Hemispherical Headforms Research Articles

A High-Order Discontinuous Galerkin Method for Solving Preconditioned Euler Equations

A high-order discontinuous Galerkin (DG) method is presented for solving the preconditioned Euler equations with an explicit or implicit time marching scheme. A detailed description is given of a practical implementation of a precondition matrix of the type of Weiss and Smith and of the DG spatial discretization scheme employed, with particular emphasis on the artificial viscosity-based shock capturing techniques. The curved boundary treatment is proposed through adopting a NURBS surface equipped with a radial basis function interpolation to propagate the boundary displacement to the interior of the mesh. The resulting methods are verified by simulating flows over two-dimensional airfoils, such as symmetric NACA0012 or asymmetric RAE2822, and over three-dimensional bodies, such as an academic hemispherical headform or aerodynamic ONERA M6 wing. Numerical results show that the present method functions for both transonic and nearly incompressible flow simulations, and the proposed treatment of curved boundaries, play an important role in improving the accuracy of the obtained solutions, which are in good agreement with available experimental data or other numerical solutions reported in literature.

Preconditioned gridless methods for solving three-dimensional Euler equations at low Mach numbers

In this paper, preconditioned gridless methods are developed for solving the three-dimensional (3D) Euler equations at low Mach numbers. The preconditioned system is obtained by multiplying a preconditioning matrix of the type of Weiss and Smith to the time derivative of the 3D Euler equations, which are discretized under the clouds of points distributed in the computational domain by using a gridless technique. The implementations of the preconditioned gridless methods are mainly based on the frame of the traditional gridless method without preconditioning, which may fail to have convergence for flow simulations at low Mach numbers, therefore the modifications corresponding to the affected terms of preconditioning are mainly addressed in the paper. An explicit four-stage Runge–Kutta scheme is first applied for time integration, and the lower-upper symmetric Gauss-Seidel (LU-SGS) algorithm is then introduced to form the implicit counterpart to have the further speed up of the convergence. Both the resulting explicit and implicit preconditioned gridless methods are validated by simulating flows over two academic bodies like sphere or hemispherical headform, and transonic and nearly incompressible flows over one aerodynamic ONERA M6 wing. The gridless clouds of both regular and irregular points are used in the simulations, which demonstrates the ability of the method presented for coping with flows over complicated aerodynamic geometries. Numerical results of surface pressure distributions agree well with available experimental data or simulated solutions in the literature. The numerical results also show that the preconditioned gridless methods presented still functions for compressible transonic flow simulations and additionally, for nearly incompressible flow simulations at low Mach numbers as well. The convergence of the implicit preconditioned gridless method, as expected, is much faster than its explicit counterpart.

An efficient calibration approach for cavitation model constants based on OpenFOAM platform

To precisely capture the cavitation process, it is essential to establish physical and mathematical models that describes the thermodynamic process between liquid and vapor. However, most mass transfer models represented by the cavitation model involves numbers of empirical constants which are case specific. The calibration for the cavitation model constants is therefore required for accurate numerical simulations. In this paper, a rapid and effective calibration approach for model coefficients using a surrogate-based approximate optimization (SAO) strategy is established based on OpenFOAM platform. Coefficients calibration has been carried out for the Kunz cavitation model using both two-dimensional and three-dimensional hydrofoil experimental data. The selection law for the Kunz model constants which defines a specific region that would guarantee low levels of errors is obtained, implying the mass transfer mechanism between gas and liquid. Furthermore, the optimized values for the model constants are gained. Validations have been carried out using the cases of the three-dimensional hemispherical head-form. Results show that the optimal constants perform much better in cavitation predictions. The proposed calibration method is validated for feasibility and efficiency for cavitation model constant calibration and can also be applied for the other models with various empirical constants.

