On the solution near the critical frequency for an oscillating and translating body in or near a free surface

1993 ◽  
Vol 254 ◽  
pp. 251-266 ◽  
Author(s):  
Yuming Liu ◽  
Dick K. P. Yue
Keyword(s):  
Free Surface ◽  
Closed Form Solution ◽  
Geometric Interpretation ◽  
Form Solution ◽  
The Body ◽  
Geometric Condition ◽  
Sufficient Condition ◽  
Gravitational Acceleration ◽  
Submerged Body ◽  
Necessary And Sufficient

We consider a floating or submerged body in deep water translating parallel to the undisturbed free surface with a steady velocity U while undergoing small oscillations at frequency ω. It is known that for a single source, the solution becomes singular at the resonant frequency given by τ ≡ Uω/g=¼, where g is the gravitational acceleration. In this paper, we show that for a general body, a finite solution exists as τ → ¼ if and only if a certain geometric condition (which depends only on the frequency ω but not on U) is satisfied. For a submerged body, a necessary and sufficient condition is that the body must have non-zero volume. For a surface-piercing body, a sufficient condition is derived which has a geometric interpretation similar to that of John (1950). As an illustration, we provide an analytic (closed-form) solution for the case of a submerged circular cylinder oscillating near τ = ¼. Finally, we identify the underlying difficulties of existing approximate theories and numerical computations near τ = ¼, and offer a simple remedy for the latter.

Nonlinear Aspects of Subcritical Shallow-Water Flow Past Two-Dimensional Obstructions

1978 ◽  
Vol 22 (04) ◽  
pp. 203-211
Author(s):  
Nils Salvesen ◽  
C. von Kerczek
Keyword(s):  
Free Surface ◽  
Shallow Water ◽  
Flow Problem ◽  
Free Stream ◽  
The Body ◽  
Two Dimensional ◽  
Third Order ◽  
Submerged Body ◽  
The Mean ◽  
The Waves

Some nonlinear aspects of the two-dimensional problem of a submerged body moving with constant speed in otherwise undisturbed water of uniform depth are considered. It is shown that a theory of Benjamin which predicts a uniform rise of the free surface ahead of the body and the lowering of the mean level of the waves behind it agrees well with experimental data. The local steady-flow problem is solved by a numerical method which satisfies the exact free-surface conditions. Third-order perturbation formulas for the downstream free waves are also presented. It is found that in sufficiently shallow water, the wavelength increases with increasing disturbance strength for fixed values of the free-stream-Froude number. This is opposite to the deepwater case where the wavelength decreases with increasing disturbance strength.

scholarly journals On 3D Anticrack Problem of Thermoelectroelasticity

2018 ◽  
Vol 12 (2) ◽  
pp. 109-114 ◽  
Author(s):  
Andrzej Kaczyński
Keyword(s):  
Arbitrary Shape ◽  
Closed Form Solution ◽  
Transversely Isotropic ◽  
Static Problem ◽  
Form Solution ◽  
Boundary Integral ◽  
The Body ◽  
Circular Rigid Inclusion ◽  
Stress Discontinuity ◽  
Suitable Potential

Abstract A solution is presented for the static problem of thermoelectroelasticity involving a transversely isotropic space with a heat-insulated rigid sheet-like inclusion (anticrack) located in the isotropy plane. It is assumed that far from this defect the body is in a uniform heat flow perpendicular to the inclusion plane. Besides, considered is the case where the electric potential on the anticrack faces is equal to zero. Accurate results are obtained by constructing suitable potential solutions and reducing the thermoelectromechanical problem to its thermomechanical counterpart. The governing boundary integral equation for a planar anticrack of arbitrary shape is obtained in terms of a normal stress discontinuity. As an illustration, a closed-form solution is given and discussed for a circular rigid inclusion.

scholarly journals Cavity dynamics in water entry at low Froude numbers

2009 ◽  
Vol 641 ◽  
pp. 441-461 ◽  
Author(s):  
HONGMEI YAN ◽  
YUMING LIU ◽  
JAKUB KOMINIARCZUK ◽  
DICK K. P. YUE
Keyword(s):  
Free Surface ◽  
Asymptotic Theory ◽  
Numerical Study ◽  
Three Dimensional ◽  
Maximum Size ◽  
Water Entry ◽  
Boundary Integral ◽  
The Body ◽  
Gravitational Acceleration ◽  
Slender Body Theory

