Lie-symmetry group and modelling in non-isothermal fluid mechanics

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Lie-symmetry group and modelling in non-isothermal fluid mechanics

Aziz Hamdouni, Dina Razafindralandy, Nazir Al Sayed

To cite this version:

Aziz Hamdouni, Dina Razafindralandy, Nazir Al Sayed. Lie-symmetry group and modelling in non- isothermal fluid mechanics. 6th AIAA Theoretical Fluid Mechanics Conference, 2011, Honolulu, United States. �hal-02086501�

Fluid Mehanis

AzizHamdouni

⋆

,DinaRazandralandy

⋆⋆

,NazirAlSayed

⋆⋆⋆

LEPTIAB,UniversitédeLaRohelle,AvenueMihelCrépeau,17042LaRohelleCedex1,Frane

E-Mail:

⋆

ahamdoununiv-lr.fr,

⋆⋆

drazanuniv-lr.fr,

⋆⋆⋆

nalsay01univ-lr.fr

1 Introdution

Thephysialpropertiesofaphenomenon whihhas beenmodelizedintoaset ofequations

are ontained in the symmetry group of these equations. This fat is well illustrated by

N÷ther's theorem [5, 7℄ whih states that for a Lagrangian system, eah symmetry of

the equations orresponds to a onservation law. For example, a system having a time-

translation symmetry isenergy onserving.

In uid mehanis, the symmetry group theory has been used tostudy, among others,

the dynamis of vorties in a uid ow [3, 4℄ and to investigate spetral properties of the

ow [9℄. Oberlak [6℄ also used the symmetry theory to dedue saling laws, suh as the

algebrai law whih has been onrmed in the enter and in the near-wall region of a

hannel ow, the usual logarithmi wall law, a linear law whih is observed in the enter

region of a ouette ow. He also found a new exponential law whih has been onrmed

later inthe mid-wakeregion of the boundary layer.

TherstpartofthispaperextendstheworkofOberlaktonon-isothermalandparallel

ow, suh as a pipe ow. Saling laws for both veloity and temperature will be given.

The seond part is devoted to the onstrution of a lass of turbulene models of a non-

isothermal ow, whih respet the physisof the ow.

Indeed,inspiteofmanydeadesofstudies,turbulenemodellingstillonstitutesanim-

portanthallengeinengineering siene. No turbulenemodel, eitherwith RANS method

or with large-eddy simulation (LES) approah, is really satisfying regarding their math-

ematial and physial properties. Moreover, the subgrid heat ux model is, most of the

time, simply dedued from the subgrid stress tensor model by a Reynolds analogy. This

analogy limitsthe sope of the models for strongly oupledproblems (natural and mixed

onvetion). Inthis paper, wepropose agenerallassof subgridstress andheat uxmod-

els based on the preservation of the symmetry group (and then the underlying physial

properties)of the non-isothermal NavierStokesequations (1).

2 Saling laws in a parallel ow

Consider the NavierStokes equations for an inompressible, non-isothermal Newtonian

uid, with kinemativisosity ν ^{and thermaldiusivity}κ^:

















∂u

∂t + div(u⊗u) + 1

ρ∇p− div τ_r−βgθe2= 0

∂θ

∂t + div(θu)− div hr= 0 div u= 0

(1)

In these expressions, τr = 2νSand hr=κ∇θ ^are respetively the visous stress tensor and the heat ux, and S = ^∇u+₂^{T∇u is the strain rate tensor.} β ^{is the thermal expansion}

and g ^{the aeleration due togravity.}

UsingtheRANSapproah,wedeomposetheveloity,thepressureandthetemperature

into amean and a utuating quantities:

f =F +f^′. ⁽²⁾

As we are seeking for stationnary bidimensionalsaling laws in aparallelow, we assume

that the mean pressure is of the form P(x2)−x1K^{, that} U2 = U3 = 0 ^{and that} U1 ^and

Θ ^{depend only on the wall normal oordinate} x2^{. The utuating quantities verify the}

followingset ofequations:







































∂u^′_i

∂t +U1

∂u^′_i

∂x1

+u^′₂

dU1

dx2

δ_i1− K+ν∂²U1

∂x²₂

!

