On a potential-velocity formulation of

Navier-Stokes equations

By Florian Marner1, Philip H. Gaskell2 and Markus Scholle1

1Heilbronn University, Institute for Automotive Technology and Mechatronics,D–74081 Heilbronn, Germany.

2School of Engineering and Computing Sciences, Durham University, Durham,DH1 3LE, UK.

Computational methods in continuum mechanics, especially those encompass-ing fluid dynamics, have emerged as an essential investigative tool in nearly everyfield of technology. Despite being underpinned by a well-developed mathematicaltheory and the existence of readily available commercial software codes, computingsolutions to the governing equations of fluid motion remains challenging: in essencedue to the non-linearity involved. Additionally, in the case of free surface film flowsthe dynamic boundary condition at the free surface complicates the mathematicaltreatment notably. Recently, by introduction of an auxiliary potential field, a firstintegral of the two-dimensional Navier-Stokes equations has been constructed [? ? ]leading to a set of equations, the differential order of which is lower than that of theoriginal Navier-Stokes equations. In this paper a physical interpretation is providedfor the new potential, making use of the close relationship between plane Stokesflow and plane linear elasticity. Moreover, it is shown that by application of this al-ternative approach to free surface flows the dynamic boundary condition is reducedto a standard Dirichlet-Neumann form, which allows for an elegant numerical treat-ment. A least squares finite element method is applied to the problem of gravitydriven film flow over corrugated substrates in order to demonstrate the capabilitiesof the new approach. Encapsulating non-Newtonian behaviour and extension tothree-dimensional problems is discussed briefly.

Keywords: Fluid Dynamics, Complex methods, Potentials, Integrability, Airystress function, Least squares finite elements

1. Motivation

As is well known, Bernoulli’s equation is obtained as a first integral of Euler’sequations in the absence of vorticity and viscosity if the velocity vector ~u is perceivedas the gradient of a scalar potential. The so-called Clebsch transformation [? ?] allows for a further extension to flows with non-vanishing vorticity. A similarmethodology has recently been reported by ? ] for the case of two-dimensionalincompressible viscous flow by making use of a representation of the fields in termsof complex coordinates. Besides a reduction of differential order the formulationof integrated Navier-Stokes equations allows for a convenient embodiment of thedynamic boundary condition as a Dirichlet-Neumann condition on the potentialfield in the case of free surface flows.

2

Initially the integration procedure is motivated and performed from a formalmathematical point of view in which a scalar potential field is introduced as anauxiliary variable to make the field equations integrable. Meanwhile, the questionof physical interpretation of this ’naturally’ occurring ’potential’ motivates a reviewof complex methods in the field of fluid dynamics in which the exploration of theclose relationship between plane Stokes flow and plane linear elasticity proves tobe illuminating. In the case of Stokes flow the new potential velocity formulationin complex form can be shown to reproduce the well-known Kolosov-Muskhelishviliformulae [? ? ? ] of plane linear elasticity, suggesting the potential to be a functionof integrated stresses.

A short review of the first integral of Navier-Stokes equations [? ] is providedin Ch. 1, followed by the derivation of the potential representation of the dynamicboundary condition in Ch. 2. Ch. 3 is devoted to the analysis and interpretationof the potential variable mentioned above. In Ch. 4 a least-squares finite elementmethod, used to solve the fully non-linear problem of gravity-driven thin film flowover corrugated topography, is presented; it shows the general and convenient nu-merical applicability of the new formulation.

2. Two-dimensional potential-velocity formulation for steadyincompressible Navier-Stokes equations

? ] developed an integration procedure for the case of two-dimensional incompress-ible viscous flow by making use of a representation of the fields in terms of complexcoordinates. For convenience the essentials of the methodology are reviewed below.

