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In the quest for a high fidelity numerical scheme for CFD it is necessary to satisfy demands on accuracy, conservation, positivity and upwinding. Recently the requirement of rotational invariance has been added to this list. In the present work we are mainly interested in upwinding and rotational invariance of Least Squares Kinetic Upwind Method (LSKUM). The standard LSKUM achieves upwinding by stencil division along co-ordinate axes which is referred to as co-ordinate splitting method. This leads to symmetry breaking and rotational invariance is lost. Thus the numerical solution becomes co-ordinate frame dependent. To overcome this undesirable feature of existing numerical schemes, a new algorithm called KUMARI (Kinetic Upwind Method Avec Rotational Invariance, 'Avec' in French means 'with') has been developed. The interesting mathematical relation between directional derivative, Fourier series and divergence operator has been used effectively to achieve upwinding as well as rotational invariance and hence making the scheme truly or genuinely multidimensional upwind scheme. The KUMARI has been applied to the test case of standard 2D shock reflection problem, flow past airfoils, then to 2D blast wave problem and lastly to 2D Riemann problem (Lax's 3rd test case). The results show that either KUMARI is comparable to or in some cases better than the usual LSKUM.

Least Squares Kinetic Upwind Method (LSKUM), a grid free method based on kinetic schemes has been gaining popularity over the conventional CFD methods for computation of inviscid and viscous compressible flows past complex configurations. The main reason for the growth of popularity of this method is its ability to work on any point distribution. The grid free methods do not require the grid for flow simulation, which is an essential requirement for all other conventional CFD methods. However, they do require point distribution or a cloud of points. Point generation is relatively simple and less time consuming to generate as compared to grid generation. There are various methods for point generation like an advancing front method, a quadtree based point generation method, a structured grid generator, an unstructured grid generator or a combination of above, etc. One of the easiest ways of point generation around complex geometries is to overlap the simple point distributions generated around individual constituent parts of the complex geometry. The least squares grid free method has been successfully used to solve a large number of flow problems over the years. However, it has been observed that some problems are still encountered while using this method on point distributions around complex configurations. Close analysis of the problems have revealed that bad connectivity of the nodes is the cause and this leads to bad connectivity related code divergence. The least squares (LS) grid free method called LSKUM involves discretization of the spatial derivatives using the least squares approach. The formulae for the spatial derivatives are obtained by minimizing the sum of the squares of the error, leading to a system of linear algebraic equations whose solution gives us the formulae for the spatial derivatives. The least squares matrix A for 1-D and 2-D cases respectively is given by (Refer PDF File for equation) The 1-D LS formula for the spatial derivatives is always well behaved in the sense that ∑∆xi2 can never become zero. In case of 2-D problems can arise. It is observed that the elements of the Ls matrix A are functions of the coordinate differentials of the nodes in the connectivity. The bad connectivity of a node thus can have an adverse effect on the nature of the LS matrices. There are various types of bad connectivities for a node like insufficient number of nodes in the connectivity, highly anisotropic distribution of nodes in the connectivity stencil, the nodes falling nearly on a line (or a plane in 3-D), etc. In case of multidimensions, the case of all nodes in a line will make the matrix A singular thereby making its inversion impossible. Also, an anisotropic distribution of nodes in the connectivity can make the matrix A highly illconditioned thus leading to either loss in accuracy or code divergence. To overcome this problem, the approach followed so far is to modify the connectivity by including more neighbours in the connectivity of the node. In this thesis, we have followed a different approach of using weights to alter the nature of the LS matrix A. (Refer PDF File for equation) The weighted LS formulae for the spatial derivatives in 1-D and 2-D respectively are are all positive. So we ask a question : Can we reduce the multidimensional LS formula for the derivatives to the 1-D type formula and make use of the advantages of 1-D type formula in multidimensions? Taking a closer look at the