Ravichandra Srinivasan1

Research Associate

Department of Aerospace Engineering,

Texas A&M University,

College Station, TX 77843

e-mail: rsrinivasan@tamu.edu

Sharath S. GirimajiProfessor

Department of Aerospace Engineering,

Texas A&M University,

College Station, TX 77843

e-mail: girimaji@tamu.edu

Partially-Averaged Navier–StokesSimulations of High-SpeedMixing EnvironmentAccurate simulation of the fuel-air mixing environment is crucial for high-fidelity scram-jet calculations. We compute the velocity fields of jet into supersonic freestream flow andcavity flow typical of scramjet flame-holding applications at different scale resolutionsusing the partially-averaged Navier–Stokes (PANS) method. We present a sequence ofvariable resolution computations to demonstrate the potential of PANS method for high-speed mixing environment calculations. [DOI: 10.1115/1.4026234]

1 Introduction

Computational fluid dynamics (CFD) is a critical tool for thedesign and development of propulsion devices in the hypersonicflow regime more than in any other Mach number range. This ismainly due to the fact that the hypersonic combustor operatingconditions cannot be easily replicated in any ground-based experi-mental facility. Premixed combustion of the type encountered inmost hypersonic propulsion devices are comprised of two impor-tant steps: turbulent mixing, followed by a chemical reaction. Inlow-Mach number flight regimes, the fuel-oxidizer mixing timescale is reasonably small compared to the mixture residencetime in the combustor. However, in the hypersonic regime, theresidence time scale is very small (in the order of a millisecond orsmaller) and consequently, mixing must be very rapid. The overallaccuracy of hypersonic turbulent combustion calculation depends,to a very large extent, on the ability to precisely compute theflow-field environment that leads to fuel-oxidizer mixing. Asuccessful hypersonic CFD tool must be capable of capturing thevarious flow processes that influence the turbulent mixing in thecombustor. Indeed some preliminary CFD calculations show thatsmall changes in the mixing models can yield vastly differentcombustor outcomes ranging from engine unstart to flame blow-out. It was also seen that the sensitivity to chemical kinetics inthese nonpremixed reacting flows is markedly lower. Therefore,while chemical kinetics is also very important, accurate modelingof the turbulent mixing flow-environment must be considered atop priority in hypersonic combustion CFD.

To precisely understand and model the various high-speedinfluences on mixing, it must be recognized that the mixing pro-cess itself is made of two distinct steps: large-scale stirring of thescalar field due to turbulent flow field, followed by molecularlevel mixing of fuel and oxidizer. In the combustor, molecularmixing is followed by a chemical reaction. Thus, large-scale stir-ring of fuel and oxidizer is the forerunner of molecular mixingand reaction. The role of the turbulent flow environment is toaccelerate molecular mixing by steepening the scalar gradientsand consequently increasing the rate of chemical conversion offuel and oxidizer to combustion products. The physics of molecu-lar mixing and chemical reaction do not change much with theflow regime, but stirring is strongly influenced by the turbulenceenvironment. In summary, the main challenge is to simulate thestirring action of the hypersonic turbulent flow field. Throughoutthe paper, the terms stirring and mixing will be used somewhatsynonymously. We will not consider chemical kinetics issues inthis paper and restrict ourselves to the task of developing a high-

fidelity tool for simulating mixing environments in high-speedpropulsion devices.

1.1 Challenges in Simulating Hypersonic TurbulentMixing. In a hypersonic propulsion device, the velocity fieldalong the flow-path going from the inlet to the isolator and onthrough to the combustor, is subject to various complicating influ-ences, such as compressibility (arising from the shock train ininlet), streamline curvature (in inlet), adverse pressure gradient,and separation (in isolator). The presence of secondary flows canalso profoundly change the mixing (or more accurately, stirring)process. Thus, compressibility and other effects further exacerbatethe already complex flow environment in hypersonic devices. Asis now well recognized, compressibility significantly reduces themixing efficiency of a turbulent field as quantified by the so-calledLangley curve. However, our ability to model this phenomenon isinadequate, due to the fact that the understanding of underlyingphysics is incomplete. An accurate hypersonic turbulent-mixingmodel suite must account for all important compressibility andhigh Mach number effects. Each of the physical effect calls forcomputation at a different level of resolution:

(1) lack of clear a priori definition of the resolved and unre-solved scales

(2) insufficient characterization of VR fluctuations, are theyreally turbulent?

