Velocity Instantaneous

With LES we get much more information than with traditional time-averaged turbulence models, since we are resolving most of the turbulence. In Fig. T1.15 the computed u velocity is shown as a function of time in two cells one cell is located in the wall jet (Fig.. 15a), and the other cell is in the middle of the room (Fig. ll.lSh). It is found the instantaneous fluctuations are very large. For example, in the region of the wall jet below the ceiling where the time-averaged velocity u)/l] ) is typically 0.5, the instantaneous velocity fluctuations are between 0.2 and 0.9. In the middle of the room, which is a low-velocity region, the variation of u is much slower, i.e., the frequency is lower. 

In the present discussion only the problem of steady flow will be considered in which the time average velocity in the main stream direction X is constant and equal to ux. in laminar flow, the instantaneous velocity at any point then has a steady value of ux and does not fluctuate. In turbulent flow the instantaneous velocity at a point will vary about the mean value of ux. It is convenient to consider the components of the eddy velocities in two directions—one along the main stream direction X and the other at right angles to the stream flow Y. Since the net flow in the X-direction is steady, the instantaneous velocity w, may be imagined as being made up of a steady velocity ux and a fluctuating velocity ut, . so that ... 

That the most likely coarse velocity is equal to the most likely terminal velocity can only be true in two circumstances either the system began in the steady state and the most likely instantaneous velocity was constant throughout the interval, or else the system was initially in a dynamically disordered state, and x was large enough that the initial inertial regime was relatively negligible. These equations are evidently untrue for x —> 0, since in this limit the most... 

Figure 8. The dimensionless thermal conductivity, b1tljcjljX,(t), at p = 0.8 and T0 = 2. The symbols are the simulation data, with the triangles using the instantaneous velocity at the end of the interval, X/(t), Eq. (277), and the circles using the coarse velocity over the interval, Eq. (278). The solid line is the second entropy asymptote, essentially Eq. (229), and the dotted curve is the Onsager-Machlup expression om(t)> Eq. (280). (Data from Ref. 6.)...

Figure A 1.2 Determination of instantaneous velocity at various points in a reaction progress curve, from the slope of a tangent line drawn to a specific time point.

Figure A1.3 Linear relationship between (A) instantaneous velocity and [5], and between (B) initial velocity and [5]0 for a first-order reaction.

When an electric field is applied, jumps of the ions in the direction of the field are somewhat preferred over those in other directions. This leads to migration. It should be noted that the absolute effect of the field on the ionic motion is small but constant. For example, an external field of 1 V m-1 in water leads to ionic motion with a velocity of the order of 50 nm s 1, while the instantaneous velocity of ions as a result of thermal motion is of the order of 100 ms-1. 

The instantaneous velocity is then used to estimate the instantaneous local static pressure using Bernoulli s equation of the following form ... 

The resulting equation for the instantaneous velocity of fluid exiting the leak is... 

A record of the axial velocity component vx for steady turbulent flow in a pipe would look like the trace shown in Figure 1.22. The trace exhibits rapid fluctuations about the mean value, which is determined by averaging the instantaneous velocity over a sufficiently long period of time. Figure 1.22 shows the case in which the mean velocity remains constant this is therefore known as steady turbulent flow. In unsteady turbulent flow, the mean value changes with time but it is still possible to define a mean value because, in practice, the mean will drift slowly compared with the frequency of the fluctuations. 

Writing the instantaneous velocity component vx as the sum of the mean value and the fluctuation... 

Writing the instantaneous velocity components vx, vy as the sums of the mean values and fluctuations, and taking the time average gives the mean momentum flux as ... 

The displacement of matter will be called u and their instantaneous velocity v = du/dt. 

The one-point joint composition PDF contains random variables representing all chemical species at a particular spatial location. It can be found from the joint velocity, composition PDF by integrating over the entire phase space of the velocity components. The loss of instantaneous velocity information implies the following. 

As discussed above, the GLM was developed in the spirit of Reynolds-stress modeling. An obvious extension is to devise large-eddy-based closures for the conditional acceleration. For this case, it is natural to decompose the instantaneous velocity into its resolved and unresolved components 42... 

Thus, the instantaneous velocity (dr/dr) is equal to the terminal velocity m0 in the gravitational field, increased by a factor of ra>2/g. 

In your previous courses in science or physics, you probably learned the difference between instantaneous velocity and average velocity. How did you use a displacement-time graph to determine instantaneous velocity and average velocity Write a memo that explains instantaneous rate and average rate to a physicist, by comparing reaction rate with velocity. 

