Turbulens – Teori ochmodellering

Introduction

Two questions:• Why did you chose this course?• What are your expectations?

Turbulence – Theory and modelling

•Understanding the phenomena that affects the transition from laminar to turbulent flow

•Knowledge about the theory for describing turbulent flow

•Knowledge about turbulence models applicability and limitations

•Ability to analyse a flow situation and chose a propper modelling approach accordingly

Goals

Turbulence – Theory and modelling

•Be able to describe the physical mechanisms of the transition from laminar to turbulent flow for a simple flow case•Be able to explain Kolmogorov’s theory, including the basic assumptions and the validity of the theory•Be able to, from a phenomenological perspective, assess if a flow is turbulent•Be able to explain some of the important and basic terms of the subject•Be able to describe the character of the turbulence in different flow situations with respect to the properties and development of the turbulence, and explain how the differences between these flow situations are reflected in the modelling

Goals

Turbulence – Theory and modelling

•Be able to analyse a flow case and suggest a method for numerical simulation with respect to governing equations, possible simplifications and choice of turbulence model, and also to compare with alternative methods.•Be able to scrutinise and from given criteria estimate the credibility of results from turbulent flow simulations•Be able to actively participate in discussion of problems relevant for the subject•Be able to present, both orally and in writing, a technical report containing analyses and choice of turbulence model

Goals (continued)

Turbulence – Theory and modelling

•To pass (grade 3) the following is required:• Approved home works, lab-report and group study (GS)• Participation in the computer exercises

•Oral exam for higher grade (grades 4 and 5)•Participation in the laboratory exercise, computer exercises and the guest lecture is mandatory.•The home works are handed in individially. However, you are alowed, even encouraged, to work in groups discussing the problems.•The groups study is to be presented both in writing as well as orally. One report per group.

Examination and requirements

Turbulence

Two questions

1. How would you describe turbulence? Think about key-words to characterise it.

2. Think about situations where turbulent flow is better than laminar and vice versa.

Turbulence

• Random• 3D• Diffusive• Dissipative• Property of the flow• High Reynolds number• Continuum

Turbulence

Big whirls have little whirlsWhich feed on their velocityLittle whirls have lesser whirlsAnd so on to viscosity – in the molecular sense

L F Richardson

outflow

Heating/Cooling

Impingement wall

nozzleMean Sherwood number Sherwood number fluctuation

0=i

i

xu∂∂

ij

ij

j

i

jij

jii Fxx

uxx

pxuu

tu

+−+−=+∂∂τ

∂∂

∂∂ν

∂∂

ρ∂∂

∂∂ 1

Models for turbulence,

combustion etc.

Geometry

Mathematical description Results

For example velocity, pressure,

temperatureNumerical methods

Question

• Why do we need to model turbulence?

Turbulence modelling

Example:

Pipe flow, turbulent Reynolds number, Re = 10000

Relation between largest and smallest scales ~ Re3/4

⇒ No. of nodes ~ Re9/4 ≈2⋅109 ⇒ ca. 30 gigabyte RAM

Conclusion: Model needed

Turbulence modelling

Isotropic turbulence in a box

Turbulence modelling

Direct simulation of isotropic turbulenceRequired time at a computing rate of 82 Gflop

Re N N3 M N3M CPU time Memory

94 104 1.1E06 1.2E03 1.3E09 14s 18 Mb

375 214 1.0E07 3.3E03 3.2E10 6.6 min 150 Mb

1500 498 1.2E08 9.2E03 1.1E12 3.8 h 2 Gb

6000 1260 2.0E09 2.6E04 5.2E13 7.3 days 30 Gb

24000 3360 3.8E10 7.4E04 2.8E15 1.1 years 565 Gb

96000 9218 7.8E11 2.1E05 1.6E17 61 years 11 Tb

N3= number of grid pointsM= number of time stepsN3M= total work required

TurbulenceInstantaneous velocity field of a round jet

Turbulence

Mean velocity

Turbulent kinetic energy

Turbulence data have meaning only in a statistical sense

Turbulence modelling

RANS models

A.Abdon och B. Sundén, Värmeöverföring, LTH

Discussion:

Are these flows laminar or turbulent? Motivate.

Turbulence

Brief history:• 15th century, da Vinci, observations of turbulence

• 18th century, Euler, equations for inviscid flow• Early 19th century, Navier and Stokes, the N-S equations• 1883, Reynolds, flow instability in pipe flow

• 1904, Prandtl, boundary layer theory• 1941, Kolmogorov, theory on turbulent scales• 1963, Smagorinsky, first sub-grid scale model for LES

• 1970, Launder et al., two-equation model for turbulence

Turbulence

Leonardo da Vinci

Turbulence

Laminar

Turbulent

Osborne Reynolds (1883)

νUL

=ReReynolds number: