Flow Of Ideal Incompressible Fluid Research Articles

On one analytical solution to the problem of flow around a tandem from two profiles by a flow of ideal incompressible fluid

On one analytical solution to the problem of flow around a tandem from two profiles by a flow of ideal incompressible fluid

Steady-State Flows of Ideal Incompressible Fluid with Velocity Pointwise Orthogonal to the Pressure Gradient

Steady-State Flows of Ideal Incompressible Fluid with Velocity Pointwise Orthogonal to the Pressure Gradient

Lagrangian reduction and wave mean flow interaction

How does one derive models of dynamical feedback effects in multiscale, multiphysics systems such as wave mean flow interaction (WMFI)? We shall address this question for hybrid dynamical systems, defined as systems whose motion can be expressed as the composition of two or more Lie-group actions. Hybrid systems abound in fluid dynamics. Examples include: the dynamics of complex fluids such as liquid crystals; wind-driven waves propagating with the currents moving on the sea surface; turbulence modelling in fluids and plasmas; and classical–quantum hydrodynamic models in molecular chemistry. From among these examples, the motivating question here is: How do wind-driven waves produce ocean surface currents?The paper first summarises the geometric mechanics approach for deriving hybrid models of multiscale, multiphysics motions in ideal fluid dynamics. It then illustrates this approach for WMFI in the examples of 3D WKB waves and 2D wave amplitudes governed by the nonlinear Schrödinger (NLS) equation propagating in the frame of motion of an ideal incompressible inhomogeneous Euler fluid flow. The results for these examples tell us that the mean flow in WMFI does not create waves, although it does transport the waves. However, feedback in the opposite direction is possible, since the 3D WKB and 2D NLS wave dynamics discussed here do in fact create circulatory mean flow.

Translationally invariant exact steady flows of gas and fluid

Abstract New exact 3-D steady translationally invariant (or z-invariant) flows of isentropic gas and ideal incompressible fluid are constructed. New exact solutions for steady magnetohydrodynamics equations are derived. We show that Euler equations for steady z-invariant flows of isentropic gas are equivalent to a coupled system of a partial differential equation of second order for the streamfunction ψ(x, y) which contains an arbitrary differentiable function H(ψ) and a transcendental equation connecting gas density ρ(x, y) with function ψ(x, y) and depending on equation of state p(ρ) = cρ γ and on the function H(ψ). We prove that functions ψ(x, y) and ρ(x, y) satisfy a universal nonlinear equation that is independent of equation of state p = p(ρ) and of the function H(ψ).

On the shape of the free-surface problem of an ideal incompressible fluid flow with a singular sink at the top of a triangular ledge at the bottom

On the shape of the free-surface problem of an ideal incompressible fluid flow with a singular sink at the top of a triangular ledge at the bottom

Nonlinear Vibrations of Laminated Cross-Ply Composite Cantilever Plate in Subsonic Air Flow

The nonlinear vibrations and responses of a laminated composite cantilever plate under the subsonic air flow are investigated in this paper. The subsonic air flow around the three-dimensional cantilever rectangle laminated composite plate is considered to be decreasing from the wing root to the wing tip. According to the ideal incompressible fluid flow condition and the Kutta–Joukowski lift theorem, the subsonic aerodynamic lift on the three-dimensional finite length flat wing is calculated by using the Vortex Lattice (VL) method. The finite length flat wing is modeled as a laminated composite cantilever plate based on Reddy’s third-order shear deformation plate theory and the von Karman geometry nonlinearity is introduced. The nonlinear partial differential governing equations of motion for the laminated composite cantilever plate subjected to the subsonic aerodynamic force are established via Hamilton’s principle. The Galerkin method is used to separate the partial differential equations into two nonlinear ordinary differential equations, and the four-dimensional nonlinear averaged equations are obtained by the multiple scale method. Through comparing the natural frequencies of the linear system with different material and geometric parameters, the relationship of 1 : 2 internal resonance is considered. Corresponding to several selected parameters, the frequency-response curves are obtained. The hardening-spring-type behaviors and jump phenomena are exhibited. The influence of the force excitation on the bifurcations and chaotic behaviors of the laminated composite cantilever plate is investigated numerically. It is found that the system is sensitive to the exciting force according to the complicate nonlinear behaviors exhibited in this paper.

