A coupled LES-ODT model for spatially-developing turbulent reacting shear layers

International Journal of Heat and Mass Transfer 127 (2018) 458–473

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

A coupled LES-ODT model for spatially-developing turbulent reacting shear layers Andreas F. Hoffie, Tarek Echekki ⇑ Department of Mechanical and Aerospace Engineering, North Carolina State University, Engineering Building III, 911 Oval Drive, Raleigh, NC 27695, United States

a r t i c l e

i n f o

Article history: Received 23 March 2018 Received in revised form 28 May 2018 Accepted 21 June 2018

2018 MSC: 00-01 99-00 Keywords: Large Eddy simulation One-dimensional turbulence LES-ODT Turbulent reacting spatially-developing shear layers

a b s t r a c t Large-eddy simulation (LES) for momentum transport combined with the one-dimensional turbulence (ODT) model for momentum and reactive scalars’ transport is designed to capture subgrid scale (SGS) turbulence-chemistry interactions. An extension of the original LES-ODT formulation is developed to capture these interactions in turbulent spatially-developing reacting shear layers. The LES-ODT results are compared to results from direct numerical simulations (DNS). Lewis number parametric variations for the variable-density simulations are carried out. The validation with DNS shows that the LES-ODT approach can qualitatively and quantitatively capture important salient features of turbulent shear layers statistics, including large-scale flow patterns, shear layer growth and mean and RMS statistics of velocity and reactive scalars. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction A principal challenge in turbulent combustion modeling is the capture of subgrid scale effects associated with the coupling of chemistry and molecular transport. A common strategy in turbulent combustion models is to exploit the separation of scales (e.g. the eddy-dissipation concept (EDC), the eddy break-up (EBU) model and the flamelet model [1–3]) or the behavior of thermochemical scalars in terms of a subset of scalars in composition space (e.g. the conditional moment closure (CMC) model, [4,5]). Other modeling approaches adopt a fundamentally different strategy by directly evaluating closure terms, which renders them more regime-independent [6]. Such modeling approaches include the PDF-transport methods [7] and the models based on coupling the linear eddy model (LEM) [8] and its extension to LES, the LEMLES approach [9,10] and the one-dimensional turbulence model (ODT) [11,12] and its extension to LES [13–18]. The LES-ODT framework attempts to combine two solutions: a 3D LES solution on a coarse grid with embedded 1D ODT solutions in the computational domain. the fine-grained ODT solutions are designed to capture unresolved subgrid scale (SGS) effects and ⇑ Corresponding author. E-mail address: [email protected] (T. Echekki). https://doi.org/10.1016/j.ijheatmasstransfer.2018.06.105 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

address the problem of closure for turbulence-chemistry interactions. The objective of the present study is to extend the LESODT framework originally developed for combustion by Cao and Echekki [13] to turbulent shear layers where velocity and scalar gradients are dominated by one principal direction. Some of the key contributions related to this study pertain to the different treatments implemented in the different directions of the flow. This includes a variable mesh implementation along the transverse direction and the implementation of different boundary conditions in the 3 directions, including inflow and outflow boundary conditions in the streamwise direction, vanishing gradients in the transverse direction and periodic boundary conditions in the spanwise direction. More importantly, a new treatmet at the so-called ‘‘nodes” is implemented to establish a robust coupling between the 3 directions of the flow. Such a treatment is important to reproduce the spatial structures associated with turbulent shear layers. The model is validated against direct numerical simulations (DNS) for turbulent spatially-developing shear layers. The paper is organized as follows. The model formulation is presented in Section 2 and its numerical implementation is discussed in Section 3. Results are presented in Section 4 for both constant density and variable density simulations based on comparison with DNS statistics.

459

A.F. Hoffie, T. Echekki / International Journal of Heat and Mass Transfer 127 (2018) 458–473


, defined at a spafilter size D, such that a generic filtered quantity u cial position, x, and time, t is expressed as

2. Model formulation Different variants of the LES-ODT method have been studied. Balasubramanian [19] and Sedhai [20] implemented a formulation, which advects the ODT domains along the flame brush in a Lagrangian fashion. In the Cao and Echekki [13] formulation, which is used as a starting point for the work, the ODT domains are fixed in space. This conforms to an Eulerian description of the flow field. More recent work by Park and Echekki [14], Ben Rejeb and Echekki [15] and Fu and Echekki [16] are based on this Eulerian formulation. The ODT formulation simulates the time evolution of velocities and scalars along a one-dimensional domain. This domain represents a notional line of sight through turbulent flow. In Cartesian coordinates and in the Eulerian context, such a one-dimensional domain can be thought to be aligned with the (x1 )-coordinate for example. Contribution along this line can be spatially and temporally resolved; whereas contributions from the second (x2 ) and third (x3 ) orthogonal directions need to be modeled. The same principle applies for notional lines parallel to the other two coordinate directions. Fig. 1 shows the embedded 1D ODT domains as a threedimensional lattice in the LES computational domain, illustrated here with one LES cell. The ODT domains are aligned with the LES cell edges and can span the extent of the computational domain. An ODT node is defined as the intersection point of three orthogonal ODT domains. In the present work, the ODT grid is aligned in such a way that the ODT node coincides with the LES node, which in the general case is not necessary. It is important to note that at the ODT node, all three orthogonal directions represented by the 2 ODT domains are resolved, such that no modeling is necessary. Outside these nodes, the ODT governing equations, as discussed below, exhibit both resolved and unresolved contributions with the latter requiring additional modeling.


ðxj ; tÞ ¼ u

Z D

uðx0j ; tÞ  Gðxj  x0j ; DÞdx0j :

ð3Þ

The symbol ‘‘” corresponds to Favre filtering

qu e ðx; tÞ ¼ u : q


ð4Þ

The subgrid scale (SGS) stress tensor,

sSGS ij , given as

~i u ~ j  ug
ðu sSGS ¼q i uj Þ: ij

ð5Þ


ij is expressed in terms of the dynamic The viscous stress tensor s viscosity l and the strain rate tensor, Sij

s
ij ¼ 2lSij ;

ð6Þ

where

Sij ¼

  ~i @ u ~j ~k 1 @u 2 @u  dij þ : 2 @xj @xi 3 @xk

ð7Þ

where l is the dynamic viscosity. In the present validation studies, sSGS given in Eq. (5) is modeled using the Smagorinsky model: ij




mT Sij ; where mT ¼ ðC S DÞ2 2Sij Sij sSGS ¼ 2q ij

1=2

:

ð8Þ

In this expression, C S is the Smagorinsky constant. Within the context of the present LES-ODT formulation, the filtered density is computed by spatially filtering the ODT density, which is evaluated on the subgrid based on an equation of state, such as the ideal gas equation. The density filtering is a substitute for the solution of the continuity equation in LES. Instead, the Poisson equation is solved to enforce continuity and resolve the pressure field. In a recent study by Fu and Echekki [16] both the continuity equation and density filtering from ODT are solved together to yield a ‘‘smooth” solution for the density field using Kalman filtering. However, this approach is not adopted here.

2.1. LES governing equations

2.2. ODT governing equations

In LES-ODT, the LES governing equations cover only the largescale motion using mass and momentum conservations (or momentum and the Poisson equation under the low-Mach number assumption). By performing a density-weighted (Favre) filtering operation on the Navier-Stokes equations the filtered LES governing equations are obtained, which are given as. LES continuity

The ODT governing equations include transport for the momentum components and the thermo-chemical scalars, expressed here in terms of the temperature and the species equations: 1D momentum


@q
u~i @q ¼0 þ @t @xi

ð1Þ

LES momentum

q


~i ~
@s
ij @ sSGS @u @u @p ij ~j i 
u þ þ ; ¼ q @t @xj @xi @xj @xj

  @ui @u 1 @ si1 ~1 i þ þ Xui ¼ u @t @x1 q @x1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} resolved    

@ui @u 1 @ si2 @ si3 ~2 ~3 i þ þ  u þu þ ; @x2 @x3 q @x2 @x3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð9Þ

modeled

1D energy

ð2Þ

assuming negligible external body forces. The symbol ‘‘-” corresponds to spatial filtering by a filter function G with a characteristic

    @T @T 1 @ q_ 001 XN _ k þ XT ~1 ¼ u þ  h x k k¼1 @t @x1 qcp @x1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} resolved

   

@T @T 1 @ q_ 002 @ q_ 003 ~2 ~3 þ þ  u þu þ ; @x2 @x3 qcp @x2 @x3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð10Þ

modeled

1D species

  _k @Y k @Y 1 @J k;1 x ~1 k þ þ þ XY k ¼ u @t @x1 q @x1 q |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} resolved

Fig. 1. Three-dimensional alignment of LES cell and ODT domains.

