26. Turbulent flow in rough pipes

In smooth pipes friction losses were completely determined by the Reynolds number. In rough pipes, however, the value of X_t depends also on the roughness of the inside pipe surface. The impor­tant point is not so much the absolute roughness size к of the projections as the so-called relative roughness for the same absolute roughness may have no effect on the resistance of a large pipe and considerably increase the resistance of a small one. Furthermore, the shape and distribution of the projections may affect the resist­ance of a pipe. The simplest case of roughness is that in which all the projections are of the same shape and size, known as uniform gran­ular wall roughness.

In the case of uniform granular roughness the friction factor λ_t depends both on Re and on the ratio :

The effect of these two parameters on pipe friction is shown on the graph in Fig. 59, which is based on Nikuradse's experiments. Niku-radse investigated the resistance of artificially roughened pipes. He coated several different sizes of pipe with sand grains which had been segregated by sieving so as to obtain different sizes of grain of uni­form diameter. In this way he obtained uniform granular roughness.

The pipes were tested in a broad range of relative roughnesses (= 1/500 to 1/15) and Reynolds numbers (Re = 500 to 10⁶). Fig. 59 contains the results of his experiments in the form of curves on a logarithmic plot relating log (1,000 λ) to log Re for various ra­tios .

The inclined straight lines A and В correspond to the resistance laws for smooth pipes, i. e., Eqs (6.7) and (7.2), which, multiplied by 1,000 and the logarithm taken, give linear equations for the given coordinate system:

and

The broken lines are curves plotted for pipes with different relative roughness. The following basic conclusions can be drawn from the graph:

1. In laminar flow roughness does not affect resistance; the broke curves corresponding to various degrees of roughness practicall coincide with line A.

2. The critical Reynolds number practically does not depend on roughness. The broken curves diverge from line A at about the same value of Re.

3. In turbulent flow when Re and are small roughness does not affect resistance; in some places the broken lines coincide with line B. However, with Re increasing the roughness begins to tell and the curves for rough pipes begin to deviate from the straight line of the resistance law for smooth pipes.

4. At high values of Re and the friction factor k_t no longer depends on Re and becomes constant for a given relative roughness. This corresponds to the horizontal portions of the broken lines after their slight rise.

Thus, for each of the curves corresponding to turbulent flow in rough pipes there are observed three distinct regions of the numbers Re and in which the behaviour of the friction factor λ_t is markedly different:

(1) Low Re and : λ_t is independent of the roughness and is controlled solely by the Reynolds number, just as in smooth pipes. Rough­ness has no maximum values.

(2) The friction factor λ_t depends on both Re and .

(3) High Re and : λ_t is independent of Re and is controlled solely by the relative roughness; the resistance law is quadratic, as X_t being independent of Re makes head losses proportional precisely to the square of the velocity [see Eq. (4.18)1; this is the "rough-law regime".

In order to gain a correct understanding of the resistance of rough pipes, it is,necessary to take into account the existence of the lami­nar sublayer mentioned in Sec. 25.

It was pointed out that with Re increasing the thickness of tha sublayer 6_t decreases. For this reason in turbulent flow in a rough pipe with a low Reynolds number the laminar sublayer covers the projections, which are contained within it and do not affect the pipe resistance. As Re increases the thickness of the sublayer decreases and the projections extend partly outside it, thus beginning to affect the resistance. At large values of Re the sublayer practically van­ishes and all the projections reach into the turbulent flow. Eddies form in the wake of each projection, which explains the quadratic resistance law in this regime.

Nikuradse carried out his experiments with artificially, uniformly granular-roughened pipes. For real rough pipes the dependence of λ_t on Re is somewhat different; notably there is no bulge in the curves following their divergence from the smooth-law curve. Fig. 60 presents a chart of some extremely precise experiments carried out by the Soviet scientist G. A. Murin.

The friction factor λ_t for real rough pipes is given as a function of Re for different values of , where k_eq is the absolute roughness equivalent to Nikuradse's granular roughness. For new steel pipes Murin suggests assuming k_eq = 0.06 mm, and for used pipes, k_eq = 0.2 mm.

The following new general formula has been suggested by the So­viet scientist A. D. Altshul for engineering calculations of the resistance of real rough pipes:

(7.4)

where d = pipe diameter;

k' = dimension proportional to absolute roughness. The limiting values of k' for different pipes are presented in Table 2.

Table 2

Pipe material	10³ k',mm
Glass tubing Drawn tubing, brass, lead, copper Seamless steel, high-grade manufacture Steel pipe Asphalt-dipped cast iron pipe Cast iron pipe	0.0 0.0 0.6-2.0 3-10 10-25 25-50

At low values of as compared with the number 7, Eq. (7.4)turns into Konakov's Eq. (7.1) for smooth pipes; at large values of it turns into the equation for the rough-law regime (the quad­ratic law of resistance):

(7.5)

Thus, a comparison of the product with the number 7 ena­bles a demarcation to be made between the different regimes of tur­bulent flow through rough pipes.