SYNTHETIC FORCING METHOD TO SUSTAINED ISOTROPIC TURBULENCE

International audienceForcing techniques in large-eddy simulations (LES) or hybrid simulations based on both Reynolds-averaged Navier-Stokes (RANS) and LES methods have gained in importance over the past decade. A large part of their development is dedicated to sustain homogeneous turbulence with good targeted statistical properties to study for instance particle dispersion cite{Bassenne} or premixed combustion cite{Klein}. Forcing techniques can also be used locally to introduce velocity fluctuations in computations cite{Breuer2017}. This allows to trigger turbulence when natural mechanisms (strong shear due to a geometrical singularity, ldots) are absent or are not sufficient to promote turbulence. In this work we provide a new linear-like forcing approach in physical space based on synthetic fluctuations. The proposed method is tested upon homogeneous and isotropic turbulence, today's state-of-start approach to benchmark and investigate forcing method's characteristics.Forcing techniques consist usually in adding a volume force to the Navier-Stokes equations. This approach has been followed in either spectral space or in physical space according to the solving method of the Navier-Stokes equations. Forcing in spectral space is restricted on a limited wave number band which is usually concentrated towards low wave numbers. This is motivated by the aim at disturbing only large-scale structures and to not hamper the inertial subrange. On the other hand, physical space forcing consists in injecting energy over the full wave number band. However, it is observed in practice that mainly the energy containing range is affected. These techniques are based on Lundgren's methodology which is a simple and efficient way to sustain turbulence cite{Lundgren}. Analysis of the turbulent kinetic energy budget shows that the term leading to turbulent production comes from $u_j'partial_joverline{u}_i$ in the transport equation for the fluctuating part of the velocity $u_i'$. This term vanishes for homogeneous isotropic flows, that motivated Lundgren to produce his “linearly forced isotropic turbulence” method in physical space by adding a forcing term proportional to the velocity. This method has two well-known main shortcomings: textit{(i)} it generates important oscillations of the turbulent kinetic energy and textit{(ii)} the resulting integral lengthscale $L_t$, depends on the domain size $L$, in practice $L_t sim 0.2 L$ . Recent improvements allow to circumvent these shortcomings by textit{(i)} introducing a control coefficient based on the turbulent kinetic energy budget that greatly reduces oscillations and substantially shortens the initial transient phase, and textit{(ii)} by employing filters that ensure a nearly unconstrained integral lengthscale which is independent of the domain size cite{Laage}.In the present work, an alternate approach is investigated that consists in introducing synthetic velocity fluctuations through a linear-like forcing technique approach. This approach is formally equivalent to the method proposed in cite{Breuer2017} in which the last term on the right hand side of Eq.(ref{NS}) plays the role of the forcing term. In cite{Breuer2017}, in order to recover the targeted two-time correlation function, the synthetic velocity is frozen and the forcing term is expressed in a very similar way as in the linear forcing approach. The resulting forcing writes as the ratio of a stochastic synthetic velocity, prescribed randomly at each time step, by the specified characteristic turbulent time scale. In the proposed approach, the targeted two-time correlations are recovered using an unfrozen synthetic velocity through the introduction of a stochastic time frequency cite{Favier}