The dynamics of the evolution of turbulence statistics depend on the structure of the turbulence. For example, wavenumber anisotropy in homogeneous turbulence is known to a ect both the interaction between large and small scales (Kida & Hunt 1989), and the non-local e ects of the pressure{strain-rate correlation in the one-point Reynolds stress equations (Reynolds 1989; Cambon et al. 1992). Good quantitative measures of turbulence structure are easy to construct using two-point or spectral data, but one-point measures are needed for the Reynolds-averaged modelling of engineering flows. Here we introduce a systematic framework for exploring the role of turbulence structure in the evolution of one-point turbulence statistics. Five one-point statistical measures of the energy-containing turbulence structure are introduced and used with direct numerical simulations to analyse the role of turbulence structure in several cases of homogeneous and inhomogeneous turbulence undergoing diverse modes of mean deformation. The one-point structure tensors are found to be useful descriptors of turbulence structure, and lead to a deeper understanding of some rather surprising observations from DNS and experiments.

The goal of the one-point Reynolds-averaged theory of turbulence, initiated by
Osborne

Rij = 2k 3 ij ? 2 Sij; = C k2=": (1.1) In the modern family of k{" models, the eddy viscosity is modelled in terms of the turbulent kinetic energy k (hereafter q2 = 2k will be used) and dissipation rate ", for which partial di erential equations (PDE) are used. Such models have proven to be very useful in predicting near-equilibrium or decaying turbulent flows, where the deformation rate S = pSijSji is small relative to (or even comparable to) the reciprocal time scale "=q2 of the turbulence.

The response of turbulence to rapid mean deformation (S q2=" 1), at least initially, is described by rapid distortion theory (RDT). Under RDT the nonlinear e ects resulting from turbulence{turbulence interactions are neglected in the governing equations, but even when linearized in this fashion, the one-point governing equations are, in general, not closed due to the non-locality of the pressure fluctuations. In flows with rapid mean deformations, the structure takes some time to respond and eddyviscosity models, which predict immediate response, are inadequate. Reynolds stress transport (RST) equations have been added to the PDE system used in turbulence models in an attempt to deal with this weakness. While RST models have enjoyed some success, they are not yet widely used in industry because they have not proven reliably better than simpler models.

The reason behind this failure has to do with the proper characterization of the anisotropy of the turbulence in a one-point theory. RST and other one-point approaches use the anisotropy of the turbulent stresses to characterize the anisotropy of the eld. However, the departure of Rij from the isotropic form q2 ij=3 is only one aspect of the anisotropy of the turbulence. Here we hope to emphasize that the turbulence can exhibit anisotropy that is dynamically signi cant (even for the transport of the one-point turbulent stresses) and which is not captured in the Reynolds stresses. Other one-point statistical measures of anisotropy, in addition to the anisotropy of the turbulent stress, are then needed.

The Reynolds stresses carry information about the componentality of the turbulence (the relative strengths of di erent velocity components). Roughly speaking, each largescale structure tends to organize spatially the fluctuating motion in its vicinity, and in so doing, to eliminate gradients of the fluctuation elds in certain directions (those in which the spatial extend of the structure is signi cant), and to enhance gradients in other directions (those in which the spatial extend of the structure is small). Thus associated with each eddy are local axes of dependence and independence. In undeformed isotropic turbulence the various axes of dependence and independence (due to individual eddies) are oriented randomly and this means that the fluctuation eld due to the ensemble of all the eddies has gradients in all three directions. Mean deformation acting on the turbulence creates structural anisotropy because it stretches and aligns the energy-containing turbulent eddies. This in turn aligns the axes of dependence and independence due to individual eddies and creates directions in which, even in a statistical sense, gradients of energy-containing fluctuations are weak. Thus by acting on the energy-containing structure mean deformation can create axes of independence that are reflected in autocorrelations of gradients of the fluctuation elds. In regions in the flow where there is one such direction of independence, the turbulence becomes two-dimensional. Note that this says nothing about the intensity of the fluctuations in that direction. Thus the anisotropy of the dimensionality of the turbulence is in general distinct from the anisotropy of its componentality.

Adequate characterization of the state of the turbulence also requires information about the dimensionality of the turbulence (the relative uniformity of structure in di erent directions), and even of additional turbulence features in some cases. Turbulence models carrying only componentality information (e.g. standard RST models) cannot possibly satisfy conditions associated with the dimensionality of the turbulence or reflect di erences in dynamic behaviour associated with structures of di erent dimensionality (nearly isotropic turbulence vs. turbulence with strongly organized two-dimensional structures).

This fundamental realization led us to introduce three second-rank and one thirdrank one-point turbulence tensors that carry the information missing from the Reynolds stresses. These new tensors, along with Reynolds stresses, provide a minimal tensorial base for a complete one-point theory of turbulence. Not all of these tensors are important under all conditions, which explains why traditional RST models (and even simpler approaches) have enjoyed considerable success in some cases.

Here we introduce these new tensors, demonstrate their e ectiveness as descriptors of the turbulence structure, and characterize the information they carry so that turbulence modellers can devise schemes for incorporating at least some of that information into their models. In x 2 we consider anisotropic homogeneous turbulence undergoing rapid mean rotation to demonstrate the need for structure information in one-point formulations. The fundamental de nitions and properties of the structure tensors are introduced and discussed in xx 3 and 4. In xx 5{9, we use a database from direct numerical simulations (DNS) to study the structure tensors for several cases of homogeneous and inhomogeneous turbulence undergoing diverse forms of mean deformation. The examples considered demonstrate the usefulness of the structure tensors in providing an accurate description of the turbulence. In certain cases, counterintuitive results can only be explained in terms of the combined componentality{dimensionality description, which again underscores the importance of these new tensors beyond the immediate scope of turbulence modelling. Finally, x 10 summarizes the results. 2. Insights from rapid distortion analysis of homogeneous turbulence

The di erence between componentality and dimensionality information is nicely
exhibited by the inviscid RDT of homogeneous turbulence, for which the evolution
equations for the Reynolds stresses, Rij = u0iu0j, are

Tirjapid = 2Gkn(Minkj + Mjnki):

Z kpkq
k2 Eij(k) d3k;
(2.1)
(2.2)
(2.3)
(2.4)
where Eij(k) u^i(k)u^j (k) is the velocity spectrum tensor, k is the wavenumber vector,
hats denote Fourier coe cients and the denotes a complex conjugate.y M is a
fourth-rank tensor that must be modelled in terms of the tensor variables in the
one-point model

Z Rij = Mijpp =

Eij(k) d3k and

Dpq = Miipq =

Z kpkq k2 Eii(k) d3k: y In homogeneous elds, discrete Fourier expansions can be used to represent individual realizations in a box of length L; then the discrete cospectrum of two elds f and g is given by X~ij(k) = (L=2 )3f^i(k)g^j (k), where the bar represents an ensemble average over the box. The cospectrum of two elds Xij(k) is the limit of the discrete cospectrum X~ij as L ! 1. Here we use Xij(k)

f^i(k)g^j (k) as a shorthand notation, but the exact de nition should be kept in mind.
As noted above, R Rij contains the componentality information in M. D describes
the distribution of energy over directions in wavenumber space, i.e. the dimensionality
information in M. A more general de nition of D will be given shortly. The importance
of dimensionality information in wavenumber space has been discussed by several
authors, including

Additional insight into the important structural information involving the
energycontaining eddies is provided by analysis of axisymmetric turbulence. Using the
methods of

