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Laminar-turbulent transition in pipe flow: wall effects and critical Reynolds number.

1. Introduction.

The notations used in this paper are given in Appendix A at the end of this paper.

1.1. The Reynolds problem. The laminar--turbulent transition in circular pipe flow is one of the fundamental problems of fluid dynamics. In particular, a major unsolved problem is to theoretically obtain the minimum critical Reynolds number [R.sub.c,min] [approximately equal to] 2050, which was first observed by Osborne Reynolds in 1883 [19]. Ever since the pioneering experimental work of Reynolds, the problem has intrigued scientists, mathematicians, and engineers alike [13].

Reynolds proposed the formula Re = [rho][DV.sub.m]/[mu], where [rho] is the density, D is the circular pipe diameter, [V.sub.m] is the mean axial velocity, and [mu] is the viscosity. He also found two critical values: an upper critical Reynolds number [R.sub.c] = 12,800 by using the color-band method and a lower critical Reynolds number [R.sub.c,min][approximately equal to] 2050, which he called the real critical value, by the pressure-loss method. The precise [R.sub.c,min] value has not yet been agreed upon unanimously; it has been reported in a range from 1760 [17] to 2320 [21]. Avila et al. reported [R.sub.c,min] = 2040 [+ or -] 10 using the mean lifetime method for decaying and spreading turbulence [2]. In this present study, [R.sub.c,min] is assumed to be about 2050, as measured by Reynolds. It was furthermore observed by us in color-band experiments that under natural disturbance conditions, [R.sub.c] takes a minimum value of about 2050 when using a sharp-edged entrance pipe [12].

Accordingly, in this paper, we address the following challenges as the Reynolds problems of laminar-turbulent transition in circular pipe flow:

(1) to theoretically determine a possible cause and reason why the Reynolds number itself is the primary parameter determining critical Reynolds numbers,

(2) to theoretically derive [R.sub.c,min] [approximately equal to] 2050, and

(3) to theoretically derive [R.sub.c] [approximately equal to] 12, 800.

1.2. Background. The colloquium "Turbulence transition in pipe flow" was held in 2008 to commemorate the 125th anniversary of O. Reynolds' original 1883 paper. Various ongoing research topics were presented at this seminar [4]: traveling waves in pipe flow, the structure and dynamics of turbulent pipe flow, the structure of a puff, aspects of linear and nonlinear instabilities leading to transitions, the optimal path to transitions, the critical threshold in transitions, the critical layer in pipe flow, edge states intermediate between laminar and turbulent dynamics, and experiments on the decay of turbulent puffs. Then there was considerable interest in determining the mechanism of transition from laminar to turbulent flow as well as in the structure of puffs and in turbulence.

Why, however, is it difficult to solve the Reynolds problems? It seems unclear what causes all the transitions in pipe flows. Regarding the primary parameter of transition, White [27] states that "Transition depends upon many effects, e.g., wall roughness or fluctuations in the inlet stream, but the primary parameter is the Reynolds number." This gives rise to a new question: why is the Reynolds number the primary parameter determining transition to turbulence?

Regarding the effects of disturbances, Schneider and Eckhardt [22] state that "Turbulence in pipe flow has to coexist with the laminar profile since the latter is linearly stable for all Reynolds numbers. Triggering turbulence hence requires not only a sufficiently high Reynolds number but also a perturbation of sufficient amplitude. The determination of this 'double threshold' in the Reynolds number and the perturbation amplitude has been the focus of many experimental, numerical, and theoretical studies."

1.3. Possible cause and objectives. First, consider the point where transition to turbulence occurs. No transition has yet occurred in the fully developed Poiseuille region under small to medium amplitude disturbances. Thus, many researchers have stated that the flow may become turbulent long before it becomes a fully developed Poiseuille flow [7, 8, 15, 28]. Taneda [26] stated that transition in pipe flow occurs only in the entrance region.

Two causes have been proposed for transition: oscillations of disturbances and direct action of bounding walls on the flow. Most theoretical investigations of transition are concerned with stability theory based on oscillations of disturbances. However, no theory based on oscillations of disturbances yields results consistent with experimental observations such as that [R.sub.c,min] [approximately equal to] 2050, or about the intermittent behavior of turbulence in transition, or about the hysteresis curve of [R.sub.c] (Figure 2.2).

On the other hand, Lindgren states that "the experiments indicate that real turbulence--both in flashes and in continuous turbulent regions--is maintained by direct action of the bounding walls [16]." Lindgren has investigated the relation between the wall roughness and the critical Reynolds number. In contrast, we investigate the smooth-surface wall effects on [R.sub.c].

More specifically, the present study is based on three assumptions: (a) transition to turbulence in pipe flow occurs in the entrance region, (b) possible causes of a transition process are wall effects, and (c) the type of disturbances is natural and not artificial. The results of this study will be shown to confirm assumptions (a) and (b).

The objectives of the present study are as follows:

(1) to consider the prerequisites for the transition problem associated with [R.sub.c,min] [approximately equal to] 2050 (Section 2),

(2) to determine a possible cause and reason why Re itself is the primary parameter determining [R.sub.c] (Sections 3-6),

(3) to review our previously calculated results for [R.sub.c,min] [10] (Section 7),

(4) to discuss the differences in [R.sub.c] (Section 8), and

(5) to consider the magnitude of artificial finite-amplitude disturbances (Section 9).

So far, we have obtained numerical values of 2040 and 2630 [10] for [R.sub.c,min] in pipe flow based on ideas of wall effects. This calculation method was confirmed by obtaining [R.sub.c,min] = 910 and 1230 for channel flow, where Re = [HV.sub.m]/v, H is the spacing between the parallel plates of the channel and v is the kinematic viscosity ([mu]/p) [11].

2. Prerequisites for the transition problem.

2.1. The characteristics of [R.sub.c].

* Assumptions and quantities.

(i) The fluid is an incompressible, isothermal, Newtonian fluid with constant viscosity and density, disregarding gravity and external forces.

(ii) The fundamental quantity of laminar-turbulent transition in pipe flow is the critical Reynolds number [R.sub.c].

(iii) The Reynolds number ranges from about 1500 to about 15,000. The value of [R.sub.c] depends greatly upon the experimental setup such as the use of a calming chamber, baffles, honeycomb, and screens. Accordingly, it is desirable for the initial analysis to avoid geometrically complex pipe entrances. Entrance shapes are limited to a sharp-edged entrance (St), quadrant-arc rounded entrances (Qa) cut at the pipe inlet, and bell-mouth rounded entrances (Be) as shown in Figure 2.1. Pipes have smooth-surface walls.

* Types of [R.sub.c] and disturbances.

(iv) The types of disturbances are simply classified into natural (N) or artificial (A) categories. Under N-disturbance conditions, there is no artificial disturbance generator in a pipe. Under A-disturbance conditions, a disturbance generator is installed in a pipe as used in [9, 15]. The magnitudes of disturbances can be classified qualitatively but unambiguously into small (S), medium (M), or large (L) categories. For example, under N-S disturbance conditions, the fluid in a reservoir tank is kept still for at least one hour before measurement in accordance with Reynolds' color-band experiments; only then does [R.sub.c] reach an upper critical Reynolds number [R.sub.c1]. In the case of N-M disturbances, for a further example, the fluid in a tank is kept still for less than 20 minutes, and the state of the fluid in the tank is slightly disturbed, but the flow state is laminar from the inlet end to a transition point downstream as seen in our color-band experiments; the [R.sub.c] type is [R.sub.c2]. Experimental conditions of type N-L were used in Reynolds' pressure-drop experiments, where he used a valve arranged at some distance upstream from the pipe inlet. The flow state was turbulent from the valve, and the type of transition was turbulent to laminar flow (reverse transition); the [R.sub.c] type is [R.sub.c3].

