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Order parameters of turbulent flows
Volume 113A, number 5
PHYSICS LETTERS
23 December 1985
O R D E R P A R A M E T E R S OF T U R B U L E N T F L O W S
G.A. K U Z ' M I N Institute of Thermophysics. 630090 Novosibirsk, USSR
and A.Z. P A T A S H I N S K Y Institute of Nuclear Physics, 630090 Novosibirsk, USSR
Received 25 July 1985; accepted in revised form 23 October 1985
Turbulent flows are described in terms of moments of the vortex momentum density. These moments describe coherent structures. Invariants of the moments give the quantitative description of the organized motion.
Both chaos and order are revealed in turbulent flows, the latter being manifested by the existence of coherent structures. An experimental investigation of these structures is difficult due to the fact that their forms, orientations and phases fluctuate [ 1 - 3 ] . Recently it has become possible to measure the velocity field simultaneously in a large number of positions [ 4 - 6 ] . There is considerable progress in modelling of turbulence on computers. It is clear that the knowledge o f the simultaneous hydrodynamic fields and of their evolution contains detailed information about the coherent structures. However the question arises, how can this information be extracted? We present a technique o f processing experimental or computer modelling data. This technique is designed to detect and analyse three-dimensional coherent motions occurring repeatedly in fluids, and implies the calculation o f tensor moments of the flow field. A similar idea for the description of condensed matter structures was used in ref. [7]. By coherent structures one usually means the clouds of coherent vorticity. To extract the organized structures of scale k one needs to decompose the vorticity field. Let us choose a spherical volume of radius k in a turbulent fluid. We decompose the vorticity to into two solenoidal components to' and to(x). The 266
field to' is assumed to coincide with to outside the chosen volume and withVx inside the volume, X being the harmonic function determined by the Neumann boundary condition (r.VX)b = (to.r)b. Here r denotes the position o f the point relative to the centre o f the sphere, and the suffLx b indicates evaluation at the boundary. The dependence on the position x of the centre is implicitly implied. The field to' does not contain any information about the field to inside the volume. This information is contained in the supplementary component to(x) = to _ to'. The field to(x) forms a vortex of finite size k because its vortex lines do not cross the boundary of the region. The extracted vortex is entirely described by the set of moments
M(n) il ...i n (x)
["
= 3 ri,"i2...ri,,_ l
+ 0 d V(,') .
The most essential features of the vortex are given by the low-rank moments. Let us consider the physical meaning o f M (n), n ~< 4. It is convenient to decompose these tensors into irreducible components [8]. The fully symmetrical component of the tensor M (n) equals the surface integral and is zero. The other irreducible tensors expressed in terms of the vorticity to are 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 113A, number 5
P={frX.
PHYSICS LETTERS (1)
o~dV,
y~ = { f (Srer/ - r26i/)¢o/dV,
f [r,(r x to)/+ r,{r
t,/=
_4
di/ - ~
(2)
dV,
(3)
f [5rir/rm - r2(riajm + r/aim + rmai/)]
X o3m d V ,
a=Afr2rX.dV,
(4)
(5)
C,/m = { f triri(r X to)m + rirm(r X to )/
+ r/rm(r X to)i ] dV -
-
](at~im + a/aim + amai/) .
