a variational multiscale higher-order finite element formulation for turbomachinery flow...

Comput. Methods Appl. Mech. Engrg. 194 (2005) 4797–4823

www.elsevier.com/locate/cma

A variational multiscale higher-order finiteelement formulation for turbomachinery flow computations

Alessandro Corsini, Franco Rispoli *, Andrea Santoriello

Dipartimento di Meccanica e Aeronautica, Universita di Roma ‘‘La Sapienza’’, Via Eudossiana, 18, I00184 Roma, Italy

Received 19 March 2004; received in revised form 5 November 2004; accepted 23 November 2004

Abstract

The Variational MultiScale approach for finite elements addresses the inclusion of the effect of fine scales of the solu-

tion in the coarse problem. In this framework advective–diffusive–reactive equations modelling turbomachinery flows

feature both advection and reaction induced instabilities to be tackled. To this end, this work deals with a new method

called V-SGS (Variable-SubGrid Scale), designed for quadratic elements, with a variable �intrinsic time� parameter.

Two-dimensional tests have been considered to compare V-SGS against SUPG and other stabilization devices, includ-

ing the flow around a NACA 4412 airfoil to assess its reliability in the handling of advanced turbulence closures on

cruder meshes.

� 2005 Elsevier B.V. All rights reserved.

Keywords: Variational multiscale method; Stabilized high order finite elements; k–e–v2–f turbulence closure model; Turbomachinery

flow

1. Introduction

The use of CFD for turbomachinery flow configurations is still 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, [22] or diffusive, [29] limits for incompressible fluid, as well

that related to the reaction dominated flow conditions, [13]. Though the existence of accurate stabilizationschemes for advection dominated conditions, the interest on this kind of equation stands in their reactive

0045-7825/$ - see front matter � 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2004.11.013

* Corresponding author. Tel.: +39 06 44585233; fax: +39 06 4881759.

E-mail address: [email protected] (F. Rispoli).

mailto:[email protected]

4798 A. Corsini et al. / Comput. Methods Appl. Mech. Engrg. 194 (2005) 4797–4823

character. This feature, that is ubiquitous in the modelling of several industrial processes, plays a key role in

turbomachinery CFD motivating the development of formulations able to work in the more general frame-

work of advection–diffusion–reaction model equations. In this ambit reaction driven effects are usually

associated to the Coriolis momentum contributions due to the blade rows rotation. Recently we have high-

lighted in [7] that significant reaction driven effects may appear in the numerical solution of turbulence dueto the structure of budget-like closure equations. In first or second moment closures absorption-like reac-

tive contributions stem from dissipation/destruction terms. Moreover, in elliptic relaxation based closures

(e.g. k–e–v2–f by Durbin [8], elliptic blending model by Manceau and Hanjalic [24]), the blending variable,

used to combine near-and far-wall effects, is modelled by means of a diffusive–reactive equation. The tur-

bulence model related reactivity is thus expected to dramatically affect the boundary layer simulation in tur-

bomachinery configurations, where the presence of stagnation, separation or adverse pressure gradient

phenomena gives rise to local reaction-to-advection ratio of order o(105) [7].

As well known, when advection dominates Galerkin finite element methods suffer from the appearanceof 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 on linear and quadratic spaces of approximation, most based on Petrov-Galerkin (PG) ap-

proaches such as SUPG [5,17,18,30], or on Residual Free Bubble (RFB) concept [3].

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 accu-

racy. In this case it is not possible to obtain a global stability estimate in the H1 norm, though it could be

evaluated in L2, coherently with the local scale of the oscillations [18]. Two alternative routes could befound in literature to build-up residual based stabilization schemes for the reactive limit. The first one in-

cludes the earlier attempts, mainly based on the extension of existing advective–diffusive stabilization con-

cepts to the reactive case. To mention but a few, the work of Tezduyar and Park [31] that used a

discontinuity capturing like operator, or the gradient GLS formulation proposed by Hughes and Harari

[13]. Idelsohn and co-workers proposed a Centered PG formulation for linear elements [20] involving

two different stabilizing parameters, the first to control advection induced instabilities and the second,

based on a second-order polynomial, for reaction induced ones. Corsini et al. developed SPG formulation

[7], which generalizes this concept to quadratic elements with the introduction of a sixth-order perturbationfor reaction effects and involving four different stabilizing parameters.

Recently, a second route has been suggested on the basis of the Variational MultiScale (VMS) method,

first proposed by Hughes [16]. This approach permits to obtain formulations with a more attractive mathe-

matical background [14–16,19,26,27], 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 [16,19]. Brezzi et al. [2] demonstrated the equivalence of the RFB and VMS approaches under cer-tain hypotheses. To the best of the authors� knowledge, the literature review confirmed that most of the

SGS stabilization schemes designed for both advective and reactive limit are proposed on linear spaces

of interpolation, with so-called intrinsic time scale (s) always proposed as element-wise constant. The value

of s is often computed by means of a spatial averaging within the element interior [19,27], while few works

proposed definitions tuned in order to fulfil the Discrete Maximum Principle (e.g. [4,10]). In this viewpoint

bubble function based formulations constitute a vital background for the design of more accurate intrinsic

time scale parameters, though their application is limited only to advective–diffusive problems [3,25], or to

linear elements [9].With respect to the presented state-of-the-art, in this paper we propose an alternative residual based SGS

stabilization device, called V-SGS (Variable-SubGrid Scale) and developed for second-order interpolation

spaces, often applied in CFD near wall turbulence modelling. The use of higher order finite element spaces

A. Corsini et al. / Comput. Methods Appl. Mech. Engrg. 194 (2005) 4797–4823 4799

guarantees the best compromise between solution stability and accuracy, as shown in Borello et al. [1]. In

particular we address the use of the Q2-Q1 element, in which the space-dependence of stabilization para-

meters needs to be exploited. This feature has been a driving criterion in the design of the proposed SGS

model. The performance of V-SGS as multiscale/stabilized approach have been tested on several scalar

problems and on real turbulent flow configurations pertinent to turbomachinery fluid dynamics. In this am-bit we propose the use of V-SGS within an elliptic relaxation based closure, the k–e–v2–f by Durbin [8], for

its challenging behaviour with respect to advection and reaction induced instabilities.

The remainder of the work is organised as follows. In Section 2 we consider the general advective–dif-

fusive–reactive problem statement, introducing the main parameters that govern the solution behaviour

and its numerical approximation with the Galerkin formulation. Section 3 deals with the design of the

sub-grid scale model, introducing the analytical and theoretical aspects of this new numerical scheme

and explaining the steps followed in order to obtain the V-SGS formulation. Moreover numerical examples

in one-dimension are included in order to deal with the basic properties of the method and to have a firstcomparison with classical stabilized formulations, and a multi-dimensional extension is proposed. In Sec-

tion 4 we assess the use of V-SGS for the RANS simulation of turbulent incompressible flows with a k–e–v2–f turbulence model, explicitly developing the variational formulation of the problem. In Section 5 we

present the numerical experiments, starting with advective–diffusive–reactive model problems and then con-

sidering the turbulent flow over a NACA 4412 airfoil at maximum incidence. The performance of V-SGS

are assessed with reference to other stabilized formulations such as SUPG, SPG and Streamline Upwind. In

the NACA 4412 airfoil flow the computations are compared to the available experimental measurements

[6]. Finally some conclusions are drawn.

