Local and Nonlocal Strain Rate Fields and Vorticity Alignment in Turbulent Flows

0.48 MB·English

by  Peter E. Hamlington

Local and Nonlocal Strain Rate Fields and Vorticity Alignment in Turbulent Flows
Peter E. Hamlington1∗, J¨org Schumacher2†, and Werner J. A. Dahm1‡
1 Laboratory for Turbulence & Combustion (LTC), Department of Aerospace Engineering,
The University of Michigan, Ann Arbor, MI 48109-2140, USA
2 Department of Mechanical Engineering, Technische Universit¨at Ilmenau, D-98684 Ilmenau, Germany
(Dated: February 2, 2008)
Local and nonlocal contributions to the total strain rate tensor Sij at any point x in a flow are
formulated from an expansion of the vorticity field in a local spherical neighborhood of radius R
centered on x. The resulting exact expression allows the nonlocal (background) strain rate tensor
SiBj(x)tobeobtainedfromSij(x). Inturbulentflows,wherethevorticitynaturallyconcentratesinto
8
relatively compact structures,thisallows thelocal alignment of vorticitywith themost extensional
0
principal axis of the background strain rate tensor to be evaluated. In the vicinity of any vortical
0
structure,therequiredradiusRandcorrespondingorderntowhichtheexpansionmustbecarried
2
are determined by the viscous lengthscale λν. We demonstrate the convergence to the background
n strain rate field with increasing R and n for an equilibrium Burgers vortex, and show that this
a resolves the anomalous alignment of vorticity with the intermediate eigenvector of the total strain
J rate tensor. We then evaluate the background strain field SB(x) in DNS of homogeneous isotropic
ij
8 turbulencewhere,evenforthelimitedRandncorrespondingtothetruncatedseriesexpansion,the
resultsshowanincreaseintheexpectedequilibriumalignmentofvorticitywiththemostextensional

] principal axis of the background strain rate tensor.
n
y
PACSnumbers: 47.27.-i,47.32.C-,47.27.De
d
-
u
l I. INTRODUCTION A key to understanding this result is that, owing to
f
. the competition betweenstrainanddiffusion, the vortic-
s
c Vortex stretching is the basic mechanism by which ki- ity in turbulent flows naturally forms into concentrated
i netic energyis transferedfromlargerto smallerscalesin vortical structures. It has been noted, for example in
s
y three-dimensional turbulent flows [1, 2, 3, 4]. An under- Refs. [7, 13, 14], that the anomalous alignment of the
h standing of how vortical structures are stretched by the vorticity with the strain rate tensor Sij(x) might be ex-
p strainratefieldS (x)isthusessentialtoanydescription plained by separating the local self-induced strain rate
ij
[ oftheenergeticsofsuchflows. Overthelasttwodecades, field created by the vortical structures themselves from
1 direct numerical simulations (DNS) [5, 6, 7] and experi- the background strain field in which these structures re-
v mentalstudies[8,9,10,11,12]ofthefine-scalestructure side. The total strain rate tensor is thus split into
8 of turbulence have revealed a preferred alignment of the
4 vorticity with the intermediate eigenvector of the strain Sij(x)=SiRj(x)+SiBj(x), (2)
2
rate tensor. This resulthas been widely regardedassur-
.1 prising. Indeedtheindividualcomponentsoftheinviscid where SiRj is the local strain rate induced by a vortical
1 vorticity transport equation, in a Lagrangianframe that structure in its neighboring vicinity, and SB is the non-
ij
0 remains aligned with the eigenvectors of the strain rate local background strain rate induced in the vicinity of
8
tensor, are simply the structure by allthe remainingvorticity. The vortical
0
structurewouldthenbeexpectedtoalignwiththeprinci-
:
v Dω1 Dω2 Dω3 palaxiscorrespondingtothemostextensionaleigenvalue
Xi Dt =s1ω1, Dt =s2ω2, Dt =s3ω3, (1) of the background strain rate tensor SiBj(x).
In the following, we extend this idea and suggest a
ar where s1, s2 and s3 are the eigenvalues of Sij. For in- systematic expansion of the total strain rate field S (x)
compressible flow, s1 +s2 +s3 ≡ 0, and then denoting that allows the background strain rate field SB(xij) to
s1 s2 s3 requires s1 0 and s3 0. As a con- ij
≥ ≥ ≥ ≤ be extracted. Our approach is based on an expansion
sequence, (1) would predict alignment of the vorticity
of the vorticity over a local spherical region of radius R
with the eigenvector corresponding to the most exten-
centered at any point x. This leads to an exact opera-
sional principal strain rate s1. Yet DNS and experimen-
tor that provides direct access to the background strain
tal studies have clearly shown that the vorticity instead
rate field. The operator is tested for the case of a Burg-
is aligned with the eigenvector corresponding to the in-
ers vortex, where it is shown that the local self-induced
termediate principal strain rate s2.
