crane j m et al_2009_laminar jet discharged into a dead end tube

8/4/2019 Crane J M Et Al_2009_Laminar Jet Discharged Into a Dead End Tube

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American Institute of Aeronautics and Astronautics1

Approximate Solution of a Laminar Jet Discharged into aDead End Tube

John M. Crane, * Yedidia Neumeier, Prasad Bhave, and Ben T. Zinn

Georgia Institute of Technology, Atlanta, Georgia, 30332

An approximate analytical solution of a laminar jet discharged into an infinitely longcylindrical cavity with a dead end is developed. The employed method assumed velocityprofiles whose parameters were determined by solving the integral forms of the continuity,momentum, and energy equations. A novel feature of the method is the use of calculus of variations to obtain a minimum dissipation solution. The solution predicted a fewfundamental characteristics of such a jet. First, the jet expands asymptotically to a finitewidth, which is 52% of the cavity width and is independent of the initial jet radius. Second,after the jet is fully expanded, the centerline velocity decays nearly linearly and reachesstagnation. Thus, the model predicted that in a realistic cavity with a finite length far largerthan the theoretical jet penetration, the flow stagnates upstream of the dead end and does

not turn around at the closed end. Significantly, the model predicted that with properscaling, the behavior of the jet for various radii and jet velocities can be collapsed to a singlecharacteristic. The model predictions were compared to CFD simulations and experimentalmeasurements and were found to be in good agreement.

Nomenclature A = System matrix B = System vector

, ,a b c = CoefficientsF = Radial shape function f = Ratio of forward and backward velocity

J = Momentum function L = Cavity length L = Jet penetration length

n = Exponent p = PressureRe = Reynolds number R = Radiusu = X component of the velocityv = Y component of the velocity

Greek = Radius ratio

= Dissipation function = Normalized radial distance = Jet width

= Viscosity

= Density= Kinematic viscosity= Dissipation functional

Scripts

[ ]

$

= Normalized variable[ ]asy = Asymptote

[ ] B = Reverse direction[ ]F = Forward direction[ ]i = Related to the inner tube[ ]ii = Normalized by inner tube radius[ ]io = Normalized by inner and outer tube radii[ ]max = Maximum[ ]o = Related to the outer tube

[ ]oo = Normalized by outer tube radius

* Graduate Research Assistant, School of Aerospace Engineering, AIAA Member Adjunct Professor, School of Aerospace Engineering, AIAA Member Graduate Research Assistant, School of Aerospace Engineering, AIAA Member David S. Lewis Jr. Chair and Regents Professor, School of Aerospace Engineering, AIAA Fellow

47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition5 - 8 January 2009, Orlando, Florida

AIAA 2009-387

Copyright 2009 by J. M. Crane, Y. Neumeier, and B. T. Zinn. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


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I. Introductionhis paper describes an approximate solution for a laminar jet discharged into a long tube. ** The motivationbehind this type of flow is the recently developed Stagnation Point Reverse Flow (SPRF) combustor 1 whose

basic features are shown in Fig. 1. In contrast to state of the art combustors where the reactants and products enterand leave the combustor through opposite ends, the reactants and products enter and leave the SPRF combustorthrough the same plane opposite a closed end. Specifically, the fuel and air are supplied to the SPRF combustor

through concentric tubes located at the center of the open end. The flame produces a stream of hot combustionproducts that move toward the closed end of the combustor where they are decelerated (in the stagnation zone) tonear zero velocity and forced to turn around and flow out of the combustor through an annular reverse flow stream.Thus, the combustor contains counter-flowing streams of outgoing products and incoming reactants that come intocontact in a shear layer where the hot products and radicals mix with the reactants (see the left schematic in Fig. 1).The combustion process in this combustor depends heavily upon the interaction of these two counter-flowingstreams. Therefore, basic understanding of the flow of a jet into a dead end tube is an essential step towardunderstanding the combustion process in the SPRF combustor.

The problem of a jet discharged into an axisymmetric dead-end channel was discussed by Abramovich, 2 whoconsidered an incompressible turbulent jet discharged into a finite tube. Abramovich assumed that the turn of the jetoccured at the dead end where the fluid was treated as ideal. Amano 3 performed a numerical (CFD) study of

turbulent axisymmetric jets flowing into closed tubes. Most of his results were obtained for relatively short tubeshaving length to injector diameter ratios between two and five. However, a case with a much longer tube was alsopresented with a length to injector diameter ratio of 52 and tube to injector diameter ratio of 5.3. For this case, it wasnoted that the centerline velocity decayed to zero near the midpoint of the tube. He concluded that the effect of theinlet flow propagated only to a certain length for a very long closed tube. This finding suggests that with a longenough tube, the flow will eventually stagnate irrelevant of the existence of a dead end. Eckmann et. al. 4 measuredthe flow of a laminar jet into a dead end tube in the context of the flow in an artificial lung. 5 They investigated jetpenetration over a range of Reynolds numbers from 50 to 400 and inlet to exit tube diameter ratios of 0.1 to 0.3

** The meaning of a long tube will be discussed in the problem formulation.

T

CombustorWall

Fuel & airinjector

Reactionzone

Reactionzone

Shear LayerAir/ProductsMixing

Hot

Products

Air Fuel

Shear LayerAir/FuelMixing

-

CombustorWall

Flow Features of the

Upper Region of Combustor

Figure 1. A schematic of the SPRF combustor (center) with the flow features of the upper region of thecombustor (left). Images of the combustor burning natural gas in both premixed and non-premixed modes of operation (right).

