25
Large-Eddy / Reynolds-Averaged Navier-Stokes Simulations of Scramjet Combustor Flow Fields Jack R. Edwards 1 , Amarnatha Potturi 2 , Jesse A. Fulton 3 North Carolina State University, Raleigh, NC, 27695 This paper describes the development of a hybrid large-eddy simulation / Reynolds-averaged Navier-Stokes (LES/RANS) turbulence modeling framework and its application to various supersonic combustion problems. The model utilizes a flow-dependent blending function, dependent on instantaneous and ensemble-averaged turbulent quantities, to shift the closure from a RANS description very near solid surfaces to an LES subgrid model in the outer parts of attached turbulent boundary layers and in free-shear and mixing regions. Other elements of the simulation framework include low-dissipation advection schemes suitable for resolving turbulence near the grid scale, various strategies for initiating and sustaining resolved turbulent fluctuations, and simple techniques that model the effects of un-resolved fluctuations on species production rates. Test cases include the Burrows-Kurkov reacting wall jet experiment, a DLR experiment involving supersonic combustion behind a wedge- shaped flame-holder, and an experiment performed in the University of Virginia’s Scramjet Combustion Facility. 1. Introduction For years, the workhorse technology for scramjet engine component simulations has been the Reynolds-averaged Navier-Stokes (RANS) model, usually closed according to Boussinesq and gradient-diffusion assumptions, with the effects of turbulent fluctuations on reaction rates either ignored or modeled by eddy-breakup or assumed PDF methods. [1] With recent increases in available computer power, there has been significant activity directed toward the application of large-eddy simulation methods and related techniques for scramjet combustion calculations. Some prior work in applying large-eddy simulation methods to supersonic combustion problems has been reported in Genin, et al. [2,3] and Berglund and Fureby [4], who simulated hydrogen-air combustion behind a wedge-shaped flameholder [5] using linear-eddy and flamelet-type subgrid combustion models. In general, reasonable agreement with experimental temperature and velocity profiles [5] was obtained. In a later study, Berglund, et al. [6] considered a range of turbulent combustion models and several simple subgrid closures (including a ‘laminar chemistry’ formulation that neglects effects of subgrid fluctuations on species production rates) in their simulations of supersonic combustion in a configuration studied at ONERA/JAXA. They found better agreement with experimental OH-PLIF images when a seven-step finite rate chemistry mechanism was employed, but the effects of the various subgrid closures considered were inconclusive. The same group has recently simulated supersonic combustion in the Hyshot II scramjet combustor.[7] Peterson and Candler [8,9] applied a detached-eddy simulation method with ‘laminar chemistry’ to the SCHOLAR supersonic-combustion experiment [10] and reported reasonable agreement with wall pressure distributions and experimental CARS images. Donde, et al. [11] applied a direct- quadrature method of moments (DQMOM) closure to several supersonic combustion problems. They used a two- environment Eulerian model to discretize the FDF governing subgrid-scale mixing and combustion. Implicit large- eddy simulation techniques were applied to the supersonic hydrogen flame of Cheng et al. [12] using reduced and detailed mechanisms under the ‘laminar chemistry’ assumption. [13,14] Flamelet-type subgrid combustion models along with wall-modeling techniques have recently been by Larsson and co-workers in simulations of the Hyshot II combustor. [15] 1 Professor, Associate Fellow AIAA 2 Graduate Research Assistant, Student Member, AIAA 3 Graduate Research Assistant, Student Member, AIAA

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Page 1: Large-Eddy / Reynolds-Averaged Navier-Stokes …...Large-Eddy / Reynolds-Averaged Navier-Stokes Simulations of Scramjet Combustor Flow Fields Jack R. Edwards1, Amarnatha Potturi2,

Large-Eddy / Reynolds-Averaged Navier-Stokes Simulations of Scramjet Combustor Flow Fields

Jack R. Edwards1, Amarnatha Potturi2,

Jesse A. Fulton3 North Carolina State University, Raleigh, NC, 27695

This paper describes the development of a hybrid large-eddy simulation / Reynolds-averaged Navier-Stokes (LES/RANS) turbulence modeling framework and its application to various supersonic combustion problems. The model utilizes a flow-dependent blending function, dependent on instantaneous and ensemble-averaged turbulent quantities, to shift the closure from a RANS description very near solid surfaces to an LES subgrid model in the outer parts of attached turbulent boundary layers and in free-shear and mixing regions. Other elements of the simulation framework include low-dissipation advection schemes suitable for resolving turbulence near the grid scale, various strategies for initiating and sustaining resolved turbulent fluctuations, and simple techniques that model the effects of un-resolved fluctuations on species production rates. Test cases include the Burrows-Kurkov reacting wall jet experiment, a DLR experiment involving supersonic combustion behind a wedge-shaped flame-holder, and an experiment performed in the University of Virginia’s Scramjet Combustion Facility.

1. Introduction

For years, the workhorse technology for scramjet engine component simulations has been the Reynolds-averaged Navier-Stokes (RANS) model, usually closed according to Boussinesq and gradient-diffusion assumptions, with the effects of turbulent fluctuations on reaction rates either ignored or modeled by eddy-breakup or assumed PDF methods. [1] With recent increases in available computer power, there has been significant activity directed toward the application of large-eddy simulation methods and related techniques for scramjet combustion calculations. Some prior work in applying large-eddy simulation methods to supersonic combustion problems has been reported in Genin, et al. [2,3] and Berglund and Fureby [4], who simulated hydrogen-air combustion behind a wedge-shaped flameholder [5] using linear-eddy and flamelet-type subgrid combustion models. In general, reasonable agreement with experimental temperature and velocity profiles [5] was obtained. In a later study, Berglund, et al. [6] considered a range of turbulent combustion models and several simple subgrid closures (including a ‘laminar chemistry’ formulation that neglects effects of subgrid fluctuations on species production rates) in their simulations of supersonic combustion in a configuration studied at ONERA/JAXA. They found better agreement with experimental OH-PLIF images when a seven-step finite rate chemistry mechanism was employed, but the effects of the various subgrid closures considered were inconclusive. The same group has recently simulated supersonic combustion in the Hyshot II scramjet combustor.[7] Peterson and Candler [8,9] applied a detached-eddy simulation method with ‘laminar chemistry’ to the SCHOLAR supersonic-combustion experiment [10] and reported reasonable agreement with wall pressure distributions and experimental CARS images. Donde, et al. [11] applied a direct-quadrature method of moments (DQMOM) closure to several supersonic combustion problems. They used a two-environment Eulerian model to discretize the FDF governing subgrid-scale mixing and combustion. Implicit large-eddy simulation techniques were applied to the supersonic hydrogen flame of Cheng et al. [12] using reduced and detailed mechanisms under the ‘laminar chemistry’ assumption. [13,14] Flamelet-type subgrid combustion models along with wall-modeling techniques have recently been by Larsson and co-workers in simulations of the Hyshot II combustor. [15]

