A ‘Synthetic Scalar’ Subgrid-Scale Model for
Large-Eddy Simulation of Turbulent Combustion
J. M. McDonough∗
Departments of Mechanical Engineering and Mathematics
University of Kentucky, Lexington, KY 40506-0108
Abstract
In this paper we present an alternative formalism for large-eddy simulation that avoids the problems arising from
filtering the governing equations. In particular, we instead propose to filter solutions leading to the ability to
directly model subgrid-scale primitive variables; i.e., model physics rather than statistics. We then demonstrate
construction of subgrid-scale models of scalar quantities within this framework.
Introduction
It is by now widely accepted that large-eddy simulation (LES) in some form probably offers the best
hope of solving practical engineering problems involving turbulent flows in the near to intermediate future.
Direct numerical simulation (DNS) is useful for theoretical studies at low Reynolds number, Re, but even
for simple incompressible flows in the absence of scalar
transport the total arithmetic requirements scale as
Re3 . Such calculations are not feasible in engineering
practice on current or even near-term foreseeable machines when “over-night turnaround” is needed unless
Re is less than ∼ O(104 ). At the same time, Reynoldsaveraged Navier–Stokes (RANS) procedures have repeatedly exhibited an inability to be truly predictive
(again, even for relatively simple flows). So although
they are currently widely used, it is recognized that
they embody fundamental flaws, and it is only a lack
of viable alternatives that leads to their present ubiquitous application.
It can easily be argued that the structure of LES
implies a total arithmetic requirement for incompressible flow that is roughly proportional to Re2 . This
suggests that in the context of currently-available parallel supercomputers simulation of flows with Re at
least as large as 106 should be nearly feasible. On the
other hand, LES possesses certain shortcomings that
will need to be remedied before it will provide a completely reliable practical simulation tool. In particular, it has been understood almost from the earliest
studies of LES that both the equation filtering process and the subgrid-scale (SGS) models lead to significant problems (see Ferziger [1] for an early review,
and Lesieur and Métais [2] for more recent discussions). Indeed, as might be expected, these problems
are exacerbated by the need to treat detailed chemistry of combustion processes, in part explaining a
somewhat heavier emphasis placed on DNS in combustion than elsewhere (see, e.g., Vervisch and Poinsot
[3]). Filtering the governing equations results in
the same types of closure problems encountered with
RANS, and these are often treated similarly in LES.
But unlike RANS, the basic structure of LES is such
that its results do converge to those of the Navier–
Stokes (N.–S.) equations as discretization step sizes are
refined. This is its significant advantage over RANS.
However, within the framework of the usual LES formalism it has been difficult to obtain high-quality results unless resolution is approaching that of DNS, precluding its use for practical problems.
In the present paper we provide an alternative formalism that specifically addresses these issues. We remark that the ideas to be presented are not completely
new, and much of what we will describe has evolved
from work first presented by McDonough et al. [4,5],
McDonough and Bywater [6,7], McDonough et al. [8]
and Hylin and McDonough [9]. It is the last two of
these that are most directly related to the present
work, and this approach has recently been concisely
summarized by Sagaut [10].
The technique we discuss here arises from a somewhat more general formalism than LES (and includes
LES as a special case), which we have typically described as additive turbulent decomposition (ATD),
and which has been subjected to rigorous mathematical analysis by Brown et al. [11]. References [8–10]
describe what we believe is the most useful form of
ATD for practical calculations, and in general such
procedures embody three main ideas that distinguish
them from typical LES: i) filter solutions rather than
equations, ii) model primitive variables (e.g., temperature, species concentrations, etc.—hence the term
“synthetic scalar”—in the present case), instead of
modeling their correlations, and iii) directly use these
* Corresponding author: jmmcd@uky.edu
Proceedings of the 2002 Technical Meeting of the Central States Section of the Combustion Institute
modeled results to impart improved resolution to the
(deliberately, by the nature of LES) under-resolved
large-scale part of the solution. We comment that all
of these have been utilized together beginning with the
work reported in [8], but formal analysis and further
development is still ongoing. In the present study we
focus on ii), specifically the SGS models, per se, and
we restrict the analysis to passive scalars. Details for
incompressible velocity fields can be found in [9] and
[10].
The remainder of this paper is organized as follows.
In the next section we provide a brief overview of all
three of the ideas mentioned above to establish the setting into which the SGS models will be incorporated.
We follow this with a section devoted to derivation of
the synthetic scalar models, and end the paper with a
short summary.
Overview of ATD Structure
In this section we will discuss the three abovementioned notions that distinguish the current method
from usual LES, emphasizing advantages of this approach for combustion simulations.
