# Computational Investigations and Control of Shock Interference +
## 1 Introduction
### Technical Approach  
Computational fluid dynamics (CFD) is a useful approach for developing an in-depth understanding of such a complicated and unique flow field observed in the scramjet isolator. There are multiple approaches to solve the Navier-Stokes equations, including  
Figure 2: Schematic of shock-train with trailing mixing region.  
Reynolds Averaged Navier-Stokes (RANS) and a scale-resolving approach called Large Eddy Simulation (LES). RANS approach has a variation called Unsteady-RANS (URANS) in which the periodic oscillations of mean turbulent flow with time can be simulated. URANS does not resolve turbulent fluctuations, for which closure models have been developed to be utilized in both RANS and URANS approaches. RANS methods resolve the mean flow field and require modeling of the complete energy cascade and turbulent dissipation by utilizing zero, one-, two-, or three-equation models for the turbulence transport and dissipation phenomena. These equations are known as turbulence closure models. Multiple turbulence closure models of increasing complexity have been developed to capture the turbulent flow field for both external and internal aerodynamics with RANS and URANS as documented on NASA Langley Turbulence Modeling repository [19]. RANS has been a computationally affordable practical tool for the design, evaluation, planning, and assessment for most hypersonic flight applications, as well as high-speed wind-tunnel tests, but no single model has shown predictive capability, especially when applied to a range of flow phenomena encountered in hypersonic flows [20].  
In order to observe turbulence scales and related characteristics of the turbulent flow, a Large Eddy Simulation (LES) method can also be used. LES is a higher fidelity and scale-resolving method, which requires a higher computational cost. However, it can resolve all turbulence above the Kolmogorov length scale (\(\eta\)) and only models the energy dissipation at the smallest turbulent scales near the Kolmogorov length scales. LES enables filtering of the Navier-Stokes equations so that scales of turbulence up to the dissipation scale can be solved by numerical solution of these equations. Turbulence closure of certain terms in the transport equations is still required, but it only models the dissipation of energy at the smallest scales of turbulence. In contrast, RANS turbulence closure models the energy cascade process in a turbulent flow.  
In order to perform LES simulation, the Kolmogorov length scale and the associated Kolmogorov time scale \(t_{\eta}\) have been calculated. The time step utilized for the simulation was set to the Kolmogorov time scale. Calculations of \(\eta\) and \(t_{\eta}\) are shown below. Equation (1) is used to find the Reynolds number, \(Re\), of the flow, where \(\rho\) is density, \(u\) is velocity, \(D\) is characteristic length, and \(\mu\) is dynamic viscosity.  
\[Re=\frac{\rho uD}{\mu} \tag{1}\]  
Equation (2) calculates the time for the flow to travel the length of the fluid domain.  
\[t_{L}=\frac{L}{u} \tag{2}\]  
Using both of those values, Equations (3) and (4) can be applied to obtain the Kolmogorov time and length scales, respectively.  
\[t_{\eta} =[Re]^{-\frac{1}{2}}t_{L} \tag{3}\] \[\eta =[\frac{1}{Re_{L}}]^{\frac{3}{2}}L \tag{4}\]