Eij(k) =

E
Here E(k; k1; t) is the energy spectrum function, A(k; k1; t) is the anisotropy spectrum
function, R(k; k1; t) is the rotational spectrum function, and H(k; k1; t) is the helicity
spectrum function. The spectrum in (2.5) di ers from the formula derived by

k = 4? 1 R; k dR dt

k
= ?4? 1 A;
k
dH
dt
= 0;
where ? ?23 = (G23 ? G32)=2 is the mean rotation rate about the axis of symmetry
x1. These show that the spectrum function R is generated by A, and will appear
even if absent initially if there is any anisotropy in the initial turbulence. The sole
contributor to the energy is E. One-point statistics are obtained by integrating over
the wavenumber space as shown in (2.3) and (2.4). Only E and A contribute to
y For example, Dij as de ned here corresponds to Cij in the notation of

The analysis above suggests that three kinds of information are important in the evolution of the turbulent stresses in homogeneous turbulence; (i) componentality information in the turbulent stresses themselves; (ii) dimensionality information available through D; and (iii) information about the breaking of reflectional symmetry by mean or frame rotation carried by the stropholysis tensor Q . With this brief motivation based on homogeneous turbulence, we next turn to general de nitions of one-point tensors that contain this and other essential information.z

We introduce the turbulence stream function vector ?i0, de ned by u0i = its?s0;t; ?i0;i = 0; ?i0;nn = ?!i0; (3.1) where !i0 denotes the turbulence vorticity vector. We require ?i0 to be divergence-free so that the last equality of (3.1) is valid. This choice is important for the physical meaning of the resulting structure tensors (see x 3.7). Note that ?i0 satis es a Poisson equation and hence, like the fluctuating pressure, carries non-local information. Using the de nition (3.1), the Reynolds stress tensor Rij, its associated normalized tensor rij, and its anisotropy tensor ~rij are

Rij = u0iu0j = ipq jts?q0;p?s0;t; rij = Rij=Rkk = Rij=q2; ~rij = rij ? ij=3: (3.2) This shows that one-point correlations of stream function gradients, like the Reynolds stresses, are dominated by the energy-containing scales. Next, we de ne several correlations of stream function gradients and then show that they carry useful information.

The structure dimensionality tensor Dij, its associated normalized tensor dij and
dimensionality anisotropy tensor d~ij are de ned by

z The reader will nd interesting the comparison of the one-point formalism developed here
with the spectral formalism of

The structure circulicity tensor Fij, and its associated normalized tensor fij and circulicity anisotropy tensor f~ij are de ned by

Fij = ?i0;n?j0;n; fij = Fij=Fkk; f~ij = fij ? ij=3: (3.4) The Fij tensor describes the large-scale structure of the vorticity eld which is most clearly seen in the case of homogeneous turbulence. In this case, the last equality in (3.1) ensures that ? 0 = 0 whenever !0 = 0; when there is no large-scale circulation about the x1-axis the turbulence stream function component ?10 = 0, and hence F11 = 0. On the other hand, when most of the large-scale circulation is concentrated about the x1-axis, f11 ! 1. Note that Fkk = Dkk.

The inhomogeneity tensor Cij, its associated normalized tensor cij and inhomogeneity anisotropy tensor ~cij are de ned by

Cij = ?i0;n?n0;j; cij = Cij=Dkk; ~cij = cij ? ckk ij=3: (3.5) Note that the normalized tensor cij is de ned in terms of the trace Dkk = Fkk, not in terms of Ckk. This choice is motivated by the following considerations. First, the diagonal components of C, unlike those of R, D and F , can be either negative or positive. This lack of positive semi-de niteness in C means Ckk can vanish in regions of a turbulent flow, thus producing an ill-de ned cij if it were used to normalize C. In addition, normalizing with Dkk = Fkk results in a tensor that provides a measure of the importance of inhomogeneity relative to the other one-point tensors. Another possibility would be to normalize with Rkk, but this choice proves problematic in wall bounded flows where Rkk vanishes at the wall.

One nds that Cij vanishes for homogeneous turbulence by using (3.1) to write the basic de nition (3.5) in the form (3.6) (3.7)

Cij = (?i0?k0;j);k:

The anisotropy tensors ~r, d~, and f~ are trace-free, and accordingly each has only two
independent anisotropy-invariants. These can be formed using the general de nitions
Ix
x~ii = 0; IIx
? 21 x~ij x~ji; IIIx
31 x~ij x~jk x~ki;
valid for any traceless second-rank tensor x~ij. One can use the independent
anisotropyinvariant coordinates (IIIx; ?IIx) to map all possible states represented by these
tensors

From the simple example of rapidly rotated homogeneous axisymmetric turbulence in x 2.2 we know that the stropholysis tensor Q carries information independent of Qkjk = 0;

Qkki = Qikk = ?(u0k?i0); k: The rst of these traces vanishes due to the divergence-free nature of the fluctuating stream function (see (3.1)). From (3.10), any bi-trace of the stropholysis Q is given by

Qiik = Qiki = Qkki = ? 32 (u0k?i0); k; which vanishes in homogeneous turbulence.

When considered together, Dij and Fij give a fairly detailed description of the turbulence structure. For example, d11 0 and f11 1 means that the dominant large-scale structures are very nearly two-dimensional eddies aligned with the x1-axis, with motion con ned in the plane normal to the eddy axis and organized in a large-scale circulation. We call structures of this type vortical eddies (see gure 1a). On the other hand, d11 0 and f11 0 corresponds to two-dimensional structures aligned with the x1-axis; motion is con ned along the eddy axis in the form of jets and wakes as opposed to circulation around the axis. We call turbulence structures of this second type jetal eddies (see gure 1b). In general, a turbulence eld includes both vortical and jetal eddies, which can be correlated or uncorrelated. We refer to structures having correlated jetal and vortical motion as helical eddies (see gure 1c). In later sections (for example see x 7) these properties of Dij and Fij are shown using DNS data.

The second-rank dimensionality and circulicity tensors represent one-point correlations that carry non-local information about the structure of a turbulent flow. This can be demonstrated by again considering the simple problem of homogeneous turbulence subjected to mean rotation. In this case, the rapid-pressure fluctuations pr are given by (see KR94 and Reynolds 1976)

1 p;rkk = ?z!z0; 12 p;rzp;rz = ?m?nFnm: (3.8) (3.9) (3.10) (3.11) (3.12) (3.13) R, D, F , and C. The general de nition of the third-rank fully symmetric structure stropholysis tensor is given by where The bi-traces of Qijk are

Qijk = 16 (Qijk + Qjki + Qkij + Qikj + Qjik + Qkji);

Qijk = ?u0j?i0;k: where ? is the mean vorticity vector. Using (3.1) and (3.12), one can show that for homogeneous turbulence with uniform density (t) Clearly in this simple case Fij carries the non-local information contained in the intensity of the rapid pressure gradient.

The normalized inhomogeneity tensor c reflects the degree of inhomogeneity of the turbulence eld and is identically zero in homogeneous turbulence. In regions of inhomogeneous flow where inhomogeneity e ects are strong one can expect at least some of the components of c to be O(1), whereas in regions where local homogeneity prevails one nds that all components of c are small. Some of the examples we consider later will help clarify the role of c.

In x 2.2, we saw that the third-rank stropholysis tensor Qijk is non-zero as a direct result of the breaking of reflectional symmetry in the velocity spectrum caused by mean rotation, hence the name stropholysis. The evolution equation for Q (see KR94 and Appendix B) and numerical simulations (see x 5) show that under irrotational mean strain, irrespectively of the rate of strain, Q will remain zero if it is initially zero. Only mean rotation can create Q and mean strain (if also present) can only act to modify Q once it has been generated.

