Transport vector field

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

In the limit of a vanishingly small time interval, this term represents the rate at which the extensive property N is transported convectively with the fluid motion across the control surfaces out of the control volume. Given that the fluid flow can be described by a vector field V, the convective transport flux across the area A of the control surface can be written as... 

Beginning with the vector form of the vorticity-transport equation (Eq. 6.47), derive the steady-state, axisymmetric, scaled vorticity transport equation (Eq. 6.50). Be careful in the evaluation of derivative operators on vector fields. 

Peclet number — The Peclet number (Pe) is a dimensionless ratio which relates the relative importance of ad-vection (transport of a scalar quantity in a vector field) and -> diffusion within a fluid. It is dependent on the heat capacity, density, - velocity, characteristic length, and heat transfer coefficient and it is defined as... 

The transport of energy in a conductive material is described by the vector field of heat flux... 

Although Eq. (6.52) is very similar to the equation for transport of the decaying passive scalar, an important difference is that in this case the flow v is not independent of the transported scalar field u>. Thus, the vorticity field is a dynamically active scalar related to the flow field through the relationship w = z (V x v), where z is the unit vector perpendicular to the plane of the flow. Nevertheless, Nam et al. (2000) have shown that the friction has a similar effect on the energy spectrum as the decay in the case of the scalar spectrum. The kinetic energy spectrum is related to the enstrophy spectrum Z(k) = (ul) by E(k) = Z(k)/k2, and therefore in the enstrophy cascade range... 

In Fig. 6 a velocity fields are shown for a system of four Rushton turbines. In addition to the velocity vector field, large arrows are used to illustrate the flow behavior. Each impeller creates a more or less independent symmetrical flow field. The multiple impeller system therefore shows very poor axial convection. The transport between the individual cells is performed mainly with the aid of axial turbulent dispersion. 

The continuous phase variables, which affect the behavior of each particle, may be collated into a finite c-dimensional vector field. We thus define a continuous phase vector Y(r, t) = [7 (r, t), 2(1, t. .., l (r, t)], which is clearly a function only of the external coordinates r and time t. The evolution of this field in space and time is governed by the laws of transport and interaction with the particles. The actual governing equations must involve the number density of particles in the particulate phase, which must first be identified. 

As an extension of the preceding discussion about the probable consequences of interactions between auxin waves in branched shoot systems, a control mechanism capable of coordinated growth of all plant parts (axes) can be proposed. Vector fields in the zones of major polar transport of auxin in axes of various orders can be envisioned as morphogenic fields specifying positional information... 

Transformations in Hilbert space, 433 Transition probabilities of negatons in, external fields, 626 Transport theory, 1 Transportation problems, 261,296 Transversal amplitude, 552 Transversal vector, 554 Transverse gauge, 643 Triangular factorization, 65 Tridiagonal form, 73 Triple product ensemble, 218 Truncation error, 52 Truncation of differential equations/ 388... 

Chapter 3 will be employed. Thus, in lieu of (x, t), only the mixture-fraction means ( ) and covariances ( , F) (/, j e 1,..., Nm() will be available. Given this information, we would then like to compute the reacting-scalar means and covariances (require additional information about the mixture-fraction PDF. A similar problem arises when a large-eddy simulation (LES) of the mixture-fraction vector is employed. In this case, the resolved-scale mixture-fraction vector (x, t) is known, but the sub-grid-scale (SGS) fluctuations are not resolved. Instead, a transport equation for the SGS mixture-fraction covariance can be solved, but information about the SGS mixture-fraction PDF is still required to compute the resolved-scale reacting-scalar fields. 

In an effort to improve the description of the Reynolds stresses in the rapid distortion turbulence (RDT) limit, the velocity PDF description has been extended to include directional information in the form of a random wave vector by Van Slooten and Pope (1997). The added directional information results in a transported PDF model that corresponds to the directional spectrum of the velocity field in wavenumber space. The model thus represents a bridge between Reynolds-stress models and more detailed spectral turbulence models. Due to the exact representation of spatial transport terms in the PDF formulation, the extension to inhomogeneous flows is straightforward (Van Slooten et al. 1998), and maintains the exact solution in the RDT limit. The model has yet to be extensively tested in complex flows (see Van Slooten and Pope 1999) however, it has the potential to improve greatly the turbulence description for high-shear flows. More details on this modeling approach can be found in Pope (2000). 

All gauge theory depends on the rotation of an -component vector whose 4-derivative does not transform covariantly as shown in Eq. (18). The reason is that i(x) and i(x + dx) are measured in different coordinate systems the field t has different values at different points, but /(x) and /(x) + d f are measured with respect to different coordinate axes. The quantity d i carries information about the nature of the field / itself, but also about the rotation of the axes in the internal gauge space on moving from x + dx. This leads to the concept of parallel transport in the internal gauge space and the resulting vector [6] is denoted i(x) + d i. The notion of parallel transport is at the root of all gauge theory and implies the introduction of g, defined by... 

Using the property of Unear independence of Fock space vectors in Eq. (31), and comparing Eqs. (31) and (24), we can see that Eq. (30) really represents the matrix elements of the parallel-transport operator. For closed paths, x(f ) = x(t") = x, Eq. (30) gives the holonomy operator Uki(x) and Ukk is the Wilson loop. Interestingly, the Wilson loop, which is supposed to describe a quark-antiquark interaction, is represented by a true quark and antiquark field, z and z, respectively. So, the mathematical trick can be interpreted physically. ... 

Consider the system and control volume as illustrated in Fig. 2.2. The Eulerian control volume is fixed in an inertial reference frame, described by three independent, orthogonal, coordinates, say z,r, and 9. At some initial time to, the system is defined to contain all the mass in the control volume. A flow field, described by the velocity vector (t, z,r, 9), carries the system mass out of the control volume. As it flows, the shape of the system is distorted from the original shape of the control volume. In the limit of a vanishingly small At, the relationship between the system and the control volume is known as the Reynolds transport theorem. 

It is also possible to consider the holonomy of the generic A in the vacuum. This is a round trip or closed loop in Minkowski spacetime. The general vector A is transported from point A, where it is denoted Aa 0 around a closed loop with covariant derivatives back to the point Aa () in the vacuum. The result [46] is the field tensor for any gauge group... 

This equation simply states that variations of energy density at a given spacetime point are due to transport along the Poynting vector plus transport along electric field. Clearly, when J = 0 all transport is along g [62, Eq. 43, p. 10]. 

A perspective on flow can be gained by examining the transport equations. Here, flow velocity v (actually a vector with a direction as well as a magnitude) adds to field-induced velocity U (also a vector) to determine the rate of change of concentration dc/dt at a given point in separation space... 

For the method of characteristics (MOC), the convective term is dealt with separately from the dipersive transport term by establishing a separate coordinate system along the convection vector for solving the dispersion problem. In most modeling programs, the convection is approximated with discrete particles. A certain number of particles with defined concentrations is used and moved along the velocity field (Konikoff and Biedehoeft, 1978). 

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