To study turbulence through the dynamics of the smscale structures which develop and their spectral counterLundgren1 introduced a model based on the intermittent fiscales of incompressible turbulent flows thought as consing in a collection of uncorrelated stretched spiral vorticrandomly oriented in space and individually subject toaxisymmetric irrotational straining field produced by largscales. The basic small scale structures are assumedcreated by large scale processes, excluded in the dynamithe model, like Kelvin–Helmholtz instabilities or vortex interaction mechanisms. These processes would produgiven quantityN of vortex length per unit time and unvolume, constant for a stationary turbulence.
This model is actually the only one which provides anlytically the famousk25/3 spectrum of Kolmogorov,2 al-though this kind of approach had already been introducethe Townsend model3 dealing with randomly oriented Burgers vortices and leading to ak21 spectrum~see the follow-ing!.
The central point of these vortex-based models is theof a spatial set of small scale structures, which are takelocal solutions of the Navier–Stokes equations. A few moels using tube-like or sheet-like structures have been intigated as well by Corrsin4 and Tennekes.5 The basic solution
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llrt,
t-,
nrbeof
a
-
in
seas-s-
has to contain the essential physics of the fine scale menism of balance between vorticity production, by the locstrain rate, and vorticity dissipation, by viscosity.
The model due to Townsend predicts ak21 scaling lawfor the energy spectrum in the case of the axisymmetric Bgers vortex and ak22 law for the plane Burgers layer fosmall scales (k@1), basically because of the singular natuof such structures. Obviously, the Kolmogorov expone~25/3! lies between the values obtained for the tube-like asheet-like structures. This suggests that, in order to obtaink25/3 scaling law, the vorticity field might be composed ofmixture of both structures. The properties of the axial straing combined with the roll-up of nonaxisymmetric vorticitstructures give rise to the most interesting model, thecalled ‘‘stretched spiral vortex’’ proposed by Lundgren;1 itproduces rich physical properties both in the inertial andthe dissipative ranges.6,7
It can thus be expected that any Navier–Stokes soluwhich includes the roll-up of fine vorticity gradients instrain field may produce ak25/3 scaling law as noted byLundgren.8 All these models have been sought considerincompressible Navier–Stokes dynamics. What happenthe compressible case? Porteret al.9,10 performed numericalsimulations at high resolution in three dimensions, for eithdecaying or forced compressible turbulent flows at a rMach number of unity, using the Piecewise ParaboMethod ~or PPM algorithm!.11 They showed that, in compressible turbulence, the solenoidal velocity spectrum ha
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avin
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2066 Phys. Fluids, Vol. 13, No. 7, July 2001 Gomez et al.
k25/3 scaling in the inertial range and, more surprisingly, ththe compressible velocity spectrum has the same behaThey also showed that the vorticity field organizes in strofilaments and weaker sheets and spirals,12 whereas theshocks are mostly planar when they appear. Thus thepose of this paper is to seek an explanation for such ahavior along the lines of the Lundgren vortex model.
In Sec. II, we extend the Lundgren spatiotemporal traformation to the dynamics of a compressible flow with pfect gas law. This transformation allows one to reducedynamics of a three-dimensional flow to that of a twdimensional one. We then describe the three-dimensionalocity spectra obtained for compressible homogeneous tulent flows by the use of a temporal integration of the twdimensional spectra of the enstrophy and of its compresscounterpart: the square velocity divergence. In Sec. III, usdirect numerical simulations in two space dimensions,deduce the spectral properties of a three-dimensional staary turbulent flow from the temporal evolution of the twdimensional flow.13 Section IV presents conclusions togethwith a discussion of the implications of these results.
II. THE COMPRESSIBLE LUNDGRENTRANSFORMATION
A. The incompressible model
For completeness, we first recall the essential stepthe Lundgren model1 in the incompressible case. One firconsiders a vortex structure parallel to thez direction, say,and independent of thez variable: (v r ,vu ,vz ,t)5(0,0,v(r ,u,t)), in the presence of an axisymmetric stra(2a(t)r /2,0,a(t)z), modeling the effect of the large scaleBy use of a spatiotemporal change of variables, the dynamfor the three-dimensional vorticity can be reduced to thenamics of a two-dimensional flow. An asymptotic vorticisolution of the two-dimensional Navier–Stokes equationthen obtained at large time and small viscosity.1 This solu-tion, v2(j,u,t), describes the roll-up of a spiral of vorticitywith an arbitrary number of branches, by an axisymmecentral core. The solution is generic in the sense thatvorticity distribution along the branches can be set arbitraand it is consistent with the existence of an infinite numof conserved moments of the vorticity field for twodimensional inviscid flows.