LS matrices, we observe that these are real and symmetric matrices with real eigenvalues and a real and distinct set of eigenvectors. The eigenvectors of these matrices are orthogonal. Along the eigendirections, the corresponding LS formulae reduce to the 1-D type formulae. But a problem now arises in combining the eigendirections along with upwinding. Upwinding, which in LS is done by stencil splitting, is essential to provide stability to the numerical scheme. It involves choosing a direction for enforcing upwinding. The stencil is split along the chosen direction. But it is not necessary that the chosen direction is along one of the eigendirections of the split stencil. Thus in general we will not be able to use the 1-D type formulae along the chosen direction. This difficulty has been overcome by the use of weights leading to WLSKUM-ED (Weighted Least Squares Kinetic Upwind Method using Eigendirections). In WLSKUM-ED weights are suitably chosen so that a chosen direction becomes an eigendirection of A(w). As a result, the multi-dimensional LS formulae reduce to 1-D type formulae along the eigendirections. All the advantages of the 1-D LS formuale can thus be made use of even in multi-dimensions. A very simple and novel way to calculate the positive weights, utilizing the coordinate differentials of the neighbouring nodes in the connectivity in 2-D and 3-D, has been developed for the purpose. This method is based on the fact that the summations of the coordinate differentials are of different signs (+ or -) in different quadrants or octants of the split stencil. It is shown that choice of suitable weights is equivalent to a suitable decomposition of vector space. The weights chosen either fully diagonalize the least squares matrix ie. decomposing the 3D vector space R3 as R3 = e1 + e2 + e3, where e1, e2and e3are the eigenvectors of A (w) or the weights make the chosen direction the eigendirection ie. decomposing the 3D vector space R3 as R3 = e1 + ( 2-D vector space R2). The positive weights not only prevent the denominator of the 1-D type LS formulae from going to zero, but also preserve the LED property of the least squares method. The WLSKUM-ED has been successfully applied to a large number of 2-D and 3-D test cases in various flow regimes for a variety of point distributions ranging from a simple cloud generated from a structured grid generator (shock reflection problem in 2-D and the supersonic flow past hemisphere in 3-D) to the multiple chimera clouds generated from multiple overlapping meshes (BI-NACA test case in 2-D and FAME cloud for M165 configuration in 3-D) thus demonstrating the robustness of the WLSKUM-ED solver. It must be noted that the second order acccurate computations using this method have been performed without the use of the limiters in all the flow regimes. No spurious oscillations and wiggles in the captured shocks have been observed, indicating the preservation of the LED property of the method even for 2ndorder accurate computations. The convergence acceleration of the WLSKUM-ED code has been achieved by the use of LUSGS method. The use of 1-D type formulae has simplified the application of LUSGS method in the grid-free framework. The advantage of the LUSGS method is that the evaluation and storage of the jacobian matrices can be eliminated by approximating the split flux jacobians in the implicit operator itself. Numerical results reveal the attainment of a speed up of four by using the LUSGS method as compared to the explicit time marching method. The 2-D WLSKUM-ED code has also been used to perform the internal flow computations. The internal flows are the flows which are confined within the boundaries. The inflow and the outflow boundaries have a significant effect on these flows. The accurate treatment of these boundary conditions is essential particularly if the flow condition at the outflow boundary is subsonic or transonic. The Kinetic Periodic Boundary Condition (KPBC) which has been developed to enable the single-passage (SP) flow computations to be performed in place of the multi-passage (MP) flow computations, utilizes the moment method strategy. The state update formula for the points at the periodic boundaries is identical to the state update formula for the interior points and can be easily extended to second order accuracy like the interior points. Numerical results have shown the successful reproduction of the MP flow computation results using the SP flow computations by the use of KPBC. The inflow and the outflow boundary conditions at the respective boundaries have been enforced by the use of Kinetic Outer Boundary Condition (KOBC). These boundary conditions have been validated by performing the flow computations for the 3rdtest case of the 4thstandard blade configuration of the turbine blade. The numerical results show a good comparison with the experimental results.

