A new Age for Turbulence: DNS, LES, URANS ... ˆ ONERA Chatillon, France, 02 December 2011
The new partially integrated transport modeling (PITM) method for continuous hybrid non-zonal RANS/LES simulations of turbulent flows
ONERA – 11/2011
Bruno CHAOUAT, ONERA, France
1
OUTLINE
From RANS to LES modeling Partial Integrated Transport Modeling (PITM) method: Hybrid RANS/LES simulations – Mathematical physics formalism developed in the spectral space – Transport equation for the subfilter scale stress – Transport equation for the subfilter dissipation rate
Engineering applications – Injection induced flows (space launchers) – Channel flow with streamwise constrictions (aeronautics industry)
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– Channel flows subjected to spanwise rotation (turbomachinery)
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FROM RANS TO LES MODELING
RANS modeling : many contributions in the past forty years – First and second order closures (Launder, Lumley, Speziale, Gatski, Rodi et al...)
Academic large eddy simulation – Smagorinsky (1963), dynamic Smagorinsky (Piomelli and Germano, 1991) – Structure-function model (Lesieur et al. , 1996) etc ....
Hybrid zonal approach – Detached-Eddy simulation DES (Spalart et al., 2000)
Hybrid continuous approach – PITM method (Schiestel, Chaouat, Dejoan 2005-2011) – TPITM method (Manceau, Gatski, Fadai-Ghotbi et al., 2007-2011)
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– Scale-adaptative simulation SAS (Menter et al., 2005-2011) – PANS method (Girimaji et al., 2006-2011 )
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TURBULENCE MODELING
Transport equation for the statistical velocity hui iRANS approach hui i 1 2 hui i hpi + hu i hu i = + t
with ij
xj
i
xi
j
xj xj
ij xj
(1)
= hui uj i hui i huj i
Transport equation for the filtered velocity ui LES and continuous HYBRID approaches ui p 1 2 ui ( ij )sf s (2) + (u u ) = + t
with
xj
i j
xi
xj xj
xj
( ij )sf s = ui uj ui uj
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Second order closure is based on the transport equation of the tensor ij or ( ij )sf s
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PARTIALLY INTEGRATED TRANSPORT MODELING (PITM) METHOD
Objective: to perform large eddy simulations of turbulent flows on relatively coarse grids Bridge between URANS and LES method with seamless coupling Self consistency of the PITM method obtained when the cutoff location continuously varied between two extreme limits (DNS/PITM/RANS)
lim !0[( ij )sf s ℄ = ( ij )RAN S
(3)
lim !1[( ij )sf s ℄ = 0
(4)
Definition of the subfilter-scale tensor ( ij )sf s = ui uj ui uj Definition of the resolved scale tensor ( ij )les = ui uj hui i huj i where h:i denotes the ONERA – 11/2011
statistical average
Definition of the Reynolds stress tensor ij including the small and large scale fluctuating velocities ij = h( ij )sf s i + h( ij )les i 5
MATHEMATICAL PHYSICS FORMALISM IN SPECTRAL SPACE
Cooperation between ONERA (Chaouat) and CNRS/IRPHE (Schiestel) – Spectral partitioning (m = number of zones), definition of filtered and averaged quantities
ui = hui i +
N X 0 (m)
=1
ui
0 (m)
; ui () =
m
Z
jj
m 1 < subgrid-scale fluctuating velocity: ui = ui
– Simulation LES (m=2) : filtered velocity:
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MATHEMATICAL PHYSICS FORMALISM IN SPECTRAL SPACE
Two-point velocity fluctuating correlation for non-homogeneous turbulence
0 0 Rij = uiA ujB (xA ; xB ) (Hinze, 1975) New independent variables xA – vector difference = xB – midway position X = 12 (xA + xB )
0 0 Transport equation for the tensor Rij = uiA ujB (X ; ) Taylor series development for the mean velocity (framework of tangent homogeneous spectral space, Schiestel, 1987; Chaouat and Schiestel, 2007)
Fourier transform of the transport equation for the tensor Rdij (; X ) Integration on a spherical shell in the wave numbers (Schiestel, 1987; Cambon et al., 1992; ONERA – 11/2011
Chaouat and Schiestel, 2007)
1 'ij (; X ) = (Rij (X ; )) = A()
ZZ A
d R ij (; X )dA()
(6)
