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Transcript of 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
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 Examples Description Reynolds number Kolmogorov cascade Summary
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
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Outline Examples Description Reynolds number Kolmogorov cascade Summary
Outline of this part
Examples of turbulent flow
Description of turbulence
The Reynolds number
The Kolmogorov Cascade
Summary
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Outline Examples Description Reynolds number Kolmogorov cascade Summary
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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Outline Examples Description Reynolds number Kolmogorov cascade Summary
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
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Outline Examples Description Reynolds number Kolmogorov cascade Summary
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
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Outline Examples Description Reynolds number Kolmogorov cascade Summary
Turbulent combustion III
Ignition simulation in an annular combustor
(Source: CERFACS)
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Outline Examples Description Reynolds number Kolmogorov cascade Summary
Outline of this part
Examples of turbulent flow
Description of turbulence
The Reynolds number
The Kolmogorov Cascade
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
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Outline Examples Description Reynolds number Kolmogorov cascade Summary
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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Outline of this part
Examples of turbulent flow
Description of turbulence
The Reynolds number
The Kolmogorov Cascade
Summary
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Notes
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Outline Examples Description Reynolds number Kolmogorov cascade Summary
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
Outline Examples Description Reynolds number Kolmogorov cascade Summary
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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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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Outline Examples Description Reynolds number Kolmogorov cascade Summary
Outline of this part
Examples of turbulent flow
Description of turbulence
The Reynolds number
The Kolmogorov Cascade
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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Outline Examples Description Reynolds number Kolmogorov cascade Summary
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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Outline Examples Description Reynolds number Kolmogorov cascade Summary
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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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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Outline Examples Description Reynolds number Kolmogorov cascade Summary
Approaches to deal with turbulence in CFD II
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Outline Examples Description Reynolds number Kolmogorov cascade Summary
Outline of this part
Examples of turbulent flow
Description of turbulence
The Reynolds number
The Kolmogorov Cascade
Summary
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Outline Examples Description Reynolds number Kolmogorov cascade Summary
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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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
94
, 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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