   

@Y @Y 1 @J k;2 @J k;3 ~2 k þ u ~3 k þ þ  u þ : @x2 @x3 q @x2 @x3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} modeled

ð11Þ

460

A.F. Hoffie, T. Echekki / International Journal of Heat and Mass Transfer 127 (2018) 458–473

Here, x1 corresponds to the ODT direction. The remaining directions on a Cartesian grid correspond to x2 and x3 . For example, in the same Cartesian coordinates, x1 , x2 and x3 can correspond to y, x and z, respectively, if the ODT is aligned with the y direction. In the above equations, q_ 00 is a heat flux, which represents molecular diffusion of heat due to temperature and species gradients as well ~ j is the large-scale advective velocity, as radiative heat transfer. u which is responsible for large-scale momentum and scalar transport. An additional advective transport contribution comes from the SGS and is modeled using stochastic stirring events. These contributions are represented by the terms: Xui , XT and XY k . The stochastic stirring events are handled as parallel processes to the remaining ‘‘deterministic” contributions to the ODT governing equations. In these equations, terms in the square brackets, ½: are fully resolved along the 1D direction. Terms within the curly brackets, f:g are terms that are not resolved; and, therefore, they must be modeled. They include large-scale advection and diffusion contributions along the remaining directions x2 and x3 . In contrast with the original formulation by Cao and Echekki [13], a special treatment is developed here at the nodes, which corresponds to intersections of ODT domains. In the original formulation, node advection was treated as another parallel process implemented at the nodes as ‘‘node advection” and then propagated between the nodes as ‘‘intra-node advection”. Here, we solve an independent set of governing equations at the nodes as 3D solutions. The motivation behind this revised node treatment is related to the need to capture the spatial structures characteristic of shear flows, especially in the presence of a dominant flow direction, such as the problem considered here. The treatment at the nodes is what imparts the 3D structure to the 1D ODT solutions when coupled at the nodes; and a 3D implementation instead of the previous 1D treatment at the nodes by Cao and Echekki [13] provides a stronger coupling between the ODT solutions in different directions in the presence of different magnitudes of the LES velocity components. The governing equations at the nodes are written as follows: Node momentum

  @ui @u 1 @ sij ~j i þ þ Xui ; ¼ u @t @xj q @xj |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð12Þ

resolved

Node energy/temperature

" ! # 00 @T @T 1 @ q_ j XN 000 _ _ ~ ¼ uj þ  h x þq þ XT ; k¼1 k k @t @xj qcp @xj |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð13Þ

resolved

This approach provides an instantaneous synchronization of the solutions at the nodes in conjunction with the intra-node processes. Stirring events are permitted at the nodes as large eddyturnover events are allowed to reach over two LES intervals. The ODT solution advancement at the subgrid is coupled with the LES time-loop. The advective large-scale velocity provided at the subgrid level is interpolated from LES and the density provided at the large-scale level is filtered from ODT. The ODT governing equations are solved by decomposing the physical contributions for advection, reaction-diffusion and stirring into separate ‘‘events”, each evolving with its own time step. The implementation of reaction, diffusion, SGS stirring and filtered advection as ‘‘parallel” events is the same strategy adopted by Cao and Echekki [13] in their original formulation of LES-ODT. The main distinction here is how the different events are implemented in both their formulation and their numerical implementation, as discussed below. The ODT solver is called from within the LES time loop (i.e. there are sub-cycles of ODT solutions within a given LES time step). Beside solving LES and ODT governing equations, the solution procedure includes downscaling of the large-scale advective velocity, correction of the ODT velocity field and in case of variable density flow, passing the filtered ODT density back to the LES solution (upscaling). As depicted in Fig. 1, the one-dimensional ODT domains are embedded into the LES domain and are aligned with the LES grid. The coupling of ODT domains is done three-dimensionally at an ODT node. Unresolved contributions are interpolated based on resolved nodal values. The overall ODT time step, which includes sub-steps for the parallel ODT process, is equal to the LES time step (Dt ODT ¼ DtLES ), but is shifted by Dt LES =2. This is done because a synchronization step is necessary between LES and ODT. The large-scale LES advection velocity is passed to ODT and the large-scale velocities of LES and ODT are synchronized at the beginning of each ODT time step (end of LES time step). In case of a variable density flow, the filtered ODT density is passed back to LES. This is illustrated in Fig. 2, with ~ i;LES and arrows between the LES and ODT time levels, labeled with u ~ ODT . q An important numerical implementation that has been incorporated in the revised LES-ODT model is a variable mesh along the transverse direction (i.e. the direction of mean flow and scalar gradients) based on the formulation of Lignell et al. [21]. A stretched grid is necessary for the shear layer cases considered, in order to resolve large gradients in the center of the domain. In the following discussion, we present in some detail the revised LES-ODT formulation and some numerical aspects of its implementation:

Node species

  _k @Y k @Y 1 @J k;j x ~j k þ þ þ XY k : ¼ u @t @xj q @xj q |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} resolved

ð14Þ

2.3. Large-scale advection in ODT The ODT large-scale advection update is based on the operator splitting method as introduced above. The update consists of two

Fig. 2. LES-ODT time advancement.

461

A.F. Hoffie, T. Echekki / International Journal of Heat and Mass Transfer 127 (2018) 458–473

steps: (1) node advection and (2) intra-node advection. As an extension of the earlier work by Cao and Echekki [13]. For a generic variable u, which in the present model corresponds to velocity components, temperature and species equations, 2 classes of equations are used to advance the large-scale advection operator. The first one is the solution at the nodes:



 @ ui @ ui @ ui @ ui ~2 ~3 ~1 ; u u ¼ u @t @x1 @x2 @x3

ð15Þ

(

!)

  @ui 1 @ si1 1 @d si2 @d si3 þ þ ; ¼ @t q @x1 q @x2 @x3 |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} resolved

modeled

1D energy diffusion

!)    ( @T 1 @ q_1 00 XN 1 @d q_2 00 @d q_3 00 _ ¼  hx þ þ ; k¼1 k k @t qcp @x1 qcp @x2 @x3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} resolved

1D species diffusion

ð16Þ

The intra-node advection equation is written for ODT domains parb corresponds to the unresolved component, which is allel to x1 . u modeled as a 1D spatial filter along the x2 and x3 directions and the filter value is evaluated at the node. Such a 1D filter (in contrast with the 3D filter adopted for the evaluation of the density) represents the contributions of the mean velocity and scalar gradients. Each resolved term on the right-hand-side of Eqs. (15) and (16) is expressed as

~j u

    þ  @ ui @ ui @ ui ~ þj ~ j ¼ u þu @xj @xj @xj

ð17Þ

and the advective velocities are obtained using:

~ þj ¼ u

1 1 ~ j þ ju ~ j  ju ~ j j and u ~ j ¼ u ~j j ; u 2 2

ð18Þ

~ j > 0) the backward difference grasuch that for the upwind flow (u   ~ j < 0) flow dient operator @ ui =@xj is used and for downwind (u  þ the forward difference operator @ ui =@xj is employed. The advection update ensures scalar boundedness and concludes with an update of the boundary conditions.

Analogous to advection, the reaction-diffusion step can be written as decomposed into its operator update. As for the advection update, the scalar boundness for the reaction-diffusion step is enforced for nodal as well as the intra-node update and is concluded by an update of the boundary conditions. The different treatments for the nodes and the remaining ODT grid cells is another novel element of the present formulation. For the node update, the gradients are fully resolved in all three dimensions for momentum, energy and species; and the reaction– diffusion operator of the governing equations are written as follows:



@ui 1 @ sij ¼ @t q @xj |fflfflfflfflffl{zfflfflfflfflffl}

ð19Þ

resolved

   @T 1 @ q_j 00 XN _k :  h x ¼ k k¼1 @t qcp @xj |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð20Þ

resolved

   @Y k 1 @J k;j _k : þx ¼ @t q @xj |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

!)    ( d d @J @Y k 1 @J k;1 1 @J k;2 k;3 _k þ þx þ : ¼ @t q @x q @x2 @x3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl1{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} resolved

ð24Þ

modeled

^

Again, the symbol ‘ ’ indicates a 1D spatial filter along the complementary orthogonal directions onto the ODT domain under consideration evaluated at the node. This model element also is different from the original implementation of LES-ODT by Cao and Echekki [13], which substituted the unresolved contributions from the x1 and x2 directions by scaling the diffusion operator in the resolved direction by a factor. Considering the ODT temperature diffusion contribution for node and intra-nodes given in Eqs. (20) and (23), substituting the expression for heat flux, as described by Fourier’s law of heat conduction

q_ 00j ¼ k

@T @xj

ð25Þ

where k is the heat conductivity, we obtain: Node energy diffusion

   X  N @T 1 @ @ _  ¼ k h x ; k¼1 k k @t qcp @xj @xj |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð26Þ

resolved

1D (intra-node) energy diffusion

2.4. Reaction-diffusion



ð23Þ

modeled

while, between the nodes, the following equation is adopted:

