Velocity Potential Formulation Research Articles

Further Investigation of Impedance Eduction in a Grazing Flow Duct Facility

An established protocol for impedance eduction is based on a search procedure in which wall impedance in a duct propagation model is varied to obtain an acoustic field converging with one measured in a grazing flow duct. Here, an acoustic velocity potential formulation for the finite element model propagation model is extended to examine the relative effectiveness of driving the model with measured acoustic pressure at the source and termination and driving with acoustic pressure at the source and closing the computational domain with measured termination impedance. Also assessed is the appropriate location of measurement microphones to satisfy plane wave requirements for the computational model. Source acoustic pressure and termination impedance driving the propagation model produce superior eduction results with lower residual error, though the model driven with pressure at both the source and termination produces acceptable spectra. The microphone location in the gazing flow test facility is judged to be adequate for the range of grazing flow Mach number and frequency considered. Examination of impedance spectra for a range of lining samples has revealed a correlation of a decrease in spectrum quality, measured by observed data scatter, with a decreasing face sheet open area ratio and therefore with increasing lining nonlinearity.

Non-linear finite element analysis of seismically excited sloped wall tank

The non-linear seismic behavior of a ground-supported sloped wall tank is analyzed in the present study using a finite element model. A combination of three-node triangular and four-node quadrilateral finite elements is used to discretize the liquid domain. The velocity potential formulation is adopted for the liquid domain with a mixed Eulerian-Lagrangian scheme. By comparing the current numerical results with the existing experimental results, the competency of the present non-linear model is verified. The fundamental sloshing frequencies obtained from the present non-linear model are found closer to the experimental results compared to the linear counterpart. Non-linear seismic characteristics of liquid sloshing inside a ground-supported sloped wall tank are investigated under six different earthquakes categorized as low-, intermediate-, and high-frequency contents. Also, the dynamic impulsive and convective response components of hydrodynamic base shear force and pressure are quantified successfully. The comparison of the non-linear hydrodynamic response with the linear counterpart shows a significant difference in results which reasonably clarifies the limited capabilities of the linear model for response prediction as well as designing sloped wall tanks subjected to seismic excitation. The developed non-linear numerical model in the present study can be utilized for tuning and designing structure-coupled sloped wall TLDs.

The mild-slope equations: a unified theory

The mild-slope equations, devised to approximate surface wave propagation over water of slowly varying depth, have hitherto been based on either the velocity potential formulation or the streamfunction formulation. By using a more general version of the governing equations, a single framework is developed that relates the existing mild-slope equations and provides new examples and derivations.

The Jordan–Moore–Gibson–Thompson Equation: Well-posedness with quadratic gradient nonlinearity and singular limit for vanishing relaxation time

In this paper, we consider the Jordan–Moore–Gibson–Thompson equation, a third-order in time wave equation describing the nonlinear propagation of sound that avoids the infinite signal speed paradox of classical second-order in time strongly damped models of nonlinear acoustics, such as the Westervelt and the Kuznetsov equation. We show well-posedness in an acoustic velocity potential formulation with and without gradient nonlinearity, corresponding to the Kuznetsov and the Westervelt nonlinearities, respectively. Moreover, we consider the limit as the parameter of the third-order time derivative that plays the role of a relaxation time tends to zero, which again leads to the classical Kuznetsov and Westervelt models. To this end, we establish appropriate energy estimates for the linearized equations and employ fixed-point arguments for well-posedness of the nonlinear equations. The theoretical results are illustrated by numerical experiments.

