Non-penetration Boundary Conditions Research Articles

Fluid-structure interaction analysis of the propeller-shafting system in a non-uniform wake

The hydroelastic analysis of the propeller-shafting system in the non-uniform wake is a key step for designing a modern propeller-shafting system. The hydroelastic responses of the propeller-shafting system in the wake of ships are predicted by applying a three-dimensional time-frequency combined panel approach in conjunction with the finite element method. A fully non-penetration boundary condition applied on the deformed blade surface is conducted. The correction of both the incoming flow velocities and the normal vectors imposed on the blade surface of the transient vibration and the equilibrium positions is considered. The added-mass and -damping matrices due to strongly coupled fluid-structure interaction are derived. By comparing the present findings to numerical solutions calculated using the commercial tool Ansys Mechanical, the accuracy of the suggested technique is verified. It is observed that the added damping is higher, and the amplitudes of bearing forces are smaller, by applying the fully non-penetration boundary condition compared with the results obtained by imposing the other two simplified non-penetration conditions. This indicates that the designers need to imply the fully non-penetration condition on the deformed surfaces to predict the hydroelastic dynamics of the propeller-shafting system, especially in the case of high skew propellers. In addition, the amplitudes of the exciting forces after considering the fluid and propeller-shaft system interaction can be larger or smaller than those ignoring the interaction. The results depend on the phase difference and the mode shape.

Dynamic responses of elastic marine propellers in non-uniform flows

This paper focuses on the development of a three-dimensional panel method in time and frequency domains combined with the finite element method for analyzing the hydroelastic responses of rotating marine propellers in the wake of ships. A fully non-penetration boundary condition imposed on the deformed blade surface is conducted, in which the corrections of both the incoming flow velocities and the normal vectors imposed on the deformed and undeformed blade surface are taken into account. The added-mass and -damping matrices due to strongly coupled fluid-structure interaction are considered. Results of the present method are compared with experimental data available in the literature. It is observed that the fully non-penetration boundary condition applied on the deformed blade surface should be imposed to predict the unsteady performance of elastic propellers, which is due to the change of the added damping predicted by considering different non-penetration boundary conditions.

Exact Beltrami flows in a spherical shell

Abstract Exact flows of an incompressible fluid satisfying the Beltrami equation inside a spherical shell are constructed in the Cartesian coordinates in terms of elementary functions. Two scale-invariant equations defining two infinite series of eigenvalues λ n and λ ̃ m ${\tilde {\lambda }}_{m}$ of the operator curl in the shell with the nonpenetration boundary conditions on the boundary spheres are derived. The corresponding eigenfields are presented in explicit form and their symmetries are investigated. Asymptotics of the eigenvalues λ n and λ ̃ m ${\tilde {\lambda }}_{m}$ at n, m → ∞ are obtained.

Numerical analysis on dynamic behaviors of coupled propeller-shafting system of underwater vehicles

The control of unsteady bearing forces generated by propellers is of great importance for the reduction of vibration and noise of underwater vehicles. This paper is concerned with the development of a numerical method for analyzing the hydroelastic behaviors of fully coupled marine propeller and shafting system immersed in water. A three-dimensional panel method for fluid modeling combined with a finite element method for modeling of the shafting system is developed. The two sets of equations of the structural system and the fluid are strongly coupled by considering the non-penetration boundary condition on the wet surface of the propeller. A modal reduction technique is employed to determine the hydroelastic responses of the coupled fluid-propeller-shafting system, which overcomes the low numerical efficiency of the method due to the asymmetric added matrices of the propeller. The validity of the proposed method is confirmed by comparing the present results with those solutions obtained from finite element analysis. It is found that for the case of the driving force frequency close to the umbrella mode frequencies of the propeller and longitudinal mode frequenices of the shaft, the coupling of the fluid, propeller, and the shaft must be taken into account for predicting the dynamic response of the system.