(3) tacit neglect of noncommutating filter effects leading toerror in the handshake regions

While the first two issues are self-evident, the third pointdeserves more explanation. When the Navier–Stokes equation issubject to a spatially (or temporally) varying filter, the filteredequations contain terms that involve the rate of change of filter-width. These terms are seldom considered and their neglect can bejustified only when the rate of change of filter-width is small,compared to the change of unresolved field statistics. Thesecommutation terms are one of the most important reasons for theinaccuracy of the models in the handshake region between zonesof different resolution. The commutation term is ignored in mostVR approaches and will not be considered in this work as well.The commutation issue is addressed in two recent papers, Wallinand Girimaji [1] and Girimaji and Wallin [2]. The credibility, reli-ability, and predictive capability of a VR calculation depends onhow well the other two issues are addressed.

The goal of this work is to demonstrate the feasibility and fidel-ity of the PANS method for simulating high-speed mixing envi-ronments. As mentioned earlier in the paper, a reliable predictivecomputational design tool is crucial for the development of high-speed propulsion devices. The suitability of any hybrid/bridgingcomputational approach for this purpose cannot possibly be estab-lished directly in the complex flow environment of a high-speed

1Corresponding author.Contributed by the Fluids Engineering Division of ASME for publication in the

JOURNAL OF FLUIDS ENGINEERING. Manuscript received December 18, 2012; finalmanuscript received December 8, 2013; published online April 28, 2014. Assoc.Editor: Ye Zhou.

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combustor. Any meaningful evaluation process must involve sev-eral steps:

(1) Analytical demonstration that the modeling approach isbased on firm theoretical grounds imparting physical char-acteristics listed above – a priori demarcation of scales andphysical character of fluctuations.

(2) Validate and demonstrate that the model behaves as desiredin simple canonical flows of relevant geometry.

(3) Validate the model in multiple high-speed propulsionconfigurations.

The objectives of this paper are to demonstrate these steps forthe PANS model. We focus on the first two steps and present pre-liminary results from the third step. We first present the PANSbridging method and discuss its attributes in the context of theabove list (Sec. 2). In Sec. 3, we present verification results fromcanonical subsonic flows and proceed to show preliminary resultsin high-speed flow over cavity and jet injection into supersonicfreestream.

2 PANS Bridging Method: Model Equations

and Attributes

The partially-averaged Navier–Stokes method offers turbulenceclosure model for any modeled-to-resolved cut-off ranging fromReynolds-averaged Navier–Stokes (RANS) to Navier–Stokes(direct numerical simulations). The general objective of thePANS, like hybrid models, is to resolve large scale structures atreasonable computational expense. However, cutoff in any spec-tral range is possible. The PANS cutoff is specified in terms oftwo parameters [3], the unresolved-to-total ratios of kinetic energy(fk) and dissipation (fe). A given combination of these two param-eters uniquely specifies the cutoff length scales and the Reynoldsnumber of the unresolved scales. The unresolved (subfilter) stressis modeled with the Boussinesq approximation and transportequations are derived for the unresolved kinetic energy and dissi-pation as a function of the cutoff parameters. In this section, wepresent a brief discussion of the PANS philosophy followed by adescription of its attributes to address objective 1 mentioned inthe Introduction.

