The class of hyperbolic conservation laws model the phenomena of non-linear wave propagation, including the presence and propagation of discontinuities and expansion waves. Such nonlinear systems can generate discontinuities in the so-lution even for smooth initial conditions. Presence of discontinuities results in break down of a solution in the classical sense and to show existence, weak for-mulation of a problem is required. Moreover, closed form solutions are di cult to obtain and in some cases such solutions are even unavailable. Thus, numerical algorithms play an important role in solving such systems. There are several dis-cretization techniques to solve hyperbolic systems numerically and Finite Volume Method (FVM) is one of such important frameworks. Numerical algorithms based on FVM are broadly classi ed into two categories, central discretization methods and upwind discretization methods. Various upwind and central discretization methods developed so far di er widely in terms of robustness, accuracy and ef-ciency and an ideal scheme with all these characteristics is yet to emerge. In this thesis, novel upwind and central schemes are formulated for various hyper-bolic systems, with the aim of maintaining right balance between accuracy and robustness. This thesis is divided into two parts. First part consists of the formulation of upwind methods to simulate genuine weakly hyperbolic (GWH) systems. Such systems do not possess full set of linearly independent (LI) eigenvectors and some of the examples include pressureless gas dynamics system, modi ed Burgers' sys-tem and further modi ed Burgers' system. The main challenge while formulating an upwind solver for GWH systems, using the concept of Flux Di erence Splitting (FDS), is to recover full set of LI eigenvectors, which is done through addition of generalized eigenvectors using the theory of Jordan Canonical Forms. Once the defective set of LI eigenvectors are completed, a novel (FDS-J) solver is for-mulated in such a manner that it is independent of generalized eigenvectors, as they are not unique. FDS-J solver is capable of capturing various shocks such as -shocks, 0-shocks and 00-shocks accurately. In this thesis, the FDS-J schemes are proposed for those GWH systems each of which have one particular repeated eigenvalue with arithmetic multiplicity (AM) greater than one. Moreover, each ux Jacobian matrix corresponding to such systems is similar to a unique Jordan matrix. After the successful treatment of genuine weakly hyperbolic systems, this strategy is further applied to those weakly hyperbolic subsystems which result on employ-ing various convection-pressure splittings to the Euler ux function. For example, Toro-Vazquez (TV) splitting and Zha-Bilgen (ZB) type splitting approaches to split the Euler ux function yield genuine weakly hyperbolic convective parts and strict hyperbolic pressure parts. Moreover, the ux Jacobian of each convective part is similar to a Jordan matrix with at least two lower order Jordan blocks. Based on the lines of FDS-J scheme, we develop two numerical schemes for Eu-ler equations using TV splitting and ZB type splitting. Both the new ZBS-FDS and TVS-FDS schemes are tested on various 1-D shock tube problems and out of two, contact capturing ZBS-FDS scheme is extended to 2-dimensional Euler system where it is tested successfully on various test cases including many shock instability problems. Second part of the thesis is associated with the development of simple, robust and accurate central solvers for systems of hyperbolic conservation laws. The idea of splitting schemes together with the notion of FDS is not easily extendable to systems such as shallow water equations. Thus, a novel central solver Convection Isolated Discontinuity Recognizing Algorithm (CIDRA) is formulated for shallow water equations. As the name suggests, the convective ux is isolated from the total ux in such a way that other ux, in present case other ux represents celerity part, must possess non-zero eigenvalue contribution. FVM framework is applied to each part separately and ux equivalence principle is used to x the coe cient of numerical di usion. CIDRA for SWE is computed on various 1-D and 2-D benchmark problems and extended to Euler systems e ortlessly. As a further improvement, a scalar di usion based algorithm CIDRA-1 is designed for v Euler systems. The scalar di usion coe cient depends on that particular part of the Rankine-Hugoniot (R-H) condition which involves total energy of the system as a direct contribution. This algorithm is applied to a variety of shock tube test cases including a class of low density ow problems and also to various 2-D test problems successfully. vi

Three-dimensional design and three-dimensional CFD analysis of a compressor rotor stage are performed. The design methodology followed is based on a mean line analysis and a radial equilibrium phase. The radial equilibrium is established at a selected number of radii. NACA 65 series airfoils are selected and stacked according to the experimental data available. The CFD methodology applied is based on a three-dimensional, finite difference, compressible flow Euler solver that includes the source terms belonging to rotational motion. The accuracy of the solver is shown by making use of two different test cases. The CFD solution of the designed geometry predicts the static pressure rises and flow turning angles to a good degree of accuracy.