2. Advection–diffusion–reaction problem statement

Let write a linear scalar advective–reactive–diffusive problem statement on the closed domain X for the

unknown U as:

F ajðUÞ;j þ F djðUÞ;j þ F rðUÞ ¼ B in X 2 Rnsd; j ¼ 1; . . . ; 3;

U ¼ UD on C;ð1:1Þ

where the structure of the operators reads as:

F ajðUÞ ¼ ujUF djðUÞ ¼ �kU ;j

F rðUÞ ¼ cU :

ð1:2Þ

In (1.1) and (1.2), nsd is the number of space dimensions, k > 0 is a constant diffusivity, uj are solenoidalvelocity components, c P 0 is a reaction coefficient, and B the source term. It is worth noting that, accord-

ing to the sign chosen for c, the solutions have an exponential behaviour.

The solution of problem (1) could be characterized by the following dimensionless numbers:

Peg ¼�udk; global Peclet number; ð2:1Þ

rg ¼cd2

k; global reaction number; ð2:2Þ

where �u and d are respectively global scales for velocity and length. A similar viewpoint could be adopted

for the numerical solution, that is characterized by the local counterpart of the introduced dimensionless

numbers, namely:


Pe ¼ kukh2k

; element Peclet number; ð3:1Þ

r ¼ ch2

k; element reaction number; ð3:2Þ

where kuk is the Euclidean norm of the velocity vector, and h represents the characteristic length scale of the

discretization. It is noticeable that as (2.1) and (2.2) give a global evaluation of advection and reaction withrespect to diffusion, so (3.1) and (3.2) do the same on a local point of view.

By introducing the linear advection–diffusion–reaction operator L, and its adjoint operator L�:

LU ¼ F ajðUÞ;j þ F djðUÞ;j þ F rðUÞ;L�U ¼ �F ajðUÞ;j þ F djðUÞ;j þ F rðUÞ:

ð4Þ

Eq. (1) can be recast in a compact form that reads as:

LU ¼ B;

U ¼ UD on C: ð5Þ

2.1. Galerkin formulation

Let consider S H1(X) as the trial solution space andW H1(X) as the weighting function space, where

H1(X) is the Sobolev space of square integrable functions with square integrable derivatives. These two setsare completely defined as follows:

S ¼ fU jU 2 H 1ðXÞ; U ¼ UD on C; UD 2 H ð1=2ÞðCÞg; ð6:1Þ

W ¼ fwjw 2 H 1ðXÞ; w ¼ 0 on Cg: ð6:2Þ
Note that the superscript (1/2) represents the restriction of the Sobolev space to the domain boundary.
The variational formulation of problem (5) could be written as:

find U 2 S such that 8w 2 W ;

aðw;UÞ ¼ ðw;BÞ; ð7Þ
where (Æ , Æ) is the L2(X) inner product, and a(Æ , Æ) is a bilinear form satisfying the following identity:
aðw;UÞ ¼ ðw; LUÞ; ð8Þ
for all sufficiently smooth w 2W,U 2 S.
Given a finite element partition of the original closed domain X into elements Xe, e = 1, . . . ,nel (nel num-

ber of elements) such that:
[e
Xe ¼ X;\e

Xe ¼ £ and[e

Xe ¼ X; ð9Þ

with the interior boundary defined as:

Cint ¼[e

Ce � C: ð10Þ

Let define the finite dimensional spaces of trial and weight functions as:

Sh ¼ fUhjUh 2 H 1hðXÞ; Uh ¼ UD on C; UD 2 H ð1=2ÞhðCÞg; ð11:1Þ


W h ¼ fwhjwh 2 H 1hðXÞ;w ¼ 0 on Cg; ð11:2Þ
where H1h is the finite dimensional function space over X and the superscript h denotes the characteristic
length scale of the domain discretization. It is thus remarkable that Sh and Wh are discrete finite-dimen-

sional subsets of S andW unable to capture the fine scales of the solution, characterized by structures which

are smaller than the grid-spacing used for the discretization of the problem.

The Galerkin formulation of the boundary-value problem (5) could be written as follows:

find Uh 2 Sh such that 8wh 2 W h;

aðwh;UhÞ ¼ ðwh;BÞ: ð12Þ

3. V-SGS stabilized formulation

In this section we present the V-SGS stabilization scheme that has been developed, for quadratic inter-

polation spaces (e.g. Q2 element), in the ambit of variational methods for the representation of multilevel or

multiscale phenomena [16]. 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 X.The sum decomposition of the solution U = Uh + U 0 permits to distinguish the resolvable or coarse scales

Uh and the unresolvable or fine or subgrid scales U 0, and in a Galerkin sense the same decomposition is

applicable to the weight functions w = wh + w 0 [16,19]. By that way the VMS approach is aimed at solving

a problem for U 0 and calculating the effect of the fine scales on the resolvable ones by means of their elim-

ination in a variational sub-grid problem, as first proposed by Hughes in [16].

Let re-write the variational formulation (7) in terms of the decomposition in coarse and fine scales as:

aðwh þ w0; Uh þ U 0Þ ¼ ðwh þ w0;BÞ 8wh 2 W h;w0 2 W 0: ð13Þ
By means of the linear independence of wh and w 0 [19], the formulation (13) splits into two sub-problems
that, due to the linearity of the L differential operator, read as:for the coarse scales

aðwh;UhÞ þ aðwh;U 0Þ ¼ ðwh;BÞ 8wh 2 W h; ð14Þ
for the sub-grid scales
aðw0;UhÞ þ aðw0;U 0Þ ¼ ðw0;BÞ 8w0 2 W 0: ð15Þ
This second sub-problem must be solved in terms of U 0 in order to describe the effect of fine scales on the
coarse ones.

