Scalar compressibility

Several criticisms of these parameters have recently been pointed out. First, they have no specific association with a material plane (i.e., they are scalar parameters), despite the fact that cracks are known to nucleate on specific material planes. With traditional parameters it is difficult to account for the effects of crack closure under compressive loading. Traditional parameters have not been successful at unifying experimental results for simple tension and equibiaxial tension fatigue tests. Finally, a nonproportional loading history can always be constmcted for a given scalar equivalence parameter that holds constant the value of the scalar parameter, but which results in cyclic loading of material planes. For such histories, scalar parameters incorrectly predict infinite fatigue life. 

Three parameters thus need to be estimated, namely the scalar factor a, the compression factor c, and the shift d. Parameter b was dropped for two reasons (1) the effect of this exponent is to be explored, so it must remain fixed during a parameter-fitting calculation, and (2) the parameter estimation decreases in efficiency for every additional parameter. Therefore the model takes on the form... 

Direct numerical simulation studies of turbulent mixing (e.g., Ashurst et al. 1987) have shown that the fluctuating scalar gradient is nearly always aligned with the eigenvector of the most compressive strain rate. For a fully developed scalar spectrum, the vortexstretching term can be expressed as... 

In the LEM, turbulence is modeled by a random rearrangement process that compresses the scalar field locally to simulate the reduction in length scales that results from turbulent mixing. For example, with the triplet map, defined schematically in Fig. 4.2, a random length scale / is selected at a random point in the computational domain, and the scalar field is then compressed by a factor of three.14 The PDF for /,... 

Moin, P., K. Squires, W. Cabot, and S. Lee. 1991. A dynamic subgrid-scale model for compressible turbulence and scalar transport. J. Physics Fluids 3(ll) 2746-57. 

Batchelor [6] and Ghil and Childress [28] examine the transformation of the governing Eulerian equations for compressible flows from an inertial to a rotational frame. As it turns out, only the momentum equation is actually effected by this transformation because the material or substantial derivative operator of scalar functions is invariant to rotation. 

The ULTIMATE ID universal limiter of Leonard [109], was later extended to multi-dimensional flows by Thuburn [188]. The multi-dimensional method was designed for advected passive scalars in either compressible or incompressible flow and on arbitrary grids. 

In the present context, however, the condition Uv — Us — 2Mb = 0 is desired for baryon anti-baryon pair creation [27] above the critical density. To achieve the higher density we gave up the self-consistent calculations and followed a crude method we compressed the Pb nucleus to 2, 4, 6, and 8.5 times normal nuclear matter density by scaling the vector density with a scaling factor c and the radius with the factor c to keep baryon number conserved. The. scalar potential was scaled with the corresponding nuclear matter factor as follows we calculated the scalar potential Us pv) for nuclear matter as a function of density (see Fig. 3, dotted curve), scaled the vector density p by a factor c to cp . Now from Fig. 3 we took the value of Us which corresponds to the density cp . For example, at p = 8po (i.e. p — 1-16 fm ) the value... 

In DNS of single-phase flows, a complete set of compressible Navier-Stokes, energy, and scalar transport equations are calculated together with the equation of state and some constitutive relations [3]. In DNS of particle-laden flows, in addition to carrier-gas equations, the Lagrangian form of particle (droplet) equations are solved via standard difference schemes [5]. 

The operator V can operate on vector functions as well as on scalar functions. An example of a vector function is the velocity of a compressible flowing fluid... 

A normalized scalar product of two spectral vectors was used during matching as the similarity measure, and sequential searches through the entire library were always performed. Thus, each spectrum in turn was treated as a query. To simulate small variances in data acquisition, and/or spectral differences for very similar compounds, 1% and 5% of random white noise, has been added. Appropriate decomposition (wavelet or PCA) was then performed, and the resulting vector was compared to each of the spectral vectors in the compressed library. 

J.N. Bradley, C.M. Brislawn and T. Hopper. The FBI Wavelet/Scalar Quantization Standard for Gray-Scale Fingerprint Image Compression, in SPIE Proc. Visual Info. Processing II, Orlando, FL, 1992, pp. 293-304. 

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