https://etd.iisc.ac.in/handle/2005/49 2026-04-15T21:26:43Z https://etd.iisc.ac.in/handle/2005/8148 A Neumerica study of laminar compressible boundary-layer problems Krishnaswami, R This thesis presents some investigations on certain problems in laminar compressible boundary-layer theory. It consists of an introductory chapter and three main chapters. A brief survey of the literature relevant to the problems considered and contributions made by the author are given at the beginning of each chapter. Chapter I is an introduction to the compressible boundary-layer theory in general. Here, several numerical techniques have also been discussed with special reference to the two-point boundary-value problems. Further, the application of the similarity variable on partial differential equations is briefly discussed. Chapter II: Axisymmetric Laminar Boundary-Layer Flow Problems Here, the role of different parameters characterizing the flow, namely, the mass injection, magnetic field, time-dependent free stream velocity and transverse curvature has been studied under three sub-divisions: Part A, Part B and Part C. Part A: The axisymmetric compressible boundary-layer flow with an applied magnetic field has been studied, with massive blowing. The difficulty arising due to massive blowing has been overcome by employing the efficient numerical method of quasilinearization in combination with finite-difference scheme. In the analysis, a realistic gas model has been employed. Further, the effect of high acceleration on the axisymmetric compressible boundary-layer flow with variable gas properties has been studied in the presence of an applied magnetic field. The difficulty that usually arises due to large acceleration is also overcome by using the above method of quasilinearization in combination with finite-difference scheme. Part B: The axisymmetric compressible boundary-layer flow with time-dependent free stream velocity and wall temperature has been studied with an applied magnetic field, variable gas properties and (time-dependent) surface mass transfer (injection and suction). Assuming two time-dependent free stream velocity distributions (constantly accelerated free stream and fluctuating free stream with a steady mean), solutions were obtained by the method of an implicit finite-difference scheme. Part C: The simultaneous effect of large injection in the presence of transverse curvature of an axisymmetric compressible boundary-layer flow has been studied with variable gas properties. The difficulty arising due to large injection in the presence of transverse curvature has been eliminated by the method of quasilinearization in combination with finite-difference scheme. Further, the effect of an applied magnetic field on the axisymmetric compressible boundary-layer flow with transverse curvature effect has also been examined. Thus, Chapter II deals with MHD compressible boundary-layer flow on the axisymmetric body with large surface mass injection, unsteady free stream velocity and transverse curvature effects. Chapter III: Three-Dimensional Stagnation-Point Boundary-Layer Flow There are two sub-divisions, namely Part A and Part B. Part A: The effect of massive blowing on the compressible laminar boundary-layer flow over a general three-dimensional stagnation-point body has been studied, for nodal and saddle point flows. In the nodal point flow, solutions were obtained using the method of quasilinearization with finite-difference scheme. In the saddle-point region, many methods including the above failed to work due to the reverse flow nature of one of the velocity components. This difficulty here is overcome by applying the method of parametric differentiation in combination with finite-difference scheme. Part B: The effect of large surface mass transfer (injection and suction) on the three-dimensional compressible stagnation-point boundary-layer flow with its second-order boundary-layer effects arising due to the curvatures of the body, boundary-layer displacements, vorticity interaction, velocity slip and temperature jump, has been investigated for nodal and saddle point flows with variable properties and cold and hot wall conditions. After solving the first-order boundary-layer equations as has been done in Part A, the second-order boundary-layer equations have been solved by the method of an implicit finite-difference scheme. Thus, Chapter III deals with three-dimensional stagnation-point boundary-layer flow with first and second-order effects and large mass transfer. Chapter IV: Non-Similar Compressible Boundary-Layer Flows There are three sub-divisions, namely Part A, Part B and Part C to study the non-similar behavior of the compressible boundary-layer flows. Part A: The axisymmetric and two-dimensional compressible non-similar boundary-layer flows were studied from the origin of the streamwise coordinate to the exact point of separation. The difficulties arising at the starting point of the streamwise coordinate and at the exact point of separation have been overcome by employing the efficient method of quasilinearization in combination with finite-difference scheme. Part B: The non-similar nature of the three-dimensional boundary-layer flow over a yawed cylinder has been studied from the starting point of the streamwise coordinate to the exact point of separation. The difficulties arising at the starting point of the streamwise coordinate and at the point of separation are overcome by using the same method as in Part A. Part C: For the sake of completeness, the non-similar boundary-layer flow over a flat plate was also discussed. Here, the shock wave propagation