Finally, note that the presence of the turbulence stream function components
in these de nitions means that the structure tensors cannot be easily measured
in experiments. On the other hand, obtaining these tensors from direct numerical
simulations is a relatively straightforward task, and this, combined with the availability
of transport equations for these tensors (see KR94 and Appendix B), can lead to a
better understanding of the dynamical role of the turbulence structure. This insight
can in turn be used as the foundation for the formulation of simpli ed structure-based
one-point models that incorporate the key physics without necessarily relying directly
on all of these tensors and their transport equations

The isotropic tensor identity (Je reys 1931; Mahoney 1985)

ipq jts = ij pt qs + it ps qj + is pj qt ? ij ps qt ? it pj qs ? is pt qj can be used to put (3.2) in the form | {Dzij }

| {Fzij } Rij + ?n0;i?n0;j + ?i0;n?j0;n ? (?i0;n?n0;j + ?j0;n?n0;i) = ijq2: | }

Cij{+zCji Note that (4.2) relates the turbulence stresses to the new one-point second-rank turbulence structure tensors introduced in x 3. Since three of the four tensors in (4.2) can be linearly independent, this constitutive equation o ers an indication that componentality alone (found in Rij) is not su cient to completely specify the state of the turbulence. Taking the trace of (4.2) and noting that

Fkk = Dkk; one obtains

Ckk = Dkk ? q2: (4.4)

A general third-rank tensor can be decomposed into the sum of six subtensors, and the tensor Q can be represented by (see KR94)

Qijk = 16 q2 ijk + 13 ikmRmj + 13 jim(Dmk ? Cmk) + 13 kjm(Fmi ? Cim) + Qijk: Using the de nitions of the second-rank tensors and (3.9), one can show that Being fully symmetric, Q makes no contribution to R, D, F , and C. Therefore, none of the second-rank tensors contains the information in Q .

In the general case, the new structure tensors and the Reynolds stress tensor are related by the full forms of (4.2) and (4.5). A number of special cases exist, where additional relationships develop between these, in e ect reducing the number of independent components that must be known for a full description of the state of the turbulence. Next we consider some of these special cases in detail.

For homogeneous turbulence the de nition of dimensionality (3.3) has an equivalent representation in terms of the velocity spectrum tensor Eij(k) u^i u^j

Dij =

Z kikj

k2 Enn(k) d3k;
where k is the wavenumber vector and u^i are the velocity Fourier components. The
dimensionality Dij as de ned in (4.7) appears also in the work of

Fij =

Z

Fij(k) d3k;

Fij(k) k2?^ i?^ j = !^ i!^ j k2 ; where Fij(k) k2?^ i?^ j is the circulicity spectrum tensor, and ?^ i and !^ i are the stream function and vorticity Fourier components. From (4.7), (4.8) follows that for (4.1) (4.2) (4.3) (4.5) (4.6) (4.7) (4.8) The inhomogeneity tensor Cij vanishes in homogeneous turbulence (see (3.6)), and the fundamental constitutive equation (4.2) takes the form

Rkk = Dkk = Fkk = q2:

Rij + Dij + Fij = q2 ij: Two tensors in (4.10) can be linearly independent. By carrying only R as a model variable, RST models e ectively lump together information found in the dimensionality D and circulicity F . Therefore, RST models are not always sensitive to di erences in dynamic behaviour associated with the di erent dimensionality in homogeneous turbulence. Two one-point tensors are needed in order to capture both componentality and dimensionality information; for example one might use ~r and d~ as in the linear M(~r; d~) model of Appendix C.

The analysis in x 2.2, shows that stropholysis Q information must also be included in the presence of mean or frame rotation, and so in Appendix C we also give a linear M(~r; d~; Q ) model. For homogeneous turbulence the third-rank tensor Q can be developed from the fourth-rank tensor M (see (2.3)) according to homogeneous turbulence

C = Using (4.3) and (4.4) with (4.15) one can show that and then using (4.16) in the (22)-component of (4.2) gives Realizing that Dkk = Fkk in (4.16), we can also write (4.9) (4.10) (4.11) (4.12) (4.13) (4.14) (4.15) (4.16) (4.17) (4.18) Note that in this case both Qijk and Qijk become bi-trace free (see (3.10) and (3.11))

Qijk = ipqMjqpk: Qiik = Qiki = Qkii = 0;

Qiik = 0; and because Cij is zero in this case, (4.5) and (4.6) simplify into and

Qijk = 16 q2 ijk + 13 ikmRmj + 13 jimDmk + 13 kjmFmi + Qijk

In later sections we consider DNS results from fully developed channel flow and temporally developing plane wakes and mixing layers, which are all parallel mean flows having a single direction of inhomogeneity. Here we show that in these flows theory predicts that special relationships develop between the structure tensors and the turbulence stresses. For the sake of clarity (and without the loss of generality), the direction of inhomogeneity is taken to be along the x2-axis. Under these conditions, one can show that the inhomogeneity tensor C must be of the form In parallel flows the invariants of the anisotropy tensor ~cij also satisfy special relationships. For example, it follows from (4.15) that ~cij must be of the form ~cij = The tensor ~cij in (4.19) is non-symmetric and therefore can be expressed in principal coordinates only under the condition which guarantees three real eigenvalues for ~cij. Note that ~c11, ~c13, and ~c31 can be positive or negative in di erent parts of the flow eld and hence (4.20) is not a trivial condition. However, we have found (4.20) to be satis ed both in fully developed channel flow and in temporally evolving plane wakes and mixing layers. Therefore, in these flows c~ can be put in principal coordinates, and in these coordinates, one can show easily that the invariants of ~c satisfy the relationship

IIc
= 6IIIc ? 316 3;
= ckk:
All of the key results in this section (including (4.16), (4.17) and (4.21)) have been
veri ed using direct numerical simulations of fully developed channel flow

In xx 8 and 9 we consider numerical simulations of self-similar plane wakes and mixing layers. An interesting feature of these flows is the approximate two-componentality (2C) (r33 r11; r22) and two-dimensionality (2D) (d33 d11; d22) that develops when external two-dimensional forcing is applied initially. In these `forced' flows the relationships between the various structure tensors assume special forms. Here we consider an idealized 2D{2C turbulence eld where the turbulent velocity components are (4.19) (4.20) (4.21) (4.22) (4.23) (4.24)

Cij = ?i0;k?k0;j = ?i0;1?10;j + ?i0;2?20;j + ?i0;3?30;j: The rst two terms on the right-hand side of (4.24) vanish because ?10 = ?20 = 0 everywhere in the eld, and hence the gradients ?10;j and ?20;j must vanish also for all j. The last term vanishes because x3 is an axis of independence and hence ?i0;3 = 0. The same facts determine the forms of the second moments, which for 2D{2C turbulence are Fij = In this case, the vector stream function ?i0 has a single non-zero component aligned with the axis of independence x3, that is ?30 = ?30(x1; x2); ?10 = ?20 = 0; and as result all the components of the inhomogeneity tensor Cij vanish even in the presence of inhomogeneity. This feature of 2D{2C turbulence can be easily veri ed by writing the de nition of Cij in an expanded form

Case AXK AXM EXO EXQ PXA PXF 5. Homogeneous turbulence subjected to irrotational strain

Irrotational mean deformation preserves reflectional symmetry, and as a result, the stropholysis Q is zero in all the flows considered in this section. RDT analysis based on the evolution equations for R and D (see KR94 and Appendix B) shows that if present initially (as for example in initially isotropic turbulence) the equality of the Reynolds stress and dimensionality anisotropies is preserved by rapid irrotational mean deformation with

~rij = d~ij = ? 21 f~ij: Here we explore the validity of (5.1) when the irrotational mean deformation is slow.