Premixed Non-Premixed


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using flow visualization and particle image velocimetry. Due to the similarity of the flow characteristics andgeometry, their measurements were used to validate the predictions of the laminar jet model described herein.

II. Theory

A. Problem Formulation

To put the fluid mechanics of the discussed problem in context, consider the three limiting cases of a jetdischarged into a long tube shown in Fig. 2. The left case shows the flow features in a closed tube with all viscousforces null. In this case, the jet penetrates the full distance of the tube, turning around near the closed end in apotential flow. The center case shows the flow features in a long open tube taking viscous forces in the fluid intoaccount while neglecting shear forces at the wall. In this case, the flow would emerge at the far end with a fullydeveloped velocity profile. In the absence of wall shear forces, the cross sectional integral of the momentum at theinlet and outlet is equal. The right case takes both fluid viscous forces and shear forces at the wall into account. Asthe incoming flow proceeds into the tube, its momentum is dissipated by the wall forces. If the tube is just longenough, the shear forces at the wall dissipate the momentum of the incoming jet, diminishing the mass flux throughthe cross sectional area. For such a long tube, the flow must reverse itself and exit near the injector. Solving thiscase is the focus of this study.

Figure 2. Three limiting cases of a jet discharged into a long tube: inviscid potential flow (left), viscous flowwith no wall shear (center), and viscous flow with wall shear (right).

Consider now the flow profile whose axial velocity component u is shown in Fig. 3. The boundary conditionsfor the axial and radial velocities, u and v , respectively, are

0

0r

ur =

= (1)

00

or r Rv v

= == = (2)

0r

u == (3)

0or R

u=

= . (4)


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It is important to note that there are no end boundary conditions sincethe end is some arbitrarily long distance from the inlet, and in fact, aspreviously discussed, may even be open.

To simplify the problem, low Mach number (thus constant density)flow is assumed. Further, we assume that the pressure is radially (butnot axially) uniform. Applying continuity and axial momentumconservation in cylindrical coordinates gives

1 ( )0

u rv x r r

+ =

(5)

1 1u u u pu v r

x r r r r x

+ =

(6)

where is the kinematic viscosity.Since all flow enters and exits through the same plane, the net flow

rate through any cross section is null. The continuity integral equationis thus

0( ) 0

o R

u r r dr = . (7)

Integrating Eq. (6) in the radial direction and using the continuity Eq.(5) yields

22

0

12

O

o

Ro

or R

Ru dpu rdr R

x r dx =

=

. (8)

It should be noted that the integral of the radial momentum equationwith uniform radial pressure distribution would provide no new information, thus is not used.

Next, the energy equation is derived. A detailed derivation of the equation is given in Appendix A with the main

steps outlined below. First, both sides of Eq. (6) are multiplied by the product ur and, using Eq. (5), it ismanipulated to

( ) ( )2

3 21 1 12 2

u u pru rvu r ru ur

x r r r r x

+ = +

. (9)

Integrating both sides of Eq. (9) with respect to r gives

23

0 0

12

O o R R uu rdr rdr

x r

= . (10)

Note that since the axial derivative of the pressure is assumed radially uniform, the pressure term disappears after

integrating Eq. (9), since the integral of the other part of that term, ur , is null from continuity. Also absent is thecontribution of the wall shear stress force. Both of these terms are accounted for in the integral momentum Eq. (8).

Since the integral energy Eq. (10) is derived from the continuity and momentum equations, the use of all threeEqs. (7), (8), and (10), is apparently superfluous. To correct this impression, consider the following. Assume that theflow field is divided into two regions containing the forward and reverse flow. Solving such a problem wouldrequire applying the integral form of the continuity and momentum equations to each domain, providing fourequations. Alternatively, the problem could be formulated with continuity and momentum in one domain andcontinuity and energy integrated over the entire cross section resulting in the same number of equations (four). Thisarrangement takes advantage of the null flow rate over a cross section and the eliminated pressure term in the energyequation. It is thus evident that a proper solution may use a combination of the different equations without over

R i

r

Ro

x

Figure 3. Schematic of the flow and thegeometry.


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defining the problem. As a side note, while this problem is nicely divided into two domains, one could divide aproblem into any number of domains and apply the governing equations to each. This would provide for an evenlarger number of possible combinations of the governing equations.