δ_i1+ ∂P

∂x2

−βgΘ−βgθ^′

! δ_i2

+∂u^′_iu^′_k

∂x_k + ∂p^′

∂x_i−ν∂²u^′_i

∂x²_k = 0

∂θ^′

∂t +∂θ^′u^′_j

∂x_j −κ∂²θ^′

∂x²_j −κ∂²Θ

∂x²₂ +u^′₂∂Θ

∂x2

+U1

∂θ^′

∂x1

= 0

∂u^′_k

∂x_k = 0

(3)

Forsimpliity,we alsodesignate system (3)by

E(t,x,U,u^′, P, p^′,Θ, θ^′, ν, κ) = 0 ⁽⁴⁾

Saling laws are self-similar solutions of (3). In what follows, we use Lie's theory to nd

the symmetries of the equationsand to dedue the self-similarsolutions.

A (Lie point-)symmetryof (3) isa transformation

(t,x,U,u^′, P, p^′,Θ, θ^′, ν, κ)7→(bt,x,b Ub,ub^′,P ,b pb^′,Θ,b θb^′,bν,bκ) ⁽⁵⁾

dependingontinuously ona parametera^{,whihmaps asolution of (3)intoanothersolu-}

tion. Knowingtransformation (5)is equivalentto knowing its generator

X =ξ_t∂

∂t+ξ_xi

∂

∂xi

+ξ_U1

∂

∂U1

+ξ_P ∂

∂P +ξΘ

∂

∂Θ+ξ_u^′_i ∂

∂u^′_i+ξ_p^′ ∂

∂p^′ +ξ_θ^′ ∂

∂θ^{′ (6)}

where

ξq= ∂qb

∂a a=0

. ⁽⁷⁾

Aording toLie's theory, transformation (5)is asymmetry of (4) if and onlyif

E = 0 ⇒ X.E = 0. ⁽⁸⁾

This last ondition permits toompute the omponents of the symmetry generators X ^of

(3), whihare

ξν = nν, ξ_κ = mκ,

ξt = [n−2m]t+c1,

ξx¹ = [n−m]x1+α1(t) +c2, ξx2 = [n−m]x2+c3,

ξx3 = [n−m]x3+α2(t) +g,

ξU¹ = mU1−α3(t) +α4(x2) + ˙α1(t),

ξP = 2mP +βgx2α5(t, x1, x3) +α6(t, x1, x3) +α9(t, x2), ξΘ = [3m−n] Θ−α7(x2) +α5(t, x1, x3),

ξu^′1 = mu^′1+α3(t)−α4(x2), ξ_u^′₂ = mu^′₂,

ξ_u^′₃ = mu^′₃+ ˙α2(t),

ξ_p^′ = 2mp^′ + [K(n−3m)−α˙3(t)]x1 −α¨2(t)x3+α8(t)−α9(t, x2), ξθ^′ = [3m−n]θ^′ +α7(x2).

(9)

where n^, m^{, the} c^′_i^{s and the} α^′_i^{s, whih are arbitrary onstants and funtions, are the}

parameters of the symmetries.

(U1,Θ)^{represents a} self-similarsolution of (3)if

X.U1 = 0 ^and X.Θ = 0,

that is,if

dU1

ξU1

= dΘ ξΘ

= dx2

ξx2

(10)

Inthe partiularasewherethe αi^{'sare onstant,theresolution of(10)leads topartiular}

saling laws.

