Ideal Incompressible Fluid Research Articles

Inertial motion of a body in an ideal fluid from the state at rest

The motion of a body in an ideal incompressible fluid flow without vortices in the absence of external forces is considered. It is demonstrated that the body can move inertially from the state at rest if its shape satisfies certain conditions.

Calculation of pressure on an airfoil contour in an unsteady separated flow

Simple formulas for calculating the pressure and the total hydrodynamic reactions acting on an arbitrarily moving airfoil are derived within the framework of the model of plane unsteady motion of an ideal incompressible fluid. Several vortex wakes may be shed from the airfoil owing to changes in velocity circulation around the airfoil contour. Cases with nonclosed and closed contours are considered.

Poisson geometry and first integrals of geostrophic equations

We describe first integrals of geostrophic equations, which are similar to the enstrophy invariants of the Euler equation for an ideal incompressible fluid. We explain the geometry behind this similarity, give several equivalent definitions of the Poisson structure on the space of smooth densities on a symplectic manifold, and show how it can be obtained via the Hamiltonian reduction from a symplectic structure on the diffeomorphism group.

Nonlinear and linear instability of the Rossby-Haurwitz wave

The dynamics of perturbations to the Rossby-Haurwitz (RH) wave is analytically analyzed. These waves, being of great meteorological importance, are exact solutions to the nonlinear vorticity equation describing the motion of an ideal incompressible fluid on a rotating sphere. Each RH wave belongs to a space H 1 ⊕ H n , where H n is the subspace of homogeneous spherical polynomials of degree n. It is shown that any perturbation of the RH wave evolves in such a way that its energy K(t) and enstrophy η(t) decrease, remain constant, or increase simultaneously. A geometric interpretation of variations in the perturbation energy is given. A conservation law for arbitrary perturbations is obtained and used to classify all the RH-wave perturbations in four invariant sets, M − n , M + n , H n , and M 0 n − H n , depending on the value of their mean spectral number χ(t) = η(t)/K(t). The energy cascade of growing (or decaying) perturbations has opposite directions in the sets M − n and M + n due to the hyperbolic dependence between K(t) and χ(t). A factor space with a factor norm of the perturbations is introduced, using the invariant subspace H n of neutral perturbations as the zero factor class. While the energy norm controls the perturbation part belonging to H n , the factor norm controls the perturbation part orthogonal to H n . It is shown that in the set M − n (χ(t) < n(n + 1)), any nonzonal RH wave of subspace H 1 ⊕ H n (n ≥ 2) is Lyapunov unstable in the energy norm. This instability has nothing in common with the orbital (Poincaré) instability and is caused by asynchronous oscillations of two almost coinciding RH-wave solutions. It is also shown that the exponential instability is possible only in the invariant set M 0 n − H n . A necessary condition for this instability is given. The condition states that the spectral number η(t) of the amplitude of each unstable mode must be equal to n(n + 1), where n is the RH wave degree. The growth rate is estimated and the orthogonality of the unstable normal modes to the RH wave are shown in two Hilbert spaces. The instability in the invariant set M + n of small-scale perturbations (χ(t) > n(n + 1)) is still an open problem.

Separating impact on an elliptic cylinder floating on the surface of an ideal incompressible finite-depth fluid

The method of nonlinear integral equations of the Hammerstein type is applied to numerically analyze the problem of the separating impact on an elliptic cylinder floating on the surface of an ideal incompressible finite-depth fluid. The bottom effect and the influence of the kinematic and geometric parameters on the zone of fluid particle separation from the cylinder surface are studied.

Separating impact on an elliptic cylinder floating on the surface of an ideal incompressible finite-depth fluid

Separating impact on an elliptic cylinder floating on the surface of an ideal incompressible finite-depth fluid

Equivalence transformations of the Clebsch equations

We consider the Clebsch equations describing the motion of an ideal incompressible fluid in the absence of mass forces. These equations contain an arbitrary element, namely a function of three variables. We find the equivalence transformations of the variables which act on the arbitrary element and preserve the structure of the equations. The equivalence transformation is pointed out that takes the arbitrary element to zero. Some examples are given.