In the discussion that follows, we use the DNS data of

Sij = dUi=dxj; S = q

SijSij=2: Z t

0 C = exp

S dt0 = eSt: Following the notation of LR, we de ne the reference total strain by and use this as the dimensionless time variable in the discussion of evolution histories that follows. (4.26) (4.27) (5.1) (5.2) 0.2 py 0 o r t o s in?0.2 A ?0.4 ? rij 1 (a) 0.2

0 ?0.2 ?0.4 1 ? dij (b) 0.8 0.4 0 f?ij (c) C 2 3

C 2 3 ?0.4 1

C 2 3 with the plus sign corresponding to the case of axisymmetric contraction and the minus sign to the case of axisymmetric expansion.

The time histories of the anisotropies ~r, d~, and f~ for the two cases of axisymmetric contraction are shown in gure 2. The e ect of the strain on the anisotropy invariants of each tensor is shown on the anisotropy invariant maps (AIM) of gure 3, where important limiting states have been identi ed. The e ect of the axisymmetric strain is to stretch and orient the eddies so that they tend to become aligned with the axial direction of positive strain (S > 0). This is accompanied by an increase in the axial large-scale circulation (f~11) and a decrease in the axial stress ~r11 and dimensionality d~11 as the axial vorticity is being stretched. At the end of the simulations, nearly all of the large-scale circulation is concentrated around the axial direction (f11 ! 1, f~11 ! 3 2 ), and the turbulence is almost independent of the axial direction (d11 ! 0, d~11 ! ? 13 ). Both of these e ects can be explained in terms of the strong alignment of elongated vortical eddies with the axial direction. The eddy stretching and alignment are most e ective in the rapid case (S q02="0 = 96:5) where the turbulence at the end reaches an approximate 2D{2C state (see gure 3b). This case belongs to the RDT regime, and (5.1) holds throughout the deformation. At slower strain rates (see gure 2), the results are qualitatively similar, but note that the RDT relation (5.1) is only approximately satis ed. Yet, despite the wide di erence in the initial S q02="0 between the slow and rapid cases, the evolution history of the anisotropy f~ is only weakly modi ed when plotted against total strain. We have found that the insensitivity of circulicity to strain 0.4 0.3

line 1C or 1D point (b) rate is a common feature of all the cases of irrotational mean deformation considered here. In fact, the insensitivity of circulicity has been observed in flows undergoing quite di erent modes of mean deformation. In gure 2 the dimensionality and Reynolds stress evolution histories are also only weakly dependent on the rate of straining, but there are cases (considered below) where these two anisotropy histories are strongly dependent on the rate of strain.

The corresponding results for two cases of axisymmetric expansion are shown
in gures 4 and 5. This flow exhibits counter-intuitive behaviour that makes it a
challenge to turbulence modelling. The e ect of the negative axial strain (S < 0) in
these cases is to concentrate the eddies in thin, disk-like regions normal to the axis of
symmetry, while the weaker positive strain causes a mild radial stretching. Sustained
axisymmetric expansion produces pancake turbulence, in which the symmetry axis
becomes the direction with the most energetic velocity fluctuations, but with very
~ 1 ). As expected, in the rapid case
little large-scale circulation around it (f11 ! ? 3
(S q02="0 = 70:7) the exact RDT result (5.1) holds true, and ~r11 d~11 ! 61 towards
the end of the simulation. However in the weak strain case, for which S q02="0 = 0:7,
there is a large disparity between the anisotropy of the dimensionality tensor, which
~
is very weak (d11 ! 01:)0.1C),leaanrdly,thteheanRisDoTtrorpeysuoltf (t5h.e1)Riesynnootldvsasltidresins tselnoswolry, wsthraicihneids
quite strong (~r11 ! 3
expansion flows, which perhaps is not surprising, but note that the level of the stress
anisotropy ~r, shown by the square symbols on the anisotropy invariant map (AIM)
in gure 5, is higher in the slower case ( gure 5a) than it is in the rapid case ( gure
5b). In fact the stress anisotropy reached in the slowly strained case exceeds the
theoretical limit (see gure 5b) placed by RDT for rapid axisymmetric expansion.
This result, which was noted by LR, is surprising since one would expect that the
more rapid mean deformation would push the stress anisotropy to higher values. We
0.4
py0.2
o
r
t
o
s
i
n 0
A
0.3
0.1
0.4
0.2
0
?
dij
(b)
3
? 0.2
1
3
line
see the same e ect in the experimental results of

rij 0.2 y p o tro 0 s i n ?A0.2 ?0.4 1 (a) 0.4 ?

dij 0.2

0 ?0.2 ?0.4 (b) the slowly strained case must be compensated by an augmented stress anisotropy. Equation (4.10) can be used to place a theoretical limit to the maximum level of stress anisotropy that can be reached in the slowly strained case, corresponding to a complete vanishing of the dimensionality anisotropy (d~ij = 0). As shown in gure 5(a) the slow theoretical limit is well beyond the RDT limit.

The mean strain rate tensor for the plane strain cases has the general form

0 0

0 ?1 0 0 1 0 A : +1 (5.4) The evolution histories of the anisotropies ~r, d~, and f~ are shown in gure 6 for the two di erent initial values of S q2=" (see table 1). Sustained plane strain produces turbulence consisting of eddies elongated in the direction of positive strain (d~33 ! ? 31 ), which have very strong circulation around their axes (f~33 ! 3 2 ), but very little motion along their axes (~r33 ! ? 13 ). Note that this corresponds to the same 2D{2C limiting state of vortical eddies aligned with direction of positive strain that is produced by axisymmetric contraction. However, plane strain di ers from the axisymmetric contraction flow in that the limiting state is reached along a di erent path in the AIM (see gure 7). As in the case of the axisymmetric expansion flows, the RDT result (5.1) is valid only in the rapid case. In all the slower cases, d~ di ers signi cantly from ~r. As in the axisymmetric strain cases, the circulicity history is insensitive to strain rate in the plane strain simulations despite the wide variation in the initial value of S q2=".

Using the simulations of LR, we have shown that the RDT equality (5.1) is not valid when the mean deformation is slow. A puzzling aspect of this result is found in the axisymmetric expansion and plane strain flows, where slow strain produces a higher 1C or 1D point (b) level of Reynolds stress anisotropy than does rapid strain, while the opposite holds true for the dimensionality anisotropy. These e ects are shown clearly in gure 8 where ?IId is plotted against ?IIr and ?IIf for each of the flows (see (3.7)). These observations raise three important questions: (i) How is it possible that ?IIr is bigger for slow rather than for rapid straining? (ii) What triggers the breaking of the equality ~r = d~ in the slow cases? (iii) What determines which anisotropy components grow at the expense of the others? ? 0.1 1 (a) C 2 To answer the rst question we take a closer look at the equation for rij , drij = pij + tirj + tisj + ij : dt | ra{pzid } | s{lozw } Here t = S t is the non-dimensional time based on the magnitude of the largest diagonal strain component, with S = 2S =p3 for axisymmetric strain, and S = S for plane strain (see (5.3)). The rapid and slow terms on the right-hand side of (5.5) are and tirj = tisj =

2 q2S

1 q2S

2 1

S Snmrnmrij ? S (Sikrkj + Sjkrki) Tisj ; ij = irj + ikj ;

