Integral Solution for the Mean Flow Profiles of Turbulent Jets, Plumes, and Wakes

Transcription

1 Amit Agrawal Ajay K. Prasad Department of Mehanial Engineering, University of Delaware, Newark, DE Integral Solution for the Mean Flow Profiles of Turbulent Jets, Plumes, and Wakes Integral methods are used to derive similarity solutions for several quantities of interest inluding the ross-stream veloity, Reynolds stress, the dominant turbulent kineti energy prodution term, and eddy diffusivities of momentum and heat for axisymmetri and planar turbulent jets, plumes, and wakes. A universal onstant is evaluated for axisymmetri and planar plumes. The ross-stream veloity profiles show that jets and axisymmetri plumes experiene an outflow near the axis and an inflow far away from it. The outflow is attributed to the deay of the enterline veloity with downstream distane, and the extent and magnitude of outflow orrelates with the streamwise deay of the enterline veloity. It is also shown that the entrainment veloity should not in general be equated to the produt of the entrainment oeffiient and the enterline veloity. It is found that, due to similar governing equations, profiles for jets and plumes are qualitatively similar. Our results show that the derived quantities are strong funtions of streamwise and rossstream positions, in ontrast to previous approahes that assumed onstant (in the rossstream diretion) eddy visosity and thermal diffusivity. The turbulent Prandtl number is approximately equal to unity whih mathes the value quoted in the literature. DOI: / Introdution Jets, plumes, and wakes are examples of free shear flows. A jet is produed when fluid exits a nozzle with some initial momentum. On the other hand, plumes are driven purely by buoyany addition at the soure. Jets and plumes spread laterally by engulfing entraining ambient fluid. The momentum ontained within the jet remains onstant at any streamwise ross setion, whereas the momentum ontained within the plume inreases monotonially with the streamwise oordinate while maintaining a onstant buoyany. The width of jets and plumes inreases at the ost of veloity. Wakes are produed behind an objet plaed in a freestream, and manifest themselves in the form of a veloity defiit profile. Jets, plumes, and wakes exhibit self-similarity beyond a ertain downstream distane suh that a harateristi length and veloity an be used to sale all distanes and veloities in the flow. Analyses and measurements have traditionally foused on the selfsimilar region beause fewer independent variables are involved making it is easier to interpret the results. For jets, plumes and wakes, the time-averaged enterline veloity is generally hosen as the harateristi veloity, the time-averaged enterline temperature differential with respet to the ambient is used as the harateristi temperature sale, whereas the ross-stream distanes are saled with the loal width. Although more analysis is available for planar ases, it is somewhat easier to setup experimentally an axisymmetri jet, plume, or wake, ompared to their planar ounterparts. Consequently, more experimental data is available for the axisymmetri ase, 1 5. In the self-similar region of the flows under onsideration, the traditional approah is to first perform an order of magnitude analysis of the Navier-Stokes equations. A boundary layer approximation is usually applied, allowing a substantial redution in the number of terms. The resulting terms are then saled using the appropriate length, veloity, and temperature sales. Further, by invoking onservation of momentum and buoyany for jets and Contributed by the Fluids Engineering Division for publiation in the JOURNAL OF FLUIDS ENGINEERING. Manusript reeived by the Fluids Engineering Division July, 00; revised manusript reeived May 6, 00. Assoiate Editor: J. S. Marshall. plumes, respetively, one an obtain the streamwise variation of width, enterline veloity and temperature, 6. Beause the number of unknowns exeeds the number of equations by one, the analysis fails to provide the ross-stream variation of these quantities. Of key interest is the funtional form of the streamwise veloity in the ross-stream diretion. One approah to lose the system of equations is to assume that the eddy visosity ( T ) is onstant in the ross-stream diretion, The equations are then solved to obtain the required funtional form. With this approah, one obtains U/ seh (A) for a plane jet where A0.706, and (1B ) for a round jet where B0.661, 7. The intent of this paper is to follow the opposite approah, i.e., we selet a funtional form for the streamwise veloity profile based on experimental data to lose the system of equations, and subsequently derive expressions for several useful quantities inluding T, whih an show a ross-stream variation depending on the hoie of the funtional form. Numerous experimental and numerial studies on free-shear flows have been undertaken by various groups in the past. Most of these studies measure or ompute the streamwise veloity profile, and fit a simple mathematial expression to the data. The literature reveals that many researhers, 1,11 16 prefer the Gaussian funtion to approximate