Experiments in momentumless turbulent wake of a jet-propelled axisymmetric body

EXPERIMENTS IN MOMENTUMLESS TURBULENT WAKE OF A JET-PROPELLED AXISYMMIETRIC BODY')

by A.I. Sirviente(2)

Department of Naval Architecture and Marine Engineering The University of Michigan, Ann Arbor, Ml 48109-2145

and V.C. Patel(3)

Department of Mechanical Engineering and Iowa Institute of Hydraulic Research,

The University of Iowa, Iowa City, IA 52242-1585 Abstract Laboratorium 'icor Scheepshydromechajca Ñchtef Meke!weg 2,2628 CD De!ft ,aL Ul-7äU _{Ft 017B133}

Experiments were performed in the turbulent boundazy layer and near-wake of an axisyrnmetric body propelled by a jet to study the evolution of the momentumless wake. Comparisons with measurements in the drag wake of the body (without the jet) and in the isolated jet provide an understanding of initial mixing between the two flows. Triple-sensor hot-wires and multi-tube pressure probes were used to measure the mean-velocity, turbulence and_{pressure fields} from the jet exit to a distance of over 15 jet diameters. It is found that the evolution of the wake takes place in three distinct stages; a zone close to the jet exit, about 4 jet diameters long, where the jet shear layer mixes with fluid from the wall region of the boundary layer; an intermediate region., about 12 jet diameters long, where there is mixing between the boundary layer and the jet up to the axis; and the third region where the two flows lose their identities to become a single

shear layer and the mean

flow acquires some of the characteristics of self-similar flows.

However, the momentumless wake does not conform to the assumptions and results of classical similarity analysis.

1. Introduction

The wake of a restrained axisymmetric body is characterized by deficit of momentumflux,

equal to the drag.

Momentumless wakes are generated by self-propelled bodies, the most common forms of propulsion being ajet or a propeller. Only a simple jet can provide streamwise momentum without swirl. This paper is concerned with this kind of momentumless wake. A previous paper1 described the turbulent drag wake (with momentum deficit) of the axisymmetric body used in these experiments, while a companion paper2 deals with the momentumless wake with swirl.

Cimbala and Park3 recently made a detailed study on two-dimensional momentumless wakes.

_{Measurements in axisymmetric, turbulent monientumless wakes were made by}

Ridjanovic,4 Wang5 and Naudascher,6 all of whom studied the flow behind _{a circular disk with a}

AllA journal, accepted pending revisions Assistant Professor, Member AIAA.

coaxial jet.

The Reynolds number based on the disk diameter was 5.5lO.

Naudascher6 concluded that in the initial regime an axisyrnrnetric momentumless wake is a free shear flow but far downstream it resembles decaying homogeneous turbulence. He also remarked on the number of scales required to define the self-preserving region of momentuniless wakes, compared to classical jets, (drag) wakes, or mixing layers which all are described by only one length and one velocity scale. Schetz and Jakubowski7 made measurements in the wake of an elongated jet-propelled body. The jet was rather large and annular in shape. More recently, the wake of a thin circular tube was experimentally investigated by Higuchi and Kubota8 for different initial momentum conditions, including one with zero net momentum. Hot-wire measurements of mean and fluctuating velocities were made up to 89 tube diameters.

Similarity analysis of a.xisymmetric, turbulent momentuniless wakes has been reported by a number of authors, but there is no consensus in the results. The predictions of similarity

analysis depend up on the turbulence model that

is adopted, and consequently, on the approximations that are made in the model equations. Finson9 used a second-order model and closed the system with an equation derived from the spectrum of turbulence kinetic energy, k. Hassid10 used a k-c model, while Ferry and Piquet" employed the k-equation and an eddy-viscosity model for the Reynolds stresses (, u2, y2) and the rate of energy dissipation, c. Recently, Sirviente'2 used a Reynolds-stress transport model, which includes a model c-equation, along with the experimental observation of negligible energy production in the transport equations. The results of the various analyses are summarized in Table 1, including those quoted by Tennekes and Lumley.13 In this table, I and U are the length and velocity scales, respectively, and the subscript m denotes scales of the other quantities, namely, their maximum values.

This paper is concerned with experiments in the momentumless wake of an axisymmetric body. The chosen elongated shape closely resembles practical self-propelled bodies. It differs from shapes used in previous studies with respect to the initial conditions of the wake. Here, the wake is generated by mixing between a well developed thick boundary layer on the body and a

co-axial jet issuing from its

tail. The present study documents the development of the momentumless wake from its origin in the mixing of these two distinct shear layers to its

evolution into a single shear layer. Understanding of this mixing is facilitated by reference to measurements in the drag wake of the body' and the isolated jet issuing from the body tail.

2. Design of Experiments

The axisymmetric body described by Sirviente and Pate!1 was also used for this study. The body has a length (L) of 143.45 cm, a maximum radius (R) of 6.95 cm, and a base diameter of 3.96 cm, allowing a coaxial jet of diameter, D = 3.90 cm, to be introduced from the tail. The boundary layer on the body was tripped by a 1.2 mm dìameter wire located at a distance of 9.5 cm from the nose. Air for the jet was supplied through a 1.24 cm diameter pipe along the body axis entering the body at the nose.

