Final version of this article can be found as… A. Corsini, C. Menichini, F. Rispoli, A. Santoriello, and T.E. Tezduyar, “A multiscale finite element formulation with discontinuity capturing for turbulence models with dominant reactionlike terms”, Journal of Applied Mechanics, 76 (2009) 021211, http://dx.doi.org/10.1115/1.3062967.
A multiscale finite element formulation with discontinuity capturing for turbulence models with dominant reaction-like terms a
a
a
b
A. Corsini , F. Menichini , F. Rispoli , A. Santoriello and T.E. Tezduyar
c
a
Department of Mechanics and Aeronautics University of Rome “La Sapienza” Via Eudossiana 18, Rome, I00184, Italy
[email protected],
[email protected],
[email protected] b
GE Oil and Gas Via Felice Matteucci 2, Firenze, 50127, Italy c
Mechanical Engineering, Rice University - MS 321 6100 Main Street, Houston, TX 77005, USA
[email protected]
Abstract A stabilization technique targeting the Reynolds-Averaged Navier-Stokes (RANS) equations is proposed to account for the multiscale nature of turbulence and high solution gradients. The objective is effective stabilization in computations with the advection-diffusion– reaction equations, which are typical of the class of turbulence scale-determining equations where reaction-dominated effects strongly influence the boundary layer prediction in the presence of non-equilibrium phenomena. The stabilization technique, which is based on a variational multiscale method, includes a discontinuity-capturing term designed to be operative when the solution gradients are high and the reaction-like terms are dominant. As test problems, we use a 2D model problem and 3D flow computation for a linear compressor cascade.
1 Introduction
Eddy Simulation (LES) techniques based on variational
Special-purpose computational fluid mechanics techniques
multiscale methods [4-6].
targeting turbomachinery are becoming more and more
The physics of turbulent flows in turbomachinery
effective in better understanding of the flow problems in this
configurations is governed by non-equilibrium phenomena that
important application area. Many challenges, including the
cannot not be addressed adequately with the Boussinesq
turbulent flow features, are still in need of improved modeling
effective viscosity concept. This is because of the presence of
techniques. Examples of the efforts in this direction include
the curvature and rotation effects and large separation and
stabilization methods for turbulence closures [1-3] and Large
recirculation regions. Even when tackled with advanced turbulence closures such as non-isotropic first order models or
1
Reynolds-stress models, flow simulations involve numerical
stabilization without affecting the accuracy in advection-
shortcomings that are not fully addressed by standard
dominated zones and in zones where the solution is smooth.
stabilization methods developed in the context of advection-
The main application area we have in mind is turbomachinery.
diffusion type equations.
We are focusing on addressing the numerical challenges posed of the non-equilibrium
by the reaction-like terms appearing in the closure equations for
phenomena is that the flow is governed by scale-determining
advanced eddy viscosity models, such as the nonlinear k-ε
equations with dominant reaction-like terms, stemming from
model [14].
The
numerical
counter-part
turbulence dissipation mechanisms involved. For example,
In Section 2, we provide an overview of the nonlinear k-ε
reaction-like terms become dominant in stagnation regions,
model and the strong formulation of the RANS problem. The
separated boundary layers and re-circulating flow cores, where
variational multiscale formulation for the RANS equations is
the flow velocity approaches zero.
described in Section 3. In Section 4, we describe the
In recent decades, a number of studies focused on stabilized
formulations
for
stabilization parameters, discontinuity capturing and the DRD
advection-diffusion–reaction
method, including the DRDJ method, which takes into account
equations. These include equations governing chemically-
the local “jump” in the solution. The model test problem is
reacting flows and equations with numerically-generated
presented in Section 5 and the 3D flow computation of a linear
reaction-like terms. As examples of such studies, we can
compressor cascade in Section 6. Concluding remarks are given
mention the DRD method by Tezduyar and Park [7, 8], studies
in Section 7.
