In order to understand whether, and to what extent, spectral representation
can effectively highlight the nonlinear interaction among different scales, it
is necessary to consider the state that precedes the onset of instabilities and
turbulence in flows. In this condition, a system is still stable, but is
however subject to a swarming of arbitrary 3D small perturbations. These can
arrive any instant, and then undergo a transient evolution which is ruled out
by the initial-value problem associated to the Navier-Stokes linearized
formulation. The set of 3D small perturbations constitutes a system of multiple
spatial and temporal scales which are subject to all the processes included in
the perturbative Navier-Stokes equations: linearized convective transport,
linearized vortical stretching and tilting, and the molecular diffusion.
Leaving aside nonlinear interaction among the different scales, these features
are tantamount to the features of the turbulent state. We determine the
exponent of the inertial range of arbitrary longitudinal and transversal
perturbations acting on a typical shear flow, i.e. the bluff-body wake. Then,
we compare the present results with the exponent of the corresponding developed
turbulent state (notoriously equal to -5/3). For longitudinal perturbations, we
observe a decay rate of -3 in the inertial range, typically met in
two-dimensional turbulence. For purely 3D perturbations, instead, the energy
decreases with a factor of -5/3. If we consider a combination of longitudinal
and transversal perturbative waves, the energy spectrum seems to have a decay
of -3 for larger wavenumbers ([50, 100]), while for smaller wavenumbers
([3,50]) the decay is of the order -5/3. We can conclude that the value of the
exponent of the inertial range has a much higher level of universality, which
is not necessarily associated to the nonlinear interaction.Comment: Proceedings of the 17th Australasian Fluid Mechanics Conference, 5-9
  December 2010, Auckland, New Zealan