Development of Efficient and Accurate Parallel Computation Algorithm Using Moving Overset Grids on Background Multi-Domains for Complex Two-Phase Flows

The goal of this study involves developing an efficient and accurate parallel computation method for two-phase flow problems including complex moving foreign bodies. The proposed parallel computing techniques are based on the moving body-fitted grids’ overset on background multidomains with grid-overlapping at their interface. First, the cavitation flow over the hemispherical head form is investigated using the two-phase flow solver, which is validated by comparing the numerical and experimental results. Subsequently, the parallel computing technique based on the multidomain method that divides the computational domain into several smaller subdomains is proposed to facilitate more efficient numerical simulations. At the interface of the subdomains, the grid-overlapping method is proposed for more accurate simulations. The illustrative computations indicate that the accuracy of the parallel computation combined with the grid-overlapping method on multidomains is identical to that of the serial computation based on a single block, albeit with a significant reduction in the computation time. Finally, the moving overset grid technique is combined with the background multidomain method and applied to simulate the gust flow that is generated by the pitching motions of the twin hydrofoils. The overset grid technique includes the following three sequential steps: hole-cutting, finding donor cells, and bilinear interpolation. The prediction results for the inflow gust generated by oscillating hydrofoils closely follow the measured results.

2차원 단순 물체의 초공동 유동에 대한 수치해석

In this paper, a numerical analysis is carried out to study the characteristics of supercavitating flows and the drag of relatively simple two-dimensional and axisymmetric bodies which can be used for supercavity generation device, cavitator, of a high-speed underwater vehicle. In order to investigate the suitability of numerical models, cavity flows around the hemispherical head form and two-dimensional wedge are calculated with combinations of three turbulence models(standard , realizable , Reynolds stress) and two cavitation models(Schnerr-Sauer, Zwart-Gerber-Belamri). From the results, it is confirmed that the calculated cavity flow is more affected by the turbulence model than the cavitation model. For the calculation of steady state cavity flows, the convergence in case of the realizable model is better than the other turbulence models. The numerical result of the Schnerr-Sauer cavitation model is changed less by turbulence model and more robust than the Zwart-Gerber-Belamri model. Thus the realizable turbulence model and the Schnerr-Sauer cavitation model are applied to calculate supercavitating flows around disks, two dimensional and wedges. In case of the disk, the cavitation number dependences of the cavity size and the drag coefficient predicted are similar to either experimental data or Reichardt`s semi-empirical equations, but the drag coefficient is overestimated about 3% higher than the Reichardt`s equation. In case of the wedges, the cavitation number dependences of the cavity size are similar to experimental data and Newman`s linear theory, and the agreement of the cavity length predicted and Newman`s linear theory becomes better as decreasing cavitation number. However, the drag coefficients of wedges agree more with experimental data than those of Newman`s analytic solution. The cavitation number dependences of the drag coefficients of both the disk and the wedge appear linear and simple formula for estimating the drag of supercavitating disks and wedges are suggested. Consequently, the CFD scheme of this study can be applied for numerical analysis of supercavitating flows of the cavitator and the cavitator design.

Preconditioned characteristic boundary conditions for solution of the preconditioned Euler equations at low Mach number flows

Preconditioned characteristic boundary conditions (BCs) are implemented at artificial boundaries for the solution of the two- and three-dimensional preconditioned Euler equations at low Mach number flows. The preconditioned compatibility equations and the corresponding characteristic variables (or the Riemann invariants) based on the characteristic forms of preconditioned Euler equations are mathematically derived for three preconditioners proposed by Eriksson, Choi and Merkle, and Turkel. A cell-centered finite volume Roe’s method is used for the discretization of the preconditioned system of equations on unstructured meshes. The accuracy and performance of the preconditioned characteristic BCs applied at artificial boundaries are evaluated in comparison with the non-preconditioned characteristic BCs and the simplified BCs in computing steady low Mach number flows. The two-dimensional flow over the NACA0012 airfoil and three-dimensional flow over the hemispherical headform are computed and the results are obtained for different conditions and compared with the available numerical and experimental data. The sensitivity of the solution to the size of computational domain and the variation of the angle of attack for each type of BCs is also examined. Indications are that the preconditioned characteristic BCs implemented in the preconditioned system of Euler equations greatly enhance the convergence rate of the solution of low Mach number flows compared to the other two types of BCs.

