- Email: [email protected]
Discrete large eddy simulation
Vol. 6, No. 1
Discrete
TAO, RAJAGOPAL
and CHEN:
Discrete large eddy simulation
17
large eddy simulation
L. TAO*, K. R. RAJAGOPAL* and G. Q. CHEN** *Department of Mechanical Engineering, Texas A&M University, USA ** Centre for Environmental Sciences, Peking e-mail: [email protected]
University,
Beijing
College Station, 100871,
Texas
77843-3123;
China;
(Received January 19, 2001) Abstract: Despite the intense effort expended towards obtaining a model for describing the turbulent flows of fluid, there is no model at hand that can do an adequate job. This leads us to look for a non-traditional approach to turbulence modeling. In this work we conjoin the notion of large eddy simulation with those of fuzzy sets and neural networks to describe a class of turbulent flows. In previous works we had discussed several issues concerning large eddy
simulation such as filtering the filtering procedure. Keywords:
and averaging.
Here, we discuss the use of fuzzy sets to improve
large eddy simulation, fuzzy set, neural network, turbulence modeling
Introduction In previous work8 T2], we discussed several issues associated with the standard version of large eddy simulation (LES) such as filtering and averaging. By the standard version, we mean the traditional practice of first constructing a set of field equations of motions for turbulent motion and then discretizing the equations to suit computational simulations. In this paper, we revisit the issue of large eddy simulation with a view towards improving the filtering procedure that is used in developing the model. The central idea behind LES is the retaining of large eddy structures in the model while ignoring the smaller structures. The filtering that is used is sharp, with eddies smaller than a certain size being modeled. However, such a sharp filter is neither warranted nor acceptable if a model that is true to an application is desired as eddies of all sizes tend to be present. It might thus be meaningful to have a model that gives more weight to the larger eddies, filters out very small esddies, and gives lesser weight to the intermediate eddies. Terms such as large and small are rels;ive and change from application to application. However, common to all applications will be the fact that different sizes of eddies are given different weights. Instead of using the weights as unil;y (1) and zero (0), with a sharp cut-off, it would be reasonable to have a “fuzzy” boundary or zone between the large eddies that are assigned the weight unity and the small eddies that are assigned the weight zero. The size of the fuzzy zone and the weights assigned to the eddies would depend on the problem on hand. This approach provides a conceptual shift in LES modeling by appealing to fuzzy sets for the weighting of the eddies. Furthermore, in this paper, we advocate the use of neural networks to develop directly discrete LES models, as neural networks are universal approximators that can have training algorithms built into them and are therefore are particularly well suited for the task of developing turbulence models that can take into account the inherent fuzziness of turbulence as well as use the learning algorithms to converge to a reasonable model.
18
Communications
in Nonlinear
Science & Numerical
Simulation
March 2001
1 Formulation Let us formulate large eddy simulation from a non-traditional point of view. According to its definition, LES tries to resolve the large scale motion structure of a turbulent flow, while filtering out and modeling the motion components associated with the small scales. Clearly LES involves fuzziness since the dividing line between large scales and small scales is vague. Even if we introduce a length scale L and a filter function for LES as a means to quantify what is large and what is small, we cannot obtain a clear-cut large scale flow field in practicelrl. To use this fuzziness to our advantage in formulating a more logically consistent scheme of LES, we can define the filter operation as follows. Consider the turbulent flows in a domain of flow V that ‘can be described by the NavierStokes equations with proper initial and boundary conditions div u = 0 dU
dt + div(uu)
= -VP + uV2u
0)
(2)
where u and pp are respectively the velocity and the dynamic pressure, and p and u the mass density and the kinematic viscosity of the fluid. Assume that we can decompose {u, p}(z, t) in 2) x [O,oo) into the components of {ii,@}(z,t;Z) in 2, x [0, oo) corresponding to length scales 1 E (0, co). That is, m{ti,i)}(+;l)dl (3) s0 where the integration can also mean the forming of summation in the case of discrete values of 1. We will adopt a length scale L (much smaller than thle characteristic scale of 2)) and a corresponding filter function PL to introduce the large scale flow field {U, P}, but here we will interpret ,LLL as a membership function as in fuzzy sets131with the following property: 0 5 AL < 1 and /JL is monotonically non-decreasing; AL := 0 if 1 < L, AL = 1 if 1 > L, and PL might vary rather steeply around 1 = L. Further, we define t) =
{%PH?
