Equivalents in Fluid Mechanics

Equivalences and Diminishments in Fluid Flow

Stuart W. Churchill

Department of Chemical and Biomolecular Engineering

University of Pennsylvania

In a presentation at the 2013 AIChE Annual Meeting and in a subsequent publication a new concept and a new methodology for the correlation of experimental data and/or numerically computed values were illustrated by means of a specific example, namely the equations that predict the same Nusselt number for free convection and for forced convection. The expressions that produce the same Nusselt number were designated as equivalents and the relationship between the Grashof number and the Reynolds number that resulted from equating these two expressions was designated an equivalence.
The initial theoretically based results were for the thin boundary layer regime on a vertical plate but the same equivalences were found to extend over the entire laminar boundary layer regime including that of thickening as the flow decreased. The equivalences were also found to be identical functionally for the thin laminar boundary layer regime on round horizontal cylinders and on isolated spheres, and for both a uniform temperature and a uniform heat flux density on the surface. The potential utility of the concept of equivalence is greatly enhanced by those generalities. Because theoretically based correlating equations for turbulent free convection are limited to a vertical isothermal plate, the only equivalence derived for the turbulent regime was for that condition.
In the version initially submitted for publication it was recommended that the applicability of the concept of equivalence for other modes of heat and mass transfer and/or related topics be investigated by teachers and practitioners of thermal science and other aspects of process design. That recommendation did not survive the process of the review, so in order to demonstrate its breadth of applicability a completely different pair of equivalents is examined herein, namely the relative rates of fully developed flow in a round tube and in a parallel- plate channel of asymptotically large aspect ratio. Several equivalences were identified as well as another new
concept for correlation, here designated a reductive. Most chemical processing is carried out in fully developed flow in a round tube so it might be presumed to be a familiar topic to chemical engineers but the current context invokes some aspects that may not be known and therefore justifies a brief review.
The application in 1926 of conventional dimensional analysis to a selective set of time-averaged variables led to the following expression for the time-mean velocity distribution in steady fully developed turbulent flow through a round tube:

1/ 2

? ? ?

? y??

? ?1/ 2 y ?

u? ?

? ? ? w , ?

(1)

? ? w ?

? ? a ?

The following notation, which was introduced for these dimensionless groupings at the time of the derivation of Eq.
1, namely

?

u+ = f ? y ? ,

y ? ?

? ?

(2)

? a ?

has remained in common usage to the present day. The velocity and distances of Eq. 2 are said to be in wall-based units. The speculation that the dependence of u+ on a+ phases out as the wall is approached led the originator of Eqs.
1 and 2 to consider the reduction of Eq. 2 to

u+ = f {y+} (3)

Although Eq. 3 would be expected to be applicable only for y+ << a+, it has been proven to be a good approximation except for very near the wall and very near the centerline. As may be inferred by the absence of any marker of shape, Eq. 3 has also been found to apply for channels of other cross-sections and accordingly it is known as the universal law of the wall.
Although turbulent flow was in mind when Eqs. 2 and 3 were derived, nothing in that derivation excludes their application to laminar flow. The following velocity distributions for fully developed laminar flow of a fluid with invariant viscosity and density in a round tube and in a parallel-plate channel were derived by integration of the respective differential force-momentum balances:

? a? ?

? ? ?r ? ?

(4)

u w ?1 ? ? ?

and

2? ??

? b? ?

? a ? ??

? ? ?z ? ?

(5)

u w ?1

? ? ?

In wall-based units Eqs. 4 and 5 become

2? ??

? b ? ??

u? ? a

??1 ? R2 ??

(6)

2

and

u? ? b

??1 ? Z 2 ??

(7)

2

y ? y ?

Here r = a - y, z = b-y, R = 1 ? , and Z = 1 ?

a ? b?

. Equations 6 and 7 are not equivalents because they involve
a distributed function. However, the corresponding integrated-mean velocities serve that role. They are, for a round tube, first in primitive variables and then in wall-based units:

a?

and

umR = w

4?

u? ? a

(8)
(9)

m R 4

The corresponding expressions for a parallel-plate channel are:
and

b?

umP = w

3?

u? ? b

(10)
(11)
It follows that for a+ = b+

m P 3

and

? ? 3 u?