1 Professor, Associate Fellow AIAA 2 Graduate Research Assistant, Student Member, AIAA 3 Graduate Research Assistant, Student Member, AIAA

Page 2: Large-Eddy / Reynolds-Averaged Navier-Stokes …...Large-Eddy / Reynolds-Averaged Navier-Stokes Simulations of Scramjet Combustor Flow Fields Jack R. Edwards1, Amarnatha Potturi2,

The focus of scramjet combustion efforts using large-eddy simulation at NCSU has been on the correct capturing (to as much of a degree as feasible) of facility-dependent effects on scramjet combustion processes, as in our view, these impact predictions as much as variations in subgrid-scale models. The overall framework is based on a hybrid large-eddy simulation / Reynolds-averaged Navier-Stokes turbulence closure strategy, augmented with low-dissipation numerics, techniques for sustaining turbulence on multiple walls, and simple subgrid-scale models (including the ‘laminar chemistry’ assumption for species production rates). Our initial effort [16] was directed toward simulations of the classical 1971 Burrows-Kurkov experiment [17]. Subsequent efforts have focused on simulations of the DLR scramjet combustor [18] (considered previously by [2-4]) and on simulations of the reactive flows within various geometries tested at the University of Virginia’s scramjet combustion facility [19], as part of our involvement with the National Center for Hypersonic Combined Cycle Propulsion. This paper summarizes the model formulation as well some of our earlier and current results in this scope.

2. Theoretical model

The computational model used in this investigation solves the Navier-Stokes equations governing a mixture of thermally-perfect gases. Several models for hydrogen oxidation have been employed in this work, including two nine-species gas models [20, 21] and a seven-species model [22], Thermodynamic curve fits from McBride, et al. [23] are used for species specific heats and enthalpies. Wilke’s law is used for the mixture viscosity, and the mixture thermal conductivity is obtained from the assumption of a constant Prandtl number (0.72). Molecular diffusion processes are described using Fick’s law, parameterized by a constant Schmidt number of 0.5. The law of mass action is used to formulate source terms describing production / depletion of chemical species. 2.1. Turbulence closure model – LES/RANS hybridization The turbulence models employed in this work invoke the Boussinesq hypothesis to relate unresolved stresses to the rate of strain of the resolved velocity field. Gradient diffusion models, parameterized by constant Prandtl (0.9) and Schmidt (0.5) numbers are used to account for heat and mass fluxes due to unresolved turbulence. Closure of the equation system is accomplished by specifying an eddy viscosity, which parameterizes both subgrid-scale and near-wall turbulence effects. The NCSU LES/RANS models are based on Menter’s baseline (BSL) k-ω formulation [24], which solves two transport equations for the turbulence kinetic energy k and specific dissipation rateω . Hybridization is accomplished by defining the eddy viscosity as follows

Γ+Γ−=

ωνρµ

ksgstt ,)1( (1)

where Γ is a time-dependent blending function that connects the RANS and LES branches. A mixed-scale algebriac model due to Lenormand, et al. [25] is used for the subgrid-scale eddy viscosity:

06.0,)( 2/34/122/1, =∆= MMSGSt CqSCν (2)

In this, an estimate of the subgrid kinetic energy is obtained by test-filtering the resolved-scale velocity data:

22 )~~(21

kk uuq −= (3)

The Generation I LES/RANS model [26] is designed to target the shift from a boundary layer’s logarithmic structure to its wake-like response as the effective RANS-to-LES ‘transition’ location. The model uses a blending function based on the ratio of the wall distance d to a modeled form of the Taylor micro-scale:

−−−=Γ ])1(5tanh[1

21 2 φηκ

µC,

χαη

1

d= (4)

with the Taylor micro-scale defined as ωνχ µC= and d being the distance to the nearest wall. An equivalence

between this modeled form of the Taylor microscale in the logarithmic layer and the wall coordinate +d is used to correlate the model constant 1α with the wall-coordinate value that defines the ‘edge’ of the logarithmic layer. This ‘edge’ value of +d is obtained by an off-line analysis based on Coles’ law of the wall/wake along with the Van Driest transformation.[26] As the wake law is not universal, but rather, dependent on the boundary layer thickness

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δ and boundary-layer edge quantities, the model constant that results is specific to a particular incoming boundary layer. The Generation I model has an obvious weakness in that a calibration is required for each problem. The Generation II model [27] seeks to avoid this difficulty by introducing an outer-layer turbulence length scale into the argument forΓ :

−−=Γ )]11(15tanh[1

21

2Nλ

, (5)

where

ωκων

κωω

ωνλ

µµ dCvkkCd

CkkC

ll R

NR

Ninner

outerN 4/12/1

10/10 ++=

++=≡ (6)

In this expression, k is the ensemble-averaged modeled turbulence kinetic energy, Rk is the ensemble-averaged resolved turbulence kinetic energy

)(21

ρρρ

ρρ kkkkR

uuuuk −= , (7)

and ω and ω are instantaneous and ensemble-averaged modeled turbulence frequencies, respectively. The combination of instantaneous and ensemble-averaged data allows the RANS-to-LES transition position 2/1=Γ to fluctuate about a mean value that is a function of the local, ensemble-averaged state of the flow. The last equivalence in Eq. 6 can be re-arranged to show that the new length-scale ratio Nλ is equivalent to that used in the Generation I model (Eq. 4) but with a spatially-varying model constant

ων

α RN

kkC ++≈ 101 . (8)

The value of NC (~1.5) was determined by comparing results from this model with those of the original for a selection of supersonic flat-plate boundary layers at different Reynolds numbers. [27] As it is dependent on both inner-layer and outer-layer length scale information, this model is more capable of adjusting to strong departures from local equilibrium, and a problem-specific selection of a model constant is not required. 2.2 Turbulence closure model – filtered species production rates We have incorporated several simple models to account for sub-cell effects in evaluating chemical species production rates. These are described as follows. ‘Laminar’ chemistry: In the baseline approach, species production terms are evaluated using filtered-mean data:

)()( jsjs VV ωω && ≈ (9) Quadrature-based SGS reconstruction: Here, SGS terms are estimated by Gaussian quadrature volume integration of polynomial re-constructions of an assumed continuous grid-filtered field. [16]. These terms actually account for the differences between an idealized, continuously-filtered field at the mesh scale and the cell averages of this field that are directly outputted during a finite-volume based large-eddy simulation.

