12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

THE QUASI-SPECTRAL VISCOSITY (QSV) CLOSURE TO SUBFILTERSCALE FLUXES: UNIFYING SHOCK CAPTURING AND LARGE-EDDY
SIMULATION

Victor C. B. Sousa
School of Mechanical Engineering
Purdue University
West Lafayette, IN, 47907
vsousa@purdue.edu

Carlo Scalo
School of Mechanical Engineering
School of Aeronautics and Astronautics
Purdue University
West Lafayette, IN, 47907
scalo@purdue.edu

ABSTRACTthe concept of a wavenumber-dependent eddy viscosity with aplateau at large scales and a cusp near the resolution limit tomodel the energy transfer across a filter cutoff while the second
proposed to inform the magnitude of such Spectral
aEddy Viscosity (SEV) from the kinetic energy at the cutoff, Epkc q{kc .
This renders the model dynamic, being only active when small
scales are present in the flow.
On the other hand, Tadmor (1989) studied the use of
Fourier-based discretization methods to solve the inviscid
Burgers’ equations and concluded that, if no regularization
term was introduced, the covergence of the numerical solution was not guaranteed. He then proposed to introduce awavenumber-dependent viscosity term concentrated on small
scales, named Spectrally Vanishing Viscosity (SVV), thatwould prevent oscillations and lead to convergence to theunique entropy solution. The idea was shown to be successful by mathematical proofs and numerical experiments.
The current manuscript analyzes the SEV and SVV
frameworks and highlights their similarities bridging a gap between methods initially developed for different purposes. Suchanalysis is carried out exploring the common features that exist
in both incompressible Navier-Stokes equations and the Burgers’ equation. This understanding allows for the specific design of a unified method for shock capturing and subfilter turbulence modeling. The QSV method is then developed in thecontext of high-order finite-difference implementations and it
is tested in situations of shock dominated problems, in predominantly hydrodynamic flows and also in shock/turbulenceinteraction cases. For example, QSV is able to simulate a Taylor Green Vortex with both sub and supersonic initial perturbations using identical structures.

The Quasi-Spectral Viscosity (QSV) method is a closure for a high-order finite-difference discretization of the filtered compressible Navier-Stokes equations capable of unifying dynamic subfilter scale (SFS) modeling and shock capturing under a single mathematical framework. It introduces aphysical-space implementation of a spectral-like SFS dissipation term by leveraging residuals of filter operations, achieving
two goals: (1) estimating the energy of the resolved solution
near the grid cutoff; (2) imposing a plateau-cusp shape to thespectral distribution of the added dissipation. The QSV approach was tested to showcase its capability to act interchangeably as: a shock capturing method, in a shock/sinusoidal wallinteractions problem; or as a SFS closure, in a subsonic Taylor
Green Vortex (TGV). QSV performs well compared to previous eddy-viscosity closures and shock capturing methods insuch test cases. In a supersonic TGV flow, a case which exhibits shock/turbulence interactions, QSV alone outperforms
the simple superposition of separate numerical treatments for
SFS turbulence and shocks. QSV’s combined capability ofsimulating shocks and turbulence independently, as well as
simultaneously, effectively achieves the unification of shock
capturing and Large-Eddy Simulation.

INTRODUCTION
Although hydrodynamic turbulence and shock formation
have been treated separately in previous literature, both arecharacterized by an energy cascade from large to small scales
due to nonlinear interactions (Gupta & Scalo, 2018). Therefore, it is proposed to treat the numerical simulation of bothphenomena in a similar fashion, as the consequence of solving
the filtered compressible Navier-Stokes equations.
The Quasi-Spectral Viscosity (QSV) method (Sousa &Scalo, 2022b) builds upon previous development in turbulencemodeling by Kraichnan (1976) and Chollet & Lesieur (1981)
and, in shock capturing by Tadmor (1989). The first introduced

FOUNDATIONS FOR THE UNIFICATION OF
SHOCK CAPTURING AND LES
The objective of this section is to highlight similarities
and establish a connection between eddy viscosity models, the1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

0.0
x

0.1

0.2

0.0

b)
0.0

0.2

0.4
0.6
k/kc

0.8

E(k)

τc
τ11max11 /cτ11−0.2 −0.10.2c)−1.0 −0.5 0.0 0.5 1.0
x

1.0

V

0.0

a)