For the sake of simplicity the development of incompressibleflow PANS equations are presented and a similar development isvalid for compressible flows with heat release as well. Incompres-sible Navier–Stokes equations for the instantaneous velocity (V)and pressure (p) fields are

@Vi

@tþ Vj

@Vi

@xj¼ � @p

@xiþ � @2Vi

@xj@xj;

@2p

@xi@xi¼ � @Vi

@xj

@Vj

@xi(1)

Here V and P represent the instantaneous velocity and pressurefields. Their filtered counterparts are Ui ¼ hVii; pU ¼ hpi, whereh i denotes an arbitrary (implicit or explicit) filter, which is con-stantly preserving and commutes with spatial and temporal differ-entiation. The PANS model development [3] entails deriving theevolution equations for filtered velocity (U) and pressure (pU)fields. The evolution equations are [4]

@Ui

@tþ Uj

@Ui

@xjþ @sðVi;VjÞ

@xj¼ � @pU

@xiþ � @2Ui

@xj@xj

� @2pU

@xi@xi¼ @Ui

@xj

@Uj

@xiþ @

2sðVi;VjÞ@xj@xj

(2)

The subfilter stress sðVi;VjÞ is given the name, generalized centralsecond moment and defined as

sðA;BÞ ¼ hABi � hAihBi (3)

Following this definition, the subfilter kinetic energy and dissipa-tion can be identified as

Ku ¼1

2sðVi;ViÞ; eu ¼ �s

@Vi

@xj;@Vi

@xj

� �(4)

Throughout the paper, the subscript u indicate PANS statistics.It is a simple matter to show that when filtering is replaced by

averaging, it is over all scales of motion (denoted by over bar),and the filtered velocity becomes the mean velocity and the SFSstress reduces to the Reynolds stress,

Ui ¼ Vi (5)

andsðVi;ViÞ ¼ RðVi;VjÞ ¼¼ ViVj � ViVj (6)

Under the same averaging operator, the PANS kinetic energy anddissipation reduce to their RANS counterparts denoted by K ande. Filtering over all scales is tantamount to averaging and thisleads Eq. (2) and sðVi;VjÞ equations to reduce to the RANSequations.

The form of the filtered Navier–Stokes equation is invariant ofthe filter when the equations are expressed in terms of the general-ized central moments [4,5]. This averaging-invariance propertyof the Navier–Stokes equation forms the basis of PANS develop-ment. The averaging-invariance property dictates that Reynoldsstress and the subfilter scale (SFS) stress sðVi;VjÞ models have thesame functional form, even though their contents are different toreflect the degree of filtering. Thus, any RANS closure modelingtechnique can be extended for PANS application. By the sametoken, any LES closure modeling technique can also be modifiedfor PANS applications. Our decision to use RANS as the basis ofPANS is dictated by the fact that the PANS bridging model is pur-ported for use at all degrees of physical resolution, including theRANS limit. Furthermore, PANS is more likely to be used closerto RANS resolution than LES-type fine resolution. For thesecoarse resolutions, LES closures that are algebraic in nature (zero-equation models) are too elementary to be useful in PANS simula-tions. Based on these considerations, we conclude that there mustbe at least as much physics incumbent in the PANS models as insome of the more advanced two-equation RANS models. RANSmodels have endured many years of testing and verification in avariety of turbulent flows and have proven to be robust. Thus,when used in the PANS context, the RANS-type closures can bemore accurate as some of the complicating large-scales influenceswill be resolved. In addition, important issues, such as tensorinvariance, realizability, and effects of extra strain rate (due torotation, buoyancy, etc.) are best addressed at the level of RANSclosure using well-tested techniques [6].