B. Solution approachAs mentioned above, the flow field is divided into two regions containing the incoming (forward) and outgoing

(backward) flow. The velocity distribution can be described as a combination of two profiles

( ) ( ) :

( ) ( ) :

F F

approx

B B oo

r u x F r

ur

u x F R r R

>(11)

where ( )F u x and ( ) Bu x are cross-sectional velocity magnitudes, is normalized radial distance, F F and BF areradial functions yet to be defined, and ( ) x is the jet spread, as shown in Figure 3. The boundary conditions givenin Eqs. (1)-(4) impose

0

0F dF d =

= (12)

1 10F BF F = == = . (13)

The following non-dimensional quantities are introduced

$

o R

(14)

0 F x

uu u

=(15)

0Re F o xou R

= (16)

0Re F i xiu R

= (17)

i

o

R R

(18)

$

Reoo

o o

x x R

(19)

$ $

$

Re Reoo

oi io

o i i o

x x x x x

R R = = = (20)


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$

$

2Reoo

ii

i i

x x x

R = (21)

0

F x

p p

u =

(22)

0

x

=

. (23)

Note the three distance scales, Eqs. (19)-(21), that arise from the possible combination of the radii andcorresponding Reynolds numbers. Later, the merits of each length scale will be discussed. Substituting Eq. (11) intoEqs. (7), (8), and (10) along with the normalized quantities, Eqs. (14)-(23), the following three equations areobtained.

The continuity equation transforms to$ $ $ $

( )$ $ $

( )22

1 2 31 1 0F B Ba u a u a u + + = , (24)

the energy equation becomes$ $ $ $

( )

$ $ $

( )

$ $ $

$

$

23 2 3 3 2 2 2

1 2 3 4 5 6

11 1

2 1F B B F B B

oo

d b u b u b u b u b u b u

dx

+ + = + +

, (25)

and the momentum equation converts to

$ $ $ $

( )$ $ $

( ) $( )$22 2 2 2

1 2 31

1 11 1

21momentumflux

BF B B B

oo oo

dF d d pc u c u c u u

dx d dx

J

=

+ + = 1444444442444444443

(26)

where the coefficients are given by

1 1 1

1 2 30 0 0

, ,F B Ba F d a F d a F d = = = (27)

1 1 13 3 3

1 2 30 0 0

, ,F B Bb F d b F d b F d = = = (28)

2 2 21 1 1

4 5 60 0 0

, ,F B BdF dF dF

b d b d b d d d d

= = =

(29)1 1 1

2 2 21 2 3

0 0 0

, ,F B Bc F d c F d c F d = = = . (30)

These coefficients express the various moments of the spatial functions, F F and BF . Equations (24) through (26)can be presented in compact form

$

$

$

[ ] [ ]

F

B

oo

ud

A u Bdx

= ! "

(31)

where the elements of the A matrix and B vector are given in the table below.


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Matrix Elements Vector Elements$ 2

11 1 A a = 10 B =

$

( )$ $

( )2

12 2 31 1 A a a = +$ $ $

$

$

2 2 2

2 4 5 61

F B B B b u b u b u

= + +

$ $ $ $

( )$ $

( )13 1 2 32 2 1 1 2F B B A a u a u a u = +$

$3

1

1 21

B B

oo

dF u dp B

d dx ==

$ $2 2

21 1

32

F A b u =

$ $

( )$

( )$

( )222 2 33 1 12 B A u b b = +$ $ $ $

( )$ $

( )3 3 323 1 2 311 1 22F B B A b u b u b u = +$ $ 2

31 12 F A c u =$ $

( )

$ $ $

( )

2

32 2 32 1 2 1 B B A c u c u = +$ $ $

( )$ $

( )2 233 1 2 32 2 1 1 2F B B A c u c u c u = +

These three equations (Eqs. (24)-(26)) contain four unknowns: the two velocity amplitudes, F u and Bu , the jet

spread, , and the pressure, p . This imbalance between the number of equations and unknowns comes not fromthe way the equations were chosen, but rather from the infinite cavity length. Consequently, the equations developedare not subjected to a boundary condition at the closed end. Instead, an additional constraint is required for closureof the problem.

To provide such closure, an additional equation is introduced by seeking the velocity flow field that minimizesdissipation. It is well known that many problems in physics can be solved using minimums on, for example, traveldistance (light propagation) and energy (free hanging chain). For the interested reader, examples of the applicationof the minimum principle in physics is given by Thornton, 6 and a detailed discussion of the fundamental theory of the calculus of variations is given by Gelfand and Fomin. 7 As a relevant example, the parabolic velocity profile of fully developed 2-D laminar flow through a circular pipe (i.e. Hagen-Poiseuille flow) can be shown to be aminimum dissipation solution as shown in Appendix B.

To see how minimum dissipation applies to our case, consider a very small radius ratio such that the return flowarea is large compared to the jet area. The resulting low velocity reverse flow will have negligible effect on thedischarging jet. Thus, initially, the jet should expand like a free jet with constant momentum, i.e. the RHS of Eq.(26), 3 0 B = . While the momentum is constant under these conditions, the kinetic energy of the jet drops

monotonically ( 2 x# ) due to the non-zero dissipation. In the free jet case, this process continues ad infinitumproviding for unlimited expansion. In a confined jet, however, the tube constricts the expansion of the jet. Thesolution attempts to calculate the value of maximum jet spread using a minimum dissipation approach as describedbelow.

