jeanflop/post_ocr_correction2 huggingface dataset:
(cf. [1]) a été retenu; c'est-à-dire que les effets linéaires dus aux ondes acoustiques sont pris en compte de façon exacte sans avoir à les introduire empiriquement dans le terme d'amortissement tourbillonnaire u. Cet amortissement caractérise ainsi exclusivement les effets non linéaires. Les fonctions réponse sont alors analytiquement résolues. Leur nature vibratoire est mise en évidence [équations (1) à (5)]. Les corrélations en un temps s'expriment linéairement à partir de celles en deux temps par le biais des fonctions réponse. La formulation de l'EDQNM est alors obtenue pour un fluide isotrope faiblement compressible. Le système d'équations donnant les corrélations doubles est obtenu sous une forme matricielle [équations (8)]. La présentation met en évidence d'une part un noyau linéaire exact prenant en compte la viscosité et les ondes acoustiques et d'autre part un second membre non linéaire représentant les transferts spectraux responsables des redistributions énergétiques entre structures. Les seconds membres [(9) à (11)] montrent la nature complexe des interactions entre les différents modes. Ils permettent de retrouver à la limite le cas d'un fluide incompressible lorsque la vitesse du son tend vers l'infini. L'étude du problème linéaire non visqueux permet de mettre en évidence un invariant incompressible et un invariant acoustique, somme de l'énergie compressible et de l'énergie potentielle de pression. Le rôle linéaire de la corrélation double vitesse compressible/pression est alors de transférer l'énergie entre les deux composantes de cet invariant acoustique. Ce système de corrélations en deux points et en un temps — formulation de l'EDQNM — sera résolu de façon numérique et les résultats seront présentés dans une prochaine publication. 1. INTRODUCTION. — In a first Note [5], the D.I.A. equations for the two-time and two-point correlations of a "weakly" (i.e. barotropic, linearized in p') compressible isotropic turbulence have been written. These equations are now used as a starting point to develop a one-time model in order to remain within reasonable Computing cost. Indeed, two-time computations require larger memory and computational time. After a presentation of the way to derive a one-time approximation from a two-time model, the procedure is applied to compressible turbulence and remarks concerning the
somme de l'énergie compressible et de l'énergie potentielle de pression. Le rôle linéaire de la corrélation double vitesse compressible/pression est alors de transférer l'énergie entre les deux composantes de cet invariant acoustique. Ce système de corrélations en deux points et en un temps — formulation de l'EDQNM — sera résolu de façon numérique et les résultats seront présentés dans une prochaine publication. 1. INTRODUCTION. — In a first Note [5], the D.I.A. equations for the two-time and two-point correlations of a "weakly" (i.e. barotropic, linearized in p') compressible isotropic turbulence have been written. These equations are now used as a starting point to develop a one-time model in order to remain within reasonable Computing cost. Indeed, two-time computations require larger memory and computational time. After a presentation of the way to derive a one-time approximation from a two-time model, the procedure is applied to compressible turbulence and remarks concerning the final set of equations are made. 2. DÉRIVATION OF A ONE-TIME APPROXIMATION FROM A TWO-TIME MODEL. — In order to do so, the usual step by step procedure is (for isotropic incompressible turbulence): (i) assume that the response function <G(t, r')> (for t < t') decays exponentially in terms of t - t': <G(K, t, t')> = exp(q(K, t). (t' - t)); (ii) assume that the two-time velocity correlations can be expressed as the product of the one-time correlation by the response function: <u>(K, t, t') = (<G(K, t, t')>. <u>(K, t') (corresponding to an extension of the "Fluctuation Dissipation" theorem); (iii) replace in the R.H.S. of the D.I.A. equations for the two-time correlation; (iv) take the limit t' -> t in this equation to obtain the master equation for one-time correlation; (v) "Markovianize": i.e. assume that <O(K, t)> has a weak dependence with respect to t in the R.H.S. of the obtained equation. (vi) introduce a phenomenological expression for the damping coefficient r)(K, t). The Eddy Damped Quasi Normal Markovian approximation — EDQNM — [2], see also [4]) is obtained if a phenomenological form compatible with the existence of a K-5/3 Kolmogorov inertial range in the energy spectrum is