strainfieldproducedbythevortexcanbesuccessfullyre-
moved,andtheunderlyingbackgroundstrainfieldcanbe
increasinglyrecoveredashigherorderterms areretained
in the expansion. The anomalous alignment of the vor-
∗email: [email protected](correspondingauthor)
†email: [email protected] ticity with respect to the eigenvectors of the total strain
‡email: [email protected] field is shown in that case to follow from a local switch-
2
From (5) and (6), S (x) can be expressed [15] as an
ij
integral over the vorticity field as
3 r
S (x)= (ǫ r +r ǫ ) kω (x′)d3x′. (7)
ij 8π ZΛ ikl j i jkl r5 l
AsshowninFig.1,thetotalstrainratein(7)isseparated
intothelocalcontributioninducedbythevorticitywithin
a spherical region of radius R centered on the point x
and the remaining nonlocal (background) contribution
induced by all the vorticity outside this sphericalregion.
The strain rate tensor in (7) thus becomes
3 3
S (x)= [ ]d3x′+ [ ]d3x′,(8)
ij
8π Z ··· 8π Z ···
r≤R r>R
SR(x) SB(x)
| ≡{zij } | ≡ {izj }
FIG. 1: Decomposition of the vorticity field in the vicinity
where [ ] denotes the integrand in (7). The nonlocal
of any point x into local and nonlocal parts; the Biot-Savart backgro·u·n·dstrain tensor at x is then
integral in (8) over each part gives the local and nonlocal
(background)contributionstothetotalstrainratetensorSij SB(x)=S (x) SR(x). (9)
at x. ij ij − ij
The total strain tensor S (x) in (9) is readily evaluated
ij
via (6) from derivatives of the velocity field at point x.
ing of the principal strainaxeswhen the vortex becomes
Thus all that is requiredto obtainthe background(non-
sufficiently strong relative to the background strain. Fi-
local) strain rate tensor SB(x) via (9) is an evaluation
nally, the operator is applied to obtain initial insights ij
ninetoousthiseotbraocpkicgrtouurnbdulesntrcaei,nanSdiBju(xse)dintoDcoNmSpaorfehtohmeovgoer-- ovfortthiceitloycfiaellsdtrωali(nxi′n)twegitrhailnSriRj(x)Rinin(8F)igp.r1o.duced by the
≤
ticity alignment with the eigenvectors of the total strain
field and of this backgroundstrain field.
A. Evaluating the Background Strain Rate Tensor
II. THE BACKGROUND STRAIN FIELD The vorticity field within the sphere of radius R can
be represented by its Taylor expansion about the center
The velocity uatanypoint x inducedby the vorticity point x as
field ω(x) is given by the Biot-Savart integral
∂ω
ω (x′) = ω (x)+(x′ x ) l (10)
u(x)= 41π ZΛω(x′)× xx−xx′′3d3x′, (3) l |r≤R + 1l(x′ xm)−(x′m x∂x)m(cid:12)(cid:12)(cid:12)(cid:12)x∂2ωl + .
| − | 2 m− m n− n ∂x ∂x (cid:12) ···
wherethe integrationdomainΛistakentobe infinite or m n(cid:12)(cid:12)x
periodic. In index notation (3) becomes Rtoecaablblirnegvitahtaettxhme −voxrt′mici≡tyrmandanidtsudsienrgivaatl,ivbelms,a(cid:12)ctlmxn,,.w.e.
1 (x x′) can write (10) as
u (x)= ǫ ω (x′) k− k d3x′, (4)
i 4π ZΛ ilk l |x−x′|3 ω (x′) a r b + 1r r c . (11)
where ǫ is the cyclic permutation tensor. The deriva- l |r≤R ≡ l− m lm 2 m n lmn−···
ilk
tive with respect to x gives the velocity gradient tensor
j Substituting (11) in the SR integral in (8) and changing
ij
the integration variable to r = x x′, the strain tensor
∂ u (x)= 1 ǫ ω (x′) δkj 3rkrj d3x′, (5) at x produced by the vorticity in−R is
∂xj i 4π ZΛ ilk l (cid:20) r3 − r5 (cid:21)
3 r
rgwarhtaeedriteeennrts,o≡nraSm|ixjel−aytxx′|isatnhde srymmm≡etxrmic−paxrt′mo.ftTheheveslotrcaitiny SiRj(x) = 8aπlZr≤rmRb(lǫmik+lrjr+mrrniǫcjlkml)nrk5 d3r.(12)
× h − 2 −···i
S (x) 1 ∂ui + ∂uj . (6) This integral can be solved in spherical coordinates cen-
ij ≡ 2(cid:18)∂xj ∂xi(cid:19) tered on x, with r1 = rsinθcosφ, r2 = rsinθsinφ, and
3
r3 = rcosθ for r [0,R], θ [0,π], and φ [0,2π). To B. Practical Implementation
∈ ∈ ∈
integrate (12) note that
r r 4π R 1 Whenusing(20)toexaminethelocalalignmentofany
Zr≤R kr5jd3r= 3 δjkZ0 rdr, (13a) cthoencbeanctkrgartoedunvdorsttircaainlsrtartuectfiuerldewSBith(xt)hienpwrhinicchipiatlraexsiedseosf,
ij
rkrjrmd3r=0, (13b) the radius R must be taken sufficiently large that the
Z r5 spherical region x′ x R encloses essentially all the
r≤R
| − | ≤
r r r r 2π vorticity associated with the structure, so that its local
Z k jr5m nd3r= 15R2(δmnδjk+ (13c) induced strain rate field is fully accounted for. Gener-
r≤R ally,as R increasesitis necessaryin (20) to retainterms
δ δ +δ δ ) .