The dynamics of the air cavity created by vertical water entry of a three-dimensional body is investigated theoretically, computationally and experimentally. The study is focused in the range of relatively low Froude numbers, Fr ≡ V(gD)−1/2 ≤ O(10) (where V is the dropping velocity of the body, D its characteristic dimension and g the gravitational acceleration), when the inertia and gravity effects are comparable. To understand the physical processes involved in the evolution of cavity, we conduct laboratory experiments of water entry of freely dropping spheres. A matched asymptotic theory for the description of the cavity dynamics is developed based on the slender-body theory in the context of potential flow. Direct comparisons with experimental data show that the asymptotic theory properly captures the key physical effects involved in the development of the cavity, and in particular gives a reasonable prediction of the maximum size of the cavity and the time of cavity closure. Due to the inherent assumption in the asymptotic theory, it is incapable of accurately predicting the flow details near the free surface and the body, where nonlinear free surface and body boundary effects are important. To complement the asymptotic theory, a fully nonlinear numerical study using an axisymmetric boundary integral equation is performed. The numerically obtained dependencies of the cavity height and closure time on Froude number and body geometry are in excellent agreement with available experiments.

A New Algorithm for Testing the Stability of a Polytope: A Geometric Approach for Simplification

1996 ◽  
Vol 118 (3) ◽  
pp. 611-615 ◽  
Author(s):  
Jinsiang Shaw ◽  
Suhada Jayasuriya
Keyword(s):  
Characteristic Equation ◽  
Robust Stability ◽  
Geometric Structure ◽  
Geometric Approach ◽  
Geometric Interpretation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Edge Theorem

Considered in this paper is the robust stability of a class of systems in which a relevant characteristic equation is a family of polynomials F: f(s, q) = a0(q) + a1(q)s + … + an(q)sn with its coefficients ai(q) depending linearly on q unknown-but-bounded parameters, q = (p1, p2, …, pq)T. It is known that a necessary and sufficient condition for determining the stability of such a family of polynomials is that polynomials at all the exposed edges of the polytope of F in the coefficient space be stable (the edge theorem of Bartlett et al., 1988). The geometric structure of such a family of polynomials is investigated and an approach is given, by which the number of edges of the polytope that need to be checked for stability can be reduced considerably. An example is included to illustrate the benefit of this geometric interpretation.

A novel method for defining the Greyhound talocrural joint axis of rotation for hinged transarticular external skeletal fixation

2013 ◽  
Vol 26 (04) ◽  
pp. 298-303 ◽  
Author(s):  
N. R. Hadley ◽  
A. M. Wallace ◽  
G. R. Colborne
Keyword(s):  
Articular Surface ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Form Solution ◽  
The Body ◽  
Transverse Axis ◽  
Talocrural Joint ◽  
Distal Articular Surface ◽  
Intertarsal Joint ◽  
Centre Of Rotation

SummaryIn order to apply hinged transarticular external skeletal fixation for stabilization of the injured canine tarsal joint, knowledge of the three-dimensional (3D) location and orientation of the transverse axis is necessary. This method of immobilization may be used as a primary or adjunctive method of stabilisation for a large number of traumatic conditions. Using pin-mounted markers in the cadaveric Greyhound crus and talus, a closed-form solution of absolute orientation was used to identify, on radiographs, the lateral and medial locations of the transverse axis by tracking the 3D excursions of the markers during flexion and extension. A line was drawn across the dorsal aspect of the calcaneus from the most dorsal point on the distal articular surface (proximal intertarsal joint: PIJ) to the most dorsal point on its proximal articulation with the body of the talus, and the location of the centre of rotation was expressed in terms of the length of that line. In seven Greyhound tarsal joints, the medial end of the axis was located 73 ± 10% proximal to the PIJ and 11 ± 7% dorsal to the line. The lateral end was 73 ± 9% proximal to the PIJ and -2 ± 3% plantar to the line.

The stability of rotating flows with a cylindrical free surface

1967 ◽  
Vol 30 (1) ◽  
pp. 127-147 ◽  
Author(s):  
T. J. Pedley
Keyword(s):  
Free Surface ◽  
Rotation Speed ◽  
Outer Boundary ◽  
Rotating Flows ◽  
Rotating Flow ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Axisymmetric Disturbances

The stability to small inviscid disturbances of a rotating flow, whose velocity components in cylindrical polars (r, 0, z) are (0, V(r), 0), is investigated when one boundary of the flow (r = b) is a free surface under the action of surface tension (γ), and the other is either at infinity, or a rigid cylinder (r = a ≠ b), or at the axis (r = 0). The free surface may be the inner or the outer boundary. A necessary and sufficient condition for stability to axisymmetric disturbances is derived, which requires that Rayleigh's criterion of increasing circulation be satisfied, and otherwise depends only on b, V(b), γ and the density of the swirling liquid. This condition may be extended to include non-axisymmetric disturbances when V ∝ 1/r and when V ∝ r although in the latter case it is no longer a necessary one. It is shown that, in the case V ∝ r, as well as V ∝ 1/r, the ‘most unstable’ disturbance on a rotating column of fluid will be non-axisymmetric if the rotation speed at the surface is sufficiently great. Several applications of the theory are suggested, and a possible experiment to test it is described.

Acoustic-Emission Wave Response to a Dislocation Source in Layered Media

2014 ◽  
Vol 598 ◽  
pp. 135-140
Author(s):  
Rui Chong Zhang
Keyword(s):  
Acoustic Emission ◽  
Wave Propagation ◽  
Free Surface ◽  
Integral Transformation ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Form Solution ◽  
Arbitrary Orientation ◽  
Dislocation Source ◽  
Time Space

This paper presents synthesis of acoustic-emission (AE) wave propagation in multi-layer materials and simulation of AE wave responses at free surface. In particular, the AE source is modelled as an arbitrary-orientation dislocation over an inclined-to-surface fault within one layer or at the layer-to-layer interface, while the materials are assumed as multi-layer media, each of which is homogeneous, isotropic and linearly elastic. With the use of the integral transformation approach, the three-dimensional wave propagation in the materials is solved in transformed or frequency-wavenumber domain. Subsequently, a closed-form solution for wave responses at free surface is found, which can then be converted in time-space domain. Numerical examples are finally provided for illustration.

Inviscid reacting flow near a stagnation point

1969 ◽  
Vol 35 (4) ◽  
pp. 799-813 ◽  
Author(s):  
Raul Conti ◽  
Milton Van Dyke
Keyword(s):  
Stagnation Point ◽  
Closed Form Solution ◽  
Molecular Transport ◽  
Basic Structure ◽  
Reaction Rates ◽  
Form Solution ◽  
The Body ◽  
Reacting Flow ◽  
Reaction Parameter ◽  
History Of

A reacting flow free of molecular transport exhibits noteworthy behaviour in the neighbourhood of a blunt, symmetrical stagnation point. A local analytical study using the Lighthill-Freeman gas model reveals the basic structure of such a flow. Chemical activity is found to affect some, but not all, of the local characteristics of the flow. Unaffected are the pressure and velocity fields near the stagnation point, where the pressure varies quadratically and the velocity varies linearly as in an inert flow. In addition, the stagnation point is found to be in chemical equilibrium for all non-zero reaction rates. On the other hand the density, temperature, and concentration fields are affected by the non-equilibrium reactions. The extent of this effect can be predicted on the basis of a reaction parameter that measures the rate of reaction in terms of the velocity gradient at the stagnation point. A rapidly reacting flow (with reaction parameter greater than unity) approaches the stagnation point with vanishing gradients of density and temperature, whereas a slowly reacting flow approaches with infinite gradients. The flow field is represented mathematically by functions that are regular along the body but non-analytic in the normal direction. Numerical computations support the validity of the local closed-form solution, and provide information on the local effects of the chemical history of the flow.

Synthesis of Acoustic-Emission Wave Propagation in Multi-Layer Media

2013 ◽  
Vol 321-324 ◽  
pp. 1321-1330
Author(s):  
Rui Chong Zhang ◽  
Alhamid Alamin
Keyword(s):  
Acoustic Emission ◽  
Wave Propagation ◽  
Free Surface ◽  
Integral Transformation ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Form Solution ◽  
Arbitrary Orientation ◽  
Numerical Examples ◽  
Time Space

This paper presents synthesis of acoustic-emission (AE) wave propagation in multi-layer materials and simulation of AE wave responses at free surface. In particular, the AE source is modelled as an arbitrary-orientation dislocation over an inclined-to-surface fault within one layer or at the layer-to-layer interface, while the materials are assumed as multi-layer media, each of which is homogeneous, isotropic and linearly elastic. With the use of the integral transformation approach, the three-dimensional wave propagation in the materials is solved in transformed or frequency-wavenumber domain. Subsequently, a closed-form solution for wave responses at free surface is found, which can then be converted in time-space domain. Numerical examples are finally provided for illustration.

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