velocity gradient invariants and local flow-field topology in compressible turbulence

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Velocity gradient invariants and localflow-field topology in compressibleturbulenceSawan Suman a & Sharath S. Girimaji aa Aerospace Engineering Department , Texas A&M University ,College Station, TX, USAPublished online: 24 Feb 2010.

To cite this article: Sawan Suman & Sharath S. Girimaji (2010) Velocity gradient invariantsand local flow-field topology in compressible turbulence, Journal of Turbulence, 11, N2, DOI:10.1080/14685241003604751

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Journal of TurbulenceVol. 11, No. 2, 2010, 1–24

Velocity gradient invariants and local flow-field topologyin compressible turbulence

Sawan Suman∗ and Sharath S. Girimaji

Aerospace Engineering Department, Texas A&M University, College Station, TX, USA

(Received 22 June 2009; final version received 7 December 2009)

The effect of compressibility on velocity gradient structure and the related flow-fieldpatterns/topology is investigated using direct numerical simulation data. To clearly iso-late compressibility effects, the behavior is investigated as a function of local level ofdilatation. Importantly, dilatation-conditioned behavior is found to be independent ofMach and Reynolds numbers. Not surprisingly, at low dilatation level, velocity gradi-ent structure and local flow topology are similar to incompressible turbulence. At highdilatation levels, however, the behavior is quite different. A recently developed veloc-ity gradient evolution equation – Homogenized Euler Equation (HEE) – qualitativelycaptures many of the observed features of compressible turbulence.

Keywords: compressible turbulence; velocity gradients; invariants; topology

1. Introduction

Velocity gradient structure and local flow field topology in a turbulent flow are inher-ently interesting and shed light on the underlying dynamics and mechanisms. Knowledgeof velocity gradient tensor invariants is particularly important for establishing variousturbulence characteristics. Invariants can be used to directly infer the local streamlinepattern/local topology of flow, thus, aiding in flow visualization [1]. Some of the mostpopular vortex eduction techniques – required for isolating and understanding coherentstructures – rely on the knowledge of velocity gradient invariants [2,3]. These invari-ants also shed light on important turbulence processes like dissipation, scalar mixing andvortex stretching. Invariant dynamics have also been used to understand important theo-retical aspects of turbulence like the possible routes to singularity in Euler flows [4] andintermittency [5].

Local streamline pattern or local topology of a turbulent flow field is useful not onlyfor flow visualization, but it also determines the level of material element deformationand scalar mixing. For example, a strain-dominated streamline pattern will deform a fluidelement and lead to increased mixing. A rotation-dominated pattern, on the other hand,will merely reorient a fluid element without much increase in mixing. Motivated by theneed of a general methodology for categorizing flow topology, Chong et al. [6] proposed ascheme based on the three invariants (P, Q R) of the velocity gradient tensor. Employingthis scheme, the local streamline pattern/topology of the velocity field could be inferred.Subsequently, Soria et al. [7] studied the joint statistical distributions of Q and R at

∗Corresponding author. Email: [email protected]

ISSN: 1468-5248 online onlyC© 2010 Taylor & Francis

DOI: 10.1080/14685241003604751http://www.informaworld.com

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2 S. Suman and S.S. Girimaji

small scales of motion in mixing layers. They found that the scatter plot of second andthird invariants has two prominent features: (i) significant amount of data lies in lower rightquadrant and (ii) the bulk of data lies in the upper left quadrant roughly distributed uniformlyover an elliptical region. The local topologies associated with these two regions are unstablenode/saddle/saddle and stable focus stretching (described in detail later). These prominenttopological features immediately attracted considerable research attention and were laterfound to be quite robust across a variety of turbulent flows: in high symmetry flow by Boratavet al. [8], boundary layer flow by Chacın et al. [9], channel flow by Blackburn et al. [10] andturbulent/non-turbulent interface by da Silva et al. [11]. In addition to inferring the dominantlocal topologies, invariant studies have also been used by several workers to develop insightinto other turbulence phenomena. Boratav et al. [8] suggested the usefulness of studyingturbulence statistics conditioned on the value of Q, as this quantity directly corresponds tothe right hand side of the Poisson equation for pressure. Thus, the role of pressure can beisolated by such conditioning. Chacın et al. [9] identified the strong association of variousproduction mechanisms with the sign of the discriminant, D (≡ 27R2/4 + Q3). Soria et al.[12] explained the preferred vorticity alignment tendencies and the preferred sign of theintermediate strain rate using volume integrals of the invariants. O’Neill et al. [13] studiedthe association between scalar dissipation and topological structures. Kobayashi et al. [14]have used a normalized form of second invariant (Q) to develop a subgrid scale (SGS)model for large eddy simulation (LES) applications.

All the above cited works are in the context of incompressible turbulence whereinthe first invariant, P, (which is the additive inverse of dilatation) is zero. Some stud-ies of topological features have been performed for compressible turbulence as well.Chen et al. [15] and Cantwell et al. [16] studied the dynamics of invariants and theassociated flow topologies in compressible wakes and flames. Chen et al. [17] exam-ined the statistics of second and third invariants in compressible mixing layer. Maekawaet al. [18] studied flow patterns/local topologies in decaying isotropic compressible tur-bulence and reported the percentage occurrence of various flow patterns conditionedon the sign of dilatation. Miura [19] performed a comparative study of compressibleand incompressible decaying turbulence in terms of (a) vortical structures and (b) rootmean square of the fluctuations in Q. Pirozzoli et al. [20] studied the invariants of thetraceless anisotropic portion of the velocity gradient tensor in compressible isotropicturbulence.

In all these studies of compressible turbulence, however, the effect of first invariant (P)on the statistics of second and third invariants has largely been ignored. The first invariantbeing the additive inverse of dilatation (trace of the velocity gradient tensor) is trivially zeroin incompressible flows. However, in compressible flows, it can be considerably differentfrom zero. A nonzero P signifies the degree of compression/expansion of a fluid element,and thus, is a very significant parameter in compressible turbulence. Maekawa et al. [18]does consider the first invariant, but is concerned only with the sign of P but not themagnitude of P.

