Inviscid Incompressible Flow

The variation-difference method is a convenient numerical method for shells of complex forms. It is enough when only cinematic boundary conditions are satisfied because the method is based on the principle of Lagrange. Another advantage of the variation-difference method is the better opportunity to create computer programs based on it. For shell analysis in orthogonal coordinate system as well as for shell analysis in principal curvatures the system of equations describing stress-strain state can be simplified. In this paper the difference between analysis in orthogonal coordinate system and analysis in principal curvatures of the surface is considered. The main distinction of the analysis of shells in orthogonal curvilinear coordinate system is the necessity of determination of components which include curvature of torsion of coordinate lines. The addition of these components in the equations of the theory of shells for the coordinate system in principal curvatures gives possibility to analyze shells in common orthogonal coordinate system. In this article shell analysis in orthogonal coordinate system is applied to shells based on normal cyclic surfaces.

The stability of the flow along the accretion line in the Bondi-Hoyle-Lyttleton-type accretion flow is studied. The WKB approximation is used to analyze short wavelength linear perturbations of the axisymmetric steady state flow. The viscosity terms in the gasdynamical equations are included in the analysis. The radial modes are studied both in two- and three-dimensional accretion flow. Since a displaced accretion line cannot be defined in three-dimensional flow, the tangential modes are only analyzed in the two-dimensional flow. The analysis of the perturbed gasdynamical equations are made simpler as a result of the radial and tangential modes being decoupled in the linear regime. As was shown before, the three-dimensional accretion flow is unstable against radial modes. The two-dimensional accretion flow is found to be unstable against tangential modes, as well as against radial modes. Using typical numerical parameters which are in common use, the wavelength which gives the maximum growth rate of a small perturbation is found to be about 0.5 r(a), where r(a) is the accretion radius.

By modifying the Rodi assumption to take account of the influence of flow curvature, a new curvature modified algebraic stress model (CMASM) is developed from the second moment closure in the generalized curvilinear coordinate system. And the explicit form of the ASM, a new curvature modified nonlineark-e model (CMNKE), is derived in the orthogonal curvilinear coordinate system. This new nonlineark-e model is further validated by a numerical simulation of a two-dimensional U-type turnaround duct flow. The results show that the CMNKE can effectively capture the main characteristic of this curvature flow and simulate the damping effect of the shear stress by a convex curvature and the enhancing effect by a concave curvature. So, this model is a rational and effective simplification to the second moment closure.

Compared with 8-node and 4-node isoparametric element, the 9-node isoparametric element is more flexible and higher rate of convergence. Based on state-space techniques, the 9-node isoparametric element of piezothermoelastic material under orthogonal curvilinear coordinate system is established. Finally, the example shows the practicability and accuracy of the 9-node isoparametric element of piezothermoelastic material under orthogonal curvilinear coordinate system.

The paper considers the regularity problem on three-dimensional incompressible Navier-Stokes equations in general orthogonal curvilinear coordinate systems. We establish one regularity criteria of the weak solutions involving only in a vorticity component ω 3 and one a priori estimate on the solution that H 3 u 3 L ∞ 0 , T ; L p ℝ 3 is bounded for 1 ≤ p ≤ ∞ to three-dimensional incompressible Navier-Stokes equations in orthogonal curvilinear coordinate systems. These extent greatly the corresponding results on axisymmetric cylindrical flow.

Microswimmers are encountered in a wide variety of biophysical settings. When interacting with flow fields, they show interesting dynamical features such as hydrodynamic trapping, clustering, and preferential orientation. One important step towards the understanding of such features is to clarify the interplay of hydrodynamic flows with microswimmer motility and shape. Here, we study the dynamics of ellipsoidal microswimmers in a two-dimensional axisymmetric vortex flow. Despite this simple setting, we find surprisingly rich dynamics, which can be comprehensively characterized in the framework of dynamical systems theory. By classifying the fixed-point structure of the underlying phase space as a function of motility and microswimmer shape, we uncover the topology of the phase space and determine the conditions under which microswimmers are trapped in the vortex. For spherical microswimmers, we identify Hamiltonian dynamics, which are broken for microswimmers of a different shape. We find that prolate ellipsoidal microswimmers tend to align parallel to the velocity field, while oblate microswimmers tend to remain perpendicular to it. Additionally, we find that rotational noise allows microswimmers to escape the vortex with an enhanced escape rate close to the system’s saddle point. Our results clarify the role of shape and motility on the occurrence of preferential concentration and clustering and provide a starting point to understand the dynamics in more complex flows.

In Part I, an analytical examination of electromagnetic boundary conditions at the interface between two different media is presented in an orthogonal curvilinear coordinate system which conforms locally and instantaneously to the interface. Subsequently, an original natural coordinate system is introduced to resolve the uncertainty in the choice of an orthogonal curvilinear coordinate system. This natural coordinate system is also appeared to reduce the number of the nonvanishing discontinuity relations, and to determine local and instantaneous natural aspects of the e.m. boundary conditions problem.

The two-dimensional potential flow on the blade surface of a centrifugal impeller can be calculated exactly. But no effective method for practical use to calculate the flow at any point in a centrifugal impeller passage seems to have been developed yet. In this paper, the authors show an approximate, but comparatively simple method applying the velocity function method which was developed previously by one of the authors to facilitate the calculation of the flow on the blade surface. Using a free impeller type apparatus, the authors measured the relative flow behind a centrifugal impeller and compared that result with theoretical one. It has been made clear that at a flow rate larger than about half of shockless entry flow rate, the theory expresses the practical flow pretty accurately.