)   ( @ ui @ ui @d ui c @d ui c ~2 ~3 ~1 ; þ u u ¼ u @t @x1 @x2 @x3

ð22Þ

   X  N @T 1 @ @T _  ¼ k h x k¼1 k k @t qcp @x1 @x1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} "

resolved

! !# d d 1 @ @T @ @T k k þ : þ @x3 @x3 qcp @x2 @x2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð27Þ

modeled

For the diffusion equations, assuming the following model for the species diffusive flux for species k, as described by Fick’s law

J k;j ¼ qDk

@Y k @xj

ð28Þ

where Dk is the kth species diffusion coefficient, the node and intranode, respectively, reaction–diffusion contributions to the species equations are written as follows: Node species diffusion

    @Y k 1 @ @Y qDk k þ x_ k ; ¼ @t q @xj @xj |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð29Þ

resolved

ð21Þ

resolved

For the one-dimensional diffusion the intra-node update needs are split into a resolved and an unresolved (modeled) part, as follows: 1D momentum diffusion

1D (intra-node) species diffusion

    @Y k 1 @ @Y qDk k þ x_ k ¼ @t q @x1 @x1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} resolved " ! !# d d 1 @ @Y k @ @Y k þ qDk qDk : þ @x3 q @x2 @x2 @x3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} modeled

ð30Þ

462

A.F. Hoffie, T. Echekki / International Journal of Heat and Mass Transfer 127 (2018) 458–473

  /i ðxj Þ ! /i Mðxj Þ :

2.5. Stirring and subgrid scale transport Stirring event emulate in 1D the rotational folding effects of 3D turbulence. The ODT governing equations, Eqs. (12)–(14) introduce the stirring event as a parallel process, which is represented by the stochastic contribution, X, for momentum, energy and species

@ ui ¼ Xui ; @t

ð31Þ

where in fact, the stirring event consists of seven sub-steps, which are (1) eddy size selection le , (2) eddy start location selection xj;0 , (3) eddy rate distribution function evaluation ke , (4) eddy acceptance probability evaluation P e , (5) eddy rejection tests, (6) execute triplet map, (7) stirring time step adjustment (Fig. 4). The eddy size is sampled from a probability density function (PDF) distribution from scales ranging from a minimum eddy size corresponding to the Kolmogorov scales to a maximum size, which is a factor of the LES filter, D, typically chosen here as 2. The rate of sampling of eddies for stirring events is governed by an eddy rate distribution function, which is expressed as (Kerstein et al. [22]):

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 X uj;K le 2 Cm u ui;K le ke ¼ 4 t þ ae T ij  Z: m m le j

ð32Þ

ui;K and uj;K the velocity after triplet mapping. ae the maximum allowable energy that can be exchanged between velocity components and usually a value of 2/3 is chosen for equi-partitioning of energy between velocity components. T ij is a transfer matrix and Z is the viscous penalty parameter, which for large numbers suppresses eddy events when the term inside the square root of Eq. (32) is negative. The probability for the occurrence of an eddy event can be estimated with the information given above as:

Pe ¼

ke  Dts ; f g

ð33Þ

where Dt s is the stirring time step. Eddies are rejected by the selection mechanism, (a) if the term under the square root of the eddy rate distribution (Eq. (32)) is negative; (b) if the acceptance probability is less than a random number sampled in the interval of [0;1]; and (c) for turbulent jets and shear layers, if the sampled eddy is ‘‘immature”, telap < bm se , where the current elapsed flow time is less than the characteristic eddy turn over time. This prevents the formation of over-sized vortices near the inlet. The elapsed time is  determined by t elap ¼ xj;0 =U m , with U m ¼ ui;max þ ui;min =2. If the sampled eddy passes all rejection tests, the triplet map is executed. Analytically, the triplet map is expressed as:

   ui ðxj Þ ! ui Mðxj Þ þ ci  xj  Mðxj Þ ;

ð34Þ

ð35Þ

For an eddy within the interval of [xj;0 ; xj;0 þ le ], the triplet mapping operation Mðxj Þ is defined as

8 3ðxj  xj;0 Þ > > > < 2l  3ðx  x Þ e j j;0 Mðxj Þ ¼ xj;0 þ > 3ðxj  xj;0 Þ  2le > > : ðxj  xj0 Þ

if xj;0 6 xj 6 xj;0 þ 1=3le if xj;0 þ 1=3le 6 xj 6 xj;0 þ 2=3le if xj;0 þ 2=3le 6 xj 6 xj;0 þ le

:

otherwise ð36Þ

The mapping procedure can be divided into three steps: (1) compression, (2) copy and (3) inversion. The compression step shrinks property profiles within the considered interval to a third of their original extent, and then fills the interval with three compressed copies of the profiles. The middle copy is inverted, which maintains the continuity of mapped properties and introduces the rotational folding effect of turbulent eddy motion, as given in Fig. 3. Transported variables outside the selected eddies are unaffected. As indicated in Eq. (34), an additional procedure is implemented  on the velocity field, which corresponds to ci  xj  Mðxj Þ . The term xj  Mðxj Þ is the kernel transformation function. Its contribution corresponds to what can be denoted as ‘‘pressure scrambling” or ‘‘return to isotropy”, which partitions kinetic energy among the 3 velocity components with ci being a parameter that controls the extent of this partitioning. This parameter is expressed as:

ci ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffii X 27 h ui;K þ sgnðui;K Þ u2i;K þ ae j T i;j u2j;K : 4l

ð37Þ

Again, ae is a free parameter as discussed earlier. In the case where there are two or three stirring events intersecting at an ODT node, the arithmetic mean is computed and reassigned to that node. In the case of a variable mesh, mapping the eddy profile from a stretched to a homogeneous grid and back is necessary. This mapping procedure is done over the eddy size only. 2.6. Model downscaling: momentum correction coupling The LES and ODT are coupled through the exchange of the largescale momentum components (from LES to ODT or downscaling) and the filtered density (from ODT to LES or upscaling). Both LES and ODT solve for the momentum equations. The LES solution captures the 3D effects associated with geometry and large scales; while, ODT is able to capture SGS transport. In the present formulation, as well as in the original formulation by Cao and Echekki [13], the large-scale ODT momemtum contribution is replaced by the LES component; while, the remaining residual component from ODT is maintained. This substitution, denoted here is model ‘‘downscaling”, is carried out every time the integration of ODT solution over one LES time step is completed. The corrected ODT velocity components are expressed as follows:

and

Fig. 3. Triplet map procedure, [23].

A.F. Hoffie, T. Echekki / International Journal of Heat and Mass Transfer 127 (2018) 458–473

~ i;LES þ u00i;ODT : ui;ODT ¼ u

ð38Þ

Here, u00i;ODT is the residual part of the ith velocity component and ~ i;LES is the ith velocity component from the momentum equation u from LES interpolated onto the ODT grid. The residual velocity component is determined as follows. First, a 1D spatial filter of the ODT velocity component ui;ODT prior to the correction is implemented ~ i;ODT . Second, the residual term is obtained by subtractresulting in u ~ i;ODT from the uncorrected ODT ing the filtered velocity component u ~ i;ODT . velocity component, i.e. u00i;ODT ¼ ui;ODT  u

463

perature field and then 3D spatially filtered as defined by Eq. (3) and passed to the LES solution. 3. Problem set-up and numerical configurations In this section, we discuss the numerical configuration used to validate our revised LES-ODT fomulation. The numerical set-up is identical to that by Mason [24] whose code is the starting codes for our DNS and LES-ODT codes. 3.1. Case descriptions

2.7. Model upscaling: variable density coupling While the LES solution provides large-scale momentum components, the ODT solution, within the low Mach number formulation adopted here, provides the filtered density. For variable density flows, the filtered density from ODT is used in the LES governing equations by implementing a 3D spatial filter of the ODT solutions. This process is denoted here as model ‘‘upscaling”. The ODT density field is calculated based on the equation of state from the ODT tem-

Fig. 5a depicts a principle thought-experimental shear layer setup. Two parallel streams of fluid at different velocities U 1 and U 2 , where U 1 > U 2 are initially separated by a plate, but then are allowed to interact at the end of the plate. The initial and inlet condition for the streamwise velocity component u1 ¼ u is specified with the non-dimensional hyperbolic tangent profile

U ¼ Um þ

  DU y  Ly =2 ; tanh 2 r

ð39Þ

where DU ¼ ðU 2  U 1 Þ, U m ¼ ðU 1 þ U 2 Þ=2 and r is the steepness of the profile. The formation of streamwise vortices, vortex roll-up and pairing is enhanced by superimposing inlet perturbations on the streamwise and spanwise velocity components through forcing of the fundamental modes calculated from linear inviscid stability theory [24–26]. Vortex roll-up occurs through perturbations corresponding to the fundamental (Kelvin-Helmholtz) mode of the inlet velocity profile, with magnitude 7.5% of DU. This magnitude is implemented for DNS; for LES, twice that magnitude is implemented to match the growth of the shear layer at constant density conditions. This adjustment can be attributed to the different resolutions of the two solutions and are not related to the specific implementation of ODT within LES. Nonetheless, for both solutions adding first, second and third subharmonics enhances subsequent vortex pairing. The growth rate of shear layers depends on this vortex pairing. In an attempt to mimic this shear layer behavior,

Fig. 4. ODT stirring algorithm.