Parametric analysis on hydroelastic behaviors of hydrofoils and propellers using a strongly coupled finite element/panel method

This paper analyzed the hydroelastic behaviors of hydrofoils and marine propellers immersed in incompressible, inviscid and irrotational fluids. Strongly coupled fluid–structure interaction analyses were performed using a three-dimensional (3-D) potential-based panel method in conjunction with a 3-D finite element method. The present method is developed for hydroelastic analyses of geometrically complex-shaped propeller blades, and the application of the method to the hydroelastic problems of hydrofoils is straightforward. The parameters dominating the added-mass and -damping matrices of the hydrofoils and propellers are examined. The effects of the translational motion of the hydrofoil and the rotational motion of the propeller on the added-mass and -damping matrices are compared based on different non-penetration boundary conditions and distributions of the inflow velocity. It is found that for hydroelastic analyses of propellers, the reduced frequency, i.e., the ratio of excitation frequency to rotational frequency, is a key parameter for determining the added-mass and -damping matrices of the propellers. The effect of the advance ratio on the added-mass and -damping matrices of the propeller blade depends upon the ratio of excitation frequency to rotational frequency. For hydrofoils, the added-damping matrix is significantly affected by the ratio of the excitation frequency multiplied by chord length and divided by axial inflow velocity.

A boundary thickening-based direct forcing immersed boundary method for fully resolved simulation of particle-laden flows

A boundary thickening-based direct forcing (BTDF) immersed boundary (IB) method is proposed for fully resolved simulation of incompressible viscous flows laden with finite size particles. By slightly thickening the boundary thickness, the local communications between the Lagrangian points on the solid boundary and their neighboring fluid Eulerian grids are improved, based on an implicit direct forcing (IDF) approach and a partition-of-unity condition for the regularized delta function. This strategy yields a simple, yet much better imposition of the non-slip and non-penetration boundary conditions than the conventional direct forcing (DF) technique. In particular, the present BTDF method can achieve a numerical accuracy comparable with other representative improved methods, such as multi-direct forcing (MDF), implicit velocity correction (IVC) and the reproducing kernel particle method (RKPM), while its computing cost remains much lower than them and nearly equivalent to the conventional DF method. The dependence of the optimum thickness value of boundary thickening on the form of the different regularized delta functions is also revealed. By coupling the lattice Boltzmann method (LBM) with BTDF-IB, the robustness of the present BTDF IB method is demonstrated using numerical simulations of creeping flow (Re=0.1), steady vortex separating flow (Re=40) and unsteady vortex shedding flow (Re=200) around a circular cylinder. The accuracy and robustness of the present method for moving particle-laden flows are performed on three benchmark cases, such as free sedimentation of single circular cylinder, DKT sedimentation of two particles and Rayleigh-Taylor instability of 504 particles sedimentation in an enclosed cavity. The computational efficiency between the present BTDF method and the MDF method is also compared at last.

Quasineutral limit of the two-fluid Euler–Poisson system in a bounded domain of [formula omitted

The quasineutral limit of the two-fluid Euler–Poisson system (one for ions and another for electrons) in a bounded domain of R3 is rigorously proved by investigating the existence and the stability of boundary layers. The non-penetration boundary condition for velocities and Dirichlet boundary condition for electric potential are considered.

Investigation of added mass and damping coefficients of underwater rotating propeller using a frequency-domain panel method

For a propeller vibrating in a fluid, the added mass and damping coefficients characterize the hydrodynamic forces and moments acting on the propeller, which are of great importance for evaluation of the vibration behaviors of submerged propellers. The present paper is concerned with the development of a numerical method for predicting the added mass and damping coefficients of a rotating marine propeller immersed in water. The three-dimensional panel method in frequency-domain is employed to establish the strongly coupled fluid-structure interaction models of the propellers to compute the added mass and damping coefficients. The relationship between the added mass and damping matrices due to the whole vibration of a rotating propeller and the local vibrations of the propeller blades is considered. Results of the present method are compared with those experimental and numerical data available in the literature. Very good agreement is achieved. The differences of the added mass and damping coefficients due to propeller vibrations of two types are analyzed. The results show that the added mass and damping coefficients of a submerged rotating propeller are functions of the ratio of oscillation frequency of rigid propeller fv to the blade frequency fb, and the advance ratio. In addition, the non-penetration boundary conditions should be imposed on the deformed blade surface for predicting the added mass and damping coefficients m32, m62, c32 and c62, where m32(c32) and m62(c62) denote mass (damping) coefficients related to the lateral force and bending moment in the z direction induced by the transversal vibration in the y direction. Absolute values of all coefficients in the added mass matrix decrease as the ratio fv/fb is increased, and the absolute values of the coefficients in the added damping matrix increases with an increase in the advance ratio.