(a) Field equations

In two-dimensions, the Navier-Stokes equations and the continuity equationgoverning a steady and incompressible flow, assuming that the external force onthe fluid is conservative with a given potential energy density U(x, y), are:

%~u · ∇~u = −∇p+ η∆~u−∇U , (2.1)

∇ · ~u = 0 , (2.2)

where ~u denotes the velocity field and p the pressure field. ? ] show that a complexvariable transformation ξ := x+iy, ξ := x− iy of the above field equations, togetherwith the introduction of a complex velocity field u := ux +iuy, yields the equivalentformulation †:

∂

∂ξ

[p+ %

uu

2+ U

]+ %

∂

∂ξ

(u2

2

)= 2η

∂2u

∂ξ∂ξ, (2.3)

Re

(∂u

∂ξ

)= 0 , (2.4)

† Note, that in contrast to [? ] the standard complex variable transformation is used hereleading to slightly different equations.

On a potential-velocity formulation of Navier-Stokes equations 3

where Re denotes the real part of the subsequent complex expression. Now theintroduction of a scalar real-valued potential Φ satisfying:

p+ %uu

2+ U = 4

∂2Φ

∂ξ∂ξ, (2.5)

allows for the integration of (??) with respect to ξ, giving:

%u2

4− η ∂u

∂ξ+ 2

∂2Φ

∂ξ2 = 0 , (2.6)

Re

(∂u

∂ξ

)= 0 . (2.7)

Note, the resulting equations contain first order derivatives only of velocity incontrast to the original Navier-Stokes equation (??)-(??) and the correspondingtwo-dimensional streamfunction version in general contains second order deriva-tives only, whereas the classical streamfunction formulation results in a forth orderequation.

A real representation of the complex system (??)-(??) in tensor notation reads:

η

[∂ui∂xj

+∂uj∂xi− ∂uk∂xk

δij

]−%[uiuj − ukuk

δij2

]= 2

[∂2Φ

∂xi∂xj− ∂2Φ

∂xk∂xk

δij2

], (2.8)

together with∂uk∂xk

= 0 , p+%

2ukuk + U =

∂2Φ

∂xk∂xk, (2.9)

in which the Einstein summation convention is used. The second equation in (??)is only relevant in applications where the recovery of pressure is of interest; thepressure can easily be computed subsequently.

(b) Boundary conditions

Common boundary conditions such as no-slip or periodic conditions can be useddirectly in connection with equations (??)-(??). In these cases just the velocity com-ponents are constrained at the boundary, whereas the potential variable remainsunconstrained. While free-surface boundary conditions are more problematic theycan be simplified substantially as shown in the following. Henceforth a simply con-nected domain with a free surface ~x = ~x(s) is considered, in which the free surfaceis assumed to be parametrized with respect to arc length s. Furthermore normaland tangential unit vectors according to:

ni(s)εji = tj(s) = x′j(s) (2.10)

are defined along the free surface, where εij denotes the Levi-Civita symbol. Intwo dimensions the surface shape is determined by one kinematic and two dynamicboundary conditions; the latter:[

(p0 − p)δij + η

(∂ui∂xj

+∂uj∂xi

)]xk=xk(s)

nj(s) = σdtids

, (2.11)

4

with σ on the right-hand side denoting the surface tension, indirectly introducethe pressure as an undesired further variable into the solution process. ? ] foundan elegant remedy for this by obtaining an integral formulation for the boundarycondition revealing (??) to be a pure condition on the potential gradient, namely:

εij

(∂Φ

∂xj

)xk=xk(s)

=σ

2ti(s)−

1

2

∫ s

s0

[U (xk(s))ni(s) + σ′(s)ti(s)] ds . (2.12)

Here, the integral limit s0 can be chosen arbitrarily. In this context it is interest-ing to note that in the two-dimensional streamfunction formulation the potentialboundary conditions on the free surface

∇Φ = ~g1(s) , (2.13)

represent a natural counterpart to the streamfunction conditions on the solid wall

∇Ψ = ~g2(s) . (2.14)

Of course, knowledge of the potential gradient on the free surface further allowsthe derivation of Dirichlet and Neumann boundary conditions for the potentialvariable in a simply connected domain if Φ is specified in a single point. By takingthe inner product of (??) with ti, its tangential component takes the form:

ni(s)

(∂Φ

∂xi

)xk=xk(s)

=σ

2− ti(s)

