Spectral Modelling of a Passive Scalar in Homogeneous Turbulence 

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The basics of the CBC dimensional analysis

In the CBC theory, the kinetic energy spectrum is given by (1.3). The original method of CBC is based on the concept of invariance of very large scale eddies (PLE) corresponding to k < kL which is notably valid for infrared slopes = 1, 2 and 3. In fact in the kinetic case, the value of the infrared spectral slope is time-independent and remains constant for = 1, 2 (Saman), 3 and 4 (Batchelor). As for the coecient A(t) of the infrared spectrum, it remains constant for values of 2 [1; 2; 3], and evolves in time as A(t) L(t)p in the case = 4, where p is the backscatter parameter. In the case of high Reynolds numbers, when there is an inertial zone, one obtains (; p) = 􀀀2 ( + 1 􀀀 p) ( + 3 􀀀 p) ; K(t) t (1.31).
in which p = 0 if 3 and p 0:55 if = 4 as computed in Lesieur et al. (1987); Eyink & Thomson (2000); Meldi & Sagaut (2012). Other kinetic exponents such as n and nL can also be determined using (1.31) and are gathered in Table 1.2.
The relevant parameter to study the dynamics of the passive scalar is not only the Reynolds number, but also the (Taylor) Peclet number Pe = Re p Pr. In this section, the emphasis is put on the case T = 4. Numerical simulations show that the scalar variance also decreases with time following a power law KT tT . Furthermore, after a transient relaxation phase, the kinetic and scalar integral scales L and LT exhibit the same behaviour and their decay exponents nL and nLT converge toward the same value. Consequently, we consider that nL = nLT . The scalar variance spectrum ET (k; t) scales similarly to E(k; t) in the infrared range, namely ET (k < kT ; t) = AT (t)kT ; T 2 [1; 4] .

Validation at large Reynolds numbers for Pr 6= 1

The emphasis is now put on the case where the initial Reynolds number is suciently large to allow the kinetic and scalar spectra E and ET to decrease according to the extended CBC exponents given on the two rst lines of Table 1.2. For the sake of brevity, only the case Pr 1 is presented here since the results for Pr 1 are very similar (for more details, see the complete paper). The initial Reynolds number is Re(0) ‘ 2:104, high enough to ensure a large Peclet number, so that there is a clear separation of scales. Since ET (k; t) is located « under » E(k; t), local energy transfers dominate, unlike the case Pr 1 where the viscous-convective range is « outside » the kinetic energy spectrum. Time exponents of T , LT and KT are investigated in Fig. 1.6 for = T = 2 and = T = 4. The scalar decay exponents clearly follow the extended CBC theory. The result is the same for any Pr 1 as soon as the Peclet number is large enough. Hence, the extended scalar CBC exponents are valid at large Reynolds numbers for both Pr 1 and Pr 1.

Transition to low Reynolds and Peclet numbers

In this section, the transition from high to low Reynolds numbers with various Prandtl numbers is investigated. A detailed comparison with experimental results is performed to provide some explanations about the scattering between existing measured scalar decay exponents.
Validation of decay exponents for Pr = 1: The numerical method based on EDQNM analysis allows to illustrate the transition from high to low Reynolds numbers. Several simulations are made until very low Reynolds numbers Re 10􀀀1, starting from Re(t = 0) = 240. This Reynolds number is high enough to capture the beginning of the transition and all the previous decay exponents are accurately recovered. According to the theory for the case = 1, the same exponents for both large and small Reynolds numbers are found. It is only from the shape of the spectrum that these two cases can be distinguished. For infrared exponents 2 and T 2, the Reynolds number decreases over time and inertial ranges of both spectra disappear. This is the low Reynolds numbers regime and the decay exponents and T converge to the values expected by the extended CBC analysis given in Table 1.2. This transition grows more rapidly for higher values of the infrared spectral slopes.

Directional and Polarization transfers TE and TZ

The aim of the EDQNM1 approximation is to provide an explicit formula for both the directional and polarization transfers TE and TZ given in (2.32) and (2.34). For this purpose, a more convenient frame (, ,) must be used, attached to the planed formed by the triad k+p+q = 0, where is perpendicular to this plane. From now, the following notations are used: 0 and 00 refer to quantities expressed in p and q respectively. Useful vectors and angles are gathered in Fig. 2.2. a, b and c are the angles formed by p and q, q and k, and k and p. Finally, x = cos a, y = cos b and z = cos c. The new frame (, ,) is obtained from Craya frame (e(1); e(2); e(3) = ) by rotations of angles , 0 and 00 around k, p and q. All the details of the computation of ij from (2.18) are given in Appendix C. The nal results for the polarization and the directional transfers are TE (k; t) = 2 Z kpqkp h (E00 + <X00) (xy + z3)(E0 􀀀 E) 􀀀 z(1 􀀀 z2)(<X0 􀀀 <X).