∑∫∫∫=Ω

′+Ω

≈ΩΩ

=nq

kkjjskjsjs xVVwdxVV

1

))((1))((1)(r

&r

&& ωωω , (10)

)())]((21[1)())]((

21[)(

1clcljj

nq

llckckjjkj xxxxVVwxxxxVVxV

rrrrrrrrr−⋅−⋅∇∇∇

Ω−−⋅−⋅∇∇+∇=′ ∑

=

Here, kw and kxr

represent Gaussian quadrature weights and locations, with the former encompassing the effects of element coordinate transformations. Implementation details may be found in [16]. It can be shown to leading order, the effective SGS model implied by the quadrature approach is

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K

444444 3444444 21

rr&&& +

′′Ω∂∂

∂=− ∑

=

SGS

nq

kkmklk

ml

sjsjs xVxVw

VVVV

1

2

)()(121)()( ω

ωω

(11)

Randomized quadrature-based SGS reconstruction: In this version of the quadrature-based reconstruction, random elements are introduced to account for effects of unresolved small-scale turbulence.

2/1

1

2

1

)(1,)(~1)(

′Ω

=+Ω

≈ ∑∑==

nq

kjkj

nl

ljljsljs VwrVwV σσωω && (12)

In the randomized model, lr is a set of mesh cell-specific random numbers defined to have zero mean and unit variance over the span of the quadrature points and σ is a vector containing estimates of the sub-cell standard deviations of each component of T

s TV ],[ρ= A potential advantage of the randomized formulation is that different quadrature rules can be applied to calculate the source term and the variance expression. In the current implementation, nl is set to two, ]1,1[−=lr , and ],[~

21

21 ΩΩ=lw , leading to a ‘bi-modal’ approximation

))()((21)( jjsjjsjs VVV σωσωω −++≈ &&& (13)

2.3. Recycling / rescaling techniques Most of our prior LES/RANS simulations have utilized recycling / rescaling techniques [28,29] to initiate and sustain turbulent fluctuations. The general approach extracts mean and fluctuating fields at a ‘recycle’ plane upstream of the inflow plane, re-scales these fields according to equilibrium boundary-layer scaling arguments and super-imposes the recycled fields onto the inflow plane. Our variants have incorporated several modifications designed to enhance the utility of this technique for very high Reynolds number flows:

a) Superposition of recycled fluctuations on a RANS mean flow [26]. This strategy helps to avoid ‘drift’ in integral boundary layer properties.

b) Klebanoff intermittency factor to damp free-stream fluctuations [26]. Here, we utilize a Klebanoff-type

intermittency function to attenuate free-stream disturbances:

16klebkleb

recycklebRANSklebinflow

recycklebRANSinflow

))(1( variablesturbulence

;)1(re temperatuvelocity,,density

;

−+=

=

+−=

=

′+=

δCdFq

qFqFqq

qFqq

(14)

c) ‘Random-walk’ recycling [30]. One drawback of recycling / rescaling methods is that they introduce an

artificial level of streamwise periodicity that can result in the ‘fixing’ of the time-averaged positions of longitudinal vortical structures within the boundary layer over long integration times. To mitigate this effect, we have developed a technique that shifts the plane containing the fluctuation fields in the spanwise (Z) direction by some fraction of the incoming boundary layer thicknessδ . The discrete shifting-distance versus time distribution is produced by sampling Gaussian distributions corresponding to an average near-wall streak length (3-5δ ) and the shifting-distance itself. This discrete distribution is converted to a continuous one by linearly interpolating between each discrete point.

d) Multi-wall recycling / re-scaling.[31] Here, recycling / rescaling methods are applied independently to

boundary layers developing on each wall and the resulting fluctuation fields averaged using an inverse-distance weighting procedure. The ‘random-walk’ procedure can also be applied by blending two sets of fluctuation fields, one shifted randomly in the wall-parallel direction and the other un-shifted. The blending process restricts the use of the shifted information to regions away from the opposite wall.

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2.4 Synthetic eddy generation techniques Synthetic turbulence-generation methods [32-34] have become increasingly popular for generating inflow data for LES, DNS, and LES/RANS calculations. In such methods, a set of randomized spatio-temporal data for velocity-component fluctuations is given single-point cross-correlations through the action of the Cholesky decomposition of a specified Reynolds stress tensor ijR . The correlated velocity components are then added to a specified mean velocity profile to complete the description:

jijiiii uaUuUu ~+=′+= ; [ ]

−−−

−=231

22122223121321131

221221121

11

)(000

RRRRRRRRRRRRR

Raij (15)

The proposals differ in the generation of the initial randomized data ju~ , which must include some level of spatio-temporal coherence for it to mimic turbulence to a reasonable degree. These methods possess several advantages over the recycling / rescaling methods used to date. First, their implementation is easier, avoiding the issues associated with transferring data among arbitrarily-defined recycling and inflow planes. Secondly, these methods do not display a pseudo-periodic behavior, which ensures that time-series analyses performed to elucidate certain frequencies of interest (such as those associated with separation-shock motion) are not affected by frequencies associated with the periodic response. However, even the best of these methods require a long transition distance (15-20 boundary layer thicknesses) before reasonable turbulence statistics begin to emerge. Further, the methods require the a priori specification of the Reynolds-stress tensor, which is not available to a high degree of precision in most experiments or in RANS closures that adopt the Boussinesq hypothesis. We are experimenting with synthetic-eddy model (SEM) of Jarrin and co-workers [33]. Here, the randomized fluctuation field is given by

