It has been shown [1, 2] that a wide class of 3D motions of in-

compressible viscous fluid in Cartesian coordinates can be de-

scribed by only one scalar function dubbed the quasi-potential.

This class of fluid flows is characterized by three-component

velocity field having two-component vorticity field; both these

fields can depend of all three spatial variables and time, in gen-

eral. Governing equations for the quasi-potential have been de-

rived and simple illustrative examples of 3D flows have been

presented. In this paper the concept of quasi-potential is fur-

ther developed for fluid flows in cylindrical coordinates. It is

shown that the introduction of a quasi-potential in curvilinear

coordinates is non-trivial and may be a subject of additional

restrictions. In the cases when it is possible, we construct il-

lustrative examples which can be of interest for some practical

applications

Ermakov, Andrei

Stepanyants, Y. A.

MURAL - Maynooth University Research Archive Library

20th Australasian Fluid Mechanics ConferencePerth, Australia5-8 December 2016Description of Vortical Flows of Incompressible Fluid in Terms of Quasi-Potential FunctionA. M. Ermakov, Y. A. StepanyantsSchool of Agricultural, Computational and Environmental Sciences,University of Southern Queensland, Queensland 4350, AustraliaAbstractIt has been shown [1, 2] that a wide class of 3D motions of in-compressible viscous fluid in Cartesian coordinates can be de-scribed by only one scalar function dubbed the quasi-potential.This class of fluid flows is characterized by three-componentvelocity field having two-component vorticity field; both thesefields can depend of all three spatial variables and time, in gen-eral. Governing equations for the quasi-potential have been de-rived and simple illustrative examples of 3D flows have beenpresented. In this paper the concept of quasi-potential is fur-ther developed for fluid flows in cylindrical coordinates. It isshown that the introduction of a quasi-potential in curvilinearcoordinates is non-trivial and may be a subject of additionalrestrictions. In the cases when it is possible, we construct il-lustrative examples which can be of interest for some practicalapplications.IntroductionA great success in the solution of fluid dynamic problems is as-sociated with the reduction of a primitive set of hydrodynamicequations to only one equation for any scalar function, e.g., ve-locity potential or stream function [3, 4, 5, 6, 7]. It has beenshown in [1, 2] that the class of exactly solvable hydrodynamicproblems can be widened by introduction one more scalar func-tion, the quasi-potential. The starting point for the introductionof the quasi-potential is the condition of incompressibility ofa fluid: divU = 0, where U is the fluid velocity. This allowsus to introduce a vector-potential A such that U = curlA au-tomatically satisfies this equation. However, there is a gaugeinvariance in the choice of the vector-potential as it is definedup to the gradient of any scalar function f (t,x,y,z), becausecurl(∇ f ) ≡ 0. Therefore, if we use another vector-potentialA′ = A+∇ f , this does not affect the velocity vector U.Due to the freedom of choice of an arbitrary function f (t,x,y,z),we can eliminate one of the components of the vector-potential.Therefore without loss of generality the vector potential A canbe chosen consisting of two components only. In the Cartesianrectilinear coordinates it does not matter which component iseliminated, because vector differential operations are symmet-rical with respect to all spatial variables x, y and z. However, it isnot the case in curvilinear coordinates, in particular, in cylindri-cal coordinates. In any case, an arbitrary 3D velocity field canbe described, in general, by two-component vector-potential,i.e. by two scalar functions – the corresponding componentsof the vector-potential. If there is any additional link betweenthese two components, then the description of a fluid flow canbe done in terms of only one scalar