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J. L. LUMLEY AND 8. KHAJEH-NOURI
inadequate models of the second moments (although preliminary results, as
we shall see later, appear quite favorable). It should not be necessary to
point out, however, that we cannot reach a rational conclusion on this
question unless we are sure that the closure used is based on a small number
of explicitly stated, readily grasped principles. and that all terms, and only
those terms, generated by these principles are used. Otherwise, we will not
know if an unsatisfactory result can be attributed to the omission of a vital
term, the inclusion of an extraneous one, or the use of an incorrect basic
principle. The model that we will present, in common with the other third
order closures, contains many undetermined constants. It has long been part
of the folk wisdom in turbulence that a model can be made to fit a flow,
given sufficiently many constants.’ While there is some justice to this, it is
not quite true. A model may be incapable of reproducing a certain qualitative behavior, regardless of the values assigned to the constants. Models with
many constants can still be intellectually satisfactory, so long as the constants are not optimized for each flow, or group of flows, and so long as the
physical interpretation of the constant is clear. It is important that the values
of the constants governing each physically distinct effect be determined by
computation in a situation in which that effect is not influenced by others.
Otherwise, one is in danger of adjusting the wrong constant for the right
reason. As an aside, if this principle is applied conscientiously, it quickly
becomes clear that despite the wealth ofexperimental data collected over the
past few decades, there is a remarkable dearth of welldocumented elemenfury turbulent flows, in which one effect at a time is carefully studied.
2. THE HIGH REYNOLDS NUMBER APPROXIMATION
AND THE DISIPAIION EQUATIONS
In Tennekes and Lumley (1972). it is shown how orders of magnitude may
be assigned to various correlations appearing in the dynamical equations.
Roughly, instantaneous quantities appearing in the correlations are of two
types, belonging either to the energy containing range of eddies or to the
dissipation range. The former has characteristic frequency u‘/l (where
k = u”/l, 3u’* is twice the mean fluctuating energy q2, and Z is the mean
dissipation of energy per unit mass), while the latter has characteristic
frequency u’/1, where 1 - 41R; ’’’, R, = u’l/v. The correlation coefficient
between two quantities from the same range may usually be taken as unity,
but the coefficient between two quantities, each from a different range, is of
the order of the time scale ratio 1/1 - 4R;’”.
* Bradshaw is credited with the remark, at the Stanford Conference on Computation of
Turbulent Boundary Layers. that with six constants he could create an elephant.
J. L. LUMLEY AND 8. KHAJEH-NOURI
inadequate models of the second moments (although preliminary results, as
we shall see later, appear quite favorable). It should not be necessary to
point out, however, that we cannot reach a rational conclusion on this
question unless we are sure that the closure used is based on a small number
of explicitly stated, readily grasped principles. and that all terms, and only
those terms, generated by these principles are used. Otherwise, we will not
know if an unsatisfactory result can be attributed to the omission of a vital
term, the inclusion of an extraneous one, or the use of an incorrect basic
principle. The model that we will present, in common with the other third
order closures, contains many undetermined constants. It has long been part
of the folk wisdom in turbulence that a model can be made to fit a flow,
given sufficiently many constants.’ While there is some justice to this, it is
not quite true. A model may be incapable of reproducing a certain qualitative behavior, regardless of the values assigned to the constants. Models with
many constants can still be intellectually satisfactory, so long as the constants are not optimized for each flow, or group of flows, and so long as the
physical interpretation of the constant is clear. It is important that the values
of the constants governing each physically distinct effect be determined by
computation in a situation in which that effect is not influenced by others.
Otherwise, one is in danger of adjusting the wrong constant for the right
reason. As an aside, if this principle is applied conscientiously, it quickly
becomes clear that despite the wealth ofexperimental data collected over the
past few decades, there is a remarkable dearth of welldocumented elemenfury turbulent flows, in which one effect at a time is carefully studied.
2. THE HIGH REYNOLDS NUMBER APPROXIMATION
AND THE DISIPAIION EQUATIONS
In Tennekes and Lumley (1972). it is shown how orders of magnitude may
be assigned to various correlations appearing in the dynamical equations.
Roughly, instantaneous quantities appearing in the correlations are of two
types, belonging either to the energy containing range of eddies or to the
dissipation range. The former has characteristic frequency u‘/l (where
k = u”/l, 3u’* is twice the mean fluctuating energy q2, and Z is the mean
dissipation of energy per unit mass), while the latter has characteristic
frequency u’/1, where 1 - 41R; ’’’, R, = u’l/v. The correlation coefficient
between two quantities from the same range may usually be taken as unity,
but the coefficient between two quantities, each from a different range, is of
the order of the time scale ratio 1/1 - 4R;’”.
* Bradshaw is credited with the remark, at the Stanford Conference on Computation of
Turbulent Boundary Layers. that with six constants he could create an elephant.