(4)
The purpose of using such a filter function is to accommodate the fuzziness mentioned above. Next, we need to derive equations that govern {U,P} on the basis of Eqs. (1) to (4). Combining (l), (3) and (4), we get div U = 0 (5) To filter Eq. (2) directly is a difficult task since we do not know the property regarding ii(z, t; Z&(X, t; 1’). We may overcome this difficulty by resorting to an approximate scheme such as the one adopted in [2]. Let us fix y E Do, where Do is the interior of V, and suppose that there is a cube C of side LO > L such that y E C” c V, where C” is the interior of C. We can represent {u,p} accurately in Co x [0, oo) in terms of a Fourier series expansion (21, PX~, t) = C {W(t;
{ii,p}*(t;m) Then, definition
= {ii,fi}(t;
m) exp (i?!T)
-m)
(4) can be replaced by (7)
Vol. 6, No. 1
TAO, RAJAGOPAL
and CHEN:
19
Discrete large eddy simulation
The filter function or membership function 61; is now defined on N3 with by = 1 if (ml is small and fin = 0 if jrnl is large (whether Irnl is large or small is determined by scale L). As mentioned in the introduction, the value of Irnl for which jl~(m) is between unity and zero depends on the application. The important fact to grasp is that it will never be a sharp cut-off, and the boundary or the zone wherein 0 < jl~(m) < 1 is at best known in a fuzzy sense. With the help of (6) and (7), Eq. (2) can be filtered to yield :U
+ divx
fiL(rn + n)ii(t;
m)ii(t;
2n(m+n)a n) exp i -Lo
1
= -VP
t vv2u
or alternately $7
+ div(UU)
+ VQ - vV2U = -f
(8)
with
f = div RR = (uu - VU)
=Cfi~(m+n)[l
i(trR)I)
-~~(m)li~(n)]ii(t;m)ii(t;n)exp
Here we have absorbed trR/S into Q and therefore we need to modify the boundary conditions involving Q in simulations to account for this fact. We should at this juncture discuss some characteristics of f defined above in order to decide how to model it. We notice that: (i) m and n in f and .?2are expected to have an upper bound as ti(t; m) = 0 for very large m resulting from the viscous effect, reflecting the fact that sufficiently small eddies tend to get destroyed due to the viscous dissipation. The spatial scale of f and R, indicated by exp [i 27r(m -I- n) . z/Lo], is bounded from below by L’ and the value of L’ cannot be much smaller than L since we have adopted fi,:(m + n) = 0 for large ]m + n]. (ii) f does not involve directly the larger scale components of U, the components ii(t; m) with [ml being small, due to TV = 1 and so 1- fi~(m)fir,(:n) = 0 in (10) when Irnl and InI are small. (iii) f contains the smaller scale components &(t; m) of U whose scales are around L since fin differs from 1 to some extent here. Also, f contains the small scale components of U, ti(t; m) with [ml being large. For example, it includes ti(t; m)ii(t; m’), ]m] being large and m’ = (1, 0,O) - m since fi~((l, 0,O)) # 0 and j.TiL(m) = fiL(m’) = 0. The velocity components of f are expected to behave irregularly compared with the large scale components of U, an idea underlying the adoption of LES to turbulence modeling. (iv) It follows from (iii) that f depends on U but is not determined completely by U, and f should be treated as a disturbing force acting on the large scale motion {U, P}. Its spatial scale should be compatible with the spatial scale of U according to (i). Meanwhile, f also involves fuzziness due to the presence of fin (see Eq. (11)). We have formulated Eq. (8) with the help of a Fourier series expansion in C” x [0, co) and we need to extend it. We will assume that Eq. (8) holds in general on the basis that (6) and (7) give us rather accurate representations of the fields of {u.,p} and the large scale motion field of {U, P} in a neighborhood of y E 2). Certainly, the filter operation (.) in (8) can pose a difficulty with regard to applications and one way to eliminate this problem is to approximate (.) with a spatial filter of the form
