ENERGY CASCADE IN LARGE-EDDY
SIMULATIONS OF TURBULENT
FLUID FLOWS
A. LEONARD'
Department of' Mechanical Engineering. Stunjbrd University
Stanford, Calgornla 91305, U S A .
1. INTRODUCTION
Computer simulations of threedimensional turbulent flows which explicitly account for the motions of eddies ranging in sizc down to the inertial
subrange are now possible. In most cases of interest, motions on the order of
the dissipation length scale cannot be treated explicitly. Modifications of the
Navier-Stokes equations must then be introduced to simulate properly the
energy cascade. Considerable damming up " of the turbulence energy in
the large scales would occur, for example, if the unmodified equations were
used with an energy-conserving finitedifference scheme on the advective
term.
One approach to the problem is to use an eddy viscosity to account for the
influence of the subgrid-scale motions on the large-scale fluctuations (Lilly,
1967). In this model, the energy cascade is then viewed solely as an energy
loss of the large-scales due to an artificial viscosity arising from subgrid-scale
motions. The advective term for the large-scak motions is unmodified.
In this paper, the derivation of smoothed or filtered momentum and
continuity equations for the large-scale, energy-mntaining eddies is reexamined. Noting that the large-scale motions vary in a nonnegligible way
over an averaging volume, we investigate a more accurate, modified advective term in the momentum equations for these motions. This term is nonconservative and is shown to lead to significant energy extraction from the
large scales due to triple correlations of these motions. The subgrid-scale
Reynolds stress term is still present but plays a reduced role as far as the
energy cascade is concerned. Similar arguments are applied to the analysis
of large-scale fluctuations of a passive scalar.
.
' frcwnr u d d r m : NASA Arnes Raearch Center, Moffett Field, California 94035, U.S.A.
237
SIMULATIONS OF TURBULENT
FLUID FLOWS
A. LEONARD'
Department of' Mechanical Engineering. Stunjbrd University
Stanford, Calgornla 91305, U S A .
1. INTRODUCTION
Computer simulations of threedimensional turbulent flows which explicitly account for the motions of eddies ranging in sizc down to the inertial
subrange are now possible. In most cases of interest, motions on the order of
the dissipation length scale cannot be treated explicitly. Modifications of the
Navier-Stokes equations must then be introduced to simulate properly the
energy cascade. Considerable damming up " of the turbulence energy in
the large scales would occur, for example, if the unmodified equations were
used with an energy-conserving finitedifference scheme on the advective
term.
One approach to the problem is to use an eddy viscosity to account for the
influence of the subgrid-scale motions on the large-scale fluctuations (Lilly,
1967). In this model, the energy cascade is then viewed solely as an energy
loss of the large-scales due to an artificial viscosity arising from subgrid-scale
motions. The advective term for the large-scak motions is unmodified.
In this paper, the derivation of smoothed or filtered momentum and
continuity equations for the large-scale, energy-mntaining eddies is reexamined. Noting that the large-scale motions vary in a nonnegligible way
over an averaging volume, we investigate a more accurate, modified advective term in the momentum equations for these motions. This term is nonconservative and is shown to lead to significant energy extraction from the
large scales due to triple correlations of these motions. The subgrid-scale
Reynolds stress term is still present but plays a reduced role as far as the
energy cascade is concerned. Similar arguments are applied to the analysis
of large-scale fluctuations of a passive scalar.
.
' frcwnr u d d r m : NASA Arnes Raearch Center, Moffett Field, California 94035, U.S.A.
237