Exact relation between spatial mean enstrophy and dissipation in confined incompressible flows

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Exact relation between spatial mean enstrophy and dissipation in confined incompressible flows

Florence Raynal

To cite this version:

Florence Raynal. Exact relation between spatial mean enstrophy and dissipation in confined in- compressible flows. Physics of Fluids, American Institute of Physics, 1996, 8 (8), pp.2242-2244.

�10.1063/1.868997�. �hal-01348399�

Exact relation between spatial mean enstrophy and dissipation in confined incompressible flows

Florence Raynal

Laboratoire de Me´canique des Fluides et d’Acoustique, UMR CNRS 5509, Ecole Centrale de Lyon and Universite´ Lyon I, 36 avenue Guy de Collongue, 69131 Ecully Ce´dex, France, and

Laboratoire de Physique, URA CNRS 1325, ENS Lyon, 46 alle´e d’Italie, 69364 Lyon Ce´dex 07, France

~Received 23 January 1996; accepted 15 April 1996!

An exact non-trivial relation between spatial mean enstrophy and dissipation in closed incompressible flows is derived analytically, which also applies to unbounded fluid domains when the velocity decays to zero sufficiently rapidly at infinity. The quantity

^

v²2s²

&

is shown to depend only on the boundary conditions, and it is pointed out that in many confined flows, the equality

^

v²

&

⁵

^

s²

&

is not satisfied. © 1996 American Institute of Physics.

@S1070-6631~96!01308-6#

It is now well known¹that in a constant density, constant viscosity fluid, by taking the divergence of the Navier-Stokes equations, one finds

¹²p5~r^/2!~v²2s²!, ~1!

where v²5(¹`vW)²52vi j

2 is the square of the vorticity modulus, also called enstrophy, ands²52si j

2 is the square of the rate of strain tensor, withvi j andsi j defined as

vi j51

2~]ivj2]jvi!, ~2!

si j51

2~]iv_j1]jv_i!. ~3!

Equation ~1! is analogous to Poisson’s law in electrostatics on the electrical potential, so thatv^2ands² behave, respec- tively, like negative and positive charges. This simple re- mark has had a renewal of interest those last years in the field of turbulence, with the direct and numerical observation of vorticity filaments^2–8which create intermittency. These fila- ments have also been characterized by low-pressure events, both experimentally and numerically.^9–11It has been empha- sized that the fluid is a ‘‘neutral’’ medium, i.e.

^

v²

&

5

^

s²

&

^{, where}

^

^X

&

denotes the volume integral*DXdt^{. We} prove in this paper that the hypothesis of neutrality is not verified for many boundary conditions.

We suppose that the fluid domain D is either infinite, with a velocity field decaying to zero sufficiently rapidly at infinity ~see later!, or finite with solid boundaries, in such a case we denote by V_fthe volume of fluid. The flow might be produced by a solid body Ds moving in the fluid, for in- stance a body in translation, or a rotating disk. We denote by V_sthe solid volume. If the domain is finite, the total volume inside the container is V_c5Vs1Vf.

Since the velocity field is generally not known exactly

~especially in turbulent flows!, we have to calculate the quantity

^

v²2s²

&

indirectly. First of all, we turn the volume integral into a surface integral, keeping in mind that in an incompressible flow, the velocity field satisfies the continuity equation]iv_i50. We get

^

v²2s²

&

⁵²²

E

D]i~vj]jv_i!dt522

E

S~vj]jv_i!nidS,

~4!

where nW is the unit normal to the element of surface, and oriented outward the fluid, as shown in Fig. 1. S5SsøSc

designates both the surface of the solid,Ss, and that of the container, Sc, which is eventually at infinity. In this latter case, we suppose that vW decays to zero like R^2a when R→`. A straightforward calculation shows that the surface integral tends to zero at infinity ifa.1/2, which seems to be generally the case in flows produced by a finite moving body. Note again that this result would not hold in open flows like for example pipe flows, since the stress boundary conditions up- and downstream ~corresponding to inlet and exit! are not known a priori.

Due to viscosity, the velocity in the fluid at the boundary is equal to the velocity of the solid. However, a problem arises concerning the continuity of the derivatives of the ve- locity, which is not satisfied a priori. We can nevertheless prove the continuity of (v_j]jv_i)n_i across the solid boundary S.