20
Communications
in Nonlinear
Science
& Numerical
March
Simulation
2001
where G is a filter function which can be continuous or discrete, the latter compatible with a finite volume, finite difference, or finite element scheme. Regarding the term f in (8), we will not follow the traditi.onal scheme of turbulence modeling, such as constructing a rather rigid continuous relation on it. We will explore the idea of discrete LES models here and thus we need to develop a method to deal with f properly. To this end, we assume that the domain 2) is discretized with a grid distribution of nodes N zz {&d : m = 1,. . . , M}. Associated with each Z( m) E N there is a control volume V(Z(“)) and there is a subset N(d”)) c N composed of all the nodes immediately neighboring yhO)
= ,(m),
N@“))
= {yhl)
: 1 = 1,. . . , mo}
The distance between any two nodes in N(s(“)) U {g(m30)} is around L, consistent with the filtering scale in LES. Next, suppose that Eq. (5) and the terms on the left-hand side of Eq. (8) have been discretized both in space and time by applying (5) and (8) to the control volume V(Z(~)) at the instant t = tie and adopting appropriate discretization schemes, and suppose that the algebraic equations that are obtained involve only the (discrete) velocities and the (discrete) pressures at the nodes of {~(~yl) : 1 = 0, 1, . . . ,me} and at the instants of tk-1 and tk. These restrictions can be relaxed, and the specifics regarding the discretization are not our concern here because our main purpose is to explain the idea behind the discrete formulation of LES. Denote the discretized equations in the forms of ’ DC”) U(y(“l”), ( D+)
(
tk), y(‘+
U(y(n”),tj),Q(y(m”),tk),y(m”),tj;l
1 = 0, 1, . . . , m. = O,l,*e* ,ma,j
>
= 0
(12)
=/C--1,/i?
>
= FCrn)
(13)
The right-hand side vector quantity F(“) is related to -f in (8) which is yet to be determined. Also, FCm) may play a role of possibly compensating for the large deficiency due to the discretization of the left-hand side terms in (8). Assume that 8+)
= B(m)(U(y(m~z),tk),Q(yO,tk),y(m.l),t~;~
= 0, I;-*
(14)
,m,)
The presence of Q in the argument is based on the fact that Q includes also the subgrid fluctuation energy (see (9)) which is expected to affect f. The dependence of F(m) on the scale L is implied though not expressed explicitly here. Requiring that FCm) meet Galilean invariance - the very invariance possessed by the Navier-Stokes equations and the left-hand side terms of (8), and considering the feature of Q (P) - adding any constant to Q will not affect U, we have
$‘(“) = 2 {G(W”) [u(y’ m, tk) _ q&4, tk) + f{h”) yh4 - ,(4
1
l=l
where G(“ll)
and
H(“y’) are [
the scalar functions of
U(y(m’i), tk) - u(d”), [U@%“),
[
yw)
_ ,w
I>
tk) ] . [ u(y @‘j), tk) ‘- u(d”),
tk) - q&4,
tk)]
tk)] . [,cm.j) -- ,w]
tk)- &(z'"', tk),i,j =1,...,mo I[. yh~)_&d1, Q(y+),
‘Here, @(z(“), tk) is intended to indicate the value and it does not represent the value of a corresponding be made close to each other, an aim of any discretization
of a discrete continuous scheme.
field 4 at space filed f#~at {zcm),
and time points of {a~(~), tk} tk}, though these two should
Vol.
6, No.