Interactions of Coherent Structures on the Surface of Deep Water

We numerically investigate pairwise collisions of solitary wave structures on the surface of deep water—breathers. These breathers are spatially localised coherent groups of surface gravity waves which propagate so that their envelopes are stable and demonstrate weak oscillations. We perform numerical simulations of breather mutual collisions by using fully nonlinear equations for the potential flow of ideal incompressible fluid with a free surface written in conformal variables. The breather collisions are inelastic. However, the breathers can still propagate as stable localised wave groups after the interaction. To generate initial conditions in the form of separate breathers we use the reduced model—the Zakharov equation. We present an explicit expression for the four-wave interaction coefficient and third order accuracy formulas to recover physical variables in the Zakharov model. The suggested procedure allows the generation of breathers of controlled phase which propagate stably in the fully nonlinear model, demonstrating only minor radiation of incoherent waves. We perform a detailed study of breather collision dynamics depending on their relative phase. In 2018 Kachulin and Gelash predicted new effects of breather interactions using the Dyachenko–Zakharov equation. Here we show that all these effects can be observed in the fully nonlinear model. Namely, we report that the relative phase controls the process of energy exchange between breathers, level of energy loses, and space positions of breathers after the collision.

Safety factor for time-dependent axisymmetric flows of barotropic gas and ideal incompressible fluid

The safety factor q(r, z, t) is proved to be a material conservation law for the time-dependent axisymmetric barotropic compressible gas flows and ideal incompressible fluid flows with constant density ρ. Infinite families of conserved quantities connected with the safety factor are derived. The existence of maximal vortex rings and vortex blobs which are frozen into the axisymmetric inviscid gas and fluid flows is demonstrated. A stratification in the space of ideal gas and fluid flows is obtained: if two axisymmetric states of the barotropic gas or fluid with constant density ρ are dynamically connected, then their total numbers of vortex rings must be equal (the same for the total numbers of vortex blobs) and the infinitely many corresponding conserved quantities must coincide.

On the Application of Shallow Water Theory to Modeling Wave Flows with Hydraulic Bores

The basic conservation laws of the shallow water theory are deduced from the multidimensional integral conservation laws of mass and total impulse describing the flow of ideal incompressible fluid over the horizontal bottom. This derivation is based on the concept of the local hydrostatic approximation which generalizes the long wave approximation and is used to justify the application of the shallow water theory to modeling wave flows of fluid with hydraulic bores.

An Inviscid Analogue of the Poiseuille Problem

We consider a plane problem of steady-state motion of an ideal incompressible fluid flow in a channel between two parallel planes under the action of a given pressure drop. The problem is considered in Cartesian coordinates. The formulation is analogous to the well-known Poiseuille problem with the difference that an ideal fluid is considered instead of a viscous one. The non-flow condition is set as the boundary ones on the channel walls. So, that the velocity vector is parallel to the bounding surfaces over the channel walls. The pressure drop is set as a given positive quantity. An approach proposed based on the use of the first integral of the Euler equations while preserving nonlinear terms. We represent the derivation of main relations for the case of 2D steady-state motion of an incompressible fluid. The solution of equations for hydrodynamic characteristics in the form of expansions in powers of the Cartesian coordinates was found out by analytical way. The standard programs of Maple package are used to determine the coefficients of decomposition for some values of defining parameters. As a result expressions for hydrodynamic characteristics are obtained and their features investigated. In particular, zones of recurrent motions and zones of intense vortex motion of fluid were revealed.

Solution of the Ovsyannikov Problem of Two-Dimensional Isothermal Motion of a Polytropic Gas

We study an overdetermined system of partial differential equations which describes the two-dimensional isothermal motion of a polytropic gas. The system is reduced to a passive form and is completely integrated. The resulting solutions are treated as ideal incompressible fluid flows bounded by a free surface or a moving solid wall.

Young Measures Generated by Ideal Incompressible Fluid Flows

In their seminal paper, DiPerna and Majda (Commun Math Phys 108(4):667–689, 1987) introduced the notion of a measure-valued solution for the incompressible Euler equations in order to capture complex phenomena present in limits of approximate solutions, such as persistence of oscillation and development of concentrations. Furthermore, they gave several explicit examples exhibiting such phenomena. In this paper we show that any measure-valued solution can be generated by a sequence of exact weak solutions. In particular this gives rise to a very large, arguably too large, set of weak solutions of the incompressible Euler equations.