by Codina [9], USFEM by Franca and Valentin [10], SPG by
2 RANS formulation for incompressible turbulent flows
Corsini et al. [1], and stabilized methods emanating from the variational multiscale (VMS) concept [11], such as the ones
Let Ω ⊂ nsd be the spatial domain with boundary Γ, and (0, T) be the time domain. The unsteady RANS equations of
described in [12, 2, 13]. In this paper we describe a stabilization technique targeting
incompressible flows can be written on Ω and ∀t ∈ ( 0,T ) as
∂u + u ⋅∇u − ℑ − ∇ ⋅ σ = 0 , ∂t
the Reynolds-Averaged Navier-Stokes (RANS) equations,
ρ
(1)
∇⋅u = 0 ,
(2)
accounting for the multiscale nature of turbulence and the high solution gradients involved. The technique is based on the
∂φ + u ⋅ ∇φ + Bkε φ − ℑ kε − ∇ ⋅ ( ρ νkε ( ∇φ ) ) = 0 , ∂t
ρ
VSGS formulation [2] and includes discontinuity capturing in the form of a new generation DRD method [3]. The objective in
(3) T
where ρ is the density, u the velocity vector, φ = ( k ,ε ) , and k
the approach we take here is to accomplish the additional
2
Table 1. Closure coefficients in k and ε equations [13]
and ε are the turbulent kinetic energy and homogeneous dissipation variable. The symbols ℑ and ℑ kε represent the
ˆ ϖˆ ) 0.3 1 − exp -0.36/exp -0.75 max ( e,
vector of external forces and the source vector of turbulent
ˆ ϖˆ ) 1 + 0.35 max ( e,
Cµ
1.5
scale-determining equations. As proposed in Corsini and Rispoli [15], ℑ accounts for the volume sources related to the second and third-order terms in the non-isotropic stress-strain constitutive relation [14]. The force vector reads as
1 3
ℑ = ∇ ⋅ −0.1ν tτ ε ( u ) ⋅ ε ( u ) − ε ( u ) : ε ( u ) I
(
Τ
+0.1ν tτ ϖ ( u ) ⋅ ε ( u ) + (ϖ ( u ) ⋅ ε ( u ) )
(4.1)
)
1 +0.26ν tτ ϖ ( u ) ⋅ϖ ( u ) − ϖ ( u ) : ϖ ( u ) I 3
(
T
−10cµ2ν tτ 2 ε ( u ) ⋅ ε ( u ) ⋅ϖ ( u ) + ( ε ( u ) ⋅ ε ( u ) ⋅ϖ ( u ) ) 2
−5cµν tτ
2
( ε ( u ) : ε ( u ) ) ε ( u ) + 5cµν t τ 2
(
T
Here: ε ( u ) = ( ∇u ) + ( ∇u )
(
)
2
0.5 2 1 − exp − ( Ret / 90 ) − ( Ret / 400 )
fµ
)
Cε1
1.44
Cε2
1.92
fε2
[1-0.3exp(- Ret2 )]
σε
1.3
σk
1
The source vector ℑ kε is defined as
(ϖ ( u ) : ϖ ( u ) ) ε ( u ) .
Pk − ρ D , Cε 1 Pk ε + E k
ℑ kε =
is twice the strain-rate tensor,
(4.2)
) is twice the vorticity tensor, ν is the
where Pk = ρ R : ∇u is the production of turbulent kinetic
turbulent kinematic viscosity defined as νt = Cµ fµ τ k, τ = k/ ε
energy with R the Reynolds stress tensor, D = 2ν∇ k ⋅∇ k is
the turbulence time scale, with Cµ and fµ and other closure
the dissipation rate value on solid boundaries, and E = 0.0022
coefficients for the turbulence model [14] used given in Table
eˆ k τνt ∇ ⋅ ( ∇u )
T
ϖ ( u ) = ( ∇u ) − ( ∇ u )
t
1.
2
is the near-wall additional source.
