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Scales of TurbulenceCourse: Turbulence (AM2205)
Akshoy Ranjan PaulDepartment of Applied Mechanics
MNNIT Allahabad
You are a fluid dynamicist visiting thefamous Louvre Museum in Paris and areasked by the Curator to comment on thepaintings below. What do you say?
Painting by Van Gogh Sketch by Leonardo da Vinci
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Turbulent illustrates by this sketch of a free water jet issuing from a square hole into a pool
Non turbulent flow, Van Goghs clouds have no small scales!
thus the water has eddying motions, onepart of which is due to the principal current,the other to the random and reversemotion.
L. da Vinci
Da Vinci provided the earliest reference tothe importance of vortices in fluid motion:Finally, da Vinci's words
"... The small eddies are almost numberless,and large things are rotated only by largeeddies and not by small ones, and smallthings are turned by both small eddies andlarge .."
Confirmed in Richardson's energy cascade, which is coherent structures, and large-eddy simulations, at least.
The world's first use of visualization as a scientific tool to study turbulence
Leonardo da Vinci
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Demonstrated by an experiment first reportedby O. Reynolds (1883)
Flow inside a pipe becomes turbulent every time a single parameter Re would increase
Dye injected on the centerline
Re=UaxialD/n
No change in time, streamlines// pipe axis
Re >2300, turbulentOccurrence of small scales.
Generated by the inertial forces and dissipated by the viscous forces.
Flowing water
From laminar to turbulent flow
Dynamics of large scale structures
Hydrodynamic stability (cf. lecture F. Gallaire) explainshow structures of a specific frequency and scale are selected and emerge.
2D cylinder (Williamson 1996)
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From Laminar to Turbulent flow
Turbulent flow: Large-scale structures + small-scale turbulence
Turbulence in a Box Lets consider a periodic box:
u(0)=u(L)
Possible to consider a Fourier expansion
where,
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Properties of Fourier Series It is a way to represent a wave-like function as
the sum of simple sine waves. If u(x) is real then Distinct Fourier modes are orthogonal
Correlations (spatially homogenous):
Fourier Decomposition in 3D In 3D space,
Physical interpretation:
Energy:
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Energy Spectrum The energy spectrum can be described as average
energy of eddies plotted against their size. This is energydensity ( ) versus wave number (k), which is defined ask = 2/.
Wavelength () characterizes eddy size. Smallwavelength corresponds to large k. Smaller eddies havelow energy; hence, the spectrum tapers off rapidly athigh k.
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E
Energy Spectrum
Figures show Energy spectra in Fourier space. Energyis normally plotted against one over the size, asabove. The axes are scaled arbitrarily.
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Energy Spectrum A plot of the vorticity spectrum would go the other way:
small eddies have high vorticity at least until some verysmall size is reached, below which there is littleturbulent motion.
The rate of energy dissipation goes like vorticity: it islargest at small scales. That is because vorticity anddissipation involve derivatives of velocity. (The rate ofenergy dissipation is |u |2 . Differentiation amplifiesat small scales.)
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Energy Spectrum: A Summary This is the picture that emerges: On average, the energy is
highest at large scales but unstable, whereas the rate ofdissipation peaks at small scales. We are led to theconcept of an energy cascade: to achieve an equilibrium,energy must flow from the large scales, where it isconcentrated, to the small scales, where it is beingdissipated. That is what is meant by the energy cascade.
Large eddies might develop instabilities that producesmaller and smaller scales of motion or straining mightstretch and fold large-scale vortices to create smaller-scale structure.
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Energy Balance for Forced Turbulence Stirring can be considered as forcing at small wave
numbers (large scales). Spoon is much larger than smallest eddies.
Eqn. for average turbulent kinetic energy:
2
stirring with spoon
( , )p tt
u u u u f x
2( , ) ( , )f
f Lt t d
f x f k
12 k u u
=Dissipation rate Power input
:t
k u u f u
Energy Balance for Forced Turbulence After statistical stationarity is reached (stirring for a
long time)
For constant power input:Decreasing Increasing
At high Reynolds number (Re) flow field has higher gradients.
=Dissipation ratePower input
0 :t
k f u u u
: u u
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Turbulent Eddy Distribution
Self Similar Energy Cascade
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Time scales in Energy Cascade
Energy Cascade in Turbulence
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Kinetic energy enters the turbulence at the largest scales atwhich energy is extracted from the mean flow. This energy is thentransferred to smaller and smaller scales. At the smallest scales, the turbulent kinetic energy is dissipatedinto heat due to viscous stresses.