• ^Ifm 6= 0^, n6=m^, n 6= 3m^{, we get the following algebraisalinglaw:}

U1 =C1

x2+ b n−m

_n^m

−m

+C3

m, Θ =C2

x2+ b n−m

^3m_n ⁻ⁿ

−m

+ C4

3m−n. ⁽¹¹⁾

• ^{A logarithmi veloity prolearises if} m = 0 ^and n6= 0^: U1 = C1

n ln

x2+ b n

+C3, Θ =C2

x2+ b

n −1

+ C4

n . ⁽¹²⁾

• ^{A linear lawappears when} m =n = 0^:

U1 =C1x2+C3, Θ =C2x2 +C4. ⁽¹³⁾

• ^{We obtain an}exponentiallawwhen m=n6= 0^: U1 =C1expmx2

b

+ C3

m, Θ = C2exp

2mx2

b

+ C4

2m. ⁽¹⁴⁾

As notied in the introdution, the orresponding saling laws in the isothermal ase

haveall been onrmed either by experimentsor by DNS data.

As weould see, the symmetry groupare ofparamountimportaneinmodellingphys-

ial phenomena. It ontains, among other, the saling laws of uid ows. In partiular,

turbulene models should alsopreserve the symmetry group of the equations. In the next

setion,webuildalass ofLESturbulenemodels,whihareompatible withthesymme-

try group of the non-isothermalNavierStokes equations. These models should naturally

reproduethe physialproperties of the ow, suh assaling laws.

Letusonsider equations(1). Usingthe sameapproahasabove,the symmetrygenerator

of these equations an be omputed. Its omponents are

ξt = c1+ 2tc3,

ξx1 = c2x2+ [c3 +c5]x1+α1(t), ξ_x2 = −c2x1+ [c3 +c5]x2+α2(t), ξx3 = [c3 +c5]x3 +α3(t), ξu1 = c2u2+ [c5−c3]u1+ ˙α1(t), ξu² = −c2u1+ [c5−c3]u2+ ˙α2(t), ξu3 = [c5 −c3]u1+ ˙α3(t),

ξp = ζ(t) + 2[c5−c3]p−ρ(xα(t)) +¨ c4βgx3, ξθ = [c5−3c3]θ+c4/ρ,

ξν = 2c5ν, ξκ = 2c5κ.

(15)

With the generator, one get the following list of symmetries of the non-isothermal

NavierStokes equations(1):

• ^{the group of time} translations: (t,x,u, p, θ) 7−→ (t+a,x,u, p, θ)

• ^{the group of pressure} translations: (t,x,u, p, θ) 7−→ (t,x,u, p+ζ(t), θ)

• ^{the group of} pressure-temperature translations:

(t,x,u, p, θ) 7−→ (t,x,u, p+a βg x3, θ+a/ρ)

• ^{the group of horizontal rotations:} (t,x,u, p, θ) 7−→ (t,Rx,Ru, p, θ)

where R isa 2D (onstant) rotationmatrix,

• ^{the group of generalized Galilean} transformations:

(t,x,u, p, θ) 7−→ (t,x+α(t),u+α(t), p˙ +ρxα(t), θ)¨

• ^{the group of the rst sale} transformations:

(t,x,u, p, θ) 7−→ (e^2at,e^ax,e^−au,e^−2ap,e^−3aθ)

• ^{and the group of the seond sale} transformations:

(t,x,u, p, θ, ν, κ) 7−→ (t,e^ax,e^au,e^2ap,e^aθ,e^2aν,e^2aκ).

Equations(1)ownotherknownsymmetries. Thesesymmetriesarenotloalontinuous

transformationslikethepreviousones andannotbededuedfromLie'stheory. Theyare:

• ^{the reetions whih are disrete}symmetries:: (t,x,u, p, θ)7→(t,Λx,Λu, p, ι3θ)

where Λis areetion matrix, Λ=diag(ι1, ι2, ι3) ^with ιi =±1^,

• ^{and the material indierene in the limit of a 2D horizontal ow [2℄ whih is a}

time-dependent rotation: (t,x,u, p)7→(t,x,b u,b p)b

with

b

x=R(t) x, ub=R(t) u+ ˙R(t) x, pb=p−3ωφ+ω²kxk²/2

where R(t) ^{is an horizontal 2D rotation matrix with angle} ωt^, ω ^{a real parameter,} φ ^{the usual 2D stream funtion suh that} (u = curl(φe3)) ^and k^•k ^{indiates the}

Eulidian norm. The material indierene onstitutes a non-loal symmetry of the

equations.

