# User:Oleg Schilling/Proposed/Turbulence: Large-Eddy Simulation

Prof. Olivier Metais, Grenoble Institute of Technology, Grenoble National Polytechnic Institute (ENSHMG) LEGI, was invited on 23 October 2009.

**Large-eddy simulation** (LES) refers to small scales elimination in the numerical simulations with computers of turbulent flows.
It is done through a proper low-pass filter applied to Navier-Stokes equations, and to equations for the energy and other
quantities transported by the fluid. This elimination, called subgrid-scale modelling, yields the appearance of subgrid-scale stresses and scalar (such as temperature) exchanges which have to be modelled appropriately. For constant-density fluids in simple geometries, Fourier representation allows to simplify the problem, and the recourse to stochastic models of three-dimensional isotropic turbulence for the small scales turns out to be helpful. Using very efficient numerical tools, one can also visualize spectacular evolutions and interactions of coherent vortices at various scales.

LES permits to simulate the evolution of incompressible isotropic turbulence, mixing layers or channels, at computing speeds increased by a factor ranging from 3 to 100 with respect to direct-numerical simulations (where all the scales are considered). It is employed also with success in geophysical (meteorology, internal geophysics, oceanography) and astrophysical (stellar atmospheres) fluid dynamics. Indeed, LES of atmospheric storms or oceanic vortices developing from baroclinic instability have been done (see Lesieur et al. 2005).

For aerospace applications and rocket-engines development, LES is used to describe quasi-deterministically species mixing by upstream forcing in coaxial jets of same density. It is also of great interest for compressible turbulence in subsonic and supersonic jets, strongly-heated curved channels in Ariane V engines, combustion (see Poinsot and Veynante 2005) and aeroacoustics (see Berland et al 2007 and Bogey and Bailly 2006). It is obvious that the exponential increase of computer ressources is going to boost the development of turbulence numerical simulations in more and more complex systems.

## Contents |

## Formalism of incompressible large-eddy simulation

Being able to properly simulate numerically on large computers turbulent flows and the associated transported species (including heat, polluting particles, chemical reactions or biological cells) is extremely important in aeronautics, combustion and aero-acoustics, hydraulics, nuclear engineering, agriculture, meteorology, internal geophysics, oceanography, astrophysics, chemistry and medical applications. In the simplest cases, this may be done with the aid of what is called a direct-numerical simulation (DNS) of the flow motion equations, called Navier-Stokes, and of related equations. Here only a monophasic Newtonian fluid is considered. In fact a given turbulent flow appears to the eye as a superposition of structures (vortices, waves), of wavelength distributed along a broad continuous spectrum. At high Reynolds number and far from boundaries, it may be shown that molecular viscosity damps out the kinetic energy at scales smaller than the so-called Kolmogorov scale \(l_D= \left(\nu^3/\epsilon\right)^{1/4}=k_D^{-1} \) (see Kolmogorov 1941 and Batchelor 1953), where \(\nu\) is the kinematic viscosity and \(\epsilon\) the kinetic-energy dissipation rate. Indeed, the local Reynolds number built on typical velocity differences at this scale falls below \(1\ .\) In a DNS of a three-dimensional flow, the space-time is discretized on a grid of typical size \(\Delta x-\Delta t\ ,\) and various partial-differential operators involving \((\overrightarrow x,t)\) are approximated by finite-differences, finite-volumes or spectral methods (French speaking readers are referred to Lesieur 1994). One advances with time starting from a given initial state, with prescribed spatial boundary conditions. But, in a DNS, \(\Delta x\) must be smaller than \(l_D\ .\) If \(L\) characterizes large scales, often prescribed by the size of the domain in which the flow evolves, the number of spatial grid points necessary for a well-resolved DNS is \(\approx (L/l_D)^3\ .\) One finds \(\approx 10^{15}\) points for a commercial-plane wing (DNS possible in \(30\approx 50\) years, see Jimenez 2002), \(10^{18}\) points for the atmospheric boundary layer, more for a fast-breader reactor core.

