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The round jet

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The round jet
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Dietrich, Donald Arthur, 1943-
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English
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x, 51 leaves. : illus. ; 28 cm.

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Subjects / Keywords:
Jets -- Fluid dynamics ( lcsh )
Laminar flow ( lcsh )
Aerospace Engineering thesis Ph. D
Dissertations, Academic -- Aerospace Engineering -- UF
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis--University of Florida, 1970.
Bibliography:
Bibliography: leaves 50-51.
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Manuscript copy.
General Note:
Vita.

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Copyright Donald Arthur Dietrich. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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Full Text
THE ROUND JET
By
DONALD ARTHUR DIETRICH
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIRED ENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA 1970




ACKNOWLEDGEMENTS
The author wishes to express his appreciation to the members of' his supervisory committee and especially to the chairman, Dr. Bernard M. Leadon, for his suggestions and criticisms during the preparation of this dissertation. The author also wishes to express his appreciation to Drs. M. H. Clarkson and K. T. Millsaps for their efforts throughout the author's graduate work. Appreciation is also expressed to the National Aeronautics and Space Administration for the granting of a traineeship which made this work possible. Finally, special appreciation is extended to the author's wife, Jan, who rendered invaluable assistance toward the completion of this dissertation.




TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS . ....................... ii
LIST OF FIGURES .... . . . . v
KEY TO SYMBOLS . . . . . . . . . . . . ... vii
ABSTRACT . . . . . . . x
CHAPTERS
1. INTRODUCTION ..... ............... . . .. 1
2. SOLUTION OF THE EQUATIONS OF MOTION ........... 3
3. SOLUTION OF THE COMPLETE ENERGY EQUATION .............. 6
3.1. THE ENERGY EQUATION FOR LAMINAR FLOW .... ......... 6
3.2. DETERMINATION OF THE SOLUTIONS G n(n) . . .... 9
3.3. SOLUTION FOR F(n) ...... .................. i.11
3.4. MATCHING THE SOLUTIONS ...... ................ ... 13
3.5. SPECIFICATION OF THE TEMPERATURE FIELD ........... ... 13
3.6. SELECTION OF THE SIMILARITY TRANSFORMATION ... ...... 16
3.7. THE TERM DUE TO DISSIPATION ... ......... 18
3.8. ADDITION OF THE NONHOMOGENEOUS TERM .. ......... ... 19
4. COMPUTED RESULTS ...................... 21
5. APPLICATION TO A TURBULENT ROUND JET .................. 24
6. SUMMARY AND CONCLUSIONS ... ................... .. 25
FIGURES ............ ............................... ....27
APPENDICES
A. SOLUTION OF THE TRANSFORMED ENERGY EQUATION BY THE
FROBENIUS TECHNIQUE ........ ................... .. 43
iii




Page
B. GENERATION OF THE SECOND SOLUTION OF THE TRANSFORM
ENERGY EQUATION . 47
BIBLIOGRAPHY . . . . 50
iv




LIST OF FIGURES
Page
1. Comparison of two centerline temperature distributions . 27
2. Illustration of the matching process by selective multiplication ......... ...................... ... 28
3. Isotherms found by N. L. Soong for the case that RN= 4 and a = 0.7 ....... ..................... 29
4. Isotherms resulting from the present study for the case that RN = 4 and a = 0.7 .............. ........29
5. Isotherms found by H. B. Squire for the case that RN = 400 and a = 0.7. ...... .................. ....30
6. Isotherms resulting from the present study for the case that RN = 400 and a = 0.7 ... ............. .... 30
7. Comparison of the azimuthal distributions G1(e) and G2(e) and the distribution due to N. L. Soong for
the case RN = 4 and a = 0.7. .... .............. .... 31
8. Comparison of the azimuthal distributions G1(e) and G2( e) for a = 0.7 and various Reynolds numbers .. ..... 32 9. G2(e) for RN = 4 and various Prandtl numbers .... ...... 33
10. G2() for RN = 40 and various Prandtl numbers .......... 34
11. G2(0) for RN = 400 and various Prandtl numbers ... ...... 35 12. G2(e) for RN = 4000 and various Prandtl numbers ... ...... 36 13. G2(e) for a = 0.7 and various Reynolds numbers ... ...... 37
14. G2(0) for a = 7 and various Reynolds numbers ... ....... 38
15. GD(e) for RN = 40 and various Prandtl numbers ... ....... 39
v




Page
16. GD(e) for RN = 400 and various Prandtl numbers. . . 40 17. GD(e) for RN = 4000 and various Prandtl numbers . . 41 18. Comparison between the temperature distribution due
to the present work and experimental data for
turbulent round Jets . ................ 42
vi




KEY TO SYMBOLS
A constant determined by boundary conditions
B constant determined by boundary conditions
C integration constant Ci/C coefficients of a series expansion C specific heat at constant pressure
P
d arbitrary constant
D(TI) function of q alone D constant used in the solution for dissipation
0
Di/D coefficients of a series expansion E Eckert number = U2/Cp T
f(rj) similarity function for the stream function F net force applied at the origin
F(,q) a transformed function of the similarity solution for the
energy equation
F(X) F(1) written in terms of X
F(w) F(-q) written in terms of w
GD(71) similarity function used in the solution for dissipation Gn(1) terms in a series of similarity functions for the energy
equation
G2(\) G2 () written in terms of X G2(w) G2(Ti) written in terms of w h total heat flux across a sphere of finite radius
HD (Q) transformed function of 71 alone
vii




H2(T) transformed function of q alone I a constant of the flow
k coefficient of thermal conductivity
K constant =(E/2)R2O
M(x) arbitrary function of x
n index used in a series
p arbitrary constant
P pressure
P pressure at infinity
P(Q) function of n alone
P(x) arbitrary function of x
Q(x) arbitrary function of x
r radius in spherical coordinate system
r perpendicular distance from jet centerline
c
R reference radius
RN Reynolds number = RU/v
R(x) arbitrary function of x
t temperature
t temperature at infinity
t reference temperature
T excess temperature = t t
T reference excess temperature
Tc local centerline excess temperature
TD excess temperature due to dissipation
u radial component of the velocity
U reference velocity
U local centerline velocity
c
viii




v azimuthal component of the velocity
w(x) function of x alone
x an independent variable
X(T) function of n alone
y(x) function of x alone
y (x) function of x alone
constant = 4/RN + 1 integration constant
ri/ ro coefficients of a series expansion Similarity variable = cos e e azimuthal coordinate
? similarity variable = n + 1
matching location
0
L coefficient of absolute viscosity
v coefficient of kinematic viscosity
p density
a Prandtl number = pCp v/k
spherical coordinate of symmetry dissipation function
stream function
similarity variable = T 1 11o matching location
2the Laplacian operator
ix




Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
The Round Jet
By
Donald Arthur Dietrich
June, 1970
Chairman: Dr. B. M. Leadon
Major Department: Aerospace Engineering
In this paper the hydrothermodynamics of an axially symmetric flow is presented and interpreted as a round laminar jet. The established solution of the complete Navier-Stokes equations for this case is used as the basis for the study of the complete energy equation.
A series of similarity transformations is introduced, and the most significant transformation is selected. Using the Frobenius technique, a series solution for the temperature profile is formed for the convective and conductive portions of the solution. In addition, a solution is determined for the contribution due to viscous dissipation. The results are interpreted as a heated round laminar jet which is produced by superimposed momentum and heat sources.
x




CHAPTER 1
INTRODUCTION
The solution of the Navier-Stokes equations has been a formidable problem since they were derived in the first half of the nineteenth century. The dozen or so solutions that now exist were obtained only after the physical problem or the differential equations were extensively simplified. The problem of the round laminar jet is one for which the complete Navier-Stokes equations and the associated energy equation have been solved. In this paper the hydrodynamic solution is reviewed and then applied to the complete energy equation which is then solved exactly.
The study of the hydrodynamics of a round laminar jet has had a logical development since its introduction by H. Schlichting (1955). Schlichting's approach was first to simplify the equations of motion with the aid of boundary layer assumptions, which are valid only for large Reynolds numbers. He introduced a stream function by a similarity transformation and solved the resulting equation in closed form. Schlichting analyzed his solution as the flow resulting from a momentum source located at an orifice in a flat plate. N. A. Slezkin (1934) showed the existence of a solution for an axially symmetric flow. This problem was further explored by L. D. Landau (1944); and V. I. Yatseyev (1950) obtained a solution to the Navier-Stokes equations for this problem. H. B. Squire (1951) also derived an exact solution to the energy equation associated with a round. laminar jet if there is a heat source located at the origin and if viscous dissipation is neglected.
1




2
There are a number of basic assumptions which are used in the solution of this problem. The flow is assumed to be axially symmetric such that the velocity and its gradients with respect to the coordinate CP are zero. This assumption simplifies the problem and yields an axially symmetric case which is related to the Jeffrey-Hamel theory (G. Birkhoff and E. H. Zarantonello 1957). The flow is considered to be a laminar flow of an incompressible fluid with its transport coefficients held constant. With this assumption, the associated energy equation m~ay be solved using the hydrodynamic solution.
This paper offers a new approach to the solution of the complete energy equation for the case of the round laminar jet. Yih (1950) extended the boundary layer solution and found a closed form solution for the heated round jet. H. B. Squire (1951) used the solution to the complete Navier-Stokes equations to obtain a closed form solution to a reduced energy equation. Both Yih's and Squire's solutions neglected the terms due to viscous dissipation. K. T. Millsaps and N. L. Soong (1965) published a similarity transformation which included both the radial conduction term and the dissipation terms. The paper by Soong (1968) presents a valid similarity transformation and studies the most important differential equation related to the energy equation; however, Soong's paper does not offer an analyt~ic solution to the energy equation. Furthermore, Soong's numerical solution appears to be valid only for one very specialized and physically questionable flow. The solution being described in this chapter is a more general approach to this problem so that the previous results become intermediate -portions of the entire solution.




CHAPTER 2
SOLUTION OF THE EQUATIONS OF MOTION
The hydrodynamic solution itself is not the objective of this paper, but., since the results of this section are used in the thermodynamic solution, an outline of the method of solving the equations of motion is included in this paper.
The continuity equation and the Navier-Stokes equations in a spherical coordinate system for a steady, axially symmetric, incompressible flow of a fluid having constant transport coefficients are (Goldstein 1965)
1 a(r2 u) + 1 c' (v sin 0) = 0 (2.1)
2 8r r sin 1 8
au2 v l[p 2 2u 2 av 2v cot ] (2.2)
r r P r 2 280 2
r r r
uv8v + v 3v + uv P+ V [2 v + 2 au v (2.3)
pr 80 r r sin 2
where p is the density, P is the pressure, v is the kinematic viscosity, and V2 is the Laplacian operator. It is customary in problems of this nature to use a stream function which is defined so that the continuity equation is automatically satisfied. Then the number of equations is reduced from three to two, but the order of the remaining two equations is increased from second to third order. The appropriate form of the stream
function assumed in this case is
i = vr f(1) (2.4)
3




4
where o = cos e and f(n) is dimensionless, so that
u (n), v=-- (2.5)
r r~l.
The pressure distribution for this flow was obtained by Squire from (2.2) and (2.3) as
P-"Po 2 C1
o = V2 + VU+ (2.6)
P T r r2
where P is the pressure at infinity, where u and v vanish, and C
is-a constant of integration.
In these terms (2.2) may be written
f,2 + ff" = 2f' + [(i n 2)f,], 2C (2.7)
Integrating twice and applying the condition of axial symmetry, whence the constants of integration are set equal to zero, the result is found to be
_ T 2(l n2 1 (2.8)
2(P 2.
where P is to be determined from one of several conditions which may be placed upon the flow.
If the velocity along the axis 'of the jet is designated U at the distance R, it is found from (2.5) and (2.8) that
-4 + 1(2.9) PIN
where
RU
is the Reynolds number. Since u varies inversely with r, the value of




5
R is immaterial and RN is a constant along the Jet axis. Reynolds numbers may be calculated for any other ray as well.
An alternative means of determining P is to calculate the force applied at the origin, regarded as known, as equal to the net outflow of axial momentum through a sphere of radius r plus the axial components of
the forces due to pressure and to viscous stresses acting on that surface. Squire has given this relationship as, in present terms,
F 8P + 2P 2 P _--_+
83 V2 Y2 2- 1)( -1
8Bnpv2 3(62- i)
For future reference these results are summarized in r, e-coordinates:
u R K I-cos 20+ 2Pcos e 1
U r (6 c s 0 2 I '
'rR [ -P Cos e) 2
v R (2)[ sin e
-P r cos e ]
() ([P -Cos e)
aP
and = /v, where U -- u(RO9 ).