t - t': <G(K, t, t')> = exp(q(K, t). (t' - t)); (ii) assume that the two-time velocity correlations can be expressed as the product of the one-time correlation by the response function: <u>(K, t, t') = (<G(K, t, t')>. <u>(K, t') (corresponding to an extension of the "Fluctuation Dissipation" theorem); (iii) replace in the R.H.S. of the D.I.A. equations for the two-time correlation; (iv) take the limit t' -> t in this equation to obtain the master equation for one-time correlation; (v) "Markovianize": i.e. assume that <O(K, t)> has a weak dependence with respect to t in the R.H.S. of the obtained equation. (vi) introduce a phenomenological expression for the damping coefficient r)(K, t). The Eddy Damped Quasi Normal Markovian approximation — EDQNM — [2], see also [4]) is obtained if a phenomenological form compatible with the existence of a K-5/3 Kolmogorov inertial range in the energy spectrum is chosen for r. For example, Following Pouquet et al. (1975) [3], it is usually assumed that: Since EDQNM gives satisfactory results for incompressible turbulence, it appears to be a good candidate for the present extension of two-point theories to compressible turbulence. The EDQNM framework is thereafter selected. When waves are present in turbulence, it has been suggested by several authors to introduce an empirical modification of the damping factor q(K, t); this is for example the case when Alfvén waves are present in MHD turbulence, or for buoyancy generated waves in stratification problems, as well as in the case of rotating fluids. A more straightforward approach has recently been derived by Cambon (1982) [1] who developed a model christened "EDQNM 2" for the case of both rotation and strain. The basic idea is that the effect of waves introducing linear terms in the response equation, can be directly accounted for in the EDQNM approach. Indeed, assuming that the fourth order cumulants are taken into account through a linear relaxation (involving the eddy damping coefficient u) in the equation governing the triple correlations, the exact solving of such correlations leads to identify the D.I.A. response equation with: d/dt <G(K, t, t')>
present in turbulence, it has been suggested by several authors to introduce an empirical modification of the damping factor q(K, t); this is for example the case when Alfvén waves are present in MHD turbulence, or for buoyancy generated waves in stratification problems, as well as in the case of rotating fluids. A more straightforward approach has recently been derived by Cambon (1982) [1] who developed a model christened "EDQNM 2" for the case of both rotation and strain. The basic idea is that the effect of waves introducing linear terms in the response equation, can be directly accounted for in the EDQNM approach. Indeed, assuming that the fourth order cumulants are taken into account through a linear relaxation (involving the eddy damping coefficient u) in the equation governing the triple correlations, the exact solving of such correlations leads to identify the D.I.A. response equation with: d/dt <G(K, t, t')> = p(K, t). <G(K, t, t')> + exact linear terms in which i stands for the effects of non-linear terms only. Cambon has successfully applied this procedure to the case of the action of rotation on an initially isotropic turbulence. He found — among other interesting features — compatibility with the results obtained after having introduced a heuristic modification of the n(K, t) formulation [1]. The case of buoyancy forces is now investigated in the same way. 3. DÉRIVATION OF THE EDQNM CLOSURE FOR COMPRESSIBLE TURBULENCE. — The procedure is now applied to the equations of weakly compressible turbulence. The first step of markovianization, (i), is to assume the role of the non-linear term in the response function equations to be a damping effect. Thus, non-linear term in the R.H.S. of the equations for the response functions are replaced by terms of the form: where a, p stand for either of the three modes I, C and P; respectively incompressible velocity, compressible velocity and pressure. To complete this step, and obtain an expression for the incompressible response function compatible with the exponential form usually retained in the case of incompressibility, a weak time dependence of ia (K, t) has to