The experiments were conducted in the 1.07-m octagonal, open test-section, return-circuit wind tunnel of the Iowa Institute of Hydraulic Research. Figure 1 shows the wind tunnel and model arrangement along with the coordinate system used to report the data. The free-stream velocity U0 was set at 16.5 mIs, resulting in a Reynolds number based on body length (Re = U01Jv) of 1.58x 106, where y is the kinematic viscosity of air. The jet velocity was adjusted to obtain the self-propelled condition, i.e., such that the axial momentum of the jet was equal to the momentum loss due to the body drag. This condition was achieved with a maximum axial velocity at the jet exit of U = l.35U0.

As shown in Fig. I, the model was mounted with a part of it extending into the tunnel contraction. This enabled measurements in the axial direction up to x/D=19.53, where x is measured from the base and up to r/R=4.5 to recover the free-stream conditions in the radial direction. Measurements were taken across the diameter (as a check for symmetry) with a triple-sensor hot-wire probe and a five-hole Pitot probe. A detailed description of the experimental equipment, instrumentation, and measurement procedures can be found in Sirviente,12 along with an analysis of the uncertainty in the data. These are also summarized in Sirviente and Patel.1

3. Measurements in Cmponent Flows 3.1 Boundary Layer and Wake of the Bare Body

A detailed study of the body boundary layer and wake can be found in Sirviente and Paid1, hence only a very brief description will be included herein. Measurements with a single

hot-wire were made at eight stations over the stern, in the region -O.180<xJL. _{At x!L=_O.180, the}

usual boundary layer

integral parameters were: momentum-thickness Reynolds number, RU0O/v=845; shape parameter, Hl.29; and friction coefficient, C=O.0045. This station ¡s located in the main test section of the wind tunnel (Fig. 1) but the boundary layer is still recovering from the favorable pressure gradient of the tunnel contraction. These features should be noted when the data are used to test calculation methods and turbulence models. In the wake, measurements were made with different instruments at 18 stations. The region of reverse flow extended up to xfD=O.551. It is well known that the radius of an axisymmetric far wake _{grows as} x113 while the maximum velocity defect, (U0-U), where U is the centerlinevelocity, decreases as x-2'3. Both laws were found to apply at the last four stations, where xfD>14. The streamwise development of the drag wake was viewed in three parts, a region of initial narrowing of the wake, the region where the similarity laws apply, and an intermediate region between the two. Similarity theory predicts that maximum values of the Reynolds stresses (and the turbulence kinetic energy) should decay as x'3, while the lengTh scales associated with their profile should grow as x1'3. It was found that the length scales follow the expected asymptotic behavior rather quickly, in about the same distance as b, but the intensities decay much more slowly and do not attain the expected power-law behavior even at the downstream stations. This confirms the often made observation that similarity of the mean flow is attained much earlier than that of turbulence.

4.2 The Jet

Measurements similar to above were made in the jet issuing from the tail of the body without any ambient flow. With the jet axial momentum adjusted to equal the bare body drag, the jet Reynolds number Re (=UD/v) was 5.7 x 1O, with U=22.27mIs.

It is well known that the streamwise development of a jet may be described in three parts; a region up to about x/D=4, where the centerline velocity and the jet radius remain essentially constant, a region farther downstream in which the half-radius, b, increases linearly with Xand the centerline velocity, U, decreases as x, as predicted by similarity analysis, and an intermediate zone between the two. The three regions are evident in the data presented in Fig. 2. The first region is traditionally referred to as that in which the potential core, if one exists, is destroyed. In

the present case, the length of the pipe leading to the jet (Fig. 1) was such that the velocity profile exiting the jet had only a small radial region of constant velocity, signifying a potential core surrounded by a thick boundary layer. In 4<x/D<z1O, termed the intermediate region, the centerline velocity decreases and the jet radius increases, indicating increased mixing all across the jet. At the end of this region, the jet adjusts to its similarity state.

From Fig. 2 it is seen that the power laws predicted by similarity theory apply beyond approximately x!D=l0. The jet virtual origin., located by extrapolation of b to the axis, is at x(jD=-0.56, and the half angle ofjet spread is 5.88°. For the mean flow, the power laws obtained from Fig. 2 are: U=A(x-x0)-' and b=B(x-x0), with A=5.50 and B=4).105. These compare well with values of A and B quoted in previous works, for example, (5.75 and 0.10) by Reichardt (1942, see Ref. 14), (7.35 and 0.097)15, and (6.39 and 0.094).16

Principal characteristics of the turbulence measurements in the jet are also shown in Fig. 2. The peak levels of turbulence measured at the station closest to the jet exit are similar to those in the near-wall region of a pipe. The jet core is associated with a near constant value of the centerline turbulence kinetic energy, k. By x!D=6, in the intermediate region, the maximum turbulence kinetic energy, k occurs at the centerline, implying that the shear layer between the jet and the surrounding flow has grown to the center of the jet. The fact that the maximum values of k and follow the x-2 power, and the radial locations where they are half their maximum values (bk,2 and bç12, respectively) follow the x'1 power simply indicates that the flow is determined by one velocity and one length scale beyond about x!D=l0.