QSV

V-2

0.2

QS

0.4

SVV-2

SV

0.4

0.6

V

SVV-1

10−2

V-1

0.6

0.8

SV

0.8

SEV

SE

DYN

10−1

2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
0.00

N
DY

1.0

DYN
SEV
SVV-1 √
(m = 5 kc )SVV-2
(m = 0)
QSV

u

1.0

1.2

10−3
10−4
10−5
10−6 0
10

d)
10

2

10

4

106

k

Figure 1. Results from a priori and a posteriori analysis of the filtered Burgers’ equation at time t “ 1 with N “ 128 grid points.
Exact SFS stress (blue) compared against SFS stresses predicted by various models (see legend), in the physical (a) and spectral (b)domain. Exact filtered solution (blue) versus a posteriori solution by various models (see legend) in the physical space (c) and theirrespective energy spectrum (d).

at a certain time instant by a sharp spectral transfer function
down to the same resolution as the numerical simulations performed. This allows the comparison between the exact SFS
stress, ui u j ´ui u j , and the output of each different model whenthe exact filtered solution ui is used as input. The intent of
this operation is to exploit the access to a reference solution
to calculate τi j explicitly, the term that needs to be modeled tosustain a filtered solution of a given equation, and assess eachmodel’s ability to generate similar effects while only having
access to the information present on the large flow scales. The
both a priori and a posteriori results are shown in physical andspectral space in figure 1.
It can be observed that the exact energy flux to subfilter
scales spans across all resolved wavenumbers and it peaks near
the cutoff (kc ), similar to Kraichnan (1976); Chollet & Lesieur(1981)’s theoretical findings for turbulent flows. Interestingly,
previous methods for either turbulence modeling or shock capturing are able to stabilize the flow with different degrees of
efficacy. The DYN model leads to a local response in physicalspace and, therefore, a flat broadband spectral response, whatcauses spurious high-wavenumber build up due to the insufficient damping of the resolved scales near the cutoff. Other
models do a better job by ramping up the dissipation at thesmall scales but either over (SEV) or underdamp (SVV) the
large scales present in the flow.
Despite the differences in the development, purpose anduse between the models, the results in figure 1 indicate thatthey can all be encompassed under a generalized format ofsubfilter scale flux model consisting of a magnitude pre-factor
composed by a length scale (`) and velocity a velocity scale(υc ) and a kernel (K) which is convolved with the strain rate
tensor (Si j ) and is responsible for the dissipation wavenumber
dependence, as indiscontinuity regularization method based on artificial addition of a spectrally vanishing viscosity (SVV) (Tadmor, 1989)
and the implementation of the Quasi-Spectral Viscosity (QSV)
method.
To begin, it is pointed out that it is possible to solve only
for the large scales present in the Navier-Stokes equations if
the original system is filtered by an operation that commuteswith the derivatives with an associated filter width (∆). That
procedure forms the mathematical framework of Large-Eddy
Simulations (LES) allowing, for example, the simulation of
turbulence in relatively coarse grids through the solution of
the filtered incompressible Navier-Stokes, constituted by theidivergence free condition Bu
Bxi “ 0, and by the conservation of
momentum equation,
Bτi j
Bui Bui u jBP
1 B 2 ui
`
“´
`
´
,
Bt
Bx j
Bxi
Re Bx j Bx j
Bx j

(1)

2 and τ “ u u ´ u u is the subfilter
where P “ p{ρrefUref
ij
i j
i j
scale (SFS) stress tensor, a remainder of the filtering operation
applied to the nonlinear governing equations. Since the SFS
term depends on the unresolved scales in the flow, it must be
modeled. The same procedure can be used to derive the filteredversion of the inviscid Burgers’ equation,

Bu1 1 Bu1 u1
`
“ 0,
Bt
2 Bx1

Bu1 1 Bu1 u1
1 Bτ11
`
“´
.
Bt
2 Bx1
2 Bx1

(2)

Because of its simplicity, its parallel with the LES frameworkand its nonlinearity that can lead to the formation of shock
discontinuities in finite time, the Burgers’ equation is used
as a way to showcase the similarities between the Dynamic
Smagorinsky (DYN) (Smagorinsky, 1963; Germano et al.,1991), the Spectral Eddy Viscosity (SEV), the SVV and QSVmodels.
The Burgers’ equation is solved in a periodic domain
x1 P r´1, 1s, starting with with initial conditions u1 px,t “
0q “ 1 ` 21 sinpπxq, until t “ 1 using a pseudo-spectral Fourier
method for the spatial derivatives and a 4th order Runge-Kutta
scheme for the time integration. The results obtained using
the different eddy viscosity closures are then compared withthe analytical solution based on the method of characteristics
in the physical and spectral domain. In the current work, theprocedure of comparing simulated results to a reference solution will be referred to as a posteriori analysis. In addition,
an a priori analysis is also conducted. It is defined as an operation that filters the exact solution and its nonlinear terms`
˘
τi j “ `υc K ˚ Si j .

