Construction of Initial Vortex-Surface Fields and Clebsch Potentials for Flows with High-Symmetry Using First Integrals Pengyu He and Yue Yang

Construction of initial vortex-surface fields and Clebsch potentials for flows with high-symmetry using first integrals Pengyu He and Yue Yang

Citation: Physics of Fluids 28, 037101 (2016); doi: 10.1063/1.4943368 View online: http://dx.doi.org/10.1063/1.4943368 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/28/3?ver=pdfcov Published by the AIP Publishing

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Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 162.105.227.139 On: Fri, 11 Mar 2016 02:35:53 PHYSICS OF FLUIDS 28, 037101 (2016)

Construction of initial vortex-surface fields and Clebsch potentials for flows with high-symmetry using first integrals Pengyu He1 and Yue Yang1,2,a) 1State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China 2Center for Applied Physics and Technology, Peking University, Beijing 100871, China (Received 2 August 2015; accepted 23 February 2016; published online 10 March 2016)

We report a systematic study on the construction of the explicit, general form of vortex-surface fields (VSFs) and Clebsch potentials in the initial fields withthe zero helicity density and high symmetry. The construction methodology is based on finding independent first integrals of the characteristic equation of a given three- dimensional velocity–vorticity field. In particular, we derive the analytical VSFs and Clebsch potentials for the initial field with the Kida–Pelz symmetry. These analytical results can be useful for the evolution of VSFs to study vortical structures in transitional flows. Moreover, the generality of the construction method is discussed with the synthetic initial fields and the initial Taylor–Green field with multiple wavenumbers. C 2016 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4943368]

I. INTRODUCTION Clebsch1 developed a Lagrangian formulation of the Eulerian velocity description for an incompressible inviscid flow. In the Clebsch representation, the incompressible fluid velocity atanytime instant can be expressed, at least locally, in terms of three Clebsch potentials, and each vortex line can be expressed locally by the intersection of iso-surfaces of the Clebsch potentials. More recently, the classical Clebsch representation has been extended to build Eulerian-Lagrangian frameworks for Navier-Stokes dynamics,2–5 and it is considered as an appealing Hamiltonian formulation as a compromise between Eulerian and Lagrangian descriptions of fluid dynamics.6 Within the Clebsch representation,1,7 Yang and Pullin8 defined a vortex-surface field (VSF) as a scalar field φ x, y, z whose iso-surfaces are vortex surfaces. In a general sense, an open or closed vortex surface can( be) defined as a smooth surface or manifold embedded within a three-dimensional velocity field u x, y, z , which has the property that the vorticity ω ≡ ∇ × u is tangent at every point on the surface. The( VSF) should be a real function and globally smooth, so that it can be evolved using its governing equation in numerical simulations and be extracted as level sets to show the geometry of vortex surfaces. The VSF and Clebsch potentials have an inherent Lagrangian nature for visualizing and investi- gating vortical structures in fluid dynamics. The VSF, as a Lagrangian-based structure identification method, is rooted in the Helmholtz vorticity theorem for inviscid flows,8 and a two-time approach was introduced as the numerical dissipative regularization for viscous flows.9,10 The VSFs for both initial Taylor–Green (TG)11,12 and Kida–Pelz (KP)13,14 velocity fields which display simple topology and high symmetry were constructed.8 Numerical results on the evolution of VSFs clarify the continuous vortex dynamics in viscous TG and KP flows including the vortex reconnection, rolling-up of vortex tubes, vorticity intensification between anti-parallel vortex tubes, and vortex stretching and twisting. This suggests a possible scenario for explaining the transition from a smooth laminar flow to turbulent flow and scale cascade in terms of topology and geometry of vortex surfaces. Furthermore, theVSFis also promising to be applied in transitional wall flows15 to elucidate the continuous temporal evolution

a)Electronic mail:[email protected]

of vortical structures from a Lagrangian perspective. In particular, the VSF method can effectively identify the vortex reconnection,16 which is a critical process for the scale cascade in transitions and is hard to be characterized via Eulerian-based vortex identification methods.17,18 Although the VSF has the potential for a paradigm of structure identification and tracking in fluid mechanics, there are open questions of its uniqueness, robustness, and computational feasibility.10,19 In particular, the existence of analytical VSFs and Clebsch potentials was rarely reported in the liter- ature,7,20,21 because challenges exist in the initial construction of VSFs and Clebsch potentials. The analytical Clebsch potentials for the TG initial field were obtained by solving the characteristic equation of vorticity.22 A methodology based on the series expansion was developed8 to construct VSFs for both TG and KP initial velocity fields, but only a numerical solution based on expansion of tailored basis functions was obtained for the KP flow. In the present study, we derive the independent first integrals of the characteristic equation of a given vorticity field. By using the first integrals, we demonstrate that the general, analytical form of VSFs and Clebsch potentials can be constructed for some given velocity–vorticity fields with the zero helicity density, e.g., the KP initial field and the TG initial field with multiple wavenumbers. These analytical results are new and can be useful as the initial conditions for the evolution of VSFs to study the generation and evolution of vortical structures in transitional flows. We begin with introducing fundamentals of the first integral, VSF, and Clebsch potentials in Sec.II. Then in Sec. III, we derive analytical VSFs for both TG and KP initial fields using first integrals. In Sec.IV, we develop a method to obtain the general form of Clebsch potentials. Some conclusions are drawn in Sec.V.