Recently Suman et al. [21] and Lee et al. [22] have demonstrated the strong dependenceof strain-rate statistics on normalized dilatation in compressible turbulence. Previous studiesof compressible velocity gradient fields at low and moderate Mach numbers conclude thatthe overall statistics of second and thirds invariants are very similar to the behaviour seen inincompressible turbulence. As discussed in [22], this behaviour may be due to the followingtwo factors: (i) the dilatational component of velocity gradient in these studies is small asthe Mach numbers of these simulations are small and (ii) the effects of positive and negativedilatation may be canceling each other, thus, showing no significant difference between the

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Journal of Turbulence 3

overall statistics of incompressible and compressible turbulence. Lee et al. [22] demonstratethat in decaying isotropic turbulence, the percentage occurrence of a nonzero dilatationvalue monotonically grows as Mach number increases. Moreover, with increasing Machnumber, more extreme values (both positive and negative) are encountered in the domain.Therefore, it is reasonable to expect that at very high Mach numbers, the overall turbulencebehaviour can be substantially different from incompressible turbulence. The studies ofSuman et al. [21] and Lee at al. [22] prompt the following questions: (i) Are the statisticsof second and third invariants also strongly dependent on normalized dilatation? (ii) Howdoes this dependence change at different levels of normalized dilatation? (iii) What are thepreferred local topologies at different nonzero dilatations? (iv) How do the initial parametersuch as Mach and Reynolds numbers influence this dependence? We attempt to answer thesequestions in this paper.

Thus, the following objectives are identified for this paper. The first set of objectivesincludes: (i) examining the dependence of joint statistics of second and thirds invariantsof velocity gradient tensor on dilatation in compressible decaying isotropic turbulence and(ii) identifying the predominant topologies at various levels of normalized dilatation. Thesecond objective is to demonstrate that the conditioned behavior investigated above is inde-pendent of Reynolds and Mach numbers. We seek to demonstrate that Reynolds and Machnumbers merely affect the level of dilatation, but the behavior of statistics conditionedon a given value of dilatation is independent of these parameters. Such independence,if found to be correct, can substantially simplify closure modeling complexity. Further-more, toward the end of the paper, we also investigate whether the invariant statisticsseen in direct numerical simulation (DNS) of compressible turbulence can be recoveredby a recently developed velocity gradient evolution model (HEE, Homogenized EulerEquation [21]).

This paper is organized into 5 sections. Section 2 includes an overview of the three-dimensional phase space of P, Q and R and the methodology of inferring the local flowtopology. Section 3, 4 include results from DNS and HEE computations, respectively. InSection 5, we summarize the important findings of this study.

2. The phase space of P-Q-R and topology of compressible turbulence

The local topology at a point in a flow field can be deduced with a knowledge of theeigenvalues (λ1, λ2, λ3) of the local velocity gradient tensor, Aij [6]. Note that Aij ≡ ∂Vi

∂xj,

where Vi is velocity and xj represents spatial coordinates. These eigenvalues satisfy thefollowing characteristic equation:

λ3 + Pλ2 + Qλ + R = 0. (1)

The quantities P, Q, and R appearing in (1) are the first, second, and third invariants ofAij :

P ≡ −tr[A] = −Sii ; (2a)

Q ≡ 1

2

(P2 − tr[A2]

) = 1

2

(P2 − Sij Sji − Wij Wji

); (2b)

R ≡ 1

3

(−P3 + 3PQ − tr[A3]) = 1

3

(−P3 + 3PQ − Sij SjkSki − 3WijWjkSki

). (2c)

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The quantities Sij and Wij represent the strain-rate and rotation-rate tensors respectively:Sij ≡ 1

2 (Aij + Aji) and Wij ≡ 12 (Aij − Aji). The characteristic Equation (1) has three

roots. The nature of solution and the local flow topology can be categorized on the basisof these roots. The possibilities are: (i) all real and distinct roots, (ii) all real with twoequal roots, (iii) all real and equal roots and (iv) one real and two complex conjugateroots. However, to infer the category of solution, we do not need to solve the characteristicequation. Topology at a point in a flow field can be directly inferred using the properties ofthe three-dimensional (P-Q-R) space and the local values of the invariants (P, Q R).

Chong et al. [6] explain that the P-Q-R space is partitioned into different spatial regionsby a set of surfaces. Each of these regions corresponds to a particular solution categoryand, hence, is associated with a particular topology. The surface that separates the regionsof real and complex roots is:

27R2 + (4P3 − 18PQ

)R + (

4Q3 − P2Q2) = 0. (3)

This surface can be split into two surfaces S1a and S1b, which osculate each other to forma cusp. The equations for S1a and S1b are:

1

3P

(Q − 2

9P2

)− 2

27

(−3Q + P2) 3

2 − R = 0; (4a)

1

3P

(Q − 2

9P2

)+ 2

27

(−3Q + P2) 3

2 − R = 0. (4b)

The region of complex roots has another dividing surface, S2, which contains the pointsassociated with purely imaginary roots:

PQ − R = 0. (4c)

It is convenient to visualize the P-Q-R space and the various regions (that determinevarious flow topologies) by considering planes of Q and R at discrete values of P. Onsuch a plane, surfaces S1a, S1b, and S2 appear simply as curves, which divide the planeinto different planar regions. In Figures 1–3, we show three representative Q − R planes atP = 0, P < 0 and P > 0. On the P = 0 plane (Figure 1), the curve S2 is coincident withthe Q axis. The curves S1a and S1b are symmetric with respect to each other about the Q

axis, thus, dividing the plane into four regions. These four regions are associated with fourdistinct flow patterns, which are schematically shown in Figure 1. These flow patterns areUFC, UN/S/S, SN/S/S and SFS. Description of these acronyms and others, which appearelsewhere in this paper, are included in Table 1. On the P = 0 plane (Figure 1), UN/S/S andSN/S/S are nonfocal structures (all real eigenvalues), whereas UFC and SFS are structureswith a focus and an out-of-plane strain (one real and two complex conjugates eigenvalues).Note that in the context of describing different topologies, the qualifiers “unstable” and“stable” imply the direction of solution trajectories on relevant eigenvector planes of thelocal velocity gradient tensor (Aij ) [1,6]. These solution trajectories can be visualizedas the local streamlines around the critical point under consideration [1]. The qualifier“stable” implies that the corresponding solution trajectories or the local streamlines aredirected toward the critical point. Similarly, “unstable” implies that the local streamlinesare directed away from the critical point. As P assumes a nonzero value, two qualitative

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Figure 1. Regions on the P = 0 (Aii = 0) plane and the corresponding flow patterns. Description ofacronyms provided in Table 1.

differences appear on the Q-R plane: (i) surface S2 is no longer coincident with the Q axisand (ii) the symmetry of surfaces S1a and S1b is lost (Figures 2 and 3). Consequently,additional regions appear on these planes. Both P < 0 and P > 0 planes are now dividedinto six regions. An increase in the number of regions results in an increase in the number ofpossible topologies as well. On both P < 0 and P > 0 planes, there are three focal and threenonfocal topologies. The schematic of these patterns and their correspondence with the sixregions are illustrated in Figures 2 and 3. On each of these figures, in addition to the flowpatterns associated with the six regions, we include an illustration of the topology associatedwith the osculation point of S1a and S1b curves. This topology is USN/USN/USN andSSN/SSN/SSN on P < 0 and P > 0 planes, respectively. The significance of these point-associated topologies in compressible turbulence will be discussed in the next section.