Let now make the quite strong assumption that subgrid scales vanish on element boundaries:

U 0 ¼ 0 on Ce e ¼ 1; . . . ; nel: ð16Þ
Eq. (16) represents a widespread hypothesis in stabilized finite elements framework (e.g. [2,14,16,19]) and
means that unresolved scales could exert their influence in the limit of the coarse grid space discretization,

thus reducing non-locality into individual elements. This corresponds with the solution on each element do-

main of a problem, whose Euler-Lagrange equations read now as:

LU 0 ¼ �ðLUh � BÞ in Xe; ð17:1Þ

U 0 ¼ 0 on Ce e ¼ 1; . . . ; nel: ð17:2Þ
As proposed by Hughes [16], (17) could be tackled introducing the element Green�s function problem
that reads as:


L�geðx; yÞ ¼ dðx; yÞ in Xe; ð18:1Þ

ge ¼ 0 on Ce e ¼ 1; . . . ; nel: ð18:2Þ
The element Green�s function permits to obtain the following coarse scales residual dependent expression
for the fine scales:

U 0ðyÞ ¼ �Z

Xe

geðx; yÞðLUh � BÞðxÞdXx 8y 2 Xe: ð19Þ

The contribution of unresolved scales could be now substituted in the problem for coarse scales and by

means of successive integrations by parts [19] reads as:

aðwh;U 0Þ ¼ ðwh;LU 0Þ ¼Xnele¼1

ZXe

wh � LU 0dXe ¼Xnele¼1

ZXe

L�wh � U 0dXe: ð20Þ

Substituting (20) into (14), the coarse scales problem turns to a subgrid scales model:

aðwh;UhÞ �Xnele¼1

ZXe

L�whðyÞZ

Xe

geðx; yÞðLUh � BÞðxÞdXxdXy ¼ ðwh;BÞ 8wh 2 W h: ð21Þ

Eq. (21) contains an alternative consistent device to build-up the residual stabilization term with respect

to classical Petrov-Galerkin formulations, which generally read as:

find Uh 2 Sh such that 8wh 2 W h;

aðwh;UhÞ þ ðw; LUh � BÞ ¼ ðwh;BÞ: ð22Þ

It is in fact evident from (20) and (21) that a SGS-like method models the residual term by means of an

adjoint based operator (L�wh, U 0) that plays the role of ðw; LUh � BÞ.The distinctive feature of SGS methods lays on the choice of a suitable approximated expression for the

coarse scales residual based integral operator containing the element Green�s function. To this end it is

widespread the use of adjoint-type stabilized methods where the subgrid scales are approximated by an

intrinsic time scale parameter s that weights the coarse scales residual. The literature review showed that

most of the proposed formulations work with element-wise constant definition of s, either computed as

average value of the exact element Green�s function (i.e. see [15,16]), or as classical in the Petrov-Galerkin

context in terms of local length and velocity scales [4]. The presented V-SGS formulation admits the follow-

ing definition for the element Green�s function:

geðx; yÞ ¼ sV-SGSðxÞdðx; yÞ; ð23Þ
where the error distributor is described by a function product including the Dirac�s delta and a space-depen-dent intrinsic time scale parameter sV-SGS. On this basis the sub-grid scales could be modelled as:
U 0ðyÞ ¼ �Z

Xe

geðx; yÞðLUh � BÞðxÞdXx ¼ �Z

Xe

sV-SGSðxÞdðx; yÞðLUh � BÞðxÞdXx

¼ �sV-SGSðyÞðLUh � BÞðyÞ; ð24Þ

where the time scale sV-SGS(y) is computed by the exact integration over each element of ge(x,y) as:

sV-SGSðyÞ ¼Z

Xe

sV-SGSðxÞdðx; yÞdXx ¼Z

Xe

geðx; yÞdXx: ð25Þ


It should be noted that the above sV-SGS(y) definition grants a-priori the suitability of the proposed ap-

proach for high order finite element interpolation spaces, such as quadratic ones. As a result of the pro-

posed V-SGS model, the stabilization integral becomes:

ðwh; LUh � BÞ ¼ ðL�wh;U 0Þ ¼ �Xnele¼1

ZXe

L�whðyÞsV-SGSðyÞðLUh � BÞðyÞdXy : ð26Þ

3.1. Further considerations on sV-SGS in one-dimension

It is interesting to give more hints on the determination of sV-SGS(y). To this end, let use a one-dimen-

sional advective–diffusive–reactive model problem with constant physical properties. In this configuration

the adjoint problem for the element Green�s function reads as:

�kge;xx � uge;x þ cge ¼ dðx; yÞ in Xe; ð27:1Þ

ge ¼ 0 on Ce e ¼ 1; . . . ; nel: ð27:2Þ
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 coor-dinate transformation [27]:

� 2

h

� �2

kge;nn �2

h

� �uge;n þ cge ¼

2

h

� �dðn; fÞ n; f 2 ð�1; 1Þ; ð28:1Þ

geð�1; fÞ ¼ 0;

geðþ1; fÞ ¼ 0ð28:2Þ

here f is the auxiliary Dirac�s delta space variable. The element Green�s function for problem (28) is found

to have an exponential behaviour:

geðn; fÞ ¼ C1ek1n þ C2ek2n � 1 6 n 6 f;

geðn; fÞ ¼ C3ek1n þ C4ek2n f < n 6 þ1ð29Þ

where k1 and k2 are the roots of the characteristic equation associated to problem (28). The four closure

constants are determined by imposing on ge the homogeneous boundary values, the continuity in f and

the value of its first derivative jump in f defined as [21]:

ge; nðfþ; fÞ � ge;nðf�; fÞ ¼ � h

2k

� �; ð30Þ

where the ge first derivative discontinuity is related to the element length and to the diffusivity.

In Fig. 1 the behaviour of element Green�s function is shown for a one-dimensional logic element focus-

ing on the effect of Pe and r magnitudes, with f set equal to zero, h = 10�1 and k = 10�4.

It is worth noting that ge(n, 0) modulates the element error distributor mechanism moving from advec-

tion dominated limit (r ! 0), where it behaves like an upwind Heaviside function, to reaction dominated

condition where it approaches a symmetric impulsive-like shape.

By expressing the integral time scale using the element Green�s function (29), a fundamental feature ofthe proposed V-SGS model becomes evident:

sV-SGSðyÞ ¼Z

Xe

sV-SGSðxÞdðx; yÞdXx ¼Z

Xe

geðx; yÞdXx ¼Z þ1

�1

geðn; fÞðdet JÞdn; ð31Þ

-1 -0.5 0 0.5 10

1

2

3

4

5

r = 0r = 10r = 100

ξ

g (ξ

,0)

-1 -0.5 0 0.5 1

0

10

20

30

40

50

Pe = 0Pe = 10Pe = 100

ξ

g (ξ

,0)

(a) (b)

Fig. 1. Element Green�s function: (a) Pe = 100, r = 0–100; (b) r = 100, Pe = 0�100.


and in the above integral the Jacobian determinant is defined:

det J ¼ h2

� �: ð32Þ

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.

One of the most remarkable criticisms on the use of a element-wise constant s lays on its ability to con-

trol only element-wise constant residuals, that are obtained on advective–diffusive problems with linear ele-

ments [19]. If reactive terms appear and/or high order elements are used, there is no agreement between aconstant stabilizing parameter and a variable residual, thus addressing the need for a space dependent s, aspursued by the designed sV-SGS.

Another important feature of the proposed sV-SGS formula is its bubble behaviour, that permits to elim-

inate the inter-element integrals related to the properties of the trial and test function spaces used, thus

allowing integration by parts for the diffusive term in the residual based operator.

In order to show these properties it is worthwhile considering the sV-SGS expression in master element

coordinates for the different combinations of reactive and advective effects.

