velocity-gradient dynamics in compressible turbulence: influence of mach number and dilatation rate

This article was downloaded by: [Chinese University of Hong Kong]On: 21 December 2014, At: 12:34Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Journal of TurbulencePublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tjot20

Velocity-gradient dynamics incompressible turbulence: influence ofMach number and dilatation rateSawan Suman a & Sharath S. Girimaji aa Department of Aerospace Engineering , Texas A&M University ,College Station , TX , 77843-3141 , USAPublished online: 23 Mar 2012.

To cite this article: Sawan Suman & Sharath S. Girimaji (2012) Velocity-gradient dynamics incompressible turbulence: influence of Mach number and dilatation rate, Journal of Turbulence, 13,N8, DOI: 10.1080/14685248.2011.649850

To link to this article: http://dx.doi.org/10.1080/14685248.2011.649850

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to orarising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/loi/tjot20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/14685248.2011.649850

http://dx.doi.org/10.1080/14685248.2011.649850

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

Journal of TurbulenceVol. 13, No. 8, 2012, 1–23

Velocity-gradient dynamics in compressible turbulence: influenceof Mach number and dilatation rate

Sawan Suman∗ and Sharath S. Girimaji

Department of Aerospace Engineering, Texas A&M University, College Station , TX,77843-3141, USA

(Received 20 July 2011; final version received 5 December 2011)

The onset of compressibility effects brings about profound changes in the nature ofpressure and its consequent influence on velocity-gradient evolution in turbulent flows.In this work we examine the changing action of pressure on velocity-gradient evolutionwith varying levels of compressibility in decaying isotropic turbulence. The degree ofdeparture from incompressibility is characterized in one of the following three ways:(1) local turbulent Mach number, which is indicative of local balance between inertialand pressure effects; (2) local relative dilatation, which is a measure of the level ofcompression/expansion rate of a fluid element and (3) global turbulent Mach number,which is a global measure of compressibility of a homogeneous turbulent flow field. It isfound that the pressure-Hessian inhibits velocity-gradient steepening at incompressibleand subsonic local Mach numbers. In contrast, at higher local Mach numbers, thepressure-Hessian becomes the dominant driver of gradient steepening. Expanding fluidelements (positive dilatation) are generally associated with gentler overall gradientsthan contracting (negative dilatation) elements. With increasing local turbulent Machnumber, the disparity increases as the gradients of contracting elements become steeper.Overall, this work provides important insight into the velocity-gradient dynamics incompressible flows.

Keywords: compressible turbulence; velocity-gradients; pressure-Hessian; direct nu-merical simulation; dilatation

1. Introduction

Pressure plays a preeminent role in velocity-gradient evolution in turbulent flows. Asvelocity-gradient dynamics is central to many important turbulence processes, such as en-ergy cascade, scalar mixing, intermittency and material-element deformation, the effect ofpressure on velocity-gradient evolution is of great interest. This effect has been reasonablywell studied in incompressible flows [1–5]. In constant-density flows, pressure is a La-grange multiplier with the sole function of imposing divergence-free condition on velocitygradients. Thus, in incompressible flows, pressure is completely determined by velocityfield. However, with the onset of compressibility, pressure changes in character becoming athermodynamic variable that is determined from energy and state equations. Pressure nowcouples thermodynamics and fluid dynamics. The change in the nature of pressure leads toa profound transformation in pressure effects on velocity-gradient evolution. In high-Mach

∗Corresponding author. Email: [email protected]

ISSN: 1468-5248 online onlyC© 2012 Taylor & Francis

http://dx.doi.org/10.1080/14685248.2011.649850http://www.tandfonline.com

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014

http://dx.doi.org/10.1080/14685248.2011.649850

http://www.tandfonline.com

2 S. Suman and S.S. Girimaji

number flows, two distinct types of pressure influences on velocity-gradient dynamics canbe identified: (1) direct/explicit compressibility effects – baroclinic and dilatation effectsthat are absent in incompressible flows and (2) indirect/implicit compressibility effects –pressure-Hessian effects that may be modified from the incompressible effects. The ob-jective of this work is to investigate how pressure affects velocity-gradient evolution anddynamics at different levels of compressibility. The study characterizes level of compress-ibility in three ways: (1) relative dilatation (dilatation normalized by the local magnitude ofvelocity-gradient tensor); (2) local turbulent Mach number and (3) global turbulent Machnumber.

Passot and Pouquet [6] performed direct numerical simulation (DNS) of 2D homoge-neous turbulence and investigated the cause and nature of baroclinic generation of vorticity.Kida and Orszag [7, 8] studied the detailed budget of enstrophy in 3D decaying isotropicturbulence at high subsonic turbulent Mach numbers and moderate Reynolds numbers. Theauthors closely examined the mechanism of enstrophy production, especially in the vicinityof shocklets. They found that across shocks, vorticity is generated by the baroclinic termand intensified by the compression term. Further, the authors studied the dependence of var-ious mechanisms on dilatation (trace of velocity-gradient tensor). Recently, Pirozzoli andGrasso [9] have revisited in detail the global statistics of enstrophy, strain-rate and dilata-tion budgets in decaying homogeneous compressible turbulence. Employing DNS resultsof different initial turbulent Mach numbers, they identified the effects of global turbulentMach number on the behavior of various production and destruction mechanisms of ve-locity gradients. The increasing importance of baroclinic and dilatational vortex-stretchingcontributions with increase in global turbulent Mach number is clearly demonstrated intheir work. Furthermore, the two-stage behavior (initial increase and then a monotonicdecay) of enstrophy and strain-rate is also shown.

Lee et al. [10] examined the vorticity budget of decaying isotropic turbulence interactingwith a weak shock employing DNS and also made comparisons with the results of LinearInteraction Analysis (LIA). Kevlahan [11] developed a new theory of propagation of weakshocks in turbulence and derived analytical expressions of vorticity jump across a shock.The author also quantified the ratio of vorticity production by shock curvature to productionby baroclinic effects. Mahesh et al. [12] also employed DNS and LIA to study vorticityjump across a normal shock with non-zero vorticity and entropy fluctuations upstream ofthe shock. Samtaney et al. [13] have performed a detailed study of vorticity and shockletstatistics in decaying turbulence.

Jaberi and Madnia [14] and Jaberi et al. [15] investigated the effects of heat of reaction onenstrophy budget in reacting homogeneous compressible flows. Like Kida and Orszag [7],they investigated vorticity budget statistics conditioned on dilatation and found that the heatrelease and initial compressibility increase the baroclinic production of vorticity. Vazquez-Semadeni et al. [16] have studied the mechanisms of vorticity in astrophysical flows,wherein compressibility is induced by cooling. In addition to the usual vorticity-enhancingmechanisms, they examined the alternative mechanisms such as the Coriolis force, localheating effects and magnetic field effects.

