Adaptive Filtering for Large Eddy Simulation of Flows with Shocks and its Application to Launch Pad Deflectors

Abstract

During lift-off of launch vehicles jets from nozzles impinge on a wedge deflector which directs the jet away from the rocket along passages. As a high Reynolds number turbulent flow the whole region has a broad spectrum of intense pressure fluctuations which can damage the launch vehicle. The deflected jet emerges from the passages and can emit acoustic radiation that can damage the payload. The low frequency unsteadiness requires a suitable method such as large eddy simulation (LES) for its prediction. LES is a viable method for the study of this flow due to its capacity to accurately compute the large scale unsteadiness at much lower computational expense than direct numerical simulation (DNS). Though DNS can provide accurate solutions, it becomes computationally prohibitive for problems of practical interest. RANS (Reynolds-averaged Navier-Stokes) methods are computationally less intensive but can sometimes yield qual- itatively incorrect solutions, especially when there is separation and/or large scale, low frequency unsteadiness. Since the jets are supersonic, LES of such flows require a method that can handle shocks. In the presence of shocks typical computations can acquire oscil- lations in the solution leading to blow-up especially when higher order methods are used, unless suitable methods are employed to mitigate this problem. The first contribution of this thesis is the development of filter order adaptation in the vicinity of shocks as a method for handling shocks. Although there have been many proposals for shock capturing for inviscid flows which have also been applied to RANS, since these are low-order methods, typically 2nd or 3rd order in general, falling to 1st- order at the shock, there is a need to design methods that complement the higher-order (4th to 8th order), high-resolution scheme used for DNS/LES. Filter-order adaptation allows the underlying numerical scheme for integrating the Navier-Stokes equations to remain the same spatial high-order everywhere. It is also a minor modifi cation of the explicit fi ltering approach to LES which requires a high resolution fi lter to be applied after every time step. In this adaptive fi ltering approach the spatial discretization remained the same everywhere and was 6th-order. The order of the spatial fi lter was reduced from 10 in smooth regions to 2 at gridpoints where a shock was detected. Filter order was increased in steps at neighboring gridpoints. To minimise dissipation due to fi lter adaptation, fi lter was adapted in coordinate directions which were almost normal to shock surface. This was achieved by adapting the lter along a coordinate direction i if ^xi rM=jrMj or ^xi r =jr j > 1= p 3 at that point, where ^xi is the unit vector along the ith coordinate direction. A requirement of the explicit filtering LES is to compute spatial derivative using high resolution schemes. A bi-diagonal split system for a 6th-order compact scheme was derived and its characteristics were con firmed by showing the error fall rate with grid re finement. The anisotropy of error of the numerical scheme was investigated in terms of numerical phase and group speeds and was observed that the anisotropy error was negligibly small for the resolved scales of the LES. The solver was made parallel for distributed memory platform using message passing interface (MPI) with the help of 2DECOMP&FFT library. The scalability of the solver was performed using the supercomputer Cray-XC40 which showed good results. The present approach for LES was validated extensively on several flows with and without shocks. The problems taken for flows without shock are 1-d linear wave prop- agation, 2-d acoustic scattering, vortex advection in curvilinear grid, viscous/inviscid Taylor-Green vortex, and turbulent subsonic jets. These test cases were chosen to demon- strate the effectiveness of the LES method for smooth flows where the shock sensor did not activate fi lter adaptation at any time unintentionally. Test cases of flows with shock include Riemann problems, interaction of 1-d and 2-d plane waves with shocks, underex- panded, overexpanded, and impinging round jets. These cases demonstrate the efficacy of the present method to capture shock at right location and strength without adding extra dissipation in the smooth regions. When turbulent fluctuations traverse a shock, they undergo changes in structure, intensity and length scales. At the same time the shock also becomes undulated due to the incoming turbulence. Before using the adaptive filtering method for LES it is necessary to understand how it performs for flows with shocks. A canonical problem is the interaction of homogeneous isotropic turbulence (HIT) with a normal shock. The second contribution of this thesis is the study of explicit filtering LES of shock turbulence interaction with the adaptive fi ltering. Homogeneous isotropic turbulence was computed

in a precursor simulation for the study of the interaction. Interaction of this turbulence with a normal shock of M=1.5 was studied in another domain where one realization of the turbulence fluctuations was injected at the in flow. Two LES with different transverse grid resolutions and a DNS were performed keeping same axial resolution of the grid for all the cases. The shock drifted slightly faster in LES grids. The jump in pressure across the shock was lower than the corresponding Rankine-Hugoniot jump, similar to findings by other researchers. The pressure jump across the shock in LES and DNS agreed closely. Though there were some differences in the axial Reynolds stresses and vorticity variance, the SPL of pressure fluctuations was same for all the LES and DNS solutions. The pressure spectrum downstream of a shock for LES was in good agreement with the DNS for a large range of large scales which implies LES should be adequate for predicting low frequency content. The third contribution of this thesis is the application of the method developed to study an ideally expanded supersonic jet at M=1.5 and Re=105 impinging on a wedge mounted on a at plate. Two impingement distances Hj=W(1 and 4) where Hj and W represent the distance between nozzle exit to wedge tip and width of the nozzle, respectively, and two wedge angles (90 and 20 ) were considered. Attached and de- tached shock waves were seen for the large and small angle wedges, respectively, near the wedge tip. A turbulent boundary layer developed on the wedge with the larger vertex angle whereas flow separated quickly with the small angle wedge. The interaction of the shock/expansion fan with the boundary layer was evident. The maximum pressure fluctuations were higher for the wedge with small angle. Lighthill sources were dominant in the jet and wall jet regions. Acoustic waves generated at the wedge tip for the large angle wedge with large impingement distance propagated and steepened to form a train of weak shock waves which were propagating sources. The time series of pressure at a location, where these waves pass by, shows the N-shaped variations which is a signature of crackle. As the wave steepened the contribution from the viscous sources became dominant. Low frequency high amplitude peaks were observed for larger separation jets. In summary a new method for LES of flows with shocks was proposed, and tested systematically on a variety of problems. Its suitability was examined by simulating a turbulent flow passing through a nominally normal shock. Next, it was applied to a new type of flow at high Reynolds number, typical of applications where shocks interacting

with structures give rise to potentially damaging low frequency unsteadiness. The method shows excellent promise for predictions needed in high speed flow applications.