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Energy Cascade in Turbulence
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The idea of energy transfer from large to subsequently smaller scaleswas introduced by Lewis F. Richardson in 1922.Energy is transferred from the large to the small scales: that is a centraltenet of turbulence theory. It is also an underpinning of all types of eddysimulation. Large eddies grind down into smaller and smaller ones, until energy isdissipation by viscous action at the smallest scales.
Big whorls have little whorls, little whorls have smaller whorls, that feed on their velocity, and so on to viscosity...
Scales of Turbulence Integral scales: Scale for largest eddies. Dissipation/Taylor scales: Scale at which
turbulence is isotropic. Kolmogorov scales: Scale for smallest eddies
(after Kolmogorov, 1941).Characteristic length scale ( )Characteristic time scale ( )Characteristic mean velocity ( )For the characteristic time, , the following ratio
holds:
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cL
ct
cU
ct
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Scales of Turbulence
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Integral scale
Taylor scale
Kolmogorov scale
Wave number (k)
Log E
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Length Scales of Turbulence Length scales:
For largest eddies, Integral length scale is (k3/2/e). At which turbulence is isotropic, Taylor microscale is
(15nu2/e)1/2. For smallest eddies, Kolmogorov length scale is
(n3/e)1/4. These eddies have a velocity scale (n.e)1/4and a time scale (n/e)1/2.
2 3
2 2
2
is the energy dissipation rate (m /s )is the turbulent kinetic energy (m /s )is the kinematic viscosity coefficient (m /s)
k
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Length Scales of Turbulence
The Taylor length scale proves always to be largerthan the Kolmogorov micro length, and thedifference between the two becomes larger withincreasing Reynolds number. From the aboverelationships, one can compute
For a Reynolds number of approximately Re=104,is approximately 10 times larger than(Following fig.).
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)1 4ReT k
Tk
Microstructure of Turbulence
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Jet at two Reynolds Numbers
Source: Tennekes & Lumley. Page 22.
Integral Scale vs. Characteristic Scale
If one considers the turbulent velocityfluctuations that occur superimposed on themean flow field, it is easy to see that the integraltime-scale of the turbulence always has to be ofthe order of magnitude of the characteristic timescale of the mean flow field, i.e. the largestvortices that the turbulent part of a flow fieldpossesses have time scales which correspond tothose of the mean flow field. Generally, thefollowing relationship holds:
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Integral Scale For integral length scale l0,
Here, is the energy dissipation rate. Theproportionality constant is of the order one.
The Reynolds number associated with these largeeddies is referred to as the turbulence Reynoldsnumber ReL, which is defined as:
Taylor Micro-scale From these scales, characterizing the smallest vortices of
a turbulent flow, the Taylor micro-scale has to bedistinguished, which is defined as follows:
Taylor micro-scale defines an eddy size which is locatedbetween the smallest viscous eddies and the large eddieshaving quantities of the dimension of the geometricextension of the mean flow. Taking this into account, itcan be shown that the following hold:
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Taylor Micro-scale vs. Characteristic Scale
Considering that the following holds:
Then inserting this relationship into theequation for and taking into account
, a further important relationship followsfrom the above derivations:
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Turbulent Length Scales
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Production vs. Dissipation Term The ratio of production term to the dissipation term of the
general turbulence flow energy equation is
where, represents the square of thecorresponding turbulence intensity (Tu).
As turbulence always occurs for large Reynolds numbers,e.g. Re = 104, above equation shows that even for Tu =20%, a comparatively large degree of turbulence, theviscous dissipation is negligible compared with theturbulence production.
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Kolmogorov Hypothesis Kolmogorovs hypothesis of local isotropy states that at
sufficiently high Reynolds numbers, the small-scale turbulentmotions (l > l >>l have a universal form that is uniquely determined by independent of .
Note: LEI is the length scale that forms the demarcation betweenthe large scale anisotropic eddies. Lc is the characteristic orIntegral length scale. l is the Kolmorogov length scale.
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Kolmogorov Scale Following a suggestion of Kolmogorov, so-called micro-
scales can be introduced to characterize the turbulentflow field:
The length, velocity and time scales introduced byKolmogorov are determined in such a way that theycharacterize that part of the spectrum of the turbulentvelocity fluctuations in which the energy production ofthe turbulent vortices is equal to the dissipation. Thus,assuming isotropic turbulence, one can introduce:
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Kolmogorov Scale
From these results, we can deduce:
From the equality of the terms for production and dissipation, it follows that:
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Kolmogorov Scale For the Kolmogorov time scale, the following
expression results:
The Reynolds number resulting on the basis of theabove-introduced micro-length scale and micro-velocity scale is:
Note that the fact that the Kolmogorov Reynoldsnumber Re of the small eddies is 1, is consistentwith the notion that the cascade proceeds to smallerand smaller scales until the Reynolds number is smallenough for dissipation to be effective.