(6)
Here integrals are taken over the chosen spherical volume. The Green function of the Neumann problem was used to find these expressions. The background vorticity to' =VX and simple convection performed by the large scale motions give no contribution to the tensors (1)-(6). The physical meaning of the irreducible tensors becomes clear when they are represented in terms of the vortex momentum density q(r) [9,10]. In our case the field q(r) is defined to be equal to the density of the force impulse required for the instantaneous generation of the vortex to(x) in the presence of the background to'. The integration of the vorticity equation
ato/at
=
V x (u x to) + v/xto + a(t)v X q
over the inf'mitesimal time interval ( - e , e) gives the relation ~(x) = V X q. We choose the gauge condition for the field q so that the extracted vortex cloud is described by the finitely distributed field q(r). After substitution of ta = VX + V X q in (1)-(6) and integration by parts one obtains the moments of the vortex momentum density. The vectors P, J are the known vortex momentum and angular vortex momentum [ 11]. The tensors ti/and ci/m describe quadrupole and octupole deformation of the field volume by the vorticity ~(x). The tensor di/gives the smoothed deformations which twist the structure along the principal axes of
23 December 1985
this tensor. The vector a describes the recirculation inside the chosen volume, associated with to (x). The choice of the most convenient representation of the moments depends on the special features of the problem. The moments can be expressed in terms of velocity u taking into account the relation to = V X u and performing the integration by parts. As a result, surface integrals appear which give rise to some difficulties. For example, let the velocity field be measured in a finite number of positions. It may happen, due to statistics, that the number of points found on the boundary is too small to calculate the surface integrals with sufficient accuracy. In order to avoid this difficulty one can renounce from integration over the region with the sharp boundary. Let us introduce the weight function 6o(X)(r) and extend the integration over the whole space. The function 6o(x) ~ 1 when r < ?~ and decreases strongly when r > X. The integrals con. sidered above correspond to the stepwise function ~(X)(r) = 0Q, - r). The extracted component of motion with the smoothed cut-off ~o(X)(r) has the scale X and is described by the tensor fields (1)-(6), provided the substitution d V-+ co(x) (r) d V is performed in their integrands. The tensor fields constructed in this paper are nonzero only in the vicinity of structures of scale X. This property is of advantage in the presence of noticeable intermittency. The combinations (1)-(6) can be used for indication of organized structures as well as for their quantitative description. The quantitative characteristics of the coherent motion which are independent of its orientation are the invariants if(u), ~t = 1, 2, ..., of these tensors. Let us assume that the tensor fields of rank n ~< n o and their invariants have been calculated from experimental data or from computer modelling. The following questions arise, (i) what information about coherent structures can be obtained, and (ii) how can it be obtained? Once the distribution of the invariants is known, one can determine the regions occupied by de£mite structures. The recognition of the structures by means of their invariants ~b(U)(x) can be done by comparing these fields with model ones, describing the specially created vortex distributions. Hence, the possibility of certain "spectroscopy" of structures by their invariants ~(U)(x) arises. The comparison of structures by their moments 267
Volume 113A, number 5
PHYSICS LETTERS
of rank n < n o is a comparison of classes to which the studied structures belong. As can be expected, the invariants qJ(v) for small numbers n ~< 4 describing the largest-scale deformations of structures are not very sensitive to small scale fluctuations inside the structures. According to the formulae given above the velocity field u(x) produces the tensor fields Mtn) which single out the motion o f the given scale ~.. In turn, the tensor fields M (n) produce the fields of the invariants $(v)(x). To reveal the coherent structures it seems promising to analyse statistically the fields $(~) rather than the velocity field. If the hypothesis that the turbulent flow is a chaos of the coherent structures is true, these structures can be revealed much easier by the fields ~k(v) than by the velocity field. We explain this by the following simple example. Let the pair correlation function, i.e. the modulus o f the Fourier harmonics Iug I, be given. The harmonics Ug = lug I exp(igk) with the given modulus, the phases 9k being arbitrary, give an extremely wide spectrum of configurations in the usual space. In particular, if the coherent structures exist for some given set of phases ¢g, then they are completely destroyed by the moderate variations of phases. The configurations behave quite differently if the invariants ~k(~) are fixed. The coherent motions o f
268
23 December 1985
the scale ~, which are described by these invariants are the same up to more subtle details contained in the tensors of higher orders. Hence, one can see that the description of turbulence in terms o f invariants facilitates the problem of the pattern recognition and it is adequate to representation of the turbulence as a stochastic system o f the coherent structures.
References [ 1] A.A. Townsend, The structure of turbulent shear flow (Cambridge Univ. Press, Cambridge, 1976). [2] B.J. CantwelL Ann. Rev. Fluid Mech. 13 (1981) 457. [3] A.K.M.F. Hussain, Phys. Fluids 26 (1983) 2816. [4] R.R. Johnson et al., Phys. Fluids 19 (1976) 1422. [5] V.V. Orlov, E.S. Mikhailov and E.M. Khabahpasheva, Metrology 3 (1970) 67. [6] B.G. Novikov and V.D. Fedosenko, in: Hydrodynamics and acoustics of near-wall and free flows (Institute of Thermophysics, Novosibirsk, 1981). [7] A.Z. Patashinsky, preprint 84-64, Institute of Nuclear Physics, Novosibirsk (1984). [8] I.M. Gerfand, R.A. Minlos and Z.Ya. Shapiro, Representations of the rotation group and of the Lorentz group (Fizmatgiz, Moscow, 1958). [9] G.A. Kuz'min, Phys. Lett. 96A (1983) 88. [10] G.A. Kuz'min, Zh. PrikL Mech. Tech. Fiz. 4 (1984) 25. [ 11 ] G.K. Batchelor, An introduction to fluid dynamics (Cambridge Univ. Press, Cambridge, 1967).