(v) Each experimental apparatus has two critical Reynolds numbers: [R.sub.c1] and [R.sub.c2]. [R.sub.c3] is assumed to be about 2050 regardless of the apparatus.

* [R.sub.c] and transition process.

(vi) Transition takes place by the appearance of an increasing number of turbulent flashes for increasing Re [16, 19]; [R.sub.c1] is assumed to be the Reynolds number at which turbulent flashes first appear. [R.sub.c2] and [R.sub.c3] are the Reynolds numbers at which a laminar color-band recovers from a disturbed turbulent state.

(vii) The transition process must be the same for pipe and channel flows since there exists [R.sub.c,min] for both, respectively. In contrast, there is no [R.sub.c,min] for flow on a flat plate, so that the transition process for flow on a flat plate is different from those for pipe and channel flows.

2.2. The minimum critical Reynolds number. Let us review the results of our earlier color-band experiments. Figure 2.1 (cf. [12, Figure 3]) shows the bell-mouth and quadrant-arc rounded entrances of a test pipe of 2.6 cm diameter and about 155 cm length, along with the experimental results for [R.sub.c1] and [R.sub.c2]. Figure 2.2 (cf. [12, Figure 5]) shows the values of [R.sub.c1] and [R.sub.c2] plotted against the contraction ratio [C.sub.b] of the quadrant-arc entrance and the bell-mouth entrance diameters to the pipe diameter.

From Figure 2.2, it can be seen that both [R.sub.c1] and [R.sub.c2] are almost completely determined by the small radius of the quadrant-arc rounded entrance (r = 1-5 mm). The values of [R.sub.c1] and [R.sub.c2] increase steadily and smoothly as [C.sub.b] increases from 1 (St) to 1.39 (Qa5), and then [R.sub.c1] and [R.sub.c2] reach approximately constant values of 12,200-12,500 and 5200-5500, respectively. The minimum values of [R.sub.c1,min] = 2200 and [R.sub.c2,min] = 2050 were obtained with a sharp-edged circular pipe. Therefore, the minimum critical Reynolds number of [R.sub.c,min] [approximately equal to] 2050 was observed for [R.sub.c2,min] [12] and [R.sub.c3,min] [19].

* Vena contracta.

Flow through a sharp-edged corner is characterized in general by a convergence of streamlines in the vicinity of the inlet. At some location downstream of the inlet, however, the streamlines again can be considered parallel, and the flow area at this location (called the vena contracta) is found in general to be less than that at the geometric opening of the pipe inlet [3]. Fox and McDonald [5] state that "If the inlet has sharp corners, flow separation occurs at the corners, and a vena contracta is formed. The fluid must accelerate locally to pass through the reduced flow area at the vena contracta."

In contrast, however, in our color-band experiments, neither a vena contracta nor an inflection point were observed at the sharp-edged inlet corner, as shown in Figure 2.3, where Re = 2265. The fluid entered the pipe smoothly, and a turbulent spot suddenly appeared 29.5 cm downstream from the pipe inlet. Similarly, another pipe inlet flow observed experimentally appeared not to have an inflection point at its sharp-edged corner. We thus conclude that the vena contracta is negligible in color-band experiments using a sharp-edged pipe in a large water tank.

2.3. Entrance length. The entrance region considered in this study includes bell-mouth entrances. Thus, the inlet (x = 0) is the pipe inlet for a sharp-edged entrance pipe and is the bell-mouth inlet end for bell-mouth entrance pipes. The entrance length [x.sub.e] is defined as the distance from the inlet to the point downstream where the center-line velocity [u.sub.c] reaches 99% of its fully developed value ([u.sub.c]/[V.sub.m] = 1.98). Then, the dimensionless entrance length [L.sub.e] is given [23] by

(2.1) [L.sub.e] = [X.sub.e]/D Re = 0.056, Re [greater than or equal to] 500,

while [L.sub.e] for [u.sub.c] reaching 99.9% of its fully developed value is 0.075 [23]. Since the bell-mouth axial length is small, [L.sub.e] in (2.1) can be used both for sharp-edged pipes and for bell-mouth entrance pipes.

2.4. Transition length. Reynolds [19] states that "Under no circumstances would the disturbance occur nearer to the trumpet than about 30 diameters in any of the pipes, and the flashes generally but not always commenced at about this distance." Reynolds used three straight circular pipes in his color-band experiments. Their diameters were D = 0.7886, 1.527, and 2.68 cm, and their length [x.sub.p] was nearly 5 feet (152 cm) giving dimensionless pipe lengths [x.sub.p]/D of 193, 100, and 57, respectively. Let us consider the transition length [x.sub.t]. The dimensionless transition length [L.sub.t] is defined as the normalized distance from the inlet to the point where transition to turbulence takes place,

[L.sub.t] = [x.sub.t]/D Re.

In Reynolds' color-band experiments, [x.sub.t] was more than 30D, but it should be less than the pipe length 152 cm. Accordingly, for the pipe with D = 2.68 cm, the transition length is bounded at Re = 12,800, so that

30D < [x.sub.t] < 57D and 0.00234 < [L.sub.t] < 0.00445.

Consider now our experimental results for the transition length. For the sharp-edged pipe, under N-S disturbance conditions, [x.sub.t] ranged from about 7D ([L.sub.t] = 7/2265 = 0.0031) to 20D ([L.sub.t] = 20/2250 = 0.0089). Under N-M disturbance conditions, [x.sub.t] ranged from about 4D ([L.sub.t] = 4/2160 = 0.0019) to 22.3D ([L.sub.t] = 22.3/2020 = 0.0110). The transition lengths under N-S and N-M disturbance conditions are approximately the same. Moreover, for the Be4 bell-mouth entrance, corresponding to Reynolds' bell-mouth with [C.sub.b] = 6 (pipe of D = 2.68 cm), under N-S disturbance conditions, [x.sub.t] ranged from about 19D ([L.sub.t] = 19/11590 = 0.0016) to 54D ([L.sub.t] = 54/11060 = 0.0049) for [R.sub.t] [approximately equal to] 10,000-13,700. These results approximately agree with Reynolds' observation of [x.sub.t] > 30D.

In short, under N-S or N-M disturbance conditions, natural transition to turbulence seems to occur only in the entrance region, particularly in the vicinity of the pipe inlet, but does not occur at the pipe inlet itself.

2.5. Pressure drop in the entrance region. Here the pressure difference and the pressure drop are defined and distinguished to avoid confusing them. Let the pressure at the inlet (x = 0, i = 1, see Figure 2.4) be zero.

(i) The axial pressure difference [([[DELTA].sub.p]).sub.x] is negative, defined as

[([[DELTA].sub.p]).sub.x] [equivalent to] p(x + [increment of x]) = p(x) = [p.sub.i+1] - [p.sub.i] < 0,

and can be used in finite difference expressions, i.e.,

[partial derivative]p/[partial derivative]x [approximately equal to] [([[DELTA].sub.p])sub.x]/[increment of x].

(ii) The axial pressure drop is positive and usually defined as

[DELTA]P(x) = p(0) - p(x) = 0 - p(x) = -p(x) > 0.

(iii) There is a significant radial pressure drop [([DELTA]p).sub.wc] in the radial direction between the pressure on the wall ([p.sub.w] = [p|.sub.r=R]) and the pressure on the centerline ([p.sub.c] = [p.sub.|r=0]) (see Figure 7.2):

[([DELTA]]p).sub.wc] = [p.sub.w] - [p.sub.c] < 0.

Note that [([DELTA]]p).sub.wc] cannot be disregarded, whereas the boundary-layer approximations disregard this radial pressure drop.