The first difference between the proposed approach
and LES as usually constructed is choosing to filter
solutions rather than the governing equations. Filtering equations presents no difficulties for linear terms,
but it is well known that averaging and filtering operators do not commute with nonlinear operators of a
typical transport equation. In the context of combustion, probably the best-known example of this is the
Arrhenius reaction rate formula for which it is clear
that
µ
¶
µ
¶
E
E
α
α
T exp
6= T exp
,
(1)
R0 T
R0 T
the consequence (and treatment) of which is discussed
in, e.g., Liñán and Williams [12]. Here, overbars denote application of a filtering (or averaging) operator.
T is temperature; E is activation energy; R0 is the
universal gas constant, and α is an exponent which
is given for each reaction (as is E). It is clear that
this is just the beginning of troubles arising from filtering, and this has been one of the main motivations
for development of the PDF methods by Pope [13] and
others.
An additional difficulty arising from filtering is that
of treating problems posed on complicated geometric
domains. If structured grids are to be used, the differential operators will now contain variable coefficients
that in general prevent commutation of the filtering
operator with even linear differential operators. If unstructured grids are employed, special grid-dependent
filters must be constructed.
Even in the absence of the above geometric problems, the consequence of Eq. (1) is that statistical
correlations representing the commutators of the differential and filtering operators must be included in
the governing equations. For example, a somewhat
simpler-to-treat case than that of Eq. (1) is
∂uρYi
∂u ρ Yi
6=
,
∂x
∂x
(2)
which requires introduction of the correlation ρuYi , in
which ρ is density, u is the x-component of velocity,
and Yi is mass fraction of species i. It is essential to
recognize that there are no fundamental equations for
these correlations, and that modeling such quantities
entails modeling statistics rather than physics. Such
models can tell us essentially nothing regarding the
instant-to-instant behavior of a turbulent combustion
process on the scales where chemical reactions occur
and interact with the turbulence.
The alternative we propose is to avoid filtering the
governing equations. This removes the need to model
statistics because no correlations arise, and it permits
direct modeling of physics. Furthermore, if the solution itself is being filtered, the filtering process is
reduced to one of signal processing. On structured
grids this is an almost trivial process although some
difficulties can arise when unstructured grids are employed. For the sake of brevity we will not provide further details regarding this here; the interested reader
is referred to Yang and McDonough [14] for more information on filtering solutions.
The form of subgrid-scale model proposed in [8] and
treated in detail in [9] is
qi∗ = Ai ζi Mi ,
(3)
where qi∗ is the ith component of the small-scale (SGS)
part of a solution vector Q ≡ (Q1 , Q2 , . . . , QM )T of M
dependent variables that has undergone the usual LES
decomposition
Qi (x, t) = qi (x, t)+qi∗ (x, t) ,
i = 1, 2, . . . , M . (4)
Here, qi is the large- or resolved-scale part that is directly computed via some discrete approximation, and
qi∗ is the unresolved, small-scale part; x is the spatial
coordinate, and t is time. We can (informally, but
not rigorously) view qi as the first terms of a Fourier
representation of Qi , with the number of terms depending on the level of resolution, and qi∗ can be considered to be the remainder. This is constructed to
directly model primitive variables at the level of the
unresolved scales. In Eq. (3) Ai is the amplitude of
the ith SGS solution component; this is derived from
Kolmogorov scalings (see Frisch [15] for a basic treatment, [9] for details in the present context and Giacomazzi et al. [16] for a somewhat similar treatment). ζi
is an anisotropy correction, the need for which arises
from the isotropy assumption of the Kolmogorov theories, and the Mi are temporal fluctuations. The latter have received considerable attention recently after it was demonstrated by McDonough and Huang
[17,18] that they could be obtained as discrete dynamical systems (DDSs) derived directly from essentially
any governing equation(s). Examples of this specifically related to combustion include [17], McDonough
and Zhang [19,20] and Zhang et al. [21].
The final aspect of the modeling approach under
consideration here is direct use of model results. Any
transport equation can be expressed in the general
form
above, it is now known how to calculate the Mi s directly from the governing equations, and we shall see
that the new formulation for the Ai s given here for the
first time obviates the need for the ζi s. Thus, under
the assumptions we will make as we proceed, we will
produce a complete SGS model of passive scalars for
LES of finite-rate kinetics descriptions of combustion.