mj kn mk jn of increasingly higher order n to maintain a sufficient
The resulting local strain rate tensor at x is then representation of ω(x′) over the spherical region. Thus
for any vortical structure having a characteristic gradi-
R2
SiRj(x)= 40clmn(ǫijlδmn+ǫjilδmn+ǫinlδmj (14) ethnet loerndgetrhsocfaλle ,λνa,nidt cnanwiblletehxepnecnteeeddthtoatbRe smuuffisctiebnetolyf
4 ν
+ǫjnlδmi+ǫimlδnj +ǫjmlδni)+O(R ), large to adequately represent the vorticity field within
thissphere. However,sincethelocalgradientlengthscale
where the contribution from the a term in (12) is zero
l
in the vorticityfield ina turbulentflow is determinedby
sinceǫ = ǫ . Forthesamereasonthefirsttwoterms
ijl jil
− an equilibrium between strain and diffusion, the vortic-
in (14) also cancel, giving
ityfieldoverthelengthscaleλ willberelativelysmooth,
SR(x)= R2c (ǫ δ +ǫ δ (15) and thus relatively low valuesνof n may suffice to give a
ij 40 lmn inl mj jnl mi usable representation of ω(x′). This is examined in the
+ǫ δ +ǫ δ )+O(R4). following Section.
iml nj jml ni
Recalling that c = c ∂2ω /∂x ∂x , and con-
lmn lnm l m n
≡
tracting with the δ and ǫ in (15), gives
III. TEST CASE: BURGERS VORTEX
R2 ∂ ∂ω
SR(x)= ǫ l (16)
ij 20 (cid:20)∂x (cid:18) iml∂x (cid:19) The equilibrium Burgers vortex [1, 3, 9, 16] is formed
j m
from vorticity in a fluid with viscosity ν by a spatially
∂ ∂ω
l 4
+ ǫ +O(R ). uniform, irrotational, axisymmetric background strain
jml
∂x (cid:18) ∂x (cid:19)(cid:21)
i m ratefieldSB thathasasingleextensionalprincipalstrain
ij
Note that ǫ ∂ω /∂x ( ω) and rate S directed along the z axis, as shown in Fig. 2.
iml l m ≡ ∇× i zz
This simple flow, often regarded as an idealized model
ω = ( u)= ( u) 2u, (17)
∇× ∇× ∇× ∇ ∇· −∇ ofthe mostconcentratedvorticalstructures in turbulent
so for an incompressible flow ( u 0) the local strain flows, provides a test case for the result in (20). The
rate tensor at x becomes ∇· ≡ combined strain rate field S (x) produced by the vor-
ij
R2 ∂u ∂u texandthe backgroundstrainflowshould,whenapplied
SR(x)= 2 i + j +O(R4). (18)
ij −20∇ (cid:18)∂x ∂x (cid:19)
j i
From (9), with SR from (18) we obtain the background
ij
strain tensor as
2
R
SB(x)=S (x)+ 2S (x)+O(R4). (19)
ij ij 10∇ ij
Theremainingtermsin(19)resultfromthehigher-order
terms in (11). The contributions from each of these can
be evaluated in an analogous manner, giving
2 4
R R
SB(x)= 1+ 2+ 2 2+ (20)
ij (cid:20) 10∇ 280∇ ∇ ···
+ 3R2n−2 2 n−1+ S (x),
(2n 2)!(4n2 1) ∇ ···(cid:21) ij
− − (cid:0) (cid:1)
wherethetermsshownin(20)correspondton=1,2,....
