Turbulent velocity distributions

Freely flowing particulate matter sliding down a vertical chute assumes a flow profile similar to a turbulent velocity distribution. Due to wall and interparticle friction there is a velocity gradient in a material layer close to the chute walls (Figure 288a). The thickness of this layer depends on particle size and cohesiveness of the material (stickiness). When the feed approaches the gripping... 

Figure 3.2 Qualitative comparison of laminar and turbulent velocity distribution.

Fig. 6. Illustration of variation of velocity of air at the oudet of a centrifugal fan and the function filled by several diameters of straight discharge duct in converting velocity head to static head and estabUshing normal turbulent dow distribution. Bends or obstmctions at the discharge oudet cause turbulence...

In some convection equations, such as for turbulent pipe flow, a special correction factor is used. This factor relates to the heat transfer conditions at the flow inlet, where the flow has not reached its final velocity distribution and the boundary layer is not fully developed. In this region the heat transfer rate is better than at the region of fully developed flow. 

The similarity of velocity and of turbulence intensity is documented in Fig. 12.29. The figure shows a vertical dimensionless velocity profile and a turbulence intensity profile measured by isothermal model experiments at two different Reynolds numbers. It is obvious that the shown dimensionless profiles of both the velocity distribution and the turbulence intensity distribution are similar, which implies that the Reynolds number of 4700 is above the threshold Reynolds number for those two parameters at the given location. 

FIGURE 12.29 Velocity distribution and turbulence intensity in the occupied zone of a room at two different Reynolds numbers. H is the height of the room. 

Velocity distributions and volumetric flowrates for turbulent flow... 

The velocity at the inner edge of the turbulent region must also be given by the equation for the velocity distribution in the turbulent region. 

The velocity distribution and frictional resistance have been calculated from purely theoretical considerations for the streamline flow of a fluid in a pipe. The boundary layer theory can now be applied in order to calculate, approximately, the conditions when the fluid is turbulent. For this purpose it is assumed that the boundary layer expressions may be applied to flow over a cylindrical surface and that the flow conditions in the region of fully developed flow are the same as those when the boundary layers first join. The thickness of the boundary layer is thus taken to be equal to the radius of the pipe and the velocity at the outer edge of the boundary layer is assumed to be the velocity at the axis. Such assumptions are valid very close to the walls, although significant errors will arise near the centre of the pipe. 

Liquid core temperature and velocity distribution analysis. BankofT (1961) analyzed the convective heat transfer capability of a subcooled liquid core in local boiling by using the turbulent liquid flow equations. He found that boiling crisis occurs when the core is unable to remove the heat as fast as it can be transmitted by the wall. The temperature and velocity distributions were analyzed in the singlephase turbulent core of a boiling annular flow in a circular pipe of radius r. For fully developed steady flow, the momentum equation is given as... 

Derive the relation between the friction factor and Reynolds number in turbulent flow for smooth pipe [Eq. (6-34)], starting with the von Karman equation for the velocity distribution in the turbulent boundary layer [Eq. (6-26)]. 

In the Lagrangian approach, individual parcels or blobs of (miscible) fluid added via some feed pipe or otherwise are tracked, while they may exhibit properties (density, viscosity, concentrations, color, temperature, but also vorti-city) that distinguish them from the ambient fluid. Their path through the turbulent-flow field in response to the local advection and further local forces if applicable) is calculated by means of Newton s law, usually under the assumption of one-way coupling that these parcels do not affect the flow field. On their way through the tank, these parcels or blobs may mix or exchange mass and/or temperature with the ambient fluid or may adapt shape or internal velocity distributions in response to events in the surrounding fluid. 

Equation 2.40 is an empirical equation known as the one-seventh power velocity distribution equation for turbulent flow. It fits the experimentally determined velocity distribution data with a fair degree of accuracy. In fact the value of the power decreases with increasing Re and at very high values of Re it falls as low as 1/10 [Schlichting (1968)]. Equation 2.40 is not valid in the viscous sublayer or in the buffer zone of the turbulent boundary layer and does not give the required zero velocity gradient at the centre-line. The l/7th power law is commonly written in the form... 

Universal velocity distribution for turbulent flow in a pipe... 

Equations 2.58, 2.70 and 2.71 enable the velocity distribution to be calculated for steady fully developed turbulent flow. These equations are only approximate and lead to a discontinuity of the gradient at y+ = 30, which is where equations 2.70 and 2.71 intersect. The actual profile is, of course, smooth and the transition from the buffer zone to the fully turbulent outer zone is particularly gradual. As a result it is somewhat arbitrary where the limit of the buffer zone is taken often the value y+ = 70 rather than j + = 30 is used. The ability to represent the velocity profile in most turbulent boundary layers by the same v+ - y+ relationships (equations 2.58, 2.70 and 2.71) is the reason for calling this the universal velocity profile. The use of in defining v+ and y+ demonstrates the fundamental importance of the wall shear stress. 

In equation 1.14, z, P/(pg), and u2/(2ga) are the static, pressure and velocity heads respectively and hf is the head loss due to friction. The dimensionless velocity distribution factor a is for laminar flow and approximately 1 for turbulent flow. 

For steady flow in a pipe or tube the kinetic energy term can be written as m2/(2 a) where u is the volumetric average velocity in the pipe or tube and a is a dimensionless correction factor which accounts for the velocity distribution across the pipe or tube. Fluids that are treated as compressible are almost always in turbulent flow and a is approximately 1 for turbulent flow. Thus for a compressible fluid flowing in a pipe or tube, equation 6.4 can be written as... 

Another important function of the impeller in three-phase systems is to generate finely dispersed and homogeneously distributed bubbles throughout the vessel. At the same time, the turbulent velocities should be sufficiently high to prevent coalescence of gas bubbles. The most important variable concerning gas dispersion is the gas holdup in the vessel. 

In the turbulent core it has been conventional to employ the velocity deficiency (B2) as a single-valued function of the relative position in the channel in order to correlate the velocity distribution outside the boundary flow. The velocity deficiency is defined by... 

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... 

Velocity distributions in turbulent flowthrough a straight, round tube vary with the flow rate or the Reynolds number. With increasing flow rates the velocity distribution becomes flatter and the laminar sublayer thinner. Dimensionless empirical equations involving viscosity and density are available that correlate the local fluid velocities in the turbulent core, buffer layer, and the laminar sublayer as functions of the distance from the tube axis. The ratio of the average velocity over the entire tube cross section to the maximum local velocity at the tube axis is approximately 0.7-0.85, and increases with the Reynolds number. 

The usual velocity distributions in a steady flow of liquid through a tube are shown in Figure 2.4. In either laminar or turbulent flow, the velocity at the tube wall is zero but is maximum at the tube axis. The ratio of the average velocity to the maximum velocity is 0.5 for laminar flow and approximately 0.8... 

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