behind the moving boundary-layer over a perforated wall has been analyzed with variable properties using the implicit finite-difference scheme. The resulting algebraic equations have been solved using the tridiagonal matrix elimination method. Thus, in Chapter IV, the non-similar compressible boundary-layer flows have been analyzed over the two-dimensional and axisymmetric bodies, a yawed cylinder and a flat plate. The books and original papers referred to in the text of the thesis are enlisted at the end of each chapter. Figures and tables relevant to each chapter are presented at the end of the chapter. The thesis is partly based on the following paper: Hypersonic stagnation-point boundary layers with massive blowing in the presence of a magnetic field (with G. Nath), The Physics of Fluids, Vol. 22, No. 9, Sept. 1979, pp. 1631–1658. Papers based on the remaining work reported in the thesis will be communicated for publication shortly. https://etd.iisc.ac.in/handle/2005/7536 A Study and development of numerical techniques for the solution of integral equations Hamsapriye We have shown that given any function f(x), there exists a unique f?(x), as defined by the relation (2.1), which exactly coincides with f(x) at the points jh. By using the package MATHEMATICA, we can choose a k, such that kh is not a zero of the function. Also, the interpolation function has been expressed as a polynomial of degree n and two correction terms. In the particular choice of U?(kx) = cos(kx) and U?(kx) = sin(kx), the results derived in this chapter tend to the mixed trigonometric interpolation formula. Thus, we have generalized the concept of mixed interpolation (see Chakrabarti and Hamsapriye [18]), by generalizing the operator of Meyer et al. [80]. We have supported the work through several numerical examples, with different choices of U?(kx) and U?(kx). The derivation given in section 3.2 and the error analysis given in section 3.3 hold for any arbitrary U?(kx) and U?(kx), which are the two linearly independent solutions of a second-order linear ODE. In Chapter 4, we have derived the Newton-Cotes quadrature formulae, based on the newly derived generalized mixed interpolation formula. Our main aim in this chapter is to derive the various quadrature rules, which integrate exactly a linear combination of a polynomial up to a certain degree and two other functions U?(kx) and U?(kx). We have established the (n+1)-point generalized Newton-Cotes quadrature formulae, which we have called GMNCF, of the closed type. We remark that the open-type formulae can also be derived on the same lines, as done for the closed type. These formulae (both open and closed) are obtained by replacing the integrand by the mixed interpolation function of the form f?(x) = aU?(kx) + bU?(kx) + ? c?x?, based on the equally spaced grid points x? = jh. It is to be noted that the quadrature formulae derived in section 4.2 and the error analysis given in section 4.3 are independent of the choice of the functions U?(kx) and U?(kx). We have worked out three examples, with two sets of functions, which are the two linearly independent solutions of linear second-order ODEs y''(x) + k²[(kx + 1)²]y(x) = 0 and y''(x) - 2ky'(x) + 2k²y(x) = 0, and which also satisfy the requirement that lim h ? 0 = Ch², for some non-zero constant C. The tables show the validity of the theory of generalized modified quadrature formulae. In the next chapter, i.e., Chapter 5, we have discussed the derivation of various Gregory quadrature rules, based on the generalized mixed interpolation formula of Chapter 3. We have derived the generalized Gregory rules, which are based on the generalized mixed interpolation theory. The interpolation function is of the form U?(kx) + U?(kx) + ? c?x?, which clearly is a combination of a polynomial of a certain degree and two other linearly independent functions U?(kx) and U?(kx). We have made the choice of these functions based on the oscillation theory of ODEs. The generalized Gregory rules are derived by considering the well-known Euler-Maclaurin formula, in which the derivatives have been replaced by the corresponding finite difference formulae. We have derived the generalized Gregory rules associated with both the composite trapezium rule, as well as the composite Simpson’s rule. We have given the error analysis in brief, for both the classes of Gregory quadrature rules. We have also discussed how to choose the appropriate k? and k?’s, which help in controlling the error. We have worked out a few numerical examples, which show the efficiency of the generalized Gregory rules over the other known rules. In these examples, we have worked with two different pairs of the functions U?(kx) and U?(kx) and in both the cases we have obtained better results. In the particular case when U?(kx) = cos(kx) and U?(kx) = sin(kx), we retrieve the results of Bodier et al. [10,11]. Also, in the limiting case as k ? 