Following a different approach, Bikkani and Girimaji [17] studied the system of dy-namical equations governing the velocity-gradient dynamics in the infinite Mach numberlimit (pressure-released turbulence or the Burgers turbulence) and analytically derived theasymptotic states of the velocity-gradient tensor in this limit. Their computations reveal thatthe Burgers turbulence exhibits bimodal behavior: contracting fluid elements are associatedwith steep gradients, whereas expanding fluid elements are associated with much gentlergradients.

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014

Journal of Turbulence 3

In context of decaying compressible turbulence, some recent studies [18–20, ] havehighlighted the significance and utility of relative dilatation as a fundamental parameterin understanding compressibility effects on turbulence by demonstrating (1) various flowstatistics (strain-rate eigenvalue, vorticity alignment, velocity-gradient invariants, etc.) con-ditioned on relative dilatation are strongly dependent on relative dilatation itself, but arereasonably independent of global compressibility parameters like global turbulent Machnumber and (2) flow statistics conditioned upon zero relative dilatation are identical tothe statistics seen in incompressible turbulence. The second finding is in line with thatof Prizzoli and Grasso [9], who demonstrated that even at high turbulent Mach numbers,scatter plot of second and third invariants of the anisotropic (traceless) portion of thevelocity-gradient tensor was very similar to that observed in incompressible turbulence.Since relative dilatation is the ratio of dilatation to the magnitude of local velocity-gradienttensor, it is indeed a more accurate measure of compressibility than absolute dilatation,which has been used earlier by Kida and Orszag [7] and Jaberi and Madnia [14].

Findings of Suman and Girimaji [19, 20] and Lee et al. [18] provide a compellingmotivation to examine the influence of relative dilatation on various other compressibleturbulence processes of interest. In this work we focus on the mechanisms of gradientsteepening, which is intimately related to energy cascade, in compressible turbulent flows.In addition, this study addresses other factors that influence velocity-gradient dynamics:local turbulent Mach number and global turbulent Mach number.

Thus, the objective of this paper is to clearly isolate, understand and characterize the ef-fects of compressibility on various mechanisms in velocity-gradient dynamics. Specifically,we investigate how the inertial (self-straining and vortex-stretching) and pressure-related(pressure-Hessian and baroclinic) mechanisms are affected by three compressibility param-eters: (1) relative dilatation, (2) local turbulent Mach number and (c) global turbulent Machnumber. Such an investigation is of great utility as it is expected to provide valuable insightson the basis of which improved Langevin and other Lagrangian mechanistic models canbe developed. Along the lines of the restricted Euler model, Suman and Girimaji [21] haverecently proposed a Lagrangian model for compressible velocity gradients. The biggestchallenge encountered in the development of such models is the unclosed pressure-Hessianterm and its dependence on compressibility. For such a model, M and aii are known, andit is desirable to develop models for other unclosed effects in terms of these quantities.An improved understanding of the effects of aii and M on various mechanisms can bereadily leveraged to further improve the Lagrangian models describing the dynamics ofcompressible velocity gradients.

This paper is organized into five sections. In Section 2, we review the evolution equationof compressible velocity gradients and present the equation in a form appropriate for thisstudy. Section 3 details the approach and also describes the DNS simulations employed inthis work. In Section 4, we present results and discuss the influence of compressibility onvarious mechanisms influencing velocity gradients. We conclude this study in Section 5with a brief summary of major conclusions.

2. Velocity-gradient evolution equation

We start with the conservation equations of mass, momentum and energy for a viscouscalorically perfect compressible medium:

∂ρ

∂t+ Vk

∂ρ

∂xk

= −ρ∂Vk

∂xk

, (1)

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


∂Vi

∂t+ Vk

∂Vi

∂xk

= − 1

ρ

∂p

∂xi

+ ∂σik

ρ∂xk

, (2)

∂T

∂t+ Vk

∂T

∂xk

= −T (n − 1)∂Vi

∂xi

− n − 1

ρR

∂qk

∂xk

+ n − 1

ρR

∂

∂xj

(Viσji), (3)

where Vi , p, ρ and T represent velocity, pressure, density and temperature, and n denotesratio of specific heats. The symbols xi and t represent spatial coordinates and time. Notethat the following caloric equation of state has been used to express energy equation (3) interms of temperature:

e = cvT , (4)

where e and cv are specific internal energy and specific heat at constant volume. For aperfect gas, the three thermodynamic variables are related through the state equation:

p = ρRT, (5)

where R is gas constant. Viscous stress (σij ) and heat flux (qk) are expressed using thefollowing constitutive relationships:

σij = µ

(∂Vi

∂xj

+ ∂Vj

∂xi

)+ δijλ

∂Vk

∂xk

, qk = −κ∂T

∂xk

, (6)

where κ and µ denote thermal conductivity and dynamic viscosity. Bulk viscosity, λ, istaken to be equal to − 2

3µ [22].Taking gradient of Equation (2), we obtain the exact evolution equation of the velocity-

gradient tensor (Aij ):

dAij

dt= −AikAkj − Pij + �ij , (7)

where

Aij ≡ ∂Vi

∂xj

, (8)

Pij ≡ ∂

∂xj

(1

ρ

∂p

∂xi

)(9)

and

�ij ≡ ∂

∂xj

[1

ρ

(∂σik

∂xk

)]. (10)

The operator ddt

(= ∂∂t

+ Vk∂xk

) is the substantial derivative operator representing therate of change following a fluid element. The tensor Pij can be expressed in terms of itsconstituents: the symmetric pressure-Hessian tensor (Hij ), and the baroclinic tensor (Bij ):

Pij = Hij

ρ− Bij

ρ2, (11)

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


where

Hij ≡ ∂2p

∂xi∂xj

and Bij ≡ ∂ρ

∂xj

∂p

∂xi

. (12)

Multiplying Equation (7) by Aij , we obtain evolution equation of the magnitude of thevelocity-gradient tensor (A):

d(A2)

2dt= −AijAikAkj − AijPij + Aij�ij , (13)

or equivalently

d(A2)

2dt= −AijAikAkj − Aij

Hij

ρ+ Aij

Bij

ρ2+ Aij�ij , (14)

where

A2 ≡ AijAij . (15)

The first term on the right-hand side (RHS) (−AikAkjAij ) of Equation (13) representsthe inertial production mechanism. It is important to point out that this term can be eitherpositive or negative. The second term (−PijAij ) represents production resulting from theinteraction of pressure and velocity fields, and we refer to this term as the pressure-stretchingmechanism. This mechanism receives contributions from two effects: (1) action of sym-metric pressure-Hessian (Hij ), and (2) action of baroclinic tensor (Bij ) (see Equation (14)).The last term on the RHS of Equation (13) represents the action of viscosity on velocity-gradient evolution. In this work we focus on inertial- and pressure-related mechanismsonly.

The next step is to non-dimensionalize Equation (14). For the normalization timescale,we propose the local timescale of fluid motion: 1

A(A is defined in Equation (15)).