Notably, the associated topology for a given Aij tensor does not depend on the magnitudeof the tensor components. It depends only on the structure of the tensor. We can define anormalized form of the velocity gradient tensor that retains all the information pertainingto the structure of Aij . The normalized form (aij ) is defined as [23]:

aij ≡ Aij√AmnAmn

. (5)

Thus, the associated topology can as well be inferred by employing the same methodologyas described in [6] but to the space of normalized invariants p, q, and r. These normalized

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Table 1. Description of acronyms.

Acronyms Description

SFS Stable Focus StretchingUFS Unstable Focus StretchingSFC Stable Focus CompressingUFC Unstable Focus CompressingSN/S/S Stable Node/Saddle/SaddleUN/S/S Unstable Node/Saddle/SaddleSN/SN/SN Stable Node/Stable Node/Stable NodeUN/UN/UN Unstable Node/Unstable Node/Unstable NodeSSN/SSN/SSN Stable Star Node/Stable Star Node/Stable Star NodeUSN/USN/USN Unstable Star Node/Unstable Star Node/Unstable Star Node

invariants are defined as :

p ≡ −tr[a] = −sii ; (6a)

q ≡ 1

2

(p2 − tr[a2]

) = 1

2

(p2 − sij sji − wij wji

); (6b)

r ≡ 1

3

(−p3 + 3pq − tr[a3]) = 1

3

(−p3 + 3pq − sij sjkski − 3wij wjkski

)(6c)

Figure 2. Regions on a negative P (or a positive Aii) plane and the corresponding flow patterns.Description of acronyms provided in Table 1.

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Figure 3. Regions on a positive P (or a negative Aii) plane and the corresponding flow patterns.Description of acronyms provided in Table 1.

where sij and wij are the normalized strain and rotation-rate tensors: sij ≡ aij +aji

2 and

wij ≡ aij −aji

2 . The advantage of using normalized invariants is that all the quantities arebounded by their known algebraic limits. In compressible flows, p(= −aii) can vary withinthe algebraic limits of −√

3 and√

3. On the other hand, the normalized invariants q and rlie in the intervals [−1/2, 1] and [−√

3/9,√

3/9], respectively.

3. DNS results of invariants and local topology

In this section, we present the statistics of velocity gradient invariants seen in direct numer-ical simulation of compressible decaying isotropic turbulence. We examine these statisticsconditioned upon normalized dilatation (aii = Aii√

AmnAmn). We also examine the dependence

of conditional statistics on initial Mach number and Reynolds number.DNS results presented in this paper are computed using a Gas Kinetic Method (GKM)

solver for compressible flows. GKM solvers have been developed and verified for variouscompressible flow applications [24,25]. These solvers are based on Kinetic Boltzmannequation, for which the computational framework is based on one-point velocity distributionfunction. The equation for pressure is the perfect gas state equation: p = ρRT , where p, ρ,T represent pressure, density, and temperature, respectively, and R is gas constant. Detailsof the theoretical aspects of GKM solvers can be found in [24–26]. Recently Kerimoet al. [26] have demonstrated the accuracy and robustness of GKM solvers for turbulent

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flow simulations. Lee et al. [27] have also employed a GKM solver to demonstrate avalidation of the Taylor’s dissipation-viscosity independence postulate in variable-viscosityturbulent fluid mixtures.

The analysis presented in this paper employs the DNS results obtained by Lee [28] fordecaying compressible isotropic turbulence. The computational domain of these simulationsis a 2563 box with periodic boundary conditions. The solver used by Lee [28] is the same asused by Lee et al. [27]. Two important nondimensional parameters characterizing decayingcompressible turbulent flows are the turbulent Mach number (Mt) and Reynolds number(Reλ) based on the Taylor microscale (λ):

Mt ≡√

2k

a; Reλ ≡

√20

3ενk (7)

where a =√

nRT represents the speed of sound (corresponding to mean temperature, T )in the fluid medium and n is the specific heat ratio of the medium. The symbols k and ε

represent turbulent kinetic energy and its dissipation rate, and ν represents the coefficient ofkinematic viscosity. The velocity field is initialized with a spectrum having the wavenumber(κ) content ranging from 1 to 8. The ratio of initial dilatational to solenoidal energy contentof the velocity field is zero. The thermodynamic field is initialized with constant density(= 1 kgm−3) and temperature (= 300 K) values. This type of compressible flow simulationis akin to what is called the constant initial condition (CIC) simulation [29]. The Prandtlnumber, n and R are chosen to be 0.7, 1.4 and 287 Jkg−1K−1. Different simulation casesused for analysis in this paper are summarized in Table 2. For a given Reynolds number,initial velocity field is amplified appropriately to attain a desired value of initial Machnumber. The value of viscosity coefficient is accordingly adjusted to maintain the desiredvalue of Reynolds number. For a given Mach number, a desired value of Reynolds numberis achieved by directly changing the viscosity coefficient (Table 2).

To study the statistics of velocity gradient invariants, we output result of each simulationat the peak of dissipation. By this time, the nonlinear turbulence processes are expectedto be in full effect. Using the velocity field at this instant, the velocity gradient field iscomputed over the entire computational domain. Subsequently, the three invariants arecomputed at each grid point. Using the values of p (= −aii) at these grid points, q − r datais then binned at various chosen levels of normalized dilatation (aii). The bin specificationsand the sample size corresponding to each bin are provided in Table 3. Note that the samplesize at extreme positive/negative dilatations sharply reduces, and the statistics obtainedwith such sample sizes can be expected to have some errors. Since extreme dilatationevents are naturally rare in compressible turbulence [22], it is not possible to obtain bettersample sizes using the parameters employed for these simulations (Table 2). Realizationof a bigger sample size at extreme dilatations would require a larger computationaldomain/higher Mach number than what has been used in the present simulations.

Table 2. DNS simulation cases.