Nonlinear and linear instability of the Rossby‐Haurwitz wave

AbstractAs is known, the large‐scale dynamics of barotropic atmosphere can approximately be described by the nonlinear barotropic vorticity equation. It is also well known that the Rossby‐Haurwitz (RH) waves, being exact solutions to this equation, represent one of the main features of meteorological fields. Therefore the stability properties of the RH wave are of considerable interest for deeper understanding of the low‐frequency variability of the atmosphere. Many works has been devoted to the barotropic instability of flows on a beta‐plane and a sphere. However, mathematically, the nonlinear stability problem of the RH wave is still far from its complete solution. Indeed, some of the stability results have been obtained numerically, and hence, contain calculation errors. Severe truncation of perturbations used in the spectral stability analysis, though leads to interesting and useful conclusions, does not allow obtaining comprehensive results. The weak point of some analytical nonlinear instability studies consists in using inappropriate norms for perturbations. It should also be noted that a necessary condition for the linear instability of the RH wave was obtained only recently (Skiba, 2000).In the present work, the nonlinear stability of the RH wave in an ideal incompressible fluid on a rotating sphere is analytically studied. Let H(n) be a subspace of homogeneous spherical polynomials of degree n. Mathematically, a RH wave of degree n is the sum of a super‐rotating flow of subspace H(1) and a homogeneous spherical polynomial of subspace H(n). First, we derive a conservation law for arbitrary RH‐wave perturbations which asserts that any perturbation evolves in such a way that its kinetic energy E(t) and enstrophy q(t) decrease, remain constant or increase simultaneously. The law is used to divide all the perturbations into three invariant sets depending on the value of their mean spectral number k(t)=q(t)/E(t) introduced by Fjortoft (1953). These sets are denoted as M where k(t)¡n(n+1) (large‐scale perturbations), N where k(t)¿n(n+1) (small‐scale perturbations), and Z where k(t)=n(n+1) (boundary surface between the sets M and N). Note that Z includes one more invariant set, namely, the subspace H(n). The existence of invariant sets of perturbations allows us to study the RH wave instability in each set separately. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Variational setting of the problem on the oscillations of a fluid in a vessel with an elastic inclusion

The problem on the oscillations of an ideal incompressible fluid in a moving rectangular vessel is studied. One wall of the vessel contains an elastic inclusion. The problem involves two free boundaries—the free surface of the fluid and the surface of the elastic inclusion. It is suggested to solve this problem by using a functional whose variation leads to differential equations with nonlinear kinematic and dynamical conditions on the free boundaries.

Behavior of aerosol particles in a suction bunker of standard design

A model is used for an ideal incompressible fluid with the discrete-vortex method to examine the behavior of a multifraction set of dust particles formed on handling loose iron-ore materials in nonstationary vortex gas flows in a suction bunker with single walls of standard design. Four models are considered for the velocity pattern within the bunker. In all cases, it is found that the large fractions of dust are deposited on the floor of the bunker.

Exact analytical solutions for steady three-dimensional inviscid vortical flows

Vortical flows with an axial (z-axis) swirl and a toroidal circulation (in the (rho,z)-plane) can be observed in a wide range of fluid mechanical phenomena such as flow around rotary machines or natural vortices like tornadoes and hurricanes. In this paper, we obtain exact analytical solutions for a general class of steady systems with such three-dimensional circulating structures. Assuming incompressible ideal fluid, a general single-variable equation, known as the Squire–Long equation, can be constructed which can uniquely describe the velocity fields with steady axial and toroidal circulations. In this paper, we consider the case where this type of flow can be analysed by solving a linear homogeneous partial differential equation. The derived equation resembles the governing equation of the hydrogen problem. As a result, we obtain a quantization relation which is similar to the expression for the quantized energy states in a hydrogen atom.For circulating flows, this formalism provides a complete set of orthogonal basis functions which are regular and localized. Hence, each of the basis solutions can be used as a simplified model for a realistic phenomenon. Moreover, an arbitrary circulating field can be expanded in terms of these orthogonal functions. Such an expansion can be potentially useful in the study of more general vortices. As illustrations, we present a few examples where we solve the linear homogeneous equation to analyse fluid mechanical systems which can be models for circulating flow in confined geometry. First, we consider three-dimensional vortices confined between two parallel planar walls. Our examples include flows between two infinite planar walls, inside and outside a vertical cylinder bounded at the ends by horizontal plates, and in an axially confined annular region. Then we describe the special way in which the basis functions should be superposed so that a complicated steady velocity-field with three-dimensional vortical structures can be constructed. Two such cases are discussed to indicate that the derived solutions can be used for complicated fluid mechanical modelling.

