Dipartimento di Meccanica e Aeronautica
Università di Roma "La Sapienza"
TMRGroup @ DMA-URLS
A new Variational Multiscale high-order finite element formulation for
turbomachinery flow computations
Alessandro Corsini, Franco Rispoli, Andrea Santoriello
Dipartimento di Meccanica e Aeronautica, Università di Roma “La Sapienza”, Via Eudossiana, 18,
I00184 Roma, Italy
1. Introduction
The use of CFD for turbomachinery flow configurations remains affected by some pacing
items, mainly related to the nature of model equations that generally appear in a complete
advective-diffusive-reactive form. Diffusion, advection and reaction respectively refer to those
terms in the PDE involving second, first and zero order derivatives of the unknowns. Their
numerical discretization must adequately tackle the instability origins that stem from the
advective (Leonard, [1]) or diffusive (Gresho, [2]) limits for incompressible fluid, as well that
related to the reaction dominated flow conditions (Hughes and Harari, [3]). Though the existence
of accurate stabilization schemes for advection dominated conditions, the interest on this kind of
equation stands in their reactive character. This feature, that is ubiquitous in the numerical
modelling of several industrial processes, plays a key role in CFD, addressing the development
of formulations able to work in the more general framework of advection-diffusion-reaction
equations used to model turbomachinery turbulent flows. In this ambit reaction driven effects are
generally encountered due to the rotation of blade rows that results in non-inertial dynamic
phenomena, such as the Coriolis force in the momentum equations. Recently, we have
highlighted in [4] that significant reaction driven effects may appear in the numerical solution of
turbulence modelling due to closure equations. For instance, in first or second moment closures
absorption-like reactive contributions stem from dissipation/destruction terms. Moreover, in
elliptic relaxation based closures (e.g. k-v2-f by Durbin [5], the elliptic blending model by
Manceau and Hanjalic [6]), the blending parameter, used to combine near- and far-wall effects,
is usually modelled by means of a diffusive-reactive equation. In turbomachinery configurations,
the turbulence model related reactivity is expected to play a critical role in the boundary layer
simulation where the presence of stagnation, separation or adverse pressure gradient phenomena
gives rise to local reaction-to-advection ratio of order o(105).
Let consider a linear advective-diffusive-reactive equation. As well known, when advection
dominates Galerkin finite element methods suffer from the appearance of global spurious
oscillations, mainly in the vicinity of discontinuities (e.g. boundary or sharp layers). Such failure
has been faced by a number of stabilized finite element methods designed for advective-diffusive
equations both for linear and quadratic spaces of approximation, most of them based on a PetrovGalerkin (PG) approach such as SUPG [7-10] schemes or on Residual Free Bubble (RFB)
approaches [11].
On the other hand, in presence of high absorption-like reaction terms the Galerkin
approximations are affected by local oscillations, even in null advection, that typically do not
degrade the global solution accuracy. In this case it is not possible to obtain a global stability
estimate in the H 1 norm, though it could be evaluated in L2 , thus explaining the local scale of
the oscillations [8]. Two alternative routes could be found in literature to build-up residual based
stabilization schemes for reactive limits. The first one includes the earlier attempts, mainly based
on the extension of existing advective-diffusive stabilization concepts to the reactive case. To
mention but a few, the work of Tezduyar and Park [12] that used a discontinuity capturing like
operator, or the gradient GLS formulation proposed by Hughes and Harari [3]. Idelsohn and coworkers developed a PG formulation for linear elements [22] based on a centered perturbation to
Galerkin weights in the form of a second-order polynomial, involving two different ‘intrinsic
time’ parameters (i.e. the first to control advection induced instabilities, the second for reaction
induced ones), whereas our SPG formulation generalizes this concept to quadratic elements [4].
Recently, a second route has been suggested on the basis of the Variational Multiscale
(VMS) method, first proposed by Hughes in [13]. This approach permits to obtain formulations
with a more attractive mathematical background [13-18], the so-called sub-grid scale models
(SGS) able to deal with multiscale phenomena and to give a theoretical foundation of stabilized
methods. The idea that lays behind the VMS methods is to obtain a residual based stabilization
device by computing analytically the effects of fine or sub-grid scale solution (i.e. the error in the
coarse-scale solution) on the resolvable one by means of element residuals, as proposed by
Hughes in [13, 15]. Brezzi et al. in [19] demonstrated under certain hypotheses the equivalence
of the RFB and VMS approaches. To the best of the authors’ knowledge, the literature review
confirmed that the SGS stabilization schemes designed for both advective and reactive limit are
uniquely proposed on linear spaces of interpolation, with so-called intrinsic time scale () always
proposed as element-wise constant. The value  is often computed by means of a spatial
averaging within the element interior [13 - 17], while few works proposed definitions tuned in
order to fulfil the Discrete Maximum Principle (e.g. [20, 21]).
With respect to the presented state-of-the-art, in this paper we propose an alternative residual
based SGS stabilization device, developed for second-order interpolation spaces. The use of a
higher order stabilized formulation, though its complexity due to the non-negligibility of second
order derivatives, guarantees the best compromise between solution stability and accuracy, as
shown by Borello et al. in [23]. In particular, in this work we address Q2-Q1 schemes, often
applied in CFD near wall turbulence modelling, that need a non-constant stabilization parameter
within the element. The method presented is called V-SGS (Variable - SubGrid Scale) and it is
presented here for the first time. It has been tested on several scalar problems and on real
turbulent flow configurations pertinent to turbomachinery fluid dynamics. In this context the
turbulence model addressed is the k--v2-f [5], for its challenging behaviour with respect to
advection and reaction induced instabilities both in turbulence closure and in the elliptic
relaxation equation.
The remainder of the work is organised as follows. In Section 2 we consider the analytical
aspects of the advective-diffusive-reactive differential operator, introducing the main parameters
that govern the solution behavior, i.e. the element Peclet number Pe and the reaction number r.
Section 3 deals with the design of the sub-grid scale model, introducing the analytical and
theoretical aspects of this new numerical approach and explaining the steps followed in order to
obtain the V-SGS formulation. Moreover a multidimensional extension is introduced. In Section
4 we use V-SGS for RANS approach to turbulent incompressible flows with a k--v2-f
turbulence model, presenting a complete variational formulation of the problem. Finally in
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Section 5 we make numerical experiments, starting with advective-diffusive-reactive model
problems and then considering the turbulent flow over a NACA 4412 airfoil at maximum lift.
The performance of V-SGS is assessed with reference to other stabilized formulations such as
SUPG, SPG and Streamline Upwind and in the NACA 4412 airfoil flow the comparison is
enriched by the available experimental measurements [27].
2.
Formulations of the advection-diffusion-reaction problem
Let write a linear scalar advective-reactive-diffusive problem statement on the closed domain
 for the unknown U as:
Fa (U ), j  Fd (U , U ), j  Fr (U )  f
in  R nsd
U ( )  U D
(1.1)
where nsd is the number of space dimensions and the structure of the operators
reads as:
Fa (U )  u jU
Fd (U , U )  kU , j
(1.2)
Fr (U )  cU
where: nsd is the number of space dimensions, k > 0 is a constant diffusivity, u j are solenoidal
velocity components, c  0 is a reaction coefficient, and f the source term. It is worth noting
that, according to the sign chosen for c, the solutions have an exponential behaviour.
The solution of problem (1) could be characterized by fundamental dimensionless numbers
such as:
ud
Peg 
, global Peclet number
(2.1)
k
cd 2
rg 
, global reaction number
(2.2)
k
where u , d, and c are respectively global scales for the velocity, length and flow reactivity.
By introducing the linear advection-diffusion-reaction operator L, and its adjoint operator L :
LU  Fa (U ), j  Fd (U , U ), j  Fr (U )
(3)
LU   Fa (U ), j  Fd (U , U ), j  Fr (U )
the eq.(1) can be recast in a compact form that reads as:
LU  f
U ( )  U D
(4)
2.1 Variational formulation
Let consider S  H 1    as the trial solution space and W  H 1    as the weighting
function space, where H 1    is the Sobolev space of square integrable functions with square
integrable derivatives. These two sets are completely defined as follows:
S   U | U  H 1 ( ),U  U D on  ,U D  H (1/ 2) ( )
TMRGroup@DMA-URLS, February 2004

