RAPID COMMUNICATIONS
PHYSICAL REVIEW E 91, 011001(R) (2015)
Passive scalar convective-diffusive subrange for low Prandtl numbers in isotropic turbulence A. Briard* and T. G om ez' Sorbonne Universites, UPMC Univ Paris 06, UMR 7190, Institut Jean Le Rond d ’Alembert, F-75005 Paris, France and CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France (Received 4 September 2014; published 6 January 2015) In this Rapid Communication, we study the behavior of a strongly diffusive passive scalar field T submitted to a freely decaying, homogeneous and isotropic turbulence with eddy-damped quasinormal Markovian simulations. We present a new subrange located between the Ar17/3 inertial-diffusive subrange and the Kolmogorov wave number k„. This subrange is generated by small-scale convection linked to k„ that balances diffusion effects. Thus, we build a typical length scale k^'D based on convection and diffusion and give an expression for the shape of the passive scalar spectrum in this subrange ET ~ v^Pr Arll/3 using physical arguments. This result unifies two different theories coming from Batchelor [G. K. Batchelor, J. Fluid. Mech. 5, 113 (1959)] and Chasnov [J. Chasnov et al., Phys. Fluids A 1, 1698 (1989)] and explains results previously obtained experimentally. DOI: 10.1103/PhysRevE.91.011001
PACS number(s): Al.21.eb, A l.27.Gs
T he dynam ic o f a passive scalar field T w hen the Prandtl num ber Pr is m uch low er than 1 is a very controversial topic. T here are four different theories regarding the shape o f the scalar spectrum E T . B atchelor [1] proposed that E r ~ k ~ xl/3, w hereas Chasnov [2] found E r ~ A:- ” / 3 for a very rapidly stirred fluid. M oreover, G ibson [3] proposed an £ > ~ A r3 evo lution by considering convection effects when scalar gradients are very w eak or zero at sm all scales. Finally, G ranatstein [4] established an E T ~ k ~ 13/3 subrange based on experim ental data in a plasm a and justified it using the H eisenberg model. Thanks to eddy-dam ped quasinorm al M arkovian (ED Q N M ) sim ulations [5,6], a large range o f Prandtl num bers P r = v /a can be explored at high Reynolds num bers, w here v is the kinem atic viscosity and a the scalar diffusivity. ED QN M sim ulations allow one to explain directly how the k ~ l3/3 could have been obtained experim entally before: T his subrange was observed for fluids w ith Pr e [0.1,0.01] and Re* ~ 160 where Rex is the Reynolds num ber based on Taylor scale X. However, there is no inertial-diffusive subrange for Pr = 0.1 at this Reynolds num ber as w e can see in Fig. 3. A nd for P r = 10~2, the inertial-diffusive subrange is not com pletely established: T his is probably the reason w hy A:- 17/3 is not observed in experim ental w orks [4], In addition to this transitional Pr state, Rex is not high enough to m atch w ith the theory (at least Rex = 200 is required). In the K olm ogorov inertial subrange k e [kL ,k n], w here k L and k,t are the wave num bers linked respectively to the integral scale and Kolm ogorov scale, the kinetic spectrum E evolves as E ( k , t ) = K 0 k ~ 5' 3 €2' 3,
(1)
w here K 0 is the Kolm ogorov constant ( ~ 1.5) and e the turbulent energy dissipation rate. In the inertial-convective subrange, the scalar spectrum E r evolves as E r (k ,t) = K Co e Te ~ '/3k ~ 5/3,
(2)
w here K co is the C orrsin-O bukhov constant ( ~ 0.66) and e r the scalar dissipation rate. The num erical values o f both
*[email protected] t thomas.gomez @upmc.fr 1539-3755/2015/91 (1 )/011001(4)
constants K q and K co are recovered in our sim ulations. These laws for E and E j are obtained using dim ensional analysis based on physical argum ents. For Pr =
I ^ - k 2a
2 .
( 12)
( 8)
The integral from 0 to Ac takes into account the main contri bution of dissipation as k » 1. The influence of small-scale dynamics on eT is modeled by the eddy conductivity aT. As a consequence, we can assume as Chasnov [2] did, that d e r / d k = 0. If we derive (8) with respect to k and we consider that E{k -> 00) = 0 we find E T(k,t) = ~ - e Te2/3k 11/3(a + aT) 2nc l .
