characterizing velocity fluctuations in partially resolved turbulence simulations
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Characterizing velocity fluctuations in partially resolved turbulence simulationsDasia A. Reyes, Jacob M. Cooper, and Sharath S. Girimaji Citation: Physics of Fluids (1994-present) 26, 085106 (2014); doi: 10.1063/1.4892080 View online: http://dx.doi.org/10.1063/1.4892080 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/26/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Reynolds number scaling of coherent vortex simulation and stochastic coherent adaptive large eddy simulation Phys. Fluids 25, 110823 (2013); 10.1063/1.4825260 Preferential concentration and settling of heavy particles in homogeneous turbulence Phys. Fluids 25, 013301 (2013); 10.1063/1.4774339 Analytical insights into the partially integrated transport modeling method for hybrid Reynolds averaged Navier-Stokes equations-large eddy simulations of turbulent flows Phys. Fluids 24, 085106 (2012); 10.1063/1.4745003 Local and nonlocal pressure Hessian effects in real and synthetic fluid turbulence Phys. Fluids 23, 095108 (2011); 10.1063/1.3638618 Multi-time multi-scale correlation functions in hydrodynamic turbulence Phys. Fluids 23, 085107 (2011); 10.1063/1.3623466
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PHYSICS OF FLUIDS 26, 085106 (2014)
Characterizing velocity fluctuations in partially resolvedturbulence simulations
Dasia A. Reyes, Jacob M. Cooper,a) and Sharath S. GirimajiDepartment of Aerospace Engineering, Texas A&M University, College Station,Texas 77843-3141, USA
(Received 30 January 2014; accepted 21 July 2014; published online 14 August 2014)
Many current turbulence model simulations follow the accuracy-on-demandparadigm to partially resolve the flow field for achieving optimal compromise be-tween accuracy and computational effort. In many such approaches, the sub-gridviscosity is modeled using kinetic energy and dissipation of unresolved scales. It is ofscientific value and practical utility to characterize the resolved velocity fluctuationsof such flow fields to: (i) establish if the computed fields exhibit the self-similarityand scaling properties consistent with physical turbulence and (ii) demonstrate ifthe flow field tends to the limit of fully-resolved turbulence in a prescribed andcontrolled manner. Toward this end, we begin by characterizing partially-resolvedflow computations as direct numerical simulations of non-Newtonian fluids. Thisparadigm permits the extension of the Kolmogorov and finite Reynolds number scal-ing to the computed fields. The statistical behavior of the intermediate and smallestcomputed scales is then postulated as a function of the degree of resolution. Thenit is demonstrated that the flow field in partially-averaged Navier-Stokes simula-tions behaves in accordance with the adapted Kolmogorov hypotheses in isotropicturbulence computations. Further, the statistics of sub-Kolmogorov fluctuations aredemonstrated to be self-similar over a range of resolutions. C© 2014 AIP PublishingLLC. [http://dx.doi.org/10.1063/1.4892080]
I. INTRODUCTION
Over the last decade, accuracy-on-demand variable-resolution (VR) turbulence simulationstrategies that partially resolve the flow field in accordance with complexity have been gainingpopularity among engineering practitioners. These methods are aimed toward achieving optimaluse of computational resources for a given flow computation by placing the cut-off length scaleat a functionally chosen location in the spectrum as dictated by flow complexity. The cut-off maybe varied in time or space affording added adaptability to the simulation on hand. The Reynoldsaveraged Navier-Stokes (RANS) method and large-eddy simulations (LES) can be regarded as thebook-ends of the VR approaches. More precisely, direct numerical simulations (DNS) constitutethe fine resolution asymptote of VR approaches. The cut-off in practical VR simulations is typi-cally much closer to the energy containing scales rather than dissipation scales as in standard LEScomputations.
The current VR approaches can be broadly classified into two categories: hybrid or zonalmethods and bridging methods. In hybrid methods, the RANS and LES equations are solved indifferent zones of the computational domain. The bridging method, on the other hand, employs thesame model in regions of different resolutions, and the variation in cut-off length scale is achievedby suitably modifying the closure coefficients in accordance with spectral cascade physics.1–4 In thebridging VR methods, the unresolved eddy viscosity is determined from unresolved kinetic energy(ku) and dissipation (εu) which are obtained by solving modeled transport equations.
