Numerical methods for high-speed ﬂows - uniroma1.it

Numerical methods for high-speed flows 1

Numerical methods for high-speed flows

Sergio Pirozzoli

Universita di Roma ‘La Sapienza’

Dipartimento di Meccanica e Aeronautica

Via Eudossiana 18, 00184 Roma, Italy;

e-mail: [email protected]

Key Words shock waves, shock-capturing schemes, energy conservation, nu-merical dissipation

Abstract We review numerical methods for DNS and LES of turbulent compressible flow in thepresence of shock waves. Ideal numerical methods should be accurate and free from numericaldissipation in smooth parts of the flow, and, at the same time, they must robustly capture shockwaves without significant Gibbs ringing, which may lead to nonlinear instability. Adapting tothese conflicting goals leads to the design of strongly nonlinear numerical schemes that dependon the geometrical properties of the solution. For low-dissipation methods for smooth flows,numerical stability can be based on physical conservation principles for kinetic energy and/orentropy. Shock-capturing requires addition of artificial dissipation, in more or less explicit form,as a surrogate of physical viscosity, to obtain non-oscillatory transitions. Methods suitable forboth smooth and shocked flows are discussed, and the potential for hybridization is highlighted.Examples of the application of advanced algorithms to DNS/LES of turbulent, compressibleflows are presented.

CONTENTS

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

METHODS FOR SMOOTH FLOWS . . . . . . . . . . . . . . . . . . . . . . . . . 5Upwinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Energy-Consistent Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Entropy-Consistent Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

METHODS FOR SHOCKED FLOWS . . . . . . . . . . . . . . . . . . . . . . . . . 10‘Classical’ Shock-Capturing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Hybrid Schemes and Nonlinear Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . 12Artificial Viscosity Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Shock-capturing through Sub-Grid Scale Models . . . . . . . . . . . . . . . . . . . . . . 16Properties of Shock-Capturing Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

FURTHER TOPICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Complex Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Discretization of Viscous Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Annu. Rev. Fluid Mech. 2011

APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1 INTRODUCTION

Numerical simulation of high-speed flows has a long history, dating back to thebeginning of the computer era (von Neumann & Richtmyer, 1950; Courant et al.,1952; Godunov, 1959). Several textbooks on numerical methods have appearedover the years (Richtmyer & Morton, 1967; Hirsch, 1988; LeVecque, 1990; Laney,1998; Toro, 2009). Important advancements have been made, but, as illustratedby a recent comparative study (Johnsen et al., 2010) computational gasdynamicshas not yet converged to an ‘optimal’ computational strategy. The purpose ofthis review is to check the status of the discipline, and systematize the enormousamount of material produced over the years, a large part of which was reviewed byEkaterinaris (2005). Given the vastness of the subject, we mostly limit ourselvesto analyze the family of finite-difference (FD) schemes that are frequently used,especially in the academic community, for direct numerical simulation (DNS)and large-eddy simulation (LES) of compressible turbulent flows. Resolving therange of scales present in these flows requires numerical schemes that are accurate,robust, and efficient in terms of CPU requirements.

The reference physical model consists in the compressible Navier-Stokes equa-tions for a calorically perfect gas, here written in integral form in Cartesiancoordinates

d

dt

∫

VudV +

3∑

i=1

∫

∂V(fi − fv

i ) ni dS = 0, (1)

where

u =

ρρuj

ρE

, fi =

ρui

ρuiuj + pδijρuiH,

, fv

i =

0σij

σikuk − qi,

, j = 1, 2, 3,

(2)are, respectively, the vector of conservative variables, and the vectors of theconvective and viscous fluxes in the i-th direction. Here ρ is the density, ui isthe velocity component in the i-th coordinate direction, p is the thermodynamicpressure, E = e + ρu2/2 is the total energy per unit mass, e = RT/(γ − 1) isthe internal energy per unit mass, H = E + p/ρ is the total enthalpy, R is thegas constant, γ = cp/cv is the specific heat ratio, σij is the viscous stress tensor,and qi is the heat flux vector. At high Mach numbers, the occurrence of verystrong shock waves and of severe viscous heating may bring to light additionaleffects that are not incorporated in (1), such as chemical reactions and non-idealthermodynamic behavior (real-gas effects). These are not addressed in the presentreview, and the interested reader should refer to specific publications (Bertin &Cummings, 2006).

Under the assumption of smooth flow, the Navier-Stokes equations can beequivalently cast in differential form

∂u

∂t+

3∑

i=1

∂fi∂xi

=3∑

i=1

∂fvi

∂xi. (3)

2


Neglect of molecular diffusion effects (i.e. setting µ = 0, k = 0) leads to theEuler equations, that only incorporate the effects of macroscopic convection andmolecular collisional effects through pressure forces. The Euler equations haveseveral important mathematical properties that are illustrated in the books ofLax (1973), Majda (1959), Hirsch (1988), and LeVecque (1990). Some usefulproperties for the development of numerical methods are briefly recalled here.

1. Hyperbolicity. The system of Euler equations can be cast in characteristicform, meaning that projection of the equations in any spatial directiongives rise to a system of coupled wave-like equations (Majda, 1959). Thismotivates the study of the model one-dimensional scalar conservation law

∂u

∂t+∂f(u)

∂x=∂u

∂t+ a(u)

∂u

∂x= 0, (4)

which is used as a prototype for the development of numerical methods.It can also be shown that the convective flux vectors are homogeneousfunctions of order one with respect to the vector of conservative variables,i.e. fi = dfi/du · u, a property that is useful for the design of upwindmethods based on the flux vector splitting technique (Steger & Warming,1981).

2. Conservation properties. The Euler equations have the obvious property(as clear from their integral form) that the integrals of ρ, ρui, and ρE overan arbitrary control volume can only vary due to flux through the bound-aries. Under the assumption of smooth flow, combining the continuity andthe momentum equations and integrating yields a balance equation for thekinetic energy ρukuk/2

d

dt

∫

Vρukuk/2 dV = −

∫

∂V(ρukuk/2 + p) uini dS +

∫

Vp∂ui

∂xidV. (5)

Equation (5) shows that the total kinetic energy only varies because of mo-mentum flux through the boundary or to volumetric work of pressure forces(which is zero for incompressible flow), whereas the convective terms donot cause any net variation. This property has inspired numerical schemesbased on the attempt to enforce ‘kinetic energy preservation’ in the dis-crete sense. Additional conservation laws can be derived from the Eulerequations for smooth flows, namely (Harten, 1983b)

d

dt

∫

Vρg(s) dV +

∫

∂Vρg(s)uini dS = 0, (6)

where g(s) is an arbitrary (but differentiable) function of the thermody-namic entropy s = log(pρ−γ), which suggests the possibility to designnumerical schemes that discretely preserve the integral of ρg(s). In thepresence of shock waves, mechanical energy is transformed into heat (withsubsequent production of entropy) even in the limit of vanishing viscosity,and energy/entropy conservation no longer holds.

3. Symmetrization. Harten (1983b) showed that the change of variables v =∂ρg(s)/∂u allows to recast the Euler equations in the ‘symmetrized’ form

P∂v

∂t+

d∑

i=1

Bi∂v

∂xi= 0, (7)

4 Pirozzoli

where P = ∂u/∂v, Bi = ∂fi/∂v are symmetric matrices, and P is positive-definite. The symmetrized form of the equations has the important propertythat continuous energy estimates can be established (Olsson, 1993; Olsson& Oliger, 1994). Indeed, referring to a 1D bounded interval [a, b], from (7)follows that

d

dt

∫ b

av · Pv dx = − [v · Bv]ba , (8)

showing that the generalized energy at the left-hand-side can only vary intime due to boundary contributions. Tadmor (1984) showed that valid-ity of the energy estimate (8) implies satisfaction of entropy conservation(6). The choice g(s) = β exp (s/(β (γ − 1))) (Harten, 1983b), leads to sim-ple transformations between conservative and entropy variables (Sandhamet al., 2002), and u, fi happen to be homogeneous functions of degree βwith respect to v,

u(θv) = θβ u(v), fi(θv) = θβ fi(v). (9)

The solutions of (1) exhibit tendency to form steep gradients (shock wavesand contact discontinuities), whose thickness is of the order of magnitude of themean-free-path, making shock ‘resolution’ with numerical methods infeasible inmost cases of practical interest. In the inviscid limit shock waves reduce to zero-measure objects, across which the Rankine-Hugoniot jump conditions must besatisfied. In order for the class of ‘weak’ solutions of the Euler equations tocoincide with that of the Navier-Stokes equations in the inviscid limit, additionalconditions (in the form of ‘entropy inequalities’) must be satisfied, to ensurethat the macroscopic effects of the small scale diffusion processes are properlyrepresented. For a thorough mathematical discussion of these issues the readermay consult Lax (1973), Majda (1959), and LeVecque (1990).

