Takahiro Nishiyama- Construction of three-dimensional stationary Euler flows from pseudo-advected vorticity equations

8/3/2019 Takahiro Nishiyama- Construction of three-dimensional stationary Euler flows from pseudo-advected vorticity equations

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doi: 10.1098/rspa.2003.1132, 2393-23984592003Proc. R. Soc. Lond. A

Takahiro Nishiyamaequations

advected vorticityEuler flows from pseudodimensional stationary

Construction of three

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10.1098/rspa.2003.1132

Construction of three-dimensional stationary

Euler ows from pseudo-advected

vorticity equations

B y T a k a h i r o N i s h i y a m a

Department of Mechanical Engineering, Fukuoka Institute of Technology,Fukuoka 811-0295, Japan ([email protected])

Received 8 July 2002; accepted 3 February 2003; published online 14 July 2003

Some vorticity equations with pseudo-advection terms are proposed. They yieldthree-dimensional stationary Euler ows at t = 1. In particular, one of them gener-ates the Beltrami ow.

Keywords: stationary Euler ows; vorticity equation; Beltrami ow

1. Introduction

The three-dimensional stationary Euler equations

u ru=

rp; r u= 0 (1.1)

give the velocity u(x) and the pressure p(x) of a steady-state inviscid incompressibleuid with unit density at x 2 ( R3).

To obtain solutions to (1.1) with non-vanishing vorticity, Vallis et al. (1989) pro-posed the following two non-stationary systems for v(x; t) and q(x; t):

vt + v rv + ! vt = rq; r v = 0; (1.2)

vt + v rv + ! r r (v !) = rq; r v = 0: (1.3)

Here is a non-zero constant and ! = r v. In the two-dimensional case, thatis, for v = (v1(x ;y;t); v2(x ;y;t); 0) and ! = (0; 0; !(x ;y;t)) at (x; y) 2 ~ R2, both(1.2) and (1.3) yield

d

dt

Z~

j!j2 d2x = 0

when v nj@~ = 0 (and !j@~ = 0 for (1.3)). Here @~ is the boundary of ~ and n is

the unit outward normal vector. Furthermore, they lead to

1

2

d

dt

Z~

jvj2 d2x

=

Z~

v (! vt) d2x

=

Z~

(v rv) vtd2x

=

Z~

jvtj2 d2x;

1

2

d

dt

Z~

jvj2 d2x =

Z~

jr (v !)j2 d2x; (1.4)

Proc. R. Soc. Lond. A (2003) 45 9, 2393{2398

2393

c 2003 The Royal Society

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2394 T. Nishiyama

respectively. These imply that vtjt=1 = 0 and r (v !)jt=1 = 0 not onlywhen > 0 but also when < 0, if ~ is bounded. Therefore, they concluded thatu = vjt=1 is a two-dimensional solution to (1.1) with p = qjt=1 . Moatt (1990)pointed out that their conclusion is also valid in the axisymmetric case. From arigorous point of view, the existence of a solution to (1.2) or (1.3) is dicult to prove

globally in time even in the two-dimensional or the axisymmetric case because thenonlinearity of the terms is too strong. Nevertheless, the idea of Vallis et al. (1989)can be justied in some rigorous way (Nishiyama 2001a; b). Here, and from now on,the word `rigorous means `rigorous in the sense of analysis with function spaces,such as the Sobolev spaces. Nishiyama (2003a) rigorously discussed

vt + ! ~P(v !) = rq; r v = 0

with vnj@~ = 0, where~P is an operator such that ~Pf = f+ rQ with Q satisfying

Q = r f and (f+ rQ) nj@~ = 0. This system has the same property as (1.2)

and (1.3).As Moatt (1990) mentioned, the method of Vallis et al. (1989) is not applicable in

the three-dimensional case becauseR

j!j2 d3x is not conserved. On the other hand,the procedure of Moatt (1985) is eective. He suggested using the relaxation of aviscous and perfectly conductive magneto-uid with the velocity v(x; t), magneticeld B(x; t), pressure q(x; t), unit density, and viscosity > 0 in a bounded ,where vj@ = 0 and B nj@ = 0. They are governed by

vt + v rv = rq + B rB rjBj2=2 + v;