First of all, we note that this quantity may be written:

~vj]jvi!ni5vj]j~vin_i!2vivj]jn_i. ~5!

The first term on the right hand-side of the above equation involves the derivatives of the normal velocity, while the second term is related to the curvature of the boundary. The continuity of the velocity field is written as

vW_f~M,t!5vW_{b~M,t!, ~6!}

where M is any point on the solid surface S, and the sub- scripts f or b are used, respectively, to denote a quantity calculated in the fluid domain or in the boundary. Equation

~6! immediately implies that at the boundary

~vivj]jn_i!f5~vivj]jn_i!b. ~7!

In order to evaluate the first term in the right hand-side of Equation~5!, we consider a point M on the boundary and we denote by nW_[eW_z the unit normal at point M . The local el- ementary surface satisfies an equation z5j(x,y ), with j^(0,0)50 at point M. Equation ~6! locally is written as:

2242 Phys. Fluids 8 (8), August 1996 1070-6631/96/8(8)/2242/3/$10.00 © 1996 American Institute of Physics

vW_f~x,y,j~x,y!,t!5vW_b~x,y,j~x,y!,t!. ~8!

However, due to the smoothness of the boundary, the partial derivatives of j ^verify]xju(0,0)5]yju(0,0)50 ~the surface is locally tangent to a plane!, and a Taylor development gives at first order

]xvW_f5]xvW_b, ~9!

]yvW_f5]yvW_b, ~10!

at point M . Now, because of the continuity equation ]ivi50, we can write that

~vj]jvz!f5~vx]xvz1vy]yvz2vz~]xvx1]yvy!!f. ~11!

Then, with Equations~8!, ~9!, and ~10!

~vj]jvz!f5~vx]xvz1vy]yvz2vz~]xvx1]yvy!!b. ~12!

Note thatvin_i5vzis not necessarily null in the general point of view chosen. Now, we use the fact that the incompress- ibility hypothesis]ivi50 is also valid in the solid, and

~vj]jvz!f5~vj]jvz!b, ~13!

which holds for any point M on the boundary. Together with Equations~5! and ~7!, we obtain

E

S~vj]jvi!fn_idS5

E

S~vj]jvi!bn_idS. ~14!

In the solid boundary,si j50, and ¹ ` vW_52VW, where VW_5VW(t) is the instantaneous rotation rate vector of the solid. We denote VW_5VW_c for the container and VW_5VW_s for the solid body. We drop the subscript b in those calculations.

Thus we have

]jv_i5vj i51

2«j ik~¹`vW_{!k5«j ikVk}, ~15!

and we finally obtain vj]jvi5(VW _{` v}W)_i. We can now turn back the surface integral into a volume integral:

E

S~vj]jvi!bn_idS 5

E

Sc

~VW_c`vW_c!•nWdS1

E

Ss

~VW_s`vW_s!•nWdS, ~16!

5

E

DøDs

¹•~VW_c`vW_c!dt2

E

D_s¹•~VW_s`vW_s!dt^, ~17!

where the sign 2 is used because nW is oriented inward the solid on Ss ~see Fig. 1!. Noting that in a solid

¹•~VW_{`v</sub>W<sub>!5v</sub>W<sub>•~¹`V}W_!2VW_•~¹`vW_!522V², ~18!

we get

E

S~vj]jvi!bn_idS52~VsVs 22VcVc

2!, ~19!

and finally, from Equations~4!, ~14!, and ~19!:

^

v²2s²

&

54~VcVc 22VsVs

2!. ~20!

The case of an unbounded domain with velocity decaying to zero more rapidly than R^21/2 at infinity is recovered by let- ting VW_c50W. Equality ~20! proves that

^

v²

&

Þ

^

s²

&

^{in con-} fined flows produced by rotating solids. Moreover, the quan- tity

^

v²2s²

&

does not depend on the dynamics of the flow

~laminar or turbulent!, but is fixed by the boundary condi- tions. If VW_c5VW_c(t) orVW_s5VW_s(t), thus

^

v²2s²

&

^{also de-} pends on time. We can introduce the local dissipation rate per mass unit,e5ns^{2, where}n is the kinematic viscosity of the fluid. Thus equation~20! implies

^

v²

&

²

K

^en

L

^{54~VcVc22VsVs2!. ~21!}

We can discuss some particular cases

~1! VW_s50W andVW_c50W: When the flow is produced by a body having a movement of translation @VW_s50W and vW( M ,t)5vW(t);M P Ds#, thus neutrality is rigorously veri- fied, i.e.