To summarize, high-speed flows typically feature regions where the flow issmooth, and the governing equations in their differential form hold, interspersedby extremely thin regions, where the flow properties vary abruptly. Therefore,it is not surprising that numerical methods for high-speed flows have specializedinto two classes, capable to deal with smooth flows and with shock waves, respec-tively, and having very different properties. Indeed, it is known that standarddiscretizations used for smooth flows cause (potentially dangerous) Gibbs oscilla-tions in the presence of shock jumps, whereas typical methods used to regularizeshock calculations exhibit excessive ‘numerical’ viscosity. Schemes of the two fam-ilies are illustrated in Sections 2 and 3, respectively. Tailored numerical methodsfor high-speed flows should be able to put together the favourable properties ofthe two basic families, which is frequently realized through their hybridization orblending, as also explained in Section 3. The following discussion will mostly dealwith the discretization of the convective derivatives that appear in (3), this beingthe main source of nonlinearity in the governing equations. As customary in thecomputational gas-dynamics community, a method-of-lines approach is assumed,whereby time integration is handled separately, using ODE dedicated methods.Time integration, as well as discretization of the viscous fluxes and specificationof boundary conditions, are addressed in Section 4.


2 METHODS FOR SMOOTH FLOWS

The most obvious choice to discretize the governing equations in smooth parts ofthe flow field is using high-order approximations of the derivative operators thatappear in (3). Straightforward semi-discretization of the model equation (4) ona grid with uniform spacing h and nodes xj = j · h yields

dvj

dt= −Dfj , (10)

where vj(t) ≈ u (xj , t), Dfj ≈ ∂f/∂x (xj , t).A class of explicit and implicit (compact) approximations for Dfj was devised

by Lele (1992),M∑

m=−M

amDfj+m =1

h

L∑

ℓ=−L

bℓ fj+ℓ, (11)

where fj = f(vj(t)), and appropriate values of the coefficients am, bℓ, are de-termined so as to maximize formal accuracy (the minimum attainable trun-cation error being O(h2(L+M))), or to shape the spectral response of the re-sulting scheme and improve the representation of the Fourier modes with thehighest wavenumbers supported by the computational mesh (Vichnevetsky &Bowles, 1982). The latter idea has produced the flourishing of the so-called DRP(Dispersion-Relation-Preserving) schemes (Tam & Webb, 1993; Zingg et al., 1993;Lui & Lele, 2002; Bogey & Bailly, 2004). Central difference approximations, re-covered for b−ℓ = −bℓ, a−m = am, have null dissipation error in a linear set-ting (Lele, 1992), and are therefore ideal candidates for computations free ofspurious numerical diffusivity.

It the context of compressible flows is useful to cast the approximation (10) in

‘locally conservative’ form, i.e. look for a numerical flux fj+1/2, such that

Dfj =1

h

(fj+1/2 − fj−1/2

). (12)

For linear representations of the first derivative of the type (11), explicit formulasfor the corresponding numerical fluxes are easily found

M∑

m=−M

am fj+1/2+m =L∑

ℓ=−L+1

cℓ fj+ℓ, (13)

where c−L+1 = −a−L, cℓ = cℓ−1 −aℓ−1, ℓ = −L+2, . . . , L. Explicit (even nonlin-

ear) approximations of the numerical flux of the type fj+1/2 = f(vj−L+1, . . . , vj+L),

that satisfy the consistency condition f(v, . . . , v) = f(v), guarantee global dis-crete conservation of the integral of u for the scalar model equation (and of totalmass, momentum and energy for the Euler equations) through the telescopicproperty, and satisfy the hypotheses of the Lax-Wendroff theorem (Lax & Wen-droff, 1960; LeVecque, 1990), which guarantees convergence to weak solutions,provided the approximation itself converges.

Central derivative approximations have been widely used in the literature, es-pecially for wave propagation problems where nonlinearities are weak (Colonius& Lele, 2004). However, it is known that application of standard central dis-cretizations to high-Reynolds-number fluid turbulence typically leads to numeri-cal instability, owing to accumulation of the aliasing errors resulting from discrete

6 Pirozzoli

evaluation of the nonlinear convective terms (Phillips, 1959). Such deficiency canalso be traced back to failure to discretely preserve quadratic invariants associ-ated with the conservation equations (Lilly, 1965; Orlandi, 2000). While finiteviscosity may help to stabilize calculations, it is usually safer to revert to al-ternative discretization techniques capable of ensuring stability in the inviscidlimit.

2.1 Upwinding

The ‘upwinding’ approach, commonly followed in the gas-dynamics community,is based on the idea that solutions of the Euler equations propagate along char-acteristics, and therefore a stable numerical method should also propagate itsinformation in the same characteristic direction (Moretti, 1979). The standardpractice (referred to as ‘flux vector splitting’ (Steger & Warming, 1981)) in theFD framework is to split the flux function into ‘positive’ and ‘negative’ partsf(u) = f+(u) + f−(u), associated, respectively, with non-negative and non-positive propagation velocities, i.e. df+/du ≥ 0, df−/du ≤ 0, which are dis-cretized by means of left- and right-biased approximations, respectively, thusensuring linear stability.

In the finite-volume (FV) framework upwinding is usually achieved throughthe ‘flux difference splitting’ (or Godunov) approach. Considering a FV semi-discretization of (4),

dvj

dt= −

1

h

(f(vj+1/2) − f(vj−1/2)

), (14)

where vj(t) = 1/h∫ xj+1/2

xj−1/2v(x, t)dx, is the spatial average of the approximate

solution over the cell Ij = (xj−1/2, xj+1/2), a suitable reconstruction operatoris used to determine approximate left and right states at the cell interfaces,v±j+1/2, and the interface flux f(vj+1/2) is replaced with the numerical flux re-

sulting from an exact (or approximate) Riemann solver (Roe, 1981; Toro, 2009),

formally fj+1/2 = R(v−j+1/2, v+j+1/2). For example, Roe’s approximate Riemann

solver (Roe, 1981) dictates

R(v−j+1/2, v+j+1/2) =

1

2

(f(v−j+1/2) + f(v+

j+1/2))−

|aj+1/2|

2

(v+j+1/2 − v−j+1/2

),

(15)where aj+1/2 = (f(v+

j+1/2)− f(v−j+1/2))/(v+j+1/2 − v

−j+1/2) is a characteristic speed

associated with the intermediate state. In the original formulation of Godunov(1959), piece-wise constant reconstructions were assumed, leading to the generalfirst-order flux

fj+1/2 = R(vj , vj+1). (16)

The extension to higher order of accuracy is achieved by replacing piece-wiseconstant with piece-wise polynomial reconstructions (van Leer, 1979). In theADER (Arbitrary accuracy DERivative Riemann problem) approach (Toro et al.,2001), the numerical flux is evaluated by solving the generalized Riemann problemusing a semi-analytical method. The approximate solution is given as a Taylortime expansion at the cell interface position, thus allowing the construction ofone-step schemes with arbitrary accuracy in both space and time.


Upwinding has the main effect of damping the Fourier modes with highestsupported wavenumbers, with subsequent stabilizing effect on the numerical so-lution. Upwind schemes have often been used for DNS of shock-free compressibleturbulence (Rai & Moin, 1993; Foysi et al., 2004; Pirozzoli et al., 2008), with agood degree of success. However, as pointed out by Park et al. (2004), the numer-ical dissipation introduced by upwinding can be harmful for LES, where properresolution of marginally resolved wavenumbers is crucial, as it may hamper theeffect of sub-grid scale models.

2.2 Filtering

Filtering the computed solution is a commonly used practice to cure nonlinearinstabilities of central schemes, while retaining high-order accuracy. High-orderfilters were introduced by Lele (1992), and used by Visbal & Gaitonde (1999)and Visbal & Gaitonde (2002) for the solution of conservation laws on stretched,curvilinear, and deforming meshes. The latter authors considered compact filtersof the form

αvj−1 + vj + αvj+1 =L∑

ℓ=0

dℓ

2(vj+ℓ + vj−ℓ) , (17)

where the coefficients dℓ at the r.h.s. are constrained to be functions of the freeparameter α through Taylor- and Fourier-series analyses, so as to achieve high-order accuracy, and selectively damp the supported Fourier modes with highestwavenumbers. A selective filtering strategy was developed by Bogey & Bailly(2004) based on optimized explicit filters of the type (17) with α = 0, L = 4, 5, 6.Those authors also suggested possible use of the selective filtering to model theeffects of the unresolved (sub-grid) flow scales, replacing classical eddy-viscositymodels. As for upwinding, filtering can eliminate spurious instabilities, but itimplies artificial energy draining that adds to the physical energy draining causedby molecular viscosity. More accurate control over diffusive effects is achieved byreverting to numerical methods capable of conserving energy in the absence ofviscosity.