2

∫ s

s0

[U (xk(s))ni(s) + σ′(s)ti(s)] ds , (2.15)

which is a Neumann boundary condition for the potential Φ. On the other handusing (??) the inner product of (??) with ni leads to the corresponding normalcomponent:

ti(s)

(∂Φ

∂xi

)xk=xk(s)

=d

dsΦ(xk(s)) =

εjitj(s)

2

∫ s

s0

fi(s)ds , (2.16)

fi(s) = U (xk(s))ni(s) + σ′(s)ti(s) , (2.17)

with a total derivative in (??). Thus, using (??), (??) and partial integration,an integrated form of a Dirichlet boundary condition for the potential Φ can beconstructed:

Φ(xk(s))− Φ(xk(s0)) =εji2

∫ s

s0

x′j(s)

∫ s

s0

fi(s)dsds (2.18)

=εji2xj(s)

∫ s

s0

fi(s)ds−εji2

∫ s

s0

xj(s)fi(s)ds (2.19)

=1

2

∫ s

s0

[U (xk(s)) δij + σ′(s)εij ] [xi(s)− xi(s)]x′j(s)ds . (2.20)

The term Φ(xk(s0)) can be set to zero without loss of generality and moreover, in thespecial case of σ′(s) = 0, that is neglecting Marangoni effects, further simplificationis possible.

The reduction of the dynamic boundary condition from its original form (??) toa standard Dirichlet-Neumann form (??, ??) is a key feature of the reformulation ofthe equations of motion in terms of the first integral of the Navier-Stokes equations(??) to allow for the construction of efficient solutions.

On a potential-velocity formulation of Navier-Stokes equations 5

3. On potentials and stresses

So far, in section ??, a potential-velocity formulation of Navier-Stokes equationswith corresponding boundary conditions has been found by a formal integrationmethod. Note, this derivation is motivated and performed in a pure mathematicalcontext, in which the scalar ’potential’ Φ appears rather as an auxiliary variable,introduced to make the field equations integrable, than as a physically meaningfulquantity. It has to be distinguished from the classical meaning of a scalar field, thegradient of which represents a given vector field. As a physical interpretation tendsto simplify analysis and numerical treatment later on, a closer look is taken at thecharacter of this potential.

In terms of the potential Φ and a streamfunction Ψ, satisfying:

−2i∂Ψ

∂ξ= u , (3.1)

equations (??)-(??) can be written as

∂2Φ

∂ξ2 + iη

∂2Ψ

∂ξ2 −

%

2

(∂Ψ

∂ξ

)2

= 0 , (3.2)

in which the continuity equation is satisfied automatically. Considering the Stokesflow case and introducing a further complex potential, χ = Φ+iηΨ, allows equation(??) to be written as the simple bianalytic equation

∂2χ

∂ξ2 = 0 , (3.3)

the solution of which can be expressed in terms of two analytic functions

χ = ξw0(ξ) + w1(ξ) , (3.4)

known as Goursat functions. On the one hand, this formula allows the problem ofStokes flow to be converted into one of complex analysis, namely that of finding w0

and w1 which satisfy appropriate conditions on the boundary. This approach leadsto the well-known Sherman-Lauricella equations, see for example [? ]. On the otherhand, using Φ = Re(χ) the representation of the potential derivatives in terms ofthe Goursat functions reproduce the well-known Muskhelishvili-Kolosov formula [?? ]:

∂Φ

∂x+ i

∂Φ

∂y= w0(ξ) + ξw′

1(ξ) + w′2(ξ) (3.5)

for the integrated components of stress in plane linear elasticity theory, revealingthe potential to be closely connected with the Airy stress function.