(5.1)
3
W  w | w  H 1 , w  0 on  
(5.2)
Note that the superscript (1/2) represents the restriction of the Sobolev space to the domain
boundary. The variational counterpart of problem (4) could be written as:
find U  S such that w W
a( w,U )  ( w, f )
(6)
where (, ) is the L2 (  ) inner product, and a (, ) is a bilinear form satisfying the following
identity
a( w,U )  ( w, LU )
(7)
for all sufficiently smooth w  W ,U  S . It is remarkable that S and W are necessarily infinitedimensional.
2.2 Coarse and fine scales; Galerkin formulation
Given a finite element partition of the original closed domain  into elements e, e =1, nel
(nel number of elements) such that:
(8)
 e   , e  e   and e  e  
e
with the interior boundary defined as:
 int 
e
e -
Let define the finite dimensional spaces of trial and weight functions as:
S h   U h | U h  H 1h ( ),U h  U D on  ,U D  H (1/ 2) h ( ) 
W h  wh | wh  H 1h , w  0 on  
(9)
(10.1)
(10.2)
where the superscript h denotes the characteristic length scale of the domain discretization. It is
remarkable that S h and W h are finite-dimensional subsets of S and W able to work only with the
coarse scales of the problem, thus it is helpful defining the infinite-dimensional spaces for the
fine scales, namely S  and W  , with the direct sum relationships S  S h  S ,W  W h  W  :
(11.1)
S    U  | U   H 1 ( ),U   0 on  
W   w | w  H 1 , w  0 on  
(11.2)
The numerical solution on the discretized domain could be related to the following
dimensionless numbers:
u h
Pe 
, element Peclet number
(12.1)
2k
r
ch 2
, element reaction number
k
(12.2)
where u is the Euclidean norm of the velocity vector, and h is defined as h = meas(e). With
(12.1) and (12.2) we can obtain a local evaluation of advection and reaction with respect to
diffusion.
The approximated variational counterpart of the boundary-value problem (4) could be written
as follows:
find U h  S h such that wh  W h
(13.1)
a(wh ,U h )  (wh , f )
For classical finite elements integration by parts formula on the diffusive term permits to obtain the
following Galerkin integral expression:
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nel

