### Abstract

The normalized Reynolds (NR-) stress is a symmetric, non-negative, dyadic-valued operator. An analysis of the hydrodynamic equation governing velocity fluctuations of a constant property Newtonian fluid shows that this operator is related to a prestress operator that is also symmetric and non-negative. The prestress operator accounts for local spatial changes in the fluctuating pressure and in the fluctuating instantaneous Reynolds stress. The Cayley-Hamilton theorem from linear algebra is used to complete the closure with a non-negative mapping of the normalized Reynolds stress into the prestress. The non-negative mapping between the prestress operator and the Reynolds stress depends on a scalar-valued turbulent transport time related to the relaxation of a Green's function associated with a convective-viscous parabolic differential operator and the relaxation of a two-point, space-time correlation related to turbulent velocity fluctuations. The preclosure equation also depends on a kinematic operator related to the average velocity gradient and a rotational operator related to the angular velocity of the frame. The resulting universal realizable anisotropic prestress (URAPS-) closure is realizable for all non-rotating and rotating turbulent flows, provided the complementary transport equations for the turbulent kinetic energy and the turbulent dissipation are formulated to yield non-negative solutions. Experimental data and DNS results previously reported in the literature for non-rotating homogeneous simple shear and for non-rotating and rotating homogeneous decay are used to determine the closure constants. For rotating homogeneous simple shear, the URAPS-closure predicts the existence of self-similar states for finite positive and negative rotation numbers. The URAPS-closure for the NR-stress predicts anisotropic states consistent with expected behavior.

Original language | English (US) |
---|---|

Pages (from-to) | 4611-4624 |

Number of pages | 14 |

Journal | Chemical Engineering Science |

Volume | 64 |

Issue number | 22 |

DOIs | |

State | Published - Nov 16 2009 |

### Profile

### Keywords

- Fluid mechanics
- Mathematical modeling
- Momentum transfer
- Reynolds stress
- Rotating frames
- Turbulence

### ASJC Scopus subject areas

- Chemical Engineering(all)
- Chemistry(all)
- Applied Mathematics
- Industrial and Manufacturing Engineering

### Cite this

*Chemical Engineering Science*,

*64*(22), 4611-4624. DOI: 10.1016/j.ces.2009.04.040

**Realizable algebraic Reynolds stress closure.** / Koppula, Karuna S.; Bénard, André; Petty, Charles A.

Research output: Contribution to journal › Article

*Chemical Engineering Science*, vol 64, no. 22, pp. 4611-4624. DOI: 10.1016/j.ces.2009.04.040

}

TY - JOUR

T1 - Realizable algebraic Reynolds stress closure

AU - Koppula,Karuna S.

AU - Bénard,André

AU - Petty,Charles A.

PY - 2009/11/16

Y1 - 2009/11/16

N2 - The normalized Reynolds (NR-) stress is a symmetric, non-negative, dyadic-valued operator. An analysis of the hydrodynamic equation governing velocity fluctuations of a constant property Newtonian fluid shows that this operator is related to a prestress operator that is also symmetric and non-negative. The prestress operator accounts for local spatial changes in the fluctuating pressure and in the fluctuating instantaneous Reynolds stress. The Cayley-Hamilton theorem from linear algebra is used to complete the closure with a non-negative mapping of the normalized Reynolds stress into the prestress. The non-negative mapping between the prestress operator and the Reynolds stress depends on a scalar-valued turbulent transport time related to the relaxation of a Green's function associated with a convective-viscous parabolic differential operator and the relaxation of a two-point, space-time correlation related to turbulent velocity fluctuations. The preclosure equation also depends on a kinematic operator related to the average velocity gradient and a rotational operator related to the angular velocity of the frame. The resulting universal realizable anisotropic prestress (URAPS-) closure is realizable for all non-rotating and rotating turbulent flows, provided the complementary transport equations for the turbulent kinetic energy and the turbulent dissipation are formulated to yield non-negative solutions. Experimental data and DNS results previously reported in the literature for non-rotating homogeneous simple shear and for non-rotating and rotating homogeneous decay are used to determine the closure constants. For rotating homogeneous simple shear, the URAPS-closure predicts the existence of self-similar states for finite positive and negative rotation numbers. The URAPS-closure for the NR-stress predicts anisotropic states consistent with expected behavior.

AB - The normalized Reynolds (NR-) stress is a symmetric, non-negative, dyadic-valued operator. An analysis of the hydrodynamic equation governing velocity fluctuations of a constant property Newtonian fluid shows that this operator is related to a prestress operator that is also symmetric and non-negative. The prestress operator accounts for local spatial changes in the fluctuating pressure and in the fluctuating instantaneous Reynolds stress. The Cayley-Hamilton theorem from linear algebra is used to complete the closure with a non-negative mapping of the normalized Reynolds stress into the prestress. The non-negative mapping between the prestress operator and the Reynolds stress depends on a scalar-valued turbulent transport time related to the relaxation of a Green's function associated with a convective-viscous parabolic differential operator and the relaxation of a two-point, space-time correlation related to turbulent velocity fluctuations. The preclosure equation also depends on a kinematic operator related to the average velocity gradient and a rotational operator related to the angular velocity of the frame. The resulting universal realizable anisotropic prestress (URAPS-) closure is realizable for all non-rotating and rotating turbulent flows, provided the complementary transport equations for the turbulent kinetic energy and the turbulent dissipation are formulated to yield non-negative solutions. Experimental data and DNS results previously reported in the literature for non-rotating homogeneous simple shear and for non-rotating and rotating homogeneous decay are used to determine the closure constants. For rotating homogeneous simple shear, the URAPS-closure predicts the existence of self-similar states for finite positive and negative rotation numbers. The URAPS-closure for the NR-stress predicts anisotropic states consistent with expected behavior.

KW - Fluid mechanics

KW - Mathematical modeling

KW - Momentum transfer

KW - Reynolds stress

KW - Rotating frames

KW - Turbulence

UR - http://www.scopus.com/inward/record.url?scp=70349485567&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70349485567&partnerID=8YFLogxK

U2 - 10.1016/j.ces.2009.04.040

DO - 10.1016/j.ces.2009.04.040

M3 - Article

VL - 64

SP - 4611

EP - 4624

JO - Chemical Engineering Science

T2 - Chemical Engineering Science

JF - Chemical Engineering Science

SN - 0009-2509

IS - 22

ER -