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Shock Waves (2016) 26:241–251DOI 10.1007/s00193-015-0580-5
ORIGINAL ARTICLE
Vorticity dynamics after the shock–turbulence interaction
D. Livescu1 · J. Ryu2
Received: 23 January 2015 / Revised: 25 June 2015 / Accepted: 29
June 2015 / Published online: 23 July 2015© Springer-Verlag Berlin
Heidelberg (outside the USA) 2015
Abstract The interaction of a shock wave with quasi-vortical
isotropic turbulence (IT) represents a basic problemfor studying
some of the phenomena associated with highspeed flows, such as
hypersonic flight, supersonic com-bustion and Inertial Confinement
Fusion (ICF). In general,in practical applications, the shock width
is much smallerthan the turbulence scales and the upstream
turbulent Machnumber is modest. In this case, recent high
resolution shock-resolved Direct Numerical Simulations (DNS) (Ryu
andLivescu, J Fluid Mech 756:R1, 2014) show that the inter-action
can be described by the Linear Interaction Approx-imation (LIA).
Using LIA to alleviate the need to resolvethe shock, DNS post-shock
data can be generated at muchhigher Reynolds numbers than
previously possible. Here,such resultswithTaylorReynolds number
approximately 180are used to investigate the changes in the
vortical structure asa function of the shock Mach number, Ms , up
to Ms = 10. Itis shown that, as Ms increases, the shock interaction
inducesa tendency towards a local axisymmetric state
perpendicularto the shock front, which has a profound influence on
thevortex-stretching mechanism and divergence of the Lambvector
and, ultimately, on the flow evolution away from theshock.
Communicated by A. Podlaskin.
This paper is based on work that was presented at the 21st
InternationalSymposium on Shock Interaction, Riga, Latvia, August
3–8, 2014.
B D. [email protected]
1 CCS-2, Los Alamos National Laboratory, Los Alamos,NM 87545,
USA
2 Department of Mechanical Engineering,University of California,
Berkeley, CA 94720, USA
Keywords Compressible turbulence · Shock–turbulenceinteraction ·
Vorticity · LIA · DNS · Lamb vector
1 Introduction
The interaction of shock waves with turbulence is an impor-tant
aspect in many types of flows, from hypersonic flight,to supersonic
combustion, to astrophysics and Inertial Con-finement Fusion (ICF).
In general, the shock width is muchsmaller than the turbulence
scales, even at low shock Machnumbers, Ms , and it becomes
comparable to the molec-ular mean free path at high Ms values. When
there is alarge scale separation between the shock and turbulence,
vis-cous effects become negligible during the interaction. If,
inaddition, the turbulentMach number,Mt , of the upstream
tur-bulence is small, the nonlinear effects can also be
neglectedduring the interaction. In this case, the interaction can
betreated analytically using the linearized Euler equations
andRankine–Hugoniot jump conditions. This is known as theLinear
InteractionApproximation (LIA) [1–3].However, dueto the high cost
of simulations for the parameter space closeto the LIA limit (and
practical applications) and difficultieswith accurate experimental
measurements close to the shock,previous studies have shown only
limited agreement withLIA [3–10]. Recently, Ryu and Livescu [11],
using high res-olution fully resolved Direct Numerical Simulations
(DNS)extensively covering the parameter range, have shown thatthe
DNS results converge to the LIA solutions as the ratioδ/η, where δ
is the shock width and η is the Kolmogorovmicroscale of the
incoming turbulence, becomes small. Theresults reconcile a long
time open question about the roleof the LIA theory and establish
LIA as a reliable predictiontool for low Mt turbulence–shock
interaction problems. Fur-thermore, when there is a large
separation in scale between
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242 D. Livescu, J. Ryu
the shock and the turbulence, the exact shock profile is
nolonger important for the interaction, so that LIA can be usedto
predict arbitrarily high Ms interaction problems, whenthe
Navier–Stokes equations are no longer valid and fullyresolved DNS
are not feasible.
The shock–turbulence interaction has been traditionallystudied
in an open-ended (shock-tube) domain, with the tur-bulence fed
through the inlet plane encountering a stationaryshock at some
distance from the inlet. Usually, turbulencehas been generated
either directly in the inlet plane or withadditional decaying
isotropic turbulence (IT) simulationsand then used in the spatial
domain by invoking the Tay-lor hypothesis. To avoid this
hypothesis, which limits themagnitude of the acoustic component and
overall turbulenceintensity to small values, Ryu and Livescu [11]
have gener-ated the inlet turbulence in separate forced
compressible ITsimulations with background velocity matching the
shockspeed and using the linear forcing method [12] (Fig. 1).
Thisforcing method has the advantage of specifying the Kol-mogorov
micro scale and ratio of dilatational to solenoidalkinetic
energies, χ , at the outset. Nevertheless, the shock-tube approach
is very expensive (with or without realisticinlet turbulence), even
when a shock-capturing scheme isused, and limited to low Taylor
Reynolds numbers, Reλ.However, the range of the achievable Reλ
values can be sig-nificantly increased if, instead, one uses the
LIA theory togenerate the post-shock fields. To be able to generate
full 3-D fields, Ryu and Livescu [11] have extended the
classicalLIA formulas, which traditionally have been used to
calcu-late second order moments only. Using this procedure,
theyshowed profound changes in the structure of post-shock
tur-bulence, with significant potential implications on
turbulencemodeling.