Bt = r (v B);r v = r B = 0;

9>=>; (1.5)for which we have

1

2

d

dt

Z

(jvj2 + jBj2) d3x =

Z

j!j2 d3x;

d

dt

Z

A B d3x = 0: (1.6)

Here A is a vector potential for B. He concluded that the system (1.5) tends to anequilibrium

B rB = r(q + jBj2=2); r B = 0; v = Bt = 0;

as t ! 1, that is, u = Bjt=1 is a non-trivial solution to (1.1) with

p = (q + jBj2=2)jt=1 :

In a rigorous sense, his theory contains two diculties. One is to prove the temporally

global existence of a solution to (1.5). The other is to obtain the decay vt ! 0 (ast ! 1), whereas v ! 0 is easy. In general, the decay of a function does not meanthe decay of its derivative. For example, limt!1 t

1 sin t2 = 0, while

limt!1

d

dt(t1 sin t2) 6= 0:

Proc. R. Soc. Lond. A (2003)



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Stationary Euler ows 2395

Although Moatt (1990) gave another magnetohydrodynamic system as a substitutefor (1.5), it contains the same two diculties. Nevertheless, Nishiyama (2002, 2003b)rigorously justied his idea by modifying (1.5) as

vt + v rv + ! vt = rq + B rB rjBj2=2 + v;

Bt = r ((v + vt) B);r v = r B = 0;

9>=>; (1.7)

and using some technique to evade discussing the temporally global solvabilityof (1.7). The constant > 0 enables us to obtain the decay of vt because we have

1

2

d

dt

Z

(jvj2 + j!j2 + jBj2) d3x =

Z

(jvtj2 + j!j2) d3x:

The aim of this paper is to propose new equations to obtain three-dimensional solu-tions to (1.1) with non-vanishing vorticity. They are given under periodic boundary

conditions in x 2. In particular, one of them generates the Beltrami ow. This isdiscussed in x 3. Non-periodic cases are also discussed there.

2. Pseudo-advected vorticity equations

Let us consider!t = r (V !) (2.1)

with V satisfying one of

V = V(1) = P;p er((r !) !);

V = V(2) = r r ((r !) !);

V = V(3) = (r !) !

and periodic boundary conditions. Here P;p er is an operator such that P;p erf =f + rQ with Q satisfying Q = r f and the periodic boundary conditions.Since (2.1) with r V = ! is the usual vorticity equation, we call (2.1) withV = V(1);V(2);V(3) `pseudo-advected vorticity equations. Taking into account thatthey are analogous to the second equation in (1.5) or (1.7) as ! $ B, we canalso call them `pseudo-advected magnetic eld equations. It should be noted that

r V(3) = 0 is not always valid, while r V(1) = r V(2) = 0.Let be a cube with periodic boundary conditions applied, using the same peri-

odicity length in each direction. Then, applying integration by parts to (2.1) andnoting thatZ

f P; p erfd3x =

Z

(P;p erf rQ) P;p erfd3x =

Z

jP;p erfj2 d3x;

we deduce that

1

2

d

dt

Z

j!j2 d3x =

8>>>>>>>>>>>:

Z

jP;p er((r !) !)j2 d3x when V = V(1);

Z

jr ((r !) !)j2 d3x when V = V(2);

Z

j(r !) !j2 d3x when V = V(3):

(2.2)




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2396 T. Nishiyama

Therefore,R

j!j2 d3x is monotonically decreasing in t. Note that the second equalityof (2.2) is similar to (1.4).