CHAPTER 3
SOLUTION OF THE COMPLETE ENERGY EQUATION 3.1. The Energy Equation for Laminar Flow
With the assumption that the incompressible fluid has constant transport properties and using the above hydrodynamic solution without modification, the energy equation becomes a linear, second-order partial differential equation involving only one dependent variable, the temperature.
In a spherical coordinate system and assuming axial symmetry, the energy equation is (Schlichting 1955)
1aT v T18 2 T1 a IT
pCpur-T + 8 k[-12 r2 aT + 1 a sin 9 -- + D (3.1)
p 8r r 891 r ar ar r2 .i 89 86
r sin 8
where
T excess temperature = t t
= terms due to dissipation Cp heat capacity at constant pressure
k coefficient of thermal conductivity
The velocity components are considered as known functions, and the relationships from Chapter 2 may be used. Using statement (2.4) which introduced a stream function and equation (2.5), the energy equation may be rewritten as Vfr() T vfr)
pC +p a r 2 a
2
1 a 8~ 2aTi 2v
k + ( 9 +2 c D(rj) (3.2)
2 r r 2 a a87 r1T
6




where
2
0=t-~ D(I)
r
and
2 2
D(T)= [(1 2)]' + + ( If(). 2)f'(11) + 4f(1) f"(O
+ (1 n12) (
Using an approach similar to that used in obtaining the hydrodynamic solution, assume
T = rn G (7) (3.3)
n
with the value of n unspecified, although it is clear that n should be a positive number so that T will approach zero asymptotically as r increases. Substituting (3.3) into (3.2) the energy equation becomes
vp Cp[f'(T) nGn() + f() Gn()]
(3.4)
Sk -n(-n + 1) Gn (I) + dI 12) Gn(1 + LV2 D(n)rn-2
where the prime denotes differentiation with respect to q. It can be seen from this equation that any value of n will be acceptable if the dissipation term is neglected; but, if the presence of the dissipation term is considered to be essential, the only allowed value of n is n = 2.
In the absence of dissipation a more general expression may be chosen instead of (3.3) such as




8
T = -)rE n G() = Go() + G(r + G-2 +
n=o r
Here n is restricted to positive integers and G0(q) = 0 because of
the boundary condition at infinity. Then for each n the basic equation (3.4) yields a corresponding ordinary differential equation for Gn(rI). In each case the solution Gn(T) includes two undetermined constants. A term due to dissipation which is a nonhomogeneous term in the differential equation for n = 2 may also be added to the series. When dissipation is present, G2(T() may be treated as the complementary solution and GD(TI) as the particular solution of the equation for n = 2. The series assumption for T may therefore be rewritten as
T= GD(0) + E r-n ( G G GD(q) + Gl1(q) + G2 (TO+(36
1 'n (~=++ .. (3.6)
T 2 GD I) + =r G2 + r 2
r r r
n;=O
where GD(,q) is due to dissipation alone, and each function Gn(,q) is the solution of a linear, second-order, homogeneous differential equation.
The solutions for the excess temperature must be bounded and continuous together with their first and second derivatives throughout the thermal field. To obtain a solution, this condition has to be relaxed in the neighborhood of the origin which is a singular point. The second requirement is that the final temperature field be axially symmetric. Thus the solutions must be even functions of e; and the temperature gradient transverse to the z-axis must be zero. Again, proper selection assures that the similarity variables will meet this second condition. The final condition is that the excess temperature field approach zero as r increases without limit.




9
3.2. Determination of the Solutions G
For n = 1 equation (3.4) becomes
pCP[vf'() G01() + vf(n) Gl(n) = k d(1 q2) G'( or
pC v Lf (TI) Gl(g) = k d( 72) Gi(n)]
This equation may easily be integrated twice and has the solution
G 1(n) = A 13- 1 (3.7)
where A is the undetermined constant and 0 is the same as that used previously in (2.9). This is the solution determined by H. B. Squire where a represents the Prandtl number, pC v/k, and the appropriate boundary conditions have already been applied.
For n = 2, the differential equation for G2() becomes
a[2G2() f'() + f(g) G2() = 2G2()+ d[(l 92) G2()] (3.8)
where the term due to dissipation has temporarily been omitted and again a is the Prandtl number.
Equation (3.8) is revised with the aid of (2.8); in addition, the definition of H2(I) is given by
G2() = [Gl(n)]p H2() (3.9)
where p is some unspecified power. With the relationships obtained from (3.7)
Gl () 2a
= n




10
and
G(n)= 2a(2a + 1)
G1 (T 2 '
(3.8) becomes
(1 n2) H2(1) 1 )(2p 1) 2 H()
t- 2 2 2
1)(l 2)2(2a + 1) + ) M (2a q 4o + 2a)
( ) + )2 -T)2
(p 1)(2(1 2 2(p 21))2 2 + 2H2() = 0 .
Inspection of this equation, in particular the H2(4) term, shows that
it would be convenient to take p = 1/2. If p is not chosen now but carried throughout the analysis, it may be shown that the only possible choice of p which will satisfy the boundary condition on the jet axis is indeed 1/2. With this value of p, the above equation reduces to the
form
(1 2) H() 2T1 H (I)
(3.10)
( 2 30 + 2)12 + 2p(30 2)] + (-C 3 + 22 ) (.
+ -0 H2(TO = 0
( PB ) 22
The first two terms of this equation are the same as the first two terms of Legendre's equation; hence (3.10).is self-adjoint. It is advantageous to put (3.10) into its normal form, eliminating the first derivative term by the substitution




11
H 2(n) = F(n)(l ni2 (3.11)
Substitution of (3.11) in (3.10) yields
F"(() + PD( 2] F(n) = 0 (3.12)
S"n)2(l n)2(l + n)
where
P(O) = (-a + 3a 2)q4 + (4p 6ap)n3 + (2a 2P2 + 3)I2
+ 6p( a 1)T 2 (-a2 3 3P2)
Retracing these steps shows that the desired solution G2(i) transforms to F( ) by the relationship
G 2QrI = 2] F(T) (3.13)
3.3- Solution for F(q)
Equation (3.12) has singularities at q = +1, -1, P, and 00; but, since i = cos e and P > 1, the only singularities of importance here are il = +1 and = -1. Examination of P(rI) shows that n = +1 is
not a root of the polynomial; therefore, the singularities in question are second-order poles. Consequently, a series expansion about each singularity may be constructed by the Frobenius technique (Ince 1956) with each series thus formed valid throughout part of the interval of interest and matched at some intermediate point.
The change of variable n = w + 1 moves the singularity at T = +1 to w = 0 after which the solution of (3.12) valid about w = 0 is
found to have the form




12
F1(u)= 01/2[ + () +C(2 2 + (C) I:) + (.h
Details of this derivation and values of the coefficient Ci/C are given in Appendix A.
Similarly, the change of variable 1 in (3.12) moves the
singularity at j = -1 to = 0, and the corresponding series solution
is found to be
[ + + ( ) + (3)?X + 3 (3.15)
where again yi/sr values are given in Appendix A.
Equations (3.14) and (3-15) offer solutions to (3.12) in two portions of the temperature field. Each portion of the temperature field has a second linearly independent solution in addition to the two above results. Using standard techniques, which depend on the nature of the solutions (Ince 1956), the second solution may be formed from the first solution. Using (3.15) a linearly independent solution F 2 () is
F2(0) = yo n ;\ Yj i+/2 + (L) \i+l/2
Details of this result and the explanation of the terms (yi/Yo)' are given in Appendix B.
The above results provide individually valid solutions in the two regions of the temperature field. The entire solution must be matched at some intermediate point to give a uniformly valid solution required for G2(r) in (3.13).




13
3.4i. Matching the Solutions
As it was stated previously, the G2 (e) solutions in the two regions must be matched. Let the solution about i~=+1 or co 0 be denoted by G 2(a)); let the other solution about =-1 or N =0 be denoted b
G ().These two solutions were determined to be
G 26c) H 1 ~2) F(u,)
where 'r0in the F(N~) solution is the constant to be determined by the
matching process.
The proper matching technique must provide both the point at which matching is to take place and the value of' Within the process it
must be provided that both the values and the slopes of' the two solutions be equal at the matching point. This process is initiated by finding the point at which the logarithmic derivatives are equal; then
G I(w,) G?(?\)
7o 20
where 7\ 0= wo+ 2 is the desired matching point. Once this point is found, 'r may be determined by equating the values of' the two solutions at the point -A0
3.5. Specification of' the Temperature Field
The solution of'G2 n entails series type solutions which must be matched. Two series are about wn = 0 or n= +1; the other two series solutions are about -A=O0 or n =-, where n = w+ 1= \-l1. The
material considered in this and following sections is presented in terms




of the first solutions since it has a simpler form; however, the second solutions may be applied in the same manner as presented in this section.
Now retrace the major steps of the solution. Using (3.9) with
p = 1/2, (3.11), (3.14), and (3.15), the solution of G2(e) may be written in terms of these two series. Without specifying the constants or the recurrence relationships, the solution is as follows:
(a) about r = +1
G2(o) = [ (2 + w)-1/2 C1 + w + 2 + (3.16)
(b) about = -1
G2(e) = a (2 /2 1 + 11 _X + L2 + .... (3.17)
Equations (3.16) and (3.17) specify G2(e) throughout the field of interest but still have two unknown constants, C0 and yo. One of these constants, say Co may be specified by the same type of boundary condition as that used in the hydrodynamic case. The only term remaining to be specified is y which may be done by matching the two series at some convenient point. By use of the Frobenius technique, (3.16) should converge uniformly (Ince 1956) for
- ) < < 0
or
4 <
'IN
Therefore, the matching of these two series should be done at some point
within this interval.