(3)Additional evidences for the generalization are provided in
Sousa & Scalo (2022b).
This demonstrates that the same principles used by largeeddy simulations of hydrodynamic turbulence, focusing on
simulating only the large, resolved or filtered scales, can beused for discontinuity capturing as well. Ultimately, this inspired the Quasi-Spectral Viscosity (QSV) closure, designed
specifically to unify the LES and shock capturing methodologies in high-order finite difference implementations, which canbe generically written as,
´τ “

2

b
´
¯
rqsv q ˚ S .
2∆Ekc p1 ´ G(4)12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
SFS modeling via the QSV method

There are two steps for implementing the method: first, estimating the cutoff energy, Ekc , and second, introducing aplateau-cusp behavior as a function of wavenumber, done via
rqsv kernel. For these to be implemented in a highthe 1 ´ Gorder finite difference setting, it was necessary for them to be
performed only using spatial operators and that was achievedby leveraging the residuals of Padé ((Lele, 1992) and Fejér(Vandeven, 1991) filtering operations. The details behind theestimation of Ekc and the construction QSV wavenumber modulation kernel can be found in Sousa & Scalo (2022b).
Although the current manuscript is focused on applications based on structured finite difference solvers, the contentsof this section may serve as the foundation for the applicationof an unified approach to shock capturing and SFS turbulencemodeling to different platforms, such as unstructured blockspectral solvers. In fact, the shock capturing capability of solving for filtered governing equations was shown for high-orderflux-reconstruction settings in Sousa & Scalo (2022a). A projection onto the Legendre basis functions was used in that work
to allow for the estimation at the energy near the cutoff and forthe modulation of the dissipation at different scales. An extension for turbulent and shock/turbulence interaction problems
in this context is the next step.

A complete model for performing compressible large
scale simulations that could include both shocks and turbulent
events is then proposed as,ˆ˙
B u: j
1
B u:i
Di j
,
` D ji
2
Bx j
Bxi
B T:
q j “ ´Cq D j j,
Bx j

τi j “ ´Cτi j

(9)
(10)

where Di j “ υi pǔ j q` j is the dissipation magnitude tensor, being ` j “ ∆ j is the subfilter length scale anddυi pǔ j q “` ˘
2Eki c ǔ j

(11)

∆i

being the subfilter velocity scale. Note that the cutoff energy
estimation operation, explained in detail in Sousa & Scalo(2022b), is carried out in the i-th spatial direction but on thej-th filtered velocity component. Additionally, the double dot
superscript in (9) and (10) indicates the filter modulated quantities, defined as

QSV CLOSURE APPLIED TO THE FILTEREDCOMPRESSIBLE NAVIER STOKES EQUATIONS¯
B u:i
B ǔi ´rqsv ,
“
˚ 1´G
Bx j
Bx j
¯
B Ť ´B T:rqsv ,
“
˚ 1´G
Bx j
Bx j

In this section, the compressible implementation of large
scale simulations focusing on the unification of the treatment
between shocks and turbulent eddies will be discussed. Firstthe governing equations will be derived from the compressible Navier-Stokes relations using filtering operations and theunclosed terms as well as their closure will be presented.

(12)
(13)

where the modulation step is performed in the three direcrqsv is a weighted average between Padé- and Fejértions and G
filtered quantities (Sousa & Scalo, 2022b). It is possible thatthe cutoff energy estimation procedure will lead to large variations in space when highly localized flow features, such as
shocks, do not align with the grid or the grid itself is deformed.
In those cases, a smoothing procedure, i.e. a gaussian filter(Cook, 2007), can be performed on the υi pǔ j q term.
The values of the pre-factors, Cq and Cτi j are given by:

Governing equations
The compressible Navier-Stokes system of equations canalso be filtered by an operation that commutes with the derivation similarly to its incompressible counterpart (1). After the
initial filtering, it is chosen to implicitly solve density related nonlinear terms by defining the Favre filter operationas, fˇ “ ρρf , and solve for the Favre filtered quantities p fˇq,
ultimately simplifying the final LES system for compressible
flows. Previous compressible LES implementations based on
the Favre-filtered equations were successfully performed withdifferent closure models by, for example, Vreman et al. (1995)
and Nagarajan et al. (2003). In this work, we follow the pathof the latter, and derive the Favre-filtered Navier-Stokes relations introducing a pressure correction based on the subfilter
contribution to the velocity advection,#
1.0, if i “ j,Cτi j “
0.6, if i ‰ j,

and Cq “ 0.8.