Fig. 5. Problem setup.

464

A.F. Hoffie, T. Echekki / International Journal of Heat and Mass Transfer 127 (2018) 458–473

randomly-walked phases are added to each of the subharmonics [27]. The initial and inlet transverse (u2 ¼ v ) and spanwise (u3 ¼ w) velocity components are given by pairs of counter-rotating, socalled Oseen vortices. The temporal and spatial resolutions are implemented to resolve the subgrid scale physics; and, therefore, the associated resolution requirements are similar to those needed for DNS. A constant density (q = 1) reacting shear layer has been examined in order to determine the appropriate ODT parameters, which correspond to the model constants C (see Eq. (32)) and bm . Once these parameters are determined, they are used to validate variable density simulations. The variable-density simulations are carried out with 3 different Lewis numbers, which measure the ratio of the thermal diffusivity to the mass diffusivities of the species, considered here as equal. The LES-ODT results are evaluated qualitatively with regards to flow field visualizations of temperature and the topology of the three-dimensional flame front. A quantitative evaluation of the LES-ODT model performance against DNS is done by comparing shear layer growth rate in the streamwise direction and by comparing first and second order statistics of the velocity and reactive scalars. The first order statistics corresponds to the Favre average (denoted as ‘‘mean”) and the second order statistic, which corresponds to the root-mean-squared (RMS) associated with that mean. 3.2. DNS and LES-ODT governing equations As stated earlier, both DNS and LES-ODT simulations are carried out to validate the model formulation for LES-ODT for turbulent shear layers. The non-dimensionalized DNS governing equations for momentum, energy and species (fuel and oxidizer) conservation are [24]:

  @ui 1 @uj uj @uj @ui 1 @p ¼  uj  þ @t 2 @xi @xi @xj q @xi "  # 1 @ 2 ui 1 @ @uj þ þ qRe @x2j 3 @xi @xj _ @T @T 1 @2T x þ uj ¼ þ q @t @xj qRePr @x2j

ð40Þ

ð41Þ

@Y F;O @Y F;O 1 @ Y F;O ¼  þ uj q @t @xj qRePrLe @x2j

ð42Þ

In the present study, we assume that the transport coefficients for heat, mass and momentum are constant for the constant and variable-density simulations. In the transport equations, Le is the Lewis number, which corresponds to the ratio of the thermal diffusivity to the mass diffusivity (considered constant and equal for all species), Pr is the Prandlt number, which corresponds to the ratio of the thermal diffusivity to the kinematic viscosity and Re is a Reynolds number based on a reference length, Lref , a velocity, U ref scales and the cold stream kinematic viscosity. Therefore, while Le and Pr represent physical attributes of the mixture, Re is closely related to the choice of the non-dimensionlization of the problem. The reaction rate is based on a single-step, irreversible reaction involving reactants, fuel (F) and oxidizer (O), and products (P):

F þ O ! P:

ð43Þ

and is expressed as:



h  a i po ¼ q 1 þ T 1a where the thermodynamic pressure, po , is considered constant and equal to 1. The DNS momentum equations are integrated by applying the fractional step method with the aid of Runge-Kutta timestepping. The fractional step method can be divided into four major steps. First, the advective and diffusive terms are integrated in time ~ j . Second, under the restriction that yielding predicted velocities u the predictor-step velocity field satisfies continuity (Eq. (1)), the Poisson equation for pressure is solved. Third, the pressure gradient is used to correct the predicted velocity. Finally, the continuity equation is monitored at each Runge-Kutta time step additionally to ensure the solution convergence for the pressure successiveover-relaxation solver. The ODT governing equations for momentum, temperature and fuel and oxidizer equations are written here in their dimensionless form for ODT grid points:

" #  ! @ui @ui 1 @ 2 ui 1 @ @u1 ~1 ¼ u þ þ þ Xui @t @x1 qRe @x21 3 @x1 @x1 ( ) d d @u @u 1 ~2 i  u ~3 i þ þ u ðvÞ @x2 @x3 qRe " # _ @T @T 1 @2T x ~1 ¼ u þ þ þ XT @t @x1 qRePr @x21 q 8 0 19 d d < d d @T @T 1 @ @ 2 T @ 2 T A= ~ ~3 þ u u þ þ : 2 @x2 @x3 qRePr @x22 @x23 ;

ð44Þ

ð45Þ

" # _ @Y F;O @Y 1 @2Y k x ~ 1 F;O þ  þ X ¼ u Yk @t @x1 qRePrLe @x21 q 8 0 19 < = 2d 2d d dk @Y @Y 1 @ Y @ Y F;O F;O F;O @ A ~2 ~3 þ u u þ þ 2 2 : ; @x2 @x3 qRePrLe @x2 @x3 ð46Þ

x_

2

x_ bð1  TÞ ¼ Da qY F Y O  exp q 1  að1  TÞ

with Damköhler number Da (an input parameter), Zel’dovich number b ¼ ðEa aÞ=ðRu T ad Þ; a ¼ ðT ad  T ref Þ=T ad the dimensionless temperature-rise. The following dimensionless equation of state is used to determine the mixture density:



where

0 1     d 2 d 1 @ d @u1  @ u 1 @ @u 2 A i v¼  þ@ 2 þ 3 @xi @x1  3 @xi @x2 @x2 i–1 0 1   d 2 d @ u 1 @ @u 3 i A þ@ 2 þ 3 @xi @x3 @x3

3.3. Run conditions Table 1 summarizes the key parameters for the DNS and LESODT simulations. The dimensionless heat release parameter a is set to 0.85, which yields, in the variable density runs, density ratios of 6.67. This value is comparable to values encountered in nondiluted hydrocarbon-air flames. The Zel’dovich parameter, b = 1.5, is relatively low (compared to practical values) to manage the stiffness of the chemistry integration in both DNS and ODT. The ODT model constants, C and bm , are tuned to 4.0 and 1, respectively, based on the constant density simulations. The same parameters are used in the variable density simulations.

A.F. Hoffie, T. Echekki / International Journal of Heat and Mass Transfer 127 (2018) 458–473

All of the simulations (DNS and LES-ODT) have been carried out on the ARC cluster at NCSU as shared memory (OpenMP) runs. Each node is equipped with 16 AMD Opteron processors, with 1.8 GHz and 32 GB RAM. Since no domain decomposition has been implemented on the original code, simulation times for DNS were approximately 30 days and those corresponding to LES-ODT were 7 days. A greater computational saving is expected if we consider a ratio between LES cells and ODT cells that are greater than the adopted 4. Given the fact that individual ODT solutions can reside entirely into single processors, implementing the LES-ODT formulation on distributed memory platforms can result in a significant computational savings and enhanced scalability. While the present formulation is not restricted to simple chemistry, more measures are needed to achieve additional computational saving. First, it is possible to prescribe different boundary conditions to the ODT domains from the LES solutions that do not extend to the LES boundaries; therefore, the ODT solutions can target only the portion of the domain that may require a sophisticated model in the SGS to capture the coupling of chemistry and transport. Second, beyond the standard strategies of chemistry reduction, additional strategies to reduce the thermo-chemical system into a subset of scalars can be explored. These include the transport of principal components that can be used to represent the composition space with the potential of significant saving in the number of transported scalars (see for example, Refs. [31,32]). 4. Results This section presents results of comparisons between LES-ODT simulations and DNS simulations for 2 cases presented here corresponding to constant-density (Section 4.1) and variable-density (Sections 4.2 and 4.3) reacting shear layers. It is important to note that statistics are constructed from the 1D ODT solutions and include the effects of molecular transport, turbulent (large-scale and SGS) transport and chemistry. 4.1. Constant density reacting shear layer