Parameter Case A Case B Case C

Reλ 55.6 80.0 55.6Mt 0.70 0.70 0.43µ × 10−4 kgm−1s−1 48 33.4 28

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Table 3. Number of samples at each dilatation level. Bin size correspond-ing to each dilatation level is ±0.05 about the median value.

Median value of aii Case A Case B Case C

1.4 657 419 —0.9 21812 13494 3000.7 66455 47592 23680.3 914684 788133 2365180.0 3866101 4208919 7108648

−0.3 786840 762275 284889−0.7 89321 69572 4811−0.9 31868 22140 1150−1.4 749 647 10

3.1. Conditional invariant statistics in compressible turbulence

We study statistics of invariants in terms of (i) joint probability density function (pdf)and (ii) probability of occurrence of various topologies shown in Figures 1–3 at variouslevels of normalized dilatations. We start the discussion with the zero dilatation case(aii = −p = 0). Subsequently, we study the statistics conditioned upon different positiveand negative dilatation values. All conditional joint pdfs presented in this paper are obtainedby considering bins of dimensions 0.008 × 0.008 on a given q-r plane. Unless specifiedotherwise, all reported DNS results are from case A simulation.

3.1.1. Zero dilatation

In Figure 4a, we plot the joint pdf of q and r conditioned upon aii = 0. The pdf shownin Figure 4 a has two prominent features: (i) significant number of points lie in the upperleft area distributed over an elliptical region and (ii) the bulk of data lies in the lower rightarea concentrated around the Vieillefosse line (q = − 3

√27r2/4). Lee et al. [22] show that

the statistics of strain-rate eigenvalues and vorticity alignment tendencies in compress-ible turbulence conditioned upon zero dilatation are very similar to the statistics seen inincompressible turbulence. We examine whether such similarity can be seen in terms ofinvariant statistics as well. In Figure 4b, we present the joint pdf of q and r seen in DNS ofincompressible decaying isotropic turbulence. Clearly, the distribution from compressibleDNS is very similar to the one from incompressible DNS. The two prominent featuresmentioned above are indeed the same features that are seen in a variety of incompressibleflows [7–11]. Further, we make quantitative comparisons between the conditional data fromcompressible turbulence and incompressible DNS results in terms of percentage of variouspossible topologies (Table 4). Indeed, the probability of occurrence of each topology seenin compressible turbulence is very close to that seen in incompressible turbulence. On thebasis of these observations, we conclude that at locations of negligible dilatation, invariantstatistics seen in compressible turbulence closely resemble the invariant statistics seen inincompressible turbulence.

Table 4. Percentage of various flow topologies conditioned upon zero dilatation.

UFC UN/S/S SN/S/S SFS

Case A 25.6 24.9 6.5 43.1Incompressible DNS 27.5 25.4 7.3 39.8

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Figure 4. (a) Joint pdf of q and r conditioned upon zero dilatation from case A simulation and (b)joint pdf of q and r from DNS of incompressible decaying isotropic turbulence.

3.1.2. Positive dilatation

In Figures 5a, 5c, 5e and 5g, we plot the joint pdfs of q and r conditioned upon variouspositive values of normalized dilatation. The chosen levels of normalized dilatation varyfrom 0.3 to 1.4. A visual inspection of these figures suggests that the pdfs are stronglydependent on the value of normalized dilatation. Both the shape and the concentration of

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Figure 5. Conditional joint pdf of q and r as seen in case A DNS simulation (left column) and HEEmodel computations (right column). Value of aii changes from top to bottom in the following order:0.3, 0.7, 0.9, and 1.4.

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Table 5. Percentage of various flow topologies conditioned upon different positive dilatation levelsfrom case A simulation.

aii UFC UN/S/S SN/S/S SFS UFS UN/UN/UN USN/USN/USN

0.3 24.4 31.2 3.2 21.2 20.0 0.0 0.00.7 22.1 33.7 0.6 5.5 37.1 0.8 0.00.9 17.7 30.6 0.0 0.9 47.5 3.2 0.01.4 0.0 1.5 0.0 0.0 60.1 38.4 0.01.7 0.0 0.0 0.0 0.0 0.0 0.0 100.0

distribution with respect to the various region-dividing curves (S1a, S1b, S2 and q axis,see Figure 2) change significantly with changing dilatation level. The shape of distributionseems to be following a trend: The shape changes from being elongated tear shaped at zerodilatation to a more localized elliptical shape as dilatation assumes higher positive values.In Table 5, we present the percentage occurrence of various topologies at different positivedilatation levels. Following is a summary of the important topological features/trends seenat positive dilatations.

(1) At low positive dilatations, the statistical behavior is similar to that in incompressibleflow.

(2) As dilatation assumes higher positive values, topologies with stable solution trajectories(local streamlines) – SFS and SN/S/S – disappear, and the favored topologies are thosethat have unstable solution trajectories (UFC or UN/S/S or UFS or UN/UN/UN). Thediverging streamlines of these favored topologies are consistent with the tendency ofa fluid element to have increase in volume and a consequent decrease in density atpositive dilatations.

(3) At the extreme positive dilatation value (aii = √3), the distribution is almost a two-

dimensional Dirac-delta distribution (not shown) with all data points located at theosculation point of S1a and S1b. As shown in Figure 2, this point corresponds to arotation free topology with all the eigenvalues being real, equal and positive. Accord-ing to the nomenclature of Chong et al. [6], the topology is USN/USN/USN. Thecorresponding eigenvalues are (1/

√3, 1/

√3, 1/

√3). It can be shown using simple

algebra that at aii = √3, this eigenvalue set is the only possibility. At this normalized

dilatation value, the velocity field is entirely dilatational and hence free of vorticity.Also, this state is one of the stable solutions of the pressure-released (Burgers) velocitygradient dynamics [30]. This topology can be visualized as a rotation-free, isotropic(spherical) expansion of a fluid element. The local streamline pattern corresponds tothat of a source flow with radially outward streamlines. Clearly, such a flow would leadto diminishing density at the source.