(iv) The total pressure drop [DELTA]]P from the pipe inlet consists of two components: (a) the pressure drop based on the fully developed flow, f (x/D), and (b) an additional pressure drop K(x) due to a momentum change [DELTA]KE flux(x) and an accumulated increment in the wall shear [K.sub.Shear](x) between developing flow and Poiseuille flow [23]; i.e., K(x) = [DELTA[KE flux(x) + [K.sub.Shear](x). Accordingly, [DELTA]]P is expressed as

(2.2) [DELTA]P(x) = (f (x)/D + [DELTA]KE flux'(x) + [K.sub.Shear](x)) (1/2 [rho][V.sup.2.sub.m]) > 0,

where the Darcy-Weisbach friction factor f is 64/Re for Poiseuille flow (see (6.10)), and a prime ' denotes a dimensionless variable; see (6.4). When using the dimensionless axial coordinate X = x/(DRe), (2.2) is not a function of Re and can be expressed as

[DELTA]P(x) = (64X + [DELTA]KE flux'(x) + [K.sub.Shear](X)) (1/2 [rho][V.sup.2.sub.m]) > 0,

K(x) increases monotonically from 0 at x = 0 to a constant value K([infinity]) in the fully developed region, and the experimental value of K([infinity]) is reported as varying from 1.20 to 1.32 [23]. Note that K([infinity]) includes an [DELTA]KE flux' of about 1 (see (6.4) and (6.6)), which is the increase in kinetic energy flux from the entrance flow to Hagen-Poiseuille flow.

3. Wall effects and pressure.

3.1. The wall model.

(i) What are wall effects for determining [R.sub.c]? We consider why the transition between laminar and turbulent flow occurs in circular pipes with smooth-surface walls. Roughness effects of pipe wall surfaces on [R.sub.c] are excluded in this study.

Panton [18] states about a wall model that "the no-slip condition at the wall means that the particles are not translating; however they are undergoing a rotation. We might imagine that the wall consists of an array of marbles, which are rotating but remain at the same location on the wall."

In Figure 2.4, the vorticities on the wall (j = J0, r = R - (1/2)[DELTA]r [approximately equal to] R) are fixed and rotating but are not moving downstream. We assume that the wall effects are due to forces, work, energies, and power acting upon fluids by the vorticities on the wall.

(ii) Since axial and radial velocities on the wall are zero, the vorticity on the wall in two dimensions is defined by

(3.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Differentiating (3.1) with respect to r gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(iii) The Navier-Stokes equation in vector form is expressed as

(3.3) [rho] ([partial derivative]V/[partial derivative]t - V x [omega]) = 1 [nabla] (p + 1/2 [rho] [V.sup.2]) - [mu][nabla] x [omega],

where V is the velocity vector and p is the difference of the actual pressure from the hydrostatic [6].

Since V = 0 on the wall, (3.3) reduces to

-[mu]([nabla] x [omega])[|.sub.r=R] = [[nabla].sub.p] < 0.

The axial component of the curl of vorticity in two dimensions is expressed from its definition:

(3.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly, the radial component of the curl of vorticity is given from its definition:

(3.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hereafter, [omega] denotes [[omega].sub.[theta]] since [[omega].sub.r] and [[omega].sub.x] vanish in two dimensions.

4. Axial force and power. In this section, we consider axial forces and powers in the fully developed Poiseuille region.

4.1. Axial wall and flow forces.

(i) By the wall we mean the fluid particles on the wall [18]. Equation (3.4) indicates that a negative force in the axial direction is active on the wall by a rotation of the particles resulting in an axial pressure difference or an axial pressure drop.

(ii) The wall shear [[tau].sub.w] is expressed as

(4.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The applied forces due to the wall shear [[tau].sub.w] and the axial pressure difference [([DELTA]p).sub.x] for the axial small distance [increment of x] are related [27] as

(4.2) 2[pi]R([increment of x])[[tau].sub.w] = [pi][R.sup.2][([DELTA]p).sub.x] < 0,

where R is the pipe radius. Equation (4.2) shows that the wall shear on the wall equals the pressure difference in the axial direction in a fluid.

Using (3.4) and (4.2), the wall shear can be expressed from the axial component of the curl of vorticity as

(43> [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(iii) We confirm (4.3) in Hagen-Poiseuille flow, where the axial velocity distribution u is

(4.4) u = 2 [V.sub.m] [1 - [(r/R).sup.2]].

Differentiating (4.4) with respect to r on the wall gives

(4.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From (3.1), (3.2), and (4.5), we have

(4.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From (4.1) and (4.6), the wall shear is expressed as

(4.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Accordingly, the axial viscous term of the N-S equations is expressed from (3.4), (4.6), and (4.7) as

(48) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus (4.3) is confirmed for Hagen-Poiseuille flow by (4.8). It is clear that the wall shear or the axial component of the curl of vorticities on the wall is one of the wall effects.

(iv) The left-hand and right-hand terms in equation (4.2) can be called A-Wall-Force and A-Flow-Force, respectively.

(4.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

A-Flow-Force = [pi][R.sup.2][([DELTA]p).sub.x] < 0,

where V is the volume for the axial distance [increment of x] in a pipe.

(v) Note that A-Wall-Force on the wall exerts an equal force A-Flow-Force on a fluid in the axial direction by Newton's Second Law of Motion.

(vi) Regarding (1/2)R in (4.3), it is noted from (4.9) that 2[pi]R[DELTA]x x (1/2)R = V, i.e., the wall surface exerts [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on all the fluids for [increment of x] in a pipe.

4.2. Axial wall and flow powers. A-Wall-Force does negative work on fluids on the wall resulting in a pressure difference in the axial direction in a fluid. This negative work can be called the axial wall power (A-Wall-Power), and the energy loss of the fluid is called the axial flow power (A-Flow-Power) since their physical dimension is that of power energy/time, as measured in watts (W = J/s, [kg [m.sup.2]/[s.sup.3]]).

The wall surfaces where A-Wall-Force is active do not move, but the fluids at 0 [less than or equal to] r < (R-[DELTA]-r) relatively travel downstream with a mean velocity [V.sub.m]. So, it is possible that A-Wall-Force does negative work on the fluid. A-Wall-Power and A-Flow-Power are derived by multiplying both sides of (4.2) by [V.sub.m]:

(4.10) A-Wall-Power = 2[pi]R([increment of x])[[tau].sub.w][V.sub.m] = -[pi][mu][R.sup.2]([increment of x])[([nabla] x [omega]).sub.x|r=R][V.sub.m] = - [mu]V [([nabla] x [omega]).sub.x|r=R] [V.sub.m],

and

(4.11) A-Flow-Power = [pi][R.sup.2][([DELTA]p).sub.x][V.sub.m] = Q([DELTA]p)x,

where Q = [pi][R.sup.2][V.sup.m] is the volumetric flux.

4.3. Dimensionless axial wall and flow powers. A-Wall-Power is made dimensionless by dividing by Q[(1/2)[rho][V.sup.2.sub.m]]; see (6.4). In the case of Poiseuille flow for [increment of x], from (4.7) and (4.10),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where 64/Re is the Darcy-Weisbach friction factor for Poiseuille flow.

Similarly, from (4.11),

A-Flow-Power' = [pi][R.sup.2][([DELTA]p).sub.x] [V.sub.m]/ Q[(1/2)[rho][V.sup.2.sub.m]] = [([DELTA]p').sub.x].

It is clear from (2.2) that A-Flow-Power' = A-Wall-Power' in the Poiseuille region, whereas A-Flow-Power' [not equal to] A-Wall-Power' in the developing entrance region. A-Flow-Power' corresponds to the left-hand term of (2.2).

5. Radial force and power.

In this section, we consider radial forces and powers in the developing entrance region.

5.1. Radial wall and flow forces.

(i) We consider the radial wall shear [[tau].sub.rw]. As [[tau].sub.w] is expressed as (4.3), [[tau].sub.rw] can be defined by multiplying both sides of (3.5) by (1/2)R:

(5.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The radial wall force (R-Wall-Force) is expressed from (5.1) as

(5.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii) R-Wall-Force must exert an equal radial flow force (R-Flow-Force) on the fluid in radial direction resulting in the radial pressure drop [([DELTA]p).sub.wc]. Then R-Flow-Force is expressed for [increment of x] as

(5.3) R-Flow-Force = 2[pi](R-[DELTA]r)([increment of x])[([DELTA]p).sub.wc] [approximately equal to] 2[pi]R[increment of x][([DELTA]p).sub.wc] < 0.

