## PRIMA

## Particle Representation in Modeling Applications (PRIMA)

PRIMA

FP7 Marie Curie Re-Integration Grant 228389

period covered: 1/9/2008 - 31/8/2011

Host-Institution: National Observatory of Athens (NOA), Greece

Scientific Coordinator: Dr. Antonis Koussis

Principal Investigator: Dr. Evangelos Akylas

Modeling the Reynolds stresses is expected to remain the dominant tool for the analysis of complex turbulent flows in environmental and engineering studies. In simple flows where the mean deformation rates are mild and the turbulence has time to come to equilibrium with the mean flow, the Reynolds stresses are determined by the applied strain rate. Hence in these flows it is often adequate to use an eddy-viscosity representation. The modern family of k-ε models has been very useful in predicting near equilibrium turbulent flows. However, in modern engineering applications as well as in numerous astrophysical and geophysical problems, turbulence models are quite often required to predict flows with very rapid deformations (shear, rotation etc.). Examples of such rapid deformations can be also found in atmospheric flow (for example in an air stream flowing over an obstacle), in engineering applications and even in astrophysics (formation of rotating stellar disks). In such flows, the turbulence structures (eddies), take some time to respond to the rapid deformation applied and thus, eddy-viscosity models are inadequate. While the k-ε approach is often adequate for mild strain rates, the response of turbulence to rapid deformations is given by the

During the last decade, it has been widely recognized that one-point models, in which the only tensor characterizing the turbulence is the Reynolds stress tensor (e.g. Reynolds-stress transport models), are fundamentally incomplete for flows with mean rotation. The basic problem is that the Reynolds stresses carry information about the fluctuating velocity components (componentality), but they lack key information about the dimensionality of the turbulence field, which has to do with the morphology of the turbulent structures (it could be realized as the shape and the orientation of the eddies). This problem has been partially overcome by the use of relatively new one-point

period covered: 1/9/2008 - 31/8/2011

Host-Institution: National Observatory of Athens (NOA), Greece

Scientific Coordinator: Dr. Antonis Koussis

Principal Investigator: Dr. Evangelos Akylas

**budget: 45000 euros****summary**Modeling the Reynolds stresses is expected to remain the dominant tool for the analysis of complex turbulent flows in environmental and engineering studies. In simple flows where the mean deformation rates are mild and the turbulence has time to come to equilibrium with the mean flow, the Reynolds stresses are determined by the applied strain rate. Hence in these flows it is often adequate to use an eddy-viscosity representation. The modern family of k-ε models has been very useful in predicting near equilibrium turbulent flows. However, in modern engineering applications as well as in numerous astrophysical and geophysical problems, turbulence models are quite often required to predict flows with very rapid deformations (shear, rotation etc.). Examples of such rapid deformations can be also found in atmospheric flow (for example in an air stream flowing over an obstacle), in engineering applications and even in astrophysics (formation of rotating stellar disks). In such flows, the turbulence structures (eddies), take some time to respond to the rapid deformation applied and thus, eddy-viscosity models are inadequate. While the k-ε approach is often adequate for mild strain rates, the response of turbulence to rapid deformations is given by the

*rapid distortion theory*(RDT). The two theories are well established and give good approximations of the two limiting cases of near-equilibrium and rapidly deformed flows. The region between those limits is critical. At the state of the art however, there is not an established approach which can describe this region successfully. A good turbulence model should have a viscoelastic character predicting turbulence stresses proportional to the mean strain rate (k-ε theory) for slow deformations, and stresses determined by the amount of strain for rapid deformations. Under RDT, the non-linear effects due to turbulence-turbulence interactions are neglected in the governing equations but, even when linearized in this fashion, the governing equations are unclosed at the one-point level due to the non-locality of the pressure fluctuations. What is needed, then, is a good one-point model for RDT, which can be used as the backbone of a more general turbulence model. Given a successful RDT model, its blending with k-ε theory is relatively straight forward. Our goal so far has been the development of one point model for engineering use with the proper viscoelastic character. We have shown that to achieve this goal one needs to include structure information in the tensorial base used in the model, because non-equilibrium turbulence is inadequately characterized by the turbulent stresses themselves. We have also argued that the greater challenge in achieving visco-elasticity in a turbulence model is posed by the matching of rapid distortion theory RDT. Given a good RDT model, its extension to flows with mild deformation rates should be relatively straight forward.During the last decade, it has been widely recognized that one-point models, in which the only tensor characterizing the turbulence is the Reynolds stress tensor (e.g. Reynolds-stress transport models), are fundamentally incomplete for flows with mean rotation. The basic problem is that the Reynolds stresses carry information about the fluctuating velocity components (componentality), but they lack key information about the dimensionality of the turbulence field, which has to do with the morphology of the turbulent structures (it could be realized as the shape and the orientation of the eddies). This problem has been partially overcome by the use of relatively new one-point

