Center for Turbulence ResearchProceedings of the Summer Program 2012

283

Towards an Accurate and Robust AlgebraicStructure Based Model

By R. Pecnik†, J. P. O.’Sullivan‡, K. Panagiotou¶, H. Radhakrishnan¶,S. Kassinos¶, K. Duraisamy AND G. Iaccarino

The algebraic structure-based turbulence model (ASBM) incorporates informationabout the structure of turbulence to provide closure to the Reynolds-averaged Navier-Stokes equations. In the past, the model has been applied to several well documentedComputational Fluid Dynamic (CFD) benchmark test cases, showing reasonable im-provement compared to standard turbulence models, but obtaining converged solutionsis not always straightforward. We therefore revisit the functional form of the ASBMand evaluate the model over a range of mean deformations of homogeneous turbulenceand compare the result with reference solutions using the Particle Representation Model(PRM). The performance of the model is assessed in detail and an improved ASBMis proposed. For a test case involving separated flow over a wall mounted hump thenew modifications are noted to yield better agreement when compared to experimentalmeasurements.

1. Introduction

Large-scale energy-containing eddies tend to organize spatially the fluctuating motionin their vicinity, and in so doing, eliminate gradients of fluctuation fields in some direc-tions (those in which the spatial extent of the structure is significant) and to enhancegradients in other directions (those in which the extent of the structure is small). Thus,associated with each eddy are local axes of dependence and independence that determinethe dimensionality of the local turbulence (Reynolds & Kassinos 1995; Kassinos et al.

2001). In non-equilibrium flows (strong rotation, separation and reattachment, etc.), thedimensionality of the turbulence structures dictates how the turbulence will respond toexternal deformation. Thus, the anisotropy of the turbulent stresses is to a large extentdetermined by dimensionality of the local turbulent structure, i.e. by the morphology ofthe turbulence eddies. This realization led to the development of the structure-based ap-proach to turbulence modeling (Reynolds & Kassinos 1995; Kassinos et al. 2001), and theengineering outcome of this effort is the Algebraic Structure Based Model (ASBM). TheASBM (Langer & Reynolds 2003; Kassinos & Langer 2005; Kalitzin et al. 2004; Kassinoset al. 2006; Radhakrishnan et al. 2008; Aupoix et al. 2009; O’Sullivan et al. 2010) has thepotential to lead to significant improvements in the performance of CFD codes used inaerospace design (Benton 2010) and offers hope for a step change in turbulence modelingbecause it provides a new route to turbulence closure that is anchored on the relationbetween the dimensionality of the turbulent eddies and the anisotropy of the turbulentstresses.

The objective of this work is to critically analyze the ASBM and identify areas where

† Delft University of Technology, The Netherlands‡ University of Auckland, Auckland, New Zealand¶ University of Cyprus, Nicosia, Cyprus

284 R. Pecnik et al.

the modeling accuracy and robustness must be improved, especially with regard to ap-plication in complex flows. The analysis was carried out using a number of new tech-niques. A parameter space map was developed that allows the state of the turbulenceto be clearly visualized with respect to the rapid distortion theory (RDT) limit andisotropic turbulence. This enabled a careful analysis of the ASBM’s representation offlows transitioning from a locally hyperbolic regime to an elliptic one and vice-versa.This helped to identify inconsistencies in the parameter blending in the ASBM. Finally anumber of reference solutions of homogeneous turbulence using the Particle Representa-tion Model (PRM) (Kassinos & Reynolds 1996) and Interactive Particle RepresentationModel (iPRM) (Kassinos & Akylas 2012) were carried out to obtain ”exact” solutionsfor the Reynolds stress tensor for various states of turbulence. These solutions were thencompared with the predictions of the ASBM and areas of improvement identified. Pre-liminary tests on a configuration involving separated flow over a wall mounted hump arethen attempted.

2. Algebraic Structure Based Model

In conventional RANS closures, the Reynolds stress tensor components are either de-rived from gradients of the mean velocity field using the eddy-viscosity concept or usingpartial differential or algebraic equations for the Reynolds stress tensors. In the ASBM,the Reynolds stress tensor is calculated in terms of one-point turbulence structure tensorsthat sensitize the model not only to the anisotropy of the turbulence componentality, butalso to the structural anisotropy.