Fig. 6 shows instantaneous 2D contours of the fitered temperature fields at a dimensionless time t = 1200 from the DNS and the LES-ODT simulations. These are two-dimensional views in the x  y plane of the three-dimensional simulation at location z = 20. However, because of the homogeneity in the spanwise z-direction, any location can be chosen. Again, the 2D contours are solely based on the 1D ODT solutions and are obtained by filtering them using a 3D box filter. The structures observed at this particular simulation time are representative of the processes that occur in shear layers. The representation of the vortex structure of the flow by relying on the 1D ODT solutions is a manifestation of the downscaling coupling between LES and ODT and the representation of large-scale advection in the model. Downstream turbulent structures in the temperature field reveal that the shear layer is well beyond the laminar region and well into the mixing transition. It is clearly visible how the shear layer develops and forms spanwise vortices, for both LES-ODT and DNS. For DNS, spanwise vortex roll-up (Kelvin-Helmholtz instability) begins at about x  5. Between x  5 and x  30, three distinct and fully developed spanwise vortices (‘‘rollers”) are visible, indicating that roll-up is completed at x  30. At this location, entrainment from the high-speed fluid stream combined with vortex pairing takes place, which grows to larger vortices downstream. Another vortex pairing combined with fluid entrainment occurs at x  50. After this location, the randomly forced subharmonic modes introduce considerable amount of threedimensionality in the flow, which goes along with a loss of symmetry of the flow in the spanwise direction. This loss of symmetry is manifested in a deflection of the shear layer toward its low-speed side resulting from a greater entrainment of fluid from the highspeed side and a greater downward spin of the rollers relative to their upward spin. This deflection is equally exhibited by the scalar fields. The asymmetry has been observed by different authors in computations and experiments (see for example, Refs. [24,25,27,30]). For LES-ODT, vortex roll-up also starts at x  5. Between x  5 and x  30, two distinct and fully developed spanwise vortices are visible. Vortex pairing occurs at x  30 and x  50, analogous

This section summarizes the results from the three-dimensional simulations of the constant density, reacting shear layer of LESODT and DNS. Constant density decouples heat release from chemical reactions from fluid dynamics and to narrow down correct ODT parameter settings.

Table 1 Summary of simulation parameters. Parameter

Description

Value

Geometry and numerics Grid DNS  N x  N y  N z DNS resolution LES resolution LES  N x  N y  N z ODT resolution ODT  N x  N y  N z

481  241  81 121  61  21 961  481  161

Velocities U 1 , U 2 , DU, U m

1.5, 0.5, 1.0, 1.0

Inlet velocities’ parameters

Non-dimensional scaling factors Re Reynolds number at inlet Pr Prandtl number Le Lewis number Da Damköhler number a Heat release parameter b Zel’dovich number

200 0.7 0.5, 1.0, 2.0 25.0 0.85 1.5

ODT parameters C Z bm

4.0 0.001 1.0

Constant for eddy rate distribution Viscous penalty parameter Beta for eddy maturing condition

465

e -field of ODT and DNS (z = 20, t = 1200, q = 1). Fig. 6. Instantaneous T

466

A.F. Hoffie, T. Echekki / International Journal of Heat and Mass Transfer 127 (2018) 458–473

to DNS. Vortex pairing goes along with fluid entrainment from the high-speed fluid stream, a strong increase in shear layer growth and a loss of symmetry beyond x  90. Next, we consider the growth rate of the shear layers based on comparisons of LES-ODT and DNS results. Fig. 7 compares LES-ODT and DNS growth rates. In this study, we use the definition of the 1% shear layer thickness, d, which can also be determined based on the ~ . d is defined as the location Favre-averaged streamwise velocity, u outside of the shear layer, where the velocity is 99% of the free stream value, where the 1% change is based on the velocity difference DU ¼ U 2  U 1 . Defining the 99% velocities as u for the upper edge and as 99 ¼ U 1  0:01  DU ~þ u 99 ¼ U 2 þ 0:01  DU for the lower edge. The 1% thickness correþ ~ sponds to the difference between the two positions at u 99 and u99 . The growth rate is monotonic and is nearly linear. This is an indication that the vortex pairings are evenly distributed downstream as a result of the randomly-walked phases imposed on the subharmonic modes. The convergence of the growth rate can be determined by computing it after several different sampling times. The Reynolds number at the exit (x ¼ 120) based on the one-percent thickness is Red  5300. In order to validate the LES-ODT model quantitatively, first and second order statistics are evaluated at three downstream locations x ¼ 30, 60, 90 and for the sampling period of eight flowthrough times. For the constant density case, Favre-averaging automatically reduces to Reynolds-averaging. The means and RMS of the u; v and w velocity component are shown in Figs. 8–10. These statistics are averaged over time and the spanwise z-direction. All statistics presented below are Favreaveraged; although, for the constant density cases, they also correspond to Reynolds-averaged statistics. The mean profile for u shows the typical hyperbolic tangent shape with a value of 0.5 in the low speed stream and a value of 1.5 for the high-speed stream. As the spread of the shear layer increases downstream, the slope of the profile decreases. LES-ODT and DNS show excellent agreement at all of three locations. This is also true for the v mean component, with small values between 0.05 and 0.05 only. The mean of w is zero and it indicates the homogeneity in the z-direction, with increasing asymmetry due to increase in small-scale turbulence toward the exit of the domain. LES-ODT is able to capture this phenomenon. The RMS for each u; v and w show a similar dome-like shape, but they are different in magnitude. The RMS values also visualize the spread of the shear layer along the downstream direction. The agreements between LES-ODT are again excellent, and agree with other studies under similar conditions from literature, such as Oster and Wygnanski [28] as well as Lowery and Reynolds [29]. The difference of the w RMS peak value is most likely due to a small 30 L DNS Le = 1.0 LES-ODT

25 20

δ

15 10 5 0

0

20

40

60

x

80

100

120

Fig. 7. Shear layer growth rate of LES-ODT and DNS, q = 1.

difference in initial conditions and interaction with forcing. The ODT initial condition is generated by interpolating DNS onto ODT. For LES-ODT, the u-velocity RMS values show a small ‘‘cup” (non-smoothness) in the peak-region on the low-speed side. Lowery and Reynolds [31] note that the ‘‘cup” is a consequence of forcing. In certain regions, vortices rotate and convect downstream without much movement in the transverse direction. Profiles without this cup therefore indicate a shear layer that behaves similar to an unforced layer. Mean and RMS values of temperature as well as fuel and oxidizer mass fractions are given in Figs. 11–13. The mean temperature profile resembles the Gaussian profile, as defined at the inlet. The peak value decreases downstream as the shear layer spreads. The mean profiles for fuel and oxidizer mass fractions are similar to the streamwise velocity component profiles with Y F ¼ 0, Y O ¼ 1 on the high-speed side and Y F ¼ 1, Y O ¼ 0 on the low-speed side. RMS values for temperature reveal the double-peak, as the turbulent intensities are higher at the edge of the shear layer. This is also confirmed by the fuel and oxidizer mass fractions with a peak on the low- and the high-speed side respectively. LES-ODT is able to match mean and RMS of the DNS values reasonably well, with slightly overpredictions of the fuel mass fraction at x ¼ 90.

4.2. Variable density reacting shear layer: Le = 1 case This section presents findings of the reacting and variable density shear layer case for Le = 1. This case is considered the base case for the variable density runs. All of the effects simulated here are closer to physical reality, as the effects of heat release due to combustion are coupled with the fluid mechanics through the equation of state by changes in pressure. As stated earlier, the Lewis number is defined as the ratio of the thermal diffusivity to mass diffusivity where the mass diffusivity is assumed to be equal for all species and the Lewis number is treated as constant. The Le = 1.0 variable density reacting shear layer case is discussed next. Two-dimensional instantaneous filtered fields of temperature are shown in Fig. 14. The figure shows that the flow develops very differently from the constant density case. Both simulations show a more distinct formation of streamwise vortices. The shear layer appears thinner over the first third of the downstream distance and also the temperature distribution is thinner between the vortices as the mass diffusivity is decreased and the thermal diffusivity increased compared to the previously discussed case. There is no vortex pairing noticeable and the streamwise vortex appears at x  20 and larger vortices continue to roll-up further downstream. Here, heat release leads to the formation of streamwise vortices; and less interaction between streamwise and spanwise vortices lead to lesser formation of vortex pairing small-scale turbulence. Fig. 15 shows the shear layer growth for the case of Le = 1.0. As it seems typical for variable density, reacting shear layers, the shear layer growth is non-linear over the first part of the domain, after which the slope of the curve becomes flat and monotonic. This trend is consistent between DNS and LES-ODT. The flattening, monotonic behavior occurs at x  20. At the end of the domain, the thickness is reduced by 30% compared to the thickness of the constant density and non-reacting case. At x  90, the shear layer thickness exhibits a non-monotonic behavior with a rapid growth towards the exit boundary. We believe that this effect is related to the interactions between his release and the forcing designed to enhance vortex roll-up and pairing in the solution. A discussion of first and second order statistics of velocities and scalars for the case of Le = 1.0 is presented next. Figs. 16–18 show mean and RMS of the three velocity components in the transverse