The unsteady evolution of the three-dimensional vority field reads
v~r ,u,t !5eatv2~j,u,t!, ~1!
v2~j,u,t!5 (n52`
`
v2~n!~j,t!exp~ inu!, ~2!
v2~n!~j,t!5 f ~n!~j !exp~2 inV~j!t2nn2L2~j!t3/3!,
~3!
v2~0!~j !5g~j!1 f ~0!~j !, ~4!
j~r ,t !5reat/2, ~5!
t~ t !5eat21
a, ~6!
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tor.g
r-e-
--e
e-u--legen-
r
of
cs-
is
ce
yr
-
where
1
j
d
dj~j2V~j!!5g~j!1 f ~0!~j !, ~7!
and
L5dV
dj. ~8!
The variables~j,t! are the stretched variables in space atime, corresponding to a purely two-dimensional evolutionais the uniform positive strain rate of the external field,n isthe kinematic viscosity,eatf (n)(j) is theu-averaged vorticityfor the vortex, andeatg(j) describes the axisymmetric background vorticity field. The spiral property is given byV(j),a monotonous decreasing function withj, which gives thetwo-dimensionalu-averaged angular velocity.
This solution, which describes the dynamics ofstretched spiral vortex subject to a constant strain ratethen used to calculate the velocity spectrum of thrdimensional homogeneous turbulence assuming severaportant hypotheses. Namely, in this ansatz, the local sttures, which are taken asN randomly oriented vortices, arsupposed to fill a box of sizeL at a timet; this means thateach vortex can be represented by the state of a uniquetex at a given age. All the vortices present the same tempevolution starting from timestn , randomly shifted, and havethe same lengthl 0 at times of creation. They do not interabetween themselves and the vorticity lies along the axisthe tube, aligned with an eigenvector of the external str~see Gibbonet al.14 for a discussion on this latter point!.Some external process creates these spiral vortices at aNc per unit time, and destroy them when the spiral branchave been dissipated. Thus a statistical equilibrium is matained with a constant number of structures.
Furthermore, an ergodic hypothesis is introducsummed up here as
(n51
n5N
@¯#5NcEt1
t2@¯#dt, ~9!
where @¯# is any physical quantity of the model, like thvelocity spectrum, wheret1 is the creation time andt2 thedestruction time of the spiral component of the local vorticstructure. The physical interpretation of this strong hypoesis is that the ensemble average over all uncorrelated vces in a stationary developed turbulence, with different agcan be replaced by a temporal integration over the histora unique vortex.
The energy spectrumE(k) is computed from the enstrophy spectrumEvv(k), namely E(k)5Evv(k)/2k2. Thethree-dimensional enstrophy spectrum is itself evaluafrom the two-dimensional vorticity using the spatiotempotransformation and the previous ergodic hypothesis. Tshell-summed velocity spectrum of the ensemble is
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2067Phys. Fluids, Vol. 13, No. 7, July 2001 Spiral vortices in compressible turbulent flows
E~k!5E0~k!14p
3Na1/3k25/3
3expF22nk2
3a G (n51
n5`
n24/3E0
` u f n~j!u2
uL~j!u4/3j dj,
~10!
whereN5Ncl 0 /L3. E0(k) is the spectral component cominfrom the axisymmetric term@Eq. ~4!#, and the second term ithe contribution of the nonaxisymmetric termsnÞ0 @Eq.~3!#.
The second term on the right-hand side of Eq.~10!dominatesE0 at small wave numbers and thus ak25/3 scalinglaw is obtained1 in the inertial range whenk(n/a)1/2!1. Thisresult is derived using the two-dimensional analytical sotion with constant strain ratea assumed to be provided blarger scales. Lundgren8 showed using numerical consideations that the spectral index is not considerably influenby taking a time-dependent strain rate, and this assumptiolargely used by different authors in the literature.
B. The compressible transformation and solutions
Similar to the Lundgren solution, we consider a problein which the fluid is strained by an axisymmetric flow of thform
ustrain5~2a~ t !r /2,0,a~ t !z! ~11!
with “"(ustrain)50 and wherea(t) is the strain rate. We arelooking for a solution in which there is only az componentof vorticity, with all the variables independent ofz. The com-pressible Navier–Stokes equations for the vorticityv5“Ãu5(0,0,v) and the velocity divergenced5“"u with aconstant dynamic viscositym can be written as
]r
]t1~u1ustrain!"“r52rd, ~12!
]v
]t1~u1ustrain!"“v5a~ t !v2vd1
1
r2 ~“rÓp!
1m
r~¹2v!1m“S 1
r DÃF2“Ãv1
4
3“dG , ~13!