nel
e 1  e
nel

wh Fah, j d   
w
e 1  e
 w, j Fd d  
h
h
e 1  e
h
nel
Frh d   
w
h
fd 
(13.2)
e 1  e
Retaining in mind the integration by parts applied, it could be re-written in the following way:
(13.3)
a(wh ,U h )  (wh , LU h )  (wh , f )
3.
V-SGS stabilized formulation
Let re-write the variational boundary-value problem (7) introducing explicitly a general
residual-based stabilization term:
find U h  S h such that w W h
h
ˆ
(14)
a( wh ,U h )  ( w,LU
 f )  ( wh , f )
In this section we present the proposed stabilization scheme that has been developed, for
quadratic interpolation spaces (e.g. Q2 element), addressing the use of an alternative consistent
h
ˆ
device to build-up the residual stabilization term ( w,LU
 f ) with respect to classical PetrovGalerkin formulations.
The followed approach is routed in the ambit of variational methods for the representation of
multilevel or multiscale phenomena. In this viewpoint two sets of overlapping scales are used to
approximate the solution of a problem governed by a general non-symmetric differential
operator on a closed domain  . The sum decomposition of the solution U  U h  U  permits to
distinguish the resolvable or coarse scales U h and the un-resolvable or fine or subgrid scales U  ,
and in a Galerkin sense the same decomposition is applicable to the weight functions w  wh  w '
[13, 15]. By that way the VMS approach is aimed at obtaining U  analytically and calculating its
effect on the resolvable scales by means of its elimination in a variational sub-grid problem (SGS
model), as first proposed by Hughes in [13], so that the residual term is of the form
h
ˆ
( w,LU
 f ) = ( L* wh ,U  ) .
3.1 V-SGS and element Green’s function
Let re-write the variational formulation (7) in terms of the decomposition in coarse and fine
scales as:
a(wh  w,U h  U )  (wh  w, f )
wh W h , w W  (15)
By means of the linear independence of wh and w , the formulation (15) splits into two subproblems that, due to the linearity of the L differential operator, read as:
for the coarse scales
wh  W h
(16)
a( wh ,U h )  a( wh ,U )  ( wh , f )
for the sub-grid scales
a(w,U h )  a( w,U )  ( w, f )
(17)
w W 
This second sub-problem must be solved in terms of U  in order to describe the effect of fine
scales on the coarse ones. This issue can be addressed by means of the fine scale Green’s
function [15]:
nel 

U ( y )     g( x, y )  LU h  f  ( x )d  x   g( x, y )  L eU h  ( x )d  x  (18)


e 1   e
e

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where L e is the boundary operator corresponding to L, x is the vector identifying a point of
the computational domain and y is the vector identifying the point of the domain in which the
Dirac’s delta is concentrated. Eq. (18) demonstrates that U  is driven by the residual of the coarse
scales, that consists of a smooth part on element interiors and jump terms on inter-element
boundaries, and the driving mechanism is the fine scale Green’s function.
The subgrid scale contribution could be now substituted in the problem for coarse scales
(16) by means of successive integration by parts [15] as:
nel
a ( wh ,U )  ( wh , LU )    wh  LU d  e 
e 1 e
nel
nel
   L w  U d  e    L w  U d  e

h
e 1 e
e 1 e
*
e
(19)
h
On a mathematical point of view, Eqs.(18) and (19) show that the sub-grid scales contribution is
non-local, namely it extends to the whole domain of integration  , and distributional effects
arise on inter-element boundaries due to the discontinuity of second order derivatives for
classical finite elements.
Let now make the quite strong, but widespread in stabilized finite elements (e.g. [13,15,17,19]),
assumption that subgrid scales vanish on element boundaries:
on  e
(20)
e  1, nel
U  0
They could exert their influence in the limit of the coarse grid space discretization. By that way
all the inter-element jump terms, arising from second order derivatives, are discarded from the
integral formulation of the problem and non-locality reduces into individual elements. This
corresponds with the solution on a single element domain of problem (18), whose EulerLagrange equations read now as:
(21.1)
LU   ( LU h  f ) in e
e  1, nel
on e
(21.2)
U  0
As proposed by Hughes [13], Eqs. (21) could be tackled introducing the element Green’s
function problem that reads as:
L g e ( x, y )   ( x, y ) in e
(22.1)
ge  0
e  1, nel
on e
(22.2)
The Green’s function dependent sub-grid scale contribution could be now written as:
U ( y )    g e ( x, y )( LU h  f )( x)d  x
y   e
(23)
e
Substituting (23) into (16), we can write the coarse scales variational problem:
nel