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Kolmogorov Scale The characteristic turbulent eddy quantities,
determined by Kolmogorovs scales of turbulence,are those that represent the viscous effects whichdamp the turbulent velocity fluctuations.
These smallest eddies are assumed to convert thekinetic energy of turbulence into heat. Because ofthese characteristic properties, the followingdefinitions are available in the literature for thesmallest scales of turbulence:
Kolmogorov scales = micro-scales = viscous eddy scales:
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Turbulent Scale: A Summary The different length scales of turbulence have
proven to be very useful in formulations ofturbulence models, which are summarized in thesubsequent sections.
The above derivations indicate the differences inthe structure of turbulent flows at small and highReynolds numbers.
For flows with the same integral dimensions, theflow at a large Reynolds number proves to bemicro-structured, i.e. the smallest eddies havesmall dimensions, whereas for the small Reynoldsnumber the flow appears macro-structured.
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In order to characterize the complex nature of turbulentflows, the following Reynolds numbers are oftenemployed:
Moreover, for the relationships of the characteristic lengthscales, the following expression holds:
These relationships are often employed when consideringturbulent flows, in order to carry out order of magnitudeconsiderations regarding the characteristic properties ofturbulence.
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Correlations for various Turbulence Scales
Here large scale refers integral scales and small scale is Kolmogorov scale.
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Ratio between large scale and small scale Ratio of Integral to Kolmogorov Length scales:
Ratio of Integral to Kolmogorov velocity scales:
Ratio of Integral to Kolmogorov Length scales:
Note: As expected, at high Reynolds numbers, thevelocity and time scales of the smallest eddies aresmall compared to those of the largest eddies.
3 40 ~ ReLl l
1 40 ~ ReLu u
1 20 ~ ReL
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A Practical Problem: Mixing
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Kolmogorov Hypothesis Kolmogorovs hypothesis of local isotropy states that at
sufficiently high Reynolds numbers, the small-scale turbulentmotions (l > l >>l have a universal form that is uniquely determined by independent of .
Note: LEI is the length scale that forms the demarcation betweenthe large scale anisotropic eddies. L0 is the Integral length scale.l is the Kolmorogov length scale.
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Introduction to Energy Spectrum
We introduce a new length scale LDI , (with LDI 60 l formany turbulent high Reynolds number flows) so that thisrange can be written as LEI > l > LDI .
This length scale splits the universal equilibrium rangeinto two sub-ranges:
The inertial sub-range (LEI > l > LDI) where motions aredetermined by inertial effects and viscous effects arenegligible.
The dissipation range (l < LDI) where motions experienceviscous effects.
Eddy Sizes The bulk of the energy is contained in the larger
eddies in the size range LEI = l0 /6 < l < 6l0 , whichis therefore called the energy containing range.
The suffixes EI and DI indicate that LEI is thedemarcation line between energy (E) and inertial(I) ranges, as LDI is that between the dissipation(D) and inertial (I) ranges.
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Energy Transfer Rate The rate at which energy is transferred from the larger
scales to the smaller scales is T(l). Under the equilibrium conditions in the inertial sub-
range this is equal to the dissipation rate , and isproportional to u(l)2/ .
Wave Numbers The wave number is defined as = 2/l Different ranges can be shown as a function of wave no. The wave no. can also be made non-dimensional by
multiplying it with the Kolmogorov length scale l (or ) toresult in the commonly used dimensionless group ( ).
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Turbulent Kinetic Energy Spectrum
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Turbulent Kinetic Energy Spectrum The energy spectrum of fully developed homogeneous turbulence is
thought to be composed of three distinct wave-number regions (seeprevious fig.).
A. In this region the large energy-containing eddies are found. Theseeddies interact with the mean flow and extract energy from the meanflow. The energy is transferred to slightly smaller scales and eventuallyinto region B.
B. This region is the inertial sub-range. In this region turbulent kineticenergy is neither produced nor dissipated. However, there is a net fluxof energy through this region from A to C. The existence of this regionrequires that the Reynolds number is high.