The analysis of small amplitude fluctuations in a com-pressible
medium performed by Kovasznay [13] showedthe existence of three
basic modes: the vorticity, acousticand entropy modes. For uniform
mean flow, in the invis-cid limit, the modes evolve independently.
The vortical andentropy modes are advected by the mean flow, while
theacoustic mode travels at the speed of sound. The vorticalmode
consists in a solenoidal velocity field only, the entropicmode only
has density and temperature fluctuationswhile theacoustic mode has
isentropic pressure and density fluctua-tions and a corresponding
irrotational (dilatational) velocitythat satisfies the acoustic
wave equation. Thus, the velocityfield has contributions from the
acoustic and vortical modes,the density and temperature fields from
acoustic and entropiccomponents and the pressure field is only
associated with theacoustic mode. The interaction with the shock
generates allthreemodes, evenwhen the upstream turbulence has only
onemode present. In many practical applications, the intensityof
the upstream turbulence fluctuations is small enough thatthe fast
interaction with a thin shock is in the linear regime
Fig. 1 Numerical setup for DNS of shock–turbulence interaction
inRef. [11]: data recorded from forced IT simulations with
backgroundvelocity matching the shock speed a are fed through the
inlet of anopen-ended domain b. The red rectangle is the plane
where the flowdata are recorded a and the location of inlet feeding
b. Eddy structuresare visualized by the Q-criterion
[11]; however, the flow evolution away from the shock occursover
much longer time and length scales and nonlinear andviscous effects
can no longer be neglected. In this case, thethree modes become
fully coupled [14,15]. Nevertheless, theevolution following the
interaction with the shock, be it puredecay or subsequent
interactions, depends on the propertiesof the post-shock
turbulence.
While recent shock–turbulence interaction studies [9,10]using
shock-capturing techniques or full DNS [11] have sig-nificantly
extended the range of Reynolds numbers achievedin such a flow (∼70
for shock capturing and ∼45 for fullDNS), the Reλ values are still
smaller than those considerednecessary to reach the transition to
fully developed turbu-lence. For example, IT is considered to be
fully developed ifReλ >100 [16], although much larger values may
be neededif higher order statistics are investigated. In addition,
previous
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Vorticity dynamics after the shock–turbulence interaction
243
studies have examined in detail various quantities related tothe
transport equations for the second ordermoments, such asReynolds
stresses (e.g. [9,10]). Ryu and Livescu [11] haveextended the
analysis of the post-shock fields to the localproperties of the
strain rate tensor, S. Yet, there are still manygaps in our
knowledge of post-shock turbulence, for exampleconcerning the
vorticity field, such as the vortex-stretchingmechanism and its
relation to the energy cascade and theLamb vector and its
connection with coherent structures.
Simulations of turbulent flowswith shocks have to contendwith
contradictory requirements for the numerical algorithmsto
simultaneously capture the turbulence and the shocks.Thus,
turbulence simulations require the minimization ofnumerical
dissipation for small scale representation, whilethe shocks require
increased local dissipation to regularizethe algorithm [17].
Explicit subgrid models also need toaccount for the presence of the
shock. Yet, available highReλ data necessary to investigate the
turbulence models arescarce.
This study aims at using the novel procedure proposedby Ryu and
Livescu [11] to generate high Reλ post-shockdata and study the
properties of post-shock turbulence, asreflected in the
characteristics and dynamics of the vorticityfield. First, the
results are shown to be consistent with ourextensive DNS database
for up to Ms = 2.2 and then usedto predict the characteristics of
the vorticity field after theshock and its downstream evolution at
high Ms values. Here,the incoming turbulence is vortical; results
with incomingturbulence having significant entropic and acoustic
modeswill be presented elsewhere.
The paper is organized as follows. Section 2 containsthe
governing equations, problem setup and the numericalmethodology, as
well as the extended LIA formulas. Section3 is the main results
section of the paper. Thus, Sect. 3.1shows the convergence of the
DNS enstrophy amplificationto the LIA prediction, Sect. 3.2
discusses the vorticity fieldand the vortex-stretchingmechanism,
Sect. 3.3 focuses on theLamb vector and its divergence and Sect.
3.4 provides somedata on the correlation between vorticity and the
thermody-namic variables. Finally, Sect. 4 provides the
conclusions.
2 Problem setup and numerical methods
The equations considered for studying the properties
ofpost-shock turbulence are the compressible Navier–Stokesequations
with the perfect gas assumption [12]. The ratio ofspecific heats is
γ = 1.4, the viscosity varies with the tem-perature as μ =
μ0(T/T0)0.75, and the Prandtl number isPr = 0.7. All simulations
use the compressible version ofthe CFDNS code [11,12,18]. The setup
for the DNS of theshock–turbulence interaction (Fig. 1) is
described in detail inRef. [11].