The stress tensor, in the momentum equation, is defined as 2
In Table 1, Ret = k /(ν ε ) is the turbulence Reynolds
2 σ ( p, u ) = − p + k I + ρν u ε ( u ) , 3
number, and eˆ and ϖˆ are, respectively, the strain-rate and vorticity invariants defined as
eˆ = τ 0.5ε ( u ) : ε ( u )
and
(5)
with νu = ν + νt, where ν is the molecular viscosity. The diffusion terms in the turbulent scale-determining
ϖˆ = τ 0.5ϖ ( u ) : ϖ ( u ) . equations depend on a diffusivity matrix defined as
3
ν kε
νt 0 ν + σk = νt 0 ν + σε
,
uh ∈ S uh , p h ∈ S ph ,φ h ∈ S φh
find (6)
∫Ω w
∂u h ⋅ + u h ⋅ ∇u h − ℑ h d Ω + t ∂
h
∫Ω ε ( w ) : σ ( p , u ) d Ω − ∫Γ ∫Ω q ∇ ⋅ u d Ω + h
The reaction terms, absorption like in Eq. (3), account for
h
the dissipation-destruction matrices and are defined as
0
,
nel
e
e =1
(7.1)
h
h
h
w h ⋅ hh d Γ +
h
∑ ∫Ω
⋅φ Bε
∀w h ∈V uh ,
that
∀q h ∈V ph and ∀ψ h ∈Vφh
with the value of the coefficients σk and σε given in Table 1.
B Bkε φ = k 0
such
(9.1)
h P stab w h ,q h ⋅ Ł p h , uh − ρℑ d Ω +
(
nel
∑ ∫Ω ν e
DCDD
)
(
)
ρ∇w h : ∇u h d Ω = 0,
e =1
with where
Bk =
ε k
, Bε = cε 2 fε 2
ε k
.
(7.2)
∂w h Ł(qh, wh) = ρ + u h ⋅∇w h − ∇ ⋅ σ q h , w h , ∂t
(
)
(9.2)
The essential and natural boundary conditions for Eqs. (1) and and (3) are represented as
u = g,
and
n ·σ = h, and
∂φ h ⋅ρ + uh ⋅ ∇φ h + Bkε φ − ℑ khε ∂ t
φ = gkε on Γg,
(8.1)
∫Ω ψ
n ⋅ ( ρ νkε ( ∇φ ) ) = 0 on Γh,
(8.2)
∫Ω ∇ψ
h
nel
h
(
(
⋅ ρ ν k ε ∇φ h
dΩ +
)) d Ω −
where Γg and Γh are the complementary subsets of the boundary
P stab ψ h ⋅ Łkε φ h − ρℑ khε d Ω + ∑ ∫ Ω e kε e =1
Γ, n is the direction normal to the boundary, and g, gkε and h are
∑ ∫Ω
( )
nel
e
(10.1)
( )
h h Κ kDC ε ρ∇ψ : ∇φ d Ω = 0,
e =1
given functions representing the essential and natural boundary where conditions.
∂φ h Łkε(φh)= ρ + uh ⋅ ∇φ h + Bkε φ h − ∇ ⋅ ρ νkε ∇φ h ∂t
(
3 Variational multi-scale formulation for RANS equations In describing the VSGS formulation of Eqs. (1) and (3), we
(
) ) . (10.2)
are, respectively, the VSGS Here P stab , Pkstab Κ kDC ε , and ε
assume that we have constructed some suitably-defined finitestabilization operators and the dissipation matrix for the dimensional trial solution and test function spaces S uh ,S ph , S φh and V uh ,V ph ,Vφh . The variational multi-scale formulation, based on the VSGS method [2], reads as
discontinuity-capturing (DC) scheme, while νDCDD is for the discontinuity-capturing
directional
dissipation
(DCDD)
stabilization [16, 6]. The definition of νDCDD is given in Section 4.