Vortex Approach for Downwash and Outwash of Tandem Rotors in Ground Effect

A vortex-based approach is employed to predict the downwash and outwash of a tandem rotor in ground effect and provide an understanding of its wake. The aerodynamic loads of the blades are represented through a panel method, and the behavior of the wake is captured by a viscous vortex particle method. The viscous effects of the ground are accounted for by a viscous boundary model satisfying the no-slip and nonpenetration boundary conditions. The method is first validated for an isolated full-scale Lynx tail rotor and a 172-mm-diameter-scale rotor in ground effect. The results show that the predicted trajectories of the tip vortices and the radial velocity profiles compare favorably with experiments and published computational fluid dynamics results. The results for model CH-47D are then compared with experiments for the downwash and outwash of the tandem rotor. As opposed to the isolated single rotor, a radial outward expansion in the overlapping area is observed, and the peak and the corresponding vertical distance of the velocity maximum of the radial outwash flow for the tandem rotor are larger. Moreover, the rotational direction of the tandem rotor leads to a wake with several vortical interactions resulting in different outwashes on the port and starboard sides.

Liquid Induced Vibrations of Truncated Elastic Conical Shells with Elastic and Rigid Bottoms

A method of estimating natural modes and frequencies of vibrations for elastic shells of revolution conveying a liquid is proposed. The vibration modes of the liquid-filled elastic shells are presented as linear combinations of their own vibration modes without liquid. The explicit expression for fluid pressure is defined using Bernoulli’s integral and potential theory suppositions. Non-penetration, kinematic, and dynamic boundary conditions are applied at the shell walls and on a free liquid surface, respectively. The solution of the hydro-elasticity problem is found out using an effective technique based on coupled finite and boundary element methods. Computational vibration analysis of elastic truncated conical shells with different fixation conditions is accomplished. Sloshing and elastic walls frequencies and modes of liquid-filled truncated conical tanks are estimated. Both rigid and elastic bottoms of shells are considered. Some examples of numerical estimations are provided to testify the efficiency of the developed method

Hydroelastic analysis of underwater rotating elastic marine propellers by using a coupled BEM-FEM algorithm

This paper focuses on the development of a numerical method for analyzing the added mass and damping of a rotating elastic marine propeller. Three-dimensional panel methods in frequency domain combined with the finite element method were employed to study the strongly coupled fluid-structure interactions of the propeller. In order to overcome the computational efficiency problem due to the asymmetric added matrices of the fluid, a mode superposition method in conjunction with Wilson-θ method was employed for calculating the structural responses of the propeller. The validity of the proposed numerical method was confirmed by comparing the present results with experimental data available in the literature and those numerical solutions computed using commercial packages ANSYS and Virtual.Lab Acoustics. The effects of the excitation frequency, inflow velocity, material parameter of propeller and the advance ratio on the added mass and damping of rotating elastic propellers were examined. The results showed that stationary flow may be sufficient for analyzing the wet modes of the propeller at a relatively high excitation frequency, and the non-penetration boundary conditions should be imposed on the deformed blade surface rather than the undeformed surface in the case of relatively lower-frequency excitations. In addition, if the inflow velocity is relatively large, the added damping due to the fluid can significantly affect the unsteady performance of the propeller.