467

A.F. Hoffie, T. Echekki / International Journal of Heat and Mass Transfer 127 (2018) 458–473

Fig. 8. Streamwise velocity first and second order statistics, q = 1.

x = 30

-0.05 20

y

60

0

−0.1 0

20

y

< vRMS >

0.3 0.2 0.1

40

60

y

40

60

y

0.3 0.2

40

60

DNS LES-ODT

0.5

0.1 20

20

0.6

0.4

0 0

0

-0.1 0

DNS LES-ODT

0.5

0.4

DNS LES-ODT

-0.05

0.6

DNS LES-ODT

0.5

< vRMS >

40

x = 90

0.05

−0.05

0.6

0 0

0.1

DNS LES-ODT



0

-0.1 0

x = 60

0.05



0.05



0.1

DNS LES-ODT

< vRMS >

0.1

0.4 0.3 0.2 0.1

20

y

40

0 0

60

20

y

40

60

Fig. 9. Transverse velocity first and second order statistics, q = 1.

x = 30 DNS LES-ODT

-0.025 20

y

40

60

-0.05 0

< wRMS >

0.05

20

y

40

60

y

40

60

-0.05 0

20

y

40

60

DNS LES-ODT

0.15

0.1

0 0

0

0.2

DNS LES-ODT

0.05

20

DNS LES-ODT

-0.025

0.15

0.1

x = 90

0.025

0

0.2

DNS LES-ODT

0.15

< wRMS >

DNS LES-ODT

-0.025

0.2

0 0

0.05



0

-0.05 0

x = 60

0.025



0.025



0.05

< wRMS >

0.05

0.1

0.05

20

y

40

60

0 0

Fig. 10. Spanwise velocity first and second order statistics, q = 1.

20

y

40

60

A.F. Hoffie, T. Echekki / International Journal of Heat and Mass Transfer 127 (2018) 458–473

x = 30 DNS LES-ODT

0.8 0.6 0.4 0.2 0 0

20

y

40

DNS LES-ODT

0.4

0.3 0.2 0.1 20

y

40

20

y

40

0.6 0.4

0 0

60

0.2 0.1 20

y

40

20

y

0.4

DNS LES-ODT

0.3

0 0

60

DNS LES-ODT

0.2

0.4

DNS LES-ODT

x = 90

0.8

0.6

0 0

60

< TRMS >

< TRMS >

1

0.2

0.4

0 0

x = 60

0.8





1



1

< TRMS >

468

60

DNS LES-ODT

0.3 0.2 0.1 0 0

60

40

20

y

40

60

DNS LES-ODT

20

y

0.6

60

DNS LES-ODT

20

y

< YF, RMS >

0.3 0.2

40

60

0.4 0.3 0.2

20

y

40

0 0

60

x = 90 DNS LES-ODT

20

y

40

60

DNS LES-ODT

0.5

0.1

0.1

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 0.6

DNS LES-ODT

0.5

0.4

0 0

x = 60

0.6

DNS LES-ODT

0.5

< YF, RMS >

40

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

< YF >

x = 30

< YF, RMS >

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

< YF >

< YF >

Fig. 11. Temperature first and second order statistics, q = 1.

0.4 0.3 0.2 0.1

20

y

40

0 0

60

20

y

40

60

DNS LES-ODT

20

y

0.6

60

DNS LES-ODT

20

y

< YO, RMS >

0.3 0.2

40

60

0.4 0.3 0.2

20

y

40

60

0 0

x = 90 DNS LES-ODT

20

y

40

60

DNS LES-ODT

0.5

0.1

0.1

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 0.6

DNS LES-ODT

0.5

0.4

0 0

x = 60

0.6

DNS LES-ODT

0.5

< YO, RMS >

40

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

< YO >

x = 30

< YO, RMS >

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

< YO >

< YO >

Fig. 12. Fuel mass fraction first and second order statistics, q = 1.

0.4 0.3 0.2 0.1

20

y

40

60

0 0

Fig. 13. Oxidizer mass fraction first and second order statistics, q = 1.

20

y

40

60

A.F. Hoffie, T. Echekki / International Journal of Heat and Mass Transfer 127 (2018) 458–473

LES-ODT

~ T

60

0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05

50

y

40 30 20 10 0

0

20

40

60 x

80

100

120

DNS

~ T

60 50

0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05

y

40 30 20 10 0

0

20

40

60 x

80

100

120

e -field of ODT and DNS (z = 20, t = 1200, Le = 1.0). Fig. 14. Instantaneous T

30 25 20

δ

15 10 DNS LES-ODT

5 0

0

20

40

60

x

80

100

120

Fig. 15. Shear layer growth rate of LES-ODT and DNS, Le = 1.0.

e and w e are in readirection at x = 30, 60, 90. The mean profiles for u sonable agreement between LES-ODT and DNS. As for RMS statistics, LES-ODT tends to overpredict the DNS values. While the DNS RMS peaks decrease when moving downe RMS , v e RMS , while w e RMS stream, the LES-ODT values increase for u does seem to follow the DNS trend here. Due to the overpredicted RMS values, there also is an overprediction in the spread of the layer by the ODT values. The overprediction in peak and spread for the case discussed can also be seen when looking at the scalar RMS values of temperature, as well as fuel and oxidizer mass fraction, given in Figs. 19– 21. The scalars also show that the LES-ODT solution tends to ‘‘ lean” toward the low-speed stream, while the means of the scalars agree fairly well with DNS. At the location of x ¼ 90, the temperature mean shows a small drop. Finally, based on the comparisons between DNS and LES-ODT statistics, we have identified both discrepancies and qualitative and quantitative agreements. Both DNS and LES-ODT for variable-density show that the qualitative development of the shear layer is different from the constant density case discussed earlier. The heat release through the density ratio is altering the

469

structure of the shear layer and impacting its growth rate. The LES-ODT exhibits a faster growth of the shear layer, which impacts the width of the mean and RMS profiles of velocity and scalars. Part of the discrepancy in the growth of the shear layer may be attributed to the LES closure for the SGS stresses, which, in our present study, is based on the Smagorinsky model with the same constant adopted for the constant density simulations. Fine-tuning the model constant can improve the comparisons of the shear layer growth in both simulations; but, this fine-tuning is not attempted here. Instead, we have chosen to‘‘freeze” the parameters adopted in the constant-density simulations and place more emphasis on establishing qualitative comparisons between the constantdensity and variable-density results as well as the comparison of the 3 different Lewis number effects. Instead of fine-tuning the Smagorinsky constant, a dynamic determination of the constant or a more complex model for the SGS stresses may be the better way to establish better quantitative comparisons between DNS and LES-ODT results. 4.3. Variable density reacting shear layer: Lewis number effects One of the key arguments advanced for methods that attempt to directly model SGS effects, such as the case for LES-ODT, is the ability of these models to capture the complex interactions between molecular processes, including diffusion and reaction, and advective transport. Such interactions can occur due to differential and preferential diffusion effects, where rates of molecular transport of either mass or heat can be different. In this study, we attempt to illustrate the LES-ODT capability to capture such effects through a comparison of 3 Lewis number cases of 0.5, 1 and 2. Here, we are still assuming that all species diffuse at the same rate; and, therefore, their Lewis numbers are equal. Therefore, we are simply contrasting the different diffusion rates of heat and mass. When Le < 1, the rate of diffusion of mass is faster than that of heat. The opposite applies to Le > 1. The consequences of different diffusion rates for heat and species can have important effects on their instantaneous profiles and the extent of chemistry in the reaction zone. Temperature plays a central role in chemistry due to its non-linear contribution to the reaction rate, especially through the exponential term in the Arrhenius expression. Figs. 22–24 compare LES-ODT and DNS RMS values of the temperature and fuel and oxidizer mass fractions with Lewis number as parameter. The profiles show that while temperature RMS peaks decrease with increasing Lewis numbers, the opposite is observed for the fuel and oxidizer mass fractions. Although, not clearly discernable from temperature RMS, the species RMS profiels also tend to be broader as the Lewis number increases. The same trends are observed in the DNS and LES-ODT statistics. Here, we attempt to put forwards an argument why the abovediscussed trends are observed with Lewis number effects. With lower rates of diffusion of heats, which correspond to a lower Le, temperature profiles tend to be thinner and the extent of reaction tends to be limited to a narrower reaction zone. These trends are expected due to the non-linear dependence of reaction to temperature through the exponential term in the Arrhenius expression. In contrast, higher Lewis numbers tend to generate broader temperature profiles and lower temperature instantaneous gradients. SGS transport is expected to play a stronger role if scalar profiles are steeper and can generate, in this process, higher RMS values. This argument can be strengthened by looking at the limit case of zero gradients of scalars, where SGS effects cannot play any role in altering these gradients in space. Based on the above argument, an increase in Le, results in a narrow instantaneous profile of temperature and broader instantaneous profiles of the fuel and oxidizer mass fractions, hence enhancing the temperature RMS and reducing the species RMS.