The rate of entrainment was found to be nearly constant all along the jet, although a constant rate is predicted only in the similarity region. The momentum balance shows that the turbulence term is small but makes an increasing contribution with downstream distance. The change of momentum flux due to pressure is negligible.

4. The Momentumless Wake 4.1 The Mean Flow

Figure 3 shows sketches of typical velocity profiles in the component flows, namely the drag wake and the jet, and the velocity profile in the combined flow, the momentumless wake. It

is clear that the latter is characterized by three velocity scales: U, U and U, and three length scales: b1, b and b2. The scale b1 corresponds to the radial location where the velocity is

(U4-U)/2, b2 to (U0-U/2, and b

to U. As the velocity outside the wake, Ue, differed from the constant free-stream velocity, U0, only in a short region close to the body, it is not considered further as a significant vaiable. Also, as the flow develops downstream, the differences in the mean velocity become small and determination of length scales from the mean-velocity profile becomes progressively inaccurate. Therefore, length scales based on turbulence profiles, defined

in section 4.2, are used.

Figure 4 shows velocity profiles at six representative streamwise stations, designated A-F, where the velocity is normalized by U0 and the streamwise distance by R and in some figures by the body length, L, to facilitate comparisons with the results of the drag (bare-body) wake. The velocity profiles clearly show the jet-like flow in the central portion of the near wake, surrounded by boundary-layer flow. The velocity excess in the jet and defect in the boundary layer interact to cause a rapid decay of the mean shear. The centerline velocity (Fig. 5) resembles that of the isolated jet (Fig. 2a). It remains constant up to x/D='4, and decreases thereafter and approaches the free-stream velocity, being 1.02U0 by the last measurement station. The minimum velocity, on the other hand, increases rapidly in the same initial region and more gradually thereafter to approach the free-stream velocity, being O.88U0 at the last station. Thus, by the last measurement station, the velocity variations across the entire wake are small. Comparisons with the maximum velocity defect in the drag wake and the centerline velocity of the isolated Jet indicate that the mean shear, and therefore the rate of turbulence production, are considerably smaller in the momenturnless wake than in the component flows. The half-radius b1 (Fig. 5) increases very slowly along the wake compared with that of the isolated jet (Fig. 2b). This is due to the confinement imposed by the boundary layer surrounding the jet. The length scale b2, which is a characteristic of the boundary layer, also increases.

Figure 5 also compares the measured evolution of the velocity and length scales with the similarity results of Sirviente12 (see Table 1). Clearly the individual velocities do not follow any of the predicted power laws. Rather surprisingly, the velocity difference (U-U)/U0, which is

more representative of the core of the wake, follows the -1.5 power law predicted by similarity theory. Of the two length scales, only the outer one, b2, appears to follow the predicted 0.25 power law. However, as already noted, determination of these lengths from experimental data in the far wake is subject to considerable uncertainty.

It is interesting to compare the growth of the drag wake with the momentumless wake. In the former, three zones were distinguished: an initial narrowing, followed by an intermediate region, and finally by a zone where the similarity laws applied. In the momentumless wake, the interaction between the jet and the wake constrains the jet from spreading radially outward, as shown by the development of b1, and destroys the original region where the wake width narrows, as shown by b2, leaving the wake width practically unchanged in the near field, x/i) <4. This

initial region, where the main characteristics of the jet are unaffected by the boundary layer outside it, is followed by an intermediate zone (4 <x/i) < 12) where the mixing between the jet and the boundary layer spreads to the wake centerline. Beyond x/i) = 12 lies the region where limited similarity is observed. Thus, the overall flow development is similar to that of the jet (Fig. 3).

Figure 6 shows the mass-flux deficit defmed as

/

U'r "r

j

1--i--d =m

_UO)R

_R)

and the three terms in the axial momentum integral equation, the convective, turbulence and pressure terms:

UU r

(r\

p r

(r

U0)U R

=0.

There is a very rapid decrease of the mass deficit in the near wake due to entrainment of the inviscid fluid by the boundary layer. Surprisingly, the mass flux deficit vanishes beyond x/D=9, indicating no entrainment in the developed region of the momentumless wake. By approximately x/D=9, all three terms in the momentum equation becQme negligible. In the near and intermediate regions, the contributions from mean-flow convection and pressure are nearly equal and opposite, with convection by turbulence remaining negligible throughout. All along the wake, the net flux of axial momentum deviates from zero by no more than 0.01.

4.2 Turbulence and Approach to Similarity

Distributions of the turbulence kinetic energy, k, and the only non-zero shear stress, _uy, are presented in Figs. 7 and 8, respectively. The characteristic intensity and length scales oftheir profiles are also shown in Fig. 9. These include the maximum values k _{UUm and uvm, and the}

radius b,2 where uu is one-half its

maximum value.