Homogeneous Shear-Released Turbulence (HSRT)

In this section, the emphasis is put on HSRT: this is an original conguration where the shear S is non zero only in the early times. During this phase, linear transfers dened in (2.76) increase the anisotropy and produce kinetic energy. Then, after the release of the velocity gradients, the velocity eld freely decays and there is a RTI mechanism: non-linear transfers dened in (2.74) tend to isotropize small scales, decreasing as a consequence both K(t) and R13(t).
This framework, rstly presented in Mons et al. (2016), has never been investigated in DNS nor in experiments. This section is divided into ve parts. Firstly, rapid distortion theory (RDT) is used to validate the model at short times, when the linear eects are dominant. Then, classical scalings for the dierent spectra involved in shear-driven turbulence are addressed. Afterwards, the RTI mechanism, once the shear is released, is investigated, with a particular attention on the dierence between Saman and Batchelor turbulence. Then, a model is proposed for the slow-part of the pressure-strain tensor in the RTI phase. Finally, eects of infrared exponent and shear rate S, along with the case of anisotropy at low Reynolds numbers, are discussed.

Validation of HSRT with Rapid Distortion Theory

In this part, the anisotropic EDQNM modelling is assessed by comparisons with RDT as done in (Mons et al., 2016). The main calculations coming from RDT are given in Appendix D. In RDT, non-linear terms are discarded: this theory is valid for short times only, when linear processes dominate the ow. Moreover, at short times (for large Reynolds numbers), viscous eects are also negligible with respect to inertial ones. Two dierent simulations are presented in Fig. 3.1: one with an initial isotropic kinetic spectrum E(k; t) in which linear transfers produce energy and anisotropy. And a second one where these linear transfers are set to zero and the initial spectral tensor ij is analytically determined thanks to RDT. It is clear that the two dierent initial conditions collapse into the same behaviour when the shear is released, for both = 2 and = 4. This validates our spectral transfers. Another point of interest is that with RDT, 13 displays an inertial k􀀀5=3 range instead of an inertial anisotropic k􀀀7=3 range, which proves that non-linear transfers are responsible for the anisotropic range.

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Table of contents :