the results obtained after having introduced a heuristic modification of the n(K, t) formulation [1]. The case of buoyancy forces is now investigated in the same way. 3. DÉRIVATION OF THE EDQNM CLOSURE FOR COMPRESSIBLE TURBULENCE. — The procedure is now applied to the equations of weakly compressible turbulence. The first step of markovianization, (i), is to assume the role of the non-linear term in the response function equations to be a damping effect. Thus, non-linear term in the R.H.S. of the equations for the response functions are replaced by terms of the form: where a, p stand for either of the three modes I, C and P; respectively incompressible velocity, compressible velocity and pressure. To complete this step, and obtain an expression for the incompressible response function compatible with the exponential form usually retained in the case of incompressibility, a weak time dependence of ia (K, t) has to be assumed. The following analytical expressions are finally found: where qn = n (K, t) = v K2 + |xn (K, t). Again, if this procedure is applied to the other response functions; and being aware of the different couplings between Gcc — Gpc and Gcp—Gpp, it is found: 1604 C. R. Acad. Sci. Paris, t. 307, Série II, p. 1601-1606, 1988 where r)Cp = qc (K) t) = v' K2 + uCp (K, t), A = Acp(K, t) = T|cp(K. 0-4K2C 2. We can notice that, assuming a weak compressibility, A is negative. Hence, the response functions display an oscillatory behaviour. Step (ii): the two-time correlations <DaP(K, t, t') have to be linked to the one-time correlations <SaP(K, t). To extend the usual incompressible expression $(K, t, £') = <G(K, t, t')>0(K, t) to the present situation, we notice that this relationship is exact in the case of a linear approximation (Rapid Distortion Theory) and we look for a similar relationship, also exact in a linear situation. We finally retain the following set of equations: (6) The next step (iii) is to plug these dependences into the correlation equations. Then, the two-time correlation equations of the DIA approach are
C. R. Acad. Sci. Paris, t. 307, Série II, p. 1601-1606, 1988 where r)Cp = qc (K) t) = v' K2 + uCp (K, t), A = Acp(K, t) = T|cp(K. 0-4K2C 2. We can notice that, assuming a weak compressibility, A is negative. Hence, the response functions display an oscillatory behaviour. Step (ii): the two-time correlations <DaP(K, t, t') have to be linked to the one-time correlations <SaP(K, t). To extend the usual incompressible expression $(K, t, £') = <G(K, t, t')>0(K, t) to the present situation, we notice that this relationship is exact in the case of a linear approximation (Rapid Distortion Theory) and we look for a similar relationship, also exact in a linear situation. We finally retain the following set of equations: (6) The next step (iii) is to plug these dependences into the correlation equations. Then, the two-time correlation equations of the DIA approach are used to extract the one-time correlation equations [step (iv)]. In step (v), the time dependence of the products <PT <Sa in the RHSs will be assumed — under Markov's hypothesis — to be weak so that: We notice that the decorrelation times r)n defined in step (i) are a priori unknown and likely different to each other. The benefit of the previous DIA procedure is to provide information on these empirical r) which would not be possible with a direct Quasi Normal approach. Now, to consider how to solve this problem, we use a matrix formalism. Our System is contracted into: where Tn(K, t), Tcc(K, t) and Tcp (K, t) are the transfer terms corresponding to the RHSs of the equations of the <SaP (K, t, t') in DIA written under the Markovian hypotheses. C. R. Acad. Sci. Paris, t 307, Série II, p. 1601-1606, 1988 1605 where "x"e"x(K, P, Q)" (already defined in the part one of this paper: DIA). 4. REMARKS. — It is relevant to notice that these equations are explicitly accounting for the linear effects on double correlations as well as on triple correlations (effects of pressure waves in the present case), which is corresponding to
unknown and likely different to each other. The benefit of the previous DIA procedure is to provide information on these empirical r) which would not be possible with a direct Quasi Normal approach. Now, to consider how to solve this problem, we use a matrix formalism. Our System is contracted into: where Tn(K, t), Tcc(K, t) and Tcp (K, t) are the transfer terms corresponding to the RHSs of the equations of the <SaP (K, t, t') in DIA written under the Markovian hypotheses. C. R. Acad. Sci. Paris, t 307, Série II, p. 1601-1606, 1988 1605 where "x"e"x(K, P, Q)" (already defined in the part one of this paper: DIA). 4. REMARKS. — It is relevant to notice that these equations are explicitly accounting for the linear effects on double correlations as well as on triple correlations (effects of pressure waves in the present case), which is corresponding to the EDQNM 2 assumptions. Let us consider the degenerated problem where transfer and viscosity are being set to equal to zero. Three eigenmodes are found: associated respectively with the eigenvalues: 0; 0 and 2 i KC0. The first two eigenmodes provide two inviscid, linear invariants: En=47iK2$n (incompressible kinetic energy) and EA=27iK2(4>cc + 1/(C0p0)2Opp) (corresponding to acoustic energy). Hence, the total energy (kinetic and potential) is given by the sum of these two invariants: with En=4iK2Un; Ecc = 2 n K2 0>cc and Epp = 27iK2<Dpp/(p0C0)2. The role of $cp is then to interchange energy from the two modes <!