1
TAO,
RAJAGOPAL
and
CHEN:
Discrete
large
Instead of such a complex though rather complete formulatioc.,
F(“) = 2 ($4
[qy’4,
eddy
21
simulation
we will take
t/J - U(&d) q]
(15)
l=l
Gh”) = &‘d) (U(y("4,
U(y(‘-‘I,&) - U(X(~),tk)l, Q(y(-"1, tk) - q&4,
tk)). (y(m)
- &4))
tk) - Q(x(~), tk), JyW)
- &4
I)
(16)
which is tractable and which may be appropriate due to the ra,ther uniform distribution of the nodes with jy(mli) - ~(~tj)) N L if i # j. For a node xcrn) which is not near a boundary, we may assume that all the G( myz)‘~have the same structure, that is,
G”+
= @m) IU(y(mT”),tk) - U(x(“), &)I, Q(y(“.*‘), tk) - Q(x’“‘, tk), ( (qyW),&) - U(x(m),tk)) * (yhu - &4), lyh”) - r&.(m) I>
07)
For a node xtrn) near a wall, G(“*“) may have different forms depending on whether ylrn’l) is near or away from the wall. In all these cases, we need to propose specific forms for G(myl) to make the model composed of (12) and (13) determinate. One way to achieve this goal is to find Gcn3”) through a neural network scheme, because neural networks as universal approximators are very suitable for such a taski4]. We now sketch how to implement such a scheme. We will first classify the nodes in N into several categories on the basis of their positions, such as whether near a wall or not, and the number of their neighboring nodes and so on, and we will adopt, the same relations for F(“) for all the nodes belon ing to the same category. Next, neural networks can be used to determine G(m,l))s in the Ji’(J’ s for all the categories. For example, for the category composed of nodes not near the boundary of D and having the same number of neighboring nodes, we have from (13), (14) and (17) that g(“) = DC”) - 2
(j(m)
[u(y(“‘z), tk) - U(&‘),
&)I = 0
l=l
Suppose that we know all the information on the discrete varia.bles U, Q and y present in the above equation, we can construct a neural network to find the structure of C?crn)through, say, supervised learningL51. We can then patch all the discrete equations of (12), (13), (15) and (16) together with the discrete relations from the initial and boundary conditions to formulate a determinate discrete model of LES with scale L on turbulent flows in any 2). This strategy is simple but it may be too simplistic since we may not have all the information on the discrete fields of {U, Q} in a turbulent flow. In this case, we can write down all the algebraic equations of motion for {U, Q} at the nodes of N in D, on the basis of (L2), (13), (15) and (16) and the initial and boundary conditions, the unknowns in the equations are the G(mJ)‘s associated with the categories mentioned above. We may then resort to the method of neural networks to fix the structures of G(m>‘)‘s.
2
Summary
We have discussed large eddy simulation from a new perspective: (i) to introduce large scale motion fields through the concept of membership function in fuzzy sets, which accounts for the ambiguity associated with largeness and smallness; (ii) to construct discrete models of
Communications
22
in Nonlinear
Science & Numerical
Simulation
March 2001
LES directly without taking the intermediate step of developi:ng a continuous model first and then discretizing the differential equations of motion, consistent with the spirit of LES to catch the large scale flow features and the practice of computational simulation; (iii) to resort to neural networks to resolve the structures of the undetermined coefficients present in discrete LES models, tapping the power of neural networks as universal approximators. Also, fuzzy neural networks may be used to resolve the problem of overfittingi61 and to accommodate the uncertainty associated with the values of {U, P}(z(“), tk). The focus of this work is to explain some basic ideas for modeling turbulence with discrete large eddy simulation. The details regarding what types of neural networks to be used and whether to train the networks supervised or unsupervised are yet to be worked out, and concrete models are to be constructed and implemented to carry out simulations. It can also be inferred from this work that a neural network may be adopted to help devise or improve numerical schemes to solve differential equations. We have explored in [7] the possibility of compensating large deficiency resulting from an improper discretization scheme of finite element methods to a differential equation and suggested the modeling of the residual of the discretized equation with a (constitutive) relation. And this relation may be found, for example, by a neural network through supervised learning, if accurate solutions to the differential equation are known in some cases.
Acknowledgment The third author would like to thank support from the Special Funds for Major State Basic Research Projects of China (grant No. G1999043605 and G1999022207), and the Trans-Century Training Foundation sponsored by the National Education Ministry of China.
References [l] Chen, G., Tao, L. and RajagopaI, K. R., Remarks on large eddy simulation, Commun. Nonlinear Sci. Numer.