Scarsoglio, Stefania

Tordella, Daniela

English

arXiv

shear instability, small pertubations collective behaviou

SCARSOGLIO, STEFANIA

TORDELLA, Daniela

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

05 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEPre-unstable set of multiple transient three-dimensional perturbation waves and the associated turbulent state in a shearflow / S. SCARSOGLIO; TORDELLA D.. - STAMPA. - 17(2010), pp. 360-363. ((Intervento presentato al convegno 17thAustralasian Fluid Mechanics Conference tenutosi a AUCKLAND, New Zealand nel December 5 - 9 2010.OriginalPre-unstable set of multiple transient three-dimensional perturbation waves and the associated turbulentstate in a shear flowPublisher:PublishedDOI:Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2367549 since:17th Australasian Fluid Mechanics ConferenceAuckland, New Zealand5-9 December 2010Pre-unstable set of multiple transient three-dimensional perturbation waves and theassociated turbulent state in a shear flowS. Scarsoglio1 and D. Tordella21Department of HydraulicsPolitecnico di Torino, Torino 10129, Italy2Department of Aeronautics and Space EngineeringPolitecnico di Torino, Torino 10129, ItalyAbstractIn order to understand whether, and to what extent, spectralrepresentation can effectively highlight the nonlinear interac-tion among different scales, it is necessary to consider the statethat precedes the onset of instabilities and turbulence in flows.In this condition, a system is still stable, but is however sub-ject to a swarming of arbitrary three-dimensional small per-turbations. These can arrive any instant, and then undergo atransient evolution which is ruled out by the initial-value prob-lem associated to the Navier-Stokes linearized formulation. Theset of three-dimensional small perturbations constitutes a sys-tem of multiple spatial and temporal scales which are subjectto all the processes included in the perturbative Navier-Stokesequations: linearized convective transport, linearized vorticalstretching and tilting, and the molecular diffusion. Leavingaside nonlinear interaction among the different scales, these fea-tures are tantamount to the features of the turbulent state.We determine the exponent of the inertial range of arbitrary lon-gitudinal and transversal perturbations acting on a typical shearflow, i.e. the bluff-body wake. Then, we compare the presentresults with the exponent of the corresponding developed turbu-lent state (notoriously equal to  5=3). For longitudinal pertur-bations – i.e. perturbations in the plane of the basic flow whichis two-dimensional – we observe a decay rate of  3 in the in-ertial range, typically met in two-dimensional turbulence. Forpurely three-dimensional perturbations, instead, the energy de-creases with a factor of  5=3. If we consider a combinationof longitudinal and transversal perturbative waves, the energyspectrum seems to have a decay of  3 for larger wavenumbers(k 2 [50;100]), while for smaller wavenumbers (k 2 [3;50]) thedecay is of the order  5=3. We can conclude that the valueof the exponent of the inertial range has a much higher level ofuniversality, which is not necessarily associated to the nonlinearinteraction.IntroductionOne very popular notion in the phenomenology of turbulence(in the sense of Kolmogorov 1941) is that a power-law scal-ing with an exponent close to  5=3 is observed for the en-ergy spectrum over a very substantial range of a few decadesof wavenumber, this range being called the inertial range. Forextensive collections of laboratory and numerical experimentalresults, see for instance [1, 2]. In fact, it is a common crite-rion for the successful production of a fully developed turbulentfield, either in the laboratory or in numerical simulations, toverify that the power spectrum has such a scaling in the inertialrange.The set of arbitrary three-dimensional small perturbations con-stitutes a system of multiple spatial and temporal scales whichare subject to all the processes included in the Navier-Stokesequations, leaving aside the nonlinear interaction among the dif-ferent scales. If it were possible to observe such a system in atemporal window and obtain the instantaneous power spectrum,it would be possible, among others, to determine the exponentof the inertial range of the arbitrary perturbation, and to com-pare it with the exponent of the corresponding developed tur-bulent state. Two possible situations can therefore appear: (a)the exponent difference is large and, as such, it is a quantita-tive measure of the nonlinear interaction in spectral terms; (b)the difference is small. This would be even more interesting,because it would indicate a higher level of universality on thevalue of the exponent of the inertial range, not necessarily asso-ciated to the nonlinear interaction.We propose building temporal observation window for the tran-sient evolution of a large number (order of 102  103) of arbi-trary small 3D (oblique) perturbations acting on a typical shearflow. In particular, we can take advantage of a recently – nu-merically obtained – set of solutions yielded by the initial-valueproblem applied to 3D perturbations of a plane bluff-body wake[3, 4]. These solutions have revealed the existence of manydifferent kinds of transient behaviour, not all of which is triv-ial. If these transients, obtained in association with arbitraryinitial conditions, are injected in a statistical way into the tem-poral observation window, we can obtain a close representationof the perturbation state that precedes the onset of instability-turbulence. In this preliminary work, we consider an ensembleof transients of asymptotically stable waves – this is becausewe want to avoid any temporal divergence – and build the en-ergy spectrum by freezing each wave at the moment it reachesa constant value of the temporal growth rate. Here, the growthis always negative since