^

v²

&

5

^

s²

&

. This is the case for example in the experiment by Villermaux et al.,⁸where the flow is produced by an oscillating grid.

~2! VW_sÞ0W and VW_c50W: when a solid is rotated in the fluid, then

^

s²

&

^.

^

v²

&

, which corresponds in the analogy to a ‘‘positive total charge.’’ This is the case for the example for the flow driven by two counter-rotating cylinders.^2,9,11 Note however that for a given

^

^«

&

^{, although}

^

vW

&

^{50W @see} Equation~28! below#, this flow produces less

^

v²

&

^{than that} studied by Villermaux et al.⁸ When the rotation rate of the disks is constant, Equation ~21! may be rewritten:

^

v²

&

5

K

^en

L

^{1Cst. ~22!}

However, it was shown experimentally for this very flow in air that

^

e

&

is a highly time-fluctuating quantity¹²; Hence, because of Equation ~22!,

^

v²

&

fluctuates correspondingly.

These fluctuations of the dissipation could thus be related to the appearance of the vorticity filaments.

FIG. 1. Orientation of the unit normal nW.

2243

Phys. Fluids, Vol. 8, No. 8, August 1996 Brief Communications

~3! VW_s50W andVW_cÞ0W: in that case, the flow is submitted to a solid body rotation, which creates an additional

^

v²

&

compared to

^

s²

&

; in terms of the electrostatic analogy, the domain is ‘‘charged negatively.’’

We are now going to prove in the general caseVW_sÞ0W and VW_cÞ0W that, in the frame R

8

of the container, rotating with angular velocity VW_c, *D_c(v

8

²22s

8

^2)dt50. In this frame,s²5s

8

² is unchanged, andvW5vW

8

^12VWc. Thus

E

D_cv^2dt5

E

D_cv

8

^2dt14VW_c•

E

D_cvW

8

^dt14VcVc 2. ~23!

First of all, we calculate*D_cvW

8

^dt. Noting that, for any fixed vector eW:

eW_•

E

DvW^dt5

E

D¹•~vW_`eW_!dt^, ~24!

5

E

S~vW_`eW_!fnWdS5

E

S~vW_`eW_!bnWdS, ~25!

5eW_•

E

D_cvWcdt2eW_•

E

D_svWsdt^, ~26!

5eW_•~2VcVW_c22VsVW_{s!, ~27!}

we get

^

vW

&

^52VcVW_c22VsVW_s. ~28!

In the solid body, we havesi j50 andvW52VW_s. Hence,

E

D_cvW

8

^dt5

^

vW

8&

¹

E

D_svW

8

^dt^, ~29!

5

^

vW

&

22VfVW_c12Vs~VW_s2VW_c!50W, ~30!

because of Equation~28!. Thus,

E

D_c~v

8

²22s

8

²!dt5

E

D_c~v²2s²!dt24VcVc 250.

~31!

Therefore neutrality is verified in the frame R

8

for the do- main $solid1fluid%: this striking result is simply due to the

continuity ofv²2s²and of its derivatives at the boundaries

@Equation ~13!#, so that the Green-Ostrogradski theorem can be applied directly in Dcfor this quantity!

In conclusion, in this paper, we have related the mean enstrophy and dissipation by the formula

K

^v22«n

L

^{54~VcVc22VsVs2!, ~32!}

where V_c and V_s are, respectively, the volume of the con- tainer and the volume of the solid, and VW_c and VW_s their instantaneous rotation rates; the case of unbounded flows with velocity decaying to zero more rapidly than R^21/2 is recovered by lettingVW_c50W. We assume that this exact result will be of interest for further experiments.

ACKNOWLEDGMENTS

I wish to thank Thierry Dauxois for helpful comments. I especially thank Ste´phan Fauve for many interesting discus- sions.

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2244 Phys. Fluids, Vol. 8, No. 8, August 1996 Brief Communications