4
A fundamental result [2] is the demonstration that VSGS stabilization can be seen as a particular form of the PetrovGalerkin operator, thus the vectors P
stab
stab kε
, P
take the
T
For u and φ = ( k ,ε ) , the time scales are defined along each parent coordinate as the product of element-wise time scale τ SCξ and the space-dependent function fξ i [2], on the i
following forms basis of directional Peclet numbers:
( )
(
)
P stab w h = τ VSGS ρ uh ⋅∇ w h + τ PSPG ∇q ,
( )
Pkstab ψh ε
0 τ VSGS -k = τ VSGS -ε 0
(11.1)
=
( Pe )
=
( Pe )
=
ξi u
ρ u h ⋅∇ ψ h ,
(
( Pe )
)
with τVSGS and τPSPG being the VSGS and PSPG stabilization the
latter
for
equal-order
,
(13.1)
uξ i hUGN ν 2 ν + t σk
,
(13.2)
,
(13.3)
velocity-pressure
approximations. In Eqs. (9.1) and (10.1), the VSGS parameters
ξi ε
are defined as the product of element-wise variable intrinsic time scale τVSGS. These are defined in Section 4.
uξ i hUGN ν 2 ν + t σε
and reaction numbers:
The dissipation terms for advection-diffusion-reaction
β k2 =
equations are defined as
κ DRDJ − k Κ kDC ε = 0
2ν u
(11.2) ξi k
parameters,
uξ i hUGN
0 . κ DRDJ −ε
(12)
Bk
ν (ν + t ) σk
2 hRGN −k , βε2 = 4
Bε
ν (ν + t ) σε
2 hRGN −ε . 4
(13.4)
In the above definitions, the element length in the advection dominated limit [7] is
Here κDRDJ-k and κDRDJ-ε are the DRDJ additional diffusivities,
hUGN = 2( ∑ s ⋅∇ N a )−1 ,
also defined in Section 4.
(14.1)
a
where s is the unit vector in direction of the velocity. 4 Stabilization parameters and discontinuity-capturing
In the diffusion-reaction dominated limit, the element 4.1 VSGS parameters
length turns to the corresponding scale [16, 17]: The intrinsic time scale τVSGS, which provides the sub-grid
hRGN − k = 2( ∑ rk ⋅∇ N a )−1 ,
scale residual modelling as proposed in Corsini et al. [2], is
hRGN −ε = 2( ∑ rε ⋅∇ N a )−1 ,
derived by the combination of one-dimensional intrinsic time scale
parameters,
associated
coordinate direction [2].
with
each
parent-domain
(14.2)
a
(14.3)
a
where rk and rε are the unit vectors in the direction of the solution gradient defined as
5
rk =
∇ k ∇ ε , rε = . ∇ k ∇ ε
(14.4)
with β = 0 for the flow equations, and time τ VSGS =
Remark 1. The reaction-like terms, not considered in most of
(16.2)
4.2 Discontinuity-capturing parameters The DCDD viscosity is defined by using the expression
the stabilized formulations found in the literature, could be important in turbulence computation. When considering the
∆t . 2
from [6], [16]
magnitude of the reaction-to-advection ratios, reaction driven
2
ν DCDD = τ DCDD u h ,
(17.1)
phenomena affect the flow in the near-wall region and are emphasized in non-equilibrium phenomena such as in the
with
stagnation, transition or separation regions. Corsini et al. [3]
τ DCDD recently demonstrated that approaching a solid wall, the
h hDCDD ∇ u hDCDD . = 2uref uref
(17.2)
reaction-to-advection ratio behaves like 1 / d w2 , with d w being
Here hDCDD is the element length scale in the diffusion
the distance from the solid wall.
dominated limit [16, 17], that reads as
hDCDD = hRGN = 2( ∑ r ⋅∇ N a )−1 ,
(18.1)
a
The multidimensional time-scale parameter, in its spacetime version, is computed by using the r-switch [18]:
= 1 space τ VSGS
τ VSGS
defined as
1 rs
1 rs
+ 1 time τ VSGS
r
s
r=
,
∇ u . ∇ u
(18.2)
(15) As far as the reaction-dominated limit is concerned, the original DRD method from Tezduyar and Park [7] was
where
space τ VSGS
and r is the unit vector in the direction of the solution gradient,
= 1 τ SCξ
proposed as a remedy for the numerical instabilities in Eqs.