15
From (3.6) a solution of the energy equation is
(/R) 1 (r/R) 2 (r/R) G() (3.18)
where G D(r) is determined in Section 3.8, R is defined in Chapter 2, and A and B are constants. As Squire has done, consider the total
heat flux, h, across a sphere of radius, r
h = 21t + [af'(n) 1]Gl(n)dn
+1
+ *f[of(1 2] [BG 2(n~) + G D( T)] d}
If the total heat flux across any sphere is solely determined by a heat source at the origin, then the heat flux is independent of the radial coordinate r; hence the second integral in the above equation is zero. Due to the complexity of the solutions, it has not been demonstrated analytically that the second integral is zero. In addition, if it is specified that T = T at (RO), then both A and B may be determined. If the G2(r) integral is zero, then 21h (3.19)
where
1
Applying the condition of a reference temperature on the axis,




16
T h + B C 0
0 21t Ek 1 1
with G D (71) 0 on the jet axis; hence B becomes
T
0 C2 h
B C N kR I T (3.20)
0 1 0 1
3.6. Selection of the Similarity Transformation
Due to the complexity of the various solutions G n (0), there should be some method by which the proper similarity transformation may be chosen prior to solving many differential equations. Certainly the analysis performed above may also be done for values of n greater than two; furthermore., solving (3.4) for values of n greater than two would produce essentially the same approach as in the n = 2 case. The major difference in each of the cases would be found only in the power series portion of the solution.
Using expression (3.5) as an approach to the problem has only mathematical significance, and the fact that there are an infinite number of solutions may not have any relevance to the physical problem. The importance of (3-5) is to recognize that there are many solutions and to approach the problem in a more systematic manner. In Section 3-5 it is shown how the use of the first three terms of equation (3.6) may be applied to the heated round jet. It would be particularly interesting if additional conditions could be placed upon the flow so that other solutions in (3.5) may be added to the final result; hence for each-added condition another solution is included in the final solution, provided that each added condition is independent of the previous conditions.
From the previous sections of this chap ter., the resulting temperature distribution using three terms of (3.6) becomes




17
T 0 21tRk1 T 011 p
211 oJk I R2I1
+ i TOl (r 4[P.T1I( 2 )( + + C 0 +
+ () GDi)(3.21)
where the above expression is applicable in the neighborhood of the jet axis. The problem which is presented by Squire is the case in which
T= h (3.22)
o 21rRk I1
If the temperature T 0has this value, then equation (3.21) reduces essentially to the Squire solution. The value of (3.21) is that it includes the case when T does not meet the condition of (3.22). The
deviation of T from expression (3.22) may include effects due to dissipation, an initial temperature distribution, or perhaps a finite heat source. One interesting result of this analysis is that the 1/r transformation dictates that each ray is a line across which there is no net heat transfer. Since conductive heat transfer exactly balances the convective heat transfer, then every cone having the origin as its vertex is a surface across which the net heat transfer is zero. The above result is true only for the 1/r transformation, and all other transformations predict some net heat transfer across the rays of the field. Examination of equation (3.4) shows that the use of the 1/r relationship automatically eliminates the radial conduction term in the energy equation. In this respect, the 1/r substitution retains all the basic terms in the differential equation. Furthermore, the dissipation term




18
requires that if a single similarity transformation is used the radial function has the form 1/r2. This is not to say that the dissipation terms in the result must be important, but it is contended that the mathematical form of the dissipation terms may have an underlying significance.
3.7. The Term Due to Dissipation
Returning to equation (3.4), the only value of n for which the dissipation term is applicable is n = 2. Then (3.4) becomes
(1- n2)G.") +
2
2[1 f -) ( (3.23)
+ k
where GD(I) is the desired solution of this nonhomogeneous differential equation. Let
K 11 2~ (R2where E = Eckert number, U2/CpTo. Using the results of the hydrodynamic solution, the form of D(TI) becomes
_(I 16 6f2 + 1) 3P( 2 2)n + (304 + lI32 + 2)] 2
)
2 + 5)TI3 + 6P2#4 PTI Once G D (1) is determined, then the temperature distribution due to dissipation, T must take the form TD _rR Il
TD = GD(T) (3.24)
0H




19
3.8. Addition of' the Nonhomogeneous Term
Given the equation:
Y"(x) + F(x) y'(x) + Q(x) y(x) =R(x) with the stipulation that
w"(x) + T(x) W'(x) + Q(x) w(X) =0. Then the particular solution y (W is
fRxep MXd x)dx (3.25)
whereEf+ePf x+J
M(x) = 2w (x)
If' GD(nj) is the desired term due to dissipation, then it may be shown that
FD)= 6 fG 2(TI) X(TI) dld .6
GD()=l ( f) ( G2(TI) 2 )6-2c 3.6
where
X(Tj) = (P2+ 1)~ 0 2) + OP4+ lip 2+ 2)Tj2
PO (2 + 5)T1 3 + 6P2 1 4 _-T and
Ti_ 1 )1 + c + 022 +.., about T +1
TO[ 1 ]G( 1 + Ll + "2 2 + about Ti=-1




20
As in the solution for G 2(01 (3.26) may be transformed by the expression w + 1 and then integrated numerically by a standard technique.




CHAPTER 4
COMPUTED RESULTS
Equation (3.21) represents the solution of' the thermodynamic problem of a heated round jet. The solution also requires that the proper selection of a series be included as outlined in Section 3.3; furthermore, the two portions of the result must be matched at the proper point as stated in Section 3.4. For convenience, the G (e) solutions for the various cases are the results of a straightforward computer program which yields the first and second solutions of the series originating at either
=+1 or qj = -1.
The selected centerline temperature distribution due to Millsaps and Soong (1965) is illustrated in figure 1 and is compared to the distribution assumed by Squire (1951).
Figure 2 demonstrates the development of the matching process for G 2(0) if a process of selective multiplication of the -q = -1 series solution is used. This figure only illustrates the matching process which should be accomplished by use of the logarithmic derivative.
Figures 3 and 4 represent two sets of isotherms for the case that RN = 4 and a = 0.7. Figure 3 is the result reported by Soong, and it is particularly interesting to note the division of the temperature field into positive excess temperature and negative excess temperature regions. Figure 4 is the result of the present work and does not exhibit the feature of a combined positive and negative excess temperature field. Figures 5 and 6 represent two sets of isotherms for the case that
21




22
RN=400 and a = 0.7. Figure 5 is due to Squire's work, and Figure 6 is produced from the results of' this paper.
Figure 7 shows a typical azimuthal distribution for the G I(e) and the G 2(0) solutions and for the solution due to Soong. This figure is the only comparison with Soong's results since in the notation of this paper this case is the only one studied by Soong. Figure 7 demonstrates the basic characteristics of all three solutions, and shows that the results due to Soong have the questionable feature of including negative excess temperature. In addition, all the cases show that the G 2(e) solution of this paper decays more rapidly to an asymptotic approach to the T = 0 axis than the G1()solution.
Figure 8 shows a further comparison between the G 1( e) and the
G 2(e) solutions. The azimuthal distributions for a =0.7 and a range of Reynolds numbers is shown. Figures 9 through 14 depict the azimuthal G 2( &) variations found by the present study for a number of cases of the Reynolds number and Prandtl number.
As presented in the previous sections, there are a number of combinations of the various G 2(0) solutions of which only one is presented in the results. Other possible combinations were discarded because of the physical significance of the result. Of the two possible solutions for the forward or n= +1 segment only the one denoted as the first solution is physically meaningful. For the rear or q= -1 segment, the first solution is used only for-Prandtl numbers greater than one, and the second solution is applied in all other cases. It is interesting to note that, if in the case of Figure 7 the first =-1 solution is
used, then the results agree with Soong's solution. After all combinations there remain two possibilities--the one shown in each case and one




23
that appears similar to Soong's result in Figure 7- Of these two, the one that includes negative excess temperature is discarded by physical reasoning.
Figures 15 through 17 show the azimuthal distributions due to G D (e) alone. These dissipation effects are shown only for a few cases studied and are presented separately due to the necessity of including the Eckert number,, which is usually very small. The G D (e) distribution in all cases is zero on the jet axis and is very small throughout the field.




CHAPTER
APPLICATION TO A TURBULENT ROUND JET~
A number of approaches have been used to analyze turbulent flow; but due to the complexity of' turbulent fluid flow, a complete theory which is free of empirical data does not exist. In the case of free jet flow, one approach is to assume that the eddy viscosity is a constant throughout the flow field. The result of this assumption, if it is true, is that the main or time-averaged characteristics of a turbulent jet flow possess the same features as a laminar jet. Unfortunately, the value of the eddy viscosity may not be determined without the use of experimental data (Squire 1951, Schlichting 1955). Using an approach analogous to the hydrodynamic problem, the experimentally determined angular displacement to the station at which the ratio T/T c is 0.5 is used to relate the analysis of a laminar flow to the results from a turbulent jet. Using a set of experimental data reported by Corrsin (19)43), it was determined that T/Tc = 0.5 at an angle of e =5.80. A variation of the analytical results must be used so that the profiles represent
the temperature distribution in a plane normal to the jet axis. This has to be done so that the experimental and analytic results represent the same area of the flow. A study was made of all the G 2(e) results, and it was determined that for all cases in which RNa = 120, the T/Tc = 0.5 point was approximately at the proper station. A comparison between the analytical results and the experimental data is shown in Figure 18.
24




CHAPTER 6
SUMMARY AND CONCLUSIONS
The solution for a heated round jet due to Squire is easily obtained from his transformation and the energy equation. Objection to Squire's approach may be raised on the basis of the necessity of omitting certain terms from the energy equation. In addition, Squire's paper presents a final answer only for large Reynolds numbers. The applicability of Squire's solution can be specified in relation to the solution of this paper by the values of the constants A and B for a given problem.
The paper by Millsaps and Soong presents the similarity transformation which in this paper is combined with Squire's transformation. Soong's work does not contain an analytic solution of the basic transformed energy equation, but rather uses a numerical scheme, a modified Runga-Kutta method, to solve equation (3.8). It should also be noted that Soong's work is restricted in all cases to a flow at a very low Reynolds number; and it is not apparent that his approach may be used for any other value of the Reynolds number. One result is that the excess temperature changes from positive to negative at some point; furthermore, the result of having a partition cone between the positive and negative excess temperature fields is unique to Soong's work. Physically this is a highly questionable result and is not supported by the results of this or other papers. His solution must actually represent some combination of a heat source and heat sink simultaneously operating at the origin.
The solution outlined in this paper is that of the hydrothermodynamics of a viscous round jet. The round jet is created in an otherwise 25.




26
quiescent fluid by the superposition of a momentum source and a heat source at the origin. Parallel to the hydrodynamic case, the strengths of the heat and momentum sources may be determined as a function of the Reynolds number, the Prandtl number, the azimuthal velocity and temperature distributions, and the reference centerline velocity U and temperature T It is particularly important to note that the solution as presented includes all values of the Reynolds number and Prandtl number and analyzes all the terms of the energy equation including dissipation.
The effect of dissipation is analyzed in a fashion similar to the
nondissipative jet. The process used to study the dissipation term shows that the effect due to dissipation is small but depends upon the Eckert number.
The last section of this paper demonstrates that the results of this paper may be applied with satisfactory results to a turbulent round jet.