Acknowledgement
Publications and conferences
Abbreviations
Symbols
Introduction
I Passive Scalar in Isotropic Turbulence & Velocity Field in Anisotropic Turbulence 
1 Passive Scalar Mixing in Homogeneous Isotropic Turbulence 
1.1 The equations of homogeneous isotropic turbulence
1.2 The inertial scaling of ET for Pr 1
1.3 Mixed-derivative skewness ST for Pr 1
1.4 Time evolution of scalar integrated quantities
1.4.1 The basics of the CBC dimensional analysis
1.4.2 Validation at large Reynolds numbers for Pr 6= 1
1.4.3 Transition to low Reynolds and Peclet numbers
1.4.4 Transition for Pr 6= 1
1.4.5 Study of the integral scales L and LT
1.5 Conclusions for a passive scalar eld in HIT
2 Spectral Modelling of the Velocity Field in Homogeneous Turbulence 
2.1 Equations in physical space
2.2 Spectral equations and transfers
2.2.1 Craya equation for ^Rij
2.2.2 Craya-Herring frame – E 􀀀 Z decomposition
2.2.3 Generalized Lin equations
2.3 The closure problem
2.3.1 The EDQNM approximation
2.3.2 Directional and Polarization transfers TE and TZ
2.4 Spherically-averaged equations
2.4.1 Spherically-averaged descriptors
2.4.2 Spherically-averaged nal Lin equations
2.4.3 Return to isotropy – Spectral tensor
3 Dynamics of the Velocity Field in Shear-driven Turbulence 
3.1 Homogeneous Shear-Released Turbulence (HSRT)
3.1.1 Validation of HSRT with Rapid Distortion Theory
3.1.2 Kinetic energy spectrum E(k; t) and spectral tensor ij(k; t)
3.1.3 Anisotropy descriptors bij(t) and H() ij (k; t)
3.1.4 Modelling of the pressure-strain tensor (s)
3.1.5 Additional remarks on HSRT
3.2 Decay of K(t) and R13(t) in Saman and Batchelor HSRT
3.3 Homogeneous Shear Turbulence
3.3.1 Exponential growth of the kinetic energy K(t)
3.3.2 Non-linear transfers and the shear wavenumber
3.3.3 Discussion on the scattering of integrated quantities in HST
3.4 Conclusion and perspectives
3.4.1 Conclusions on HST and HSRT
3.4.2 Exponential growth rate
3.4.3 Perspectives
II Transport and Mixing in Homogeneous Anisotropic Turbulence 
4 Spectral Modelling of a Passive Scalar in Homogeneous Turbulence 
4.1 Scalar and scalar ux generalized Lin equations
4.2 EDQNM closure for ET and Fi
4.3 Final spherically-averaged scalar Lin equations
4.3.1 Modelling of ET and Fi
4.3.2 Spherical average of the passive scalar and scalar ux
4.4 Cospectrum for an uniform mean scalar gradient
5 Dynamics of a Passive Scalar in Homogeneous Turbulence 
5.1 Homogeneous shear-driven turbulence
5.1.1 Scalar spectrum ET (k; t) and non-linear transfers
5.1.2 Scalar decay laws and RTI in HSRT
5.1.3 Sustained shear (HST)
5.1.4 Decay and growth laws for the passive scalar in HSRT and HST
5.2 Isotropic Turbulence with a mean Scalar Gradient
5.2.1 Spectra and transfers
5.2.2 Comparisons with experimental and numerical results
5.2.3 Decay and growth laws for the cospectrum and passive scalar
5.2.4 Return to isotropy in HITSG
5.3 Homogeneous Shear Turbulence with Scalar Gradient
5.3.1 Denitions and transfers
5.3.2 Comparisons with experimental and numerical results
5.3.3 Growth of K, KT , KF and KSF
5.3.4 Streamwise ux spectrum FS(k; t)
5.3.5 Return to isotropy in HSTSG
5.4 Conclusions for the passive scalar at Pr = 1
6 Prandtl Number Eects on Passive Scalar Dynamics 
6.1 Prandtl number eects in HITSG
6.1.1 Inertial scalings for ET (k; t) and F(k; t) – Comparisons
6.1.1.1 Highly diusive passive scalar Pr 1
6.1.1.2 Weakly diusive passive scalar Pr 1
6.1.1.3 Spectral transfers and conclusions for the inertial scalings
6.1.2 Numerical results – Time evolution and anisotropy
6.1.2.1 Prandtl eects on the decay and growth of < u3 > and < 2 >
6.1.2.2 Cospectrum correlation w, pressure-scalar correlation F, and Nusselt number Nu
6.1.2.3 Return to isotropy of small scales
6.1.3 Conclusions for Pr 6= 1 in HITSG
6.2 Prandtl number eects in shear-driven turbulence
6.2.1 Homogeneous shear-released turbulence
6.2.2 Sustained shear ow
6.2.3 Homogeneous Shear Turbulence with a mean Scalar Gradient
6.2.4 Conclusions about shear-driven turbulence for Pr 6= 1
7 Spectral Modelling for Unstably Stratied Homogeneous Turbulence 
7.1 Evolution equations in USHT
7.1.1 Additional coupling terms
7.1.2 Spherically-averaged Lin equations for USHT
7.2 Spectral scaling and infrared dynamics
7.2.1 Spectral scaling of E, ET and F
7.2.2 Infrared dynamics