]d
]t1~u1ustrain!"“d5a~ t !d2“u:“u2
3
2a2~ t !
21
r¹2p1
1
r2 “r"“p14
3
m
r¹2d
1m“S 1
r D •F2“Ãv14
3“dG ,
~14!
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-
dis
]e
]t1~u1ustrain!"“e52~g21!ed1
m
r S t:D2ad
13
2a~ t !2D1
k
r¹2T, ~15!
wherer is the mass density ande the internal energy per unimass. The temperatureT is related to the internal energye bythe relatione5CvT, assuming constant specific heatsCvandCp . Let us then add the perfect gas law
p
r5RT, ~16!
whereR is the specific gas constant. The viscous stresssor is defined ast i j 522/3“"ud i j 12Di j , whered i j is theKronecker symbol and the strain tensor is defined asDi j
51/2(] jui1] iuj ). These equations conserve mass, momtum, and total energy.
Note that in Eq.~14! for the velocity divergence, there ia term of divergence production by the strain,a(t)d, similarto the term of vorticity production in Eq.~13! for the vortic-ity.
In order to reduce the three-dimensional dynamics ttwo-dimensional one, the change of variables in spacetime is defined, similarly to the incompressible case, as
S~ t !5expE0
t
a~ t8!dt8, ~17!
j~ t !5S1/2~ t !t, ~18!
t~ t !5E0
t
S~ t8!dt8, ~19!
and, in the compressible case, the spatiotemporal transfotions between the three- and two-dimensional fields and tmodynamic variables take the form
v~r ,u,t !5S~ t !v2~j,u,t!, ~20!
d~r ,u,t !5S~ t !d2~j,u,t!, ~21!
u~r ,u,t !5S1/2~ t !u2~j,u,t!, ~22!
r~r ,u,t !5r2~j,u,t!, ~23!
e~r ,u,t !5S~ t !e2~j,u,t!, ~24!
p~r ,u,t !5S~ t !p2~j,u,t!, ~25!
where the subscript 2 denotes the two-dimensional flow.The behavior of the compressible velocity spectrum
the vortex structures, which describe the local dynamicsturbulent flows, can be investigated under the two followiassumptions. First, there exists a scale separation betwthe large scales of the external strain and the internal fltuations of the vortex structures at scales comparable to tcross-sectional diameters. Thus, we assume that the intefluctuations in planes perpendicular to the local structuresprovided only by the velocity field of the structures themselves. A similar hypothesis has been introduced by Pull15
to evaluate the pressure spectrum of the incompress
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r
2068 Phys. Fluids, Vol. 13, No. 7, July 2001 Gomez et al.
FIG. 1. Temporal evolution of the ratiox(t)5Ed(t)/Es(t) at left; and of the rms Mach numbeM rms(t) at right.
o
m
onenq
nao
w
aed
. Aht-
lltheis
ticog-ingss-
a-nu-er,
Lundgren vortex. Second, the dynamical solutions are csidered for large rescaled timet and small viscosity.
Under these assumptions, the spatiotemporal transfortion enables us to find solutions of Eqs.~12!–~15! based onthe solutions of the transformed equations which correspto the dynamics of a purely two-dimensional flow. Whtransforming the three-dimensional equations, using E~17!–~25!, the nonlinear terms, proportional toa2, resultingfrom the self-interaction of the large scales of the exterstrain field are neglected in the two-dimensional dynamicsthe local structure, as well as a term of order 1/t on theright-hand side of the equation of the internal energy, asare looking for solutions at larget. Starting from the three-dimensional dynamics, we obtain the following set of equtions involving the two-dimensional fields in the rescaltime and space variables:
]r2
]t1u2"“jr252rd2 , ~26!
]v2
]t1u2"“jv252v2d21
1
r22 ~“jr2Ójp2!1
m
r2~¹j
2v2!
1m“jS 1
r2DÃF2“jÃv21
4
3“jd2G , ~27!
]d2
]t1u2"“jd252“ju2 :“ju22
1
r2¹j
2p211
r22 “jr2"“jp2
14
3
m
r2¹j
2d21m“jS 1
r2D "F2“jÃv2
14
3“jd2G , ~28!