(14)These values are obtained by carrying out a similar a priori
analyses as the one shown in figure 1. The magnitude ofthese constants are informed, though, through the applicationof such procedure to test cases such as the 1D Riemann shocktube problem (Sod, 1978) and the Taylor Green Vortex (TGV).
The details of such analyses are gathered in Sousa & Scalo(2022b). Ultimately, the suggested constants (all in the order of unitary value) work well for a broad set of test cases
as shown below. However, if different numerical schemes orfilters are used, adjustments might be necessary.

Bρ Bρ ǔ j`
“ 0,(5)Bt
Bx j
Bρ ǔi Bρ ǔi ǔ j
B p Bµ σ̌i j Bρτi j
`
“´
`
´
,(6)
Bt
Bx j
Bxi
Bx j
Bx j
ˆ
˙
Bµ σ̌i j ǔi BρC p q j
BE BpE ` pqǔ j
B
B Ť
k
`
`
“
´
,(7)
Bt
Bx j
Bx j
Bx j
Bx j
Bx j
p
1
1
“ E ´ ρ ǔi ǔi ´ ρτii .(8)
γ ´1
2
2

DEMONSTRATING QSV’S SHOCK CAPTURINGCAPABILITY
It is chosen to simulate a challenging test case for a shock
capturing model to assess how the QSV model behaves andcompare it against the established Localized Artificial Diffusivity (LAD) (Cook, 2007; Kawai et al., 2010). Inspiration is

The non negligible terms that contribute to the energy flux
from large to small scales are, the SFS stress tensor, τi j “}j ´ Ť ǔ j .
u}
i u j ´ ǔi ǔ j , and the SFS temperature flux, q j “ Tu3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 2. Numerical Schlieren comparison between QSV- and LAD-based simulations of a M “ 5.0 shock in a fluid with γ “ 1.15
after it is reflected from a sinusoidal wall performed with Nx “ 1024 and Ny “ 256 grid points.
taken from experiments reported by Denet et al. (2015) andnumerical simulations by Lodato et al. (2016), where a planar
shock wave impinges on a sinusoidal wall with 1.0 mm amplitude and wavelength of 2.0 cm. These tests were designed toconfirm the theoretical results by Clavin (2013) that predicted
the formation of a lasting pattern of triple points after a shockreflection off a smooth sinusoidally-perturbed wall.
Following Tonicello et al. (2020), a computational domain encompassing one wavelength and 10 cm in the wall normal direction is used to simulate a M “ 5.0 shock propagating
in a fluid with a lower specific heat ratio of γ “ 1.15, which increases the strength of the reflected shock and approaches the
Newtonian limit (Lodato et al., 2017). Moreover, the shock is
initialized 7.5 cm away from the wall and the Sutherland’s law
for air was used to model dynamic viscosity.
The simulations are solved with periodic conditions at the
top and bottom boundaries together with a no slip adiabaticwall on the left. Moreover, this test case confirms the capability of the QSV model to solve the filtered equations in distorted grids. The grid-transformations necessary for such calculations is described in Sousa & Scalo (2022b).
Figure 2 shows a comparison between the resolution capability of QSV- and LAD-based simulations for the same test
case. The QSV approach is able to accurately simulate theflow dynamics, e.g. the presence of triple points at the intersection of the incident shock (IS), the Mach stem (MS), the reflected shock and the slip line (SL) and the formation of strong
counter-rotating vortex pairs after the collision and subsequentdetachement of slip lines, as well as preserve symmetry. On
the other hand, when the LAD approach is used to solve thecurrent test case, a high level of spurious oscillations was observed. Additionally, an early symmetry breaking behavior
is present. Similar post shock oscillations were reported to
be a numerical challenge during in the analysis performed in
Lodato et al. (2017).

