Assessment of large-scale forcing in isotropic turbulence using a closed Karman-Howarth equation

We attempt to model the effects of large-scale forcing on the statistical behavior of small scales in isotropic turbulence. More specifically, the effect of large-scale forcing on the second-order velocity structure function, S-2, in the region beyond the dissipative range, is analyzed via the transport equation for S-2 where a closure model for S-3, the third-order velocity structure function, is introduced. The model [L. Djenidi and R. A. Antonia, Fluid Turbulence Applications in Both Industrial and Environmental Topics, Marseille, 9-11 July, 2019, https://fab60.sciencesconf.org/] is based on a gradient type with an eddy-viscosity formulation and has the following expression: S3=-CS3(S2)2Epartial derivative S2 partial derivative r, where E is the mean rate of the turbulent kinetic energy dissipation, r is the spatial increment, and CS3 is a constant. The closed S-2-transport equation is further exploited to derive a model for S-2 for scales beyond the dissipative range. The model for S-2 takes the form S2=CK(Err)23 with Er=E1-Br/Re lambda1/2, where C-K is a constant, Re-lambda is the Taylor microscale Reynolds number, and the function B-r accounts for the effect of the large scales. The numerical solutions of the S-2 equation and the predictions based on the model for S-2 agree very well with direct numerical simulation data for steady-state forced homogeneous and isotropic turbulence. The solutions of the S-2-transport equation without large-scale forcing show that S-2 behaves like (Er)2/3. When forcing is applied, S-2 deviates from this behavior. However, increasing the Reynolds number tends to restore this behavior over an increasing range of scales. This is also observed in the predictions of the model for S-2.