For this work we propose a Boussinesq constitutive relation forthe subfilter stress. When the Boussinesq approximation isinvoked in conjunction with an averaging-invariance requirementfor arbitrary filters, we get

sðVi;VjÞ ¼ ��uSij; where �u ¼ ClK2

u

eu(7)

Fixed-point analysis [6] has clearly shown that the asymptoticenergetics of unresolved scales does not depend on the value ofCl. Hence, for the sake of simplicity, we maintain the PANS Clat the RANS value. Thus, to complete the PANS closure evolu-tion, equations for the unresolved kinetic energy (ku) and dissipa-tion (eu) must be derived. The challenge is to develop theunresolved energy and dissipation equations that are filter-invariant in form and consistent with the RANS equations in func-tion. Further requirements include (i) clear a priori demarcationbetween resolved and resolved scales and (ii) smooth transitionfrom RANS to DNS as a function of decreasing cut-off scale mustbe analytically guaranteed.

2.1 Resolution Control Parameter. First, we establish [3,7]that due to compelling physical reasons, the cut off must be

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parameterized by the fraction of unresolved kinetic energy anddissipation, (fk) and dissipation (fe),

fk ¼Ku

K; fe ¼

eu

e(8)

Parameterizing the cutoff in terms of these fractions (rather thancutoff length) makes it easier to deduce the unresolved PANSkinetic energy and dissipation from their RANS counterpart moredirect [3,7].

The resolution control parameters can take any value betweenzero and unity. Turbulence spectral scaling laws dictate that0 � fk � fe � 1. This is due to the fact that most of the energycontent is in the large scales and much of the dissipation occurs inthe small scales. Therefore, the fraction of unresolved dissipationcan never exceed that of kinetic energy. As the parameter fk deter-mines the unresolved-to-total kinetic energy, it plays a crucial rolein determining the cut-off between resolved and unresolvedscales. The smaller the value of fk, the smaller the cut-off length;fk ¼ 1 corresponds to RANS and fk ¼ 0 yields DNS.

While fk controls the cutoff, fe determines the Reynolds numberof the unresolved scale. It depends upon the extent of overlapbetween the energy-containing and dissipation ranges in the givensimulation. If the cutoff is in the large scales, such that none ofthe dissipation range eddies are resolved (typical of high Reynoldsnumber simulations), then fe ¼ 1. If on the other hand, there is nosignificant separation between energy-containing and dissipationranges (low Reynolds number flow), then fe � fk. For most highReynolds number simulations with cutoff in the inertial range orlarger scales, it is reasonable to set fe to unity, implying that theRANS and PANS unresolved scales dissipate an equal amount ofenergy. Throughout this study, we will assume that the cutoff is inthe energy-containing or inertial scales leading to fe ¼ 1. Thus fk

is the sole the resolution-control parameter in the simulations pre-sented in this study.

In order to perform a PANS computation, the resolution-controlparameter fk must be specified. Clearly, fk must be consistent withgrid resolution. Kolmogorov scaling laws are used to determinethe relationship between fk and grid size. It is found that the small-est fk a grid can support at a given location is

fkðxÞ �1ffiffiffiffiffiffiCl

p DK

� �2=3

� 3DK

� �2=3

(9)

where D is the smallest local grid size and K is the Taylor micro-scale given by K ¼ K1:5=e. It is important to note that this rela-tionship is an inequality. The implication is clear that larger fk

simulations can be performed on the same grid. In LES, this isakin to the statement that the cutoff length can be larger than thegrid size but not smaller.

From a practical application viewpoint, it is generally desirableto have the physical resolution vary as a function of space as inDES. However, as mentioned in the Introduction, spatially-varying filters introduce commutation error. Thus, from a theoreti-cal viewpoint, constant filter size is preferable. Most VR methodsignore the commutation error. It can be argued that this makestheir results somewhat unreliable as, at the very minimum, thesize of the commutation error relative to the results must be estab-lished. Several PANS computations have been performed withspatially varying resolution. However, for the purpose of modelverification, it is more appropriate to perform constant fk simula-tions. If the model can be shown to yield a desired cutoff, physi-cally correct fluctuations and good comparison against data forsuch constant fk simulations, then one can proceed with confi-dence to variable fk computations. As the purpose of this paper isestablish the fidelity and reliability of PANS for high-speed appli-cations, we will restrict ourselves to constant fk simulations only.

