Lec3 Intro Turb

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Outline Examples Description Reynolds number Kolmogorov cascade Summary

DEN403/DENM010

Computational Fluid DynamicsPart 3: Introduction to turbulence

Dr. Jens-Dominik Muller

School of Engineering and Materials Science,

Queen Mary, University of London

[email protected]

Room: Eng 122office hours: any reasonable time

c Jens-Dominik Muller, 2011

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Organisation of the lectures on turbulence

1. Introduction

• Motivating examples, description of turbulence• The Reynolds number• The Kolmogorov cascade

2. Reynolds-Averaged Navier-Stokes

• Averaging the Navier-Stokes equations• Reynolds stresses, closure• Modelling the Reynolds stresses

3. Using RANS• The near-wall structure of turbulent boundary layers• Mesh spacing requirements, wall functions• Limits of applicability

4. Alternative approaches

• DNS, LES, DES

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Outline of this part

Examples of turbulent flow

Description of turbulence

The Reynolds number

The Kolmogorov Cascade

Summary

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Notes

Notes

Notes







The Reynolds number


Summary

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On Turbulence

Benoit Mandelbrot:

”The techniques I develo ped

for studying turbulence, like

weather, also apply to the

stock market.”

Werner Heisenberg:

”When I meet God, I am

going to ask him two questions: Why relativity?

And why turbulence? I really

believe he will have an

answer for the first.”

(Source: Great-Quotes.com, Wikipedia)

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An artist’s view of turbulence

Leonardo da Vinci

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Notes

Notes

Notes




Turbulence on a global scale

Flow around Selkirk island

(Source: NOAA)

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Reynolds experiment

(Source: Reynolds, 1883)

Turbulence in smooth pipes

typically occurs above

Re = 2000.

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Effect of Reynolds number

Re = 15, 000 Re = 30,000

(Source: van Dyke: Album of fluid motion)

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Notes

Notes

Notes




Laminar and turbulent flow II

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Turbulent combustion I

Turbulent mixing downstream of a swirler

(Source: CERFACS)

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Turbulent combustion II

Turbulent combustor

(Source: CERFACS)

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Notes

Notes

Notes




Turbulent combustion III

Ignition simulation in an annular combustor

(Source: CERFACS)

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The Reynolds number


Summary

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Characteristics of turbulence

• Turbulence is inherently unsteady and 3-dimensional.

• Turbulence is dominated by chaotic - but not random

motion of swirling structures, the eddies .

• There is a cascade of eddies, largest eddies determined

e.g. by the geometry.

• Largest scales take their energy from mean flow.

• Larger eddies break up, passing their energy to smaller

scales.

• Smallest scales dissipate their energy into heat.

• Is always dissipative, i.e. increases mixing, disorder.

• The Reynolds number will play a major role.

• Nearly all relevant industrial flows are turbulent!

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Notes

Notes

Notes




How does turbulence arise?

A flat plate boundary layer

• starts out laminar

• transitions from laminar

to turbulence after

some running length

• remains turbulent

downstream of

transition

• transition modelling is

very complex: in CFD

typically full turbulence

is assumed.

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Laminar vs. turbulent boundary layers

• The laminar b.l. profile

has a lower velocity

gradient ∂ u ∂ y

near the

wall, hence a lower

wall shear stress

• The turb. b.l. profile has

a higher velocity near

the wall, hence is more

resistant to separation• The turb. b.l. has more

mixing, hence heat

transfer or surface

reactions are

enhanced.

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The Reynolds number


Summary

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Notes

Notes

Notes




The Reynolds number in the momentum equationsThe momentum equ. incompr. Navier-Stokes equations in

vector notation:

∂ u

∂ t + u · u

= −p + µ2u,

where u is the vector of velocities. The unit of the equation, as

stated above, is force per volume: F /V = m a /V = a . Dividing

the equation by this factor of this dimension, u 2/D , which is

equivalent to normalising the variables by

u =u

U , p = p

1

U 2,

∂

∂ t =

D

U

∂

∂ t , = D

makes the equ. nondimensional:

∂ u

∂ t + u · u = −p +

1

Re2u,

Note: for Re→ 0 the effect of the viscous term vanishes, but

the no-slip condition at the wall may remain! 19/37


Reynolds number and Turbulence

Re =inertial forces

viscous forces=

momentum of the flow

viscous stress

=u 2

µ∂ u ∂ y

=u 2

µu l

=ul

µ=

ul

ν

• When Re→ 1 the flow is very viscous (creeping flow).

• As Re→∞ the flow becomes less dominated by viscosity,

and boundary-layers confined to small region near

surfaces.

• The Reynolds number depends on the choice of

length-scale!

• Choosing an overall length-scale, e.g. aerofoil chord

length, provides only analysis of ‘overall’ effects.

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Reynolds number based on length

• Most simply: base Reynolds number on the length of the

body L,

• but a boundary-layer grows with distance (δ ∝ x 0.5 for a

laminar boundary-layer) L and δ are inter-related.

• Instead, base Re on distance from the L.E.:

Re L = UL/µ

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Notes

Notes

Notes




Reynolds number for a boundary layer

• Reynolds number:

Re =momentum of the flow

viscous sresses

• momentum of the flow: ≈ U 2

• viscous sresses:

τ = µ (du /dy ) ≈ µ (U /δ)

• Reynolds number based on

b.l. thickness:

Re δ =U 2

µ (U /δ)=

U δ

µ=

U δ

ν

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The Reynolds number


Summary

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Effect of Reynolds number on small scales

med.