Simul.,
2000, 5(3): 85-90
[2] Rajagopal, K. R., Tao, L. and Chen, G., A formulation of large eddy simulation, Commun. Nonlinear Sci. Numer. Simul., 1999, 4(4): 245-248 [3] Zimmermann, H. J., Fuzzy Set Theory and Its Applications, Kluwer, Boston, 1996 [4] White, H., Artificial Neural Networks: Approximation and Learn.ing Theory, Blackwell, Cambridge, Mass, 1992 [5] Haykin, S., Neural Networks: A Comprehensive Foundation, Macmillan, New York, 1994 [6] Buckley, J. L. and Feuring, New York, 1999
T., Fuzzy and Neural:
Interaction
and Applications,
Physica-Verlag,
[7] Tao, L., Chen, G. Q. and Rajagopal, K. R., Notes on finite element method and its generalization, J. Hydrodynamics,
1996, B3: 97-101
Discrete
TAO, RAJAGOPAL
and CHEN:
Discrete large eddy simulation
17
large eddy simulation
L. TAO*, K. R. RAJAGOPAL* and G. Q. CHEN** *Department of Mechanical Engineering, Texas A&M University, USA ** Centre for Environmental Sciences, Peking e-mail: [email protected]
University,
Beijing
College Station, 100871,
Texas
77843-3123;
China;
(Received January 19, 2001) Abstract: Despite the intense effort expended towards obtaining a model for describing the turbulent flows of fluid, there is no model at hand that can do an adequate job. This leads us to look for a non-traditional approach to turbulence modeling. In this work we conjoin the notion of large eddy simulation with those of fuzzy sets and neural networks to describe a class of turbulent flows. In previous works we had discussed several issues concerning large eddy
simulation such as filtering the filtering procedure. Keywords:
and averaging.
Here, we discuss the use of fuzzy sets to improve
large eddy simulation, fuzzy set, neural network, turbulence modeling
Introduction In previous work8 T2], we discussed several issues associated with the standard version of large eddy simulation (LES) such as filtering and averaging. By the standard version, we mean the traditional practice of first constructing a set of field equations of motions for turbulent motion and then discretizing the equations to suit computational simulations. In this paper, we revisit the issue of large eddy simulation with a view towards improving the filtering procedure that is used in developing the model. The central idea behind LES is the retaining of large eddy structures in the model while ignoring the smaller structures. The filtering that is used is sharp, with eddies smaller than a certain size being modeled. However, such a sharp filter is neither warranted nor acceptable if a model that is true to an application is desired as eddies of all sizes tend to be present. It might thus be meaningful to have a model that gives more weight to the larger eddies, filters out very small esddies, and gives lesser weight to the intermediate eddies. Terms such as large and small are rels;ive and change from application to application. However, common to all applications will be the fact that different sizes of eddies are given different weights. Instead of using the weights as unil;y (1) and zero (0), with a sharp cut-off, it would be reasonable to have a “fuzzy” boundary or zone between the large eddies that are assigned the weight unity and the small eddies that are assigned the weight zero. The size of the fuzzy zone and the weights assigned to the eddies would depend on the problem on hand. This approach provides a conceptual shift in LES modeling by appealing to fuzzy sets for the weighting of the eddies. Furthermore, in this paper, we advocate the use of neural networks to develop directly discrete LES models, as neural networks are universal approximators that can have training algorithms built into them and are therefore are particularly well suited for the task of developing turbulence models that can take into account the inherent fuzziness of turbulence as well as use the learning algorithms to converge to a reasonable model.
18
Communications
in Nonlinear
Science & Numerical
Simulation
March 2001
1 Formulation Let us formulate large eddy simulation from a non-traditional point of view. According to its definition, LES tries to resolve the large scale motion structure of a turbulent flow, while filtering out and modeling the motion components associated with the small scales. Clearly LES involves fuzziness since the dividing line between large scales and small scales is vague. Even if we introduce a length scale L and a filter function for LES as a means to quantify what is large and what is small, we cannot obtain a clear-cut large scale flow field in practicelrl. To use this fuzziness to our advantage in formulating a more logically consistent scheme of LES, we can define the filter operation as follows. Consider the turbulent flows in a domain of flow V that ‘can be described by the NavierStokes equations with proper initial and boundary conditions div u = 0 dU
dt + div(uu)
= -VP + uV2u
0)
(2)
where u and pp are respectively the velocity and the dynamic pressure, and p and u the mass density and the kinematic viscosity of the fluid. Assume that we can decompose {u, p}(z, t) in 2) x [O,oo) into the components of {ii,@}(z,t;Z) in 2, x [0, oo) corresponding to length scales 1 E (0, co). That is, m{ti,i)}(+;l)dl (3) s0 where the integration can also mean the forming of summation in the case of discrete values of 1. We will adopt a length scale L (much smaller than thle characteristic scale of 2)) and a corresponding filter function PL to introduce the large scale flow field {U, P}, but here we will interpret ,LLL as a membership function as in fuzzy sets131with the following property: 0 5 AL < 1 and /JL is monotonically non-decreasing; AL := 0 if 1 < L, AL = 1 if 1 > L, and PL might vary rather steeply around 1 = L. Further, we define t) =
{%PH?