2.0
T
T
c
1.0
Millsaps and Soong Squire
0 I I
0 1.0 2.0 3.0 4.o
r/R Figure 1. Comparison of two centerline temperature distributions




2.0
RN = 4o
a = 0.5
= 0.03675 \
T= 0.2
T
c 1.0 r = 0.1
N N
01
Tc 1. ---._.._N % = 0.
"~~'. -'
0.5 o
o I | |I- I ,
0 8 16 24 32
e(aeg)
Figure 2. Illustration of the matching process by selective multiplication




29
-0.0625 -o.o625
-0.125
0.25 T/To = 0.0625
0.125
0.25
8 4 0 4 8 12
r/R
Figure 3. Isotherms found by N. L. Soong for the case that RN 4
and a = 0.7
T/T = 0.02
0.05
0o.1
0.25
2 1 0.5
0 1 2 3 4
r/R
Figure 4. Isotherms resulting from the present study for the casethat RN = 4 and a = 0.7




30
r/R
Figure 5. Isotherms found by H. B. Squire for the case that
= I400 and a = 0.7
T/T = 0.01 0.2
0.5
0 2 4 6
r/R
Figure 6. Isotherms resulting from the pSquiresent study for the case that
400that Rand400 a =0 .7
T/T 0= 0.01
0 2 4 6
r/R
Figure 6. Isotherms resulting from the present study for the case
that RN = 400 and cr = 0.7




1.0
0.6
T G (e)
Tc
0.2 Ga
0 1 1 1 1 1 e(deg)
20 40 6o 80 100 120 14o
-0.2
" Soong
-0.6
Figure 7. Comparison of the azimuthal distributions G (e) and G2(e) and the distribution due to N. L. Soong for the case R =4 and a = 0.7




1.0
a= 0.7
0.8 G (e), = 4o
G2(2), e) = 4o
0.6
G(e) -)
o.4 4ooo
G2(), = 400 G e) ,
0.2 = 4000
0
0 2 4 6 8 10 12 14 i6
e(deg)
Figure 8. Comparison of the azimuthal distributions G1(0) and G2(e) for a = 0.7 and various
Reynolds numbers




1.0
HN4
0.8
0.6
G2( e)
0.4 = 0.3
= 0 .7
0.2
a = 3.0 a = 7.0
0
0 8 16 24 32 40 48 56 64
e(deg) Figure 9. G2(0) for = 4 and various Prandtl numbers




1.0
RN = 4o
0.8
= 0.3
0.6
G2(e) a = 0.7
0.4- a= 1.0
= .=3.0
0.2
0
0 2 4 6 8 10 12 14
e(deg) Figure 10. G2( e) for RN = 40 and various Prandtl numbers




1.0
RN= 400oo
0.8
0.6
a =0.2 G ( e)
0.4
a = 0-.7
0.2
a=1.0
- a= 7.0 a 3.0
0
0 1 2 3 4 5 6 7 8
e(deg) Figure 11. G2(e) for RN= 400 and various Prandtl numbers




1.0
= 4ooo
0.8
0.6 G e)
o.4"
a= 0.3
0.2
a = 0.7 a = 7.0 a = 1.0
0 I
0 1 2 3 4
e(deg) Figure 12. G2( e) for RN = 4000 and various Prandtl numbers




1.0
a= 0.7
0.8
0.6
G2(e)
0.4
R= 100
0.2
= 4000 = 40o
0
0 II I III IIII II I II
0 2 4 6 8 10 12 14 16
e(deg) Figure 13. G2(0) for a = 0.7 and various Reynolds numbers




1.0
a = 7.0
0.8
0.6
G2(e)
0.4
=N 40 = 100
0.2
RN = 4000 = 400
0
o 1 2 3 4 5 6 7 8
e(deg) Figure 14. G2(9) for a = 7 and various Reynolds numbers




0.5
RN 40
0.4
a = 0.7
/ ==1.0
0.3
Cr1
a=3.0
0.2
= 7.0
0.1
0 t- A III
0 4 8 12 16 20 24 28 32
e(deg) Figure 15. G D(e) for RN 40= and various Prandtl numbers




0.5
RN= 400
0.4 0.7
1.0
0.3 7 3
-4
O
0
x
f=1
0
0.2
0.1
0
0 .I I I I I I
0 1 2 3 4 56 7 8
e(deg) Figure 16. GD(0) for R = 400 and various Prandtl numbers 0




0.5
= 40oo00
0.4
c = 0 .7
0.3
a=1.0
o
'4
x
0.2
=3
0.1
0
o I I. J !
0 1 2 3 4
e(deg) Figure 17. GD( e) for RN = 4000 and various Prandtl numbers




1.0
A Corrsin and Uberoi (1951)
0 0 Corrsin (1943)
0 Ruden (1933)
0.8
0.6
T
T
c
0.4 A
w9
0
0.2
A A
0 II IlIIII
0 2 4 6 8 10 12 14
Arcsine (r /r) Figure 18. Comparison between the temperature distribution due to the present work and
experimental data for turbulent round jets




APPENDIX A
SOLUTION OF THE TRANSFORMED ENERGY EQUATION BY THE FROBENIUS TECHNIQUE As stated in the body of this paper, it is appropriate to solve (3.12) with the use of a Frobenius series expansion. Using the transformation w = + 1, (3.12) becomes
,, Al4 + + Ap + A+ ( l)2
Fl () + 2[4+ 3 2 2 Fl(W) =0 (A.1)
+ Am + A 63 62 + A7 + 4(B 1)J
where
A1 = -(a 2)(a i)
A2 = -42 + 2(3a- 2)(2 0)
2
A = -4c2 + 18 9 + 12P 18ap 2p2
3
A4 = -2(p 1)(2p + 6a 1)
A5 = 2(3 B)
A6 = 2 10P + 13
A7 = 4(p 1)(P 3).
If (A.1) is written in the form F I(w) + Q(w) Fl(6) = O, it may be seen that Q(w) has a second-order pole at w = 0. Then in the use of the Frobenius technique, the point w = 0 is a regular singular point of the differential equation, and Q(W) may be written as a Laurent expansion about w = 0. This expansion has the form
43




44
Q~w)= ~-~)+ 0(1) + o(i) + O(w) +
where the first term, (1/4)(i/w2), is the most significant term. The Frobenius technique assumes a series expansion of F(w), of the form
F1(W) = Z Ciai+k (A.2)
where here k is determined solely by the first term of the expansion for Q(w). In this case, the first derivative term in the differential equation is missing; therefore, for a differential equation in its reduced form, the indicial equation for k becomes i-2
k =2 =1/2 (A.3)
where Q-2 is the coefficient of the i/w2 term in the expansion for Q(w). This result is a proper one because a study of equations (3.13) and (A.2) shows that k = 1/2 is the only value that will yield a finite, nonzero value of the temperature along the jet axis. Assumption (A.2) may be used in equation (A.1) yielding a set of complicated infinite series. Putting the coefficient of each power of w equal to zero yields the following:
for W 0: Co ( 1)(-l + 1) = 0 .< C unspecified (A.4)
for w[(/4) + A4 5P + 12a 5 (A.5)
C 4(p 1)2 4(p-i)
C -[A6- + A3] (C 3 +
for 2 2 l6( T 3l)2 o 7 2 (A.6)
0 16(1 1) 16(p 1)




45
w:C 3-1C1 J6+A
for 3 A 2 + 6 A
0 36(2 -5 )o
(A.7)
+ 02 [ + A
Co 7
And for other Ci+4/C when i 0
Ci+4 -1 C i
+ -(i + 1/2)(1 1/2) + Al
Co [(2i + 9)(2i + 7) + 1(p- l1)2 C
Ci+l 3)1l+ ci+2
+ o (1 + 3/2)(1 + 1/2) + A o [A6(,i + 5/2)(1i + 3/2) + A3
C r
+ 3 jA7(i + 7/2)(i + 5/2) + Ak .(A.8)
OO
The result is that (A.4) through (A.8) leave the unspecified constant Co as a factor of the entire series and the ratios Ci/Co determined in
terms of the Reynolds number RN and the Prandtl number a. These ratios, Ci/Co, may be written in terms of RN and a if desired. Finally, Fl(m)
may be written
Fl( m) = Co ) 2 . (A.9)
This same procedure may be carried out for the singularity at n = -1.
In this case, let n = A 1. Now (3.24) becomes
S+ Bl\ + B2\3 + B3 2+ B4 + (P + 1)
F() 2[ + B3 + B62 + B + (B + )2 F()= (A.O)




46
where
B = -(o 1)(c 2)
B2 2(3a- 2)(2+ 0)
B3 = -4a 2 + 18a- 9 12P+ 18ao- 2P2
B4 = 2(p + 1)(2P 6a + 1)
B5 = -2( + 3)
B6 = P2 + 10 + 13
B7 = -4(0 + 3)(P + 1). Again, let
FI(00 = l i+k (A.11)
where again k = 1/2. The result is very similar to that previously done. The only differences are that in this case )2 is replaced by (P + 1)2 and each A is replaced by B .
n n




APPENDIX B
GENERATION OF THE SECOND SOLUTION OF THE TRANSFORMED ENERGY EQUATION
To obtain a second independent solution of (3.12), both (A.1) and (A.lO) must be solved for their alternate solutions.
The solution of F(n) in Appendix A is performed by using the
Frobenius technique, and now the second solution may be determined by a variation of the method previously used. Equation (A.2) specifies the Frobenius technique with the additional power being determined by the indicial equation. The result of applying the indicial equation in this case is specified in expression (A.3); and it is important to note that the two roots of the indicial equation are equal. The technique to determine a second linearly independent solution depends upon the relationship of the two roots, and the special case of the roots being equal is the only case considered in this paper.
Let the differential equation (A.lO) be denoted by the operator L[FI( )] so that
F1)]= o
is the equation to be solved. Substitution of (A.1O) into (A.ll) yields
L[F=\,k] = (k 1/2)2 T0Nk 0 (B.1)
where the higher order terms are zero by the proper selection of each yi,i>O, in terms of ro. It is obvious from (B.1) that k = 1/2 is a double root of the indicial equation, and the process of finding Fl(\)
47




48
is to state at this time that k is 1/2. This process to specify Fl(), the first solution, may be written symbolically as
LF(A\)] = L([F(Xtk)Ik=1/2) = 0
But consider (B.1) to be differentiated with respect to k before selecting the value of k. Then
T [F(A,k)]= 2(k 1/2)yok + (k 1/2)2 ok? 1 = 0 .
Again, it can be seen that k = 1/2 yields a valid statement. Since Fl(X) satisfies (B.1), then the alternate solution may be determined from the expression
a 8i 1______kLF(,k) = L k =0
where the order of differentiation has been exchanged. This process may also be written symbolically as
LF2(\) = L = 0
k=1/2
Then the relationship between the two solutions is
[ E V l (A k )]
= k (B.2)
Using (A.11) and (B.2) the second solution becomes
F() = Fl(X)ln + i[r(k) Ai+1/2 (B.3)
2=o1 I k=1/2
where 7i(k) represents the differentiation of the coefficients previously determined for F1(?) with respect to k. It is interesting to note 'that




49
the combination of (B.3) and (3.13) yields a function that behaves properly at the singularity X = 0. The precise form of the above equation is somewhat complicated but may be determined in a straightforward manner. Finally, the second solution may be written
F20 $k (")i ?+1/2 + x~ () i+ 1/2} (B .4)
where
[d r(k))
T ro ] =/
The result is that the unspecified constant r remains as a factor of the entire function F() and the various ratios (ri/ro) and are determined in terms of the Reynolds number RN and the Prandtl number a.




BIBLIOGRAPHY
Birkhoff, G. and Zarantonello, H. 1957 Jets, Wakes, and Cavities. New
York: Academic Press, Inc.
Corrsin, S. 1943 Wartime Report 3L23. Corrsin, S. and Uberoi, M. S. 1951 NACA TR-1040. Goldstein, S. 1965 Modern Developments in Fluid Dynamics. New York:
Dover Publications.
Ince, E. L. 1956 Ordinary Differential Equations. New York: Dover
Publications.
Landau, L. D. 1944 "A New Solution of the Navier-Stokes Equations."
Dokl. Ak. Nauk S. S. S. R. 43, 286-288.
Millsaps, K. T. and Soong, N. L. 1965 "Thermal Distributions in a Round
Laminar Jet." The Physics of Fluids. 8.
Ruden, P. 1933 "Turbulente Aurbreitungsvorg'dnge im Freistrahl." Die
Naturwissenshafter. 21, 375-378.
Schlichting, H. 1955 Boundary Layer Theory, 4th edition. New York: McGrawHill Book Company.
Slezkin, N. A. 1934 "On an Exact Solution of the Equations of Viscous Flow."
Uch. Zap. WU Sci. Rec. 2.
Soong, N. L. 1968 The Hydrothermodynamics of a Round Laminar Jet. Ph.D.
Dissertation, Dept. of Aerospace Engineering, University of Florida.
50




51
Squire, H. B. 1951 "The Round Laminar Jet." Quart. J. Mech. 4., 321-329. Yatseyev, V. I. 1950 "On a Class of Exact Solutions of the Equations of
Motion of a Viscous Fluid." NACA TM-1349.
Yih, C. S. 1950 "Temperature Distribution in a Steady Laminar, Preheated
Air Jet." J. App. Mech. 17, 381-382.