In fact large-eddy simulation allows to reduce drastically the number of computational points. The small wavelengths arising in the flow are eliminated by applying a low-pass filter. First, incompressible LES in physical space will be presented. This section parallels and complements the very detailed review on turbulence subgrid-scale modelling carried out by C. Meneveau (2010) in Scholarpedia. The latter restricts to constant-density flows, with essentially applications to isotropic turbulence. He discusses also other concepts in subgrid-scale modelling (such as deconvolution models) than the eddy-viscosity models (mainly based on a Fourier-space approach) used in the present article. It will be shown below (see also Lesieur et al. 2005) how this approach permits to go to LES of varying-density turbulence in complex domains, with extremely practical applications.

The flow is first assumed to be of constant density \(\rho=\rho_0\ ,\) and \(\Delta x\) is a fixed length characterizing the spatial grid mesh (\(l_D<\Delta x<L\)). \(G_{\Delta x}(\overrightarrow x)\) is a low-pass spatial filter of width \(\Delta x\ ,\) chosen in order to eliminate *subgrid scales* of wavelength \(<\Delta x\ .\) For a given function \(f(\overrightarrow x,t)\ ,\) its associated filtered function is

\[\tag{1} \bar f(\overrightarrow x,t)=f* G_{\Delta x}= \int f(\overrightarrow y,t)G_{\Delta x}(\overrightarrow x-\overrightarrow y) d\overrightarrow y\ .\]

The filter commutes with spatial and temporal partial derivatives if the grid is uniform.
Applying the filter to Navier-Stokes equations

\[\tag{2} {\partial u_i \over \partial t} + {\partial \over \partial x_j} (u_i u_j) =-{1 \over \rho_0}\ {\partial p \over \partial x_i} + {\partial \over \partial x_j} (2 \nu S_{ij})\ \ {\rm with}\ \ S_{ij}={1 \over 2} \left({\partial u_i \over \partial x_j} + {\partial u_j \over \partial x_i}\right)\ . \]

yields

\[\tag{3} {\partial \bar u_i \over \partial t}+ {\partial \over \partial x_j} (\bar u_i \bar u_j) =-{1 \over \rho_0}\ {\partial \bar p \over \partial x_i} + {\partial \over \partial x_j} (2 \nu \bar S_{ij}+{\bar u_i \bar u_j}- \overline {u_iu_j})\ . \]

\(T_{ij}={\bar u_i \bar u_j} - \overline {u_iu_j}\ \ \) is called here the *subgrid-scale tensor*.
Remark that its sign is opposite to the more classical notation of Meneveau (2010),
in order to have a quantity of same sign as the molecular-viscous term. The difference disappears when the tensor is modelled.
In an eddy-viscosity assumption, and by analogy with molecular-viscous effects,
\(\ T_{ij}=2\nu_t(\overrightarrow x,t)\ \bar S_{ij} + {\textstyle\frac{1}{3}} T_{ll}\ \delta_{ij}\ .\)
The LES equation for the linear-momentum rate (Navier-Stokes) is

\[\tag{4} {\partial \bar u_i \over \partial t}+ {\partial \over \partial x_j} (\bar u_i\bar u_j) =-{1\over \rho_0}\ {\partial \bar P\over \partial x_i} + {\partial \over \partial x_j} [2 (\nu +\nu_t)\bar S_{ij}]\ ,\ \nu_t= (\Delta x) V_{\Delta x}\ ,\ \bar P=\bar p-{1\over 3}\rho_0T_{ll}\ . \]

\(\bar P\) is a modified pressure, determined using \(\partial\bar u_j/\partial x_j=0\ \ .\)
The large-eddy simulation of a scalar \(T(\overrightarrow x,t)\) satisfying a Lagrangian heat Fourier equation writes

\[\tag{5} {\partial T\over dt}+{\partial \over \partial x_j}(T\ u_j)={\partial \over \partial x_j}\left\{ \kappa{\partial{T} \over \partial x_j}\right\}\ ,\ {\partial \bar T \over dt}+{\partial \over \partial x_j}(\bar T\ \bar u_j)={\partial \over \partial x_j} \left\{(\kappa+\kappa_t){\partial \bar{T}\over\partial x_j}\right\}\ , \]