∑=

−=N

k

kkjj f

Nu

1)( )(1~ xxxσε , (16)

where k represents the kth eddy of a collection defined at random locations kx within a bounding box placed just

upstream of the actual inflow. The shape function )()(kf xxx −σ is the velocity distribution of the eddy located at

kx which provides spatial coherence (eddy structures) to the fluctuations. Jarrin, et al. used an isotropic tent function to define )()(

kf xxx −σ

∏=

−=−

3

1)(

1)(j

kjj

Bk xx

fVfσσσ xxx ,

<−

=otherwise,0

1|| if|),|1(5.1)( xxxf , (17)

where )(xσ is a representative turbulence length scale that is limited to be no larger than a representative filter width. After each time step the new location of each eddy is calculated. This is done by interpolating the velocity for a particular eddy location using the reference velocity field. If the new eddy location is out of the bounds of the bounding box, then the eddy is replaced with a new eddy whose location is randomly generated within the bounds of the bounding box. The number of eddies ( N ) randomly distributed within the bounding box is calculated as follows:

3max/σbN Ω= (18)

where bΩ is the volume of the bounding box and maxσ is the largest length scale. Each eddy is also assigned a random intensity of +1 or -1. The whole process of interpolation and eddy tracking is straight-forward for a serial code. The implementation is slightly more complicated in a parallel code as the entire bounding box can span over multiple blocks which are distributed among multiple processors. In this case it is not necessary that all eddies be located on a particular processor as it may not have the reference data spanning over the entire bounding box. This means multiple data transfers between processors. If the blocks containing the SEM surfaces are distributed smartly, then the number of data transfers can be kept to a minimum.

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In our current work, this synthetic-eddy-method (SEM) is used to generate turbulence in the boundary layers of the upper and lower surfaces of a wedge-shaped flame-holder and within the fuel injecting holes for a scramjet combustor configuration experimentally tested by Waidmann and co-workers [5] and previously simulated by our group [18], among others [2-4]. Synthetic eddies are added to create fine-scale turbulence in the shear layers behind the flame-holder and also to aid in mixing the hydrogen jets. For the upper and lower surfaces of the flame-holder, turbulence generated using the SEM is introduced through an interface between two blocks. Fuel injection ports are treated with a supersonic-inlet boundary condition. The turbulence is introduced through these surfaces (Figure 1). The implementations for both the flame-holder boundary layer and the fuel-injection ports differ in the way the reference velocity field and Reynolds stresses are calculated and in the way that the bounding boxes that contain the generated eddies are defined Flame-holder boundary layers: The steady state RANS solution is used for the reference velocity field and to calculate the Reynolds stresses. The length scales are calculated as shown below:

),max( 1 jj l ∆=σ ,

=

δκ

δµ

µ CyCl dtanh1 (19)

where j∆ is the grid spacing in the jth direction, 09.0=µC , δ is the boundary layer thickness, 41.0=κ , and dy is the distance from the wall, 4.0=K is a constant and L is the characteristic length. The bounding box spans over the entire width of the combustor. In the direction perpendicular to the boundary layer, the height of the bounding box is large enough to have the entire boundary layer well within its bounds. Velocity interpolations are determined as follows. First, each eddy is located between the eight adjacent nodes forming a cell in the reference grid. The velocity at this particular location is interpolated between these eight nodes using either tri-linear or inverse distance methods. Fuel injection ports:: DNS databases for pipe flow [35] at two different Reynolds numbers (5300 and 44000) are used to calculate the reference velocity field and Reynolds stresses. Given the injection velocity (1200 m/s) and the diameter of the hole (1 mm), the Reynolds number is 11765. The required reference values are then interpolated for the current Reynolds number (11765) using the Re = 5300 and Re = 44000 databases. The length scale is calculated as shown below:

))max(),,max(min( 21 jll ∆=σ ,

−−=

+

26exp11

yyl dκ , DCl 22 = (20)

where +y is the non-dimensional wall distance defined as ντ dyu

, 07.02 =C , and D is the pipe diameter. Due to the

coarseness of the grid representing the surface of the hole, the grid spacing is always chosen as the length scale. The bounding box is in the shape of a cylinder which covers the entire hole surface and has a width of the order of largest length scale. Calculating the eddy velocities involves one-dimensional interpolation using the reference grid and the reference velocity fields, as the properties vary only in the radial direction. Figure 2 shows a snapshot of eddy locations within the bounding boxes for each case.

3. Numerical methods The studies described herein were performed using NCSU’s REACTMB code, which solves the Navier-Stokes equations discretized using a finite-volume method on simply-connected, multi-block grids. The code utilizes MPI message passing for parallel communication. Edwards’ low-diffusion flux-splitting scheme, extended to higher-order as described next, is used for discretization of the inviscid terms, while central differences are used for the viscous terms. Time integration is facilitated by a planar relaxation sub-iteration procedure based on a Crank-Nicholson - type discretization of the unsteady equations. The specific form used is

0)()1(21 )()1(

21 ,1

,1

=−+++∆−

Ω ++

nknnkn

VRVRt

UU θθ (21)

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where Ω is the cell volume, t∆ is the time step, U is the vector of conserved variables, and R is the residual vector. The function θ is defined a

ttt ddd

ddd

2.0,m101)),tanh(1(21 4 =∆×=

∆−

−= −θ (22)

Here, d is the distance to the nearest solid surface. The function θ switches the time discretization from Crank-Nicholson to Euler implicit for mesh cells essentially within the laminar sub-layer. Some loss of temporal accuracy results, but this approach appears necessary to suppress oscillations in the pressure and transverse-velocity fields for mesh cells with a very high aspect ratio. Jacobian matrix elements are stored over the number of blocks mapped to a particular processor, allowing the “freezing” of the matrix elements and their factorization over the duration of the sub-iterations. This reduces the computational workload significantly. 3.1. Higher-order extension To resolve turbulence near the mesh scale as well as to capture shock waves monotonically, a hybrid central / upwind differencing scheme is used. Given that V represents the primitive-variable vector T

s kTwvupV ],,,,,,[ ω= , where sp are species partial pressures, left and right states at a general cell interface i+1/2 are calculated as follows:

112/1,12/1,

2/1,2/1,

)1(

)1(

+++++

++

+−=

+−=

iP

iH

iRP

iiR

iP

iHiL

PiiL

VfVfV

VfVfV (23)

Here, Pif is a pressure limiter that serves to reduce the scheme’s accuracy to first order where the curvature in the

pressure field is large:

)2(,2],2.0)2.0,||

||[max(25.1 1141

112

2

2

−+−+ ++=+−=∆−+∆

∆= iiiiii

Pi pppppppp

pppf (24)

The higher-order interpolants are constructed as a blend of an average value ciV 2/1+ and monotone interpolants

MiR

MiL VV 2/1,2/1, , ++ , which may be obtained from a TVD, PPM, or WENO scheme:

)(

)(

2/12/1,2/12/12/1,

2/12/1,2/12/12/1,

ci

MiR

Di

ci

HiR

ci

MiL

Di

ci

HiL

VVfVV

VVfVV

+++++

+++++

−+=

−+= (25)

The blending function Dif 2/1+ is based on the well-known vorticity / divergence function of Ducros, et al. [36]:

i

iii

iDucrosi

Ducrosi

Ducrosi

pi

pi

Di zyxV

VVV

ffffff ),,max(/101,||)(

)(),,,,max( 8

222

2

112/1 ∆∆∆×=+×∇+⋅∇

⋅∇== ∞

−+++ ε

εrr

r

(26)

In regions of high vortical content, the Ducros function Ducrosif approaches zero, whereas in shock-dominated regions,

Ducrosif approaches one. In most of our work, we have used the following averaging operator, which yields a fourth-

order central difference method on uniform meshes: )(

121)(

127

1212/1 −+++ −−+= iiiic

i VVVVV (27)

4. Results

In this section, several results from previous and current LES/RANS simulations of supersonic combustion processes are described. The interested reader is encouraged to consult the published references for further details of these cases. 4.1. Burrows and Kurkov reacting wall jet experiment In this study [17], the LES/RANS methodology was applied to the Burrows and Kurkov reacting hydrogen wall jet experiment, a commonly-used test case for evaluating RANS scramjet combustion models. Simulations were conducted with turbulence on all walls of the domain sustained using the multi-wall recycling / rescaling method (statistically 3D) and with periodic boundary conditions in spanwise direction assumed (statistically 2D). Other factors examined in this study include the effects of the choice of chemical kinetics mechanism (Jachimowski 1992 nine-species model vs. abridged Jachimowski seven-species model), the effects of using the Gaussian quadrature

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model for the filtered species production rates, and the effects of axial mesh refinement. The largest mesh considered in this study contained upwards of 28 M cells. Figure 3 shows a snapshot of temperature along the domain centerline for non-reacting and reacting cases. A lifted flame is evident. Time-averaged species mole fraction profiles are presented in Figure 4. None of the variations tested captures the peak in water production, and the effect of considering turbulent boundary layers along all walls is to shift the reaction zone further away from the lower wall of the combustor. Additional comparisons and some discussion of the structure of the lifted flame may be found in [17]. 4.2. DLR scramjet combustor The LES/RANS methodology has also been used to simulate combustion downstream of a wedge-shaped flame-holder, behind which hydrogen is injected at sonic conditions through fifteen round holes.[18] In this case, no attempt was made to resolve turbulence in the boundary layers of the combustor geometry. Rather, a blending function that is fixed in space was used to shift the closure from unsteady RANS within these boundary layers to LES in the regions away from the wall. A schematic of the combustor is shown in Figure 5. This case has been simulated using a mesh containing about 33 M cells. Axial velocity and temperature predictions for this case (from [18]) are shown in Figure 6. Reasonable agreement with experimental LDV and CARS data is indicated, with the seven-species reaction model yielding noticeably better predictions. The failure of the models to predict the temperature rise observed at the first measurement station (X = 120 mm, Figure 6) has led to subsequent simulations that utilize the SEM described above in an attempt to enhance turbulence levels in the shear layers and to accelerate the breakdown of the hydrogen jet. Though the procedures are successful in generating some small-scale turbulence, they do not solve the problem of delayed ignition, as shown in the temperature contours of Figure 7 and in close-up views of H2 and HO2 mass fraction also shown in Figure 7. Current work is focused on developing a new mesh that should resolve shear layer development behind the flame-holder better. 4.3. University of Virginia scramjet combustion experiment The last cases described in this paper relate to on-going simulations of the reactive flow in a test rig mounted in the University of Virginia’s Scramjet Combustion Facility. A side view of the combustor, termed ‘Configuration A’, is sketched in Figure 8. As part of the National Center for Hypersonic Combined-Cycle Propulsion, an extensive measurement campaign has been completed for this configuration. Measurements include stereoscopic particle-image velocimetry (PIV), coherent anti-Stokes Raman scattering imagery (CARS), focused Schlieren imagery, hydroxyl planar laser-induced fluorescence (OH-PLIF) imagery, wavelength modulation spectroscopy (WMS), and wall pressure and wall temperature measurements. Initial comparisons with most of this data were presented in [19].Only wall pressure surveys were available for comparison prior to completing the simulations, but nevertheless, good agreement with experimental measurements was obtained in general. Since the publication of [19], we have performed several other calculations designed to reduce uncertainties in our previous simulations (including correcting mistakes made). The simulations use a 35 M cell mesh that includes a ‘dump’ region downstream of the extender outflow plane. An initial RANS solution of the nozzle was performed to provide an inflow boundary condition for the combustor. Specific items addressed in the latest set of simulations include:

1. assessment of the effects of the different quadrature approximations to the filtered species production terms 2. fixing a mistake in the fuel flow rate – the original calculation was run at an equivalence ratio of Ф =

0.1875 instead of the target Ф = 0.172. 3. accounting for measured temperature asymmetry in the combustor inflow plane 4. assessment of the effects of the choice of chemical kinetics mechanism (Jachimowski, 1992 vs.

O’Connaire, et al. , 2004) 5. comparing predictions with all available CARS data – in the original paper, only mean temperature and

oxygen, hydrogen, and nitrogen mole fraction measurements were compared with; the CARS database includes standard deviations of fluctuations in these quantities, as well as higher-order moments.