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085106-2 Reyes, Cooper, and Girimaji Phys. Fluids 26, 085106 (2014)
Despite the clear engineering motivation in practical applications, analysts have approachedVR methods with trepidation citing a lack of mathematical rigor of the closure model development.Important efforts have been made to place the methodology on firmer theoretical foundations espe-cially in the context of bridging methods. The spectral (both wavenumber and frequency) partitioningproof of Schiestel3 and the physical space decomposition derivation of Girimaji2 have addressedsome of these concerns by highlighting the theoretical foundations of bridging VR methods such asPartially-averaged Navier Stokes (PANS) and Partially-integrated turbulence method (PITM). Thedevelopment of many of these bridging methods is based on consistency with the zeroth law of tur-bulence, which requires statistically uniform flux through the inertial scales.3–5 In addition, the fixedpoint analysis of Girimaji et al.1 demonstrated that the PANS bridging approach recovers the correctenergetics in homogeneous turbulence. Furthermore, the computational study of Lakshmipathy andGirimaji6 demonstrated that PANS a posteriori results are consistent with a priori length-scalecut-off and viscosity reduction specifications. Since its initial formulation, the PANS method hasbeen enhanced to include near-wall effects7 and added fidelity in regions of resolution variation toaddress commutation error.8 At this stage of development, the PANS approach has been reasonablywell tested in a variety of canonical and complex flows: e.g., flow past square cylinder,9, 10 circularcylinder,11–14 backward-facing step,15, 16 automotive flows,17 and landing-gear flow.18
With practical utility clearly established, much of the remaining challenges facing bridging VRmethods are theoretical in nature. One of the main questions pertains to the fidelity of the fluctuationfield. At the present time, most VR methods are evaluated only on the basis of comparing the firstfew moments of velocity fluctuations against those of DNS or experiments. Higher-order momentsof velocity or statistics of velocity-gradient field are rarely examined. For establishing a higher levelof compliance with turbulence physics, it is highly desirable to characterize higher-order moments,distribution functions, and multi-point statistics.
The overarching goal of the study is to develop a framework for characterizing the VR fluctuatingflow field as a function of resolution. Such description is not merely of intrinsic value, but it is alsoof practical importance. The characterization can be used to (i) establish if the computed VR fieldsexhibit the self-similarity and scaling laws consistent with physical turbulence and (ii) demonstrateif the VR flow field tends to the limit of fully-resolved turbulence in a prescribed and controlledmanner.
As a first step toward the stated goal, we invoke the paradigm that VR computations areeffectively DNS of non-Newtonian turbulence19–21 at lower Reynolds number. The reduction inthe Reynolds number is due to the fact that the effective viscosity in a VR simulation is muchhigher than the molecular viscosity value. Accordingly, a VR computation must exhibit the flowfeatures of a physical (DNS) field of a lower Reynolds number. The next step is to characterize thefluctuating flow field of the lower Reynolds number physical field or equivalent DNS. Kolmogorovhypotheses22, 23 represent the central tenets of turbulence theory for describing fluctuations in inertialand dissipative scales. However, full compliance with Kolmogorov hypotheses may require that theequivalent DNS be of a very high Reynolds number. As mentioned earlier, the effective Reynoldsnumber is likely to be too low to invoke the Kolmogorov hypotheses. Recently, important lowReynolds number corrections to the Kolmogorov scaling laws have been introduced based onexperimental observations and DNS data.32 The resulting semi-empirical scaling laws are ideallysuited for examining VR fluctuating field. Furthermore, it has been recently established that thesub-Kolmogorov scales of fluctuations exhibit a great degree of universality even at low Reynoldsnumbers.39 The statistics and probability distribution functions of sub-Kolmogorov fluctuations alsooffer a reasonable basis for characterizing the velocity gradient field of VR fluctuations.
The specific objectives of this paper are: (i) adapt Kolmogorov hypotheses to characterize thefluctuations of partially-resolved bridging VR flows, (ii) examine the behavior of PANS fluctuatingfields for conformity with the Kolmogorov hypotheses and low-Reynolds number semi-empiricalscaling, and (iii) examine the probability distribution of sub-Kolmogorov fluctuations in partially-resolved simulations.
The remaining sections are arranged as follows. In Sec. II we present the non-Newtonian VRparadigm. In Sec. III we present the details of the PANS VR closure equations and explain the testcase. In Sec. IV the non-Newtonian turbulence paradigm is used to adapt the Kolmogorov hypotheses
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085106-3 Reyes, Cooper, and Girimaji Phys. Fluids 26, 085106 (2014)
to VR simulations. Section V examines the PANS velocity field at different resolution levels and theconsistency with physical turbulence. Section VI concludes with a brief summary.
II. THE NON-NEWTONIAN TURBULENCE PARADIGM
The objective of partially-resolved flow simulations such as VR methods and LES is to providea reasonable facsimile of large scales of turbulent motion. Toward this end, the velocity and pressurefields are decomposed into two parts − large resolved and smaller unresolved scales. The length scaleseparating the resolved and unresolved scales is the so-called cut-off scale. Germano24 demonstratedthat the resolved flow equations are identical in form to the Navier-Stokes equation with the exceptionof an additional stress due to the unresolved motion. The added stress is the generalized secondmoment of the total velocity field. In the RANS context, the unresolved stress is the Reynolds stress.Much like in RANS method, in VR and LES approaches there are many levels at which closuremodeling of the unresolved or sub-grid stress can be achieved. Two of the most popular sub-gridstress closure approaches are grid-based and viscosity-based methods.