(4) At high positive dilatations (aii = 1.4 in Table 5), two shapes dominate: UN/UN/UNand UFS. The vorticity-deficient UN/UN/UN (unstable node on all the three eigen-vector planes [6]) topology is similar to the USN/USN/USN topology seen in theextreme dilatation limit (aii = √

3). The UFS topology contains significant vorticityand appears to emerge from the two focal topologies – SFS and UFC – seen in nearlyincompressible limit. Clearly, UFS represents a vortex being axially stretched whilethe streamlines spiral radially outwards. This can be visualized as a cylindrical fluidelement undergoing both axial and radial expansion leading to an increase in the fluidelement volume. Correspondingly, density in this flow region must reduce. Note that

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Table 6. Percentage of various flow topologies conditioned upon different negative dilatation levelsfrom case A simulation.

aii UFC UN/S/S SN/S/S SFS SFC SN/SN/SN SSN/SSN/SSN

−0.3 22.4 14.6 17.1 30.3 15.4 0.0 0.0−0.7 11.1 2.7 33.5 21.8 30.1 0.8 0.0−0.9 2.7 0.2 37.6 18.3 37.9 3.3 0.0−1.4 0.0 0.0 2.7 0.5 45.4 51.4 0.0−1.7 0.0 0.0 0.0 0.0 0.0 0.0 100.0

in incompressible flows, a fluid element never undergoes a deformation that allowsboth radial and axial expansions. By virtue of the incompressibility constraint (volumeconservation), a focal topology in incompressible flow is either SFS (radial contractionand axial stretching) or UFC (radial expansion and axial compression).

3.1.3. Negative dilatation

We present conditional joint pdfs of q and r at various negative dilatation values inFigures 6a, 6c, 6e and 6g. In these figures, aii varies from −0.3 to −1.4. In Table 6,we present the percentage occurrence of various topologies at different negative dilatationlevels. The inferences are now summarized.

(1) At low negative dilatations, the statistical behavior is once again similar to that inincompressible flow.

(2) As dilatation assumes higher negative values, topologies with unstable solution tra-jectories (UFC and UN/S/S) become less important, and the favored topologies arethose that have stable solution trajectories (SFS or SN/S/S or SFC or SN/SN/SN). Thisbehavior is in direct contrast with that at positive dilatations, which favor topologieswith unstable solution trajectories. The converging streamlines of topologies with sta-ble solution trajectories are consistent with the tendency of a fluid element to havedecreasing volume/increasing density at negative dilatation values.

(3) At extreme negative dilatation (aii = −√3), normalization dictates SSN/SSN/SSN

topology with the three eigenvalues being equal, real, and negative(−1/

√3,−1/

√3,−1/

√3). However, unlike the state at aii = √

3, this unique eigen-value set is not one of the stable solutions of pressure-released velocity gradientdynamics [30]. Physically, the SSN/SSN/SSN topology represents a sink flow withall streamlines converging radially to the center leading to a decrease in volume andincrease in density.

(4) At high negative dilatation levels, the topologies that dominate incompressible turbu-lence regime disappear and SN/SN/SN and SFC are the predominant flow patterns. Thevorticity-deficient SN/SN/SN is similar to the source flow topology (SSN/SSN/SSN).The SFC flow pattern corresponds to a vortex undergoing both radial and axial contrac-tion. The predominant SFC topology at negative dilatations is in a way exact oppositeof the UFS topology that becomes dominant at high positive dilatations. Again, notethat the deformation pattern represented by SFC topology at negative dilatations is notrealizable in incompressible flows, wherein the volume conservation dictates either ax-ial expansion with radial contraction (SFS) or axial contraction with radial expansion(UFC).

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Figure 6. Conditional joint pdf of q and r as seen in case A simulation (left column) and HEE modelcomputations (right column). Value of aii changes from top to bottom in the following order: −0.3,−0.7, −0.9, and −1.4.

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(5) As dilatation approaches −1 (Figure 6e), we observe that the joint pdf of q and rdevelops a substantial concentration of data points near the origin of q − r axes. Onthe aii = −1 plane, the eigenvalue set associated with the origin is (0, 0,−1). A stateof aij with two equal and small (in magnitude) eigenvalues and one large negativeeigenvalue is suggestive of a compression front or a shocklet like structure. Thus, inFigure 6e, the high concentration of particles in the vicinity of the origin is clearlysuggestive of the enhanced tendency of the flow to form shocklets. Note that a majorportion of this high concentration region in Figure 6e is associated with the SN/S/Stopology (region left of q axis and below S1a curve, see Figure 3). Thus, it seemsplausible to associate the increasing percentage of SN/S/S topology – as aii changesfrom 0 to −1 (see Table 6) – with the enhanced tendency of the flow to form shocklets.In contrast to Figure 6e is Figure 5e, which shows the joint pdf of q − r with aii beingclose to +1. In Figure 5e, there is no significant preference of data points to accumulatenear the origin of q − r axes. At aii = 1, the eigenvalue set associated with the originis (0,0,1) and the corresponding deformation pattern would be an expansion front.No significant tendency of the points to accumulate near the origin in Figure 5e is anevidence that an expansion front is not a highly favored flow structure in compressibleturbulence. No significant preference for an expansion front at positive dilatations,whereas a high preference for contraction front (shocklet) at negative dilatations isan important difference between the topological trends seen at positive and negativedilatations.

On the basis of this discussion, we conclude that the joint pdfs of second and thirdinvariants of normalized velocity gradient tensor are strongly dependent on normalizeddilatation value. At zero dilatation, the distribution has a tear drop shape and is very similarto the distribution seen in incompressible turbulence. As the dilatation level increases(positive or negative), we see a drastic departure from the incompressible behaviour. Ateither extreme of normalized dilatation (−√

3 and√

3), the distribution shrinks to the cuspof surfaces S1a and S1b and is associated with a unique topology with all the eigenvaluesbeing real and equal. In general, positive dilatations kill topologies with stable solutiontrajectories, whereas negative dilatations kill topologies with unstable solution trajectories.At intermediate and high positive dilatations, a vortex expanding in both radial and axialdirections (UFS) is the predominant focal topology, whereas at intermediate and highnegative dilatations a vortex contracting in both radial and axial direction (SFC) is thedominant focal topology. As dilatation changes from zero to −1, SN/S/S topology becomesincreasingly more important and seems to be associated with the enhanced tendency ofcompressible flows to form shocklets.

3.2. Dependence of conditional statistics on Mach and Reynolds number

Lee et al. [22] demonstrate that conditional statistics of strain-rate eigenvalues and vorticityalignment tendencies in compressible turbulence remain largely independent of Reynoldsnumber and Mach number of the flow. In this subsection we further examine the influenceof initial Reynolds number and Mach number on the conditional statistics of invariants. Wemake comparisons between different cases listed in Table 2. Case A is compared with caseB to examine the effects of Reynolds number. Case A is compared with case C to infer theeffects of initial Mach number.

Using figures similar to Figures 4–6, conditional joint distributions in cases B and Chave also been qualitatively examined. In case C, which has a lower initial Mach number

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Table 7. Percentage of flow topologies conditioned upon zero dilatation from different simulationcases. In each cell below, percentages are presented in the following order: (UFC, UN/S/S, SN/S/S,SFS).