2.1. Structure tensors

The turbulence tensors, as defined in Kassinos & Reynolds (1994) and Kassinos et al.

(2001), are Rij , the Reynolds stress tensor, Dij , the dimensionality structure tensor, andFij , the circulicity structure tensor. Dij and Fij contain information about the large-scale, energy-bearing structures that is not conveyed by Rij alone.

In the case of homogeneous turbulence, the contractions of the structure tensors areall twice the turbulence kinetic energy, i.e. Rii = Dii = Fii = q2 = 2k. Thus, normalizedstructure tensors can be defined as rij ≡ Rij/q2, dij ≡ Dij/q2, fij ≡ Fij/q2 , withq2 being twice the turbulent kinetic energy. An exact constitutive relation between thethree normalized tensors can be derived and it shows that in homogeneous turbulenceonly two of them are linearly independent:

rij + dij + fij = δij . (2.1)

The eddy-axis concept (Kassinos & Reynolds 1994) is used to relate the Reynoldsstress rij and structure tensors to parameters of a hypothetical turbulent eddy field.Each eddy represents a two-dimensional turbulence field, and is characterized by aneddy-axis vector ai. The turbulent motion and its energy content along this axis canbe decomposed into a jettal component along the eddy axis and a vortical componentperpendicular to the eddy axis. Additionally, if the motion around the eddy axis is notaxisymmetric the eddy is flattened, as it is in the case of turbulence subjected to rotation.Averaging over an ensemble of such eddies gives statistical quantities representative ofthe eddy field. The eddy axis tensor aij = 1

q2 〈V2aiaj〉 (where V is the velocity) is the

energy-weighted average direction-cosine of the ensemble that characterizes its structure.Exact constitutive equations are then derived to relate the Reynolds stresses rij and the

Algebraic structure based model 285

eddy axis tensor aij as follows:

rij =u′

iu′

j

2k= (1 − φ)

1

2(δij − aij) + φaij

+ (1 − φ)χ[1

2(1 − anmbmn)δij −

1

2(1 + anmbmn)aij − bij + ainbnj + ajnbni]

−γΩT

k

ΩT(ǫiprapj + ǫjprapi)

1

2[1 − χ(1 − anmbmn)]δkr + χbkr − χaknbnr ,

(2.2)

with δij the Kronecker delta, ǫijk the Levi-Civita tensor and ΩT the total rotation vector(sum of mean rotation and frame rotation). φ, χ, γ are three additional structure param-eters and bij is the flattening tensor (normalized direction-cosine of flattening vector).The flattening tensor is modeled in terms of mean rotation Ωi and frame rotation vectorΩf

i as,

bij =(Ωi + CbΩ

fi )(Ωj + CbΩ

fj )

(Ωk + CbΩfk)(Ωk + CbΩ

fk)

, Cb = −1.0. (2.3)

The three structure parameters φ, χ, γ are discussed in more detail below.

2.2. Structure parameters

Modeling the structure parameters φ, χ and γ is a crucial part in the construction of themodel. The eddy jetting parameter φ represents the amount of energy in the jetal mode,with φ = 0 purely vortical and φ = 1 purely jettal eddies, as in the case for turbulencesubjected to mean shear in the limit of RDT. The eddy helix vector γ correlates thejettal and the vortical mode and is the key parameter in setting the shear stress level inthe turbulent field. The eddy flattening parameter, together with the direction-cosine ofthe flattening vector, defines the deformation of the eddy due to rotation. These threeparameters are defined in terms of the ratio of mean rotation to mean strain ηm, theratio of frame rotation to mean strain ηf and a2 the second invariant of the eddy axistensor as a measure of anisotropy of turbulence. Since in this work only cases withoutframe rotation have been considered, the discussion of the parameterization is restrictedto two parameters. They are,

ηm =

√

Ω2

S2, a2 = amnamn , (2.4)

with:

Ω2 = −aijΩikΩkj , S2 = −aijSikSkj . (2.5)