BIOGRAPHICAL SKETCH
Donald Arthur Dietrich was born on October 17, 1943, in Pittsburgh, Pennsylvania. In June, 1961, he was graduated from Seacrest High School in Delray Beach, Florida. He received the degree of Bachelor of Science in Aerospace Engineering with high honors in December, 1965. In 1966, he enrolled in the Graduate School of the University of Florida. For his graduate work, he was granted a National Aeronautics and Space Administration traineeship. In October, 1969, he joined the staff at the NASA, Lewis Research Center in Cleveland, Ohio.
Donald Arthur Dietrich is married to the former Janet Ruth Love. He is a member of Tau Beta Pi and Sigma Tau.




This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all members of that committee. It was submitted to the Dean of the College of Engineering and to the Graduate Council, and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy. June, 1970
,College of Engineering
Dean, Graduate School Supervisory Committee:
k /"
S- Z1 a ,J-




Full Text

PAGE 1

THE ROUND JET By DONALD ARTHUR DIETRICH A DISSERTATION PRESENTED TO THE GRADUATE COUNOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIRE ME NTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1970

PAGE 2

ACKNOWLEOOEMENTS The author wishes to express his appreciation to the members of his supervisory committee and especially to the chairman, Dr. Bernard M. Leaden, for his suggestions and criticisms during the preparation of this dissertation. The author also wishes to express his appreciation to Drs. M. H. Clarkson and K. T, Millsaps for their efforts throughout the author's graduate work. Appreciation is also expressed to the National Aeronautics and Space Administration for the granting of a traineeship which made this work possible. Finally, special appreciation is extended to the author's wife, Jan, who rendered invaluable assistance toward the completion of this dissertation. ii

PAGE 3

ACKNOWLEIGEMENTS LIST OF FIGURES. KEY TO SYMBOLS TABLE OF CONTENTS . . . . . . . . ABSTRACT CHAPTERS . . . . . . 1. INTRODUCTION. 2 SOLUTION OF THE EQUATIONS OF MOTION 3. SOLUTION OF THE COMPLETE ENERGY EQUATION .. 3 .1. THE ENERGY EQUATION FOR LAMINAR FLOW .. DETERMINATION OF THE SOLUTIONS G (11) n 3.3. SOLUTION FOR F(11) 3.4. MATCHING THE SOLUTIONS .. 3. 5 SPECIFICATION OF THE TEMPERATURE FIELD. 3.6. 3. 7 3.8 SELECTION OF THE SIMILARITY TRANSFORMATION THE TERM DUE TO DISSIPATION ADDITION OF THE NONHOMOGENEOUS TERM 4. COMPUTED RESULTS 5. APPLICATION TO A TURBULENT ROUND JET. 6. SUMMARY AND CONCLUSIONS FIGURES APPENDICES A. SOLUTION OF THE TRANSFORMED ENERGY EQUATION BY THE Page ii V vii X 1 3 6 6 9 11 13 13 16 18 1 9 21 24 25 27 FROBENIUS TECHNIQUE . . 4 3 iii

PAGE 4

B. GENERATION OF THE SECOND SOLUTION OF THE TRANSFORMED ENERGY EQUATION BIBLIOGRAPHY . . . . . iv . Page

PAGE 5

LIST OF FIGURES 1. Comparison of two centerline temperature distributions 2, Illustration of the matching process by selective multiplication 3. Isotherms found by N. L. Soong for the case that = 4 and cr = 0, 7 4. Isotherms resulting from the present study for the case that = 4 and a = 0.7. 5. Isotherms found by H.B. Squire for the case that = 400 and cr = 0.7 6. Isotherms resulting from the present study for the case that = 4oo and a= 0.7. 7. Comparison of the azimuthal distributions G 1 (e) and G 2 (e) and the distribution due to N. L. Soong for the case = 4 and a= 0.7 8. Comparison of the azimuthal distributions G 1 (e) and G 2 ( e) for a= 0.7 and various Reynolds numbers 9. G 2 ( e) for = 4 and various Prandtl numbers 10. G 2 ( e) for = 4o and various Prandtl numbers 11. G 2 ( e ) for = 4oo and various Prandtl numbers 12. G 2 ( e) for = 4ooo and various Prandtl numbers. 13. G 2 ( e) for (1 = 0.7 and various Reynolds numbers 14. Gie) for (1 = 7 and various Reynolds numbers 15. GD( e) for = 4o and various Prandtl numbers. V Page 27 28 29 29 30 30 31 32 33 34 35 36 37 38 39

PAGE 6

Page 16. GD( 0) for = 4oo and various Prandtl numbers. . 4o 17. GD( 0) for = 4ooo and various Prandtl numbers 41 18. Comparison between the temperature distribution due to the present work and experimental data for turbulent round jets. 42 vi

PAGE 7

A B co c.jc l. 0 C p d KEY TO SYMBOLS constant determined by boundary conditions constant determined by boundary conditions inte g ration constant coefficients of a series expansion specific heat at constant pressure arbitrary constant D(~) function of alone D 0 D./D l. 0 E F constant used in the solution for dissipation coefficients of a series expansion Eckert number= u 2 /c T p 0 similarity function for the stream function net force applied at the origin a transformed function of the similarity solution for the ener gy equation F(A) F(~) written in terms of A F(~) written in terms of m similarity function used in the solution for dissipation terms in a series of similarity functions for the energy equation G 2 (~) written in terms of A G 2 (~) written in terms of m h total heat flux across a sphere of finite radius transformed function of alone vii

PAGE 8

transfonned function of alone a constant of the flow K M(x) coefficient of thermal conductivity constant= -(aE/~)R~ 0 arbitrary function of x n index used in a series p arbitrary constant P pressure P pressure at infinity 00 P(~) function of alone P(x) arbitrary function of x Q(x) arbitrary function of x r radius in spherical coordinate system r perpendicular distance from jet centerline C R reference radius Reynolds number= RU/v R(x) arbitrary function of x t temperature t temperature at infinity 00 t reference temperature 0 T excess temperature= t t 00 T reference excess temperature 0 T local centerline excess temperature C TD excess temperature due to dissipation u radial component of the velocity U reference velocity U local centerline velocity C viii

PAGE 9

V azimuthal component of the velocity w(x) function of x alone X an independent variable X(~) function of alone y(x) function of x alone y (x) p function of x alone constant= 4/~ + 1 integration constant e coefficients of a series expansion similarity variable= cos e azimuthal coordinate similarity variable=~+ 1 A matching location 0 coefficient of absolute viscosity v coefficient of kinematic viscosity p density a Prandtl number= pC v/k p spherical coordinate of symmetry dissipation function w stream function similarity variable=~ 1 m 0 matching location the Laplacian operator ix

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Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy The Round Jet By Donald Arthur Dietrich June, 1970 Chairman: Dr. B. M. Leadon Major Department: Aerospace Engineering In this paper the hydrothermodynamics of an axially symmetric flow is presented and interpreted as a round laminar jet. The established solution of the complete Navier-Stokes equations for this case is used as the basis for the study of the complete energy equation. A series of similarity transformations is introduced, and the most significant transformation is selected. Using the Frobenius technique, a series solution for the temperature profile is formed for the convective and conductive portions of the solution. In addition, a solution is deter mined for the contribution due to viscous dissipation. The results are in terpreted as a heated round laminar jet which is produced by superimposed momentum and heat sources. X

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CHAPTER 1 INTRODUCTION The solution of the Navier-Stokes equations has been a formidable problem since they were derived in the first half of the nineteenth century. The dozen or so solutions that now exist were obtained only after the physical problem or the differential equations were extensively simplified. The problem of the round laminar jet is one for which the complete Navier-Stokes equations and the associated energy equation have been solved. In this paper the hydrodynamic solution is reviewed and then applied to the complete energy equation which is then solved exactly. The study of the hydrodynamics of a round laminar jet has had a logi cal development since its introduction by H. Schlicht:i.ng (1955). Schlich ting' s approach was first to simplify the equations of mot ioo. with the aid of boundary layer assumptions, which are valid only for large Reynolds numbers. He introduced a stream function by a similarity transformation and solved the resulting eg_imtion in closed form. Schlichting analyzed his solution as the flow resulting from a momentum source located at an orifice in a flat plate. N. A. Slezkin (1934) showed the existence of a solution for an axially symmetric flow. This proble:n was further explored by L. D. Landau (1944); and v. I. Yatseyev (1950) obtained a solution to the Navier-Stokes equations for this problem. H.B. Squire (1951) also derived an exact solution to the energy equation associated with a round laminar jet if there is a heat source located at the origin and if viscous dissipation is neglected. 1

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2 There are a number of ba s ic assumptions which are used in the solu tion of this problem. The flow is assumed to be axially symmetric such that the velocity and its gradients with respect to the coordinate cp are zero. This assumption simplifies the problem and yields an axially symmetric case which is related to the Jeffrey-Hamel theory (G. Birkhoff and E. H. Zarantonello 1957). The flow is considered to be a laminar flow of an incompressible fluid with its transport coefficients held con stant. With this assumption, the associated energy equation ri ay be solved using the hydrodynamic solution. This paper offers a new approach to the solution of the complete energy equation for the case of the round laminar jet. Yih (1950) ex tended the boundary layer solution and found a closed form solution for the heated round jet. H.B. Squire (1951) used the solution to the com plete Na.vier-Stokes equations to obtain a closed form solution to a re duced energy equation. Both Yih's and Squire's solutions neglected the terms due to viscous dissipation. K. T. Millsaps and N. L. Soong (1965) published a similarity transformation which included both the radial con duction term and the dissipation terms. The paper by Soong (196 8 ) pre sents a valid similarity transformation and studies the most importa n t differential equation related to the energy equation; however, Soong's paper does not offer an analytic solution to the energy equation. Fur thermore, Soon g 's numerical solution appears to be valid only for one very specialized and physically questionable flow. The solution being described in this chapter is a more general a pproach to this problem so that the previous results become intermediate portions of the entire solution.

