Numerical Calculation of Marine Propeller Hydrodynamic Characteristics in Unsteady Flow by Boundary Element Method

In this paper, a low order potential based on surface panel method is used for the analysis of marine propellers in unsteady flow. A linear propeller wake model is employed and its geometry is assumed to be independent of the time. The calculation in time domain is carried out from a moment when the rotation of the propeller becomes steady instead of from the moment when the rotation starts from stationary condition. At every time step a linear algebraic equation established on a key blade is solved numerically combined with the Kutta pressure condition. The calculated results by developed code indicate good convergency and effectiveness of present algorithm for conventional propellers and highly skewed propellers.

Boundary Element Analysis (Laplace Transform Solution) of Groundwater Unsteady Flow to a Multiple Well System in a Confined Aquifer

The calculations of unsteady flow to a multiple well system with the application of boundary elementmethod (BEM) are discussed. The mathematical model of unsteady well flow is a boundary value problem ofparabolic differential equation. It is changed into an elliptic one by Laplace transform to eliminate time varia-ble. The image function of water head H can be solved by BEM. We derived the boundary integral equation ofthe transformed variable H and the discretization form of it, so that there is no need to discretize the bounda-ries of well walls and it becomes easier to solve the groundwater head H by numerical inversion.

A simplified two-dimensional boundary element method with arbitrary uniform mean flow

To reduce computational costs, an improved form of the frequency domain boundary element method（BEM） is proposed for two-dimensional radiation and propagation acoustic problems in a subsonic uniform flow with arbitrary orientation. The boundary integral equation（BIE） representation solves the two-dimensional convected Helmholtz equation（CHE） and its fundamental solution, which must satisfy a new Sommerfeld radiation condition（SRC） in the physical space. In order to facilitate conventional formulations, the variables of the advanced form are expressed only in terms of the acoustic pressure as well as its normal and tangential derivatives, and their multiplication operators are based on the convected Green＇s kernel and its modified derivative. The proposed approach significantly reduces the CPU times of classical computational codes for modeling acoustic domains with arbitrary mean flow. It is validated by a comparison with the analytical solutions for the sound radiation problems of monopole,dipole and quadrupole sources in the presence of a subsonic uniform flow with arbitrary orientation.

DISCONTINUITY-CAPTURING FINITE ELEMENT COMPUTATION OF UNSTEADY FLOW WITH ADAPTIVE UNSTRUCTURED MESH

A discontinuity-capturing scheme of finite element method(FEM)is proposed.The unstructured-grid technique combined with a new type of adaptive mesh approach is developed for both compressible and incompressible unsteady flows,which exhibits the capability of capturing the shock waves and/or thin shear layers accurately in an unsteady viscous flow at high Reynolds number. In particular,a new testing variable,i.e.,the disturbed kinetic energy E,is suggested and used in the adaptive mesh computation,which is universally applicable to the capturing of both shock waves and shear layers in the inviscid flow and viscous flow at high Reynolds number.Based on several calculated examples,this approach has been proved to be effective and efficient for the calculations of compressible and incompressible flows.

Application of dual reciprocity boundary element method to predict acoustic attenuation characteristics of marine engine exhaust silencers

In marine engine exhaust silencing systems, the presence of exhaust gas flow influences the sound propagation inside the systems and the acoustic attenuation performance of silencers. In order to investigate the effects of three-dimensional gas flow and acoustic damping on the acoustic attenuation characteristics of marine engine exhaust silencers, a dual reciprocity boundary element method (DRBEM) was developed. The acoustic governing equation in three-dimensional potential flow was derived first, and then the DRBEM numerical procedure is given. Compared to the conventional boundary element method (CBEM), the DRBEM considers the second order terms of flow Mach number in the acoustic governing equation, so it is suitable for the cases with higher Mach number subsonic flow. For complex exhaust silencers, it is difficult to apply the single-domain boundary element method, so a substructure approach based on the dual reciprocity boundary element method is presented. The experiments for measuring transmission loss of silencers are conducted, and the experimental setup and measurements are explained. The transmission loss of a single expansion chamber silencer with extended inlet and outlet were predicted by DRBEM and compared with the measurements. The good agreements between predictions and measurements are observed, which demonstrated that the derived acoustic governing equation and the DRBEM numerical procedure in the present study are correct.

MOTION AND DEFORMATION OF VISCOUS DROP IN STOKES FLOW NEAR RIGID WALL

A boundary integral method was developed for simulating the motion and deformation of a viscous drop in an axisymmetric ambient Stokes flow near a rigid wall and for direct calculating the stress on the wall. Numerical experiments by the method were performed for different initial stand-off distances of the drop to the wall, viscosity ratios, combined surface tension and buoyancy parameters and ambient flow parameters. Numerical results show that due to the action of ambient flow and buoyancy the drop is compressed and stretched respectively in axial and radial directions when time goes. When the ambient flow action is weaker than that of the buoyancy the drop raises and bends upward and the stress on the wall induced by drop motion decreases when time advances. When the ambient flow action is stronger than that of the buoyancy the drop descends and becomes flatter and flatter as time goes. In this case when the initial stand-off distance is large the stress on the wall increases as the drop evolutes but when the stand-off distance is small the stress on the wall decreases as a result of combined effects of ambient flow, buoyancy and the stronger wall action to the flow. The action of the stress on the wall induced by drop motion is restricted in an area near the symmetric axis, which increases when the initial stand-off distance increases. When the initial stand-off distance increases the stress induced by drop motion decreases substantially. The surface tension effects resist the deformation and smooth the profile of the drop surfaces. The drop viscosity will reduce the deformation and migration of the drop.

