BASIC ANALYSIS OF SOME SECOND MOMENT CLOSURES PART II : INCOMPRESSIBLE TURBULENT FLOWS INCLUDING BUOYANT EFFECTS

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TURBULENT FLOWS INCLUDING BUOYANT EFFECTS

Jean-Marc Hérard

To cite this version:

Jean-Marc Hérard. BASIC ANALYSIS OF SOME SECOND MOMENT CLOSURES PART II : INCOMPRESSIBLE TURBULENT FLOWS INCLUDING BUOYANT EFFECTS. 1994. �hal- 02007060�

BASIC ANALYSIS OF SOME SECOND MOMENT CLOSURES

PART II : INCOMPRESSIBLE TURBULENT FLOWS INCLUDING BUOYANT EFFECTS

Jean-Marc Hérard^1,2

1Electricité de France - Recherche et Développement Département Mécanique des Fluides et Transferts Thermiques

6, quai Watier. 7840&. Chatou Cedex. France.

Jean-Marc.Herard@edf.fr

2Université de Provence

Centre de Mathématique et d’Informatique 39, rue Joliot Curie.13453. Marseille Cedex 13. France

Herard@cmi.univ-mrs.fr

Abstract :

The suitability of some second moment closures is discussed herein. A general frame of strongly realisable models is exhibited, and the ability of so-called "slow" terms to provide return-to-isotropy is examined. Hyperbolicity of first order differential systems is then investigated. When focusing on a simple so-called Gaussian closure, the solution of the Riemann problem associated to the generalised convection system is detailed, which confirms that realisability still holds when non regular solutions are involved. A priori suitable numerical consequences are then discussed. These results should be related to recent work pertaining to the numerical modelling of compressible flows using second order closures.

Nomenclature

Introduction

I Basic set of equations and main requirements.

I.1 Governing equations I.2 Realisability requirements

I.3 Maximum principle for the mean temperature and scalar variance II Some more about isothermal turbulent flows

III A simple Gaussian closure for non isothermal turbulent flows

IV The eigenvalue problem

V Analysis of the generalised convection system

VI Discussion

Conclusion

Références

Appendix I

Appendix II

Appendix III

Appendix IV

Nomenclature :

f,i Partial derivative of f with respect to xi

f,t Partial derivative of f with respect to t

f = < f > + f' Reynolds decomposition

D_t f =f_,t + <U_i>f_,i Convective derivative of f function

<p> Mean pressure

<T> Mean temperature

<U_i> Mean velocity in xi direction

R_ij = <u_i u_j> (∈R3x3) Reynolds stress tensor

X_j= <θ u_j> Turbulent heat flux

<θ²> Scalar (temperature) variance

Z_ij = <θ²> R_ij - X_i X_j (∈R3x3)

I R = 2 K = trace ( R ) Turbulent kinetic energy

II R = trace ( R2) Second invariant of R

III R = trace ( R3) Third invariant of R

B_ij=U_i,j S_ij= U_i,j+U_j,i

G_ij= <u_i u_k>U_k,j+ <u_j u_k>U_k,i= (R B + B^t R)_ij P_ij= <u_i u_k >U_j,k + <u_j u_k>U_i,k= (R B^t+ B R)_ij P = 1

2 P_ii= 1 2 G_ii

ε = ν<u_i,ku_i,k> Dissipation rate of K

ε_θ=λ<θ_,k θ_,k > Dissipation rate of <θ²>

τK= (ε / K)^-1 Turbulent mechanical time scale

τ_θ= (ε_θ/ <θ²>)^-1 Turbulent scalar time scale

For i=1-> 3 : δ1

i = R_ii First fundamental minor of R (no summation) For i=1-> 3 :

δ₂ ⁱ = R_αα R_ββ - (R_αβ)²

Second fundamental minor of R (no sum- -mation), α and β non equal, non equal to i

δ3 = det ( R_ij ) Third fundamental minor of R

For i=1-> 3 :

