Nonequilibrium jet boundary layer in a polyatomic gas

The two-dimensional nonequilibrium hypersonic free jet boundary layer gas flow in the near wake of a body is studied using a closed system of macroscopic equations obtained (as a thin-layer version) from moment equations of kinetic origin for a polyatomic single-component gas with internal degrees of freedom. (This model is can be used to study flows with strong violations of equilibrium with respect to translational and internal degrees of freedom.) The solution of the problem under study (i.e., the kinetic model of a nonequilibrium homogeneous polyatomic gas flow in a free jet boundary layer) is shown to be related to the known solution of the well-studied simpler problem of a Navier-Stokes free jet boundary layer, and a method based on this relation is proposed for solving the former problem. It is established that the gas flow velocity distribution along the separating streamline in the kinetic problem of a free jet boundary layer coincides with the distribution obtained by solving the Navier-Stokes version of the problem. It is found that allowance for the nonequilibrium nature of the flow with respect to the internal and translational degrees of freedom of a single-component polyatomic gas in a hypersonic free jet boundary layer has no effect on the base pressure and the wake angle.     

Kinetic model of the Boltzmann equation for a diatomic gas with rotational degrees of freedom

A system of model kinetic equations is proposed to describe flows of a diatomic rarefied gas (nitrogen). A conservative numerical method is developed for its solution. A shock wave structure in nitrogen is computed, and the results are compared with experimental data in a wide range of Mach numbers. The system of model kinetic equations is intended to compute complex-geometry three-dimensional flows of a diatomic gas with rotational degrees of freedom.     

Stabilization of von Kármán plate in the presence of thermal effects in a subsonic potential flow of gas

We discuss the problem of nonlinear oscillations of a clamped plate in the presence of thermal effects in a subsonic gas flow. The dynamics of the plate is described by von Kármán system in the presence of thermal effects, in which rotational inertia is accounted for. To describe influence of the gas flow we apply the linearized theory of potential flows. Our main result states that each weak solution of the problem considered tends to the set of the stationary points of the problem.     

Solution of a kinetic equation for diatomic gas with the use of differential scattering cross sections computed by the method of classical trajectories

A collision integral is constructed taking into account the rotational degrees of freedom of the gas molecules. Its truncation error is shown to be second order in the rotational velocity mesh size. In the solution of the kinetic equation, the resulting collision integral is directly computed using a projection method. Preliminarily, the differential scattering cross sections of nitrogen molecules are computed by applying the method of classical trajectories. The resulting cross section values are tabulated in multimillion data arrays. The one-dimensional problems of shock wave structure and heat transfer between two plates are computed as tests, and the results are compared with experimental data. The convergence of the results with decreasing rotational velocity mesh size is analyzed.     

Analytic Solution of the Kinetic Equation in the Problem of Pointlike Sources in a Diatomic Gas

We obtain an analytic solution of the kinetic equation in the problem of pointlike sources of heat or particles in a gas of diatomic molecules. We analyze the distributions of energy and temperature over separately excited translational and rotational degrees of freedom of molecules.     

Mathematical model of a kinetic boundary layer

The two-dimensional (plane) problem of a hypersonic kinetic boundary layer developing on a thin body in the case of a homogeneous polyatomic gas flow with no dissociation or electron excitation is considered assuming that energy exchange between translational and internal molecular degrees of freedom is easy. (The approximation of a hypersonic kinetic boundary layer arises from the kinetic theory of gases and, within the thin-layer model, takes into account the strong nonequilibrium of the hypersonic flow with respect to translational and internal degrees of freedom of the gas particles.) A method is proposed for constructing the solution of the given kinetic problem in terms of a given solution of an equivalent well-studied classical Navier-Stokes hypersonic boundary layer problem (which is traditionally formulated on the basis of the Navier-Stokes equations).     

A quadrilateral plate bending element based on deformation modes

In this paper, a quadrilateral element is proposed for the analysis of thin plate bending. This element is non-conforming and consists of four-nodes and twelve degrees of freedom. A third-order field for the element displacement is written in terms of the deformation modes. Moreover, the rotational fields are obtained by utilizing the first-order Jacobean matrix. All interpolation functions are explicitly found by the presented formulation. The stiffness matrix of the element is then computed by using these functions. Finally, the accuracy of the suggested element is evaluated by solving some thin plate bending structures. Numerical findings reveal the new quadrilateral element MKQ12 is robust and accurate for analysis of thin plates.     