Figure 9 shows that the peak _{turbulence kinetic energy increases}

_{up to x/D = 2 and}

gradually decreases thereafter. The centerline value was found to remain unchanged up to x/D = 4. The latter results are similar to those in an isolated jet and suggest that the jet core is preserved for about 4 diameters. On the other hand, the earlier decrease of km indicates _{more rapid mixing} in the shear layer surrounding the jet. _{By approximately xJD = 12, km occurs at the centerline} indicating mixing of the shear layer across the wake. The normal stress (not shown) distributions were found to be similar to those of k, but the maximum of _{did not occur at the centerline until} the last measurement station. This underscores _{the difficulty of precisely defming the origin of} the momentumless wake as a single shear flow. The shear stress distributions of Fig. 9 clearly show the shear layer in the near wake, and sign changes expected from the changes in the _mean velocity gradients. The maximum value starts to decay from approximately the station where the turbulence kinetic energy is maximum. Although bk/2 would appear to be a good choice fora length scale, especially in view of the importance _{of k in turbulence models and in similarity}

analysis, previous studies have employed b,7

_{because its determination does not require}

measurement of all three normal stresses.

Figure 9 shows not only the various velocity and length_{scales of the turbulence profiles in} logarithmic plots, but also their comparisons with some of the power laws of Table 1. It is seen that all scales follow these power laws beyond x/i) = 12 somewhat better than the scales of mean-velocity profiles. _{Secondly, the data are in better agreement with the power laws deduced by} Sirviente12 than those of Ferry and Piquet.11 _{Production of turbulence was neglected in the} former analysis, while the latter was based on the assumption of equilibrium between production and dissipation. Comparison of the results for the momentuinless wake (Fig. 9) with those of the

drag wake1 and the jet (Fig. 2) clearly shows a more rapid decay of turbulence in the

momentumless wake due to the decrease in production.

Comparisons with the data of Naudascher6 and Higuchi and Kubota8 are also provided in Fig. 9. The differences in the near field are obviously related to the vastly different initial conditions in the experiments; Naudascher used a circular disk as the wake generator while Higuchi and Kubota used a thin-wall tube. Similarity theory requires the data to follow the same power laws in the far wake, however. While comparison of Figs. 9 with 6 shows that the intensity and length scales of turbulence show somewhat better correlation than those of mean velocity, there persist systematic differences between Naudscher's data and the other two sets. The fact that the present data are in agreement with those of Higuchi and Kubota in almost all respects, and both are in better agreement with the power laws predicted by similarity analysis, suggests that Naudascher's results continue to be influenced by initial conditions (and much longer distance may be required for them to achieve similarity).

The approach to similarity of the mean velocity and turbulence profiles at the most downstream stations is explored in Fig. 10. The radial distance is made dimensionless by b72, following previous studies in momentumless wakes. The velocity profiles are plotted twice, showing the distributions of

-Uj/(U-U) and (U-U)/(U0-U), while

and are normalized by their maximum values. The velocity profiles reveal the central difficulty of similarity analysis, which assumes a single velocity scale, whereas the results indicate that the inner and outer parts of the momentumless wake scale on different velocity scales, namely, (U-UJ and (U0-U), respectively. The remarkable collapse of the profiles separately in the two regions, matching at the location of minimum velocity, approximately at r=1.2b11, is perhaps

the most interesting and intriguing result from Fig.

10. Clearly, this would require a reconsideration of analytical studies based on similarity theory under the usual assumption of a single charaçteristic velocity scale. Note that both velocity scales) (U-U) and (Uo-Umn), tend to zero with axial distance and their experimental determination is subject to considerable error. The profiles of the normal stress in Fig. lOe appear to indicate a tendency toward similarity but this is somewhat deceptive. Greater insight is provided by the shear stress profiles of Fig. lOd.

There are two peaks in the shear stress, _{one in the inner region and another of opposite sign in the} outer region. Figure lOd uses the maximum value occurring in the inner region. Therefore, the collapse of the profile from the centerline to the point of _{zero stress (roughly coincident with} UJ simply confirms the similarity of the inner region noted from Fig. lOa.. _{When the profiles} were normalized by the peak value in the outer region, they collapsed in the outer region. Thus, these profiles also allude to the need for two velocity scales _{to describe the momentumless wake.}

5. Mixing Between the Jet and Boundary Layer

In the previous section attention was focused on major features of streamwise flow development. With this accomplished, it is useful to review some (due to space restrictions) of the data at a few selected stations to further investigate the mixing between the jet and the boundary layer. Figures 11 through 13 show the measurements at stations A, B, D and F. Each figure is in four parts, showing profiles of the mean-velocity _{components (U, V), turbulence} kinetic energy (k), and the shear stress . Also, each figure shows the measurements at the same

locations in the drag wake, the jet, and their combination, i.e., the _{momentumless wake. Ali} quantities are normalized by the constant scales U0 and R, although it is recognized _{that the} free-stream velocity has no significance in the isolated jet. Data obtained from a vertical traverse of the probes, on both sides of the flow axis, are plotted to show the level of_{symmetry achieved in} the experiments. With the exception of the pressure at all stations, which _{was measured with the} five-hole Pitot probe, and the velocity and turbulence data at station A in the drag wake_measured with the LDV, all data were obtained with the triple-sensor hot-wire probe.