II. FIRST INTEGRAL, VSF, AND CLEBSCH POTENTIALS In this section, the concept of the first integral, VSF, and Clebsch potentials will befirst reviewed, and then the relation between them will be discussed.

A. First integral We consider the general form of nth-order ordinary differential equations (ODEs). Suppose n n fi : R → R is a continuous function. Then for any initial value yi0 = y10, y20,. . . , yn0 in R , the system ( )

dyi = fi x, y1,. . . , yn , i = 1,...,n (1) dx ( ) n must have a unique solution yi x = yi x, yi0 with yi 0, yi0 = yi0. A function Φ : D ⊂ R → R is called a first integral of Eq.(1) in( region) ( D ⊂ )Rn, when( Φ is) continuous together with its first partial derivatives, and Φ y1 x ,. . . , yn x = constant is along the solution of Eq.(1) and is not constant on any open set in(D.( It) is an open( )) question if there exists a general method to find the global and nonconstant first integrals of Eq.(1), but the reward of finding these first integrals can be significant in the rare cases where they exist.23

B. VSF Based on the definition, the constraint ω · ∇φ = 0 (2) for VSF8 is a first-order linear homogeneous partialff di erential equation (PDE). Consider a general form of Eq.(2) ∂Q ∂Q A1 x1, x2,..., xn + ··· + An x1, x2,..., xn = 0, (3) ( ) ∂x1 ( ) ∂xn where x1,..., xn are independent variables, Q = Q x1,..., xn is an unknown function, and Ai ⊂ D are known nontrivial functions. The characteristic equation( of) Eq.(3) is dx dx dx 1 = 2 = ··· = n . (4) A1 x1, x2,..., xn A2 x1, x2,..., xn An x1, x2,..., xn ( ) ( ) ( ) Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 162.105.227.139 On: Fri, 11 Mar 2016 02:35:53 037101-3 P. He and Y. Yang Phys. Fluids 28, 037101 (2016)

From the theory of ODEs, it is necessary and sufficient that any solution Q x1, x2,..., xn of Eq.(3) is a first integral of Eq.(4). Since Eq.(4) contains n − 1 first integrals,( namely,) Φ1,Φ2,...,Φn−1, the general solution of Eq.(3) can be expressed as Q = ψ Φ1,Φ2,...,Φn−1 , where ψ : Rn−1 → R is an arbitrary continuously differentiable function. ( ) In principle, one can obtain the numerical solution of Q by solving hyperbolic PDE(3). Nev- ertheless, it is shown that a necessary condition for the existence and uniqueness of solutions to equations of the type Eq.(3) is not satisfied. 24 Therefore, the regular numerical approach, such as the spectral method for solving Eq.(2) in a periodic domain, fails for the computation. 8 Instead of solving Eq.(3) directly, we construct the general solution of Eq.(3) by finding the independent first integrals of Eq.(4). Additionally, although the first integrals of ω satisfy the VSF constraint Eq.(2), they may have singularities. Hence, a appropriate smooth function of the first integrals is required to carefully remove the singularities.

C. Clebsch potentials In the Clebsch representation,1 the incompressible fluid velocity at any time instant can be expressed locally in terms of three scalar fields as

u = ϕ1∇ϕ2 − ∇ϕ3, (5)

where ϕ1 and ϕ2 are Clebsch potentials, and ϕ3 is used for the solenoidal projection. It follows that the vorticity can be expressed by

ω = ∇ϕ1 × ∇ϕ2. (6) 8 It is noted that both ϕ1 and ϕ2 satisfy the VSF constraint Eq.(2), but they may have singularities. From the vector analysis and Eq.(5), the vanishing helicity density h ≡ u · ω = 0 (7)

is equivalent to ∇ϕ1 × ∇ϕ2 · ∇ϕ3 = 0 locally, which is also referred as the Frobenius condition of integrability. A( velocity field) with Eq.(7) everywhere, which is also called the complex-lamellar 25 field, can be expressed by Clebsch potentials globally as Eq.(5) where ϕ3 must be a first integral of ω or zero. On the other hand, the global existence of the Clebsch potentials in a control volume

Ω implies that the (total) helicity H ≡ Ω h dΩ is vanishing within the volume whose boundary is a vortex surface.26 For the initial field with h , 0, global existence of the Clebsch potentials is not guaranteed, but it is still possible to find exact first integrals and VSFs in some particular flows, such as the integrable Arnold–Beltrami–Childress flow.8,27

III. CONSTRUCTION OF INITIAL VSFS In this section, we construct VSFs for a given vorticity field ω via finding first integrals of the characteristic equation of ω. First, we will apply this method to the initial TG field and compare the initial VSFs with those obtained using the Fourier expansion method.8 Then, the proposed method will be applied to a challenging case, the initial KP field, in which only a numerical solution8 was obtained using the series expansion method for solving Eq.(2).