At first glance, the analogy to linear elasticity theory provides an interpretationfor the potential Φ in the linear Stokes case only, but rewriting the full Navier-Stokesequations (??) in terms of the streamfunction and Airy stress function actuallyreproduces the real version of equation (??). Accordingly, the constitutive law ofa Newtonian fluid involving the convective momentum flux density Rij = uiuj† is

† by averaging Reynolds stress tensor is obtained from Rij

6

adopted to account for inertial effects:

σij = −pδij + η

(∂ui∂xj

+∂uj∂xi

)− %Rij , (3.6)

with σij being a symmetric stress tensor, p the pressure and ui the velocity com-ponents. Now introducing an Airy stress function such that:

σ11 =∂2Φ

∂y2, σ22 =

∂2Φ

∂x2, σ12 = − ∂2Φ

∂x∂y, (3.7)

leads to

−%uxuy + η

[∂ux∂y

+∂uy∂x

]+

∂2Φ

∂x∂y= 0 , (3.8)

%

2

[u2x − u2

y

]+ η

[∂uy∂y− ∂ux

∂x

]+

1

2

[∂2Φ

∂y2− ∂2Φ

∂x2

]= 0 , (3.9)

which agrees with equation (??) except for a constant scaling of Φ. This result is notsurprising, considering that the dynamic boundary condition (??) only constrainsΦ on the one hand while essentially constituting a condition on the stresses on theother hand.

4. Numerical formulation and solution

Complex variable methods have been known for a long time in hydrodynamics[? ] and have contributed a great deal to the development of analytic and semi-analytic solutions of linear flow problems in simple two-dimensional geometries [?? ? ]. Also various forms of complex boundary integral methods have entered thearena of computational fluid dynamics which nowadays are ranked among the mostefficient methods in their range of applicability [? ? ? ? ]. This development waspartly due to realizing the close connection to the field of plane linear elasticity [?? ? ] which also allowed for a transfer of solution methods (see for example ? ]).

Complex variable methods are essentially based on rewriting the field equationsin terms of the analytic Goursat functions given in equation (??) and solving thecorresponding complex system. Though very efficient for plane Stokes flow, espe-cially in the case of stress boundary conditions [? ], their applicability remainslimited. Note, that this classical approach results as a special case of the more gen-eral first integral procedure of section ?? and that the tensor version (??)-(??) cannaturally be extended to three dimensions as shown by ? ].

Since the overall goal is to construct a fairly general and flexible numericalmethod allowing for inertial effects and potentially being extendible to three dimen-sions, below the transformed real version of Navier-Stokes equations (??)-(??) isdiscretized directly. Due to the close connection between the integrated real versionof Navier Stokes equations and a streamfunction / Airy stress function formulationas shown in section ??, a review of the corresponding, though limited, literature [?? ? ? ? ? ] was undertaken.

For the purposes of the work reported here a least squares finite element method,inspired by the contribution to the field of ? ? ? ], proved to be adequate. This

On a potential-velocity formulation of Navier-Stokes equations 7

way the highly efficient semi-analytic Ritz method developed by ? ] for the inte-grated Stokes equations is complemented by a more flexible method, allowing forthe incorporation of inertial effects and more general applicability. For a compre-hensive review of least squares methods, including a special treatment of Stokesand Navier-Stokes equations, the reader is referred to [? ].

(a) Least-squares finite element method

The least-squares FEM has gained great popularity for the numerical solution offlow problems, allowing for the use of simple equal order elements as well as highlyefficient multigrid solvers due to symmetry and positive definiteness of the resultingsystem matrices [? ]. Arguments of practicality suggest rewriting the tensor equa-tions (??)-(??) in terms of velocity variables and first derivatives of the Airy stressfunction. Thus a system of four equations is obtained, including first order deriva-tives only, which is covered by the first order system least squares methodology.

By introducing the two variables Φx = ∂Φ∂x and Φy = ∂Φ

∂y and a further equation

∂Φx

∂y− ∂Φy

∂x= 0 , (4.1)

equations (??, ??) are transformed into the system:

−%uxuy + η

(∂ux∂y

+∂uy∂x

)− ∂Φx

∂y+∂Φy

∂x= 0 , (4.2)

%

2

(u2x − u2

y

)+ η

(∂uy∂y− ∂ux

∂x

)+∂Φx

∂x− ∂Φy

∂y= 0 , (4.3)

∂ux∂x

+∂uy∂y

= 0 , (4.4)

∂Φx

∂y− ∂Φy

∂x= 0 . (4.5)