*structure tensors*, carrying key information on the morphology of the turbulent structures. A modern model making use of the morphology of the turbulent structures (structure-based model), is the*particle representation model*(PRM). PRM was introduced in 1994-1999 at Stanford University by Profs. Kassinos and Reynolds, as a innovative method for computing the linear RDT one-point statistics efficiently. The PRM provided a completely new method for executing RDT calculations for homogeneous turbulence. At the same time it provides the conceptual foundation for turbulence modeling based on the characterization of the type of the turbulent structures (structure-based modeling). In a particle representation method a number of key properties and their evolution equations are assigned to hypothetical particles. The idea is to follow an ensemble of particles, determine the statistics of the ensemble, and use those as the representation for the one-point statistics of the corresponding field. The key innovation in the original PRM approach lies in the recognition that the linearity of the RDT governing equations makes it possible to emulate exactly the RDT for homogeneous turbulence without any modeling assumptions. The non-local pressure effects can be evaluated within the framework of the PRM itself with no modeling assumptions. The PRM can be used to evaluate all the one-point tensors needed in turbulence modeling, including the new structure tensors but, unlike spectral methods, provides no two-point information. However, the PRM does provide information about the directional dependence of the real part of the spectrum of homogeneous turbulence. In this sense, this method provides closure of RDT at minimum additional expense relative to a one-point approach. When the time scale of the mean deformation is large compared to that of the turbulence, the non-linear turbulence-turbulence interactions become important in the governing field equations. In the context of the PRM, these non-linear processes should be represented by particle-particle interactions. As in the case of the one point field equations, the non-linear processes cannot be evaluated directly and modeling is required. By doing so, the linear PRM version for the solution of RDT was extended to what we have termed, in this project, the*Interacting Particle Representation Method*(IPRM), which takes into account the non-linear effects. Due to the introduction of modeling, the emulation of the field equations by the IPRM is no longer exact, which was the case for the PRM emulation of RDT. As it was concluded with a relatively simple model for the non-linear turbulence-turbulence interactions, the IPRM is able to handle quite successfully a surprising wide range of flows. Some of these flows involve paradoxical effects and the fact that the IPRM is able to reproduce them suggests that the model captures a significant part of the underlying physics. In this sense, the IPRM is a viscoelastic structure-based model that bridges successfully RDT with k-ε theory.## basic references

Kassinos S.C. and Akylas E.:
Advances in Particle Representation Modeling of Homogeneous Turbulence. From the Linear PRM Version to the Interacting Viscoelastic IPRM.

Gravanis E. and Akylas E.: Stationarity of linearly forced turbulence in finite domains,

Akylas E., Kassinos S.C. and Langer C.: Analytical Solution for a Special Case of Rapidly Distorted Turbulent Flow in a Rotating Frame, Phys. Fluids, 18, 085104 (2006)

Akylas E., Kassinos S.C and Langer C.A.: Rapid shear of initially anisotropic turbulence in a rotating frame, Phys. Fluids, 19, 025102, 10.1063/1.2675939 (2007)

Kassinos, S.C. and Reynolds W. C.: A structure-based model for the rapid distortion of homogeneous turbulence. Report TF-61, Thermosciences Division, Department of Mechanical Engineering, Stanford University (1994)

Kassinos S.C. and Reynolds W.C.: A particle representation model for the deformation of homogeneous turbulence. In Annual Research Briefs 1996, 3150. Stanford University and NASA Ames Research Center: Center for Turbulence Research (1996)

S.C. Kassinos, W.C. Reynolds and Rogers M.M.: One-point turbulence structure tensors, J. Fluid Mech., 428, 213248 (2001)

*New Approaches in Modeling Multiphase Flows and Dispersion in Turbulence*,*Fractal Methods and Synthetic Turbulence*, ERCOFTAC Series, 2012, Volume 18, 81-101, DOI: 10.1007/978-94-007-2506-56 (2012).Gravanis E. and Akylas E.: Stationarity of linearly forced turbulence in finite domains,

*Physical Review E*84 (4), 046312 (2011).Akylas E., Kassinos S.C. and Langer C.: Analytical Solution for a Special Case of Rapidly Distorted Turbulent Flow in a Rotating Frame, Phys. Fluids, 18, 085104 (2006)

Akylas E., Kassinos S.C and Langer C.A.: Rapid shear of initially anisotropic turbulence in a rotating frame, Phys. Fluids, 19, 025102, 10.1063/1.2675939 (2007)

Kassinos, S.C. and Reynolds W. C.: A structure-based model for the rapid distortion of homogeneous turbulence. Report TF-61, Thermosciences Division, Department of Mechanical Engineering, Stanford University (1994)

Kassinos S.C. and Reynolds W.C.: A particle representation model for the deformation of homogeneous turbulence. In Annual Research Briefs 1996, 3150. Stanford University and NASA Ames Research Center: Center for Turbulence Research (1996)

S.C. Kassinos, W.C. Reynolds and Rogers M.M.: One-point turbulence structure tensors, J. Fluid Mech., 428, 213248 (2001)

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