∆2

i = Z_αα Z_ββ - (Z_αβ)²

Second fundamental minor of Z (no sum- -mation), α and β non equal, non equal to i

∆3 = det ( Z_ij ) Third fundamental minor of Z

Introduction

Statistical modelling of turbulence has generated the need for computation of complex sets of partial differential equations. This is mainly due to the highly non linear terms present in instantaneous Navier Stokes equations on which is based such a modelling process. This is true of course for both compressible and incompressible flow patterns. In both cases, numerical simulation requires applying for stable and accurate enough schemes in order to ensure convergence towards the right solution. When focusing on compressible flows, it is now well admitted that upwinding techniques have enabledgaining sufficient stabilizing effects to allow computation of flows including strong shocks, even when viscous effects are of small amplitude or even vanishing . It does not necessarily mean that these provide the ultimate numerical approach for such a purpose, since the balance between stability and accuracy is clearly in favour of the former, and consequently may penalize improvement of accuracy. Nonetheless, use of standard reconstruction techniques (such as MUSCL approach, or ENO schemes,...) has proved to be an efficient way to handle the whole. From a technical point of view, these upwinding techniques rely on the theory of hyperbolic systems of conservation laws, and have been recently extended to the framework of non linear hyperbolic systems which do not have any conservation form. Clearly, approximate Riemann solvers are almost overwhelmingly present in the frame of compressible flows, but are seldomly used in the frame of incompressible flows. They have been mainly used in the framework of conservative Euler -or Navier Stokes- equations for gas dynamics. More recently, they also have been applied for investigation of magneto hydrodynamics, and also to deal with averaged Navier Stokes equations with statistical Favre averaging. A first step has been devoted to the K or K-e type closures ( ? ? ? ?°), and in a second step to realisable second-moment closures ( ? ? ? ?). A unifying approach based on the entropy concept was also proposed (see ? ?) in order to cope with first or second order closures. These methods have been successfully implemented in industrial codes using unstructured Finite Volume approach (N3S-NATUR ? ?, PLEXUS ? ,Code_Saturne ? ?). It has also been demonstrated that standard techniques based on Euler type algorithms lead to unstable computations (when focusing on impninging jets on walls for instance, see ? ? ? and ? ? among others) whiwh lead to blow up of codes. These techniques may also be used to enforce the coupling between Eulerian and Lagrangian approaches ( see ? ?, ? ?, ? ? and ? ? for instance). They are robust enough to be implement

We examine here the suitability of some second moment (so-called Gaussian) closures involved in systems of partial differential equations which aim at predicting the behaviour of incompressible turbulent flows including buoyant effects. As in /5/, special emphasis is given on the realisability conditions and to the link between realisability requirement and hyperbolicity constraints. Present contribution is organised as follows. In section I, governing equations and realisability requirements are briefly recalled. In section II, the class of strongly realisable closures initially defined in /5/ is

concerning return to isotropy process are also recalled. In section III, a simple closure is proposed, which enables to achieve strong realisability in the non isothermal case. Section IV is devoted to the examination of the general eigenvalue problem, which confirms that realisability is compulsary to gain hyperbolicity. Since the result stated in section III requires the boundedness of two different tensors, which might be violated, especially when dealing with "non-viscous" Gaussian closures, the Riemann problem associated with the initial value problem obtained by getting rid off continuity constraint and eliminating mean pressure variable is examined in section V ; results obtained show that over-realisability still holds (in a standard sense) but that the maximum principle for mean temperature is no longer valid. Some numerical consequences of these results are given in the last section.