ulation started from the initial conditions against a reference
solution, in this case a Direct Numerical Simulation (DNS)
and is shown hereafter. The prior was the source of valuable
information regarding the coefficients used in the QSV closure
and it is gathered in Sousa & Scalo (2022b).
The subsonic TGV test case is initialized in a cubic domain Ω P r´πL, πLs3 with a Reynolds number Re “ ρ0V0 L{µ0equal to 5000 and a Mach number Mt “ 0.1. This choice
avoids compressibility effects with the objective to focus only
on QSV’s performance when applied to hydrodynamic turbulence.
An a posteriori study is performed with 963 grid points,
comparing the QSV model, the Smagorinsky model (SMAG),
the Dynamic procedure (DYN) and the Coherent vorticity Preserving (CvP) method against the DNS sharp-spectrally filtered down to the LES grid resolution. Details on the implementation of the latter method, such as constants and test filter
strength, can be found in Chapelier et al. (2018).
The state of the turbulence of the different TGV simulations is monitored by analyzing the evolution of volumeaveraged kinetic energy, Ekin “ x 21 ρu2 y, and its dissipation
rate, defined as, εkin “ ´ BEBtkin , with results gathered in figure 3. It can be observed that simulating a Mt “ 0.1 TGVflow with 963 points without any turbulence model leads to anumerically unstable run. In the absence of a model, εkin becomes positive around tV0 {L « 10, indicating a spurious generation of kinetic energy, ultimately leading to diverging numerical results. Nonetheless, the results without the additionof extra dissipation to account for the energy flux to subfilter
scales can be used to assess the performance of SFS models in
the early stages of the simulation, when only large scales exist
and the models should be inactive.
In figure 3, a zoomed region focused on the periodtV0 {L “ 0 ´ 5 shows that all the models considered, apart fromthe plain Smagorinsky model, are able to mitigate the additionof excess dissipation in the early stages of the flow, following both filtered DNS and no-model results closely. The overattenuation induced by the plain Smagorinsky model persists
as the flow develops and ends up leading to a smaller dissipation peak, in comparison with the other models considered.
Furthermore, QSV’s results, when compared to ones obtained
via DYN or CvP, are closer to the filtered DNS results fromtV0 {L « 6 onwards, introducing a slight over dampening in
the prior period. Similar levels for the peak in εkin are recovered for these three models but, only the QSV-based results
recover the dissipation plateau existent in the DNS results af-

DEMONSTRATING QSV’S SUBFILTER SCALE(SFS) TURBULENCE MODELING CAPABILITY
The complete QSV closure relations (9) - (10) for the filtered compressible Navier-Stokes are now tested by assessing
their capability of modeling the energy flux from large to small
scales in a three-dimensional turbulence problem. The evolution of a Taylor-Green vortex (TGV) is studied via a priori anda posteriori analyses.
The latter consists in comparing the result of a coarse sim-

4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

a)b)

Figure 3. Evolution of volume-averaged kinetic energy (a) and dissipation (b) of the LES of the Taylor-Green Vortex with initial
amplitude of Mt “ 0.1. Different subfilter models are shown and compared against sharp spectral filtered DNS data up to the same grid
resolution.
ter its peak.

Despite achieving numerical stability, all the LAD-aided
runs overestimate the dissipation rate around tV0 {L « 6.5, during the presence of the largest shock discontinuities in the flow,
and underestimate the dissipation rate peak, in comparison
with the reference results. In comparison with the LAD-aided
turbulence models, the QSV-obtained curve remains closest
to the reference results throughout the whole evolution of theflow, introducing less dissipation in the shock-dominated period and predicting better the magnitude of the dissipation
rate peak. Additionally, it can be observed that, in the initialstages of the flow, when only large scales are present QSV is
the model that is closest to the reference results, showcasing
its ability to dynamically modulate the added SFS dissipation
magnitude.
These results ultimately support the claim that the proposed QSV model is a genuine unified approach for turbulencemodeling and shock capturing. On top of being able to perform each task separately, it outperforms the simple additionof separate turbulence and shock capturing models in flow setups where both hydrodynamic turbulence and shock discontinuities are happening simultaneously.