Partial integrations on the wavenumbers to return in the physical space 7
MATHEMATICAL PHYSICS FORMALISM IN SPECTRAL SPACE
Resulting equation in the spectral space by mean integrations over spherical shells, 'ij (X ; ) = (Rij (X ; )) (Chaouat and Schiestel, 2007) D'ij (X ; ) = P (X ; ) + T (X ; ) + (X ; ) + J (X ; ) E (X ; ) Dt
ij
ij
ij
ij
ij
(7)
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Production term Pij , Transfer term Tij , Redistribution term ij , Diffusion term Jij , Dissipation term Eij
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MATHEMATICAL PHYSICS FORMALISM IN SPECTRAL SPACE
Integration in the spectral space for recovering physical space quantities – Statistical one-scale models in the physical space
Rij =
1
Z
0
'ij (X ; )d =>
DRij = Pij + ij + Jij Dt
ij
(8)
– Statistical multiple-scale models in the physical space, (Schiestel, 1987)
R(m) = ij
Z
m
m
DRij(m) = Pij(m) +Fij(m 1) Fij(m) +(ijm) +Jij(m) (ijm) 'ij (X ; )d => Dt 1 (9)
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– Subfilter-scale models in the physical space, (Chaouat and Schiestel, 2005)
ij )sf s i (2) = Pij(2) +Fij(1) Fij(2) +(2) + J h( ij )sf s i = u>i u>j = Rij(2) => D h( Dt ij ij (10) D( ij )sf s by analogy : (11) = ::::
Dt
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MATHEMATICAL PHYSICS FORMALISM IN PHYSICAL SPACE
Resulting equation in the physical space, (Chaouat and Schiestel, 2007) DRij(m) = Pij(m) + Fij(m 1) Fij(m) + (ijm) + Jij(m) (ijm) Dt where
P (m) = ij
Z
m
m 1
huj i R(m) hui i ; Pij (X ; )d = Rik(m) x jk x k
Fij(m) = Fij(m) 'ij (X ; ) m t and
(m) =
Z
ij
J (m) = ONERA – 11/2011
ij
m
m 1
Z
ij
(m) =
and
Z
m
m 1 m m 1
F (m) = ij
Z
0
(12)
(13)
k
m
Tij (X ; )d;
(14)
ij (X ; )d;
(15)
Jij (X ; )d;
(16)
Eij (X ; )d:
(17)
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PARTIALLY INTEGRATED TRANSPORT MODELING (PITM) Production zone
E
Inertial zone
Dissipation zone
0
kc
kd
8
0
kc
kd
8
0
kc
kd
8
k-5/3
Fe
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T
Figure 1: Turbulent processes in the spectral space
Wavenumber ranges such as [0; ℄, [ ; d ℄ and [d; 1[ 11
PARTIALLY INTEGRATED TRANSPORT MODELING (PITM) METHOD
Transport equation for the partial turbulent energy
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Dk(m) = P (m) + F (m 1) F (m) + J (m) (m) Dt Wavenumber ranges such as [0; ℄, [ ; d ℄ and [d; 1[ (k hksf s i) = P (1) F (1) ( ) t hksf s i = P (2) F (2) (d ) + F (1) ( ) t 0 = F (2) (d) (3) where (3) = sf s . Equation (21) indicates that the dissipation rate can indeed be
(18)
(19) (20) (21)
interpreted as a spectral flux.
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SPECTRAL SPLITTING Ln E
F (κ c )
F (κ d ) k − k sgs
k sgs κc
κd
κ
Figure 2: Turbulent processes in the spectral space The splitting wavenumber d is related to the cutoff wavenumber by the relation:
d =
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In the case of full statistical modeling where
hksf s i3=2
(22)
= 0, equation (22) is reduced to the equation:
d = d 3=2 k
(23)
where d is located after the inertial range.
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FIRST FORMULATION OF THE DISSIPATION-RATE The dissipation rate equation is then obtained by taking the derivative of equation
d =
hksf s i3=2
One can easily obtain:
(2) 2 (1) = P + F ( ) sf s2 t sf s1 hksf s i hksf s i
(25)
= 3=2 and hksf s i F (d ) F (d E (d) F ( ) F ( ) 3
sf s2 = 2 (d ) E (d ) E ( )
where sf s1
(26)
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Setting E
( ) E (d ) and d , then 3 h ksf s i F (d ) F (d
sf s2 2 dE (d )
(27)
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SECOND FORMULATION OF THE DISSIPATION-RATE In the case of full statistical modeling where
= 0, Taking the time derivative of equation
d = d 3=2 k
yields another formulation of the dissipation rate equation:
(1) (2) = P +P t 1 k where 1
2
2 k
(29)
= 3=2 and k F ( d ) F (d ) 3
2 = 2 d E (d)
(30)
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This is in fact the usual equation used in statistical closures.