2.3 Energy-Consistent Schemes

Several attempts have been made to design nonlinearly stable numerical schemesby replicating the energy preservation properties of the governing equations inthe discrete sense. Most efforts are based on the idea of recasting the convectiveterms in skew-symmetric form, that loosely amounts to expanding the convectivederivatives found in (3) as either

∂ρuiϕ

∂xi=

1

2

∂ρuiϕ

∂xi+

1

2ϕ∂ρui

∂xi+

1

2ρui

∂ϕ

∂xi, (18)

as proposed by Feiereisen et al. (1981), or as

∂ρuiϕ

∂xi=

1

2

∂ρuiϕ

∂xi+

1

2ui∂ρϕ

∂xi+

1

2ρϕ

∂ui

∂xi, (19)

as proposed by Blaisdell et al. (1996), where ϕ stands for a generic transportedscalar property, being unity for the continuity equation, uj for the momentum

8 Pirozzoli

equation, H for the total energy equation. Discretization of the mass and mo-mentum equations in the split form (18) implies kinetic energy preservation at thesemi-discrete level (Honein & Moin, 2004), provided the difference operators sat-isfy the summation by parts (SBP) property (Strand, 1994) that automaticallyholds in the case of periodic problems for central (both explicit and implicit)approximations of any order (Mansour et al., 1979). From another viewpoint,discretization of (19) guarantees minimization of the aliasing error (Blaisdellet al., 1996). Additional robustness in the presence of strong density variationsis gained (Kennedy & Gruber, 2008) by expanding the convective derivatives inthe generalized form

∂ρuiϕ

∂xi= α

∂ρuiϕ

∂xi+ β

(ui∂ρϕ

∂xi+ ρ

∂uiϕ

∂xi+ ϕ

∂ρui

∂xi

)

+ (1 − α− 2β)

(ρui

∂ϕ

∂xi+ ρϕ

∂ui

∂xi+ uiϕ

∂ρ

∂xi

). (20)

This arrangement leads to semi-discrete energy conservation in the case α = β =1/4. Ducros et al. (2000) showed that the skew-symmetric forms (18) and (19)give rise to locally conservative schemes, when the derivative operators are dis-cretized with explicit central formulas (with order up to six). Pirozzoli (2010)has recently proved that this holds true for explicit central formulas of arbitraryorder of accuracy, and presented computationally inexpensive numerical fluxesalso for the skew-symmetric form (20). Apparently, compact derivative approxi-mations applied to the skew-symmetric split form of the convective terms do notlead to locally conservative schemes. For a formal mathematical formulation offully conservative, skew-symmetric splitting of the compressible Euler equations,the reader may refer to Morinishi (2010).

2.4 Entropy-Consistent Schemes

Stabilization of numerical schemes can also be achieved by enforcing, at the dis-crete level, the conservation properties associated with entropy. Tadmor (1987)developed a second-order FV, locally conservative discretization of the Eulerequations, that discretely satisfies (6), resulting in the formulation of the nu-merical flux

fj+1/2 =

∫ 1

0f(vj + ξ (vj+1 − vj)) dξ, (21)

which was also considered by Jameson (2008b). Note that Equation (21) is ex-pensive, as it requires the evaluation of the entropy variables associated withthe left and right states, and the use of quadrature formulas to approximate theintegral. More general expressions for the numerical flux of entropy-consistentschemes can be found in Tadmor (2003).

An alternative strategy to ensure entropy stability was proposed by Gerritsen& Olsson (1996), who split the flux vector into conservative and non-conservativeparts, exploiting the homogeneity property (9)

∂f

∂x=

β

β + 1

∂f

∂x+

1

β + 1B∂v

∂x. (22)

This splitting, referred to in the literature as ‘canonical splitting’ or ‘entropy split-ting’, leads to a discrete analogue of the energy estimate (8) (with consequent


benefits in terms of nonlinear stability) if the first derivatives in (22) are replacedwith SBP operators. The resulting schemes are referred to as split high-order en-tropy conserving (SHOEC), since discrete entropy conservation in the sense (6)follows (Tadmor, 1984). The performance of SHOEC schemes is apparently af-fected by the choice of the splitting parameter β. Sandham et al. (2002) suggestedthat a value of β ≈ 4 is appropriate for numerical simulation of compressible tur-bulence, whereas β ≈ 2 is more appropriate for numerical simulation of isotropicturbulence at zero viscosity (Honein & Moin, 2004). As all formulations of theEuler equations that rely on the use of entropy variables, SHOEC schemes suf-fer for reduced computational efficiency, owing to swapping from conservative toentropy variables, that requires CPU-intensive transcendental operations.

Honein & Moin (2004) developed entropy-consistent schemes by applying theskew-symmetric splitting to the Euler equations, upon replacement of the totalenergy equation with the entropy equation. This approach preserves the integralsof ρs and ρs2. An equivalent alternative is to use the internal or total energyequations, rearranged in such a way that the skew-symmetric form of the entropyequation automatically follows.

2.5 Numerical Tests

The robustness of the numerical methods considered so far can be appreciatedfrom numerical simulations of isotropic compressible turbulence at zero viscos-ity on a 323 grid, as proposed by Honein & Moin (2004). The calculations areinitialized with a field of synthetic isotropic solenoidal turbulence with assignedwavenumber spectrum E(k) ∼ (k/k0)

4 exp[−2(k/k0)

2], peaking at k0 = 6, and

fluctuations of temperature and density are initially set to zero. Time integra-tion is performed by means of a third-order SSP Runge-Kutta algorithm (Shu &Osher, 1989) (see Section 4.2). This computational arrangement causes energyto be quickly transferred to the smallest scales, posing a significant challenge tonumerical algorithms, and leading to nonlinear instability in the absence of thealiasing control mechanisms discussed above.

The results are reported in Figure 1 for initial turbulent Mach numbers Mt0 =0.07, 0.3. In the former case, the ‘exact’ solution, computed by means of a de-aliased spectral method (Honein & Moin, 2004), has nearly constant total kineticenergy (K =

∫ρu2 dV ), constant total entropy (S =

∫ρsdV ), and the r.m.s.

density levels off to ρ′/ρ0/Mt20 ≈ 0.35, after an initial transient. As seen from

Figure 1, central discretization of the equations cast in divergence form (schemesD2, D6) leads to divergence of the numerical solution on a time scale shorterthan one eddy turnover time, regardless of the order of accuracy. Divergence isassociated with increase of kinetic energy, and loss of total entropy. Upwinding(scheme D7) curves the divergence, but dissipates most of the kinetic energyby ten eddy turnover times, and leads to an entropy increase. The results ofsimulations performed at Mt0 = 0.07 show that skew-symmetric splitting of theconvective terms (schemes S6, K6) helps conserve kinetic energy and entropy,although spurious growth of density fluctuations is observed. This is also trueof Tadmor entropy conserving scheme (T2) that exhibits larger errors over longtimes not leading to divergence though. The entropy-consistent scheme of Honein& Moin (2004) (H6) succeeds in correctly predicting the limiting inviscid behaviorof the system for this specific test case. The results at Mt0 = 0.3 show failure ofall methods at large times, except the skew-symmetric scheme based on (20) and

10 Pirozzoli

Tadmor scheme.

3 METHODS FOR SHOCKED FLOWS

All methods presented in the previous section suffer from spurious Gibbs oscil-lation near shock jumps, that may lead to nonlinear instabilities. The onset ofoscillations can be avoided following two strategies. In the ‘shock-fitting’ ap-proach, shock waves are treated (separately from the rest of the flow) as genuinediscontinuities, whose dynamics is governed by their own algebraic equations,and the Rankine-Hugoniot relations are used as a set of nonlinear boundary con-ditions to relate the states on the two sides of the discontinuity (Moretti, 1987).In the ‘shock-capturing’ approach the same discretization scheme is used at allpoints, and regularization is achieved through addition of numerical dissipation,that prevents (or at least limits) the onset of Gibbs oscillations. While the for-mer approach very often guarantees more accurate representations of shockedflows (Johnsen et al., 2010), it is only feasible in cases where the shock topologyis extremely simple, and no shock wave forms during the calculation. In thisreview, only the shock-capturing approach is discussed.

3.1 ‘Classical’ Shock-Capturing Methods

Historically, most shock-capturing schemes have been constructed having in mindthe mathematical properties of entropy solution of the model scalar conservationlaw (4), including:

1. monotonicity preservation (MP), i.e. initially monotone data remain mono-tone for all times (Godunov, 1959);

2. total variation diminishing (TVD), i.e. solutions of (4) satisfy TV (u(·, t2)) ≤TV (u(·, t1)), ∀t2 ≥ t1, where TV (u(·, t)) denotes the total variation ofu (Harten, 1983a);

3. monotonicity, i.e. let u(x, t), v(x, t) be any two entropy solutions of (4),with u(x, 0) ≤ v(x, 0),∀x, then u(x, t) ≤ v(x, t),∀x, t (Harten et al., 1986).