The influence of helmet size and shape on peak linear decelerations when impacting crash pads

During training and competition in short track speed skating, skaters commonly fall on the ice and slide into crash pads that line the boards of the rink. Skaters wear helmets to protect their heads from such impacts. Nevertheless, concussion injuries are not uncommon, especially from impacts into the crash pads. Basic mechanical principles suggest that, all other things being equal, smaller sized and rounder shaped helmets should reduce peak impact forces when hitting relatively soft crash pads. This study validates these assumptions and determines the magnitude of these effects using drop tests and a 3D accelerometer. Hemispherical head forms of various radii, each weighing approx. 4.5kg, were dropped from four heights (0.3-4.0 m) onto a crash pad. Peak linear decelerations were recorded. In one set of tests, complete hemispheres were used, highlighting the effect of helmet size (radius). In a second set of tests, another set of hemispheres of various radii were sliced to produce caps each with a diameter of 8” but each with a different radius of curvature. Impact tests at four drop heights using these caps revealed the effect of helmet shape. Size was found to be more important than shape, with the greatest effects being in the 10-20cm radius range, a range which is relevant to helmets used in the sport today.

An Algorithm for Estimating Acceleration Magnitude and Impact Location Using Multiple Nonorthogonal Single-Axis Accelerometers

Accelerations of the head are the likely cause of concussion injury, but identifying the specific etiology of concussion has been difficult due to the lack of a valid animal or computer model. Contact sports, in which concussions are a rising health care concern, offer a unique research laboratory environment. However, measuring head acceleration in the field has many challenges including the need for large population sampling because of the relatively low incidence of concussions. We report a novel approach for calculating linear acceleration that can be incorporated into a head-mounted system for on-field use during contact sports. The advantages of this approach include the use of single-axis linear accelerometers, which reduce costs, and a nonorthogonal arrangement of the accelerometers, which simplifies the design criteria for a head-mounted and helmet compatible system. The purpose of this study was to describe the algorithm and evaluate its accuracy for measuring linear acceleration magnitude and impact location using computer simulation and experimental tests with various accelerometer configurations. A 10% error in magnitude and a 10 deg error in impact location were achieved using as few as six single-axis accelerometers mounted on a hemispherical headform.

EXPERIMENTAL EVALUATION OF THE IMPACTABSORPTION PROPERTIES OF RUBBER TILES FOR THE PLAYGROUND

Rubber tiles are popular in playgrounds as protective surfacing to reduce the incidence of head injuries caused by children falling from equipment. However, Taiwan has not yet established a test code for assessment of the shock-absorption properties of such surfacing. For this study, an experimental model was established to evaluate the behavior of various rubber tiles. A hemispherical headform was dropped from a set height to strike the center of the specimen tile. The peak acceleration and Head Injury Criterion (HIC) were measured to assess the impact absorption of and critical height for a given rubber tile. The results show that utilization of the HIC index provides a more conservative assessment of the shock absorption and, ultimately, protection from head injuries than peak acceleration. The maximum critical heights of the rubber tiles used in this study for tile thicknesses of 45, 60 and 80 mm were 1.6, 2.0 and 2.2 m, respectively. Two-part rubber tiles with a base structure consisting of a box-like core offer superior protection from head injuries relative to analogous cylindrical, square pillar and solid structures. The maximum differences in peakacceleration and HIC values comparing the box-like core and solid structures at a thickness of 45 mm were 21% and 44%, respectively. The results of this study suggest a minimum of rubber thickness of 60 mm, based on probable maximum fall heights of more than 1.6 m. Moreover, incorporation of an appropriate cushioning structure in the base of the rubber tile could further improve protection.

Suppression of Sheet Cavitation Inception by Water Discharge Through Slit

This paper studies experimentally the suppression effect of water discharge on incipient and developed sheet cavitation on a hemispherical headform. The water is discharged tangentially from a slit upstream of a cavitating region into the boundary layer on the headform. Increasing the discharge rate, the length of the sheet cavity becomes shorter and then the cavity is suppressed completely. Flow visualization test shows the generation of wavy motion in the boundary layer which disturbs the separated zone, resulting in disappearance of the sheet cavitation.

