Chapter VII turbulent flow

25. Turbulent flow in smooth pipes

It was mentioned in Sec. 19 that turbulent flow is characterised by a mixing of the fluid and fluctuations of velocity and pressure. A sensitive recorder of velocity fluctuations would draw a graph like the one in Fig. 52. The magnitude of the velocity varies erratically about a certain mean value which is constant for a given condition.

The pathlines traced by fluid particles through a fixed point in space are curves of different shape, despite the straightness of the pipe. The streamlines also vary greatly at any given instant (Fig. 53). Thus, strictly speaking, turbulent flow is unsteady as the velocity, pressure intensity and pathlines of the fluid particles change with time.

Nevertheless, turbulent flow can be regarded as steady if the mean velocity and pressure and the total discharge do not change with time. This kind of flow is frequently encountered in engineering practice.

As the fluid is not laminar and it mixes continuously, Newton's friction law is not applicable to turbulent flow. Because of mixing of the fluid and continuous transport of momentum across the flow, the intensity of shear at the pipe boundary is much greater than in laminar flow with the same Re and dynamic pressure as computed for the mean velocity and density of the fluid. ,

The velocity distribution (averaged over a period of time) ^across a section of turbulent flow differs markedly from velocity distribu­tion in laminar flow.

A comparison of the velocity profiles in the same pipe and with the same discharge (same mean; velocity) in conditions of laminar and turbulent flow reveals a pronounced difference (Fig. 54). Velocity distribution in turbulent flow is somewhat more uniform and the increase in velocity at the boundary is steeper than in laminar flow, in which, it will be recalled, the velocity profile is a parabola.

Consequently, the a coefficient, which takes account of nonuni-form velocity distribution in Bernoulli's equation (see Sec. 16), is much smaller in turbulent flow. Unlike laminar flow, where a does not depend on Re (see Sec. 23), in turbulent flow a is a function of Re, decreasing with the latter increasing from 1.13 at Re = Re_r to 1.025 at Re = 3 X 10^е. As the graph in Pig. 55 shows, thecurve of a as a function of Re approaches unity asymptotically. In most cases of turbulent flow a can be assumed to be unity.

Energy losses in turbulent flowthrough pipes of uniform cross-section (i. e., head losses due to friction) also differ from the case of laminar flow. In the former friction losses are much greater for the same rate of discharge, viscosity and pipe size.

This increased loss is due to the for­mation of eddies, mixing and curving of pathlines. Whereas in laminar flow friction losses increase as the first power of the velocity (and the discharge), in transition to turbu­lent flow a jump is observed in the resistance and then a steeper increase of h_f along a curve approaching a parabola of the second power (Fig. 56).

In view of the complexity of turbulent flow and the difficulty of analytical study, the theory of turbulent flow lacks precision. The so-called semiempirical, approximate turbulence theories of Frandol, von Karman and others are applied, but we shall not consider them here.

In most cases engineering calculations of turbulent flow in pipos are based on purely experimental data systematised on the basis of hydrodynamic similarity.

The fundamental equation for turbulent flow in circular pipes is the experimental formula (4.18), which follows directly from similarity considerations and has the form

or

where λ_t = friction factor for turbulent flow.

This basic formula is valid for both turbulent and laminar flow (see Sec 22), the difference lying in the value of λ.

As in turbulent flow friction losses vary approximately as the square of the velocity (and the square of the rate of discharge), the friction factor in Eq. (4.18) can, to a first approximation, be assumed constant.

However, it follows from the similarity law (Sec. 20) that λ_t, like λ_t should be a function of the fundamental criterion of similar­ity, the Reynolds number, which includes velocity, diameter and kinematic viscosity, i. e.,

There are a number of empirical and semiempirical formulas which express this function for turbulent flow through smooth pipes;

one of the most convenient and commonly used is a formula devel­oped by the Russian scientist P. K. Konakov:

(7.1)

which holds good for values ranging from Re = Re_cr to several milions.

When 2,300 < Re < 10⁵ the old Blasius equation can be used:

(7.2)

It will be observed that with Re increasing K_t decreases somewhat, though to a much smaller degree than in laminar flow (Fig. 57).

This difference in the change of X is due to the fact that the effect of viscosity on friction is much less in turbulent than in laminar flow. Whereas in the latter friction losses vary directly as the viscosity (see Sec. 22), in turbulent flow, as is evident from Eqs (4.18) and (7.2), they vary as the l/4th power of viscosity. This is because of mixing and momentum transport in turbulent flow.

Eqs (7.1) and (7.2) for determining the friction factor X_t in terms of Re are valid for so-called technically smooth pipes, the roughness of which is small enough so as practically not to affect resistance. To a sufficient degree of accuracy, as technically smooth can be re­garded drawn nonferrous (including aluminium alloy) tubing and special seamless high-grade steel pipes. Thus-, the pipes used in aircraft fuel systems and hydraulic transmissions can be regarded as smooth and the foregoing equations used in calculating them. Since steel water pipes and cast iron pipes offer greater resistance, they cannot be treated as smooth, and Eqs (7.1) and (7.2) are not valid for them.

The resistance of rough pipes will be examined later on.

It is known from similarity theory, and confirmed by experiments carried out by a number of researchers (Nikuradse, G. G. Gurzhiyenko and others) that in turbulent flow a laminar sublayer usually forms at the pipe walls (Fig. 58). This is a very thin layer in which the fluid moves much slower and without mixing.

Within the sublayer the velocity increases steeply from zero at the wall to a certain value v_t at the sublayer boundary. The sublayer is very thin; the Reynolds number computed for 6_P the velocity v_t and the kinematic viscosity v, was found to be a constant quantity:

(7.3)

where δ = sublayer thickness.

This quantity is a universal constant, like the critical Reynolds number for flow through pipes. When the velocity of motion and, consequently, the Reynolds number increase, the velocity v_t in­creases as well and the sublayer thickness 6, decreases. At large values of Re the laminar sublayer practically disappears.