It should be noted that PANS is an “accuracy-on-demand”approach, i.e., the fidelity of the PANS simulation can be adjusted

to accommodate available grid resolution. For grids with suffi-cient resolution, lowering the fk value (filter-width) is purported toincrease prediction accuracy.

2.2 Unresolved Kinetic Energy and Dissipation Equations.The transport equations for the unresolved kinetic energy and dis-sipation are derived by posing the question [3], “If the total kineticenergy evolution is dictated by the RANS model equation, what isthe corresponding equation for a fraction of the kinetic energy?The starting point of the derivations are

dKu

dt

� �PANS

¼ fkdK

dt

� �RANS

anddeu

dt

� �PANS

¼ fededt

� �RANS

(10)

The RANS evolution equations are substituted on the right-handside and the PANS transport equations are determined after invok-ing the averaging-invariance property [4]. The two-equationPANS model can be summarized as

dKu

dt¼ Pu � eu þ

@

@xj

�u

rkuþ �

� �@Ku

@xj

� �

deu

dt¼ fk Ce1

Pueu

Ku� C�e2

e2u

Ku

� �þ @

@xj

�u

reuþ �

� �@eu

@xj

� � (11)

The various modified model coefficients are

C�e2 � Ce1 þfkfeðCe2 � Ce1Þrk;eu � rk;e

f 2k

fe(12)

The values for the various model constants used in our works areCe1 ¼ 1:44;Ce2 ¼ 1:92;rk ¼ 1:0;re ¼ 1:3. Overall, the form ofthe equations are clearly invariant to averaging. The resolution-control parameter manifests only via the modified modelcoefficients.

2.3 Fixed-Point Analysis of PANS. We will now directlyaddress objective 1 given in the Introduction. The formal basis ofthe PANS derivation from RANS enables us to establish someimportant properties of the former analytically. The most impor-tant question is whether the model can produce decreased levelsof kinetic energy with decreasing fk. The answer to this hingesdirectly on the asymptotic production-to-dissipation ratio, Pu=eu

[7]. The larger the value of Pu=eu, the more energetic the unre-solved scales. In a bridging model, the energy content of the unre-solved fluctuations should decrease as the resolution goes fromRANS (fk ¼ 1) to DNS (fk ¼ 0). When the implied cutoff is in thedissipation range, Pu=eu should go to unity as all energy cascadedinto the dissipation scales is immediately expended without anyaccumulation. The long-time global behavior Pu=eu as a functionof resolution can be best evaluated from a fixed point analysis ofEqs. (11) in the absence of transport (which does not create ofdestroy energy). A detailed analysis is given by Girimaji et.al. [3]and only the important results are presented here. At the weak-equilibrium turbulence fixed point, we have only one nontrivialfixed point,

Pu

eu

� �lim t!1

¼ C�e2 � 1

Ce1 � 1¼ 1þ fk

fe

� �Ce2 � Ce1

Ce1 � 1(13)

Asymptotically, Pu=eu varies linearly with fk, going from theRANS value to the DNS value of unity. The PANS SFS velocityfield progressively gets weaker with smaller fk as required. On thecontrary in URANS, wherein the model coefficients are constantsat RANS values, the only nontrivial fixed point is the same as inRANS. Hence, the URANS unresolved scales will be as energeticas the RANS scales irrespective of implied resolution. It is shownthat the trivial fixed point of RANS leads to K¼ e¼ 0. This

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corresponds to DNS. Such a clear analytical proof that the modelproduces decreasing kinetic energy and stress is very useful forestablishing the fundamental characteristics of the model.