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FUNCTION
Csfs2
Analytical expression of the function sf s2 3 h ksf s i F (d ) F (d )
sf s2 2 dE (d ) k F ( d ) F (d ) 3
2 = 2 d E (d)
= + hksf s i ( ) sf s2
1
k
2
1
(31)
(32)
(33)
Equations (31) and (32) show that the coefficients sf s2 and 2 are functions of the spectrum shape
ONERA – 11/2011
Note that the fluxes F and F can be analytically computed in some particular situations Computation of ratio hksf s i =k 2 L3e k 2 3 E () = [1 + ( Le )3 )℄11=9
(34)
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PITM METHOD
Instantaneous transport equations and practical formulations Dksf s = Psf s sf s + Jsf s Dt
–
D( ij )sf s = (Pij )sf s (ij )sf s + (ij )sf s + (Jij )sf s Dt 2sf s sf s (Pmm )sf s Dsf s = 1 k
sf s2 ( ) + (J)sf s Dt 2 ksf s sf s “Exact ” coefficient 2 where = (k 3=2 )=(sf s + < )
sf s2 ( ) = 1 +
2 1 [1 + 3 ℄2=9
– Dynamic procedure (Friess et al., 2010) with rEq
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(35)
Æ sf s2 = sf s2 1
(36)
(37)
(38)
= (hksf s i =k)Eq rCF D rEq
(39)
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NUMERICAL METHOD
Three-dimensional compressible code for solving (5+7) transport equations (Chaouat, 2010) – –
, u1 , u2 , u3 , E ( 11 )sf s , ( 12 )sf s , ( 13 )sf s , ( 22 )sf s , ( 23 )sf s , ( 33 )sf s , sf s
Finite volumes technique: fluxes conservative method X U 1 = v( ) (F Fv)A + S t
(40)
F and Fv represent respectively the convective and viscous fluxes through the surfaces A around the control volume v ( ), n is the unit vector normal to the surface A and S is the source term.
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where
Fourth order Runge-Kutta scheme in time discretization Implicit scheme in time for the treatment of the turbulent equations Second and fourth order centered schemes in space discretization (MUSCL scheme) CPU time : the subfilter-scale stress model (7 transport equations) only requires 25 % more time than the viscosity model (2 transport equations)
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INJECTION INDUCED FLOWS
No slip condition for the upper surface Injection condition for the lower surface Numerical simulations performed on different grids – PITM1 (400 32 80) 1:0 106 – PITM2 (400 44 80) 1:4 106 Comparison with highly resolved LES (Apte and Yang, 2003) performed on a refined grid 8:4 106 (NLES =NP IT M 2 = 6) x
3 x2 x1
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δ
U1 U3
Figure 3: Schematic of channel flow with fluid injection.
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INJECTION INDUCED FLOWS
Figure 4: Isosurfaces of instantaneous filtered vorticity
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direction (i=2)
!i = ijk uk =xj
j!2 j = 3000 (1/s). (Chaouat and Schiestel, 2007)
in the spanwise
Transitional laminar-turbulent flow The three-dimensional structures are squeezed upward in the normal direction to the wall
20
INJECTION INDUCED FLOWS 150
100
50
0
0
0.2
0.4
0.6
0.8
1
X3/δ
Figure 5: Mean velocity profiles in different cross sections.
x1 = 12 cm: O; 22 cm: ; 35 cm:
; 40 cm : +; 45 cm: 2; 50 cm: ; 57 cm: Æ. —–: PITM; Symbols: experimental data (Avalon, ONERA – 11/2011
2000)
21
INJECTION INDUCED FLOWS 0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0
0
0.2
0.4 X3/δ
(a)
0.6
0.8
1
0
0
0.2
0.4
0.6
0.8
X3/δ
(b)
1
0
0
0.2
0.4
0.6
X3/δ
(c)
0.8
1
0
0
0.2
0.4
0.6
0.8
1
X3/δ
(d)
( 11 ) 21 > =um in different cross sections (a) x1 = 40 cm; (b) 45 cm; (c) 50 cm; (d) 57 cm. + : PITM ; : experimental data (Avalon, 2000)
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Figure 6: Streamwise turbulent stresses