Figure 1(a) shows the two-dimensional parameter plane used to parameterize the struc-ture parameters. It is limited on the left at ηm = 0 by plane strain, at the bottom witha2 = 1/3 by isotropic turbulence and at the top with a2 = 1 by RDT. The vertical lineat ηm = 1 corresponds to shear flows, which separates hyperbolic and elliptic stream-line flows. For ηm > 1 the flow is dominated by rotation. In Figure 1(b) the computedReynolds stress r11 is plotted as a function of ηm and a2 as obtained using the ASBMmodel from Langer & Reynolds (2003). Along the plane strain line ηm = 0 and fora2 = 1/3 the Reynolds stress tensor is isotropic. At the RDT limit of pure shear, forwhich an analytic solution exists, the turbulence consists of purely jettal eddies withr11 = 1. A discontinuity is clearly visible at the boundary between elliptic and hyper-bolic flow. Since the asymptotic state in the elliptic region is oscillatory, prior modeling

286 R. Pecnik et al.

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

ηm

a2

rapid distortion limit

isotropic turbulence

shea

r

pla

ne

stra

in

hyperbolic elliptic streamline flow

(a) ηm-a2 parameter plane.

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

R11: 0.3 0.4 0.5 0.6 0.7 0.8 0.9

ηm

a2

(b) Normalized Reynolds stress r11 plot-ted as a function of ηm and a2.

Figure 1. Parameter plane used to construct the structure parameters φ, χ and γ.

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

ηm

a2

Figure 2. Post-processed results from the periodic channel flow with separation (Frohlich et al.

2005) plotted in the parameter map. Dots represent the whole flow field, while the tringles areextracted within the flow separation indicated by the line in the inset below.

of the ASBM was limited by simply making the Reynolds stress tensor isotropic. Thediscontinuity also typically leads to convergence issues in CFD solvers and thus motivatesthe present improvements of the ASBM at the shear line.

in Figure 2 the results of a large eddy simulation (Frohlich et al. 2005) are post-processed and plotted in the ηm-a2 parameter plane. The black dots correspond to pointsin the entire flow domain, while the red dots indicate the points within the separationbubble. It is clearly seen that many points are located near the shear line, where thediscontinuity of the Reynolds stresses computed by the ASBM can potentially lead tonumerical instabilities in CFD simulations.

Algebraic structure based model 287

3. Assessment of the ASBM in homogeneous turbulence

In the original formulation as detailed in Langer & Reynolds (2003), the equations ofthe structure parameters are found by comparing target turbulent states obtained at theRDT limit, given a local mean deformation rate, and then these results are extrapolatedfor slower mean deformations (away from RDT). During the summer program, we revis-ited the ASBM functional forms and evaluated them over a range of mean deformations.

To establish reference solutions, we have made extensive use of the Particle Representa-tion Model (PRM) and the Interactive Particle Representation Model (iPRM) (Kassinos& Reynolds 1996; Kassinos & Akylas 2012). The PRM is essentially a reduced Fourierrepresentation retaining the minimum information beyond one-point that allows an ex-act closure of the Rapid Distortion Theory (RDT) governing equations without using amodel. From a different point of view, the PRM formalism can be thought of as a realspace Monte Carlo type of approach, and this duality offers a valuable framework fordeveloping one-point structure-based closures. The iPRM, or Interactive Particle Repre-sentation Model, is an extension of the PRM method for flows with weak deformationrates, but unlike the PRM, involves an approximate model for slower mean deformations.The iPRM has been successfully applied to all standard benchmark cases for homoge-neous turbulence. The iPRM particularly utilizes the concept of Effective Gradients. It ispostulated that the nonlinear turbulence-turbulence interactions can be represented byan effective deformation rate, which acts on each eddy or particle in addition to the meangradients as a result of the deformation caused by the sea of all other eddies (particles).It has to be mentioned that the iPRM transport equations retain the same form as theRDT (PRM) equations, with just the mean velocity gradient tensor replaced by the sumof the mean and effective gradient tensors. The effective gradient tensors are modeled interms of the one-point structure tensors thus providing the route for closure even at theone-point level.

Figure 3 compares the Reynolds stress components from the ASBM with iPRM resultsfor homogeneous turbulence subjected to planar hyperbolic mean deformation (meanstrain of magnitude S12 = S21 = S in the 12-plane) combined with mean rotation alongthe 3-axis with magnitude ηmS (i.e. Ω13 = −Ω31 = ηmS). The results confirm that forhigh Sq2/ǫ, i.e., near the RDT limit, the ASBM results match the iPRM results extremelywell as they were designed to. However, for slower deformation rates, significant deviationsare seen in the ASBM results.