the streamwise veloity and temperature profiles. Our own measurements of axisymmetri jets also reveal that overall the Gaussian profile is a superior fit to the data. As an example, we plot in Fig. 1 the normalized streamwise veloity data for an axisymmetri jet of water issuing from a -mm orifie into a large tank; data were aquired using twodimensional PI at 110z/d175 see 17 for experimental details. The omputed profile for Reynolds stress is ompared with experimental measurements made by the authors later in the paper to provide additional validation for the hoie of the Gaussian streamwise veloity profile. For these reasons, we preferred the Gaussian profile to the other hoies as the input funtional form for our analysis. Some previous researhers have employed an approah similar to ours to verify ertain diffiult measurements for example, Wygnanski and Fiedler 5 and Gutmark and Wygnanski 18 verified their measurements of Reynolds stress against values omputed from their streamwise veloity profile using an intergal approah. Journal of Fluids Engineering Copyright 00 by ASME SEPTEMBER 00, ol. 15 Õ 81

2 Fig. Coordinate system for planar x,z and axisymmetri r,z jets, plumes, and wake. For wakes, U represents the veloity defet. Turbulent Jets Fig. 1 Streamwise veloity profiles for jets axisymmetri jet from Gaussian profile, -- axisymmetri jet from polynomial profile, 7, dots from experiment, 7 Perhaps beause the primary aim of these researhers was to verify measurements, they did not exploit integral theories to its full extent, nor did they disuss the properties of the expressions that they derived. In fat, expliit expressions for even basi quantities are not readily available in the literature. The aim of this paper is to extrat a omplete set of results from the integral method. We derive a host of useful expressions and explore their properties in physial terms. As we show, several interesting features of the flow an be obtained, and some probable misoneptions an be orreted in this manner. Further, upon ontrasting appropriate solutions for example jets versus plumes, axisymmetri versus planar, our approah yields some new insights. The basi approah adopted here omprises the following three steps. First, we assume an analytial expression for the mean streamwise veloity U and temperature. Seond, the expression for U is substituted into the ontinuity equation along with the assumption of the enterline veloity variation, and integrated to determine the mean ross-stream veloity profile. Third, the expressions for U and are substituted into the simplified momentum and energy equations, and integrated to determine the Reynolds stress and veloity-temperature orrelations. These expressions an then be employed to derive a number of useful quantities. Integral methods are shown here to be suessful in reproduing experimental results for standard jets, plumes and wakes axisymmetri and planar whih are ommonly used as model flows in a variety of situations. Our results should serve as a useful referene for suh studies. However, it should be remembered that integral methods may have restrited appliation to more omplex turbulent flows, 1, where it is diffiult to assign profile shapes, and relate entrainment rates to loal influenes in omplex environments. The relevant governing equations for jets and wakes are the ontinuity and the streamwise momentum equation. The two additional momentum equations, ross stream and azimuthal, relate pressure with the flutuating omponents of veloity, and the flutuating omponents of veloity with themselves, respetively; in this paper, we are not interested in exploring these relationships. For plumes, an additional equation for temperature is required. By symmetry onsiderations, both the ross-stream veloity and the Reynolds stress are zero at the enterline. We have used this ondition throughout this paper to evaluate the onstant of integration. Axisymmetri Jets. The ontinuity equation for the timeaveraged veloities in ylindrial oordinates see Fig. for the oordinate system used in this paper is 1 r 0. (1) r r z In the self-similar region, the simplified streamwise momentum equation an be obtained using an order of magnitude analysis as, 6, r z 1 r r 0 () r where the overbar denotes time-averaged quantities. For the self-similar axisymmetri jet, varies as z 1, while b inreases linearly with z, 6. By approximating the streamwise veloity at any downstream loation by a Gaussian,,11 1,17, see also Fig. 1 Ur,z zexpr /b z zexpr / z zexp, () one an solve for by substituting Eqs. into 1: 1exp exp. (4) ()/ is plotted in Fig. and reveals that, ontrary to the onnotation of an inflow implied by the term entrainment, the ross-stream flow in the viinity of the enterline is atually outward, i.e., ()/ 0 for (()/ 0 at the enterline, reahes a maximum of at 0.54 and delines bak to zero at 1.1. The reason for suh an outflow is that the enterline veloity dereases as z 1, and not beause of the Gaussian streamwise veloity profile assumption. ()/ 0 i.e., inward for 1.1, reahes a minimum of 0.0 at.08 and asymptotes to 0 as. It should be pointed out that 0 muh more slowly than U, i.e., although the entral region is dominated by the axial omponent of veloity, the ross-stream flow predominates far away from it. Aording to 11, the entrainment oeffiient an be defined using the inremental volume flux as d dz b, (5) where for axisymmetri jets is given by 0 rurdr. 814 Õ ol. 15, SEPTEMBER 00 Transations of the ASME