Thus

(5.4) [[tau].sub.rw] [approximately equal to] [([DELTA]p).sub.wc] and R-Wall-Force[approximately equal to]R-Flow-Force.

(iii) Accordingly, from (3.3), (5.2), and (5.3), it is possible to assume the following process for the acceleration of a fluid in axial direction:

(a) R-Wall-Force on the wall exerts an equal force R-Flow-Force on the fluid in the radial direction by Newton's Second Law of Motion.

(b) R-Wall-Force and R-Flow-Force cause a fluid near the wall to move towards the centerline in the radial direction, as displayed in Figure 2.4, resulting in an increase in kinetic energy; cf. Section 6.1.

(c) An increase in kinetic energy means that the pressure of the fluid is transformed into an increase in kinetic energy of the fluid, as can be seen form the first term on the right-hand side of (3.3). R-Flow-Force shows the result of this change.

(d) R-Flow-force snows mat me pressure on me centerline is mgner man mat on me wail. As explained in (b) above, however, the fluid near the centerline does not move towards the wall.

(e) Thus R-Wall-Force and R-Flow-Force act as acceleration forces together with the continuity equation. Hence, the onset of the transition should depend on whether or not the acceleration power provided by R-Wall-Force exceeds a required value of [DELTA]KE flux.

5.2. Radial flow work. R-Wall-Force yields a radial velocity [upsilon] and a pressure drop [([DELTA]p).sub.wc] in the radial direction. Using the pressure drop, the radial flow work (R-Flow-Work) done on a fluid by R-Wall-Force is considered.

Here, on the basis of thermodynamics [14], the variation of the enthalpy H with pressure p at a fixed temperature can be obtained from the definition

H [equivalent to] [U.sub.int] + pV,

where [U.sub.int] is the internal energy and V is the volume. For most solids and liquids, at a constant temperature the internal energy [U.sub.int](T, V) does not change since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where T is the temperature. Since the change in volume is rather small unless changes in pressure are very large, a change in enthalpy [DELTA]H resulting from a change in pressure [DELTA]p can be approximated by

(5.5) [DELTA]H [approximately equal to] [DELTA](pV) [approximately equal to] p[DELTA]V + V[DELTA]p [approximately equal to] V([DELTA]p).

Equation (5.5) can be applied to incompressible flow as well. R-Flow-Work is expressed from (5.5) as

(5.6) R-Flow-Work = V[([DELTA]p).sub.wc] = [pi][R.sup.2]([increment of x])[([DELTA]p).sub.wc].

Note that the dimension of V[([DELTA]p).sub.wc] is physically equivalent to energy, and by multiplying by frequency (the inverse of the period [omega][|.sub.r=R]), the physical dimension of V[([DELTA]p).sub.wc]([omega][| .sub.r=R]) becomes power; see (5.12).

5.3. Radial flow power.

(i) We begin by calculating the work done on the fluid from (5.6) for the space between [x.sub.i] and [x.sub.i+1], [increment of x], as seen in Figure 2.4. The volume V on which R-Wall-Force acts is simply expressed as

(5.7) V = [pi][R.sup.2][increment of x].

Next, the pressure drop in the radial direction is approximated by the mean difference in pressure between [x.sub.i] and [x.sub.i+1],

(5.8) [([DELTA]pi).sub.wc] = 1/2[([p.sub.i,j0] + [p.sub.i+1,j0]) - ([p.sub.i,1] + [p.sub.i+1,1])].

(ii) The period during which R-Wall-Force acts on the flow passing along [[omega].sub.i,J0] and [[omega].sub.i+1,J0] on the wall is considered. The period [DELTA]t* (i) may be given by dividing the axial mesh space [increment of x] by the mean velocity at two points (i, J1) and (i + 1, J1),

[DELTA][t.sup.*](i) = [increment of x]/(1/2)([u.sub.i,J1] + [u.sub.i+1,j1]) [approximately equal to] [increment of x]/[u.sub.i+1/2,J1].

However, if this [DELTA]t* (i) is the correct period, an inconsistency is encountered. Three simple cases are considered as examples to illustrate this inconsistency.

(a) First, if the mesh aspect ratio is [increment of x] = 2[DELTA]r (m = 2), as observed in Figure 2.4, where [increment of x] is constant, then R-Flow-Work(a) and R-Flow-Power(a) can be expressed as

(5.9) R-Flow-Work(a) = V [([DELTA]p).sub.wc],

R-Flow-Power(a) = v[([DELTA]p).sub.wc]/[increment of x]/[u.sub.i+1/2,J1] = v [([DELTA]p).sub.wc] [u.sub.i+1/2,J1]/2[DELTA]r

(b) Next, if [increment of x] and V are equally divided into two parts, then [DELTA]r + [DELTA]r = [increment of x] and V1 + V2 = V. R-Flow-Work(b) in V is calculated by adding the work in V1 and the work in V2,

(5.10) R-Flow-Work(b) = V1[([DELTA]p1).sub.wc] + V2[([DELTA]p2).sub.wc] [approximately equal to] V[([DELTA]p).sub.wc].

R-Flow-Power(b) in V is calculated by adding the power in V1 and the power in V2,

R-Flow-Power(b) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

assuming that [([DELTA]p1).sub.wc] [approximately equal to] [([DELTA]p2).sub.wc] [approximately equal to] [([DELTA]p).sub.wc] and [u.sub.i+1/4,J1] [approximately equal to] [u.sub.i+3/4,J1] [approximately equal to] [u.sub.i+1/2,J1]. From (5.9) and (5.10), R-Flow-Work(a) equals R-Flow-Work(&). Comparing R-Flow-Power(a) and (b), however, R-Flow-Power(b) is two times R-Flow-Power(a), although the volume and position are the same.

(c) In numerical calculations, usually [increment of x] = m[DELTA]r (m = 1,2,3, ...). To avoid the inconsistency between (a) and (b) above, the following period is required for a general mesh system of [increment of x] = m[DELTA]r:

(5.11) [DELTA][t.sub.i] = [DELTA]r/(1/2) ([u.sub.i,j1] + [u.sub.i+1,j1]) [approximately equal to] 1/(1/2)([[omega].sub.i,J0] + [[omega].sub.i+1,J0])

This period in (5.11) is based on the following assumptions: a rotation of a fluid particle on the wall yields a vortex and a vorticity. Then the curl of vorticity yields R-Wall-Force from (5.2). The diameter of vorticities on the wall is [DELTA]r. Accordingly, R-Wall-Force is produced between two continuous vortexes or per [DELTA]r.

(iii) R-Flow-Power is derived from (5.7), (5.8), and (5.11):

(5.12) R-Flow-Power = [pi][R.sup.2][increment of x][([DELTA]p).sub.wc] [omega][|[sub.r=R] = V[([DELTA]p).sub.wc] [omega][|.sub.r=R].

5.4. Radial wall work and wall power.

(i) The radial wall work (R-Wall-Work) approximately equals R-Flow-Work. >From (3.5), (5.1), (5.4), and (5.6), R-Wall-Work can be expressed from R-Flow-Work by replacing [([DELTA]p).sub.wc] with - (1/2)[mu]R[([nabla] x [omega]).sub.r]|[sub.r=R]:

R-Wall-Work = [pi][R.sup.2]([increment of x])[[tau].sub.rw] = -1/2[pi][mu][R.sup.3]([increment of x])[([nabla] x [omega]).sub.r]|[sub.r=R] = 1/2 [pi][mu][R.sup.3] [([DELTA][omega]).sub.x][|.sub.r=R].