(12)
8 ReR = 3 ReP (13)
Equations 9 and 11, as well as a+ = b+, can be interpreted as equivalents, and Eqs. 12 and 13 as the equivalence in
terms of u ?
and Re, respectively. The expression in terms of the Reynolds number depends on the choice of a
characteristic length – here the equivalent diameter, namely 2a for a round tube and 4b for a parallel-plate channel.
The conjecture that the congruence in the velocity distributions for a round tube and a parallel-plate channel in fully developed laminar flow might apply for fully developed turbulent flow was confirmed by a plot of experimental values and correlating equations for the extreme case of the centerline.
The possibility that law of the center, namely

u? ? u? ? ?{y ? / a? }

(14)
and the law of the wall might have some region of overlap in which they were both acceptable approximations was subsequently speculated and it was recognized that

u+ = A + B ln{y+} (15)

was the only expression satisfying that condition.Equation 15 with the same coefficients A and B may be presumed on the basis of its derivation to be applicable for both a round tube and a parallel-plate channel.
The integration of u+ from Eq. 15 over the entire radius results in the following approximation for the mean velocity in a round tube

? ? A ? 3 B ? B ln{a ? }

m R 2

(16)
The corresponding integration across a parallel-plate channel results in

? ? A ? B ? B ln{b? }

(17)
Equations 16 and 17 may be considered to be equivalents for the mixed-mean velocity. The corresponding equivalence is
( ? ? B

um )P = ( um )R +

2

or
(18)
2ReR = ReP + B (19)

ReR and ReP in Eq.19 are expressed in terms of the hydraulic diameter.

Equations 18 and 19 are invaluable resources because they allow the more extensive and accurate experimental data and correlating equations for a parallel-plate channel, as well as the numerical solutions obtained by direct numerical simulation to be utilized for prediction for a round tube. On the one hand, the integrations of Eq.
15 that result in Eqs. 16 and 17 are exact. On the other hand, Eq.15, with any arbitrary pair of numerical coefficients, as well as all of its direct applications such as Eqs. 16 – 19, are in error structurally, and very near the wall numerically as well, because of the failure of Eq. 15 to take into account the reduced flow in the boundary layer and to a lesser extent near the centerline.
Values of A = 5.5 and B = 2.5, which were first determined experimentally in 1932, have stood the test of time although many alternative values pairs of numerical values have appeared in the literature, presumably reflecting different rates of flow and starting lengths. The corresponding value of A - (3/2)B is 1.75, of A - B is 3.0, and of B/2 is 1.25, lead to
and

? ? 1.75 ? 2.5ln{a? }

? ? 3.0 ? 2.5ln{a? }

(20) (21)
( ? ?

In1996, the quantity ?u' v'??? ?

um )P = ( um )R +1.25 (22)

?? ?u' v'? /? , which can be noted to be the local fraction of the total

shear stress that is due to turbulence, was introduced as a dimensionless variable for the quantitative description of the intensity of fully developed turbulent flow. It is defined so as always to be positive. This dimensionless variable, which constitutes a ratio of physically measureable quantities, is superior in every respect to its empirical and mechanistically based counterparts such as the eddy viscosity and the mixing length. The differential force- momentum balance for fully developed, time-averaged flow in a round tube in terms of the local fraction of the total shear stress due to turbulence is

? y ? ??

? ??? ?

du?

?1?

?

Equation 23 can be integrated formally to obtain

? 1? u' v'

a ?

? ?

dy ?

1

(23)

u ? ? a

?1? R2 ?? a

?u' v'??? dR2

(24)

R ?

2 2 2

and then that expression for u+ can be integrated over the cross-section and by parts to obtain

u? ? a

1

?1?

?u' v'??? dR4 ]

(25)

mR ?

4 0

The combination of Eq. 25 and its analog for a parallel-plate channel (not shown because of limitations of space) constitutes another equivalence for a round tube and a parallel-plate channel.
Equation 24 reveals that turbulence diminishes the rate of flow at all values of the radius as well as on the mean, as could be reasoned and as predicted by Eq. 25. The resulting relationship, namely

1

?

mR lam

? (u?

)turb

? ? ?u' v'??? dR4

0

(26)
can be designated a diminishment and thereby another new concept with potential generality for correlation and interpretation.

Summary and Recommendation

The equivalence examined here, namely that between fully developed flow in a round tube and a parallel- plate channel has received limited attention although not in this context. The diminishment between fully developed laminar and turbulent flow has apparently received no attention. It is recommended that teachers of fluid mechanics and related subjects introduce these concepts and be on the alert for other applications.

Key words: Fluid mechanics, fully developed flow, mechanisms for turbulent transport, models for correlation