First, we introduce the dissipation function, , which is defined by Schlichting, 8 (see Chap. XII Sec. a) as

2

0

o R urdr

r

. (32)

It worth mentioning that the dissipation constitutes the RHS of the energy Eq. (10). Next, using the continuity Eq.(24), a relationship between the forward and backward centerline velocities is formed

$ $

F B Qu u f = (33)


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where$

( )$ $

( ) ( )2

22 3 1

1 1Q f a a a +

. Next, from the energy and momentum Eqs. (25) and (26),

respectively, the dissipation, , and momentum, J , are

$ $ $

$

$

2 2 2

4 5 6 1F B Bb u b u b u

= + + (34)$ $ $ $

( )$ $ $

( )22 2 2 2

1 2 31 1F B B J c u c u c u = + + . (35)

Finally, using Eq. (33)-(35), the ratio of the dissipation to momentum is expressed by the functional,$

$

$ $

( )$ $

( )

24 5 6

2221 2 3

1

1 1

Q

Q

b f b b

J c f c c

+ += =

+ +. (36)

The objective is to find a bounding asymptote for the jet spread,$

asy = that minimizes this functional, .

Assuming such a

$

1asy < is found and substituted into the energy equation (25), and using Eq. (33), one arrives at aconstant gradient of the velocity amplitude when

$ $

asy = ,

$

$ $

( )$ $

( )$

$

223

1 2 3

2

4 5 6

1 12 1

3

1

asy asy asy asyQ asyF

oo asyQasy asyQ asy

asy

b f b bdudx f

b f b b

+ + =

+ +

. (37)

It is important to note that the constant gradient of the velocity amplitude implies that the axial centerline velocity

decays linearly along the centerline once$ $

asy = . Consequently, at a certain distance downstream, after$

reaches

the value of $

asy , the flow stagnates.

When$ $

asy = , the LHS of the momentum Eq. (26) can also be calculated using Eqs. (37) and (33) yielding

$

$

( )$ $

( )$ $

( )$ $

( )$ $

( )

222 24 5 6 1 2 3

2231 2 3

1 114

3 1 1

asyasy asy asy asyQ Q

asyF

oo Qasy asy asy asy asyasy Q

b f b b c f c cudJ

dx f b f b b

+ + + +

=+ +

. (38)

Equations (37) and (38) describe the flow field in the asymptotic region. The region between the jet discharge andthe asymptotic region is solved next.

As previously discussed, if the radius ratio, , is much less than 1, the reverse flow near the discharge zone isnegligible and the jet will spread as a free jet in the near field with a zero momentum gradient with an initially linear

jet spread. The jet will continue to expand until it approaches the asymptote. In contrast, if the radius ratio$

asy = ,then the jet boundary should progress along the asymptote from the outset, implying a non-zero momentum gradientright from the inlet.

A solution that evolves gradually and captures the two scenarios described above is proposed

( )( )$ $

( )$

asy

n

oooo

asyoo oo

xdJ dJ x

dx dx

=

=

(39)


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where the positive exponent n is to be determined. It is important to confirm that the proposed solution adequatelydescribes the two scenarios. At the discharge, 0oo x = , the jet width equals the injector radius, thus the jet spread

equals the radius ratio, = . If 1


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III. Results and DiscussionThe solution employs spatial functions whose boundary conditions were defined (Eqs. (12) and (13)) without

specifying their form. First, a specific form for these functions should be chosen. Since their selection is arbitrary, itshould be demonstrated that the solution is not sensitive to the specifics of the shape functions. To this end, flowfield solutions using two different sets of shape functions will be compared. The two shape functions, named Shape1 and Shape 2 (for later reference), are given in Eqs. (42) and (43), respectively.

cos2

1cos

2

F

B

r F r

r F R r

R

%

%

= <

= + > >

(42)

( )

( )( )( )

sin

sin 1

1

F

B

r F r

r F R r

R

%

%

%

%

= >

+

(43)

The corresponding coefficients (moments) calculated by Eqs. (27)-(30) for these functions are given in Table 1. Although the coefficients of the two profile functions differ, sometimes significantly, the resultsobtained with these two velocity function were nearly identical, as will beshown.

First, the minimum dissipation principle is used to determine the asymptote of the jet spread, asy . Figure 4

shows the variation of the dissipation functional with$

for the two velocity profile functions, Eqs. (42) and (43).It is seen that the two profiles show nearly identical behavior for , suggesting solution independence to the chosen

shape function. The figure indicates that minimum

dissipation is achieved at$

0.52 = . To keep thepaper within reasonable length, further

investigations will use the Shape 2 profile (Eq. (43))save a final comparison of the jet penetrationobtained with both shape functions.