<p>Combustion instability, a complex phenomenon observed in combustion chambers is due to the coupling between heat release and other unsteady flow processes. Combustion instability has long been a topic of interest to rocket scientists and has been extensively investigated experimentally and computationally. However, to date, there is no computational tool that can accurately predict the combustion instabilities in full-size combustors because of the amount of computational power required to perform a high-fidelity simulation of a multi-element chamber. Hence, the focus is shifted to reduced fidelity computational tools which may accurately predict the instability by using the information available from the high-fidelity simulations or experiments of single or few-element combustors. One way of developing reduced fidelity computational tools involves using a reduced fidelity solver together with the flame transfer functions that carry important information about the flame behavior from a high-fidelity simulation or experiment to a reduced fidelity simulation.</p> <p> </p> <p>To date, research has been focused mainly on premixed flames and using acoustic solvers together with the global flame transfer functions that were obtained by integrating over a region. However, in the case of rockets, the flame is non-premixed and distributed in space and time. Further, the mixing of propellants is impacted by the level of flow fluctuations and can lead to non-uniform mean properties and hence, there is a need for reduced fidelity solver that can capture the gas dynamics, nonlinearities and steep-fronted waves accurately. Nonlinear Euler equations have all the required capabilities and are at the bottom of the list in terms of the computational cost among the solvers that can solve for mean flow and allow multi-dimensional modeling of combustion instabilities. Hence, in the current work, nonlinear Euler solver together with the spatially distributed local flame transfer functions that capture the coupling between flame, acoustics, and hydrodynamics is explored.</p> <p> </p> <p>In this thesis, the approach to extract flame transfer functions from high-fidelity simulations and their integration with nonlinear Euler solver is presented. The dynamic mode decomposition (DMD) was used to extract spatially distributed flame transfer function (FTF) from high fidelity simulation of a single element non-premixed flame. Once extracted, the FTF was integrated with nonlinear Euler equations as a fluctuating source term of the energy equation. The time-averaged species destruction rates from the high-fidelity simulation were used as the mean source terms of the species equations. Following a variable gain approach, the local species destruction rates were modified to account for local cell constituents and maintain correct mean conditions at every time step of the nonlinear Euler simulation. The proposed reduced fidelity model was verified using a Rijke tube test case and to further assess the capabilities of the proposed model it was applied to a single element model rocket combustor, the Continuously Variable Resonance Combustor (CVRC), that exhibited self-excited combustion instabilities that are on the order of 10% of the mean pressure. The results showed that the proposed model could reproduce the unsteady behavior of the CVRC predicted by the high-fidelity simulation reasonably well. The effects of control parameters such as the number of modes included in the FTF, the number of sampling points used in the Fourier transform of the unsteady heat release, and mesh size are also studied. The reduced fidelity model could reproduce the limit cycle amplitude within a few percent of the mean pressure. The successful constraints on the model include good spatial resolution and FTF with all modes up to at least one dominant frequency higher than the frequencies of interest. Furthermore, the reduced fidelity model reproduced consistent mode shapes and linear growth rates that reasonably matched the experimental observations, although the apparent ability to match growth rates needs to be better understood. However, the presence of significant heat release near a pressure node of a higher harmonic mode was found to be an issue. This issue was rectified by expanding the pressure node of the higher frequency mode. Analysis of two-dimensional effects and coupling between the local pressure and heat release fluctuations showed that it may be necessary to use two dimensional spatially distributed local FTFs for accurate prediction of combustion instabilities in high energy devices such as rocket combustors. Hybrid RANS/LES-FTF simulation of the CVRC revealed that it might be necessary to use Flame Describing Function (FDF) to capture the growth of pressure fluctuations to limit cycle when Navier-Stokes solver is used.</p> <p> </p> <p>The main objectives of this thesis are:</p> <p>1. Extraction of spatially distributed local flame transfer function from the high fidelity simulation using dynamic mode decomposition and its integration with nonlinear Euler solver</p> <p>2. Verification of the proposed approach and its application to the Continuously Variable Resonance Combustor (CVRC).</p> <p>3. Sensitivity analysis of the reduced fidelity model to control parameters such as the number of modes included in the FTF, the number of sampling points used in the Fourier transform of the unsteady heat release, and mesh size.</p> <p> </p> <p>The goal of this thesis is to contribute towards a reduced fidelity computational tool which can accurately predict the combustion instabilities in practical systems using flame transfer functions, by providing a path way for reduced fidelity multi-element simulation, and by defining the limitations associated with using flame transfer functions and nonlinear Euler equations for non-premixed flames.</p> <p> </p><br>