The basic idea underlying construction of formulas
for the Ai s is that qi∗ represents behavior not resolved
in the large-scale calculation, and thus the corresponding amplitude should be the square root of some measure of the “energy” summed over all wavenumbers
higher than the highest one supported by the largescale discretization. We denote this wavenumber by
kh for a grid of spacing h. Then
Qt + ∇·F (Q) = ∇·G(Q) + S(Q) ,
k=kh +1
A2i ∼
=
(5)
with Q as given above. The subscript t denotes partial differentiation with respect to time, and ∇· is the
divergence operator. The functions F and G are associated with, respectively, advection and diffusion. S is
a general nonlinear source term which in the present
case corresponds to species production and the resulting heat release. In usual LES Eq. (5) would be filtered, and (4) would be used along with models of
terms related to (1) and (2) to produce equations for
the qi s. Thus, except for the adjustment of qi due
to these models, information associated with the qi∗ s
would be lost. No direct effect of small-scale turbulent
fluctuations can be imparted to the large-scale part
of the solution, and it is precisely on the small scales
that important interactions between turbulence and
combustion occur.
In the modeling technique treated here, we substitute Eq. (4) into (5) to obtain
(q + q ∗ )t + ∇·F (q + q ∗ ) = ∇ · G(q + q ∗ )
+ S(q + q ∗ ) , (6)
similar to usual LES—but we do not filter this equation. Instead, we use (3) to evaluate q ∗ , substitute
this into (6), and solve for q. This result will be under
resolved, and probably aliased; so we filter it, that is,
filter the solution q. Then we again use the qi∗ s in Eq.
(4) with the filtered qi s to obtain an approximation
to the complete solution Q. This sequence of steps is
executed for each discrete time step of the numerical
integration of Eq. (5).
The Primitive-Variable SGS Model
In the remainder of this paper we will concentrate
on deriving formulas for the Ai s of Eq. (3). As noted
∞
X
Ei (k) ,
(7)
where Ei is the abovementioned characterization of
energy associated with the ith dependent variable.
We remark that Eq. (7) is actually global in space
because it has arisen from Fourier analysis of the energy. Namely, by the Parseval identity, (7) is simply
the square of the (spatial) L2 norm of the small-scale
part of the ith dependent variable. But in applications
we will make of Ai , this value must be local.
If we consider a local (re-)expansion of, say Yi
(= yi + yi∗ ), on the interval [0, 2h] (or any translate of
this interval) then wavenumbers of this expansion can
be related to those of the global expansion through the
scaling N kh where N is the number of points used in
discretizing the physical domain. Thus, the size of the
local wavenumbers increases more rapidly than does
that of the global ones, and on a 2h-interval we should
not need very many wavenumbers to adequately represent yi∗ . If we denote these local wavenumbers as knloc ,
we have that the lowest wavenumber in the Fourier
representation of yi∗ should be k0loc = kh , and these
are related to global wavenumbers via knloc = nN kh .
If we denote restriction of Ei to [0, 2h] by Ei∗ , we have
A2i ∼
=
∞
X
Ei∗ (knloc ) .
(8)
n=1
As noted above, we expect to need only a few terms
in this series to obtain an adequate representation because knloc increases very rapidly with n.
Our next task is to determine the Ei∗ s. There are
several ways by which this might be done, and we will
present only one herein. The present approach will
be based on the second-order structure function arising in the Kolmogorov-Oboukov-Corrsin (KOC) treatment of passive scalars, as described by Warhaft [22],
and it will utilize a form of the scale similarity hypothesis widely used in dynamic SGS models (see Germano
et al. [23] and Domaradzki and Saiki [24]). We will not
here give details of this latter aspect of the model beyond explaining the manner in which it enters.
The second-order structure function of a scalar yi∗
is defined as
S2 (yi∗ , `) = h(δyi∗ )2 i
≡ h(yi∗ (x + `) − yi∗ (x))2 i ,
(9)
where h · i denotes a chosen form of averaging (local
spatial in the present case), and ` is a spatial displacement from the point x. Here, we will average over
a selected number of points in the `-neighborhood of
x. For example, on a uniform finite-difference grid in
2D we would define this
q neighborhood as those points
such that ` ≡ |`| = h2x + h2y , where hx and hy are
grid spacings in the indicated directions. Then for
any given point x we calculate (δyi∗ )2 for each of the
nearest neighbors at a distance ` from this point and
average the results.