e 1
e
nel

e 1
e
nel
wh LU h d e   L*wh ( y )  g e ( x, y )( LU h  f )( x)d  x d  y 
e 1
w fd e
h
e

w W
h
(24)
h
As shown in (24), the distinctive feature of SGS-like methods lays on the choice of a suitable
approximated expression for the coarse scale residual based integral operator containing the
element Green’s function. In order to obtain an adjoint-type stabilized formulation equivalent to
the subgrid model, this integral operator must be approximated by an intrinsic time scale
parameter τ that weights the coarse scales residual. The literature review showed that most of the
proposed formulations work with element-wise constant definition of τ, either computed as
average value of the exact element Green’s function (i.e. see [13,16]), or as classical in the
Petrov-Galerkin context in terms of local length and velocity scales [20]. The presented V-SGS
formulation admits the following definition for the element Green’s function:
TMRGroup@DMA-URLS, February 2004
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g e ( x, y )   v  sgs ( x) ( x, y )
(25)
where the error distributor is described by a function product including the Dirac’s delta and a
space-dependent intrinsic time scale parameter  v  sgs . On this basis the sub-grid scales could be
modelled as:
U ( y )    g e ( x, y )( LU h  f )( x)d  x 
e
(26)
   v  sgs ( x) ( x, y )( LU h  f )( x)d  x    v  sgs ( y )( LU h  f )( y )
e
where the time scale  v  sgs (y) is computed by the exact integration over each element of ge(x,y)
as:
 v  sgs ( y )    v  sgs ( x) ( x, y )d  x   g e ( x, y )d  x
(27)
e
e
It should be noted that the above 
(y) definition grants a-priori the suitability of the
proposed approach model for high order finite element interpolation spaces, such as quadratic
ones. As a result of the proposed V-SGS model, the stabilization integral becomes:
v  sgs
nel
( wh , LU h  f )  ( L wh ,U )    L wh ( y ) ( y )( LU h  f )( y )d  y
e 1
e
(28)
It is interesting to give more hints on the determination of  v  sgs (y). To this end, we use a
one-dimensional advective-diffusive-reactive model problem with constant physical properties.
In this configuration the adjoint problem for the element Green’s function reads as:
in e
(29.1)
kge , xx uge , x cge   ( x, y)
on e
(29.2)
ge  0
e  1, nel
For both linear and quadratic isoparametric finite elements, the above problem could be
reformulated, in element parent domain taking into account the invariance of the properties of
the Dirac’s delta in the coordinate transformation [14]:
2
2
2
2
   kg e ,    ug e , cg e     ( ,  )
h
h
h
g e (1,  )  0
 ,   (1,1)
(30.1)
(30.2)
g e (1,  )  0
here is the auxiliary Dirac’s delta space variable. The element Green’s function associated to
problem (30) is found to have an exponential behaviour:
1    
ge ( ,  )  C1e1  C2e2
(31)
1
2
    1
ge ( ,  )  C3e  C4e
where 1 and 2 are the roots of the characteristic equation associated to problem (30). The four
closure constants are determined by imposing on g e the homogeneous boundary values, the
continuity in  and the value of its first derivative jump in  defined as [24]:
u
2
(32)
ge , (  ,  )  ge , (  ,  )     k 
Pe
h
where the g e first derivative discontinuity is related to the inverse of local Pe.
By expressing the integral time scale using the element Green’s function (31), an additional
feature of the proposed V-SGS model becomes evident:

v  sgs
( y)   
e
v  sgs
( x) ( x, y)d  x 
1
 g ( x, y)d    g ( ,  )(det J )d
e
x
e
e
(33)
1
and in the above integral the Jacobian determinant is defined:
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h
det J   
(34)
2
It could be, thus, inferred that the V-SGS formulation does not depend on the choice between
quadratic or linear finite elements, thus being suitable for both these types of formulation.
In Fig. 1 the behaviour of element Green’s function is shown for a one-dimensional logic
h2
element with varying Pe and r magnitudes and
 102 . For instance  is set equal to zero that
k
is the centre of the element.
g(  )
g(  )
0.5
0.05
Pe = 0
Pe = 1
Pe = 10
0.4
0.04
r=0
r = 10
r = 100
0.03
0.3
0.02
0.2
0.01
0.1
0
0
-1
-0.5
a)
0
0.5
1