Closure to “Discussion of ‘The Onset of Bubble-Ring Cavitation on Hemispherical Headforms’” (1982, ASME J. Fluids Eng., 104, p. 123)

Closure to “Discussion of ‘The Onset of Bubble-Ring Cavitation on Hemispherical Headforms’” (1982, ASME J. Fluids Eng., 104, p. 123)

The Onset of Bubble-Ring Cavitation on Hemispherical Headforms

An approximate theory is developed to predict the onset of cavitation on hemispherical headforms for Reynolds numbers at which laminar separation is known to occur. Insofar as it is possible, the theory is based upon first principles. Fairly good agreement is obtained between the cavitation desinence trends recently measured by Holl and Carroll and the present theory. It is also found that the onset cavitation number should be less than the magnitude of the pressure coefficient at the laminar separation point and that the cavitation number increases with freestream velocity. As long as there is an appreciable concentration of dissolved air in the water, it is also found, in agreement with experiment, that the onset of bubblering cavitation is practically independent of air content. Moreover, the observed occurrence of a lowest speed for “bubble-ring” cavitation, which is the only cavitation form considered here, and the range of “cutoff” speeds predicted by the present asymptotic theory show very encouraging agreement. The present theory suggests that this cutoff speed and its accompanying cutoff cavitation number can also depend on the temperature of the water, provided that the initial size attributed to a “typical” spherical free-stream air bubble nucleus also varies with the temperature. At 80° F (26.6°C) it is estimated that the typical nucleus from which bubble-ring cavitation originates has a radius of about seven μm. At higher temperatures the nucleus radius decreases from this value while at lower temperatures the initial radius exceeds the value noted.

The Initiation Of Gaseous Microbubble Growth In Laminar Separation Bubbles

Flow conditions surrounding bubble-ring cavitation inception on hemispherical headforms are analyzed with respect to the initiation of air diffusion into microbubbles as is observed to occur at fixed positions in the boundary layer. Fairly recent observations have shown this phenomenon to occur in the laminar separation bubble on the body. The analysis shows, in agreement with the body of experimental evidence now available, that gaseous growth must be preceded by a period of vaporous growth starting in regions of low pressure upstream of the laminar separation bubble. It also appears that the most favorable condition for the initiation of gaseous growth should occur when a typical vapor bubble reaches its maximum radius as it enters the laminar separation bubble. The conditions for the initiation of subsequent gaseous growth, once the cavitation bubble is stabilized in the laminar separation zone, are more demanding. Nevertheless, it is found that the liquid in the water surrounding the bubble in the separation zone is definitely supersaturated for most flows of experimental or practical interest. Therefore, gaseous growth, as well as vaporous growth, is definitely to be associated with the onset of bubble-ring cavitation on both theoretical and experimental grounds.

A Possible Criterion for Cavitation Inception on Hemispherical Headforms

The present paper reviews several pertinent papers about the onset of bubble-ring cavitation on hemispherical headforms in which a laminar bubble is present. From this review, a likely sequence of events for cavitation inception, or desinence, can be discerned. When this sequence is postulated, the observational basis for each event is discussed. Once the physical aspects of the inception process have been noted, a mathematical formulation of key relationships required to establish an inception criterion can be undertaken. From these results, a quantitative criterion for the inception of bubble ring cavitation is stated and we give a discussion which interprets this possible inception criterion for future use. The use of this criterion in a theory of inception and a comparison of analytical and experimental results is left to a future publication.

Cavitation Inception Tests on Axisymmetric Headforms

Six axisymmetric headforms were tested in the 350-mm dia water tunnel at China Ship Scientific Research Center (CSSRC) for observation and measurement of cavitation inception. Among them, three of the flow patterns were displayed and analyzed by holography. Results of the tests show that cavitation inception on the headforms was closely related to the structure of the near-wall flow past the body with different flow regimes corresponding to different features of cavitation inception. By carefully fitting a trip wire of suitable height on a hemispherical headform, it was observed that under conditions of low air content, this artificial means of boundary transition may bring about a significant reduction in cavitation inception number.