Furthermore, using a function v which satises r v = !, r v = 0,Rvd3x = 0

and the periodic boundary conditions, we have the conservation of helicity:

d

dtZ v ! d

3

x = 0: (2.3)

The helicity is a quantity measuring the degree of knottedness of !-lines, that is,the ux lines of!. By the analogy ! $ B, it corresponds to the magnetic helicityRA B d3x in (1.6). See Berger & Field (1984) and Moatt (1969), in which they

investigated the relation between the uid or magnetic helicity and the topologicalstructure of!- or B-lines. Representing v and ! in the form of the Fourier series,we can prove

Z jvj

2

d

3

x6

CZ(j!j

2

+ (r v)

2

) d

3

x = CZ j!j

2

d

3

x; (2.4)

where C is a positive constant depending only on the lengths of edges of. Therefore,(2.3) leads to

Z

v0 !0 d3x

=

Z

v ! d3x

6

Z

jvj2 d3x

Z

j!j2 d3x

1=26 C1=2

Z

j!j2 d3x;

where v0 = vjt=0 and !0 = !jt= 0. If we give v0 and !0 so thatR v0 !0 d

3x 6= 0,then

R

j!j2 d3x does not go to zero as t ! 1. (According to Freedman (1988), evenifRv0 !0 d

3x = 0, there exists a case in whichR

j!j2 d3x is bounded from belowby a positive constant.)

Since (2.2) yields

1

2Z

j!0j2 d3x >

8>>>>>>>>>>>:

Z1

0

dt

Z

jP;p er((r !) !)j2 d3x when V = V(1);

Z1

0

dt Z

jr ((r !) !)j2 d3x when V = V(2);Z1

0

dt

Z

j(r !) !j2 d3x when V = V(3);

we obtainP;p er((r !) !) = 0;

r ((r !) !) = 0;

(r !) ! = 0;

9>=>; (2.5)

respectively, at t = 1. Clearly, each of them is rewritten in the form

! r! = rp

with some p(x).These facts imply that u = !jt=1 is a solution to (1.1) with p(x) and is non-

trivial ifRv0 !0 d

3x 6= 0. We remark that the pseudo-advected vorticity changes




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Stationary Euler ows 2397

into the stationary velocity. By the analogy ! $ B, we can also regard !jt=1 asa magnetostatic equilibrium. Furthermore, using a term of Moatt (1985), we cansay that it is `topologically accessible from the initial state !0 because of (2.3).The concept of topological accessibility was introduced by him to represent a kindof preservation of ux-line congurations. Note that r u = r !jt=1 is non-

vanishing if!jt=1 is. Indeed, in the same way as (2.4), we can prove thatZ

j!j2 d3x 6 C

Z

jr !j2 d3x:

3. Discussion

In x 2, we proposed using the relaxation of the pseudo-advected vorticity equa-tions (2.1) with V = V(1);V(2);V(3) to obtain three-dimensional solutions to (1.1)with non-vanishing vorticity under periodic boundary conditions. These equations

are based on the ideas of Moatt (1985) and Vallis et al. (1989) in the sense thatthe second equality of (2.2) is similar to (1.4), and (2.3) has the same form as (1.6).It is an open problem how dierent the solutions constructed with V(1);V(2) andV(3) are, from the same initial data.

From a rigorous point of view, our theory is justied by using spaces of the Radonmeasures in the same way as in Nishiyama (2002). For this, a set of eigenfunctions forthe operator r with periodic boundary conditions, which was used by Constantin& Majda (1988), is useful. The spaces of the Radon measures contain elements whichare singular with respect to the Lebesgue measure, such as Diracs -measure. This

implies that some of the steady ows generated by (2.1) may have singularities.When is not a cube but an arbitrarily bounded domain in R3, we obtain ananalogous result for (2.1) with

V = V(4) = P((r !) !)

by imposing ! nj@ = 0 instead of periodic boundary conditions. Here P is thethree-dimensional version of the operator ~P introduced in x 1. Moreover, if!j@ = 0is imposed, then V(2) and V(3) seem to work well because integrals on @ vanishin applying integration by parts. However, the author does not know whether or not

our theory with !j@ = 0 is justied in a rigorous manner.Lastly, let us remark on the third equality of (2.5). It means that (2.1) with

V = V(3) yields a solution u = !jt=1 to

(r u) u = 0; or u ru = rjuj2=2: (3.1)