2
irj = ? q2S (ui;kuj;k);
ikj = ? rmm rij :
Here tirj and tisj are the familiar rapid and slow pressure{strain-rate terms (see (A 1)),
pij is a production-rate tensor, and ij a dissipation-rate tensor. For the sake of clarity,
we limit our discussion to the case of axisymmetric expansion and consider only the
axial component ( )11 of (5.5). The balance of the rapid and slow contributions in (5.5)
is shown in gure 9(a) for both the slowly strained (EXO) and rapidly strained (EXQ)
runs. Note that the contribution of the rapid terms is relatively insensitive to the rate
of straining, but that of the slow terms is quite sizable in the slowly strained run and
practically negligible in the rapidly strained case. Figure 9(b) shows the individual
contributions of the various slow terms for the weak-strain run (EXO). Note that the
slow pressure{strain-rate term is negligible, and hence the slow contribution is solely
due to the dissipation term ij . The contribution due to r11 is negative as expected,
but the positive contribution due to k11, which arises from the trace-normalization of
the Reynolds stress tensor, is slightly larger, so that the net e ect of the dissipation
term is to increase r11. This additional contribution due to the dissipation term in
case EXO, acting to complement the rapid terms, explains how IIr can be larger in
this case than in the rapid case. However, this e ect does not explain why ~r11 grows
at the expense of d~11. The problem is that the term k11, which is the term that helps
push ~r11 to higher values, has a counterpart in the d~11 equation (Appendix B), which
is exactly equal to k11 as long as rij = dij . To answer the remaining questions we need
to look at the evolution equation for the anisotropy di erence ~ij = ~rij ? d~ij. The
transport equations for r, d , and f (see Appendix B) can be used to write down the
evolution equation for ~ij for the case of homogeneous turbulence. The nonlinear
terms that appear in the evolution equation for ~ij are found to be negligible in the
simulations of LR (see Appendix B), an observation also made by

rapid slow 1 [?z 57 (Sik ~kj + Sjk ~ki) + 1201 Snm ~nm ij + 2Snm~rnm ~i{j ? zrmm}S| ~i{j]

}| S } + ( irj ? ij)

d | tri{gzger } (5.8) axisymmetric expansion, the ~11 component of (5.8) simpli es to where idj = ?2 (u0k;iu0k;j)=(q2S ) and dmm = rmm = ?2 (u0k;iu0k;i)=(q2S ). For the case of d~11 = [?(~r11 ? 2215 ) ? rmmS ]~11 + ( r11 ? 1d1) : dt | expo{nzential } | tri{gzger } (5.9) The terms within square brackets contribute to exponential growth. Note that the last of these terms arises from the decay of the turbulent kinetic energy and complements the e ect of the rapid terms. The exponential terms, however, cannot create ~ij if it is initially zero. Only the last term, representing the di erential dissipation of ~r and d~, can initially generate ~ij. The trigger term, representing the di erence between r11 and d11, determines the sign of ~11. That is to say, whether ~r11 will grow at the expense of d~11 or vice versa is determined by the sign of this last term. A comparison of the exponential terms and the trigger term is shown in gure 9(c) for case EXO. Note that r11 ? d11 > 0, which correctly implies that ~11 = ~r11 ? d~11 > 0. Note however, that the exponential term contributes more to the subsequent evolution of ~11 and in the axisymmetric expansion case, an exponential growth of ~11 is inevitable since ?(~r11 ? 2215 ) ? rmmS > 0 at all times.

These basic principles can help explain the di erence between the various ~rij components and d~ij in the case of slow plane strain and slow axisymmetric contraction. For example, in the slow axisymmetric contraction flow the exponential term is given by (~r11 ? 2215 )~11, and because (~r11 ? 2215 ) ? rmmS < 0 at all times the exponential term suppresses any ~ij produced by the di erent rates of dissipation for ~rij and d~ij.

Interesting behaviour is also encountered upon removal of the mean strain following
axisymmetric expansion and plane strain. In their simulations LR found a return to
isotropy of the Reynolds stresses upon removal of the mean straining following an
axisymmetric contraction. However, for turbulence that had been previously distorted
by an axisymmetric expansion (pancake turbulence) they found a slight increase in
the stress anisotropy following removal of the mean straining. Similarly, for the case
of plane strain, they found that some of the components of the stress anisotropy
increased while others decreased after the removal of the mean straining. These
results are consistent with the analysis of the previous section: stress anisotropy
components that under slow straining grow beyond their corresponding RDT limit
are also the ones which diverge from isotropy upon removal of the straining. This
suggests that essentially the same structure-related mechanism that was described
in the previous section is active both during the slow mean straining mode and
after its removal. In the later case, the larger scales can be expected to provide an
e ective strain rate on smaller scales. As explained by

In the discussion that follows, we use a time-evolving hydrodynamic eld (case
C128U) from the direct numerical simulations of

The evolution histories for the normalized second-rank tensors r, d , and f and the third-rank normalized stropholysis tensor q = Q =Rnn are shown in gure 10. The r11 component of the normalized stress tensor grows at the expense of the other two normal stresses. The dominant circulicity component is f22, with f11 and f33 maintaining comparable levels throughout the evolution history. In the case of the dimensionality tensor, both d22 and d33 dominate over d11.

The flow appears to be reaching equilibrium towards the end of the simulation,
with approximately constant S q2= and P= and the components of r approaching
values that are very close to the experimentally observed asymptotic equilibrium
values

0 ?0.2 (d ) 123 113

333
15
20
hairpin-like structures with statistically axisymmetric legs observed by

A comparison of the structures in this flow to those found in the RDT limit of inviscid in nite total rapid shear is instructive. In the RDT case, the nal state consists of jetal eddies (no large-scale circulation around the eddy axes) completely aligned with the streamwise direction so that r11 ! 1, d11 ! 0 and f11 ! 0. So there is a ? II 0.10 0.10

(b) ?IId 0.05

IIr

IIf 0.01 0.02 0 marked di erence in the turbulence structure in these two situations. As will be shown in x 7, the structure tensors indicate that the turbulence structure near the centreline of fully developed channel flow is similar to that found in equilibrium homogeneous shear flow.

As shown in gure 11, ~r and f~ reach anisotropy levels that are considerably higher than those reached by d~. Figure 11(b) shows a comparison of the second invariant ?IId with ?IIr and ?IIf. Clearly, the dimensionality remains more isotropic than both the Reynolds stresses and the circulicity. These results are in general agreement with what was observed in the case of irrotational strain (x 5) where, with the exception of the RDT cases, ?IId was found to be lower than both ?IIr and ?IIf.

Stropholysis plays an important role in flows with strong mean rotation because it modi es the rapid pressure{strain-rate term. As shown in the plot for the normalized stropholysis tensor qijk = Qijk=Rnn (see gure 10d) only q113, q123, and q333 (out of the nine independent components of the fully-symmetric q tensor) are signi cantly energized. These components contribute to the rapid pressure{strain-rate term (see (A 4)). Note that q113 is roughly equal to the negative of q333, indicating a transfer of energy from r33 to r11. One can decompose the rapid pressure{strain-rate term into three parts (see Appendix A) given by

Tirjapid=q2 = (TiSj + Ti!j + TiQj)=q2: (6.1) TiSj is the contribution of the mean strain, Tiwj involves the explicit contribution of mean rotation, and TiQj involves the contribution of mean rotation through stropholysis e ects. Each of the three contributions TiSj , Ti!j, and TiQj is trace-free and represents a separate intercomponent energy transfer mechanism. In gure 12 we show a comparison of the three contributions. The main role of TiSj is to drain roughly equal amounts of energy out of the R11 and R22 components and transfer it to R33. The o -diagonal component T1S2 tends to decrease the magnitude of the shear stress R12. The main role of Ti!j is simply to increase the magnitude of the shear stress R12. The stropholysis part TiQj, on the other hand, returns a third of the energy that TiSj (d ) 0.3 0.2 2q /S 0 3 3 T ?0.1 ?0.2 ?0.3 0 0 transfers from R11 and R22 into R33 back to R11. In this sense, T1S1 and T1Q1 play competing roles. The o -diagonal component T1Q2 acts to decrease the magnitude of R12, thereby reinforcing the e ect of T1S2. This example underscores the challenge faced by turbulence modellers who have to devise ways to capture some stropholysis e ects, even though these are not completely parametrized by second moments like r and d .