Normalization with the local timescale is of utility for understanding the local balancebetween various processes and closure modeling in the context of the Langevin, restrictedEuler [5, 23] and Homogenized Euler models [19, 21] for velocity-gradient evolution. Inthese approaches, we need closure models of individual mechanisms. Thus motivated, weconsider a locally normalized form of velocity-gradient growth rate, which can be derivedfrom Equation (14) by multiplying this equation by 1/A3:

1

2A2

d(AijAij )

dt′ = −aikakj aij − aij

Hij

A2ρ+ aij

Bij

A2ρ2+ aij

�ij

A2, (16)

where dt ′ in Equation (14) is normalized time:

dt ′ = dtA. (17)

The quantity aij appearing on the RHS of Equation (16) is the normalized form of Aij :

aij ≡ Aij

A. (18)

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


For a more insightful analysis, we further decompose (see for example [9]) aij intonormalized anisotropic strain-rate, rotation-rate and isotropic dilatation tensors:

aij = s∗ij + wij + app

δij

3, (19)

where

s∗ij = aij + aji

2− app

δij

3, wij = aij − aji

2, (20)

and s∗ij and wij are the normalized anisotropic strain- and rotation-rates of a fluid element,

and aii represents the normalized rate of change of volume of a fluid element. UsingEquation (19), we rewrite Equation (16) in terms of s∗

ij , wij and app:

1

A2

d(AijAij

)2dt

′′ ≡ − s∗kj s

∗ij s

∗ik −

(apps∗

iks∗ik + a3

pp

9

)

− s∗ijωkiωkj − appwijwij

3− Hij s

∗ij

ρA2

− Hmmapp

3ρA2+ Bijaij

ρ2A2+ �ijaij

A2. (21)

For the sake of algebraic brevity, we refer to various terms on the RHS of Equation (21)by symbols and names that are suggestive of the physics associated with these terms. Thecomplete nomenclature is listed in Table 1.

Various mechanisms can be easily classified into purely inertial, pressure- and viscous-related mechanisms (Table 2). Alternatively, these mechanisms can also be classified as“incompressible” and “compressible” mechanisms (see Table 3). “Incompressible” mecha-nisms are those that have an exact algebraic analogue in the budget of strictly incompressiblevelocity-gradient dynamics. In Section 4, we will examine the influence of compressibility

Table 1. Nomenclature and symbols for various mechanisms appearing in Equation (21).

Term Symbol Nomenclature

1. −s∗kj s

∗ij s

∗ik s

ss Solenoidal self-straining

2. −[apps∗iks

∗ik + a3

pp

9 ]d

ss Dilatational self-straining3. −s∗

ijωkiωkj svs Solenoidal vortex-stretching

4. − appwij wij

3 dvs Dilatational vortex-stretching

5. −Hij s∗ij

ρA2 sph Solenoidal pressure-Hessian

production

6. −Hmmapp

3ρA2 dph Dilatational pressure-Hessian

production7.

Bij aij

ρ2A2 bc Baroclinic production

8.�ij aij

A2 dvd Viscous destruction

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


Table 2. Inertial, pressure-related and viscous mechanisms.

Inertial mechanisms −s∗kj s

∗ij s

∗ik, −

(apps∗

iks∗ik + a3

pp

9

),

−s∗ijωkiωkj , − appwij wij

3

Pressure mechanisms −Hij s∗ij

ρA2 , −Hmmapp

3ρA2 ,Bij aij

ρ2A2

Viscous mechanism�ij aij

A2

on these “incompressible” mechanisms (implicit effect of compressibility). “Compressible”mechanisms are those that appear in the budget equation only if the flow is compressible.Non-zero dilatation and density gradient are responsible for the creation of these mecha-nisms. A study of the effect of compressibility on such mechanisms will help us identifythe explicit effect of compressibility on velocity-gradient dynamics.

3. Approach: parameterization of compressibility effects

We characterize compressibility in terms of three parameters each of which highlights adifferent aspect of high-speed flow: (1) relative dilatation; (2) local turbulent Mach numberand (3) global turbulent Mach number.

Relative dilatation is defined as

aii ≡ Aii

A. (22)

While a negative value of aii indicates decrease in volume (compression), positive aii im-plies increase in fluid volume (expansion). A value of zero implies a volume-conservingfluid element. The quantity aii is algebraically bounded between ±√

3. Note that a largemagnitude of Aii does not necessarily represent large compression/expansion of a fluidelement. The degree of expansion/contraction of a fluid element actually depends on theratio of dilatation (Aii) to the overall magnitude of the tensor (

√AmnAmn), and this ratio

is precisely represented by relative dilatation (aii). This measure of dilatation is more ap-propriate for investigating velocity-gradient evolution than the more traditional Helmholtzdecomposition of the velocity field into solenoidal and non-solenoidal parts. Firstly, sucha decomposition is not always possible in inhomogeneous flows and secondly, it does notinvolve length scale information critical for velocity-gradient dynamics.

The second compressibility parameter in our study is the local turbulent Mach number:

M ≡√

V ′′i V ′′

i

nRT, (23)

Table 3. “Incompressible” and “compressible” mechanisms (viscous mechanism not included).

“Incompressible” mechanisms −s∗kj s

∗ij s

∗ik, −s∗

ijωkiωkj , −Hij s∗ij

ρA2

“Compressible” mechanisms −(apps∗iks

∗ik + a3

pp

9 ),− appwij wij

3 ,Bij aij

ρ2A2 , −Hmmapp

3ρA2

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


where Vi and T are local velocity and temperature and (.′′) represents the Favre fluctuation.Local turbulent Mach number (M) represents the local balance between inertia and pressureforces and further governs the nature of pressure evolution.

The third compressibility parameter in our study is global turbulent Mach number,

MT =√

˜V ′′i V ′′

i

nRT̃. (24)

In context of homogeneous turbulence, MT is a statistical (volume-averaged) measure of thebalance between fluctuating inertial and mean pressure forces. The utility of MT as a usefulmodeling parameter has been earlier demonstrated by Zeman [24], Sarkar et al. [25,26] andRistorcelli [27]. In this work, we examine the statistical behavior of various contributingterms on the RHS of Equation (21) conditioned on M and aii .