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

An unstructured explicit, Reynolds averaged Navier-Stokes solver is developed to operate on inviscid flows, laminar flows and turbulent flows and one equation Spalart-Allmaras turbulence modeling is implemented to the solver. A finite volume formulation, which is cell-center based, is used for numerical discretization of Navier-Stokes equations in conservative form. This formulation is combined with one-step, explicit time marching upwind numerical scheme that is the first order accurate in space. Turbulent viscosity is calculated by using one equation Spalart-Allmaras turbulence transport equation. In order to increase the convergence of the solver local time stepping technique is applied. Eight test cases are used to validate the developed solver,for inviscid flows, laminar flows and turbulent flows. All flow regimes are tested on NACA-0012 airfoil. The results of NACA-0012 are compared with the numerical and experimental data.

No description available.

This Ph.D. thesis aims to develop numerical models for two-dimensional and three-dimensional shallow water systems over mobile beds. In order to accomplish the goal of this dissertation, the following sub-projects are defined and completed. 1: The first sub-project consists in developing a robust two-dimensional coupled numerical model based on an unstructured mesh, which can simulate rapidly varying flows over an erodible bed involving wet–dry fronts that is a complex yet practically important problem. In this task, the central-upwind scheme is extended to simulation of bed erosion and sediment transport, a modified shallow water system is adopted to improve the model, a wetting and drying scheme is proposed for tracking wet-dry interfaces and stably predict the bed erosion near wet-dry area. The shallow water, sediment transport and bed evolution equations are coupled in the governing system. The proposed model can efficiently track wetting and drying interfaces while preserving stability in simulating the bed erosion near the wet-dry fronts. The additional terms in shallow water equations can improve the accuracy of the simulation when intense sediment-exchange exists; the central-upwind method adopted in the current study shows great accuracy and efficiency compared with other popular solvers; the developed model is robust, efficient and accurate in dealing with various challenging cases. 2: The second sub-project consists in developing a novel numerical scheme for a coupled two-dimensional hyperbolic system consisting of the shallow water equations with friction terms coupled with the equations modeling the sediment transport and bed evolution. The resulting 5*5 hyperbolic system of balance laws is numerically solved using a Godunov-type central-upwind scheme on a triangular grid. A spatially second-order and temporally third-order central-upwind scheme has been derived to discretize the conservative hyperbolic sub-system. However, such schemes need a correct evaluation of local wave speeds to avoid instabilities. To address such an issue, a mathematical result by the Lagrange theorem is used in the proposed scheme. Consequently, a computationally expensive process of finding all of the eigenvalues of the Jacobian matrices is avoided: The upper/lower bounds on the largest/smallest local speeds of propagation are estimated using the Lagrange theorem. In addition, a special discretization of the bed-slope term is proposed to guarantee the well-balanced property of the designed scheme. 3: The third sub-project consists in designing a novel scheme to estimate bed-load fluxes which can produce more accurate results than the previously reported coupled model. Using a pair of local wave speeds different from those used for the flow, a novel wave estimator in conjunction with the central upwind method is proposed and successfully applied to the coupled water-sediment system involving a rapid bed-erosion process. It was demonstrated that, in comparison with the decoupled model, applying the proposed novel scheme to approximate the bed-load fluxes can successfully avoid the numerical oscillations caused by simple and less stable schemes, e.g. simple upwind methods; in comparison with the coupled model using same flux-estimator for both hydrodynamic and morphological systems, the proposed numerical scheme successfully prevents excessive numerical diffusion for prediction of bed evolution. Consequently, the proposed scheme has advantages in terms of accuracy which are shown in several numerical tests. In addition, analytical expressions have been provided for calculating the eigenvalues of the coupled shallow-water-Exner system, which greatly enhances the efficiency of the proposed method. 4: The fourth sub-project consists in developing a three-dimensional numerical model for the simulation of unsteady non-hydrostatic shallow water flows on unstructured grids using the finite volume method. The free surface variations are modeled by a characteristics-based scheme which simulates sub- and super-critical flows. Three-dimensional velocity components are considered in a collocated arrangement with a sigma coordinate system. A special treatment of the pressure term is developed to avoid the water surface oscillations. Convective and diffusive terms are approximated explicitly, and an implicit discretization is used for the pressure term. The unstructured grid in the horizontal direction and the sigma coordinate in the vertical direction facilitate the use of the model in complicated geometries. 5: The fifth sub-project consists in developing a well-balanced three-dimensional shallow water model which is able to simulate shock waves over dry bed. Due to the hydrostatic simplification of the vertical momentum equation, the governing system of equations is not hyperbolic and can not be solved using standard hyperbolic solvers. That is, one can not use a high-order Godunov-type scheme to compute all fluxes through cell-interfaces. This may cause the model to fail in simulations of some unsteady-flows with discontinuities, e.g., dam-break flows and floods. To overcome this difficulty, a novel numerical scheme for the three-dimensional shallow water equations is proposed using a relaxation approach in order to convert the system to a hyperbolic one. Thus, a high-order Godunov-type central-upwind scheme based on the finite volume method can be applied to approximate the numerical fluxes. The proposed model can also preserve the ``lake at rest'' state and positivity of water depth over irregular bottom topographies based on special reconstruction of the corresponding parameters. 6: The sixth sub-project consists in extending the result of the fifth sub-project to development of a three-dimensional numerical model for shallow water flows over mobile beds, which is able to simulate morphological evolutions under shock waves, e.g. dam-break flows. The hydrodynamic model solves the three-dimensional shallow water equations using a finite volume method on prismatic cells in sigma coordinates based on the scheme prposed in sub-project 5. The morphodynamic model solves an Exner equation consisting of bed-load sediment transportation. The performance of the proposed model has been demonstrated by several laboratory experiments of dam-break flows over mobile beds.