Shock-capturing schemes implementing the above conditions in the discrete senseare abundant in the literature, the main reason being the availability of theoremsthat guarantee convergence to weak solutions for TVD schemes (LeVecque, 1990)and to the (unique) entropy solution for monotone schemes (Harten et al., 1986).However, monotone schemes are at most first-order accurate (Harten et al., 1986),which explains the wide success enjoyed by TVD schemes, especially in the 1980’s.An excellent overview of the contributions to TVD schemes was reported by Yee(1989).

Classical TVD schemes were found to suffer from loss of accuracy at bothsmooth and non-smooth extrema, which stimulated researchers to pursue alter-natives for constructing uniformly high-order accurate shock-capturing schemes.The very successful family of essentially-non-oscillatory (ENO) schemes (Hartenet al., 1987), was based on the idea of determining the numerical flux fromhigh-order (say r) reconstruction over an adaptive stencil that is selected toavoid as much as possible interpolation across discontinuities, thus minimizingGibbs oscillations. ENO reconstructions of order r guarantee that TV (vn+1) ≤TV (vn)+O(hr), which implies uniform boundedness of the total variation. ENO


schemes were found to suffer problems of convergence to steady solutions andloss of accuracy, due to free stencil adaptation (Rogerson & Meiburg, 1990). Theproblem was cured by Shu (1990) by biasing towards the ‘best’ central stencil inthe adaptation procedure. An alternative technique to circumvent the accuracyreduction phenomenon of TVD schemes, based on the enforcement of the MPconstraint at discrete level, was proposed by Suresh & Huynh (1997).

The class of the weighted essentially-non-oscillatory (WENO) schemes, firstintroduced by Liu et al. (1994), and generalized and improved by Jiang & Shu(1996), is based on the idea of constructing a high-order numerical flux from aconvex linear combination of lower order polynomial reconstructions over a setof staggered stencils, with weights selected to achieve maximum formal order ofaccuracy in smooth regions, and assign (nearly) zero weight to reconstructions onstencils crossed by discontinuities. As illustrated in Figure 2, L+ 1 sub-stencils,each including L points, Sℓ = {xj−L+ℓ+1, . . . , xj+ℓ} , ℓ = 0, . . . , L, are consideredout of the overall reconstruction stencil S = {xj−L+1, . . . , xj+L}, that includesa total of 2L points. Assuming for now f(u) = f+(u), and suppressing thesuperscripts, the FD WENO numerical flux is constructed as

fj+1/2 =

L∑

ℓ=0

ωℓ f(ℓ)j+1/2, (23)

where f(ℓ)j+1/2 is the numerical flux resulting from polynomial reconstruction over

the stencil Sℓ,

f(ℓ)j+1/2 =

L−1∑

m=0

cℓm fj−L+1+ℓ+m, (24)

and the weights, defined as

ωℓ =αℓ∑L

m=0 αm

, αℓ =dℓ

(ǫ+ βℓ)2, (25)

are functions of the ‘smoothness measurements’ associated with the sub-stencils

βℓ =L−1∑

m=1

(L−1∑

n=1

γℓmn fj−L+1+ℓ+n

)2

. (26)

The key to the success of WENO reconstructions is the proper selection of the‘linear’ weights dℓ, so as to achieve maximum order of accuracy (2L), and thedesign of the ‘nonlinear’ weights ωℓ, such that if the stencil Sℓ contains a jump,ωℓ ≈ 0 follows. Upwinding is achieved by suitable biasing in the reconstructionprocedure. Referring to the ‘positive’ part of the flux, in the original approach ofJiang & Shu (1996) a left-biased reconstruction was obtained discarding the right-most sub-stencil (i.e. dL = 0), thus reducing the overall stencil width (and theformal order of accuracy) to 2L−1. Weirs & Candler (1997) suggested the use ofa central stencil to reduce the numerical viscosity of WENO schemes, and biasedthe reconstruction by re-defining the smoothness measurement for the right-mostsub-stencil, setting βL = max0≤ℓ≤L βℓ, thus preventing the selection of the fullydownwinded stencil (SL). Formulas for the ‘negative’ split flux are easily obtainedby symmetry about xj+1/2. Comprehensive compilations of WENO coefficientscℓm, dℓ, γℓmn, can be found in Weirs & Candler (1997), Balsara & Shu (2000)and Gerolymos et al. (2009).

12 Pirozzoli

Provable properties of WENO schemes are limited to the scalar case, andinclude convergence for smooth solutions (Jiang & Shu, 1996), and existence andstability of discrete shock profiles for the third-order version of the scheme (Jiang& Yu, 1996). Nevertheless, WENO schemes are extremely robust in practice,and have been applied successfully (both in their FD and FV versions) to a widenumber of physical problems. For an overview of the wide-ranging ramificationsof the WENO idea the reader may consult the review paper by Shu (2009).

Many investigators have tried to improve the properties of WENO schemes,especially in terms of their behavior in smooth regions. The mainstream of re-search (Weirs & Candler, 1997) has been focused on the attempt to modify thelinear weights and the polynomial reconstruction coefficients to improve the be-havior in Fourier space and extend the range of well resolved wavenumbers, in thespirit of the DRP approach. Follow-up studies of the ‘optimized’ WENO strat-egy have been reported by Hill & Pullin (2004), Martın et al. (2006), and Tayloret al. (2007). An alternative approach to conjugate the spectral-like resolutionproperties of compact schemes with the shock-capturing properties of WENO wasundertaken by Deng & Zhang (2000), who developed weighted compact nonlin-ear schemes (WCNS) by coupling WENO interpolation of conservative variablesto the intermediate nodes and compact conservative approximation of the fluxderivatives at the grid nodes. In the author’s experience, WENO schemes andtheir variants are highly accurate, and quite robust. However, they are compu-tationally expensive, mostly because of the large number floating point opera-tions required for the evaluation of the smoothness measurements. Furthermore,spectral-like resolution is never achieved owing to the noxious effect of the nonlin-ear weights, unless the weights are locally frozen in smooth regions, as suggestedby Hill & Pullin (2004) and Taylor et al. (2007).

A visual impression of the performance of WENO schemes is gained from Fig-ure 3, where the results of numerical simulations of the test case 13 of Lax &Liu (1998) are reported. The solution of the two-dimensional Riemann probleminvolves a complex shock diffraction pattern leading to the formation of a Machstem, and Kelvin-Helmholtz roll-up of the vertical slip-line. Figure 3 shows thatboth the TVD and the WENO methods robustly capture the shock wave pattern,without significant spurious oscillations. Significant differences are observed inthe predicted structure of the slip line, that exhibits finer scale roll-up for theWENO schemes, and the formation of secondary vortex cores for the seventh-order scheme.

3.2 Hybrid Schemes and Nonlinear Filtering

This class of methods is based on the idea of endowing a baseline spectral-likescheme with shock-capturing capability through local replacement with a classicalshock-capturing scheme, or through controlled addition of the dissipative part ofa shock-capturing scheme, that is made to act as a ‘nonlinear filter’. A key rolein this class of schemes is played by ‘shock sensors’, that must to be defined insuch a way that numerical dissipation is effectively confined in shocked regions,not to pollute smooth parts of the flow field.

Lee et al. (1997) used a hybrid discretization made up of a sixth-order compactscheme and a sixth-order shock-capturing ENO scheme, to analyze the interac-tion of a (nearly) planar shock wave with a field of isotropic turbulence. Sincethe shock location in that configuration is approximately known, the switch be-


tween the two schemes was established a-priori, and ENO was only activatedin the shock-normal direction, over twelve points upstream and twelve pointsdownstream of the mean shock location. Adams & Shariff (1996) first considereda truly adaptive hybrid discretization, consisting of a baseline compact upwindscheme coupled with a fifth-order ENO scheme. A simple switch, based on thelocal gradient of the flux vector components, was used to mark critical cells, thatwere subsequently padded with buffer cells on each side, to prevent the onset ofoscillations arising from the coupling of schemes with different properties. Themethod was expanded by Pirozzoli (2002), who developed a fully conservativeformulation by hybridizing a fifth-order compact upwind numerical flux (of theform (13)) with a seventh-order WENO flux, the switch being based on the localdensity gradient. Figure 3d shows the results obtained with this method for theLax-Liu Riemann problem. Compared to baseline WENO schemes, the hybridformulation yields more accurate representation of smooth flow features, withthe formation of additional discrete vortices in the shear layer, but it also yieldssome spurious wiggles near shocks. For this specific test case, the reduction ofcomputational cost with respect to standard seventh-order WENO is about 50%.Further improvements to hybrid WENO schemes were introduced by Ren et al.(2003) and Hill & Pullin (2004). In particular, the latter authors used a baselinecentral DRP-like discretization (tuned for LES of compressible turbulence) anddesigned a shock sensor based on the ratio of the largest to the smallest WENOsmoothness indicators λ = (maxℓ βℓ)/(minℓ βℓ + ε), that becomes large wheneverone (or more) WENO sub-stencils are crossed by discontinuities.

