1332
Ind. Eng. Chem. Res. 1995,34, 1332-1341
Theoretically Based Correlating Equations for the Local Characteristics of Fully Turbulent Flow in Round Tubes and between Parallel Plates Stuart W. Churchill* and Christina Chant Department of Chemical Engineering, University of Pennsylvania, 31 1A Towne Bldg., 220 South 33rd St., Philadelphia, Pennsylvania 191 04
Comprehensive correlating equations that incorporate all of the known theoretical structure, namely, the asymptotic behavior near the wall, the asymptotic behavior near the center line, and the speculative intermediate behavior, have been devised for the local time-averaged velocity and the local time-averaged turbulent shear stress in smooth round tubes. These expressions are postulated to apply directly for parallel plates on the basis of the analogy of MacLeod. The expression for the local time-averaged velocity for smooth surfaces is extended for naturally rough surfaces on the basis of the analogy of Colebrook for the friction factor, while that for the local turbulent shear stress is directly applicable without modification. Separate correlating equations for the time-averaged velocity and the turbulent shear stress are suggested on grounds of convenience and accuracy despite the one-to-one correspondence of these quantities. Correlating equations for the eddy viscosity and the mixing length follow directly but are shown to be redundant.
Introduction Experimental data for turbulent flows have generally been expressed and correlated in terms of the timeaveraged velocity, sometimes in terms of the mixing length or the eddy viscosity, but almost never in terms of the turbulent shear stress. Separate correlating equations are developed herein for each of these four quantities. They are then examined individually with respect to accuracy, functionality, and generality and comparatively with respect to utility. The several correlating equations presented herein are all based upon interpolation between asymptotes in closed form. For turbulent flows the appropriate asymptotes for this purpose cannot be derived by exact analyses alone. They must instead be obtained in major part by a combination of techniques such as dimensional analysis, trial-and-error elimination of variables or parameters, recognition of physical constraints (such as symmetry), and the postulate of a region of functional overlap between two bounding asymptotes. The asymptotes obtained by these various techniques or their combination must of course be tested functionally, and any unknown coefficients or experiments must be evaluated numerically, using experimental data or the results of “exact” numerical solutions. The recent accomplishment of essentially exact numerical solutions by direct simulation is the most important development in turbulent fluid mechanics of the past several decades. These solutions are as yet limited t o unconfined flow along a flat plate and t o confined flow between parallel plates and in both instances to values of the Reynolds number within, but barely within, the regime of complete development of the turbulence. Since the results are in the form of discrete values, they do not provide any direct information of a functional nature. Their precision and presumed accuracy are, however, sufficient to provide more reliable tests of functionality and more certain values
* Author to whom correspondence should be sent.
+ Current address: E.I. du Pont Marshall Laboratory, 3500 Grays Ferry Road, Philadelphia, PA 19146.
for the coefficients and exponents of tentative asymptotes and interpolating equations than experimental data alone. This new source of information has not directly affected the functional form of any of the expressions presented herein but has raised the level of confidence in which they are held, and it has contributed to the numerical values of the coefficients and exponents. Extension of the range of direct simulation to higher values of the Reynolds number and to other geometries may eventually provide the basis for fine-tuning of the asymptotes and correlating equations presented herein but is unlikely to change their structure significantly. Unanimity does not appear to exist in the fluid mechanical community with respect to the exact location of the dividing line between the use of the terms theoretical and empirical. To avoid this problem of semantics, the asymptotes devised and utilized herein will not be characterized as theoretical, semitheoretical, or empirical. Instead their genesis, as well as the means of evaluating their coefficients and exponents, will simply be described as precisely as possible. All of the correlating (interpolating) equations presented herein consist of an arbitrary root-mean-power of the asymptotes as proposed by Churchill and Usagi (1972). Such expressions are usually very accurate numerically because they only need to represent the variance of the dependent variable relative to the prediction of the nearest asymptote. As contrasted with the asymptotes, these interpolating expressions are, with a few exceptions not encountered herein, purely empirical and nonunique. As a result of their empirical nature, functional discrepancies that are not important with respect to prediction of the dependent variable itself may result in completely unacceptable behavior for its derivatives. It is for this reason that separate correlating equations are developed herein for the four different but closely related variables.