3 PANS Calculations of Canonical Flows and

Ramjet/Scramjet Mixing Environment

As mentioned in the Introduction, the main objective of this pa-per is to establish the fidelity and feasibility of PANS for comput-ing mixing environment in high-speed propulsion devices. In theprevious section, we analytically established the connectionbetween RANS and PANS. Further it was shown using fixed pointanalysis that the unresolved kinetic energy progressively decreaseas the cutoff scale becomes smaller and smaller. While such ana-lytical proof is very encouraging, it is important that the physical“correctness” of the model be demonstrated in actual flow calcula-tions. In this section we first present results to demonstrate thefidelity of the model in canonical flow calculations (objective 2).Then, we present some preliminary calculations in high-speedflows addressing objective 3.

3.1 Verification in Subsonic Flows. Detailed verification ofPANS in subsonic flows have been completed [8–11]. In theseworks, flows past square cylinder, circular cylinder, backward-facing step, and cavity are considered. Quantitative demonstrationthat PANS performs as intended going smoothly from RANS toDNS/LES/experiment with decreasing fk value is provided. Wenow present two crucial results from these studies to address thefirst two VR-method issues raised in the Introduction. In Fig. 1 wepresent the spatial distribution of viscosity reduction (over compa-rable RANS calculation) achieved in a PANS calculation andcompare it with the prespecified value from the backward-facingstep calculation. It is clear that the a posteriori viscosity-reductionfactor agrees well with the a priori value very well over most ofthe domain. Indeed, over 85% of the domain, the calculated reduc-tion is within 5% of the specified value. This implies that PANS iscapable of clearly identifying the resolved and unresolved scales apriori. In Fig. 2, we show that the fluctuations in the PANSscale according to turbulence theory. We define the computationalKolmogorov scale as ð�3

u=eÞ0:25

. This represents the smallest scalein the computation, just as the Kolmogorov scale (ð�3=eÞ0:25

)represents the smallest scale in DNS. The local fluctuation scale isdefined from local strain rate and eddy viscosity. The pdf of thespatial distribution of local scale normalized by Kolmogorov scaleis shown in Fig. 2 for various fk calculations. The spatial distribu-tion is similar for all fk values. Moreover, this distribution is iden-tical to the DNS (of decaying isotropic turbulence) distribution.Thus, the PANS fluctuations at various cut-offs are self-similarand this self-similar form is identical to that of DNS. This is

crucial proof that PANS fluctuations at various resolutions arephysical.

In Fig. 3 we compare RANS and PANS calculations against ex-perimental data for the case of subsonic 3D driven cavity flow.(Simulations are performed using FLUENT.) The mean stream-wise velocity at the symmetry plane is plotted as a function ofcavity height. The RANS model produces reasonably good agree-ment, although PANS predictions are much more accurate. It mayappear that RANS is adequate for engineering computations.However, to probe further, we present RANS and PANS vorticityisocontours in Fig. 4. This figure paints a completely different pic-ture. With decreasing fk, more and more of the flow features areresolved. In fact, the fk ¼ 0:2 computation captures the number ofvortex pairs on the cavity floor very accurately. Detailed compari-son between PANS calculations and data is presented in [9] andthe agreement is excellent. It is clear that the mixing characteris-tics obtained from PANS simulations is clearly of the level of ac-curacy required in high-speed propulsion applications.

3.2 High-Speed Mixing Environments. Here we show pre-liminary PANS calculations of (i) two cases of diamond jet injec-tion into the supersonic freestream and (ii) high-speed flow overcavity. In the jet-injection case, the flow structures generated byPANS simulations at two different resolutions are comparedagainst DES. Detailed experimental data for turbulent quantitiesare not available, but qualitative comparison against someongoing experiments is made for the DES simulation results. Inthe cavity case, PANS data are shown at only one resolution toqualitatively demonstrate the unsteady features.

The flow field generated by a transverse underexpanded jet exit-ing normally into a supersonic freestream has been investigated invarious studies [12–14]. Briefly, the transverse injection of fluid

Fig. 1 Subsonic backward-facing step. Recovery of the pre-specified level of viscosity reduction.

