a closure model for shallow water equations of thin liquid...

Context Thin film simulation Results Thin film modelling Conclusions & perspectives

A closure model for Shallow Water equations ofthin liquid films flowing down an inclined plane

Gianluca Lavalleand

Jean-Paul Vila

Supervisor: François CharruCo-Supervisor: Claire Laurent

ONERADMAE/MH

Gianluca Lavalle ONERAA closure model for Shallow Water equations of thin liquid films flowing down an inclined plane


Plan

1 Context

2 Thin film simulation

3 Results

4 Thin film modelling

5 Conclusions & perspectives



Industrial applicationsThin liquid films

cooling materials, cleaning tubes, paintingwater ingestion in aircraft enginespre-filming primary atomization for fuel injectionliquid film of alumina in solid rocket motorsde-icing of aircraft systems



Thin liquid film instabilitiesSheared films

The instabilities of thin liquidfilm manifest themselves incertain conditions, as 2D and3D waves, with several shapesof ribs and crests.The presence of an unstablethin liquid film flowing on awall strongly alter the nature ofthe exchanges between the gasflowing over the film and thewall



Plan

1 Context


3 Results





Thin liquid film simulation

simulation of wall liquid filmcode CEDRE, solver FILMmodeling: Shallow water equations (NS equations integrated overthe thickness)numerical method: finite volume method, 3D complex surface



The FILM solver

State of the artfilm formation (drop impact, condensation)film progression (gravity, shear effect)heat exchanges (with gas and drops)

Targets

instabilities (for falling and sheared film)surface tension (pincement, ruisselet)film passing (atomization, evaporation)



The FILM solverShallow Water equations

Neglecting source terms (drop impact, condensation and movingreference frame):∂h∂t +

∂q∂x = 0

ρ∂q∂t + ρ

∂

∂x

(q2

h −h2 gy2

)=− h ∂

∂x pg + τg + ρgxh −(3µq

h2 −12τg︸︷︷︸

τFILMw

)

constant velocity profileparabolic velocity profile



Plan

1 Context


3 Results





Inclined planeExperience of Liu and Gollub

β = 6.4◦

u0 = 0.0459 m/sh0 = 0.00089 mν = 6.28 ·10−6 m2/s

f = 1.5 Hzl = 2 m1600 cells

The velocity in the x-direction is given by

u = u0[1 + 0.05 cos(2πft)]

For the equations of the solver FILM, the linear stability analysis leads to

Recr =34 cotβ = 6.69



ResultsFilm thickness profiles

Re = 6.5 < Recrt = 60 s

Re = 8.5 > Recrt = 40 s



Solitary wavesFilm thickness profile

Re = 12without surfacetensiont = 40 s



Plan

1 Context


3 Results





Modelling

We look for a model being both consistent (correct Recr ) andconservative (important for finite volume method).

Shkadov model (neither conservative nor consistent)V. Ya. Shkadov, Wave Flow Regimes of a Thin Layer of Viscous Fluid subject to Gravity

Ruyer-Quil & Manneville model (consistent)C. Ruyer-Quil and P. Manneville, Modeling film flows down inclined planes

Luchini & Charru model (consistent)P. Luchini and F. Charru, Consistent section-averaged equations of quasi-one-dimensionallaminar flow

Vila model (both consistent and conservative)M. Boutounet, L. Chupin, P. Noble, J.P. Vila, Shallow Water viscous flow for arbitrarytopography



The Vila’s modelThe Galilean invariance principle 1/3

Galilean transformation:Coordinates transformation oftwo reference frames differingby a constant relative motion

x = x − wtt = tu(x , t) = u(x , t)− w

Navier-Stokes equations applied to a volume of fluid are invariantunder Galilean transformation

Velocity gradients and all the quantities depending on are Galileaninvariant




NS equations are not invariant under Galilean transformation whenboundary conditions are also considered

example: with a wall velocity vp, the boundary condition at the wallis u(x , 0) = vp − w in the moving reference frame

In this case we can state that the solution in the moving reference frameu(x , y , t, vp) coincides with the solution of the inertial reference framewhen replacing the wall velocity vp with vp − w , which leads tou(x , y , t, vp − w).




Are the Shallow Water equations Galilean invariant? Yes!

ExampleChoosing a moving reference frame moving with w in respect of theinertial one, the Galilean invariance principle reads:

F (h,U) = F (h, U + w)− 2hUw − hw2

Considering the SW flux F (h,U) = αhU2 + p(h), it comes out that it isGalilean invariant ⇔ α = 1.



The Vila’s model

The idea of this model is to impose the Galilean invariance with wallvelocity to the Shallow Water equations, since it works for the NSequations.

1 Dimensionless quasi-linear Shallow Water equations2 Introduction of a wall velocity vp3 Applying the Galilean transformation4 Imposing the Galilean invariance with wall velocity vp



The Vila’s model1. Dimensionless quasi-linear Shallow Water equations

By integration of NS equations over the film thickness,

∂h∂t +

∂q∂x = 0

∂q∂t +

125

qh∂q∂x −

65

q2

h2∂h∂x +

1F 2 h cos β ∂h

∂x −λ2h415

∂h∂x︸︷︷︸

τ(1)w

=1εRe

(λh − 3q

h2)

where Re = UHν , 1

F 2 = gHU2 , ε = H

L , λ = ReF 2 sinβ.

Note that τ (1)w is the correction of the shear stress proposed byRuyer-Quil & Manneville and then found by Luchini & Charru by anothermethod. It ensures the consistence.