Re

higher

Re

(Source: van Dyke: Album of fluid motion)

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Notes

Notes

Notes




Reynolds number based on eddy diameter d

Re =inertial forces

viscous forces=

momentum of the flow

viscous stress

=u 2

µ∂ u ∂ y

=u 2

µ u d

=ud

µ=

ud

ν

• For Re >> 1, inertial forces dominate. The flow keepsswirling, energy is passed down to smaller scales.

• When Re→ 1, viscous forces become equal in magnitude

to inertial forces, eddies dissipate.

• There is a smallest length scale for turbulent eddies!

Smaller eddies are dissipated by viscosity.

• Rotational energy∼ d 2, hence smaller eddies contain less

energy.

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The Kolmogorov Cascade I

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The Kolmogorov Cascade II

• Eddy structure is fractal

• with higher Re, we find

smaller scales

• smaller scales have

smaller diameters, hence

in a flow with the same

speed lead to fluctuations

with higher frequencies

• in turbulent flow literature,

rather than higher

frequency, the term higher

wavenumber k is used.

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Notes

Notes

Notes




Spectrum of turbulent kinetic energy

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Spectrum of turbulent kinetic energy

• There is a peak of overall

turbulent kinetic energy E

at some wavenumber

k = O (L−1), i.e. some

diameter L given by the

geometry.

• In isotropic turbulence

energy drops at a rate of

k −53 with increasing

wavenumber k .

• There is a smallest

wavenumber/scale which

increases with Re.

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How small is the smallest eddy? I

• The largest eddies depend on the geometry scale, e.g. b.l.

thickness or pipe diameter.

• The scale of the smallest eddies, the scale at which

dissipation occurs, is independent of the scale of the

largest eddies or the mean flow.

• At the smallest scales there is an equilibrium between

• energy supplied by larger scales• energy dissipated by viscosity

• This is known as Kolmogorovs Universal Equilibrium Theory

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Notes

Notes

Notes




How small is the smallest eddy? II

• Define the dissipation rate per unit mass ε [m2sec−3],

• and use the kinematic viscosity ν [m2sec−1],

• using dimensional analysis we can then we can derive theKolmogorov microscales :

• Kolmogorov length scale: η = ν 3

ε

1/4

• Kolmogorov time scale: τ η =ν ε

1/2

• Using dimensional analysis we can approximate ε ∼ U 3/L

• hence the ratio of typical length L to smallest eddy size η is

Lη = L/

ν 3

ε

1/4= (UL/ν )

34 = Re

34

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Implications of eddy scaling for CFD

• size of smallest scales: Lη = Re

34

• To resolve an eddy we need at least two mesh points to

represent the velocity fluctuations: ∆x = h ∼ η.

• hence the number of meshpoints in one direction isLh

= Lη = Re

34

• for a three-dimensional calculation we need this many

mesh points in each direction, hence the overall number of

nodes N scales withRe

as N = (L/h )

3

=Re

94

• Resolving all turbulent structures is only possible for low

Re, but is prohibitive for high Re.

• DNS of a complete aircraft will require at least an exaflop

(1018 flops) computer. The best performance currently is a

around 500 tera flops (500 · 1012 flops)

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Approaches to deal with turbulence in CFD I

• Simulation of the finest scales means: resolving these

scales with mesh points such that we can accurately model

them in the conservation equations.

• This approach is called Direct Navier-Stokes (DNS), but is

not affordable even in the mid-term future.

• We are typically not interested in the fine scale

fluctuations, in engineering we care for the long-term

time-averages as they would affect the flight of an aircraft.

• Hence, we could approximate the average effect of these

fluctuations with an additional model that embodies our

knowledge of turbulent flows.

• This approach is called Reynolds-averaged Navier-Stokes

(RANS), and is the most popular approach to CFD for

turbulent flows.

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Notes

Notes

Notes




Approaches to deal with turbulence in CFD II

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The Reynolds number


Summary

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Introduction to turbulence: summary I

• Turbulence is unsteady and three-dimensional.

• The chaotic motion of turbulent flow is fully described by

the conservation equations.

• There is a cascade of eddies, largest scales determined by

geometry.

• Turbulence increases skin friction, but also increases

mixing.

• Nearly all flows of industrial interest are turbulent.

• They Reynolds number can describe turbulent effects, but

care needs to be taken to choose the correct length scale.

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Notes

Notes

Notes




Introduction to turbulence: summary II

• There is a smallest eddy scale, the Kolmogorov scale. For

smaller scales viscosity becomes dominant and eddies are

dissipated.

• The ratio of smallest to largest scales is Re−

34 .

• If we were to resolve the smallest scale in a numerical flow

simulation, a DNS, the required number of mesh points

would scale withRe

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, an unsteady computation wouldrequire an exaflop computer.

• For lower computational cost, we need to model the

time-averaged effect of turbulent fluctuations, the

Reynolds-averaged Navier Stokes approach (RANS).

• There is also an intermediate approach, the Large Eddy

Simulation (LES), where the largest scales are resolved

(simulated) and the “sub-grid” scales are modelled.

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Notes

Notes

Notes

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Lec3 Intro Turb