Characteristic scales velocity

To generate characteristic velocities and bring a molecular system toequillbrium at th e sim illation temperature, atom s are allowed to in teract W ith each other through the equation s of motion. For isothermal simulations, a temperature bath" scales velocities to drive the system towards the simulation temperature,. Scaling occurs at each step of a simulation, according to equation 2S. 

Here f denotes the fraction of molecules diffusely scattered at the surface and I is the mean free path. If distance is measured on a scale whose unit is comparable with the dimensions of the flow channel and is some suitable characteristic fluid velocity, such as the center-line velocity, then dv/dx v and f <<1. Provided a significant proportion of incident molecules are scattered diffusely at the wall, so that f is not too small, it then follows from (4.8) that G l, and hence from (4.7) that V v° at the wall. Consequently a good approximation to the correct boundary condition is obtained by setting v = 0 at the wall. ... 

Most of the energy dissipation occurs on a length scale about 5 times the Kolmogorov eddy size. The characteristic fluctuating velocity for these energy-dissipating eddies is about 1.7 times the Kolmogorov velocity. 

Using the system (9.15-9.17) we determine the distribution of velocity, temperature and pressure within the liquid and vapor domains. We render the equations dimensionless by the following characteristic scales l,o for velocity, 7l,o for temperature, Pl,o for density, Pl,q for pressure, Pl,o Lo f " force and L for length... 

Assuming steady state in Eqs. (10.8-10.10) and (10.18-10.20), we obtain the system of equations, which determines steady regimes of the flow in the heated miero-channel. We introduce values of density p = pp.o, velocity , length = L, temperature r = Ti 0, pressure AP = Pl,o - Pg,oo and enthalpy /Jlg as characteristic scales. The dimensionless variables are defined as follows ... 

According to measurements made in the atmosphere, the Lagrangian time scale is of the order of 100 sec (Csanady, 1973). Using a characteristic particle velocity of 5 m sec", the above conditions are 100 sec and L > 500 m. Since one primary concern is to examine diffusion from point sources such as industrial stacks, which are generally characterized by small T and L, it is apparent that either one (but particularly the second one) or both of the above constraints cannot be satisfied, at least locally, in the vicinity of the point-like source. Therefore, in these situations, it is important to assess the error incurred by the use of the atmospheric diffusion equation. 

This particular nondimensionalization and the small-parameter expansion in Mach-number is useful in showing that the pressure-gradient term may neglect acoustic disturbances. For the purposes of modeling real flow problems, however, the accoustic-scale nondimensionalization is not particularly useful. Rather, the velocities and pressure should be scaled using a characteristic flow velocity. Of course, for high-speed flows, where fluid velocities are comparable to the sound speed, the full pressure variation must be retained. 

When particles are accelerated in a gas, their motion is governed by the balance between inertial, viscous, and external forces. An important characteristic scale is the time for an accelerated particle to achieve steady motion. To find this parameter, the deceleration of a particle by friction in a stationary gas is considered. In the absence of external forces, the velocity of a particle (q) traveling in the x direction is calculated by ... 

Designs based on such a value of Re may well not adequately describe the true hydrodynamics of the system. Re is defined as pvA1 2/pij where p and p are the density and viscosity of the fluid phase, and v and A1 2 are a characteristic fluid velocity and length-scale, respectively, of the system under study. Reproduced with permission from Johns et al. (2000). Copyright (2000), A.I.Ch.E. 

To be consistent with our preceding analyses, we begin by nondimensionalizing. For this purpose, we identify characteristic scales. Because the perturbation to the channel shape is assumed to be very small, it is apparent that the relevant velocity scale is... 

Following our usual custom, we now nondimensionalize. The physically obvious characteristic scales are the length scales for variations of the velocity and perturbation pressure in the x and v directions, and the characteristic magnitude of the velocity in the x direction,... 

Furthermore, the length scale characteristic of velocity gradients in the thin gap is just the characteristic distance across the gap,... 

One final remark should be made about the nondimensionalization that led to (7-2) and (7 3), and thence for Re creeping-flow equations, (7-6). This concerns the scale factor chosen to nondimensionalize the pressure. Assuming for simplicity that the geometry and boundary conditions lead to a single characteristic length, velocity, and time scale, the nondimensionalization of u, t, and V in (7-1) is unambiguous. However, even for this simplest case there is a second possible choice for nondimensionalization of p, namely, pU2, which is another combination of dimensional parameters with dimensions of... 

We have carried out all of the analysis of this section with the equations and boundary conditions in dimensional form. However, in view of the rather large number of dimensional parameters that appear in (12-82)-(12-85), there is a definite advantage in nondimension-alizing. The problem is to choose characteristic scales because these must be intrinsic, and formed from the available dimensional parameters, rather than explicitly evident, The available parameters are p. p2. m, m, g, and y. Given that the mechanism for motion is the body force that is due to gravity, it would be surprising if the characteristic velocity did not involve g. In fact, a combination of dimensional parameters with units of velocity is (gv)1/3. The decision to use v or v2 in this definition is completely arbitrary. Hence we take... 

In Section 1.1 we briefly described the velocity field for some cases of gradient flows with nonuniform structure. For particles whose size is much less than the characteristic scale of spatial nonuniformity of the flow, the velocity distribution (1.1.15) can be used as the velocity distribution remote from the particle in problems about mass transfer to a particle in a linear shear flow. 

In most cases one is interested in fluid flows at scales that are much larger than the distance between the molecules. The value of the molecular mean free path in air at room temperature and 1 atm of pressure is A = 6.7 x 10-8 m and in water A = 2.5 x 10-10 m. When the Knudsen number - defined as the ratio of the molecular mean-free-path to a characteristic length scale of the flow (e.g. the size of the smallest eddies) - is small, the fluid can be described as a continuous medium in motion. In this continuum approximation the flow can be characterized by the velocity field v(x, t) representing the instantaneous velocity of infinitesimal fluid elements at time t and at position x. Fluid elements represent small volumes of fluid that are much smaller than the smallest characteristic scale of the flow, but sufficiently large to contain a large number of molecules so that a well defined local velocity exists and molecular fluctuations can be neglected. 

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