3.1. DNS and conditional averages

We employ results from DNS of compressible decaying isotropic turbulence for this study.These DNS simulations have been performed by Kumar and Girimaji [28] using a GasKinetic Method (GKM)-based solver. GKM solvers have been demonstrated to be a viablemethod of simulating highly compressible flows by Xu et al. [29, 30] and have beensubsequently applied to simulate decaying compressible turbulence [18–20, 31–33]. GKMsolvers are based on the Boltzmann equation of one-point velocity distribution function anduse the Bhatnagar–Gross–Krook (BGK) model for collision operator. The thermodynamicquantities (p, ρ and T ) follow the perfect gas state equation. The solver employed for thepresent computations uses a finite volume framework integrated with weighted essentiallynon-oscillating (WENO) scheme for interpolating flow variables. All results discussed inthe paper have been obtained over a computational domain with 2563 grid points andperiodic boundary conditions in all three directions. Simulations start from a statisticallyhomogeneous, isotropic and divergence-free velocity field with the initial energy spectrumin the Fourier space given by

E(κ, 0) = V̂i V̂∗i

4πκ2= Aκ4e−Bκ2

, κ =√

κ2x + κ2

y + κ2z , (25)

where κ represents wave vector, and (.̂) represents the Fourier amplitude. Initially, onlya narrow band of wave numbers are energized: κ ∈ [1, 8]. Two important parametersdescribing decaying isotropic turbulence are the initial global turbulent Mach number(MT ) (Equation (24)) and Reynolds number (Reλ) based on the Taylor microscale (λ):

Reλ ≡√

20

3ενk, (26)

where k is the turbulent kinetic energy,

k =˜V

′′i V

′′i

2, (27)

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


Table 4. DNS simulation cases.

Parameter Case A Case B Case C

Reλ 55.6 55.6 55.6MT 1.2 0.7 0.4

and ε and ν represent kinetic energy dissipation and kinematic viscosity. The initial velocityfield is chosen to be purely solenoidal such that the ratio of dilatational to total kinetic energyof the domain (V dilatational

rms /Vrms, [25]) as well as the relative dilatation (aii) at each locationis zero. However, at later times these quantities evolve in accordance with the governingequations. In this study we employ three DNS cases. Description of these cases are includedin Table 4.

We examine various flow statistics of interest at 0.4 eddy turnover time. By this timeall transient effects have passed and all non-linear processes are in full effect. To study theinfluence of MT , flow fields from cases A and B (obtained at 0.4 eddy turnover time) aresimultaneously considered. Unless specified otherwise, results discussed in the next sectionare from case A simulation. Sample sizes corresponding to different bins in simulation casesA and B are shown in Tables 5 and 6.

4. Results and discussion

In Section 4.1, we present a brief discussion on the relationship between the global flowvariables (MT , 〈A2〉) and the local flow variables (M , aii and A). Subsequently in Section4.2, we discuss influence of compressibility on the normalized growth rate of velocitygradients. Note that 〈.〉 implies a globally averaged quantity.

4.1. Relationship between global and local variables in compressible turbulence

Figure 1 shows the temporal evolution of turbulent kinetic energy (k) and pseudo-dissipationεp(≡ 〈µ〉〈A2〉). These quantities are closely related to global turbulent Mach number (MT )

Table 5. Case A: sizes of conditioned sample sets. Each bin size is ±0.05 about the median valuesof M and aii .

M = 0.3 M = 0.6 M = 0.9 M = 1.2

aii = 0.7 3045 11,244 17,345 12,433aii = 0.5 11,545 35,261 40,255 28,313aii = 0.3 36,951 98,269 79,127 37,800aii = 0.1 78,351 145,546 94,282 35,600aii = 0.0 67,500 121,580 77,873 31,392aii = –0.1 47,887 89,244 62,582 26,428aii = –0.3 22,629 47,780 38,478 17,979aii = –0.5 10,573 26,101 24,844 12,175aii = –0.7 5100 14,232 16,321 8585

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


Table 6. Case B: sizes of conditioned sample sets. Each bin size is ±0.05 about the median valuesof M and aii .

M = 0.3 M = 0.6 M = 0.9 M = 1.2

aii = 0.7 1569 5320 3956 271aii = 0.5 7911 21,168 10,035 533aii = 0.3 48,878 78,339 23,610 981aii = 0.1 267,283 240,527 38,360 1188aii = 0.0 337,930 258,344 38,373 1233aii = –0.1 204,386 187,790 32,202 1114aii = –0.3 46,371 68,320 17,075 745aii = –0.5 10,599 23,940 8593 450aii = –0.7 2565 8339 3936 263

and⟨A2⟩, respectively:

k(t)

k(0)= M2

T (t)

M2T (0)

T (t)

T (0), and (28)

⟨A2⟩ = εp

〈µ〉 . (29)

Since at moderate global turbulent Mach numbers, the change in temperature is notas significant as that in kinetic energy itself, evolution of kinetic energy in decaying tur-bulence is a good representation of global turbulent Mach number. The quantity εp canbe considered as a good measure of global average of the velocity-gradient magnitude〈A〉. The overall trends in the three simulations in Figures 1(a) and (b) are the same.Pseudo-dissipation (or approximately 〈A〉) first undergoes steepening followed by a decay-ing stage. Turbulent kinetic energy (or approximately MT ) undergoes a monotonic decaywith time.

In order to investigate the effect of MT on the distribution of local turbulent Machnumber (M) in compressible turbulence, we present the probability density function (pdf)of M normalized by the instantaneous value of global turbulent Mach number MT (t) inFigure 2(a). Clearly, the pdf of M

MT (t) is not very sensitive to MT (0) (case A, B or C). InFigure 2(b), we present the pdf of relative dilatation (aii) from three different simulations.There is a significant difference between the three distributions. As global turbulent Machnumber increases, probability of occurrence of higher relative dilatation levels increases.In Figure 2(c), we show the pdf of A normalized by 〈A(t)〉. As in the case of local turbulentMach number, this pdf is nearly insensitive to global turbulent Mach number. Finally,we investigate the dependence of velocity-gradient magnitude on aii . In Figure 3, wepresent mean value of A conditioned upon relative dilatation (aii). Figure 3 demonstrates asignificant disparity that exists between the velocity-gradient magnitudes of expanding andcontracting fluid elements in simulations with high initial MT . Contracting fluid elementsare associated with steeper gradients as compared with expanding fluid elements. Thisobservation is in line with the behavior seen in highly compressible turbulence by Bikkaniand Girimaji [17]. In their analysis of Burgers velocity-gradient dynamics, they find thatinfinitely high Mach number turbulence shows a similar bimodal behavior: contracting fluidelements undergo gradient steepening, while expanding fluid elements undergo gradient

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

k(t)

/k(0

)

tεp(0)/k(0)

Case ACase BCase C

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

ε p(t

)/ε p(0

)

tεp(0)/k(0)

Case ACase BCase C

(b)

Figure 1. Evolution of (a) turbulent kinetic energy (k), and (b) pseudo-dissipation (εp) in varioussimulations.

smoothening. Figure 3 clearly demonstrates that the disparity between the magnitudes ofexpanding and contracting fluid elements increases in a simulation with a higher initialglobal turbulent Mach number.

4.2. Locally normalized statistics

In this subsection, we study the influence of compressibility on various inertial and pressuremechanisms (Equation (21)). In Section 4.2.1, we discuss the influence of local compress-ibility parameters (aii and M). Influence of global turbulent Mach number is discussed inSection 4.2.2.

