1
Modified Theories of Turbulent Two-Dimensional and Axi-Symmetric Jets and Turbulent Velocity Discontinuity
and Free Jet Boundary
SIAVASH H. SOHRAB
RobertMcCormickSchool of Engineering and Applied Science
Department of Mechanical Engineering
Northwestern University, Evanston, Illinois60208
UNITED STATES OF AMERICA
Abstract: The modified form of the equation of motion at laminar eddy-dynamic scale is solved for the classical problems of turbulent (a) axi-symmetric jet (b) two-dimensional jet (c) smoothing out of a velocity discontinuity (d) free jet boundary. For (a), (b) and (c), the predicted velocity profiles are found to be in excellent agreement with the experimental data of Reichardt and Förthmann without any empirically adjustable constant. For (c), contrary to the classical theory of Prandtl, the predicted velocity in the mixing region does go over to the free-stream velocities asymptotically.
Key-Words: Theory of turbulent axi-symmetric and wo-dimensional jets. Turbulence.
1
1 Introcuction
Turbulence is a common feature of physical phenomena across diverse spectrum of spatio-temporal scales from the exceedingly small scale of stochastic quantum fields, to the intermediate scale of classical hydrodynamic fields, to the extremely large scale of cosmology. This universality of turbulent phenomena led to the recent introduction of a scale-invariant model of statistical mechanics and its application to the field of thermodynamics [4]. The implications of the model to a modified statistical theory of turbulence and invariant forms of conservation equations have also been addressed [5, 6]. As examples of exact solutions of the modified form of equation of motion in laminar flows, the Blasius problem of flow over a flat plate [7], and the problems of axi-symmetric and two-dimensional jets [8] where investigated. In the present study, the modified form of the equation of motion is applied to four classical problems of free turbulent flows. First, following Tollmien [9] and Schlichting [2, 10], the problems of turbulent axi-symmetric and two-dimensional jets are investigated. Next, following Prandtl [11], the problem of smoothing out of a velocity discontinuity is studied. Finally, following the classical studies of Tollmien [9] and Görtler [12], the problem of turbulent free jet boundary is examined. The predicted velocity profiles are found to be in excellent agreement with theexperimental observations of Reichardt [13] and Förthmann [14].
2. Invariant Form of Conservation Equations for Reactive Fields
Following the classical methods [1-3], the invariant
definitions of the density , and the velocity of atomu, element v, and systemw at the scale are given as [4]
u = v(1)
w= v (2)
Also, the invariant definitions of the peculiarand the diffusion velocities are given as [4]
, 3
Following the classical methods [1-3], the scale-invariant forms of mass, thermal energy, and linear momentum conservation equations at scale are given as [5, 6]
(4)
(5)
(6)
An important feature of the modified equation of motion (6) is that it involves a convective velocity w that is different from the local fluid velocity v. Because the convective velocity w is not locally-defined it cannot occur in differentialform within the conservation equations [5]. To determine w, one needs to go to the next higher scale (+1) where w = vbecomes a local velocity. However, at this new scale one encounters yet another convective velocity w which is not known, requiring consideration of the higher scale (+2). This unending chain constitutes the closure problem of the statistical theory of turbulence discussed earlier [5].
3 A Modified Theory of Axi-Symmetric Turbulent Jets
The problem of axi-symmetric turbulent jet has been subject of theoretical studies by Tollmien [9] and Schlichting [2, 10], experimental studies by Zimm [15], Ruden [16], Wuest [17], Reichardt [13], and numerical study by Howarth [18]. The local axial and radial velocities (v'x, v'r) are along the corresponding coordinates (x', r') and one introduces the dimensionless quantities
,
, (7)
The initial axial convective velocity w'o at the origin of the jet is assumed to be known and signifies the strength of the jet. At the scale of laminar eddy-dynamics LED [5], the (atom, element, system) velocities (1)-(2) are (ue, ve, we) and the corresponding length scales are of the order (le = 105 m, e = 103 m, Le = 101 m). Also, = e = leue/3 = cvc/3 is the kinematic viscosity at eddydynamic scale [5] and lH is the hydrodynamic thickness.
