The 14th International Ship Stability Workshop (ISSW), 29th September- 1st October 2014, Kuala Lumpur, Malaysia Design and Construction of Computer Experiments in Fluid Mechanics and Ship Stability Alexander Degtyarev*, Vasily Khramushin and Vladimir Mareev Dept. of Computer Modelling and Multiprocessor Systems, Faculty of Applied Mathematics and Control Processes, St.Petersburg State University, Russia Abstract: The paper considers a generalized functional and algorithmic construction of direct computational experiments in fluid dynamics. Tensor mathematics naturally embedded in the finite- operation in the construction of numerical schemes. As an elementary computing object large fluid particle which has a finite size, its own weight, internal displacement and deformation is considered. The proposed approach focuses on the use of explicit numerical schemes. The numerical solution of the problem is divided into several stages that are a combination of Lagrange and Euler methods. Key words: Fluid mechanics, direct computational experiment, computational efficiency 1. Introduction infinitesimal elements [12], which basically do not allow direct control of internal state of measurable In the paper mathematical basis for direct fluid volumes. At the same time the proposed computational hydroaeromechanic experiment approach differs from the well known smooth particle formation is considered. In contrast to the traditional hydrodynamics (SPH) simulation [13-16], which is a approach of finite difference numerical schemes purely Lagrangian method. For better numerical construction that are output from analytical models in realization we combine Lagrangian and Eulear the form of partial differential equations [1], the approaches at different stages. proposed techniques are focused on the construction Strict and mutually reversible mathematical and use of direct computational experiments. For these definition of properties and description of mechanics purposes fundamental laws of motion [2] are applied of finite fluid volumes transformations are possible to large fluid particles [3], which have a finite size, using classical tools of tensor calculus. This their own weight, internal displacement and instrument sufficiently specifies transformation of deformation [4, 5]. Each particle is represented in complex fluid flows through first-order spatial world (global) and local coordinate systems [6, 7]. It approximations. gives opportunity to examine them as free particles It is shown in the paper that with the proposed with strictly defined laws of neighbor interaction and approach hydromechanics problems can be reduced to with alternation of modeling stages of independent the use of explicit numerical schemes. At the same internal transformation processes [8, 9]. Such time tensor form of state control of three-dimensional modeling is carried out in accordance with the basic computational objects and processes allows to tailor conservation laws of energy, mass and fluid continuity the solution to the real laws of motion or to the [10, 11]. With this approach mathematical description empirical and the asymptotic dependences. Apparatus of physical processes in aerohydrodynamics is greatly of three-dimensional tensor mathematics in a natural enhanced. It is the better than the traditional way is embedded in the finite-difference operations of mathematical models based on differential calculus of © Marine Technology Centre, UTM 187 The 14th International Ship Stability Workshop (ISSW), 29 September-1 October 2014, Kuala Lumpur, Malaysia large particles (final volume) method. This is done in Firstly, equation change-type at its finite-difference a strict and an unambiguous representation of the representation is possible [22, 23]. Secondly, the physical laws in the nearest vicinity of an elementary hydrodynamic nature of the studied phenomena is far particle continuum. In the paper carrying out of from concept of infinitesimals with which we work at numerical experiments in a natural way comes down consideration of any differential equations [12, 17]. In to three conditionally independent physical processes. contrast to the problems of strength and elasticity of This fact, combined with predominant use of explicit solid body, where deflections, shifts, turns may be numerical schemes, enables natural parallel computing considered in the finite-difference representation as with the ability to dynamically select appropriate smalls, shift of particles in continuous medium hydrodynamics laws. This choice is carried out hydrodynamic problems even with a small impact depending on the characteristics of transformation and may be finite. interaction of considered computational objects Thus, as a result we not only have fundamentally (particles). wrong equation as a model, but we often incorrectly In practice, the constructions of direct numerically solve it. Therefore our task, in essence, computational experiments are usually obtained from consists in tearing off calculations from representation close analogs of the numerical schemes from systems of model of physical system. For the solution of this of partial differential equations. However, these problem methods of the direct computing experiment analogs differ as short canonical result expressions in based on the modern computer architecture are the final difference form [17]. For them, the results of developed in the paper. the calculations are more appropriate for comparisons 2. Numerical construction of continuous with physical or full scale experiments than for medium objects analytically accurate but simplistic solutions of classical mathematical physics. Direct numerical experiments in continuum It should be noted that the above considerations are mechanics using digital discrete computers are based not new or unexpected. This work is focused on on a limited set of numeric objects which interpolate overcoming of two "eternal" questions in parameters of the state of the physical fields in time. computational fluid mechanics: Computational processes with such numerical objects 1. incomplete adequacy of the Navier-Stokes have to take place in accordance with physical laws in equations; the mesh areas (including nonregularized ones). At the 2. problems arising in discretization of the equation. same time each mesh cell is represented as The essence of the first question is that the independent corpuscle actively interacting with the Navier-Stokes equation is not closed [19]. Therefore surrounding cell particles [12]. at the solution of these equations in different cases Let us call one mesh cell as elementary various closing ratios are put into practice [1, 22]. computational object (large particle of continuous These ratios have character of conservation laws. medium of finite volume). All internal transformation Thus, the first problem of hydrodynamics is isolation of such particle within linear approximations is strictly of physical model of considered system from the and uniquely determined by the rules of tensor actual situation. arithmetics. This is a convenient tool for geometric The problems arising at discretization of the model and kinematic description of a large particle. Apart equations of hydrodynamics, are also quite serious. from its position in space, classical tensor calculus © Marine Technology Centre, UTM 188 The 14th International Ship Stability Workshop (ISSW), 29 September-1 October 2014, Kuala Lumpur, Malaysia describes more complex transformation: rotation, a – value measured in a local basis, it refers to small compression, elastic deformation etc. Its functional volume or contiguous particles only (differential apparatus is sufficient for development of strong differences, can be scalar, vector or tensor) forward and reverse mathematical description of R,r – values projected on global basis physical processes of fluid mechanics in the finite R,r – values projected on local basis ∧ mesh area. r – local tensor in projections of global system ∨ For description of large mobile elementary particles r – local tensor in projections of local system in a three-dimensional space we introduce two Detailed notation is in appendix 1. coordinate systems: absolute and mobile local With this alphabet, capital letters for values in (associated with the particle) (Fig. 1). global coordinate system are used. Lower letters are used for small quantities at local bases projections in Y r 2 ( y ) a spatial location and current time. Basic mathematical r 3 (z ) r operations are tensors products and products of j R tensors and vectors. They define the ratio of local r (x ) A 1 reactions of the fluid particles to external influences of k the environment. Formally possibility of rank Ω i X increasing of tensor-vector objects is excluded. They Z have not immediate physical interpretation. Fig. 1 – Local basis ri is formed by triad of basis vectors, ijk – Absolute or full velocity vector of a large particle is unit vectors of global coordinate system (XYZ); R – radius introduced as a shift of the center of mass in the global vector of the moving system; A - radius vector of the point in coordinate system: global coordinate system; a - the same point in local → → → → → → V⋅t = ∆ R = t R− 0 R = T −t R− T R (1) coordinate system Tensor of instantaneous velocities relative to the Let us initially restrict our consideration of distant conditional center of large particle in projections on mechanical interference. Then mechanical laws for absolute coordinate system is assembled by direct local