In order to generate high Reλ post-shock data,
forcedcompressible IT simulations are performed first. The forc-ing
procedure, proposed in Ref. [12], is the same as thatused for the
DNS of the shock–turbulence interaction inRef. [11]. Then the
turbulent fields, instead of being fedthrough the inlet of the
shock-tube, are passed through theLIA formulas. Thus, by
alleviating the need to resolve theshock in the shock-tube
simulations, much higher Reynoldsnumber turbulence data can now be
used. For this paper,the forced turbulence simulations are
performed on 5123
domains, with η/Δx = 0.8 (for which the differentia-tion error
is small compared to a spectral simulation withηkmax = 1.5 [12]), χ
= 0.01 (quasi-vortical upstream tur-bulence) and Mt = 0.05. For
this upstream turbulent Machnumber, the downstream ratio between
the turbulent, Mt2,and mean flow, Ms2, Mach numbers is less than
0.1, indi-cating small nonlinear effects during the interaction
[11].Also, the ratio δ/η � 7.69Mt/(Re0.5λ (Ms − 1)) is around0.14
for the smallest (Ms = 1.2) and 0.003 for the largest(Ms = 10) Ms
value considered. The forced turbulence sim-ulations yield a
Reynolds number Reλ � 180, which is muchlarger than previous
shock-tube simulations (with or withoutshock capturing) and above
the transition to fully developedturbulence.
2.1 Linear interaction analysis
TraditionalLIAonly requires information about the
upstreamturbulence spectrum shape and provides solutions for
thesecond-moment statistics behind the shock wave. To com-pute full
post-shock flow fields, which are necessary forhigher order
statistics, one needs full flow fields in front ofthe shock as
well. These fields are taken from separate three-dimensional forced
IT calculations described above. First,the velocity components are
Fourier transformed in all threephysical directions (x, y, z), and
each Fourier mode with(kx , ky, kz) wavenumber component is related
to the two-dimensional plane wave in the traditional LIA by a
spherical
coordinate transformation: k=√k2x + k2y + k2z , kx = k cos
ψ , and kz = ky tan φ. Here, the wavenumber kx is in
thedirection perpendicular to the shock wave and the
two-dimensional plane wave lies in the plane formed by the
xdirection and thewave vectork. Then, k,ψ , andφ are,
respec-tively, the wavelength of the plane wave, the angle
betweenthe wave vector and x direction and the angle between they
axis and the plane of the wave. In this study, we focus onthe
interaction of a shock wave with vortical turbulence. TheHelmholtz
decomposition [15] is used to remove the smalldilatational part of
the upstream velocity, which is less than1% of the total kinetic
energy (χ = 0.01). This small magni-tude component does not affect
the overall numerical results;however, the vortical LIA formulas
rely on zero divergence
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244 D. Livescu, J. Ryu
of velocity. In order to be able to apply the LIA procedure,the
velocity vector is decomposed into a component lying inthe plane of
the wave, u�, and a component perpendicular tothe plane of the
wave, u⊥. For non-interacting plane waves,the latter acts as a
constant background velocity, parallel tothe shock wave, which
should pass unchanged through theshock [1]. Thus, for small Mt
values, the full 3-D turbulencefields can be decomposed into a
collection of non-interactingplane waves, which follow the LIA
theory [1,2], each withan additional velocity component which
remains unchangedthrough the shock.
The velocity vector in the plane of the wave is perpendic-ular
to the wave vector and has a complex velocity amplitudeAv = ûsx/
sinψ , where ûsx is the Fourier coefficient of thesolenoidal
streamwise velocity disturbance. The complexamplitude ofu⊥ can
bewritten as u⊥ = −ûsy sin φ+ûsz cosφ,where ûsy and û
sz are the Fourier coefficients of the solenoidal
transverse velocity fluctuations.Then, for given Ms values, the
post-shock velocity distur-
bances are
u′x =∞∫
k=0
π∫
ψ=0
2π∫
φ=0Av
{F̃eik̃x + G̃eikxr x
}�k2dVs, (1)
u′y =∞∫
k=0
π∫
ψ=0
2π∫
φ=0
{Av cosφ
[H̃eik̃x + Ĩeikxr x
]
−u⊥ sin φeikxr x}
�k2dVs, (2)
u′z =∞∫
k=0
π∫
ψ=0
2π∫
φ=0
{Av sin φ
[H̃eik̃x + Ĩeikxr x
]
+ u⊥ cosφeikxr x}
�k2dVs, (3)
where k̃ is the post-shock acoustic wavenumber, r is the ratioof
the upstream and downstreammean streamwise velocities,� = ei(ky
y+kz z), dVs = sinψdφdψdk, and F̃ , G̃, H̃ and Ĩare the LIA
coefficients. Note that ψ and φ are varied from0 to π and from 0 to
2π , respectively, to consider the fullflow field; whereas ψ is
varied from 0 to π/2 and the φvariation is not considered in the
traditional LIA due to thesymmetry and homogeneity of the
second-moment statisticswith the angles. Also, the u⊥ contribution
does not appear inthe final formulas for the Reynolds stresses but
it needs to beincluded when considering the full flow fields. The
formulasfor the density and pressure fluctuations behind the shock
canbe found in Ref. [3]. k̃ is the root of the quadratic
equationwhich is derived from the pressure wave equation behindthe
shock wave. The root which corresponds to the physicalsolution is
chosen; the other root implies either exponentiallygrowing or
upstream propagating acoustic wave behind the
shock wave. For ψ < ψc and ψ > π − ψc, k̃ is real; the+
sign is chosen for the former and − sign for the latter. ψcis the
critical angle at which the term under the square rootis zero. The
derivatives with respect to ψ have an infinitediscontinuity at ψc,
due to the divergence of the k̃ derivative.The solutions themselves
are continuous with a cusp at ψc,which leads to a much larger
amplification at ψc. Physically,as k̃ changes from real to complex
atψc, the acoustic solutionacquires a decaying component in the
streamwise directionand the downstream velocity in a frame of
reference movingalong the shock with velocity V changes from
supersonic tosubsonic [1]. The moving velocity V is chosen such
that thedisturbance velocity in the plane of the wave does not
changeits direction through the shock.
