Velocity profiles in pipes

Fig. 6. Gas velocity profile in pipe cross section — DEM/CFD simulation (at 0.4 s).

The turbulent flow velocity profile for Newtonian fluids is arbitrarily divided into three regions the viscous sublayer, the buffer layer, and the turbulent core. To represent velocity profiles in pipe flow, friction velocity defined as... 

From the DSMC results and solutions of the linearized Boltzmann equation, it is evident that the velocity profiles in pipes, channels and ducts remain approximately parabolic for a large range of Knudsen number. This is also consistent with the analysis of the Navier-Stokes and Burnett equations in long channels, as documented in Ref [1]. Based on this observation, we model the velocity profile as parabolic in the entire Knudsen regime, with a consistent slip condition. We write the dimensional form for velocity distribution in a channel of height h. 

The flow inside a pipe is not uniform because the fluid at the center of the pipe moves faster than the fluid near the pipe walls. The velocity profile in pipes is a subject of fluid mechanics. Here, we avoid such complicating factors by considering the average velocity of the fluid over the entire cross section of the pipe. 

For a Bingham plastic, qualitatively, what unusual feature must a velocity profile in pipe flow exhibit In a cup-and-bob viscometer, what discontinuity might be expected in the flow field for the same model In a cone-plate viscometer, would any discontinuities be expected ... 

Velocity Profiles In laminar flow, the solution of the Navier-Stokes equation, corresponding to the Hagen-PoiseuiUe equation, gives the velocity i as a Innction of radial position / in a circular pipe of radius R in terms of the average velocity V = Q/A. The parabolic profile, with centerline velocity t ce the average velocity, is shown in Fig. 6-10. 

On the assumption that the velocity profile in a fluid in turbulent flow is given by the Prandtl one-seventh power law, calculate the radius at which the flow between it and the centre is equal to that between it and the wall, for a pipe 100 mm in diameter,... 

It may be noted that equations 3.167 and 3.170 are identical to equations 3.136 and 3.148 derived earlier. Although these derivations are simpler to carry out, the method does not allow the velocity profile in the pipe to be obtained. 

In general, the velocity profile will be curved but as equation 1.33 contains only the local velocity gradient it can be applied in these cases also. An example is shown in Figure 1.13. Clearly, as the velocity profile is curved, the velocity gradient is different at different values of y and by equation 1.32 the shear stress r must vary withy. Flows generated by the application of a pressure difference, for example over the length of a pipe, have curved velocity profiles. In the case of flow in a pipe or tube it is natural to use a cylindrical coordinate system as shown in Figure 1.14. 

Thus, for a parabolic velocity profile in a pipe, the volumetric average velocity is half the centre-line velocity and the equation for the velocity profile can be written as ... 

Orifice meters, Venturi meters and flow nozzles measure volumetric flow rate Q or mean velocity u. In contrast the Pitot tube shown in a horizontal pipe in Figure 8.7 measures a point velocity v. Thus Pitot tubes can be used to obtain velocity profiles in either open or closed conduits. At point 2 in Figure 8.7 a small amount of fluid is brought to a standstill. Thus the combined head at point 2 is the pressure head P/( pg) plus the velocity head v2/(2g) if the potential head z at the centre of the horizontal pipe is arbitrarily taken to be zero. Since at point 3 fluid is not brought to a standstill, the head at point 3 is the pressure head only if points 2 and 3 are sufficiently close for them to be considered to have the same potential head... 

The laminar flow velocity profile in a pipe for a power law liquid in steady state flow is given by the equation... 

For rough pipes, the velocity profile in the turbulent core is given by... 

Therefore, as was the case with fully developed pipe flow, the velocity profile in fully developed plane duct flow is parabolic. 

If the velocity profile is known together with the distribution of e then, for any assumed relationship between the distributions of c and e, Eq. (7.49) can be used to deduce the temperature profile. Once this has been obtained the relation between the Nusselt and Reynolds numbers can be derived. Before illustrating this procedure, there is a simplifying assumption that can be introduced without any significant loss of accuracy. Because the velocity profile in turbulent pipe flow is relatively flat over a large portion of the pipe cross-section, it is usually sufficiently accurate to replace u in the integral in the expression for I by the constant value um, i.e., to write ... 

In turn, the values of d can be converted to the radial location in the pipe so that the velocity profile in a pipe can be obtained. The velocity profile is used to calculate velocity gradients (shear rates), (dv/dr), at specific locations using an even-order polynomial curve fit to the velocity data. 

The developmeni of ihe velocity profile in a circular pipe. V V(r, z) and thus the flow is two-dimensional in the entrance region, and becomes one-dimensional downstream when the velocity profile fully develops and remains unchanged in the flow direction, = V r). 

The velocity profile in fully developed I uminar flow in a circular pipe of inner radius / - 10 cm, in m/s, is given by it(r) = 4(1 - r //( ). Determine the mean and maximum velocities in the pipe, and the volume flow rale. 

Unfortunately, it is not possible to derive an analogue velocity profile for turbulent flow in an anal dical manner based on the generalized momentum equations. However, a number of entirely empirical relations of similar simplicity exist for the velocity profile in turbulent pipe flow. One such relation often found in introductory textbooks on engineering fluid flow is the power law velocity profile. ... 

When a gas enters a smooth pipe from a large reservoir through a well-faired entry, a laminar boundary layer forms along the walls. The velocity profile in the main body of the How remains flat. The velocity boundary layer thickens with distance downstream from the entry until it eventually fills the pipe. If the Reynolds number based on pipe diameter is less than 2100, the pipe boundary layer remains laminar. The flow is said to be fully... 

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