Applying a Newton linearisation, with ux and uy denoting the velocity componentsof the previous iteration step, leads to the following linearised system:

−%(uxuy + uxuy) + η

(∂ux∂y

+∂uy∂x

)− ∂Φx

∂y+∂Φy

∂x= −%uxuy =: f1 , (4.6)

%(uxux − uyuy) + η

(∂uy∂y− ∂ux

∂x

)+∂Φx

∂x− ∂Φy

∂y=%

2

(u2x − u2

y

)=: f2 , (4.7)

∂ux∂x

+∂uy∂y

= 0 =: f3 , (4.8)

∂Φx

∂y− ∂Φy

∂x= 0 =: f4 , (4.9)

which is written in condensed form as L(W ) = f , with W = (ux, uy,Φx,Φy)T , hereon in. Note, that here the linearisation is done before least squares minimization (cp.[? ]). The corresponding least-squares functional to be minimized in each iterationstep, can be written as:

J (W ; f) =

4∑i=1

‖LiW − fi‖2L2(Ω) . (4.10)

8

For convenience a simply connected domain Ω ⊂ R2 is assumed with prescription ofeither ux and uy or Φx and Φy on each part of the boundary ∂Ω, where the potentialboundary conditions are set according to (??) in case of problems involving a freesurface. Note, that in this formulation a free surface flow problem leads to pureDirichlet boundary conditions on the velocity vector and on the potential gradient.Additionally in the case of pure velocity boundary conditions, ∇Φ and ∆Φ have tobe specified at a single point in order to achieve a unique solution.

In the following Hm(Ω) denotes the standard Sobolev space of functions havingsquare integrable derivatives of order up to m over Ω. Now, introducing test andsolution spaces W , V ⊂ [H1(Ω)]4, where the boundary conditions are assumed tobe incorporated into the solution space, as well as corresponding finite elementsubspaces Wh, Vh, the problem of minimizing functional (??) can be rewritten in avariational formulation, namely, find Wh ∈ Wh such that:∫

Ω

L(Wh)L(Vh)dΩ =

∫Ω

L(Vh)fdΩ , ∀ Vh ∈ Vh . (4.11)

In producing the numerical results presented in the following subsection contin-uous finite element spaces of piecewise quadratic polynomials are employed for alltest and solution spaces. Well-known problems associated with mass conservation inconjunction with least-squares methods [? ] are addressed by an appropriate weight-ing of the continuity equation as suggested in [? ] or by an augmented least-squaresmethod in severe cases (cp. [? ]).

(b) Treatment of free surfaces

In two dimensions, three boundary conditions are required along the free surface,a kinematic condition (??) and two dynamic conditions (??). As two conditions aresufficient to determine a problem with fixed domain, the shape of the free surfacecan be found by iterating over one of the conditions while solving a sequence of flowproblems with the other two conditions set on a fixed domain. The problem for-mulation provided suggests solving such problems with a given dynamic boundarycondition (??) and to iterate over the kinematic condition

ui(xk(s))ni(s) = 0 . (4.12)

A correction H(n+1)(x) of the previous free surface shape, given by a height functionH(n)(x), is found by solving a differential equation of the form:

d

dxH(n+1)(x)

∣∣∣∣x=x(s)

=u

(n)y (x, y)

u(n)x (x, y)

∣∣∣∣∣(x,y)=(x(s),H(n)(x(s)))

, (4.13)

via an explicit Runge-Kutta integration method. The update algorithm is supple-mented by an appropriate damping in the case of strong variations.