A. Taylor-Green flow The initial velocity field of the TG flow11,12 is

ux = ax sin x cos y cos z, uy = ay cos x sin y cos z, (8)  uz = az cos x cos y sin z,  2  2π 2 2π 2 where the parameters ax = √ sin θ + , ay = √ sin θ − , and az = √ sin θ with an arbitrary 3  3 3 3 3 real multiplicative constant of θ. The( corresponding) vorticity( is)

ωx = bx cos x sin y sin z, ωy = by sin x cos y sin z, (9)  ωz = bz sin x sin y cos z, √  √ where b = − 3 sin θ + cos θ , b = 3 sin θ − cos θ , and b = 2 cos θ. x y  z The characteristic( equation) of the( vorticity field is ) dx dy dz = = . (10) bx cos x sin y sin z by sin x cos y sin z bz sin x sin y cos z From the first two terms, we obtain

by ln cos x = bx ln cos y + constant, (11) and then cosb y x/cosbx y = constant. (12)

TG b y bx Thus Φ1 = cos x/cos y is one of the first integrals of the initial TG vorticity field. By repeating the manipulations above for the first term and the third term in Eq.(10), we obtain TG bx bz TG TG another first integral Φ2 = cos z/ cos x. The Jacobian matrix J x, y, z for Φ1 and Φ2 has Rank J = 2, which implies that ΦTG and ΦTG are two independent first( integrals.) ( ) 1 2 Then, the initial VSF φTG of the TG flow can be constructed from the first integrals in various approaches to remove singularities. For example, if bx ∈ N,

TG TG bx φTG = Φ /Φ = cos x cos y cos z , (13) 2 1 ( ) is a group of VSFs for the initial TG field. This initial VSF is similar to that developed using the Fourier expansion method for the TG flow.8 Furthermore, for a general form of initial fields with TG symmetries and high wavenumbers,28

ux = ax sin mx cos ny cos pz, uy = ay cos mx sin ny cos pz, (14)  uz = az cos mx cos ny sin pz,  where m, n, and p are integers, we obtain the corresponding VSFs as  bx φTG = cos mx cos ny cos pz , (15) ( ) where bx ∈ N. It is still an open question if there exists a general approach to find explicit first integrals or VSFs for any given vorticity with h = 0 and perhaps under particular symmetries. A turbulent-like velocity can involve Fourier modes with a range of wavenumbers rather than a small portion of low wavenumbers in Eqs.(8) and(14). An attempt of finding first integrals and VSFs for the initial field with TG symmetries and multiple wavenumbers is presented in Appendix A.

B. Kida-Pelz flow The initial velocity field of the KP flow13,14 is defined as

ux = sin x cos 3y cos z − cos y cos 3z , ( ) uy = sin y cos 3z cos x − cos z cos 3x , (16)  ( )  − uz = sin z cos 3x cos y cos x cos 3y .  ( ) The corresponding vorticity is  ωx = −2 cos 3x sin y sin z + 3 cos x sin 3y sin z + sin y sin 3z , ( ) ωy = −2 cos 3y sin z sin x + 3 cos y sin 3z sin x + sin z sin 3x , (17)  ( )  − ωz = 2 cos 3z sin x sin y + 3 cos z sin 3x sin y + sin x sin 3y ,  ( ) and its characteristic equation is  Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 162.105.227.139 On: Fri, 11 Mar 2016 02:35:53 037101-5 P. He and Y. Yang Phys. Fluids 28, 037101 (2016)