Let us return for the moment to Eq. (2.2). In atmospheric problems it is impossible to solve the equations of motion analytically. Under these conditions information about the instantaneous velocity field u is available only from direct measurements or from numerical simulations of the fluid flow. In either case we are confronted with the problem of reconstructing the complete, continuous velocity field from observations at discrete points in space, namely the measuring sites or the grid points of the numerical model. The sampling theorem tells us that from a set of discrete values, only those features of the field with scales larger than the discretization interval can be reproduced in their entirety (Papoulis, 1%5). Therefore, we decompose the wind velocity in the form... 

A detailed investigation of the flow, using PIV, with and without control is carried out. The instantaneous velocity field is obtained by the method described in section 18.2, with interrogation regions of 6 x 6 pixels corresponding to a physical dimension of 0.36 X0.36 mm. Typical mean and instantaneous velocity fields of the near region of a flame without suction at U =3.9 m/s and = 2.0 are shown in Figs. 18.6a and 18.66, respectively. 

The mean velocity field is obtained by averaging 60 instantaneous velocity fields. For these conditions the flame is anchored to the collar lip. Superimposed on... 

The instantaneous velocity field measurements indicate that a secondary stream, travelling in the direction opposite to the primary flow, is established within the collar to create the countercurrent shear layer. The dynamics of the countercurrent shear layer is conducive to the stabilization of the premixed jet flame up to... 

Mungal, M.G., L. Lourenco, and A. Krothapalh. 1995. Instantaneous velocity measurements in laminar and turbulent premixed flames using on-line PIV. Combustion Science Technology 106 239-65. 

Aybers and Tapucu (A4, A5) measured trajectories of air bubbles in water. When surface-active agents continue to accumulate during rise, the terminal velocity may never reach steady state (A4, Bl) and may pass through a maximum (W4). Five types of motion were observed, listed in Table 7.1 with Re based on the maximum instantaneous velocity. Secondary motion of fluid par-... 

The total drag on the sphere may be obtained, as in steady flow, by integrating the normal and shear stresses over the surface. In terms of the instantaneous velocity U the result is (L4) ... 

The first term of Eq. (11-11) is the Stokes drag for steady motion at the instantaneous velocity. The second term is the added mass or virtual mass contribution which arises because acceleration of the particle requires acceleration of the fluid. The volume of the added mass of fluid is 0.5 F, the same as obtained from potential flow theory. In general, the instantaneous drag depends not only on the instantaneous velocities and accelerations, but also on conditions which prevailed during development of the flow. The final term in Eq. (11-11) includes the Basset history integral, in which past acceleration is included, weighted as t — 5) , where (t — s) is the time elapsed since the past acceleration. The form of the history integral results from diffusion of vorticity from the particle. 

The first term again represents drag in steady motion at the instantaneous velocity, with Cd an empirical function of Re as in Chapter 5. The other terms represent contributions from added mass and history, with empirical coefficients, Aa and Ah, to account for differences from creeping flow. From measurements of the drag on a sphere executing simple harmonic motion in a liquid, Aa and Ah appeared to depend only on the acceleration modulus according to ... 

The instantaneous drag on a rigid spherical particle moving with velocity Lp in a fluid whose instantaneous velocity in the vicinity of the particle is follows from an extension to Eq. (11-30) ... 

There are also many ways of expressing the velocity of a chemical species present in a flow system. We do not concern ourselves here with the instantaneous velocity of the individual molecules of a species, but rather with the average macroscopic velocities with which the species travel. These may be measured from a stationary coordinate system, but for flow systems the velocities of individual species are frequently measured from a coordinate frame moving with (a) the mass-average velocity of the stream, (b) the molar average velocity of the stream, or (c) the velocity of one particular component. The mass- and the molar-average velocity are defined in Table II and the notation for the various velocities of an individual species is given. 

In the description of turbulent fluctuations it has been useful to employ the nomenclature and approach developed by Reynolds (R2). In this instance the instantaneous velocity is made up of two terms, the time-average velocity and the fluctuating velocity as indicated in the following expressions ... 

Finally, the plot in Fig 11 shows their calculations for the ratio of instantaneous velocity to terminal as a function of flight path distance in terms of expl thickness ie, 0.5 on the plot is a flight distance equal to 0.5 of the expl thickness, etc... 

Under the steady-state conditions that usually apply (see p. 449), the rate of increase of product will be the same as the rate of decrease of substrate. The units of velocity are moles per liter per second (M s 1) or more traditionally in enzymology moles per liter per minute. We are interested in the instantaneous velocity, which at... 

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