The steady dimensionless forms of (4) and (6) for incompressible fluid in the absence of reactions = 0 under the usual boundary layer assumptions at LED scale reduce to
(8)
(9)
subject to the boundary conditions
, (10)
(11)
To solve equation (9), the axial convective velocity wx which is the average of the local axial velocity vx is needed. Because the jet momentum is initially only in the axial direction, the radial dispersion of the jet is entirely caused by diffusion of axial momentum in the radial direction. Therefore, the diameter of the jet d' at any axial position will be given by the Einstein-Smoluchowski relation
(12)
where the local diffusion time t' is related to the axial position x' and the local convective velocity w'x by the expression
(13)
The jet cross sectional area A' is given by
(14)
that in dimensionless form becomes
(15)
Hence, the mass flow rate at any axial position through the jet cross sectional area will vary as
(16)
where the symbol (in (16) denotes proportionality. On the other hand, the total axial momentum along the jet must remain constant [2] and hence
constant (17)
leading to the dimensionless convective velocity
(18)
The choice of the factor (1/2) for the constant in (18) will be justified in the sequel. Solving the global continuity equation
(19)
after substitution from (18) results in
(20)
Following Schlichting [2], one introduces the stream function and the similarity variable
, (21)
leading to the axial velocity
(22)
Substitutions from (18), (20), (21), and (22) into (9) results in
(23)
(24)
(25)
where and primes denote differentiation with respect to . The bounded solution of (23)-(25) is
(26)
that by (22) gives
(27)
This solution is exactly the same as that found for laminar circular jet [8]. Similar correspondence was also found in the classical theory of Schlichting [2].
From (21) and (27), the stream function that satisfies = 0 at = 0 is obtained as
(28)
that in turn gives the radial velocity component
(29)
From (27), the ratio of the axial velocity to its maximum, i.e. centerline, value vxc becomes
(30)
such that (30) may be expressed as
(31)
The predicted velocity profile calculated from (32) using Mathematica [19] is in excellent agreement with the experimental data of Reichardt [13] as shown in Fig.1 . It is important to note that, as opposed to the classical theories [9, 10], the agreement with experimental data shown in Fig.1 is achieved without any empirically adjustable constants.
Fig.1 Comparisons of predicted velocity profile with data [13] for turbulent axi-symmetric jet.
varies from unity at = 0 to zero at , and vxc has no radial dependence, it is clear that the average of (30) over the cross sectional area of the jet will lead to the convective velocity
, (32)
This result provides a posteriori justification for the choice of the factor of (1/2) made in the derivation of (18) from (17).
Because the velocity rationin (30) always varies from unity at h = = 0 to zero at , and vxc has no radial dependence, it is clear that the average of (30) over the cross sectional area of the jet will lead to the convective velocity
, (33)
This result provides a posteriori justification for the choice of the factore of (1/2) made in the derivation of (18) from (17).
4 A Modified Theory of Two-Diemnsional Tuirbulent Jet
In this section the problem of two-dimensional turbulent jet studied theoretically by Tollmien [9], Schlichting [2, 10], Görtler [12], Reichardt [13], and experimentally by Reichardt [13] and Förthmann [14] is considered. The relevant dimensionless quantities are
,
, (34)
where the initial convective velocity w'o at the jet origin is assumed as the known measure of the jet strength. The steady forms of the continuity equation (4) and the modified equation of motion (6) for incompressible fluid under the usual boundary-layer assumptions in the absence of reactions = 0 at laminar eddy-dynamic LED scale reduce to
(35)
(36)
that are subject to the boundary conditions
, (37)
(38)
Similar to the treatment of axi-symmetric jet discussed in the previous section, the solution of system (36)-(38) requires the knowledge of the convective velocities wx and wy. Following the same steps as (12)-(18) of the previous section, one arrives at the convective velocity [8]
(39)
that with the global continuity equation
(40)
results in
(41)
Following the classical studies [10, 20], one introduces for the two-dimensional turbulent jet the stream function and a similarity variable as
, (42)
leading to the linear ordinary differential equation
(43)
(44)
(45)
where and primes denote differentiation with respect to z. The solution of (43)-(45) is [8]
(46)
that by (42) leads to the stream function
(47)
The solution (46)-(47) is exactly the same as that found for laminar two-dimensional jet [8], in accordance with the classical findings of Schlichting [2].The velocity components from (47) are given as
(48)
(49)
From (48) the ratio of axial velocity to the centerline velocity vxc is found as
(50)
Fig.2 Comparisons of predicted velocity profile with data [14] for turbulent two-dimensional jet.
