CALTECH ASCI TECHNICAL REPORT 073 caltechASCI/2000.073
Lagrangian methods for the tensor-diffusivity subgrid model Piet Moeleker and Anthony Leonard
Lagrangian methods for the tensor-di usivity subgrid model Piet Moeleker and Anthony Leonard Graduate Aeronautical Laboratories California Institute of Technology, Pasadena, CA 91125 email:
[email protected]
Abstract We consider the ltered scalar advection-di usion equation. An in nite series expansion for the advection term was found for closure. By retaining only the rst two terms in the expansion, the tensor-di usivity subgrid model is obtained. In order to avoid problems associated with the negative di usion of the model in one or more spatial directions, the scalar eld is represented as a sum of anisotropic Gaussian particles. Numerical results in two dimensions will be presented.
1 Tensor-di usivity subgrid model Consider the scalar advection-di usion equation. In order to perform numerical computations for small values of the di usivity without large CPU e orts, a ltering operation is used to obtain an equation for the large scale structures in which the e ects of the small scales have to be modeled. These simulations are known as large eddy simulations (LES) and the models as subgrid models. Our research has used a Gaussian spatial lter with a characteristic length scale . In order to close the ltered equation, an in nite series expansion was found in terms of known variables[1, 3], uc (x; t) =
1 1 2 n X b @ nu @n b ; n=0 n! 2 @xi1 @xi2 : : : @xin @xi1 @xi2 : : : @xin
(1)
where a hat indicates a ltered quantity, u(x; t) is an incompressible velocity eld, and (x; t) the (unknown) scalar function. Einstein's summation convention is implied. By retaining only the rst two terms in the expansion, we end In
conference proceedings of the 8th European Turbulance Conference in Barcelona, 2000
2 up with the ltered advection-di usion equation closed by the tensor-di usivity subgrid model, @b @t
b r b = r2 b +u
2
2
@2 b
bij S
@xi @xj
(2)
;
where S is the strain rate tensor. The extra term can be interpreted as an 2 b S , which depends on the added di usivity with an e ective di usivity 2 ij spatial direction. It is straight forward to show that this model is material frame indi erent and will allow for backscatter. b = r ub = 0, at least one of the eigenvalues of the strain Since trace(S) rate tensor has to be greater than zero indicative of a direction where the subgrid model acts as negative di usion. Computations using a nite di erence or spectral method show that this can lead to growing instabilities in the scalar eld.
2 Lagrangian particle method On occasion mathematical models of physical processes lead to ill-posedness requiring some kind of regularization. Care needs to be taken in the choice of regularization, since the results can depend strongly on the approach used. Our work will regularize the problem of negative di usion in the stretching directions by decomposing the scalar eld in a collection of Lagrangian particles each of them well-behaved for large wave numbers. Assume the scalar function can be written as a collection of N anisotropic Gaussian particles, N a pdet(M ) X kp k exp (x xk )T Mk (x xk ) ; b(x; t) = (3) d Æk2 k=1 ( Æk ) where each particle k is centered at xk and d is the spatial dimension of the problem. The positive de nite matrix Mk gives the shape of each particle. Equations for the time evolution of xk and Mk are obtained after substituting (3) in (2), expanding each term in a series of Hermite polynomials, and setting the lowest order coeÆcients equal to zero. For particle k , we end up with dxk dt
dMk dt
2 uk ; r 2
2
= uk
ruk Mk
= +
2
2
kT
Mk ru
4 Æ2
k
Mk Mk +
rr2 uk Mk + Mk rr2 ukT
2 Æ2
k
Mk
ruk + rukT
(4a) Mk
;
(4b)
3 where an overline indicates a weighted average over anisotropic particle k . To obtain realistic results, we need to choose the core size Æk larger than or equal to the ltering constant . Core sizes smaller than can lead to growing instabilities and do not correspond to a physically realizable un ltered eld. 6
6 4
0.001
0.0001
2
y
y
0.0001
0.01
4
0.001
0.01
0.01
2
0.0001 0.1 0.01
0
0 0.01
-2 -4
0.1
0.01 0.0001
-2
0 x
2
4
-2 -4
0.001
-2
(a) ltered DNS
0 x
2
4
6 0.01
0.01
4
4 0.0001
2
y
y
4
(b) no subgrid model
6
0.0001
2
0.1
0
0.1
0 0.01
-2 -4
2
0.01 0.0001
0.0001
-2
0 x
2
(c) tensor-di usivity model
4
-2 -4
-2
0 x
(d) Smagorinsky model
Figure 1: Scalar contour lines (0.0001, 0.001, 0.01, 0.1, and 0.3) at t = 9. Our scheme solves the di usion term exactly (core-expansion method), which necessitates the implementation of a particle splitting or remeshing scheme to obtain accurate results[2, 5]. A novel remeshing scheme has been implemented making use of the properties of the particle method. The new particles are assumed to be all identical, axisymmetric, and spread out on a regular equidistant mesh. A least square error method is used to obtain explicit expressions for the
4 unknown amplitudes. Numerical experiments show good agreement between the scalar elds before and after the remeshing operation, while keeping the number of total particles reasonably small as well.
3 Numerical results The two-dimensional incompressible velocity eld with a time-dependent perturbation, u sin(x) sin(y ) = (5) v cos(x) cos(y ) + 0:5 sin(t) has been used to test di erent aspects of the subgrid model and the numerical implementation. A Gaussian exponential located initially at (0:3; 0:4) with core size 0.5 was used as an initial condition. The di usivity was set equal to = 0:001 and the ltering constant = 0:15. In order to assess the accuracy of the model, a ltered DNS solution has been computed, which is given in gure 1(a) at t = 9. The initial Gaussian has been approximated by 6504 anisotropic particles. In absence of any subgrid model, the result is given in gure 1(b). When the tensordi usivity model is used, the solution tracks the ltered DNS solution very well as shown in gure 1(c). This solution was remeshed every time unit. Without remeshing, the solution will break down after t = 3. To assess the eÆciency of this model, the results using a Smagorinsky subgrid model (Cs = 0:20) are given in gure 1(d). Work is currently underway to implement the model and particle method in a forced turbulence code to study the e ects of mixing for high Schmidt numbers.
References [1] Bedford, K. and W. Yeo: 1993, `Conjunctive ltering procedures in surface water ow and transport'. In: B. Galperin and S.A. Orszag (ed.): Large eddy simulation of complex engineering and geophysical ows. CUP, pp. 513{537. [2] Greengard, C.: 1985, `The core spreading vortex method approximates the wrong equation'. J. Comput. Phys. 61, 345{348. [3] Leonard, A.: 1997, `Large-eddy simulation of chaotic convection and beyond'. AIAA 97-0204. [4] Moeleker, P. and A. Leonard: 2000, `Lagrangian methods for the tensordi usivity model'. submitted to J. Comput. Phys. [5] Rossi, L.: 1996, `Resurrecting core spreading vortex methods: a new scheme that is both deterministic and convergent'. SIAM J. Sci. Comput. 17(2), 370{397.