(4)
The purpose of using such a filter function is to accommodate the fuzziness mentioned above. Next, we need to derive equations that govern {U,P} on the basis of Eqs. (1) to (4). Combining (l), (3) and (4), we get div U = 0 (5) To filter Eq. (2) directly is a difficult task since we do not know the property regarding ii(z, t; Z&(X, t; 1’). We may overcome this difficulty by resorting to an approximate scheme such as the one adopted in [2]. Let us fix y E Do, where Do is the interior of V, and suppose that there is a cube C of side LO > L such that y E C” c V, where C” is the interior of C. We can represent {u,p} accurately in Co x [0, oo) in terms of a Fourier series expansion (21, PX~, t) = C {W(t;
{ii,p}*(t;m) Then, definition
= {ii,fi}(t;
m) exp (i?!T)
-m)
(4) can be replaced by (7)
Vol. 6, No. 1
TAO, RAJAGOPAL
and CHEN:
19
Discrete large eddy simulation
The filter function or membership function 61; is now defined on N3 with by = 1 if (ml is small and fin = 0 if jrnl is large (whether Irnl is large or small is determined by scale L). As mentioned in the introduction, the value of Irnl for which jl~(m) is between unity and zero depends on the application. The important fact to grasp is that it will never be a sharp cut-off, and the boundary or the zone wherein 0 < jl~(m) < 1 is at best known in a fuzzy sense. With the help of (6) and (7), Eq. (2) can be filtered to yield :U
+ divx
fiL(rn + n)ii(t;
m)ii(t;
2n(m+n)a n) exp i -Lo
1
= -VP
t vv2u
or alternately $7
+ div(UU)
+ VQ - vV2U = -f
(8)
with
f = div RR = (uu - VU)
=Cfi~(m+n)[l
i(trR)I)
-~~(m)li~(n)]ii(t;m)ii(t;n)exp
Here we have absorbed trR/S into Q and therefore we need to modify the boundary conditions involving Q in simulations to account for this fact. We should at this juncture discuss some characteristics of f defined above in order to decide how to model it. We notice that: (i) m and n in f and .?2are expected to have an upper bound as ti(t; m) = 0 for very large m resulting from the viscous effect, reflecting the fact that sufficiently small eddies tend to get destroyed due to the viscous dissipation. The spatial scale of f and R, indicated by exp [i 27r(m -I- n) . z/Lo], is bounded from below by L’ and the value of L’ cannot be much smaller than L since we have adopted fi,:(m + n) = 0 for large ]m + n]. (ii) f does not involve directly the larger scale components of U, the components ii(t; m) with [ml being small, due to TV = 1 and so 1- fi~(m)fir,(:n) = 0 in (10) when Irnl and InI are small. (iii) f contains the smaller scale components &(t; m) of U whose scales are around L since fin differs from 1 to some extent here. Also, f contains the small scale components of U, ti(t; m) with [ml being large. For example, it includes ti(t; m)ii(t; m’), ]m] being large and m’ = (1, 0,O) - m since fi~((l, 0,O)) # 0 and j.TiL(m) = fiL(m’) = 0. The velocity components of f are expected to behave irregularly compared with the large scale components of U, an idea underlying the adoption of LES to turbulence modeling. (iv) It follows from (iii) that f depends on U but is not determined completely by U, and f should be treated as a disturbing force acting on the large scale motion {U, P}. Its spatial scale should be compatible with the spatial scale of U according to (i). Meanwhile, f also involves fuzziness due to the presence of fin (see Eq. (11)). We have formulated Eq. (8) with the help of a Fourier series expansion in C” x [0, co) and we need to extend it. We will assume that Eq. (8) holds in general on the basis that (6) and (7) give us rather accurate representations of the fields of {u.,p} and the large scale motion field of {U, P} in a neighborhood of y E 2). Certainly, the filter operation (.) in (8) can pose a difficulty with regard to applications and one way to eliminate this problem is to approximate (.) with a spatial filter of the form