Dans le contexte du futur système de gestion du trafic aérien, un des objectifs consiste à réduire l’impact environnemental du trafic aérien. Pour respecter ce but, le concept de “free-route”, introduit dans les années 1990, semble bien adapté aujourd’hui. Les avions ne seraient plus contraints à voler le long de routes aériennes, mais pourraient suivre des trajectoires optimales en terme de consommation. L’objectif de cette thèse est d’introduire une nouvelle méthode de planification du trafic à l’horizon pré-tactique avec des objectifs quelques fois contradictoires, c’est-à-dire avec pour but de minimiser la consommation ou de façon équivalente la durée de trajet en tenant compte des conditions météorologiques et de minimiser l’encombrement de l’espace aérien.La méthode a été mise au point en deux étapes. La première étape a été consacrée au calcul d’une seule trajectoire optimale en terme de temps de vol en tenant compte du vent et de contraintes celles des zones interdites de survol. Cette optimisation est basée sur une adaptation de l’algorithme Ordered Upwind. La deuxième étape introduit un algorithme hybride développé, basé sur un algorithme de recuit simulé et sur l’algorithme déterministe développé dans la première étape, afin de minimiser un compromis entre la congestion et la consommation. L’algorithme combine ainsi la capacité d’atteindre la solution optimale globale via une recherche locale qui permet d’accélérer la convergence.Des simulations numériques avec des prévisions de vent sur du trafic européen donnent des résultats encourageants qui démontrent que la méthode globale est à la fois viable et bénéfique en terme du temps de vol total comme de la congestion globale donc de la diminution des conflits / In the context of the future Air Traffic Management system (ATM), one objective is to reduce the environmental impact of air traffic. With respect to this criterion, the “freeroute” concept, introduced in the mid 1990’s, is well suited to improve over nowadays airspace based ATM. Aircraft will no longer be restricted to fly along airways and may fly along fuel-optimal routes. The objective of this thesis is to introduce a novel pretactical trajectory planning methodology which aims at minimizing airspace congestion while taking into account weather conditions so as to minimize also fuel consumption.The development of the method was divided in two steps. The first step is dedicated to compute a time-optimal route for one aircraft taking into account wind conditions. This optimization is based on an adaptation of the Ordered Upwind Method on the sphere.The second step introduces a hybrid algorithm, based on simulated annealing and on the deterministic algorithm developed in the first step, in order to minimize congestion. Thus the algorithm combines the ability to reach a globally-optimal solution with a local-search procedure that speeds up the convergence.

No / Owing to unpredictable bed topography conditions in natural shallow flows, various numerical methods have been developed to improve the treatment of source terms in the shallow water equations. The surface gradient method is an attractive approach as it includes a numerically simple approach to model flows over topographically-varied channels. To further improve the performance of this method, this study deals with the numerical improvement of the shallow-flow source terms. The so-called surface gradient upwind method (SGUM) integrates the source term treatment in the inviscid discretization scheme. A finite volume model (FVM) with the monotonic upwind scheme for conservative laws is used. The Harten–Lax–van Leer-contact approximate Riemann solver is used to reconstruct the Riemann problem in the FVM. The proposed method is validated against published analytical, numerical, and experimental data, indicating that the SGUM is robust and treats the source terms in different flow conditions well.

With increasing demand for computational solutions of fluid dynamical problems, researchers around the world are working on the development of highly robust numerical schemes capable of solving flow problems around complex geometries arising in Aerospace engineering. Also considerable time and effort are devoted to development of convergence acceleration devices, for reducing the computational time required for such numerical solutions. Reduction in run times is very vital for production codes which are used many times in design cycle. In this present work, we consider a numerical scheme called LSKUM capable of operating on any arbitrary distribution of points. LSKUM is being used in CFD center (IIsc) and DRDL (Hyderabad) to compute flows around practical geometries and presently these LSKUM based codes are explicit- It has been observed already by the earlier researchers that the explicit schemes for these methods are robust. Therefore, it is absolutely essential to consider the possibility of accelerating explicit LSKUM by making it LSKUM-Implicit. The present thesis focuses on such a study. We start with two kinetic schemes namely Least Squares Kinetic Upwind Method (LSKUM) and LSKUM based on entropy variables (q-LSKUM). We have developed the following two implicit schemes using LSKUM and q-LSKUM. They are (i)Non-Linear Iterative Implicit Scheme called LSKUM-NII. (ii)Linearized Beam and Warming implicit Scheme, called LSKUM-BW. For the purpose of demonstration of efficiency of the newly developed above implicit schemes, we have considered flow past NACA0012 airfoil as a test example. In this regard we have tested these implicit schemes for flow regimes mentioned below •Subsonic Case: M∞ = 0.63, a.o.a = 2.0° •Transonic Case: M∞ = 0.85, a.o.a = 1.0° The speedup of the above two implicit schemes has been studied in this thesis by operating them on different grid sizes given below •Coarse Grid: 4074 points •Medium Grid: 8088 points •Fine Grid: 16594 points The results obtained by running these implicit schemes are found to be very much encouraging. It has been observed that these newly developed implicit schemes give as much as 2.8 times speedup compared to their corresponding explicit versions. Further improvement is possible by combining LKSUM-Implicit with modern iterative methods of solving resultant algebraic equations. The present work is a first step towards this objective.