Also, from (50), the transverse position where vx is half its maximum value vxc is given by
(51)
such that (4.17) may be expressed as
, (52)
The calculated velocity profile from (52) is in excellent agreement with the experimental data of Förthmann [14] as shown in Fig.2. The significance of such close agreement (Fig.2) is to be emphasized since the modified theory described herein does not contain any empirically adjustable constant as opposed to the classical studies [9, 10, 13].
5 A Modified Theory of Smoothing Out of a Velocity Discontinuity
The unsteady problem of smoothing out of a velocity discontinuity was first treated by Prandtl [11]. The schematic diagram of the flow is shown in Fig.3 and the modified equation of motion (6) reduces to
(53)
With the similarity variable and dimensionless velocity vx
, (54)
One obtains from (53)
(55)
vx = 1 (56)
vx = 0 (57)
The solution of (55)-(57) is
(58)
and the calculated velocity profile is shown in Fig.3(b).
(a) (b)
Fig.3 The smoothing out of a velocity discontinuity after Prandtl [11]; a) Initial pattern, b) Pattern at a later instant.
The solution (58) may be compared with the classical solution of Prandtl [11]
(59)
where
(60)
involving and empirical constant = l/b that must be determined from experimental data. As stated by Schlichting [2] an esthetical deficiency of the solution (59) is that the velocity in the mixing region does not go over to the free-stream velocities asymptotically. Instead, the transition occurs at a finite distance y' = b with a discontinuity in . However, the solution (58) of the modified theory presented herein is free of this deficiency.
6 A Modified Theory of Turbulent Free Jet Boundary
The problem of turbulent jet boundary was treated by Tollmien [9], Görtler [12], and Reichardt [13]. The modified form of the equation of motion (6) at laminar eddy-dynamic scale reduces to
(61)
Since the convective velocity at all axial positions changes from free-stream velocity w'1 to w'2, the velocity w'x in (61) may be expressed as the constant mean velocity defined as
(62)
where V' = (w'1 + w'2) is the constant total velocity. Introducing the dimensionless coordinates
, , (63)
and the similarity variable and dimensionless velocity vx
, (64)
in (61) one obtains
(65)
vx = 1 (66)
vx = 0 (67)
The solution of (65)-(67) is
(68)
The predicted velocity profile with w'2 = 0 calculated from (68) using Mathematica [19] is in excellent agreement with the experimental data of Reichardt [13] as shown in Fig.4 .
Fig.4 Comparisons between the predicted velocity distribution in the mixing zone of a jet from equation (68) and measurements of Reichardt [13].
The solution (68) is now related to the classical solution of Görtler [12, 2]
(69)
where
(70)
It is important to note that the solution (6.9) involves a semi-empirical constant and an assumed expression for the virtual kinematic viscosity
(71)
as discussed by Schlichting [2].
In view of the definitions of , , and in (70)-(71), if one modifies the definition of the empirical constant in (70) as
(72)
then by (62), (70), (71), and (72), the similarity coordinate in (69) becomes
(73)
when the virtual kinematic viscosity in (71) is identified as the eddy-viscosity e . With the modified value of the empirical constant in (72) given by
(74)
the results (64) and (73) lead to
(75)
thus establishing an exact correspondence between the solutions of the modified (68) and the classical (69) theories. It is noted however that while (68) is an exact solution of the modified form of the equation of motion (61), the classical solution (69) was obtained by Görtler [12] as an approximate solution of the non-linear Blasius equation using the method of power-series expansion [2].
CONCLUDING REMARKS
The modified form of the equation of motion at laminar eddy-dynamic scale was solved for the classical problems of steady turbulent axi-symmetric and two-dimensional jets. The predicted velocity profiles, without involving any empirical constants, were found to be in excellent agreement with the experimental data of Reichardt [13] and Förthmann [14]. In addition, the modified form of the equation of motion at laminar eddy-dynamic scale was solved for the problems of smoothing out of a velocity discontinuity and free jet boundary. For the former problem, as opposed to the classical theory of Prandtl [11], the predicted velocity profile in the mixing region was found to go over to the free-stream velocities asymptotically as expected. For the latter problem, the predicted velocity profile was found to be in excellent agreement with the experimental data of Förthmann [14], without involving any empirically adjustable constant. It was shown that an exact correspondence between the solution of the modified theory and that of the classical theory of Görtler [12] is established if the semi-empirical constant in the classical theory is redefined as .
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