The Vila’s model2. Introduction of a wall velocity vp

Applying a wall velocity vp, the mass flow built on the average velocitychanges as

q =

∫ h

0u dy =

13λh3 + hvp

By substitution, the quasi-linear form of SW equations becomes∂h∂t +

∂q∂x = 0

∂q∂t + a(h,U, vp)

∂h∂x + b(h,U, vp)

∂q∂x =

1εRe

(λh − 3U

h +3vph

)a(h,U, vp) = − 6

5U2 + 1F 2 h cos β − λ2h4

15 −215λh2vp − 1

5v2p

b(h,U, vp) = 125 U



The Vila’s model3. Applying the Galilean transformation

x = x − wt t = t u(x , t) = u(x , t)− wy = y v = v p = p h = h

With this coordinates transformation,

∂h∂ t +

∂q∂x = 0

∂q∂ t + (a(h, U + w , vp) + b(h, U + w , vp)w − w2)

∂h∂x

+ (b(h, U + w , vp)− 2w)∂q∂x =

1εRe

(λh − 3 U − vp

h

)



The Vila’s model4. Imposing the Galilean invariance with wall velocity vp 1/2

As seen for Navier-Stokes equations, we want to extend the idea ofGalilean invariance with wall velocity vp to the system of Shallow Waterequations.We consider the equations written in the inertial reference frame, whenreplacing the wall velocity vp with the quantity vp − w :∂h∂t +

∂q∂x = 0

∂q∂t + a(h,U, vp − w)

∂h∂x + b(h,U,vp − w)

∂q∂x =

1εRe

(λh − 3U

h + 3vp − wh

)



The Vila’s model4. Imposing the Galilean invariance with wall velocity vp 2/2

We impose that the new system of equations must coincide with thefixed system when replacing vp with vp − w since the boundary conditionat the wall is the only quantity to change.For the quasi-linear form we have

a(h, U + w , vp) + b(h, U + w , vp)w − w2 = a(h,U, vp − w)

b(h, U + w , vp)− 2w = b(h,U, vp − w)

For the conservative form of SW equations, the expression to imposereads

F (h, U + w , vp)− 2hwvp − hw2 = F (h,U, vp − w)



The Vila’s modelResolution 1/2

The idea consists in expressing ∂q∂t in 6 different ways Mi , i = 1 . . . 6,

using the equation of mass conservation and the expressions of q and U.We have:

M1 =∂q∂t

M2 = −(λh2 + vp)∂q∂x

M3 = −(3U − 2vp)∂q∂x

M4 = −(λh2 + vp)2∂h∂x

M5 = −(λh2 + vp)(3U − 2vp)∂h∂x

M6 = −(3U − 2vp)2∂h∂x

For 6 real numbers αi , such that∑6αi=1 αi = 0 (only 5 of them

independent) it is clear that

6∑αi=1

αiMi ∝ O(ε)



The Vila’s modelResolution 2/2

We can add these quantities to the equations, without changing the finalresults.∂h∂t +

∂q∂x = 0

∂q∂t + a(h,U, vp)

∂h∂x + b(h,U, vp)

∂q∂x +

6∑αi=1

αiMi =1εRe

(λh − 3U − vp

h

)For the sake of semplicity, we express the 5 independent coefficients αi asfunctions of Ai , i = 1 . . . 5 and imposing the

Galilean invariance principle with wall velocityConservative formHyperbolicity



The Vila’s modelGalilean invariance principle with wall velocity

As stated before, we impose the condition of Galilean invariance when awall velocity vp is considered.We found other 2 conditions, which read

A3 = −254 A1 − 3A2 + 6

A5 = −3A2 − 6A4 +45A1



The Vila’s modelConservative form

Following the Schwarz’s theorem, the condition to be conservative reads

∂2F∂h∂q =

∂2F∂q∂h

Since ∂F∂h = a(h, q, vp) and ∂F

∂q = b(h, q, vp), it is necessary to imposethat

∂a∂q =

∂b∂h

This condition provides the expression for two on five parameters, that are

A1 =56

A2 =215 −

65A4



The Vila’s modelHyperbolicity

We have now a flux F (h, q, vp) depending only on the coefficient A4.

A hyperbolic system of equations is characterized by real and distincteigenvalues of teh Jacobian matrix. Now 2 conditions drive to the finalresult

19 ≤ A4 ≤

23

Choosing A4 = 19 , the flux does not depend on the wall velocity vp. This

result leads to the final expression for the flux as

F (h, q) =q2

h +145λ

2h5 +512

h2F 2 cos β



Dimensional SW equations

Finally, the dimensional conservative and consistent SW equations, whichfulfill the Galilean invariance principle when a wall velocity vp is alsoconsidered, are

∂h∂t +

∂q∂x = 0

∂q∂t +

∂

∂x

(q2

h +145

g2

ν2h5(sinβ)2 +

512gh2 cos β

)=

56

(gh sinβ − 3ν q

h2 + 3ν vph

)

Considering also the shear of the gas at the free surface of the film τg , the analysis is similar. Inthis case only A4 = 1

9 gives the hyperbolicity of the system for any values of τg .



Plan

1 Context


3 Results





Conclusions & perspectives

Description of the CEDRE code and solver FILMResults for falling films

good answer for stable and unstable regimelack of surface tension for solitary wavesverify the good accordance with Recr

Description of the Vila’s modelgood for ONERA applications, with rotative componentsflux does not depend on the wall velocity vp

Understanding and improving the model in order to implement insidethe codeVerifying numerical and physics results for falling and sheared filmsComparing with test-cases in litterature



Thank youfor the attention!


a closure model for shallow water equations of thin liquid...

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