The definitions of the LIA coefficients are presentedbelow. The
complete derivation of these coefficients can befound in Refs.
[1–3]. Here, the final formulas are shown asrequired by the
extended procedure.
F̃ = αD1(l − L̃), (4)G̃ = L̃(1 − B1) − F̃ + B1l, (5)H̃ = βD1(l −
L̃), (6)Ĩ = −mr
l(1 − B1 + αD1)L̃ + mrαD1 − mr B1, (7)
where
α = 1γ r2M2s2
k̃/k
m − k̃/(kr) , (8)
β = 1γ r2M2s2
l
m − k̃/(kr) , (9)
B1 = (γ − 1)M2s − 2
(γ + 1)M2s, (10)
D1 = 4γ M2s
2γ M2s − (γ − 1), (11)
E1 = 2(M2s − 1)
(γ + 1)M2s, (12)
L̃ = −m − βD1l − mrαD1 + mr B1E1l2 − βmlD1 − m2(1 − B1 + αD1)ml,
(13)
m = cosψ , l = sinψ , γ is the ratio of specific heats, Ms2
isthe mean Mach number behind the shock, which is given bythe
Rankine–Hugoniot relations, and, finally,
k̃ =−m ± 1/Ms2
√m2 − (1/M2s2 − 1)l2/r21/M2s2 − 1
kr. (14)
It is noted that the formulas above do not require isotropy,
sothey can be applied to anisotropic turbulence as well.
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Vorticity dynamics after the shock–turbulence interaction
245
3 Results and discussion
The properties of post-shock turbulence, as related to sev-eral
aspects of the vorticity field, are examined below. First,a
discussion is provided for the convergence of the DNSresults for
enstrophy amplification to the traditional LIA pre-diction. These
results use theDNS database generated in Ref.[11]. Then, highReλ
post-shock turbulence data are analyzedusing the new forced
compressible IT fields and extendedLIA formulas described
above.
Below,Shock-LIAandShock-DNS refer to the post-shockfields
computed using the extended LIA theory and DNS,respectively. The
Shock-DNS results for the vorticity ampli-fication are calculated
at the shock, for comparison withprevious studies. Those results
are averaged in time andover the transverse directions. However,
note that enstro-phy remains constant after the shock in Shock-LIA.
TheShock-LIA results correspond to the plane of
maximumamplification of the streamwise Reynolds stress, whichoccurs
approximately at k0x = π . Here, k0 = 1 is thewavenumber of the
peak of the kinetic energy spectrum (theforcing wavenumber for the
forced turbulence simulations).At k0x = π , the variations in
themean fields become small ina corresponding shock-tube DNS, so
that the contributionsfrom the mean flow to the turbulence
quantities discussedhere are small. Physically, this location
corresponds to theend of the inviscid adjustment of the acoustic
component, fol-lowing the shock–turbulence interaction, after which
the LIAstatistics become spatially constant. The region of
agreementbetween DNS and LIA can be extended into this
constantregime, provided that δ/η and Mt are small enough, sincethe
eddy turnover time and, consequently, the decay distanceincrease
with decreasing Mt at fixed Reλ [11]. Note that fea-tures of the
evolution away from the shock, like return toisotropy, cannot be
captured by the LIA solutions. However,such effects due to
nonlinear interactions can be made arbi-trarily small by decreasing
Mt . The Taylor Reynolds numberfor Shock-LIA is Reλ = 180 and for
shock-DNS it variesbetween 10 and 45.
3.1 Enstrophy amplification
As the scale separation between the shock wave width andthe
turbulence scales becomes large, for small upstream Mtvalues, the
viscous and nonlinear effects become negligibleduring the
interaction process, even at relatively low Reλvalues. In this
case, the DNS results should be close to theLIA prediction. Ref.