General Relationships The time-averaged equation for the conservation of momentum in the radial direction in fully developed,
0888-5885/95/2634-1332$09.00/0 0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995 1333 fully turbulent flow of a fluid of constant density and viscosity in a round tube can be expressed as
where here P and U symbolize time-averaged values and u and u instantaneous fluctuations about the timeaveraged values of U and V. Integrating eq 1from the center line, where UV is zero, and replacing the pressure gradient with the shear stress on the wall through an overall force balance gives
r
(11)
-
dU dr
= -p - - QUV
-z,
a
and
The left-hand side of eq 2 represents the total local shear stress, and the first and second terms on the right-hand side represent the viscous and turbulent contributions, respectively. For convenience, eq 2 can be expressed in dimensionless form as
(3) where
Y
(-eUV)/t,,
It is worthy of note that the contribution of the turbulent fluctuations to the time-averaged velocity distribution, as represented by the integral in eq 8, is simply subtractive from the terms correspondingto the velocity distribution for purely laminar flow. On the other hand, the dimensionless velocity gradient can be eliminated from eqs 4 and 5 by means of eq 3 to obtain the following algebraic expressions for the mixing length and the eddy viscosity in terms of the turbulent shear stress only:
= y/a = y+/a+, U+ = U (Q/Z,)~/~, (UV)’ = and y+ = (y(rw@)l’z)/p.The time-averaged
product UV is negative, but the dimensionless form (E)’is arbitrarily defined so as to be positive in accordance with the turbulent shear stress itself. The mixing length of Prandtl (1925) and the eddy viscosity of Boussinesq (1877) are defined in terms of the dimensionless turbulent shear stress (UV)’ and the dimensionless velocity gradient dU+ldy+as (4) and
- _-
(E)+
%4
P
(5)
dpldy’
The resulting representations of eq 3 are dVt dY
l-Y=_+
+dP2 1 7 ( d y )
and d P 1 - Y = I + Pt - -
(
(7)
P)dy+
Equations 3,6, and 7 can each be integrated formally to obtain the following expressions for the velocity distribution itself
+
)
[l 4(1 - Y)(1+)21’/2 -1 2(1+12
dy+
(9)
and
(10)
(12) Equations 3 and 8-12 indicate that a one-to-one correspondence exists between (E)+, 1+, and pJp, while U itself is related to each of these three quantities by integration. U and UV are directly measurable, whereas I+ and pt/p are arbitrarily defined quantities determined from (E)+ and/or dU+ldy+. MacLeod (1951) postulated that the time-averaged velocity distribution, when expressed in terms of U+, might be the same function of y+ and b+ in both laminar and fully turbulent flow between parallel plates as it is of y+ and a+ in laminar and fully turbulent flow in round tubes. With parallel plates the transition from laminar flow begins at b+ 45, and fully turbulent flow is attained at b+ = 150. With round tubes the corresponding values are a+ = 65 and a + = 150. The combined limits are therefore a + = b+ 45 for similarity in laminar flow and a + = b+ > 150 for similarity in fully turbulent flow. In the latter range, dU+/dy+,(E)’,I + , and ptlp may be inferred, on the basis of eqs 3,6, and 7, to have the identical functional behavior in these two geometries insofar as the analogy of MacLeod is valid. This generalization of the analogy of MacLeod appears to be supported within the uncertainty of the various sets of experimental data for U+ and (E)’. It is therefore implied t o be valid in the developments which follow in order that results for both geometries can be utilized in constructing and testing the correlating equations.
Asymptotic and Speculative Expressions Although exact solutions in closed forms are presumably not possible for the velocity distribution in fully turbulent flow in smooth round tubes and parallel plates, the necessary structural form can be deduced for individual regions and regimes by dimensional, asymptotic, and speculative arguments. For example, the postulate that near the wall the distribution of the time-averaged velocity, for a specified shear stress on the wall, depends negligibly on the radius of the tube or the spacing between the plates yields the functional asymptote
u+= f l y + )
(13)
Equation 13, which is known as “the law of the wall,” is confirmed beyond question by both experimental data and numerical solutions as a good representation for y+ < O.la+.
1334 Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995
A number of slightly differing asymptotic analyses all lead to the following expression for the region very near the wall
u'= y+ - ab+)"+ ...