We begin our study of the structure tensors in inhomogeneous flows with the direct numerical simulations of fully developed channel flow by Kim, Moin & Moser (1987), and Kim (1992). These simulations have Reynolds numbers based on the wall shear 100 200 300 400 100 200 300

400 (d ) 2c22, 2c33 200 y +

0.8 r i j 0.4 0

0 200 y + velocity u of Re = 180 and Re = 385. In both cases, we take x1 and x3 to be the homogeneous streamwise and spanwise directions, and x2 to be the wall-normal direction.

The pro les of the normalized Reynolds stress, dimensionality, circulicity, and symmetrized inhomogeneity tensors are shown in gure 13 for Re = 385. The corresponding results from the Re = 180 case were qualitatively similar and are not shown here, but some minor di erences that were observed are discussed. Because the flow is statistically symmetric about the channel centreline, the statistical sample in these pro les was e ectively doubled by averaging the two channel halves together. The dominant component of the Reynolds stress tensor r11, corresponding to the fraction of turbulent kinetic energy found in streamwise fluctuations, reaches a maximum at y+ 8. The streamwise component d11 is smaller than r11 throughout the channel, Channel centreline ij 11 22 33 12 rij 0:84 0:00 0:16 ?0:02 dij 0:03 0:38 0:59 0:00

fij 0:09 0:62 0:29 0:03 ij 11 22 33 12 rij 0:44 0:29 0:27 0:00 dij 0:23 0:34 0:43 0:00 fij 0:22 0:42 0:36 0:00 indicating the presence of large-scale structures that are preferentially elongated in the streamwise direction.

The components of the three normalized tensors in the near-wall region (at y+ =
3:5) and at the channel centreline are shown in table 3 for the Re = 385 case (with
almost identical values found in the Re = 180 case). Based on these values one can
argue that the near-wall structures are nearly two-dimensional (d11 0) and roughly
aligned with the wall (r12 0, d12 0), have a strong jetal character (r11 r22; r33),
very little circulation around their axes (f11 1), and roughly a circular cross-section
since d22 d33. These results are indicative of the streaky structures in the near-wall
region of the channel, which have been observed directly both experimentally

The pro les of the symmetrized inhomogeneity tensor, normalized by Dkk to give
a measure of the relative importance of inhomogeneity, are shown in gure 13(d). As
expected, cij +cji is large in the near-wall region (y+ . 30), indicating a strong degree of
inhomogeneity there. A similar inhomogeneity pro le was obtained in the Re = 180
case, but in that case inhomogeneity values were somewhat more pronounced in the
region 10 . y+ . 30. A small residual inhomogeneity also seems to exist at the
channel centreline in both cases, which can be attributed to the gradual vanishing of
S q2= (with increasing y+) as the channel centreline is approached. Note, however,
that the flow in the log-region is locally homogeneous, with the higher Re case reaching
a local homogeneity closer to the wall. Local homogeneity in the log-region has been
noted in the past, for example by

The AIMs for d~ij and ~rij are shown in gure 14. The large di erence between the structure of the wall-region turbulence and that of the turbulence in the central core is reflected in the AIM for d~ij. The jetal character of the wall-region structure is also apparent from the AIM for ~rij. Consistent with the trend observed in homogeneous

IIId ? IId (c) (b) y+ = 0 0.10

(a) ?IId 0.05 y+ = 0 0 ?0.010

2D structure Viscous region y+ = 3.5 y+ = 3.5 iscous V Centreline 0.02 IIIr flows, the anisotropy of the dimensionality tensor is smaller than that of the stress tensor (see gure 14c) everywhere in the flow, except at the channel centreline where the two anisotropies are equal but quite small.

We consider two simulations of time-developing turbulent wakes presented by

Following

0 0.8 0.6 0.4 0.2

0 ?0.2 0 0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2

1.5 d i j (d ) 1.2 1.5 0 0.3 0.6 0.9 1.2

1.5 y / b ?0.2 0 0.3 0.6

0.9 y / b half-width b(t). The half-width is taken to be the distance between the y-locations at which the mean velocity is half of the maximum velocity de cit magnitude U0. The non-dimensionalization of the time variable is based on the mass-flux de cit m_ and the initial magnitude of the velocity de cit Ud, and is given by = tUd2= m_.

The normalized Reynolds stress, dimensionality, circulicity and symmetrized inhomogeneity tensors for the unforced case are shown in gure 15 for a time during the self-similar period. Since the wake is statistically symmetric, the statistical sample in these pro les was e ectively doubled by averaging the two sides of the wake together. The distribution of the various tensor components is similar to what was obtained in the case of homogeneous shear flow (see gure 10). The pro le for the normalized dimensionality tensor dij remains relatively constant across the wake. The streamwise component d11 is slightly smaller than the cross-stream and spanwise components (d22 and d33) indicating the existence of structures that are somewhat elongated in the streamwise direction. As in the case of homogeneous shear, f12 attains a relatively r i j 0.8 0.6 0.4 0.2

0 0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2

1.5 d i j (d ) 1.2 1.5 0 0.3 0.6 0.9 1.2

1.5 y / b ?0.2 0 0.3 0.6

0.9 y / b constant positive value across the wake. The non-zero value of f12 suggests that the structures are inclined to the streamwise direction.

In gure 15(d) the components of the symmetrized inhomogeneity tensor are shown normalized with Dkk to obtain a measure of the relative signi cance of inhomogeneity in this flow. The pro les of cij + cji suggest that the e ects of inhomogeneity are moderate across most of the wake except in the potential far eld where c22 and c33 grow signi cantly, but where all terms of the non-normalized Cij (and the other tensors) are small.

The pro les of the components of the normalized tensors in the case of the forced
wake are shown in gure 16, also during the (approximate) self-similar period. The
striking di erence between these pro les and those for the unforced wake (see gure
15) is a manifestation of the di erences in the structure of the turbulence in these
two flows.

rIId= IIro

IIf

IIr 0.02 0.04 ? IIr or ? IIf

IIf (b)

IIf IId=

IIr as `a slab of turbulence with undulating boundaries'. In contrast, in the forced wake one can observe coherent `concentrations of large-scale spanwise circulation'. These di erences are nicely reflected in the pro les of d33 and f33 for the two cases. Note that the two-dimensional forcing has suppressed d33 and augmented f33, as one would expect in a flow with the principal energy-containing structure consisting of vortical structures aligned with the spanwise (x3) axis.