-1
-0.5
0
0.5
1

b)
Fig. 1. Element Green’s function: a) Pe =100, r = 0  100; b) r =100, Pe = 0  10.
It is worth noting that ge(, 0) modulates the element error distributor mechanism moving
from advection dominated limit (r  0), where it behaves like an upwind Heaviside function, to
reaction dominated condition where it approaches a symmetric impulsive-like shape.
3.2 Further considerations on  v  sgs
One of the most remarkable criticisms on the use of a element wise constant  lays on its
ability to control only element wise constant residuals, that are obtained on advective-diffusive
problems with linear elements [15]. If reactive terms appear and/or high order elements are used,
there is no agreement between a constant stabilizing parameter and a variable residual, thus
addressing the need for a space dependent .
Another important feature of the proposed  v  sgs formula is its bubble behaviour, that permits
to eliminate the inter-element integrals without considering the functional properties of the trial
and test functions used, thus addressing the use of integration by parts for the diffusive term also
in the residual based operator.
In this viewpoint it is worthwhile considering the  v  sgs expression in master element
coordinates for the different combinations of reactive and advective effects.
3.2.1 Advective-diffusive problem
The V-SGS stabilizing parameter in this limit case reads as:
u
 

Pe

u



u
e
1

avdf
 V  SGS ( )   SC
1   u   sinh( Pe)  Pe   coth( Pe)  1 Pe   ,

 


 

h
avdf
 SC
  SUPG 
 coth( Pe)  1 Pe 
2u
TMRGroup@DMA-URLS, February 2004
(35)
8
avdf
Expression (35) shows that  v  sgs could be seen as a sum: the first term  SC
, which assigns
the order of magnitude to the stabilizing parameter, is exactly the element wise constant  SUPG
intrinsic time scale (e.g. see [8]) as obtained for SGS methods proposed in literature (e.g. see
[13]), and the second one exploits the space dependence through a zero mean value function. The
dependence on the element coordinate permits to capture non-constant residuals, at least for one
dimensional linear problems.

0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
-1

0.5
-0.5
0
0.5

1
0
-1
-0.5
0
0.5

1
Fig. 2. Stabilizing parameter  v  sgs for advective-diffusive problems in parent coordinates: a) for u>0 and b) for
avdf
u<0. Solid line =  v  sgs ; dashed line =  SUPG =  SC
.
Fig.2 shows that the  SUPG intrinsic time scale is indifferent to the velocity orientation
whereas  v  sgs is able to create a correct upwind effect. The behaviour is shown for Pe=100 and
h u  0.5 .
3.2.2 diffusive-reactive problem
In this case the analytical expression of  v  sgs reads as follows:
  ( r 2)(1 )  r ( r 2)(1 )  r ( r 2)(1 ) ( r 2)(1 ) 2
e
e
e

1  e
  e
r
dfrt
1  
 V  SGS ( )   SC
2
2



2 r

1  e r 
1  e

r






dfrt
SC


2 1  e r
1
 1 
c
r 1  e 2



2
r


r
  
2
,








(36)
where, according to the Eq.(30), c is the reaction coefficient. Again the resulting  v  sgs is
dfrt
obtained as a sum of a term  SC
which assigns the magnitudo of the stabilizing parameter and a
zero mean function. In Fig. 3 it is shown the behaviour of  v  sgs for c  10 and r  103 .
TMRGroup@DMA-URLS, February 2004
9

0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
-1
-0.5
0
0.5

1
Fig. 3. Stabilizing parameter  v  sgs for diffusive-reactive problems in parent coordinates: solid line =  v  sgs ;
dfrt
dashed line =  SC
.
3.2.3 Advective-diffusive-reactive problem
In the more general case of non-zero velocity and dissipation,  v  sgs has a richer behaviour
and reads as:

V  SGS
1 1  e 2 (1 )  e2( 2 1 )  e 2 2 1 (1 )  e2 1 2 (1 )  e1 (1 )  1  
h2


( ) 
4k (2  1 )  e2( 2 1 )  1   1 2  e 2 (1 )  e2( 2 1 )  e2 2  1 (1 )  e2 1 2 (1 )  e1 (1 )  1 


(37)
where the notation agrees with (30,31). Even in this more genereal case 
could be
decomposed in its element mean value that gives the scale of the stabilizing parameter, namely
avdfrt
 SC
, and a zero mean function which gives the spatial dependence. In Fig. 4 are shown the
v  sgs
behaviours of  v  sgs for different combinations of Pe and r, with Pe=100 and r varying from
dominant advection to dominant reaction effects. The profiles are obtained for h k  103 .