1 rs
1 rs
+ 1 τ SC η
r
s + 1 τ SC η
r
s
⋅
(10.1). The DRD method was obtained for two limit cases: (16.1)
advection-reaction and diffusion-reaction. For both cases the analytical expression for the additional DRD term depends on dimensionless numbers that relate respectively the reaction rate
(1 + f (ξ ,Pe ,β )) ⋅ (1 + f (η ,Pe ,β )) ⋅ (1 + f (ζ ,Pe ,β )) , 2
ξ
ξ
2
η
η
2
ζ
ζ
6
to the advection and diffusion-rates, taking into account the
where t and v are two unit vectors orthogonal to s and to each
quality of the grid used.
other. We note that along the t and v directions the numerical
Recently [3], a new stabilization technique named DRDJ
diffusion is the one associated with the one-dimensional
was formulated, and this takes into account the local variation
diffusion-reaction
(“jump”) in the solution, turning to a DRD-like discontinuity-
characteristic length measure is provided by the hRGN [16, 17].
capturing scheme.
Remark 3. The parameter Je is defined as follows:
The DRDJ additional diffusivity reads, for advection-
Je = reaction limit, as
1 2
1 + 4α , (19.1) sinh 2 γ φ
case.
(φmax )e − (φmin )e φ
For
this
reason,
,
the
element
(22)
e
where (φmax)e and (φmin)e are the maximum and minimum values
κ AR (γ φ , J e ) = uhUGN J e − coth γ φ + γ φ
while for the diffusion-reaction limit it is defined as
(19.2)
corresponding to k and ε, with γk and γε defined as Bk hUGN , u 2
represents a local
scaling. Turbulent flow parameters are characterized by different orders of magnitude for different zones of the flow
where the subscript φ = k, ε generates the expressions
γk =
e
scaling for the unknown, which could be set equal to a global
2
hRGN −φ 1 1 − 2 , J e 4α + 2 sinh βφ βφ 2
κ DR ( βφ , J e ) = Bφ
of the variable φ for element e. Here φ
field, thus for turbulence computations φ
e
in a way takes into
account the local features of the problem and is chosen equal to (20.1)
the maximum value of the unknown in the element. With such a choice, as it is done for the calculations presented here, we
γε =
Bε hUGN . u 2
(20.2)
assure that Je ranges from 0 to 1, thus leading to a diffusivity that is everywhere limited.
Here, Je is a normalized measure of the variation (“jump”) in the solution over an element. In Eqs. (19.1) and (19.2), α is a parameter for the integration rule of the element reaction coefficient matrix (e.g. α equal to 1/6 for two-point Gaussian quadrature and 0 for the “lumped” case). The resulting DRDJ could be generalized to a multidimensional diffusivity tensor:
κ DRDJ -φ = κ AR ( γ φ ,J e )ss + κ DR ( βφ ,J e )( tt + vv ) ,
(21)
7
5 Scalar test case The scalar test problem was first proposed in [7] as a test for the original DRD formulation. The square domain is discretized with of a non-isotropic Cartesian grid with 41 × 21 linear finite elements, and the problem statement is given in Fig. 1.
b)
c)
d)
φ,y=0
1
5φ + uφ , x = 0, u ( y ) = umax (1 − y 2 ),
φ=1
umax = 1.
0.5
Fig. 2. Scalar test case. a) exact solution, b) SUPG, c) SUPG + DRDJ, d) VSGS + DRDJ.