\(\kappa\) and \(\kappa_t\) being the molecular and eddy-diffusivity. The latter is determined
thanks to a turbulent Prandtl (resp. Schmidt, Péclet) number
\(\nu_t/\kappa_t\ \ .\) This problem is crucial in geophysical turbulence and in combustion modelling.
In *Smagorinsky's model* (Smagorinsky, 1963), \(V_{\Delta x} \sim \Delta x \sqrt{\bar S_{ij}\bar S_{ij}}\ \ .\) The model
is too dissipative close to walls, and improvements with a local dynamic evaluation of the constant
by double filtering have been brought by Germano, Piomelli, Moin and Cabot (1991).

Now another eddy-viscosity approach in Fourier space is presented. Turbulence is assumed to extend in an infinite domain, with statistical
homogeneity (see Lesieur, 2008). Taking a Spatial Fourier transform (FT)
\[\tag{6}
\hat f(\overrightarrow k,t) = \left({1\over 2\pi}\right)^3 \int e^{- i \overrightarrow k.\overrightarrow {x}}\ \ f(\overrightarrow x,t)\ d \overrightarrow x\ \ ,
\]

the chosen filter is a sharp cutoff low-pass filter
\(\overline{\hat f}=\hat f\ \ {\rm for}\ \ k=\vert \overrightarrow k\vert<k_C=\pi/\Delta x\ ;\ \overline{\hat f}=0\ \ {\rm for}\ \ k>k_C\ .\)
The kinetic-energy spectrum \(E(k,t)\) in isotropic turbulence is such that \(E(k,t)\delta k\) is the mean kinetic energy per unit mass in a spatial frequency band \([k, k+\delta k]\) (*mean*, in the sense of a statistical average on an ensemble of realizations \(<>\)).

To write Navier-Stokes in Fourier space, pressure is eliminated by projection in the incompressibility plane
(plane perpendicular to \(\overrightarrow k\)) of the advection term, giving \(ik_j\ {\rm FT
}\ \{u_iu_j\}\ \ .\) Remark that \( {\rm FT\ }\{ {\rm dissipative\ term}\}=-\nu k^2\hat u_i(\overrightarrow k,t)\ .\) Non-linear interactions involve *resonant triads* with \(\overrightarrow k=\overrightarrow p +\overrightarrow q\ .\) Sub-grid modelling turns out to evaluate transfers such that \(k<k_C\ {\rm ,} \ p\ {\rm or}\ q>k_C\ .\) In the LES of Navier-Stokes in Fourier space, a spectral eddy viscosity \(\nu_t(k\vert k_C)\) is added to \(\nu\ .\) It is evaluated thanks to kinetic-energy transfers across \(k_C\) given by the Eddy-Damped Quasi-Normal Markovian theory of turbulence (EDQNM, see Orszag 1970, André and Lesieur 1977, Lesieur and Schertzer 1978 and Lesieur 2008), with

\[\tag{7} \nu_t(k\vert k_C)=0.441\ {C_K}^{-3/2}\ \left[{E(k_C)\over k_C}\right]^{1/ 2}\ X\left({k\over k_C}\right)\ \ , X\left({k\over k_C}\right)\approx 1\ {\rm for}\ \ {k\over k_C}<{1\over 3}\ , \]