function, equation for whichfollows from the primitive Navier–Stokes equation. This ap-proach has been exploited in [1, 2] in Cartesian coordinates andnontrivial examples of fluid motions have been found.Below we consider fluid flow in cylindrical coordinates andshow how the quasi-potential can be introduced when one ofthe components of the vector-potential A is eliminated. An il-lustrative example is constructed.Governing equations and quasi-potential in cylindrical co-ordinatesCase 1 – Derivation of basic equations when Ar = 0Consider first the case when the vector-potential does not con-tain the radial component Ar and reads: A = (F1/r)eϕ−F2ez,where F1 and F2 are functions of time and all spatial variables.The velocity and vorticity fields for such vector-potential read:U = F3er +∂F2∂reϕ+1r∂F1∂rez, F3 =−1r(∂F2∂ϕ+∂F1∂z), (1)ω =(1r2∂2F1∂r∂ϕ− ∂2F2∂r∂z)er +[∂F3∂z− ∂∂r(1r∂F1∂r)]eϕ +1r[∂∂r(r∂F2∂r)− ∂F3∂ϕ]ez. (2)Consider a particular class of fluid flows having only two com-ponents of the vorticity. To eliminate the first component of thevorticity let us introduce such function P that1r2∂F1∂r=∂P∂z,∂F2∂r=∂P∂ϕ. (3)Thus, the assumption that the r-component of vorticity is zeroimplies that P is the potential function of ϕ and z.It is convenient to introduce further a quasi-potential Φ suchthat P = Φr (here and below the indices of function Φ stand forpartial derivatives with respect to the corresponding variables).Substitute then expressions for U and ω into the Navier–Stokesequation in the Helmholtz form [3]:∂ω∂t+ curl [ω×U] = ν∆ω. (4)Bearing in mind that thankful to the condition (3) the r-component of the vorticity is zero, in the case of perfect fluid(ν = 0) the r-component of this vector equation reduces to:∆Φ+1r∂∂r[r(H− ∂Φ∂r)]= 0. (5)where H(Φr) is an arbitrary function of Φr.Two other components (ϕ and z) of Eq. (4) after simple mani-pulations reduce to one equation:(H ′−1) ∂∂r[∂Φ∂t+(∇Φ)22]+(H ′−1)[(H− ∂Φ∂r)∂2Φ∂r2+1r3(∂Φ∂ϕ)2]= Q(t, r), (6)where Q(t, r) is an arbitrary function of its arguments, and H ′stands for a derivative of function H with respect to its argu-ment.In terms of function Φ the velocity and vorticity fields read:U = ∇Φ+[H(Φr)−Φr]er , (7)ω = (H ′−1)(∂2Φ∂r∂zeϕ−1r∂2Φ∂r∂ϕez). (8)In the particular case of H(Φr)≡Φr, the quasi-potential Φ be-comes the conventional hydrodynamic velocity potential. Equa-tion (5) reduces to the Laplace equation, Eq. (6) disappears, andvorticity vanishes.In general the main equations to be solved simultaneously areEqs. (5) and (6) with given functions H(Φr) and Q(t, r). Asthere is a freedom in the choice of these function, we obtain agood perspective to construct exact solutions to hydrodynamicequations and accommodate them to practical needs.Unfortunately, in the case of a viscous fluid (ν 6= 0) the intro-duction of a quasi-potential does not help to simplify the basicequation (4).Case 2 – Derivation of basic equations when Aϕ = 0Consider now the case when the vector potential has only r- andz-components: A = −r F1 er + r F2 ez. In this case the velocityand vorticity fields become:U =∂F2∂ϕer +F3reϕ +∂F1∂ϕez , (9)where F3 =−r[∂(r F1)∂z+∂(r F2)∂r],ω =1r(∂2F1∂ϕ2− ∂F3∂z)er +∂∂ϕ(∂F2∂z− ∂F1∂r)eϕ +1r(∂F3∂r− ∂2F2∂ϕ2)ez. (10)Let us eliminate now the second component of the vorticity. Tothis end introduce such function P that∂2F2∂ϕ2− ∂F3∂r=∂P∂r,∂2F1∂ϕ2− ∂F3∂z=∂P∂z. (11)In the case of a perfect fluid the second ϕ-component of vectorequation (4) reduces to:∆[P+ r2H(P)]=1r2∂2P∂ϕ2, (12)where H(P) is an arbitrary function of P.This equation can be presented in terms of a quasi-potential Φwhich is defined as P+ r2H(P) = ∂Φ/∂ϕ. Alternatively, onecan think that P is an arbitrary function G of Φϕ and r, viz.P = G(Φϕ, r). Then Eq. (12) reads:∆Φ− 1r2∂G(Φϕ, r)∂ϕ= R(t, r, z), (13)where R(t, r, z) is an arbitrary function of its arguments.Two other components (r and z) of Eq. (4) after simple manip-ulations can be reduced to the following one equation:∂G∂t+∇Φ ·∇G− Gr2∂G∂ϕ= Q(t, ϕ), (14)where Q(t, ϕ) is an arbitrary function of its arguments.In terms of quasi-potential