φ,y=0 0
a)
0
0.5
φ,x=0
1
1
φ
φ
Fig. 1. Scalar test case. Problem statement, grid and boundary conditions. Figure
2
shows
the
solutions
with
the
0.75
SUPG, 0.5
SUPG+DRDJ, and VSGS+DRDJ. Figure 3 shows some solution profiles extracted from near the boundaries where the
exact MLLumping SUPG + Mass SUPG + newDRDJ DRD DRDJ V-SGS + new DRD
0.25
reaction dominates advection. The exact and SUPG plus mass 0
lumping (SUPG+ML) solutions are also plotted to provide a)
-0.1
comparison.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
1
1
φφ
The SUPG+DRDJ solution exhibits a 0.05 undershoot
exact ML SUPG + Mass Lumping DRDJ SUPG + new DRD DRDJ V-SGS + new DRD
0.75
along the second row of nodes (see Fig. 2.c and 3.a). This local 0.5
oscillation is eliminated in SUPG+ML solution but with an over-diffusive layer, and in VSGS+DRDJ solution, which
0.25
produces the sharpest undisturbed solution layer. For the third 0
row of nodes (Fig. 3.b), the DRDJ solutions are all very close to b)
the exact solution but SUPG+ML still shows an over-diffusive behaviour.
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
1
Fig. 3. Scalar test case. Profiles extracted at a) second and b) third rows of nodes.
8
From these results we conclude that stabilization methods designed to remedy instabilities due to dominant advection NACA65 cascade geometry
terms cannot control instabilities due to dominant reaction-like
profile family aspect ratio
terms. Some additional stabilization, different from mass lumping, which is over-diffusive, is needed. The DRD
NACA65 – 1810 1.0
chord
200 mm
spacing
180 mm
solidity
1.111
stagger angle
10°
formulation serves that purpose while retaining the solution accuracy. Fig. 4 Cascade geometry and computational grid. 6 Turbulent flow test case: NACA65 compressor cascade
The solution domain encompasses the upper half of the
We considered the tip leakage flow in a linear compressor
blade span (i.e. 104 mm from the casing to midspan, expecting
cascade of NACA65-1810 profile with flat ends. The main
the flow to be symmetric in the spanwise direction), bounded by
cascade design parameters are shown in Fig. 4 together with the
the two symmetry planes in the pitchwise direction, and
computational grid. The cascade has been experimentally
stretches 40% of the chord length upstream from the blade
investigated by Kang and Hirsch [19, 20]. The cascade has a tip
leading edge and one chord downstream from the trailing edge.
clearance of 2% of the chord length, and the inlet flow angle is equal to 29.3° with respect to the streamwise direction.
The coordinate system used is orthogonal, with x, y and z denoting the streamwise, pitchwise and spanwise directions,
The cascade flow is simulated in near-design incidence
respectively. An embedded H-topology computational mesh
condition, with the flow regarded as incompressible and steady.
with a total of about 0.8 million nodes with Q1-Q1 elements is
In accordance with the experimental findings, the flow at the
adopted. There are 141 nodes in the stream-, 73 nodes in the
inlet is fully turbulent, i.e. the measured shape factor is about
pitch- and 81 nodes in the span-wise directions, respectively.
1.22 at 40% of the chord length upstream of the leading edge.
The mesh is clustered around the blade walls, leading and
The Reynolds number, based on the chord length and the inlet
trailing edges and in the wake. The first wall-adjacent y+ values
bulk velocity is 3×105. The experimental free-stream turbulence
were everywhere lower than 1.6. A uniform discretisation (with
intensity is 3.4%. The dissipation length scale lε is set equal to
21 points) is used to resolve the tip gap.
5% of the chord length.
The position of the inlet is regarded as being sufficiently distant to eliminate any upstream effect of the outflow conditions on the solution both in the cascade passage and in the wake region.