assuming that \(k_C\) belongs to a Kolmogorov inertial range (Kolmogorov 1941) \(E(k)=C_K\epsilon^{2/3} k^{-5/3}\ .\) It is the *Plateau-peak model* of Chollet and Lesieur (1981),
extending to the eddy diffusivity a behaviour found by Kraichnan (1976) for the eddy viscosity.
The two coefficients normalized by \(\sqrt{E(k_C)/k_C}\) with \(C_K=1.4\) are presented on #figcusp.
Now two arguments validating the EDQNM model in 3D isotropic turbulence will be given. First,
the theory predicts for the helical case (no statistical invariance in plane symmetry)
a simultaneous \(k^{-5/3}\) Kolmogorov cascade for the kinetic-energy and helicity spectra (see André and Lesieur 1977).
This double cascade was recovered by Borue and Orszag (1997) with LES using a hyperviscosity (Laplacian raised to a
higher order). Second, the plateau-cusp shape of the eddy viscosity has been recoved by Métais and Lesieur (1992) through LES in the following way as explained in Lesieur (2008). A fictitious cutoff \(k'_C\) is defined (for instance \(k'_C=k_C/2\)). Transfers between \(k<k'_C\) and the range \([k'_C,k_C]</MATH> are evaluated directly in the LES, while transfers between <math>k\) and the range \([k_C,+\infty]\) come from the EDQNM approximation. This is, at an energetic level, the philosophy followed by Germano et al (1991) dynamic model in physical space for the velocity.
In their so-called *Spectral dynamic model*, Lamballais, Métais and Lesieur (1998) account for a \(k^{-m}\) spectrum at the cutoff.

#figcritq presents an application to the decay of isotropic turbulence at zero molecular viscosity in a periodic box. The initial velocity field is Gaussian, and pseudo-spectral numerical methods are used with an initial peak at \(k_i =4\ .\) The resolution is \(128^3\) grid points. Various animations presented on the site of the book Lesieur et al. 2005 show the formation and evolution of spaghetti-type vortices, visualized by iso-surfaces at a positive threshold of \(Q=\frac{1}{2} (\Omega_{ij}\Omega_{ij}-S_{ij}S_{ij})=\frac{1}{2} (\nabla^2 p/\rho)\quad\ ,\) where \(\Omega_{ij}\) is the antisymmetric part of the velocity gradient tensor \(\partial u_i/\partial x_j\ .\)

It is the *\(Q\) criterion* of Hunt, Wray and Moin (1988) proposed for isotropic turbulence. It corresponds in fact to
local regions in the flow where rotation dominates shear, and also low-pressure zones. It was checked by Dubief and Delcayre (2000) and in Lesieur et al. (2005) that the criterion is also good to capture vortices in shear and compressible flows. It is possible that the existence of these vortices is responsible for turbulence multifractal character at small scales reviewed by Frisch (1995). A complete review of these effects from an experimental viewpoint are given by Sreenivasan and Antonia (1997). The sharp cutoff filter creates oscillations (damped by a Gaussian function) of the fields when carrying an inverse Fourier transform, but this does not seem to affect results.

Most practical situations in engineering (aerospace, hydraulics, nuclear, combustion) and environmental studies (meteorology, oceanography, internal geophysics) involve domains of a too large complexity to allow for the use of spectral methods. It is therefore necessary to work in physical space for directions where the flow is not periodic. So one returns to LES of Navier-Stokes in physical space, and introduce the LES models of the so-called *structure-function family* using a formulation where the peak in the spectral eddy viscosity is erased in Fourier space by subgrid-scale kinetic-energy conservation arguments.
Details can be found in Lesieur and Métais (1996),Lesieur et al. (2005) and Lesieur (2008).
In fact, there is no need for subgrid-scale modelling in regions of space where the flow is calm or transitional, and it is essential to dissipate in the subgrid scales local bursts of turbulence if they become too intense. Assuming that turbulence in small scales may not be too far from isotropy, it is proposed to come back to the classical formulation in physical space, the eddy viscosity being determined with \(E(k_C,\overrightarrow x)\ ,\) a local kinetic energy spectrum (\(k_C=\pi/\Delta x\)) calculated in terms of the local second-order velocity structure function of the filtered field