PAGE 13

CHAPTER 2 SOLUTION OF THE EQUATIONS OF MOTION The hydrodynamic solutio n itself is not the objective of this paper, but, since the re s ults of thi s section are used in the ther m odynamic solu tion, an outline of the method of solving the equations of motion is in cluded in this paper. The continuity equation and the Navier-Stokes equations in a spherical coordinate system for a steady, axially symmetric, incompressible flow of a fluid having constant transport coefficients are (Goldstein 1965) l:_ .E__( r 2 u) + 1 a ( El) o 2 e ae v sin = ar r sin r ( 2 .1) au+ 2 1 aP + v[if u 2u 2 av 2v cot e] V au V ur ae = 2 2 ae 2 ar r par r r r (2.2) u av+ V av + UV _!_ aP + v[if V + 2 au s:n 2 e] r ae = 2 a e ar r pr ae 2 r r (2.3) where p is the density, P is the pressure, v is the kinematic viscosity, and ef is the Laplacian operator. It is customary in problems of this nature to use a stream function which is defined so that the continuity equation is automatically satisfied. Then the number of equations is re duced from three to two, but the order of the remaining two equations is increased from second to third order. The appropriate form of the stream function assumed in this case is iv= vr f(ri) (2.4) 3

PAGE 14

4 where ~=cos e and f(~) is dimensionless, so that U = f'(~), V = (2.5) The pressure distribution for this flow was obtained by Squire from (2.2) and (2.3) as p .. p 0 p = (2.6) where P 0 is the pressure at infinity, where u and v vanish, and c 1 is-a constant of integrat i on. In these terms (2.2) may be written ( 2. 7) Integrating twice and applying the condition of axial symmetry, whence the constants of inte g ration are set equal to zero, the result is found to be (2.8) where is to be determined from one of several conditions which may be placed upon the flow. If the velocity along the axis of the jet is designated U at the distance R, it is found from (2.5) and (2.8) that (2.9) where is the Reynold s number Since u varies ihversely with r, the value of

PAGE 15

5 R is immaterial and is a constant along the jet axis. Reynolds num bers may be calculated for any other ray as well. An alternative means of determining is to calculate the force ap plied at the origin, regarded as known, as equal to the net outflow of ax ial momentum through a sphere of radius r plus the axial components of the forces due to pressure and to viscous stresses acting on that surface. Squire has given this relationship as, in present terms, F 2 8rc pv = 8~ + 2~ ~2 ln1~ + 1) 3(~2 1) 1 For future reference these results are summarized in r, e-coordinates: = I _g_) [cos 2 e + 2~ co; e J.] L ( cos e) J (~)~ ~ 1 :a: e] -\-:-)= 1~r(~)[f /: 8 c:s-e~ 2 ] V u = and = RU/v, where U = u(R,O).

PAGE 16

CHAPTER 3 SOLU'l'ION OF THE COMPLETE ENERGY EQUATION 3 .1. T'ne Energy Equation for Laminar Flow With the assumption that the incompressible fluid has constant trans port properties and using the above hydrodynamic solution without modi fication, the energy equation becomes a linear, second-order partial differ ential equation involving only one dependent variable, the temperature. In a spherical coordinate system and assuming axial symmetry, the energy equation is (Schlichting 1955) pC [u aT + !. aT] = kl.l.. __I r2 p ar r ae 2 ar r where T excess temperature= t t 00 terms due to dissipation Cp heat capacity at constant pressure sin k coefficient of thermal conductivity a I e ae sin (3.1) The velocity components are considered as known functions, and the relation ships from Chapter 2 may be used. Using statement (2.4) which introduced a stream function and equation (2.5), the energy equation may be rewritten as pCpf ~;c~) I ;;J + ~~~)I:~)} {~2 8 ~ (r 2 :~ I + :2 :, 1 f 1 ll + 6 ( 3 .2)

PAGE 17

where 2 = T D(11) r and Using an approach similar to that used in obtaining the hydrodynamic solution, assume T = r-n G (T)) n 7 ( 3. 3) with the va-lue of n unspecified, although it is clear that n should be a positive number so that T will approach zero asymptotically as r in creases. Substituting (3.3) into (3.2) the energy equation becomes (3.4) where the prime denotes differentiation with respect to T) It can be seen from this equation that any value of n will be acceptable if the dissipa tion term is neglected; but, if the presence of the dissipation term is considered to be essential, the only allowed value of n is n = 2, In the absence of dissipation a more general expression may be chosen instead of (3.3) such as

PAGE 18

00 T = [r-n Gn(11) n=O 8 (3.5) Here n is restricted to positive integers and G 0 (11) = 0 because of the boundary condition at infinity. Then for each n the basic equa tion (3.4) yields a corresponding ordinary differential equation for G (11). In each case the solution G (T}) includes two undetermined conn n stants. A term due to dissipation which is a nonhomogeneous term in the differential equation for n = 2 may also be added to the series. When dissipation is present, G 2 ( Tl) may be treated as the complementary solution and GD(11) as the particular solution of the equation for n = 2. The series assumption for T may therefore be rewritten as where 00 GD( Tl) 2 r G 1 (11) G 2 (11) + --r+ --2+ r (3.6) GD(11) is due to dissipation alone, and each function G (11) is the n solution of a linear, second-order, homogeneous differential equation. The solutions for the excess temperature must be bounded and continu ous together with their first and second derivatives throughout the thermal field. To obtain a solution, this condition has to be relaxed in the neighborhood of the origin which is a singular point. The second require ment is that the final temperature field be axially symmetric. Thus the solutions must be even functions of e; and the temperature gradient trans verse to the z-axj_s must be zero. Again, proper selection assures that the similarity variables will meet this second condition. The final condition is that the excess temperature field approach zero as r increases without limit.

PAGE 19

or 3.2. Determination of the Solutions GJ,.!il For n = 1 equation (3.4) becomes This equation may easily be integrated twice and has the solution 9 (3.7) where A is th e undetermined constant and is the same as that used previously in (2. 9 ). This is the solution determined by H.B. Squire where a repre s ents the Prandtl number, pC v/k, and the appropriate p boundary conditions have already been applied. For n = 2, the differential equation for G 2 (ri) becomes (3.8) where the term due to dissipation has temporarily been omitted and again cr is the Prandtl number. Equation (3.8) is revised with the aid of (2.8); in addition, the definition of H 2 (ri) i s given by (3. 9 ) w here p is some unspecified power. With the relationships obtained from (3.7) 2cr TJ

PAGE 20

10 and 20(20 + 1) (f3 11) 2 (3.8) becomes 2 2 2 1)(1 1) )2o(2o + 1) + p(p 1)(1 1) )4o 2 2 (f3-,i) (f3-,i) ( 2o 11 2 4o t3T) + 2o) (f3 ,.. 11)2 Inspection of this equation, in particular the H;(11) term, shows that it would be convenient to take p = 1/2. If p is not chosen now but carried throughout the analysis, it may be shown that the only possible choice of p which will satisfy the boundary condition on the jet axis is indeed 1/2. With this value of p, the above equation reduces to the form 2 2 2 2~ + (o 30 + 2)1) + 2(3(30 2)1) + (-o 30 + 2f3) () = 2 H2 Tl O (f3 11) (3.10) The first two terms of this equation are the same as the first two terms of Legendre s equation; hence (3.10) is self-adjoint. It i s advanta geous to put (3.10) into its normal form, eliminating the first derivative term by the substitution

PAGE 21

11 (3.11) Substitution of (3.11) in (3.10) yields F"(T)) + [ 2 P(T)) 2 ) F(T)) = 0 (t3 T)) (1 T)) (1 + T)) (3.12) where 2 2 + 6t3(a l)T) + (-cr 3cr + 3t3) Retracing these steps shows that the desired solution G 2 (T)) transforms to F(T)) by the relationship ( 3 .13) 3.3. Solution for F(T)) Equation (3.12) has singularities at T) = +l, -1, ~, and oo; but, since T) = cos e and > 1, the only singularities of importance here are T) = +l and T) = -1, Examination of P(T)) shows that T) = is not a root of the polynomial; therefore, the singularities in question are second-order poles. Consequently, a series expansion about each singu larity may be c onstructed by the Frobenius technique (Ince 1956) with each series thus formed valid throughout part of the interval of interest and matched at some intermediate point. The change of variable T) = ru + 1 moves the singularity at T) = +l to ru = 0 after which the solution of (3.12) valid about ru = 0 is found to have the form

PAGE 22

Details of this derivation and values of the coefficient given in App e ndix A. c./c are l 0 12 (3.1h) Similarly, th e chan ge of variable ~=A 1 in (3.12) mo v es the singularity at = -1 to A= o, and the correspondin g series solution is found to be Fl(A) = yo ,1/2r +(~;)A+ (~:)A2 + (~!)A3 + -] (3.15) where again yi/y 0 values are given in Appendix A. Equations (3.14) and (3.15) offer solutions to (3.12) in two portions of the temperature field. Each portion of the temperature field has a second linearly i n depende n t solution in addition to the two above results. Using standard te c hniques, which depend on the nature of the solutions ( Ince 1956), the second solution may be formed from the first solution. Using (3.15) a linearly indepe n dent solution F 2 (A) is Details of this result and the explanation of the terms (y./y ) are l 0 given in Appendix B. The above resul t s provide individually valid solutions in the two regions of the temperature fi.eld. The entire solution must be matched at some intermediate point to give a uniformly valid solution required

PAGE 23

13 3.4. Matching the Solutions As it was stated previously, the G 2 (e) solutions in the two regions must be matched, Let the solution about = +l or ro = 0 be denoted by G 2 (ro); let the other solution about = -1 or A= 0 be denoted by G 2 (A), These two solutions were determined to be Giro) = ~-~l r2 1 ~2 F(ro) G 2 (A) = -f(l t2 1 ~2 F(A) where in the F(A) solution is the constant to be determined by the matching process The proper matching technique must provide both the point at which matching is to take place and the value of r. Within the process it 0 must be provided that both the values and the slopes of the two solutions be equal at the matching point. This process is initiated by finding the point at which the logarithmic derivatives are equal; then where A = m + 2 is the desired match i ng point. Once this point is 0 0 found, y 0 may b e determined by equatin g the values of the two solutions at the point A 0 3.5. Specification of the Temperature Field The solution of G 2 (~) entails series type s ol u tions whi c h mu s t be matched. Two series are about m = 0 or = +l; the other t wo series solu t ions are about A= 0 or = -1, where = m + 1 = A 1, The material c onsidered in this and following sections is presented in terms

PAGE 24

14 of the first solutions since it has a simpler form; however, the second solutions may be applied in the same manner as presented in this section. Now retrace the major steps of the solution. Using (3.9) with p = 1/2, (3.11), (3.14), and (3.15), the solution of G 2 (e) may be writ ten in terms of these two series. Without specifying the constants or the recurrence relationships, the solution is as follows: (a) about = +l ) (3.16) (b) about = -1 (J G 2 ( 0) = = ,] (2 (3.17) Equations (3.16) and (3.17) specify G 2 (e) throughout the field of in terest but still have two unknown constants, C and y. One of these 0 0 constants, say C, may be specified by the same type of boundary condio tion as that used in the hydrodynamic case. The only term remaining to be specified is y 0 which may be done by matching the two series at some convenient point. By use of the Frobenius technique, (3.16) should con verge uniformly (Ince 1956) for or -(~ 1) < m 0 4 Therefore, the matching of these two series should be done at some point within this interval.