QSV AS A UNIFIED APPROACH FOR TURBULENCE MODELING AND SHOCK CAPTURING
After having established the capability of the QSV modelto act separately as a shock capturing and a SFS turbulencemodeling closure, we now test a case exhibiting shock-wave
turbulence interaction such as a supersonic TGV flow withinitial Mach number Mt “ 1.2. At this level, the initial perturbations rapidly induce wave steepening and shocks in thesolution before the initial vortex breaks down into turbulence
and the hydrodynamics start to govern the flow. The objectiveof this test case is to assess how the different models cope withthe presence of both shocks and turbulence in the simultaneously.
To study the evolution of a supersonic TGV, an exact
compressible energy norm (Myers, 1991) that accounts for energy stored in both hydrodynamic and thermodynamic fields
is used. As in the subsonic TGV case, the exact compressible
norm can be volume-averaged,
1
Ec “ xρph ´ h0 q ` ρu2 ´ ρT0 ps ´ s0 q ´ pp ´ p0 qy,
2

CONCLUSION(15)

A novel technique named the Quasi-Spectral Viscosity model (QSV), was introduced. It is designed to simulate accurately the large scales present in both shock and turbulence
dominated flows by exploiting the residual of filter transfer
functions to estimate both the amplitude of fluctuations near
the grid cutoff and modulate the viscosity magnitude for different wavenumbers. This feature allows for an implementation using only spatial operators, applicable to finite-difference
solvers.
The QSV mathematical framework is based on an extension of LES closures and a parallel between these and spectral
vanishing viscosity (SVV) based models. The 1D Burgers’
problem is used to showcase the connection between previous LES models and how they can be understood as a wayof solving shock dominated solutions. Moving forward, the
QSV model was shown to perform well in both low-speed
and highly compressible flow setups. For example, the same
QSV framework can be used to solve a Taylor-Green Vortex
with both sub and supersonic initial conditions. Moreover, the
QSV model is flexible, being able to be applied in curved andstretched domains by using grid transformations. The collection of satisfactory results across intrinsically different flow
setups supports the claim that the QSV method can simultaneously capture shocks and act as a subfilter turbulent closure.

and its dissipation rate can be determined by a time derivativec
as εc “ ´ BE
Bt .
Again, an a posteriori study is performed with 963 grid
points, comparing the QSV model and various eddy viscosity
methods when augmented by the addition of the LAD shock
capturing approach against a QSV-based Mt “ 1.2 TGV simulation with 3843 grid points, used here as a reference solution
since the presence of a shock in the flow field renders a DNS
is not strictly possible. In Sousa & Scalo (2022b), though, it
is shown that the solution is essentially converged at that resolution level. Additionally, if the current test case is simulated
without the inclusion of a shock capturing approach, it leads
to numerical instabilities regardless of the turbulence model
considered (Sousa & Scalo, 2022b).
The results from such a posteriori analysis are gatheredin figure 4. The base formulation of the LAD model (Kawai
et al., 2010) induces the addition of artificial dynamic viscosity (µart ), which ultimately contributes to the stabilization of
the run performed with LAD as the only active model. The
results from the LAD model are the most inaccurate and become unstable if µart is deactivated and only the artificial bulk
viscosity and artificial conductivity components are active.

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

0.200

0.150

963 No Model+LAD
963 SMAG+LAD
963 DYN+LAD
963 CvP+LAD

0.125

3843 QSV

1.6

b)

1.2

963 QSV

0.100
0.075

1.0
0.8
0.6
0.4

0.050

0.2

0.025
0.0

×10−2

1.4

c L/ρ0 V03

0.175

Ec /ρ0 V02

a)

0.0
2.5

5.0

7.5

10.0 12.5
tV0 /L

15.0

17.5

20.0

0.0

2.5

5.0

7.5

10.0 12.5
tV0 /L

15.0

17.5

20.0

Figure 4. Evolution of the volume-averaged exact compressible energy norm (a) and its dissipation rate (b) in a Mt “ 1.2 TGV
simulation. Results obtained by different sub-filter models using 963 grid points are compared against a reference solution computed
using QSV model with 3843 points.