the velocity and vorticity fields read:U = ∇Φ− 1rG(Φϕ, r)eϕ , (15)ω =1r(∂G∂zer−∂G∂rez). (16)In the particular case of G(Φϕ, r)≡ 0, the quasi-potential Φ be-comes ϕ-independent conventional hydrodynamic velocity po-tential. Equation (13) reduces to the Laplace equation withR(t, r, z) ≡ 0, Eq. (14) disappears, provided that Q(t, ϕ) ≡ 0,and vorticity vanishes. The velocity field becomes plane anddepends only of r and z.In general the main equations to be solved for Φ simultaneouslyare Eqs. (13) and (14) with given functions G(Φϕ, r), Q(t, ϕ),and R(t, r, z). Despite of the more complex character of theseequations, the freedom of choice of arbitrary functions allowsus to obtain again a good perspective to construct exact solu-tions to hydrodynamic equations and accommodate them to thepractical needs.Unfortunately, in the case of a viscous fluid (ν 6= 0) the intro-duction of a quasi-potential does not help to simplify the basicequation (4).Case 3 – Derivation of basic equations when Az = 0Consider at last the case when the vector potential has only r-and ϕ-components: A = (F1/r)er−F2 eϕ.In this case the velocity and vorticity fields become:U =∂F2∂zer +1r∂F1∂zeϕ +F3 ez , (17)where F3 =−1r(∂r F2∂r+1r∂F1∂ϕ),ω =1r(∂F3∂ϕ− ∂2F1∂z2)er +(∂2F2∂z2− ∂F3∂r)eϕ , (18)where we have eliminated already the third component of vor-ticity assuming that there is a relationship between functions F1and F2: ∂F1/∂r = ∂F2/∂ϕ. This relationship is satisfied auto-matically if we introduce a quasi-potential Φ such that∂F1∂z=∂Φ∂ϕ,∂F2∂z=∂Φ∂r.Substituting the expressions for U and ω into Eq. (4), we obtainfrom the third, z-component, of this vector equation even withthe viscous term accounted for:∆Φ+∂∂z(H− ∂Φ∂z)= R(t, r, ϕ). (19)where H(Φz) is an arbitrary function of Φz and R(t, r, ϕ) is anarbitrary function of its arguments.Two other components (r and ϕ) of Eq. (4) with ν 6= 0 after sim-ple manipulations can be reduced to the following one equation:∂∂t(H− ∂Φ∂z)+∇Φ ·∇(H− ∂Φ∂z)+12∂∂z(H− ∂Φ∂z)2−ν∆(H− ∂Φ∂z)= Q(t, z), (20)where Q(t, z) is an arbitrary function of its arguments.In terms of quasi-potential the velocity and vorticity fields read:U = ∇Φ+(H− ∂Φ∂z)ez , (21)ω =1r∂∂ϕ(H− ∂Φ∂z)er−∂∂r(H− ∂Φ∂z)eϕ. (22)In the particular case of H(Φz)≡Φz, the quasi-potential Φ be-comes the conventional hydrodynamic velocity potential. Equa-tion (19) reduces to the Laplace equation with R(t, r, ϕ) ≡ 0,Eq. (20) disappears, and vorticity vanishes. In the case of vis-cous fluid there is no additional restrictions.Below we present an example of non-trivial fluid flow describedby these Eqs. (19) and (20).ExamplesWe managed to construct two examples illustrating the devel-oped theory.Example of a vertical flow in Case 2Consider first an example for the case when the vector potentialhas only r- and z-components in cylindrical coordinates. Let usassume that G(Φϕ, r)≡ (1−λ2)Φϕ, where λ 6= 1 is a constant.One can readily prove that the following quasi-potentialΦ =(√r2 + z2 + zr)λsinϕ (23)satisfies Eqs. (13) and (14). Then the velocity and vorticityfields read:U =(√r2 + z2 + zr)λλr√r2 + z2[−zsinϕer +λ√r2 + z2 cosϕeϕ + r sinϕez], (24)ω =(√r2 + z2 + zr)λλ(1−λ2)cosϕr2√r2 + z2(r er + zez) . (25)This solution describes stationary vortical flow periodic in azi-muthal coordinate ϕ and having two components of vorticity.When λ = 0, the velocity and vorticity fields vanish, whereaswhen λ = 1, the flow becomes potential with the zero vorticity.In Cartesian coordinates both the velocity and vorticity fieldsare three-component and read:U =(R+ z√x2 + y2)λλR(x2 + y2)3/2 ×[−xy(λR+ z) i+(λx2R− y2z)j+ y(x2 + y2)k], (26)where R =√x2 + y2 + z2;ω =(R+ z√x2 + y2)λλ(1−λ2)xR(x2 + y2)3/2 (x i+ y j+ zk) . (27)Figures 1 and 2 illustrate the velocity and vorticity fields as perEqs. (26) and (27).Figure 1: A fragment of velocity field as per Eq. (26).Figure 2: A fragment of vorticity field as per Eq. (27).The modulus of the velocity field is:|U|= λ(R+ z√x2 + y2)λ √λ2x2 + y2(x2 + y2) . (28)And the modulus of the vorticity field is:|ω|= λ(1−λ2)( R+ z√x2 + y2)λx(x2 + y2)3/2 . (29)Example of a vertical flow in Case 3Consider now an example of the vortical flow for the third casewhen the vector potential has only r- and ϕ-components incylindrical coordinates. Let us assume that H(Φz) ≡ −λ2Φz,where λ 6= 1 is a constant.One can readily prove that the following quasi-potentialΦ =T (t)λcos(λr sinϕ)cosz, (30)where T (t)=C exp[−(1+λ2)ν t], satisfies Eqs. (19) and (20).Then the velocity and vorticity fields read:U =−T (t)[cosz sin(rλsinϕ)(sinϕer + cosϕeϕ)−λsinzcos(rλsinϕ)ez] , (31)ω =(1+λ2)T (t)sinz cosϕsin(rλsinϕ)(−er + r eϕ). (32)This solution describes exponentially decaying with time velo-city and vorticity fields. The vector fields are periodic in ϕand z. In Cartesian coordinates the velocity field becomes two-component:U =−λT (t)(sinλycosz j−λcosλysinzk) , (33)whereas the vorticity field has only one component:ω =−λ(1+λ2)T (t)sin(λy) sinz i. (34)Both these fields do not depend on x and therefore canbe described by the conventional stream function ψ =−λT (t)sinλysinz, so that the velocity components are:Uy =∂ψ∂z, Uz =−∂ψ∂y.For purely imaginary λ = i, the quasi-potential reduces to theconventional hydrodynamic potential, and the fluid field be-comes potential with the zero vorticity.Thus, this example describes a vortical flow double periodic inthe (y, z) plane. Figure 3 illustrates the velocity field for t = 0and λ = 1 when y and z vary in the range of [0, π]. There is amirror symmetry with respect to planes y = 0 and z = 0. Figure4 illustrates the corresponding vorticity field for t = 0 and λ= 1.As one can see from these two examples, the introduction ofquasi-potentials allows us to construct exact solutions for fairlycomplex vortical flows in cylindrical coordinates. The transfor-mation of a two-component vorticity field in cylindrical coordi-nates can lead to either on-component or three-component vor-ticity field in Cartesian coordinates. There is no regular methodto construct three-component vortical flows in Cartesian coor-dinates, whereas the developed theory allows us to find exactsolutions in terms of quasi-potential.ConclusionIt has been demonstrated that introduction of quasi-potentialis possible in cylindrical geometry, whereas it is not a triv-ial generalisation of quasi-potential theory developed in [1, 2].Quasi-potential approach helps us to construct exact solutionsdescribing fairly complicated three-dimensional fluid field witha multi-component vorticity field. In the particular cases thequasi-potential theory reduces to the conventional potential the-ory and to the theory based on introduction of a stream-function.It is believed that the quasi-potential approach can be used alsoin other curvilinear coordinates, in particular, in a spherical ge-ometry. The results for spherical coordinates will be publishedelsewhere.References[1] Stepanyants, Y.A. and Yakubovich, E.I., Scalar descriptionof three-dimensional vortex flows of incompressible fluid,Doklady, Physics, 56, 2011, 130–133.[2] Stepanyants, Y.A. and Yakubovich, E.I., The Bernoulli in-tegral for a certain class of non-stationary viscous vorticalflows of incompressible fluid, Stud. Appl. Math., 135, 2015,295–309.[3] Lamb, H., Hydrodynamics, 6th edn. Cambridge, Cam-bridge University Press, 1932.[4] Milne-Thomson, L.M., Theoretical Hydromechanics, 4thedn. London, Macmillan and Co LTD, 1960.[5] Kochin, N.E., Kibel, I.A. and Roze, N. V. Theoretical Hy-dromechanics, 6th edn. New York, Interscience Publishers,1964.[6] Batchelor, G.K., An Introduction to Fluid Mechanics Cam-bridge, Cambridge University Press, 1967.[7] Landau, L.D. and Lifshitz, E.M., Fluid Mechanics Oxford,Pergamon Press, 1993.Figure 3: A fragment of velocity field as per Eq. (33).Figure 4: A fragment of vorticity field as per Eq. (34).

Description of vortical flows of incompressible fluid in terms of quasi-potential function

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Description of vortical flows of incompressible fluid in terms of quasi-potential function

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