all waves are asymptotically dampedin time. When this happens, the wave may be considered outof its transient and in the asymptotic condition. When build-ing the power spectrum, the energy of each wave is normalizedover its initial energy. In this way, we are considering the spec-tral dynamics of a sort of white noise perturbation made up ofasymptotically stable waves.FormulationThe energy spectrum behaviour is studied using the initial-valueproblem formulation. The base flow is approximated at a fixedlongitudinal station, x0 = 10, through an analytical expansionsolution [5] of the Navier-Stokes equations (see Fig. 1). TheReynolds number is set to a value of 40, in order to consider sta-ble evolutive configurations. The viscous perturbative equationsare written in terms of the vorticity and the transversal velocity[6] and then transformed through a Laplace-Fourier decompo-sition [3, 4] in the plane (x;z) which is normal to the base flowplane (x;y),¶2vˆ¶y2  (k2 a2i +2iarai)vˆ= Gˆ; (1)x yzγαr φkcylinder   axis −5 0 50.60.81yUFigure 1: Perturbation geometry scheme, mean flow U in thewake at Re= 40 and x0 = 10 (blue curve), symmetric and asym-metric initial conditions in terms of vˆ(t = 0;y) (red curves).¶Gˆ¶t= (iar ai)(d2Udy2vˆ U Gˆ)++1Re[¶2Gˆ¶y2  (k2 a2i +2iarai)Gˆ]; (2)¶wˆy¶t=  (iar ai)Uwˆy  igdUdy vˆ++1Re[¶2wˆy¶y2  (k2 a2i +2iarai)wˆy]: (3)The transversal velocity and vorticity components are indicatedas vˆ and wˆy respectively, while Gˆ is defined through the kine-matic relation eG = ¶xewz  ¶zewx that in the physical plane linkstogether the perturbation vorticity components in the x and z di-rections (ewx and ewz) and the perturbed velocity field. Equations(2) and (3) are the Orr-Sommerfeld and Squire equations re-spectively, from the classical linear stability analysis for three-dimensional disturbances in the phase space. We define k asthe polar wavenumber, ar = kcos(f) as the wavenumber in xdirection, g = ksin(f) as the wavenumber in z direction, f asthe angle of obliquity with respect to the physical plane, and aias the spatial damping rate in x direction (see the perturbationscheme in Fig. 1). The measure of the perturbation growth canbe defined through the disturbance kinetic energy density in theplane (a;g):e(t;a;g) =Z +yd yd(juˆj2+ jvˆj2+ jwˆj2)dy= (4)=1ja2+ g2jZ +yd yd ¶vˆ¶y2+ ja2+ g2jjvˆj2+ jwˆyj2!dy;where uˆ and wˆ are the streamwise and spanwise components ofthe perturbation velocity, respectively, while 2yd is the exten-sion of the spatial numerical domain. The amplification factorG(t) can be introduced in terms of the normalized energy den-sityG(t;a;g) =e(t;a;g)e(t = 0;a;g): (5)0 100 200 300 400100101102103104tαi=0.02τinter=35τfar=100 G (a) x0=10 (intermediate)x0=50 (far)Re=100k=0.6φ = pi/40 10 20 30 40 5004812αi=0.05k t G (b)Re=50φ = pi/2x0=100 20 40 60 80 100−0.0200.020.040.06 αi=0.05k t r (c)Re=50φ = pi/2x0=10Figure 2: (a) The amplification factor G, asymmetric initialcondition, intermediate (x0 = 10, solid curves) and far field(x0 = 50, dashed curves) wake configurations. The periodstinter;t f ar are the periods of the modulation visible on G, inthe intermediate and far field, respectively. (b) The amplifica-tion factor G and (c) the temporal growth rate r as function oftime, symmetric initial condition, k = 0:5;1;1:5;2.The temporal growth rate on the kinetic energy rr(t;a;g) =logje(t;a;g)j2t; t > 0 (6)is introduced in order to evaluate both the early transient as wellas the asymptotic behaviour of the perturbations.Examples, in terms ofG and r, of significant transient behaviourand asymptotic fate of the three-dimensional perturbations [3]are shown in Fig. 2. In part (a) the amplification factor Gof an asymmetric wave is shown for two typical intermediate(x0 = 10) and far (x0 = 50) wake configurations. For x0 = 10a local maximum, followed by a minimum, is visible in the en-ergy density, then the perturbation is slowly amplifying and thetransient can be considered extinguished only after hundreds oftime scales. For x0 = 50 these features are less marked, but stillpresent. For both configurations, the function G shows a mod-ulation in time in the first part of the transient (t f ar = 100 andtinter = 35), which is always observed in the case of asymmet-ric longitudinal or oblique instability waves. In parts (b) and (c)of Fig. 2, the amplification factor G and the temporal growthrate r of purely three-dimensional symmetric perturbations arereported, respectively. Such waves may become damped by in-creasing their wavenumber (k = g). Before the asymptotic sta-ble states are reached, these configurations yield maxima of theenergy density (e.g. when k = 1:5;G 3 at t  30) in the tran-sients. This trend is also typical of oblique and longitudinalwaves, and it can be considered a general feature in the contextof the stability of near parallel shear flows.ResultsComputations to evaluate the energy spectrum are made, at afixed wavenumber, by integrating the equations forward in timeuntil the temporal growth rate r asymptotes to a constant value,i.e. when the transient can be considered as extinguished. Weconsider a range of values for the polar wavenumber k in the in-terval [0.05, 100], and three angle of obliquity f = 0;p=4;p=2.For simplicity’s sake, the spatial damping rate ai is here takenequal to 0, thus no spatial damping is considered in the lon-gitudinal direction. We account for symmetric and asymmet-ric initial conditions (see red curves in Fig. 1) in terms of thetransversal velocity vˆ, while the transversal vorticity wˆy is ini-tially equal to zero.The normalized energy density G at the asymptotic state (blueand black circles for symmetric and asymmetric initial con-ditions, respectively) and the initial energy density e(t = 0)(blue and black triangles for symmetric and asymmetric ini-tial conditions, respectively) are shown – as function of thepolar wavenumber k – in parts (a) and (b) of Fig. 3 for alongitudinal (f = 0) and a transversal (f = p=2) perturbativewave, respectively. In Fig. 3c we sum, at a fixed wavenumber,asymmetric and symmetric perturbations with angle of obliq-uity f = 0;p=4;p=2, and then we calculate the energy densityof the resulting perturbation obtained by combining six differ-ent waves. We report  5=3 and  3 slopes with red and blackcurves, respectively.In the case of longitudinal waves (two-dimensional base andperturbed flow configurations, see Fig. 3a), the normalized en-ergy density G at asymptotic state has a decay of 3 in the iner-tial range (k 2 [2;100]) for both symmetric and asymmetric per-turbations (see blue and black circles). For smaller wavenum-bers, instead, the decay is faster. For purely transversal waves(Fig. 3b), we observe a decay of  5=3 in the inertial range(k 2 [2;100]) for both symmetric and asymmetric perturbations(blue and black circles). Longer waves have a deeper decay,reaching a maximum of energy at about k= 0:5. If we sum sym-metric and asymmetric waves with different angles of obliquity(f = 0;p=4;p=2), the normalized energy density G (blue cir-cles) of the resulting perturbation at the asymptotic state has adecay of  3 for k 2 [25;100], while the decay is of the order 5=3 for k 2 [2;25]. For longer waves, the decay is faster and amaximum of energy can be found for k 2 [0:1;0:5].From Eq. (4) and recalling that the transversal vorticity wˆy isinitially imposed equal to zero, one can observe that the initialenergy density has a well-defined decay, given by:e(t = 0;k) =c1k2+ c2; (7)10−1 100 101 10210−2010−1510−1010−5100G kφ=0symasymG   e(t=0) −3 slope(a)10−1 100 101 10210−2010−1510−1010−5100kGG   e(t=0)symasymφ=pi/2(b)−5/3 slope10−1 100 101 10210−2010−1510−1010−5100kG−5/3 slope−3 slopeSum of sym and asym waves: φ=0,pi/4, pi/2(c)G   e(t=0)Figure 3: Spectrum of the amplification factor of a collec-tion of stable perturbation waves, Re = 40. The normalizedenergy density G at the asymptotic state (circles) and the initialenergy density e(t = 0) (triangles). The energy of each wavein asymptotic conditions is normalized over the value ownedat the initial instant. Blue and black symbols for symmetricand asymmetric initial conditions, respectively. (a) f = 0, (b)f = p=2, and (c) sum of symmetric and asymmetric perturba-tions with angle of obliquity f = 0;p=4;p=2. Red and blackcurves indicate 5=3 and 3 slopes, respectively. Note that theintermediate waves are stable and highly damped perturbations,that reach the asymptotic state after a long lapse of time wherethey progressively lose their kinetic energy. This explain thesharp fall of energy density at the transition between the longand the intermediate wave ranges.wherec1 =Z +yd yd¶vˆ(y; t = 0)¶y2 dy; c2= Z +yd yd jvˆ(y; t = 0)j2dy: (8)This analytical behaviour is confirmed for the three cases weconsidered in Fig. 3. In parts (a) and (b) blue and black triangles(for symmetric and asymmetric initial conditions, respectively)represent the initial energy density and show a constant valuefor large wavenumbers (k 2 [2;100]), while a decay of about 2 is visible for longer waves (k 2 [0:05;2]). It should be notedthat e(t = 0) does not depend on the perturbation obliquity, thusthe energy density values at t = 0 are exactly the same for bothf = 0 and f = p=2. In case (c), where six different waves aresummed and then the energy density of the resulting perturba-tion is computed, e(t = 0) (blue triangles) still displays a trendin agreement with Eq. (7).ConclusionsThis is a preliminary study of the behaviour of a collection oftransient waves seen throughout their power spectrum. To avoidthe complexity introduced by the temporal divergence associ-ated to asymptotically unstable waves, we considered an en-semble of transients of asymptotically stable waves. Then, wehave built the energy spectrum by freezing each wave at themoment it reaches a constant (negative) value of the temporalgrowth rate. When this happens, the waves are dynamically outof their transient and in the asymptotic condition. In the powerspectrum, the energy of each wave is normalized over its owninitial value. In practice, it is the same as considering the spec-tral dynamics of a white noise perturbation, a kind of model forthe swarm of small perturbations that affects any system in alinear way.We observe that, whether the waves are aligned with the basesheared flow or not, the energy of the intermediate range ofwavenumbers in the spectrum decays with the same exponent( 5=3 for 3D oblique waves,  3 for 2D aligned waves) thatis observed in the spectrum of the velocity fluctuation of fullydeveloped turbulent flows. Where the nonlinear interaction isconsidered dominant.At the moment, we can conclude by observing that the spectralpower-law scaling of intermediate/inertial waves (with an expo-nent close to  5=3 in 3D and to  3 in 2D) is a general dynam-ical property which encompasses the nonlinear interaction. Inother words, it seems to us that the strength of the nonlinearityhas been overestimated in the determination of a few turbulenceproperties.References[1] U. Frisch, Turbulence: the legacy of A. N. Kolmogorov,Cambridge University Press (1995).[2] K. R. Sreenivasan and R. A. Antonia, The phenomenologyof small-scale turbulence, Annu. Rev. Fluid Mech. 29, 35-72 (1997).[3] S. Scarsoglio, D. Tordella, and W. O. Criminale, An ex-ploratory analysis of the transient and long term behav-ior of small 3D perturbations in the circular cylinder wake,Stud. Applied Math. 123, 2 (2009).[4] S. Scarsoglio, D. Tordella, and W. O. Criminale, The roleof long waves in the stability of the plane wake, Phys. Rev.E 81, 036326/1-9 (2010).[5] D. Tordella and M. Belan, A new matched asymptotic ex-pansion for the intermediate and far flow behind a finitebody, Phys. Fluids 15 (2003).[6] W. O. Criminale and P. G. Drazin, The evolution of lin-earized perturbations of parallel shear flows, Stud. AppliedMath. 83 (1990).