dx dy dz = = . (18) ωx ωy ωz With the triple-angle formula, the first term in Eq.(18) becomes dx d sin x d sin x . = = 2 2 2 2 (19) ωx ωx cos x 1 − sin x sin y sin z 16 + 8sin x − 12sin y − 12sin z ( ) ( ) Similar manipulations are repeated for the other terms in Eq.(18) to re-express the equation only in terms of sin x, sin y, and sin z. By multiplying all the terms in Eq.(18) by 2 sin x sin y sin z, and setting α = cos2 x, β = cos2 y, and γ = cos2 z with the identity 1 − sin2 θ = cos2 θ, we can obtain dα dβ dγ = = . (20) α 3β + 3γ − 2α β 3γ + 3α − 2β γ 3α + 3β − 2γ ( ) ( ) ( ) Thus characteristic equation(18) has been rewritten from a trigonometric form to a polynomial form in order to facilitate finding of the first integrals. Note that Eq.(20) is homogeneous. By letting β = αU and γ = αV, it becomes dα αdU + Udα V dα + αdV = = . (21) α2 3U + 3V − 2 α2U 3 + 3V − 2U α2V 3 + 3U − 2V ( ) ( ) ( ) From the first two terms of Eq.(21), after some algebra we obtain dα dU = . (22) α 3U + 3V − 2 5U − 5U2 ( ) Similarly, applying the same change of variables from the first term and the third term in Eq.(21) yields dα dV = . (23) α 3U + 3V − 2 5V − 5V 2 ( ) By comparing Eq.(22) with Eq.(23), we obtain an ODE of U and V dU dV = , (24) 5U − 5U2 5V − 5V 2 which can be solved as U − 1 V ( ) = C, (25) U V − 1 ( ) where C is a constant independent with U and V. Hence, a first integral of Eq.(20) can be written as

KP U − 1 V α − β γ Φ = ( ) = ( ) . (26) 1 U V − 1 β α − γ ( ) ( ) Next, we seek another independent first integral of the KP initial field. From Eq.(25), U can be re-expressed by V and C as V U = . (27) V + C − CV Substituting Eq.(27) into Eq.(23) yields dα −2 + 3V 1 + 1/ V + C − CV = ( ( )) dV, (28) α 5V − 5V 2 and its solution is 4 2 3 ln α = − ln 1 − V − ln V + ln CV − V − C + constant. (29) 5 ( ) 5 5 ( ) Then, substituting Eq.(25) into Eq.(29) with U = β/α and V = γ/α yields β3 α − γ 4 = constant. (30) ( αγ ) Therefore, we obtain another first integral of Eq.(18).

With α = cos2 x, β = cos2 y,γ = cos2 z, and extracting the square root, the final result of the first integrals of Eq.(20) is

2 2 2 KP cos y − cos x cos z Φ = ( ) , (31) 1 cos2 z − cos2 x cos2 y ( ) 3 2 2 2 KP cos y cos x − cos z Φ = ( ) . (32) 2 cos x cos z KP KP KP KP The Jacobian matrix J x, y, z for Φ1 and Φ2 has Rank J = 2, so Φ1 and Φ2 are independent. According to the rotational( ) symmetry of the KP flow,( ) we obtain three groups of first integrals

2 2 2 3 2 2 2 KP cos y − cos x cos z KP cos y cos x − cos z Φ = ( ) , Φ = ( ) , (33) 11 cos2 z − cos2 x cos2 y 21 cos x cos z ( ) 2 2 2 3 2 2 2 KP cos z − cos y cos x KP cos z cos y − cos x Φ = ( ) , Φ = ( ) , (34) 12 cos2 x − cos2 y cos2 z 22 cos y cos x ( ) 2 2 2 3 2 2 2 KP cos x − cos z cos y KP cos x cos z − cos y Φ = ( ) , Φ = ( ) . (35) 13 cos2 y − cos2 z cos2 x 23 cos z cos y ( ) From the results above, we also construct nontrivial, globally smooth VSFs φKP for the KP initial field. For example, the simplest one is

KP KP KP KP KP KP φKP = Φ11 Φ12 Φ13 Φ21 Φ22 Φ23 = cos x cos y cos z cos2 x − cos2 y 2 cos2 y − cos2 z 2 cos2 z − cos2 x 2. (36) ( ) ( ) ( ) The iso-surfaces of φKP in the initial KP field are shown in Fig.1 within a subdomain −π/2 ≤ x ≤ π/2 and 0 ≤ y, z ≤ π. Some vortex lines are integrated and plotted on the vortex surfaces. We remark that the geometry of this explicit initial VSF is very similar to the numerical solution.8,9 The particular initial VSF in Eq.(36) implies a general form of VSFs for the initial KP field

a b c 2 2 a+b+2c 2 2 2a+b+c 2 2 a+2b+c φ˜KP = Acos xcos ycos z cos x − cos y 2 cos y − cos z 2 cos z − cos x 2 , ( ) ( ) ( ) (37)

2a+b+c a+2b+c a+b+2c where A is an arbitrary real number and the parameters a, b,c, 2 , 2 , 2 ∈ N, which can be verified by Eq.(2).

FIG. 1. Iso-surfaces of the VSF in Eq.(36) at φKP = 0.002 (blue) and φKP = −0.002 (yellow) in the initial KP field. Some vortex lines are integrated and plotted on the vortex surfaces.