The final result in (20) is an operator that extracts the
nonlocal background strain rate tensor SB at any point
ij
x from the total strain rate tensor S . For the Taylor FIG. 2: Equilibrium Burgers vortex with circulation Γ and
ij
expansion in (10), this operator involves Laplacians of strain-limited viscous diffusion lengthscale λν in a uniform,
the total strain rate field S (x). irrotational,axisymmetricbackgroundstrainratefieldSiBj(x).
ij
4
in (20), produce the underlying background strain field S ,andthecorrespondingprincipalstrainaxiswillpoint
zz
(SB,SB,SB) = ( 1, 1,1)S at all x when R in the zˆ direction. The vorticity is then aligned with the
rr θθ zz −2 −2 zz → ∞
and all orders n are retained. For finite R and n, the most extensional eigenvector of S . This remains the
ij
resultingSB(x)willreflectthe convergencepropertiesof case until the vortex becomes sufficiently strong relative
ij
(20). to the background strain rate that s>S , namely
zz
3
Sv S , (27)
A. Strain Rate Tensor | rθ|≥ 2 zz
which from (25) occurs wherever
TheequilibriumBurgersvortexalignedwiththeexten-
sional principal axis of the background strain rate field 2 −1
1 1 3π Γ/λ
has a vorticity field α+ exp( αη2) ν . (28)
(cid:18) η2(cid:19) − − η2 ≥ 2 (cid:18) S (cid:19)
zz
α Γ
ω(x)=ω (r)ˆz= exp αη2 ˆz, (21)
z πλ2 − Atany η forwhich (28) is satisfied,the mostextensional
ν (cid:0) (cid:1) principal axis of the combined strain rate tensor S (x)
ij
where Γ is the circulation, λν is the viscous lengthscale will switch from the ˆz direction to instead lie in the r-
that characterizes the diameter of the vortex, η r/λν θ plane. Since the vorticity vector everywhere points in
≡
is the radial similarity coordinate, and the constant α the zˆ direction, wherever (28) is satisfied the principal
reflects the chosen definition of λν. Following [9], λν is axis of Sij that is aligned with the vorticity will switch
taken as the full width of the vorticalstructure at which fromthemostextensionaleigenvectortotheintermediate
ωz has decreasedto one-fifth of its peak value, for which eigenvector. This alignment switching is purely a result
α = 4ln5. When diffusion of the vorticity is in equilib- of the induced strain field Sv(x) locally dominating the
ij
rium [9] with the background strain, then backgroundstrain field SB(x).
ij
The dimensionless vortex strength parameter
1/2
ν
λ =√8α . (22)
ν (cid:18)S (cid:19) Γ/λ2 πω
zz Ω ν = max (29)
≡(cid:20) S (cid:21) α S
Thecombinedvelocityfieldu(x)producedbythevortex zz zz
and the irrotational background strain is given by the
on the right-hand side of (28) characterizes the relative
cylindrical components
strengthofthe backgroundstrainandtheinducedstrain
fromthevorticalstructure,whereω is obtainedfrom
S max
zz
ur(r,θ,z) = r, (23a) (21) at η =0. For
− 2
Γ 1
u (r,θ,z) = 1 exp αη2 , (23b) Ω<Ω∗ 2.45, (30)
θ 2πλ η − − ≈
ν (cid:2) (cid:0) (cid:1)(cid:3)
u (r,θ,z) = S z. (23c) the backgroundstrainrateS iseverywherelargerthan
z zz zz
the largest s in (26), and thus no alignment switching
The combined strain rate tensor for such a Burgers occurs at any η. For Ω > Ω∗, alignment switching will
vortex is thus occuroverthelimited rangeofη valuesthatsatisfy(28).
S /2 Sv 0 With increasing values of Ω, more of the vorticity field
Sij(x)= −Szrvzθ −Szrzθ/2 0  , (24) wthiellcboemabilnigendedstrwaiitnhrtahtee tinentesromr,eedviaetnetphroiungchipaalllaoxfisthoef
0 0 S
zz
  vorticity field remains aligned with the most extensional
where Sv is the shear strainrate induced by the vortex, principal axis of the background strain rate tensor.
rθ
given by Figure 3 shows the vorticity ωz and the induced shear
straincomponent Sv asafunctionofη. Thehorizontal
Sv (x)= Γ α+ 1 exp αη2 1 . (25) dashed lines corre−spornθd to three different values of Ω,
rθ πλ2 (cid:20)(cid:18) η2(cid:19) − − η2(cid:21) and indicate the range of η values where the alignment
ν (cid:0) (cid:1)
switching in (28) occurs for each Ω. Wherever Sv is
From (24), Sij(x) has one extensional principal strain above the dashed line for a given Ω, the vorticity−wirllθbe
rate equal to Szz along the zˆ axis, with the remaining aligned with the local intermediate principal axis of the
two principal strain axes lying in the r-θ plane and cor- combined strain rate field.