As mentioned previously, the exponent, n, in Eq.(39) is a free parameter, thus, it should beestablished that n can assume a large range of valueswithout significantly affecting the salient features of the solution. Figures 5 and 6 show the jet spread andcenterline velocity distribution, respectively, alongthe tube for various values of n. Figure 5 shows thatfor 0,1n = , the jet spread approaches two differentasymptotes; these asymptotes, however, are not theone that minimizes the dissipation. For the other

cases ( 1n > ), though, the solutions predict a spreadwith the same asymptote that is proven to minimizethe dissipation. Interestingly, for very large valuesof n, the model predicts a free jet expansion along astraight line that sharply transitions to the

asymptote. Figure 6 shows that the centerline velocity distribution is almost identical for all values of n, specificallypredicting that velocity will decay to zero at 3 8ii x & . This justifies the use of the ad hoc exponential law in Eq.(39). For further studies presented in this paper, 2n = was used.

Shape 1 Shape 2a1 0.231335 0.202642a2 -0.31831 -0.06456a3 -0.63662 -0.13808b1 0.109192 0.088275b2 -0.21221 -0.00195b3 -0.42441 -0.00434b4 0.868084 0.719326b5 2.467401 0.094307b6 4.934802 0.243795c1 0.148679 0.123493c2 0.25 0.010724c3 0.5 0.023558

Table 1. The coefficients defined inEqs. (27)-(30) for the two shapefunctions, Eqs. (42) and (43).

25

35

45

0 0.2 0.4 0.6 0.8 1

Figure 4. Dependence of the dissipation functional, ,upon the jet spread of the flow,

$

, obtained for twovelocity shape functions; ' Shape 1, Eq. (42) and Shape2, Eq. (43).


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Figure 5. Variation of normalized jet spread,$

, with normalized axial distance ii x for radius ratio 0.2 =

and various exponent values; n = 0 , n = 1 , n = 2 , n = 10 , n = 100 .

Figure 6. Variation of normalized centerline velocity, F u , with normalized axial distance, ii x , for a radiusratio 0.2 = and various exponent values; n = 0 , n = 1 , n = 2 ,n = 10 ,n = 100 .

At this point, the model can be used to predict the effects of geometric and flow parameters on the flow field.Figures 7 and 8 show the jet expansion and centerline velocity distribution for various radius ratios, , as a function

of axial distance normalized by the inner radius, ii x . Figure 7 shows that the jet spread angle increases withincreasing radius ratio. Assuming, for the sake of discussion, that the radius of the inner tube is unchanged and theouter tube radius is increased (radius ratio decreases), for small ratios the solution should approximate a free jetexpansion with a jet spread angle inversely proportional to the jet Reynolds number. 8 Thus, as 0 $ the jet spread,

, should converge to a single straight line (single spread angle). However, the non-dimensional jet spread, ,which is normalized by the outer radius (that increases for decreasing ), converges to straight lines with everdecreasing slopes, as shown in Fig. 7. Figure 8 shows that the centerline velocity decays faster with increasingradius ratio. Figures 7 and 8 together show that as the jet expansion nears the asymptote, the centerline velocity

0

0.5

1

0 0.2 0.4 0.6 0.8 1ii x

$

u

0

0.5

1

0 0.2 0.4 0.6 0.8 1

ii x

$


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decays linearly as discussed previously, see Eq. (37). Notably, Fig. 8 shows that as 0 $ , the velocity distributionapproaches the solution of free jet of the form 9 ( ) 1 1u x= + .

Figure 7. Variation of normalized jet spread$

with the normalized axial distance ii x for different radiusratios; = 0.05 , = 0.1 , = 0.2 , = 0.3 , = 0.4 , = 0.5 .

Figure 8. Variation of centerline velocity in the forward direction, F u with normalized axial distance ii x , fordifferent radius ratios; = 0.05 , = 0.1 , = 0.2 , = 0.3 , = 0.4 , = 0.5 .

0

0.5

1

0 0.5 1 1.5 2

$

F u

ii x

0

0.5

1

0 0.5 1 1.5 2

$

$

ii x


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Figures 9 and 10 show the same data as shown in Figs. 7 and 8 save for the axial scale, which is io x instead of ii x , see Eq. (20). Significantly, Fig. 9 shows that with such scaling, the jet approaches the asymptote at roughly the

same axial location of about 0.075 for all radius ratios. A similar trend is also apparent in Fig. 10, which shows thatthe stagnation point only changes from 0.055 to 0.08 when the radius ratio changes by a factor of 10 from 0.5 = to0.05, respectively. This is a most important finding since, thus scaled, all geometries collapse (approximately) to onesignificant distance, providing a very powerful measure of a laminar jet discharged into a long dead end tube.Validation of this finding will be the focus of the next section, where the model predictions will be compared toCFD simulations.

Figure 9. Variation of the normalized jet spread$

with normalized axial distance io x for different radiusratios; = 0.05 , = 0.1 , = 0.2 , = 0.3 , = 0.4 , = 0.5 .

Figure 10. Variation of normalized centerline velocity in the forward direction F u with normalized axial

distance io x , for different radius ratios; = 0.05 , = 0.1 , = 0.2 , = 0.3 , = 0.4, = 0.5 .