In Figure 4 the Reynolds stresses and the structure parameters φ and γ are compared tothe PRM at the RDT limit for 0 < ηm < 3.5. As mentioned previously, the discontinuityat ηm = 1 is clearly visible for the Reynolds stresses as given by the ASBM (Langer &Reynolds 2003) (dashed line). New functional relations for the structure parameters inthe elliptic flow regime are proposed to avoid these discontinuities and to better matchthe PRM results (solid line). The details of the changes to these functional forms for thestructure parameters are not outlined in this paper, as they are preliminary - representinga possible progress towards a more accurate model. Furthermore the description of allthe equations involved would not be possible within this paper format. However, as anexample the functional forms of the equation for φ in the previous version and the newversion of the ASBM are given as:

φold =1

3+

2

3(

1 + ηm

2 (1−A2)5/2

) ; φnew =1

3+

2

3(

1 + ηm

2 (4/3−A2)5/2

) ; (3.1)

These equations are representative of the simplified case of a non-rotating frame in the

288 R. Pecnik et al.

ηm

0 0.2 0.4 0.6 0.8 1

-0.2

0

0.2

0.4

0.6

0.8

r

r

r

11

22

12

(a) Sq2/ǫ = 100

ηm

0 0.2 0.4 0.6 0.8 1

-0.2

0

0.2

0.4

0.6

0.8

r

r

r

11

22

12

(b) Sq2/ǫ = 10

ηm

0 0.2 0.4 0.6 0.8 1

-0.2

0

0.2

0.4

0.6

0.8

r

r

r

11

22

12

(c) Sq2/ǫ = 1

Figure 3. Comparison of ASBM (lines) and iPRM (symbols) computed normalized Reynoldsstress components for homogeneous turbulence undergoing hyperbolic mean deformation

elliptic regime and the interpolation away from the RDT limit is not included. Noticethat the only change is in the denominator of the second term. This small change ensuresthat the second term has a non-zero value for ηm = 1 at the RDT limit where A2 = 1.The resulting functional forms are shown in Figure 4(e) and it can be seen that thediscontinuity in φ has been removed by this change also improving the predictions of rij

in plots Figures 4(a) to 4(a).Figure 5 shows the same quantities, but for slower deformation rates of Sq2/ǫ = 10.

The new ASBM (solid line) results in a better agreement with the iPRM for the turbulentshear stress in the elliptic region, but still large disagreements are observed for the normalturbulent stresses. This has been identified as a key area of improvement of the ASBMin the future.

4. Test results

The improved functional forms of the structure parameters were tested on a highReynolds number flow over a two-dimensional bump that has been investigated as part ofan ongoing NASA research project (Greenblatt et al. 2006). The flow has a large separatedregion similar in size to the periodic channel flow discussed in Section 2.2 and a good setof experimental data for comparison. The Mach number and the Reynolds number basedon the chord length of the bump were Ma= 0.1 and Re = 9.36×105, respectively. The gridused for the simulations fulfilled the standard requirement of y+

1 < 1.0 and had 108 cellsin the wall-normal direction and 280 in the streamwise direction. Symmetry conditionswere applied at the top of the domain, an outlet boundary condition downstream of thebump and freestream conditions at the inlet. The domain was sufficiently long for theboundary layer to develop fully and the profiles of the mean and turbulent quantitiesagreed well with those measured upstream of the bump.

Figure 6 shows that the modifications to the functional forms of the structure param-eters significantly improves the agreement between the flow calculated using the ASBMand the experimental data. Comparing plots Figure 6(a) and Figure 6(c) it can be seenthat the new ASBM no longer overestimates the size of the separated region. The profileof the streamwise velocity shown in Fig. 7(a) more clearly demonstrates the improve-ment in the mean flow predicted by the new ASBM. Previous studies have shown thatthe ASBM underestimated the turbulent shear stress in separated regions (O’Sullivanet al. 2010). Figure 7(b) shows that the new ASBM accurately estimates r12 for most ofthe flow. Close to the turbulent shear layer, the new ASBM still underestimates r12 but