On the convergence of a dual-primal substructuring method

Summary.   In the Dual-Primal FETI method, introduced by Farhat et al. [5], the domain is decomposed into non-overlapping subdomains, but the degrees of freedom on crosspoints remain common to all subdomains adjacent to the crosspoint. The continuity of the remaining degrees of freedom on subdomain interfaces is enforced by Lagrange multipliers and all degrees of freedom are eliminated. The resulting dual problem is solved by preconditioned conjugate gradients. We give an algebraic bound on the condition number, assuming only a single inequality in discrete norms, and use the algebraic bound to show that the condition number is bounded by for both second and fourth order elliptic selfadjoint problems discretized by conforming finite elements, as well as for a wide class of finite elements for the Reissner-Mindlin plate model. Received January 20, 2000 / Revised version received April 25, 2000 / Published online December 19, 2000     

Symmetric Cauchy stresses do not imply symmetric Biot strains in weak formulations of isotropic hyperelasticity with rotational degrees of freedom.

We show that symmetric Cauchy stresses do not imply symmetric Biot strains in relaxed weak formulations of finite isotropic hyperelasticity with exact rotational degrees of freedom. This is contrary to claims in the literature. It is noted that in the linear isotropic case with skew symmetric infinitesimal rotational degrees of freedom a similar discrepancy does not arise. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Active damping of rotational load oscillations at container cranes by individual rope length actuation

A particular challenge in the field of container crane control are rotations of the load about its vertical axis when neither special rotational actuators nor variable configurations of the rope directions are available. It is possible to enhance damping of rotational load oscillations by changing the rope lengths individually. Under the assumption of rigid ropes, a model representing the approximate dynamical behavior of the oscillator with three degrees of freedom is developed which reflects the nonlinear coupling effects between the degrees of freedom. This model is used for a stabilizing controller design with only the position of the trolley and the rope lengths actuated. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On dynamics of an aeroelastic system with two degrees of freedom

An aeroelastic system with one translational and one rotational degrees of freedom is considered. Evolution of the set of equilibrium positions of the system (including both trivial and oblique ones) is examined depending on parameters of the system. Stability criteria are obtained for these equilibria. Numerical simulation of behavior of the system is performed for different values of parameters, including in the area of large angles of attack. Limit cycles existing in the system and their domains of attraction are studied depending on the flow speed.     

Analytic solution to the problem of the temperature jump in a gas with rotational degrees of freedom

An exact solution is obtained for the first time for the problem of the temperature jump in a gas with allowance for internal (rotational) degrees of freedom. The treatment is based on a model collision integral proposed by the authors. The problem reduces to the solution of a boundary-value problem for a linear vector transport equation with matrix kernel. Separation of the variable leads to a characteristic equation for which eigenvectors are found in a space of generalized functions and the eigenvalue spectrum is investigated. An expansion of the solution to the problem with respect to eigenvectors of the continuous and discrete spectra is established. On the basis of the conditions of solvability of the vector Riemann-Hilbert boundary-value problem which arises in the process of the proof, an exact (in closed form) expression is obtained for the temperature jump.Moscow Pedagogical University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 3, pp. 530–540, June, 1993.     

Approximation of a circular cylindrical shell by Clough-Johnson flat plate finite elements

Summary In this paper, we study the approximation of a right circular cylindrical shell by a nonconforming method using Clough-Johnson flat plate finite elements. Compatibility conditions which have to be satisfied by the degrees of freedom at every node of the triangulation are given. Then, we prove that convergence is insured for shallow shells when using particular families of triangulations which are practically easy to implement. Finally, we propose a new approximation method by flat plate finite elements which assures the convergence for any kind of circular cylindrical shell.     

Smoluchowski's problem for molecular gases taking the accommodation coefficients of the translational and rotational energy of the molecules into account

The effect of the accommodation coefficients of the energy of translational and rotational motion of molecules on the temperature jump in the classical Smoluchowski problem is investigated. Using the extension of the well-known kinetic BKW (Boltzmann-Krook-Welander) equation to the case of molecular gases, an analytical solution of this problem is obtained for diatomic and polyatomic gases taking the rotational degrees of freedom into account. A numerical analysis of the results obtained is carried out.     