Analysis of unsteady flows is a very challenging topic of research. A decade ago, potential flow equations were used to predict unsteady pressures on oscillating bodies. Recognising the fact that nonlinear aerodynamics is essential to analyse unsteady flows accurately, particularly in transonic and supersonic flows, different Euler formulations operating on moving grids have emerged recently as important CFD tools for unsteady aerodynamics. Numerical solution of Euler equations on moving grids based on upwind schemes such as the ones due to van Leer and Roe have been developed for the purpose of numerical simulation of unsteady transonic and supersonic flows. In the present work, Euler computations based on yet another recent robust upwind scheme (for steady flows) namely Kinetic Flux Vector Splitting (KFVS) scheme due to Deshpande and Mandal is chosen for further development of a time accurate Euler solver to operate on problems involving moving boundaries. The development of an Euler code based on this scheme is likely to be highly useful to analyse problems of unsteady aerodynamics and computational aeroelasiticity especially when it is noted that KFVS has been found to be an extremely robust scheme for computation of subsonic, transonic, supersonic and hypersonic flows. The KFVS scheme, basically exploits the connection between the linear scalar Boltzmann equation of kinetic theory of gases and the nonlinear vector conservation law, that is, Euler equations of fluid dynamics through moment method strategy. The KFVS scheme has inherent simplicity in splitting the flux even on moving grids due to underlying particle model. The inherent simplicity of KFVS for moving grid problems is due to its relationship with the Boltzmann equation. If a surface is moving with velocity w and a particle has velocity v, then it is quite reasonable to do the splitting based on (v-w)<0 or >0. Only particles having velocity v greater than w will cross the moving surface from left to right and similar arguments hold good for particles moving in opposite direction. It is therefore quite natural to extend KFVS by splitting the Maxwellian velocity distribution at Boltzmann level based on the sign of the normal component of the relative velocity. The relative velocity is the difference between the molecular velocity (v) and the velocity of the moving surface(w). This inherent simplicity of the Kinetic Flux Vector Splitting scheme on Moving Grids (KFMG) method has prompted us to extend the same ideas to 2-D and 3-D problems leading to the present KFMG method. If w is set to zero then KFMG formulation reduces to the one corresponding to KFVS. Thus KFMG formulations axe generalisation of the KFVS formulation. In 2-D and 3-D cases, in addition to the KFMG formulation, the method to move the grids, the appropriate boundary conditions for treating moving surfaces and techniques to improve accuracy in space and time are required to be developed. The 2-D and 3-D formulations based on Kinetic Flux Vector Splitting scheme on Moving Grids method have been developed for computing unsteady flows. Between two successive time steps, the body changes its orientation in case of an oscillation or it deforms when subjected to, aerodynamic loads. In either of these cases the grid corresponding to the first time step has to be moved or regenerated around the displaced or deformed body. There are several approaches available to generate grids around moving bodies. In the present work, the 'spring analogy method' is followed to obtain grid around deflected geometries within the frame work of structured grid. Using this method, the grids are moved from previous time to the current time. This method is capable of tackling any kind of aeroelastic deformation of the body. For oscillating bodies, a suitable boundary condition enforcing the flow tangency on the body needs to be developed. As a first attempt, the body surface has been treated as an 1-D piston undergoing compression and expansion. Then, a more general Kinetic Moving Boundary Condition(KMBC) has been developed. The KMBC uses specular reflection model of kinetic theory of gases. In order to treat fixed outer boundary, Kinetic Outer Boundary Condition(KOBC) has been applied. The KOBC is more general in the sense that, it can treat different type of boundaries (subsonic, supersonic, inflow or out flow boundary). A 2-D cell-centered finite volume KFMG Euler code to operate on structured grid has been developed. The time accuracy is achieved by incorporating a fourth order Runge-Kutta time marching method. The space accuracy has been enhanced by using high resolution scheme as well as second order scheme using the method of reconstruction of fluxes. First, the KFMG Euler code has been applied to standard test cases for computing steady flows around NACA 0012 and NACA 64AQ06 airfoils in transonic flow. For these two airfoils both computational and experimental results are available in literature. It is thus possible to verify (that is, prove the claim that code is indeed solving the partial differential equations + boundary conditions posed to the code) and validate(that is, comparison with experimental results) the 2-D KFMG Euler code. Having verified and validated the 2-D KFMG Euler code for the standard test cases, the code is then applied to predict unsteady flows around sinusoidally oscillating NACA 0012 