A.F. Hoffie, T. Echekki / International Journal of Heat and Mass Transfer 127 (2018) 458–473

DNS LES-ODT

20

0.6

y

60

DNS LES-ODT

20

y

< uRMS >

0.3 0.2 0.1

40

60

0.3 0.2 0.1

20

y

40

60

x = 90

DNS LES-ODT

20

y

40

60

DNS LES-ODT

0.5

0.4

0 0

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 0.6

DNS LES-ODT

0.5

0.4

0 0

x = 60

0.6

DNS LES-ODT

0.5

< uRMS >

40

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0



x = 30

< uRMS >

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0





470

0.4 0.3 0.2 0.1

20

y

40

0 0

60

20

y

40

60

Fig. 16. Streamwise velocity first and second order statistics, Le = 1.0.

DNS LES-ODT

40

60

DNS LES-ODT

20

y

0.5

DNS LES-ODT

< vRMS >

0.2 0.1

40

60

y

40

0.3 0.2

0 0

60

DNS LES-ODT

20

y

40

60

DNS LES-ODT

0.4

0.1 20

x = 90

0.5

DNS LES-ODT

0.4

0.3

0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0

< vRMS >

y

0.4

0 0

x = 60

20

0.5

< vRMS >

0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0



x = 30



0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0

0.3 0.2 0.1

20

y

40

0 0

60

20

y

40

60

Fig. 17. Transverse velocity first and second order statistics, Le = 1.0.

x = 30 DNS LES-ODT

-0.05

y

0.1

0

-0.1 0

60

20

y

0.04

40

20

y

40

60

20

y

0.1

0.04

40

60

DNS LES-ODT

0.08

0.06

0 0

0

-0.1 0

60

0.02

0.02

DNS LES-ODT

0.05

DNS LES-ODT

0.08

0.06

x = 90

-0.05

0.1

< wRMS >

< wRMS >

40 DNS LES-ODT

0.08

0 0

DNS LES-ODT

-0.05 20

0.1



0

x = 60

0.05





0.05

-0.1 0

0.1

< wRMS >

0.1

0.06 0.04 0.02

20

y

40

60

0 0

Fig. 18. Spanwise velocity first and second order statistics, Le = 1.0.

20

y

40

60

471

A.F. Hoffie, T. Echekki / International Journal of Heat and Mass Transfer 127 (2018) 458–473

x = 30 DNS LES-ODT



0.2 0.1 20

40

y

60

0.2

20

y

< TRMS >

0.2 0.1

40

40

y

60

0.2

20

y

0.2

40

60

DNS LES-ODT

0.4

0.1 20

0.3

0.5

0.3

0 0

0.4

0 0

60

DNS LES-ODT

0.4

0.3

DNS LES-ODT

0.1

0.5

DNS LES-ODT

0.4

< TRMS >

0.3

0 0

x = 90

0.5

0.1

0.5

0 0

0.6

DNS LES-ODT

0.4

0.3

0 0

x = 60



0.4



0.5

< TRMS >

0.5

0.3 0.2 0.1

20

y

40

0 0

60

20

y

40

60

0

DNS LES-ODT

20

40

y

0.6

DNS LES-ODT

20

y

< YF, RMS >

0.3 0.2 0.1

40

60

0.3 0.2

40

y

60

DNS LES-ODT

20

y

40

60

DNS LES-ODT

0.4 0.3 0.2 0.1

0.1 20

x = 90

0.5

0.4

0 0

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 0.6

DNS LES-ODT

0.5

0.4

0 0

x = 60

0.6

DNS LES-ODT

0.5

< YF, RMS >

60

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

< YF >

x = 30

< YF, RMS >

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

< YF >

< YF >

Fig. 19. Temperature first and second order statistics, Le = 1.0.

20

y

40

0 0

60

20

y

40

60

DNS LES-ODT

20

y

0.6

60

DNS LES-ODT

20

y

< YO, RMS >

0.3 0.2 0.1

40

60

0.3 0.2 0.1

20

y

40

60

x = 90 DNS LES-ODT

20

y

40

60

DNS LES-ODT

0.5

0.4

0 0

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 0.6

DNS LES-ODT

0.5

0.4

0 0

x = 60

0.6

DNS LES-ODT

0.5

< YO, RMS >

40

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

< YO >

x = 30

< YO, RMS >

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

< YO >

< YO >

Fig. 20. Fuel mass fraction first and second order statistics, Le = 1.0.

0.4 0.3 0.2 0.1

20

y

40

60

0 0

Fig. 21. Oxidizer mass fraction first and second order statistics, Le = 1.0.

20

y

40

60

A.F. Hoffie, T. Echekki / International Journal of Heat and Mass Transfer 127 (2018) 458–473

< TRMS >

0.2 0.1 0 0 0.3

< TRMS >

0.3

Le = 0.5 Le = 1.0 Le = 2.0

20

y

DNS

40

0.2

0 0

20

y

40

0.1

0.3

0.1

20

y

DNS

40

20

y

40

Le = 0.5 Le = 1.0 Le = 2.0

0.1 0 0 0.3

0.1

x = 90 LES-ODT

0.2

60

Le = 0.5 Le = 1.0 Le = 2.0

0.2

0 0

60

0.3

Le = 0.5 Le = 1.0 Le = 2.0

0.2

0 0

60

Le = 0.5 Le = 1.0 Le = 2.0

x = 60 LES-ODT

< TRMS >

x = 30 LES-ODT

< TRMS >

< TRMS >

0.3

< TRMS >

472

20

y

DNS

40

60

Le = 0.5 Le = 1.0 Le = 2.0

0.2 0.1 0 0

60

20

y

40

60

Fig. 22. Temperature RMS for different Le.

0.4

0.5

0.3 0.2

< YF, RMS >

0.4

0.5

0.3 0.2

20

y

40

DNS

60

0.6

Le = 0.5 Le = 1.0 Le = 2.0

0.4

0.5

0.3 0.2 0.1

20

y

40

20

y

40

DNS

0.3 0.2 0 0 0.6

Le = 0.5 Le = 1.0 Le = 2.0

0.4

0.5

0.3 0.2

20

y

40

DNS

60 Le = 0.5 Le = 1.0 Le = 2.0

0.4 0.3 0.2 0.1

0 0

60

Le = 0.5 Le = 1.0 Le = 2.0

0.4

60

0.1

0 0

x = 90 LES-ODT

0.1

0 0

< YF, RMS >

0 0

0.5

0.6

Le = 0.5 Le = 1.0 Le = 2.0

0.1

0.1

0.6

x = 60 LES-ODT

< YF, RMS >

0.6

Le = 0.5 Le = 1.0 Le = 2.0

< YF, RMS >

< YF, RMS >

0.5

x = 30 LES-ODT

< YF, RMS >

0.6

20

y

40

0 0

60

20

y

40

60

Fig. 23. Fuel mass fraction RMS for different Le.

0.4

0.5

0.3 0.2

< YO, RMS >

20

y

40

DNS

60

0.5

0.3 0.2

0.6

Le = 0.5 Le = 1.0 Le = 2.0

0.4

0.5

0.3 0.2 0.1 0 0

0.4

20

y

40

DNS

y

40

60

0.3 0.2 0 0 0.6 0.5

0.3 0.2 0.1

20

Le = 0.5 Le = 1.0 Le = 2.0

0.4

60 Le = 0.5 Le = 1.0 Le = 2.0

0.4

0 0

x = 90 LES-ODT

0.1

0 0

< YO, RMS >

0 0

0.5

0.6

Le = 0.5 Le = 1.0 Le = 2.0

0.1

0.1

0.6

x = 60 LES-ODT

< YO, RMS >

0.6

Le = 0.5 Le = 1.0 Le = 2.0

< YO, RMS >

< YO, RMS >

0.5

x = 30 LES-ODT

< YO, RMS >

0.6

20

y

40

DNS

60 Le = 0.5 Le = 1.0 Le = 2.0

0.4 0.3 0.2 0.1

20

y

40

60

0 0

20

y

40

60

Fig. 24. Oxidizer mass fraction RMS for different Le.