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

M/MT

(t )

pdf

Case ACase BCase C

(a)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

10−4

10−2

100

aii

pdf

Case ACase BCase C

(b)

0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

A/<A(t )>

pdf

Case ACase BCase C

(c)

Figure 2. Pdf of (a) M

MT (t) , (b) aii and (c) A/〈A(t)〉 in various simulations at 0.4 eddy turnover time.

4.2.1. Influence of aii and M

In Figure 4(a) we present contours of mean values of all inertial mechanisms [sss(=

−s∗kj s

∗ij s

∗ik) + s

vs(= −s∗ijωkiωkj ) + d

ss(= −apps∗iks

∗ik − a3

pp

9 ) + dvs(= − appwij wij

3 )] condi-tioned on M and aii . Similarly, Figure 4(b) presents contours of conditional means of

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


−1.5 −1 −0.5 0 0.5 1 1.50.2

0.4

0.6

0.8

1

1.2

1.4

1.6

aii

<A

|aii>

/<A

(t)>

Case ACase BCase C

Figure 3. 〈A|aii〉 in various simulations at 0.4 eddy turnover time.

the sum of all pressure mechanisms [sph(= −Hij s

∗ij

ρA2 ) + dph(= −Hmmapp

3ρA2 ) + bc(= Bij aij

ρ2A2 )].Note that 〈.|M,aii〉 denotes a mean value jointly conditioned on M and aii . To clearlyidentify the inherent trends in these contour plots, we present conditional means at somerepresentative values of M and aii in Figure 5. Figure 5(a) clearly shows that the overallinertial mechanism in Equation (21) is fairly independent of local turbulent Mach number,but highly dependent on relative dilatation (aii). On the other hand, pressure-related mech-anisms show significant dependence on both local turbulent Mach number and relativedilatation (Figure 5(b)), especially in contracting and volume-preserving fluid elements(aii ≤ 0). Figure 5(b) is a clear indication that it is the changing action of pressure thatcauses the turbulence characteristic in different Mach regimes to differ. We examine theinfluence of dilatation and local turbulent Mach number on these mechanisms in furtherdetail in the following subsections.

4.2.1.1. Inertial mechanisms. We observe that as relative dilatation assumes higher neg-ative values (intense contraction), inertial mechanisms get more positive, indicating theintensifying gradient steepening action of inertia on these fluid elements. This tendencymonotonically reduces with the magnitude of aii and reverses at aii ≈ +0.2. For fast ex-panding fluid elements (aii > 0.2), the inertial mechanism is negative, indicating that therole of inertia is completely reversed leading to gradient smoothening (decay of velocitygradients).

We next examine contributions of the four individual inertial mechanisms in Fig-ure 6: solenoidal self-straining (s

ss = −s∗kj s

∗ij s

∗ik), solenoidal vortex-stretching (s

vs =−s∗

ijωkiωkj ), dilatational self-straining (dss = −[apps∗

iks∗ik + a3

pp

9 ]) and dilatational vortex-

stretching (dvs = − appwij wij

3 ). At low dilatation levels (−0.2 < aii < 0.2), dilatationalself-straining and dilatational vortex-stretching mechanisms play negligible role. This is ex-pected as these terms have explicit dependence on aii (Table 1). Thus, at low dilatation levels,solenoidal self-straining and solenoidal vortex-stretching mechanisms are the only signif-icant inertial mechanisms. Further comparing contributions of these two solenoidal mech-anisms, it is clear that at low dilatations, solenoidal self-straining is more dominant thansolenoidal vortex-stretching. However, as the level of dilatation increases (positive or nega-tive), dilatational self-straining (Figure 6(c)) clearly emerges as the dominant inertial mech-anism. In contracting fluid elements it causes gradient steepening (positive production),

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


Figure 4. Contours of conditional means of various mechanisms. (a) Case A: 〈(sss + s

vs + dss +

dvs)|M,aii〉; (b) case A: 〈(s

ph + dph + bc)|M,aii〉; (c) case B: 〈(s

ss + svs + d

ss + dvs)|M,aii〉;

and (d) case B: 〈(sph + d

ph + bc)|M,aii〉.

whereas in expanding fluid elements it induces gradient smoothening (negative production).Thus, we conclude that in compressible turbulence, the most effective inertial mechanismdepends strongly on relative dilatation. At low dilatation levels, solenoidal self-straining(s

ss) is the dominant mechanism, whereas at higher dilatation levels, dilatational self-straining (d

ss) is the most dominant mechanism of inertial production of velocity gradients.

4.2.1.2. Overall pressure mechanism at low turbulent Mach number. We refer to Figure5(b) to better understand the overall role of pressure mechanism [s

ph(= −Hij s∗ij

ρA2 ) + dph(=

Hmmapp


ρ2A2 )] in velocity-gradient evolution. The figure suggests that the pres-sure effects depend strongly on both local turbulent Mach number and dilatation. We firstsummarize the conditional averages at low subsonic Mach numbers (M < 0.6). At lowturbulent Mach numbers, the velocity field can be considered predominately solenoidal.

(1.) For expanding fluid elements (aii > 0), total pressure contribution is positive. Thus, theoverall action of pressure is to favor gradient steepening in expanding fluid elements.In Figure 5(a), we compare this behavior with the overall action of inertia at M = 0.3.In expanding fluid elements, the inertial mechanisms lead to gradient smoothening.

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

aii

Φsss

+Φ

vss+

Φssd

+Φ

vsdM = 0.3M = 0.6M = 0.9M = 1.2

M = 0.3M = 0.6M = 0.9

M = 0.3M = 0.6M = 0.9

M = 0.3M = 0.6M = 0.9

M = 1.2

(a)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

aii

Φphs

+Φ

phd+

Φbc

(b)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

aii

Φsss

+Φ

vss+

Φssd

+Φ

vsd

(c)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

aii

Φphs

+Φ

phd+

Φbc

(d)

Figure 5. Mean values of various mechanisms jointly conditioned on M and aii . (a) Case A:〈(s

ss + svs + d

ss + dvs)|M,aii〉; (b) case A: 〈(s

ph + dph + bc)|M,aii〉, (c) Case B: 〈(s

ss +s

vs + dss + d

vs)|M,aii〉; and (d) Case B: 〈(sph + d

ph + bc)|M,aii〉.

Thus, clearly the action of pressure in expanding fluid elements at low turbulent Machnumber is to resist the action of inertia.

(2.) In contracting and volume-preserving fluid elements at low M , pressure-related con-tribution is negative. Again, this is exactly opposite to the action of inertia (see Figure5(a)). On the basis of these observations, we conclude that at low turbulent Machnumbers, the overall action of pressure is to resist the action of inertia on all fluidelements.

(3.) Further, at low turbulent Mach numbers, irrespective of the sign of dilatation, magni-tude of pressure contribution is smaller than that of inertia. Thus, despite the tendencyof pressure to have an action opposite to that of inertia, the overall direction of velocity-gradient evolution (steepening/smoothening) is still dictated by inertial mechanisms.