(ii) The radial wall power (R-Wall-Power) is obtained by multiplying R-Wall-Work by the vorticity [omega][|.sub.r=R]:

(5.13) R-Wall-Power = 1/2 [pi][mu][R.sup.3][omega][([DELTA][omega]).sub.x|r=R].

Equation (5.13) approximately equals (5.12):

R-Wall-Power [approximately equal to] R-Flow-Power.

6. A criterion for determining [R.sub.c].

6.1. Kinetic energy flux.

(i) Consider the kinetic energy flux (KE flux) at the inlet. The KE flux varies with inlet shapes such as bell-mouths and with radial velocities. If the velocity profile is the mean velocity [V.sub.m] only, then the kinetic energy flux across the inlet is given by

(6.1) KE [flux.sub.Mean] = [[integral].sup.R.sub.0] (2[pi]rdr)[V.sub.m] (1/2 [rho][V.sup.2.sub.m]) = 1/8 [pi][rho][D.sup.2][V.sup.3.sub.m] = Q (1/2 [rho][V.sup.2.sub.m]).

Note that the physical dimension of KE flux is that of power. In Hagen-Poiseuille flow, the axial velocity distribution is given by (4.4), and the kinetic energy flux of Hagen-Poiseuille flow is

(6.2) KE [flux.sub.Poiseuille] = [[integral].sup.R.sub.0] [(2[pi]rdr) (1/2[rho]) {2[V.sub.m][1 - [(r/R).sup.2]]}.sup.3] dr = 2Q (1/2 [rho][V.sup.2.sub.m]).

Let the kinetic energy flux of a fluid at the pipe inlet be KE [flux.sub.Iniet]. Then, the [DELTA]KE flux is expressed as

[DELTA]KE flux = KE [flux.sub.Poiseuille] - KE [flux.sub.Inlet].

(ii) In numerical calculations, dimensionless variables denoted with a prime (') are used:

(6.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where t' is the time, [psi]' is the stream function, and X' is the axial coordinate . Note that the dimensionless axial coordinate x' (= x/D) is used for the calculations, and X' (= x/(D Re)) is used in our figures and tables .

(iii) KE flux and [DELTA]KE flux are made dimensionless by dividing by KE [flux.sub.Mean]. From (6.1) and (6.2), KE [flux'.sub.Mean] = 1 and KE [flux'.sub.Poiseuille] = 2. The dimensionless [DELTA]KE flux is given by

(6.4) [DELTA]KE flux' = KE [flux.sub.Poiseuille] - KE [flux.sub.Inlet]/KE [flux.sub.Mean] = [DELTA]KE flux/Q[(1/2)[rho][V.sup.2.sub.m].

6.2. Determination of [R.sub.c].

(i) It appears likely that laminar-turbulent transition always occurs in the developing entrance region, where flow develops into Hagen-Poiseuille flow. In that case, KE flux increases. On the other hand, there is no R-Wall-Power or [DELTA]KE flux in the Poiseuille region. Hence, it is assumed that R-Wall-Force is a possible cause for the flow development and R-Wall-Power is used for [DELTA]KE flux in the developing region.

Let the total R-Wall-Power be T-R-Wall-Power. Thus, the criteria for determining [R.sub.c] are

(6.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii) When [absolute value of T-R-Wall-Power] > [DELTA]KE flux, the difference between [absolute value of T-R-Wall- Power] and [DELTA]KE flux might be maintained in the internal energy of the fluid, restoring to the pressure of the fluid.

(iii) Let the total R-Flow-Power be T-R-Flow-Power. Since R-Flow-Power approximately equals R-Wall-Power, the criterion for determining [R.sub.c] are also

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

6.3. Sharp-edged inlet pipe and [R.sub.c,min].

(i) [R.sub.c] is [R.sub.c,min] when using a sharp-edged inlet pipe under natural disturbance conditions in a water tank. For a sharp-edged pipe, it is assumed that KE [flux.sub.Inlet] = KE [flux.sub.Mean]. [DELTA]KE flux' is calculated from (6.1), (6.2), and (6.4) as

(6.6) [DELTA]KE flux' = 2Q[(1/2)[rho][V.sup.2.sub.m] - Q[(1/2)[rho][V.sup.2.sub.m]/Q[(1/2)[rho][V.sup.2.sub.m] = 1.

If KE [flux.sub.Inlet] [not equal to] KE [flux.sub.Mean], then [DELTA]KE flux' [not equal to] 1.

(ii) For a sharp-edged inlet pipe, R-Wall-Power' is reduced from (5.13) to

R-Wall-Power' = (1/2)[pi][mu][R.sup.3][omega][([DELTA][omega]).sub.x]/Q[(1/2)[rho][V.sup.2.sub.m]] = [omega]' [([DELTA][omega]).sub.'x]/2Re[|.sub.=R],

and

(6.7) [absolute value of T-R-Wall-Power'] = 1/2Re [I1.summation over (i=2)] {absolute value of [omega]'[([DELTA][omega])'.sub.x|r=R],

where if I1(= 10 - 1) indicates the Poiseuille region, then [omega]'[|.sub.r=R] = 8 from (4.6) and (6.3). If [absolute value of T-R-Wall-Power'] = 1, then [R.sup.c,min] is obtained from (6.7) as

(6.8) Re = [R.sub.c,min] = 1/2 [I1.summation over (i=2)] [absolute value of [omega]'[([DELTA][omega])'.sub.x|r=R].

It is noted [27] that "Equations (6.7) and (6.8) show a possible answer to why Re is the primary parameter for laminar-turbulent transition in pipe flow."

(iii) The dimensionless R-Flow-Power is obtained from (5.12) as

[absolute value of R-Flow-Power'] = [absolute value of [pi][mu][R.sup.3][omega][([DELTA][omega]]).sub.x]/Q [(1/2)[rho][V.sup.2.sub.m] = [omega]' [([DELTA][omega])'.sub.x]/2Re|[sub.r=R],

and

(6.9) [absolute value of T-R-Flow-Power'] = [I1.summation over (i=2) [absolute value of ([increment of x]') [omega]' [([DELTA]p').sub.wc]].

If [absolute value of T-R-Flow-Power'] = 1, then Re = [R.sub.c,min].

Equations (6.7) and (6.9) can be solved by an interpolation method with varying Re.

6.4. Stability of Poiseuille flow.

(i) We consider the question why Poiseuille flow is stable by using the shear stress [mu](du/dr). Let the shear forces exerted on a fluid by shear stresses at r and (r + [DELTA]r) for [increment of x] be [[tau]-force.sub.1] and [[tau]-force.sub.2], respectively. From (4.4),

[tau]-[force.sub.1] = (2[pi]r[increment of x])[mu]du/dr = - 8[pi][mu][V.sub.m][[DELTA].sub.x]([r.sup.2])/[R.sup.2]/[R.sup.2],

[tau]-[force.sub.2] = [2[pi](r + [DELTA]r) [increment of x]] [mu]du/dr = 8[pi][mu][V.sub.m][increment of x](r + [DELTA]r).sup.2]/[R.sup.2].

Then the shear force per unit volume is expressed by subtracting [tau]-[force.sub.1] from [tau]-[force.sub.2] resulting in the constant axial pressure difference:

(6.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From (6.10), the Darcy-Weisbach friction factor f = 64/Re in (2.2) is obtained.

Thus the constant shear force of (-8[mu][V.sub.m]/[R.sup.2]) is active across the radius in the entire Poiseuille region, so that transition to turbulence will not occur in the Poiseuille region.

(ii) Our earlier calculated results of the axial velocity development [25] show that the velocity distribution is concave in the central portion for X < 0.0002 at Re = 2000 and appears approximately flat in the central portion for 0.0003 < X < 0.004. It is clear from the calculated results that the magnitude of shear force per unit volume near the wall is larger than that in the central portion resulting in a difference in energy of the fluid in the radial direction. Therefore, differences in the shear force and in the energy of a fluid in radial direction trigger a transition to turbulence in the developing entrance region.