A TIME DOMAIN BOUNDARY ELEMENT METHOD FOR WATER-SOLID IMPACT ANALYSIS

A reciprocal theorem of dynamics for potential flow problems is first derived by means of the Laplace transform in which the compressibility of water is taken into account. Based on this theorem, the corresponding time-space boundary integral equation: is obtained. Then, a set of time domain boundary element equations with recurrence form is immediately formulated through discretization in both time and boundary. After having carried out the numerical calculation two solutions are found in which a rigid semicircular cylinder and a rigid wedge with infinite length suffer normal impact on the surface of a half-space fluid. The results show that the present method is more efficient than the previous ones.

Analysis of Unsteady Flow over Offshore Wind Turbines in Combination with Different types of Foundations

Environmental effects have an important influence on Offshore Wind Turbine (OWT) power generation efficiency and the structural stability of such turbines. In this study, we use an in-house Boundary Element (BEM)-panMARE code-to simulate the unsteady flow behavior of a full OWT with various combinations of aerodynamic and hydrodynamic loads in the time domain. This code is implemented to simulate potential flows for different applications and is based on a three-dimensional first-order panel method. Three different OWT configurations consisting of a generic 5 MW NREL rotor with three different types of foundations (Monopile, Tripod, and Jacket) are investigated. These three configurations are analyzed using the RANSE solver which is carried out using ANSYS CFX for validating the corresponding results. The simulations are performed under the same environmental atmospheric wind shear and rotor angular velocity, and the wave properties are wave height of 4 m and wave period of 7.16 s. In the present work, wave environmental effects were investigated firstly for the two solvers, and good agreement is achieved. Moreover, pressure distribution in each OWT case is presented, including detailed information about local flow fields. The time history of the forces at inflow direction and its moments around the mudline at each OWT part are presented in a dimensionless form with respect to the mean value of the last three loads and the moment amplitudes obtained from the BEM code, where the contribution of rotor force is lower in the tripod case and higher in the jacket case and the calculated hydrodynamic load that effect on jacket foundation type is lower than other two cases.

Solution of Two-Dimensional Stokes Flow Problems Using Improved Singular BoundaryMethod

In this paper,an improved singular boundarymethod(SBM),viewed as one kind of modified method of fundamental solution(MFS),is firstly applied for the numerical analysis of two-dimensional(2D)Stokes flow problems.The key issue of the SBM is the determination of the origin intensity factor used to remove the singularity of the fundamental solution and its derivatives.The new contribution of this study is that the origin intensity factors for the velocity,traction and pressure are derived,and based on that,the SBM formulations for 2D Stokes flow problems are presented.Several examples are provided to verify the correctness and robustness of the presented method.The numerical results clearly demonstrate the potentials of the present SBM for solving 2D Stokes flow problems.

A Spectrally Accurate Boundary Integral Method for Interfacial Velocities in Two-Dimensional Stokes Flow

We present a new numerical method for solving two-dimensional Stokes flow with deformable interfaces such as dynamics of suspended drops or bubbles.The method is based on a boundary integral formulation for the interfacial velocity and is spectrally accurate in space.We analyze the singular behavior of the integrals(single-layer and double-layer integrals)appearing in the equations.The interfaces are formulated in the tangent angle and arc-length coordinates and,to reduce the stiffness of the evolution equation,the marker points are evenly distributed in arc-length by choosing a proper tangential velocity along the interfaces.Examples of Stokes flow with bubbles are provided to demonstrate the accuracy and effectiveness of the numerical method.

A SPLIT-CHARACTERISTIC FINITE ELEMENT MODEL FOR 1-D UNSTEADY FLOWS

An efficient and accurate solution algorithm was proposed for 1-D unsteady flow problems widely existing in hydraulic engineering. Based on the split-characteristic finite element method, the numerical model with the Saint-Venant equations of 1-D unsteady flows was established. The assembled f＇mite element equations were solved with the tri-diagonal matrix algorithm. In the semi-implicit and explicit scheme, the critical time step of the method was dependent on the space step and flow velocity, not on the wave celerity. The method was used to eliminate the restriction due to the wave celerity for the computational analysis of unsteady open-channel flows. The model was verified by the experimental data and theoretical solution and also applied to the simulation of the flow in practical river networks. It shows that the numerical method has high efficiency and accuracy and can be used to simulate 1-D steady flows, and unsteady flows with shock waves or flood waves. Compared with other numerical methods, the algorithm of this method is simpler with higher accuracy, less dissipation, higher computation efficiency and less computer storage.