4.2.1.3. Overall pressure mechanism at high turbulent Mach number. We now considerthe flow behavior [s

ph(= −Hij s∗ij



ρ2A2 )] at high local tur-bulent Mach numbers (M ≥ 0.6). At such high Mach numbers, the velocity field consistsof more dilatational fluctuations.

(1.) For expanding fluid elements, the influence of pressure at high Mach numbers remainslargely unchanged from low Mach numbers (Figure 5(b)).

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

aii

Φsss

M = 0.3M = 0.6M = 0.9M = 1.2

M = 0.3M = 0.6M = 0.9M = 1.2

M = 0.3M = 0.6M = 0.9M = 1.2

M = 0.3M = 0.6M = 0.9M = 1.2

(a)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

aii

Φvss

(b)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

aii

Φssd

(c)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

aii

Φvsd

(d)

Figure 6. Mean values of various inertial mechanisms jointly conditioned on M and aii . (a)Solenoidal self-stretching 〈s

ss |M,aii〉, (b) solenoidal vortex-stretching 〈svs |M,aii〉, (c) dilatational

self-straining 〈dss |M,aii〉 and (d) dilatational vortex-stretching 〈d

vs |M,aii〉.

(2.) For contracting fluid elements, increase in Mach number significantly changes the roleof pressure. Note that for contracting fluid elements at high Mach numbers (M = 1.2),pressure contribution in Figure 5(b) is positive and larger than the correspondinginertial contribution in Figure 5(a). Thus, the role of pressure transforms from beingresistive to inertial action at low turbulent Mach numbers (discussed earlier) to beingthe primary driver of intense gradient steepening at high Mach numbers. Furthermore,note that as the role of pressure transforms, there exists a regime – around M ≈ 0.7 –wherein the overall action of pressure is very insignificant as compared to the inertialaction (Figure 5(a)). Thus, in this regime, the process of gradient steepening is drivenalmost solely by inertial mechanisms (akin to the pressure-less Burgers dynamics).

In order to further understand the role of individual pressure mechanisms [sph(=

−Hij s∗ij

ρA2 ), dph(= −Hmmapp

3ρA2 ) and bc(= Bij aij

ρ2A2 )], we refer to Figures 7(a), (c), (e) and (g) alongwith Figures 8(a), (c), (e) and (g). The first set of figures presents contours of conditionalmeans, whereas the latter set identifies the inherent trends by plotting conditional means atsome representative M and aii values. Comparing Figure 5(b) with Figures 8(a), (c) and (e),it is clear that almost all the Mach number dependence of the overall pressure contributionin Figure 5(b) can be entirely attributed to the effects of the symmetric pressure-Hessian

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


Figure 7. Contours of conditional means of various mechanisms. Figures in the left column are fromcase A and those in the right column are from case B simulation. From top to bottom: 〈s

ph|M, aii〉,〈d

ph|M,aii〉, 〈bc|M,aii〉 and 〈(sph + d

ph)|M,aii〉

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

aii

Φphs

M = 0.3M = 0.6M = 0.9M = 1.2

M = 0.3M = 0.6M = 0.9M = 1.2

M = 0.3M = 0.6M = 0.9M = 1.2

M = 0.3M = 0.6M = 0.9

M = 0.3M = 0.6M = 0.9

M = 0.3M = 0.6M = 0.9

M = 0.3M = 0.6M = 0.9

M = 0.3M = 0.6M = 0.9

M = 1.2

(a)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

aii

Φphs

(b)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

aii

Φphd

(c)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

aii

Φphd

(d)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

aii

Φbc

(e)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

aii

Φbc

(f)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

aii

Φphs

+Φ

phd

(g)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

aii

φ phs+

φ phd

(h)

Figure 8. Mean values of various pressure-related mechanisms jointly conditioned upon M and aii .Figures in the left column are from case A and those in the right column are from case B simulation.From top to bottom: 〈s

ph|M,aii〉, 〈dph|M, aii〉, 〈bc|M,aii〉 and 〈(s

ph + dph)|M,aii〉.

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


tensor (sph and d

ph). The baroclinic production bc (Figure 8(e)) shows little dependenceon local turbulent Mach number. Thus, to further understand the changing role of pressurewith respect to local turbulent Mach number, we first focus on the pressure-Hessian-relatedmechanisms (s

ph and dph) only. The baroclinic production mechanism will be discussed

later.

4.2.1.4. Solenoidal pressure-Hessian, sph(= −Hij s

∗ij

ρA2 ). In Figure 8(a) we observe thatwith increasing M , s

ph tends to become more positive for both expanding and contracting

fluid elements. Note that −Hij s∗ij

ρA2 is algebraically the same term that would appear inthe budget of velocity gradients in strictly incompressible turbulence. Thus, s

ph is an“incompressible” mechanism. Any change in s

ph can be interpreted as an implicit effectof compressibility. Furthermore,

(1.) in expanding fluid elements, this tendency to attain higher positive values impliesthat s

ph resists the gradient smoothening action of inertia even more intensely ascompared to its behavior at low Mach numbers. Thus, the solenoidal pressure-Hessianproduction weakens the tendency of an expanding fluid element to undergo gradientsmoothening at high Mach numbers.

(2.) For contracting and volume-preserving fluid elements, the tendency of sph to attain

increasingly more positive values has a totally different implication. At negative dilata-tions, as Mach number increases, we observe that s

ph changes from being negativeto positive. Since the inertial action is insensitive to local turbulent Mach number (seeFigure 5(a)), this implies that the action of s

ph changes from being resistive of inertialaction at low Mach number to being supportive of inertial action at high Mach number.Figure 8(a) is clear evidence that local turbulent Mach number induces a huge impliciteffect on compressible velocity-gradient dynamics.

4.2.1.5. Dilatational pressure-Hessian, dph(= −Hmmapp

3ρA2 ). As in the case of the solenoidalpressure-Hessian production, M has profound effects on d

ph (Figure 8(c)) as well. Forexpanding fluid elements, an increase in M makes d

ph more negative. On the other hand,for contracting fluid elements, an increase in M makes d

ph more positive. ComparingFigure 8(c) with Figure 5(a), we can conclude that the general effect of an increase in M isto make this mechanism more effective in supporting the inertial action in both expandingand contracting fluid elements.

4.2.1.6. Baroclinic mechanism, bc(= Bij aij

ρ2A2 ). Conditional mean values of the barocliniccontribution (bc) to velocity-gradient growth is shown in Figure 8(e).

(1.) For expanding fluid elements, irrespective of M , baroclinic mechanism leads to gradi-ent steepening, especially at large positive values of aii . However, the contribution ofthis mechanism to the overall growth of velocity gradients at positive dilatations seemsto be minimal. Unlike highly contracted fluid elements (large negative aii), which tendto show shocklet-like behavior [20] and associated sharper density gradients, expand-ing fluid elements are expected to be associated with gentler gradients in density. Suchgentler gradients, in turn, imply low baroclinic production.