4.3.1. Quadrature-approximation assessment Several simple estimates for the filtered species production rates have been tested through the course of this research. As shown in Figure 4, the use of the Gaussian-quadrature model (Eq. 10) produces a modest increase in reactivity for the Burrows-Kurkov case. This model can be quite expensive, and for large reaction mechanisms, the

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cost of evaluating the production term in this way can dominate the calculation, as eight source-term evaluations per mesh point are required. The simpler bi-modal approximation (Eq. 13) requires two source-term evaluations. This model has been tested for the ‘Configuration A’ experiment at the same conditions considered in [19] (that is, before the equivalence-ratio mistake was uncovered and with a symmetric inflow plane). Figure 9 compares temperature distributions for the laminar chemistry and bimodal approximations at X/H = 6, 12, and 18 planes within the combustor. The bimodal approximation leads to slightly higher temperatures and a broader reactant plume, factors that are indicative of an increase in average heat release. The CARS database is rather sparse, as problems were encountered in the stepping motor used to position the lasers over the plane of interest. As such, the values have been interpolated to a uniform 200 ×200 mesh to aid in the visual comparison. Figure 10 shows snapshots of

)~,(OH2Tkρω& at the X-Y centerplane (top), and the differences between the Gaussian-quadrature and bi-model

approximations and the ‘laminar chemistry’ approximations (middle and bottom, respectively). White contours correspond to positive quantities, while black contours correspond to negative quantities. It is clear that both the Gaussian quadrature and bi-model approximations mostly provide a positive ‘correction’ to the laminar-chemistry model for the water production rate, with the bi-model approximation having a stronger effect. The CARS database indicates lower temperatures at all stations, and it is clear that the effects of the quadrature model, though small, lead to higher temperatures. 4.3.2. Inflow-plane temperature asymmetry Recent CARS measurements of the flow upstream of the combustor section [37] have revealed that the flow exiting the facility nozzle is in thermal non-equilibrium and is asymmetric. The CARS measurements provide separate distributions for vibrational and rotational temperatures of molecular oxygen and nitrogen. The vibrational temperatures are near the design stagnation temperature, meaning that vibrational modes are frozen as the flow moves from the stagnation chamber into the isolator. As the current computational approach assumes thermal equilibrium, it is necessary to convert the CARS temperature measurements to a thermally-equilibrated static temperature and to a stagnation temperature consistent with the thermodynamic curve fits used in the state description. Assuming that the translational temperature is equal to the measured rotational temperature, the following nonlinear equation is solved at each of the CARS measurement points for the thermally-equilibrated static temperature T :

)1)/exp(2

5(Y))()((Y,

,

O,Nk

O,Nk

2222−

+=−− ∑∑== vkv

kvrot

kkkrefkk

k TTRTRThTh

θθ

(28)

Here, )(Thk is the species enthalpy curve fit, refT =298.15 K is a reference temperature, kR is the species gas constant, and the characteristic temperatures for nitrogen and oxygen respectively are K3390

2N, =vθ and K2270

2N, =vθ . Following this step, a stagnation temperature consistent with the thermodynamic curve fits is

determined by solving the non-linear equation 2

O,Nk ||

21))()((Y

22

VThTh kokk

r=−∑

=

, where the velocity field Vv

is

determined by interpolating values from an initial solution determined using uniform inflow values for the stagnation pressure and temperature to the CARS data points. A large number of streamlines (~30000) are also extracted from this initial solution, and the locations where they intersect the nozzle inflow planes and the CARS data plane (X/H = 13.8, measured from the nozzle throat) are determined. Stagnation-temperature data for each of the CARS points is then interpolated to the locations where the streamlines intersect the CARS data plane. This data is assumed constant along each streamline and is mapped to the corresponding points on the nozzle inflow planes to form a non-uniform stagnation-temperature inflow boundary condition. The stagnation pressure at the nozzle inflow planes is still assumed to be a uniform value of 295 kPa. Temperature distributions in Figure 11 illustrate several stages in the procedure. Figures 11a and 11b are the measured vibrational and rotational temperatures at the data plane, interpolated to a uniform 200 ×200 grid. Figure 11c is the static temperature obtained by solving Eq. 28, and Figure 11d is the corresponding stagnation temperature. Finally, Figure 11e is the stagnation temperature at the data plane determined by computing the nozzle flow field based on the non-uniform stagnation-temperature distribution at the inflow plane. Excepting near the wall, where there are no CARS data points and the interpolation is inaccurate, good agreement between Figures 11d and 11e can be noted. It should be mentioned that the velocity and pressure fields do not respond strongly to the stagnation-temperature variation at the inflow plane. Additional velocity measurements will be required to determine whether other sources of flow asymmetry might be present.