For these axisymmetric flows, the tangential velocity (W) and two (uw and ) of the three shear stresses should be zero. Also, the radial velocity (V) and the remaining shear stress (i) should be antisymmetric across the flow. Not all of these conditions were precisely realized in the experiments. _{However, the observed departures from the expected conditions are, in} general, within the limits of experimental uncertainty, especially if probe interference and measuring-volume effects in the shear layers in the near field (stations A and B) are taken into account Lack of perfect symmetry of the flows also cannot be discounted. Therefore, the observed departures from the requirements of axial symmetly of the three flows should be viewed

as a measure of the overall reliability of the complete data base. With this in mind, no further discussion is necessary of the quantities thatshould be zero.

Station A (Fig. 11), at x!D = 0.38 from the body tail and the jet exit, lies in the separated

flow region of the bare-body wake and in

the core region of the isolated jet. A cursory

examination of Fig. i la would suggest that the momentumless wake at this station is a simple superposition of the two component flows, with measurements in the momentumless wake agreeing with those in the jet in the region nR <0.3, and with the pure wake in r/R > 0.3. This concept, while useful, hides differences that are not immediately apparent from Fig. lia. For example, the profile of the radial component of velocity (Fig. lib) and the pressure coefficient (not shown) would suggest that the effect of the jet in the core of the momentumless wake is felt

farther than that of the separated flow

region in the bare-body wake, implying different

entrainment and growth rates of the shear layeremanating from the lip of the jet orifice located at

nR

0.35. The pressure decrease in the bare-body wake in nR > 0.8 was attributed to thefact that the velocity at the edge of the wake, Ue,is less than U0. The pressure distribution across the momentumless wake suggested that the change in the flow near the base is felt deeper into the mviscid region, as is seen from the velocity profiles in Fig. lia.

The shear layers in the three flows are easily identified by the peaks in the turbulence quantities shown in Fig. lic, d. The shear layerof the isolated jet and that of the momentumless wake are, of course, considerably more intense than that of the bare-body wake, as isevident from the peak values of all turbulence quantities. Once again, the concept of superposition is useful in that the turbulence in the wake core resembles the flow in the jet while that in the outer part resembles the flow in the bare-body wake. The dividing line is around nR = 0.3.

The measurements at station B, at xiD = 2.28, are shown in Fig. 12. This station corresponds to the recovery from separation in thebare-body wake and the end of the core region of the isolated jet. The axial velocity profile shows a behavior similar to that at the previous station insofar as the inner and outer regions ofthe momentumless wake agree, respectively, with the jet and wake dat& However, the centerline velocity is increased in comparison with that of

turbulence data, which show that the jet-like region is considerably narrower, as a result of the confinement imposed by the surrounding boundary layer. This narrowing of the jet is apparent from the location of the minimum _{axial velocity (Fig. 12a) as well as the distributions of the} kinetic energy (Fig. 12c) and the shear stress (Fig. l2d). _{The maximum level of turbulence} kinetic energy measured in the momentumless wake is comparable to that found at station A, but much smaller than in the isolated jet. On_{the other hand, the centerline value is the same as that in} the jet, implying that the mixing between the boundary layer and the jet has _{not reached the} centerline. The maximum of the axial normal stress is significantly reduced in comparison_{to the} maximum values of the radial and tangential _{normal stresses, indicating different anisotropy} levels in the shear layers of the two flows. The considerably lower intensity of the shear layer of the bare-body wake is also apparent from the_{peak values of all turbulence quantities.}

Station D, at x!D = 9.12, lies in what was termed the transition or intermediate regions of the two component flows, downstream of the core region of the jet and the near-field of the bare-body wake. The data presented in Fig. 13 clearly show that the flow structure is vastly different from that at the previous two stations. The _{centerline velocity in the momentumless wake is} 1.2U0 compared to O.8U0 in the jet and O.56U0 in_{the drag wake. The mean shearacross the wake} is considerably smaller than in the component flows, indicating a lower level of turbulence production. _{By this station the static pressure variation makes negligible contribution to the} momentum balance. The radial velocity component is small in the _{three flows suggesting that} boundary-layer approximations apply equally well to all of them.

The turbulence profiles (Fig. 13c, d) show that the diameter _{of the momentumless wake is} comparable to that of the drag wake and much smaller than the isolated jet. Also, the magnitudes of the various turbulence quantities are intermediate to those in the two component flows, but closer to those in the jet. The turbulence kinetic energy continues to show a dip at the centerline while such dip is no longer seen in the jet data. Th.s suggests that the shear layer between the jet and the boundary layer in the momentumless wake is not fully merged at the centerline. In other words, this flow continues to show_{some of the features of the jet at the center.}

Measurements at the last station (station F at x!D = 19.53), (not shown), indicate greatly reduced mean shear compared to the drag wake and the jet at the same location. The shear stress was comparable with that in the drag wake but considerably smaller than in the jet. The turbulence kinetic energy showed single maxima at the centerline, and its magnitude is generally similar to those in the drag wake but smaller than in the jet. The mixing of the boundary layer and the jet was practically completed, with the diameter of the momentumless wake being comparable with that of the drag wake, and almost half of the isolated jet. Note that the drag wake as well as the jet indicated similarity properties at this measurement station while the momentumless wake showed limited similarity with respect to the shapes of the velocity and turbulence profiles.

