- •Chapter I introduction
- •1. The subject of hydraulics
- •2. Historical background
- •3. Forces acting on a fluid. Pressure
- •4. Properties of liquids
- •Chapter II hydrostatics.
- •5. Hydrostatic pressure
- •6. The basic hydrostatic equation
- •7. Pressure head. Vacuum. Pressure measurement
- •8. Fluid pressure on a plane surface
- •Fig. 12. Pressure distribution on a rectangular wall
- •9. Fluid pressure on cylindrical and spherical surfaces. Buoyancy and floatation
- •Fig. 18. Automatic relief valve.
- •Relative rest of a liquid
- •10. Basic concepts
- •11. Liquid in a vessel moving with uniform acceleration in a straight line
- •12. Liquid in a uniformly rotating vessel
- •The basic equations of hydraulics
- •13. Fundamental concepts
- •14. Rate of discharge. Equation of continuity
- •15. Bernoulli's equation for a stream tube of an ideal liquid
- •16. Bernoulli's equation for real flow
- •17. Mead losses (general considerations)
- •18. Examples of application of bernoulli's equation to engineering problems
- •Chapter V flow through pipes. Hydrodynamic similarity
- •19. Flow through pipes
- •20. Hydrodynamic similarity
- •21. Cavitati0n
- •Chapter VI laminar flow
- •22.Laminar flow in circular pipes
- •23. Entrance conditions in laminar flow. The α coefficient
- •24. Laminar flow between parallel boundaries
- •Chapter VII turbulent flow
- •25. Turbulent flow in smooth pipes
- •26. Turbulent flow in rough pipes
- •27. Turbulent flow in noncircular pipes
- •Chapter VIII local features and minor losses
- •28. General considerations concerning local features in pipes
- •29. Abrupt expansion
- •30. Gradual expansion
- •31. Pipe contraction
- •32. Pipe bends
- •33. Local disturbances in laminar flow
- •34. Local features in aircraft hydraulic systems
- •Chapter IX flow through orifices, tubes and nozzles
- •35. Sharp-edged orifice in thin wall
- •36. Suppressed contraction. Submerged jet
- •37. Flow through tubes and nozzles
- •38. Discharge with varying head (emptying of vessels)
- •39. Injectors
- •Relative motion and unsteady pipe flow
- •40. Bernoulli's equation for relative motion
- •41. Unsteady flow through pipes
- •42. Water hammer in pipes
- •Chapter XI calculation of pipelines
- •43. Plain pipeline
- •44. Siphon
- •45. Compound pipes in series and in parallel
- •46. Calculation of branching and composite pipelines
- •47. Pipeline with pump
- •Chapter XII centrifugal pumps
- •48. General concepts
- •49. The basic equation for centrifugal pumps
- •50. Characteristics of ideal pump. Degree of reaction
- •51. Impeller with finite number of vanes
- •52. Hydraulic losses in pump. Plotting rated characteristic curve
- •53. Pump efficiency
- •54. Similarity formulas
- •55. Specific speed and its relation to impeller geometry
- •56. Relation between specific speed and efficiency
- •57. Cavitation conditions for centrifugal pumps (according to s.S. Rudnev)
- •58. Calculation of volute casing
- •59. Selection of pump type. Special features of centrifugal pumps used in aeronautical and rocket engineering
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 Xt 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 λt 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 kt 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 λt is markedly different:
(1) Low Re and : λt 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 λt depends on both Re and .
(3) High Re and : λt is independent of Re and is controlled solely by the relative roughness; the resistance law is quadratic, as Xt 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 6t 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 λt 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 λt for real rough pipes is given as a function of Re for different values of , where keq is the absolute roughness equivalent to Nikuradse's granular roughness. For new steel pipes Murin suggests assuming keq = 0.06 mm, and for used pipes, keq = 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.