Turbulent flow, transition from laminar

Hwang and Kim (2006) investigated the pressure drop in circular stainless steel smooth micro-tubes ks/d <0.1%) with inner diameters of 244 pm, 430 pm and 792 pm. The measurements showed that the onset of flow transition from laminar to turbulent motion occurs at the Reynolds number of slightly less than 2,000. It... 

Experiments were conducted to measure flow and heat transfer characteristics of gaseous flows in microchannels in [12]. Their experimental result of the Poiseuille number is 118 for laminar flow, which is higher than the expected value. They also reported that the flow transition from laminar to turbulent occurs at Reynolds numbers around 400 to 900, which is lower than the conventional value of... 

Re <2.1 X 10 is laminar Re >4 x 10 is turbulent. The transition from laminar to turbulent flow in porous channels and tubes, e.g. tubular membranes, Re<2 x 10. Flow is considered turbulent because of mesh spacers. 

In turbulent flows. Re is important because it can be used to determine when the flow transitions from laminar to turbulent flow and also when the flow achieves a fully turbulent state. For macroscale pipe and channel flows, transition to turbulence can occur in the range 1,800 < Re < 2,300, depending of flow conditions, and the flow becomes fully turbulent at a Reynolds number of approximately 4,000. As will be shown later in this chapter, whether flows in microchannels agree with this macroscale behavior has long been a topic of debate, as some researchers have reported transitional Reynolds numbers in microchannel flows far lower than the 1,800-2,300 reported for macroscale flows. However, recent studies using newly developed experimental techniques and also careful reexamination of the results of some of the earlier studies strongly suggest that microchannel flows... 

In physical as well as mathematical modeling of transport phenomena, it is important to consider the existence or otherwise in the real process, any possible flow transition from laminar to turbulent flow, natural convection to forced convection, or subsonic to supersonic flow. This is because of the significant impact that such transitions may have on the process. 

The transition from laminar to turbulent flow occurs at Reynolds numbers varying from ca 2000 for n > 1 to ca 5000 for n = 0.2. In the laminar region the Fanning friction factor (Fig. 2) is identical to that for Newtonian fluids. In the turbulent region the friction factor drops significantly with decreasing values of producing a family of curves. 

Reynolds Number. The Reynolds number, Ke, is named after Osborne Reynolds, who studied the flow of fluids, and in particular the transition from laminar to turbulent flow conditions. This transition was found to depend on flow velocity, viscosity, density, tube diameter, and tube length. Using a nondimensional group, defined as p NDJp, the transition from laminar to turbulent flow for any internal flow takes place at a value of approximately 2100. Hence, the dimensionless Reynolds number is commonly used to describe whether a flow is laminar or turbulent. Thus... 

No simple equation exists for accomplishing a smooth mathematical transition from laminar flow to turbulent flow. Of the relationships proposed, Hausen s equation [Z. Ver Dtsch. Ing. Beih. Veifahrenstech., No. 

Roughness may also affect the transition from laminar to turbulent flow (Schhchting). 

The critical Reynolds number for transition from laminar to turbulent flow in noncirciilar channels varies with channel shape. In rectangular ducts, 1,900 < Re < 2,800 (Hanks and Ruo, Ind. Eng. Chem. Fundam., 5, 558-561 [1966]). In triangular ducts, 1,600 < Re < 1,800 (Cope and Hanks, Ind. Eng. Chem. Fundam., II, 106-117 [1972] Bandopadhayay and Hinwood, j. Fluid Mech., 59, 775-783 [1973]). 

The transition to turbulent flow begins at Re R in the range of 2,000 to 2,500 (Metzuer and Reed, AIChE J., 1, 434 [1955]). For Bingham plastic materials, K and n must be evaluated for the condition in question in order to determine Re R and establish whether the flow is laminar. An alternative method for Bingham plastics is by Hanks (Hanks, AIChE J., 9, 306 [1963] 14, 691 [1968] Hanks and Pratt, Soc. Petrol. Engrs. J., 7, 342 [1967] and Govier and Aziz, pp. 213-215). The transition from laminar to turbulent flow is influenced by viscoelastic properties (Metzuer and Park, J. Fluid Mech., 20, 291 [1964]) with the critical value of Re R increased to beyond 10,000 for some materials. 

The basis for single-phase and some two-phase friction loss (pressure drop) for fluid flow follows the Darcy and Fanning concepts. The exact transition from laminar or dscous flow to the turbulent condition is variously identified as between a Reynolds number of 2000 and 4000. 

Flow is fully turbulent with Re>3500 (the transition from laminar flow starts at about Re = 2000) ... 

The critical value of the Reynolds number (Remit) for the transition from laminar to turbulent flow may be calculated from the Ryan and Johnson001 stability parameter, defined earlier by equation 3.56. For a power-law fluid, this becomes ... 

In Chap. 3 the problems of single-phase flow are considered. Detailed data on flows of incompressible fluid and gas in smooth and rough micro-channels are presented. The chapter focuses on the transition from laminar to turbulent flow, and the thermal effects that cause oscillatory regimes. 

Transition from laminar to turbulent flow occurs when the friction factor exceeds the low ARe range. In Fig. 2.20a the results obtained for a mbe of diameter 705 pm by Maynes and Webb (2002) are compared against the value accepted for laminar flow A = 64/Re. Based on the above data, one can conclude that the transition occurs at Tie >2,100. 

