Velocity potential

The Champ-Sons model has been developed to quantitatively predict the field radiated by water- or solid wedge- eoupled transdueers into solids. It is required to deal with interfaces of complex geometry, arbitrary transducers and arbitrary excitation pulses. It aims at computing the time-dependent waveform of various acoustical quantities (displacement, velocity, traction, velocity potential) radiated at a (possibly large) number of field-points inside a solid medium. 

The transmission coefficient Cl (Qj,t), considering transient (broadband) sources, is time-dependent and therefore accounts for the possible pulse deformation in the refraction process. It also takes account of the quantity actually computed in the solid (displacement, velocity potential,...) and the possible mode-conversion into shear waves and is given by... 

Equation 7 should not be constmed to mean that the constant is the same for all stieamlines. Once the velocities are known, this immediately gives the piessuie. For a constant density, itiotational flow, the velocity components themselves are detivable from a velocity potential, ( ) ... 

For an irrotational, incompressible, and frictionless fluid flow there exists a scalar velocity potential 4> such that the velocity vector V is... 

The analytical solution for an infinitely flanged slot can be obtained by assuming that the inlet is composed of elemental point sinks.The velocity field of an infinitely flanged slot can be obtained by assuming the velocity to be uniformly distributed across the opening. The contribution to the velocity potential at point (x, y) due to the elemental line sink of length and located at (0, Q is given by... 

When calculating the velocity potential at a point near an infinitely flanged exhaust opening, the hood face can be assumed to be divided into many area sinks, each of them contributing to the potential at a point in space. The overall velocity potential is then obtained by integrating over the inlet area. The velocity component in the. v direction, for example, is then... 

The velocity potential for the flow field in front of an expanding piston surface can now be derived from the boundary condition so that at its surface the medium velocity equals the piston velocity. In this way, Taylor (1946) found... 

These results were analytically reproduced by Taylor (1985), who employed a velocity potential function for a convected monopole. This concept makes it possible to model an elongated vapor cloud explosion by one single volume source which is convected along the main axis at burning velocity, and whose strength varies proportionally to the local cross-sectional cloud area. 

The function is known as the velocity potential. In equation A.36 the minus sign is arbitrary but is usually incorporated so that flow is from a high value of the velocity potential to a low value. 

For potential flow, ie incompressible, irrotational flow, the velocity field can be found by solving Laplace s equation for the velocity potential then differentiating the potential to find the velocity components. Use of Bernoulli s equation then allows the pressure distribution to be determined. It should be noted that the no-slip boundary condition cannot be imposed for potential flow. 

Moreover further developments will be restricted to linear acoustic. Therefore, space variations are small enough to approximate p(r) p0. Regarding the velocity field v as the gradient of the velocity potential v = — V we can write... 

Just as there arc many types of fluids, so there arc. partly as a result, many types of fluid flow. Uniform flow is steady in lime, or the same at all points in space. Steady flow is flow of which the velocity at a point fixed with respect to a fixed system of coordinates is independent of lime. Many common types of flow can be made steady by a suitable choice of coordinates. Rotational flows have appreciable vorticily, and they cannot he described mathematically by a velocity potential function. Turbulent flow is flow in which the fluid velocity at a fixed point fluctuates with lime in a nearly random way. The motion is essentially rotational, and is... 

The equations above can be represented in terms of the velocity potential to yield... 

A specific solution for the velocity field satisfying the modified equations above is given by the velocity potential function below ... 

The V may be interpreted as a gravitational potential, velocity potential or, in the present context, as a circulation potential. The differential equation may be solved by separation of the variables under the assumption that the potential may be defined as the product of three one-dimensional potentials, or cartesian components of V = X (x) -Y(y) Z(z). This solution is substituted into the equation and after differentiation, divided by V, to give... 

At t = 0 the surfaces S = a, b coincide with W = a, b respectively. However, at time df the surfaces S = a, b now coincide with surfaces for which W = (a, b) + Edt. The surface S = a has therefore moved from W = a to W = a + Edt, i. e. dW = Edt. To emphasize the parallel between Sommerfeld s quantization rules and the HJ equation, the latter is reformulated [25] as a differential field equation of the action potential, W, in the same way that fluid motion is described by a velocity potential, or the propagation of a wave front. The surfaces of constant S may thus be considered as wave fronts propagating in configuration space. Let s measure the distance normal to the moving surface. Writing dW = VlT d.s, gives the velocity of the wave front... 

Gaussian half peak widths (time units) of eluting peaks complex conductivity angular velocity -potential... 

The causal interpretation of quantum theory as proposed by De Broglie and Bohm is an extension of the hydrodynamic model originally proposed by Madelung and further developed by Takabayasi [36]. In Madelung s original proposal R2 was interpreted as the density p(x) of a continuous fluid with stream velocity v= VS/rri. Equation (5) then expresses conservation of fluid, while (6) determines changes of the velocity potential S in terms of the classical potential V, and the quantum potential... 

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