The most striking e ect of the forcing is the emergence of relationships among the components of the four tensors that are valid across the entire wake, and which can be summarized as follows: d11 t r22; d22 t r11; d33 t r33

d11; d22; d12 t ?r12; f33 f11; f22; f12 t 0 and cij + cji t 0 for i; j = 1; 2; 3: (8.1) Note that these results are just an approximate form of the exact results discussed in x 4.3 for 2D{2C turbulent flow. Hence, the forcing of the two-dimensional modes has produced a turbulent flow where the dominant energy-containing motion is very close to being 2D{2C; as a result, the e ects of inhomogeneity are much less dominant in this case than in the unforced wake. Note that ckk 0, and hence Fkk q2 (see (4.3) and (4.4)), which is consistent with both the fluctuating velocity and the largescale circulation elds being generated primarily by the same large spanwise vortical structures.

It should be emphasized that the suppression of cij + cji is due to the strong 2D{2C character of this flow and does not necessarily imply vanishing of all spatial gradients of statistical moments. The suppression of the inhomogeneity tensor means that all the relationships that exist between the remaining structure tensors in homogeneous turbulence (for example (4.9) and (4.10)) are also valid locally in this case. This suggests that it should be relatively easy to extend good models for homogeneous turbulence to handle cases like this. In essence, forcing of two-dimensional modes results in a reduction in the number of independent tensor components that are required for the description of the turbulence in this flow.

Figures 17(a) and 17(b) show a comparison of ?IId to ?IIr and ?IIf for the unforced and forced wakes. In the unforced wake, the anisotropies of r and f are very close to each other and considerably higher than that of d , except near the centreline of the wake, where all anisotropies are small. Note, however, that two-dimensional forcing makes the anisotropies of both r and f roughly uniform across the entire wake, and IIr approximately equals IId, as expected for nearly 2D{2C turbulence.

We also studied the structure tensors in self-similar turbulent mixing layers. For
this purpose we used the direct numerical simulations by

We have introduced a systematic framework for studying the e ects of turbulence structure on the evolution of one-point statistics using four one-point statistical measures of structure: the dimensionality D, circulicity F , inhomogeneity C and stropholysis Q . These one-point structure tensors provide useful information about di erent aspects of the energy-containing turbulence structure and are useful as diagnostic tools and in turbulence modelling.

These ideas were explored using a number of numerical simulations of both homogeneous and inhomogeneous turbulence undergoing diverse modes of mean deformation. Several general trends were observed (with some variation) across all the simulations examined.

Rapid irrotational mean deformation of initially isotropic homogeneous turbulence produces identical componentality and dimensionality anisotropies (~r = d~). The largescale circulation is concentrated around the axes of positive mean strain rate. This is consistent with the idea that the primary mode of deformation in rapidly strained flows is one of vortex stretching along the principal directions of positive mean strain rate.

As the rate of the irrotational mean deformation decreases, the dimensionality
tends to remain closer to isotropy. However, circulicity anisotropy is relatively
insensitive to the rate of mean deformation. Since the sum of the componentality,
dimensionality, and circulicity anisotropies must vanish in homogeneous turbulence,
one can encounter flows in which slower deformation produces larger Reynolds
stress anisotropy than. This is con rmed by the DNS of LR and the experiments
of

The tendency of the anisotropy of the dimensionality to remain small under weak strain and to become signi cant under strong strain has important implications for turbulence modelling. It explains why turbulence models based solely on r (ignoring d ) work well for weakly distorted turbulence, but fail for strongly distorted turbulence. A model based on the standard return-to-isotropy assumption is insensitive to the role of structure in the return-to-isotropy problem and simply cannot be calibrated consistently for all irrotational flows.

Stropholysis plays an important role in flows with strong mean rotation by providing a separate intercomponent transfer mechanism within the rapid pressure{strain-rate term. We believe that good models for turbulence subjected to mean or frame rotation must include stropholysis information in some (perhaps simpli ed) form.

The inhomogeneity tensor c appears to be an e ective diagnostic tool for local
homogeneity in inhomogeneous turbulence. The local homogeneity in the log-region
of fully developed channel flow

The inhomogeneity tensor c vanishes in 2D{2C turbulence. The implication of this is that these flows can be treated as locally homogeneous, in the sense that all the constitutive relations among various one-point statistics that exist in homogeneous turbulence are also valid in 2D{2C turbulence.

We hope that these ideas will stimulate some innovative schemes for incorporating structure information in one-point turbulence closures. We will report our progress in this direction in due course.

The authors wish to thank Peter Bradshaw for scrutinizing the manuscript as well as the referees for suggesting a number of valuable improvements. This work has been supported by the Air Force O ce of Scienti c Research (Drs James McMichael, Mark Glauser and Thomas Beutner) and by the Center for Turbulence Research. Appendix A. An exact decomposition of the rapid pressure{strain-rate term

The inviscid RDT evolution equations for the Reynolds stresses in homogeneous turbulence are given by (2.1). Closure of these equations requires modelling of the fourth-rank tensor M appearing in the rapid pressure{strain-rate term Tirjapid (see (2.2) and (2.3)), which results from the familiar splitting of the pressure fluctuations 1 p;0mm = ?2Gmnu0n;m | ra{pzid } ?u0m;nu0n;m + u0m;nu0n;m: | s{lozw } Group theory allows the decomposition of a general fourth-rank tensor into ve subtensors satisfying speci ed symmetries and antisymmetries. This decomposition is analogous to the splitting of a second-rank tensor into its symmetric and antisymmetric parts. KR94 carried out the exact decomposition of M to isolate contributions from individual structure tensors. Based on this decomposition of M, one can write the rapid pressure{strain-rate term (2.2) as

Tirjapid = TiSj + Ti?j (A 1) (A 2) with and

TiSj = 4SknMijnk + 23 q2[Sij ? Skk( ij ? fij) + ijSknfnk ? (Sikfkj + Sjkfki)] Ti?j = Tiwj + TiQj;

Tiwj = 23 q2[?im(rmj ? dmj) + ?jm(rmi ? dmi)]; TiQj = ?2?zQkij: Here Sij = (Gij + Gji)=2 and ?ij = (Gij ? Gji)=2 are the mean strain and rotation rate tensors, and Mijnk is a fully symmetric tensor that can be constructed from M according to

Mijnk = 16 (Mijnk + Minkj + Mikjn + Mjkni + Mjnki + Mnkij):
Equations (A 3) and (A 4) highlight the role played by the dimensionality d and
the stropholysis Q in the rapid pressure{strain-rate term. In x 6, the simulations of

The transport equations for the dimensionality D, circulicity F , and stropholysis Q can be obtained starting from the fluctuating momentum equation. Because the basic de nitions for the structure tensors involve correlations between gradients of the stream function, the necessary manipulations that lead to the desired transport equations are most conveniently done in a frame of reference deforming with the mean motion, in which all the dependent turbulence statistics are homogeneous. Details of the derivation procedure can be found in KR94 and will not be repeated here. The resulting evolution equations for the dimensionality and circulicity tensors in homogeneous turbulence are

DDij = ?DikGkj ? DjkGki + 2SmnLimnj ? 2SmnMmnij Dt ?Tisjlow ? (?i0;m?n0;ju0m;n + ?j0;m?n0;iu0m;n) ? 2 u0m;iu0m;j; (B 1) (A 3) (A 4) (A 5) (B 2) (B 3) (B 4) DFij = FikGkj + FjkGki + 2Dik?kj + 2Djk?ki ? 2SkkFij Dt +2SnmDnm ij ? 2SnmLinmj ? 2SnmMijnm ? Ti?j +(?i0;m?n0;ju0m;n + ?j0;m?n0;iu0m;n) ? 2 !i0!j0 : Here Ti?j is the rotational part of the rapid pressure{strain-rate term (see (A 4)) and Tisjlow is the slow pressure{strain-rate term,

Tisjlow = 2 pss0ij; ps = ?2u0m;nu0n;m; s0ij = 12 (u0i;j + u0j;i): The fourth-rank tensor L is de ned by

Lijpq =

Z kikjkpkq k4

Enn(k) d3k: The triple correlations in (B 1) and (B 2) are trace-free and represent nonlinear intercomponent energy transfer between the Dij and Fij.