>cc and <6PP. Dielectric slowing down of charge carriers moving along a conducting polymer, in a polar solvent Abstract — A carrier (electron or hole) is assumed to move along a conjugate chain dissolved in a polar solvent (e.g. pure water or water + salt). The moving charge induces in the solvent a polarization cloud which is slightly delayed: the cloud creates a field Ey which slows down the carrier. This effect would be described by standard formulas giving the stopping power for charged particles in radiation, if the carrier moved along a straight line. Here, however, the carrier follows the chain
EA=27iK2(4>cc + 1/(C0p0)2Opp) (corresponding to acoustic energy). Hence, the total energy (kinetic and potential) is given by the sum of these two invariants: with En=4iK2Un; Ecc = 2 n K2 0>cc and Epp = 27iK2<Dpp/(p0C0)2. The role of $cp is then to interchange energy from the two modes <!>cc and <6PP. Dielectric slowing down of charge carriers moving along a conducting polymer, in a polar solvent Abstract — A carrier (electron or hole) is assumed to move along a conjugate chain dissolved in a polar solvent (e.g. pure water or water + salt). The moving charge induces in the solvent a polarization cloud which is slightly delayed: the cloud creates a field Ey which slows down the carrier. This effect would be described by standard formulas giving the stopping power for charged particles in radiation, if the carrier moved along a straight line. Here, however, the carrier follows the chain (a random walk) and is pulled by a fixed voltage applied at the ends of the chain. We analyze its curvilinear velocity (w) assuming that w = Const. (a) If the polymer coil is not under tension, the polarization cloud remains (on the average) centered on the instantaneous position of the carrier, and Ey=0. (b) If the coil is (weakly) stretched, Ey≠0. In pure water we even expect (within this approximation) a vanishing mobility when the applied field (Ec) is below a certain threshold E*. In salt solutions, these effects are less singular and give only a renormalization of the mobility u. Abridged English Version — Certain conjugated polymers (polythiophenes with suitable side groups) may exist in the form of melts or solutions. In the latter case, the mobility of carriers along one chain may be reduced by dielectric lags in the solvent. The basic role of these lags was first pointed out by Born, and elaborated by many others, for the motion of ions in water. Here we are concerned with (quasi) ideal coils, where the motion is a random walk. A voltage U is applied at the ends of the chain (of curvilinear length L), and produces a
the carrier, and Ey=0. (b) If the coil is (weakly) stretched, Ey≠0. In pure water we even expect (within this approximation) a vanishing mobility when the applied field (Ec) is below a certain threshold E*. In salt solutions, these effects are less singular and give only a renormalization of the mobility u. Abridged English Version — Certain conjugated polymers (polythiophenes with suitable side groups) may exist in the form of melts or solutions. In the latter case, the mobility of carriers along one chain may be reduced by dielectric lags in the solvent. The basic role of these lags was first pointed out by Born, and elaborated by many others, for the motion of ions in water. Here we are concerned with (quasi) ideal coils, where the motion is a random walk. A voltage U is applied at the ends of the chain (of curvilinear length L), and produces a field Ec = U/L. We assume that the curvilinear velocity w of the carrier is nearly constant (an approximation of the mean field type, which is extremely tentative). The first case considered corresponds to a pure polar solvent with a certain rotational relaxation time τr for the dipoles. The field Ef due to dielectric effects is analyzed in detail, but can be understood by the following qualitative argument. The distance over which memory effects contribute is of order λ=(τrτD)1/2, where a is a monomer size. The fields produced are of order eλ2 where e is the unit charge, s the dielectric constant. In our mean field approximation the field, measured at time t at the carrier position r(t), would vanish by symmetry: the earlier positions r(t - x) being distributed isotropically around r(t). Thus we obtain interesting effects only if the chain is stretched, with a (small) extension parameter < cos θ > = r (where θ is the angle between one link and the stretching direction). Then the linear distance of flight during a time τr is d=rwτr, and the field at point r is of order (e/s) (d/l). We still need to project this field over the local