7. Calculation of [R.sub.c,min].

Part of this section refers to our earlier calculations [10]. The notational primes denoting dimensionless expressions are hereafter elided for simplicity.

7.1. Governing equations. We introduce the stream function and vorticity formulae in two-dimensional cylindrical coordinates for the governing equations to avoid the explicit appearance of the pressure term. Accordingly, the velocity fields are determined without any assumptions concerning pressure. Subsequently, the pressure distribution is calculated using values of the velocity fields.

Let [psi] be the stream function. The dimensionless transport equation for the vorticity [omega] is expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The Poisson equation for [omega] is derived from the definition of [omega], i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where only the angular ([theta]) component of [omega] in a two-dimensional flow field is effective, and thus [omega] denotes [[omega].sub.[theta]]. The axial velocity u and radial velocity v are defined as derivatives of the stream function, i.e.,

(7.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The pressure can be calculated from the steady-state form of the N-S equations. The pressure distribution for the x-partial derivative is

(7.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and that for the r-partial derivative is

(7.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since u and [upsilon] are known at every point from (7.1), a smooth pressure distribution that satisfies both (7.2) and (7.3) is calculated using the Poisson equation for the pressure [20],

(7.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Initial values are determined using (7.2), and then (7.4) is used to obtain better solutions.

7.2. Numerical method and the mesh system. The finite difference equations for both the stream function-vorticity and the pressure are solved by the Gauss-Seidel iterative method. This computational scheme uses the Forward-Time, Centered-Space (FTCS) method. The scheme has second-order accuracy in the space variables and first-order accuracy in time. The rectangular mesh system used is schematically illustrated in Figure 2.4, where 10 and J0 are the maximum coordinates for axial and radial mesh points, respectively, and 11 = 10 - 1 and J1 = J0 - 1. J0 is located on the wall, and J1 is located at the wall.

To calculate R-Wall-Power and R-Flow-Power, two mesh systems (b) and (c) are used for four Reynolds numbers, 1000, 2000, 4000, and 10,000 (cf. [10]): (b) 10 = 1001, J0 =51, [DELTA]X = 0.00002, [increment of x] = Re [DELTA]X, and max X = 0.02, and (c) J0 =101, with other parameters the same as for (b).

7.3. Vorticity on the wall. The vorticity boundary condition on no-slip walls is derived from (3.1) as [omega]|[sub.r=R] = -du/dr. A three-point, one-sided approximation for (3.1) is used to maintain second-order accuracy,

(7.5) [omega][|.sub.r=R] = [[omega].sub.i,J0] [approximately equal to] - [3u.sub.i,J0] [4u.sub.i,J1] + [u.sub.i,J2]/2[DELTA]r = [4u.sub.i,J1] - [u.sub.i,J2]/2[DELTA]r.

The boundary conditions for the axial velocity at the pipe inlet (i = 1) are approximated as

(7.6) [u.sub.1,j] = 1, 1 [less than or equal to] j [less than or equal to] J1, and [u.sub.1,J0] = 0.

Table 7.1 and Figure 7.1 show the vorticities on the wall. Our major conclusions for the vorticity distribution are:

(i) A large value of vorticity may appear at a pipe inlet edge. According to (7.5) and (7.6), [[omega].sub.1,j0] at the pipe inlet is 150 for J0 = 51 ([DELTA]r = 0.01) and 300 for J0 = 101 ([DELTA]r = 0.005); i.e., if [DELTA]r [right arrow] 0, then [[omega].sub.1,J0] [right arrow] [infinity]. In the FTCS method, [[omega].sub.1,J0] is not used, so that reasonable values of [omega]|[sub.r=R] can be observed in Table 7.1 and Figure 7.1.

(ii) It is clear from Table 7.1 that for X [greater than or equal to] 0.00002, the vorticity on the wall is approximately the same for J0 = 51 and 101 but varies somewhat with Re. For X [greater than or equal to] 0.0001 the vorticity on the wall is independent of Re and the size of [DELTA]r.

7.4. Radial pressure drop along the pipe. A natural transition occurs where X is less than 0.01 experimentally, so the pressure drop for X [less than or equal to] 0.02 was calculated. To verify the accuracy of the calculations, the calculated results for the pressure drop are compared with Shapiro's experimental results [24] as displayed in Figure 7.2(a) through (d) (cf. Figures. 7-10 in [10]), where zz = X.

The major conclusions for the radial pressure distribution are as follows:

(i) From Figure 7.2, the radial pressure difference [absolute value of [([DELTA]p).sub.wc]] decreases as Re increases; i.e., [absolute value of R-Flow-Power] drops with Re.

(ii) Consider the above item in detail. So far, three major aspects have been studied regarding phenomena in the entrance region [6]: (a) the pressure difference between any two sections in axial direction, (b) the velocity distribution at any section, and (c) the length of the entrance region [L.sub.e]. According to many previous investigations, variables such as velocity and pressure distributions become similar and independent of the Reynolds number when they are plotted against the dimensionless distance X (= x/(DRe)).

Accordingly, it is important to find variables which decrease in the X coordinate as Re increases since transition occurs as Re increases. For that purpose we identified R-Wall-Force, R-Wall-Power, R-Flow-Force, R-Flow-Work, and R-Flow-Power, presented here for the first time.

(iii) [([DELTA]p).sub.wc] can be used to calculate R-Flow-Power.

(iv) It is necessary to numerically calculate [([DELTA]p).sub.wc] with varying mesh systems and with varying inlet boundary conditions to obtain more precise values of [([DELTA]p).sub.wc].

7.5. Calculation of [R.sub.c,min]. [absolute value of T-R-Wall-Power] and [absolute value of T-R-Flow-Power] are obtained from (6.7) and (6.9), respectively, and the calculated results are displayed in Figure 7.3, where the minimum critical Reynolds number [R.sub.c,min] is obtained via linear interpolation.

For [absolute value of T-R-Wall-Power] and J0 = 51,

[R.sub.c,min] - 1000/200 - 1000 = 1 - 2.494/0.859 - 2.494 and [R.sub.c,min] = 1910.

Similarly, [R.sub.c,min] = 1950 for J0 = 101. For [absolute value of T-R-Flow-Power], [R.sub.c,min] = 2840 for J0 = 51, and 1910 for J0 =101.

In the case of [absolute value of T-R-Wall-Power], both calculated values of [R.sub.c,min] are close to Reynolds' experimental value of 2050. When using [absolute value of T-R-Flow-Power], the calculated value for J0 = 51 is somewhat higher than the experimental value, although the value for J0 = 101 is close to the experimental value. Approximations for the pressure field in a pipe needs to be reconsidered in future investigations.

In summary, [absolute value of T-R-Wall-Power] and [absolute value of T-R-Flow-Power] vs. [DELTA]KE flux are possible methods for calculating [R.sub.c].

8. Discussion of the difference in [R.sub.c].

8.1. Difference between [R.sub.c1,min] and [R.sub.c2,min]. Let the flow state be laminar. It is assumed that [DELTA]KE [flux.sub.Variance] is the variance from KE [flux.sub.Mean] if the inlet velocity is not the mean velocity [V.sub.m] and that [DELTA]KE flux includes the inlet loss [K.sub.inlet] due to inlet disturbances. Accordingly, the required kinetic energy flux for the development into Hagen-Poiseuille flow is described as

(8.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where KE [flux.sub.Poiseuille] = 2, KE [flux.sub.Mean] = 1 (see Section 6.1).

There are two possible terms in (8.1) to which the difference between our experimental values [R.sub.c1,min] [approximately equal to] 2200 and [R.sub.c2,min] [approximately equal to] 2050 might be ascribed: [DELTA]KE [flux.sub.Variance] and [K.sub.Inlet]. Recall from (6.5) that [DELTA]KE flux is equal to [absolute value of T-R-Wall-Power] when Re is [R.sub.c].