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n-
a-
d
s.
lf
e
-
]e2
]t1u2"“je21~g21!e2d22
m
r2~t2 :D2!2
k
r2¹j
2T2
52a
S~t! S e21m
r2d2D ~29!
with the perfect gas law
p2
r25RT2 , ~30!
where“j denotes the gradient in the stretched variablessupplementary simplifying assumption is to discard the righand side of the internal energy equation~29!: the 1/S(t)coefficient, of order 1/t, is likely to render those terms smacompared to the left-hand side of the equation, since, inspirit of the Lundgren analysis, the temporal integrationcarried out for long rescaled times (t;50– 200).8 In thecontext of the incompressible Lundgren model, an analysolution can be found, as recalled in Sec. II, and the Kolmorov spectrum emerges from an integration over time usthe approximation of the stationary phase. In the compreible case, in view of the obvious complexity of the equivlent model coupling all variables, we shall seek here americal approach to be given in the next section. Howevthe definition of the velocity spectrum is first introduced.
C. The compressible velocity spectrum
The velocity is decomposed as usual into two parts
u5us1ud, ~31!
where
“Ãud50, “"us50; ~32!
FIG. 2. Two-dimensional spectra at timet530: for theincompressible square velocityEs(k) at left, and for thecompressible square velocity divergenceEd(k) at right.
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lee
2069Phys. Fluids, Vol. 13, No. 7, July 2001 Spiral vortices in compressible turbulent flows
FIG. 3. Temporal evolution of the incompressibsquare velocityEs(t) at left, of the compressible squarvelocity Ed(t) at right.
ibinitydee
br-
r-
eo
ra-
ther-at aein
te-he
is
the
us is the solenoidal velocity andud the dilatational velocity.The velocity decomposition~31! led Moyal16 to introducethe decomposed spectra
E~k!5Es~k!1Ed~k!, ~33!
where
Es~k!5E E us"us* dSk , ~34!
and
Ed~k!5E E ud"ud* dSk , ~35!
where an asterisk denotes complex conjugates andu the Fou-rier transform of the velocity.
The method developed by Lundgren8 to obtain the three-dimensional kinetic energy spectrum of an incompressflow from two-dimensional numerical simulations consistsa time integration of the two-dimensional square vortic~enstrophy! spectrum. In the compressible case, we canfine the corresponding physical quantities, with the addingredient of the velocity divergenced5“"u.
The power spectrum of the velocity divergence canwritten in terms of the Fourier integral of the velocity divegence correlation function
Rdd~r,t !51
L3 E d~r ,t !"d~r1r,t !dr , ~36!
whereL is the length of the box, and of its Fourier transfomation
Fdd~k,t !51
2p3 Eall r
exp~2 ik"r!Rdd~r,t !dr. ~37!
In the case of three-dimensional compressible homogenturbulence, the compressible velocity spectrum is
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le
-d
e
us
Ed~k,t !5F~k,t !/2k2 ~38!
with F the velocity divergence spectrum obtained by integtion in spherical shellsSk of radiusk in wave number space
F~k,t !5E Fdd~k,t !dSk . ~39!
If we consider a stationary turbulence, and assume thatvelocity divergence is concentrated in the vicinity of the votex filaments which are themselves isolated and createdconstant rateNc , with the same structure and the samstrength, we can invoke the ergodic hypothesis asLundgren to transform a space integration into a time ingration on the temporal evolution of a single structure. Tcompressible velocity spectrum is thus expressed as
Ed~k!52p2Nc
L3
1
k2 E0
tcutl ~t!F~k,t!dt, ~40!
wheretcut is the lifetime of the spiral structure andl (t) thefilament length.
III. NUMERICAL EXPERIMENTS
A. Numerical setup
We consider a medium of characteristic lengthL, ofmean densityr0 , and mean velocityu0 . These quantities areused to normalize to unity the densityr and the velocityu,and the spatial scale~size of the computational box! is 2p.The internal energy is normalized byu0
2 and is related to thetemperature as usual bye5CvT with the nondimensional-ized constant parameterCv51/(g(g21))M0
2; the initialMach number isM05u0 /c0 where the speed of sounddefined asc0
25gRT0 with g the adiabatic index andR theperfect-gas constant. The normalization temperatureT0 onwhich the sound velocity is based, is set to ensure that
FIG. 4. Temporal evolution of enstrophyVs(t) at left,and of its compressible counterpartVd(t)
512^(“"u)2& at right.
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i-ys
2070 Phys. Fluids, Vol. 13, No. 7, July 2001 Gomez et al.
FIG. 5. Temporal evolution of the ratio of the maxmum to the minimum of the mass densitrmax(t)/rmin(t) at left, and temporal evolution of the rmvelocity at right.
akeun
stwm
p-hrdreckum
bea
ths-
ln
frgyiblethetheinandhise-otaltiotr, inith
ghar
ini-and
alia-
he
nsdr,its
nalw-al
temperature variable is of order unity. Furthermore, we tthe eddy turnover timeL/u0 as the dependent unit. In thcode, the equations for conservation of mass, momentand internal energy are written in the following nondimesionalized form:
]r
]t1u"“r52r“"u, ~41!