Pressure-field permeable-surface integral formulations for sound scattered by moving bodies

This paper presents two novel integral formulations for the prediction of sound scattered by moving bodies, derived from the Lighthill and the Ffowcs Williams and Hawkings equations for the pressure field. These are expressed in the frequency-domain, over a permeable (fictitious) surface surrounding the scatterer(s), and are numerically evaluated through application of a boundary element technique. The aims of the paper are the assessment of the influence of the nonlinear terms of Lighthill and Ffowcs Williams and Hawkings equations on sound scattering prediction in the presence of nonuniform mean-flow due to scatterer motion, the assessment of the corresponding limits of applicability of the widely-used linear formulations for solid-wall boundaries, and the development of integral formulations capable to predict accurately and efficiently the noise scattered in the far field by moving bodies. The numerical investigation concerns a non-lifting wing in uniform translation impinged by an acoustic disturbance generated by a co-moving source, and includes the comparison of the results obtained through the proposed scattering formulations with those provided by a boundary-field velocity-potential approach recently validated for moving-body problems. Its main outcomes reveal that, when the proposed pressure-field formulations include the nonlinear terms through application of a suitable permeable-surface, their predictions match those provided by the boundary-field velocity-potential solver, whereas the fully linear versions of both pressure-field approaches yield underestimated scattered noise predictions, significantly less accurate than those given by the linear version of the velocity-potential formulation.

Application of reciprocal absorbing boundary condition to transient analysis of acoustic wave propagation

The Reciprocal Absorbing Boundary Condition (RABC) was recently introduced as an approach toward solution of elastic wave propagation problems in unbounded layered media. The idea underlying RABC is the application of time-domain reciprocity theorems which are valid not only for bounded but also for unbounded domains. Since these theorems involve the convolution integral of complete response time histories on the entire edge of the computational domain, the RABC is considered a “global” absorbing boundary condition in space and time. The “exact” nature of this approach offers accurate and effective treatment of wave propagation in infinite domains.We present in this paper an extension of RABC to the transient analysis of acoustic wave propagation. For the purpose of computations, we use a velocity-potential formulation of the wave equation. Multiple segments of the reciprocal absorbing boundary can be employed in arbitrary orientations, thus facilitating truncation of the unbounded domain.The performance of RABC when applied to acoustic media is demonstrated via numerical examples, including wave propagation and excitation of a cylindrical cavity embedded in an infinite channel.

Liquid sloshing in a circular cylindrical container containing a two-layer fluid

In this work, a semi-analytical approach is employed to investigate sloshing in an immiscible, incompressible and inviscid two-layer fluid. We evaluate natural sloshing frequencies of the two-layer fluid of different layer-wise densities $$\rho _1 < \rho _2$$ with an interface and a surface in a vertical circular cylinder. On the basis of the velocity potential formulation of the fluid motion inside the container, an infinite system of homogeneous linear equations is obtained. We demonstrate the effects of the parameters such as fluid heights and density ratio on the natural sloshing frequencies. The results are supported by relevant graphs.

Stability‐preserving model order reduction for time‐domain simulation of vibro‐acoustic FE models

SummaryThis work proposes novel stability‐preserving model order reduction approaches for vibro‐acoustic finite element models. As most research in the past for these systems has focused on noise attenuation in the frequency‐domain, stability‐preserving properties were of low priority. However, as the interest for time‐domain auralization and (model based) active noise control increases, stability‐preserving model order reduction techniques are becoming indispensable. The original finite element models for vibro‐acoustic simulation are already well established but require too much computational load for these applications. This work therefore proposes two new global approaches for the generation of stable reduced‐order models. Based on proven conditions for stability preservation under one‐sided projection, a reformulation of the displacement‐fluid velocity potential (u − ϕ) formulation is proposed. In contrast to the regular formulation, the proposed approach leads to a new asymmetric structure for the system matrices which is proven to preserve stability under one‐sided projection. The second approach starts from a displacement‐pressure (u − p) description where the system level projection space is decoupled for the two domains, for which we also prove the preservation of stability. Two numerical validation cases are presented which demonstrate the inadequacy of straightforward model order reduction on typical vibro‐acoustic models for time‐domain simulation and compare the performance of the proposed approaches. Both proposed approaches effectively preserve the stability of the original system. Copyright © 2016 John Wiley &amp; Sons, Ltd.