3 Fig. Cross-stream veloity profiles for jets axisymmetri jet, -- planar jet It is readily seen that d/dz is the inremental volume flux entering the jet through a irular ontrol surfae at large r, i.e., d lim r. dz r From Eq. 4, / for large an be approximated as /. Our result gives d dz r b. (6) By equating Eqs. 5 and 6,weget/0.055, whih is the same value as obtained by 11. From Fig., it is seen that Turner s 11 statement: the inflow veloity at the edge of the flow is some fration of the maximum mean upward veloity, is not stritly orret, beause / never reahes for any value of. However, it is easily shown, 19, that the inward extension of the urve / the asymptoti urve for / at large intersets the jet edge 1 with a value of /. Equation 4 an also be derived using a ontrol volume approah by equating the differene of volume flux at two suessive downstream stations to the inoming fluid volume. As seen in Fig. the maximum value of is just % of, making it rather diffiult to measure preisely, and therefore it is less frequently presented in the literature, 4,5. The profile an be used to diretly determine the spread rate,. Researhers in the past have determined by urve fitting experimental data for U. However, determining from the experimental profile for may be advantageous sine the loation and height of the extrema an be unambiguously determined. However, as stated above, the experimental unertainty in an be substantial. A omparison of the derived ross-stream veloity profile with the experimental data is provided in 0. One an insert the time-averaged profiles for U and into Eq., to obtain the time-averaged profile for Reynolds stress as U exp exp. (7) The maximum for Reynolds stress lies at 0.58 Fig. 4. Wygnanski and Fiedler 5 provide a plot for Reynolds stress by integrating their streamwise veloity profile. However, they do not provide an expliit expression for it. Figure 4 also shows our experimentally measured Reynolds stress for a water jet at Re000 using PI. The experimental data an be ompared with the analytial result based on the Fig. 4 Reynolds stress profiles for jets axisymmetri jet from Gaussian profile, -- axisymmetri jet from polynomial profile, 7, -.. planar jet, dots from experiment Gaussian profile and the one derived using the (1B ) profile, 7, where B We see that the latter result does not math as well, justifying the seletion of the Gaussian profile for our analysis. Additionally, our analytial result mathes very well with the Reynolds stress data of 5,18. Planar Jets. It is worthwhile to ompare the results for an axisymmetri jet against a planar jet. For planar jets the ontinuity and momentum equations are, respetively, 6: x x 0, (8) z z 0. (9) x Like axisymmetri jets, plane jets have a linear spread rate and are well approximated by a Gaussian veloity profile, 1,15, given again by Eq. where is now x/b(z)), but now the enterline veloity deays as z 1/, 6. Integrating Eq. 8 using we obtain 4 4 exp erf. (10) This expression has also been obtained by 15. Similar to the axisymmetri ase, we find an outflow near the jet axis, and an inflow far away from it. The maximum outflow ours around 0.5 whih is very lose to the axisymmetri ase and the flow turns inward for this value is slightly smaller than for axisymmetri jets Fig.. Similar to the axisymmetri ase we an define the oeffiient of entrainment following Turner 11: d dz, (11) where is now given by Udx. Inserting Eq. and differentiating with respet to z, we obtain Journal of Fluids Engineering SEPTEMBER 00, ol. 15 Õ 815

4 Fig. 5 jets Dominant kineti energy prodution term profiles for axisymmetri jet, -- planar jet Fig. 6 Cross-stream variation of eddy visosity for jets axisymmetri jet, -- planar jet d dz. (1) Equating Eqs. 11 and 1, weget/ from Fig. 5 of Gutmark and Wygnanski 18, we estimate 0.09). It should be pointed out that the ross-stream veloity does beome equal to for large. Turner s 11 statement that the inflow veloity at the edge of the jet equals is thus seen to apply stritly in the ase of planar free-shear flows. Ramaprian and Chandrasekhara 15 measured for planar jets as e / The slight disrepany might be due to the fat that their measurements, 15, were made rather lose to the nozzle (5z/d60) and therefore, the jet might not have ahieved omplete self-similarity. As before, the momentum Eq. 9 along with the expressions for U and an be used to solve for the Reynolds stress as 4 exp erf. (1) The plot looks similar to that of axisymmetri jets with a maximum at 0.6 and 0 for 0 and large Fig. 4. In fat, the maxima lie at almost the same and are of nearly the same magnitude. Although U profiles for the two ases onsidered above are similar, their and the governing Eqs. and 9 are very different. Thus it is not expeted that would look so similar. This similarity is due to the fat that U/z is the dominating term, and behaves similarly for