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Turbulence intensity

7.17

7.18

7.19

7.20



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Turbulence scales /1

7.22



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Turbulence scales /2

τK lK

lT

Symbols also used here:

7.23

7.24



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Turbulence Reynolds numbers /1

ReT

Rel

t2

7.26



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Turbulence Reynolds numbers /2

7.27

7.28

7.29

7.30



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Direct simulation – prospects /1

7.31



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7.32

7.18



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See previous

slide

7.18



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Two-point correlations /1

7.33

7.34



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L11 L22L22

7.35

7.36



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223

22

21 2

3

2

1uuuuk

k7.24.

7.20



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The turbulent energy spectrum /1

7.37

7.38

7.41



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7.42

7.43

7.44



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7.21

See alsoslide 13




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2.6 Turbulent transport phenomena


Note: most of section 8.1 and 8.2 NOT part of this course



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Example situations



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Convective processes


oktober 2019 Åbo Akademi Univ - Thermal and Flow Engineering -Biskopsgatan 8, 20500 Turku

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2.7 Turbulence modelling: The closure problem

See also HKTJ07Section 9.1



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Turbulence modelling /1 The closure problem

The Reynolds averaged Navier-Stokes (RANS) equations:

10 unknowns: p, ux, uy, uz + 6 Reynolds stresses ui’uj’only 4 equations a turbulence model is needed

i

j

j

i

ij

jijii

i

i

x

u

x

u

x

p

x

uuuu

t

u

x

u

)''()(

0)(

F97



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The closure problem /1 The RANS equations give unknown second moments of

the type Φuj

Solutions for these, i.e. additional model equations must be provided the Turbulence Closure Problem

In principle, for example, multiplying the N-S equations for ui and uj gives terms of the form uiuj, but at the same time higher order terms arise like uiujuk (one may wonder what for example uiujukul means...)



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The closure problem /2 Fortunately, the higher order terms become less and less

important and allow for simple approximations– Eddy viscosity models (first order models)– Reynolds stress models (second order models)



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RANS



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2.8 Eddy viscosity / eddy diffusivity(zero-equation) models

See also HKTJ07Section 9.2,9.3



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Eddy viscosity / eddy diffusivity /1

See lecture 1

9.3



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Eddy viscosity / eddy diffusivity /2

λeff = λ + λt 9.5



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Turbulence modelling /2 Zero-equation models (Boussinesq 1877, Prandtl 1925)

with eddy viscosity µt, and turbulent kinetic energy k:

turbulence can be characterised by k and a length scale L (mixing length):

kx

u

x

uuu ij

i

j

j

i

tji 3

2''

)''''''½(''½ zzyyxxii uuuuuuuuk

09.0 and ckLct F97



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Turbulent viscosity, mixing length

Assuming that large scale eddies are most important

9.6

!!



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Mixing length

See bookSection 8.2

(not part of the course)

9.7

9.8

9.9



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2.9 One-equation (k), two-equation(k-ε, etc.) models

See also HKTJ07Section 9.4, 9.5



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Shortcomings mixing length concept

9.6



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One- and two-equation models

Note fluctation u = Û – UÛ = Reynold decomposition of U

9.12



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One-equation models

The k-equation



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Turbulence modelling /3 One-equation models (Taylor 1935, Kolmogorov, 1942,

Prandtl 1945)

dissipation vs. turbulence length scale

with turbulent Prandtl number k ~ 1

j

i

i

j

j

i

tjk

t

jj

j

x

u

x

u

x

u

x

k

xx

ku

t

k

L

kc

L

k

)(

~2

3

2

3

F97

See p. 66



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Two-equation models: kpLq

Whatever p and q: length scale or time scale follows from kpLq, and eddy viscosity from µt = Cµρk½L = Cµρk2/ε 9.14



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The k-equation /1 See book p. 215

9.15



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The k-equation /2



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The k-ε model /1

Compare with (9.15) 9.29

9.36



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The k-ε model /2

9.39



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Wall functions; low Re numbers

.

SeeSection 8.2



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Turbulence modelling /4 Two-equation models: k- model (Launder and Spalding 1972)

Turbulent kinetic energy, k :

Turbulent kinetic energy dissipation, (= - dk/dt)

with µt= Cµk²/ Cµ = 0.09 C1 = 1.44 C2 = 1.92 k = 1.0 = 1.3

j

i

i

j

j

i

tjk

t

jj

j

x

u

x

u

x

u

x

k

xx

ku

t

k)(

kC

x

u

x

u

x

u

kC

xxx

u

t j

i

i

j

j

i

tj

t

jj

j2

21

F97In the k-ε model, eddy viscosity as µt = Cµρk2/ε, with length scale k3/2/ε



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Example k - ε



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Example k - ε

Source: TUD exam FT2-2002

see p. 20

See also book p. 224 (9.40 & 9.41)



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2.10 Other two-equation models

See also HKTJ07Section 9.6



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Other two-equation models /1

9.64



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Not convenient: k-τ, k-L, k-kL use k-ε or k-ω

9.36



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See Section 8.1, 8.2


Limitations of 2-eq. linear EVMs


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2.11 Reynolds stress models



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Reynolds stress models

The eddy viscosity model has limitations, for example, in 3-D, µt may not be a scalar.