Significant differences were found in quantities such as eddy viscosity, intermittency, triple-velocity correlations, and turbulence length scales at the last measuring station in the three flows. Limitations of space preclude a discussion of these aspects in the present paper.

7. Summary and Conclusions

This paper reports experiments in the momentumless wake of an axisymmetric body propelled by a concentric circular jet. The principal results and conclusions are summarized below.

I. These experiments provide a detailed set of data describing the mixing ofajet and a thick boundary layer, and their evolution toward a single flow, the axisymmetric momentumless wake. This is one of a very few experiments that relate to mixing of two turbulent shear layers with different velocity and length scales, and as such, should provide valuable insights in the development and validation of turbulence models. This is facilitated by corresponding data in the two component flows.

2. The mixing between the jet and the boundary layer takes place in three stages: in the near field, xiD<4, the two flows preserve their respective characteristics in the inner and outer regions, with a growing shear layer separating them; an intermediate region, where the shear layer penetrates and mixes up to the centerline; and the final zone of developed flow, xjD>12, where

the flow acquires some of the characteristics of a single canonical free shear layer - the

Comparisons with previous experiments in axisymmetric momentumless wakes show some similarities in the far wake in spite of significant differences in their _origins.

Several characteristic scales of the flow in the developed _{region exhibit power-law} behaviors predicted by classical similarity analysis. _{However, the velocity and turbulence profiles} clearly indicate the need for two separate velocity scales to describe the inner and outer regions, and quite possibly two length scales, although the evidence for _{the latter was not conclusive.} These results are important for two reasons. First, they suggest that the two component flows do not completely mix. In other words, the momentumless wake retains a memory of its origin far longer than the more common free shear layers. This tends to support the characterization of this flow, by some previous workers4'5'6, _{as decaying homogeneous turbulence. Second, the need for} two velocity scales casts serious doubt on classical similarity analysis which is based on a single velocity scale. It is possìble that this flow requires overlap type arguments used in equilibrium boundary layers, in which an inner layer is matched to an outer layer in a common region.

The mean shear in the developed-flow region of the momentumless wake

_is

considerably smaller than that in the drag wake and the isolated jet at comparable locations, indicating diminishing rates of turbulence energy production. _{Predictions of similarity theory} based on negligible production are in somewhat better agreement with experiments but the role of such theory in this flow remains questionable.

References

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14Schlichting, H., "Boundary Layer Theory," McGraw-Hill, New York, NY, 1968.

15Abramovich, G.N., "The Theory of Turbulent Jets," M.I.T. Press, Cambridge, MA, 1963.

16Hinze, J.O. and Zijnen, B.G., "Transfer of Heat and Matter in the Turbulent Mixing Zone of an Axially Symmetrical Jet," Journal of Applied Scientific Research, Vol. Al, 1949, _pp. 435-461.

Table 1. Power Laws for an Axisymmetric Momentumless Wake 16 U

()

km Sm (U2_V Termekes & Lumley5

0200

-0.800 Fcny& Piquet33 0.200 -0.800 -1.600 -1.600 -2.600 Hassid12 0.270 -1.270 -0.730 Finson11 0.276 -1.636 -2.364 -1.455 -1.636 Sirviente14 0.250 -1.500 -2.250 -1.500 -5.500 -1.500

Figure Captions

Figure 1. Wind Tunnel and Model Arrangement Figure 2. Jet Characteristics

Velocity and Turbulence Intensity Scales Length Scales

Figure 3. Typical Axial (U) Velocity Profile in a Momentumless Wake Component Flows

Combined Flow

Figure 4. Mean Axial Velocity Profiles Figure 5. Wake Velocity and Length Scales

Velocity Scales Length Scales

Figure 6. Mass Flux and Momentum Balance Mass Flux

Momentum Balance

Figure 7. Turbulence Kinetic Energy Profiles Figure 8. Shear Stress, uv/U, Profiles

Figure 9. Decay of Turbulence Intensity and Length Scales in Momentumless Wakes Turbulence Kinetic Energy

Shear Stress Length Scale

Figure lO. Similarity of Velocity and Turbulence Profiles

Figure 11. Measurements at Station A of Wake, Jet and Momentumless Wake. Wake (circles); Jet (triangles); Momentumless Wake (squares).

Axial Velocity Radial Velocity

Turbulence Kinetic Energy uv/U

Figure 12. Measurements at Station B of Wake, Jet and Momentumless Wake. Wake (circles); Jet (triangles); Momentumless Wake (squares).