A new doubly discrete analogue of smoke ring flow and the real time simulation of fluid flow

Modelling incompressible ideal fluids as a finite collection of vortex filaments is important in physics (super-fluidity, models for the onset of turbulence) as well as for numerical algorithms used in computer graphics for the real time simulation of smoke. Here we introduce a time-discrete evolution equation for arbitrary closed polygons in 3-space that is a discretization of the localized induction approximation of filament motion. This discretization shares with its continuum limit the property that it is a completely integrable system. We apply this polygon evolution to a significant improvement of the numerical algorithms used in computer graphics.

Hamiltonian description of shear and gravity shear waves in an ideal incompressible fluid

Methods of studying the dynamics of wave disturbances in st;ratified shear flows of an ideal incompressible fluid are considered. The equations governing the motions of interest represent Hamilton equations and are derived by writing the velocity field in terms of Clebsch potentials. Equations written in terms of semi-Lagrangian variables are integrodifferential equations, which make it possible to consider both continuous and discontinuous solutions, as well as the cases where the parameters of the undisturbed medium are step functions. Two dynamic systems are presented. The first, canonical system of equations is most suitable for describing gravity waves in a shear flow in the case where the undisturbed medium is characterized by sharp gradients of density and flow velocity. The simplest model in which disturbances obey this system of equations is the well-known Kelvin-Helmholtz model. The second dynamic system describes, in particular, gravity-shear waves and, in the case of a homogeneous medium, shear waves in a two-dimensional flow. This system is most suitable for studying the dynamics of disturbances in models with sharp gradients of vorticity. On the basis of the approach developed in this study, the problem of the dynamics of disturbances in a flow with a continuous distribution of vorticity in a finite-thickness layer is solved. If the thickness of this layer is small compared to the characteristic wavelength and the gradient of the undisturbed vorticity in this layer is large, the solution has the form of a mode whose frequency is close to the frequency of the shear wave on a vorticity jump that would be obtained by letting the layer’s thickness approach zero. The results obtained allow, in particular, the estimation of the range of validity of finite-layer approximations for models with smooth profiles of flow and density. In addition, these results can be interpreted as the basis for the development of nonlinear aspects of the theory of hydrodynamic stability.

Elementary fluid mechanics

Fundamentals. Fluid Statics. Kinematics of Fluid Motion. Systems, Control Volumes, Conservation of Mass, and The Reynolds Transport Theorem. Flow of an Incompressible Ideal Fluid. The Impulse--Momentum Principle. Flow of a Real Fluid. Similitude, Dimensional Analysis and Normalization of Equations of Motion. Flow in Pipes. Flow in Open Channels. Lift and Drag--Incompressible Flow. Introduction to Fluid Machinery. Flow of Compressible Fluids. Fluid Measurements. Appendices. Index.