(2.) For contracting fluid elements, the baroclinic contribution is negative at all dilatationlevels, and it gets increasingly more significant at higher negative dilatations (intensecontractions). Note that at low subsonic Mach numbers, the total pressure-Hessian

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


contribution [sph(= −Hij s

∗ij


3ρA2 )] (see Figure 8(g)) works in tandemwith the baroclinic mechanism resisting the gradient-steepening action of inertia.However, as discussed before, with increasing M , the pressure-Hessian-related pro-duction undergoes a transformation and tends toward more positive values at largeMach numbers. Figures 8(e) and (g) clearly reveal the counteracting natures of thesymmetric pressure-Hessian and baroclinic mechanisms in contracting fluid elementsas Mach number changes. As M increases from low subsonic to high subsonic val-ues, the algebraic sum of the pressure-Hessian (s

ph + dph) and baroclinic (bc)

contributions changes from a high negative value to a high positive value. Followingthis trend, at some intermediate subsonic M (≈ 0.7), the pressure-Hessian and baro-clinic contributions completely cancel each other, leaving inertia as the sole driverof velocity-gradient growth. Thus, the reason behind the disappearance of the overallpressure-related production in contracting fluid elements at a subsonic M (discussedearlier, Figure 5(b)) is actually caused by the mutually opposing actions of the sym-metric pressure-Hessian and baroclinic mechanisms.

4.2.2. Influence of MT

In order to identify the influence of initial global turbulent Mach number (MT ) on theinertial and pressure-related mechanisms, we perform comparisons between case A andcase B simulations. These two simulations differ only in terms of the initial global turbulentMach number. In Figures 4(c) and (d) we present contour plots of conditional means ofoverall inertial and pressure mechanisms from case B simulation. Case A and case Bcontour plots do appear very similar. We further compare the two simulation cases inFigure 5, wherein the conditional means are presented at some chosen M and aii values.Note that the means conditioned upon M = 1.2 have not been included in Figures 5(c)and (d) because of the extremely low sample size at this Mach number (see Table 6). Theoverall trends seen in case B are the same as in case A. At low subsonic Mach numbers,irrespective of the level of dilatation, there is little difference between the two simulationresults. Thus, global turbulent Mach number does not seem to have any significant effecton pressure mechanism at low subsonic Mach numbers. At higher local turbulent Machnumbers, for expanding fluid elements, again the effect of MT is minimal. However, forcontracting fluid elements at a high local turbulent Mach number (M = 0.9), the overallpressure-related production is somewhat more intense in case B simulation as comparedwith case A simulation. This trend is consistently seen in all individual components ofpressure-related mechanisms (Figures 8(b), (d), (f) and (h)). The corresponding contourplots from case A and case B can be compared in Figure 7. Notwithstanding the detailedquantitative differences, we summarize that the overall behavioral trends of inertial andpressure mechanisms are fairly independent of initial global turbulent Mach number – atleast over the range investigated in this work.

5. Summary

The complex phenomenon of turbulence is further exacerbated by the compressibilityeffects in high speed flows. Compressibility alters the dynamics of all scales of turbulentmotion but its effect on the velocity gradients is particularly profound. Pressure is themain agency of change as its role transforms from that of a mere Lagrange multiplierto preserve dilatation-free constraint on velocity gradients to that of a thermodynamic

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


variable governed by energy conservation equation. At high speeds, the unconstrainedvelocity gradients can develop significant amount of dilatation substantially altering thevelocity-gradient dynamics. Understanding and closure modeling of compressibility effectson velocity gradients are of much importance for a variety of engineering applications,including high-speed mixing and material element deformation. While statistical modelingof turbulence is of eventual interest, in this study we focus on characterizing the effect ofcompressibility on key physical mechanisms – inertia and pressure. Thus, we emphasize thedependence of these mechanisms on local flow conditions rather than global statistics. Sucha study can directly be of relevance to the Langevin and Lagrangian modeling of the process,which is the preferred choice of modeling for mixing and material element deformationproblems. Specifically, a clear identification of the dependence of the pressure-Hessianmechanism on local compressibility parameters (Mach number and dilatation) is of utmostinterest to improve the recently proposed models of compressible velocity gradients [19,21].

In this paper, we study the effects of compressibility on the interplay between inertial andpressure mechanisms in velocity-gradient dynamics employing direct numerical simula-tions of compressible decaying isotropic turbulence. It is found that pressure and inertial ef-fects can be influenced by several local flow conditions. To clearly categorize and character-ize the effects of compressibility on velocity-gradient dynamics, we parameterize the effectsin terms of both local compressibility parameters (local turbulent Mach number and localrelative dilatation) and global compressibility parameter (global turbulent Mach number).Furthermore, the relationship between these local and global compressibility parameters indecaying isotropic turbulence is also discussed in this work (Section 4.1). We study variousstatistics (jointly conditioned upon local turbulent Mach number and relative dilatation)of the inertial (self-straining and vortex-stretching) and pressure-related (pressure-Hessianand baroclinic) mechanisms instrumental in the evolution of velocity gradients.

The major findings of this study are now summarized:

(1.) Appropriately scaled inertial and pressure mechanisms are fairly independent of initialglobal turbulent Mach number, but dependent on relative dilatation and local turbu-lent Mach number. Thus, global turbulent Mach number, which plays a central rolein the Reynolds-averaged Navier–Stokes (RANS)-based modeling, does not seem tobe of much utility in context of the Langevin and Lagrangian mechanistic equationmodeling. According to this study, local turbulent Mach number along with relative di-latation seems to be the appropriate modeling parameters for the Langevin/Lagrangianmechanistic models.

(2.) Inertial mechanisms are independent of local turbulent Mach number as well, butstrongly dependent on relative dilatation level of a fluid element.

(3.) At low dilatation levels, solenoidal self-straining is the major inertial mechanism,whereas at higher dilatations, dilatational self-straining is the dominant inertial mech-anism. Irrespective of relative dilatation, vortex-stretching mechanism plays a minorrole. This implies that the influence of inertial production mechanisms is more signif-icant in strain-dominated regions than in vorticity-dominated regions.

(4.) Unlike inertial mechanisms, pressure mechanism is dependent on relative dilatationas well as local turbulent Mach number – especially for contracting and volume-preserving fluid elements. Thus, future modeling attempts to capture the physics ofthe unclosed pressure-Hessian term in the Langevin/Lagrangian mechanistic modelingmust be made appropriately sensitive to both local turbulent Mach number and relativedilatation. Specifically, following are the effects of local turbulent Mach number andrelative dilatation on pressure mechanism:

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


(i) At low local turbulent Mach numbers, pressure mechanism offsets the inertialproduction at all dilatations but with varying degree, which is dependent on relativedilatation.

(ii) As local turbulent Mach number increases, role of pressure drastically changesfrom being an inhibitor of gradient steepening to being the dominant driver ofgradient steepening.