]u
]t1u"¹u52
1
r“p1
1
r ReS ¹2u11
3“~“"u! D , ~42!
]e
]t1u"“e52~g21!e“"u1
1
r Re~t:D!1
g
Pr Re¹2T,
~43!
wherer is the mass density,u the velocity, ande the internalenergy per unit mass;t and D are, respectively, the strestensor and the strain tensor defined as in Sec. II B. Thedimensionless parameters that arise are the Reynolds nuRe5r0u0L0 /m and the Prandtl number Pr5mCp /k5n/hwherek is the constant thermal conductivity,n the kinematicviscosity, andh the thermal diffusivity. The perfect gas lawin nondimensional form writesp5(g21)re.
Using periodic boundary conditions with a Fourier reresentation, the numerical simulations are performed witpseudospectral code. The temporal scheme is a third-oRunge–Kutta scheme, the intermediate steps of whichon an Euler scheme for the nonlinear terms and a CranNicholson scheme for the dissipative ones. The wave nbers vary fromkmin51 to kmax5N/2, whereN is the numberof grid points in each direction, withN up to 512.
B. Initial conditions
As stated before, the velocity fluctuations canuniquely distributed via the Helmholtz decomposition intosolenoidal part whose Fourier transform is orthogonal towave vectork, and a dilatational part with its Fourier tran
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e
m,-
ober
aerly–-
e
form collinear tok.16 The incompressible part of the initiavelocity distribution is given by the Lundgren spiral solutioat a given nondimensional time.8 The compressible part othe velocity is taken as a Gaussian noise with an enespectrum at small scales analogous to the incompresspart, and set up at wave numbers corresponding tobranches of the spiral in a range excluding essentiallyfive first modes linked to the vortex core. This is doneorder to favor the interactions between the compressiblespiral incompressible components of the velocity field. Tcondition allows for an intensification of the interactions btween incompressible and compressible modes. The tcompressible kinetic energy level is given through the rax5Ed/Es, initially equal to 0.1 in the simulations. Note thax is a second free parameter, besides the Mach numbecompressible flows; the value chosen here is in keeping wmost numerical simulations of supersonic flows, althouhigh values ofx can arise in the context of the interstellmedium when energy is injected through supernov, blastwaves or through heating by incoming cosmic rays. Thetial density and temperature fields are set to be uniformequal to unity. The initial internal energye5CvT defines theinitial rms Mach number; we take it equal to 0.23 with locvalues up to 0.9. The Prandtl number is unity and the adbatic index isg51.4. The Reynolds number, based on trms velocity and the integral scale, is Re5700 with a viscos-ity such that 1/n520 000.
C. Two-dimensional dynamical evolutions
We now examine the numerical results for simulatiowith an initial ratio x50.1. This ratio is stabilized aroun0.03 att5100 as shown in Fig. 1. The rms Mach numbeinitially equal to 0.23, has a quasiconstant value duringtime evolution, with a decrease of the order of 1% at the fitime t5200, as shown on the right-hand side of Fig. 1. Hoever, the maximum of the local Mach number, initially equ
i-
FIG. 6. Temporal evolution of the maximum and minmum of the vorticity production terms; left:2v“"u(t); right: the baroclinic term “rÓp/r2(t).
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of
2071Phys. Fluids, Vol. 13, No. 7, July 2001 Spiral vortices in compressible turbulent flows
FIG. 7. Map of the total velocity square as a functiontime for time ranging fromt55 to t5150;3tac. Themaximum values of velocity squareu2 are, respec-tively, 1.2331022, 1.0831022, 8.0231023, 7.1131023, 5.8931023, 4.4831023 at times 5, 10, 15, 50,100, 150.
tse
--
th
-ibne--
iarsin
or
th
av-of
ag-ction
ueentacaness-tialt at
om-own
he
to 0.9, decreases along the simulation and stabilizes iaround 0.55 at a time aroundt5100. Note that it takes asound wave a timetac52p/crms;47 to cross the computational box of sizeL52p; this time is the characteristic compressible time for the interaction of a sound wave withcentral spiral vortex.