English

The paper presents a numerical method for analyzing the potential flow around two dimensional body such as single circular cylinder, NACA0012 hydrofoil and double circular cylinders by finite element method. The numerical technique is based upon a general formulation for the Laplace’s equation using Galerkin technique finite element approach. The solution of the systems of algebraic equations is approached by Gaussian elimination scheme. Laplace’s equation is expressed in terms of both steam function and velocity potential formulation. A finite element program is developed in order to analyze the result. The contours of stream and velocity potential function are drawn. The contour of stream function exhibits the characteristics of potential flow and does not intersect each other. The calculated pressure co-efficient shows the pressure decreasing around the forwarded face from the initial total pressure at the stagnation point and reaching a minimum pressure at the top of the cylinder. Key words: Stream function, velocity potential, number of nodes (NDE), number of elements (NEL).

Analytical and Numerical Development of a Baffled Liquid Rocket Combustion Stability Code

A predictive model for assessing the stabilization impact of radial blades, hub blade, and acoustic cavities in liquid propellant rocket engin es was developed, including nozzle, mean flow, and distributed combustion effects. A full t hree dimensional linear stability analysis was couched in terms of a pseudo velocity potential formulation, and the solution found through an integral Fourier series eigenfunction ex pansion. The non-homogeneous terms of the governing equation are dependent on the velocit y potential and the derivatives of the potential; therefore a successive approximation tec hnique was employed using the acoustic wave shape as the zeroth level approximation. The complex geometry that arises from the inclusion of the blade system requires separate sol utions for each compartment and a matching technique that allows for the velocity pot ential as well as the gradient of the potential to match at these interfaces. It will be shown that non-iterative techniques that do not allow for distortion of the wave shape from the acoustic solution may predict stable operation for a given combustion response while the iterative technique presented in this paper predicts an instability due to modification o f the potential field by mean flow, combustion, and cavity effects.

A numerical analysis of a class of generalized Boussinesq‐type equations using continuous/discontinuous FEM

SUMMARYAn improved class of Boussinesq systems of an arbitrary order using a wave surface elevation and velocity potential formulation is derived. Dissipative effects and wave generation due to a time‐dependent varying seabed are included. Thus, high‐order source functions are considered. For the reduction of the system order and maintenance of some dispersive characteristics of the higher‐order models, an extra O(μ2n+2) term (n ∈ ℕ) is included in the velocity potential expansion. We introduce a nonlocal continuous/discontinuous Galerkin FEM with inner penalty terms to calculate the numerical solutions of the improved fourth‐order models. The discretization of the spatial variables is made using continuous P2 Lagrange elements. A predictor‐corrector scheme with an initialization given by an explicit Runge–Kutta method is also used for the time‐variable integration. Moreover, a CFL‐type condition is deduced for the linear problem with a constant bathymetry. To demonstrate the applicability of the model, we considered several test cases. Improved stability is achieved. Copyright © 2011 John Wiley &amp; Sons, Ltd.

Investigation of sloshing damping in baffled rectangular tanks subjected to the dynamic excitation

In this research, an analytical model is developed to estimate the hydrodynamic damping ratio of liquid sloshing for wall bounded baffles using the velocity potential formulation and linear wave theory. Here, an analytical solution approach and experimental investigations are conducted for describing the hydrodynamic damping which is provided by vertical and horizontal baffles in partially filled rectangular liquid tanks. In order to evaluate the accuracy of the analytical solution which is developed in present work, a series of experiments are carried out with a rectangular liquid tank excited by harmonic oscillation. The parametric study is conducted on the damping efficiencies of both vertical and horizontal baffles with various dimensions and locations. According to the results of the present investigations, the hydrodynamic damping is significantly affected by the size and location of baffles. Furthermore, the validity of the developed analytical approach as well as the effectiveness of various baffle configurations are discussed. Finally, a simple approach is proposed for estimating the damping ratios of the baffles during earthquake motions.