C. This is the dissipative region where turbulent kinetic energy isdissipated into heat. Eddies in this region are isotropic and the scalesare given by the Kolmogorov scales.
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Turbulent Kinetic Energy Spectrum
However, the small eddies, which are of very highfrequency, in the region of , dissipate energy. Here,turbulent kinetic energy is dissipated into heat bymolecular viscosity. The viscous stresses prevent thegeneration of eddies with higher frequency.
In wave-number space, the energy of eddies from tocan be expressed as
The total turbulent kinetic energy, k, which is the sumof the kinetic energies of the three fluctuating velocitycomponents, i.e. is obtained byintegrating over the whole wave-number space
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d
d )E d
)0
k E d
Turbulent Kinetic Energy Spectrum One important tool for analysing the different regions of
turbulence is the energy spectrum. It is common practice to use wave numbers ( ) instead of
length scales. The dimension of a wave number is oneover length, thus we can think of the wave number asinversely proportional to the eddy radius, i.e.
This means that large wave numbers correspond to smalleddies and small wave numbers to large eddies. Eddieswith wave numbers in the region of , the previous figurecontains the largest part of the energy and contribute littleto the energy dissipation.
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1 r
e
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Turbulent Kinetic Energy Spectrum The energy contained in eddies with wave numbers
between A and B is then
The full spectrum is given by: E () in Inertial sub-range: According to the second similarity hypothesis E () will
solely depend on k and .
The last equation describes the famous Kolmogorov 5/3 spectrum. C is the universal Kolmogorov constant, which experimentally was determined to be C = 1.5.
Full Energy Spectrum E() The full spectrum is given by:
The production range is governed by fL (which goes to unity for large ( l0):
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Full Energy Spectrum E() The dissipation range is governed by f (which
goes to unity for small ( ):
The model constants were determinedexperimentally and based on the constraint thatE() integrate to . Their values are:
For given values of , , and ; the full spectrumcan now be calculated based on theseequations.
Normalized Energy Spectrum It is, however common to normalize the spectrum in one of
two ways: Based on Kolmogorov scale:
Measure of length scale becomes ( ). E() is made dimensionless as E() /( u2)
Based on Integral scale: Measure of length scale becomes (l0 ). E() is made dimensionless as E() /(l0 )
Instead of having three adjustable parameters (, , ), thenormalized spectrum then has only one adjustable parameter:Taylor-scale Reynolds number R.
Note: R can be related to the turbulence Reynolds number asfollows:
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The normalized energy spectrum for R = 500
The energy spectrum as a function of R
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The energy spectrum as a function of R
Measurements of Turbulent Energy Spectra The figure shows exp.
measured 1D spectra (one velocity component was measured only, as indicated by the 1 and 11 subscripts). The number at the end of the reference denotes the value of R for which the measurements were done.
Source: Pope, page 235. Energy Containing Range:
Most of the energy (~80%) iscontained in eddies of lengthscale lEI = l0/6 < l < 6l0.
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Measurements of Turbulent Energy Spectra Determination of the spectrum requires
simultaneous measurements of all velocitycomponents at multiple points, which is usuallynot possible.
It is common to measure one velocity componentat one point over a certain period of time andconvert the time signal to a spatial signal using x= Ut with U being the time averaged velocity.
This is commonly referred to as Taylorshypothesis of frozen turbulence. It is only validfor u/U
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Energy Spectrum captured by DNS & LES The direct numerical simulation (DNS) of turbulent flows at
technically relevant Reynolds numbers will remain unfeasibledue to its huge computational cost in the foreseeable future.
Large Eddy Simulation (LES) offers here an attractivecompromise. As shown in figure, the basic idea of LES is that itresolves numerically only the large structures of the turbulentmotion, which are associated with small wave numbers range,while it models unresolved small so-called sub-grid scale (SGS)structures.
Dissipation Rate Spectrum We now know which eddies contain most of the energy. The
question remains, which eddies exactly dissipate the energy? This question can be answered by constructing a dissipation
rate spectrum D(). The integral of D() over the fullwavelength range is by definition the energy dissipation rate :
Furthermore, with being defined as the multiple of thekinematic viscosity and squared velocity gradients [of order(du/dx)2 ~ k/l2 ~ k 2 ~ 2 E() ] we can then deduce:
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Dissipation Rate Spectrum Therefore, dissipation rate would be
Here (0, ) is the cumulative dissipation; the energy dissipatedby eddies with a wavelength between 0 and .
The unit of D() is m3/s3 and it can thus be normalized with avelocity scale cubed, typically the Kolmogorov velocity scale.