Algebraic structure based model 289

0 0.5 1 1.5 2 2.5 3 3.5-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

ηm

r 11

(a) r11

0 0.5 1 1.5 2 2.5 3 3.5-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

ηm

r 22

(b) r22

0 0.5 1 1.5 2 2.5 3 3.5-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

ηm

r 33

(c) r33

0 0.5 1 1.5 2 2.5 3 3.5-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

ηm

r 12

(d) r12

0 0.5 1 1.5 2 2.5 3 3.5

-0.2

0

0.2

0.4

0.6

0.8

1

ηm

φ

(e) φ

0 0.5 1 1.5 2 2.5 3 3.5

-0.2

0

0.2

0.4

0.6

0.8

1

ηm

γ

(f) γ

Figure 4. Reynolds stresses and structure parameter at RDT limit. Symbols: PRM, solid line:new ASBM, dashed line: ASBM (Langer & Reynolds 2003).

the improvement is significant. The comparison of the skin friction coefficient shown inFigure 7(c) highlights the effect of the higher turbulent shear stress seen by the betteragreement reattachment point, but also the need for further research as the new ASBMimproves the predictions of cf in the separated region but is still inaccurate close toseparation.

5. Conclusions

The Algebraic Structure Based Model has been critically evaluated and a number ofproblems have been identified. By analyzing flows using novel parameter space maps, spe-cific areas have been identified where the ASBM does not agree well with DNS and LESresults. In particular the ASBM performs poorly at the transition between hyperbolicand the elliptic flow regimes. Investigations show that this occurs often in many flows

290 R. Pecnik et al.

0 0.5 1 1.5 2 2.5 3 3.5

-0.2

0

0.2

0.4

0.6

ηm

r 11

(a) r11

0 0.5 1 1.5 2 2.5 3 3.5

-0.2

0

0.2

0.4

0.6

ηm

r 22

(b) r22

0 0.5 1 1.5 2 2.5 3 3.5

-0.2

0

0.2

0.4

0.6

ηm

r 33

(c) r33

0 0.5 1 1.5 2 2.5 3 3.5

-0.2

0

0.2

0.4

0.6

ηm

r 12

(d) r12

0 0.5 1 1.5 2 2.5 3 3.5

-0.2

0

0.2

0.4

0.6

0.8

1

ηm

φ

(e) φ

0 0.5 1 1.5 2 2.5 3 3.5

-0.2

0

0.2

0.4

0.6

0.8

1

ηm

γ

(f) γ

Figure 5. Reynolds stresses and structure parameter at Sq2/ǫ = 10. Symbols: iPRM, solidline: new ASBM, dashed line: ASBM (Langer & Reynolds 2003).

and the treatment of the functional form of the structure paramters must be improvedin this region. PRM and iPRM simulations have also been used to identify issues withthe ASBM and more importantly they have been used to determine corrections to thefunctional forms of the structure parameters. The corrections have been implementedin a new version of the ASBM and preliminary results from a representative test caseinvolving flow separation show improvements in the prediction of the mean flow and theturbulent quantities.

Acknowledgements

The authors would like to thank Prof. Paul Durbin for discussions during the 2012summer program.

Algebraic structure based model 291

0.7 0.8 0.9 1 1.1 1.2 1.30

0.05

0.1

0.15

0.2

0.25

x/c

y/c

(a) new ASBM

0.7 0.8 0.9 1 1.1 1.2 1.30

0.05

0.1

0.15

0.2

0.25

x/c

y/c

(b) ASBM (Langer & Reynolds 2003)

0.7 0.8 0.9 1 1.1 1.2 1.30

0.05

0.1

0.15

0.2

0.25

x/c

y/c

(c) experiment

Figure 6. Streamlines for wall-mounted hump case of Greenblatt et al. (2006)(no flow-controlcase).