Pre-unstable set of multiple transient three-dimensional perturbation waves and the associated turbulent state in a shear flow

PORTO Publications Open Repository TOrino

17th Australasian Fluid Mechanics Conference
Auckland, New Zealand
5-9 December 2010
Pre-unstable set of multiple transient three-dimensional perturbation waves and the
associated turbulent state in a shear flow
S. Scarsoglio1 and D. Tordella2
1Department of Hydraulics
Politecnico di Torino, Torino 10129, Italy
2Department of Aeronautics and Space Engineering
Politecnico di Torino, Torino 10129, Italy
Abstract
In order to understand whether, and to what extent, spectral
representation can effectively highlight the nonlinear interac-
tion among different scales, it is necessary to consider the state
that precedes the onset of instabilities and turbulence in flows.
In this condition, a system is still stable, but is however sub-
ject to a swarming of arbitrary three-dimensional small per-
turbations. These can arrive any instant, and then undergo a
transient evolution which is ruled out by the initial-value prob-
lem associated to the Navier-Stokes linearized formulation. The
set of three-dimensional small perturbations constitutes a sys-
tem of multiple spatial and temporal scales which are subject
to all the processes included in the perturbative Navier-Stokes
equations: linearized convective transport, linearized vortical
stretching and tilting, and the molecular diffusion. Leaving
aside nonlinear interaction among the different scales, these fea-
tures are tantamount to the features of the turbulent state.
We determine the exponent of the inertial range of arbitrary lon-
gitudinal and transversal perturbations acting on a typical shear
flow, i.e. the bluff-body wake. Then, we compare the present
results with the exponent of the corresponding developed turbu-
lent state (notoriously equal to  5=3). For longitudinal pertur-
bations – i.e. perturbations in the plane of the basic flow which
is two-dimensional – we observe a decay rate of  3 in the in-
ertial range, typically met in two-dimensional turbulence. For
purely three-dimensional perturbations, instead, the energy de-
creases with a factor of  5=3. If we consider a combination
of longitudinal and transversal perturbative waves, the energy
spectrum seems to have a decay of  3 for larger wavenumbers
(k 2 [50;100]), while for smaller wavenumbers (k 2 [3;50]) the
decay is of the order  5=3. We can conclude that the value
of the exponent of the inertial range has a much higher level of
universality, which is not necessarily associated to the nonlinear
interaction.
Introduction
One very popular notion in the phenomenology of turbulence
(in the sense of Kolmogorov 1941) is that a power-law scal-
ing with an exponent close to  5=3 is observed for the en-
ergy spectrum over a very substantial range of a few decades
of wavenumber, this range being called the inertial range. For
extensive collections of laboratory and numerical experimental
results, see for instance [1, 2]. In fact, it is a common crite-
rion for the successful production of a fully developed turbulent
field, either in the laboratory or in numerical simulations, to
verify that the power spectrum has such a scaling in the inertial
range.
The set of arbitrary three-dimensional small perturbations con-
stitutes a system of multiple spatial and temporal scales which
are subject to all the processes included in the Navier-Stokes
equations, leaving aside the nonlinear interaction among the dif-
ferent scales. If it were possible to observe such a system in a
temporal window and obtain the instantaneous power spectrum,
it would be possible, among others, to determine the exponent
of the inertial range of the arbitrary perturbation, and to com-
pare it with the exponent of the corresponding developed tur-
bulent state. Two possible situations can therefore appear: (a)
the exponent difference is large and, as such, it is a quantita-
tive measure of the nonlinear interaction in spectral terms; (b)
the difference is small. This would be even more interesting,
because it would indicate a higher level of universality on the
value of the exponent of the inertial range, not necessarily asso-
ciated to the nonlinear interaction.
We propose building temporal observation window for the tran-
sient evolution of a large number (order of 102  103) of arbi-
trary small 3D (oblique) perturbations acting on a typical shear
flow. In particular, we can take advantage of a recently – nu-
merically obtained – set of solutions yielded by the initial-value