While hybrid schemes are now frequently used, it appears that the issues relatedto coupling of the underlying schemes have not been thoroughly investigated.A notable exception is the work of Larsson & Gustafsson (2008), who appliedKreiss stability theory to the analysis of hybrid methods. Interestingly, it wasfound that the coupling of two separately stable schemes may, in some instances,give rise to an unstable system. However, it was also found that, if either schemeis dissipative, the coupled system is strongly stable, thus endorsing the use ofupwind shock-capturing schemes around discontinuities.

The nonlinear filtering approach was introduced by Yee et al. (1999), who de-signed a low-dissipative, shock-capturing algorithm based on SHOEC discretiza-tion, augmented with the artificial dissipation of a TVD scheme. Referring tothe scalar conservation law, the following semi-discretization was proposed

dvj

dt= −Dfj −

1

h

(f∗j+1/2 − f∗j−1/2

), (27)

where the filter numerical flux is defined as

f∗j+1/2 = kΨj+1/2 dj+1/2. (28)

Here k is a (problem-dependent) constant, dj+1/2 is the artificial flux of a second-order TVD scheme, and Ψj+1/2 is the Harten switch (Harten, 1978), which isdesigned to be near unity near shocks, and near zero in smooth parts of theflow. As a consequence, the algorithm automatically localizes numerical dissi-pation around discontinuities. The extension of the methodology to the multi-dimensional Euler equations involves application of the nonlinear filter to eachcharacteristic field, hence the frequently used terminology of ‘characteristic-basedfilters’ also given to this class of schemes. An advantage of the nonlinear filtering

14 Pirozzoli

approach over hybrid methods is that, in the case of explicit multi-stage time in-tegration, the filter numerical flux can be applied at the end of the full time step,rather than at each sub-step, with substantial saving in terms of CPU time (butwith some loss in terms of robustness). Ducros et al. (1999) modified the originalnonlinear filtering approach by adopting the artificial flux of the JST scheme (asfrom Equation (30)), and replacing in Equation (28) the Harten switch with theproduct of the Jameson sensor (32) and another shock indicator

Θ =(∇ · u)2

(∇ · u)2 + (∇× u)2 + ǫ, 0 ≤ Θ ≤ 1, (29)

frequently referred to as the Ducros sensor. Further modifications of the methodwere introduced by Garnier et al. (2001), who replaced the TVD artificial fluxwith the dissipative part of ENO and WENO schemes, and showed that theDucros sensor is capable to distinguish turbulent fluctuations from shocks betterthan the Harten switch. The idea of adaptively filtering the numerical solution forshock-capturing also underlies the work of Visbal & Gaitonde (2005) and Bogeyet al. (2009), who developed computational strategies based on high-accuracycentral discretization of spatial derivatives, supplemented with selective filteringof the computed solution. Shock-capturing is achieved by locally decreasing thefilter bandwidth in critical regions.

A key ingredient in the formulation of hybrid and nonlinear filtering schemesconsists in the proper specification of the shock sensor. The Ducros sensor rep-resents a simple and frequently used choice. Alternative simple shock sensorsinclude those proposed by Hill & Pullin (2004), Visbal & Gaitonde (2005), andLarsson et al. (2007) (the latter being a modification of the Ducros sensor). Moreelaborate sensors for shock waves and shears, based on multi-resolution waveletanalysis were proposed by Sjogreen & Yee (2000) and Yee & Sjogreen (2007).

The performance of shock sensors in practical computations can be appreciatedfrom inspection of Figure 4, where we report results of application of some of theindicators previously mentioned to the instantaneous flow field obtained fromDNS of transonic shock/boundary layer interaction (Pirozzoli et al., 2010). Thethreshold values for the various sensors have been selected in such a way thatsimilar representation of the shock system is obtained. Note that the Ducrossensor in its original formulation does not perform properly outside of the walllayer (where ∇× u ≈ 0), due to excessive sensitivity to dilatational fluctuations.The sensor can be conveniently adapted to this case by setting ǫ = (u∞/δ0)

2

in Equation (29), so that it is only activated when the local dilatation becomeslarger than a typical large-scale velocity gradient. Also note that, for evaluationpurposes, all other sensors are based on the pressure field. The data reported inFigure 4 suggest that the modified Ducros sensor is capable to selectively isolate‘genuine’ shocks, whereas other simple sensors also mark as critical zones smoothregions populated by vortical structures.

3.3 Artificial Viscosity Methods

The basic idea of this class of methods is to explicitly introduce the amount of nu-merical dissipation needed to stabilize shock computations through the additionof diffusive terms that adaptively adjust to the local regularity of the solution.In the original approach of von Neumann & Richtmyer (1950), artificial diffu-sive terms with viscosity coefficient µ∗ = cµρh

2 |∂u/∂x| (where cµ is a tunable


constant), were added to the momentum and energy equations to spread shockwaves to a thickness comparable to the local mesh spacing. The artificial vis-cosity idea was expanded by Jameson et al. (1981), who designed a high-ordernon-oscillatory numerical flux by augmenting the numerical flux of a second-ordercentral scheme (fC), with explicit diffusive fluxes resulting from discretization ofsecond- and fourth-derivative operators, as follows

fj+1/2 = fCj+1/2 − dj+1/2,

dj+1/2 = aj+1/2

[ǫ(2)j+1/2 (vj+1 − vj) − ǫ

(4)j+1/2 (vj+2 − 3vj+1 + 3vj − vj−1)

], (30)

where in the case of the Euler equations the characteristic speed aj+1/2 is replacedwith the spectral radius of the Jacobian matrix of the inviscid flux at Roe’saverage state (Roe, 1981). The diffusion coefficients , defined as

ǫ(2)j+1/2 = k(2)ψj+1/2, ǫ

(4)j+1/2 = max

(0, k(4) − ǫ

(2)j+1/2

), (31)

are functions of the local regularity of the solution through

ψj+1/2 = max (ψj , ψj+1) , ψj =|vj+1 − 2vj + vj−1|

|vj+1 + 2vj + vj−1|, 0 ≤ ψj ≤ 1, (32)

and k(2) ≈ 1, k(4) ≈ 0.01 − 0.05. The Jameson-Schmidt-Turkel (JST) artificialflux is designed in such a way that second-order dissipation is activated only nearshocks, whereas fourth-order dissipation only acts in smooth regions to suppresshigh-frequency oscillations. In the case of the Euler equations, regularity of thesolution is gauged by applying (32) to the pressure field.

von Neumann’s original idea was extended by Tadmor (1989) and Guo et al.(2001), who introduced a spectrally accurate vanishing viscosity to augment theapproximation of nonlinear conservation laws in spectral space. It was shownthat a suitable choice of the viscosity kernel in Fourier space recovers spectralconvergence for weak solutions of Burgers equation. Cook & Cabot (2004, 2005)carried over the idea of selectively damping the highest resolved wavenumbers tophysical space, through addition of an artificial stress tensor

σ∗ij = 2µ∗ Sij + (β∗ − 2/3µ∗)uk,k δij , (33)

where Sij = 1/2 (ui,j + uj,i). The artificial shear and bulk viscosity coefficientsare defined as

µ∗ = crµ ηr, β = crβ ηr, ηr = ρ∆2(r+1) |∇2rS|, (34)

where crµ, crβ are user-specified constants, ∆ is a measure of the local mesh spac-

ing, S = (2SijSij)1/2, ∇2r denotes the polyharmonic operator (i.e. the r-th power

of the Laplace operator), and the caret denotes a truncated Gaussian filter, de-signed to have smooth ηr. A set of coefficients used by most investigators is r = 2,c2β = 1, c2µ = 0.002. Compared to the von Neumann formulation (that is recovered

for r = 1), the choice (34) imparts on the numerical viscosity a higher wavenum-ber bias. Inclusion of the bulk viscosity term is the key to capture shock waveswithout affecting the vorticity field, whereas the shear viscosity has the effect ofa Smagorinsky-like sub-grid scale model for unresolved vortical fluctuations. Asobserved by Mani et al. (2009), the definition (34) introduces unnecessary damp-ing into resolved dilatational motions (acoustic waves). To cure such deficiency,

16 Pirozzoli

Mani et al. (2009) replaced S with uk,k in (34), in the definition of crβ, and set

crβ to zero whenever uk,k > 0 (i.e. in expansion zones). Additional selectivity in

the definition of crβ was introduced by Bhagatwala & Lele (2009), in order thatit effectively becomes zero in the presence of resolved acoustic motions.