TCD =
(9)
In Batchelor’s paper [1], a j is introduced differently. The characteristic time n~l appears to be (ak2)~1 which is a diffusion time. This makes sense when diffusion dominates
In Fig. 4 we clearly see that in the IBS, for k > A:Cd , n ~ l becomes constant. In other words, we find the result predicted by Chasnov but in a more general case: Indeed, we did not have to consider the case of a rapidly stirred fluid. The other point of interest is that the constant reached by n ~[ is the Kolmogorov time scale rn. This result is consistent with the fact that the characteristic time of the convection is given by the Kolmogorov time scale when we approach Kolmogorov wave number kn: The IBS is generated by convection effects coming from small scales, as said before. Moreover, one can observe that in the inertial-diffusive subrange, n ~1 is propor tional to Ac-2, which is what was predicted by Batchelor [1] when he introduced the eddy conductivity a j ■
011001-3
RAPID COMMUNICATIC
PHYSICAL REVIEW E 91, 011001(R) (2015)
A. BRIARD AND T. GOMEZ TABLE 1. Variation in Pr of the compensated scalar spectrum £ r -d-e~16~|/6a 3/2^n/3 fo rpr = i0 “4 10_5,and 10-6 when the k~ ]1/3
number kcDaT = ^ 0Pr1/3c 1/3.
and k~ u/3 subranges are completely formed. Pr 10~1234 10“5
io - 67
VPr 2
Compensated E r
0.005 0.0016 0.0005
0.0053 0.0017 0.00053
Now that nc is available, we can write the scalar spectrum in the inertial-balanced subrange as E T(k,t)
=
j f r
e'/6VPra-V2r u/\
(13)
We note that &-ll/3 is in good agreement with the £—3.55 spectrum found in numerical simulations in Fig. 3. Now, if we consider the compensated scalar spectrum 1e~ 1/6a 3/,2£ 11/,3 = x/Pr in the IBS k e [^co.^cd] spanning on almost two decades in wave numbers from Pr = 10-4 to Pr = 10-6 , we assess that we obtain the a/P t dependance with EDQNM simulations. Table I shows that the compensated scalar spectrum is indeed proportional to VFr, more precisely equal to -v/Pr/2 within 1% error. So in addition to the k ~]l|/3 spectrum, we have assessed the VFr behavior. Moreover, we can also verify the hypothesis aj/a 1 by evaluating the eddy conductivity at wave
[1] [2] [3] [4]
G. K. Batchelor, J. Fluid. Mech. 5, 113 (1959). J. Chasnov et al., Phys. Fluids A 1, 1698 (1989). C. H. Gibson, Phys. Fluids 11, 2316 (1968). V. L. Granatstein and S. J. Buchsbaum, Phys. Rev. Lett. 16, 504 (1966). [5] P. Sagaut and C. Cambon, Homogeneous Turbulence Dynamics (Cambridge University Press, Cambridge, 2008). [6] M. Lesieur, Turbulence in Fluids (Springer, New York, 2008). [7] A. A. Townsend, The Structure o f Turbulence Shear Flow (Cambridge University Press, Cambridge, 1976).
(14)
The simulations show that for Pr = 10~4, when both the k ~ '7/3 and k~n , i ranges are completely formed, one has a j / a ~ 10-2, and this ratio decreases with Pr. In conclusion, we have shown that the theories of both Chasnov and Batchelor can be merged into a single one. Thanks to EDQNM, a large range of Pr and ReAnumbers has been explored and our theory has been assessed. We recall that the two main results are the following ones. First, for a strongly diffusive scalar Pr -C 1, a new k ~u/3 spectrum appears in the range [kernel where kCo = VPr^r, is a characteristic wave number based on diffusion and convection. This subrange that we called the inertial-balanced range, is generated by convection effects that balance diffusion effects of the k ~ xl/3 inertial-diffusive subrange. These convection effects come from small scalar gradients predicted by Gibson [3] and smallscale eddies of order k ~ ]. This range appears with the k-17/3 spectrum when both the Reynolds and the Prandtl numbers are, respectively, high enough (Rex > 2 x 104) and small enough (Pr ^ 1CT3). Finally, we determined explicitly the scalar spectrum in the inertial-balanced subrange using the eddy conductivity E T ~ VPr k_n/3. Both variations in k ~ u/3 and a/P t are verified by EDQNM simulations. Using analytical and numerical approaches, we have brought a general result regarding the homogeneous and isotropic turbulence for the passive scalar. This result is a further step forward in the field of passive scalar dynamics.
[8] G. K. Batchelor, The Theory o f Homogeneous Turbulence (Cambridge University Press, Cambridge, 1959). [9] M. Larcheveque et al., Turbulent Shear Flows II (Springer, New York, 1980). [10] J. R. Herring etal., J. Fluid. Mech. 124, 411 (1982). [11] S. Corrsin, J. Aeronaut. Sci. 18, 417 (1951). [12] S. B. Pope, Turbulent Flows (Cambridge University Press, Cambridge, 2000). [13] J. H. Rust and A. Sesonske, Int. J. Heat Mass Transf. 9, 215 (1966).
011001-4
Copyright of Physical Review E: Statistical, Nonlinear & Soft Matter Physics is the property of American Physical Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.
