
DNS of Detonation Wave/Homogenous Isotropic Turbulence Interaction 

A current project involving Profs. Frank Lu and Luca Massa with M.S. student Hari Nagarajan is aimed at exploring the interaction between shock and detonation waves with homogenous isotropic turbulence (HIT) using direct numerical simulation (DNS). Understanding the physics of this interaction is important for various turbulent mixing and propulsion applications. Homogenous isotropic turbulence is defined by the following properties: all spatial derivatives of the mean turbulent quantities are negligible; Also, the properties remain invariant with respect to direction. HIT decays freely because there is no mean velocity gradient to generate new turbulence. 

Two and threedimensional simulations of HIT decay have been completed by using incompressible NS code which employs finitedifference discretization, 4th order Arkawa scheme and 3rd order explicit RK integrator. Below, the energy spectra for the 2D HIT is shown along with a video of the decay which shows the evolution and interaction of vortices . The 2D and 3D incompressible turbulence results were obtained by using the logic and algorithms given in the book Fluid Flow Phenomena by Prof. Paolo Orlandi, Department of Mechanics and Aeronautics, University of Rome 'La Sapienza, Italy. 



Development of 3DEuler and 3DNavierStokes code has been completed using 5th order WENO (Weighted Essentially Nonoscillatory) scheme and 3rd order explicit Runge Kutta integrator. The 3DEulerWENO code is obtained by modifying the 1DWENOCLAW package and also using the logics explained in CLAWPACK package. The 3DEuler code is validated using a shock propagation problem. Animation results obtained using the shock propagation validation test is shown below. The shock Mach number is 3.0. The preshock condition shows the intentional sinusoidal oscillation in the density plot while the post shock condition captures the oscillations caused by Gibbs phenomena. The 3DNavier stokes code is validated by simulating the compressible HIT. The simulation uses periodic boundary condition on a computational domain of 2π x 2π x 2π. The initial energy model is obtained by using the MansourWray model. The wave number can be corresponded with physical space using a Fourier transformation and vice versa. The energy decay spectra shows that the flow evolves with Kolmogorov decay as shown in the figure below. The results of the compressible HIT corroborate with contemporary literature. 



The simulation of interaction of shockwave and turbulence is in progress. The simulation of interaction of shockwave and turbulence is carried out using three different grid resolutions and three different shock Mach numbers. The following four parameters govern the generation of the initial turbulent field, viz., density, turbulent Mach number, pressure, and the Reynolds number based on the Taylor length scale. Initialization of isotropic turbulence begins with uniform density and pressure and completely random velocity values. It is deemed that a field has reached a proper state of isotropic turbulence when the skewness of the velocity derivatives become the desired value observed in the experimental data. Once the turbulence simulation reaches the desired skewness value, a normal shock wave is allowed to propagate through the cube. 





The interaction resulted in higher total energy at high wave numbers as also observed by Lele et al. Variation in the shock Mach number revealed that, for an increase in shock strength, the dissipation of the energy is more rapid and an increase in turbulence length scales is observed. The effect of turbulence on the detonation wave resulted in corrugation of the shock wave, as shown in the animation. 


The interaction of detonation wave and turbulence is accomplished with the same procedure used for shockturbulence case. Here, simulation of detonation wave is accomplished by the use of reactive NavierStokes code with a one step Arrhenius chemical reaction. The effect of heat release and scales were studied by varying the value of heat release (Q), activation energy (Ea), and N number. N number is a new nondimensional number which is the ratio of the reaction length scale to the turbulence length scale. Higher total energy at wave numbers corresponding to moderate to small scales was observed. Much higher amplification in turbulence statistical parameters was observed with detonation and turbulence interaction when compared with what was seen in shock and turbulence interaction. Moreover, the amplification in the turbulence statistical parameters was proportional to the heat release. There is rapid energy decay, with the decay being more rapid with higher heat release. The effect of turbulence on the detonation wave resulted in deformation of the detonation wave; see the animation. The higher N is, the higher the amplification in turbulence statistical parameters is (except vorticity, Reynolds stresses, and more rapid energy decay). 






The simulation of detonation wave with turbulence using detailed chemistry is envisaged as the future work which will help us to understand deflagrationtodetonation transition and causes of detonation instabilities. 