Further improvements of the artificial viscosity method have led to the inclusionof the effects of strong temperature gradients (contact discontinuities), as well asspecies diffusion in multi-component mixtures (Fiorina & Lele, 2007; Cook, 2007).The extension of the method to generalized curvilinear coordinates was presentedby Kawai & Lele (2007).

3.4 Shock-capturing through Sub-Grid Scale Models

Methods of this class are based on the attempt to regularize weak solutions ofthe conservation equations through the addition of sub-grid scale models thatdrain energy from the unresolved scales of motion, in analogy to what is done forLES of smooth flows. This idea is related to the artificial viscosity method, andwas elaborated by Adams & Stolz (2002). Those authors developed a methodbased on explicit filtering of the flow variables, whereby the closure terms arederived from an approximate deconvolution of the flow scales represented onthe computational grid, and entropy regularization is achieved by addition of arelaxation term driven by the amount of energy found at the smallest resolvedscales. Referring to the scalar conservation law (4), the method of Adams & Stolz(2002) can be formulated as an evolution equation

∂u

∂t+∂f(u)

∂x= G1 + R, (35)

for the filtered variable u = G ⋆ u, where G is a (primary) low-pass filteringoperator. The term G1 accounts for the effect of the computed flow scales (i.e.those represented on the computational grid) on the resolved ones (i.e. on thefiltered field), and is approximated as

G1 =∂f(u)

∂x−∂f(u)

∂x, (36)

where u = Q⋆u, Q being a regularized approximation of the exact deconvolutionoperator G−1. The effect of the interaction between the non-represented scalesand the resolved scales is further modeled through the relaxation term

R = −χ (u−G2 ⋆ u) , (37)

where G2 is a secondary (more selective) low-pass filter, and χ is an adjustablerelaxation constant. The modified equation (35) is then discretized through high-order compact differencing. A weak point of the formulation is the specificationof a suitable value for the relaxation constant χ, that apparently cannot be es-tablished a-priori. A dynamic procedure to estimate χ as a function of space andtime was described by von Kaenel et al. (2004).

3.5 Properties of Shock-Capturing Schemes

The analysis of shock-capturing schemes is made difficult by their inherent non-linearity, which prevents straightforward application of the tools extensively used


to evaluate linear schemes, especially Fourier analysis (Vichnevetsky & Bowles,1982). Nonlinearity also makes comparison of the relative quality of differentmethods, other than a case-by-case check, almost impossible. Very often, classi-cal shock-capturing schemes have linear counterparts, that are used as a guidanceto improve their spectral behavior. For example, freezing the nonlinear weightsturns WENO schemes into linear upwind schemes, whose approximate disper-sion relation has been used in the development of the optimized-WENO schemes.However, numerical tests show that the nonlinear features have a dramatic impacton the performance of numerical schemes, and their actual behavior may be verydifferent from linear predictions. For instance, going back to Figure 1, one can seea substantially poorer performance (in terms of increased numerical dissipation)of the seventh-order WENO scheme (W7) compared to its ‘linearized’ seventh-order upwind version (D7). For a discussion of the impact of nonlinearities on thebehavior of shock-capturing schemes in Fourier space see Pirozzoli (2006). Usefulinformation related to the performance of shock-capturing schemes for shock-freeturbulent flows can be found in Garnier et al. (1999) and Larsson et al. (2007).

A major flaw of shock-capturing schemes, often disregarded, is the reductionof accuracy near shocks. Indeed, even (nominally) uniformly high-order schemesyield first-order accurate solutions downstream of moving shocks, the main rea-son being that shocks are objects with zero measure, and as such, their location isonly known to O(h) on a finite grid. The loss of accuracy phenomenon was high-lighted in model scalar shock-sound interaction problems (Casper & Carpenter,1998; Engquist & Sjogreen, 1998), and it is likely to be the cause of slow conver-gence of apparently simple shock/turbulence interaction calculations (Larsson &Lele, 2009). For an interpretation of the pathology in terms of numerical viscositythe reader may refer to Efraimsson & Kreiss (1999). Shock-capturing schemesapplied to systems of conservation laws also suffer from spurious post-shock os-cillations, especially in the case of slowly moving shock (Arora & Roe, 1997),which may prevent the accurate prediction of shock/sound and shock/turbulenceinteractions, as observed by Johnsen et al. (2010). These pathologies are ap-parently unavoidable, unless one reverts to special techniques, such as sub-cellresolution (Harten, 1989), or even to shock-fitting.

4 FURTHER TOPICS

All methods presented so far are focused on the discretization of the spatialderivatives that appear in the governing equations. Other issues, related to timeintegration and specification of numerical boundary conditions, as well as theextension to complex geometries, are briefly addressed in the present Section.

4.1 Complex Geometries

The extension of FD schemes to multi-dimensional systems of conservations lawsin Cartesian domains is easily obtained through component-by-component (orcharacteristic-wise), line-wise application of one-dimensional algorithms. Rela-tively complex geometries, that can be smoothly mapped to a Cartesian mesh,can be handled exploiting the ‘strong conservation form’ of the equations, (Vi-nokur, 1974) that allows extension of locally conservative approximations derivedfor uniformly spaced Cartesian meshes, thus allowing convergence to weak solu-tions. The ‘immersed boundary’ approach (Mittal & Iaccarino, 2005) promises

18 Pirozzoli

simple handling of complex geometries in the FD framework, representing a vi-able alternative to the adoption of body-fitted grids. However, relatively fewapplications of the method to compressible flows have appeared so far (Hu et al.,2006; de Tullio et al., 2007; Ghias et al., 2007).

Finite-volume methods provide greater flexibility than FD in dealing with com-plex geometries, since locally conservative FV schemes can be easily designedfor both structured and unstructured meshes. The FV framework also allowsthe construction of one-step methods in time (as opposed to multi-stage Runge-Kutta time integration commonly used for FD, see Section 4.2) having high-order accuracy in both space and time, which makes ADER schemes competitivewith typical shock-capturing FD schemes having comparable accuracy (Titarev& Toro, 2005). Furthermore, it appears that a FV formulation, with use ofsuitable positivity-preserving approximate Riemann solvers, is necessary in someinstances, such as the computation of compressible multi-component flows, toavoid oscillations in the presence of material interfaces (Johnsen & Colonius,2006). With regard to FV methods for structured meshes, it is known thatstraightforward dimensional splitting gives rise to second-order errors (regard-less of the accuracy of the underlying reconstructions), unless computationallyexpensive quadratures are used to evaluate the flux integrals transverse to the di-rection being reconstructed. High-order accurate, quadrature-based FV schemeshave been developed both in the classical method-of-lines approach with stan-dard Riemann solvers (Titarev & Toro, 2004), and in the one-step ADER frame-work (Titarev & Toro, 2005). However, as observed by Ducros et al. (2000) andGerolymos et al. (2009), the splitting error is usually small, and line-wise ap-plication of 1D reconstructions is quite successful in practice, yielding accuraterepresentation of smooth flow features and good shock-capturing properties. Un-structured meshes mandate the use of both high-order flux quadratures at cellinterfaces and of genuinely multi-dimensional reconstructions (Hu & Shu, 1999),thus making high-order FV schemes highly expensive. An important step in thedirection of improving the computational efficiency of high-order FV schemeswas accomplished by Dumbser et al. (2007), who succeeded in designing one-stepnon-oscillatory FV schemes for unstructured tetrahedral meshes with arbitraryorder of accuracy, without the need of quadratures. Their strategy exploits acharacteristic WENO reconstruction yielding the whole polynomial informationin each cell, and a Cauchy-Kovalewski procedure to provide a space-time Taylorseries for the conserved quantities and the physical fluxes. This information isused to construct highly accurate upwind numerical fluxes, that are subsequentlyintegrated analytically in space and time. A comprehensive review of modern FVmethods is reported in the book of Toro (2009).

Alternative strategies to achieve uniform high accuracy for flows in complex ge-ometries, especially in an unstructured mesh framework, include the classes of thespectral volumes (SV) and discontinuous Galerkin (DG) methods (Wang, 2007).However, these methods require the storage of several degrees-of-freedom per cell,and therefore seem to be currently too demanding in terms of memory require-ments for large-scale turbulence computations. A novel approach was proposedby Dumbser et al. (2008), who developed a unified framework for the constructionof fully-discrete and very high-order accurate quadrature-free one-step FV andDG schemes on unstructured tetrahedral meshes (the so-called PNPM schemes),where the data are represented in each cell by piecewise polynomials of degree N ,and the fluxes are computed in using piecewise polynomials of degree M ≥ N .