3.1.1. Advective–diffusive problem

The V-SGS stabilizing parameter in this limit case reads as:

sV-SGSðnÞ ¼ savdfSC 1þ ukuk n � e

ukukPen

sinhðPeÞ þ1

Pe

!" ,ðcothðPeÞ � 1=PeÞ

#;

savdfSC ¼ sSUPG ¼ h2kuk ðcothðPeÞ � 1=PeÞ:

ð33Þ

Expression (33) shows that sV-SGS could be seen as a sum of two terms: the first one, called savdfSC , which

assigns the magnitude of the stabilizing function, is exactly the element wise constant sSUPG intrinsic timescale (e.g. see [18]) as obtained for SGS methods proposed in literature (e.g. see [16]), and the second one

exploits the space dependence through a zero mean value function. The dependence on the element coor-

dinate permits to capture non-constant residuals, at least for one dimensional linear problems.

-1 -0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

ξ

τV-S

GS

-1 -0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

ξ

τV-S

GS

(a) (b)

Fig. 2. Stabilizing parameter sV-SGS for advective–diffusive problems in parent coordinates: (a) for u > 0 and (b) for u < 0 (solid

line = sV-SGS; dashed line = sSUPG = savdfSC ).


Fig. 2 shows that the sSUPG intrinsic time scale is indifferent to the velocity orientation whereas sV-SGS is

able to create a correct upwind effect. The behaviour is shown for Pe = 100, h = 10�1 and k = 10�4.

3.1.2. Diffusive–reactive problem

In this case the analytical expression of sV-SGS reads as follows:

sV-SGSðnÞ¼sdfrtSC � 1þ�e�ð

ffiffir

p=2Þð1þnÞþe�

ffiffir

p�ðffiffir

p=2Þð1�nÞþe�

ffiffir

p�ðffiffir

p=2Þð1þnÞ�e�ð

ffiffir

p=2Þð1�nÞþ 2ffiffi

rp 1�e�

ffiffir

p �2� �1�e�2

ffiffir

p� 2ffiffi

rp 1�e�

ffiffir

p �2� �0BB@

1CCA;

sdfrtSC ¼1

c1�

2 1�e�ffiffir

p� �2ffiffir

p1�e�2

ffiffir

pð Þ

!:

ð34Þ

Even in this case the resulting sV-SGS is obtained as a sum of a term sdfrtSC , which assigns the magnitude of

the stabilizing parameter, and a zero mean function. In Fig. 3 it is shown the behaviour of sV-SGS for

r = 102, h = 10�1 and k = 10�4.

3.1.3. Advective–diffusive–reactive problem

In the more general case of non-zero velocity and dissipation, sV-SGS has a richer behaviour and reads

as:

sV-SGSðnÞ ¼ savdfrtSC �

1þ Aavdfrt1 =Aavdfrt

3 �e�k2ð1þnÞ þ e�k1ð1þnÞ�2ðk2�k1Þ þ ek2ð1�nÞ�2ðk2�k1Þ � ek1ð1�nÞ� �þ

þAavdfrt2 =Aavdfrt

3 e�k2ð1þnÞ � e�k1ð1þnÞ�2ðk2�k1Þ � ek2ð1�nÞ�2ðk2�k1Þ þ ek1ð1�nÞ� �þ

� 8ð1þ r=Pe2Þ=Aavdfrt3 ðcothðk1Þ � cothðk2ÞÞr2=Pe3

2666437775;ð35:1Þ

-1 -0.5 0 0.5 1

0

0.2

0.4

0.6

0.8

1

ξ

τV-SG

S

Fig. 3. Stabilizing parameter sV-SGS for diffusive–reactive problems in logic coordinates (solid line = sV-SGS; dashed line = sdfrtSC ).


where the notation agrees with (28) and (29) and

savdfrtSC ¼ hkuk 2

Per

1þ 4Per

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r=Pe2

pðcothðk1Þ � cothðk2ÞÞ

" #;

Aavdfrt1 ¼ 1

ð�sgnðuÞ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r=Pe2Þ

qðe�2ðk2�k1Þ � 1Þ

;

Aavdfrt2 ¼ 1

ð�sgnðuÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r=Pe2

pÞðe�2ðk2�k1Þ � 1Þ

;

Aavdfrt3 ¼ 2


pr=Pe2

1þ 4Per


pðcothðk1Þ � cothðk2ÞÞ

" #:

ð35:2Þ

Even in this more general case sV-SGS could be decomposed in its element average value that gives the

scale of the stabilizing parameter, namely savdfrtSC , and a zero mean function which assigns the spatial depen-

dence. In Fig. 4 are shown the behaviours of sV-SGS for different combinations of Pe and r, with Pe = 100

and r varying from dominant advection to dominant reaction effects. The profiles are again obtained for

h = 10�1 and k = 10�4.

3.1.4. One-dimensional advective–diffusive–reactive examples

It is a matter of fact that consistent stabilized formulations are generally designed in order to fulfil the

super-convergence feature for one-dimensional problems [4,20]. In this viewpoint it is worthwhile comput-

ing some simple one-dimensional tests, where effective considerations could be done on differences and sim-

ilarities between the results obtained with V-SGS and with a classical stabilized formulation, i.e. the SUPG

method. Moreover a question that could be addressed with simple numerical experiments is the suitability

of V-SGS for both linear and quadratic elements.

According to these, the linear scalar advective–reactive–diffusive problem statement (1) in one-dimensionon the closed domain X = [0,1] has been considered, in order to make calculations for both linear and qua-

dratic finite elements. The computations have been performed both on a grid of 10 linear elements (h = 0.1)

-1 -0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

ξ

r

10

100

1000

V-SG

Sτ

-1 -0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

ξ

r

10

100

1000

V-SG

Sτ

(a) (b)

Fig. 4. Parameter sV-SGS for advective–diffusive–reactive problems in logic coordinates: (a) u > 0 and (b) u < 0.


and on a grid of 5 quadratic elements (h = 0.2), considering two different sets of Dirichlet boundary

conditions, namely U(0) = 0, U(1) = 1 and U(0) = 1, U(1) = 0 as in [14]. The governing parameters Pe

and r have been chosen, respectively, equal to 103 and 2 · 102, with B = 0 and the velocity u = 1 oriented

according to the positive semi-axis x.

The results on linear finite elements, shown in Figs. 5 and 6, reveal a general tendency to good stability of

both V-SGS and SUPG solutions, while the Galerkin method is unable to give an adequate solution. The

x0 0.25 0.5 0.75 1

-1.5

-1

-0.5

0

0.5

1 GSUPGV-SGS

U

Fig. 5. One-dimensional advective–diffusive–reactive problem on linear elements with U(0) = 0, U(1) = 1.

x0 0.25 0.5 0.75 1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1 GSUPGV-SGS

U

Fig. 7. One-dimensional advective–diffusive–reactive problem on quadratic elements with U(0) = 0, U(1) = 1.

x0 0.25 0.5 0.75 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6GSUPGV-SGS

U

Fig. 6. One-dimensional advective–diffusive–reactive problem on linear elements with U(0) = 1, U(1) = 0.


situation is different for quadratic elements, as shown in Figs. 7 and 8, where V-SGS solution is consider-ably sharper than the SUPG one.

x0 0.25 0.5 0.75 1

0

0.5

1

1.5

2

2.5GSUPGV-SGS

U

Fig. 8. One-dimensional advective–diffusive–reactive problem on quadratic elements with U(0) = 1, U(1) = 0.