It is clear from (9) that S2 is directly related to the
energy of yi∗ . Indeed, for high-Re, turbulence the wellknown k −5/3 inertial subrange scaling follows directly
from this and the Kolmogorov 23 -law (see [15]). Then
the KOC theory gives
2/3
S2 (yi∗ , `) = C2,Yi ε−1/3 εYi `2/3 ,
(10)
where C2,Yi is a universal constant, ε is dissipation rate
of turbulence kinetic energy, and εYi is dissipation rate
of the scalar (in this case, mass fraction Yi ). We have
and
ε = νk∇u∗ k2 ,
(11)
εYi = DYi k∇yi∗ k2
(12)
for the dissipation rates. In (11) u∗ is the small-scale
velocity vector, and ν is kinematic viscosity; in (12)
DYi is molecular mass diffusivity of the species whose
mass fraction is Yi .
Several remarks regarding these formulas are in order. First, Eq. (10) holds for high-Re flows in the
inertial-convective subrange with respect to the scalar,
i.e., in an inertial range in which the scalar is merely
convected. There are other inertial subranges (see,
e.g., Tennekes and Lumley [25]), so (10) is not general. On the other hand, Eqs. (11) and (12) are general, and simpler forms are widely used. Furthermore,
in combustion studies εYi is usually expressed in terms
of mixture fraction.
To obtain a suitable formula for the Ai s under
general flow conditions we will work locally to remove
needed assumptions associated with homogeneity, calculate separate Ai s for every variable (especially, the
velocity components) to remove the need for isotropy,
and use the expression
S2 (yi∗ , `) = C2,Yi ε−α (εYi `)
β
(13)
obtained as a generalization of (10). In this formula α
will have been determined from an independent analysis of the velocity field similar to what we present here
for passive scalars.
It is important to note that ε and εYi contain
small-scale information, and this is not immediately
available—it is what is being modeled. So to estimate these quantities we will employ the scale similarity hypothesis as indicated above. Scale similarity
utilizes the assumption that flow and scalar behavior
at the lowest modes of the small, unresolved scales is
the same as that of the highest modes of the resolved
scales. In dynamic SGS models [23] use of this idea
is embodied in a double filtering procedure that ultimately leads to local in space and time values for the
Smagorinsky constant. In [24] scale similarity is used
to “estimate” subgrid-scale synthetic velocities similar
to what we do here.
But it is important to recognize the fundamental
flaw in this reasoning. While the indicated relationship
between behaviors on opposite sides of the maximum
grid-scale wavenumber is reasonable, applications of
this in highly under-resolved regions of the flow do
not provide the correct information. Namely, at high
Re the small-scale behavior predicted by scale similarity might not even correspond to that in the inertial
subrange; yet, it would be employed as if it were at
the Kolmogorov scales. This problem is addressed by
Domaradzki and various co-workers, e.g., [24] and Domaradzki and Loh [26]. The basic idea is to employ
an extrapolation of wavenumber information to inexpensively gain higher resolution for application of scale
similarity. In [26] this is modified to utilize interpolation in physical space.
The methods employed in [24], [26] and elsewhere
provide only a factor of two increase in wavenumbers
in each spatial direction. Hence, at very high Re, or
from the standpoint of scalars, high Péclet number,
P e, this will not be sufficient. Moreover, because the
high-wavenumber information produced in this way is
not actually a solution of the governing equations, recursive application of this is not possible.
The present author and co-workers have recently
developed a physical-space interpolation procedure
based on the discrete solution operator of the governing equations which produces interpolants on any desired scale that are solutions to the discrete governing
equations (see Xu et al. [27]). This permits local arb-
itrarily high-wavenumber representations (exactly
what is needed in the present context), and thus valid
application of the scale similarity approach. Hence, we
can suppose that (11) and (12) are approximated with
reasonable accuracy. Then calculation of S2 (yi∗ , `) as
a function of ` at several values of ` using these approximate values in the right-hand side of (9) provides
sufficient information to determine C2,Yi and β in (13).
We then directly calculate the required energy
¢β−1
¡
EYi (k loc ) = C2,Yi ε−α εβYi k loc
,
(14)
and this is used in (8) to predict AYi .
There are several important observations to be
made regarding use of Eq. (14) in (8) to obtain AYi .
The first is that this must be done for every dependent variable: all species concentrations, temperature
and the velocity components. Second, it is done at every large-scale grid point during every time step, except when local subgrid-scale Reynolds and/or Péclet
numbers indicate that turbulence cannot be present
(see [9] for details of this). Despite this, the amount of
arithmetic is not excessive. Indeed, it scales as O(N )
where N is the total number of grid points of the largescale calculation. But in addition, these computations
are highly parallelizable. Our experience with earlier
implementations of models with similar structure has
shown that even without parallelization, run times for
executions utilizing the turbulence model do not exceed those for laminar flow on the same grid by more
than a factor of two. Yet, the details seen in the fluctuating flow fields approach those of DNS. Finally, we
note that there are no adjustable constants in the construction of the amplitude factors. This differs from
the model presented in [9].