This is the trend that is observed in the present simulations in both the DNS and LES statistics. The broadening of the profiles for the

species RMS with increasing Le can be attributed to the fact that, although the species gradients are not steep, the spatial range over

A.F. Hoffie, T. Echekki / International Journal of Heat and Mass Transfer 127 (2018) 458–473

which species gradients are present is still broader because of their enhanced diffusion. Therefore, Lewis number effects are reproduced, at least qualitatively, with the LES-ODT approach. Finally, it is important to comment on the overall shapes of the RMS profiles for temperature and the fuel and oxidizer mass fractions. The temperature RMS is characterized by 2 peaks on the fuel and oxidizer sides. The profiles are not as clear at x ¼ 90 for the LES-ODT statistics. The mean temperature peaks in the reaction zone and decays on the fuel and oxidizer sides. Peak gradients are established on both sides, which are also reflected in 2 peaks for the RMS. Fuel and oxidizer profiles on the other hand, peak on opposite sides of the shear layer and single peaks for their gradients are established around the location of the peak temperature from the fuel and oxidizer sides, respectively. Therefore, only one peak for the RMS values is established for the fuel and oxidizer. 5. Conclusions Spatially developing turbulent reacting shear layers are analyzed with LES in conjunction with ODT and with direct numerical simulation (DNS). Direct comparison of flow fields and characteristic flow quantities are used to validate and advance the LES-ODT concept. While momentum and pressure are solved on the coarse LES grid, reacting scalars and momentum are solved on the ODT subgrid. The chemical reaction is simulated as a global singlestep with an Arrhenius reaction rate. Reduction in resolution compared to DNS is compensated by introducing a one-dimensional triplet map, which emulates eddy turn-over events allowing for different eddy sizes on the subgrid level. A benchmark constant density reacting shear layer case is used to adjust 2 ODT model parameters and the forcing implemented for the LES and DNS. The flow field exhibits a deflection toward the lowspeed side. This asymmetric behavior is also reported in the literature [24,25,27,30] and is due to the entrainment of high-speed fluid. The growth of the shear layer is nearly linear in both the DNS and LES-ODT solutions and mean and RMS statistics for the velocity components, temperature and fuel and oxidizer mass fractions are well reproduced by the LES-ODT solutions compared to DNS. Variable density simulations are carried out to validate the mode. Parametric simulations by varying the Lewis numbers are carried out as well to assess the model capability to capture one key SGS effect associated with the competition of the various process at scales not resolved by LES. Comparisons with DNS show the LES-ODT approach establishes a non-linear growth of the shear layer; although, more work is needed to maintain an adequate growth further downstream of the inlet. The velocity and scalar first and second order statistics are reasonably reproduced by the LES-ODT approach as shown by comparisons for the case of Le = 1. Lewis number effects also are established through trends of peaks and the broadening of RMS spatial profiles due to the coupling of instantaneous profiles and the SGS turbulent transport effects. Conflict of interest None. Acknowledgments This work was partially supported through a grant from the National Science Foundation through grant DMS-1217200. Computations were carried out using the NCSU ARC cluster. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.ijheatmasstransfer. 2018.06.105.

473

References [1] N. Peters, Turbulent Combustion, Cambridge, New York, 2000. [2] H. Pitsch, Large-eddy simulation of turbulent combustion, Annu. Rev. Fluid Mech. 38 (2006) 453. [3] M. Ihme, C.M. Cha, H. Pitsch, Prediction of local extinction and re-ignition effects in non-premixed turbulent combustion by a flamelet/progress variable approach, Proc. Combust. Inst. 30 (2005) 793. [4] A.Y. Klimenko, Multicomponent diffusion of various scalars in turbulent flow, J. Fluid Dyn. 25 (1990) 327. [5] R.W. Bilger, Conditional moment closure for turbulent reacting flows, Phys. Fluids A5 (1993) 436. [6] T. Echekki, The emerging role of multiscale methods in turbulent combustion, in: T. Echekki, E. Mastorakos (Eds.), Turbulent Combustion Modeling: Advances, New Trends and Perspectives, Springer, 2011, p. 249. [7] S.B. Pope, Computations of turbulent combustion: progress and challenges, Proc. Combust. Inst. 23 (1990) 591. [8] A.R. Kerstein, Linear-eddy model for turbulent scalar transport and mixing, Combust. Sci. Technol. 60 (1988) 391–421. [9] P.A. McMurtry, T.D. Gansauge, A.R. Kerstein, S.K. Krueger, Linear eddy simulations of mixing in a homogeneous turbulent flow, Phys. Fluids A5 (4) (1993) 1024. [10] S. Sannan, T. Weydahl, A.R. Kerstein, Stochastic simulation of scalar mixing capturing unsteadiness ans small-scale structure based on mean-flow properties (LEM3D), Flow, Turbul. Combust. 90 (2013) 189–216. [11] A.R. Kerstein, One-dimensional turbulence: model formulation and application to homogeneous turbulence, shear flows and boyant stratified flows, J. Fluid. Mech. 392 (1999) 277–334. [12] T. Echekki, A.R. Kerstein, J.C. Sutherland, The one-dimensional turbulence model, in: T. Echekki, E. Mastorakos (Eds.), Turbulent Combustion Modeling: Advances, New Trends and Perspectives, Springer, 2011, p. 249. [13] S. Cao, T. Echekki, A low-dimensional stochastic closure model for combustion large eddy simulation, J. Turbul. 9 (2) (2008) 1–35. [14] J. Park, T. Echekki, LES-ODT study of turbulent premixed interacting flames, Combust. Flame 159 (2012) 609–620. [15] S. Ben Rejeb, T. Echekki, Thermal radiation modeling using the LES-ODT framework for turbulent combustion flows, Int. J. Heat Mass Trans. 104 (2017) 1300–1316. [16] Y. Fu, T. Echekki, Upscaling and downscaling approaches in LES-ODT for turbulent combustion flows, Int. J. Multiscale Comput. Eng. 16 (2018) 45–76. [17] S. Srivastava, T. Echekki, A novel Kalman filter based approach for multiscale reacting flow simulations, Comp. Fluids 81 (2013) 1–9. [18] S. Srivastava, T. Echekki, Particle-filter based upscaling for turbulent reacting flow simulations, Int. J. Multisc. Comput. Eng. 15 (2017) 1–17. [19] S. Balasubramanian, A Novel Approach for the Direct Simulation of SubgridScale Physics in Fire Simulations, Master’s Thesis, North Carolina State University, 2010. [20] S.R. Sedhai, A Multiscale Flame-Embedding Framework for Turbulent Combustion Using LES-ODT, Master’s Thesis, North Carolina State University, 2011. [21] D.O. Lignell, A.R. Kerstein, G. Sun, E.I. Monson, Mesh adaptation for efficient multiscale implementation of one-dimensional turbulence, Fluid Dyn. 27 (2013) 237–295. [22] A.R. Kerstein, W.T. Ashurst, S. Wunsch, V. Nilsen, One-dimensional turbulence: vector formulation and application to free shear flows, J. Fluid. Mech. 447 (2001) 85. [23] R.J. McDermott, Toward One-Dimensional Turbulence Subgrid Closure for Large-Eddy simulation, Ph.D. Dissertation, University of Utah, 2005. [24] S.D. Mason, Turbulence Transport in Spatially Developing Reacting Shear Layers, Ph.D. Dissertation, University of Wisconsin, Madison, 2000. [25] S.D. Mason, C.J. Rutland, Turbulent transport in spatially developing reacting shear layers, Proc. Comb. Inst. 28 (2000) 505–513. [26] A. Michalke, On spatially growing disturbances in an inviscid shear layer, J. Fluid Mech. 23 (1965) 521–544. [27] N.D. Sandham, W.C. Reynolds, Some inlet-plane effects on the numerically simulated spatially-developing mixing layer, in: Turbulent Shear Flows 6, Sixth Symposium on Turbulent Shear Flows, 1989. [28] D. Oster, I. Wygnanski, The forced mixing layer between parallel streams, J. Fluid. Mech. 123 (1982) 91–130. [29] P.S. Lowery, W.C. Reynolds, Numerical Simulation of a Spatially Developing Forced, Plane Mixing Layer, Tech. rep., Report No. TF-26, Thermoscienes Division, Dept. of Mechanical Engineering, Stanford University, Stanford, CA, 1986. [30] P.E. Dimotakis, Two-dimensional shear-layer entrainment, AIAA J. 24 (1984) 1791–1795. [31] T. Echekki, H. Mirgolbabaei, Principal component transport in turbulent combustion: a posteriori analysis, Combust. Flame 162 (2015) 1919–1933. [32] O. Owoyele, T. Echekki, Toward computationally efficient combustion DNS with complex fuels via principal component transport, Cobust. Theo. Model. 21 (2017) 770–798.