Case A Case B Case Caii (Reλ = 55.6, Mt = 0.70) (Reλ = 80.0, Mt = 0.70) (Reλ = 55.6, Mt = 0.43)

0 (26, 25, 7, 42) (25, 26, 7, 42) (27, 25, 6, 42)

(Mt = 0.43), very high positive or very high negative dilatations are rarely seen [22]. Inthe absence of adequate sample size, statistics at aii > 1 and aii < −1 are not availablefor this case. Otherwise, various qualitative features of the distributions and general trendsseen in cases B and C are very similar to what we observed in case A. We do not includethe joint distribution contours from cases B and C here. However, we present quantitativecomparisons between the three cases in terms of conditional percentage occurrence ofvarious topologies. In Table 7, we present the conditional percentages at zero dilatation. InTables 8 and 9, percentages at positive and negative dilatations are included.

3.2.1. Effect of Reynolds number on topology

In Tables 7–9, we compare case A and case B to infer the influence of Reynolds number onthe percentage occurrence of flow patterns. At zero and almost all positive dilatations, thepercentage values from case B simulation are very close to those from case A simulations,except some differences in the percentages of UN/S/S and UFS topologies at aii = 0.7 andaii = 0.9. At various negative dilatations in Table 9, the agreement is generally good exceptat aii = −1.4. While SN/SN/SN is the most dominant topology at lower Reynolds number(case A), at higher Reynolds number (case B), SFC emerges as the most dominant topology.Since the sample size at aii = −1.4 is quite small, this difference may be a statistical error.Over all, Reynolds number (in the range investigated) appears to have negligible effect onthe conditional percentage occurrence of various topologies.

3.2.2. Effect of Mach number on topology

We compare case A and case C results in Tables 7–9 to infer the effects of initial Machnumber. As mentioned above, case C simulation does not offer adequate samples at highpositive/negative dilatations. Thus, this comparison is naturally restricted to low and mod-erate dilatation levels. In Table 7, we observe that Mach number has little effect on the

Table 8. Percentage of flow topologies conditioned upon various positive dilatation levels fromdifferent simulation cases. In each cell below, percentages are presented in the following order: (UFC,UN/S/S, SN/S/S, SFS, UFS, UN/UN/UN).


0.3 (24, 32 3, 21, 20, 0) (23, 34, 4, 22, 17, 0) (27, 32, 5, 24, 12, 0)0.7 (22, 34, 1, 6, 36, 1) (21, 39, 1, 7, 31, 1) (27, 37, 1, 8, 26, 1)0.9 (18, 31, 0, 1, 47, 3) (17, 38, 0, 2, 40, 3) (23, 25, 0, 3, 48, 1)1.4 (0, 2, 0, 0, 60, 38) (0, 1, 0, 0, 60, 39) -

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Table 9. Percentage of flow topologies conditioned upon various negative dilatation levels fromdifferent simulation cases. In each cell below, percentages are presented in the following order: (UFC,UN/S/S, SN/S/S, SFS, SFC, SN/SN/SN).


−0.3 (22, 15, 17, 30, 15, 0) (22, 15, 20, 28, 15, 0) (26, 14, 20, 27, 13, 0)−0.7 (11, 3, 33, 22, 30, 1) (11, 3, 35, 22, 28, 1) (10, 2, 36, 20, 31, 1)−0.9 (3, 0, 38, 18, 38, 3) (3, 0, 40, 17, 37, 3) (2, 0, 36, 12, 47, 3)−1.4 (0, 0, 3, 1, 45, 51) (0, 0, 0, 0, 57, 43) -

topology of the flow field at zero dilatation. Even at nonzero dilatations, the difference be-tween case A and case C results are not very significant. On the basis of these observations,we can conclude that turbulent Mach number (in the range investigated) does not appear tosignificantly influence the conditional percentage occurrence of various topologies.

4. HEE model computations of invariants and local topology

Velocity gradient evolution is a highly nonlinear process. Understanding this process is ofparamount importance, as it holds the key to many turbulence processes of practical interestlike mixing, intermittency, cascading, etc. While DNS results do provide an almost time-continuous, domain-spanning data of this evolution, it is often difficult to develop in-depthphysical insights with such large volumes of data. Thus, there has always been a need tohave simple Lagrangian dynamical models that can offer an approximate, nevertheless, amore direct way to probe the evolution of velocity gradients. However, such models requireadequate validation before they can be employed for wider purposes. Restricted Euler equa-tion (REE) is one such model [31] for incompressible velocity gradient dynamics. It hasbeen used with considerable success [4,5] and being continuously improved [23,32–35].However, REE is not useful for compressible velocity gradients. The REE model hinges onthe Poisson equation, which is true for incompressible flows only. To the authors’ knowl-edge, the only available dynamical model for compressible velocity gradient dynamics isthe Homogenized Euler equation model [21]. The model shows considerable success inpredicting strain-rate statistics in compressible turbulence [21]. In this section, we examinewhether the model can capture various characteristics of compressible velocity gradientinvariants discussed in §3.

4.1. The HEE model

Before discussing the HEE results, we present a brief overview of the model. A completedescription of the model can be found in [21]. The HEE model is based on conservationof mass, momentum and energy equations of an inviscid homentropic calorically perfectcompressible medium. For such a flow, the state equation follows the polytropic relationship:p = Cρn; where p, ρ and n denote pressure, density, and the specific heat ratio, respectively,and C is a constant both in time and space. Substituting this expression for pressure in themomentum equation and subsequently taking the spatial derivative of the equation, leadsto the following equation of the velocity gradient tensor, Aij :

dAij

dt= −AikAkj − Cn

n − 1

∂2g

∂xi∂xj

(8)

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where g ≡ ρn−1 and t represents time. The second term on the right hand side of (8)is the pressure Hessian tensor of the compressible flow under consideration. The exactevolution equation of pressure Hessian tensor is derived using the continuity equation forcompressible flows:

d

dt

(∂2g

∂xi∂xj

)= −Akj

∂2g

∂xi∂xk

− Aki

∂2g

∂xk∂xj

− ∂Aki

∂xj

∂g

∂xk

− (n − 1) Akk

∂2g

∂xi∂xj

− (n − 1)∂Akk

∂xi

∂g

∂xj

− (n − 1)∂Akk

∂xj

∂g

∂xi

− (n − 1)∂2Akk

∂xi∂xj

g. (9)

Though (9) is exact, the obvious problem – from a computational view point – is a set ofunclosed terms containing higher order derivatives of velocity gradients on the right handside of (9). Taking a cue from the foundational assumption of the REE model, HEE modelsimplifies (9) by assuming the velocity gradients to be constant in space [31]:

∂Aij

∂xk

≡ 0. (10)

This assumption results into a closed set of equations for the velocity gradient and pressureHessian tensors:

dAij

dt= −AikAkj − Pij ; (11)

dPij

dt= −PikAkj − Pkj Aki − (n − 1) Pij Akk; (12)

where

Pij ≡ Cn

n − 1

∂2g

∂xi∂xj

(13)

is the pressure Hessian tensor. This closed set of 15 Equations (11)–(12) is referred to as theHomogenized Euler equation (HEE), and it models the evolution of velocity and pressureHessian tensors in a compressible inviscid calorically perfect medium.