0, it is verified that the generalized Gregory rules reduce to the corresponding classical Gregory rules. In Chapter 6, we have discussed the utility of these Gregory rules in solving integral equations of the second kind, Fredholm type, following an iterative procedure. We have presented here a numerical method for solving Fredholm integral equations of the second kind. This method is based on the generalized Gregory quadrature rules, which in turn are generalizations of the modified Gregory quadrature rules based on the mixed-trigonometric interpolation formula. The iterative methods are explained and the iterative method II is a modified version of the Fox and Goodwin iterative approach. The condition for the convergence of the iterative scheme is also discussed. The truncation error involved in the approximate solution of the integral equation is also explained. A method of choosing values for the free parameters k? and k? (originally present in the generalized mixed interpolation formula) has also been explained. Several numerical examples are studied, which exhibit both the advantages as well as the disadvantages of the numerical method. Though the method is a little more time-consuming, at times it may give better results. In Chapter 7, we have taken up the study of a singular integro-differential equation, which is of practical interest. We have discussed four different methods for handling the equation. https://etd.iisc.ac.in/handle/2005/5205 Algorithmic and Hardness Results for Fundamental Fair-Division Problems Rathi, Nidhi The theory of fair division addresses the fundamental problem of dividing a set of resources among the participating agents in a satisfactory or meaningfully fair manner. This thesis examines the key computational challenges that arise in various settings of fair-division problems and complements the existential (and non-constructive) guarantees and various hardness results by way of developing efficient (approximation) algorithms and identifying computationally tractable instances. • Our work in fair cake division develops several algorithmic results for allocating a divisible resource (i.e., the cake) among a set of agents in a fair/economically efficient manner. While strong existence results and various hardness results exist in this setup, we develop a polynomial-time algorithm for dividing the cake in an approximately fair and efficient manner. Furthermore, we identify an encompassing property of agents’ value densities (over the cake)—the monotone likelihood ratio property (MLRP)—that enables us to prove strong algorithmic results for various notions of fairness and (economic) efficiency. • Our work in fair rent division develops a fully polynomial-time approximation scheme (FPTAS) for dividing a set of discrete resources (the rooms) and splitting the money (rents) among agents having general utility functions (continuous, monotone decreasing, and piecewise-linear), in a fair manner. Prior to our work, efficient algorithms for finding such solutions were known only for a specific set of utility functions. We com- plement the algorithmic results by proving that the fair rent division problem (under general utilities) lies in the intersection of the complexity classes, PPAD (Polynomial Parity Arguments on Directed graphs) and PLS (Polynomial Local Search). • Our work respectively addresses fair division of rent, cake (divisible), and discrete (in- divisible) goods in a partial information setting. We show that, for all these settings and under appropriate valuations, a fair (or an approximately fair) division among n agents can be efficiently computed using only the valuations of n 1 agents. The nth (secretive) agent can make an arbitrary selection after the division has been proposed and, irrespective of her choice, the computed division will admit an overall fair allocation. https://etd.iisc.ac.in/handle/2005/1927 An Algorithmic Approach To Crystallographic Coxeter Groups Malik, Amita Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. It turns out that the finite Coxeter groups are precisely the finite Euclidean reflection groups. Coxeter studied these groups and classified all finite ones in 1935, however they were known as reflection groups until J. Tits coined the term Coxeter groups for them in the sixties. Finite crystallographic Coxeter groups, also known as finite Weyl groups, play a prominent role in many branches of mathematics like combinatorics, Lie theory, number theory, and geometry. The computational aspects of these groups are of great interests and play a very important role in representation theory. Since it’s enough to study only the irreducible class of groups in order to understand any Coxeter group, we discuss irreducible crystallographic Coxeter groups here. Our goal is to try to deal with some of the fundamental computational problems that arise in working with the structures such as Weyl groups, root system, Weyl characters. For the classical cases, especially type A, many of these problems are not very subtle and have been solved completely. However, these solutions often do not generalize. In this report, our emphasis is on algorithms which do not really depend on the classifications of root systems. The canonical example, we always keep in mind is E8. In chapter 1, we fix the notations and give some basic results which have been used in this report. In chapter 2, we explain algorithms to various Weyl group problems like membership problem; how to find the length of an element; how to check if two words in a Weyl group represent the same element or not; finding the coset representative for an element for a given parabolic subgroup; and list all the expressions possible for an element. In chapter 3, the main goal is to write an algorithm to compute the weight multiplicities of the irreducible representations using Freudenthal’s formula. For this, we first compute the positive roots and dominant weights for a given root system and then finally find the weight multiplicities. We argue this mathematically using the results given in chapter 1. The crystallographic hypothesis is unnecessary for much of what is discussed in chapter 2. In the last chapter, we give codes of the computer programs written in C++ which implement the algorithms described in the previous chapters in this report. 2013-02-14T00:00:00Z