\[\tag{8} F_2(\overrightarrow x,\Delta x)=\left\langle\Vert\overline{\overrightarrow u}(\overrightarrow x,t)- \overline{\overrightarrow u}(\overrightarrow x+\overrightarrow r,t)\Vert^2\right\rangle_{\Vert \overrightarrow r \Vert=\Delta x}= 4 \int_{0}^{k_C} E(k) \left(1- {\sin(k\Delta x)\over k\Delta x } \right) dk \]

where Batchelor's formula (Batchelor, 1953) has been used, with a modification of the
spectral integration upper bound \(k_C\) instead of \(\infty\ .\)
This yields, for a Kolmogorov spectrum (see Métais and Lesieur, 1992)

\[\tag{9} \nu_t^{SF}(\overrightarrow x, \Delta x)=0.105\ C_K^{-3/2}\ \Delta x\ [F_2(\overrightarrow x, \Delta x)]^{1/2}\ \ . \]

\(F_2\) is determined making a local statistical average of square velocity differences between \(\overrightarrow x\) and the six (or four) closest points surrounding \(\overrightarrow x\) on the computational grid. For a transported scalar, one
takes a turbulent Prandtl number \(\nu_t/\kappa_t=0.6\ .\) The structure-function model (SF) works well for isotropic turbulence,
where it yields a fairly good Kolmogorov spectrum at the cutoff (Métais and Lesieur, 1992). It gives also good qualitative results for free-shear flows, but as with Smagorinsky's model, however, it is too sensitive to large-scale shears. To overcome the difficulty, two improved versions have been developed: the selective structure-function model (SSF), and the filtered structure-function model (FSF).

In the SSF (David, 1993) the eddy viscosity is switched off when the flow is not three-dimensional enough. The constant arising in #1254 is multiplied by \(1.56\ \ :\) one requires in a LES of decaying isotropic turbulence that the eddy viscosity averaged over the computational domain should be the same in SSF and SF model simulations. A discussion on this choice can be found in Ackerman and Métais (2001) and in Lesieur et al. (2005). However, it seems in practice that the original version should be preferred, as stressed by Münch et al. (2004) in LES of heated channels. The SSF model works very well for isotropic turbulence, free-shear flows, and wall flows (without or with thermal transfers). The reader is referred to Lesieur et al. (2005) for more details.

The FSF model was developed by Ducros et al. (1996) and applied to transition in a spatially-developing boundary layer on an adiabatic flat plate at Mach 0.5. Here, the filtered field \(\bar u_i\) is submitted to a high-pass filter in order to get rid of low-frequency oscillations which affect \(E(k_C,\overrightarrow x)\ .\) The high-pass filter is a Laplacian discretized by second-order centered finite differences and iterated three times. The analysis yields

\[\tag{10} \nu_t^{FSF}(\overrightarrow x, \Delta x)=0.0014\ C_K^{-3/2}\ \Delta x\ [\tilde F_2(\overrightarrow x,\Delta x)]^{1/2}\ \ . \]

This works well for both isotropic turbulence and transition in a spatially-developing boundary layer. This simulation (see Ducros et al., 1996) was done in a weakly-compressible case at \(M_{\infty}=0.5\ ,\) for an adiabatic plate. More recent results on the same problem with animations may be found in Lesieur et al. (2005).

Finally, an alternative approach to large-eddy simulation should be mentioned, the dynamic multilevel method (see Dubois, Jauberteau and Temam, (1999)). It assumes time-scale separation between resolved and subgrid scales, and restricts to low-Reynolds numbers. It reduces the computing time by a factor of two with respect to a DNS.