At this point it is pertinent to highlight the following salient features of the HEE model:(i) The evolution of pressure (Hessian) in HEE is governed by the exact state equation of acalorically perfect gas, rather than the Poisson equation, and (ii) Unlike the restricted Eulerequation, HEE model does not simplify the pressure Hessian tensor to a mere isotropictensor. In the REE model, pressure Hessian is approximated as [31]:

∂

∂xj

(1

ρ

∂p

∂xi

)= 1

ρ

∂2p

∂xi∂xj

= −AmnAnm

3δij . (14)

The HEE model in (11)–(12), on the other hand, includes both the isotropic and theanisotropic portions of the tensor. This is a major advancement as compared to the originalREE model.

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4.2. Nature of HEE solution and statistics

Like REE, HEE Equations (11)–(12) exhibit finite time singularity: velocity gradient mag-nitudes evolve to very large values in finite time. This behavior poses computationaldifficulty. Following the strategy adopted by Girimaji et al. [23] to circumvent the finitetime singularity problem in REE, we recast the HEE equations in terms of the normalizedvelocity gradient tensor: aij (≡ Aij√

AmnAmn). This procedure requires inclusion of an evolution

equation for τ (≡ (AmnAmn)−1/2). Equations (11)–(12) are recast as:

daij

dt ′= aij amnaknamk + τ 2aij amnPmn − aikakj − τ 2Pij ; (15)

dτ

dt ′= τamnaknamk + τ 3amnPmn; (16)

dPij

dt ′= −Pikakj − Pkj aki − (n − 1) Pij akk (17)

where dt′ = dt/τ . Equations (15)–(17) are equivalent to Equation (11)–(12) and have the

advantage of being computationally more amenable. We employ a fourth order Runge–Kuttamethod to integrate the equation set (15)–(17).

Like Vieillefosse’s REE model [31], HEE model is a set of autonomous ordinarydifferential equations describing the nonlinear evolution of velocity gradients in a turbulentflow field. Like REE, the normalized velocity gradients in HEE show the tendency to attaincertain asymptotic states. Computations of Suman et al. [21] show that a typical solutiontrajectory starts with a specified initial condition, evolves with a rapid change in normalizeddilatation and finally settles down to one of the two asymptotic states. These asymptoticstates are [21]:

(α, β, γ, ωα, ωβ, ωγ

) ≈(

1√3, 1√

3, 1√

3, 0, 0, 0

); (18)

(α, β, γ, ωα, ωβ, ωγ

) ≈ (0, 0,−1, 0, 0, 0) . (19)

The quantities α, β, γ represent the eigenvalues of the normalized strain-rate tensor (sij ),and ωα , ωβ , ωγ represent the normalized vorticity components along the three eigenvectorsof sij . Note that the normalized dilatation (aii) at these two asymptotic states are

√3 and

−1, respectively.Notably, again like REE, HEE solution does not show any statistical stationarity in terms

of the absolute velocity gradient components, Aij . In the absence of viscosity, velocitygradient magnitudes in both these models quickly evolve to unrealistic values. However,as demonstrated by Vieillesfose [31], Cantwell [4] and Ashurst et al. [36], the physicalfidelity of REE is evident in terms of the structure of velocity gradient tensor (normalizedvelocity gradient components, aij ). For example, in REE, though the velocity gradientmagnitude grows rapidly in time and no time stationarity is seen in terms of Aij , the modelquickly captures the orientation tendencies of the vorticity vector as seen in DNS results[36]. A similar study of the structure of velocity gradient tensor can be performed with theHEE solution as well. Our interest in compressible flows requires examination of velocitygradient structure conditioned upon normalized dilatation. Relying only on the asymptoticsolution will confine such a study to merely two discrete values of normalized dilatation:√

3 and −1 (18–19). Development of further insights into velocity gradient tensor at otherintermediate dilatation values requires examination of the transient solution as well. This is

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the approach adopted by Suman et al. [21] to study the dependence of strain-rate statisticson normalized dilatation. We will adopt exactly the same strategy in this paper to studythe dependence of invariant statistics on dilatation. We gather normalized velocity gradientstatistics conditioned upon various levels of normalized dilatation. To reasonably mitigatethe effects of initial conditions on the studied HEE results, we delay the gathering ofstatistics by an initial time lapse of, T :

T = 1(τamnaknamk + τ 3amnPmn

)t ′=0

. (20)

As pointed out in [21], the time lapse T can be seen as one velocity gradient turn-over time.We numerically solve the HEE equations employing the same initial conditions as used

in [21]. The nine components of the velocity gradient tensor are generated using a randomnumber generator, which generates uniformly distributed numbers between −1 and 1. Initialvalue of

√Aij Aij is set to unity for all particles. The initial Pij components are chosen as

in the REE model:

Pij (t=0) = −AmnAnm

3δij . (21)

Despite this initially isotropic structure of Pij , both the anisotropic and isotropic portionsof the tensor evolve as dictated by Equation (17) at subsequent times. Computations areperformed with a large ensemble (100,000) of initial conditions or “particles” to obtainaccurate HEE statistics, which are then compared against the DNS results. Before we moveon to discuss the HEE results, we re-emphasize that our plan of comparing the HEE resultsof velocity gradient invariants against the DNS results is exactly the same as the approachused by Suman et al. [21] and Lee et al. [22] to study the conditional statistics of strain-rateeigenvalues.