IV. CONSTRUCTION OF CLEBSCH POTENTIALS A. Construction of Clebsch potentials using first integrals The explicit Clebsch potentials can be constructed from multiple independent VSFs or first integrals, if the latter exist, for a given velocity field, and this approach has been successfully applied to the TG initial field.8 Given two known independent first integrals Φ1 and Φ2 of ω, the two vectors ω and ∇Φ1 × ∇Φ2 are colinear. Assume they point in the same direction as

ω = µ ∇Φ1 × ∇Φ2 . (38) ( ) Here, the scalar

µ = ω / ∇Φ1 × ∇Φ2 (39) | | | | is also a first integral of ω, which is proved in Appendix B. To seek particular Clebsch potentials, we rewrite Eq.(38) in the form of

ω = µ ∇Φ1 × ∇Φ2 = ∇ µ Φ1 × ∇Φ2 − Φ1 ∇µ × ∇Φ2 (40) ( ) ( ) ( ) without loss of generality. In the special cases with ∇µ × ∇Φ2 = 0, e.g., µ is a constant, we can simply construct the particular Clebsch potentials as ϕ1 = µ Φ1 and ϕ2 = Φ2. In the general cases with ∇µ × ∇Φ2 , 0, we have to seek a particular scalar ν that satisfies

ω = ∇ ν Φ1 × ∇Φ2 = ν ∇Φ1 × ∇Φ2 + Φ1 ∇ν × ∇Φ2 (41) ( ) and is a first integral of ω. If ν can be solved from Eq.(41), the corresponding Clebsch potentials are ϕ1 = ν Φ1 and ϕ2 = Φ2.

B. Construction of Clebsch potentials for the KP initial field The analytical Clebsch potentials for the simple TG initial field Equation(8) have been developed,22 but those for the relatively sophisticated cases, e.g., the KP initial field Equation(16) and the TG initial field with multiple wavenumbers (Eq.(14)), are still unknown. In this subsection, we will construct the explicit Clebsch potentials from multiple independent first integrals of the initial KP field obtained in Sec. III B. For the initial KP field with Eqs.(31) and(32), the scalar µ in Eq.(39) is reduced to a constant KP KP KP KP as µ = 2. By setting ϕ1 = µ1Φ1 and ϕ2 = µ2Φ2 with µ1µ2 = µ for constants µ1 and µ2, we obtain two particular Clebsch potentials

2 2 2 KP cos y − cos x cos z ϕ = µ1 ( ) , (42) 1 cos2 z − cos2 x cos2 y ( ) 3 2 2 2 KP cos y cos x − cos z ϕ = µ2 ( ) (43) 2 cos x cos z KP KP satisfying Eq.(6) for the initial KP field. The typical iso-surfaces of ϕ1 and ϕ1 in the initial KP field are shown in Fig.2 within a subdomain 0 ≤ x, y, z ≤ π. The geometry of vortex lines can be expressed by the intersections of the iso-surfaces. KP It is straightforward to obtain the irrotational Clebsch potential ϕ3 = 0 from Eq.(5) with the KP KP particular solutions ϕ1 and ϕ2 . Finally, the initial KP velocity can be expressed by the Clebsch potentials as

KP KP u = ϕ1 ∇ϕ2 . (44)

As an additional note, the analytical Clebsch potentials ϕ1 and ϕ2 given in Sec. 7 in Ref.5 have the similar form of Eq.(43) but they are incorrect. They fail to satisfy the definition of Clebsch potentials Eq.(5) and their derivation was omitted. We remark that in general cases where µ is not constant, the construction of Clebsch potentials by solving Eq.(41) may be challenging. For example, Appendix A presents the construction of Clebsch potentials with the non-constant µ in the TG initial field with multiple wavenumbers.

√ KP KP ± FIG. 2. Iso-surfaces of the Clebsch potentials in Eqs.(42) and(43) with µ1 = µ2 = 2 at ϕ1 = 0.1 (yellow) and ϕ2 = 0.5 (blue) in the initial KP field. Some vortex lines are integrated and plotted on the iso-surfaces.

C. General solution of Clebsch potentials The general solution of Clebsch potentials for a given velocity–vorticity field can be obtained KP KP by seeking all the ϕ1 and ϕ2 satisfying Eq.(6), with given particular solutions such as ϕ1 and ϕ2 for the KP initial field. This can be restated in a more general context, i.e., how to obtain allthe solutions ϕ1 and ϕ2 satisfying Eq.(6), with given independent particular solutions ξ and η? From Eq.(6), we notice

∇ξ × ∇η = ∇ϕ1 × ∇ϕ2, (45)

which implies that ∇ϕ1 · ∇ξ × ∇η = 0 and ∇ϕ2 · ∇ξ × ∇η = 0. Since ξ and η are independent, ( ) ( ) they can be two bases of functional space for ϕ1 and ϕ2. Then the general Clebsch potentials can be expressed as ϕ1 = ϕ1 ξ,η and ϕ2 = ϕ2 ξ,η . Then Eq.(45) can be rewritten as ( ) ( ) ∂ϕ ∂ϕ ∂ϕ ∂ϕ 2 1 − 2 1 = 1. (46) ∂η ∂ξ ∂ξ ∂η