A fast and automatic full-potential finite volume solver on Cartesian grids for unconventional configurations

To meet the requirements of fast and automatic computation of subsonic and transonic aerodynamics in aircraft conceptual design, a novel finite volume solver for full potential flows on adaptive Cartesian grids is developed in this paper. Cartesian grids with geometric adaptation are firstly generated automatically with boundary cells processed by cell-cutting and cell-merging algorithms. The nonlinear full potential equation is discretized by a finite volume scheme on these Cartesian grids and iteratively solved in an implicit fashion with a generalized minimum residual (GMRES) algorithm. During computation, solution-based mesh adaptation is also applied so as to capture flow features more accurately. An improved ghost-cell method is proposed to implement the non-penetration wall boundary condition where the velocity-potential of a ghost cell is modified by an analytic method instead. According to the characteristics of the Cartesian grids, the Kutta condition is applied by specially computing the gradients on Kutta-faces without directly assigning the potential jump to cells adjacent wake faces, which can significantly improve the solution converging speed. The feasibility and accuracy of the proposed method are validated by several typical cases of sub/transonic flows around an ONERA M6 wing, a DLR-F4 wing-body, and an unconventional figuration of a blended wing body (BWB). The validation cases demonstrate a fast convergence with fully automatic grid treatment and computation, and the results suggest its capacity in application for aircraft conceptual design.

CONVERGENCE ANALYSIS ON GIBOU-MIN METHOD FOR THE SCALAR FIELD IN HODGE-HELMHOLTZ DECOMPOSITION

The Hodge-Helmholtz decomposition splits a vector field into the unique sum of a divergence-free vector field (solenoidal part) and a gradient field (irrotational part). In a bounded domain, a boundary condition needs to be supplied to the decomposition. The decomposition with the non-penetration boundary condition is equivalent to solving the Poisson equation with the Neumann boundary condition. The Gibou-Min method is an application of the Poisson solver by Purvis and Burkhalter to the decomposition. Using the L 2 -orthogonality between the error vector and the consistency, the convergence for approximating the divergence-free vector field was recently proved to be O(h 1.5 ) with step size h: In this work, we analyze the convergence of the irrotattional in the decomposition. To the end, we introduce a discrete version of the Poincare inequality, which leads to a proof of the O(h) convergence for the scalar variable of the gradient field in a domain with general intersection property.

Convergence of the 2D Euler-[formula omitted] to Euler equations in the Dirichlet case: Indifference to boundary layers

In this article we consider the Euler-α system as a regularization of the incompressible Euler equations in a smooth, two-dimensional, bounded domain. For the limiting Euler system we consider the usual non-penetration boundary condition, while, for the Euler-α regularization, we use velocity vanishing at the boundary. We also assume that the initial velocities for the Euler-α system approximate, in a suitable sense, as the regularization parameter α→0, the initial velocity for the limiting Euler system. For small values of α, this situation leads to a boundary layer, which is the main concern of this work. Our main result is that, under appropriate regularity assumptions, and despite the presence of this boundary layer, the solutions of the Euler-α system converge, as α→0, to the corresponding solution of the Euler equations, in L2 in space, uniformly in time. We also present an example involving parallel flows, in order to illustrate the indifference to the boundary layer of the α→0 limit, which underlies our work.

EFFECT OF LIQUID VISCOSITY ON INSTABILITY OF HIGH-SPINNING PARTIALLY-FILLED SHELL ROTORS

In this study, the instability of spinning cylindrical shells partially filled with viscous liquid is investigated. Based on the Navier–Stokes equations for the incompressible flow, a 2D model is developed for liquid motion at each section of the cylinder. The governing equations of the cylinder vibrations are obtained based on the first-order shear deformable shell theory. The nonpenetration and no-slip boundary conditions of the flow on the wetted surface of the cylinder relate the liquid motion to the shell vibrations. Also the liquid pressure exerted on the cylinder wall combines the vibrations of the rotary cylinder to the liquid motion. By using the obtained coupled liquid-structure model, the instability conditions of the system are determined. The governing parameters of the rotor vibrations are introduced and their influence on the stability boundaries of the rotor is studied.