(c) Numerical results

Consider now the case of gravity-driven free surface film flow over a corrugatedsubstrate inclined at angle of α to the horizontal, as shown in Fig. 1. Assume Ω to bea symmetric part of the domain and impose periodic velocity boundary conditions

On a potential-velocity formulation of Navier-Stokes equations 9

Figure 1. On the left side: Principle set-up for thin gravity-driven film flow over corrugatedsubstrate with inclination angle α. On the right side: Streamlines with isotach lines ascontour colours for the case of Re = 50.

to the left and right, no-slip conditions on the lower fixed boundary and Φx = gx,Φy = gy on a fixed approximation to the upper free surface with functions gx, gyconstructed via (??). The potential energy density U in (??) is given by

U(x, y) = %g [cos(α)y − sin(α)x] (4.14)

where % denotes the density and g the gravitational constant.Numerical results are found using a structured criss-cross grid in combination

with a moving mesh method for grid adaptation. In the finite element space, piece-wise continuous quadratic functions are employed. Figure ?? shows the impactof varying film thickness on the resulting flow structure. As the film thickness in-creases, the free surface becomes smoother and eddies form in the corrugations. Thesolutions obtained are in accord with comparable experimental and computationalresults [? ? ? ].

Figure 2. Varying film thickness effect. Constant substrate geometry and fluid dataλ = 1 cm, a = 0.2 cm, α = π/8, η = 5.78 Pa·s, σ = 0.074 N·m−1, g = 9.81 m·s−2 and% = 972 kg·m−3; the film thickness from left to right is h = 0.15 cm, h = 0.3 cm andh = 0.8 cm.

In a further parameter study the Reynolds number is varied in order to demon-strate the capability of the new approach (see Figure ??). From the global perspec-tive, the appropriate choice of h as reference length leads to the following definition

10

Figure 3. Varying Reynolds number effect. Constant substrate geometry and fluid dataλ = 6 cm, a = π/2 cm, α = π/4, η = 5.78 Pa·s, σ = 0.074 N·m−1, g = 9.81 m·s−2 andh = 12π /25 cm; the Reynolds number from left to right is Re = 0.3/10/30/50/100.

of a global Reynolds number (cp. [? ]):

Re =%2gh3 sin(α)

2η2(4.15)

which is important as far as the stability of the flow and dynamics of the freesurface are concerned. Although thin gravity-driven film flow appears unstable atthe surface at a sufficiently high Reynolds number, as indicated in ? ], the regionbelow the free surface in the vicinity of the eddy is heuristically stable and resolvedcorrectly. These results are in accordance with ? ? ? ]. Although problems of massconservation in conjunction with least-squares methods tend to increase when otherthan pure velocity boundary conditions are imposed, the example flows consideredshows the method to produce accurate results even in the case of periodic and freeboundary conditions.

5. Summary and outlook

It is shown by use of complex variables that a first integral of the two-dimensionalincompressible and steady Navier-Stokes equations can be established, the orderof which is lower than that of the original Navier-Stokes equations. The proce-dure results in either a single complex valued equation of second order dependingon a potential and the streamfunction or a system of two equations in the casewhen velocities are used. Alternatively in terms of Cartesian coordinates a tensorformulation can be given.

The potential field is formally introduced as an auxiliary variable to make thefield equations integrable, while posing the question of physical interpretation. Alook at the integrated equations in the Stokes flow case and the analogy betweenplane Stokes flow and plane linear elasticity for a complex formulation, on the onehand shows that the new formulation reproduces the well-known complex Kolosov-Muskhelishvili formulas of linear elasticity, while on the other hand revealing the

On a potential-velocity formulation of Navier-Stokes equations 11

potential to be equivalent to an Airy stress function except for constant scaling.This interpretation is shown to be cogent also in the non-linear case.

Motivated by the above integration procedure a representation of the dynamicboundary condition as a pure Dirichlet-Neumann condition on the potential is de-rived. This formulation gives rise to the development of new numerical methods forfree surface flows. An efficient least-squares finite element method is developed andshown to accurately solve a fully non-linear problem; that of gravity-driven filmflow over corrugated substrate for different film thickness and different Reynoldsnumber.

As indicated in [? ] the tensor representation (??)-(??) of the first integral ofNavier-Stokes equations gives rise to a natural generalization to three dimensions.Apart from this, a natural generalization to non-Newtonian materials is obvious byreplacing the stress tensor for a Newtonian fluid by the stress tensor for arbitrarymaterials. The presentation of a generalized theory will be the subject of subse-quent publications.

Acknowledgement The authors are grateful to the Deutsche Forschungsgemein-schaft (DFG) for their financial support.

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