PHYSICAL REVIEW E 91, 011001(R) (2015)
Passive scalar convective-diffusive subrange for low Prandtl numbers in isotropic turbulence A. Briard* and T. G om ez' Sorbonne Universites, UPMC Univ Paris 06, UMR 7190, Institut Jean Le Rond d ’Alembert, F-75005 Paris, France and CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France (Received 4 September 2014; published 6 January 2015) In this Rapid Communication, we study the behavior of a strongly diffusive passive scalar field T submitted to a freely decaying, homogeneous and isotropic turbulence with eddy-damped quasinormal Markovian simulations. We present a new subrange located between the Ar17/3 inertial-diffusive subrange and the Kolmogorov wave number k„. This subrange is generated by small-scale convection linked to k„ that balances diffusion effects. Thus, we build a typical length scale k^'D based on convection and diffusion and give an expression for the shape of the passive scalar spectrum in this subrange ET ~ v^Pr Arll/3 using physical arguments. This result unifies two different theories coming from Batchelor [G. K. Batchelor, J. Fluid. Mech. 5, 113 (1959)] and Chasnov [J. Chasnov et al., Phys. Fluids A 1, 1698 (1989)] and explains results previously obtained experimentally. DOI: 10.1103/PhysRevE.91.011001
PACS number(s): Al.21.eb, A l.27.Gs
T he dynam ic o f a passive scalar field T w hen the Prandtl num ber Pr is m uch low er than 1 is a very controversial topic. T here are four different theories regarding the shape o f the scalar spectrum E T . B atchelor [1] proposed that E r ~ k ~ xl/3, w hereas Chasnov [2] found E r ~ A:- ” / 3 for a very rapidly stirred fluid. M oreover, G ibson [3] proposed an £ > ~ A r3 evo lution by considering convection effects when scalar gradients are very w eak or zero at sm all scales. Finally, G ranatstein [4] established an E T ~ k ~ 13/3 subrange based on experim ental data in a plasm a and justified it using the H eisenberg model. Thanks to eddy-dam ped quasinorm al M arkovian (ED Q N M ) sim ulations [5,6], a large range o f Prandtl num bers P r = v /a can be explored at high Reynolds num bers, w here v is the kinem atic viscosity and a the scalar diffusivity. ED QN M sim ulations allow one to explain directly how the k ~ l3/3 could have been obtained experim entally before: T his subrange was observed for fluids w ith Pr e [0.1,0.01] and Re* ~ 160 where Rex is the Reynolds num ber based on Taylor scale X. However, there is no inertial-diffusive subrange for Pr = 0.1 at this Reynolds num ber as w e can see in Fig. 3. A nd for P r = 10~2, the inertial-diffusive subrange is not com pletely established: T his is probably the reason w hy A:- 17/3 is not observed in experim ental w orks [4], In addition to this transitional Pr state, Rex is not high enough to m atch w ith the theory (at least Rex = 200 is required). In the K olm ogorov inertial subrange k e [kL ,k n], w here k L and k,t are the wave num bers linked respectively to the integral scale and Kolm ogorov scale, the kinetic spectrum E evolves as E ( k , t ) = K 0 k ~ 5' 3 €2' 3,
(1)
w here K 0 is the Kolm ogorov constant ( ~ 1.5) and e the turbulent energy dissipation rate. In the inertial-convective subrange, the scalar spectrum E r evolves as E r (k ,t) = K Co e Te ~ '/3k ~ 5/3,
(2)
w here K co is the C orrsin-O bukhov constant ( ~ 0.66) and e r the scalar dissipation rate. The num erical values o f both
*[email protected] t thomas.gomez @upmc.fr 1539-3755/2015/91 (1 )/011001(4)
constants K q and K co are recovered in our sim ulations. These laws for E and E j are obtained using dim ensional analysis based on physical argum ents. For Pr =
I ^ - k 2a
2 .
( 12)
( 8)
The integral from 0 to Ac takes into account the main contri bution of dissipation as k » 1. The influence of small-scale dynamics on eT is modeled by the eddy conductivity aT. As a consequence, we can assume as Chasnov [2] did, that d e r / d k = 0. If we derive (8) with respect to k and we consider that E{k -> 00) = 0 we find E T(k,t) = ~ - e Te2/3k 11/3(a + aT) 2nc l .