7.3 One-point statistics
7.3.1 The Froude number Fr
7.3.2 The mixing intensity
7.3.3 Growth of the kinetic energy K(t)
7.3.4 Global anisotropy
7.3.5 Comparison with Burlot et al. (2015b)
7.3.6 Conclusions on one-point statistics
7.3.7 Eddy-damping constants
7.4 Scale by scale anisotropy and structure of the ow
7.5 Pressure spectra and high Schmidt numbers
7.5.1 Pressure spectra
7.5.2 Cospectrum at high Schmidt numbers
7.6 Conclusion on USHT
7.7 Perspective – Variable stratication N(t)
7.7.1 Evolution equations with variable stratication
7.7.2 Prediction of the growth rate RT
7.7.3 Numerical results
8 Dynamics of Helicity in Skew-Isotropic Turbulence 
8.1 Spectral modelling of helicity
8.1.1 The E-H decomposition
8.1.2 Spherically-averaged helical Lin equations for E(k; t) and H(k; t)
8.2 Numerical results on the helical and kinetic elds
8.2.1 The importance of initial conditions H(k; t = 0)
8.2.2 Helical spectrum H(k; t) and non-linear transfers
8.2.3 Infrared dynamics and non-local transfers
8.2.4 Decay laws in helical ows
8.2.5 Robustness of the decay exponents – Altered infrared dynamics
8.3 Structure functions in helical turbulence
8.3.1 Inertial scaling for S(r) and D(uu!)(r)
8.3.2 Evolution equation of H
8.4 Eect of helicity on the scalar ux
8.4.1 Modelling of the quadrature spectrum
8.4.2 Decay of < !3 > and inertial scaling of Q(k; t)
8.5 Conclusion on homogeneous skew-isotropic turbulence
9 General Conclusions and Perspectives 
A Statistics and Structure Functions 
A.1 Evolution equations and denitions
A.2 Tensorial relations for homogeneous turbulence
A.2.1 Dissipation and enstrophy < !2 >
A.2.2 Identities for the velocity eld
A.2.3 Evolution equations of Wij and < !2 >
A.2.4 Evolution equation of ij
A.2.5 Evolution equations of < ij > and < 2 >
A.2.6 Cospectrum in isotropic turbulence with mean scalar gradient
A.3 Homogeneous isotropic turbulence
A.3.1 Spectral formalism
A.3.2 Second and third-order statistics
A.3.3 Results for the velocity eld
A.3.4 Results for the passive scalar eld
A.4 Structure functions and auto-correlations
A.4.1 Second-order longitudinal correlation and structure function
A.4.2 Third-order longitudinal correlation and structure function
A.4.3 Towards the Karman-Howarth equation
A.4.4 Yaglom and Corrsin equations
B Non-local Expansions of the Non-Linear Transfers 
B.1 Non-local uxes
B.2 Expansions for q k p
B.3 Expansions for k p q
B.4 Applications of the isotropic non-local transfers
C Details on the Spherically-Averaged Lin Equations 
C.1 Spectral evolution equations
C.1.1 Craya equation
C.1.2 Generalized Lin equations for E and Z
C.1.3 Evolution equation of Sijk(k; p; t)
C.2 Calculations of TE and TZ
C.2.1 Relations between frameworks
C.2.2 Computation of TQN
C.3 Spherically-averaged non-linear transfers
C.3.1 -integration
C.3.2 Spherical integration
C.4 Spherically-averaged linear transfers
C.4.1 Spherical integration
C.4.2 Computation of TL
C.4.3 Return to isotropy
C.4.4 Rotation
C.5 Kinetic quadratic anisotropic contributions
C.6 Fourth-order expansion for E and Z
C.6.1 Fourth order linear transfers
C.6.2 Fourth order non-linear transfers
C.6.3 Fourth-order nal spherically-averaged equations
D Additional Results for the Velocity Field in Homogeneous Turbulence 
D.1 Rapid Distortion Theory
D.2 Homogeneous Axisymmetric Turbulence
D.3 Homogeneous Plane Distortion
D.4 Pressure uctuations in HAT
D.4.1 Evolution equation of the pressure correlation EP
D.4.2 Spectrum and pressure variance
D.5 Details on helical turbulence
D.5.1 Non-linear helical transfer TH
D.5.2 Non-linear purely helical transfer
D.5.3 Details on the evolution equation of H
D.5.4 Re-interpretation of the helical viscous cuto kH
E Details on Spherically-Averaged Scalar Lin Equations 
E.1 Scalar-scalar correlation
E.1.1 Scalar Craya equation
E.1.2 EDQNM closure for ET
E.1.3 Spherically-averaged scalar Lin equations
E.1.4 Scalar quadratic anisotropic contributions
E.2 Scalar-velocity correlation F
E.2.1 Craya equation for the cospectrum
E.2.2 Quasi-normal approximation for Fi
E.2.3 Computation of the non-linear transfers of Fi
E.2.4 Spherically-averaged cospectrum Lin equations
E.2.5 Alternative modelling for F
E.2.6 Scalarux quadratic anisotropic contributions
E.2.7 Scalarux in HHTSG
Bibliography 

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