Grid-based closure paradigm: In grid-based closures such as the Smagorinsky model, thesub-grid viscosity is prescribed as a function of the cut-off length scale (�) and local strain rate (S):
νL E S ∝ |S|�2. (1)
The premise of this closure is that there is local equilibrium between production and dissipation atthe small scales:
Pu = εu . (2)
Such specification of eddy viscosity ensures that scales of motion smaller than the cut-off aredissipated by viscous action in the LES calculation and therefore the cut-off length-scale (�) istaken to be of the order of local grid size. An important feature of this closure is that it entails nodescription of the unresolved scales of motion.
Viscosity-based closure paradigm: In this approach sub-grid viscosity is first specified basedon some knowledge of unresolved flow field:
νu = νu(ku, εu), (3)
where ku and εu are the kinetic energy and dissipation of the unresolved scales. The cut-off lengthscale of the resolved flow calculation is not precisely predetermined, and it emerges from thecalculations as a consequence of the magnitude of the specified viscosity. Such a computationis much like DNS wherein the dissipative scales are determined by fluid viscosity and Reynoldsnumber. In such a case, it is important to ensure that the numerical resolution is commensurate withthe (sub-grid) viscosity as in DNS.
Analogy with non-Newtonian flow simulations: Both grid-based and viscosity-based closuremodel calculations may be considered non-Newtonian flow calculations. Computational economyin both cases is achieved by the fact that the equivalent Reynolds number of the calculation issubstantially smaller than the corresponding DNS as the effective viscosity in the former is largerthan that in the latter: νu � ν.
Muschinski19 proposed that the numerical integration of the LES equations is tantamount toa DNS computation of a hypothetical “LES fluid.” The LES equations are identical to the Navier-Stokes equations, except for the addition of a Reynolds stress term which is a function of the LESviscosity (νLES). Since LES viscosity is variable and dependent upon shear, the “LES fluid” caneffectively be considered non-Newtonian. However, the analogy between LES and non-Newtonianfluid turbulence has certain limitations. Yakhot and Wanderer25 point out that LES model closuresbased on local energetic equilibrium bear good semblance to Newtonian turbulence in the intervalr ≥ 10�, where r is any flow scale of interest. For r < 10�, the fidelity of the LES field tothe corresponding DNS is not entirely precise. On the other hand, the viscosity-based approachesthat do not invoke local equilibrium do not preclude intermittency of dissipation. In Navier-Stokesturbulence, the small-scale intermittency can be characterized by analyzing the local/instantaneousvariations in length-scale compared to the statistical Kolmogorov scale.26 It has been established in
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085106-4 Reyes, Cooper, and Girimaji Phys. Fluids 26, 085106 (2014)
recent literature26 that the so-called sub-Kolmogorov fluctuations are more universal in characterthan other statistical features. In fact, the sub-Kolmogorov fluctuations exhibit very similar statisticaldistributions in homogeneous and inhomogeneous flows.
Overall, viscosity-based bridging VR closures bear closer similarity to DNS of non-Newtonianfluid turbulence than grid-based methods. In this paper, we demonstrate that the PANS VR ap-proach, at various resolution levels, exhibits velocity-gradient moments and structure functions thatbehave according to Kolmogorov hypotheses (subject to finite Reynolds number corrections) anddemonstrates sub-Kolmogorov self-similarity.26
III. PANS CLOSURE EQUATIONS AND SIMULATIONS
In the PANS methodology, as in all bridging VR methods, the flow field is decomposed intoresolved and residual components as follows:
Vi = Ui + u′i , (4)
p = P + p′. (5)
The resolved velocity and pressure fields are Ui and P, respectively, and the residual velocity andpressure terms are u′
i and p′. The decomposition of the flow field is achieved with an arbitrary filter(〈 〉),24 such that
〈Vi 〉 = Ui ;⟨u′
i
⟩ �= 0, (6)
〈p〉 = P;⟨p′⟩ �= 0. (7)
When the filter is applied to the Navier-Stokes equations, the resolved velocity evolution equationtakes on the following form:
∂Ui
∂t+ U j
∂Ui
∂x j= −∂ P
∂xi+ ν
∂2Ui
∂x j∂x j− ∂τ (Vi , Vj )
∂x j− Fi , (8)
∇2 P = −∂Ui
∂x j
∂U j
∂xi+ ∂2τ (Vi , Vj )
∂xi∂x j, (9)
where ν is the kinematic velocity, and the sub-filter stress (τ (Vi , Vj )) is defined according toGermano24 as
τ (Vi , Vj ) = 〈Vi Vj 〉 − 〈Vi 〉〈Vj 〉. (10)
The commutation term (Fi) arises from a filter that does not commute with spatial and temporaldifferentiation. The closure of this term is not the focus of the present work and for brevity we donot include the details of its treatment. The reader is referred to Girimaji and Wallin8 and Wallinand Girimaji27 for further discussion on the closure modeling of the commutation terms. It sufficeshere to mention that this effect can also be modeled with a commutation viscosity that can reduce oraugment the sub-grid viscosity. Through the remainder of the work we will restrict our considerationto a commuting filter: Fi = 0. The treatment is valid even when Fi is non-zero.