If an arbitrary differentiable function ϕ1 = ϕ1 ξ,η is given, then Eq.(46) is a first-order quasi-linear ODE. As discussed in Appendix C, Eq.((45) can) be solved by finding the first integrals of its characteristic equation dξ dη dϕ = = 2 . (47) ∂ϕ1 ∂ϕ1 1 − ∂η ∂ξ If we can find two independent first integrals ofEq.(47)

Φ1 = Φ1 ξ,η, ϕ2 , (48) ( ) Φ2 = Φ2 ξ,η, ϕ2 , (49) ( ) then the implicit general solution of ϕ2 is

Ψ Φ1 ξ,η, ϕ2 ,Φ2 ξ,η, ϕ2 = 0, (50) ( ( ) ( )) where Ψ is an arbitrary differentiable function. For example, with given ϕ1 = ξ + η, it is straightforward to obtain the implicit general solution from Eq.(47) as

Ψ ϕ2 + ξ, ϕ2 − η = 0. (51) ( ) Without loss of generality, assuming Ψ = ϕ2 − η in Eq.(51), another particular solution of Clebsch potentials is ϕ1 = ξ + η and ϕ2 = η.

D. VSFs and Clebsch potentials for synthetic initial fields We have demonstrated how to obtain the VSFs and Clebsch potentials for several given velocity–vorticity fields including the well-known TG and KP initial fields. In order to investigate the generality of this technique in more initial fields, a synthetic vorticity field ωϑ, where the VSF and Clebsch potentials exist, can be constructed from a given smooth scalar field ϑ as

ωϑ = ∇ϑ × ∇ϑ0, (52)

where ϑ0 is an arbitrary scalar function. Consequently ϑ is the VSF of ωϑ from the identity ∇ϑ × ∇ϑ0 · ∇ϑ = 0. Additionally, ϑ and ϑ0 can be considered as a pair of Clebsch potentials for ( ) ωϑ. As an example, we use ϑ = sin x + sin y + sin z and ϑ0 = cos x + cos y + cos z to construct an initial vorticity field

ωϑ = sin y − z ,sin z − x ,sin x − y (53) ( ( ) ( ) ( )) from Eq.(52). This initial field was proposed by Ohkitani 5 to study an Euler flow evolution with a very weak energy transfer. On the other hand, assuming we do not know ϑ and ϑ0 a priori, we can solve the VSF and Clebsch potentials by finding the first integrals of ωϑ. The characteristic equation of Eq.(53) is dx dy dz = = , (54) sin y − z sin z − x sin x − y ( ) ( ) ( ) and it can be re-expressed as cos x dx cos y dy cos z dz = = . (55) cos x sin y − z cos y sin z − x cos z sin x − y ( ) ( ) ( ) Applying Componendo and the trigonometric identity cos x sin y − z + cos y sin z − x + cos z sin x − y = 0 (56) ( ) ( ) ( ) in Eq.(55) yields cos x dx + cos y dy + cos z dz = d sin x + sin y + sin z = 0, (57) ( ) Oh which shows that ϑ1 = sin x + sin y + sin z is a first integral for Eq.(54). Using the similar proce- Oh dure, we can derive another first integral ϑ2 = cos x + cos y + cos z, then we obtain the VSF and Clebsch potentials for Eq.(53) using first integrals. Although there appears to be no general technique of solving first integrals, the method of first integrals has been applied successfully in several given initial velocity–vorticity fields.A generalized method is expected to solve the first integrals of Eq.(3) numerically, though it can be challenging to develop a suitable regularization method owing to the existence and uniqueness issues8,24 of solutions of Eq.(3). The TG and KP initial fields with the known, explicit first integrals can be good candidates to verify the proposed numerical methods. Besides these fields with high-symmetry, a more challenging verification case can be developed using a synthetic initial vorticity field ωϑ generated from a turbulent-like scalar field ϑ. For example, the synthetic vorticity field can be generated as ( ) ∂ϑ ∂ϑ ∂ϑ ∂ϑ ∂ϑ ∂ϑ ω = − , − , − , (58) ϑ ∂ y ∂z ∂z ∂x ∂x ∂ y

with a simple nontrivial ϑ0 = x, y, z . ( )

V. CONCLUSIONS We report a systematic study on the construction of VSFs and Clebsch potentials for given initial three-dimensional velocity–vorticity fields with high-symmetry. Instead of solving thePDE for the VSF constraint directly,8 we seek the first integrals of the characteristic equation of vorticity.