I - Basic set of equations and main requirements :

I.1 Governing equations

For a wide variety of problems, temperature field is assumed not to have any significant influence on the velocity field, except through external gravity effects, which stands for usual Boussinesq approximation. Hence, instantaneous velocity field remains divergence free and the governing equation for the instantaneous temperature partially decouples with the remaining equations. Thus, no compressibility effect is accounted for in the present study, and the basic set of equations used to construct turbulent model writes :

U_i,i= 0 (1.1)

U_{i ,t}+ U_j U_{i , j}= (-p ρ0

δ_ij +ν (U_{i , j}+U_{j, i}))

,j

+β_i (T - T₀)

(1.2) T_,t+ U_j T_{, j}= (λ T_{, j} )_,j (1.3)

Considering standard statistical approach and focusing on one point closures, the latter enable to derive the well-known unclosed set of equations to describe mean velocity, pressure and temperature fields, i.e. :

<Ui >,i= 0 (2.1)

<U_i>_,t+ <U_j> <U_i>_{, j}= (<Σij>)_,j- (R_ij)_,j+βi(<T > - T₀) (2.2)

<T >_,t+ <U_j> <T >_{, j}= (λ <T >_{, j} )_,j- X_j,j (2.3) where :

<Σ_ij> = -<p >

ρ₀ δ_ij +ν ( <U_i>_,j+ <U_j>_,i)

In order to close previous set, governing equations for second moment unknowns Rij, Xj and

<θ²> are needed ; these are listed below :

(R_ij )_,t + <U_k> (R_ij)_,k + P_ij - (βi X_j+βj X_i) =Φij- (d_ijk )_,k (2.4) (Xi)_,t + <Uk> (Xi)_,k + Xk <Ui>,k + Rik <T >_,k- βi<θ²> =Φi- (dik )_,k (2.5)

<θ²>_,t + <U_k> <θ²>_,k +2 X_k <T >_,k =Φ - (d_k )_,k (2.6) while noting :

Φij = < u_jΣik,k

' + u_iΣjk,k

' > d_ijk = <u_iu_ju_k > ₍₃₎

Φ_i = < θ Σ_ik^' _,k > +λ<u_iθ_,kk > d_ik = <u_iθ u_k >

Φ = 2λ <θ θ_,kk > _{dk = <}θ θ u_k >

Hence, set (2) requires the modelling of the Φij,Φi, Φ _, d_ijk ,d_ik, d_k contributions.

I.2 Realisability requirements

When dealing with non isothermal flows, the following realisability condition should hold (see /8/), which is:

A =

R₁₁ R₁₂ R₂₁ R₂₂

R₁₃ X₁ R₂₃ X₂

R₃₁ R₃₂ X₁ X₂

R₃₃ X₃

X₃ <θ²>

= R X

X^t <θ²>

must be positive half definite. Provided that < θ2> is non zero, the latter may be premultiplied on the left by the non singular matrix :

B = I -<θ²>^-1 X

0 1

and on the right by Bt ; thus :

∀ Z ∈ R4, Z^t A Z ≥0

is achieved as soon as (see /8/ for similar statement) :

∀ Y ∈ R₄, Y^t R-<θ²>^-1X X^t 0

0 <θ²>

Y ≥0 Hence, provided that < θ2> is non zero :

(i) Z= (<θ²>R-X X^t) must be half definite positive (4.1)

(ii)<θ²> must be positive (4.2)

Moreover, if < θ2> is zero :

(i)' R must be half definite positive (ii)'X^t X must be zero

Note that (i) implies :

trace (Z) = <θ²> trace (R)-X^tX ≥ 0

Thus, (ii)' is contained in (i). Note also that for non vanishing value of < θ2> which fulfills (ii), (i)' holds since :

R = <θ²>^-1(Z+X X^t)

From now on, the limit case of vanishing value of scalar variance will be omitted, and (4.1) and (4.2) will be accounted for while ensuring that < θ2> and the three fundamental minors of Z (namely ∆₁¹, ∆₂³, ∆₃) remain positive (which stands for over-realisability concept as defined in /5/). Moreover, closures will be chosen in such a way that :

f = 0 ( D_t f = 0 and : D_t(D_t f ) ≥0) for : f =∆₁¹, ∆₂³, ∆_{3 (5)} Obviously, closures involving time scales τKand τ_θ should be such that :