responding to the principal strain rates In principle, regardlessof the vortex strength parame-
terΩ,atanyηtheresultin(20)canrevealthealignment
1
s=−2Szz ±|Srvθ|. (26) of the vorticity with the most extensional principal axis
ofthebackgroundstrainfieldSB. However,thisrequires
ij
As long as the largest s in (26) is smaller than S , the Rtobesufficientlylargethataspherewithdiameter2R,
zz
most extensional principal strain rate s1 of Sij will be centered at the largest η for which −Srvθ in Fig. 3 is still
5
1 the approximation becomes increasingly poorer, but the
(cid:58)=(cid:58)*=2.45 0.3 1/r2 decrease in the Biot-Savart kernel in (3) neverthe-
0.9
lessrendersitadequatetoaccountformostofthevortex-
0.8 (cid:358)Sr(cid:84) 0.25 induced strain rate field. At the largest radial location,
0.7 correspondingtothebottomrightpanelofFig.4,theap-
0.2 proximation becomes relatively poor, however at large η
0.6
ωz −Srθ valuesthe vortex-inducedstrainis sufficiently smallthat
ωmax 0.5 2(cid:58)* 0.15 ωmax itis unlikely to leadto alignmentswitching for typicalΩ
0.4 values.
0.3 3(cid:58)* 0.1 Figure 5 shows the shear component SrBθ(η) of the
background strain rate tensor obtained via (20) for var-
0.2
(cid:90) 0.05 ious n and R˜ as a function of η. In each panel, the
z
0.1 blackcurveshowsthetotalstrainrateS (η)andthecol-
ij
0 0 oredcurvesshow the backgroundstrainrateSB(η) from
(cid:358)1.5 (cid:358)1 (cid:358)0.5 0 0.5 1 1.5 ij
η (20) for the (n,R˜) combinations listed. The horizontal
dashedlinecorrespondingtoΩ=(3/2)Ω∗reflectstherel-
FIG. 3: Similarity profiles of ωz(η) and Srθ(η) for any equi- ativevortexstrength,andshowstherangeofηwherethe
libriumBurgersvortex;wherever−S exceedsthehorizontal anomalousalignmentswitchingoccursduetothevortex-
rθ
line determined by the relative vortex strength parameter Ω inducedstrainfield. Whereverthe SB curvesareabove
− rθ
in (29) the most extensional principal axis of thetotal strain thisline,thevorticitytherewillbealignedwiththeinter-
rate Sij(x) switches from thezˆ-axis to lie in ther-θ plane. mediateprincipalaxisofthecombinedstrainratetensor
S . Figure 5(a) examines the effect of increasing the
ij
radius R˜ of the spherical region for fixed order n = 6.
above the horizontal dashed line, will enclose essentially It is apparent that with increasing R˜ the resulting SB
allofthe vorticityassociatedwiththe vorticalstructure. − rθ
converges toward the correct background strain field in
As Ω increases, the required R will increase accordingly
(31). For the value of Ω shown, it can be seen that for
as dictated by (28), and as R is increasedthe required n R&0.5λ theresultingSB iseverywherebelowthehor-
in (20) also increases. ν rθ
izontal dashed line, indicating that the vorticity every-
Irrespective of the value of Ω, when (20) is applied to
whereis alignedwith the mostextensionalprincipalaxis
the combined strain rate field Sij(x) in (24) and (25), oftheresultingbackgroundstrainratetensorSB(x)from
if R˜ (R/λν) and all orders n are retained then (20). ij
there≡sultingSB→(x)∞shouldrecoverthebackgroundstrain
ij
field, namely
SB 0 (31) 1.25 1.25
rθ →
1 1
for all x, and the vorticity should show alignment with
themostextensionalprincipalaxisofSiBj. ForfiniteR/λν ωz 0.75 0.75
andvariousordersn,theconvergenceofSB from(20)to ωmax 0.5 0.5
ij
this background strain field is examined below.
0.25 0.25
0 0
(cid:358)1 (cid:358)0.5 0 0.5 1 (cid:358)1 (cid:358)0.5 0 0.5 1
B. Convergence of the Background Strain
1.25 1.25
The accuracy with which (20) can recover the back- 1 1
ground strain field SB(x) that acts on a concentrated
ij ω 0.75 0.75
vortical structure depends on how well the expansion in z
ωmax 0.5 0.5
(10) representsthe vorticityfieldwithin the localspheri-
calneighborhoodR. Figure4showstheresultsofalocal 0.25 0.25
sixth-order Taylor series approximation for the vorticity
0 0
in(21)atvariousradiallocationsacrosstheBurgersvor- (cid:358)1 (cid:358)0.5 0 0.5 1 (cid:358)1 (cid:358)0.5 0 0.5 1
η η
tex. In each panel, the blue square marks the location x
atwhichthesphereiscentered,andthereddashedcurve
shows the resulting Taylor series approximation for the FIG.4: AccuracyoftheTaylorexpansionforthelocalvortic-
vorticity. On the axis of the vortex, the approximated ityin (10)foraBurgersvortex,showingresultsfor6thorder
vorticity field correctly accounts for most of the circula- approximation. Ineach panel,solid black curveshows actual
tioninthe vortex,andthusthe induced strainfieldfrom vorticity profile, and red dashed curve gives approximated
thevortexwillbereasonablyapproximated. Offtheaxis, vorticity from derivativesat location marked by square.