problem applied to 3D perturbations of a plane bluff-body wake
[3, 4]. These solutions have revealed the existence of many
different kinds of transient behaviour, not all of which is triv-
ial. If these transients, obtained in association with arbitrary
initial conditions, are injected in a statistical way into the tem-
poral observation window, we can obtain a close representation
of the perturbation state that precedes the onset of instability-
turbulence. In this preliminary work, we consider an ensemble
of transients of asymptotically stable waves – this is because
we want to avoid any temporal divergence – and build the en-
ergy spectrum by freezing each wave at the moment it reaches
a constant value of the temporal growth rate. Here, the growth
is always negative since all waves are asymptotically damped
in time. When this happens, the wave may be considered out
of its transient and in the asymptotic condition. When build-
ing the power spectrum, the energy of each wave is normalized
over its initial energy. In this way, we are considering the spec-
tral dynamics of a sort of white noise perturbation made up of
asymptotically stable waves.
Formulation
The energy spectrum behaviour is studied using the initial-value
problem formulation. The base flow is approximated at a fixed
longitudinal station, x0 = 10, through an analytical expansion
solution [5] of the Navier-Stokes equations (see Fig. 1). The
Reynolds number is set to a value of 40, in order to consider sta-
ble evolutive configurations. The viscous perturbative equations
are written in terms of the vorticity and the transversal velocity
[6] and then transformed through a Laplace-Fourier decompo-
sition [3, 4] in the plane (x;z) which is normal to the base flow
plane (x;y),
¶2vˆ
¶y2
  (k2 a2i +2iarai)vˆ= Gˆ; (1)
x 
y
z
γ
α
r φ
k
cylinder
   axis 
−5 0 5
0.6
0.8
1
y
U
Figure 1: Perturbation geometry scheme, mean flow U in the
wake at Re= 40 and x0 = 10 (blue curve), symmetric and asym-
metric initial conditions in terms of vˆ(t = 0;y) (red curves).
¶Gˆ
¶t
= (iar ai)(d
2U
dy2
vˆ U Gˆ)+
+
1
Re
[
¶2Gˆ
¶y2
  (k2 a2i +2iarai)Gˆ]; (2)
¶wˆy
¶t
=  (iar ai)Uwˆy  igdUdy vˆ+
+
1
Re
[
¶2wˆy
¶y2
  (k2 a2i +2iarai)wˆy]: (3)
The transversal velocity and vorticity components are indicated
as vˆ and wˆy respectively, while Gˆ is defined through the kine-
matic relation eG = ¶xewz  ¶zewx that in the physical plane links
together the perturbation vorticity components in the x and z di-
rections (ewx and ewz) and the perturbed velocity field. Equations
(2) and (3) are the Orr-Sommerfeld and Squire equations re-
spectively, from the classical linear stability analysis for three-
dimensional disturbances in the phase space. We define k as
the polar wavenumber, ar = kcos(f) as the wavenumber in x
direction, g = ksin(f) as the wavenumber in z direction, f as
the angle of obliquity with respect to the physical plane, and ai
as the spatial damping rate in x direction (see the perturbation
scheme in Fig. 1). The measure of the perturbation growth can
be defined through the disturbance kinetic energy density in the
plane (a;g):
e(t;a;g) =
Z +yd
 yd
(juˆj2+ jvˆj2+ jwˆj2)dy= (4)
=
1
ja2+ g2j
Z +yd
 yd
 ¶vˆ¶y
2+ ja2+ g2jjvˆj2+ jwˆyj2
!
dy;
where uˆ and wˆ are the streamwise and spanwise components of
the perturbation velocity, respectively, while 2yd is the exten-
sion of the spatial numerical domain. The amplification factor
G(t) can be introduced in terms of the normalized energy den-
sity
G(t;a;g) =
e(t;a;g)
e(t = 0;a;g)
: (5)
0 100 200 300 400
100
101
102
103
104
t
αi=0.02τinter=35
τfar=100
 G
 (a) x0=10 (intermediate)
x0=50 (far)
Re=100
k=0.6
φ = pi/4
0 10 20 30 40 50
0
4
8
12
αi=0.05
k 
t 
G 
(b)
Re=50
φ = pi/2
x0=10
0 20 40 60 80 100
−0.02
0
0.02
0.04
0.06 αi=0.05
k 
t 
r 
(c)
Re=50
φ = pi/2
x0=10
Figure 2: (a) The amplification factor G, asymmetric initial
condition, intermediate (x0 = 10, solid curves) and far field
(x0 = 50, dashed curves) wake configurations. The periods
tinter;t f ar are the periods of the modulation visible on G, in
the intermediate and far field, respectively. (b) The amplifica-
tion factor G and (c) the temporal growth rate r as function of
time, symmetric initial condition, k = 0:5;1;1:5;2.
The temporal growth rate on the kinetic energy r
r(t;a;g) =
logje(t;a;g)j
2t
; t > 0 (6)
is introduced in order to evaluate both the early transient as well
as the asymptotic behaviour of the perturbations.
Examples, in terms ofG and r, of significant transient behaviour
and asymptotic fate of the three-dimensional perturbations [3]
are shown in Fig. 2. In part (a) the amplification factor G