0

0.5

1

0 0.02 0.04 0.06 0.08 0.1

F u

ii x

0

0.5

1

0 0.02 0.04 0.06 0.08 0.1

$

$

io x


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Figure 11 shows the jet penetration as a function of radius ratio scaled with the three different length scales asdefined in Eqs. (19)-(21). It shows

1. the ii-scaled jet penetration varies from infinite as 0 $ (not shown, but inferred from a free jet) toapproximately 0.1 when asy = , and

2. the oo-scaled penetration varies from zero as 0 $ to approximately 0.028 when asy = .

Thus, the ratio of maximum to minimum penetration for both of these scales is infinite. In contrast, when using themixed scaling, io, the penetration distance is relatively constant varying only about plus or minus 15% around thevalue of 0.07. This once again illustrates the usefulness of the mixed scaling to represent the behavior of the jet.Particularly, a rough estimate of the jet penetration would be given as a single value (of 0.07) for any combination of geometries and inlet velocities.

Figure 11. Variation of three different normalized jet penetration lengths, , p oo L , , p io L , and , p ii L ,

normalized by Re o o R , Re o i R , and Re i i R , respectively, as a function of radius ratio, ; , p oo L , , p oi L , , ii L

IV. Model Validation

A. CFD SimulationsThe model predictions were compared with results from CFD simulations. The latter were performed using the

commercially available software package, FLUENT 6.2. The problem was simulated as a 2-D axisymmetric laminarflow, and results were obtained for two Reynolds numbers, Re 50, 100i = , and various jet to tube radius ratios,

0.05,0.1 0.6 = in steps of 0.1.Figure 12 shows the geometry of the problem studied. The inlet velocity, at the injector discharge, was assumed

to have a profile of fully developed laminar pipe flow, thus,

2

inlet max 21i

r u U

R

=

(44)

where U max is the centerline velocity and r is the radial distance measured from the axis. The other boundaryconditions were no-slip at the walls (i.e., zero velocity), zero gradients along the axis, and constant ambient pressure

0

0.02

0.04

0.06

0.08

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6

0

2

4

6

8

10

, p oo L , , p oi L , p ii L


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at the outlet. The inlet was offset from the outlet an arbitrary distance, H, to avoid flow reversal (i.e. vortices)crossing the imposed pressure boundary.

Figure 12. Geometry and boundary conditions used for CFD simulations.

B. Numerical Solution SchemeFLUENT solves the steady axi-symmetric continuity and Navier-Stokes equations using the finite volume

method with a segregated, implicit, steady-state, laminar, double-precision solver. For pressure-velocity coupling,the SIMPLE algorithm was used, and for discretization, a first order upwind scheme was used. The grid for the CFDstudy comprised a uniform 2D orthogonal mesh, see Fig. 13.

Figure 13. Grid for the radius ratio = 0.4 with grid spacing / , / 0.067i i x R r R' ' = .

First, simulations with various mesh spacings were performed to ascertain that the CFD results were not griddependent. An example of the results from three different mesh spacings are shown in Table 2, which lists jetpenetration and mass imbalance between the inlet and outlet for each case. It should be noted that the CFD-based jetpenetration differs from the model definition in that it measures the distance for the center line velocity to decay to5% of the inlet velocity; since, in reality, the velocitymay assume a finite very small (non-zero) valuewhere it is negligible. The data in suggest that theresults are almost independent of the grid employedfor the normalized grid spacings equal to or smallerthan 0.067 (second and third cases). The resultspresented in the following were obtained with anormalized grid spacing of 0.067.

C. CFD ResultsFigure 14 shows the radial distribution of the axial velocity at different cross sections as predicted by the CFD

simulation for a case whose parameters are given in Table 3.

Ro Ri

L

Wall

AxisVelocity inlet

Pressure outlet

rH

x

Grid Size( / , / i i x R r R' ' )

Jet Penetration( / i L R )

Mass Imbalance[ / kg s ]

0.1 8.696 4.86 x10 -12

0.067 8.428 2.83 x10 -12

0.05 8.5 2.20 x10 -12

Table 2. Grid independence study results.


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As discussed, the boundary condition at 0 x = imposes a parabolic velocityprofile in the region0 ir R< < , see Eq. (44). Figure 14 shows that the numericalscheme did not handle the flow in this region properly. Instead of having aparabolic profile that reaches zero at / 1ir R = , the profile does not reach zeroand, moreover, exhibits a discontinuity. Significantly, this discontinuity skewsthe profile, pushing the boundary between the forward and reverse flow outwardto around 0.4, instead of 0.2 as prescribed by the boundary condition. Note thatthe discontinuity is absent from subsequent profiles farther downstream.Consequently, it is expected that the jet spread, measured by the location of thetransition between the forward and reverse flow (i.e. the zero crossing) will beover-predicted by the CFD simulation.

Figure 14: Velocity profiles at different axial locations plotted versus normalized radial distance, / or R ,obtained from CFD simulations at various axial locations; 0 x = , 0.05 x = , 0.1 x = , 0.14 x = for aradius ratio of 0.2 = .