6
most extensional principal axis of the background strain
tensor throughout the entire field.
0.30 (a)
Figure 5(c) shows the combined effects of increasing
0.25 both R˜ and n, in accordance with the expectation that
3π
2αΩ larger R˜ should require a higher order n to adequately
0.20
−SB represent the vorticity field within the spherical region.
ω rθ 0.15 The shear strainrate field shows convergenceto the cor-
max (cid:358)S
r(cid:84) rect background strain field in (31). The convergence of
0.10
(6, 0.20) the shear strain rate SB(x) to zero in the vicinity of the
(6, 0.35) rθ
0.05 (6, 0.50) vortex core is of particular importance. The systematic
(6, 0.65) reduction in the peak remaining shear stress indicates
0
(cid:358)1.5 (cid:358)0.75 η0 0.75 1.5 that, even for increasingly stronger vortices or increas-
ingly weaker backgroundstrain fields as measured by Ω,
the resulting SB(x) from (20) will reveal the alignment
0.30 (b) rθ
of all the vorticity in such a structure with the most ex-
0.25 3π tensional principal strain axis of the background strain
2αΩ
0.20 field.
−SB
rθ
ω 0.15
max
0.10 (cid:358)S
r(cid:84) IV. VORTICITY ALIGNMENT IN TURBULENT
0.05 (3, 0.65)
FLOWS
(4, 0.65)
0 (5, 0.65)
(6, 0.65)
(cid:358)0.05 Having seen in the previous Section how (20) is able
(cid:358)1.5 (cid:358)0.75 0 0.75 1.5
η to reveal the expected alignment of vortical structures
with the most extensional eigenvector of the background
strainrate in which they reside, in this Section we apply
0.30 (c)
it to obtain insights into the vorticity alignment in tur-
0.25 3π bulent flows. In particular,we examine the alignmentat
2αΩ every point x of the vorticity ω relative to the eigenvec-
0.20
−SB tors of the total strain rate tensor field Sij(x) and those
ω rθ 0.15 of the backgroundstrain field data SB(x). This analysis
max ij
0.10 (cid:358)S uses data from a highly-resolved, three-dimensional, di-
r(cid:84)
rect numerical simulation (DNS) of statistically station-
(2, 0.45)
0.05
(4, 0.55) ary, forced, homogeneous, isotropic turbulence [17, 18].
0 (6, 0.65) Thesimulationscorrespondtoaperiodiccubewithsides
(cid:358)1.5 (cid:358)0.75 η0 0.75 1.5 oflengthof2π resolvedby20483gridpoints. TheTaylor
microscale Reynolds number R is 107.
λ
The DNS data were generated by a pseudospectral
FIG.5: ConvergenceofbackgroundstrainratefieldSB(x)for method with a spectralresolutionthat exceeds the stan-
ij
a Burgers vortex, obtained from total strain rate field Sij(x) dard value by a factor of eight. As a result, the highest
using (20) for various (n,R˜) combinations, where R˜ ≡R/λν. wavenumber corresponds to kmaxηK = 10, and the Kol-
Shown are effects of increasing R˜ for fixed n = 6 (top), in- mogorov lengthscale ηK = ν3/4/ ǫ 1/4 is resolved with
h i
creasing n for fixed R˜ =0.65 (middle),and increasing n and three grid spacings. This superfine resolution makes it
R˜ simultaneously (bottom). The dashed horizontal lines fol- possible to apply the resultin (20) for relativelyhighor-
low from (27) and (29). ders n, which require accurate evaluation of high-order
derivatives of the DNS data. In Schumacher et al. [17]
it was demonstrated that derivatives up to order six are
statistically converged. More details on the numerical
In Fig. 5(b) similar results are shown for the effect of simulations are given in Refs. [17, 18].
increasing the order n of the expansion for the vorticity Figure6givesarepresentativesampleoftheDNSdata,
field for fixed R˜ = 0.65. It is apparent that the effect of wheretheinstantaneousshearcomponentS12ofthetotal
nissomewhatsmallerthanforR˜ inFig.5(a). Moreover, strain rate tensor field S (x) is shown in a typical two-
ij
the results suggestthat the seriesin (20)alternates with dimensional intersection through the 20483 cube. The
increasing order n. For this Ω and R˜, even n = 3 is datacanbeseentospannearly700η ineachdirection.