PAGE 25

15 From (3.6) a solution of the energy equation is T = A ( + B 2 G2(D) + 1 ( ) (r/R) Gl DJ (r/R) (r/R)2 GD D ( 3 .18) where GD(D) is determined in Section 3.8, R is defined in Chapter 2, and A and B are constants. As Squire has done, consider the total heat flux, h, across a sphere of radius, r If the total heat flux across any sphere is solely determined by a heat source at the origin, then the heat flux is independent of the radial coordinate r; hence the second integral in the above equation is zero. Due to the complexity of the solutions, it has not been demonstrated ana lytically that the second integral is zero. In addition, if it is spec ified that T = T at (R,o), then both A and B may be determined. 0 If the G 2 (D) integral is zero, then A= h 2:rc Rk I 1 ( 3 .19) where Applying the condition of a reference temperature on the axis,

PAGE 26

with 16 h C T = + B o 0 2rc Rk 11 GD(T)) = 0 on the jet axis; hence B becomes B = To-if'2 [1 C 2rc 0 (3.20) 3.6. Selection of the Similarity Transformation Due to the complexity of the various solutions G (e), there should n be some method by which the proper similarity transformation may be chosen prior to solving many differential equations. Certainly the ana lysis performed above may also be done for values of n greater than two; furthermore, solving (3.4) for values of n greater than two would produce essentially the same approach as in the n = 2 case. The major difference in each of the cases would be found only in the power series portion of the solution. Using expression (3.5) as an approach to the problem has only math ematical significance, and the fact that there are an infinite number of solutions may not have any relevance to the physical problem. The impor tance of (3.5) is to recognize that there are many solutions and to ap proach the problem in a more systematic manner. In Section 3.5 it is shown how the use of the first three terms of equation (3.6) may be ap plied to the heated round jet. It would be particularly interesting if additional conditions could be placed upon the flow so that other solu tions in (3.5) may be added to the final result; hence, for each added condition another solution is included in the final solution, provided that each added condition is independent of the previous conditions. From the previous sections of this chapter, the resulting tempera ture distribution using three terms of (3.6) becomes

PAGE 27

17 (3.21) where the above expression is applicable in the neighborhood of the jet axis. The problem which is presented by Squire is the case in which If the temperature h T = --=---=-=o 2n Rk I 1 (3.22) T has this value, then equation (3.21) reduces 0 essentiall y to the Squire solution. The value of (3.21) is that it in cludes the case when T does not meet the condition of (3.22). The 0 deviation of T from expression (3.22) may include effects due to diso sipation, an initial temperature distribution, or perhaps a finite heat source. One interesting result of this analysis is that the 1/r trans formation dictates that each ray is a line across which there i s no net heat transfer. Since cond u ctive heat transfer exactly balances the co n vective heat tran s fer, then every cone having the ori g in as its vertex is a surface a c ro ss which the net heat transfer is zero. The a b ove re sult is true onl y for the 1/r transformation, and all o t her transforma tions predict s ome net heat transfer across the rays of the field. Ex amination of equa t ion (3.4) shows that the use of the 1/r relation s h i p automatically el i minates the radial conduction term in the ener g y equa tion. In this re s pe c t, the l/r 2 substitution retains all th e ba s ic terms in the diff e rential equation. Furthermore, the di s sipation te~m

PAGE 28

requires that if a single similarity transformation is used the radial function has the form l/r 2 This is not to say that the dissipation terms in the result must be important, but it is contended that the mathematical form of the dissipation terms may have an underlying sig nificance. 3.7. The Term Due to Dissipation Returning to equation (3.4), the only value of n for Which the dissipation term is applicable is n = 2. Then (3.4) becomes 18 (3.23) where GD(~) is the desired solution of this nonhomogeneous differential equation. Let K ( oE)(R~ ) 0 where E = Eckert n umber, u 2 /c T Using the results of the hydrodynamic p 0 solution, the form of D(~) becomes D(~) = 16 rp 2 + 1) 3P(P 2 + 2)~ + (W 4 + np 2 + 2)~ 2 (p ~)6 l -P( 9P 2 + 5)~ 3 + 6p 2 ~ 4 llr}J Once GD(~) is determined, then the temperature distribution due to dissipation, TD,' must take the form ( 3 .24)

PAGE 29

3.8. Addition of the Nonhomogeneous Term Given the e~uation: y"(x) + P(x) r' (x) + Q(x) y(x) = R(x) with the stipulation that w"(x) + P(x) w'(x) + Q(x) w(x) = 0, Then the particular solution Y p(x) [ fxp(-f(x)dx)] [Jw~ exp(JM(x)dx)dx]} dx where M(x) = 2wi()) + P(x) wx 19 (3.25) I f GD(~) is th e de si red term due to dissipation, then it may be shown that where and [ ~ 1 1( 2 ) 1 / 2 ( c 1 c 2 2 ) 1 + 1 + c 0 ill+ Co ill + a bout~ = +l

PAGE 30

20 As in the solution for G 2 (~), (3.26) may be transformed by the expres sion = m + l and then integrated numerically by a standard technique.

PAGE 31

CHAPTER 4 COMPUTED RESULTS Equation (3.21) represents the solution of the thermodynamic problem of a heated round jet, The solution also requires that the proper selec tion of a series be included as outlined in Section 3.3; furthermore, the two portions of the result must be matched at the proper point as stated in Section 3.4. For convenience, the G 2 (e) solutions for the various cases are the results of a straightforward computer program which yields the first and second solutions of the series originating at either = +l or = -1. The selected centerline temperature distribution due to Millsaps and Soong (1965) is illustrated in figure 1 and is compared to the dis tribution assumed by Squire (1951). Figure 2 demonstrates the development of the matching process for a 2 (e) if a process of selective multiplication of the = -1 series solution is used. This figure only illustrates the matching process which should be accomplished by use of the logarithmic derivative. Figures 3 and 4 represent two sets of isotherms for the case that = 4 and cr = 0 .7. Figure 3 is the result reported by Soong, and it is particularly interesting to note the division of the temperature field into positive excess temperature and negative excess temperature regions. Figure 4 is the result of the present work and does not exhibit the fea ture of a combined positive and negative excess temperature field. Figures 5 and 6 represent two sets of isoth~rms for the case that 21

PAGE 32

22 = 400 and a= 0.7. Figure 5 is due to Squire's work, and Figure 6 is produced from the results of this paper. Figure 7 shows a typical azimuthal distribution for the G 1 (e) and the G 2 (e) solutions and for the solution due to Soong. This figure is the only comparison with Soong's results since in the notation of this paper this case is the only one studied by Soong. Figure 7 demonstrates the basic characteristics of all three solutions, and shows that the re sults due to Soong have the questionable feature of including negative excess temperature. In addition, all the cases show that the G 2 (e) solution of this paper decays more rapidly to an asymptotic approach to the T = O axis than the G 1 (e) solution. Figure 8 shows a further comparison between the G 1 (e) and the G 2 (e) solutions. The azimuthal distributions for a= 0.7 and a range of Reynolds numbers is shown. Figures 9 through 14 depict the azimuthal G 2 (e) variations found by the present study for a number of cases of the Reynolds number and Prandtl number. As presented in the previous sections, there are a number of com binations of the various G 2 (e) solutions of which only one is presented in the results. Other possible combinations were discarded because of the physical significance of the result. Of the two possible solutions for the forward or = +l segment only the one denoted as the first solution is physically meaningful. For the rear or = -1 segment, the first solution is used only for Prandtl numbers greater than one, and the second solution is applied in all other cases. It is interesting to note that, if in the case of Figure 7 the first = -1 solution is used, then the results agree with Soong's solution. After all combinaI tions there remain two possibilities--the one shown in each case and one

PAGE 33

that appears similar to Soong's result in Figure 7. Of these two, the one that includes negative excess temperature is discarded by physical reasoning. 23 Figures 15 through 17 show the azimuthal distributions due to GD(e) alone. These dissipation effects are shown only for a few cases studied and are presented separately due to the necessity of including the Eckert number, which is usually very small. The GD(e) distribution in all cases is zero on the jet axis and is very small throughout the field.

PAGE 34

CHAPTER 5 APPLICATION TO A TURBULENT ROUND .00 A number of approaches have been used to analyze turbulent flow; but due to the complexity of turbulent fluid flow, a complete theory which is free of empirical data does not exist In the case of free jet flow, one approach is to assume that the eddy viscosity is a constant throughout the flow field. The result of this assumption, if it is true, is that the main or time-averaged characteristics of a turbulent jet flow possess the s ame features as a laminar jet. Unfortunately, the value of the eddy viscosity may not be determined without the use of ex perimental data (S4uire 1951, Schlichting 1955). Using an approach anal o g ous to the hydrodynamic problem, the experimentally determined angular displacement t o the station at which the ratio T/T C is 0.5 is u sed to relate the analysi s of a laminar flow to the results from a turbulent j et. Usin g a set of experimental data reported by Corrsin (1943), it was determined that T/T = 0.5 a t an angle of e = 5.8. A variation C of the analytical results must be used so that the profiles represent the temperature distribution in a plane normal to the jet axis. This has t o be done so t hat the experimental and analytic results represent t he same area of t he flow. A study was made of all the G 2 (e) results, and it was determined that for all cases in which ~a= 120, the T/T = 0.5 point was approximately at the proper station. A comparison C between the analytical results and the experimental data is shown in Fi g ure 18. 24

PAGE 35

SUMMARY AND CONCLUSIONS The solution for a heated round jet due to Squire is easily obtained from his transformation and the energy equation. Objection to Squire's approach may be raised on the basis of the necessity of omitting certain terms from the energy equation. In addition, Squire's paper presents a final answer only for large Reynolds numbers. The applicability of Squire's solution can be specified in relation to the solution of this paper by the values of the constants A and B for a given problem. The paper by Millsaps and Soong presents the similarity transforma tion which in this paper is combined with Squire's transformation. Soong's work does not contain an analytic solution of the basic transformed energy equation, but rather uses a numerical scheme, a modified Runga-Kutta method, to solve equation (3.8). It should also be noted that Soong's work is restricted in all cases to a flow at a very low Reynolds number; and it is not apparent that his approach may be used for any other value of the Reynolds number. One result is that the excess temperature changes from positive to negative at some point; furthermore, the result of having a partition cone between the positive and negative excess temperature fields is unique to Soong's work. Physically this is a highly question able result and is not supported by the results of this or other papers. His solution must actually represent some combination of a heat source and heat sink simultaneously operating at the origin. The solution outlined in this paper is that of the hydrothermody namics of a viscous round jet. The round jet is created in an otherwise

PAGE 36

26 quiescent fluid by the superposition of a momentum source and a heat source at the origin. Parallel to the hydrodynamic case, the strengths of the heat and momentum sources may be determined as a function of the Reynolds number, the Prandtl number, the azimuthal velocity and tempera ture distributions, and the reference centerline velocity U 0 and ternperature T 0 It is particularly important to note that the solution as presented includes all values of the Reynolds number and Prandtl number and analyzes all the terms of the energy equation including dissipation. The effect of dissipation is analyzed in a fashion similar to the nondissipative jet. The process used to study the dissipation term shows that the effect due to dissipation is small but depends upon the Eckert number. The last section of this paper demonstrates that the results of this paper may be applied with satisfactory results to a turbulent round jet.