PHYSICS LETTERS
23 December 1985
O R D E R P A R A M E T E R S OF T U R B U L E N T F L O W S
G.A. K U Z ' M I N Institute of Thermophysics. 630090 Novosibirsk, USSR
and A.Z. P A T A S H I N S K Y Institute of Nuclear Physics, 630090 Novosibirsk, USSR
Received 25 July 1985; accepted in revised form 23 October 1985
Turbulent flows are described in terms of moments of the vortex momentum density. These moments describe coherent structures. Invariants of the moments give the quantitative description of the organized motion.
Both chaos and order are revealed in turbulent flows, the latter being manifested by the existence of coherent structures. An experimental investigation of these structures is difficult due to the fact that their forms, orientations and phases fluctuate [ 1 - 3 ] . Recently it has become possible to measure the velocity field simultaneously in a large number of positions [ 4 - 6 ] . There is considerable progress in modelling of turbulence on computers. It is clear that the knowledge o f the simultaneous hydrodynamic fields and of their evolution contains detailed information about the coherent structures. However the question arises, how can this information be extracted? We present a technique o f processing experimental or computer modelling data. This technique is designed to detect and analyse three-dimensional coherent motions occurring repeatedly in fluids, and implies the calculation o f tensor moments of the flow field. A similar idea for the description of condensed matter structures was used in ref. [7]. By coherent structures one usually means the clouds of coherent vorticity. To extract the organized structures of scale k one needs to decompose the vorticity field. Let us choose a spherical volume of radius k in a turbulent fluid. We decompose the vorticity to into two solenoidal components to' and to(x). The 266
field to' is assumed to coincide with to outside the chosen volume and withVx inside the volume, X being the harmonic function determined by the Neumann boundary condition (r.VX)b = (to.r)b. Here r denotes the position o f the point relative to the centre o f the sphere, and the suffLx b indicates evaluation at the boundary. The dependence on the position x of the centre is implicitly implied. The field to' does not contain any information about the field to inside the volume. This information is contained in the supplementary component to(x) = to _ to'. The field to(x) forms a vortex of finite size k because its vortex lines do not cross the boundary of the region. The extracted vortex is entirely described by the set of moments
M(n) il ...i n (x)
["
= 3 ri,"i2...ri,,_ l
+ 0 d V(,') .
The most essential features of the vortex are given by the low-rank moments. Let us consider the physical meaning o f M (n), n ~< 4. It is convenient to decompose these tensors into irreducible components [8]. The fully symmetrical component of the tensor M (n) equals the surface integral and is zero. The other irreducible tensors expressed in terms of the vorticity to are 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 113A, number 5
P={frX.
PHYSICS LETTERS (1)
o~dV,
y~ = { f (Srer/ - r26i/)¢o/dV,
f [r,(r x to)/+ r,{r
t,/=
_4
di/ - ~
(2)
dV,
(3)
f [5rir/rm - r2(riajm + r/aim + rmai/)]
X o3m d V ,
a=Afr2rX.dV,
(4)
(5)
C,/m = { f triri(r X to)m + rirm(r X to )/
+ r/rm(r X to)i ] dV -
-
](at~im + a/aim + amai/) .