This solution is called the Beltrami ow or a force-free eld. It is represented in theform

r u = (x)u (3.2)

with a scalar function (x) which satises ur = 0 and may depend on u. Recently,(3.2) was rigorously or numerically investigated by Boulmezaoud & Amari (2000,2001) and Kaiser et al. (2000). As a special case of (3.2), the ow u satisfyingr u = const:u has been studied by many authors in the context of uid mechanics,magnetohydrodynamics, astrophysics, etc. (see Boulmezaoud et al. 1999; Constantin& Majda 1988; Moatt & Tsinober 1992, and references therein). In particular,




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2398 T. Nishiyama

Dombre et al. (1986) showed that the Arnold{Beltrami{Childress (ABC) ow, thatis,

u = r u = (A sin z + Ccos y; B sin x + A cos z; Csin y + B cos x)

with some constants A, B, C, has chaotic streamlines. We conjecture that solutionsto (3.1) generated by (2.1) with V = V(3) have complex streamlines like the ABC

ow if we give initial data !0 with complex ux lines.

This work was partly supported by a Grant-in-Aid for Scientic Research from the Japan Min-istry of Education, Culture, Sports, Science and Technology.

References

Berger, M. A. & Field, G. B. 1984 The topological properties of magnetic helicity. J. Fluid Mech.147, 133{148.

Boulmezaoud, T. Z. & Amari, T. 2000 On the existence of nonlinear force-free elds in three-

dimensional domains. Z. Angew. Math. Phys. 51, 942{967.Boulmezaoud, T. Z. & Amari, T. 2001 A nite-element method for computing nonlinear force-

free elds. Math. Comput. Modelling 34, 903{920.

Boulmezaoud, T. Z., Maday, Y. & Amari, T. 1999 On the linear force-free elds in bounded andunbounded three-dimensional domains. Math. Modelling Numer. Analysis 33, 359{393.

Constantin, P. & Majda, A. 1988 The Beltrami spectrum for incompressible uid ows. Com-mun. Math. Phys. 115, 435{456.

Dombre, T., Frisch, U., Greene, J. M., Henon, M., Mehr, A. & Soward, A. M. 1986 Chaoticstreamlines in the ABC ows. J. Fluid Mech. 167, 353{391.

Freedman, M. H. 1988 A note on topology and magnetic energy in incompressible perfectly

conducting uids. J. Fluid Mech. 194, 549{551.Kaiser, R., Neudert, M. & von Wahl, W. 2000 On the existence of force-free magnetic elds with

small nonconstant in exterior domains. Commun. Math. Phys. 211, 111{136.

Moatt, H. K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35,117{129.

Moatt, H. K. 1985 Magnetostatic equilibria and analogous Euler ows of arbitrarily complextopology. 1. Fundamentals. J. Fluid Mech. 15 9, 359{378.

Moatt, H. K. 1990 Structure and stability of solutions of the Euler equations: a Lagrangianapproach. Phil. Trans. R. Soc. Lond. A 333, 321{342.

Moatt, H. K. & Tsinober, A. 1992 Helicity in laminar and turbulent ow. A. Rev. Fluid Mech.24, 281{312.

Nishiyama, T. 2001a Pseudo-advection method for the two-dimensional stationary Euler equa-tions. Proc. Am. Math. Soc. 129, 429{432.

Nishiyama, T. 2001b Pseudo-advection method for the axisymmetric stationary Euler equations.Z. Angew. Math. Mech. 81, 711{715.

Nishiyama, T. 2002 Magnetohydrodynamic approach to solvability of the three-dimensionalstationary Euler equations. Glasgow Math. J. 44, 411{418.

Nishiyama, T. 2003a Construction of solutions to the two-dimensional stationary Euler equationsby the pseudo-advection method. Arch. Math. (In the press.)

Nishiyama, T. 2003b Magnetohydrodynamic approach to measure-valued solution of the two-dimensional stationary Euler equations. (Submitted.)

Vallis, G. K., Carnevale, G. F. & Young, W. R. 1989 Extremal energy properties and constructionof stable solutions of the Euler equations. J. Fluid Mech. 207, 133{152.


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