Velocity fields

Let Ap, Au and AT denote the deviations of the mass density, p, the velocity field, u, and the temperature, T, fiom their fiill equilibrium values. The fluctuating, linearized Navier-Stokes equations are... 

The third application is to velocity field fluctuations. For an equilibrium fluid the velocity field is, on average, zero everywhere but it does fluctuate. The correlations trim out to be... 

It is convenient to analyse tliese rate equations from a dynamical systems point of view similar to tliat used in classical mechanics where one follows tire trajectories of particles in phase space. For tire chemical rate law (C3.6.2) tire phase space , conventionally denoted by F, is -dimensional and tire chemical concentrations, CpC2,- are taken as ortliogonal coordinates of F, ratlier tlian tire particle positions and velocities used as tire coordinates in mechanics. In analogy to classical mechanical systems, as tire concentrations evolve in time tliey will trace out a trajectory in F. Since tire velocity functions in tire system of ODEs (C3.6.2) do not depend explicitly on time, a given initial condition in F will always produce tire same trajectory. The vector R of velocity functions in (C3.6.2) defines a phase-space (or trajectory) flow and in it is often convenient to tliink of tliese ODEs as describing tire motion of a fluid in F with velocity field/ (c p). 

If a flow satisfies the condition of zero vorticity, that is, the velocity field v is such that V X V = 0, then there exists a function v such that v = Vv. In that case, one can describe the fluid mechanical system with the following Lagrangean density... 

By using the relationship between the fluid current and its veloeity field, J = Pv, a quantum fluid velocity field of... 

At Che opposite limit, where Che density Is high enough for mean free paths to be short con ared with pore diameters, the problem can be treated by continuum mechanics. In the simplest ease of a straight tube of circular cross-section, the fluid velocity field can easily be obtained by Integrating Che Nsvler-Stokes equations If an appropriate boundary condition at Che... 

Maxwell considered the motion of a gas in the neighborhood of a plane solid wall, in che presence of a temperature gradient. In particular, when Che velocity field is one dimensional and everywhere parallel to the wall, and the temperature gradient is parallel to the velocity field, he found that... 

The well-known inaccuracy of numerical differentiation precludes the direct calculation of pressure by the insertion of the computed velocity field into Equation (3.6). This problem is, however, very effectively resolved using the following variational recovery method Consider the discretized form of Equation (3.6) given as... 

In the decoupled scheme the solution of the constitutive equation is obtained in a separate step from the flow equations. Therefore an iterative cycle is developed in which in each iterative loop the stress fields are computed after the velocity field. The viscous stress R (Equation (3.23)) is calculated by the variational recovery procedure described in Section 1.4. The elastic stress S is then computed using the working equation obtained by application of the Galerkin method to Equation (3.29). The elemental stiffness equation representing the described working equation is shown as Equation (3.32). 

If the reference time f and the current time t coincide then the reference and current positions will also coincide and the right-hand side of Equation (3.77) can be replaced by the reference position defined as x in Equation (3.76). In a velocity field given as = u x,t ) the motion of a material point can be described... 

The right-hand side of Equation (3,87) is set to zero considering that DA//Dt, DFIDt and the divergence of the velocity field in incompressible fluids are all equal to zero. Therefore, after integration Equation (3.87) yields... 

Step 5 - the obtained velocity field is used to solve the energy equation (see Chapter 3, Section 3). 

Initial distribution and the predicted free surface boundaries within the twin-blade mixer represented by the mesh configurations shown in Figure 5.4, after 30, 60 and 90° rotation of the left blade are presented in Figures 5.5a to 5.5d, respectively. Samples of the predicted velocity fields after 30 and 45 ° rotation of the left rotor are shown in Figures 5.6a to 5.6b, respectively. The finite element... 

Figure 5.6 (a.b) The predicted velocity fields after 30 and 45 rotation of the left blade in the partially filled twin-blade mixer... 

Predicted velocity fields in a segment adjacent to the tip of the blade in the single-blade mixer, described in a previous sub-section, before and after imposition of the wall slip are shown in Figures 5.14a and 5.14b, respectively. As expected, momentum transfer from the rotating wall to the fluid is significantly affected by the imposition of the wall slip. In Figures 5.15a and 5.15b temperature contours corresponding to these velocity fields are shown. 

Figure 5.14 (a) The predicted velocity field corresponding to no-slip wall boundary conditions, (b) Tlie predicted velocity field corresponding to partial slip boundary conditions... 