The extent of the filter or cut-off is controlled by specifying by the ratios of unresolved-to-totalkinetic energy and dissipation2
fk = ku
k; fε = εu
ε, (11)
where k is the total kinetic energy and ε is the specific kinetic energy dissipation. For modeling thesub-filter stress, several options are available. In the context of PITM, differential stress models havebeen used. With PANS, algebraic stress models and non-linear constitutive relations have been usedin literature.38 The Boussinesq closure is the simplest two-equation constitutive relation appropriate
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085106-5 Reyes, Cooper, and Girimaji Phys. Fluids 26, 085106 (2014)
for examining the consequences of the non-Newtonian turbulence paradigm. Therefore, in this paperwe use the Boussinesq approximation for a partially-averaged field as presented in Girimaji:2
τ (Vi , Vj ) = −νu
(∂Ui
∂x j+ ∂U j
∂xi
)+ 2
3kuδi j ; where νu = Cμ
k2u
εu. (12)
Here, ku, εu, and νu are the unresolved kinetic energy, dissipation, and viscosity, respectively, andCμ (=0.09) is a k − ε modeling coefficient.28 The closure equations for ku and εu are derived froma parent RANS k − ε model. The derivation procedure most notably ensures the consistency withthe zeroth law of turbulence.3–5 The PANS ku-εu models are given by
∂ku
∂t+ U j
∂ku
∂x j= Pu − εu + ∂
∂x j
(νu
σku
∂ku
∂x j
), (13)
∂εu
∂t+ U j
∂εu
∂x j= Cε1
Puεu
ku− C∗
ε2ε2
u
ku+ ∂
∂x j
(νu
σεu
∂εu
∂x j
), (14)
where
C∗ε2 = Cε1 + fk
fε(Cε2 − Cε1); σku = f 2
k
fεσk ; σεu = f 2
k
fεσε. (15)
The degree of resolution is controlled with suitable specification of fk and fε . It can be shown that fkis related to the grid size as29
fk(x) ≥ 1√Cμ
(�
) 23
≈ 3
(�
) 23
. (16)
The inequality clearly indicates that it is permissible to use larger fk values than that in the RHS ofEq. (16) for a given grid size. Further details of the development of the PANS ku-εu model can befound in Girimaji2 and Girimaji et al.1
IV. KOLMOGOROV HYPOTHESES FOR PARTIALLY-RESOLVED FIELDS
Consider a turbulent flow field with a typical energy spectrum as shown in Fig. 1. Let U, L,and T = L/U be the characteristic large turbulence scales as given in Table I. The smallest scalesof motion – the Kolmogorov scales uη, η, and tη – can be determined from the dissipation rate andviscosity (also given in Table I). These scales will subsequently be referred to as the DNS or physicalturbulence scales.
FIG. 1. Full DNS spectrum with integral and Kolmogorov scales.
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085106-6 Reyes, Cooper, and Girimaji Phys. Fluids 26, 085106 (2014)
TABLE I. Physical (DNS) scales of turbulence.
Scale Length Time Velocity
Largest L = k3/2
εT = k/ε U = k1/2
Smallest η = (ν3/ε)1/4 tη = (ν/ε)1/2 uη = (νε)1/4
We now commence the characterization of resolved and unresolved spectra in a partially-resolved simulation. Although the development is presented in the context of PANS, the char-acterization is more general and valid for all simulation approaches that demarcate resolved andunresolved scales using fractions of unresolved kinetic energy (ku) and unresolved dissipation (εu).Now consider the spectrum of the resolved velocity field obtained from a PANS calculation. Theideal PANS resolved spectrum is shown schematically in Fig. 2. Clearly, this ideal spectrum may notbe achieved due to closure model limitations. The difference between the DNS and PANS resolvedspectrum is designated the PANS unresolved spectrum. Given that PANS does not yield a cleanspectral cut-off, there will be a certain degree of overlap between the PANS resolved and unresolvedspectra. We will now examine the implications of the Kolmogorov hypotheses on the resolved andunresolved PANS scales subject to the non-Newtonian turbulence paradigm.
First similarity hypothesis: The first Kolmogorov similarity hypothesis states that at sufficientlyhigh-Reynolds number, the smallest scales of motion are determined uniquely by the value of spectralcascade rate (ε) and viscosity (ν). Therefore, we formulate the first VR similarity hypothesis as: “Atsufficiently high Reynolds number, the smallest scales of resolved motion are determined uniquelyby the value of the PANS cascade rate (εu) and unresolved viscosity (νu).”