Figure 15 compares the model and CFD prediction of the jet spread and centerline velocity along the tube for thesame case shown in Fig. 14. The velocities in the figure are normalized by the discharge velocity maxU from Eq.(44). The distribution of the jet spread predicted by the CFD simulation clearly shows the abnormality discussedabove; the jet spread is grossly over-predicted at the inlet discharge, 0 x = , followed by an unrealistic narrowing atthe adjacent point. Remarkably, in spite of the discontinuity error in the CFD scheme, it still predicted theexistence of an asymptote at / 0.52or R = , which is identical to the value predicted by the minimum dissipationmethod. This confirms the fundamental hypothesis upon which the model was established.

Referring now to the centerline velocity in Fig. 15, the CFD prediction closely follows the general shapepredicted by the model. Significantly, the CFD corroborates the linear velocity decay into the stagnation zonepredicted by the model. It should be noted that while the axial dependence of the centerline and reverse velocitiespredicted by both the model and CFD simulation follow similar trends, the predicted jet penetration differs by about10%. It is quite likely, though, that this discrepancy is the result of the discontinuity error in the CFD simulation.

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.2 0.4 0.6 0.8 1 1.2

u

o

r R

Parameter Value

o R 0.07 m

i R 0.014 m

L 0.5 m

maxU 0.104 m/sec

Re i 50 0.2

Table 3. Simulationparameters for the datapresented in Figs. 14 and 15.


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Figure 15. Comparison between predictions by the theory and CFD results for the normalized centerline

forward velocity, F u , normalized centerline backward velocity, Bu and normalized jet spread$

as a

function of normalized axial distance ii x for radius ratio of 0.2;$

from CFD,$

from theory; F ufrom CFD, from theory; from CFD, from theory.

Figure 16. Variation of the normalized jet penetration, , Reoi p o i L L R, with radius ratio, ; Shape 1,

Eq. (42), Shape 2, Eq. (43), and CFD simulations.

, p oi L

0

0.02

0.04

0.06

0.08

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

$

F u ,$

Bu and$

ii x


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Figure 16 compares predictions of the jet penetration obtained by the CFD simulation and model as a function of the radius ratio. In this comparison, two model predictions are given corresponding to the two shape functionsdefined in Eqs. (42) and (43). Figure 16 shows that the use of the two different functions yielded identical jetpenetrations, further demonstrating the robustness of the model.

Figure 16 also shows that the jet penetration predicted by the CFD is in general lesser than that predicted by themodel, including the case where 0.2 = . This is at odds with the stagnation point location predicted by the CFD

shown in Fig. 15, which shows a longer penetration distance as compared with the model. This is because, asdiscussed earlier, the jet penetration calculated from the CFD results was determined by the location where thecenterline velocity reaches 5% of the inlet velocity rather than zero. Taking this into account, the discrepancybetween the model and CFD is even smaller than seen in Fig. 16.

D. Comparison with Experimental DataFigure 17 shows a comparison of the jet penetration predicted by the model with experimental results obtained

by Eckmann et. al. 4 It should be noted that the normalization procedure used to present the model prediction wasmodified to allow proper comparison with the measurements. Figure 17 shows that at low Reynolds number the datapoints closely follow the linear trend predicted by the theory. At higher Reynolds numbers, the measured dataindicate lesser penetration. This is expected as the ideal laminar nature of the flow breaks down with increasingReynolds number. Noticeably, the trend of the dependence of the penetration upon the radius ratio is confirmed bythe measured data in the entire regime of Reynolds numbers.

0

2

4

6

8

10

12

14

16

18

0 50 100 150 200 250 300 350 400

Figure 17. Comparison between experimental data 4 and present theoretical predictions of the variation of the jet penetration normalized by the outer tube diameter, / p o L D , with Reynolds number, Re o , for different

radius ratios; = 0.1, predicted , measured ( ; =0.2, predicted , measured ; =0.3, predicted, measured .

V. ConclusionAn analytical model that predicts the behavior of a jet discharged into a dead end tube was developed. At the

core of the model was the hypothesis that the jet expansion reaches an asymptote, and this asymptote could bededuced using a minimum dissipation approach similar to the principle of least action in mechanics. The model usedassumed velocity profiles whose parameters were determined by continuity, momentum and energy equations. Itwas shown that the laminar jet penetration, appropriately normalized, was approximately constant for all geometriesand inlet jet velocities. The predictions of the model were found to be in good agreement with both CFD simulationsand experimental measurements.