On the convergence of a vortical numerical method for three-dimensional Euler equations in Lagrangian coordinates

In the present paper, we investigate the problems of regularization and numerical solution of a three-dimensional vorticity transport equation for an ideal incompressible fluid in Lagrangian coordinates (the Euler equation in the vortex–velocity variables). The numerical discrete vortex method used below was originally developed for modeling plane flows. The mathematical proof of the convergence of the method for the two-dimensional vorticity transport equations was given in [1, 2] for boundary-free domains. Numerical methods for threedimensional problems for flows past bodies in which body surfaces and vortical domains inside the flow are approximated by systems of closed vortical frames or systems of vortical segments joined in closed cells were developed by Belotserkovskii and his students [3, 4]. Here much attention was paid to ensure that the property of vortical lines (lines of rotor of the vector field) to be closed is reflected in a discrete scheme. Obviously, the last requirement leads to some difficulties in practical computations, which are caused by strong stretching of the vortical elements. In the present paper, we consider an alternative numerical scheme with isolated vortical elements. Problems of the approximation of vector fields by such elements as well as their use in practical computations were considered in [5], and the convergence of the numerical velocity fields in the discrete lp norm was proved in [6]. In the present paper, we first perform the regularization of the vorticity transport equations by smoothing the singularity in the integral representation of the velocity fields via the vorticity according to the Biot–Savart law. Under the assumption of the existence of a smooth solution of original equations on some finite time interval, we prove the solvability of the smoothed equations on the same time interval and the convergence of their solutions to the solutions of the original �� ���

Oblique entry of a wedge into an ideal incompressible fluid

A new approach to the solution of the self-similar problem of the entry of a wedge into an ideal fluid at an arbitrary angle to the free surface is proposed. The method is based on the construction of the expressions for the complex velocity and the derivative of the complex potential in a parametric variable domain. An integral and an integro-differential equation are obtained for determining the absolute magnitude and the angle of the velocity vector at the free boundary. The calculated results for the free surface shape, the angles of contact between the free surface and the wedge, and the coefficients of the hydrodynamic forces are presented.

Optimization of the shapes of obstacles in jet-separation flow

The model of an ideal incompressible fluid is used to study the solvability of optimal control problems for the shape of a nozzle which discharges free-boundary fluid flow with and without accounting for gravity (internal aerodynamics) and shape optimization problems for an obstacle with jet separation (external aerodynamics). The qualitative properties of such flows are studied.

수중폭발 충격파와 가스구체 압력파를 함께 고려한 구조물의 동적응답해석

Two typical impact loadings, shock wave and gas bubble pulse, due to UNDEX(UNDerwater EXplosion), should be considered together for the closest response analysis of structure subjected to UNDEX to a reality. Since these two impact loadings have different response time bands, however, their response characteristics of structure are different from each other. It is impossible to consider these effectively under the current computational environment and the mathematical model has not yet been developed. Whereas Hicks model approximates the fluid-structure interaction due to gas bubble pulse as virtual mass effect, treating the flow by the response of gas bubble after shock wave as incompressible ideal fluid contrary to the compressible flow due to shock wave, Geers-Hunter model could make the closest response analysis of structure under UNDEX to a real one as a mathematical model considering the fluid-structure interaction due to shock wave and gas bubble pulse together using acoustic wave theory and DAA(Doubly Asymptotic Approximation). In this study, the application and effectiveness of integrated dynamic response analysis of submerged structure was examined with the analysis of the shock wave and gas bubble pulse together.

To formulating a nonlinear initial-boundary problem of an unsteady separated flow around an airfoil

A general formulation of a nonlinear initial-boundary problem of an unsteady separated flow around an airfoil by an ideal incompressible fluid is considered. The problem is formulated for a complex velocity. Conditions of shedding of vortex wakes from the airfoil are analyzed in detail. The proposed system of functional relations allows constructing algorithms for solving a wide class of problems of the wing theory.

On stability of steady-state plane–parallel shearing flows in a homogeneous in density ideal incompressible fluid

A problem on linear stability of stationary plane–parallel shearing flows in a homogeneous in density inviscid incompressible fluid between two immovable impermeable solid parallel infinite plates is studied. With the use of the direct Lyapunov method it is shown that all sufficient conditions (by Rayleigh, Fjørtoft, Arnol’d) known to this moment for stability of these flows with respect to small plane perturbations are the necessary ones as well. An a priori lower estimate is constructed; the estimate displays exponential in time growth of the considered perturbations if these conditions are not affected. An analytical example of steady-state plane–parallel shearing flows and superimposed small plane perturbations growing in time in accordance with the constructed estimate is given.