Ef due to dielectric effects is analyzed in detail, but can be understood by the following qualitative argument. The distance over which memory effects contribute is of order λ=(τrτD)1/2, where a is a monomer size. The fields produced are of order eλ2 where e is the unit charge, s the dielectric constant. In our mean field approximation the field, measured at time t at the carrier position r(t), would vanish by symmetry: the earlier positions r(t - x) being distributed isotropically around r(t). Thus we obtain interesting effects only if the chain is stretched, with a (small) extension parameter < cos θ > = r (where θ is the angle between one link and the stretching direction). Then the linear distance of flight during a time τr is d=rwτr, and the field at point r is of order (e/s) (d/l). We still need to project this field over the local direction of motion, and this gives another factor q [Eq. (9)]. The net result is a force tending to slow down the carrier, and proportional to eλ1/2. This gives an anomalous relation between velocity and field (see Fig.). A similar discussion can be carried out for salty water, where the fields are screened out (by ion transport) over a time τd (the dielectric relaxation time) and over spatial distances 1 (the Debye screening length). Here two diffusion constants add their effects: one of them (D1=am/6) describes the random walk of the carrier, the other D = 1/2τd describes brownian motion of the ions. The length λ is now [D+D1)τd]1/2 and is finite for w → 0. Then Ey is linear in rv [equation (12)], and we find a simple renormalization of the mobility. The main conclusion is that, in solutions or gels containing conducting chains, we may expect some spectacular mecano-electric effects. I. NATURE DU PROBLÈME. — On dispose actuellement de chaînes conjuguées conductrices (après dopage) en phase solide. Plus récemment, il s'est avéré possible de solubiliser certaines de ces chaînes par l'adjonction de groupes latéraux bien choisis : en particulier les polythiophènes paraissent garder dans ces conditions une conductivité
fields are screened out (by ion transport) over a time τd (the dielectric relaxation time) and over spatial distances 1 (the Debye screening length). Here two diffusion constants add their effects: one of them (D1=am/6) describes the random walk of the carrier, the other D = 1/2τd describes brownian motion of the ions. The length λ is now [D+D1)τd]1/2 and is finite for w → 0. Then Ey is linear in rv [equation (12)], and we find a simple renormalization of the mobility. The main conclusion is that, in solutions or gels containing conducting chains, we may expect some spectacular mecano-electric effects. I. NATURE DU PROBLÈME. — On dispose actuellement de chaînes conjuguées conductrices (après dopage) en phase solide. Plus récemment, il s'est avéré possible de solubiliser certaines de ces chaînes par l'adjonction de groupes latéraux bien choisis : en particulier les polythiophènes paraissent garder dans ces conditions une conductivité honorable. Les effets de solvant sont multiples : ici nous en discutons un seul, présent avec les solvants polaires : le freinage diélectrique, dû au retard de phase entre la polarisation (induite dans le solvant) et le champ électrique d'un porteur en mouvement. 1. Le freinage diélectrique. — Son rôle en électrophorèse a été souligné en premier par Born, et développé par Zwanzig, Hubbard et Onsager. L'approximation utilisée au départ consistait à attribuer un mouvement rectiligne uniforme (de vitesse v) à l'ion, entraîné par un champ macroscopique E0. On peut alors calculer le nuage de polarisation résultant au moyen de la constante diélectrique généralisée s(q, ω) au vecteur d'onde q et à la fréquence ω. Le résultat final est une force de freinage (linéaire en v à basse vitesse) formellement identique à celle qui intervient dans le pouvoir de ralentissement des particules chargées. Ce freinage peut être important pour la prévision de certaines mobilités ioniques, malgré diverses complications : (a) sphère d'hydratation des ions à dipôles plus ou moins gelés (b) mouvement brownien de l'ion, qui se superpose au mouvement rectiligne d'entraînement, et qui peut modifier considérablement le nuage. 2. Le cas de chaînes flexibles. Nous voulons transposer ici ces