Then the initial VSFs and Clebsch potentials can be constructed from the first integrals. In addi- tion, the implicit form of the general solution, relying on an arbitrary continuously differentiable function, can be obtained from a group of particular solutions of VSFs and Clebsch potentials. Particularly, we derive the explicit general forms of VSFs and Clebsch potentials for the KP initial field and for the TG initial field with multiple wavenumbers. This methodology based on first integrals can be applicable to obtain explicit VSFs and Cleb- sch potentials in other flows with high-symmetry and the zero helicity density, though adhoc methods may be required to manipulate the characteristic equation to solve first integrals. These analytical VSFs and Clebsch potentials can be valuable for further investigations on the Lagrangian vortex dynamics via their temporal evolution, particularly in inviscid flows5 and in viscous transitional flows,9,10 and for the development of the general numerical methods of solving VSFs from an arbitrary velocity field as benchmark solutions.

ACKNOWLEDGMENTS The authors thank D. I. Pullin, W.-D. Su, and S. Xiong for helpful comments. This work has been supported in part by the National Natural Science Foundation of China (Nos. 11342011, 11472015, and 11522215) and the Thousand Young Talents Program of China.

APPENDIX A: INITIAL FIELD WITH TG SYMMETRIES AND MULTIPLE WAVENUMBERS The generalized Fourier representation12 of velocity in the evolution of the TG flow is ∞ ∞ ∞ ux = uˆx sin mx cos ny cos pz, m=0 n=0 p=0  ∞ ∞ ∞     uy = uˆy cos mx sin ny cos pz, (A1)   m=0 n=0 p=0  ∞ ∞ ∞   uz = uˆz cos mx cos ny sin pz,   m=0 n=0 p=0  whereu ˆx,u ˆy, andu ˆz are Fourier coefficients under constraints of TG symmetries. The explicit VSFs of the generalized TG field may exist and valuable, but it appears to be challenging to solvethem from Eq.(A1) with infinite wavenumbers. As an attempt to solve first integrals of a velocity field with multiple wavenumbers, we set fixed m and n, and equal coefficients asu ˆx = uˆy = uˆz = 1. Then Eq.(A1) is reduced to a solvable case as

p 2p+1 z z  sin ( ) − sin u = sin mx cos ny cos lz = sin mx cos ny 2 2 , x 2 sin z  l=1 2  p * 2p+1 z − z +   sin ( 2 ) sin 2  u = cos mx sin ny cos lz = sin mx cos ny , (A2)  y , 2 sin z -  l=1 2  p * z − 2p+1 z +   cos 2 cos ( 2 ) uz = cos mx cos ny sin lz = sin mx cos ny ,  , 2 sin z -  l=1 2  * + where p is an arbitrary positive integer or p → ∞, and the helicity density of Eq.(A2) is vanishing.  The characteristic equation of the vorticity field of Eq.(A2) is , - 2 tan mx dx − z 2p+1 z z 2p+1 z 2n cos + 1 − 2n + 2p cos ( ) − cot sin ( ) 2 ( ) 2 2 2 2 tan ny dy = z 2p+1 z z 2p+1 z 2m cos + 1 − 2m + 2p cos ( ) − cot sin ( ) 2 ( ) 2 2 2 dz = , (A3) z 2p+1 z m − n sin − sin ( ) ( )( 2 2 ) Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 162.105.227.139 On: Fri, 11 Mar 2016 02:35:53 037101-11 P. He and Y. Yang Phys. Fluids 28, 037101 (2016)

which is equivalent to two variable separable ODEs

z 2p+1 z z 2p+1 z 2n cos + 1 − 2n + 2p cos ( ) − cot sin ( ) tan mx dx = 2 ( ) 2 2 2 dz, (A4) z 2p+1 z −2 m − n sin − sin ( ) ( ) 2 2 z 2p+1 z z 2p+1 z 2m cos + 1 − 2m + 2p cos ( ) − cot sin ( ) tan ny dy = 2 ( ) 2 2 2 dz. (A5) z 2p+1 z 2 m − n sin − sin ( ) ( ) 2 2 Two independent first integrals of Eq.(A3) can be solved directly as

TGM − m−n 1+p m −1+2n−p 1 + p z m 1+p z −m 1+p pz Φ = cos ( )( )mx cos ( ) ( ) sin ( ) sin ( ) , (A6) 1 2 2 2

TGM m−n 1+p n −1+2m−p 1 + p z n 1+p z −n 1+p pz Φ = cos( )( )ny cos ( ) ( ) sin ( ) sin ( ) . (A7) 2 2 2 2 Then the VSF for can be constructed as