Just as the normalized E() only depended on Re, so does thenormalized D() depend only on Re.
Dissipation Rate Spectrum
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Dissipation Rate Spectrum Most of the dissipation (~ 90%) occurs in eddies of length
scales LDI / = 60 > l/ > 8 . This means that most of the dissipation occurs at scales
that are larger than the Kolmogorov scale . TheKolmogorov scale should be interpreted as a measure ofthe smallest eddies that are present in a turbulent flow athigh Reynolds numbers.
How long does it take for a large scale eddy to break up andbe dissipated? The spectra can be further analyzed to showthat eddies spend about 90% of their total lifetime =k/ inthe production range, and that once eddies enter theinertial sub-range it takes only about /10 before theenergy is being dissipated. This time /10 is also referred toas the cascade timescale.
Intermittency Neither k nor are constant in time or space. Within a turbulent flow field, k and may vary
widely in space, sometimes by orders ofmagnitude.
Also, at a given point in space the instantaneousvalues of may vary in time. This is calledintermittency. The peak values of relative to themean tend to increase with Reynolds number.
Peak values may be of the order of 15 times theaverage in laboratory scale flows and 50 timesthe average in atmospheric flows.
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Validity of Kolmogorovs Theory Kolmogorovs theory is an asymptotic theory: it has been
shown to work well in the limit of very high Reynolds no. The exact shape of the normalized spectra may deviate from
Kolmogorovs model spectra for intermediate Reynoldsnumbers. E.g. for many laboratory scale flows which haveReynolds numbers on the order of 10,000 with Re~ 250, theexponent of E() ~ p in the inertial subrange is oftenmeasured to be p ~ 1.5 instead of 5/3 (~1.67).
Kolmogorovs theory assumes that the energy cascade is oneway: from large eddies to small eddies. Experimental studieshave shown that energy is also transferred from smaller scalesto larger scales (a process called backscatter), albeit at a muchlower rate and the dominant energy transfer is indeed fromlarge to small.
Validity of Kolmogorovs Theory The theory assumes that turbulence at high Reynolds
numbers is completely random. In practice, large scalecoherent structures may form.
Research into the fundamental aspects of turbulencecontinues, both experimentally and by means of largecomputer simulations using DNS (direct numericalsimulation); and the theory continues to be refined.
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71
Vorticity and Vortex Stretching Existence of eddies implies rotation or vorticity. Vorticity concentrated along contorted vortex lines or bundles. As end points of a vortex line move randomly further apart the vortex
line increases in length but decreases in diameter. Vorticity increasesbecause angular momentum is nearly conserved. Kinetic energyincreases at rate equivalent to the work done by large-scale motion thatstretches the bundle.
Viscous dissipation in the smallest eddies converts kinetic energy intothermal energy.
Vortex-stretching cascade process maintains the turbulence anddissipation is approximately equal to the rate of production of turbulentkinetic energy.
Typically energy gets transferred from the large eddies to the smallereddies. However, sometimes smaller eddies can interact with eachother and transfer energy to the (i.e. form) larger eddies, a processknown as backscatter.
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Vortex Stretching The interaction between vorticity ( )and velocity
gradients is an essential ingredient to create andmaintain turbulence.
Consider an fluid element rotates on x-y plane aboutz-axis in addition to a stretching along z-axis.
Therefore, an extension in z-direction can decreasethe length scales and increase the velocitycomponents in the x-y plane.
Neglecting viscous dissipation, Conservation ofangular momentum says that
According to Kelvins Circulation theorem:
),i iu
2r Constant
2r ConstantCirculation,
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Vortex Stretching Vorticity will cause stretching along z-direction. Thus, there will
be decrease in length in x and y directions. This will increase vorticity in these directions, hence stretching will occur in x and y-directions.
In this fashion, stretching inone direction will promote stretchingIn other directions. Soon stretching will be rapidly promote d in all directions making the turbulence isotropic in nature.(See family chart in next slide).
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r
z
Vortex stretching on a circular cylinder
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Vortex Stretching Thus, the cascade of energy of turbulent motion
estimates to smaller scales.(i.e., larger and larger velocity gradients)
Vortex stretching tendsto make the smaller eddieswhich is isotropic, hence, their contribution to Reynolds shear stressis zero.
z
yx
zy z x
x yxz
x y y z
Family Tree showing how vortexstretching produces small-scale isotropy
)uv
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75
t2 t3
t4t5 t6
t1
(Baldyga and Bourne, 1984)
Vortex Stretching