All along the dynamical evolution of the twodimensional flow, the wave numbers of the compresssquare velocity, when compared to the incompressible oare dominant at small scales (k.5) while they are subdominant at large scales~Fig. 2!, as observed in numerous numerical simulations using, for example, random initconditions.17 Hence, when integrated over wave numbethis leads to an incompressible square velocity remainlarger than its compressible counterpart~see Fig. 3!, as wellas to a ratiox decreasing from its initial value of 0.1~Fig. 1!.One can distinguish three different regimes on the tempevolution of the enstrophyVs(t)5 1
2^(v)2& and of its com-pressible counterpart, the square velocity divergenceVd(t)5 1
2^(“•u)2& ~Fig. 4!. The first one, extending fromt50 tot;50, corresponds to an acoustic time and starts withrapid generation of weak shocks up tot;15, followed by a
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lf
e
les,
l,g
al
e
decrease ofVd(t) up to t;50 during an interaction timebetween the spiral vortex and the sound waves. This behior is also visible on the temporal evolution of the ratiomaximal to minimal mass densityrmax(t)/rmin(t) which ischaracteristic of the flow compressibility, because of thegregation and condensation on the one hand, and rarefawaves on the other hand~Fig. 5 at the left-hand side!. Duringthis phase, there is no creation of enstrophy;Vs(t) decaysrapidly from t50 to t;15, displaying then a small platea~from t;15 to t;40!. This is similar to the incompressiblcase: all the incompressible small scales are initially presfor the incompressible part of the initial velocity which issolution of the incompressible equations; thus enstrophyonly decrease through dissipation. Moreover, the comprible modes of the velocity do not produce any substanenstrophy at large scales, as they are mostly prevalensmall scales. However, the interactions between the incpressible and compressible modes are very intense as shon the temporal evolution of the maxima and minima of tnonlinear terms of the vorticity equation~Fig. 6!; up to t;50 for the 2v“•u term, and up tot;80 for the baro-clinic term“r3“p/r2.
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2072 Phys. Fluids, Vol. 13, No. 7, July 2001 Gomez et al.
FIG. 8. Velocity square contours and its profile aty5256 at timet530;0.60tac. The maximum values ofvelocity squareu2 is 7.4131023.
r:h
ttein,
mon
av
cnblono
-.al
o
octemf t
eity,toof
har-s ofle
svis-e
c-n-anr-hex;
e-citydi-
hyughup-
Note that the ratiox(t) ~Fig. 1, on the left-hand side!and the compressible square velocityEd(t)5^(ud)2& ~Fig. 3,on the right-hand side! present altogether a similar behavioa rapid increase up tot;10, followed by a decrease whicsaturates fromt;15 to t;40.
The second temporal regime, fromt;50 tot;100, cor-responds to a second interaction of the weak shocks withcentral spiral vortex, and with other weak shocks creasimultaneously. The weak shocks dissipate less energy, smost of it has already been dissipated at small scalesvisible on the slowing down of the decrease ofVd(t) ~Fig.4!, measuring the viscous dissipation of the kinetic copressible energy, to within a pressure term. The fluctuatiof the ratiormax(t)/rmin(t) ~Fig. 5! stabilize around 1.9 withshort weak oscillations. They reveal the presence of smturbulent fluctuations of the compressible velocity that hanot yet dissipated.
The third regime, fromt;100 onward, is an acoustione, during which the compressible velocity fluctuatiotravel through the computational box, with almost no sizainteractions with the incompressible part. This is visiblethe strong diminution of the maximum and the minimumthe vorticity production terms ~Fig. 6!. The ratiosrmax(t)/rmin(t), x(t) andEd(t) are quasiconstant, with values, respectively, oscillating around 1.5, 0.03, and 0.001
We now turn to the dynamics of the flow in physicspace. We show the square velocity as a function of timeFig. 7, fromt55 to t5150, corresponding approximately tthree acoustic times.
The intense fine compressible structures are weak shwhich interact strongly with the spiral branches of the vorduring a characteristic time comparable to the acoustic tiOne can observe these interactions on the iso-contours o
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hedceas
-s
lle
se
f
in
ksxe.he
velocity square~Fig. 7!; note as well the persistence of thspiral structure of the solenoidal component of the veloceven though the maximum initial Mach number is closeunity. This is due to the fact that the energetic structuresthe compressible velocity are at smaller scales than the cacteristic scale of the spiral arms. The energetic structurethe square velocity field are exemplified by plotting a profi~Fig. 8, on the right-hand side! of the field at timet530;0.6tac ~Fig. 8, on the left!. In fact, the most intense shockare localized along the branches of the spiral vortex, asible on the profiles plotted in Fig. 9 for, respectively, thvorticity field ~Fig. 10, on the left-hand side! and the velocitydivergence field~Fig. 10, on the right-hand side! shown attime t530;0.6tac. Moreover, these compressible strutures locally—in the vicinity of the spiral arms—are perpedicular to them, leading globally to what can be called‘‘ortho-spiral’’ structure; the divergence field is locally pependicular to the vorticity and is then carried along in tglobal rotation of the flow enticed by the strong vortehence, the complex structure for“•u whose skeleton is thespiral of vorticity.