[11] showed that the results can becomefully converged for the
streamwise variation of the Reynoldsstresses and enstrophy in a
region close to the shock wave,even at Reλ ≤ 45. The extent of this
region increases as thescale separation increases. Since for
upstream IT the ratioof the shock width to Kolmogorov microscale is
given by
Fig. 2 Convergence of �tr amplification to the LIA solution for
dif-ferent values of Reλ. In this figure only, the results are
calculated at theshock, for comparison with previous studies.
Symbols along the verticalaxis represent the LIA solution with the
shape and color matched for thesymbol-lines of corresponding Ms .
Higher Reλ cases are located abovethe corresponding lower Reλ
cases. Additional results can be found inRef. [11]
δ/η � 7.69Mt/(Re0.5λ (Ms − 1)), the scale separation canbe
arbitrarily increased at a fixed Reλ value by decreasingthe
turbulent Mach number. Figure 2 shows the transverseenstrophy
amplification, �tr = 〈ω2y + ω2z 〉d/〈ω2y + ω2z 〉u ,where the
exponents d and u represent the values immedi-ately downstream and
upstream of the shock and ω = ∇ × uis the vorticity, for some of
the DNS cases discussed in Ref.[11]. The amplifications, as well as
the streamwise variationimmediately following the shock (not shown
here), becomefully converged to the LIA solutions for theReλ values
acces-sible inDNS.The convergence region includes the location
ofthe streamwise Reynolds stress maximum amplification andextends
into the region where the LIA statistics become spa-tially
constant. When the enstrophy amplification convergesto the LIA
solutions, it no longer changes as the Reynoldsnumber is increased
(Fig. 2).
The traditional LIA procedure calculates second ordermoments of
the turbulence fields, which require informationabout the incoming
turbulence spectra only. Thus, higherorder correlations
characterizing the turbulence fields andtheir change through the
shock cannot be predicted by theusual LIA formulas. In principle,
formulas to predict higherorder moments could be derived from
(1)–(3); however,these formulas would require knowledge about
higher ordermoments upstream of the shock as well and involve
increas-ingly cumbersome convolution products. The procedure
usedhere, with full flow fields ahead of the shock, provides
fullinformation downstream of the shock. Figure 3 shows
theProbability Density Function (PDF) of the transverse veloc-ity
component, normalized by the corresponding enstrophy
123
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246 D. Livescu, J. Ryu
Fig. 3 PDF of transversal vorticity component. All Shock-LIA
PDFsare normalized by
√�tr , so that they have the same variance as the IT
PDF
jump. Any departure from the IT profile after the normaliza-tion
is reflective of the contributions from the higher
ordercorrelations. The largest differences occur around the peak
ofthe PDF and at lower Ms values. After Ms = 6, the changesin the
PDF are small.
3.2 Vorticity field and the vortex-stretching mechanism
While the DNS results converge to the LIA predictions evenat low
Reynolds numbers, investigating post-shock turbu-lence properties
requires Reynolds numbers large enoughthat the upstream turbulence
is fully developed. The resultspresented below are obtained using
the Shock-LIA proce-dure, with Reλ ≈ 180. Some comparisons are made
with thedatabase from Ref. [11] using both Shock-DNS and the
cor-responding Shock-LIA results from the M = 1.8 run withReλ � 30,
which had δ/η � 0.3.
The LIA relations show that the shock interaction ampli-fies
preferentially the transverse components of the rotationand strain
stress tensor [3,11]. This leads to an increase inthe correlation
between the two quantities as Ms increases[11,19]. Ref. [11]
provides some information based on thejoint PDF of the strain and
rotation tensors magnitudes. Inorder to investigate this behavior
in more detail and alsoassess the Reynolds number influence, Fig. 4
shows the PDFof the strain-enstrophy angle, Ψ , defined as
[20]:
Ψ = tan−1 Si j Si jWi jWi j
(15)
where the strain and rotation tensors components are givenby Si
j = 12 (Ai j +Ai j ) andWi j = 12 (Ai j −Ai j ), respectively,with
Ai j = ∂ui/∂x j . By definition, large values ofΨ (45◦)correspond
to strain dominance and small values (�45◦) cor-respond to rotation
dominance. The regions with Ψ ∼ 45◦are the highly correlated
regions. In IT, the PDF of Ψ peaks
Fig. 4 PDF of the strain-enstrophy angle Ψ (degrees). The Reλ =
30results are obtained from the database of Ref. [11]. The present
resultshave Reλ = 180
at large values, consistent with previous results [20,21].
Thisbehavior can be seen even at relatively low Reynolds num-bers.
However, after the shock interaction, some differencesin low and
highReynolds number behavior start to appear. AstheMach number
increases, the PDF becomesmore symmet-rical, with a stronger peak
atΨ = 45◦. Nevertheless, the tailsof the PDF remain asymmetric, as
there are still more regionsof strain dominance compared to
rotation dominance. Thelow Reynolds number results with Ms = 1.8,
while showinggood agreement between Shock-DNS and Shock-LIA, tendto
underestimate these regions (Fig. 4).