K
seentobesufficienttoremovemostofthevortex-induced The 5122 box at the lower left of Fig. 6 is used here to
shearstrain,andthusreduceSB(x)belowthehorizontal obtain initial results for alignment of the vorticity with
rθ
dashed line. For these parameters, the SB field from the eigenvectors of the backgroundstrain rate tensor.
rθ
(20)wouldthusrevealalignmentofthevorticitywiththe The background strain rate tensor field SB(x) is first
ij
7
numerous other DNS studies [5, 6, 7] and experimental
studies [8, 9, 10, 11, 12], which show the vorticity to be
predominantly aligned with the eigenvector correspond-
ing to the intermediate principal strain rate. However,
the resultsfor the Burgersvortexinthe previous section
show that such anomalous alignment with the eigenvec-
tors of the total strain rate tensor is expected when the
local vortex strength parameter Ω is sufficiently large to
cause alignment switching.
By comparison, the results in Fig. 8 (b) and (c) ob-
tained for the alignment cosines of the vorticity vector
with the background strain rate tensor SB from (20)
ij
show a significant decrease in alignment with the inter-
mediate eigenvector, and an increase in alignment with
the most extensional eigenvector. While data in panel
(b) show only a slight change compared to those in (a),
the resultsinpanel(c)demonstratethatourdecomposi-
FIG. 6: Instantaneous snapshot of total strain rate com-
tioncanindeeddiminishtheanomalousalignmentsignif-
ponent field S12(x) in a two-dimensional slice through a
highly-resolved three-dimensional (20483) DNS of homoge-
neous, isotropic turbulence [17, 18]. Axes are given both in
grid coordinates (i = 1...2048) and normalized by the Kol-
mogorovlengthηK. Boxindicatesregioninwhichbackground
strain rate field SB(x) is computed in Fig. 7.
ij
extracted via (20) from S (x) for n = 3 and vari-
ij
ous (R/η ). Higher-order evaluation of the background
K
strainrate is notfeasible,as the results inRef. [17]show
that only spatial derivatives of the velocity field up to
order six can be accurately obtained from these high-
resolutionDNSdata. Forn=4,theexpansionin(20)in-
volvesseventh-orderderivativesof the velocity field, and
thebackgroundstrainevaluationbecomeslimiteddue to
the gridresolution. Theresultsareshownandcompared
inFig.7,wheretheshearcomponentS12ofthefullstrain
rate tensor is shown at the top, and the corresponding
nonlocal (background) component SB and local compo-
12
nent SR are shown, respectively, in the left and right
12
columns for (R/η ) = 2.5 (top row), 3.5 (middle row),
K
and 4.5 (bottom row). Consistent with the results from
the Burgers vortex in Fig. 5, as (R/η ) increases the
K
magnitude of the extracted local strain rate in the right
columnincreases. However,forthe largest(R/η )=4.5
K
case, n = 3 appears to be too small to adequately rep-
resent the local vorticity field. This leads to truncation
errorswhicharemanifestedasstrongripplesintheback-
ground and local strain fields (see panels (f) and (g)).
The results in Fig. 7 thus indicate that radii up to
(R/η )=3.5 in combination with n=3 can be used to
K
assessalignmentofthevorticityvectorwiththeeigenvec-
tors of the background strain rate field. Figure 8 shows
the probability densities of the alignment cosines for the
vorticityvectorwiththetotalstrainratetensorandwith
the background strain rate tensors from (20). We com-
FIG.7: TotalstrainratecomponentfieldS12(x)(a),withcor-
pareS (Fig.8a) with SB for(R/η )=2.5,n=3(Fig.
ij ij K responding results from (20) for nonlocal (background) field
8b) and SiBj for (R/ηK) = 3.5,n = 3 (Fig. 8c). The re- S1B2(x) (left) and local field S1R2(x) (right) for (R/ηK)=2.5
sults for alignment with the total strain rate tensor are (b,c), (R/ηK) = 3.5 (d,e), and (R/ηK) = 4.5 (f,g), all with
essentially identical to the anomalous alignment seen in n=3.