Axial Velocity Radial Velocity

Kinetic Energy

iO0

101

102

io-3

100

10-i

A a

o

o

_{o 00000}

o

A o J t J- _¡ I

Slope -1

o

oQDO

AA

A

Slope -2

Slope -2

A A

io'

a) Velocity and Turbulence Intensity Scales

ô

bID

b112/D

Dbi /2uv

Slope i

AAAO

o

e

000°

D D

e

₀₀₀₀₀

o

J_0_i

100

_10'

x/D

b) Lgth ScaI

U

Combined Flow

Wake

Jet

Component Flows

C 8 X t-4

Q

Q

I

Q

Q

o

r i t

...»»»1':4

xi

D2

.280

w

4444444e 4dlIIIiJI!IIIfthIIIlII =

x/D=14

.345

t'i

o

x/D=0

.368

x/D=5

.517

DD000000

C)

x/D=9

.122

U

x/D=19.

531

nR

o

tsj

100

10'

100

lO_i

0

0 00000

A _A

AA

0dt

A A

°

_{Slope -1.5}

G o

b1/R

b2!?

Slope 0.25

00

o

0

0000

D D D

_DDDDO

-D

a) Velocity Scales

Figure 5. Wake Velocity and La*gth Scales

D

o

G A

(U-U) lu0

10'

100

101

x/D

b) Length Scales

101

_10°

lOi

x/D

0.0

m

X

-0.2

-0.4e

₅

₁₀

x/D

u

u

u

a

O O 00 00

0

I I I a) Mass Flux

x/D

b) Momaitum Ba1nce

15

20

X

s ,. I-4

III

%

ab&ç-

..

¿,

"A

I)

_VIIIIII

u _-I

i

_'R

_a.

b

RIf- J

'1

)Nȏ.

-

..ii ;"&'

t,

xj I I

nR

x/D=O .368

x/D=5 .517

00000 00 D

x/D=9. 122

x/D=14.345

x/D=19. 531

Lo

Q

Q

Q

'

«««/'

q ft u

s,.

(((((Wqf

O8U/X

44444444«««444;

_4IL44

_{<4C<«<444444444}

89o=a/x

$

1I

4 4 4

vvvvvvvv

TES 6Ta/X

OflE DO D OD

LIS

co

I I I I I L I I I t I I i I I

N

if

O

iI

vvvvvvv

N

I _I 1 C) CN

O

O

Present

O Naudascher'

Slope -1.6

O

O'

c) Length Scale

Figure 9 Decay of Turbulence Intensity and Length Scales in Momentumless Wakes

100

_10"

_{'o_4}

a

Present O Naudascher' A Higuchi and s Kubota U Slope -1.6 _O

a

O i Slope -2.25

0

A

0

A

O1

i

0 100 101

x/D

a) Turbulence Kinetic Energy

101 1°_1 100

10'

x/D

b) Shear Stress U Present _O Naudsacher'

Q

s-A Higuchi and Kubota' Slope 0.25 0 I

10

'JA

e

U

a

Slope 0.2

101

101

100

101

x/D

1.0

I:j

H

_0.5

0.Oo

r/b,2

x/D=12.36 _O xID=16.29 x/D=19.53

UO

, r

-0.5

1.0

-0.5

Figure 10. Similarity of Velocity and Turbulezìce Profilez

-L001

r/b2

E ri

i

O . O

'

, V V x/D=12 .36 x/D-14.34 _D x/D-16.29

_o

x/D=].9.35 _y x/D-1236 O x/D16.29 O x/D=19.53 y 2

r /

b,2

o

1.0

1.5

1.0

o

0.5

0.0

0.04

0.03

0.02

0.01

0.00

b) Tuibulence KinicEnergy

-0.01

_-1.0

_0.0

nR

c) uv/U

1.0

Figure 11. Measurements at Station A of Wake Jet and Momenturniess Wake. Wake (circles); Jet (triangles); Momeitumless Wake (squares).

0.0

nR

-1.0

1.0

0.0

nR

a) Axial Velocity

-1.0

1.0

1.5

1.0

o

0.5

0.05

0.04

03

0.02

0.01

a Axial Velocity O

_{. 0f}

AAffff

L.

.

-1.0

0.0

1.0

nR

I -J A _A

ODA

A L

°°°r

-0.01

_-1.0

nR

b) Turbulence Kinetic Energy

0.01

(NO

0.00

£

t' A

D jjj)ji111

Figure 12. Measuremtz at Station B of Wake, Jet and Moinentumless Wake. Wake (circles); Jet (triangles); Monituinls Wake (squares).

0.0

1.0

nR

1.5

1.0

o

0.5

0.0

nR

a) Axial Velocity

0.02

0.01

0.001%

-0.01

nR

b) Turbulence Kinetic Energy

Figure 13. Measuremts at Staticm D of Wake, Jet and Momoetumless Wake. Wake (circles); Jet (triangles); Momentumless Wake (squares).