Second similarity hypothesis: The Kolmogorov second similarity hypothesis states that atsufficiently high Reynolds number, there is an inertial range of scales at which the statistics canbe uniquely described by ε independent of ν.30 We state the second similarity hypothesis for VRmethods as follows: “At sufficiently high Reynolds number, there is an inertial range of scales atwhich the statistics can be uniquely described by εu independent of νu.”
It should be reiterated here that Kolmogorov hypotheses are valid only at sufficiently highReynolds numbers. By their very design, VR simulations are unlikely to achieve such high Reynoldsnumbers. In recent years, semi-analytical corrections to Kolmogorov hypotheses have been devel-oped from experimental and numerical data. The non-Newtonian paradigm implies that the VRfluctuations must be consistent with the finite Reynolds number (FRN) scalings as well.
Now we will investigate the implications of the two Kolmogorov hypotheses on the partially-resolved flow fields. By definition, the resolved VR simulations share the same large scales as a DNSsimulation. The PANS resolved Kolmogorov scales are taken to be the smallest scales of the PANSresolved spectrum and are characterized by νu and εu. The resulting characteristic PANS scales for
FIG. 2. PANS resolved and unresolved spectrum.
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085106-7 Reyes, Cooper, and Girimaji Phys. Fluids 26, 085106 (2014)
TABLE II. PANS resolved scales of turbulence.
Scale Length Time Velocity
Largest L = k3/2
εT = k/ε U = k1/2
Smallest ηu ∼ (ν3u/εu )1/4 tηu ∼ (νu/εu )1/2 uηu ∼ (νuεu )1/4
the resolved fields are given in Table II. We will refer to ηu, tηu , and uηu prescribed in Table II as thePANS resolved Kolmogorov length, time, and velocity scales, respectively.
The relationship between the PANS and RANS turbulence viscosities is defined as
νu = Cμ
k2u
εu= Cμ
f 2k
fε
k2
ε= νR AN S
f 2k
fε. (17)
From the definitions of νu, fk, and fε , the PANS Kolmgorov scales can now be related to the physicalintegral scales:
ηu ∼ C3/4μ
f 3/2k
fε
k3/2
ε∼ C3/4
μ
f 3/2k
fεL , (18)
tηu ∼ C1/2μ
fk
fε
k
ε∼ C1/2
μ
fk
fεT, (19)
uηu ∼ C1/4μ fk
1/2k1/2 ∼ C1/4μ fk
1/2U. (20)
Thus, the ratio of PANS smallest to largest scales is given by
ηu
L∼ C3/4
μ
f 3/2k
fε;
tηu
T∼ C1/2
μ
fk
fε;
uηu
U∼ C1/2
μ f 1/2k . (21)
Equation (21) highlights the advantage of partial resolution of the flow field. When the cut-off is inthe inertial range fε can be taken to be unity as all dissipation will occur in the unresolved scales.Then, ηu/L will scale as f 3/2
k independent of Reynolds number. So long as the cut-off is not in thedissipative range, the computational effort depends only upon the chosen value of fk, irrespective ofthe Reynolds number.
Similarly, the ratio of the PANS resolved Kolmogorov scales and DNS Kolmogorov scales canbe estimated as
ηu
η∼ C3/4
μ
f 3/2k
fεRe3/4
L ;tηu
tη∼ C1/2
μ
fk
fεRe1/2
L ;uηu
uη
∼ C1/4μ f 1/2
k Re1/4L ; where ReL = k2
νε. (22)
When the cut-off is the dissipation range then we have from Eq. (22)
ηu
η∼ 1 and
ηu
L∼ Re−3/4
L . (23)
As can be seen from Eq. (23), as the cut-off approaches dissipation range, the computational effortwill be similar to that of DNS.
Turning now to the PANS unresolved spectrum, it is straightforward to assert the smallestunresolved PANS scales are ideally those of the DNS Kolmogorov scales. The largest unresolvedscales must be a function of unresolved kinetic energy and dissipation. From scaling argumentssimilar to those used for total integral length scales, we obtain the largest unresolved scales as
L ∼ k3/2u
εu∼ f 3/2
k
fε
k3/2
ε, (24)
T ∼ ku
εu∼ fk
fε
k
ε, (25)
U ∼ k1/2u ∼ f 1/2
k k1/2. (26)
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085106-8 Reyes, Cooper, and Girimaji Phys. Fluids 26, 085106 (2014)
TABLE III. PANS unresolved scales of turbulence.
Scale Length Time Velocity
Largest L ∼ k3/2u /εu T ∼ ku/εu U ∼ k1/2
u
Smallest η = (ν3/ε)1/4 tη = (ν/ε)1/2 uη = (νε)1/4
These scalings are summarized in Table III.Now we can quantify the extent of overlap between the resolved and unresolved spectrum.