20
Communications
in Nonlinear
Science
& Numerical
March
Simulation
2001
where G is a filter function which can be continuous or discrete, the latter compatible with a finite volume, finite difference, or finite element scheme. Regarding the term f in (8), we will not follow the traditi.onal scheme of turbulence modeling, such as constructing a rather rigid continuous relation on it. We will explore the idea of discrete LES models here and thus we need to develop a method to deal with f properly. To this end, we assume that the domain 2) is discretized with a grid distribution of nodes N zz {&d : m = 1,. . . , M}. Associated with each Z( m) E N there is a control volume V(Z(“)) and there is a subset N(d”)) c N composed of all the nodes immediately neighboring yhO)
= ,(m),
N@“))
= {yhl)
: 1 = 1,. . . , mo}
The distance between any two nodes in N(s(“)) U {g(m30)} is around L, consistent with the filtering scale in LES. Next, suppose that Eq. (5) and the terms on the left-hand side of Eq. (8) have been discretized both in space and time by applying (5) and (8) to the control volume V(Z(~)) at the instant t = tie and adopting appropriate discretization schemes, and suppose that the algebraic equations that are obtained involve only the (discrete) velocities and the (discrete) pressures at the nodes of {~(~yl) : 1 = 0, 1, . . . ,me} and at the instants of tk-1 and tk. These restrictions can be relaxed, and the specifics regarding the discretization are not our concern here because our main purpose is to explain the idea behind the discrete formulation of LES. Denote the discretized equations in the forms of ’ DC”) U(y(“l”), ( D+)
(
tk), y(‘+
U(y(n”),tj),Q(y(m”),tk),y(m”),tj;l
1 = 0, 1, . . . , m. = O,l,*e* ,ma,j
>
= 0
(12)
=/C--1,/i?
>
= FCrn)
(13)
The right-hand side vector quantity F(“) is related to -f in (8) which is yet to be determined. Also, FCm) may play a role of possibly compensating for the large deficiency due to the discretization of the left-hand side terms in (8). Assume that 8+)
= B(m)(U(y(m~z),tk),Q(yO,tk),y(m.l),t~;~
= 0, I;-*
(14)
,m,)
The presence of Q in the argument is based on the fact that Q includes also the subgrid fluctuation energy (see (9)) which is expected to affect f. The dependence of F(m) on the scale L is implied though not expressed explicitly here. Requiring that FCm) meet Galilean invariance - the very invariance possessed by the Navier-Stokes equations and the left-hand side terms of (8), and considering the feature of Q (P) - adding any constant to Q will not affect U, we have
$‘(“) = 2 {G(W”) [u(y’ m, tk) _ q&4, tk) + f{h”) yh4 - ,(4
1
l=l
where G(“ll)
and
H(“y’) are [
the scalar functions of
U(y(m’i), tk) - u(d”), [U@%“),
[
yw)
_ ,w
I>
tk) ] . [ u(y @‘j), tk) ‘- u(d”),
tk) - q&4,
tk)]
tk)] . [,cm.j) -- ,w]
tk)- &(z'"', tk),i,j =1,...,mo I[. yh~)_&d1, Q(y+),
‘Here, @(z(“), tk) is intended to indicate the value and it does not represent the value of a corresponding be made close to each other, an aim of any discretization
of a discrete continuous scheme.
field 4 at space filed f#~at {zcm),
and time points of {a~(~), tk} tk}, though these two should
Vol.
6, No.