In Spite of the existence of numerous experimental and theoretical investigations, a number of principal problems related to micro-fluid hydrodynamics are not well-studied. There are contradictory data on the drag in micro-channels, transition from laminar to turbulent flow, etc. That leads to difficulties in understanding the essence of this phenomenon and is a basis for questionable discoveries of special microeffects (Duncan and Peterson 1994 Ho and Tai 1998 Plam 2000 Herwig 2000 Herwig and Hausner 2003 Gad-el-Hak 2003). The latter were revealed by comparison of experimental data with predictions of a conventional theory based on the Navier-Stokes equations. The discrepancy between these data was interpreted as a display of new effects of flow in micro-channels. It should be noted that actual conditions of several experiments were often not identical to conditions that were used in the theoretical models. For this reason, the analysis of sources of disparity between the theory and experiment is of significance. 

We consider the problem of liquid and gas flow in micro-channels under the conditions of small Knudsen and Mach numbers that correspond to the continuum model. Data from the literature on pressure drop in micro-channels of circular, rectangular, triangular and trapezoidal cross-sections are analyzed, whereas the hydraulic diameter ranges from 1.01 to 4,010 pm. The Reynolds number at the transition from laminar to turbulent flow is considered. Attention is paid to a comparison between predictions of the conventional theory and experimental data, obtained during the last decade, as well as to a discussion of possible sources of unexpected effects which were revealed by a number of previous investigations. 

This chapter has the following structure in Sect. 3.2 the common characteristics of experiments are discussed. Conditions that are needed for proper comparison of experimental and theoretical results are formulated in Sect. 3.3. In Sect. 3.4 the data of flow of incompressible fluids in smooth and rough micro-channels are discussed. Section 3.5 deals with gas flows. The data on transition from laminar to turbulent flow are presented in Sect. 3.6. Effect of measurement accuracy is estimated in Sect. 3.7. A discussion on the flow in capillary tubes is given in Sect. 3.8. 

Glass and silicon tubes with diameters of 79.9-166.3 iim, and 100.25-205.3 am, respectively, were employed by Li et al. (2003) to study the characteristics of friction factors for de-ionized water flow in micro-tubes in the Re range of 350 to 2,300. Figure 3.1 shows that for fully developed water flow in smooth glass and silicon micro-tubes, the Poiseuille number remained approximately 64, which is consistent with the results in macro-tubes. The Reynolds number corresponding to the transition from laminar to turbulent flow was Re = 1,700—2,000. 

Wu and Cheng (2003) measured the friction factor of laminar flow of de-ionized water in smooth silicon micro-channels of trapezoidal cross-section with hydraulic diameters in the range of 25.9 to 291.0 pm. The experimental data were found to be in agreement within 11% with an existing theoretical solution for an incompressible, fully developed, laminar flow in trapezoidal channels under the no-slip boundary condition. It is confirmed that Navier-Stokes equations are still valid for the laminar flow of de-ionized water in smooth micro-channels having hydraulic diameter as small as 25.9 pm. For smooth channels with larger hydraulic diameters of 103.4-103.4-291.0pm, transition from laminar to turbulent flow occurred at Re = 1,500-2,000. 

Lelea et al. (2004) investigated experimentally fluid flow in stainless steel microtubes with diameter of 100-500 pm at Re = 50-800. The obtained results for the Poiseuille number are in good agreement with the conventional theoretical value Po = 64. Early transition from laminar to turbulent flow was not observed within the studied range of Reynolds numbers. 

The existence of roughness leads also to decreasing the value of the critical Reynolds number, at which transition from laminar to turbulent flow occurs. The character of the dependence of the friction factor on the Reynolds number in laminar flow remains the same for both smooth and rough micro-channels, i.e., X = const/Re. 

A study of forced convection characteristics in rectangular channels with hydraulic diameter of 133-367 pm was performed by Peng and Peterson (1996). In their experiments the liquid velocity varied from 0.2 to 12m/s and the Reynolds number was in the range 50, 000. The main results of this study (and subsequent works, e.g., Peng and Wang 1998) may be summarized as follows (1) friction factors for laminar and turbulent flows are inversely proportional to Re and Re ", respectively (2) the Poiseuille number is not constant, i.e., for laminar flow it depends on Re as PoRe ° (3) the transition from laminar to turbulent flow occurs at Re about 300-700. These results do not agree with those reported by other investigators and are probably incorrect. 

The hypothesis on the earlier transition from laminar to turbulent flow in micro-tubes is based on analysis of the dependence of pressure gradient on Reynolds number. As shown by the experimental data by Mala and Li (1999), this dependence may be approximated by three power functions AP Re (Re < 600),... 

For the most part of the experiments one can conclude that transition from laminar to turbulent flow in smooth and rough circular micro-tubes occurs at Reynolds numbers about RCcr = 2,000, corresponding to those in macro-channels. Note that other results were also reported. According to Yang et al. (2003) RCcr derived from the dependence of pressure drop on Reynolds number varied from RCcr = 1,200 to RCcr = 3,800. The lower value was obtained for the flow in a tube 4.01 mm in diameter, whereas the higher one was obtained for flow in a tube of 0.502mm diameter. These results look highly questionable since they contradict the data related to the flow in tubes of diameter d> mm. Actually, the 4.01 mm tube may be considered... 

The transition from laminar to turbulent flow in micro-channels with diameters ranging from 50 to 247 pm was studied by Sharp and Adrian (2004). The transition to turbulent flow was studied for liquids of different polarities in glass micro-tubes having diameters between 50 and 247 pm. The onset of transition occurred at the Reynolds number of about 1,800-2,000, as indicated by greater-than-laminar pressure drop and micro-PIV measurements of mean velocity and rms velocity fluctuations at the centerline. 

Hao et al. (2007) investigated the water flow in a glass tube with diameter of 230 Lim using micro particle velocimetry. The streamwise and mean velocity profile and turbulence intensities were measured at Reynolds number ranging from 1,540 to 2,960. Experimental results indicate that the transition from laminar to turbulent flow occurs at Re = 1,700—1,900 and the turbulence becomes fully developed at Re > 2,500. 

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