Steady turbulence

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. 

An approximate equation for the profile of the time-averaged velocity for steady turbulent flow of a Newtonian fluid through a pipe of circular... 

Equations 6.5, 6.7 and 6.12 all relate to the energy changes involved for a fluid in steady turbulent flow. The most appropriate equation is selected for each particular application equation 6.12 is a convenient form from which a basic flow rate-pressure drop equation will be derived. 

A centrifugal pump is used to pump a liquid in steady turbulent flow through a smooth pipe from one tank to another. Develop an expression for the system total head A/t in terms of the static heads on the discharge and suction sides zd and zs respectively, the gas pressures above the tanks on the discharge and suction sides Pd and Ps respectively, the liquid density p, the liquid dynamic viscosity p, the gravitational acceleration g, the total equivalent lengths on... 

Fig. 17. Generalized temperature distribution in uniform, steady, turbulent flow.

Figure 2.4b shows, conceptually, the velocity distribution in steady turbulent flow through a straight round tube. The velocity at the tube wall is zero, and the fluid near the wall moves in laminar flow, even though the flow of the main body of fluid is turbulent. The thin layer near the wall in which the flow is laminar is called the laminar sublayer or laminar film, while the main body of fluid where turbulence always prevails is called the turbulent core. The intermediate zone between the laminar sublayer and the turbulent core is called the buffer layer, where the motion of fluid may be either laminar or turbulent at a given instant. With a relatively long tube, the above statement holds for most of the tube length, except for... 

In a steady or mean steady turbulent regime, (d J cT2 dV)/dt = 0 exactly or approximately over time intervals which are large compared to the pulsation period. We will show that JTk(VT)n dS may be considered a measure of the heat transfer. 

Apropos of the analogy noted by Batchelor between vortex velocity and a magnetic field, it should be noted that for the realization of truly steady turbulence a supply of mechanical energy is necessary. The supply of energy comes about either through nonpotential volume forces or through the motion of the surfaces bounding the fluid. With these factors taken into consideration, the set of equations and boundary conditions for a vortex does... 

Flow regime in applications Steady, turbulent Pseudolaminar to unsteady and stratified... 

Because the flow variables are fluctuating with time in this way it is necessary to clearly define what is meant by steady turbulent flow. To do this, mean values of the variables are defined in the following way [3],[7],[10],[11],[12] ... 

Here, t is some interval of time. The term steady turbulent flow implies that, provided a sufficiently long interval of time, t, is taken for this averaging process, constant values of the mean quantities u, v, w, p, and T will be obtained at all points in the flow. These mean values are thus often termed time-averaged values. 

First a steady turbulent two-phase flame is calculated. The 15 pm droplet motion follows the carrier phase dynamics so that the Centered Recirculation Zones (CRZ) are similar for gas and liquid, as illustrated on Fig. 10.4, showing the instantaneous backflow lines of both phases, plotted in the vertical central cutting plane. Maintained by this CRZ, the droplets accumulate and the droplet number density, presented with the liquid volume fraction field on Fig. 10.4, rises above its initial value a zone where the droplet number density n is larger than 2n j j (where is its value... 

An aerosol issuing from a point source is dispersed in a steady turbulent plume in the atmosphere. Derive an expression for the variation of the extinction coeflicieni, h (Chapter 5), with position in the plume assuming that (a) the only mechanism affecting the light-scattering portion of the size distribution is turbulent diffusion and (b) the only mechanisms are turbulent diffusion and growth. 

In turbulent flow, the fluid particles do not travel in a well-ordered pattern. These particles possess velocities with macroscopic fluctuations at any point in the flow field. Even in steady turbulent flow, the local velocity components transverse to the main flow direction change in magnitude with respect to time. Instantaneous velocity consists of time-average velocity and its fluctuating component. When heat transfer is involved in turbulent flow, the instantaneous temperature is composed of the time-average temperature and its fluctuating components. 

Let us assume a steady turbulent shear flow in which u = u i (xj) and 2 = 3 = 0. We first consider turbulent momentum transport, that is, the Reynolds stresses. The mean flux of xi momentum in the X2 direction due to turbulence is pu. Let us see if we can derive an estimate for this flux. 

We are no more able to calculate the pressure drop in steady, turbulent flow in a noncircular conduit than we are in a circular one. However, it seems reasonable to expect that we could use the friction-loss results for circular pipes to estimate the results for other shapes. Let us assume that the shear stress at the wall of any conduit is the same for a given average fluid flow velocity independent of the shape of the conduit. Then, from a force balance on a horizontal section like that leading to Eq. 6.3, we conclude that in steady flow... 

THE STEADY, TURBULENT BOUNDARY LAYER ON A FLAT PLATE... 

There are no known analytical solutions for turbulent boundary layers that are analogous to Blasius solution for the laminar boundary layer on a flat plate. Prandtl, to describe the steady, turbulent boundary layer on a flat plate, made... 

The shear stresses over the flow boundaries can be rigorously derived as an integral part of the solution of the flow field only in laminar flows. The need for closure laws arise already in single-phase, steady turbulent flows. The closure problem is resolved by resorting to semi-empirical models, which relate the characteristics of the turbulent flow field to the local mean velocity profile. These models are confronted with experiments, and the model parameters are determined from best fit procedure. For instance, the parameters of the well-known Blasius relations for the wall shear stresses in turbulent flows through conduits are obtained from correlating experimental data of pressure drop. Once established, these closure laws permit formal solution to the problem to be found without any additional information. 

The results of these experiments indicated that the Sherwood number for an oscillating flow was about threefold greater than that for a steady turbulent flow at the same Reynolds number. Clearly, the pressure amplitude and frequency of oscillations of the gas flow are dominant factors influencing the heat and mass transfer rates. Similar conclusions were presented by Dec and Keller (1989). 

Concept of average in the turbulent sense, steady turbulence, and homogeneous turbulence... 

For the uniform one-dimensional isotropic steady turbulent flow and mass transfer, the equations of k, s, c and Sd are reduced to the following forms ... 

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