4.3. HEE model results

In Figures 5b, 5d, 5f, 5h and 6b, 6d, 6f, 6h we present the conditional joint pdfs of qand r obtained from HEE computations. The dilatation level ranges from −1.4 to 1.4. Atzero dilatation, HEE model accurately captures the tear-drop shape of the distribution atzero dilatation (discussed in detail in [21]). As the dilatation level changes, HEE distri-butions show change in shape and location with respect to the various partitioning curves(Figures 1–3). The underlying trend of these changes is similar to the DNS behaviour inFigures 5 and 6. As normalized dilatation assumes a higher positive/negative value, theHEE distributions shrink and clearly show the tendency to get concentrated near the cuspof surface S1a and S1b. At moderate/high negative dilatations, HEE distributions show apronounced tendency to concentrate near the origin (Figures 6d and 6f). This is consistent

Table 10. Percentage of various flow topologies conditionedupon zero dilatation from HEE computations.

UFC UN/S/S SN/S/S SFS

44.3 13.2 4.2 38.2

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Table 11. Percentage of various flow topologies conditioned upon different positive dilatation levelsfrom HEE computations.

aii UFC UN/S/S SN/S/S SFS UFS UN/UN/UN USN/USN/USN

0.3 28.2 17.1 1.8 18.5 32.6 1.8 0.00.7 17.5 16.4 0.2 2.3 61.0 2.6 0.00.9 13.2 13.0 0.0 0.2 69.8 3.8 0.01.4 0.0 0.0 0.0 0.0 86.3 13.6 0.01.7 0.0 0.0 0.0 0.0 0.0 0.0 100.0

with the behaviour observed in DNS at moderate/high negative dilatations (see Figure 6e).However, HEE clearly overestimates this behaviour.

Next, we compare the HEE computations against DNS results in terms of the probabilityof occurrence of various topologies. In Tables 10–12, we present the percentage of varioustopologies from HEE computations at different levels of dilatation. First we consider thetopologies at zero dilatation (Table 10). The percentage occurrence of SN/S/S and SFSpredicted by HEE are close to the corresponding percentage seen in case A DNS simulation(Table 4). The model also correctly predicts UFC and SFS as the most dominant topologies.However, it overestimates the probability of UFC.

In Table 11, we present the HEE results at positive dilations. A comparison againstthe corresponding DNS results (Table 5), clearly suggests the failure of HEE model tocapture the probabilities accurately. However, the model does capture some features. HEEcorrectly predicts UFS as the dominant topology at all moderate/high positive dilatations.At high dilatation (aii = 1.4), the model correctly predicts UN/UN/UN as the second mostdominant topology.

Table 12 includes the percentage of topologies at various negative dilatations obtainedwith HEE computations. We compare this table against the DNS data in Table 6. At allnegative dilatation levels, the failure of HEE to estimate the correct percentages is veryapparent. However, at high negative dilatation (aii = −1.4), the model correctly identifiesSN/SN/SN and SFC to be the only existing topologies. At extreme negative dilatation(aii = −√

3), HEE does not offer any statistics for comparison.Thus, we conclude that the HEE model does not recover the percentage occurrence of

topologies very accurately. However, it captures many of the qualitative trends seen in DNSdata. In general, the model performance at zero and positive dilatations is better than thatat negative dilatations. HEE computations show that particles at negative dilatations, ingeneral, have higher velocity gradient magnitudes. Thus, the viscous action (not includedin the HEE model) on such particles can be expected to be significant. Further, the

Table 12. Percentage of various flow topologies conditioned upon different negative dilatation levelsfrom HEE computations.

aii UFC UN/S/S SN/S/S SFS SFC SN/SN/SN SSN/SSN/SSN

−0.3 52.5 10.0 7.5 22.4 7.4 0.0 0.0−0.7 60.3 7.6 10.2 9.0 12.5 0.3 0.0−0.9 60.8 10.5 10.5 4.2 13.2 0.7 0.0−1.4 0.0 0.0 0.0 0.0 72.2 27.8 0.0−1.7 — — — — — — —

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22 REFERENCES

homentropic assumption of the model can also be a potential reason of inaccuracy becauseof the substantial irreversible action of viscosity on such particles. Thus, improvingthe HEE model performance – especially at negative dilatations – may require furtherenhancements such as modeling viscous effects and relaxing the homentropic assumption.

5. Conclusions

We study the dependence of velocity gradients and flow topology on dilatation in compress-ible decaying isotropic turbulence, using direct numerical simulation. Exact probabilitiesof occurrence of various flow patterns/local topologies are computed conditional upon thedilatational state of velocity gradient tensor. We find that the joint statistics of second andthird invariants are highly dependent on normalized dilatation. However, the dilatation-conditioned statistics is found to be fairly independent of Reynolds number and Machnumber.

Invariant statistics when conditioned upon zero dilatation are very similar to the be-havior seen in incompressible turbulence but completely changes as dilatation assumeshigher positive/negative values. While at zero dilatation/incompressible flows, topologieswith both stable and unstable solution trajectories are important; topologies with unstablesolution trajectories are favored at positive dilatations. On the other hand, at negative di-latations, topologies with stable solution trajectories are favored. At extreme normalizeddilatations (±√

3), unique topologies are seen. While at√

3, the only existing topology isunstable star node – unstable star node – unstable start node (isotropic radial expansion),stable star node – stable star node – stable star node topology (isotropic radial contraction)is the only topology seen at −√

3. At intermediate/high positive dilatations, the predom-inant focal topology is unstable focus stretching, which is essentially a stretched vortexundergoing radial expansion. On the other hand, at intermediate/high negative dilatations,stable focus compressing – an axially compressed and radially contracting vortex – is thedominant focal topology. These focal structures are very different from those seen at zerodilatation/incompressible turbulence, wherein (owing to the incompressibility constraint) astretched vortex always undergoes radial contraction (stable focus stretching) and a com-pressed vortex always undergoes radial expansion (unstable focus compressing). Anotherinteresting feature seen in the negative dilatation regime is the increasing significance ofstable node/saddle/saddle topology as normalized dilatation changes from 0 to −1. Thistrend seems to be associated with the enhanced tendency of compressible flows to formshocklets.

Additionally, we compare the performance of a recently developed velocity gradientmodel – Homogenized Euler equation (HEE) – against the invariant statistics seen inDNS. Unlike the restricted Euler model, HEE includes both isotropic and anisotropicpressure Hessian components and is shown to qualitatively capture many features seen incompressible DNS. However, the quantitative performance is not very accurate, especiallyin the negative dilatation regime.

AcknowledgementsThis work was supported by AFOSR (MURI) Grant No. FA9550-04-1-0425 (Program Manager: Dr.John Schmisseur).

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velocity gradient invariants and local flow-field topology in compressible turbulence

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