(6)
Here integrals are taken over the chosen spherical volume. The Green function of the Neumann problem was used to find these expressions. The background vorticity to' =VX and simple convection performed by the large scale motions give no contribution to the tensors (1)-(6). The physical meaning of the irreducible tensors becomes clear when they are represented in terms of the vortex momentum density q(r) [9,10]. In our case the field q(r) is defined to be equal to the density of the force impulse required for the instantaneous generation of the vortex to(x) in the presence of the background to'. The integration of the vorticity equation
ato/at
=
V x (u x to) + v/xto + a(t)v X q
over the inf'mitesimal time interval ( - e , e) gives the relation ~(x) = V X q. We choose the gauge condition for the field q so that the extracted vortex cloud is described by the finitely distributed field q(r). After substitution of ta = VX + V X q in (1)-(6) and integration by parts one obtains the moments of the vortex momentum density. The vectors P, J are the known vortex momentum and angular vortex momentum [ 11]. The tensors ti/and ci/m describe quadrupole and octupole deformation of the field volume by the vorticity ~(x). The tensor di/gives the smoothed deformations which twist the structure along the principal axes of
23 December 1985
this tensor. The vector a describes the recirculation inside the chosen volume, associated with to (x). The choice of the most convenient representation of the moments depends on the special features of the problem. The moments can be expressed in terms of velocity u taking into account the relation to = V X u and performing the integration by parts. As a result, surface integrals appear which give rise to some difficulties. For example, let the velocity field be measured in a finite number of positions. It may happen, due to statistics, that the number of points found on the boundary is too small to calculate the surface integrals with sufficient accuracy. In order to avoid this difficulty one can renounce from integration over the region with the sharp boundary. Let us introduce the weight function 6o(X)(r) and extend the integration over the whole space. The function 6o(x) ~ 1 when r < ?~ and decreases strongly when r > X. The integrals con. sidered above correspond to the stepwise function ~(X)(r) = 0Q, - r). The extracted component of motion with the smoothed cut-off ~o(X)(r) has the scale X and is described by the tensor fields (1)-(6), provided the substitution d V-+ co(x) (r) d V is performed in their integrands. The tensor fields constructed in this paper are nonzero only in the vicinity of structures of scale X. This property is of advantage in the presence of noticeable intermittency. The combinations (1)-(6) can be used for indication of organized structures as well as for their quantitative description. The quantitative characteristics of the coherent motion which are independent of its orientation are the invariants if(u), ~t = 1, 2, ..., of these tensors. Let us assume that the tensor fields of rank n ~< n o and their invariants have been calculated from experimental data or from computer modelling. The following questions arise, (i) what information about coherent structures can be obtained, and (ii) how can it be obtained? Once the distribution of the invariants is known, one can determine the regions occupied by de£mite structures. The recognition of the structures by means of their invariants ~b(U)(x) can be done by comparing these fields with model ones, describing the specially created vortex distributions. Hence, the possibility of certain "spectroscopy" of structures by their invariants ~(U)(x) arises. The comparison of structures by their moments 267
Volume 113A, number 5
PHYSICS LETTERS
of rank n < n o is a comparison of classes to which the studied structures belong. As can be expected, the invariants qJ(v) for small numbers n ~< 4 describing the largest-scale deformations of structures are not very sensitive to small scale fluctuations inside the structures. According to the formulae given above the velocity field u(x) produces the tensor fields Mtn) which single out the motion o f the given scale ~.. In turn, the tensor fields M (n) produce the fields of the invariants $(v)(x). To reveal the coherent structures it seems promising to analyse statistically the fields $(~) rather than the velocity field. If the hypothesis that the turbulent flow is a chaos of the coherent structures is true, these structures can be revealed much easier by the fields ~k(v) than by the velocity field. We explain this by the following simple example. Let the pair correlation function, i.e. the modulus o f the Fourier harmonics Iug I, be given. The harmonics Ug = lug I exp(igk) with the given modulus, the phases 9k being arbitrary, give an extremely wide spectrum of configurations in the usual space. In particular, if the coherent structures exist for some given set of phases ¢g, then they are completely destroyed by the moderate variations of phases. The configurations behave quite differently if the invariants ~k(~) are fixed. The coherent motions o f
268
23 December 1985
the scale ~, which are described by these invariants are the same up to more subtle details contained in the tensors of higher orders. Hence, one can see that the description of turbulence in terms o f invariants facilitates the problem of the pattern recognition and it is adequate to representation of the turbulence as a stochastic system o f the coherent structures.
References [ 1] A.A. Townsend, The structure of turbulent shear flow (Cambridge Univ. Press, Cambridge, 1976). [2] B.J. CantwelL Ann. Rev. Fluid Mech. 13 (1981) 457. [3] A.K.M.F. Hussain, Phys. Fluids 26 (1983) 2816. [4] R.R. Johnson et al., Phys. Fluids 19 (1976) 1422. [5] V.V. Orlov, E.S. Mikhailov and E.M. Khabahpasheva, Metrology 3 (1970) 67. [6] B.G. Novikov and V.D. Fedosenko, in: Hydrodynamics and acoustics of near-wall and free flows (Institute of Thermophysics, Novosibirsk, 1981). [7] A.Z. Patashinsky, preprint 84-64, Institute of Nuclear Physics, Novosibirsk (1984). [8] I.M. Gerfand, R.A. Minlos and Z.Ya. Shapiro, Representations of the rotation group and of the Lorentz group (Fizmatgiz, Moscow, 1958). [9] G.A. Kuz'min, Phys. Lett. 96A (1983) 88. [10] G.A. Kuz'min, Zh. PrikL Mech. Tech. Fiz. 4 (1984) 25. [ 11 ] G.K. Batchelor, An introduction to fluid dynamics (Cambridge Univ. Press, Cambridge, 1967).