In most fully developed 3-D turbulent flows, there is
apreferential alignment between the vorticity vector and
theeigenvectors of the strain rate tensor [22–24]. This is dueto
the local dynamics of vorticity and strain rate tensor, andcan be
affected by several mechanisms, e.g. the formationof distinct
spatial structures [25] or by heat release due tothe enhancement of
dilatational motions or local decrease inReynolds number [21,26].
Thus, the vorticity vector tendsto align with the intermediate (β-)
eigenvector and thereis no preference with respect to the most
extensive (α-)eigenvector. The eigenvectors correspond to the
eigenval-ues α, β and γ , denoted with the usual convention thatα
> β > γ . For quasi-vortical vortical upstream turbulence,α +
β + γ = Aii � 0. Figure 5 shows that IT turbulenceresults are
consistent with the previous studies. However, inpost-shock
turbulence, the alignmentwith theβ-eigenvectorsstrengthens and
there is a tendency towards a local alignmentwith the vorticity
perpendicular to the α- and γ - eigenvectorsas Ms increases. Here,
the angles ζ1, ζ2 and ζ3 correspondto the α-, β- and γ
-eigenvectors, respectively. The enhance-ment of the alignment with
the β-eigenvector was also foundin Ref. [19]. This change in
alignment is due to a preferentialamplification of the transverse
vorticity and strain rate ten-sor components due to the compression
in the shock normaldirection.
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Vorticity dynamics after the shock–turbulence interaction
247
Fig. 5 PDF of cosines of the angles between ω and the
eigenvectors ofthe strain rate tensor, S: cos ζ1, dash dotted
lines, cos ζ2, dashed lines,cos ζ3, solid lines
Indeed, Fig. 6 shows that both vorticity and the β-eigenvector
are increasingly aligned at a 90◦ angle with theshock normal
(streamwise) direction, as Ms increases. Atthe same time (not
shown) the other two eigenvectors tendto align in the shock normal
direction. In fully developedturbulence, there is no preferential
orientation of the inertialrange structures with the coordinate
directions, consistentwith the IT results in Fig. 6. However, the
interaction withthe shock changes the turbulence at all scales and
vorticityand strain rate eigenvectors acquire a strong
directionalitywith the coordinate directions. Post-shock turbulence
is nolonger fully developed 3-D turbulence, there is a
tendency,amplified as Ms increases, towards an axisymmetric
(2-D)local state.
The state and structure of post-shock turbulence are
veryimportant for the evolution away from the shock.The changesin
the orientation of vorticity and strain rate eigenvectors willgive
rise to various transients, until a fully developed stateis again
reached. Some consequences of these changes canbe highlighted by
considering the transport equation for theenstrophy:
∂〈�〉∂t
+ 〈∇ · (v�)〉 = 〈ω · S · ω〉 − 〈�∇ · v〉
−〈ω ·
(∇ p × ∇ρρ2
)〉+
〈ω ·
(∇ ×
[∇ · τρ
])〉, (16)
where� = |ω|2/2 and τ represents the stress tensor.
[4,5,27]analyzed this equation to explain the evolution of the
vortic-ity through the shock. Here, the focus is on the
consequencesof the changes in the turbulence structure behind the
shockfor the evolution downstream of the shock. The terms on
theright hand side (RHS) of (16) represent
vortex-stretching,vorticity-expansion, production due to baroclinic
torque andviscous dissipation. During the evolution through the
shock,the variations in the mean fields give most of the
contribu-
Fig. 6 PDF of cosine of the angle between a ω and b
β-eigenvectorwith the streamwise direction
tions to the terms in (16). Thus, vortex-stretching and
viscousterms are negligible and the advection,
vorticity-expansionand baroclinic terms are dominant [27]. However,
after theshock interaction, for the case of upstream vortical
turbu-lence, these terms become small and are not discussed
here,although we note that both are amplified as Ms increases,
inboth absolute magnitude and relative to the upstream fields.
The two important terms in (16) after the shock interac-tion
with quasi-vortical turbulence are vortex-stretching andviscous
terms. The vortex-stretching term is a fundamen-tal aspect of 3-D
turbulence and is intimately related to theenergy cascade to small
scales. Due to the change in the ori-entation of both vorticity and
strain rate eigenvectors andtendency towards a local axisymmetric
state, an importantquestion is about the effect on the
vortex-stretching mecha-nism. This term can be expressed using the
eigenvectors andeigenvalues of S as:
〈ω · S · ω〉 =〈|ω|2
(α cos2 ζα + β cos2 ζβ + γ cos2 ζγ
)〉
(17)
As both cos2 ζα and cos2 ζγ are larger after the shock (Fig.5),
there is an increasing cancelation between the first and last
123
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248 D. Livescu, J. Ryu
Fig. 7 Amplification of transverse vorticity variance and
vortex-stretching term in the enstrophy equation, normalized by�tr
, turbulencetime scale and the corresponding IT value
contributions to vortex-stretching, while the increased
align-ment with the β-eigenvector does not play a role since the
βeigenvalue has small magnitude. Nevertheless, the enstrophyand α
and γ eigenvalues increase substantially in magni-tude due to the
compression in the shock normal direction.This leads to an
amplification in the absolute value of thevortex-stretching term
after the shock interaction. However,after the normalization by the
enstrophy and turbulence timescale, τ = K/�, where K = Rii/2 is the
turbulent kineticenergy, Ri j = 〈uiu j 〉 are the Reynolds stresses
and � is thedissipation, Fig. 7 shows that vortex-stretching
becomes sub-stantially lower than in IT. As a result, on the time
scale ofthe turbulence, the flow may take a much longer time,
com-pared to a similar non-shocked flow, to return to a 3-D,
fullydeveloped state. This tendency is stronger at higher Ms
val-ues, indicating a slower rate of return; however, the
changesare less significant as Ms increases above 6.