0.5

0.5
0.45
0.45
0.4
0.4
10
10
0.35
0.35
0.3
0.3
100
100
0.25
0.25
0.2
0.2
r
0.15
1000
0.1
1000
0.1
0.05
0
r
0.15
0.05
-1
-0.5
0
0.5

1
0
-1
-0.5
0
0.5

1
Fig. 4. Stabilizing parameter  v  sgs for advective-diffusive-reactive problems in parent coordinates: a) u>0 and b)
u<0.
3.3
Multi-dimensional stabilized V-SGS formulation
TMRGroup@DMA-URLS, February 2004
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The multi-dimensional formulation of the proposed stabilized method must face the lack of
an established solution for the multi-dimensional integral of element Green’s functions in case of
advective-diffusive-reactive problem. This forces the definition of V-SGS model for nsd > 1 on
the basis of multi-dimensional generalization of time scale  v  sgs . The proposed method
combines the 1D intrinsic time scale parameters computed from the element Green’s functions
associated to each parent domain coordinate direction. In the 2D case, we solve the Green’s
function problem for  and  directions, which differ due to the velocity components magnitudes
that unbalance the advective phenomena on the parent domain. On the basis of the directional
Peclet numbers the time scale is:
1
 v  sgs ( x, y)   v  sgs i ( x, y)    ge ( , i )(det J )d 
i
i

v  sgs
SC i


 1  fi i , Pei , r

(38)
1
The two directional intrinsic time scales, namely  v sgs ,v sgs , computed in each element node
must be composed in order to obtain the  v  sgs to be used for the sub-grid scales model (28). In
this respect, the composing criterion consists in using a combination between the directional
 vsgs and vsgs , namely:
 v  sgs ( x, y ) 
1
1
v  sgs
 SC

4.

1


 1  f  , Pe , r   1  f , Pe , r 

(39)
v  sgs
 SC

RANS formulation for incompressible turbulent flows
4.1 Problem statement
The dynamic response of incompressible turbulent flows is modeled by using a RANS
approach. Each quantity U is then decomposed into its conventional average (denoted by an
overbar) and the fluctuation with respect to the latter (denoted by a prime), as U  U  U ' .
The turbulence model for the closure of the system of equations is the k    v 2  f (Durbin,
[5]), for its improved performance in simulating transition phenomena with respect to standard
k- model, due to the use of v2 as velocity scale for turbulent transport toward the wall instead
of k.
The Boussinesq approximation is used for the stress-strain relation:
2
ui'u 'j  k ij   t Sij
(40.1)
3
where  ij is the Kronecker tensor, the eddy viscosity is given by
 t  c v2 k 
(40.2)
Sij  (ui , j  u j ,i )
(40.3)
and it is used twice the strain tensor
The value used for the coefficient c  could be found in Table 1. The computations have been
performed with the code-friendly version of the model [26], in which homogeneous boundary
conditions are imposed on solid walls for the elliptic relaxation variable, namely f . The RANS
complete formulation is obtained in terms of: momentum components  u i (i=1,2,3) (where  is
the density, and u i the Cartesian averaged velocity components), static pressure p , turbulent
TMRGroup@DMA-URLS, February 2004
11
kinetic energy k, dissipation variable     2 ( k / xi )2 , that replaces  , average of the square
of the turbulence fluctuation in the wall-normal direction, namely v2 , and modified elliptic
relaxation variable f . The boundary value problem reads as:
Fa U   Fd U , U   Fr (U )   f  0 in  R nsd
,j
,j
U UD
on  D
Fd , n   N
on  N
(41)
where    D   N , with  D   N   and U is the vector of the averaged unknowns
related to U by
U   u1 , u 2 , u 3 ,

and
could
p, k ,  ,
be
interpreted
U p  u1 , u 2 , u 3 , 0, k ,  , v , f 


2
T
f   U  0, 0, 0,

v 2 ,
T
in
terms
of
and constrained variable
p  1, 0, 0,
0,
0 
T
primary-turbulent
(42)
flow
T
U c  0,0,0, p,0,0,0,0 .
properties
The boundary
conditions, specified along the computational domain boundary, generally include inflow
Dirichlet conditions ( U D ) and outflow Neumann conditions (  N ). On solid boundaries,
homogeneous Dirichlet conditions are imposed for U p . The flux vectors appearing in (41) are
defined as:
Fa U   u j  u1 , u j  u2 , u j  u3 , u j , u j  k , u j  , u j  v 2 , 0 


T





 
 
 
Fd U , U    1 j ,  2 j ,  3 j ,0,     t  k , j ,     t   , j ,     t  v2 , j ,  L2 f , j 
k 
 
k 





(43.1)
T
(43.2)
where the stress tensor is:
 ij  p *  ij     t  Sij
(43.3)
The non-linear Newtonian like turbulent stress terms are thus included, affecting the
molecular kinematic viscosity with t, whereas the modified pressure ( p *) includes the
isotropic part of the turbulent stress tensor. The reactive terms are described by
Fr (U )  0, 0, 0, 0, ck k , c  , cv2 v 2 , c f 