Triple correlations of the velocity correlations ui’uj’uk’ and correlations between velocity and pressure uj’p’ occur.

Modelling these is complicated but can give very goodresults, especially in cases where k-ε fails: swirlingflows, flows with curvatures or curved surfaces

See book section 9.9 p. 236-238F97



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2.12 Large-eddy simulations (LES)

See S10 section 11.4 See also B01



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LES /1

B01

compared to DNS

Piomelli, AIAA paper 98-0534, 1998



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LES /2 spatial filtering

B01

D


Filters, effect of filtering


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S10


LES and turbulent energy spectrum


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S10



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LES /3 Incompressible N-S

B01The detailed grid must be linked to the less-detailed grid in

a physically correct way.



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LES /4 Sub-grid scale modelling

Domaradzki & Saiki, 1997

B01



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LES /5 Sub-grid scale modelling

B01



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LES /6 Wall models

B01


LES and near-wall region


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S10



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2.13 Turbulent reacting flows(with k-ε turbulence) modelsfor example: combustion

See also P92, P97



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Turbulent reacting flow modelling (k- ) /1

Favre averaged balance equations:

with perpendicular coordinates and P92



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Favre averaged Reynolds’ stresses:

P92



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Favre averaged turbulent kinetic energy and dissipation:

with, in the standard k- model, 1.3, C1 1.44, C2 1.9

P92



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Temperature and mass fractions for n species:

JT, = heat diffusion (conduction), Ji, = mass diffusionYi = mass fraction, hi = enthalpy , mi = chemical reaction source term

P97



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Turbulent reacting flow modelling /5

Direct numerical simulation (DNS): only possible for ReT ~ few 100with one or two chemical reactions

Probabilistic representations of turbulence, and turbulence/chemistry interactions (PDF methods)

Large eddy simulations (LES) : DNS for the large scales, a simpler sub-model for the smaller scales. Accurate for cell size > 30 lK.

Eddy break-up (EBU) models (based on "mixed = converted") : the reaction rate is determined by a typical turbulence time (/k) and the mean square fluctuations of the reactants concentration.



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Modelling the chemicals’ source terms (i.e. dm/dt):the Eddy Break-Up concept EBU.

“Small scale mixing down to the molecular scales controls the chemistry”Turbulent reaction rate (formation of products):

where Y “pr2 is the Favre variance of the

product mass fraction, andCEBU = 0.35 is the Eddy Break-Up constant


~

P97



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Sources / further reading #2

B92: Banerjee, S. “Turbulence structures” Chem.Eng.Sci. 47(8) (1992) 1973-1817B01: J. Blazek ”Computational fluid mechanics: principles and applications” Elsevier (2001)BSL60: Bird, R.B., Stewart, W.E., Lightfoot, E.N. “Transport phenomena”, John Wiley & Sons, New York (1960) Chapter 6vD82: van Dyke, M. “An album of fluid motion”, The Parabolic Press, Stanford (CA) (1982)F97: J.H. Ferziger, M. Perić "Computational methods for fluid dynamics", Springer, Berlin (1997) Chapter 9HKTJ07: K. Hanjalić, S. Kenjereš, M.J. Tummers, H.J.J. Jonker “Analysis and modelling of physical transport phenomena” VSSD, Delft, the Netherlands (2007, 2011)ÅA library 13 hardcopies (ASA): https://abo.finna.fi/Record/alma.896997

H75: Hinze, J. O. “Turbulence” (2nd Ed.) New York: McGraw-Hill (1975) Chapters 1,3,5

…………………. continues on next page



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Sources / further reading #2

P92: N. Peters "Fifteen lectures on Laminar and Turbulent Combustion" RWTH Aachen, 1992 https://www.itv.rwth-aachen.de/fileadmin/Downloads/Summerschools/SummerSchool.pdf

P97: N. Peters "Four lectures on Turbulent Combustion" RWTH Aachen, 1997 https://itv.rwth-aachen.de/fileadmin/Downloads/Summerschools/SummerSchool97_ueberarbeitet.pdf

S10: O. Zikanov ”Essentional Computational fluid dynamics” Wiley & Sons (2010)ÅA library: https://ebookcentral.proquest.com/lib/abo-ebooks/detail.action?docID=819001

T06: S.R. Turns ”Thermal – Fluid Sciences”, Cambridge Univ. Press (2006)TYL12: Jiyuan Tu, Guan Heng Yeoh, Chaoqun Liu Computational Fluid Dynamics: A Practical Approach. (2nd Ed.) Elsevier (2012) ÅA library: https://ebookcentral.proquest.com/lib/abo-ebooks/detail.action?docID=1012531

W93: Wilcox, D.C. "Turbulence modelling for CFD", DCW Industries Inc., La Cañada (CA), (1993)

2. turbulence ; turbulence modellingusers.abo.fi/rzevenho/icfd19-rz2.pdf · 2019-10-10 ·...

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