8
5 (a) 5 (b) 5 (c) e1·eω
e2·eω
4 4 4 e3·eω
)|)
θ( 3 3 3
s
o
c
(| 2 2 2
P
1 1 1
0 0 0
0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1
|cos(θ)| |cos(θ)| |cos(θ)|
FIG. 8: Probability densities of alignment cosines for the vorticity with the eigenvectors of the strain rate tensor, showing
results for Sij (a) and for SiBj using (R/ηK)=2.5 with n=3 (b) and (R/ηK)=3.5 with n=3 (c).
icantly. ThisisconsistentwiththeresultsfortheBurgers anomalousalignmentswitchingthroughouttheflowfield.
vortex in the previous Section, and with the hypothesis This conclusionis expected to alsoapply to the morere-
thatthealignmentswitchingmechanismduetothelocal alistic case of a non-uniformly stretched vortex where
contributionSR to the totalstrainrate tensoris the pri- S =f(z) [16, 19, 20, 21].
ij zz
mary reasonfor the anomalous alignment seen in earlier
Consistent with the results for the Burgers vortex,
studies. It is also consistent with the expected equilib-
when (20) was used to determine the background strain
riumalignmentfrom(1). While a moredetailedstudy is
ratetensorfieldSB(x)inhighly-resolvedDNSdatafora
neededtoexaminepossiblenonequilibriumcontributions ij
turbulentflow,theanomalousalignmentseeninprevious
to the alignment distributions associated with eigenvec-
DNSandexperimentalstudieswassubstantiallyreduced.
tor rotations of the background strain field, as well as
Weconcludethat(20)allowsthelocalbackgroundstrain
to definitively determine the R and n convergenceof the
rate tensor to be determined in any flow. Furthermore,
backgroundstrainrate tensor in Fig.7, the presentfind-
we postulate that the vorticity vector field in turbulent
ings support both the validity of the result in (20) for
flows will show a substantially preferred alignment with
extracting the backgroundstrain rate tensor field SB(x)
ij the most extensional principal axis of the background
fromthe totalstrainratetensorfieldS (x), andthe hy-
ij strain rate field, and that at least much of the anoma-
pothesis that at least much of the anomalous alignment
lous alignment found in previous studies is simply a re-
ofvorticityinturbulentflowsisduetothedifferencesbe-
flection of the alignment switching mechanism analyzed
tween the total and background strain rate tensors and
in Section III and conjectured by numerous previous in-
the resulting alignment switching noted herein.
vestigators.
Lastly,theresultin(20)isbasedonaTaylorseriesex-
pansion of the vorticity within a spherical neighborhood
V. CONCLUDING REMARKS ofradiusRaroundanypointx. Suchanexpansioninher-
ently involves derivatives of the total strain rate tensor
We have developed a systematic and exact result in field,whichcanleadtopotentialnumericallimitations. If
(20)thatallowsthelocalandnonlocal(background)con- largerR andcorrespondinglyhigher ordersn are needed
tributions to the totalstrainratetensorS atanypoint to obtain accurate evaluations of backgroundstrain rate
ij
xinaflow tobe disentangled. The approachis basedon fields, then otherwise identical approaches based on al-
aseriesexpansionofthevorticityfieldinalocalspherical ternative expansions may be numerically advantageous.
neighborhood of radius R centered at the point x. This Forinstance,anexpansionintermsoforthonormalbasis
allows the background strain rate tensor field SB(x) to functions allows the coefficients to be expressed as inte-
ij
be determined via a series of increasingly higher-order gralsoverthevorticityfieldwithinr R,ratherthanas
Laplacians applied to the total strain rate tensor field derivatives evaluated at the center p≤oint x. (For exam-
S (x). For the Burgers vortex, with increasing radius ple, wavelets have been used to test alignment between
ij
R relative to the local gradient lengthscale λ and with the strain rate eigenvectors and the vorticity gradient in
ν
increasing order n, we demonstrated convergence of the two-dimensionalturbulence [22].) This wouldallowa re-
resulting background strain tensor field to its theoreti- sultanalogousto(20)thatcanbecarriedtohigherorders
cal form. We also showed that even with limited R and withless sensitivity to discretizationerror. The key con-
n values, the local contribution to the total strain rate clusion,however,ofthepresentstudyisthatitispossible
tensor field can be sufficiently removed to eliminate the to evaluate the background strain tensor following the
9
general procedure developed herein, and that when such CAS),andbytheNationalAeronautics&SpaceAdmin-
methodsareappliedtoassessthebackgroundstrainrate istration (NASA) Marshall and Glenn Research Centers
fieldsinturbulentflowstheyrevealasubstantialincrease and the Department of Defense (DoD) under the NASA
intheexpectedalignmentofthevorticityvectorwiththe Constellation University Institutes Project (CUIP) un-
most extensional principal axis of the backgroundstrain der Grant No. NCC3-989. JS acknowledges support by
rate field. the German Academic Exchange Service (DAAD) and
by the Deutsche Forschungsgemeinschaft (DFG) under
grant SCHU 1410/2. The direct numerical simulations
Acknowledgments havebeencarriedoutwithintheDeepComputingInitia-
tive of the DEISA consortium on 512 CPUs of the IBM-
PH and WD acknowledge support from the Air Force p690 cluster JUMP at the John von Neumann Institute
ResearchLaboratory(AFRL)undertheMichiganAFRL forComputingattheResearchCentreJu¨lich(Germany).
Collaborative Center for Aeronautical Sciences (MAC-
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