The class of FV methods is then recovered for N = 0, whereas DG methods arerecovered for N = M . The intermediate classes of schemes with M > N andN 6= 0 were found to produce computationally more efficient algorithms thanconventional FV and DG methods.

Some recent studies have addressed the possibility of constructing low-dissipative,nonlinearly stable schemes for complex geometries. Kok (2009) designed fourth-order accurate schemes suitable for smooth structured meshes, that preserve themathematical skew symmetry of the convective terms. Jameson (2008a) andSubbareddy & Candler (2009) have developed second-order FV methods suitablefor unstructured meshes, that discretely preserve kinetic energy. In particular,in the latter study, fully discrete energy conservation was obtained through adensity-weighted Crank-Nicholson-like time integration, and shock-capturing wasincorporated in the method in the form of a TVD filter controlled by the Ducrossensor.

4.2 Time Integration

Time integration in the method-of-lines framework is usually performed by meansof explicit Runge-Kutta multi-stage schemes (Kennedy et al., 2000). Special typesof so-called strong-stability-preserving (SSP) Runge-Kutta schemes are frequentlyused (Shu & Osher, 1989; Gottlieb et al., 2001), having the remarkable propertythat if a semi-discretization of the scalar conservation law (4) is stable in a certainnorm (e.g. the TV norm) with first-order Euler forward time stepping, the multi-stage time discretization will maintain stability in that norm, under a suitableCFL restriction. It has been argued (Shu, 1997) that the use of SSP Runge-Kutta time discretizations is safer for solving hyperbolic problems in the presenceof shock waves. In the author’s experience, and as confirmed by inspection ofthe literature on the subject, standard Runge-Kutta time integration will workout just as fine for most of typical DNS and LES applications. Optimized low-dissipation and low-dispersion Runge-Kutta algorithms, originally introduced forcomputational aeroacoustics (Hu et al., 1996), especially in their low-storageimplementation (Stanescu & Habashi, 1998), can also be fruitfully adopted forthe purpose. Recent developments on the subject are reported in Bogey & Bailly(2004), Berland et al. (2006) and Bernardini & Pirozzoli (2009).

4.3 Boundary Conditions

The correct enforcement of boundary conditions in compressible flow simulationsis of utmost importance for accuracy and stability of numerical algorithms. Anexhaustive account of the subject can be found in Colonius (2004). A criticalissue in the simulation of spatially evolving wall-bounded compressible flows,not covered in that review, is the specification of suitable inlet conditions, thatmust guarantee a fully developed turbulent state of the boundary layer at theshortest possible distance from the inflow. A frequently used technique is therescaling-recycling procedure developed by Lund et al. (1998), and extended tothe compressible case by Sagaut et al. (2004) and Xu & Martın (2004). The basicidea consists of extracting a cross-stream slice of the flow field and recycling it tothe inflow, after suitable rescaling. This approach produces a realistic turbulentboundary layer within a short distance from the inflow, and it allows to controlthe skin friction and the thickness of the simulated boundary layer. Alterna-

20 Pirozzoli

tive techniques rely on the specification of synthetic disturbances at the inletconstructed so as to mimic coherent boundary layer structures (Sandham et al.,2003; Pamies et al., 2009), or on the attempt to synthesize quasi-deterministicdisturbances having desired first- and second-order velocity statistics at the in-flow (Klein et al., 2003).

4.4 Discretization of Viscous Fluxes

Discretization of the viscous terms in the compressible Navier-Stokes equationsinvolves consecutive derivative operations. As observed by Lele (1992), Sandhamet al. (2002) and Nagarajan et al. (2003), better accuracy and robustness isachieved by expanding the diffusive terms as

∂

∂x

(µ∂u

∂x

)= µ

∂2u

∂x2+∂µ

∂x

∂u

∂x, (38)

and applying dedicated second derivative discretization to the first term at ther.h.s. This treatment is justified by spectral analysis of the approximation, show-ing significantly improved representation of the diffusive effects at the highestresolved wavenumbers, compared to straightforward repeated application of dis-crete first derivative operators, that may result in odd-even decoupling.

5 APPLICATIONS

Applications of the methods of computational gasdynamics to the analysis offlow physics are innumerable. We limit ourselves to highlight the most recentapplications to high-fidelity simulation of flows involving the interaction of shockwaves with turbulence.

The simplest setting consists of the interaction of a (nominally) normal shockwave with a field of isotropic turbulence. Canonical shock/turbulence interac-tions were first investigated through DNS by Lee et al. (1997), using a hybridcompact/ENO scheme. LES of the same problem was performed by Ducroset al. (1999), by means of a characteristic-based nonlinear filtering scheme. Theproblem was revisited by Larsson & Lele (2009), who carried out high-resolutioncalculations using a hybrid central skew-symmetric/WENO discretization.

Substantial efforts in the last decades have been devoted to the analysis ofshock / boundary layer interactions (SBLI) (Delery & Marvin, 1986). The firstDNS study of SBLI was reported by Adams (2000), who investigated the flowover a 18◦ ramp at free-stream Mach number M∞ = 3, using the hybrid com-pact/ENO method of Adams & Shariff (1996). DNS of a 24◦ compression rampconfiguration at M∞ = 2.9, was performed by Wu & Martın (2007), using thebandwidth-optimized WENO algorithm of Taylor et al. (2007). LES of the su-personic ramp flow was carried out by Rizzetta et al. (2001), using the adaptivefiltering technique of Visbal & Gaitonde (2005), and by von Kaenel et al. (2004),using the regularization method of Adams & Stolz (2002).

Another frequently used prototype SBLI consists of the reflection of an obliqueshock wave from a flat plate where a boundary layer is developing. The first LESof impinging shock interaction was performed by Garnier et al. (2002), who useda baseline central fourth-order discretization augmented with a nonlinear WENOfilter, whose local activation was controlled by the Ducros sensor. Pirozzoli &Grasso (2006) performed a DNS study with flow conditions similar to Garnier


et al. (2002), using a seventh-order WENO scheme. LES of the impinging shockinteraction have also been performed by Touber & Sandham (2009). Their nu-merical method relied on a baseline SHOEC scheme, and shock-capturing wasachieved through a TVD filter based controlled by the Ducros sensor. Sampleresults of the calculations of Touber & Sandham (2009) are reported in Figure 5,showing a comparison between the mean LES field (in terms of streamwise veloc-ity and turbulent shear stress), and reference experimental data (Dupont et al.,2008). Overall, the LES results are in very good agreement with experimentalPIV data, the most apparent difference being the size of the separation bubble.However, the boundary layer thickening is very well captured, as well as the am-plification of the Reynolds shear stress past the interaction zone, associated withthe shedding of vortices.

Shock/boundary layer interactions also occur under transonic conditions. Thefirst LES of transonic SBLI over a circular-arc bump was performed by Sandhamet al. (2003), using the SHOEC scheme with nonlinear TVD filtering. Pirozzoliet al. (2010) have recently reported DNS results of transonic SBLI at M∞ =1.3 over a flat plate using a hybrid approach, whereby smooth flow regions arehandled by means of conservative sixth-order central discretization of the skew-symmetric split form (20), and shock waves are captured through a seventh-orderWENO scheme, the switch being based on the Ducros sensor. An illustrativeresult of that calculation is reported in Figure 6, where the shock system iscaptured through the modified Ducros sensor, and vortical structures throughiso-surfaces of the ‘swirling strength’ (Zhou et al., 1999). The figure highlights thethree-dimensional nature of the lambda-like shock pattern, whereby the vorticalstructures in the incoming boundary layer cause the spanwise wrinkling of theupstream compression fan. Numerous hairpin-shaped vortex loops, resemblingthose found in incompressible boundary layer DNS, are observed both in theupstream boundary layer and past the interacting shock.

The turbulent mixing resulting from the injection of an under-expanded sonicjet in a supersonic cross-flow was studied by Kawai & Lele (2010) by means ofLES. Those authors used the artificial viscosity method in the version of Kawai &Lele (2007), employing sixth-order compact approximations of the spatial deriva-tives, coupled with eight-order low-pass filtering. The flow visualizations reportedin Figure 7 demonstrate the capability of the numerical method to capture thefront bow shock, the upstream separation shock, the barrel shock, and the Machstem, all without spurious wiggles, and at the same time to accurately resolve abroad range of turbulence scales.

Low-dissipative shock-capturing methods have also been used by Hill et al.(2006) to analyze Richtmyer-Meshkov instability with reshock. Those authorsused a DRP-like central approximation of the skew-symmetric split equationsin smooth regions, and switched to the tuned-WENO scheme of Hill & Pullin(2004) near shock waves, the switch being controlled by the local ‘curvature’ ofthe pressure and density field, in a fashion similar to the Jameson sensor.

