Turbulence, point velocity

In homogeneous isotropic turbulence, the two-point velocity correlation function can be expressed (Pope 2000) in terms of the longitudinal (/) and transverse (g) auto-correlation functions ... 

Transported PDF methods combine an exact treatment of chemical reactions with a closure for the turbulence field. (Transported PDF methods can also be combined with LES.) They do so by solving a balance equation for the joint one-point, velocity, composition PDF wherein the chemical-reaction terms are in closed form. In this respect, transported PDF methods are similar to micromixing models. 

It should be recognized that the boundary conditions of the problem will establish the value of the hydrodynamic velocity, u. In the case of most turbulent flows the indirect influence of molecular diffusion on the hydro-dynamic velocity can be neglected. It should be emphasized that the hydrodynamic velocity is the time-average point velocity in Reynolds sense (R2). Under unsteady, nonuniform conditions of flow between parallel plates the material balance may be expressed for turbulent flow in the following form ... 

The point is that no unambiguous turbulent burning velocity can be measured at present. However, it is perfectly possible to determine burning rates with reference to arbitrarily chosen surfaces in time-exposed photographs these rates must serve at present to characterize turbulent flames. 

Figure 2 presents three typical flow patterns were observed at narrow annulus, which are the flow with small bubbles whose size is less than a channel width (see Fig. 2a), the flow with large Taylor bubbles (see Fig. 2b) and the flow with the cell structure of liquid plugs, (see Fig. 2c). The flow pattern map is presented on Fig. 3. The first type of the flow is observed at the superficial liquid velocities greater than 2 m/s when the flow becomes turbulent (point 1 and line A in Fig. 3). At such velocities the flow is turbulent and small bubbles...

Fig. 7.16. Computed loci of changes in turbulent burning velocity and related parameters during engine combustion at four different engine speeds. Ignition occurs at the lowest point on each curve. As engine speed increases, combustion moves away from the continuous laminar flame sheet regime. From [132].

FIGURE 3.1 Typical point velocity behavior in turbulent flows. 

This turbulence quantity is defined as the covariance of the difference in velocity between two points in physical space. The two-point velocity structure function should not be confused with the normal component of the Re3molds stresses, which is a one-point, one time, covariance of the velocity. 

In the case of turbulent advection velocity, the transported quantity in the PBE (i.e. the NDF) fluctuates around its mean value. These fluctuations are due to the nonlinear convection term in the momentum equation of the continuous phase. In turbulent flows usually the Reynolds average is introduced (Pope, 2000). It consists of calculating ensemble-averaged quantities of interest (usually lower-order moments). Given a fluctuating property of a turbulent flow f>(t,x), its Reynolds average at a fixed point in time and space can be written as... 

Taylor (95) pointed out that in order to quantify special structure of turbulence, the velocity fluctuations at two neighbouring points, 1 and 2, in flow flelds have to be observed. This will define correlation function R as... 

A somewhat less computationally demanding approach for calculating the composition field is based on the one-point joint composition PDF, f ), instead of the one-point joint velocity-composition PDF, f v,yf). With this approach, information on the turbulent flow / velocity field must be provided by appropriate flow, turbulence, scalar-flux and micro-mixing models. The reaction rate can still be exactly dealt with. A one-point joint composition PDF transport equation similar to the one-point joint velocity-composition PDF transport equation, (12.4.2-2), can be derived. For statistically stationary flow ... 

This example is a bit unrealistic in that the flame will most likely blow out due to the high exit velocity of the jet. As the flow velocity of the jet is increased, the flame moves downstream to a new location where the turbulent burning velocity equals the flame velocity. As the velocity is increased, a point is eventually reached where the burning location is so far downstream that the fuel concentration is below the lower flammability limit due to air entrainment. Mudan and Croce (1988) provide flame blowout criteria. 

Figure 2.5 shows a pipe with a flow restriction in the form of an orifice. The flow through the orifice is turbulent. The velocity will increase fi om point A to point B. 

The one-point velocity correlations known from the turbulence model (or experiments) must be satisfied. 

The results obtained are presented in Fig. 3.10. No pressure effect on the turbulent flame velocity St was observed. The experimental points obtained, at 5 times initial pressure, are chaotically distributed in the area between the two dashed lines. 

A low Reynolds number indicates laminar flow and a paraboHc velocity profile of the type shown in Figure la. In this case, the velocity of flow in the center of the conduit is much greater than that near the wall. If the operating Reynolds number is increased, a transition point is reached (somewhere over Re = 2000) where the flow becomes turbulent and the velocity profile more evenly distributed over the interior of the conduit as shown in Figure lb. This tendency to a uniform fluid velocity profile continues as the pipe Reynolds number is increased further into the turbulent region. 

The basic concepts of a gas-fluidized bed are illustrated in Figure 1. Gas velocity in fluidized beds is normally expressed as a superficial velocity, U, the gas velocity through the vessel assuming that the vessel is empty. At a low gas velocity, the soHds do not move. This constitutes a packed bed. As the gas velocity is increased, the pressure drop increases until the drag plus the buoyancy forces on the particle overcome its weight and any interparticle forces. At this point, the bed is said to be minimally fluidized, and this gas velocity is termed the minimum fluidization velocity, The bed expands slightly at this condition, and the particles are free to move about (Fig. lb). As the velocity is increased further, bubbles can form. The soHds movement is more turbulent, and the bed expands to accommodate the volume of the bubbles. 

Circulating fluidized beds (CFBs) are high velocity fluidized beds operating well above the terminal velocity of all the particles or clusters of particles. A very large cyclone and seal leg return system are needed to recycle sohds in order to maintain a bed inventory. There is a gradual transition from turbulent fluidization to a truly circulating, or fast-fluidized bed, as the gas velocity is increased (Fig. 6), and the exact transition point is rather arbitrary. The sohds are returned to the bed through a conduit called a standpipe. The return of the sohds can be controUed by either a mechanical or a nonmechanical valve. 

In a free jet the absence of a pressure gradient makes the momentum flux at any cross section equal to the momentum flux at the inlet, ie, equations 16 and 17 define jet velocity at all points. For a cylindrical jet this leads to a center-line velocity that varies inversely with (x — aig), whereas for slot jets it varies inversely with the square root of (x — Xq As the jet proceeds still further downstream the turbulent entrainment initiated by the jet is gradually subordinated to the turbulence level in the surrounding stream and the jet, as such, disappears. 

In turbulent mixing, the Hquid velocity at any point u can be considered the sum of an average velocity ul and a fluctuating (with time) velocity u ... 

Equation (5-62) predicts the point of maximiim velocity for laminar flow in annuli and is only an approximate equation for turbulent flow. Brighton and Jones [Am. Soc. Mech. Eng. Basic Eng., 86, 835 (1964)] and Macagno and McDoiigall [Am. Inst. Chem. Eng. J., 12, 437 (1966)] give more accurate equations for predicting the point of maximum velocity for turbulent flow. 

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