D. Three-dimensional spectral properties
In this section, we compute numerically the thredimensional spectrum of the compressible square velofrom the two-dimensional spectra of the square velocityvergence according to relation~40!, as well as its incom-pressible counterpart from the two-dimensional enstropspectra.8 The three-dimensional spectra are obtained throa temporal integration of the two-dimensional ones; theper time limit of this integration,tcut, is chosen according tothe dynamical evolution of the two-dimensional flow.
FIG. 9. Profiles aty5256 of vorticity ~left! and“"u~right! at timet530;0.60tac.
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2073Phys. Fluids, Vol. 13, No. 7, July 2001 Spiral vortices in compressible turbulent flows
FIG. 10. At timet530;0.60tac, contour plot of thevorticity v ~left! with minimum and maximum valuesof 20.08 and 0.25, and, on the right, contours of“"uwith values ranging from23.68 up to 1.25.
te
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bye
thTma
n
,
p
ulcit
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olf-asi
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thelylt ofe in-ofingofig. 6
ained
For the compressible part of the flow, the temporal ingration is carried up totcut
d 550;tac to take into accountonly one interaction between the shocks and the vortex stture. In the model, after this time, the compressible strtures, which have interacted with the spiral vortex, leaveinfluence domain of the local vortex; they travel away whweakening because of dissipation until another intense svortex is encountered which re-energizes them again. Duthis travel between intense vortices, the compressible fltuations are assumed to have nonrelevant contributions tothree-dimensional compressible kinetic spectrum.
The resulting compressible spectrum, compensatedfactor k5/3, is plotted in Fig. 11 for different values of thexternal strain rate, namelya51, a54, anda510; note thata higher value of the parametera, that is related to an in-crease of the strain intensity, allows for an acceleration ofenergy cascade mechanism from large to small scales.procedure thus enables us to extend the range of the doof the three-dimensional wave numbers toward higher vues, from 1 toS1/2(t)3kmax5A11at3kmax, wherekmax isthe highest wave number reached in the two-dimensionumerical simulation, withkmax5256 here. Fora51, ak25/3
inertial range is obtained and extends roughly fromk535 tok570. For a54 anda510, this k25/3 range extends fromk570 to k5130 andk5110 tok5210, respectively. Thusthe increase of the strain rate enlarges thek25/3 range of thespectrum, without a significant change of the spectral slothus clearly pointing out to the correlation between thek25/3
range and the strain intensity. This physical property cobe due to a self-similar behavior of the square of the velodivergence analogous to that of the enstrophy as shownLundgren8 in the purely incompressible case. However, ohas to realize that increasinga enables one to enlarge thinertial range, but does not change the total number of tdimensional wave numbers used to construct the thdimensional spectrum. Note that in the case of a substantlower Mach number than that taken here, the 5/3 spectdoes not obtain; for example, for an initial rms Mach numbof order 1022, with local maxima of the Mach number up t3.931022, our model does not provide a well-defined sesimilar range for the spectrum, probably due to the weenergy exchanges, in that case, between the compresand incompressible velocity scales.
The three-dimensional incompressible velocity spectris computed from the the temporal integration of the twdimensional enstrophy spectra, up to eithertcut
s 550 or tcuts
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-
c--e
alg
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a
ehisainl-
al
e,
dybye
-e-llymr
kble
-
5100;2tac; the second choice oftcuts is done in order to
check the influence of the temporal upper limit. Indeed,lifetime of the spiral vortex is shorter than in the pureincompressible case; this can be interpreted as the resuthe energy exchanges between the compressible and thcompressible flow components leading to the formationinhomogeneities inside the spiral branches, thus reinforctheir dissipation. The temporal evolution of the intensitythese energetic exchanges, as can be observed in F
FIG. 11. Compressible three-dimensional compensated spectra obtwith a time integration betweent50 andtcut
d 550;tac for various externalstrain ratesa51, a54, anda510.
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n-en
2074 Phys. Fluids, Vol. 13, No. 7, July 2001 Gomez et al.
FIG. 12. Incompressible three-dimensional compesated spectra obtained with time integrations betwet050 and tcut
s 550 ~left!, and tcuts 5100 ~right! for a
unity external strain rate.
on
oteThersib
erve
ruetiothe
al
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l t
hes
peobheac-
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iblensen a-
ulde,ayser-edng
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is-
R.