The ombinationof all these symmetriesonstitutes the symmetry group of (1).

In LES approah, one does not solve diretly equations (1). Instead, one omputes an

approximate solution (u, p, θ) ^{whih ontains only the large sales of the atual veloity} u^{, pressure} p ^and temperature θ ^{of the uid. The separation of large and small sales is}

dened by a ltering. The equationsof (u, p, θ)^are:























∂u

∂t + div(u⊗u) + 1

ρ∇p− div(τ_r−τ)−βgθe2= 0

∂θ

∂t + div(θu)− div(hr−h) = 0 divu= 0.

(16)

In these equations, τ =u⊗u−u⊗u ^{is the subgrid stress tensor and} h=θu−θu ^the

subgridheat ux. Inorder tolose the equations, thesetwoterms must bemodelled,i.e.,

replaed by funtions of (u, p, θ)^{, alled turbulene model. A good turbulene model is}

one with whih (u, p, θ) ^{has the same properties as} (u, p, θ) ^{from ertain point of view.}

In our approah, we require that (u, p, θ) ^{has the same symmetry properties as} (u, p, θ)^.

Thisrequirementisimportantbeause, asunderlinedearlier,thesymmetrygroupontains

importantphysialinformationon the ow.

More preisely, themodelshouldbesuh thateahsymmetryof(1)appliedto(u, p, θ)

is alsoasymmetry of (16) appliedto (u, p, θ)^{. When itis the ase, the modelwillbesaid}

invariant. Unfortunately, as analyzed in ([8℄), the major part of existing LES turbulene

models do not omply with this requirement. In what follows, we propose new models

based onthe preservation of the symmetry groupof the equations.

Time, pressure, and pressure-temperature translations, and the galilean transforma-

tions remain symmetriesof (16) if τ ^and h ^{depend onlyon} Sand T=∇θ^:

−τ =−τ(S,T), −h=−h(S,T) ⁽¹⁷⁾

Next,rotations,reetionsandthematerialindierenearesymmetriesof(16)ifthemodel

has the following form







−τ^d=E1S +E2( AdjS)^d +E3(T⊗T)^d +E4

S(T⊗T)d

+E5

S(T⊗T)Sd

−h=E6T +E7S T +E8S²T

(18)

where the oeients Ei ^{are salar funtionsof:}

X = trS², ξ = detS, ϑ=T², ω1 =TS T, ω2 =S TS T ⁽¹⁹⁾

and Adj ^{is the adjoint operator. Next, equations (16) are invariant under the rst sale}

transformations if

τb= e^−2aτ ^and Tb = e^−4aT. ⁽²⁰⁾

Ei(X, ξ, ϑ, ω1, ω2) =X^siE_i^′(v1, v2, v3, v4) ⁽²¹⁾

where the vi^{'sare the} invariants:

v1 = ξ

X^3/2, v2 = ϑ

X², v3 = ω1

X^5/2, v4 = ω2

X^{3. (22)}

and

s1 = 0, s2 =−¹₂, s3 =−³₂, s4 =−2, s5 =−⁵₂, s6 = 0, s7 =−¹₂, s8 =−1

Finally,the seondsale transformation are symmetries of equations(16) if

E_i^′ =νFi, i= 1, ...,5 ^and E_i^′ =κFi, i= 6, ...,8.

To sum up, we get the following lass of subgrid models whih are onsistent with the

symmetry group of (1):















−τ^d=νF1S+νX^−1/2F2Adj^dS+νX^−3/2F3(T⊗T)^d +νX⁻²F4[S(T⊗T)]^d] +νX^−5/2F5S[(T⊗T)S]^d]

−h=κ

F6+X^−1/2F7S+X⁻¹F8S² T

(23)

In order toredue the degree offreedom of the model,wepropose torestrainthe lass

(23) to models whih derive from a potential. This restrition is legitimated by the fat

that, like τr ^andhr^, τ ^and h ^represent respetivelya (subgrid)stress and a(subgrid) heat ux. Moreover,τr ^andhr ^{arederived fromthepotentials}ν trS² andκ||T||²/2^,inthesense

that

τ_r= ∂ν trS²

∂S

and hr = ∂

∂T 1 2κ||T||²

!