We compute the ratio between the smallest resolved scales and the largest unresolved scales fromTables II and III:
ηu
L ∼ C3/4μ ∼ 0.164, (27)
uηu
U ∼ C1/2μ ∼ 0.3, (28)
tηu
T ∼ C1/4μ ∼ 0.548. (29)
It can be seen that the PANS resolved-field Kolmogorov scales are of the same order as the unresolved-field integral scales, but not identical. The overlap is about a decade or smaller. As PANS is aviscosity-based closure rather than a grid-based closure, this overlap is to be expected. For cleanspectral cut-offs, the overlap will be negligible.
In this paper, we examine if the PANS approach yields velocity fields that exhibit the self-similarity and scaling laws developed in this section. In future works, we will develop avenues forimproving the PANS closure based on the characterization derived here. Specifically, a techniquefor determining the appropriate fε for a given fk and effective Reynolds number using the scalingsgiven in Eqs. (24)–(29) is currently under development.
V. CHARACTERIZATION OF THE VR FLUCTUATING FIELD
The scaling expressions of various statistics with the degree of resolution (fk) developed inSec. IV can be used to determine the fidelity of VR closure models. Such an investigation is mostconveniently achieved in decaying isotropic turbulence. Accordingly, we will perform DNS andPANS simulations of decaying isotropic turbulence. At each level of resolution (parameterized byfk), the PANS velocity field will be examined for compliance with the different similarity hypothesesand empirical scaling laws.
A. PANS simulations
As mentioned in the Introduction, VR methods in general, and PANS in particular, are purportedfor practical turbulence simulations. Most such simulations are performed on robust, industrial CFDcodes with the cut-off in scales only marginally smaller than the dynamically active energy containingrange. If the cut-off is at the large inertial scales, the effective non-Newtonian turbulence will beof relatively low Reynolds numbers. Therefore, any theoretical developments must be verified withPANS simulations performed on industrial codes at low effective Reynolds numbers. Thus motivated,for PANS test calculations, we chose decaying isotropic turbulence simulations performed at smallto moderate Reynolds numbers: Reλ ≈ 35 − 70. The industrial CFD solver EDGE is used in thesimulations. EDGE is a compressible unstructured CFD solver. The governing equations are solvedwith a finite-volume cell-centered method. Spatial discretization is achieved with a central schemewith artificial viscous dissipation or a first- or second-order Roe flux difference splitting upwindscheme. Time integration is performed either explicitly with a global timestep or implicitly withsubiterations. The time-integration can also be achieved explicitly with a second-order Runge-Kutta.
Decaying isotropic turbulence simulations are performed using DNS (1283 and 2563) and PANS(643, 1283, and 2563) schemes. Various PANS model resolutions are used: fk = 0.3, 0.4, 0.5, and
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085106-9 Reyes, Cooper, and Girimaji Phys. Fluids 26, 085106 (2014)
100
101
10−3
10−2
10−1
100
κ/(2π)
E(κ
)
100
101
10−3
10−2
10−1
100
κ/(2π)
E(κ
)
(a) (b)
FIG. 3. Normalized energy spectrum vs normalized wave number: DNS (dots), fk = 0.3 (heavy dash), fk = 0.4 (light dash),fk = 0.5 (solid), fk = 0.7 (dashed-dotted), and −5/3 law (dashed black line).
0.7. According to the proposed characterization, PANS simulations can be considered DNS atprogressively smaller Reynolds numbers. The DNS and PANS results are then compared at differenteddy turn-over times ( ε
k = T > 1). The initialization of the DNS and PANS fields are discussed indetail in Girimaji and Wallin,8 and Wallin, Reyes, and Girimaji.31
B. Examination of PANS velocity field
The characteristics of the PANS fluctuations at all scales are now systematically investigated.
(1) To evaluate the large scales, PANS spectra are compared against that of DNS at two differenttimes of decay.
(2) The intermediate scales are examined using second and third order structure functions. TheFRN correction to the Kolmogorov hypotheses are used as appropriate.
(3) The dependence of PANS Kolmogorov scales on fk is examined for conformity with Eq. (22).(4) The sub-Kolmogorov fluctuations are examined by characterizing the probability distribution
function of the velocity gradient field.
Large-scale fluctuations: In Fig. 3 we plot the normalized energy spectrum against normalizedwavenumber at T = 3 and T = 5. The DNS spectra are compared against those of PANS for variousfk values. At low wavenumbers (low κ/2π ), which corresponds to the largest scales of motion, thereis a clear collapse of the various spectra. As expected, the lower the fk value the closer the predictedspectrum is to DNS. Also shown for comparison is the −5/3 slope. Clearly, the spectra do not exhibitan inertial range, indicative of a very low Reynolds number. Recall that our objective is indeed toexamine low Reynolds number behavior. Overall, the PANS large scales behave as anticipated.