and NACA 64A006 airfoils in transonic flow. The computational and experimental unsteady results are available in literature for these airfoils for verification and validation of the present results. The unsteady lift and normal force coefficients have been predicted fairly accurately by all the CFD codes. However there is some difficulty about accurate prediction of unsteady pitching moment coefficient. Even Navier-Stokes code could not predict pitching moment accurately. This issue needs further in depth study and probably intensive computation which have not been undertaken in the present study. Next, a two degrees of £reedom(2-DOF) structural dynamics model of an airfoil undergoing pitch and plunge motions has been coupled with the 2-D KFMG Euler code for numerical simulation of aeroelastic problems. This aeroelastic analysis code is applied to NACA 64A006 airfoil undergoing pitch and plunge motions in transonic flow to obtain aeroelastic response characteristics for a set of structural parameters. For this test case also computed results are available in literature for verification. The response characteristics obtained have showed three modes namely stable, neutrally stable and unstable modes of oscillations. It is interesting to compare the value of airfoil-to-air mass ratio (Formula) obtained by us for neutrally stable condition with similar values obtained by others and some differences between them are worth mentioning here. The values of \i for neutral stability are different for different authors. The differences in values of (Formula) predicted by various authors are primarily due to differences which can be due to grid as well as mathematical model used. For example, the Euler calculations, TSP calculations and full potential calculations always show differences in shock location for the same flow problem. Changes in shock location will cause change in pressure distribution on airfoil which in turn will cause changes in values of \L for conditions of neutral stability. The flutter speed parameter(U*) has also been plotted with free stream Mach number for two different values of airfoil - to - air mass ratio. These curves shown a dip when the free stream Mach number is close to 0.855. This is referred as "Transonic Dip Phenomenon". The shock waves play a dominant role in the mechanism of transonic dip phenomenon. Lastly, cell-centered finite volume KFMG 3-D Euler code has been developed to operate on structured grids. The time accuracy is achieved by incorporating a fourth order Runge-Kutta method. The space accuracy has been enhanced by using high resolution scheme. This code has 3-D grid movement module which is based on spring analogy method. The KMBC to treat oscillating 3-D configuration and KOBC for treating 3-D outer boundary have also been formulated and implemented in the code. The 3-D KFMG Euler code has been first verified and validated for 3-D steady flows around standard shapes such as, transonic flow past a hemisphere cylinder and ONERA M6 wing. This code has also been used for predicting hypersonic flow past blunt cone-eylinder-flare configuration for which experimental data are available. Also, for this case, the results are compared with a similar Euler code. Then the KFMG Euler code has been used for predicting steady flow around ogive-cylinder-ogive configuration with elliptical cross section. The aerodynamic coefficients obtained have been compared with those of another Euler code. Thus, the 3-D KFMG Euler code has been verified and validated extensively for steady flow problems. Finally, the 3-D KFMG based Euler code has been applied to an oscillating ogive-cylinder-ogive configuration in transonic flow. This test case has been chosen as it resembles the core body of a flight vehicle configuration of interest to DRDO,India. For this test case, the unsteady lift coefficients are available in literature for verifying the present results. Two grid sizes are used to perform the unsteady calculations using the present KFMG 3-D Euler code. The hysteresis loops of lift and moment coefficients confirmed the unsteady behaviour during the oscillation of the configuration. This has proved that, the 3-D formulations are capable of predicting the unsteady flows satisfactorily. The unsteady results obtained for a grid with size of 45x41x51 which is very close to the grid size chosen in the reference(Nixon et al.) are considered for comparison. It has been mentioned in the reference that, a phase lag of (Formula) was observed in lift coefficients with respect to motion of the configuration for a free stream Mach number of 0.3 with other conditions remaining the same. The unsteady lift coefficients obtained using KFMG code as well as those available in literature are plotted for the same flow conditions. Approximately the same phase lag of (Formula) is present (for (Formula)) between the lift coefficient curves of KFMG and due to Nixon et al. The phase lag corrected plot of lift coefficient obtained by Nixon et al. is compared with the lift coefficient versus time obtained by 3-D KFMG Euler code. The two results compare well except that the peaks are over predicted by KFMG code. It is nut clear at this stage whether our results should at all match with those due to Nixon et al. Further in depth study is obviously required to settle the issue. Thus the Kinetic Flux Vector Splitting on Moving Grids has been found to be a very good and a sound method for splitting fluxes and is a generalisation of earlier KFVS on fixed grids. It has been found to be very successful in numerical simulation of unsteady aerodynamics and computational aeroelasticity.