An iterative penalty method for a nonlinear problem of equilibrium of a Timoshenko-type plate with a crack

A variational problem of equilibrium of an elastic Timoshenko-type plate in a domain with a slit is considered. A nonpenetration boundary condition in the form of an inequality is specified on the edges of the slit. A penalized equation and an iterative linear equation in integral and differential forms are constructed. Some results on solution convergence and an error estimate are obtained.

Ghost-cell method for analysis of inviscid three-dimensional flows on Cartesian-grids

The present paper deals with the implementation of non-penetration boundary conditions at solid walls for three-dimensional inviscid flow computations on Cartesian grids. The crux of the method is the curvature-corrected symmetry technique (CCST) developed by the present authors for body-fitted grids. The method introduces ghost cells near the boundaries whose values are developed from an assumed flow-field model in vicinity of the wall consisting of a vortex flow, with locally symmetric distribution of entropy and total enthalpy. In three dimensions this procedure is implemented in the so-called “osculating plane”. This method was shown to be substantially more accurate than traditional surface boundary condition approaches. This improved boundary condition is adapted to a Cartesian mesh formulation, which we have termed the “ghost-cell method”. In this approach, all cell centers exterior to the body are computed with fluxes at the six surrounding cell faces, without any cut cell. A multiple-valued point technique is used to compute sharp edges. The merits of the ghost-cell method for three-dimensional inviscid flow computations are established by computing compressible and transonic flows about a sphere, an oblate and a prolate spheroid, a cylindrical wing with an end-plate, the ONERA M6 wing and detailed comparison to body-fitted grid computations and to published data. The computed results show the surface non-penetration condition to be satisfied in the limit of vanishing cell size and the method to be second-order accurate in space. The comparison with body-fitted results proves that the accuracy is comparable to the accuracy of CCST computations on body-fitted grids and remarkably superior to body-fitted computations based on traditional pressure extrapolation, non-penetration boundary conditions. In addition, we prove that the results are independent of the position of the body with respect to the grid. Finally, we show that the ONERA M6 wing results compare very well with published data.

Ghost-Cell Method with far-field coarsening and mesh adaptation for Cartesian grids

Cartesian grid methods for inviscid computational fluid dynamics offer great promise for the development of very rapid conceptual design tools. The present paper deals with a number of new features for Cartesian grid methods which appear to be particularly well suited for this application. A key ingredient is the implementation of non-penetration boundary conditions at solid walls which is based upon the curvature-corrected symmetry technique (CCST) developed by the present authors for body-fitted grids. The method introduces ghost cells near the boundaries whose values are developed from an assumed flow-field model in vicinity of the wall. This method was shown to be substantially more accurate than traditional surface boundary condition approaches. This improved boundary condition has been adapted to a Cartesian mesh formulation, which we termed the Ghost-Cell Method. In this approach, all cell centers exterior to the body are computed with fluxes at the four surrounding cell edges, without any cut cells which complicate other Cartesian mesh methods. Another typical drawback of non-adaptive Cartesian grid methods is that any Cartesian grid clustering near the body must be maintained to the far field boundary. To address this issue, we have introduced a far-field grid coarsening, based on the iblanking approach, which maintains the structured nature of the grid while computing only the active cell centers. In addition, to highlight the advantages of grid adaptation in connection with Cartesian mesh methods, we have introduced a rudimentary procedure which detects the shock position and automatically refines the mesh by locally reducing the grid dimension to one fourth of its original dimension. The merits of the Ghost-Cell Method are established by the computation of the compressible flow about circular cylinders. The results show the surface non-penetration condition to be satisfied in the limit of vanishing cell size and the method to be second-order accurate in space. The results of the far-field grid coarsening indicate that the numerical solutions are unaffected by the coarsening, while the number of the computed cells and the CPU time are reduced to less than 50% of the uncoarsened solution values. The results computed using the mesh adaptation at the shock indicate that the method is effective in reducing the shock transition region thickness, without modifying the flow solution away from the shock. Finally, some test cases with airfoils located at different positions have been considered and the results are proven to be practically independent of the position of the body with respect to the grid. Although this paper is limited to two-dimensional applications, the methodology is well-suited for general three-dimensional geometries, which will be the subject of future research.