of an asymmetric wave is shown for two typical intermediate
(x0 = 10) and far (x0 = 50) wake configurations. For x0 = 10
a local maximum, followed by a minimum, is visible in the en-
ergy density, then the perturbation is slowly amplifying and the
transient can be considered extinguished only after hundreds of
time scales. For x0 = 50 these features are less marked, but still
present. For both configurations, the function G shows a mod-
ulation in time in the first part of the transient (t f ar = 100 and
tinter = 35), which is always observed in the case of asymmet-
ric longitudinal or oblique instability waves. In parts (b) and (c)
of Fig. 2, the amplification factor G and the temporal growth
rate r of purely three-dimensional symmetric perturbations are
reported, respectively. Such waves may become damped by in-
creasing their wavenumber (k = g). Before the asymptotic sta-
ble states are reached, these configurations yield maxima of the
energy density (e.g. when k = 1:5;G 3 at t  30) in the tran-
sients. This trend is also typical of oblique and longitudinal
waves, and it can be considered a general feature in the context
of the stability of near parallel shear flows.
Results
Computations to evaluate the energy spectrum are made, at a
fixed wavenumber, by integrating the equations forward in time
until the temporal growth rate r asymptotes to a constant value,
i.e. when the transient can be considered as extinguished. We
consider a range of values for the polar wavenumber k in the in-
terval [0.05, 100], and three angle of obliquity f = 0;p=4;p=2.
For simplicity’s sake, the spatial damping rate ai is here taken
equal to 0, thus no spatial damping is considered in the lon-
gitudinal direction. We account for symmetric and asymmet-
ric initial conditions (see red curves in Fig. 1) in terms of the
transversal velocity vˆ, while the transversal vorticity wˆy is ini-
tially equal to zero.
The normalized energy density G at the asymptotic state (blue
and black circles for symmetric and asymmetric initial con-
ditions, respectively) and the initial energy density e(t = 0)
(blue and black triangles for symmetric and asymmetric ini-
tial conditions, respectively) are shown – as function of the
polar wavenumber k – in parts (a) and (b) of Fig. 3 for a
longitudinal (f = 0) and a transversal (f = p=2) perturbative
wave, respectively. In Fig. 3c we sum, at a fixed wavenumber,
asymmetric and symmetric perturbations with angle of obliq-
uity f = 0;p=4;p=2, and then we calculate the energy density
of the resulting perturbation obtained by combining six differ-
ent waves. We report  5=3 and  3 slopes with red and black
curves, respectively.
In the case of longitudinal waves (two-dimensional base and
perturbed flow configurations, see Fig. 3a), the normalized en-
ergy density G at asymptotic state has a decay of 3 in the iner-
tial range (k 2 [2;100]) for both symmetric and asymmetric per-
turbations (see blue and black circles). For smaller wavenum-
bers, instead, the decay is faster. For purely transversal waves
(Fig. 3b), we observe a decay of  5=3 in the inertial range
(k 2 [2;100]) for both symmetric and asymmetric perturbations
(blue and black circles). Longer waves have a deeper decay,
reaching a maximum of energy at about k= 0:5. If we sum sym-
metric and asymmetric waves with different angles of obliquity
(f = 0;p=4;p=2), the normalized energy density G (blue cir-
cles) of the resulting perturbation at the asymptotic state has a
decay of  3 for k 2 [25;100], while the decay is of the order
 5=3 for k 2 [2;25]. For longer waves, the decay is faster and a
maximum of energy can be found for k 2 [0:1;0:5].
From Eq. (4) and recalling that the transversal vorticity wˆy is
initially imposed equal to zero, one can observe that the initial
energy density has a well-defined decay, given by:
e(t = 0;k) =
c1
k2
+ c2; (7)
10−1 100 101 102
10−20
10−15
10−10
10−5
100
G 
k
φ=0
sym
asym
G   e(t=0) −3 slope
(a)
10−1 100 101 102
10−20
10−15
10−10
10−5
100
k
G
G   e(t=0)
sym
asym
φ=pi/2
(b)
−5/3 slope
10−1 100 101 102
10−20
10−15
10−10
10−5
100
k
G
−5/3 slope
−3 slope
Sum of sym 
and asym 
waves: φ=0,
pi/4, pi/2
(c)
G   e(t=0)
Figure 3: Spectrum of the amplification factor of a collec-
tion of stable perturbation waves, Re = 40. The normalized
energy density G at the asymptotic state (circles) and the initial
energy density e(t = 0) (triangles). The energy of each wave
in asymptotic conditions is normalized over the value owned
at the initial instant. Blue and black symbols for symmetric
and asymmetric initial conditions, respectively. (a) f = 0, (b)