τK> 0 and :τ_θ> 0

I.3 Maximum principle for mean temperature and scalar variance

Going back to the original system (1.1, 1.2, 1.3), and considering micro-gravity effects for sake of simplicity, we may expect that slightly different initial (or boundary) conditions will provide different solutions for (k) or (k+1) experiments ; denoting by U(k)(x,t), T(k)(x,t), P(k)(x,t) solutions associated to (k) experiment, it occurs that T(k)(x,t) will be solution of : T_,t^(k)+ U_j^(k) T_{, j}^(k)= (λ T_{, j}^(k) )_,j

where : (x, t )∈Ω x (0,T )

with given initial and boundary conditions, which may be of Dirichlet type i.e. : T^(k)(x, t = 0) = T₀^(k)(x ) for :x ∈Ω

T^(k)(x, t > 0) = T_δΩ^(k)(x,t ) for : (x,t)∈ δΩ x (0,T )

Provided that there exists two real constant values (independant of statistics) such that : T_low≤ T_δΩ^(k)(x,t ) ≤T_up and : T_low ≤ T₀^(k)(x) ≤T_up

then T(k)(x,t) will be such that : T_low≤ T^(k)(x,t ) ≤T_up

Consequently, the following constraint for the average value <T> (x,t) should hold :

T_low≤<T>(x,t ) ≤ T_{up (6)}

Moreover, (<T>- T(k))(x,t) will be such that : 0≤(T^(k)-<T>)²(x,t ) ≤ (T_low -T_up)² Hence, scalar variance <θ2> should fullfill :

0≤<θ²>(x,t ) ≤ (T_low -T_up)² (7)

It must be emphasized that the latter still holds when external (buoyancy) forces are accounted for. Note anyway that constraints (6) and (7) are not intrinsic to mean temperature and scalar variance entities, unlike (4.1) and (4.2) ; they occur in fact to stand for some counterpart of (5), in the sense that they arise while focusing on the instantaneous governing equations (1.1) to (1.3).

II Some more about isothermal turbulent flows

Before going ahead, let us go back to the isothermal case. In a previous work, it was suggested that the following closure of Reynolds stress governing equations enable to achieve strong realisability :

Φ_ij=α₂ R_ij +α₃ R_ij² +α₅( R_ik <U_j>_,k+ R_jk <U_i>_,k ) (8.1) +α6(R_ik <U_k >_,j+R_jk <U_k >_,i)

However, a few questions naturally arise. First of all, do "slow terms" enable to retrieve

"return to isotropy" process which appears to be an important feature actually occuring in basic experimental apparatus ? Second, this closure obviously no longer contains tensorial forms proportionnal to either (δij) or ( <U_i>_,j+ <U_j>_,i) ; does that mean that the latter may not appear in any formal development ? Third, and though it also does not contain any (δij ) or ( <U_i>_,j+ <U_j>_,i) terms, another proposal described in /11/ (see /12/ also), which fulfils realisability requirement writes :

Φij

SL= -6

5 q² ( R_lk <U_l>_,k ) R_ij + 3

5 ( R_ik <U_j>_,k+ R_jk <U_i>_,k )

+ 2

5 q² ( R_ik² <U_j>_,k+ R_jk² <U_i>_,k - (R_jk R_li+ R_ik R_lj)<U_l>_,k)

(8.2) which actually differs from the one above ; does the whole make emerge any contradiction ? As far as second and third items are concerned, it seems that the answer dwells in a broadened frame of strongly realisable closures :

φij = 2β0 R_ij + 2β1 R_ij ² + 2β2 R_ij ³ +β3(RB + B^tR)_ij +β₄ (B R + RB^t)_ij +β₅ (R²B + B^tR²)_ij

+(β₆+β₈)(R(B + B^t)R)_ij +β₇(R²B^t+ B R²)_ij +β₉(R³B + B^tR³)_ij +β₁₀ (R²B^t R + R B R²)_ij +β₁₁ (R³B^t + B R³)_ij +β₁₂ (R²B R + R B^t R²)_ij +β13 (R²B R + R B^t R²)_ij +β14 (R²B^t R + R B R²)_ij (8.3) where β_i 's functions should obey some constraints and B should remain bounded (see appendix 1 for proof). Using Cayley-Hamiltonian identity provides a counterpart of the latter which makes "explicitely" arise (δij ) or ( <U_i>_,j+ <U_j>_,i) contributions. Even more, it must be noted that both (8.1) and (8.2) are included in the latter.