3.3 The Lamb vector
The advection, vortex-stretching and vortex-expansion termsin
(16) can be grouped together using the Lamb vector, l ≡ω × v:
ω · ∇ × l = ∇ · (v�) − ω · S · ω + �∇ · v (18)
The Lamb vector also appears in the momentum equation,if the
advection term is re-written as:
(v∇) · v = ω × v + ∇2K (19)
and in the transport equation for the divergence of velocity,Δ ≡
Ai,i :
∂〈Δ2〉∂t
= −〈Δ∇ · l〉 − 〈Δ∇2K 〉
+〈Δ
(∇ p · ∇ρρ2
)〉+
〈∇ ·
[∇ · τρ
]〉(20)
In incompressible turbulence, the Lamb vector and itsdivergence
have been intensely studied, e.g. since it is solelyresponsible for
the total force acting on a moving body ordue to the connection to
the description of coherent structures[28]. Negative values of∇·l
are interpreted as spatially local-ized motions that have
accumulated the capacity to introducea time rate of change in
momentum. On the contrary, positivevalues represent motions with a
depleted such capacity. Thedivergence of the Lamb vector can be
written as:
∇ · l = u · ∇ × ω − ω · ω, (21)
so that ∇ · l can be positive only when the flexion product,F ≡
u · ∇ × ω is positive. Since the Lamb vector acts asa vortex force,
the Lamb vector divergence identifies inho-mogeneities in the
momentum transport surrounding a fluidelement, or a flux of energy,
that propagates or concentrateslocal energy curvature. In a region
of flow where the flexionproduct is positive, the enstrophy acts as
a storagemechanismand the flexion product behaves like a release
mechanism ofthe momentum flux and kinetic energy. In addition, the
inter-action between the flexion product and enstrophy gives rise
toan energy curvature interpretation and minimization processfor
interactions occurring in many incompressible flows. Incompressible
flows,while some interesting phenomena, suchas the connection with
the Bernoulli equation, are lost, wenote the additional connection
between ∇ · l and the produc-tion of dilatational motions (see
equation 20). In addition,∇ · l plays a key role in the production
of jet noise wheneverits mean is different than zero [29].
Compressible general-izations for the force acting on a body using
the Lamb vectorhave also been attempted (e.g. [30]).
In both IT and post-shock turbulence, the flexion producthas
both positive and negative values, but the PDF is skewedto the
right (Fig. 8). As the Mach number increases, themagnitude of F
also increases considerably. Perhaps moreinteresting is the
connection between the flexion productand velocity divergence (Fig.
9). In IT, the two quantitiesare uncorrelated, as reflected in the
joint PDF. However,in post-shock turbulence, the regions with the
largest flex-ion product values occur predominantly in the
compressionregions (Δ < 0). Following the transport equation for
thesquare of the divergence, it is likely that the strongest
com-pression regions will be further amplified during the
initialstages of the evolution away from the shock. The
energyreleased from these regions should continue to enhance
thesmall scale activity, in addition to the decrease of the
Kol-
123
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Vorticity dynamics after the shock–turbulence interaction
249
Fig. 8 PDF of the flexion product
Fig. 9 Joint PDF of the flexion product and divergence of
velocity aIT and b post-shock turbulence with Ms = 10
mogorov microscale due to the direct interaction with
theshock.
3.4 Thermodynamic variables
One characteristic of the interaction with the shock is
that,even if only one of the three compressible modes is
present
Fig. 10 PDF of cosine of the angle between the pressure and
densitygradients
Fig. 11 PDF of cosines of the angles between∇ρ and the
eigenvectorsof S: cosχ1, dotted lines, cosχ2, dashed lines, cosχ3,
solid lines
in the upstream turbulence, all modes are generated by
theinteraction. The upstream turbulence data used here
arequasi-vortical, with a small dilatational component of lessthan
1% kinetic energy. In this case, the density and pres-sure
fluctuations in the upstream fields are correlated, whilethe
temperature fluctuations are smaller. Thus, the pres-sure and
density gradients are mostly aligned (Fig. 10).However, as entropic
fluctuations are generated through theshock, this alignment weakens
in post-shock turbulence. Asa result, the baroclinic contribution
to the enstrophy equa-tion increases, while the contribution to the
square dilatationequation decreases.
