The transition from laminar to turbulent flow

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

From equation 3.178, the critical Reynolds number for a Newtonian fluid is 2100. As n decreases, R mr)c rises to a maximum of 2400, and then falls to 1600 for n = Ori, Most of the experimental evidence suggests, however, that the transition occurs at a value close to 2000. 

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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... 

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

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. 

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. 

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. 

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. 

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 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. 

The behavior of the flow in micro-channels, at least down to 50 pm in diameter, shows no difference with macro-scale flow. For smooth and rough micro-channels with relative roughness 0.32% turbulent flow occurs between 1,800 < Recr < 2,200, in full agreement with flow visualization and flow resistance data. In the articles used for the present study there was no evidence of transition below these results. 

For smooth micro-channels the transition from laminar to turbulent flow occurs at Re = 1,500—2,200. For turbulent flows the friction factor maybe calculated as... 

The transition from laminar to turbulent flow will normally occur at a Reynolds number of 100 to 400, depending on the plate design. With some designs, turbulence can be achieved at very low Reynolds numbers, which makes plate heat exchangers very suitable for use with viscous fluids. 

The transition from laminar to turbulent flow on a rotating sphere occurs approximately at Re = 1.5 4.0 x 104. Experimental work by Kohama and Kobayashi [39] revealed that at a suitable rotational speed, the laminar, transitional, and turbulent flow conditions can simultaneously exist on the spherical surface. The regime near the pole of rotation is laminar whereas that near the equator is turbulent. Between the laminar and turbulent flow regimes is a transition regime, where spiral vortices stationary relative to the surface have been observed. The direction of these spiral vortices is about 4 14° from the negative direction of the azimuthal angle,. The phenomenon is similar to the flow transition on a rotating disk [19]. 

In turbulent flow, the edge effect due to the shape of the support rod is quite significant as shown in Fig. 6. The data obtained with a support rod of equal radius agree with the theoretical prediction of Eq. (52). The point of transition with this geometry occurs at Re = 40000. However, the use of a larger radius support rod arbitrarily introduces an outflowing radial stream at the equator. The radial stream reduces the stability of the boundary layer, and the transition from laminar to turbulent flow occurs earlier at Re = 15000. Thus, the turbulent mass transfer data with the larger radius support rod deviate considerably from the theoretical prediction of Eq. (52) a least square fit of the data results in a 0.092 Re0 67 dependence for... 

When running a CFD simulation, a decision must be made as to whether to use a laminar-flow or a turbulent-flow model. For many flow situations, the transition from laminar to turbulent flow with increasing flow rate is quite sharp, for example, at Re — 2100 for flow in an empty tube. For flow in a fixed bed, the situation is more complicated, with the laminar to turbulent transition taking place over a range of Re, which is dependent on the type of packing and on the position within the bed. 

The distribution of the gas velocity across the profile of a moving column of gas changes in the transition from laminar to turbulent flow. In the case of flow through a pipe of radius ro, the laminar flow variation is given by... 

Measurements with different fluids, in pipes of various diameters, have shown that for Newtonian fluids the transition from laminar to turbulent flow takes place at a critical value of the quantity pudjp in which u is the volumetric average velocity of the fluid, dt is the internal diameter of the pipe, and p and p. are the fluid s density and viscosity respectively. This quantity is known as the Reynolds number Re after Osborne Reynolds who made his celebrated flow visualization experiments in 1883 ... 

A stability analysis made by Ryan and Johnson (1959) suggests that the transition from laminar to turbulent flow for inelastic non-Newtonian fluids occurs at a critical value of the generalized Reynolds number that depends on the value of The results of this analysis are shown in Figure 3.7. This relationship has been tested for shear thinning and for Bingham... 

One may also note from Eq. (11) that the exponent of the Schmidt number is independent of the exponents of the macro- and microscales. In other words, the exponent of v/D should be the same for both laminar and turbulent flows. Experiment indeed indicates that the value of 1/3 for this exponent is valid for many laminar and turbulent flows along solid interfaces and that the value of 1/2 is valid for laminar and turbulent motions along fluid-fluid interfaces. It is interesting to note that there is a jump from 0.5 to 0.75 in the value of the bound of the exponent m with the transition from laminar to turbulent flow, a result which is in agreement with experimental observations [2],... 

It is well known in fluid flow studies that below a certain critical value of the Reynolds number the flow will be mainly laminar in nature, while above this value, turbulence plays an increasingly important part. The same is true of film flow, though it must be remembered that in thin films a large part of the total film thickness continues to be occupied by the relatively nonturbulent laminar sublayer, even at large flow rates (N e ARecr J- Hence, the transition from laminar to turbulent flow cannot be expected to be so sharply marked as in the case of pipe flow (D12). Nevertheless, it is of value to subdivide film flow into laminar and turbulent regimes depending on whether (Ar6 5 Ar u). 

The momentum, which is a certain amount of mass moving at a certain velocity (v), takes into account the fluid s density (p) and the diameter of the tube (Df). The momentum of the fluid can be increased by increasing the velocity or the diameter of the tube or both. The resistance to flow is expressed as the absolute or dynamic viscosity of the fluid (rj), which is in units of grams per centimeter-second, or centipoise (cP). The transition from laminar to turbulent-flow occurs as Re increases past a critical value between 2000 and 3000 in a straight tube having a smooth internal surface. 

Often it is useful to combine variables that affect physical phenomenon into dimensionless parameters. For example, the transition from laminar to turbulent flow in a pipe depends on the Reynolds number, Re = pLv/p, where p is the fluid density, I is a characteristic dimension of the pipe, v is the velocity of flow, and // is the viscosity of the fluid. Experiments show that the transition from laminar to turbulent flow occurs at the same value of Re for different fluids, flow velocities, and pipe sizes. Analyzing dimensions is made easier if we designate mass as M, length as L, time as t, and force as F. With this notation, the dimensions of the variables in Re are ML 3 for p, (L) for L, (L/t) for v, and (FL 2t) for //. Combining these it is apparent that Re = pLu/p, is dimensionless. 

The most important case of this transition for chemical engineers is the transition from laminar to turbulent flow, which occurs in straight bounded ducts. In the case of Newtonian fluid rheology, this occurs in straight pipes when Re = 2100. A similar phenomenon occurs in pipes of other cross sections, as well and also for non-Newtonian fluids. However, just as the friction factor relations for these other cases are more complex than for simple Newtonian pipe flow, so the criteria for transition to turbulence cannot be expressed as a simple critical value of a Reynolds number. 

The above discussion is for laminar flows. In practice, the transition from laminar to turbulent flow in free convection typically occurs when the Rayleigh number Ra = GrPr = 109. 

Initially, the boundary-layer development is laminar, but at some critical distance from the leading edge, depending on the flow field and fluid properties, small disturbances in the flow begin to become amplified, and a transition process takes place until the flow becomes turbulent. The turbulent-flow region may be pictured as a random churning action with chunks of fluid moving to and fro in all directions. The transition from laminar to turbulent flow occurs when... 

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