Using the described algorithm the flow domain inside the cone-and-plate viscometer is simulated. Tn Figure 5.17 the predicted velocity field in the (r, z) plane (secondary flow regime) established inside a bi-conical rheometer for a non-Newtonian fluid is shown. 

Step 3 - using the calculated velocity field, find the shear rate and update viscosity using the power law model. 

Natural convection occurs when a solid surface is in contact with a fluid of different temperature from the surface. Density differences provide the body force required to move the flmd. Theoretical analyses of natural convection require the simultaneous solution of the coupled equations of motion and energy. Details of theoretical studies are available in several general references (Brown and Marco, Introduction to Heat Transfer, 3d ed., McGraw-HiU, New York, 1958 and Jakob, Heat Transfer, Wiley, New York, vol. 1, 1949 vol. 2, 1957) but have generally been applied successfully to the simple case of a vertical plate. Solution of the motion and energy equations gives temperature and velocity fields from which heat-transfer coefficients may be derived. The general type of equation obtained is the so-called Nusselt equation hL I L p gp At cjl... 

The theoretical velocity field was experimentally verified Sparrow, and Goldstein (Jnt. J. Heat Mass Transfer, 10,... 

Although direct numerical simulations under limited circumstances have been carried out to determine (unaveraged) fluctuating velocity fields, in general the solution of the equations of motion for turbulent flow is based on the time-averaged equations. This requires semi-... 

Isolated Droplet Breakup—in a Velocity Field Much effort has focused on defining the conditions under which an isolated drop will break in a velocity field. The criterion for the largest stable drop... 

The results of computations of T o for an isolated fiber are dhistrated in Figs. 17-62 and 17-63. The target efficiency T t of an individual fiber in a filter differs from T o for two main reasons (Pich, op. cit.) (1) the average gas velocity is higher in the filter, and (2) the velocity field around the individual fibers is influenced by the proximity of neighboring fibers. The interference effect is difficult to determine on a purely theoretical basis and is usually evaluated experimentally. Chen (op. cit.) expressed the effecd with an empirical equation ... 

Braconnier, R. 1988. Bibliographic review of velocity field in the vicinity of local exhaust hood openings. American Industrial Hygienists Association Journal, vol. 49 no. 4, 18.5-198. 

FIGURE 10.10 Velocity field outside circular (left) and flanged circular (right) opening. Velocities are given on a plane of the diameter and in percentages of opening velocity. 

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... 

Superposition of Flows Potential flow solutions are also useful to illustrate the effect of cross-drafts on the efficiency of local exhaust hoods. In this way, an idealized uniform velocity field is superpositioned on the flow field of the exhaust opening. This is possible because Laplace s equation is a linear homogeneous differential equation. If a flow field is known to be the sum of two separate flow fields, one can combine the harmonic functions for each to describe the combined flow field. Therefore, if d)) and are each solutions to Laplace s equation, A2, where A and B are constants, is also a solution. For a two-dimensional or axisymmetric three-dimensional flow, the flow field can also be expressed in terms of the stream function. 

Flow Past a Point Sink A simple potential flow model for an unflanged or flanged exhaust hood in a uniform airflow can be obtained by combining the velocity fields of a point sink with a uniform flow. The resulting flow is an axially symmetric flow, where the resulting velocity components are obtained by adding the velocities of a point sink and a uniform flow. The stream function for this axisymmetric flow is, in spherical coordinates. 

Semi-Theoretical and Empirical Velocity Fields Since the use of formulas to calculate the velocities outside an arbitrary opening could be very tedious, only some examples of these formulae are given. These calculations are best done on computers and there are some dedicated programs to calculate and visualize the flow fields outside exhaust openings. There could sometimes be problems when calculating the velocity field outside an opening close to... 

Circular Flanged Openings- This velocity field has a circular symmetry and it suffices to calculate the velocity directed into the opening ... 

Rectangular Flanged Openings- C i Jhe velocity fields for these openings must be calculated for each direction individually. Here and are the nondimensional velocities parallel to rhe opening plane and v. is the nondimensional velocity perpendicular to the opening plane directed toward the opening plane ... 

IJnflanged Circular, Rectangular, and Slot Openings For unflanged openings no explicit equations exist for the flow fields. However, it is possible to calculate the velocity field outside a specific BEO using computers. These... 

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