PAGE 37

T T C 2.0 1.0 0 0 1.0 2.0 r/R Figure 1. Comparison of two centerline temperature distributions 3.0 4.o

PAGE 38

T T C 2.0 = 4 0 (J = 0.5 I o .03675 Yo = 1.5 -i 1.0 \ \ \ \ \ \ ' ' ..... ' \ \ y = 0.1 0 ..... y = 0 .2 0 ' ..... ..... ..... ..... ..... ..... ..... ......... ..... ........................ ..... ...., y 0 = 0.05 .......... ..... ........... ...... ...... ...... -.......... ........ ---------..... ..... .......... ..... ..... 0 L..------L-----"'---------L------'-------------'--------------0 8 16 e(deg) 24 Figure 2. Ill ustration of the mat ch in g process by selective multiplication 32 I\) co

PAGE 39

-0.0625 -0.125 = 0.0625 8 4 0 4 8 12 r/R Fi g ure 3. Isotherms found by N. L. Soong for the case that = 4 and a= 0.7 0 1 2 r/R T/T = 0,02 0 3 4 Fi g ure 4. Iso t herms resulting from the present stud y for the c ase tha t = 4 and a= 0.7 29

PAGE 40

r/R Figure 5. Isotherms found by H.B. Squire for the case that = 400 and a= 0.7 0.1 0 2 r/R 4 T/T = 0 .01 0 0.05 6 Fi g ure 6. Isotherms resulting from the present study for the case tha t = 4 0 0 and a= 0.7 30

PAGE 41

o.6 T T C 0.2 0 -0 .2 -o.6 G 1 ( 0) 20 4o 100 120 140 Soong Figure 7. Comparison of the azimuthal distributions G 1 (e) and G 2 (e) and the distribution due to N. L. Soong for the case = 4 and cr = 0.7 0(deg)

PAGE 42

1.0 a= 0.7 o.8 o.6 G( e) o.4 0.2 0 0 2 4 6 8 10 12 14 16 e( deg) Figure 8. Compari son of the azimuthal distributions a 1 (e) and a 2 ( e) for a~ 0.7 and various w I\) Reynolds numbers

PAGE 43

1.0 o.8 o.6 o.4 0.2 0 0 8 16 32 e(deg) Figure 9. G 2 (e) for = 4 and various Prandtl n umbers 48 64

PAGE 44

LO o.8 o.6 o.4 0.2 0 2 4 cr = 7.0 6 8 e(deg) Figure 10. G 2 ( e) for = 40 and various Prandtl numbers = 4o 10 12 14 16

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1.0 o .8 o.6 o.4 0.2 0 0 1 2 3 4 e(deg) Figure 11. G 2 (e) for = 4 00 and various Prandtl num bers 5 6 7 = 400 8 w V1

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o .8 o.6 o.4 0.2 0 1 2 e(deg) Figure 12. G 2 ( e) for = 4 000 and various Prandtl numbers = 4000 3 4

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1.0 o .8 o.6 o.4 0.2 0 2 4 6 8 0(deg) Fi gure 13. G 2 ( e ) for a= 0 .7 and various Reynolds numbers a= 0.7 1 0 12 14 16

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1.0 o.8 o.6 o.4 0.2 0 = 4000 0 1 2 3 4 e(deg) Figure 14. G 2 (e) for a= 7 and various Reynolds numbers a= 7.0 5 6 7 8

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(Y') 0 X r,::i "A c!) 0.5 ------------------------------------, o .4 0.3 0.2 0.1 0 0 4 8 12 16 e( deg) Fi gure 15. GD( e) for = 4 o an d various Prandtl nu m b er s = 40 20 24 28 32

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0.5 -----------------------------------------, = 400 0.3 0 r-l X p.c'.j ........... C, 0.2 0 1 2 3 4 5 6 7 8 e( deg) Figure 16 GD( e ) for = 4 00 and various Prandtl numbers

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0.5 o .4 0.3 L00 r-i X Alr,::l 0 0.2 0.1 0 0 1 2 e(deg) Figure 17, GD( e) for = 4 000 and v arious Prandtl numbers RN= 4 000 3 4

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T T C 1.0 o 8 o.6 o.4 0.2 0 0 2 4 6 8 Arcsine (r /r) C A C 0 10 Corrsin and Uberoi (1951) Corrsin (1943) Ruden (1933 ) El 12 14 Figure 18. Comparison between the tempe rature distribution due to th e ~resent .rork and experimental data for turbulent round jets

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APPENDIX A SOLUTION OF THE TRANSFORMED ENERGY EQUATION BY THE FROBENIUS TECHNIQUE As stated in the body of this paper, it is appropriate to solve (3.12) with the use of a Frobenius series expansion. Using the trans formation ~=ill+ 1, (3.12) becomes {~ ill 4 + A,tn 3 + A-jD 2 + A 4 ill + (t3 1) 2 Fl (ill) + -2-r ...... 4 ___ 3 _______ 2 -----1 -)---=21 F 1 (ill) ill L ill + Arf" + AfP + + 4( t3 = 0 (A,1) where Al = -( a 2)( a 1) 2 2( 3a 2)(2 t3) A2 = -4a + 2 18a 9 + 12t3 2 A3 = -4a + 18at3 2t3 A4 = -2(~ 1)(2~ + 6a 1) A5 = 2(3 ~) A6 = ~ 2 10~ + 13 = 4(~ 1)(~ 3), If (A,1) is WTitten in the form it may be seen that Q(ill) has a second-order pole at ill = O Then in the use of the Frobenius technique, the point ill= 0 is a regular singular point of the differential equation, and Q(ill) may be written as a Laurent expansion about ill = O. This expansion has the form

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44 Q(m) "(JI + o(~I + 0(1) + O(m) + where the first term, (1/4)(1/m 2 ), is the most significant term. The Frobenius technique assumes a series expansion of F(m), of the form (A.2) where here k is determined solely by the first term of the expansion for Q(m), In this case, the first derivative term in the differential equation is missing; therefore, for a differential equation in its reduced form, the indicial equation for k becomes (A.3) where Q_ 2 is the coefficient of the l/m 2 term in the expansion for Q(m). This result is a proper one because a study of equations (3.13) and (A.2) shows that k = 1/2 is the only value that will yield a finite, non zero value of the temperature along the jet axis. Assumption (A.2) may be used in equation (A.1) yielding a set of complicated infinite series. Putting the coefficient of each power of m equal to zero yields the fol lowing: for 0 C (~ 1)(-1 + 1) 0 C unspecified m: = .. 0 0 (A.4) for m : Cl -[~(-1/4) + A4] 5~ + 12cr 5 = = C 4(~ 1) 2 4(~ 1) 0 (A.5) for 2 c2 -h ( A3] (cl) i~ + A4 m : c= 0 16(~ 1) 2 co 16(~ 1)2 (A.6)

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for w 3 : c3 -1 {iA + A + ccol[3A6 + A3] co 36(~ 1)2 5 2 4 (A.7) And for other Ci+ 4 /c 0 when i O Ci+4 -l ] 2 r~i f( i + l/2)(i 1/2) == [<21 + 9)(21 + 1) + 1 <~ 1) to l +~ (A.8) The result is that (A.4) through (A.8) leave the unspecified constant C 0 as a factor of the entire series and the ratios ci/C 0 determined in terms of the Reynolds number and the Prandtl number o, These ratios, ci/C 0 may be written in terms of and o if desired. Finally, F 1 (m) may be written -] (A.9) This same procedure may be carried out for the singularity at D == -1. In this case let D == A 1. Now (3.24) becomes (~ + 4(~ + == 0 (A.10)

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46 where Bl = -( o 1)( o 2) 2 2)(2 + t3) B2 = 4o 2(3o 2 9 12t3 + l8ot3 2t3 2 B3 = -4o + 180 B4 = 2(t3 + 1)(2t3 60 + 1) B5 = -2(t3 + 3) B6 = t3 2 + 10t3 + 13 B7 = -4(t3 + 3)(t3 + 1). A g a i n, le t 00 Fl(l\) = L r/,i+k l.=O (A.11) where a g ain k = 1/2 The result is very similar to that previously done. 2 2 The only differences are that in this case (t3 1) is replaced by (t3 + 1) and each A is replaced by B n n

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APPENDIX B GENERATION OF THE SECOND SOLUTION OF THE TRANSFORMED ENERGY EQUATION To obtain a second independent solution of (3.12), both (A.1) and (A,1O) must be solved for their alternate solutions. The solution of F(~) in Appendix A is performed by usin g the Frobenius technique, and now the second solut i on may be determined by a variation of the method previously used. Equation (A,2) specifies the Frobenius technique with the additional power being determined by the indicial equation. The result of applying the indicial equation in this case is spe c if i e d in expression (A,3); and it is important to note that the two roots of the indicial equation are equal, The technique to de t ermine a seco n d linearly independent solution depends upon the relation ship o f the two roots, and the special case of the roots being equal is the only case considered in this paper, Let the di f ferential equation (A,1O) be denoted by the operator L[F l ( t-.)] so that is the equation t o be solved, Substitution of (A,1O) into (A,11) yields (B,1) where the hi g her order terms are zero by the proper selection of each y.,i)O, i n term s of y. It is obvious from (B,1) that k = 1/2 is a i 0 double root of t he indicial equation, and the process of findin g F 1 (t-.)

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is to state at this time that k is 1/2. This process to specify F 1 (A), the first solution, may be written symbolically as But consider (B.1) to be differentiated with respect to k before selecting the value of k. Then at [F(A,k)1} = 2(k 1/2)r 0 Ak + 2 k ( k 1/ 2) r I\ lM = 0 0 Again, it can be seen that k = 1/2 yields a valid statement Since F 1 (A) satisfies (B.1), then the alternate solution may be determined from lhe expression 48 where the order of differentiation has been exchanged. This process may also be written symbolically as 0 Then the relationship between the two solutions is (B.2) Using (A.11) and (B,2) the second solution becomes (B,3) where r'.(k) represents the differentiation of the coefficients previously l determined for F 1 (A) with respect to k. It is interesting to note that

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49 the combination of (B,3) and (3.13) yields a function that behaves prop erly at the sin g ularity A= o. The precise form of the above equation is somewhat complicated but may be determined in a strai g htforward manner. Finally, the sec o nd solution may be written (B.4) where The result is that the unspecified constant y 0 remains as a factor of the entire function F 2 (A) and the various ratios (y./y) and (yi/y )' l O 0 are determined in terms of the Reynolds number and the Prandtl number a

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BIBLIOGRAPHY Birkhoff, G. and Zarantonello, H. 1957 Jets, Wakes, and Cavities. New York: Academic Press, Inc. Corrsin, S. 1943 Wartime Report 3123. Corrsin, S. and Uberoi, M. S. 1951 NACA TR-1O4O. Goldstein, S. 1965 Modern Develo~ments in Fluid Dynamics. New York: Dover Publications. Ince, E. L. 1956 Ordinary Differential Equations. New York: Dover Publications. Landau, L. D. 1944 "A New Solution of the Navier-Stokes Equations." Dokl. .Ak. Nauk S.S. S. R. 43, 286-288. Millsaps, K. T. and Soong, N. L. 1965 "Thermal Distributions in a Round Laminar Jet." 'l'he Physics of Fluids ._. Ruden, P. 1933 "Turbulente Aurbreitungsvorgange im Freistrahl." Die Natu.rwissenshafter. 21, 375-378. Schlichting, H. 1955 Boundary Layer Theory, 4th edition. New York: McGraw Hill Book Company. Slezkin, N. A. 1934 "On an Exact Solution of the Equations of Viscous Flow." Uch. Zap. MGU Sci. Rec. 2. Soong, N. L. 1968 'l'he Hydrothermodynamics of a Round Laminar Jet. Ph.D. Dissertation, Dept. of' Aerospace Engineering, University of Florida.

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51 Squire, H.B. 1951 "The Round Laminar Jet." Quart. J. Mech.!, 321-329. Yatseyev, v. I. 1950 "On a Class of Exact Solutions of the Equations of Motion of a Viscous Fluid." NACA TM-1349. Yih, c. S. 1950 "Temperature Distribution in a Steady Laminar, Preheated Air Jet." J. ApP Mech. KI., 381-382.

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BIOGRAPHICAL SKETCH Donald Arthur Dietrich was born on October 17, 1943, in Pittsburgh, Pennsylvan i a. In June, 1961, he was graduated from Seacre s t High S c hool in Delray Beach, Florida. He received the degree of Bachelor of Science in Aerospace En g ineering with high honors in December, 1965. In 1966, he enrolled in the Graduate School of the University of Florida. For his graduate work, he was granted a National Aeronautics and Space Administration traineeship. In October, 1969, he joined the staff at the NASA Lewis Research Center in Cleveland, Ohio. Donald Arthur Dietrich is married to the former Janet Ruth Love. He is a member of Tau Beta Pi and Sigma Tau.

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This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been ap proved by all members of that committee. It was submitted to the Dean of the College of Engineering and to the Graduate Council, and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy. June, 1970 Dean, Graduate School Supervisory Committee: