Eulerian homogeneity

There is a potential pitfall in the use of the Legendre transformation of Eulerian homogeneous equations. We start from the representation of the energy [/ of a single phase in natural variables as a total differential. We use purposely a sloppy notation, as sometimes common,... 

The expression for the energy will only be finite, if we have k — 1 > 0. Observe that the Eulerian homogeneity relation still holds for an ideal gas, if y/n is replaced by n/T. 

Question 6.4. Is the form U S, V, n, h) thermodynamically similar, or in other words, is it Eulerian homogeneous of first order ... 

In addition to overall mass conservation, we are concerned with the conservation laws for individual chemical species. Beginning in a way analogous to the approach for the overall mass-conservation equation, we seek an equation for the rate of change of the mass of species k, mk. Here the extensive variable is N = mu and the intensive variable is the mass fraction, T = mk/m. Homogeneous chemical reaction can produce species within the system, and species can be transported into the system by molecular diffusion. There is convective transport as well, but it represented on the left-hand side through the substantial derivative. Thus, in the Eulerian framework, using the relationship between the system and the control volume yields... 

In the recent years different numerical models for the conversion of wood in a packed bed have been presented, e. g, [3-6], Existing models mostly describe the as a porous media by an Eulerian approach, with the cons vation equations for the solid and the gas phase solved with the same mesh. This approach implies that heat and mass transfer can only be taken into account according to the dimensions of the bed but not within the particles itself. Temperature and species distributions are assumed to be homogenous over the fuel particles. Thus, the influence of the particle dimensions on the conversion process can only be captured by simplified assumptions or macrokinetic data. 

The Taylor h rpothesis has been further discussed by [144] (pp 5-7) and [59] (pp. 31-81) considering theories of double correlations between turbulence-velocity components, the features of the double longitudinal and lateral correlations in homogeneous turbulence, integral scales of turbulence, and Eulerian, Lagrangian and mixed Eulerian-Lagrangian correlations. 

Some attempts have been made to determine these time scales, in one set of experiments with a homogeneous grid-generated turbulence, the ratio of the Lagrangian to the Eulerian scales was reported to be three (Snyder and Lumley. 1971). Thus the ratio of the eddy diffusivity of a panicle that is too heavy to follow the gas motion to the eddy dilTusivity of the gas would he 1/3. What is meant by a bemy particle in this context One measure is the ratio of the characteristic particle time to the smallest time scale ofthe fluid... 

Batchelor, G. K. 1949. Diffusion in a Field of Homogeneous Turbulence 1. Eulerian Analysis, Australian Journal of Scientific Research, vol. 2, pp. 437-450. 

Druzhinin, O.A. and Elghobashi, S.E. (1999). A Lagrangian-Eulerian mapping solver for direct numerical simulation of a bubble-laden homogeneous turbulent shear flow using the two-fluid formulation. J. Comput. Phys., Vol. 154, pp. 174-196. 

Figure 12. Orientation of a solute molecule in the liquid crystal matrix. U, V, W is a space fixed axes system. The optical axis L of the homogeneously oriented liquid crystal is assumed to be parallel to W. The solute molecule fixed coordinate system x, y, z is rotated by the Eulerian angles

By extrapolation of the computed autocorrelation coefficient to zero time the square of the llagrangian fluctuating velocity can be obtained. The square root of this quantity can then be compared to the Eulerian rms velocity for the same column compartment. The two velocities should be equal in case of homogeneous turbulence. Devanathan (1991) shows that this is not the case in bubble columns but that homogeneous turbulence is approached at the highest gas velocities in the largest diameter column. 

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