T
(43.4)
with
ck 

k
c  c 2 f 2
cv 2  6


k
(43.5)
k
cf 1
TMRGroup@DMA-URLS, February 2004
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Finally, the source vector is defined as:
 PM 1 , PM 2 , PM 3 , 0,  Pk    D, c 1 Pk  / k ,


f  
 kf ,
 (C1  1)(v 2 k  2 3)  T  5( k ) v 2 k  C2 Pk k 




T
(43.6)
Concerning the momentum source components PMi , they account for volume sources (i.e.
those originating from non-inertial frame of reference). The closure coefficients are recalled in
Table 1.
Table 1. k    v2  f closure coefficients.
k
1

1.3
C1
k v2 )
1.4(1+0.05
c 2
1.9
f2
[1-0.3exp(- Re t )]
Ret
k 2 / 
C
0.22
Pk
uiuk ui ,k
2
2( 
D
k / xi )2

CL max k 3 2  ,C 3 4  1 4
L




max k  ,6   
C1

C2

CL

C

4.2 V-SGS Variational formulation for RANS equations
Let define, by using the notation introduced in (10.1, 10.2), the finite dimensional spaces of
trial and test functions for primary and constrained variables as:

S ph  U p U p  H 1h    , U p   D
h
h
h

Wph  whp whp  H 01h    , whp  0 on  D


on Γ D ,  D  H ( 1 2 )h   D 

(44)

Sch  Wch  U c U c  H 01h    , wch wch  H 01h   
h
h
where the Galerkin test functions for mixed elements are adopted, namely quadratic for primary
TMRGroup@DMA-URLS, February 2004
13
variables and linear for constrained ones. The associated weights and adjoint based stabilization
functions could be written in vector form as:
T
(45.1)
wh   whp , whp , whp , wch , whp , whp ,
whp ,
whp 
   u ,  u ,  u , 0,  k ,   ,
1
2
3
v ,
2
 f 
T
(45.2)
In (45.2) for each component, a product is done multiplying two factors: the first one is the
intrinsic time scale obtained as described in Sect.3 for each degree of freedom while the second
is, for the same variable, the associated adjoint operator acting on the weight, that according to
(3) reads as:
(45.3)
L* whp  Fa (whp ), j Fd (whp , whp ), j  Fr (whp )
The approximated variational formulation of differential problem (41) now reads for each
variable as follows:
h
find U  H 1h whp Wph , wch  Wch , such that

 
 
 


c u h ,U ,wh  s U ,wh  r U ,wh   (U , f ),    f ,wh    N ,w|h N
h
h
h
h

N
(46)
with use of bi-linear and tri-linear forms


s U ,wh    w,hj Fdh d 
h

  f ,w   
h


N
,w|h N


N
wh  f d 
(47)
  w|h N  N d 


c u h ;U ,wh   wh Fah , j d 
h



r U , wh   wh Fr d 
h
h

Finally, the stabilization integrals are defined as


 (U , f ),      Fa,h j  Frh   f    , j Fdh d 
h
nel
(48)
e 1  e
and the stabilizing diffusive contributions could be integrated by parts due to the bubble
nature of intrinsic time scale parameters.
5.
Numerical examples
In this section we assess the numerical performance of the proposed V-SGS formulation for
model problems and configurations pertinent to turbomachinery fluid dynamics. In these
TMRGroup@DMA-URLS, February 2004
14
validation studies the improvement are commented with respect to the classical stabilization
schemes, such as the Streamline Upwind Petrov-Galerkin (SUPG), the Discontinuity Capturing
(DC) and the Streamline Upwind (SU), and with respect to our recent Spotted Petrov-Galerkin
(SPG) formulation. It is remarkable that, since all the consistent stabilization schemes usually
share the optimal property in 1D, all the investigated test cases violate one of the superconvergence conditions (i.e. multidimensional domain, problems with source terms).
Remark: the adopted element length scale h = meas(e) is geometrical.
In Subsection 5.1 we investigate on the numerical performance of V-SGS on a model
problem with a rich solution behavior, involving several conditions with advective-diffusive and
advective-diffusive-reactive flow phenomena. Comparisons are done with respect to Galerkin
formulation and SPG on quadratic finite elements. After that, in Subsection 5.2 we add a relevant
source term to the previous problem statement, focusing on the performance with respect to
Galerkin, SUPG and SPG.
5.1
Advective-diffusive-reactive problems on a unit square domain
5.1.1 Advection skew to the mesh
The TC1 test case deals with a classical problem (i.e. see [7]), namely the advection skew to
the mesh of a scalar unknown on the unit square domain already used for the previous two test
cases. The grid is uniform with 100 quadratic elements, while the problem is described
analytically imposing c=0, f=0 and k=10-4 in (1). The Pe number is approximately 6  102, and
the problem statement is shown in Fig. 5.
/n=0
=0
y
u
ux/uy=2
/n=0
=1
x
=1
Fig. 5. TC1 problem statement.
The solutions predicted by Galerkin (GQ2), SPG improved with the addition of Discontinuity
Capturing (SPG+DC) and V-SGS are compared in Fig. 6, where is evident the superior behavior
of V-SGS in predicting strongly advective fields, as confirmed in Fig. 7 with the x-constant
profiles of the solutions for x=0.1 and x=0.9.
TMRGroup@DMA-URLS, February 2004
15


1.2
1.2
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
0
0.2
0.2
0.4
Y
0
0
0
0.2
0.4
0.6
0.6
0.8
0.2
0.4
Y
X
0
0
0
0.2
0.4
0.6
0.8
1

1.2
0.6
0.8
1
Y
0.8
1
0.2
0.4
X
0.4
0.6
0.6
0.8
X
0.8
1
1
1
Fig. 6. TC1 comparison of solution fields: a) Galerkin GQ2, b) SPG+DC and c) V-SGS.