We have already indicated that the fluctuating factors, the Mi s, of Eq. (3) have received considerable attention in recent studies, but for the sake of completeness we will provide a brief overview of this important
aspect of the SGS models; more details are available in
the cited references. We noted above that these coupled DDSs can be derived from essentially any governing equations. This is done via a Galerkin procedure
applied locally in space (in the same spirit as construction of the Ai s), and thus with the same expectation
that only a few terms in a Fourier representation will
be needed to accurately represent small-scale temporal behavior. In fact, Yang et al. [28] have shown that
a linear combination of only three realizations of the
DDS modeling the 2-D N.–S. equations [18] is sufficient to provide a very good model of an experimental
turbulent flow field with Re = 105 . Furthermore, in
[19] and [20] it is shown that a single realization of the
complete DDS presented below is sufficient to closely
mimic temporal behavior of major species mass fraction fluctuations in H2 –O2 and H2 –air, respectively,
reactions.
Here we will present the general form of this DDS
and indicate how it is incorporated into the overall
model of the qi∗ s. As is fairly common, we assume low
Mach number and require the hydrodynamic pressure
to be divergence free. Then we recover density from
an equation of state. Within this context the DDS
leading to the fluctuations Mi is the following.
³
´
a(n+1) = βu a(n) 1 − a(n) − γu a(n) b(n) ,
(15a)
³
´
b(n+1) = βv b(n) 1 − b(n) − γv a(n) b(n) + αT c(n) ,
c(n+1) =
"Ã N
s
X
(15b)
(n+1)
αT di di
− γuT a(n+1) −
i=1
(n+1)
γvT b
´
(n)
c
−
Ns
X
#
Hi ω̇i /(1 + βT ) + c0 ,
i=1
(n+1)
di
³
(n+1)
= − βYi + γuYi a
+ γvYi b
(n+1)
´
(15c)
(n)
di
+ ω̇i + di,0 i = 1, 2, . . . , Ns ,
(15d)
with
ω̇i =
Nr
X
j=1
"
Cf,ij
Ns
Y
`=1
ν0
d` j,`
− Cb,ij
Ns
Y
#
ν 00
d` j,`
.
`=1
Here, superscripts (n) denote time step (or map iteration) index; a, b, c and the di s can be viewed (heuristically) as Fourier coefficients of the two velocity components, temperature and the species concentrations,
respectively; the subscripted αs, βs, etc., are bifurcation parameters of the DDS, all of which are related
to the various physical bifurcation parameters. For
example, βu and βv are (the same) functions of the
flow Reynolds number; αT is related to the Rayleigh
or Grashof number; αT di contains Schmidt and Lewis
number information, and the Hi are associated with
specific enthalpies for each species i; the Cf,ij , Cb,ij
can be related to Kolmogorov-scale Damköhler numbers, DaK . The various γs correspond to velocity,
temperature and species concentration gradients (as
would be available from resolved-scale results). For
example, γu ∼ uy , γuT ∼ Tx (subscripts x and y indicate partial differentiation), etc. The di,0 s and c0 are
high-pass filtered species concentrations and temperature, respectively (obtained from the resolved-scale
calculation), about which the subgrid-scale behavior
fluctuates.
For DaK > 1, the case considered in [17,19–21], an
iteration step is performed with Eqs. (15a,b) to obtain
fluctuating velocity components. Then Eqs. (15d) are
iterated a number of times based on their respective
Damköhler numbers, and finally Eq. (15c) is evaluated. This process is repeated a number of times determined by the ratio of integral-scale to Kolmogorovscale times, tL /tK , for the velocity field. The results
from the final such step are normalized by the maximum value over all steps calculated during the current
large-scale time step, and these are inserted into Eqs.
(3) as the Mi s. We remark that DaK ≤ 1 can also be
treated, and Eqs. (15) extend to three space dimensions in a natural way.
Summary
In this paper we have presented a derivation of a SGS
model for LES that represents a rather distinct departure from previous approaches, and we have indicated
some of its potential advantages with regard to combustion simulations. Among these are its ability to
directly simulate fluctuating behavior of physical variables, and thus interaction of turbulence with chemical
kinetics on appropriate length and time scales, providing a distinct advantage over RANS methods and
usual LES.
Acknowledgements
The author expresses his gratitude to the AFOSR for
partial support of this work and to NASA/EPSCoR for
the funding that permitted initiation of these studies.
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