3.2. Multi-dimensional stabilized V-SGS formulation

The multi-dimensional formulation of the proposed stabilized method must face the difficulties arising in

the treatment of the multi-dimensional integral of element Green�s functions in case of advective–diffusive–

reactive problem [21,27]. This forces the definition of the V-SGS model for nsd > 1 on the basis of a multi-

dimensional generalization of the time scale sV-SGS. The proposed method combines the 1D intrinsic time

scale parameters computed from the element Green�s functions associated to each parent domain coordi-

nate direction. In the 2D case, we solve the Green�s function problem for n and g 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:

sV-SGSni

ðx; yÞ ¼ sV-SGSni

ðniðx; yÞÞ ¼Z þ1

�1

geðf; niÞðdet JÞdf ¼ sV-SGSSCni

� ð1þ fniðni; Peni ; rÞÞ: ð36Þ

The two directional intrinsic time scales, namely sV-SGSn ; sV-SGS

g , computed in each element node must be

composed in order to obtain the sV-SGS to be used for the sub-grid scales contribution (26). In this respect,

the composing criterion, aimed to both a correct scale evaluation of the resulting stabilizing parameter and

a simple extension to three-dimensional problems, consists in using the following combination between the

directional sV-SGSn and sV-SGS

g :

sV-SGSðx; yÞ ¼ 1

1

sV-SGSSCn

þ 1

sV-SGSSCg

� ð1þ fnðn; Pen; rÞÞ � ð1þ fgðg; Peg; rÞÞ: ð37Þ


4. RANS formulation for incompressible turbulent flows

4.1. Problem statement

The dynamic response of incompressible turbulent flows is modelled by using a RANS approach. Eachquantity 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 ¼ U þ U 0.

The turbulence model for the closure of the system of equations is the k–e–v2–f (Durbin, [8]), for its im-

proved performance in simulating transition phenomena with respect to standard k–e model, due to the use

of v02 as velocity scale for turbulent transport toward the wall instead of k. As a matter of fact, recent LES

suggested that the turbulence fluctuation in the wall-normal direction, namely v 0, plays a fundamental role

in the evolution of transition [23], thus requiring a specific modelling equation. Moreover k–e–v2–f turbu-lence model features a rich background for testing the performance of a stabilized formulation in control-ling both advection and reaction dominated instabilities.

In the adopted model, the Boussinesq approximation is used for the stress–strain rate relation:

Table

k–e–v2

rkre

ce1ce2fe2RetclPk

D

ks

Ts

C1

C2

CL

Cg

u0iu0j ¼

2

3kdij � mtSij; ð38:1Þ

where dij is the Kronecker tensor, the eddy viscosity is given by

mt ¼ clv02k=e; ð38:2Þ
and it is used twice the strain rate tensor
Sij ¼ ð�ui;j þ �uj;iÞ: ð38:3Þ
The value used for cl could be found in Table 1, which describes all the coefficients and parameters nec-
essary to complete the model definition. The homogeneous boundary conditions are imposed on solid walls

for the modified elliptic relaxation variable, namely ef [23]. The RANS complete formulation is obtained in

terms of: momentum components q�ui ði ¼ 1; . . . ; 3Þ (where q is the density, and �ui the Cartesian averaged

velocity components), static pressure �p, turbulent kinetic energy k, homogeneous dissipation variableee ¼ e � 2mðoffiffiffik

p=oxiÞ2, average of the square of turbulence fluctuation in the wall-normal direction, namely

v02, and modified elliptic relaxation variable ef . The boundary value problem reads as:

1

–f closure coefficients

1

1.3

1:4 1þ 0:05

ffiffiffiffiffiffiffiffiffiffik=v02

q� �þ 0:4 exp �0:1Retð Þ

1.9

1��0:3 expð�Re2t Þ !k2/me0.22

qu0iu0k�ui;k

2m offiffiffik

p=oxi

� �2CLmax k3=2=e;Cgm3=4=e1=4

�max k=e; 6

ffiffiffiffiffiffiffim=e

p �1.4

0.3

0.23

70


Fajð eU Þ;j þ Fdjð eU Þ;j þ Frð eU Þ � B ¼ 0 in X 2 Rnsd; j ¼ 1; . . . ; 3;

U ¼ UD on CD; ð39Þ

Fdn ¼ hN on CN ;

where C = CD [ CN, with CD \ CN = B, n is the direction normal to the boundary, and U is the vector of

the averaged unknowns related to eU by

U � �u1; �u2; �u3; �p; k; ee; v02; ef !T ¼ eU þ 0; 0; 0; �p � 1; 0; 0; 0½ �T; ð40Þ

and could be interpreted in terms of primary-turbulent flow properties Up � �u1; �u2; �u3; 0; k;ee; v02; efh iTand

constrained variable Uc � ½0; 0; 0; �p; 0; 0; 0; 0�T. The boundary conditions, specified along the computational

domain boundary, generally include inflow Dirichlet conditions (UD) and outflow Neumann conditions

(hN). On solid boundaries, homogeneous Dirichlet conditions are imposed for Up. The flux vectors appear-

ing in (39) are defined as:

Fajð eU Þ ¼ ½�ujq�u1; �ujq�u2; �ujq�u3; �uj; �ujqk; �ujqee; �ujqv02; 0�T; ð41:1Þ

Fdjð eU Þ ¼ �r1j; �r2j; �r3j; 0;�q m þ mtrk

� �k;j;�q m þ mt

re

� �ee;j;�q m þ mtrk

� �v02;j;�L2s ef ;j$ %T

; ð41:2Þ

where Ls is the turbulent length scale, defined in Table 1, and the stress tensor is:

�rij ¼ �p#dij � qðm þ mtÞSij: ð41:3Þ

The non-linear Newtonian like turbulent stress terms are thus included, affecting the molecular kine-matic viscosity with mt, whereas the modified pressure (�p# ¼ �p þ qð2=3Þk) includes the isotropic part of

the turbulent stress tensor. The reactive terms are described by

Frð eU Þ ¼ 0; 0; 0; 0; ckk; ceee; cv2v02; c~fh iT; ð41:4Þ

with

ck ¼ qeek; ce ¼ qce2fe2

1

T s;

cv2 ¼ 6qek; c~f ¼ 1:

ð41:5Þ

It is thus possible to calculate the reaction numbers for the k–e–v2–f equations, that read as:

rk ¼ckh

2

q m þ mtrk

� � ; re ¼ceh

2

q m þ mtre

� � ; ð42:1Þ

rv2 ¼cv2h

2

q m þ mtrk

� � ; r~f ¼c~f h

2

L2s: ð42:2Þ

It is remarkable that reaction effects, not considered in most of the stabilized formulations available in

literature, could be relevant for all the four equations of the turbulence closure. In example, considering the

magnitude of reaction-to-advection ratios ðrk=Pek; re=Pee; rv2=Pev2Þ for the advective–diffusive–reactive equa-tions, it becomes relevant in the near-wall region, mainly within the viscous and buffer sub-layers. Moreover


reaction-driven effects on these equations are emphasized in presence of non-equilibrium phenomena such

as stagnation region, transition or separation. To this end it is possible to express the relative magnitude of

reaction with respect to advection in terms of a time scale ratio:

rkPek

� rePee

� rv2Pev2

� ekh

k�uk � Tsm

; ð43Þ

where T ¼ h=k�uk is the element mean time scale and sm = ke is the modeled turbulence time scale of thestandard k–e model. For instance, in case of a fully developed plane channel flow it is easy to show that

approaching the wall T/sm behaves as:

Tsm

� 1

d2w; ð44Þ

where dw is the distance from the solid bound. Finally, the source vector is defined as:

B �0; 0; 0; 0; Pk � qD; ce1Pkee=k;

qkef ;�½ðC1 � 1Þðv02=k � 2=3Þ�=T s þ ð5=T sÞv02=k þ ðC2=qÞPk=k

" #T: ð45Þ

The closure coefficients for the k–e–v2–f turbulence model [23] are recalled in Table 1.The non linear problem (39) is solved by using a fixed point technique, and the original problem is refor-

mulated into a generalized Oseen like form:

Fajð eU Þ;j þ Fdjð eU Þ;j þ Frð eU Þ � B ¼ 0 in X 2 Rnsd; j ¼ 1; . . . ; 3; ð46Þ

where Fajð eU ) is linearized for the primary variables by means of a given velocity field �V h(i.e. the velocity

field itself evaluated at the preceding equilibrium iteration). Moreover, Fdjð eU ), Frð eU ) and B are the lin-

earized versions of (41.2), (41.4) and (45) respectively.

4.2. V-SGS formulation for RANS equations

By introducing the following vector function spaces for n degrees of freedom:

H1hðXÞ ¼ ½H 1hðXÞ�n;

H1h0 ðXÞ ¼ ½H 1h

0 ðXÞ�n;

H ð1=2ÞhðXÞ ¼ ½H ð1=2ÞhðXÞ�n;

ð47Þ

the finite dimensional spaces of trial and test vector functions, for primary and constrained variables, are

defined as:

Shp ¼ Uhp U

hp

&&& 2 H1hðXÞ;Uhp ¼ UD on CD; UD 2 H ð1=2ÞhðCDÞ

n o;

Whp ¼ whp w

hp

&&& 2 H1h0 ðXÞ; whp ¼ 0 on CD

n o;

ð48Þ

Shc ¼ W hc ¼ U

hc U

hc

&&& 2 H 1h0 ðXÞ;whc

&&whc 2 H 1h0 ðXÞ

n o;

where the Galerkin test functions for mixed elements are adopted, namely quadratic for primary variables

and linear for constrained ones. The associated weights and adjoint based stabilization functions could be

written in vector form as:

wh ¼ whp; whp; whp; whc ; whp; whp; whp; whp !T

; ð49:1Þ


p ¼ p�u1 ; p�u2 ; p�u3 ; 0; pk; p~e; pv02; p~f

!T: ð49:2Þ

In (49.2) for each primary variable, a product is done multiplying two factors: the first one is the intrinsic

time scale obtained as described in Section 3, while the second is the associated adjoint operator acting on

the weight, that according to (4) reads as:

L�whp ¼ �FajðwhpÞ;j þ FdjðwhpÞ;j þ FrðwhpÞ: ð50Þ

The approximated variational formulation of the linearized differential problem (46) now reads as

follows:

find Uh 2 H1h; 8whp 2 W h

p; 8whc 2 W hc ; such that

cðVh; eU h

;whÞ þ sð eU h;whÞ þ rð eU h

;whÞ þ Pðð eU h;BÞ; pÞ ¼ ðB;whÞ � ðhN ;whjCN ÞCN ; ð51Þ

with use of bi-linear and tri-linear forms,

sð eU h;whÞ ¼ �

ZXwh

;j � FhdjdX;

ðB;whÞ ¼Z

Xwh � BdX;

ðhN ;whjCN ÞCN ¼Z

CN

whjCN � hNdC;

cðVh; eU h

;whÞ ¼Z

Xwh � Faj;jdX;

rð eU h;whÞ ¼

ZXwh � Fh

rdX;

ð52Þ

Finally, the stabilization integrals are defined as:

Pðð eU h;BÞ; pÞ ¼

Xnele¼1

ZXe

½p � ðFaj;j þ Fhr � BÞ � p;j � Fh

dj�dX; ð53Þ

where the stabilizing diffusive contributions have been integrated by parts according to the bubble nature ofintrinsic time scale parameters.

5. Numerical examples

In this section we assess the numerical performance of the proposed V-SGS formulation for model prob-

lems and configurations pertinent to turbomachinery fluid dynamics. In these validation studies the

improvement are commented with respect to the classical stabilization schemes, such as the Streamline Up-wind Petrov-Galerkin (SUPG), the Discontinuity Capturing (DC) [31] and the Streamline Upwind (SU),

and with respect to our recent Spotted Petrov-Galerkin (SPG) formulation [7]. It is remarkable that, since

all the consistent stabilization schemes fulfil the super-convergence feature for linear one-dimensional prob-

lems with homogeneous source and uniform grid [20], all the test cases have been designed in order to vio-

late at least one of the super-convergence conditions, i.e. they are all two-dimensional. In this respect the

multi-dimensional element characteristic length used to compute the stabilizing parameters for all the for-

mulations considered is purely geometrical, namely h = meas(Xe).


In Section 5.1 we investigate on the numerical performance of V-SGS on a model problem with a rich

solution behaviour, involving several conditions with advective–diffusive and advective–diffusive–reactive

flow phenomena. In the second case we consider first a homogeneous and then a non-constant source,

in order to violate one more super-convergence condition. Comparisons are done with respect to Galerkin

formulation, SUPG and SPG on quadratic finite elements.After that, in Section 5.2 we consider a real turbulent flow configuration, namely the flow around a

NACA 4412 airfoil, where super-convergence conditions are violated also with respect to both non-unifor-

mity of the grid and non-linearity of the problem. The performance of V-SGS are compared with respect to

the Streamline Upwind and to experimental data from Coles and Wadcock [6].

5.1. Advective–diffusive–reactive problems on a unit square domain

The three test cases proposed in this Subsection deal with the numerical solution of model problem (1)with different combinations of advective, diffusive, reactive and source term coefficients on a 2D unit square

domain with a uniform grid of 100 quadratic elements, thus consisting of 441 nodes. The test cases are

labelled as TC1, TC2, TC3.