Numerical study of the Couette flow of a diatomic rarefied gas

The Couette flow is numerically studied using a model kinetic equation for a diatomic rarefied gas (nitrogen). The boundary condition set on the wall takes into account that the molecular rotational energy passes into translational energy when the molecule interacts with the wall. For comparison purposes, the Couette flow is computed using the classical diffuse model of the gas-wall interaction. A comparison of the results obtained with both types of boundary conditions shows that the computed parameters of the Couette flow coincide only for sufficiently low Knudsen numbers. This suggests that transitions between rotational and translational energy in the gas-wall interaction have to be taken into account in the boundary condition.     

On the stabilizability of a structural acoustic model which incorporates shear effects in the structural component

In this paper we consider a linear three-dimensional structural acoustic model which takes account of displacement, rotational inertia and shear effects in the flat flexible structural component of the model. Thus the deflections of the structural component of the structure are governed by the Reissner–Mindlin plate equations. We show strong stabilization of the coupled model without incorporating viscous or boundary damping in the equations for the gas dynamics and without imposing geometric conditions. It turns out that damping is needed in the interior of the plate, to which end Kelvin–Voigt damping is introduced in the plate equations. As our main tool we use a resolvent criterion for strong stability due to Tomilov.     

Dynamics of a thermoelastic von Kármán plate in a subsonic gas flow

We discuss the problem of non-linear oscillations of a clamped thermoelastic plate in a subsonic gas flow. The dynamics of the plate is described by von Kármán system in the presence of thermal effects. No mechanical damping is assumed. To describe the influence of the gas flow we apply the linearized theory of potential flows. Our main result states that each weak solution of the problem considered tends to the set of the stationary points of the problem. A similar problem was considered in [27], but with rotational inertia accounted for, i.e. with the additional term −αΔu_tt,α > 0, and the same result on stabilization was obtained. There was introduced the decomposition of the solution such that the one term tends to zero and the other is compact in special (“local energy”) topology. This decomposition enables us to prove the main result. But the case of rotational inertia neglected (α = 0) appears more difficult. Low a priori smoothness of u_t in the case α = 0 prevents us to construct such a decomposition. In order to prove additional smoothness of u_t we use analyticity of the corresponding thermoelastic semigroup proved in [25]. The isothermal variant of this problem with additional mechanical damping term −εΔu_t , ε > 0 was considered in [13] and stabilization to the set of stationary solutions to the problem was proved. The problem, considered in the present work can also be regarded as an extension of the result of [18] to the case when gas occupies an unbounded domain.     

On the plane couette flow of a viscous compressible fluid with transpiration cooling

The exact solution for the plane couette flow of a viscous compressible, heat conducting, perfect gas with the same gas injection at the stationary plate and its corresponding removal at the moving plate has been studied. It is found that the gas injection is very helpful in reducing the temperature recovery factor. Effects of injection on the shearing stress at the lower plate, longitudinal velocity profiles and the enthalpy are shown graphically.     

Magnetic effect on the formation of longitudinal vortices in natural convection flow over a rotating heated flat plate

A numerical study of magnetic effect on the formation of longitudinal vortices in natural convection flow over a rotating heated flat plate is presented. The onset position characterized by the local Grashof number, depends on the rotational Reynolds number, the Prandtl number, the Hartmann number, and the wave number. The Coriolis force and the Lonertz force have significant effects on the formation of longitudinal vortices and the associated instability. Positive rotation stabilizes the flow on the rotating flat surface. On the contrary, a negative rotation destabilizes the flow. The flow is found more stable as the value of Hartmann number increases. The numerical data show reasonable agreement with the experimental results with the case of thermal instability in natural convection over a flat plate heated from below.     

Dynamic responses of Euler–Bernoulli beam subjected to moving vehicles using isogeometric approach

Dynamic analysis of beam structures subjected to moving vehicles using an isogeometric Euler–Bernoulli formulation is presented in this paper. The method utilizes B-Splines or Non-Uniform Rational–Splines (NURBS) as the basis functions for both geometric and analysis implementation. The rotation-free technique has been incorporated into the formulation by using only one deflection variable with excluding the rotational degrees of freedom adopted for each control point. Then, it enables to use a few degrees of freedom (Dofs) to achieve a highly accurate solution. The validations of the proposed method included a complicated moving vehicle and rough pavement effects are compared to the precisely analytical results. Compared with most existing methods of finite element method (FEM) and readily analytical solutions, the present technique indicated the effectiveness of present isogeometric method and its well accurate prediction for suitable simulating the interaction model of the bridge structures and complicated vehicles.