1.2
1
1
0.8
0.8
G Q2
SPG + DC
V-SGS
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
0

1.2
0.2
0.4
0.6
0.8
y
-0.2
1
G Q2
SPG + DC
V-SGS
0
0.2
0.4
0.6
0.8
y
1
Fig. 7. TC1 comparison of solution profiles: a) x=0.1 and b) x=0.9.
5.1.2 Advective-diffusive-reactive problem with non-uniform velocity field
The second test case (labelled TC2) concerns with the numerical solution of the linear scalar
advective-diffusive-reactive model problem (1), in a unit square domain without source term.
The mesh is again uniform with 100 quadratic elements, thus consisting of 441 nodes. The
problem statement is resumed in Fig. 8. The known velocity field u is assumed to have a
parabolic profile (e.g. u(x,y) = 2y – y2, v(x,y) =0), with maximum value equal to 1. The
coefficients are: k = 10-5, c = 5  102. The maxima for dimensionless numbers are: Pe = o(103)
and r = o(105).
/n=0
u
=1
TC2
y
/n=0
x
/n=0
Fig. 8. TC1 Scalar advective-diffusive-reactive problem statement.
SPG and V-SGS solutions for TC2 are compared with quadratic Galerkin (G Q2) and SUPG
Q2 ones in Fig. 9. It is worth noting that the Streamline Upwind (SUQ2) solution field is not
shown because of its inability to improve on the Galerkin one. As clearly appears, the new
proposed formulations are able of controlling the instability origins in the near- and far-wall
TMRGroup@DMA-URLS, February 2004
16
regions, where the SPG completely eliminates any kind of oscillation without assuming the overdiffusive behavior of the SUPG solution.


y
1
1
0.75
0.75
0.5
0.5
0.25
0.25
0
0
1
0.8
x
0.75
0.6
1
y
1
1
0.8
0.4
0
0.5
0.4
0.25
0.2
0.25
0.2
0
0
a)
x
0.75
0.6
0.5
0
b)

y

1
1
0.75
0.75
0.5
0.5
0.25
0.25
0
0
1
1
0.8
0.75
0.6
1
y
x
1
0.8
0.75
0.6
0.5
0.4
0.25
0.2
0
0.25
0.2
0
0
c)
x
0.5
0.4
0
d)
Fig. 9. TC2 comparison of solution fields: a) GQ2, b) SUPGQ2, c) SPG, d) V-SGS.
Fig. 10 deals with the streamwise profiles of the solutions for y=0 and y=0.05, where
reaction dominates and numerical methods find more difficulties, and shows a comparison of the
nodal values in the last five stations before the Dirichlet boundary.
, G Q2
, SUPG Q2
, SPG
, V-SGS
, G Q2
, SUPG Q2
, SPG
, V-SGS
2.94 e-02
-2.51 e-02
1.71 e-01
-1.46 e-01
1.
1.38 e-02
5.63 e-02
1.17 e-01
4.80 e-01
1.
2.30 e-04
3.82 e-03
1.73 e-02
2.19 e-01
1.
3.24e-03
1.03 e-02
7.29 e-02
9.11 e-02
1.
3.04 e-02
-2.60 e-02
1.74 e-01
-1.49 e-01
1.
1.37 e-02
5.60 e-02
1.17 e-01
4.78 e-01
1.
1.08 e-03
7.10 e-03
3.34 e-02
2.12 e-01
1.
2.34e-02
-7.97 e-03
1.53 e-01
-5.16 e-02
1.


1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
GQ2
SUPGQ2
SPG
V-SGS
0.4
0.3
0.2
GQ2
SUPGQ2
SPG
V-SGS
0.4
0.3
0.2
0.1
0.1
0
0
-0.1
-0.1
0
0.25
0.5
0.75
1
x
a)
0
0.25
0.5
0.75
b)
1
x
Fig. 10. TC2 comparison of streamwise profiles: a) in y=0 and b) in y=0.05.
TMRGroup@DMA-URLS, February 2004
17
It is evident how an excellent accuracy is obtained by the SPG, closer to the exact sharp
exponential solution. Moreover the V-SGS manages to damp the oscillations of the Galerkin
solution although the incomplete control of instability effects.
5.3 Turbulent flow over a NACA4412 airfoil
References
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
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