TGM m−n 1+p ( )( ) Φ1 1 + p z pz z φTGM = = cos mx cos ny cos ( ) sin csc , (A8) TGM 2 2 2 Φ2 * + which only has removable singularities. This VSF in Eq.(A8) is distinguished with the TG VSFs Eqs.(13) and(14) because, Eq.-(A8) has multiple characteristic scales. In other words, the corresponding vortex surfaces, as iso-surfaces of Eq.(A8) with a single iso-contour value, have di fferent sizes. For example, the iso-surfaces of φTGM in Eq.(A8) with m = 1, n = 2, and p = 4 is shown in Fig.3 within a subdomain −π/2 ≤ x ≤ π/2, −π/4 ≤ y ≤ π/4, and −π ≤ z ≤ π. Some vortex lines are integrated and plotted on the vortex surfaces. It is expected to obtain the explicit VSF of a more complex, or perhaps turbulent-like field in future work. TGM TGM Based on the independent first integrals φ and Φ1 , we constructed the Clebsch potentials for Eq.(A2) by solving ν from Eq.(41) as

TGM TGM TGM TGM ω = ν ∇φ × ∇Φ1 + φ ∇ν × ∇Φ1 . (A9) We obtain the solution ( m−n 1+p m 1−2n+p 1 + p z 1−m 1+p z ν = cos( )( )mx cos ( ) ( ) sin ( ) 2 2 ( ) )  m 1+p pz z 1 + p z 1 + p sin ( ) −2n cos + 2n − 1 − 2p cos z + cot sin z 2 2 ( ) 2 2 2 ( ( pz 1 + p z z 2mn 1 + p p cos cos ( ) sin ( ) 2 2 2 ) pz ( z 1 + p z z 1 + p z )) − sin cos cos ( ) + 1 − 2n + p sin sin ( ) , (A10) 2 2 2 ( ) 2 2

FIG. 3. Iso-surfaces of the VSF in Eq.(A8) with m = 1, n = 2, and p = 4 at φTGM = 0.5 (blue) and φTGM = −0.5 (yellow) present different characteristic scales of the vortex surfaces in the initial TG field. Some vortex lines are integrated and plotted on the vortex surfaces.

TGM where the derivation is omitted. Thus, the corresponding particular Clebsch potentials are ϕ1 = TGM TGM TGM νφ and ϕ2 = Φ1 .

APPENDIX B: PROOF OF µ AS A FIRST INTEGRAL OF VORTICITY It is interesting that a nontrivial µ in Eq.(39) is also a first integral of vorticity ω. From Eq.(38), we obtain ( ) ( ) ( ) 1 1 ∂Φ ∂Φ ∂Φ ∂Φ 1 ∂Φ ∂Φ ∂Φ ∂Φ 1 ∂Φ ∂Φ ∂Φ ∂Φ = 1 2 − 1 2 = 1 2 − 1 2 = 1 2 − 1 2 , µ ωx ∂ y ∂z ∂z ∂ y ωy ∂z ∂x ∂x ∂z ωz ∂x ∂ y ∂ y ∂x (B1) which implies ( ) ∂µ ∂2Φ ∂Φ ∂2Φ ∂Φ ∂2Φ ∂Φ ∂2Φ ∂Φ ∂ω ω = −µ2 1 2 + 2 1 − 1 2 − 2 1 + µ x . (B2) x ∂x ∂x∂ y ∂z ∂x∂z ∂ y ∂x∂z ∂ y ∂ y∂z ∂x ∂x ∂µ ∂µ This is the first term of ω · ∇µ, and other two terms ωy ∂y and ωz ∂z have the similar form. The summation of the three terms results in that the first term in the right hand side of Eq.(B2) is canceled, and then yields ( ) ∂ω ∂ωy ∂ω ω · ∇µ = µ x + + z = µ ∇ · ω = 0. (B3) ∂x ∂ y ∂z ( ) Hence, µ is a first integral of ω.

APPENDIX C: FIRST-ORDER QUASI-LINEAR PARTIAL DIFFERENTIAL EQUATION We consider a first-order quasi-linear PDE ∂Q ∂Q A1 x1, x2,..., xn,Q + ··· + An x1, x2,..., xn,Q = R x1, x2,..., xn,Q , (C1) ( ) ∂x1 ( ) ∂xn ( )

where Ai and R are known, continuously differentiable functions in D. Solving Eq.(C1) is similar to solving Eq.(3). The characteristic equation of Eq.(C1) is dx dx dQ 1 = ··· = n = . (C2) A1 x1, x2,..., xn,Q An x1, x2,..., xn,Q R x1, x2,..., xn,Q ( ) ( ) ( ) Suppose n independent first integrals of Eq.(C2) are

Φ1 x1, x2,..., xn,Q ,...,Φn x1, x2,..., xn,Q (C3) ( ) ( ) and ψ : Rn → R is an arbitrary continuously differentiable function. Then

ψ Φ1 x1, x2,..., xn,Q ,...,Φn x1, x2,..., xn,Q = 0 (C4) ( ( ) ( )) is the implicit general solution of Eq.(C2).

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