A Parallel Boundary Element Formulation for Tracking Multiple Particle Trajectories in Stoke’s Flow for Microfluidic Applications

A new formulation for tracking multiple particles in slow viscous flow for microfluidic applications is presented.The method employs the manipulation of the boundary element matrices so that finally a system of equations is obtained relating the rigid body velocities of the particle to the forces applied on the particle.The formulation is specially designed for particle trajectory tracking and involves successive matrix multiplications for which SMP(Symmetric multiprocessing)parallelisation is applied.It is observed that present formulation offers an efficient numerical model to be used for particle tracking and can easily be extended for multiphysics simulations in which several physics involved.

Least Square Finite Element Method for Viscous Splitting of Unsteady Incompressible Navier–Stokes Equations

In order to solve unsteady incompressible Navier–Stokes(N–S) equations, a new stabilized finite element method,called the viscous-splitting least square FEM, is proposed. In the model, the N–S equations are split into diffusive and convective parts in each time step. The diffusive part is discretized by the backward difference method in time and discretized by the standard Galerkin method in space. The convective part is a first-order nonlinear equation.After the linearization of the nonlinear part by Newton’s method, the convective part is also discretized by the backward difference method in time and discretized by least square scheme in space. C0-type element can be used for interpolation of the velocity and pressure in the present model. Driven cavity flow and flow past a circular cylinder are conducted to validate the present model. Numerical results agree with previous numerical results, and the model has high accuracy and can be used to simulate problems with complex geometry.

A General Boundary Solution Method for Potential Flow around a 2D Semi-infinite Body

By using Cauchy's integral formula of analytical complex function and the third order complex spline function, a general boundary solution method for solving the complex potential field of the flow field around a 2D semi infinite body is presented in this paper. The pressure coefficients obtained by the present method agree well with those given by Acrivous, showing the validity of our method.

A Hybrid Immersed Interface Method for Driven Stokes Flow in an Elastic Tube

We present a hybrid numerical method for simulating fluid flow through a compliant,closed tube,driven by an internal source and sink.Fluid is assumed to be highly viscous with its motion described by Stokes flow.Model geometry is assumed to be axisymmetric,and the governing equations are implemented in axisymmetric cylindrical coordinates,which capture 3D flow dynamics with only 2D computations.We solve the model equations using a hybrid approach:we decompose the pressure and velocity fields into parts due to the surface forcings and due to the source and sink,with each part handled separately by means of an appropriate method.Because the singularly-supported surface forcings yield an unsmooth solution,that part of the solution is computed using the immersed interface method.Jump conditions are derived for the axisymmetric cylindrical coordinates.The velocity due to the source and sink is calculated along the tubular surface using boundary integrals.Numerical results are presented that indicate second-order accuracy of the method.

Unsteady Spillway Flows by Singular Integral Operators Method

The Singular Integral Operators Method (S.I.O.M.) is applied to the determination of the free-surface profile of an un-steady flow over a spillway, which defines a classical hydraulics problem in open channel flow. Thus, with a known flow rate Q, then the velocities and the elevations are computed on the free surface of the spillway flow. For the numerical evaluation of the singular integral equations both constant and linear elements are used. An application is finally given to the determination of the free-surface profile of a special spillway and comparing the numerical results with corresponding results by the Boundary Integral Equation Method (B.I.E.M.) and by using experiments.

An Iteration Method to Solve the Boundary Layer Flow past a Flat Plate

An iteration method similar to the thin-wing-expansion method for the compressible flow has been proposed to solve the boundary layer flow past a flat plate. Using such an iteration, the first step of which is Oseen’s approximation, the boundary layer past a flat plate is studied. As proceeding from the first approximation to the second and third approximations, it is realized that our solution approaches to a well known Howarth’s bench mark one gradually. Hence, it is concluded that the usefulness of the present method has been confirmed.

Generation of Dynamic Grids and Computation of Unsteady Transonic Flows around Assemblies

Algebraic methods and rapid deforming techniques are used to generate three-dimensional boundary-fitted dynamic grids for assemblies. The conservative full-potential equation is solved by a time-accurate approximate factorization algorithm and internal Newton iterations. An integral boundary layer method based on the dissipation integral is used to account for viscous effects. The computational results about unsteady transonic forces on wings, bodies and control surfaces are in agreement with experimental data.

TIME DOMAIN SOLUTION OF UNSTEADY FLOW AROUND A SHIP

The unsteady flow around a ship advancing at a constant speed was studied in time domain by using a higher-order boundary element method. The Rankine source was employed. The ship hull and free surface were discretized into 8-node isoparametric elements. A modified Green formula was used to overcome the numerical difficulty arising from singular or quasi-singular integral equation. In the case of τ>0.25, a rigid wall condition was applied to the upstream and a numerical beach was applied to the downstream to absorb the outgoing waves. The numerical results are satisfactory compared with the experimental results.

Unsteady Inverse Problem of Type B for 2-D Transonic Flow: A Variational Formulation

A family of variational principles (VP) has been developed for the unsteady inverse problem of the second type I B. It opens new ways for the inverse shape design of unsteady airfoils and can serve as key basis of multipoint inverse shape design of steady airfoils and cascades.