Fig. 2 Subsonic backward-facing step. Length scale distribu-tion at different fk s. DNS data from decaying isotropic turbu-lence also shown for comparison.

Fig. 3 Cavity flow: centerline mean velocity profiles, zero-transport model, clustered grid. RANS, LES, and PANS compar-ison against experimental data.

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into the high-speed freestream flow leads to the formation of aninteraction shock upstream of the injector port. A separationregion is created in the boundary layer upstream of the injector insituations, where d=d is Oð1Þ [13]. The separation region creates aweak k-shaped shock that merges with the interaction shock. Thejet expands to an effective back pressure as it exits into the free-stream. In the case of diamond injector ports, this expansion ter-minates in a wedge-shaped shock. The upstream separation regioncreates a horseshoe-shaped vortex that wraps around the injectorexit along the wall. Interaction between the freestream and the in-jector fluid around the port leads to the formation of an axialcounter-rotating vortex pair (CVP) in the plume region. Theupwash of the boundary freestream boundary layer fluid creates alateral counter-rotating vortex pair (LCVP) just downstream of

the injector. A shear layer develops starting at the injector locationdue to the action of freestream fluid on the jet plume. Vortices arealso formed in the wake region behind the injector.

3.2.1 Calculations of Mach 5 Test Case. Numerical simula-tions were performed using DES and PANS on the high-speed jetinteraction flow field studied previously. The flow involves a crossflow jet exiting normally at Mach 1 from a diamond shaped orificeinto a Mach 5 freestream. The simulations were performed usingthe Cobalt flow solver with PANS modifications added by JamesForsythe at Cobalt Solutions, LLC. The DES variant of Menter’sSST turbulence model [15] was used in the current simulations.The base SST model incorporates the compressibility correctionof Suzen and Hoffmann [16]. A freestream boundary layer

Fig. 4 Unsteady RANS and PANS vorticity contours

Fig. 5 Mach 5 case: Pitot pressure comparison of experimental and DES results at z=deff 5 8:0. The left half of the image corre-sponds to simulation results and the right half is from experiments.

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was specified at the inlet to match experimental conditions. Nosymmetry planes were used to accommodate asymmetries in theflow domain. The injector port was also modeled to account forlosses. The total number of cells (hexahedrons) was approxi-mately 10� 106.

A comparison of the transient and averaged DES results [17]with mean-flow experimental data [14] was performed to evaluate

the model capability. The contour plots in Fig. 5 show pitot pres-sure measurements at a distance of x=deff ¼ 8:0 from the center ofthe injector. The left half of the images show numerical resultsand the right half are from experiments. A line plot of the compar-ison along the center line (z=deff ¼ 0:0) is shown in Fig. 6. Resultsfrom a prior RANS simulation is also included for reference. Theresults show that DES performs adequately in predicting these

Fig. 6 Mach 5 case: Line plot at z=deff 5 8:0 for pitot pressurefor RANS, DES, and experimental results

Fig. 7 Mach 5 case: DES and PANS simulations of jet in cross-stream

Fig. 8 Mach 5 case: Observed flow structures in experiments,DES, and PANS simulations. (a) Injector seed particles showeddy structures in the shear layer (Bowersox, Fan, and Lee[14]). (b) Transient DES results do not resolve these eddies. (c)Eddy formation observed in PANS with fk ¼ 0:35. (d) Moreresolved eddies seen in PANS results with fk ¼ 0:20.

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types of flows. Also, the performance of DES is relatively betteras compared to the RANS results.

The results from the DES and PANS (fk ¼ 0:35 and 0:2) simu-lations are shown in Fig. 7. The freestream flow is from bottomleft to top right. The images represent isosurfaces of entropy at anidentical scale. The isosurface is colored by temperature. Theresults show that the flow exhibits more structures when the modelis switched from DES to PANS (fk ¼ 0:35). At fk ¼ 0:20, evenmore structures are seen, including more shear layer vortices. Thedevelopment of these shear layer structures have been observed in

experiments. A Mie scattering image with visible eddies is shownin Fig. 8(a). This is compared against transient results from DESand PANS (Figs. 8(b)–8(d)). For this flow field and grid,detached-eddy simulations do not resolve these structures. Resultsfrom PANS clearly show the resolved eddies.