Velocity potential formulations of highly accurate Boussinesq-type models

The highly accurate Boussinesq-type equations of Madsen et al. (Madsen, P.A., Bingham, H.B., Schäffer, H.A., 2003. Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: Derivation and analysis. Proc. R. Soc. Lond. A 459, 1075–1104; Madsen, P.A., Fuhrman, D.R., Wang, B., 2006. A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry. Coast. Eng. 53, 487–504); Jamois et al. (Jamois, E., Fuhrman, D.R., Bingham, H.B., Molin, B., 2006. Wave-structure interactions and nonlinear wave processes on the weather side of reflective structures. Coast. Eng. 53, 929–945) are re-derived in a more general framework which establishes the correct relationship between the model in a velocity formulation and a velocity potential formulation. Although most work with this model has used the velocity formulation, the potential formulation is of interest because it reduces the computational effort by approximately a factor of two and facilitates a coupling to other potential flow solvers. A new shoaling enhancement operator is introduced to derive new models (in both formulations) with a velocity profile which is always consistent with the kinematic bottom boundary condition. The true behaviour of the velocity potential formulation with respect to linear shoaling is given for the first time, correcting errors made by Jamois et al. (Jamois, E., Fuhrman, D.R., Bingham, H.B., Molin, B., 2006. Wave-structure interactions and nonlinear wave processes on the weather side of reflective structures. Coast. Eng. 53, 929–945). An exact infinite series solution for the potential is obtained via a Taylor expansion about an arbitrary vertical position z = zˆ. For practical implementation however, the solution is expanded based on a slow variation of zˆ and terms are retained to first-order. With shoaling enhancement, the new models obtain a comparable accuracy in linear shoaling to the original velocity formulation. General consistency relations are also derived which are convenient for verifying that the differential operators satisfy a potential flow and/or conserve mass up to the order of truncation of the model. The performance of the new formulation is validated using computations of linear and nonlinear shoaling problems. The behaviour on a rapidly varying bathymetry is also checked using linear wave reflection from a shelf and Bragg scattering from an undulating bottom. Although the new models perform equally well for Bragg scattering they fail earlier than the existing model for reflection/transmission problems in very deep water.

Dynamic analysis of conical shells conveying fluid

A formulation, based on the semi-analytical finite element method, is proposed for elastic conical shells conveying fluids. The structural equations are based on the shell element proposed by Ramasamy and Ganesan [Finite element analysis of fluid-filled isotropic cylindrical shells with constrained viscoelastic damping, Computers & Structures 70 (1998) 363–376] while the fluid model is based on velocity potential formulation used by Jayaraj et al. [A semi-analytical coupled finite element formulation for composite shells conveying fluids, Journal of Sound and Vibration 258(2) (2002) 287–307]. Dynamic pressure acting on the walls is derived from Bernoulli's equation. By imposing the requirement that the normal component of velocity of the solid and fluid are equal leads to fluid–structure coupling. The computer code developed has been validated using results available in the literature for cylindrical shells conveying fluid. The study has been carried out for conical shells of different cone angles and for boundary condition like clamped–clamped, simply supported and clamped free. In general, instability occurs at a critical fluid velocity corresponding to the shell circumferential mode with the lowest natural frequency. Critical fluid velocities are lower than that of equivalent cylindrical shells. This result holds good for all boundary conditions.

Unsteady Wing-Body Aerodynamics for Aeroelastic Applications at Mach One

Renewed interest in the sonic-box method is prompted by the possibility of near-sonic flights. For aeroelastic applications, the sonic-box methods, however, would be hindered by their drawbacks/confinements. They are based on a velocity-potential formulation, thus the wake domain must be modeled for planforms with swept trailing edges or for multiple lifting surfaces; their nonplanar lifting surface application has not been demonstrated, nor has its flutter applicability; their extensions for bodies remain unexplored. To this end, an unsteady sonic methodology employing sonic acceleration potential for multiple lifting surfaces (wings) and sonic source velocity potential for bodies has been developed. The present unsteady sonic method should be recognized as the sonic counterpart of our previous development of the unsteady subsonic and supersonic wing-body methods. It will be shown that unsteady sonic methodology employing sonic acceleration potential is an expedient method for flutter/aeroelastic applications for wing-body configurations in the sonic/near-sonic Mach number range.