0.01

0.00

-LO

0.0

1.0

FLOW

18.7

Suppoct Wires

Nose Suppty air for jet

22cm Afte rbody

Ar

X

100

10_1

1O2

iü

100

101

o

0

0 0O0Oo

o

U/tJ3

D

UV/U

Slope -1

Slope -2

.

t'

o

bID

b112ID

°

b ID

lI2uv A

o

Slope i

£ h,, a, a, I'.,,

o

o A

₀₀₀₀

00

e

00000

o

Figure 2. Jet Charteristics

101

100

101

x/D

a) Velocity and Turbulence Intensity Scales

100

101

x/D

Component Flows

U

C Combined Flow

Wake

Jet

Figure 3. Typical Axial (U) Velocity Profile in a Momentumless Wake i

X t-1 r ¡ r

x/D=0

.368

x/D=2

.280

x/D=5.

517

DDDEJJ

r)

x/D=14

.345

t'i

x/D=19

.531

AAAAALA1

tttIt&t

Lt

AALA

zj

C

C

o

w

I I

nR

Q

H1

H

t\)

100

101

100

10'

D

0

0 00000

DD

D D

Slope -1.5

o U/U0

Du/u

mn o

(U-U) lu0

Figure 5. Wake Velocity_{and Length Scales}

b1/R

b2/

Slope 0.25

00

o

0

0000

D _{D D}

D°DDD

D

101

100

101

x/D

a) Velocity Scales

101

_10°

₁₀₁

x/D

b) Length Scales

0.2

0.1

0.0

0.1

0.2

0

5

1_O

x/D

5 a) Mass Flux Convection Pressure Turbulence

10

x/D

b) Momentum Balance

Figure 6. Mass Flux and Momentum Balaixe

N)

_U

r'i

-.

n

Q

r)

DODD DO OD

t'i

'1

t

oJ

4444444 I T

x/D=O.368

x/D=5 .517

00000000

x/D=9

.

122

x/D=14 .345

ST

I

s

x/D=19. 531

o

o

t\) N)

r/R

N)

_Q

w

-

-T

SI))))))á

_Í4_,

Q

.

t\)

I I I I T I ( T J 1 I

nR

Q

Spil II.,

,.,.-

II!pmP

hi

1JJ)

i

4

.

-

,IflhlIL.4444444« 4

Q

DDEG1JD

I1

AAtLAAA

AItj1f

Obi T

x/D=5.

517

DD0000

E D

x/D=9

.

122

x/D=19

.531

Q

Q

Q

u,

Q

)

ï

100

₁₀₂

_iO

100

_10'

c) Length Scale

Figure 9. Decay of Turbulence Intensity

and Length Scales in Momentumless Wakes

Present Naudascher' i Higuchi and Kubota'

u

u Slope -1.6

_o

u

Slope -2.25 --e

o

o

£'

OL

a) Turbulence Kinetic Energy

101 TN

¡10o

-Q 10_1 b) Shear Stress Slope 0.25 Slope 0.2 Present O Naudascher' L Higuchi and Kubota'

Ipt

10'

10°

10'

x/D

10_1

10°

10'

10'

100

101

x/D

x/D

Present Q Naudascher' Slope -1.6

o

o'

...

Slope -1.5

1.

. 00

1.0

0.5

Figure 10. Similarity of Velocity andTurbulence Profiles

O

r / b,2

x/D=12.36 x/D=14.34

_o

xID=16.29 K xID=49.35 c E

0.5ì

y. 'y

I.

0.0

x/D=12.36 _O x/D=16.29 O x/D=19.53 y 1 2

r/b,2

7 3 1 2 3

r/b,2

1.5

1.0

O

0.5

0.0

0.04

0.03

(NO r

0.01

0.00

-1.0

0.0

nR

1.0

b) Turbulence Kinetic Energy

0.01

(NO

0.00

.2 L

- --

_{k A} -

_é

-iiIIt Li

s.

D

-0.01

_-1.0

0

O

nR

c) UV/U

Figure 11. Measurements at Station A of Wake, Jet and Momentumless Wake. Wake (circles); Jet (triangles); Mornentumless Wake (squares).

-1.0

0.0

1.0

nR

1.5

1.0

o

0.5

0.0

0.05

0.04

03

0.02

0.01

0.00

a Axial Velocity L

ODL

o

A'I

*i

l, i)Va:t

I

o.

nR

O

b) Turbulence Kinetic Energy

0.01

(NO

0.00

£ o

E

t!

-0.01

_bd

nR

c)

Figure 12. Measurements at Station B of Wake, Jet and Momentumless Wake. Wake (circles); Jet (triangles); Monientutniess Wake (squares).

ib

-1.0

0.0

1.0

1.5

1.0

O

0.5

0.0

0.05

0.04

(NO

-0 03

0.02

0.01

0.00

a

a) Axial Velocity

-1.0

0.0

nR

b) Turbulence Kinetic Energy

Figure 13. Measurements at Station D of Wake, Jet and_{Momentumless Wake. Wake}

(circles); Jet (triangles); Momentumless Wake (squares).

(NO

0.01

0.00

nR

-0.01

_o:o

_1:6

-1.0

0.0

1.0

nR