f = p=2, and (c) sum of symmetric and asymmetric perturba-
tions with angle of obliquity f = 0;p=4;p=2. Red and black
curves indicate 5=3 and 3 slopes, respectively. Note that the
intermediate waves are stable and highly damped perturbations,
that reach the asymptotic state after a long lapse of time where
they progressively lose their kinetic energy. This explain the
sharp fall of energy density at the transition between the long
and the intermediate wave ranges.
where
c1 =
Z +yd
 yd
¶vˆ(y; t = 0)¶y
2 dy; c2= Z +yd yd jvˆ(y; t = 0)j2dy: (8)
This analytical behaviour is confirmed for the three cases we
considered in Fig. 3. In parts (a) and (b) blue and black triangles
(for symmetric and asymmetric initial conditions, respectively)
represent the initial energy density and show a constant value
for large wavenumbers (k 2 [2;100]), while a decay of about
 2 is visible for longer waves (k 2 [0:05;2]). It should be noted
that e(t = 0) does not depend on the perturbation obliquity, thus
the energy density values at t = 0 are exactly the same for both
f = 0 and f = p=2. In case (c), where six different waves are
summed and then the energy density of the resulting perturba-
tion is computed, e(t = 0) (blue triangles) still displays a trend
in agreement with Eq. (7).
Conclusions
This is a preliminary study of the behaviour of a collection of
transient waves seen throughout their power spectrum. To avoid
the complexity introduced by the temporal divergence associ-
ated to asymptotically unstable waves, we considered an en-
semble of transients of asymptotically stable waves. Then, we
have built the energy spectrum by freezing each wave at the
moment it reaches a constant (negative) value of the temporal
growth rate. When this happens, the waves are dynamically out
of their transient and in the asymptotic condition. In the power
spectrum, the energy of each wave is normalized over its own
initial value. In practice, it is the same as considering the spec-
tral dynamics of a white noise perturbation, a kind of model for
the swarm of small perturbations that affects any system in a
linear way.
We observe that, whether the waves are aligned with the base
sheared flow or not, the energy of the intermediate range of
wavenumbers in the spectrum decays with the same exponent
( 5=3 for 3D oblique waves,  3 for 2D aligned waves) that
is observed in the spectrum of the velocity fluctuation of fully
developed turbulent flows. Where the nonlinear interaction is
considered dominant.
At the moment, we can conclude by observing that the spectral
power-law scaling of intermediate/inertial waves (with an expo-
nent close to  5=3 in 3D and to  3 in 2D) is a general dynam-
ical property which encompasses the nonlinear interaction. In
other words, it seems to us that the strength of the nonlinearity
has been overestimated in the determination of a few turbulence
properties.
References
[1] U. Frisch, Turbulence: the legacy of A. N. Kolmogorov,
Cambridge University Press (1995).
[2] K. R. Sreenivasan and R. A. Antonia, The phenomenology
of small-scale turbulence, Annu. Rev. Fluid Mech. 29, 35-
72 (1997).
[3] S. Scarsoglio, D. Tordella, and W. O. Criminale, An ex-
ploratory analysis of the transient and long term behav-
ior of small 3D perturbations in the circular cylinder wake,
Stud. Applied Math. 123, 2 (2009).
[4] S. Scarsoglio, D. Tordella, and W. O. Criminale, The role
of long waves in the stability of the plane wake, Phys. Rev.
E 81, 036326/1-9 (2010).
[5] D. Tordella and M. Belan, A new matched asymptotic ex-
pansion for the intermediate and far flow behind a finite
body, Phys. Fluids 15 (2003).
[6] W. O. Criminale and P. G. Drazin, The evolution of lin-
earized perturbations of parallel shear flows, Stud. Applied
Math. 83 (1990).


A new matched asymptotic expansion for the intermediate and far ﬂow behind a ﬁnite body,

An exploratory analysis of the transient and long term behavior of small 3D perturbations in the circular cylinder wake,

The evolution of linearized perturbations of parallel shear ﬂows,

The phenomenology of small-scale turbulence,

The role of long waves in the stability of the plane wake,

Turbulence: the legacy of A.

https://iris.polito.it/retrieve/handle/11583/2367549/52868/Scarsoglio_Tordella_AFMC17_2010.pdf

Pre-unstable set of multiple transient three-dimensional perturbation waves and the associated turbulent state in a shear flow

Abstract

Similar works

Full text

Available Versions

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

PORTO Publications Open Repository TOrino