e,’’
of
a
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nd
showing the vorticity production terms, leads to the choicethe values fortcut
s used to perform the temporal integratioproviding the incompressible spectrum. In fact, aftert;100, the spiral branches of the Lundgren vortex have csiderably faded out and, as discussed before, we have enan acoustic regime unrelated to the present problem.resulting compensated spectra are plotted in Fig. 12. Tdisplay a lesser agreement with ak25/3 behavior on a shorterange of wave numbers than in the case of an incompresflow at the same Reynolds number.8 However, at the sameMach number, one can obtain a better agreement and a mextended scaling zone for flows at higher Reynolds numb
In fact, the numerical simulations we performed hashown that when increasing the Mach number, thek25/3
spectral range is extended for the compressible spectwhile, when increasing the Reynolds number, this rangextended for the incompressible spectrum. At the resoluconsidered here, some compromise has to be found. Inwork, we make the choice to favor the compressible aspin considering a range of Mach numbers approaching locunity. This explains, in part, the shorterk25/3 range obtainedfor the incompressible spectrum.
On the other hand, one has to notice that the specdecrease at small scale that we observe here can be comwith a similar behavior obtained by Lundgren8 for the in-compressible spectrum, when considering an initial conditconsisting in an axisymmetric central vortex surroundedeight identical smaller circular vortices. Such an initial codition is not a solution of the Navier–Stokes equations athe interactions among its different substructures feed allscales, including the smallest ones. This could influenceself-similar behavior of the temporal decrease of the ensphy, lead to the presence of inhomogeneities, and enforcedissipation, thus explaining the observed spectral decreon the compensated plots.
IV. CONCLUSION
We have extended the Lundgren spiral vortex modecompressible flows and shown that it is compatible withk25/3 Kolmogorov scaling range for the spectrum of tcompressible square velocity, as well as for the incompreible one, although with a lesser agreement on a shorter stral extent for the latter. Such spectral scaling laws areserved in three-dimensional numerical simulations of eitforced or decaying compressible turbulence, at rms Mnumbers of order unity.9,10 Note that, although our simula
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f
n-redhey
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m,isnisctly
alred
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tions are at rms Mach numbers of 0.23, they display lovalues of the Mach numbers up to the order of unity.
The model is based on geometrical considerations linto the roll-up of the velocity divergence fluctuations by tstrong vorticity structures of the flow. We have thus shothe crucial role of the intense vorticity structures, includiin the case of compressible flows. Indeed thek25/3 scalinglaw satisfied by the energy spectrum of the compressvelocity component could have as its origin the most intevorticity structures which drive the compressible modes iglobal roll-up motion. This differential roll-up of compressible fluctuations combined with a large scale straining cothus explain thek25/3 spectrum by a process similar to thone described by Gilbert18 for incompressible flows. Indeedone observes that the divergence of the velocity displwhat can be called an ortho-spiral organization: the divgence field, locally perpendicular to the vorticity, is carrialong in the global rotation of the flow enticed by the strovortex; hence, the complex structure for“•u leads as well toa 5/3 spectrum~to within intermittency corrections, which isa topic not considered by these types of models!, and for thesame basic reasons as exemplified in the model developeGilbert: it is a combination of stretching by a strain field aa differential rotation by a vorticity field which leads to suca balance.
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2A. Kolmogorov, ‘‘The local structure of turbulence in incompressible vcous fluid for very large Reynolds number,’’ Dokl. Akad. Nauk SSSR30,9 ~1941!.
3A. A. Townsend, ‘‘On the fine scales structures of turbulence,’’ Proc.Soc. London, Ser. A208, 534 ~1951!.
9D. H. Porter, A. Pouquet, and P. R. Woodward, ‘‘Kolmogorov-like specin decaying three-dimensional supersonic flows,’’ Phys. Fluids6, 2133~1994!.
10D. H. Porter, A. Pouquet, and P. R. Woodward, ‘‘Inertial range structuin decaying compressible turbulent flows,’’ Phys. Fluids10, 237 ~1998!.
11P. Woodward and P. Colella, ‘‘The numerical simulation of twdimensional fluid flow with strong shocks,’’ J. Comput. Phys.54, 115~1984!.
12D. H. Porter, A. Pouquet, and P. R. Woodward, ‘‘Compressible flows a
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ed
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vortex stretching. Small-scale structures,’’ inThree-dimensional Hydrody-namics and Magnetohydrodynamics Turbulence, edited by M. Meneguzzi,A. Pouquet, and P. L. Sulem@ Lect. Notes Phys.462 51 ~1995!#.
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14J. D. Gibbon, A. S. Fokas, and C. R. Doering, ‘‘Dynamically stretchvortices as solutions of the 3D Navier–Stokes equations,’’ Physica D132,497 ~1998!.
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16J. E. Moyal, ‘‘The spectra of turbulence in a compressible fluid; Edturbulence and random noise,’’ Proc. Cambridge Philos. Soc.48, 329~1951!.
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