The evolution equations of the normalized tensors dij and fij, and for the corresponding anisotropies d~ij and f~ij follow easily from the de nitions (3.3) and (3.4) and (B 1) and (B 2). Using the evolution equation for the dij and that for the normalized Reynolds stress tensor rij, one can write down the equation for the tensorial di erence ~ij = ~rij ? d~ij. For the case of irrotational mean deformation, this equation is D ~ Dt ij = ?(Sik ~kj + Sjk ~ki) + 2Snmrnm ~ij +2[Snm(Mimnj + Mjmni + Mmnij ? Lmnij) +Tisj ? 21 (?i0;m?n0;ju0m;n + ?j0;m?n0;iu0m;n) ? 2 (u0i;ku0j;k ? u0k;iu0k;j)]=q2: (B 5) When the mean strain rate is rapid (inviscid RDT), (B 5) reduces to D ~ Dt ij = ?(Sik ~kj + Sjk ~ki) + 2Snm[rnm ~ij + (Mimnj + Mjmni + Mmnij ? Lmnij)=q2]: (B 6) In addition, one can show that when the velocity spectrum tensor satis es reflectional symmetry, for example during rapid irrotational deformation of initially isotropic turbulence,

Mimnj + Mjmni + Mmnij ? Lmnij = 0; (B 7) and ~ij remains zero when initially so. When the mean strain rate is slow both the nonlinear and viscous terms are in general signi cant, but special cases exist where the nonlinear e ects are relatively weak. For example, in the slowly strained irrotational cases from the simulations of LR the slow pressure{strain term is found to be insigni cant relative to the viscous and rapid terms. While information was not reported by LR for the triple correlations in (B 5) there is strong evidence of the insigni cance of these terms as well. For example, the insensitivity of the circulicity evolution histories to strain rate, which is particularly strong in the axisymmetric expansion and plane strain cases, suggests that the intercomponent energy transfer between d~ and f~ is negligible and of the same level as the slow pressure{strain-rate term. Otherwise, one would expect a stronger dependence of the f~ histories on the rate of straining in these flows, because the evolution of d~ is strongly dependent on the rate of strain. The rapid terms in (B 5) involve the fourth-rank tensors M and L, but for a simpli ed analysis these may be modelled with a linear representation in d~ and ~r (see Appendix C). KR94 have shown that when these two linear models are used together, the resulting representation of the rapid terms is exact for weakly anisotropic turbulence. In fact for the irrotational flows considered here, this linearized representation of the rapid e ects is quite accurate for C . 3. With these simplifying assumptions, we obtain d~ij = ? 75 (Sik ~kj + Sjk ~ki) + 1201 Snm ~nm ij + 2Snm~rnm ~ij + 2 kkS ~ij dt +2 (u0i;ku0j;k ? u0k;iu0k;j)]=q2: (B 8) The evolution equation for the third-rank tensor Qijk in inviscid RDT is D Dt Qijk = ?GmkQijm ? GjmQimk ? itsGsmMjmtk ? itsGmtMjsmk +?zHizjk + 2Szm(Qijkzm + Qizmjk); (B 9) Hijpp = Fij;

Hiipq = Dpq; (B 10) (B 11) and where ?i = inm?mn is the mean vorticity vector,

Hijpq = ?i0;p?j0;q =

Z kpkq

k2 Fij(k)d3k;

Qijrpq = its

Z ktkrkpkq k4

Esj(k) d3k;

Qijrpp = Qijr: Details on the de nitions (B 10) and (B 11) and the properties of these higher-rank tensors are given in KR94. The stropholysis Q can be obtained from Q using (3.8). Alternatively, the RDT transport equation for Q can be obtained by a complete symmetrization of (B 9). For the special case of homogeneous turbulence undergoing rapid mean rotation (with no strain), the stropholysis equation is

D Dt Qijk = 118 [?i(Rkj ? Fkj) + ?j(Rki ? Fki) + ?k(Rij ? Fij)] + 118 [ kj?r(Fir ? Rir) + ki?r(Fjr ? Rjr) + ij?r(Fkr ? Rkr)] +?z(Mijkz ? Hijkz) + 16 ?z(2 pkz Qpij + pjz Qpki ? pki Qpjz): (B 12) Here, M is the fourth-rank, fully symmetric, sub-tensor of M de ned in (2.3). A relation analogous to (2.3) relates H and its fully symmetric sub-tensor H . Even in the simple case of rapid mean rotation, the stropholysis equation involves a closure problem, because of the presence of the M and H terms (see KR94 for a detailed discussion).

Appendix C. Linear structure-based models for weakly anisotropic turbulence

Two simple models that make use of the new tensors are given here because
they provide additional insight, but one must keep in mind that more sophisticated
structure-based models have been constructed

A model for the fourth-rank tensor M (see (2.3)) occurring in the rapid pressure{ strain-rate term in terms of ~r alone is fundamentally wrong in non-equilibrium turbulence. For weakly anisotropic turbulence, one can easily construct a model for M that is linear in the anisotropy tensors ~r and d~. We rst write the most general fourth-rank linear tensor function of two second-rank tensors,

Mijpq=q2 = C1 ij pq + C2( ip jq + iq jp) +C3 ij~rpq + C4 pq~rij + C5( ip~rjq + iq~rjp + jp~riq + jq~rip) +C6 ijdpq + C7 pqd~ij + C8( ipdjq + iqdjp + jpd~iq + jqd~ip): ~ ~ ~ (C 1) For homogeneous turbulence, continuity and the de nitions x all of the coe cients in the linear model, and one nds

Mijpq=q2 = 125 ij pq ? 310 ( ip jq + iq jp) + 241 ij~rpq + 1211 pq~rij ? 7 ( ip~rjq + iq~rjp + jp~riq + jq~rip)

1 + 1211 ijdpq + 241 pqdij ? 7 ( ipdjq + iqdjp + jpd~iq + jqd~ip): ~ ~ 1 ~ ~ (C 2) Similar analysis leads to a model for the fourth-rank tensor L (see (B 4)), that is linear in the anisotropy d~, and with all its numerical coe cients determined by analysis: Lijpq=q2 = 115 ( ij pq + ip jq + iq jp) + 17 ( ij dpq + ipdjq + jpdiq + iqdjp + jqdip + pqd~ij ): (C 3)

~ ~ ~ ~ ~
It is interesting to note that if d~ = 0 then the model above reduces to the linear
M(~r) model with a coe cient C5 = ?1=7 = ?0:143 in (C 1), almost exactly the value
found by

A detailed discussion on the performance of the linear model (C 2) and (C 3) is
given in KR94 and

Mijpq=q2 =

1 ? 4 ( ipkQkqj + jpkQkqi + iqkQkpj + jqkQkpi)=q2: (C 4) Again all coe cients in the linear model (C 4) are determined by analysis.

y Note that the tensor almji appearing in Equation (8) in LRR corresponds to 2Mimlj. Then, 21 = ?(2 + 3c2)=22 with c2 = 0:4 is the LLR coe cient that corresponds to C5 in (C 1) above. Thus, the LRR model corresponds to a value of C5LRR = ?0:145 in (C 1).