First, in the case that [DELTA]KE flux varies and [K.sub.Inlet] = 0, if [DELTA]KE flux is assumed to be inversely proportional to Re around Re = 2000-2300, then [DELTA]KE flux at Re = 2200 is approximated by

[DELTA]KE [flux.sub.Rc1,lmin] x 2200 = [DELTA]KE [flux.sub.Rc2,min] x 2050.

Then,

[DELTA]KE [flux.sub.Rc2,min] = 1, [DELTA]KE [flux.sub.Rc1,min] [approximately equal to] 0.932, and [approximately equal to]KE [flux.sub.Variance] [approximately equal to] 0.068.

The velocity distribution for turbulent flow in a pipe is flatter than that for laminar flow. Thus, the value of 0.068 for [DELTA]KE [flux.sub.Variance] is possible for the difference between [R.sub.cl,min] and [R.sub.c2,min]. Figure 8.1(a) conceptually shows the difference between [R.sub.cl,min] and [R.sub.c2,min] in the first case.

Second, if [DELTA]KE [flux.sub.Variance] = 0 and [K.sub.Inlet] [not equal] 0, then

[DELTA]KE [flux.sub.Rc1,min] x 2200 = ([DELTA]KE [flux.sub.Rc2,min] + [K.sub.Inlet]) x 2050.

Then,

(8.2) [DELTA]KE [flux.sub.Rc2,min] = 1 and [K.sub.Inlet] [approximately equal to] 0.073, [DELTA]KE [flux.sub.Rc1,min] = 1.

Here, both velocity distributions at the inlet are uniform and [DELTA]KE flux = 1. The assumption of [K.sub.Inlet] = 0.073 causes the difference between [R.sub.cl,min] and [R.sub.c2,min]. Exact calculations and experiments will determine which case is operative in future investigations.

8.2. Effects of the bell-mouth entrance on Rc. R-Wall-Power is generated on the walls of a bell-mouth entrance and a pipe, indicating that R-Wall-Power depends upon Re when the entrance shape is fixed. Here, it is considered referring to Figure 8.1(b) why and how [R.sub.c2] is about 5500 for the Be4 bell-mouth entrance with [C.sub.b] = 6 as displayed in Figure 2.2. Let the average velocity at the bell-mouth inlet end, [V.sub.bell], be [V.sub.m]/[C.sup.2.sub.b]. The increase in kinetic energy flux between the bell-mouth inlet end and the Poiseuille region is approximated in dimensional form by

(8.3) [DELTA]KE [flux.sub.Bell] = [pi]/4[rho][D.sup.2][V.sup.3.sub.m] - [pi]/8 [rho]1/[C.sup.4.sub.b][D.sub.2][V.sup.3.sub.m] = Q(1/2[rho][V.sup.2.sub.m]) (2 - 1/[C.sup.4.sub.b]].

The dimensionless form of (8.3) is given by

[DELTA]KE [flux.sub.Bell] = 2 - [C.sup.4.sub.b].

From this relation, the [DELTA]KE flux value increases from 1 to 2 as [C.sub.b] increases.

9. Questions about disturbances.

9.1. Disturbance amplitude.

(i) Flow in the entrance region is sensitive to N-S disturbances, whereas A-L disturbances are required to trigger transition in a Hagen-Poiseuille flow. Hence, are there double disturbance standards for transitions in the entrance flow and Hagen-Poiseuille flow?

(ii) Consider the disturbance loss coefficient. A pipe system has many fitting losses, including entrance shape, bends, elbows, valves, expansions, and contractions. They can be aggregated into a single total system loss using the pressure drop equation (2.2). The pressure drop [DELTA]P, head loss [DELTA]h, and fitting loss coefficient K for a valve, device, or fitting are related by

(9.1) [DELTA]P = K(1/2 [rho][V.sup.2.sub.m]) = [rho]g([DELTA]h).

The loss coefficient K is derived from (9.1) as

(9.2) K = [DELTA]P/(1/2)[rho][V.sup.2.sub.m] = [DELTA]h/[V.sup.2.sub.m]/(2g).

Next, the power dissipation of a valve generating continuous disturbances is considered.

Power dissipation = Q([DELTA]P) = QK(1/2[rho][V.sup.2.sub.m]),

where the dimensions are those of power. On the other hand, the energy of a pulse disturbance is given by

Energy = Q([DELTA]t)K(1/2 [rho][V.sup.2.sub.m]) = Q([DELTA]t)([rho]g[DELTA]h).

To compare the magnitude of a pulse disturbance with that of a continuous disturbance, the width [DELTA]t of the pulse disturbance must be a single second so that the integrated magnitude of the pulse becomes power. That is, continuous disturbances are divided into a series of single-second discrete disturbances, and each one-second discrete disturbance is compared with the one-second pulse disturbance using the same units of power. If [DELTA]t for a pulse disturbance is less than one second, then the value of [DELTA]t can be used as a non-unit weight [bar.[DELTA]t] of the pulse disturbance.

As a result, the magnitude of disturbances expressed in the dimension of power is given as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In a pipe system, since Q and (1/2) [rho][V.sup.2.sub.m] are constant, the disturbance magnitude reduces to a coefficient:

(9.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(iii) Consider Hof et al.'s disturbance amplitude [9]. Let [member of] = [member of] (Re) denote the minimal amplitude of all finite perturbations that can trigger transition. If [member of] scales with Re according to [member of] = O([Re.sup.[gamma]]) as Re [right arrow] [infinity], then what is the value of the exponent [gamma]? Hof et al. obtained [gamma] = -1 for 2000 < Re < 18,000 using a single "boxcar" rectangular pulse of fluid, injected tangentially into a flow via a ring of six equally spaced holes, from which the pressure trace of the perturbation was observed. Then, D = 20 mm, [DELTA]h = 37 mm [H.sub.2]O, [DELTA]t = 1.2 s (length = 6D), and [V.sub.m] = 0.1 m/s at Re [approximately equal to] 2000. Thus, the loss coefficient K for the pulse disturbance is calculated from (9.2) and (9.3) as

K = [37 x [10.sup.-3]]/[0.[1.sup.2]/(2 x 9.8)] = 72.5.

For example, the values of K in a screwed 20-mm-pipe-fitting system are 10 for a glove valve, 0.28 for a gate valve, and 6.1 for an angle valve; cf. [1, Table 1 in ASHRAE Handbook 22.2]. The amplitude of the injected-pulse disturbance of K = 72. 5 is much larger than that of these valves. Therefore, it is difficult to compare N-S disturbances in the inlet flow and A-L disturbances injected into Hagen-Poiseuille flow.

(iv) What is the magnitude of natural disturbances? The disturbance loss coefficient [K.sub.Dist] may be expressed as

(9.4) [K.sub.Dist] = [([V.sub.m] + [u'.sub.m]).sup.2] - [V.sup.2.sub.m]/[V.sup.2.sub.m],

where [u'.sub.m] is the axial velocity perturbation in [V.sub.m].

The perturbations in turbulent flow were observed as [u'.sub.c]/[V.sub.c] [approximately equal to] 0.035 for x/D > 60, where the subscript 'c' denotes centerline; see [29, Figure 5]. If [u'.sub.c]/[V.sub.c] [approximately equal to] [u'.sub.m]/[V.sub.m], then [K.sub.Dist] is obtained from (9.4): [K.sub.Dist] = [1.035.sup.2] - 1 [approximately equal to] 0.071. The disturbance loss of [K.sub.Dist] = 0.071 is much less than the value of [absolute value of T-R-Wall-Power] [approximately equal to] 1 around Re = 2000, so that R-Wall-Force can depress disturbed flow and change it into laminar flow.