5.1.1. Advection skew to the mesh

The TC1 test case deals with a classical problem (i.e. see [17]), namely the advection skew to the mesh of a

scalar unknownUon theunit square domain already introduced.Theproblem is described analytically impos-

ing c = 0, B = 0 and k = 10�5 in (1). The Pe number is 6 · 103, and the problem statement is shown in Fig. 9.The solutions predicted by Galerkin (GQ2), SPG improved with the addition of Discontinuity Capturing

(SPG + DC) and V-SGS are compared in Fig. 10, where is evident the superior behaviour of V-SGS in pre-

dicting strongly advective fields, as confirmed in Fig. 11 with the x-constant profiles of the solutions for

x = 0.1 and 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) without source term. The problem statement is outlined in Fig. 12. Theknown velocity field u is assumed to have a parabolic profile (e.g. u(x,y) = 2y � y2, v(x,y) = 0), with max-

imum value equal to 1. The coefficients are: k = 10�5, c = 0.75, B = 0. The maxima for dimensionless num-

bers are: Pe = o(103) and r = o(103).

V-SGS solution for TC2 is compared with SUPGQ2 and SPG ones in Fig. 13. It is worth noting that the

Galerkin GQ2 solution is not shown due to its too much oscillating behaviour. As clearly appears, the new

U=1

U=0

∂U/∂ n=0

∂U/ ∂n=0

u

ux /uy =2

y

x

U=1

Fig. 9. TC1 problem statement.

-0.2

0

0.2

0.4

0.6

0.8`

1

1.2

00.2

0.40.6

0.81

00.2

0.40.6

0.81

U U U

y x

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

00.2

0.40.6

0.81

00.2

0.40.6

0.81

y x

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

00.2

0.40.6

0.81

00.2

0.40.6

0.81

y x

(a) (b) (c)

Fig. 10. TC1 comparison of solution fields: (a) Galerkin GQ2, (b) SPG + DC and (c) V-SGS.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

GQ2SPG+DCV-SGS

U U

y0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

GQ2SPG+DCV-SGS

U

y(b)(a)

Fig. 11. TC1 comparison of solution profiles: (a) x = 0.1 and (b) x = 0.9.

u

∂U/∂n=0

∂U/∂n=0

U=1

∂U/∂n=0x

y

Fig. 12. TC2 Scalar advective–diffusive–reactive problem statement.


0

0.2

0.4

0.6

0.8

1

00.2

0.40.6

0.81

00.2

0.40.6

0.8

y x

U U U

0

0.2

0.4

0.6

0.8

1

00.2

0.40.6

0.81

00.2

0.40.6

0.81

y x

0

0.2

0.4

0.6

0.8

1

00.2

0.40.6

0.81

00.2

0.40.6

0.81

y x

(a) (b) (c)

Fig. 13. TC2 comparison of solution fields: (a) SUPGQ2, (b) SPG, (c) V-SGS.


proposed formulation is able of quite well controlling the instability origins in the near- and far-wall re-

gions, where the SPG gives the best results completely eliminating any kind of oscillation without assuming

the over-diffusive behaviour of the SUPGQ2 solution.Fig. 14 deals with the streamwise profiles of the solutions for y = 0 and y = 0.05, where reaction domi-

nates on both advection and diffusion, and shows a comparison of the nodal values in the sharp layer near

the Dirichlet boundary.

It is evident how an excellent accuracy is obtained by the SPG, closer to the exact sharp exponential solu-

tion. Moreover the V-SGS manages to damp the oscillations and, although the incomplete control of insta-

bility effects, shows a sharper layer than the one given by SUPGQ2.

5.1.3. Advective–diffusive–reactive problem with relevant source term

In the third test case (TC3) we focus on the V-SGS performance in solving equations with complex ana-

lytical structure, adding a non-zero source term B to the TC2 problem statement. The behaviour of this

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SUPGQ2SPGV-SGS

U U

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SUPGQ2SPGV-SGS

x

U, SUPGQ2 U, SPG

U, V-SGS

1.26 e-02 2.84 e-04 2.41e-05 4.71 e-02 2.92 e-03 9.26 e-03 1.13 e-01 1.69 e-02 2.63 e-02 4.17 e-01 1.73 e-01 1.26 e-01

1. 1. 1.

U, SUPGQ2 U, SPG

U, V-SGS

1.42 e-02 2.64 e-04 2.70e-04 4.51 e-02 1.72 e-03 -7.87 e-03 1.19 e-01 1.62 e-02 -1.49 e-02 3.78 e-01 1.06 e-01 5.14 e-01

1. 1. 1.

(a) (b)

Fig. 14. TC2 comparison of streamwise profiles: (a) in y = 0 and (b) in y = 0.05.

u

U=0

U=0

U=1

U/ n=0

x

y B∂ ∂

Fig. 15. TC3 Scalar advective–diffusive–reactive with source problem statement.

0

0.2

0.4

0.6

0.8

1

00.2

0.40.6

0.81

00.2

0.40.6

0.81

U U U

y x

0

0.2

0.4

0.6

0.8

1

00.2

0.40.6

0.81

00.2

0.40.6

0.81

y x

0

0.2

0.4

0.6

0.8

1

00

0.40.6

01

00.2

0.40.6

01

y x.8

.2

.8

(a) (b) (c)

Fig. 16. TC3 solution fields comparison: (a) SUPGQ2, (b) SPG and (c) V-SGS.


term is approximated with a linear function, as suggested by Codina et al. in [5], and reaches its maximum

at y = 0.1, with Bmax(y = 0.1) = 0.25. The problem statement of TC3 is shown in Fig. 15.

In Fig. 16 the solution fields obtained with SUPGQ2, SPG and V-SGS are compared. As in TC2, Galer-

kin solution is not shown due to its completely oscillating behaviour. It is worthwhile noting that SPG gives

an excellent prediction of the zone near the non-homogeneous Dirichlet boundary, where also SUPGQ2

features a better control of oscillations than V-SGS.Nonetheless, by comparing in Fig. 17 the crosswise profiles in the inflow boundary of the computational

domain, the solution obtained with V-SGS shows a completely sharp behaviour not achievable with the

other two formulations.

In Fig. 18 streamwise profiles are provided for the zone in which the source term features the sharpest

growth, namely for y 6 0.1, and a list of the nodal values in the Dirichlet layer is given. The comparison

between the listed values shows the ability of V-SGS in predicting sharp solutions layers without being af-

fected by high source terms, and its lower diffusivity with respect to SUPG Q2.

5.2. Turbulent flow over a NACA4412 airfoil

The last test case concerns with the numerical study of the flow over the NACA4412 airfoil at maximum

lift with incidence angle 13.87�, experimentally studied by Coles and Wadcock [6]. The Reynolds number

based on the chord length (lc) is: Relc ¼ ureflc=v ¼ 1:52� 106.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

SUPGQ2SPGV-SGS

y

B/c

U

Fig. 17. TC3 comparison of crosswise profiles in x = 0.0.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SUPGQ2SPGV-SGS

UU

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SUPGQ2SPGV-SGS

x

U, SUPG Q2 U, SPG U, V-SGS

1.79 e-01 1.67 e-01 1.67 e-01 2.05 e-01 1.68 e-01 1.62 e-01 2. 69 e-01 1.81 e-01 1.57 e-01 4.98 e-01 2.63 e-01 5.95 e-01

1. 1. 1.