## Non-rotating and rotating plane channels

Some LES results of a plane turbulent channel (either incompressible or weakly compressible) are now given. In order to understand blood flow, the medical doctor Poiseuille (1841) first derived analytically in the laminar case the laws of fluid motion in pipes and channels between flat plates. This profile is \( \bar u(y)=\frac{3}{2} U_b[1-(y^2/h^2)]\ ,\) where \(U_b\) is the bulk velocity across the channel of width \(2h\ .\) In the turbulent case, one recalls the importance of the viscous thickness
\(\delta_v=\nu/ v_*\ ,\) defined by the friction at the wall \(\rho v_*^2=\rho\nu\ d\bar u/dy\ .\)
The microscopic Reynolds number is \(h^+=h/\delta_v\ .\)
Dubief and Delcayre (2000) have carried out with the FSF model a LES of a weakly-compressible turbulent channel
(Mach 0.3) at \(h^+=160\ .\) Some details on the numerical code COMPRESS will be given below. The resolution is
\(200\times 128\times 64\) spatial grid points. One of the walls is equipped with two small spanwise square cavities. At this low Reynolds, the LES reduces the computing time by a factor of 3 with respect to the DNS. Animations of vortices
(see Lesieur et al. (2005)) seen by Q isosurfaces indicate very neatly the longitudinal propagation of asymmetrical hairpin vortices which creep along the wall. The calculations are done at a constant \(U_b\) and with longitudinal and spanwise periodicity conditions.
Hairpin vortices are associated with a system of low- and high-speed streaks close to the wall.
The latter produce high drag. #hairp is a view of the vortices in this LES, represented by the Q-criterion.
In the simulations, the mean longitudinal velocity profile \(\bar u(y)\) has a range with the well-known logarithmic shape.
In Lesieur et al. (2005) are presented LES of a constant-density channel at \(h^+=389\ .\) These LES use the spectral-dynamic model discussed above. Numerical methods are pseudo-spectral in the longitudinal and spanwise directions, and sixth-order compact finite differences in the normal direction. This is a quite high Reynolds number, and first and second-order statistics
are in good agreement with the DNS presented in Antonia et al. (1992).
The first LES grid point at the wall is one wall unit, with a stretching away. There are no-slip conditions at the boundaries, and \(128\times97\times64\) grid points. In this channel LES, about 70 times faster than the DNS,
there are still longitudinal vortices and velocity streaks. A few words on passive control of turbulence by
longitudinal triangular riblets on planes, boats and "*shark-skin*" swimming costumes should be said,
knowing empirically that the optimal spanwise distance of crests is \(\lambda_z^+=10 \approx 20\) wall units.
In their basic DNS work of an incompressible channel with one wall flat and one ribbed,Choi et al. (1993)
stress the essential role played by the diameter \(d^+\approx 25\) of longitudinal vortices shown above. In the calculation
with a rib distance of \(40\ ,\) vortices stay within the valleys and the drag is increased. With a distance
of \(20\ ,\) vortices travel above the peaks and drag is reduced of 8%. The first LES (using the SSF model) of this problem was
performed by Hauët (2003) in the compressible case with uniform bulk velocity \(U_b\ ,\) density \(\rho_b\) and fixed wall temperature \(T_w\ .\) The Mach number is \(U_b/ c_w\ ,\) \(c_w=\sqrt{\gamma T_w}\ .\)
The LES uses immersed boundary methods on the obstacles. Two riblet sizes (22 et 44) have been studied. At Mach 0.33, Hauët recovers the incompressible results of Choi et al. (1993), which provides a good validation of the LES. For an olympic swimmer in water, the velocity at infinity is 2 m/s, with \(\nu=10^{-6}m^2/s, \delta_v\approx 10^{-5}m\ ,\)
and the riblet size is \(2\times 10^{-4}\) m (see Lesieur(2010)).
Another interesting approach to turbulence control for drag reduction in incompressible channels was developed by
Temam and coworkers using control tools of distributed systems. The first problem solved by Abergel and Temam (1990)
consists in minimizing the drag on the lower wall by velocity control on the upper wall through a feedback law to be determined. The development of this work was carried out with Bewley et al. (2001).
It uses also intensive turbulence DNS. In fact the practical use of these methods requires an industrial development
of Micro-Electro-Mechanical-Systems (MEMS) and compliant materials on bodies. It was slowed down with the
present crisis encountered by aeronautics and petrol shortage.