Least Square Kinetic Upwind Method (LSKUM) belongs to the class of mesh-less method that solves compressible Euler equations of gas dynamics. LSKUM is kinetic theory based upwind scheme that operates on any cloud of points. Euler equations are derived from Boltzmann equation (of kinetic theory of gases) after taking suitable moments. The basic update scheme is formulated at Boltzmann level and mapped to Euler level by suitable moments. Mesh-less solvers need only cloud of points to solve the governing equations. For a complex configuration, with such a solver, one can generate a separate cloud of points around each component, which adequately resolves the geometric features, and then combine all the individual clouds to get one set of points on which the solver directly operates. An obvious advantage of this approach is that any incremental changes in geometry will require only regeneration of the small cloud of points where changes have occurred. Additionally blanking and de-blanking strategy along with overlay point cloud can be adapted in some applications like store separation to avoid regeneration of points. Naturally, the mesh-less solvers have advantage in tackling complex geometries and moving components over solvers that need grids. Conventionally, higher order accuracy for space derivative term is achieved by two step defect correction formula which is computationally expensive. The present solver uses low dissipation single step modified CIR (MCIR) scheme which is similar to first order LSKUM formulation and provides spatial accuracy closer to second order. The maximum time step taken to march solution in time is limited by stability criteria in case of explicit time integration procedure. Because of this, explicit scheme takes a large number of iterations to achieve convergence. The popular explicit time integration schemes like four stages Runge-Kutta (RK4) are slow in convergence due to this reason. The above problem can be overcome by using the implicit time integration procedure. The implicit schemes are unconditionally stable i.e. very large time steps can be used to accelerate the convergence. Also it offers superior robustness. The implicit Lower-Upper Symmetric Gauss-Seidel (LU-SGS) scheme is very attractive due to its low numerical complexity, moderate memory requirement and unconditional stability for linear wave equation. Also this scheme is more efficient than explicit counterparts and can be implemented easily on parallel computers. It is based on the factorization of the implicit operator into three parts namely lower triangular matrix, upper triangular matrix and diagonal terms. The use of LU-SGS results in a matrix free implicit framework which is very economical as against other expensive procedures which necessarily involve matrix inversion. With implementation of the implicit LU-SGS scheme larger time steps can be used which in turn will reduce the computational time substantially. LU-SGS has been used widely for many Finite Volume Method based solvers. The split flux Jacobian formulation as proposed by Jameson is most widely used to make implicit procedure diagonally dominant. But this procedure when applied to mesh-less solvers leads to block diagonal matrix which again requires expensive inversion. In the present work LU-SGS procedure is adopted for mesh-less approach to retain diagonal dominancy and implemented in 2-D and 3-D solvers in matrix free framework. In order to assess the efficacy of the implicit procedure, both explicit and implicit 2-D solvers are tested on NACA 0012 airfoil for various flow conditions in subsonic and transonic regime. To study the performance of the solvers on different point distributions two types of the cloud of points, one unstructured distribution (4074 points) and another structured distribution (9600 points) have been used. The computed 2-D results are validated against NASA experimental data and AGARD test case. The density residual and lift coefficient convergence history is presented in detail. The maximum speed up obtained by use of implicit procedure as compared to explicit one is close to 6 and 14 for unstructured and structured point distributions respectively. The transonic flow over ONERA M6 wing is a classic test case for CFD validation because of simple geometry and complex flow. It has sweep angle of 30° and 15.6° at leading edge and trailing edge respectively. The taper ratio and aspect ratio of the wing are 0.562 and 3.8 respectively. At M∞=0.84 and α=3.06° lambda shock appear on the upper surface of the wing. 3¬D explicit and implicit solvers are tested on ONERA M6 wing. The computed pressure coefficients are compared with experiments at section of 20%, 44%, 65%, 80%, 90% and 95% of span length. The computed results are found to match very well with experiments. The speed up obtained from implicit procedure is over 7 for ONERA M6 wing. The determination of the aerodynamic characteristics of a wing with the control surface deflection is one of the most important and challenging task in aircraft design and development. Many military aircraft use some form of the delta wing. To demonstrate the effectiveness of 3-D solver in handling control surfaces and small gaps, implicit 3-D code is used to compute flow past clipped delta wing with aileron deflection of 6° at M∞ = 0.9 and α = 1° and 3°. The leading edge backward sweep is 50.4°. The aileron is hinged from 56.5% semi-span to 82.9% of semi-span and at 80% of the local chord from leading edge. The computed results are validated with NASA experiments