The paper considers three-dimensional isentropic flows of a polytropic gas under gravity. A system of gas dynamics equations for a polytropic gas is used as a mathematical model. With the polytropic exponent $ \gamma = 7.02 $, this system describes the motion of water. An orthogonal curvilinear coordinate system is introduced into the system. To formulate the problem of the decay of a special discontinuity in the system, a degenerate change of variables is made, namely: the dependent and independent variables swap roles. In the new variables, an initial-boundary value problem with data on the sound characteristic and an additional condition is formulated for the system. The solution to the initial-boundary value problem is constructed in the form of power series. The convergence of the constructed series in the region from the surface of the weak discontinuity to the gas-vacuum boundary inclusive is proved. To determine the law of motion of the gas-vacuum boundary, a quasi-linear system of partial differential equations is written out, which is reduced to a system of ordinary differential equations using a characteristic parameter.

In this study, a time-domain numerical method based on three-dimensional potential flow was developed to analyze the hydrodynamic characteristics of an inclined oscillating-water-column (OWC) wave energy converter (WEC). A finite element method was applied to solve the potential flow around and inside the OWC chamber. A turbine–chamber interaction was considered to take into account the pressure drop inside the OWC chamber, which is a nonlinear function of airflow speed via turbine operation. The instantaneous pressure drop was updated on the free-surface boundary condition inside the chamber in the time-domain to account for the coupling effect between the turbine and the chamber. The present numerical method was verified by comparing it with the model test results. The hydrodynamic characteristics of an inclined OWC chamber in terms of potential flow, such as the water column motion and the three-dimensional flow distribution around the chamber, were investigated. In terms of hydrodynamic performance, the energy conversion efficiency of the chamber showed a nonlinear response characteristic dependent on the incident wave height. In addition, numerical calculations were carried out to clarify the relationship between the main geometric parameters and the hydrodynamic response of the inclined OWC chamber.

This paper is concerned with the stability of steady inviscid flows with closed streamlines. In increasing order of complexity we look at two-dimensional planar flows, poloidal (r, z) flows, and swirling recirculating flows. In each case we examine the relationship between Arnol’d's variational approach to stability, Moffatt's magnetic relaxation technique, and a more recent relaxation procedure developed by Valliset al.We start with two-dimensional (x, y) flows. Here we show that Moffatt's relaxation procedure will, under a wide range of circumstances, produce Euler flows which are stable. The physical reasons for this are discussed in the context of the well-known membrane analogy. We also show that there is a close relationship between Hamilton's principle and magnetic relaxation. Next, we examine poloidal flows. Here we find that, by and large, our planar results also hold true for axisymmetric flows. In particular, magnetic relaxation once again provides stable Euler flows. Finally, we consider swirling recirculating flows. It transpires that the introduction of swirl has a profound effect on stability. In particular, the flows produced by magnetic relaxation are no longer stable. Indeed, we show that all swirling recirculating Euler flows are potentially unstable to the extent that they fail to satisfy Arnol’d's stability criterion. This is, perhaps, not surprising, as all swirling recirculating flows include regions where the angular momentum decreases with radius and we would intuitively expect such flows to be prone to a centrifugal instability. The paper concludes with a discussion of marginally unstable modes in swirling flows. In particular, we examine the extent to which Rayleigh's original ideas on stability may be generalized, through the use of the Routhian, to include flows with a non-zero recirculation.

This chapter starts by formulating the Euler equations and the boundary condition for inviscid flow, namely the impermeability condition on a rigid-body surface. The Bernoulli integral for incompressible flows is derived, as is the Cauchy–Lagrange integral, and Kelvin’s Circulation Theorem is proved. This theorem is used to identify situations where a flow can be treated as potential. Two examples of three-dimensional potential flows are considered: flow past a motionless sphere and flow produced by a sphere moving through stagnant fluid. In this chapter, attention is principally paid to two-dimensional potential flows, for which the flow analysis reduces to finding a suitable analytic function as the complex potential in the complex plane. The method of conformal mapping is discussed in detail and is used to analyse a number of flows, including flows past Joukovskii aerofoils. The chapter concludes by describing Kirchhoff flow, for which purpose free-streamline theory is used.

This study is conducted to investigate the thermal stability of a reactive Hydromagnetic Poiseuille fluid flow through a channel. It is assumed that the reaction is exothermic under different chemical kinetics: Sensitized, Arrhenius and Bimolecular neglecting the concentration of the material. The analytical solutions of the nonlinear dimensionless equations governing the fluid flow are obtained using Adomian Decomposition Method (ADM) together with Pade Approximation technique to determine the thermal stability of the fluid flow and important properties of temperature fields including bi-furcations and thermal criticality conditions are discussed.

A pedagogically pleasing and mnemonically useful method for expressing the velocity and acceleration vectors in orthogonal systems of coordinates is shown. The method relies on expressing the time derivatives of the unit vectors in matrix form. It is also noted that the spatial derivatives of the unit vectors obey a similar rule.

Methods to correct and compute confidence and prediction intervals of models neglecting sub-parameterization heterogeneity – From the ideal toward practice