In addition, the shock interaction also changes the align-ment
between the density gradient and the eigenvectors of thestrain rate
tensor (Fig. 11). In IT, the density gradient pointsmostly in the
direction of the most compressive (γ -) eigen-vector, with no
correlation with the other two eigenvectors.This is similar to
passive scalar alignment in compressive andincompressible turbulent
flows [21,22]. After the interactionwith the shock, the density
gradient tends to align at a 90◦angle with the direction of the
α-eigenvector and the align-
123
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250 D. Livescu, J. Ryu
mentwith the γ -eigenvectorweakens. Interestingly, at higherMs
values, as the acoustic field becomes stronger, ∇ρ startsto have an
angle different than zero with the γ -eigenvector.
4 Conclusions
Direct Numerical Simulations (DNS) of shock waves inter-acting
with turbulence are restricted to low Reynolds num-bers due to the
extremely large meshes required to resolveboth the turbulence and
the shock. Experimental realizationsof this problem are also very
challenging, due to problemswith controlling the shockwave and the
small time and lengthscales involved in the measurements especially
close to theshock front. However, recent high resolution DNS
exten-sively covering the parameter space show that, when thereis a
large scale separation between the turbulence and theshock width
and the turbulence intensity is small, the inter-action between a
shock wave and isotropic turbulence (IT)can be described by the
Linear Interaction Approximation(LIA). Such interaction conditions
occur in many practicalapplications.
In order to study the properties of high Reynolds
numberpost-shock turbulence, LIA was used to generate post-shock
fields starting from a forced compressible IT data-base. This
procedure was named Shock-LIA. The databasewas generated using a
5123 mesh and a Taylor Reynoldsnumber, Reλ = 180, which is much
larger than thoseattained in previous shock–turbulence interaction
studies.Here, the case of quasi-vortical turbulence was
consid-ered. Since traditional LIA addresses second order
momentsonly, in order to calculate full flow fields, necessary
forthe higher order moments, the detailed procedure to calcu-late
these fields was given. The main theme of the paperis related to
properties of the vorticity field, as a centralfeature of turbulent
flows, and various related quantities.Most of the results presented
are in terms of probabilitydensity functions (PDFs) of various
quantities, which can-not be inferred from the traditional LIA
formulas sincethey require the knowledge of all higher order
moments,beyond the variance. The properties of these higher
ordermoments are one of the central open questions in
turbulenceresearch.
First, using theDNSdatabase fromRef. [11], it was shownthat the
vorticity variance from DNS converges to the LIAresults as the
scale separation increases. The convergencecan be obtained even at
low upstream Reynolds numbers;however, the properties of the
post-shock turbulence changewith the Reynolds number. Indeed, the
PDF of the strain-enstrophy angle, which changes significantly
compared toIT, shows a good match between the DNS and LIA
resultsusing the corresponding IT database at Reλ = 30, but
dif-ferences compared to LIA results using the Reλ = 180
ITdatabase.
In general, the shock interaction significantly changes
theproperties of upstream turbulence. Thus, the orientations
ofvorticity and eigenvectors of the strain rate tensor point to
alocal axisymmetric state, with a reduced
vortex-stretchingmechanism on the time scale of the turbulence. In
addi-tion, the flexion product becomes inversely correlated tothe
dilatation in the regions of positive Lamb vector diver-gence.
These changes point to a shock Mach number (Ms)dependent slowing of
the return to a fully developed stateand increased small scale
activity as the turbulence evolvesaway from the shock. On the other
hand, the thermody-namic quantities are also strongly affected by
the interactionwith the shock. Both acoustic and entropic
components aregenerated even for upstream vortical turbulence and
thesecomponents propagate with different velocities. The
specificcorrelations between the thermodynamic quantities and
theorientations of their gradients depend on the relative
strengthof these components. Thus, at high Ms values, the
orienta-tion between the density gradient and the eigenvectors of
thestrain rate tensor is very different from that in IT.Again,
theseare structural changes in post-shock turbulence expected
tohave a significant effect on the evolution away from
theshock.
Finally, we would like to mention that, while shock-resolved DNS
remains the gold standard, the results fromRef. [11] highlight the
applicability of shock-capturedturbulence-resolved simulations and
their importance as anaccurate tool for shock–turbulence
interaction problems,when the scale separation is large enough.
However, dueto computational limitations, Shock-LIA still can
access aregion of the parameter space not available to either
toolsand provide an understanding of the properties of
post-shockturbulence in those regimes.
Acknowledgments Los Alamos National Laboratory is operated byLos
Alamos National Security, LLC for the US Department of EnergyNNSA
under Contract No. DE-AC52-06NA25396. Computationalresources were
provided by the LANL Institutional Computing (IC)Program and
Sequoia Capability Computing Campaign at LawrenceLivermore National
Laboratory.
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123
Vorticity dynamics after the shock--turbulence
interactionAbstract1 Introduction2 Problem setup and numerical
methods2.1 Linear interaction analysis
3 Results and discussion3.1 Enstrophy amplification3.2 Vorticity
field and the vortex-stretching mechanism3.3 The Lamb vector3.4
Thermodynamic variables
4 ConclusionsAcknowledgmentsReferences