9
Figure 5 shows the static pressure coefficient Cp computed
compared to the flow physics as interpreted by Kang and Hirsch
by using SUPG+DRDJ and VSGS+DRDJ. The figure shows
[19]. The figure shows the streamline patterns at various
that the two numerical solutions exhibit similar behaviors, being
significant locations, obtained by the SUPG+DRDJ and
both in qualitative agreement with experimental results. When
VSGS+DRDJ methods. The two solutions are very close, and
comparing the pressure fields in the region where the leakage
compare well to the experimental interpretation.
flow develops and rolls-up, the VSGS+DRDJ provides a nicer and sharper representation of the isobar troughs tracing the
Sp
vortex core path. This is due to the reaction limit control in the vortex defect region.
PV TL HSV a) Sp
a)
PV TL HSV
b)
Sp
b) PV TL HSV
c)
c)
Fig. 5 Static pressure coefficient isolines in the tip gap, a) experiments [19], b) SUPG+DRDJ and c) VSGS+DRDJ.
Fig. 6 Streamlines and flow patterns in the tip gap, a) experiments [19], b) SUPG+DRDJ and c) VSGS+DRDJ. (Sp: saddle point, PV: passage vortex, TL: tip leakage, HSV: horse-shoe vortex).
Figure 6 compares the predictions of the secondary flow
Figure 7 shows the chordwise evolution of the normalized
phenomena developing at the endwall, under the influence of tip
turbulent intensity TI on cross-flow planes. The TI isolines are
leakage vortex. The streamline behaviours predicted are
visualized with the tip leakage vortex streamlines. When
10
comparing the vortex cores, we see that both numerical 150
formulations predict fairly well the complex multiple-vortex 100
aerodynamics at the tip [19]. They successfully detect i) the main tip leakage vortex, ii) a tip separation vortex travelling
10
from the pressure to the suction side of the blade, and iii) the separation vortex at the leading edge. Figure 8 shows the turbulent viscosity νt (normalized by the
a)
b)
Fig. 8 Normalized turbulent viscosity νt/ν distribution, a) SUPG+DRDJ and b) VSGS+DRDJ.
molecular viscosity ν) in the vicinity of the blade tip. The νt
7 Conclusion
contours are shown on the crossflow planes to describe its axial
In this paper we presented a variational multiscale method for
evolution within the swirling region at the tip. The comparison
turbulence modeling based on the RANS approach. The method
shows that the VSGS+DRDJ formulation exhibits less diffusive
addresses the numerical issues related to dominant reaction-like
behavior in the region with the largest solution gradient.
terms involved in the turbulence model. A discontinuitycapturing term provides the additional stabilization needed in parts of the flow domain where the reaction-like terms become large, without introducing excessive dissipation in other parts of the domain. The test computations presented are for a 2D model problem and 3D flow computation for a linear compressor cascade, and they establish the effectiveness of the method. Acknowledgements
a)
The authors acknowledge MIUR support under the projects Ateneo and Visiting Professor Programme at University of
Rome “La Sapienza”.
References [1] A. Corsini, F. Rispoli, A. Santoriello, “A new stabilized b)
Fig. 7 Turbulence intensity TI distribution, and tip leakage vortices detection, a) SUPG+DRDJ and b) VSGS+DRDJ.
finite
element
method
for
advection-diffusion-reaction
equations using quadratic elements”, in: Lajos, T. et al. editors, Modelling Fluid Flow, Springer Verlag, 2004.
11
[2] A. Corsini, F. Rispoli, and A. Santoriello, “A variational
[9] R. Codina, “Comparison of some finite element methods for
multiscale
for
solving the diffusion–convection–reaction equation”, Computer
turbomachinery flow computations”, Computer Methods in
Methods in Applied Mechanics and Engineering, 156, (1998)
Applied Mechanics and Engineering, 194 (2005) 4797–4823.
185–210.
[3] A. Corsini, F. Rispoli, A. Santoriello, T.E. Tezduyar,
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