Intermediate-scale fluctuations: The definition of a velocity structure function of order n is
Dxn (r, x, t) = ⟨(Ui (x + r, t) − Ui (x, t))2⟩n . (30)
In locally isotropic turbulence,30 the velocity structure function can be completely determined interms of the longitudinal correlation:
Dxn (r, x, t) = ⟨(Ui (x + r , t) − Ui (x, t))2
⟩n. (31)
According to the Kolmogorov second similarity hypothesis, in the inertial range the second orderstructure function (Dxx), assumes a self-similar form when normalized appropriately by ε and r:
Dxx (r, t) = C2(εr )2/3. (32)
It is important to mention here that there are no analytical FRN corrections to the Kolmogorovestimate of the second-order structure function. Rather, our goal is to demonstrate self-similar PANS
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085106-10 Reyes, Cooper, and Girimaji Phys. Fluids 26, 085106 (2014)
100
101
102
0
0.2
0.4
0.6
0.8
1
100
101
102
0
0.05
0.1
0.15
0.2
0.25
(a) (b)
FIG. 4. Normalized structure functions: DNS (dots), fk = 0.3 (heavy dash), fk = 0.4 (light dash), fk = 0.5 (solid), andfk = 0.7 (dashed-dotted).
statistical approach to DNS with improving resolution. In Fig. 4(a), the second order structurefunction of various PANS simulations are compared against DNS data. Most importantly, the PANSstructure function shapes appear self-similar and approach the DNS structure function as fk → 0.Moreover, the profiles collapse onto a single line at small and intermediate separations.
It has been established that the value of C2 is approximately two32, 33 at high Reynolds numbersin the presence of a distinct inertial range. The current investigations involve very low Reynoldsnumber simulations, and therefore the peak value is expected to be less than two. The computedvalue of C2 is comparable to the low Reynolds number experimental data of Antonia and Burattini32
and Lavoie et al.34
The Kolmogorov second similarity hypothesis postulated that the third order structure function(Dxxx) scales in the inertial range as
Dxxx (r, t) = −4
5rε. (33)
It should be recalled that this scaling is valid only for very high Reynolds number. Antonia andBurattini32 use the full Karman-Horwath equation including the unsteady and viscous terms toestimate the departure of the third-order structure function from the above scaling. More recentlyTchoufag et al.35 use the more complete Lin’s equation to obtain corrections that are valid over alarger range of Reynolds numbers. In Fig. 4(b) the third order structure functions are plotted forPANS (fk values of 0.3, 0.4, 0.5, and 0.7) and DNS at a given time.
The third order PANS structure functions are nearly identical to those of DNS for small andintermediate separations. The smaller the fk value, the larger the overlap with DNS, both for secondand third order structure functions. This is to be expected as lower fk simulations correspond to highereffective Reynolds numbers. To complete the assessment, we next compare the PANS peak valueof the third order structure function against experimental data at corresponding Reynolds numbers.The peak value of the third order structure function is defined as Au = max(Dxxx(r, t)/(εr)).
In Fig. 5, Au from DNS and various PANS simulations are plotted versus effective TaylorReynolds number (Reλ). The solid line represents the semi-analytical curve fit for Au presented inAntonia and Burattini,32 and the open symbols represent experimental measurements from Lavoieet al.34 The DNS and PANS simulations agree well with experiments and the empirical fit at thecorresponding Reynolds number. Only the lowest Reynolds number case shows sizable deviation.The deviation of the fk = 0.5 case is possibly due to the strong dependence on initial conditions atthese low Reynolds numbers. To confirm this, a DNS of a corresponding initial state is performedat that low Reynolds number and that result is also displayed in Fig. 5. The PANS and DNS valuesagree at the indicated Reynolds number leading to further verification of the analogy.
Kolmogorov scales: Adapting the Kolmogorov first similarity hypothesis to partially-resolvedcomputations, we developed the relationship (Eq. (22)) between the DNS and PANS Kolmogorov
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085106-11 Reyes, Cooper, and Girimaji Phys. Fluids 26, 085106 (2014)
FIG. 5. Variation of Au against Rλ: Experiments by Lavoie et al.,34 and semi-empirical fit by Antonia and Burattini.32
scales as a function of fk. In Fig. 6, the variation of ηu with fk is presented at time T = 5 for fk = 0.3,0.4, 0.5, and 0.7. Equation (22) dictates the ratio of ηu should scale as f 3/2
k and this slope is depictedin Fig. 6 by the dashed line. The DNS value of η is shown with the solid line for reference. Thevariation in ηu with fk follows the prescribed slope of f 3/2
k to a good degree even at the relativelylow Reynolds numbers.
As the Kolmogorov velocity and time scales represent different moments of the small-scalefield, we examine their scaling as well. We present the ratios of PANS to DNS Kolmogorov timeand velocity scales in Figs. 7 and 8. The values of tηu and uηu scale reasonably well according toEq. (22) as fk and f 1/2
k , respectively.Sub-Kolmogorov scales: Very recently, it has been established26, 36 that sub-Kolmogorov scales
exhibit a greater degree of universality than Kolmogorov scaling laws. Sub-Kolmogorov scales havebeen shown to be self-similar over a range of inhomogeneous flows and even at low Reynoldsnumbers. While there are many statistics that characterize sub-Kolmogorov fluctuations, here we
0.4 0.5 0.7 0.9
10−2
10−1
η u
fk
FIG. 6. ηu Variation with fk (black dots): f 3/2k Scaling (Eq. (22)) for PANS shown by dashed line along with DNS value of
η shown by solid line.