Now a constant-density plane channel rotating about a spanwise axis with a rotation vector \(\overrightarrow\Omega\) (angular rotation velocity \(\Omega\)) is looked at. Coriolis acceleration is included in Navier-Stokes, and the irrotational centrifugal accélération in the pressure gradient. This problem is important in industry (turbomachinery in hydraulics and aerospace engineering) and environment. One defines a *local Rossby number* \(R_o(y,t)=-(1/ 2\Omega) (d\bar u(y,t)/ d y)\) characterizing
the relative importance of non-linear upon Coriolis terms. Let \(R_o^{(i)}\) be the initial minimal Rossby number.
It was checked experimentally by Johnston et al (1972) that, for \(R_o^{(i)}<-1\ ,\) the flow evolves to create a range \(R_o(y,t)=-1\) in the anticyclonic region (\(\overrightarrow\omega . \overrightarrow \Omega < 0 \)), which replaces and extends the logarithmic range (one recalls that \(\overrightarrow\omega\) is the curl of the relative velocity).
The behaviour was recovered in LES, in particular by Lamballais et al (1998). It
corresponds to a longitudinal alignment of the *absolute vorticity* \(\overrightarrow\omega+ 2\overrightarrow\Omega\ .\)
Such rotation regimes \(\vert R_o^{(i)}\vert >1\) are moderate, compared with low \(\vert R_o\vert\) limits
for which Proudman-Taylor theorem applies with a two-dimensionalization of the flow.
In fact the above results have analogies with anticyclonic mixing layers for \(R_o^{(i)}<-1\ :\) Kelvin-Helmholtz vortices are replaced by strong longitudinal vortices (analogy with centrifugal instabilities). This can be seen on a DNS of absolute-vortex filaments evolution presented in Métais et al (1995). Still in the anticyclonic case, the mixing layer becomes purely two-dimensional for \(R_o^{(i)}\ge -1\ .\)
In the cyclonic case (\(R_o^{(i)}>0\)), DNS and LES indicate that it remains
strictly two dimensional up to \(R_o^{(i)}\approx 20\ .\) Above, various types of coherent
billows (undergoing helical pairing or pairing while stretching thin hairpins) develop.

## Active control of incompressible co-axial jets and species mixing

We now consider the flow dynamics and mixing properties of incompressible co-axial jets. They are composed of an inner single round jet surrounded by an outer annular jet. This flow configuration constitutes an efficient way to mix two different fluid streams which finds numerous industrial applications (combustion, chemistry...). It is studied numerically using LES techniques. The spatial discretization is based on a sixth-order compact finite difference scheme in the streamwise direction, combined with pseudo-spectral methods in transverse and spanwise directions. The time advancement is a third order Runge-Kutta scheme. In the flow equations, the influence of subgrid scales on the grid scale variables is modelled with the classical eddy-viscosity and eddy-diffusivity assumptions. The subgrid-scale model is the FSF model with a turbulent Schmidt number of \(0.6\) in the scalar transport (#scales).

The numerical grid consists of \(231\times480\times480\) points which allows to simulate a domain size of \(10.8D_1\times13.3D_1\times13.3D_1\ .\) The Reynolds number value is varied
in the range \(3000<Re=U_2 D_1/\nu<30000\) (\(D_1\) and \(U_2\) are the inner jet diameter
and the outer jet velocity respectively) and in these simulations the molecular Schmidt number is taken equal to \(1\ .\)
The inlet velocity profile is constructed with two *hyperbolic tangent* profiles (see balarac, 2006)
to which is superimposed of a weak-amplitude perturbation (\(3\%\) of the outer velocity).

#critereQ shows the spatial evolution of coherent vortices for a moderate and a high Reynolds number with
the same velocity ratio between the outer and the inner jets. As beginning of the
transition, both inner and outer shear layers roll up into axisymmetric vortex rings due to Kelvin-Helmholtz instability. The inner
primary structures (i.e. from the inner shear layer) are trapped in the free spaces between two consecutive outer
structures characterizing the *locking* phenomenon (see balarac, 2004). Further downstream, the flow undergoes an
azimuthal instability leading to the formation of several pairs of streamwise vortices.
Finally, the flow becomes fully turbulent with the appearance of an intense small scale turbulent activity.
For the larger value of the Reynolds number, we can observe these vortices already from the beginning of the jet. The precocity of the azimuthal instability development corroborates the observations for
the single round jet performed by Dimotakis (2000) consisting of a flow three-dimensionalization from the
jet origin when \(Re > 10000\) called the *mixing transition*.

