I74
J. 1.. I.UMLEI’ A N D 1%. KHAJEH-NOURI
procedurc, howcver. it will be insrructiw lo carry out a physical analysis of
the right-hand side of (I). to see how a production term can be retained at
infinite Reynolds number.
The right-hand side of ( I ) represents a balance between stretching and
dissipation. and it must be possible for the (relatively small) mismatch to bc
of either sign. That is, consider the stretching of a single vortex to equilibrium: if the stretching is suddenly increased, momentarily the first term will
dominate the second, setting up more vorticity until equilibrium is again
attained, at a higher level; if the stretching is reduced, the process is reversed.
Put in statistical terms, if the spectral energy flux increases, the first term
should dominate the second until the dissipation has been increased to
match the flux, and vice versa. There is, in addition an unsteady effect: in a
fluctuating turbulent velocity field, equilibrium is never attained since the
strain rate changes before it can be achieved. Hence. there is always a fluctuating mismatch; although to first order, we would expect this to average to
zero, we would expect nonlinear effects to produce a small net loss.
The response to these effects should depend on the time scale ratio. The
time scale of the dissipative eddies is (v/E)”’; if the turbulence were isotropic
and decaying, a time scale descriptive of the unsteady stretching would be
q2/2r;. If there is an input to the spectral flux, there will be another time scale
associated with this input. In a homogeneous flow, the input is characterized
by P, the production, and the time scale will be q’l2P. In an inhomogeneous
situation, it is more difficult to tind a simple way of characterizing the input,
since part of P is transported. I n the next section we will obtain by formal
means an appropriate expression in the inhomogeneous situation ; here we
will retain P, which may be thought of as an approximation for small inhomogeneity. This is effectively what was done by Daly and Harlow (1970).
Jones and Launder (1972) and Ng and Spalding (1972). The inverse of the
time scale thus may be written as (2E/q2)F(P/~), where F is an unknown
function. If the production (and hence the anisotropy) is small, we may
expand to obtain (2E/q2)( 1 - aP/F:). It is, of course, not legitimate to use such
an expression for values of PIE - O( 1); the expression will serve at least to
predict qualitative behavior, however, since it reverses sign as we have
reasoncd it must.
The difference on the right-hand side of (1) might consequently be
modeled as
(4)
-E(E/V)“’{O + h(2E/qZ)[I - u(p/i!)](v/c)”2 + O( I&)}
where thc 0 symbolizes the equality of the terms at infinite Reynolds
number. To second order in time, the direct response to the mean flow
distortions will appcar through Y,,, etc.; these cannot appear to first order
hecause they are of the wrong tensor rank; a scalar term is needed and can
J. 1.. I.UMLEI’ A N D 1%. KHAJEH-NOURI
procedurc, howcver. it will be insrructiw lo carry out a physical analysis of
the right-hand side of (I). to see how a production term can be retained at
infinite Reynolds number.
The right-hand side of ( I ) represents a balance between stretching and
dissipation. and it must be possible for the (relatively small) mismatch to bc
of either sign. That is, consider the stretching of a single vortex to equilibrium: if the stretching is suddenly increased, momentarily the first term will
dominate the second, setting up more vorticity until equilibrium is again
attained, at a higher level; if the stretching is reduced, the process is reversed.
Put in statistical terms, if the spectral energy flux increases, the first term
should dominate the second until the dissipation has been increased to
match the flux, and vice versa. There is, in addition an unsteady effect: in a
fluctuating turbulent velocity field, equilibrium is never attained since the
strain rate changes before it can be achieved. Hence. there is always a fluctuating mismatch; although to first order, we would expect this to average to
zero, we would expect nonlinear effects to produce a small net loss.
The response to these effects should depend on the time scale ratio. The
time scale of the dissipative eddies is (v/E)”’; if the turbulence were isotropic
and decaying, a time scale descriptive of the unsteady stretching would be
q2/2r;. If there is an input to the spectral flux, there will be another time scale
associated with this input. In a homogeneous flow, the input is characterized
by P, the production, and the time scale will be q’l2P. In an inhomogeneous
situation, it is more difficult to tind a simple way of characterizing the input,
since part of P is transported. I n the next section we will obtain by formal
means an appropriate expression in the inhomogeneous situation ; here we
will retain P, which may be thought of as an approximation for small inhomogeneity. This is effectively what was done by Daly and Harlow (1970).
Jones and Launder (1972) and Ng and Spalding (1972). The inverse of the
time scale thus may be written as (2E/q2)F(P/~), where F is an unknown
function. If the production (and hence the anisotropy) is small, we may
expand to obtain (2E/q2)( 1 - aP/F:). It is, of course, not legitimate to use such
an expression for values of PIE - O( 1); the expression will serve at least to
predict qualitative behavior, however, since it reverses sign as we have
reasoncd it must.
The difference on the right-hand side of (1) might consequently be
modeled as
(4)
-E(E/V)“’{O + h(2E/qZ)[I - u(p/i!)](v/c)”2 + O( I&)}
where thc 0 symbolizes the equality of the terms at infinite Reynolds
number. To second order in time, the direct response to the mean flow
distortions will appcar through Y,,, etc.; these cannot appear to first order
hecause they are of the wrong tensor rank; a scalar term is needed and can