On the transient modelling of hydro-elastic interaction of gravity waves with an underwater elastic structure

Fluid structure interaction (FSI) of offshore structures is an important area of research. The behaviour of offshore structures in Numerical Wave Tanks possessing FSI capabilities is of significant industrial value. In FSI problems, it is necessary to interactively solve and analyse both the structural and fluid dynamics as a coupled system, because the structural response and fluid loading are mutually dependent. We present a methodology for analysing a two-dimensional fluid-structure interaction problem in the time domain. The methodology is based on a nonlinear velocity potential formulation and linear elastic theory. The transient forces exerted by the fluid on the structure vary due to the elastic response of the structure. The dynamics of the free surface and the structure are also analysed for selected scenarios. We demonstrates that it is important to develop similar tools capable of solving such FSI problems in order to perform reliable and complete analysis leading to better and safer design.

Finite element analysis for acoustic characteristics of a magnetostrictive transducer

This paper presents a finite element analysis for a magnetostrictive transducer by takinginto account the nonlinear behavior of the magnetostrictive material and fluid interaction.A finite element formulation is derived for the coupling of magnetostrictive and elasticmaterials based upon a separated magnetic and displacement field calculation and a curvefitting technique of material properties. The fluid and structure coupled problem is takeninto account based upon pressure and velocity potential fields formulation. Infinitewave envelope elements are introduced at an artificial boundary to deal with theinfinite fluid domain. A finite element code for the analysis of a magnetostrictivetransducer is developed. A magnetostrictive tonpilz transducer is taken as anexample and verification for the developed program is made by comparing with acommercial code. The acoustic characteristics of the magnetostrictive tonpilztransducer are calculated in terms of radiation pattern and transmitted currentresponse.

Aspects of acoustics in MEMS devices

In this talk we will present an overview of aspects of structural acoustics in MEMS devices at Sandia. Recent results concerning viscous wave motion in a rotational micromechanical system will be presented. In addition, experimental approaches for the characterization of acoustic effects in MEMS devices will be discussed. We will also present an overview of massively parallel numerical simulation capabilities for large-scale and small-scale (MEMS) structural acoustic analysis. Since large numbers of degrees of freedom are typically present in structural acoustic analysis, massively parallel computations are essential in solving practical application problems. Two formulations for the fluid will be considered: a standard linear velocity potential formulation, and a nonlinear wave formulation. The applications of the two formulations will be discussed. A parallelization scheme will be presented that allows for arbitrary decomposition of the wet interface. Finally, numerical results of application problems will be presented. [Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000.]

A new complementary mild-slope equation

A new depth-integrated equation is derived to model a time-harmonic motion of small-amplitude waves in water of variable depth. The new equation, which is referred to as the complementary mild-slope equation here, is derived from Hamilton's principle in terms of stream function. In the formulation, the continuity equation is satisfied exactly in the fluid domain. Also satisfied exactly are the kinematic boundary conditions at the still water level and the uneven sea bottom. The numerical results of the present model are compared to the exact linear theory and the existing mild-slope equations that have been derived from the velocity-potential formulation. The computed results give better agreement with those of the exact linear theory than the other mild-slope equations. Comparison shows that the new equation provides accurate results for a bottom slope up to 1.

Preliminary investigations on non-linear fluid-structure interaction of an offshore structure

Fluid-structure interaction (FSI) of offshore structures is an important area of research. It is also of significant industrial value, where such capabilities could be used for simulating the structural response in numerical wave tanks. The current research focuses on the development of a numerical wave tank with FSI capabilities. This requires solving the non-linear fluid and structure behaviour in the time domain considering a non-linear velocity potential formulation for the wave flow and a linear elastic formulation for the structure. The fluid and structural behaviour are coupled in time through a mixed Lagrangian-Eulerian procedure and the differential equations are discretized using the finite element method. Unstructured triangular meshes are used to discretize the fluid and structural domain, augmented by mesh smoothing algorithms. Based on the studie, it is shown that the transient force developed by the fluid on the structure varies due to the elastic nature of the structure. The perturbations in the free-surface profile due to the structural response are also captured. It is also shown that transient non-linear studies are important as they can help in identifying critical design limitations of traditional guides. It is thus demonstrated that the present work is capable of solving such FSI problems in order to obtain a reliable and complete analysis leading to a better understanding and safer design of compliant structures. The effect on the structure due to prolonged interaction with the fluid needs to be further studied for a better understanding.