(24)

Hene, τ ^and h ^{should also be derived from} potentials. This ondition leads to the followinglass of models (see [8℄):































−τ^d=ν

"

2g_m−3v1

∂g_m

∂v1

−4v2

∂g_m

∂v2

−5v3

∂g_m

∂v3

−6v4

∂g_m

∂v4

# S

+ν

"

X^−1/2∂g_m

∂v1

Adj^dS+X^−3/2∂g_m

∂v3

(T⊗T)^d+ 2X⁻²∂g_m

∂v4

[S(T⊗T)]^d

#

−h=κ ∂gt

∂v2

I3+X^−1/2∂gt

∂v3

S+X⁻¹∂gt

∂v4

S²

! T

(25)

where gm ^and gt ^{are arbitrary funtions of the} vi^{'s. Note that the assumption that the}

modelderives froma onvex potential ensures the stability of the model[8℄.

Further simpliationsan be done on lass (25) aording to the type of modelthat

we wish toobtain. Forexample,

• ^if gm ^and gt ^{are only funtionsof} v1 ^and v2,^{we get a strongly oupledmodel:}

−τ^d=ν 2gm−3v1

∂g_m

∂v1

−4v2

∂g_m

∂v2

!

S+ν 1

||S||

∂g_m

∂v1

Adj^dS, −h=κhtT ⁽²⁶⁾

whereht= ∂g_t

∂v2

.

• ^{Adeoupled modelan be obtainedif} gm ^{isfuntiononlyof} v1 ^andgt^{funtion of}v2^:

−τ^d=ν(2g_m−3vg˙_m)S+ν 1

||S|| g˙_m Adj^dS, −h=κh_t T. ⁽²⁷⁾

• ^{Finally,we get a linear modelwhen} gm ^and ht ^{are linearfuntions of} v^:

−τ^d=νC_m −detS 1 S

³S+ Adj^dS 1 S

!

, −h=κC_tdetS S

³ T ⁽²⁸⁾

where Cm ^are Ct ^{are the onstants of the model, whih may depend on the grid}

size. These onstants may also be evaluated dynamially, using the Germano-Lilly

proedure.

The simpliest model (28) (alled invariant model) and its dynamial version (alled

dynami invariant model) have been implemented for the simulation of an air ow in a

ventilated 3D room (see Figure 1). The oor is heated at 35^◦C ^{and the other walls are}

maintainedat15^◦C^{. The ode used forthe simulationis basedona nitevolumesheme,}

with Crank-Niholson sheme in time [1℄. The grid size is (86×86×12)^.

Figure 1: Ventilated andheated room.

Themeanveloityandtemperatureprolesalongthevertialline(x= 0.502, z = 0.35)

are shown on Figure2. A goodagreement with experimentaldata [10℄, and better results

ompared to Smagorinsky and its dynamial version an be observed. These results are

partiularlygoodnear the walls(no wallmodelwere used). This may beexplainedby the

preservation of saling laws in the onstrution of the invariantmodels.

4 Conlusions

Weshowed that the symmetrytheory providesapowerfullmodellingtoolinuid mehan-

is, as well for theoretial as for numerial developpements. In partiular, we saw that

it ould be used to establish saling laws, whih are essential for the understanding of

turbulene. The symmetry approahhas alsobeenused forthe onstrutionof turbulene

models whih naturallyhave goodphysialproperties, sine they preserve the symmetries

of the equations. The numerialtest onrms this goodbehaviour.

Figure 2: Mean streamwise veloity(left)and meantemperature (right)at x ^=0.502m.

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