If we start with Einstein’s beautiful hypothesis that test particles in gravitational field move along geodesic lines of some Riemannian spaces V4, it is natural to expect that the differential equations for finding the metric tensor gαβ(x) for a given distribution of matter also should be invariant under any transformations at which the geodesic equations remain invariant. However, the geodesi...

In this paper, a Schwarz-Christoffel method for generating two-dimensional grids for a variety of complex internal and external flow configurations based on the numerical integration procedure of the Schwarz-Christoffel transformation has been developed by using Mathematica, which is a general purpose symbolic-numerical-graphical mathematics software. This method is highly accurate (fifth order...

In geometric function theory, generalized elliptic integrals and functions arise from the Schwarz-Christoffel transformation of the upper half-plane onto a parallelogram and are naturally related to Gaussian hypergeometric functions. Certain combinations of these integrals also occur in analytic number theory in the study of Ramanujan’s modular equations and approximations to π. The authors stu...

The flow structure about a segmented projectile in hypersonic turbulent flow has been studied. The Reynoldsaveraged Navier-Stokes equations were solved numerically by application of the implicit monotonized scheme of second-order accuracy, the modified Newton's method, and the Christoffel-Schwarz grid-transformation technique. A strong interference effect has been found. The effect can be chara...

in this paper, a schwarz-christoffel method for generating two-dimensional grids for a variety of complex internal and external flow configurations based on the numerical integration procedure of the schwarz-christoffel transformation has been developed by using mathematica, which is a general purpose symbolic-numerical-graphical mathematics software. this method is highly accurate (fifth order...

We address the problem of two-dimensional heat conduction in a solid slab embedded with an isothermal, symmetric pipe of general cross section. Similar formulations have applications in continuum mechanics and electricity. The main objective of this work is to develop a Shape Optimization algorithm that will reveal the optimal shapes of the pipe such that the conduction rate is maximized or min...

In this paper, two-dimensional turbulent flows in different and complex geometries are simulated by using an accurate grid generation method. In order to analyze the fluid flow, numerical solution of the continuity and Navier-Stokes equations are solved using CFD techniques. Considering the complexity of the physical geometry, conformal mapping is used to generate an orthogonal grid by means of...

Given a quasi-definite linear functional u in the space of polynomials with complex coefficients, let us consider corresponding sequence monic orthogonal (SMOP short) (Pn)n≥0. For canonical Christoffel transformation u˜=(x−c)u SMOP (P˜n)n≥0, we are interested to study relation between u˜ and u(1)˜, where u(1) is for associated first kind (Pn(1))n≥0, u(1)˜=(x−c)u(1) its transformation. This prob...

In this paper, a Schwarz-Christoffel method for generating two-dimensional grids for a variety of complex internal and external flow configurations based on the numerical integration procedure of the Schwarz-Christoffel transformation has been developed by using Mathematica, which is a general purpose symbolic-numerical-graphical mathematics software. This method is highly accurate (fifth order...

In this research we have implemented the Random Vortex Method to calculate velocity fields of fluids inside open cavities in both turbulent and laminar flows. the Random Vortex Method is a CFD method (in both turbulent and laminar fields) which needs the Schwarz-Christoffel transformation formula to map the physical geometry into the upper half plane. In some complex geometries like the flow in...