Re o

p

o

L

D


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Appendix A: Derivation of the Energy Equation, Eq. (10)Starting with the continuity and axial-momentum equations

1 ( )0

u rv

x r r

+ =

(A.1)

1 1u u u pu v r x r r r r x

+ = (A.2)

and multiplying both sides of the momentum equation (A.2) by ur , the left and right hand sides of the expressionbecome

( ) ( )2 2 2 21 12 2

u u u LHS ru ruv ru rvu u rv

x r x r r

= + = +

(A.3)

21 1 1u p u u p

RHS ur r ur r ur ur r r r x r r r x

= = +

. (A.4)

Also, from continuity,

( )rv ur

r x

=

, (A.5)

which, when substituted into Eq. (A.3), gives

( ) ( )3 21 12 2

LHS ru rvu x r

= +

(A.6)

and thus

( ) ( )2

3 21 1 1

2 2

u u pru rvu r ru ur

x r r r r x

+ = +

. (A.7)

Integrating both sides of Eq. (A.7) with respect to r between the limits r = 0 and r = R o and noting that the pressure(thus density) does not vary with r, the following expression is obtained:

( ) ( )2

3 2

00 0 00

1 1 12 2

ooo o o R R R R Ru u pru dr rvu r dr ru urdr

x r r x

+ = + . (A.8)

At 0r = , 0ur

= and 0v = , while at or R= , 0u = , thus ( )20

0

0o

o

R R u

rvu rur

= =

. Also, since the density is

constant at every cross section and the net mass flux through the cross section is zero, it follows that0

0o R

urdr =

. Eq.

(A.8) thereby reduces to

23

0 0

12

O o R R uu rdr rdr

x r

= (A.9)

which gives Eq. (10) used in the theory.


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Appendix B: Application of Minimum Dissipation to Hagen-Poiseuille FlowFully developed, steady, laminar flow in a round tube is considered, so called Hagen-Poiseuille flow. Solving the

steady-state 1-D momentum equation, Eq. (6), for fully developed flow readily provides the classical parabolicvelocity profile (see e.g., Schlichting 8 Ch. I Sec. d). It will be shown that the same profile can be obtained fromminimum dissipation considerations using the fundamental theorem from calculus of variations. 7

Let the velocity ( )u r satisfy boundary conditions at points 0r = and or R= . The velocity is subjected to a

constraint, K , of the form

0

( , , ')o R

K G r u u dr = (B.1)

where 'du

udr

. Consider further the functional, J , whose extremum is to be found, has the form

0

( , , ')o R

J F r u u dr = . (B.2)The theory of calculus of variations implies that a necessary condition for an extremum is the fulfillment of

' ' 0u u u ud d

F F G Gdr dr

+ = (B.3)

where uF

F u

; ' 'uF

F u

and is a constant, a so-called Lagrange multiplier.

In our case the boundary conditions are0

' 0r

u=

= from symmetry and 0or R

u=

= from no slip at the wall. The

constraint, K , for the problem is the flow rate, Q, thus,

0

o R

K urdr Q= = . (B.4)

The functional that represents the dissipation is 2

0

' R

J u rdr = , which constitutes the RHS of the energy Eq. (A.9),

and is defined in Eq. (32).Accordingly,2

'

'

'

0

2 '

0

u

u

u

u

F u r

G u r

F

F u r

G r

G

=

=

=

=

=

=

(B.5)

Substituting Eq. (B.5) into Eq. (B.3) gives

( )2 ' 0d u r r dr

+ = (B.6)

which can then be integrated to give a parabolic velocity profile,

2

1o

r u c

R

= . (B.7)

Eq. (B.7) satisfies the boundary conditions for any constant c. The specific value of c (and thus ) is determined bythe prescribed flow rate Q.


21/21


References1 Neumeier, Y., Kenny, J., Weksler, Y., Seitzman, J., Jagoda, J., and Zinn, B. T., Ultra Low Emissions Combustor with Non-

Premixed Reactants Injection, AIAA 2005-3775, 41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Tucson, AZ,2005.

2 Abramovich, G. N., The Theory of Turbulent Jets, translated by Scripta Technica with technical editing by L. H. Schindel,MIT press, Cambridge, MA, 1963, Chap. 10.

3 Amano, R. S., A Numerical Study of Turbulent Axisymmetric Jets Flowing Into Closed Tubes, ASME J. Energy Resour.Technol. , Vol. 108, No. 4, 1986, pp. 286-291.

4 Eckmann, D., M., Frerichs, T. A., Fogg, N. R. and Lueptow, R., M., Laminar Jet Flow into a Dead-End Tube, ASME FED , Vol. 237, 1996, pp. 667-672.

5 Eckmann, D. M., and Gavriely, N., Intra-airway CO 2 distribution during airway insufflation in ventilatory failure, J. Appl.Physiol. , Vol. 78, 1995, pp. 546-554 .

6 Thornton, M., Classical Dynamics , 4 th ed., Harcourt College Publisher, Fort Worth, TX, 1995, Section 7.2.7 Gelfand, I. M., and Fomin, S. V., Calculus of Variations , translated and edited by R. A. Silverman, Dover, New York, 2000.8 Schlichting, H., Boundary Layer Theory , 7 th ed. McGraw-Hill, New York, 1979.9 Andrade, E. N. da C., and Tsien, L. C., The velocity distribution in a liquid into liquid jet, Proc. Phys. Soc., Vol. 49, No.

4, 1937, pp. 381-391.

crane j m et al_2009_laminar jet discharged into a dead end tube

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