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4.3.3. Flame structure and CARS data comparisons The remainder of this paper discusses the results of recent simulations of Configuration A that utilize the asymmetric inflow solution along with the Jachimowski [20] and O’Connaire, et al. [21] hydrogen oxidation models. As in the previous simulations [19], no attempt is made to sustain turbulence within the thin nozzle wall boundary layers. We have conducted initial tests of the multi-wall recycling strategy for this problem and plan to utilize this approach in the next set of simulations. Hydroxyl mass fraction and temperature snapshots in Figures 12 and 13 show the general structure of the flame. Counter-rotating vortices generated by the ramped injector rapidly mix fuel and air, leading to a highly turbulent, distorted flame structure that is generally oriented along the outer edges of the vortex pairs. High intensity small-scale turbulence is generated as these large structures break down, leading to a transition from a flamelet-type response downstream of the ramp to a region characterized by distributed reaction zones. Peak heating occurs in the region between X/H = 10 and X/H = 20. A continuous band of high-temperature fluid is observed near the upper wall for the O’Connaire mechanism, whereas the Jachimowski solution indicates the presence of an intermittent region with cooler temperatures at X/H from ~6 to 12. Animations of the process reveal a progressive raising and lowering of the temperature levels in this region for both mechanisms. The flame stabilization location is different between the models, as shown in Figure 14, which plots iso-surfaces of OH mass fraction (0.005) and HO2 mass fraction (0.0001). Significant OH production takes place within the counter-rotating vortex pair located just upstream of the ramp trailing edge for the O’Connaire solution. In contrast, OH radical growth is delayed in the Jachimowski solution. Figures 15-18 compare mean and rms values for temperature, H2 mole fraction, N2 mole fraction, and O2 mole fraction with CARS data at X/H = 6, 12, and 18. As before, the CARS data is interpolated to a uniform 200 ×200 mesh. Statistics were collected over a total of five combustor transit times, with one transit time being that required for a particle at the nominal free-stream velocity of 1050 m/s to pass through the 0.375 m length of the combustor. Ensemble averages were developed from 105 discrete snapshots extracted over the five transit times. Continuous ensemble averages were also collected, but in view of the fact that the CARS ensemble averages were developed over similar numbers of samples, the results from discrete snapshot approach are shown instead. Each figure contains mean values in the top set and rms values in the bottom set. Contour levels at each axial station (X/H = 6, 12, 18) were scaled independently for each quantity, with the maximum and minimum values determined over all three data sets – Jachimowski (left), O’Connaire (middle), and CARS (right). While this scaling does not provide a representation of how each quantity varies with downstream location, it does provide the best means of comparing the models with the data. Temperature contours in Figure 15 reveal that both models generally over-predict measured mean and rms levels, with the Jachimowski results being closer to the CARS data overall. The effect of inflow asymmetry can be noted in the inviscid core region outside the reacting plume, but the overall shape of the plume is not significantly distorted. Measured temperature levels in the inviscid core at X/H = 6 and 12 are higher than predicted, possibly indicating the presence of a stronger shock system in this region. In all cases, the rms temperature peaks toward the outer edge of the reacting plume. This is the expected response, as significant heat release occurs in this region along with intermittent entrainment of colder fluld. Both models over-predict the peak rms fluctuation levels. It is likely that the rendering of the CARS data is affected somewhat by the sparse sample space (see the small squares in Figure 9 for the exact locations of the CARS measurement points), but it is also possible that the dynamics of the instantaneous plume / core fluid juncture is inhibited somewhat by the activity of the Ducros ‘switch’ in this region. Mean and rms contours of nitrogen and oxygen mole fraction are shown in Figures 16 and 17, respectively. The plume size is well-predicted by both models except for the X/H = 12 station, where the measured plume is slightly broader in the vertical direction. The Jachimowski model yields slightly better predictions for the mean quantities and captures the trends of the rms distributions better, though the peaks are again over-predicted. Hydrogen mean and rms mole-fraction contours in Figure 18 reveal more noticeable discrepancies. At X/H = 6, the rate of hydrogen depletion is less in the computations than in the data, but this trend is reversed at X/H = 12 and 18. At X/H = 18, larger hydrogen mole-fraction levels persist near the center of the plume; the corresponding temperature contours (Figure 15) show lower values in this region. The computations predict more rapid hydrogen mixing and

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consumption, leading to a more uniform temperature distribution over the plume. The general shape of the hydrogen plume is better predicted by the Jachimowski model, which again captures the trends in the rms distributions, if not the peak values, reasonably well. Wall pressure distributions for the Jachimowski and O’Connaire cases are compared with experimental data in Figures 19 and 20, respectively. To indicate the degree of variation in the pressure levels, all 105 realizations of the wall pressure are plotted. It is clear that the rms variation in the pressure signal downstream of the wedge backplane is mostly uniform and is quite large: ~13% for Jachimowski and 16% for O’Connaire. The O’Connaire distribution is in better agreement with the experimental distribution, particularly just downstream of the wedge back plane. This trend indicates that the degree of volumetric expansion predicted by the O’Connaire model is higher, on the average, in this region than that predicted by the Jachimowski model, a result that is consistent with the temperature distributions described earlier. All of these results, taken together, reveal a clear dilemma. The reactive scalar trends are better predicted by the Jachimowski model, yet the measured pressure distribution implies that more heat release is generated just downstream of the wedge back plane – an effect captured (but still under-estimated) by the O’Connaire model. It should be mentioned that there are several variants of the O’Connaire model that are in use. The forward rate coefficients for the one implemented in this work comes from a Chemkin input file included as an addendum to their journal article [21] (both the article and the Chemkin file may be found at https://www-pls.llnl.gov/?url=science_and_technology-chemistry-combustion-hydrogen). The Chemkin implementation differs from that presented in the article in that several reactions are reversed and that explicit backward rate-coefficient information is given for all reactions. As REACTMB currently does not support separate backward rates, we implemented the forward rate coefficients from the Chemkin file and used detailed balancing to get the backward rates. We have also tested the exact implementation given in the Journal article (which also used detailed balancing for the backward rates) in our RANS solver and have obtained results that are much closer to those provided by the Jachimowski mechanism. This variant has yet to be tested in the LES/RANS code.

5. Conclusions and Future Work

This paper has described NCSU’s LES/RANS turbulence modeling framework and its use in several supersonic-combustion studies. A high level of predictive capability is offered by the framework, particularly in comparison with un-tuned RANS solutions, though the expense is considerable (~6+ times the cost of a RANS calculation on the same mesh). None of the tested approaches for accounting for the effects of un-resolved fluctuations on the species production terms have any real benefit, relative to the baseline ‘laminar chemistry’ model, and there is a clear need for more accurate, yet cost-effective, models for this effect. However, the predictions can be at least as sensitive to other parameters (presence or absence of resolved turbulence, inflow / outflow boundary conditions, numerics, mesh topology, mesh size, wall-blockage effects, among many others). It is difficult to separate discrepancies due to inaccuracies in various subgrid models from those due to these other factors, even if high-quality, non-intrusive data is available for comparison. One component of the current work is focused on resolving the discrepancy in the flame-anchoring position for the DLR combustor. Here, we are developing a new mesh that better resolves shear-layer development downstream of the flame-holder and also will be employing either the SEM or the multi-wall recycling strategy to sustain turbulence on the combustor walls. The next set of simulations for the University of Virginia ‘Configuration A’ experiment will incorporate inflow turbulence generation using the multi-wall recycling method and will utilize the O’Connaire model as described in their article. Inflow asymmetry in the University of Virginia tunnel is expected to be more significant for the ‘Configuration C’ experiment, which adds a long isolator upstream of the ramped injector. Simulations of the ‘Configuration C’ experiment on a mesh containing upwards of 70 M cells are underway.