U, SUPG Q2 U, SPG U, V-SGS

3.39 e-01 3.29 e-01 3.31 e-01 3.57 e-01 3.30 e-01 3.28 e-01 4.04 e-01 3.39 e-01 3.26 e-01 5.48 e-01 3.78 e-01 6.50 e-01

1. 1. 1.

(a) (b)

Fig. 18. TC3 comparison of streamwise profiles: (a) in y = 0.05 and (b) in y = 0.1.


Several industrial relevant physical phenomena are involved in the study of such an airfoil flow config-

uration, which is governed on suction side by oscillatory trailing-edge stall (OTE stall) with moderate loss

of lift when incidence is increased beyond the maximum value [11]. The modelling challenge is thus related

to the transitional boundary layer, experimentally located at 0.023 6 x/lc 6 0.1, and to the trailing edge sep-

aration bubble, strictly related to turbulence anisotropy and curvature effects [12].

The present investigation has been carried out by using the k–e–v2–f turbulence model [8], as described in

Section 4. The stabilization schemes are respectively V-SGS and Streamline Upwind on a Q2-Q1 interpo-

lation basis. A Generalized Minimal Residual Solver with a Flexible preconditioning [28] is used, and theKrylov space basis dimension is 10. The convergence thresholds are set to 10�4 for both the solution error

and its residual. The flow region has been modelled with a C-grid of 325 · 71 nodes and 5670 Q2-Q1

0 0.05 0.1

-0.05

0

0.05

y / lc

x / lc

Fig. 19. NACA 4412 airfoil mesh.


elements, with 175 nodes on the airfoil surface and a minimum grid spacing at the wall of 0.5 · 10�4lc. The

computational domain has an inlet section located lc upstream the airfoil and the outflow one is located

2 Æ lc downstream the trailing edge. The top and the bottom boundaries are lc away from the airfoil. Fig.

19 shows a grid detail close to the blade leading edge.

Concerning the boundary conditions, at the inflow section uniform profiles have been applied for the

velocity components and the turbulent variables (i.e. k, ee and v02). The inlet turbulence intensity (TI) is

set to 5% and the dissipation length scale is le = 6.5 · 10�2lc. No slip conditions have been imposed onthe airfoil profile, including homogeneous Dirichlet values on modified elliptic relaxation variable efand homogeneous dissipation ee. At the outflow, top and bottom boundaries homogeneous Neumann

-0.5 0 0.5 1 1.50

0.05

0.1

0.15

0.2

u/uref

0 0.5 1 1.5u/uref

0 0.5 1 1.5u/uref

dist

ance

(a) (b) (c)

Fig. 20. Comparison of streamwise velocity profiles: (a) x/lc = 0.786, (b) x/lc = 0.842, (c) x/lc = 0.953 (long-dashed lines: SUQ2; solid

lines: V-SGS; symbols: experiments).


conditions have been imposed on primary variables. Moreover, no transition triggering has been adopted

to obtain the results.

In Fig. 20 the streamwise velocity profiles are shown for three measurement sections, namely x/lc =

0.786, x/lc = 0.842, x/lc = 0.953. These probing traverses fall into the trailing edge separation region where,

as underlined by Hanjalic et al. [12], the computational difficulties are more evident.The predicted streamwise velocity distributions obtained with V-SGS agree with the experiments quite

better than Streamline Upwind (SUQ2) results. This solution appears to be affected by an anticipated sep-

aration that turns into a larger trailing edge bubble. This behaviour is confirmed in Fig. 21 by comparing

the chordwise evolution of turbulence intensity profiles. The disturbance related to the separation extends

to a larger distance from the solid wall for the SUQ2 results, shifting the position of the turbulence intensity

0 0.05 0.1 0.15

0

0.05

0.1

0.15

0.2

TI0 0.05 0.1 0.15

x / lc = 0.786x / lc = 0.842x / lc = 0.953

TI

dist

ance

(a) (b)

Fig. 21. Comparison of turbulence intensity profiles at different locations: (a) SUQ2, (b) V-SGS.

0.9 0.95 1 1.05 1.1 1.15 1.2

0

0.05

0.1

y / lc

x / lc0.9 0.95 1 1.05 1.1 1.15 1.2

0

0.05

0.1

y / lc

x / lc(a) (b)

Fig. 22. Comparison of streamtraces at trailing edge: (a) SUQ2, (b) V-SGS.

0.6 0.7 0.8 0.9 1

0

25

50

75

100

Pev2

rv2

x / lc

Fig. 23. Reaction-to-advection comparison on suction side rear portion.


peak farther from the airfoil than in the case of V-SGS. The same conclusions could be drawn by observing

Fig. 22, that shows a close-up view of the modelled separating bubbles, starting just beyond the experimen-

tal separation origin at x/lc = 0.9.

The differences in the simulated behaviours could be explained in view of the sensitivity of V-SGS solu-

tion to reaction effects. In this respect, Fig. 23 compares the variation of reaction and Peclet numbers forthe v02 budget on the suction side rear portion of the airfoil (x/lc > 0.5) at a constant distance from the wall

set to 10�3lc. As evident, the reaction dominance could be associated here to the presence of the adverse

pressure gradient that starts to govern the boundary layer development just downstream the laminar to tur-

bulent transition. The closure equations reactivity is such that the r-to-Pe ratio is of order 10 for k, ee and v02at the onset of the separation. This appears to be a counter-proof of the key role played by reaction effects,

that must be adequately considered in the design of stabilized formulations for the numerical simulation of

complex turbulent flows.

6. Conclusions

In this work we have presented a new Variational MultiScale finite element formulation, called V-SGS

(Variable-SubGrid Scale), for advective–diffusive–reactive problems arising in modelling complex turbulent

flows, tackling both advection and reaction induced numerical instabilities. The method is based on the

well-known element Green�s function approach, with some original features:

• the intrinsic time scale parameter sV-SGS is variable within the element and is obtained in view of suit-

ability for both linear and quadratic finite element spaces of interpolation;

• sV-SGS behaves �naturally� as a bubble, being zero on element boundaries and upwinding or centering its

maximum value according to advection-to-reaction relationship;

• the one-dimensional examples show that the proposed V-SGS formulation improves the stability with

respect to the state-of-art Petrov-Galerkin stabilization methods designed for advection dominated

problems, on linear and quadratic finite elements;

• the multi-dimensional extension is pursued with a scale combination of one-dimensional stabilizingparameters;


• a complete stabilized weak formulation is proposed for the RANS approach to incompressible turbulent

flows with an advanced turbulence closure model, namely the k–e–v2–f, with particular emphasis on the

parameters governing advection and reaction induced instabilities.

Numerical tests have been performed with quadratic finite elements in two dimensions for both modelproblems and a real turbulent flow configuration, namely the flow over a NACA 4412 airfoil. Good results

have been obtained and compared with those given by other stabilized formulations, such as SUPG and

Streamline Upwind, and with available experimental data.

Acknowledgment

The authors acknowledge MIUR under the projects COFIN 2003.

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