1
TAO,
RAJAGOPAL
and
CHEN:
Discrete
large
Instead of such a complex though rather complete formulatioc.,
F(“) = 2 ($4
[qy’4,
eddy
21
simulation
we will take
t/J - U(&d) q]
(15)
l=l
Gh”) = &‘d) (U(y("4,
U(y(‘-‘I,&) - U(X(~),tk)l, Q(y(-"1, tk) - q&4,
tk)). (y(m)
- &4))
tk) - Q(x(~), tk), JyW)
- &4
I)
(16)
which is tractable and which may be appropriate due to the ra,ther uniform distribution of the nodes with jy(mli) - ~(~tj)) N L if i # j. For a node xcrn) which is not near a boundary, we may assume that all the G( myz)‘~have the same structure, that is,
G”+
= @m) IU(y(mT”),tk) - U(x(“), &)I, Q(y(“.*‘), tk) - Q(x’“‘, tk), ( (qyW),&) - U(x(m),tk)) * (yhu - &4), lyh”) - r&.(m) I>
07)
For a node xtrn) near a wall, G(“*“) may have different forms depending on whether ylrn’l) is near or away from the wall. In all these cases, we need to propose specific forms for G(myl) to make the model composed of (12) and (13) determinate. One way to achieve this goal is to find Gcn3”) through a neural network scheme, because neural networks as universal approximators are very suitable for such a taski4]. We now sketch how to implement such a scheme. We will first classify the nodes in N into several categories on the basis of their positions, such as whether near a wall or not, and the number of their neighboring nodes and so on, and we will adopt, the same relations for F(“) for all the nodes belon ing to the same category. Next, neural networks can be used to determine G(m,l))s in the Ji’(J’ s for all the categories. For example, for the category composed of nodes not near the boundary of D and having the same number of neighboring nodes, we have from (13), (14) and (17) that g(“) = DC”) - 2
(j(m)
[u(y(“‘z), tk) - U(&‘),
&)I = 0
l=l
Suppose that we know all the information on the discrete varia.bles U, Q and y present in the above equation, we can construct a neural network to find the structure of C?crn)through, say, supervised learningL51. We can then patch all the discrete equations of (12), (13), (15) and (16) together with the discrete relations from the initial and boundary conditions to formulate a determinate discrete model of LES with scale L on turbulent flows in any 2). This strategy is simple but it may be too simplistic since we may not have all the information on the discrete fields of {U, Q} in a turbulent flow. In this case, we can write down all the algebraic equations of motion for {U, Q} at the nodes of N in D, on the basis of (L2), (13), (15) and (16) and the initial and boundary conditions, the unknowns in the equations are the G(mJ)‘s associated with the categories mentioned above. We may then resort to the method of neural networks to fix the structures of G(m>‘)‘s.
2
Summary
We have discussed large eddy simulation from a new perspective: (i) to introduce large scale motion fields through the concept of membership function in fuzzy sets, which accounts for the ambiguity associated with largeness and smallness; (ii) to construct discrete models of
Communications
22
in Nonlinear
Science & Numerical
Simulation
March 2001
LES directly without taking the intermediate step of developi:ng a continuous model first and then discretizing the differential equations of motion, consistent with the spirit of LES to catch the large scale flow features and the practice of computational simulation; (iii) to resort to neural networks to resolve the structures of the undetermined coefficients present in discrete LES models, tapping the power of neural networks as universal approximators. Also, fuzzy neural networks may be used to resolve the problem of overfittingi61 and to accommodate the uncertainty associated with the values of {U, P}(z(“), tk). The focus of this work is to explain some basic ideas for modeling turbulence with discrete large eddy simulation. The details regarding what types of neural networks to be used and whether to train the networks supervised or unsupervised are yet to be worked out, and concrete models are to be constructed and implemented to carry out simulations. It can also be inferred from this work that a neural network may be adopted to help devise or improve numerical schemes to solve differential equations. We have explored in [7] the possibility of compensating large deficiency resulting from an improper discretization scheme of finite element methods to a differential equation and suggested the modeling of the residual of the discretized equation with a (constitutive) relation. And this relation may be found, for example, by a neural network through supervised learning, if accurate solutions to the differential equation are known in some cases.
Acknowledgment The third author would like to thank support from the Special Funds for Major State Basic Research Projects of China (grant No. G1999043605 and G1999022207), and the Trans-Century Training Foundation sponsored by the National Education Ministry of China.
References [l] Chen, G., Tao, L. and RajagopaI, K. R., Remarks on large eddy simulation, Commun. Nonlinear Sci. Numer.
Simul.,
2000, 5(3): 85-90
[2] Rajagopal, K. R., Tao, L. and Chen, G., A formulation of large eddy simulation, Commun. Nonlinear Sci. Numer. Simul., 1999, 4(4): 245-248 [3] Zimmermann, H. J., Fuzzy Set Theory and Its Applications, Kluwer, Boston, 1996 [4] White, H., Artificial Neural Networks: Approximation and Learn.ing Theory, Blackwell, Cambridge, Mass, 1992 [5] Haykin, S., Neural Networks: A Comprehensive Foundation, Macmillan, New York, 1994 [6] Buckley, J. L. and Feuring, New York, 1999
T., Fuzzy and Neural:
Interaction
and Applications,
Physica-Verlag,
[7] Tao, L., Chen, G. Q. and Rajagopal, K. R., Notes on finite element method and its generalization, J. Hydrodynamics,
1996, B3: 97-101