(14)
However, these several analyses, when examined collectively, leave some doubt as to whether the proper power of the first higher-order term in y+ is 4 or 5. Notter and Sleicher (1971) concluded from a detailed analysis of the experimental data for heat transfer at high Prandtl numbers as well as for momentum transfer that 4 was the proper choice. The recent numerical simulations of Kim et al. (19871, Rutledge (19881, and others confirm the fourth-power dependence and suggest a value of about 1.75 x lo4 for the coefficient a. The resulting first complete asymptotic component of a correlating equation for the velocity distribution is thus
u'= y+ - 1.75 ~ y + i i o ) ~
(15)
Experimental data and the simulated values suggest, as shown subsequently, that eq 15 provides a good representation for y+ -= 10 for all a+ or b+ > 150. In the limit as y+ 0, eq 15 reduces to
-
ut = y +
(16)
-
which also follows directly from eq 3 if (E)' and U+ are postulated t o approach zero as y+ 0. The postulate that near the center line the derivative of the time-averaged velocity with respect toy depends negligibly on the viscosity leads to the following counterpart of eq 13:
C-u'=fiY)
(18)
Equation 18 is not unique, but with a coefficient E = 7.5, it is found to provide an excellent representation for the extensive experimental data for both round tubes and parallel plates for 0.5 < Y I 1. The second complete asymptotic component of a correlating equation for the velocity distribution is thus
- u'= 7.5(1 - n2
(19)
The speculation that an intermediate regime exists in which both eqs 13 and 17 are satisfied requires that
u'= A + B ln(yf}
u'= 2.5 ln{9.025yf}
(21)
It may be noted that eq 21 has no purported range of applicability for a+ < 300. Because of this restriction, the direct simulations for parallel plates, which are effectively limited to b+ 5 180, do not provide a test of the functionality of eq 20 nor contribute to the numerical evaluation of the coefficients. Equation 21 is the third complete asymptotic component of a correlating equation for the velocity distribution. The restriction to a+ > 300 does not prove to be an impediment in that role. Asymptotic expressions for (E)',1+, and ptlp corresponding to eqs 15, 16, 19, and 21 follow directly from differentiation. Such expressions will be considered in context rather than here.
A General Correlating Equation for the Velocity Distribution An ideal overall correlating equation for the velocity distribution in smooth round tubes would conform to eqs 15, 19, and 21 in at least an asymptotic sense and would be convenient for use in various applications, including heat transfer. Apparently, no expression has been previously proposed that satisfies all of these criteria. Reichardt (1951) made a first step in the direction of an overall correlating equation by suggesting the following expression for the entire turbulent core:
(17)
The function of Y in eq 17 must satisfy three conditions at Y = 1: U+ = as required by definition; dU+/dy+ = 0 as required by symmetry; and a finite value of (1Y)dY+/dU+as required by the unquestionable experimental determination of finite values of the eddy viscosity. The simplest functional relationship conforming t o these conditions is
q - u'= E(1- n2
The most reliable experimental data for round tubes suggest that eq 20 with A = 5.5 and B = 2.5 provides a very good representation per 30 y+ < a+/10. For convenience, eq 20, with these coefficients, can be expressed in the alternative, more compact form.
(20)
Prandtl (1933) derived eq 20 directly from eq 6 by neglecting the variation in the shear stress, neglecting the viscous shear stress, and postulating the equivalent of I+ = y+/B. It is probable that these two simplifications and this postulate were actually made in response to the observation of a linear relationship between experimentally measured velocities and the logarithm of the distance from the wall.
-
-
Equation 22 conforms to eqs 18 and 20 functionally in the limits of Y 1 and Y 0, respectively, but does not reduce to eq 15 or even to eq 16 as y+ 0. Finley et al. (1966) proposed an expression for fully turbulent flow of a thin film down an inclined plane that can be rewritten for a round tube as
u'= A + B ln(y+) + B P ( 1 -
-
Y)
+ 2BnY?3
- 2Y) (23)
If A is chosen as 5.5 and B as 2.5 to conform to eq 21, and if Il is chosen as 1/12 to conform to eq 19 in the limit of Y 1, eq 23 becomes
-
u'= 2.5 ln(9.025~') + T 15P
-7 10Y
3
(24)
Equation 24 has the same asymptotic merits and demerits as eq 22 in that it conforms to eqs 19 and 21 but fails to conform to eqs 15 and 16. However, on the whole, eq 24 appears to represent the data for the entire turbulent core (30 I y+ I a+) better than eq 22. It may be interpreted as a correlating equation that interpolates between eqs 19 and 21. Spalding (1961) devised an expression that incorporates the equivalent of eqs 15 and 21, but not eq 19, and therefore fails in the central regime near the center line. Also, his expression has the inverse form of y+ = f{ U+}and hence is awkward for indirect applications. Dean (1976) extended the velocity distribution of Spal-
Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995 1336 ding to include the region of the wake in both confined and unconfined turbulent flows by incorporating the equivalent of the two right-most terms of eq 23. He originally attributed these terms to Granville (1976) rather than t o Finley et al. but later acknowledged the priority of the latter. He proposed a different value of ll for different geometries. The extended expression is comparable in many respects to the final one achieved herein but retains the shortcoming of that of Spalding in being implicit in U+. Equations 15 and 24 might presumably be combined in terms of the model of Churchill and Usagi (19721, namely (f{x>)P = ( f 0 b - P
+ (fm{xl)P
(25)
to obtain an expression meeting all of the previously mentioned criteria. However, this combination fails to meet some of the additional conditions imposed by eq 25 itself. First, eqs 15 and 24 do not intersect in the range of 0 Iy+ Ia+; second, eq 15 gives negative values of U+for y+ > (10411.75)1/3= 17.88; and third, eq 24 gives negative values of U+for y+ < 119.025 = 0.1108 for all values of a+. The first and second of these shortcomings can be eliminated by approximating eq 15 with
u+=
(r+>2 1 y+ - exp{ - 1.75(y+/10)4}
+
(26)
-
Equation 26 reduces to eq 15 and then to eq 16 as y+ 0 and intersects eq 24 at y+ = 12.8. Equation 26 does not have a physical rationalization, but such a deficiency is unimportant because its role is merely to introduce eq 15 into eq 25 without itself affecting the predictions of the final correlating equation. The third shortcoming can be eliminated by approximating eq 24 with
P =2.5 ln{9.025yf
15 10 + l}+ ~p - -Y3 3
(27)
Combination of eqs 26 and 27 in the form of eq 25 with an exponent p = -3, as chosen on the basis of various representative sets of experimental data for both round tubes and parallel plates, results in
+
(1 y+ - exp{-1.75(y+/10)~)>~
v=[
(-
2.5 ln(9.025y'
+
Cr+T
+ 1) + F p - y$
) 3 ] u 3 (28)
The small errors introduced due t o the approximation of eq 15 by eq 26 and of eq 24 by eq 27 are almost completely eliminated in eq 28 by virtue of the relative contributions of eqs 26 and 27 therein as a function of y+. The success of eq 28 in representing experimental data and theoretically simulated values in both round tubes and parallel plates is illustrated in Figure 1. These particular sets of data and computed values were chosen on the basis of range, apparent precision, and direct accessibility. The three asymptotic expressions upon which eq 28 is ultimately based, namely eqs 15, 19, and 21, are apparent in Figure 1only outside their range of applicability where they deviate significantly from eq 28. Equation 28 does not improve significantly upon eq 24 with respect to substitution of y+ = a+ to obtain
-10
-I
I
0
I
I
1
0
H
8
Y+
Figure 1. Comparison of eq 28 with experimental local velocity
data for round tubes and parallel plates: *, Laufer (1964)round tubes; 0,Schlinger and Sage (1952)parallel plates; 0,Sherwood et al. (1968)round tubes; A, Deissler (1950)round tubes; V, Deissler and Eian (1952)round tubes; 0, Eckelmann (1974) parallel plates; - - -,eq 28;-, eq 21; -,eq 19,u+ = 10 000;-.-, eq 15;- -, eq 19,a+ = 1000.
--
-
since the added terms are negligible in this instance. The simpler and wholly adequate expression resulting from eq 24 is = 5.92
+ 2.5 ln{af)
(29)
It should be noted that the coefficient and the fourthpower dependence of the right-most term of eq 15 and its counterpart in eq 28 are of critical importance in predicting rates of heat and mass transfer for large Prandtl numbers and Schmidt numbers, respectively. Insofar as the analogy of MacLeod is valid, eqs 28 and 29 are adaptable for smooth parallel plates simply by substituting b+ for a+. A General Correlating Equation for the Turbulent Shear Stress Apparently the only correlating equations that have been proposed for the turbulent shear stress for round tubes and parallel plates are those of Pai (1953a,b). Since these expressions are only for particular values of a+ and b+,respectively, and since they do not conform to the limiting behavior described by eq 15 they will not be discussed herein. In principle, eq 28 could simply be differentiated and the resulting expression for dU+ldy+ substituted in eq 3 t o produce a correlating equation for (E)'. However, owing to the combining power of -3, the resulting expression would be excessively complex. Also, it would be singular at y+ = 2.5 - 119.025 = 2.39. As a rule, simpler and more accurate correlating equations are obtained when the Churchill-Usagi model is applied directly for the function of interest. For these two reasons the latter procedure is followed here. For the region very near the wall where y+