3.2.2 Calculations of Mach 2 Test Case. Numerical simula-tions were performed using DES and PANS on the sonic injectioninto a supersonic freestream studied previously. The flow involvesa cross flow jet exiting normally at Mach 1 from a diamond

Fig. 9 Mach 2 case: DES and PANS simulations of jet in cross-stream

Fig. 10 Mach 2 case: Observed flow structures in experiments, DES, and PANS simulations. (a)NO PLIF image shows eddy structures in the shear layer [18]. (b) DES results show correspond-ing eddy structure. (c) PANS with fk ¼ 0:35 does not resolve these structures. (d) PANS withfk 5 0:20 is comparable to DES in the shear layer.

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shaped orifice into a Mach 2 freestream. A freestream boundarylayer was specified at the inlet to match experimental conditions.No symmetry planes were used to accommodate asymmetries inthe flow domain. The injector port was also modeled to accountfor losses. The total number of cells (hexahedrons) was approxi-mately 4� 106. The results from the DES and PANS (fk ¼ 0:35and 0:2) simulations are shown in Fig. 9. The freestream flow isfrom bottom left to top right. The images represent isosurfaces ofentropy at an identical scale. The isosurface is colored by temper-ature. The results show that for this grid resolution, the DESresults show more structures than the PANS with fk ¼ 0:35. Theresults from PANS with fk ¼ 0:2 are more comparable to DES.

A comparison of the flow field structures was made againstavailable experimental NO-PLIF data [18]. Similar to the trendsseen in the isosurface plots, DES results were comparable toPANS with fk ¼ 0:2 Fig. 10. The direction of rotation of shearlayer structures was opposite to that seen in the Mach 5 test case.

Additional work is required in both experiments and simula-tions to evaluate the performance of PANS in capturing the turbu-lence structure of these flows.

4 Conclusion

In this paper, we present the verification of the PANS method-ology for canonical test cases and preliminary PANS computa-tions of two mixing environments typical of scramjet combustors.First, it is demonstrated in subsonic flows (circular cylinder andbackward-facing step) that PANS is a high-fidelity variable reso-lution scheme in which the resolution can be controlled in a cleara priori fashion and the fluctuations scale according to turbulencephysics. Then, it is demonstrated in subsonic cavity flow thatPANS captures important vortical structures crucial for accuratesimulation of mixing. Quantitative comparison with experimentaldata is made (in references provided) in all subsonic cases.Finally, we present PANS results from cross-jet in supersonicfreestream and high-speed flow over cavity cases. In the jet injec-tion case, comparison of PANS data at two resolutions againstDES results shows that PANS captures more details in the certainregions of flow where DES employs the RANS model. Overall,the present work demonstrates that potential of the PANSvariable-resolution method for scramjet applications.

Acknowledgment

Support from NASA under NRA Project No. NNX08AB44A(Program monitors, Aaron Auslender and Dennis Yoder) is grate-fully acknowledged.

Nomenclature

fk ¼ fraction of unresolved turbulent kinetic energyfe ¼ fraction of unresolved dissipationi ¼ x, y or z

K ¼ turbulent kinetic energy, m2=s2

p ¼ instantaneous pressure, N/m

P ¼ turbulence productionu ¼ unresolved quantity

U ¼ filtered velocity, m/sV ¼ instantaneous velocity, m/sxi ¼ Cartesian directions

Greek Symbols

D ¼ smallest local grid dimensione ¼ dissipation, m2=s3

� ¼ viscositys ¼ subfilter stressK ¼ Taylor microscaler ¼ turbulent Prandtl number

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