To investigate the mixing properties of coaxial jets, we characterize the mixing by the evolution of the mixture fraction \(f\) here considered as a passive tracer. We seed the tracer in the outer annular jet (\(f=1\) in the outer jet and \(f=0\) elsewhere as upstream condition). In the early transition stage near the inlet, the mixing process is dominated by molecular diffusion. Further downstream, the turbulent mixing process combines radial pulsations caused by the Kelvin-Helmholtz vortices and fluid ejections caused by the counter-rotating streamwise vortices. When the Reynolds number is increased, the molecular diffusion stage is reduced. Moreover, the *mixing transition* allows ejections of the tracer in the inner jet and in the ambient outer fluid from the
beginning of transition. The early appearance of pairs of streamwise vortices is indeed noticeable through the formation of mushroom shape structures characterizing ejections from the outer jet.
Thus, a high Reynolds number co-axial jet favours the tracer mixing. This mixing process can be seen on the animations (movie 1 and movie 2) showing contours of the mixture fraction in the central plane and in different radial sections for a moderate and a high Reynolds number, respectively.

To improve the mixing properties of moderate Reynolds number co-axial jet, the following active controls are proposed. Two different types of inflow forcing are then considered based on the information provided by the natural coaxial jet: first, a purely axisymmetric excitation and second, combined axisymmetric and azimuthal excitations all of moderate amplitude. These excitations are applied to the outer shear layer with a frequency corresponding to the periodic passage of the outer vortical structures. The goal of these excitations is to trigger the vortices formation and to control their dynamics to improve the mixing properties of the jet. With the purely axisymmetric excitation, the outer and inner Kelvin-Helmholtz vortices appear earlier than for the natural jet and the transition process is faster. This early three-dimensionality growth is due to a rapid appearance of streamwise vortices, stretched between consecutive vortex rings, which lead to enhanced mixing. For the combined axisymmetric and azimuthal excitations, the outer Kelvin-Helmholtz rings appear moreover with an azimuthal deformation from the beginning of the jet. This allows for the early generation of streamwise vortices. Scalar ejections from the outer jet thus appear sooner and with a larger intensity. Animation (movie 3) shows contours of the mixture fraction in the central plane and in different radial sections for a moderate Reynolds number co-axial jet with a combined axisymmetric and azimuthal excitation. We can see the mixing enhancement by comparison of this animation with the previous animations (movie 1 and movie 2).

## LES of subsonic and supersonic jets

One summarizes first simplified equations used for LES of compressible turbulence in a perfect gas (neglecting gravity).
The reader is referred to Lesieur et al. (2005) for details. Let \(\rho e= \rho\ C_v\ T + \frac{1}{2}\rho( u_1^2 + u_2^2 + u_3^2)\quad\) be the total energy. One writes the equations in a flux form for mass, linear momentum and total energy.
The usual *bar-filter* in spatial space is still applied. The problem is much simplified by introducing Favre filters weighted by density, and analogous to Favre averaging in turbulence (see Favre, 1965) \(\overline{\rho f}=\bar\rho \tilde f\ .\)
A *macro-pressure* \(\varpi\) and *macro-temperature* \(\vartheta\) are also introduced. Eddy coefficients are not changed. Equations read :

## Curved heated ducts in rocket engines

## References

- Abergel, F. and Temam, R. (1990) On some control problems in fluid mechanics, Theor. Comp. Fluid Dyn.,
**1**, 1990, 303-325.

- Ackermann, C. and Métais, O. (2001) A modified selective structure function subgrid-scale model. J. Turbulence,
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