Accordingly, it may be stated that a possible cause of the onset of transition is the wall effects exerted by R-Wall-Force rather than an oscillation of disturbances.

Conclusions. A definite and fundamental problem of fluid dynamics is to theoretically obtain Reynolds' findings of [R.sub.c,min] [approximately equal to] 2050 and [R.sub.c1] = 12,800 for the transition to turbulence in circular pipe flow. It seems that a transition in circular pipe flows always occurs in the developing entrance region, where the axial velocity distribution develops from an uniform flow at the inlet to the Poiseuille profile, and the kinetic energy of the flow increases, i.e., that a [DELTA]KE flux exists.

In this paper we have studied this flux by the introduction of R-Wall-Power, which arises due to the radial component of the viscous term (R-Wall-Force) in the Navier-Stokes equations on the wall.

Accordingly, the hypothesized criterion for laminar-turbulent transition can be tersely expressed as follows: transition occurs if and only if [absolute value of T-R-Wall-Power] < [DELTA]KE flux. The criterion simply implies that if a pipe flow develops into Hagen-Poiseuille flow, then transition does not occur, and if not, transition occurs.

R-Wall-Power clarified that under natural disturbance conditions, (i) R-Wall-Power and [DELTA]KE flux are effective only in the entrance region, so that the transition occurs only in the entrance region, (ii) R-Wall-Force is a possible cause of a transition process, (iii) the Reynolds number becomes a critical Reynolds number, i.e., Re = [R.sub.c] when [absolute value of T-R-Wall-Power] = [DELTA]KE flux, and (iv) the [R.sub.c] value depends upon both the entrance shape and the flow conditions at the inlet. Thus, [R.sub.c] takes a minimum value of about 2050 only when using a sharp-edged entrance pipe.

Future investigations will be (i) to numerically calculate T-R-Wall-Power with varying mesh systems and with varying inlet boundary conditions, and (ii) to determine why and how [R.sub.c] takes values of about 5500 and 12,800 when using a bell-mouth entrance.

Acknowledgments. The author wishes to express his sincere appreciation to Professors M. Cohen and M. Honma of the University of Aizu, Dr. K. Shimomukai of SGI Japan, Mr. T. Yanagiya, Professor Emeritus F. Stenger of the University of Utah, and the ETNA referees for encouragement and valuable technical advice.

Appendix A. Notation.

A = artificial disturbance condition

Be = bell-mouth entrance pipe

[C.sub.b] = contraction ratio ([D.sub.b]/D)

D = pipe diameter

[D.sub.b] = bell-mouth and quadrant-arc inlet diameter

f = Darcy-Weisbach friction factor, (2.2)

H = enthalpy

i = axial point of mesh system

[I.sub.0] = maximum axial mesh point

J = radial point of mesh system

[J.sub.0] = maximum radial mesh point

K = minor pressure loss

[K.sub.Dist] = inlet disturbance loss, (9.4)

[K.sub.Iniet] = inlet disturbance loss, (8.2)

[L.sub.e] = dimensionless entrance length ([x.sub.e]/(DRe))

N = natural disturbance condition

p = pressure

P = pressure (P = -p)

Q = volumetric flux (([pi]/4)[D.sup.2][V.sub.m])

r = radial coordinate; radius of quadrant-arc rounded entrance

R = pipe radius

[R.sub.c] = critical Re for laminar-turbulent transition

[R.sub.c1] = [R.sub.c] under S disturbance conditions

[R.sub.c2] = [R.sub.c] under M disturbance conditions

[R.sub.c3] = [R.sub.c] under L disturbance conditions

Re = Reynolds number ([DV.sub.m]/v)

St = sharp-edged entrance pipe

u = axial velocity

[U.sub.int] = internal energy of a fluid

v = radial velocity

[V.sub.m] = mean axial velocity

V = volume

x = axial coordinate

x' = dimensionless axial coordinate (x/D)

[x.sub.e] = entrance length

[x.sub.t] = transition length from inlet

X = dimensionless axial coordinate (x/(DRe))

[theta] = cylindrical coordinate

[mu] = viscosity coefficient

v = kinematic viscosity (y/p)

[rho] = density

[[tau].sub.w] = wall shear, (4.1)

[[tau].sub.rw] = radial wall shear, (5.1)

[omega] = vorticity

[DELTA]KE flux = difference in kinetic energy flux, (6.4)

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* Received March 14, 2015. Accepted July 17, 2017. Published online on October 9, 2015. Recommended by F. Stenger.

HIDESADA KANDA ([dagger])

([dagger]) Department of Computer Science and Engineering, University of Aizu, Aizu-Wakamatsu, Fukushima 965-8580, Japan (hidesada.kanda@gmail.com).

Table 7.1
Vorticity vs. X and Re, (b) J0 = 51 and (c) J0 = 101.

X         1000-b    2000-b    4000-b    10000-b

0.00002   100.2      83.3      70.3      60.8
0.00004    66.0      49.8      41.5      38.8
0.00006    46.7      36.0      32.6      33.5
0.00008    36.4      30.3      29.6      31.6
0.0001     31.0      27.7      28.4      30.3
0.0002     23.1      24.1      25.4      25.8
0.0005     19.1      20.0      20.2      20.1
0.001      16.3      16.6      16.6      16.6
0.005      11.0      11.0      11.0      11.0
0.01        9.6       9.6       9.6       9.6
0.02        8.8       8.8       8.8       8.8

X         1000-c    2000-c    4000-c    10000-c

0.00002    112.4     85.0      69.8      59.6
0.00004    55.6      42.8      37.9      37.3
0.00006    38.5      32.6      31.5      33.5
0.00008    31.8      28.9      29.5      31.8
0.0001     28.4      27.1      28.4      30.4
0.0002     22.7      24.1      25.5      25.9
0.0005     19.1      20.1      20.3      20.2
0.001      16.4      16.7      16.7      16.6
0.005      11.0      11.0      11.0      11.0
0.01        9.6       9.6       9.6       9.6
0.02        8.8       8.8       8.8       8.8

Fig. 2.1. (a) Bell-mouth entrance and (b) quadrant-arc entrance.
Sizes often pipe entrance shapes (D = 2.6 cm, L2 = 150 cm),
and experimental results for [R.sub.c1] and [R.sub.c2]; cf.
[12, Figure 3 and Table 6].

Entrance   [D.sub.b]    r    [L.sub.b]

St            2.6      0.0       -
Qa1           2.8      0.1      0.1
Qa2           3.0      0.2      0.2
Qa3           3.2      0.3      0.3
Qa4           3.4      0.4      0.4
Qa5           3.6      0.5      0.5
Bel           4.0       -      1.05
Be2          6.07       -       2.6
Be3          10.4       -      5.85
Be4          15.6       -      9.75

Entrance   [C.sub.b]   [R.sub.c1]   [R.sub.c2]

St             1          2200         2050
Qa1          1.08         3600         3150
Qa2          1.15         5000         4100
Qa3          1.23         6700         4650
Qa4          1.31         8750         4850
Qa5          1.39        12,200        5200
Bel          1.54        12,500        5200
Be2          2.34        12,200        5450
Be3            4         12,500        5500
Be4            6         12,200        5500

FIG. 7.3. [R.sub.c,min] based on (a) [absolute value of
T-R-Wall-Power] (TRWP) and [absolute value of T-R-Flow-Power]
(TRFP) for mesh systems (b) J0 = 51 and (c) J0 = 101.

Re-mesh         1000-b    2000-b    4000-b    10000-b

TRWP             2.494     0.859     0.305     0.091
[R.sub.c,min]              1910

TRFP             1.789     1.183     0.747     0.427
[R.sub.c,min]              2840

Re-mesh         1000-c    2000-c    4000-c    10000-c

TRWP             3.143     0.895     0.301     0.087
[R.sub.c,min]              1950

TRFP             1.497     0.952     0.574     0.265
[R.sub.c,min]              1910
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