Construction of an approximate solution of the problem of a cavitational flow around a disk

An approach to analytical analysis of a steady axisymmetric cavitation flow of a potential ideal incompressible fluid is developed. An approximate analytical solution of Ryabushinskii’s problem for a disk-shaped cavitator is constructed. All integral and local characteristics of the flow calculated by the proposed approximation formulas in a wide range of cavitation numbers are demonstrated to differ insignificantly from the numerical solution of the problem in the exact nonlinear formulation.

Kinematic problem of an ideal incompressible fluid flow around an arbitrarily moving airfoil

An unsteady kinematic problem for arbitrary two-dimensional motion of an airfoil in an ideal incompressible fluid with formation of one and two vortex wakes is solved. The problem is solved by the method of conformal mapping of the flow domain onto a circle exterior; solution singularities in the vicinity of a sharp edge are analyzed, and the initial asymptotics of the solution is taken into account. The calculated results are found to be in good agreement with available experimental data on visualization of the flow pattern. The necessity of correct modeling of the initial stage of vortex-wake formation is demonstrated. A regular flow pattern is found to form after three and more periods of oscillations.

Symmetries and exact solutions of the shallow water equations for a two-dimensional shear flow

This paper considers nonlinear equations describing the propagation of long waves in two-dimensional shear flow of a heavy ideal incompressible fluid with a free boundary. A nine-dimensional group of transformations admitted by the equations of motion is found by symmetry methods. Two-dimensional subgroups are used to find simpler integrodifferential submodels which define classes of exact solutions, some of which are integrated. New steady-state and unsteady rotationally symmetric solutions with a nontrivial velocity distribution along the depth are obtained.

Nonstationary ideal incompressible fluid flows: Conditions of existence and uniqueness of solutions

The problem of formulating minimal conditions on input data that can guarantee the existence and uniqueness of solutions of the boundary value problems describing non-one-dimensional ideal incompressible fluid flow is considered using as an example the initial boundary value problem in a space-time cylinder constructed on a bounded flow domain with the nonpenetration condition on its boundary (which corresponds to fluid flow in a closed vessel). The existence problems are considered only for plane flows, and the uniqueness issues for three-dimensional flows as well. The required conditions are obtained in the form of conditions specifying that the vorticity belongs to definite functional Orlicz spaces. The results are compared with well-known results. Examples are given of admissible types of singularities for which the obtained results are valid, which is a physical interpretation of these results.

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.

Numerical modeling of ideal incompressible fluid flow over a step

Results of calculations of fluid flow over a step located on a channel bottom are given. Numerical modeling is performed for the model of free-boundary potential flows of an ideal incompressible fluid using a finite-difference method with dynamically adaptive grids. The behavior of the free surface in the neighborhood of the step is studied as a function of the incident-flow velocity. The results are compared with experimental data.

On unsteady aerodynamic airfoil loads in the design of a control system for minimizing the effect of wind perturbations using the rudder on the trailing edge of the wing

The problem of minimizing wind perturbations using the control on the trailing edge of the airfoil is considered for an airfoil passing through a short wind gust. It is considered that the airfoil movement is bounded by two degrees of freedom, viz. in pitch and height, and each channel has an elastic spring that simulates torsional and bending stiffness of the wing. The study is held on the basis of the developed mathematical model for unsteady aerodynamic loads (the lift coefficient and the pitching moment) when a thin two-dimensional airfoil is flowed without separation by a flow of ideal incompressible fluid. The results obtained using this mathematical model and the conventional quasi-steady model on the basis of aerodynamic derivatives are compared. It is shown that the stability properties for the quasi-steady and unsteady aerodynamics models differ in a wide range of parameters of the airfoil movement considered. Moreover, taking into account the unsteadiness of aerodynamics when designing a control system allows us to suppress the wind perturbations more efficiently and counts when determining the requirements on the actuator rates necessary to suppress wind gusts of high amplitude.

On solving the problem of an ideal incompressible fluid flow around large-aspect-ratio axisymmetric bodies

Based on Babenko’s fundamental mathematical ideas, principally new (unsaturated) algorithms are developed for the numerical solution of problems of a potential axisymmetric ideal fluid flow around bodies of revolution, in particular, an ellipsoid of revolution with an aspect ratio equal to 1000.