If we turn now to return-to-isotropy topic, it may be checked that the behaviour of (realisable) solutions of :

(R_ij)_,t=α2 R_ij +α3 R_ij² ₍₉₎

is such that return to isotropy is achieved in the purely two-dimensional case, and in a weaker sense in the three-dimensional case, provided that first and second (negative) co-factor

functions are chosen in a suitable way. Previous result still holds in the three dimensional case, when dealing with the most general expansion of "slow" part :

(R_ij)_,t=α₂ R_ij +α₃ R_ij²+α₄ R_ij³

provided that third co-factor function remains negative. Proof is given in appendix 3. Note that the present analysis also confirms the need for boundedness of the inverse of turbulent (mechanical) time scale τ_k. The anisotropy of the Reynolds stress tensor is indeed a very important feature since traceless part of R which writes :

a_ij = R_ij- I 3 δ_ij

explicitely contributes to the budget of rotational of mean velocity field :

Ωi,t+ <U_j>Ωi, j +Ωl<U_i>_{, l} -ν Ωi, ll= -εijk (a_lj)_,lk (noting : Ωi = εijk (<U_j> )_,k)

III A simple Gaussian closure for non isothermal turbulent flows

We examine here a very simple so-called Gaussian closure, setting : d_ijk = <u_iu_ju_k> = 0 _; d_ik = <u_iθ u_k > = 0 _; d_k = <θ θ u_k > = 0 and :

Φ_ij=α₂ R_ij +α₃ R_ij² +α₅( R_ik <U_j >_,k+ R_jk <U_i>_,k ) +α₆( R_ik <U_k >_,j+ R_jk <U_k >_,i) Φ_i=(α2+β1)

2 X_i+α₃(R_ik X_k - (X_kX_k) 2 <θ²>

X_i) +α₅ X_k <U_i>_,k +α₆ X_k <U_k >_,i

Φ =β1<θ²>

with :

α2 =α₂^' (I_R, II_R, III_R ) ε

I_R ; α3 =α₃^' (I_R, II_R, III_R) ε II_R αi =α_i^' (I_R , II_R , III_R) for : i = 5, 6

β₁ =β₁^' (I_R , II_R , III_R ) ε_θ

<θ²>

where all primed functions stand for non dimensional bounded functions. We have the following result, defining :

C_ij= X_i<T >_,j/ <θ²> ₍₁₀₎

Prop. III :

Provided that the inverses of turbulent mechanical and scalar time scales, and that Bij's and Cij's remain bounded, set (2) associated to above Gaussian closure is strongly realisable.

This represents a straightforward extension of the one stated in /5/. The proof may be obtained as follows. First, (2.4, 2.5, 2.6) may be rewritten in :

(Z_ij )_,t + <U_k> (Z_ij)_,k +Z_ik <U_j>_,k +Z_jk <U_i>_,k + 2 Z_ij (X_k <T>_,k

<θ²>

) - Z_ik (X_j<T>_,k

<θ²>

) - Z_jk (X_i<T>_,k

<θ²>

) = RHS_ij

<θ²>_,t + <U_k> <θ²>_,k +2 X_k <T >_,k =Φ

(X_i)_,t + <U_k> (X_i)_,k + X_k <U_i>_,k+ R_ik <T >_,k - βi<θ²> =Φi

noting :

RHS_ij = α5(Z_ik <U_j>_,k +Z_jk <U_i>_,k ) +α6(Z_ik <U_k>_,j+Z_jk <U_k>_,i) + (α2+β1) Z_ij +α3 Z_ij²