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085106-12 Reyes, Cooper, and Girimaji Phys. Fluids 26, 085106 (2014)
0.4 0.5 0.7 0.910
−3
10−2
10−1
t η u
fk
FIG. 7. tηu Variation with fk (black dots):fk Scaling (Eq. (22)) is shown by dashed line along with DNS value of η shown bysolid line.
will examine the probability distribution function (pdf) of the normalized velocity gradients. Weseek to investigate if the pdf is self-similar at different resolutions of VR simulations and comparethat with the pdf obtained from DNS.
Invoking the small-scale isotropy assumption, the dissipation rate of the resolved scales can bewritten as
εu = 〈ε〉 = 15νu
⟨(∂U
∂x
)2⟩
, (34)
where 〈 〉 represents ensemble averaging. Thus, the statistics of ∂u∂x are of much importance. We
investigate the distribution of the normalized velocity gradient statistics:(
∂u∂x
)n/(
∂u∂x
)2n2
. In Fig. 9 thepdf of the normalized velocity gradients are plotted for powers n = 1 and n = 2 at T = 2.5 for DNSand PANS. The PANS pdf’s of different resolutions collapse on the DNS distribution function over awide range of values. Indeed, the DNS and PANS distribution functions are nearly indistinguishable.The excellent agreement between DNS and PANS reveals similarity of the smallest scales of motion.Thus, it is evident that the PANS resolved small scales are similar in character to that of DNS.
In Lakshmipathy and Girimaji,37 it is shown that the PANS sub-Kolmogorov fluctuations areself-similar at different resolutions even in more complex flows such as flow past a circular cylinderand flow over a backward-facing step.
0.4 0.5 0.7 0.91
2
3
4
u η u
fk
FIG. 8. uηu Variation with fk (black dots): f 1/2k Scaling (Eq. (22)) for PANS shown by dashed line along with DNS value of
η shown by solid line.
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085106-13 Reyes, Cooper, and Girimaji Phys. Fluids 26, 085106 (2014)
FIG. 9. PDF of(
∂u∂x
)n/(
∂u∂x
)2n2
. DNS (dots), PANS simulations (lines) are same as those in Fig. 3.
VI. CONCLUSIONS
Partially-resolved turbulence simulations such as PANS and PITM are derived from the zerothlaw of turbulence3–5 and hence can be justifiably considered DNS of non-Newtonian turbulence.Such a paradigm permits the adaptation of the Kolmogorov hypotheses to characterize the simulatedresolved and modeled unresolved fluctuating flow fields. In these methods, the unresolved eddyviscosity is calculated using unresolved kinetic energy and dissipation which are obtained fromsolving modeled evolution equations. The objective of this work is to clearly demonstrate that thePANS subgrid closure model produces the requisite degree of viscosity reduction and yields flowfield fluctuations that behave according to well-known turbulence scaling laws.
We first investigate the implications of Kolmogorov hypotheses to partially-resolved flow fields.Computational Kolmogorov scales are estimated using unresolved viscosity and unresolved dissi-pation. Then the first and second similarity hypotheses are adapted to partially-resolved flow fields.In the second part of the work, decaying isotropic turbulence simulations using PANS equationsof different resolutions are performed. One- and two-point PANS fluctuating field statistics areinvestigated for compliance with the Kolmogorov hypotheses and their finite Reynolds number sur-rogates. Sub-Kolmogorov fluctuations are examined for self-similarity over a range of resolutions.The results can be summarized as follows:
(1) The DNS and PANS spectra are quite similar at the largest scales of fluctuations.(2) Second and third order two-point correlations calculated from PANS simulations of different
resolutions scale according to DNS statistics.(3) The PANS Kolmogorov scales depend upon resolution (fk) as dictated by the Kolmogorov first
similarity hypothesis.(4) The probability distribution functions of the PANS sub-Kolmogorov fluctuations are self-
similar at different resolutions when normalized by the corresponding Kolmogorov lengthscale. More importantly, the self-similar shape is identical to that of DNS. The self-similarityis evident even in circular cylinder and backward-facing step flows.37
Overall, this work develops a framework – using Kolmogorov hypotheses, their finiteReynolds number modifications, and sub-Kolmogorov scale self-similarity – for characterizingpartially-resolved turbulent flow field characteristics. This characterization is of intrinsic
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085106-14 Reyes, Cooper, and Girimaji Phys. Fluids 26, 085106 (2014)
scientific value and of practical importance for assessing the physical fidelity of VR closure modelingapproaches.
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