(iii) Baroclinic production does not show any dependence on local turbulent Mach num-ber, and all the Mach number dependence of pressure mechanisms is attributableto the symmetric pressure-Hessian tensor (implicit effect of compressibility).

(iv) Baroclinic effects play a significant role in highly contracted regions of the flow.

AcknowledgementThis work was supported by NASA NRA Grant (Technical Monitor: Dennis Yoder)

References[1] K. Ohkitani, Eigenvalue problems in 3D Euler flows, Phys. Fluids A. 5 (1993), pp. 2570–2572.[2] K. Ohkitani and S. Kishiba, Non-local nature of vortex stretching in an inviscid fluid, Phys.

Fluids 7 (1995), pp. 411–421.[3] A. Ooi, J. Martin, J. Soria, and M.S. Chong, A study of the evolution and characteristics of the

invariants of the velocity-gradient tensor in isotropic turbulence, J. Fluid Mech. 381 (1999),pp. 141–174.

[4] C. Kalelkar, Statistics of pressure fluctuations in decaying isotropic turbulence, Phys. Rev. E.73 (2006), pp. 046301–1–10.

[5] L. Chevillard, C. Meneveau, L. Biferale, and F. Toschi, Modeling the pressure Hessian and vis-cous Laplacian in turbulence: Comparisons with direct numerical simulation and implicationson velocity gradient dynamics, Phys. Fluids 20 (2008), pp. 101504–1–101504–15.

[6] T. Passot and A. Pouquet, Numerical simulation of compressible homogenous flows in turbulentregime, J. Fluid Mech. 181 (1987), pp. 441–466.

[7] S. Kida and S.A. Orszag, Enstrophy budget in decaying compressible turbulence, J. Sci. Comput.5 (1990), pp. 1–34.

[8] S. Kida and S.A. Orszag, Energy and spectral dynamics in decaying compressible turbulence,J. Sci. Comput. 7 (1992), pp. 85–125.

[9] S. Pirozzoli and F. Grasso, Direct numerical simulations of isotropic compressible turbulence:Influence of compressibility on dynamics and structures, Phys. Fluids 16 (2004), pp. 4386–4407.

[10] S. Lee, S.K. Lele, and P. Moin, Direct numerical simulation of isotropic turbulence interactingwith a weak shock wave, J. Fluid Mech. 251 (1993), pp. 533–562.

[11] N.K.R. Kevlahan, The vorticity jump across a shock in a non-uniform flow, J. Fluid Mech. 341(1997), pp. 371–384.

[12] K. Mahesh, S. Lele, and P. Moin, The influence of entropy fluctuations on the interaction ofturbulence with a shock wave, J. Fluid Mech. 334 (1997), pp. 353–379.

[13] R. Samtaney, D.I. Pullin, and B. Kosovic, Direct numerical simulation of decaying compressibleturbulence and shocklet statistics, Phys. Fluids 13 (2001), pp. 1415–1430.

[14] F.A. Jaberi and C.K. Madnia, Effects of heat of reaction on homogenous compressible turbulence,J. Sci. Comput. 13 (1998), pp. 201–228.

[15] F.A. Jaberi, D. Livescu, and C.K. Madnia, Characteristics of chemically reacting compressibleturbulence, Phys. Fluids 12 (2000), pp. 1189–1209.

[16] E. Vazquez-Semadeni, T. Passot, and A. Pouquet, Influence of cooling-induced compressibilityon the structure of turbulent flows and gravitational collapse, Astrophys. J. 473 (1996), pp.881–893.

[17] R. Bikkani and S.S. Girimaji, Role of pressure in non-linear velocity gradient dynamics inturbulence, Phys. Rev. E 75 (2007), pp. 036307–1–8.

[18] K. Lee, S.S. Girimaji, and J. Kerimo, Effect of compressibility on velcoity gradients and small-scale structure, J. Turbul. 10 (2009), pp. 1–18.

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014


[19] S. Suman and S.S. Girimaji, Homogenized Euler equation: A model for compressible velocitygradient dynamics, J. Fluid Mech. 620 (2009), pp. 177–194.

[20] S. Suman and S.S. Girimaji, Velocity gradient invariants and local flow-field topology in com-pressible turbulence, J. Turbul. 11 (2010), pp. 1–24.

[21] S. Suman and S.S. Girimaji, Dynamical model for velocity-gradient evolution in compressibleturbulence, J. Fluid Mech. 683 (2011), pp. 289–319.

[22] F. White, Viscous Fluid Flow, McGraw Hill, Boston, MA, 2006.[23] P. Vieillefosse, Local interaction between vorticity and shear in a perfect incompressible fluid,

J. Phys. (Paris) 43 (1982), pp. 837–842.[24] O. Zeman, Dilatation dissipation: The concept and application in modeling compressible mix-

ing, Phys. Fluids 7 (1990), pp. 178–188.[25] S. Sarkar, G. Erlebacher, M.Y. Hussaini, and H.O. Kreiss, The analysis and modelling of

dilatational terms in compressible turbulence, J. Fluid Mech. 227 (1991), pp. 473–493.[26] S. Sarkar, G. Erlebacher, and M.Y. Hussaini, Direct simulation of compressible turbulence in

shear flow, Theor. Comput. Fluid Dyn. 2 (1991), pp. 291–305.[27] J.R. Ristorcelli, A pseudo-sound constitutive relationship for the dilatational covariances in

compressible turbulence, J. Fluid Mech. 347 (1997b), pp. 37–70.[28] G. Kumar and S.S. Girimaji, WENO enhanced gas-kinetic scheme for direct simulations of

compressible transition and turbulence, under review in Journal of Computational Physics,(2011).

[29] K. Xu, Gas kinetic schemes for unsteady compressible flow simulation, Lecture series, 29thComputational Fluid Dynamics, von Karman Institute for Fluid Dynamics, Belgium, 1988(February).

[30] K. Xu, A gas kinetic scheme for the Navier–Stokes equation and its connection with artificialdissipation and Godunov method, J. Comput. Phys. 171 (2001), pp. 289–335.

[31] J. Kerimo and S.S. Girimaji, Boltzmann–BGK approach to simulating weakly compressible3D turbulence: Comparison between lattice Boltzmann and gas kinetic methods, J. Turbul. 8(2007), pp. 1–16.

[32] W. Liao, Y. Peng, and L. Luo, Gas-kinetic schemes for direct numerical simulations of com-pressible homogenous turbulence, Phys. Rev. E 80 (2009), pp. 046702–1–26.

[33] W. Liao, Y. Peng, and L. Luo, Effects of multitemperature non-equilibrium on compressiblehomogenous turbulence, Phys. Rev. E 81 (2010), pp. 046704–1–19.

Dow

nloa

ded

by [

Chi

nese

Uni

vers

ity o

f H

ong

Kon

g] a

t 12:

34 2

1 D

ecem

ber

2014

velocity-gradient dynamics in compressible turbulence: influence of mach number and dilatation rate

Documents