<θ²>

Introducing :

H_kj = (α₅ -1) <U_j>_,k +α₆ <U_k>_,j+ C_jk + (α₂+β₁

2 - C_ll)δ_jk +α₃ Z_kj 2 <θ²>

enables to rewrite :

(Z_ij )_,t + <U_k> (Z_ij)_,k =Z_ik H_kj +Z_jk H_ki and :

<θ²>_,t + <U_k> <θ²>_,k = (β1- 2 C_ll) <θ²>

The latter enables to ensure that the scalar variance will remain positive, due to the boundedness of both inverse of turbulent scalar time scale and Cij 's components. Moreover, applying for appendix 1 permits to conclude that the model is strongly realisable, provided that, in addition, inverse of turbulent mechanical time scale and the Bij 's remain bounded, since :

α3

Z_kj 2 <θ²>

=α₃^' (I_R, II_R , III_R ) (ε I_R) ( I_R²

2II_R) (R_kj

I_R - X_j X_k I_R<θ²>

)

with : 1≤ I_R²

II_R ≤3 and : abs ( X_j X_k I_R<θ²>

)≤R_jj^1/2 R_kk^1/2 I_R ≤1

2

It must be underlined anyway that the boundedness of Cij components does not arise naturally since Schwarz inequalities only provide :

abs (C_ij)≤ R_ii^1/2 abs ( <T >_,j)

<θ²>^1/2

This will be implicitely revisited in section V.

IV - The eigenvalue problem

We focus now on the following non viscous realisable system while neglecting zeroth order terms associated to gravity, which writes :

<U_i >_,i= 0 (11)

<U_i>_,t+ <U_j> <U_i>_{, j}+ (R_ij)_,j+ <p >_{, i}= 0 (R_ij )_,t + <U_k> (R_ij)_,k + P_ij- Φij

r= 0

(X_i)_,t + <U_k> (X_i)_,k + X_k <U_i>_,k+ R_ik <T >_,k - Φi r= 0

<θ²>_,t + <U_k> <θ²>_,k +2 X_k <T >_,k = 0

<T >_,t+ <U_j> <T >_{, j}+X_j,j= 0 noting :

Φij

r= α5( R_ik <U_j>_,k+ R_jk <U_i>_,k ) +α6 ( R_ik <U_k>_,j+ R_jk <U_k>_,i) Φi

r=α5 X_k <U_i>_,k + α6 X_k <U_k >_,i We assume here that :

α_i=α_i^' (I_R, II_R, III_R) for : i = 5, 6

We consider two-dimensional turbulence i.e. :

<U₃> (x, y, z, t ) = 0 (H1)

<u₁u₃> (x, y, z, t ) = <u₂u₃> (x, y, z, t ) = <θu₃> (x, y, z, t ) = 0 (H2)

<φ>_,3 (x, y, z, t ) = 0 , whatever φ stands for. (H3)

Provided that zero Dirichlet boundary conditions and initial conditions are retained, both (H1) and (H2) are consistent with (11), which may be rewriten in the form :

A W_,t+A_xW_,x +A_yW_,y = 0 while defining :

W^t= (<U₁>,<U₂>,<u₁u₁>,<u₂u₂>,<u₁u₂>,<u₃u₃>,<p >, <θu₁>,<θu₂>,<θ²> ,<T> )∈ R11

A ∈ R11x 11 , A_x ∈ R11x 11 , A_y ∈ R11x 11

This enables to state :

Prop. IV :

System (11) is such that :

∀(nx, n_y)∈ R²/ n_x²+ n_y²= 1, det (n_xA_x + n_yA_y-λ A) = 0 λ∈ R if and only if :

∀( n_x , n_y)∈ R2, ( n_x , n_y) <u₁u₁> <u₁u₂>

<u₁u₂> <u₂u₂>

n_x n_y ≥0 and :

∀ n^t= (n_x, n_y)∈ R² , n^tCn ≥0 noting :