Lele, Sanjiva K. 1992 Compact Finite Difference Schemeswith Spectral-Like Resolution. Journal of Computational
Physics 103 (1), 16–42.
Lodato, Guido 2019 Characteristic modal shock detection for
discontinuous finite element methods. Computers & Fluids
179, 309–333.
Lodato, G., Vervisch, L. & Clavin, P. 2016 Direct numericalsimulation of shock wavy-wall interaction: analysis of cellular shock structures and flow patterns. J. Fluid Mech. 789,
221–258.
Lodato, Guido, Vervisch, Luc & Clavin, Paul 2017 Numerical
study of smoothly perturbed shocks in the Newtonian limit.
Flow, Turbulence and Combustion 99 (3), 887–908.
Myers, Michael K 1991 Transport of energy by disturbancesin arbitrary steady flows. Journal of Fluid Mechanics 226,
383–400.
Nagarajan, Santhanam, Lele, Sanjiva K & Ferziger, Joel H
2003 A Robust High-Order Compact Method for LargeEddy Simulation. Journal of Computational Physics
191 (2), 392–419.
Smagorinsky, J. 1963 General circulation experiments with theprimitive equations. Monthly Weather Review 91 (3), 99 –164.
Sod, Gary A 1978 A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws.
Journal of computational physics 27 (1), 1–31.
Sousa, Victor C. B. & Scalo, Carlo 2022a A Legendre spectral viscosity (LSV) method applied to shock capturing forhigh-order flux-reconstruction schemes. Journal of Computational Physics p. 111157.
Sousa, Victor C. B. & Scalo, Carlo 2022b A unified QuasiSpectral Viscosity (QSV) approach to shock capturing andlarge-eddy simulation. Journal of Computational Physics
459, 111139.
Tadmor, Eitan 1989 Convergence of spectral methods for nonlinear conservation laws. SIAM Journal on Numerical Analysis 26 (1), 30–44.
Tonicello, Niccolò, Lodato, Guido & Vervisch, Luc 2020 Entropy preserving low dissipative shock capturing with wavecharacteristic based sensor for high-order methods. Computers & Fluids 197, 104357.
Vandeven, Hervé 1991 Family of spectral filters for discontinuous problems. Journal of Scientific Computing 6 (2), 159–
192.
Vreman, B., Geurts, B. & Kuerten, H. 1995 A priori tests ofLarge-Eddy Simulation of the compressible plane mixing
layer. Journal of engineering mathematics 29 (4), 299–327.

As a final remark, although the current implementation is
aimed at finite-difference solvers, it is possible to extend the
QSV approach to unstructured solvers based on spectral numerics. Due to the opportunity of projecting the solution ofeach element onto a hierarchical set of orthogonal basis functions, a spectrally based implementation would be able to easily gage the magnitude the energy near the cutoff and would
be able to modulate freely the amplitude of the viscosity kernel for different wavenumbers. This fact renders the use of
global filtering operations unnecessary and could lead to simpler and more flexible implementations. An initial implementation of a shock capturing method that exploits the Legendre
basis in discontinuous flux reconstruction settings was introduced in Sousa & Scalo (2022a). Its extension to turbulenceand shock/turbulence interaction modeling will be explored infuture research.

REFERENCES
Chapelier, J-B, Wasistho, B & Scalo, C 2018 A coherent vorticity preserving eddy-viscosity correction for large-eddy
simulation. Journal of Computational Physics 359, 164–
182.
Chollet, J.P. & Lesieur, M. 1981 Parameterization of small
scales of three-dimensional isotropic turbulence utilizing
spectral closures. J. Atmo. Sci. 38 (12), 2747–2757.
Clavin, Paul 2013 Nonlinear analysis of shock–vortex interaction: Mach stem formation. Journal of Fluid Mechanics
721, 324–339.
Cook, Andrew W 2007 Artificial Fluid Properties forLarge-Eddy Simulation of Compressible Turbulent Mixing.
Physics of fluids 19 (5), 055103.
Denet, B, Biamino, L, Lodato, G, Vervisch, L & Clavin, P2015 Model equation for the dynamics of wrinkled shockwaves: comparison with DNS and experiments. Combustion Science and Technology 187 (1-2), 296–323.
Germano, M., Piomelli, U., Moin, P. & Cabot, W.H. 1991 A
dynamic subgrid-scale eddy viscosity model. Phys. Fluids
A 3, 1760–1765.
Gupta, P. & Scalo, C. 2018 Spectral energy cascade and decay
in nonlinear acoustic waves. Phys. Rev. E 98, 033117.
Kawai, Soshi, Shankar, Santhosh K & Lele, Sanjiva K2010 Assessment of localized artificial diffusivity scheme
for large-eddy simulation of compressible turbulent flows.
Journal of Computational Physics 229 (5), 1739–1762.
Kraichnan, R.H. 1976 Eddy viscosity in two and three dimensions. J. Atmo. Sci. 33 (8), 1521–1536.

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