-0.5 0 0.5 10

0.05

0.1

0.15

0.2

0.25

y/c

U/U∞

(a) U/U∞

-0.024 -0.016 -0.008 00

0.05

0.1

0.15

0.2

0.25

y/c

r12

(b) r12

0.5 0.75 1 1.25 1.5 1.75

-2.0E-03

0.0E+00

2.0E-03

4.0E-03

6.0E-03

x/c

skin

fric

tion

c f

(c) skin friction cf

Figure 7. Streamwise velocity U/U∞ (a) and turbulent shear stress u′v′/U2

∞ (b) at x/c = 0.9,and wall skin friction coefficient cf (c) for wall-mounted hump. Symbols: experiment, solid line:new ASBM, dashed line: ASBM (Langer & Reynolds 2003).

REFERENCES

Aupoix, B., Kassinos, S. & Langer, C. 2009 An easy access to the structure-basedmodel technology. In 6th Int. Symp. on Turbulence and Shear Flow Phenomena, pp.22–24. Seoul, Korea.

Benton, J. J. 2010 Evaluation of v2-f and ASBM turbulence models for transonicaerofoil RAE2822. In Progress in Wall Turbulence: Understanding and Modeling

(ed. M. Stanislas, J. Jimenez & I. Marusic), pp. 76–80. ERCOFTAC Series, Vol. 14.

Frohlich, J., Mellen, C. P., Rodi, W., Temmerman, L. & Leschziner, M. A.

2005 Highly resolved large-eddy simulation of separated flow in a channel withstreamwise periodic constrictions. Journal of Fluid Mechanics 526, 19–66.

292 R. Pecnik et al.

Greenblatt, D., Paschal, K. B., Yao, C.-S., Harris, J., Schaeffler, N. W.

& Washburn, A. E. 2006 A separation control cfd validation test case, part 1:Baseline and steady suction. AIAA 44 (12), 2820–2830.

Kalitzin, G., Iaccarino, G., Langer, C. & Kassinos, S. 2004 Combining eddy-viscosity models and the algebraic structure-based reynolds stress closure. In Pro-

ceedings of the Summer Program, pp. 171–182. Stanford.

Kassinos, S. & Akylas, E. 2012 Advances in Particle Representation Modeling of ho-mogeneous turbulence. From the linear PRM version to the interacting viscoelasticIPRM. In New Approached in Modelling Multiphase Flows and Dispersion in Tur-

bulence, Fractal Methods and Synthetic Turbulence (ed. F. Nicolleau, C. Cambon,J.-M. Redondo, J. Vassilicos, M. Reeks & A. Nowakowski), pp. 81–101. ERCOFTACSeries, Vol. 18.

Kassinos, S. & Langer, C. 2005 A new algebraic structure-based model with properhandling of strong rotation. In Int. Congress on Computational Mechanics. Limassol,Cyprus.

Kassinos, S. & Reynolds, W. C. 1996 A particle representation model for the de-formation of homogeneous turbulence. Tech. Rep. Annual Briefs 1996. Center forTurbulence Research, Stanford/NASA-Ames Research Center.

Kassinos, S. C., Langer, C. A., Kalitzin, G. & Iaccarino, G. 2006 A simplifiedstructure-based model using standard turbulence scale equations: computation ofrotating wall-bounded flows. International Journal of Heat and Fluid Flow 27 (4),653–660.

Kassinos, S. C. & Reynolds, W. C. 1994 A structure-based model for the rapiddistortion of homogeneous turbulence. Tech. Rep. TF-61. Mech. Eng. Department,Stanford University.

Kassinos, S. C., Reynolds, W. C. & Rogers, M. 2001 One-point turbulence struc-ture tensors. Journal of Fluid Mechanics 428, 213–248.

Langer, C. A. & Reynolds, W. C. 2003 A new algebraic structure-based turbulencemodel for rotating wall-bounded flows. Tech. Rep. TF-85. Mech. Eng. Department,Stanford University.

O’Sullivan, J. P., Pecnik, R. & Iaccarino, G. 2010 Investigating turbulence inwind flow over complex terrain. In Proceedings of the Summer Program 2010 , pp.129–139. Stanford University.

Radhakrishnan, H., Pecnik, R., Iaccarino, G. & Kassinos, S. C. 2008 Compu-tation of separated turbulent flows using the ASBM model. In Proceedings of the

Summer Program 2008 , pp. 365–376. Stanford University.

Reynolds, W. & Kassinos, S. 1995 One-point modelling of rapidly deformed homo-geneous turbulence. Proceedings - Royal Society of London, A 451 (1941), 87–104.