Acknowledgments

This research was sponsored by the National Center for Hypersonic Combined Cycle Propulsion grant FA 9550-09-1-0611 (monitored by Chiping Li (AFOSR) and Aaron Auslender and Rick Gaffney (NASA)), by NASA under Cooperative Agreement NNX07AC27A-S01 (monitored by Robert Baurle), and by the Army Research Office under grant W911NF-08-1-0430 (monitored by Dr. Frederick Ferguson).Computing time was obtained from NASA’s NAS supercomputing resource and by the DoD’s HPC modernization program. The authors wish to acknowledge significant contributions to this work made by Dr. Robert Baurle of NASA Langley Research Center, Dr. Hassan Hassan of North Carolina State University, and Mr. Daniel Gieseking of Honda Aircraft.

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[31] Boles, J.A., Choi, J.-I., Edwards, J.R., and Baurle, R.A. “Multi-wall Recycling / Rescaling Method for Inflow Turbulence Generation,” AIAA Paper 2010-1099, Jan. 2010.

[32] Subbareddy, P, Peterson, D, Candler, G., and Marusic, I, “A Synthetic Inflow Generation Method using the Attached Eddy Hypothesis”, AIAA Paper 2006-3672, June, 2006.

[33] Jarrin, N., Prosser, R., Uribe, J.-C., Benhamadouche, S. and Laurence, D. “Reconstruction of Turbulent Fluctuations for Hybrid RANS/LES Simulations using a Synthetic-Eddy Method”, International Journal of Heat and Fluid Flow, Vol. 30, 2009, pp.435-422

[34] Pamies, M., Weiss, P.-E., Garnier, E., Deck, S., Sagaut, P. “Generation of Synthetic Turbulent Inflow Data for Large-Eddy Simulation of Spatially-Evolving Wall-Bounded Flows”, Physics of Fluids, Vol. 21, 2009.

[35] Wu, X. and Moin, P. “A Direct Numerical Simulation Study on the Mean Velocity Characteristics in Turbulent Pipe Flow” Journal of Fluid Mechanics, Vol. 10, 2008, pp. 81-112.

[36] Ducros, F., Ferrand, V., Nicaud, F., Weber, C., Darracq, D., Gachareiu, C., Poinsot, T. “Large-Eddy Simulation of the Shock / Turbulence Interaction” Journal of Computational Physics, Vol. 152, No. 2., 1999, pp. 517-549.

[37] Cutler, A.D., Magnotti, G., Cantu, L.M.L., Gallo, E.C.A., Daheny, P.M., Baurle, R., Rockwell, R., Goyne, C., and McDaniel, J. “Measurement of Vibrational Non-equilibrium in a Supersonic Freestream using Dual-Pump CARS”, AIAA Paper 2012-3199, June, 2012.

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Figure 1: Turbulence-generating surfaces on the upper and lower surfaces of the flame-holder and the fuel-injection holes.

Figure 2: Eddy locations with flame-holder top boundary layer (left) and injection holes (right)

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Vitiated air stream

Hydrogen jet

Lifted turbulent flame

Figure 3: Temperature snapshots at X-Y centerplane: Burrows and Kurkov reacting wall jet experiment

Figure 4: Species mole fraction profiles at combustor exit - Burrows and Kurkov reacting wall jet experiment

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Figure 6: Axial velocity (top) and temperature (bottom) profiles at different axial stations within the DLR combustor

Figure 5: Schematic of DLR combustor

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Figure 7: Snapshots of DLR combustor flowfield (top: temperature, bottom left: HO2 mass fraction, bottom right: H2O mass fraction)

Figure 8: Schematic of U. Virginia’s ‘Configuration A’ scramjet combustor

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Figure 9: Effect of ‘bi-modal’ quadrature approximation on mean temperature profiles (from left to right: interpolated CARS data, laminar chemistry, bi-modal approximation; from top to bottom: X/H = 6, X/H = 12, X/H = 18

Figure 10: Effect of various quadrature approximations on water production rate (top: laminar chemistry, middle: difference between Gaussian quadrature production rate and laminar chemistry production rate; bottom: difference between bimodal approximation production rate and laminar chemistry production rate; scales from -5e-9 (black) to 5e-9 (white))

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a.) b.) c.)

d.) e.)

Figure 11: Temperature distributions at X/H = -13.8 ( a. – vibrational temperature (interpolated CARS data), b. – rotational temperature (interpolated CARS data), c. – static temperature (thermal equilibrium), d. – stagnation temperature (from thermal equilibrium data), e. – stagnation temperature (predicted from nozzle solution)

Figure 12: Temperature contours at centerplane and at X/H = 12 (top: Jachimowski , bottom: O’Connaire, et al.)

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Figure 13: OH mass fraction contours at centerplane and at X/H = 12 (top: Jachimowski; bottom: O’Connaire, et al.)

Figure 14: Iso-surface of OH mass fraction (0.005) colored by HO2 mass fraction (top: O’Connaire, bottom: Jachimowski

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Figure 15: Temperature (top set) and standard deviation (botttom set) (from top to bottom on each set: X/H = 6, X/H = 12, X/H = 18; from left to right in each set: Jachimowski, O’Connaire, et al., interpolated CARS data)

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Figure 16: Nitrogen mole fraction (top set) and standard deviation (botttom set) (from top to bottom on each set: X/H = 6, X/H = 12, X/H = 18; from left to right in each set: Jachimowski, O’Connaire, et al., interpolated CARS data)

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Figure 17: Oxygen mole fraction (top set) and standard deviation (botttom set) (from top to bottom on each set:

X/H = 6, X/H = 12, X/H = 18; from left to right in each set: Jachimowski, O’Connaire, et al., interpolated CARS data)

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Figure 18: Hydrogen mole fraction (top set) and standard deviation (botttom set) (from top to bottom on each set: X/H = 6, X/H = 12, X/H = 18; from left to right in each set: Jachimowski, O’Connaire, et al., interpolated CARS data)

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Figure 19: Wall pressure distributions (Jachimowski hydrogen-air model)

Figure 20: Wall pressure distributions (O’Connaire, et al. hydrogen-air model)