TCD =
(9)
In Batchelor’s paper [1], a j is introduced differently. The characteristic time n~l appears to be (ak2)~1 which is a diffusion time. This makes sense when diffusion dominates
In Fig. 4 we clearly see that in the IBS, for k > A:Cd , n ~ l becomes constant. In other words, we find the result predicted by Chasnov but in a more general case: Indeed, we did not have to consider the case of a rapidly stirred fluid. The other point of interest is that the constant reached by n ~[ is the Kolmogorov time scale rn. This result is consistent with the fact that the characteristic time of the convection is given by the Kolmogorov time scale when we approach Kolmogorov wave number kn: The IBS is generated by convection effects coming from small scales, as said before. Moreover, one can observe that in the inertial-diffusive subrange, n ~1 is propor tional to Ac-2, which is what was predicted by Batchelor [1] when he introduced the eddy conductivity a j ■
011001-3
RAPID COMMUNICATIC
PHYSICAL REVIEW E 91, 011001(R) (2015)
A. BRIARD AND T. GOMEZ TABLE 1. Variation in Pr of the compensated scalar spectrum £ r -d-e~16~|/6a 3/2^n/3 fo rpr = i0 “4 10_5,and 10-6 when the k~ ]1/3
number kcDaT = ^ 0Pr1/3c 1/3.
and k~ u/3 subranges are completely formed. Pr 10~1234 10“5
io - 67
VPr 2
Compensated E r
0.005 0.0016 0.0005
0.0053 0.0017 0.00053
Now that nc is available, we can write the scalar spectrum in the inertial-balanced subrange as E T(k,t)
=
j f r
e'/6VPra-V2r u/\
(13)
We note that &-ll/3 is in good agreement with the £—3.55 spectrum found in numerical simulations in Fig. 3. Now, if we consider the compensated scalar spectrum 1e~ 1/6a 3/,2£ 11/,3 = x/Pr in the IBS k e [^co.^cd] spanning on almost two decades in wave numbers from Pr = 10-4 to Pr = 10-6 , we assess that we obtain the a/P t dependance with EDQNM simulations. Table I shows that the compensated scalar spectrum is indeed proportional to VFr, more precisely equal to -v/Pr/2 within 1% error. So in addition to the k ~]l|/3 spectrum, we have assessed the VFr behavior. Moreover, we can also verify the hypothesis aj/a 1 by evaluating the eddy conductivity at wave
[1] [2] [3] [4]
G. K. Batchelor, J. Fluid. Mech. 5, 113 (1959). J. Chasnov et al., Phys. Fluids A 1, 1698 (1989). C. H. Gibson, Phys. Fluids 11, 2316 (1968). V. L. Granatstein and S. J. Buchsbaum, Phys. Rev. Lett. 16, 504 (1966). [5] P. Sagaut and C. Cambon, Homogeneous Turbulence Dynamics (Cambridge University Press, Cambridge, 2008). [6] M. Lesieur, Turbulence in Fluids (Springer, New York, 2008). [7] A. A. Townsend, The Structure o f Turbulence Shear Flow (Cambridge University Press, Cambridge, 1976).
(14)
The simulations show that for Pr = 10~4, when both the k ~ '7/3 and k~n , i ranges are completely formed, one has a j / a ~ 10-2, and this ratio decreases with Pr. In conclusion, we have shown that the theories of both Chasnov and Batchelor can be merged into a single one. Thanks to EDQNM, a large range of Pr and ReAnumbers has been explored and our theory has been assessed. We recall that the two main results are the following ones. First, for a strongly diffusive scalar Pr -C 1, a new k ~u/3 spectrum appears in the range [kernel where kCo = VPr^r, is a characteristic wave number based on diffusion and convection. This subrange that we called the inertial-balanced range, is generated by convection effects that balance diffusion effects of the k ~ xl/3 inertial-diffusive subrange. These convection effects come from small scalar gradients predicted by Gibson [3] and smallscale eddies of order k ~ ]. This range appears with the k-17/3 spectrum when both the Reynolds and the Prandtl numbers are, respectively, high enough (Rex > 2 x 104) and small enough (Pr ^ 1CT3). Finally, we determined explicitly the scalar spectrum in the inertial-balanced subrange using the eddy conductivity E T ~ VPr k_n/3. Both variations in k ~ u/3 and a/P t are verified by EDQNM simulations. Using analytical and numerical approaches, we have brought a general result regarding the homogeneous and isotropic turbulence for the passive scalar. This result is a further step forward in the field of passive scalar dynamics.
[8] G. K. Batchelor, The Theory o f Homogeneous Turbulence (Cambridge University Press, Cambridge, 1959). [9] M. Larcheveque et al., Turbulent Shear Flows II (Springer, New York, 1980). [10] J. R. Herring etal., J. Fluid. Mech. 124, 411 (1982). [11] S. Corrsin, J. Aeronaut. Sci. 18, 417 (1951). [12] S. B. Pope, Turbulent Flows (Cambridge University Press, Cambridge, 2000). [13] J. H. Rust and A. Sesonske, Int. J. Heat Mass Transf. 9, 215 (1966).
011001-4
Copyright of Physical Review E: Statistical, Nonlinear & Soft Matter Physics is the property of American Physical Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.