Many other applications of low-dissipative shock-capturing algorithms are col-lected in the review paper of Ekaterinaris (2005).

22 Pirozzoli

6 SUMMARY AND CONCLUSIONS

We have presented (some of the) currently available numerical methods for high-speed flows, with emphasis on FD schemes of current use for LES and DNS ofcompressible flows with shocks. Owing to their great computational efficiency,FD methods are preferable for large-scale computations whenever the physicalproblem allows use of uniform Cartesian or smooth curvilinear meshes.

Two classes of methods have been discussed, specifically designed for appli-cation to smooth and shocked flows, respectively. With regard to the formerclass, several mathematical principles can be exploited for the design of stable,low-dissipative schemes that are superior to classical upwinding or filtering. Inparticular, nonlinearly stable schemes can be designed by enforcing at the discretelevel properties inherent to the Euler equations in the absence of shocks, namelypreservation of the total kinetic energy and conservation properties related to thethermodynamic entropy. The results of several investigators, and numerical testsherein reported, support the effectiveness of these methods in providing nonlinearstability for the calculation of high-Reynolds-number flows, without addition ofnumerical viscosity or filtering. A survey of the current literature leads me to givesome favour to energy-preserving schemes that rely on expansion of convectivederivatives to skew-symmetric form. Numerical schemes stemming from centralexplicit discretization of equations in skew-symmetric split form are computation-ally efficient, have good nonlinear stability properties, and can be formulated inlocally-conservative form. In particular, the variant of skew-symmetric splittingproposed by Kennedy & Gruber (2008) seems to be promising for smooth flowswith strong density variations.

Regarding methods for shocked flows, the class of the WENO schemes (andtheir variants) seem to have superseded other shock-capturing methods in thelast decade, having proven to be extremely accurate and robust in the pres-ence of strong shock waves and complex shock interactions. WENO schemes,however, are quite CPU intensive, and suffer from excessive numerical damp-ing in smooth zones of the flow field. Therefore their application to LES andDNS is suggested only in hybrid form, i.e. in conjunction with a non-dissipativealgorithm to treat smooth flow zones. Effective strategies for coupling shock-capturing and non-dissipative methods include hybridization and filtering. Bothstrategies rely on shock sensors that have to be as simple and effective as possi-ble. One such choice is the Ducros sensor, that was found to perform reasonablywell in many shock/turbulence interaction problems, particularly with some mi-nor case-by-case modification and tuning of the threshold value for activation ofshock-capturing. With regard to the hybrid methods, a suggested best practiceis marking critical nodes using a shock sensor, and then pad a sufficient numberof nodes around the critical ones to make sure that the stencil of the underlyingnon-dissipative scheme does not cross shocked zones.

An attractive alternative to hybrid shock-capturing methods is offered bythe nonlinear artificial viscosity methods, especially in their most recent vari-ants (Mani et al., 2009). These methods have the advantage to be applicablein the same form throughout the computational domain, without the need ofhybridization, and that their implementation in existing compressible Navier-Stokes solvers for shock-free flows is straightforward. Applications of the methodare recent and further testing is required to demonstrate comparable reliabilityto WENO for a range of applications.


In all cases, a locally conservative discretization is encouraged. Although itis frequently stated that non-conservative methods are capable to handle shockwaves properly, a locally conservative formulation (based on numerical fluxes)automatically guarantees global conservation of integral properties through thetelescopic property, and applicability of the Lax-Wendroff theorem. Note thereis no extra cost incurred with the use of a locally conservative discretization, andnumerical fluxes are available for skew-symmetric and entropy-consistent schemes(see Sections 2.3,2.4). Furthermore, hybridization with shock-capturing schemes(always formulated in conservation form) is more natural.

Some open issues remain. First, one must be aware that the global order ofaccuracy of shock-capturing schemes in unsteady problems is always reduced tounity, and shock-capturing is the cause of spurious oscillations, especially down-stream of slowly moving shocks. These limitations, related to the misrepresen-tation of discontinuities on a mesh with finite spacing, can only be overcome bysome form of shock-fitting. A detailed study of the effect of shock-capturing os-cillations on the prediction of shock/sound and shock/turbulence interactions islacking, which would be highly desirable. Second, even though hybrid schemes arefrequently used, a systematic quantitative analysis of the coupling between shock-capturing and non-dissipative schemes has not been carried out so far (however,hints are found in the work of Larsson & Gustafsson (2008)). Third, a compar-ative efficiency analysis of numerical algorithms (in terms of CPU cost for givenerror tolerance) for problems involving shock waves is not available at present,and cost figures are seldom reported in computational studies. Fourth, it ap-pears that efficient, low-dissipative methods suitable for compressible turbulencesimulation on unstructured meshes are lacking in the literature, one notable ex-ception being the recent work of Subbareddy & Candler (2009). In this respect,alternative avenues to FV are also worth being explored (Dumbser et al., 2008).Further efforts are needed before computational gasdynamics can reach a fullymature stage, and cope with the growing demand for DNS and LES of high-speedturbulent flows for configurations of technological relevance.

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0 10 20 30 400

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0/M

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D2 D6 D7 W7 S6 K6 H6 T2

Figure 1: Simulations of isotropic turbulence at zero viscosity at Mt0 = 0.07 (leftcolumn) and Mt0 = 0.3 (right column): time evolution of r.m.s. density fluctua-tions, total kinetic energy and total entropy (τ is the eddy turnover time). Dn: n-th order discretization of divergence form; W7: seventh-order WENO scheme; S6:sixth-order discretization of skew-symmetric split form (19); K6: sixth-order dis-cretization of skew-symmetric split form (20); H6: sixth-order entropy-consistentscheme of Honein & Moin (2004); T2: second-order entropy-conserving schemeof Tadmor (1984) (β = 4). Interrupted lines indicate numerical divergence.

S

S0

S1

S2

S3

h

j + 1/2

j − 2 j − 1 j j + 1 j + 2 j + 3

Figure 2: Definition of stencils for WENO reconstruction (L = 3).


Figure 3: Numerical simulation of test case 13 of Lax & Liu (1998) on a meshwith hx = hy = 1/1200 (only the central part of the domain is shown). Timeintegration is performed with a third-order SSP Runge-Kutta algorithm (theCourant number was set to 0.6). Fourteen equally spaced density contours areshown, from 0.6 to 2.3. (a) second-order TVD scheme with Van Leer limiter; (b)fifth-order WENO scheme; (c) seventh-order WENO scheme; (d) hybrid com-pact/WENO scheme (Pirozzoli, 2002). In the latter case, WENO is activated atthe intermediate node xj+1/2 (and two neighboring nodes left and right of it),whenever |ρj+1 − ρj | ≥ 0.2.

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Figure 4: Performance of shock sensors for DNS of transonic shock/boundarylayer interaction. Black lines denote pressure iso-lines (24 equally spaced levels,1 ≤ p/p∞ ≤ 1.85). Red lines indicate iso-lines of shock sensors. (a) Ducrossensor (Ducros et al., 1999); (b) Jameson sensor (Jameson et al., 1981); (c) Hill& Pullin sensor (Hill & Pullin, 2004); (d) Visbal & Gaitonde sensor (Visbal &Gaitonde, 2005). δ0 denotes the incoming boundary layer thickness.


240 260 280 300 320 340 360 380 400

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Figure 5: LES of shock/boundary layer interaction (Touber & Sandham, 2009):comparison with PIV experimental data (Dupont et al., 2008). Top: meanstreamwise velocity; bottom: Reynolds shear stress. The same contour levelsare used for PIV (filled contours) and LES (solid lines). The yellow/red linesdenote the mean separation line as from PIV (dashed) and LES (solid). Figurecourtesy of E. Touber and N. D. Sandham.

36 Pirozzoli

Figure 6: DNS of transonic shock/boundary layer interaction (Pirozzoli et al.,2010). The shock system (in gray shades) is identified as an iso-surface of themodified Ducros sensor (Θ = 0.0025). Vortical structures are educed as iso-surfaces of the swirling strength (λci = 1.6u∞/δ0), and are coloured with thelocal Mach number (levels from 0 to 1.3, colour scale from blue to violet). Axesare scaled with respect to the incoming boundary layer thickness (δ0).


Figure 7: LES of jet in supersonic cross-stream (Kawai & Lele, 2010). Visualiza-tions in streamwise/wall-normal plane are shown in the top row: density gradient(left), and jet fluid (right). Visualizations in wall-parallel plane are shown in thebottom row: streamwise velocity (left), and jet fluid (right). D is the jet diameter.Figure courtesy of S. Kawai and S.K. Lele.

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