the two ases. The dilution rates an be omputed from 1 d dz b axisymmetri jets (14) 1 d dz planar jets. (15) b Beause the spread rates for the two ases are virtually idential, it is obvious that axisymmetri jets dilute twie as rapidly as their planar ounterparts axisymmetri jets entrain irumferentially, while planar jets entrain from the two sides only. Greater mixing in the axisymmetri jets is onsistent with the faster deay of their enterline veloity with down-stream distane. It should be pointed out that, sine bz, 1/(d/dz)z 1, i.e., the dilution rate keeps dereasing with downstream distane. The omplete turbulent kineti energy equation an be found in the literature e.g., for axisymmetri jets see 5. The dominant kineti energy prodution term is /. Using Eqs., 7, and 1 we an write exp exp axisymmetri jets exp erf planar jets. The prodution term has a maximum around 0.6 lose to the maximum of the term and redues to zero for large Fig. 5. Again, the maxima of the prodution terms for axisymmetri and planar jets lie at almost the same. The magnitude of the turbulent kineti energy term for axisymmetri jets is slightly larger than planar jets. The similarity of these terms is due to the fat that both and / resemble eah other losely for the two ases onsidered here. One an derive orresponding expressions for T for the axisymmetri and planar jets using as and Taxisym. T r, (16) b 4 Tplanar 1exp, erf. b 8 Beause T b produt of the integral veloity and length sales, we an expet Taxisym. z 0 (bz, and z 1 for axisymmetri jets and, Tplanar z 1/ (bz, and z 1/ for planar jets, i.e., not only is T a funtion of radius, it an also be a funtion of z. Lessen 1 predited idential z-dependenes for T based on the priniple of marginal instability. While the streamwise variation of T is well known, T is generally assumed to be independent of, The results presented next indiate that this need not be true. The eddy visosity T for the axisymmetri and planar ases are plotted in Fig. 6. T is a maximum at the enter of the jet and deays to 0 for large. For the axisymmetri ase we find that 816 Õ ol. 15, SEPTEMBER 00 Transations of the ASME

5 T (0) is. times greater than T (). & represents the e point of the Gaussian. For the planar ase, this ratio is about 1.7. Townsend 8 also observes that T should diminish as the jet edge is approahed beause the measured veloity profiles approah zero more rapidly than the profiles alulated on the basis of onstant T. Pope 9 states that T is within 15% of the value 0.08 over bulk of the axisymmetri jet, and therefore T an be assumed onstant, independent of. Turbulent Plumes Axisymmetri Plumes. The ontinuity equation remains the same as Eq. 1 for axisymmetri plumes, whereas a buoyany term appears in the simplified momentum Eq., 1,14: r z 1 r r g. (17) r The g term in Eq. 17 differentiates the plume from the jet. The temperature equation an be obtained starting from r z v r u z v r r 1 r r, z using saling arguments similar to jets, to finally arrive at 1,14 r z 1 r rv 0. (18) r A Gaussian profile is ommonly used in the literature to approximate the distributions for veloity and temperature in plumes, 1,,11,1,14,16. From similarity onsiderations, the enterline veloity and temperature for axisymmetri plumes vary, respetively, as z 1/ and z 5/ while the width inreases linearly, 6. Therefore, the streamwise veloity is given by Eq., while the temperature in the plume an be written as r,z zexpr / T z zexph, (19) where T, i.e., temperature and veloity need not spread at idential rates. In fat / T H is 1. for axisymmetri plumes, 11. Again, from the ontinuity Eq. 1 and using the deay rate of the enterline veloity, one an derive. (0) exp exp The plot of / losely resembles axisymmetri jets plumes experiene outflow near the enterline, and an inflow far away from it. However, the maximum positive is now just 0.4% of for 0., while the maximum negative value of.8% lies at 1.87 Fig. 7. The ross-stream veloity hanges sign at muh earlier than for jets. This is due to the presene of buoyany. In fat, the presene of buoyany in the plume reates ontrasting effets near the enter and far away from it when ompared to the axisymmetri jet. Buoyany auses the enterline veloity to deay less slowly than the jet. Consequently, the magnitude of the outflow near the enterline is smaller for the plume resulting in a smaller positive. Seondly, beause buoyany is ontinuously inreasing the momentum of the plume, there is a larger volume influx from its lateral surfae. Hene, larger inflow veloities are seen far away from the enterline. This larger inflow is responsible for greater mixing in plumes. Sreenivas and Prasad have proposed a model based on vortex dynamis to explain the greater entrainment in the plumes. It is again found that the inflow veloity never equals for axisymmetri plumes. The expression for / for axisymmetri plumes an also be found in 1 and the plot in 1,. Fig. 7 Cross-stream veloity profiles for plumes axisymmetri plume, -- planar plume Similar to axisymmetri jets, for large we an approximate / as 5/6. The inremental volume flux entrained by the plume is again given by Eq. 5, while our result gives d lim rr 5 dz r 6 U 5 b 6. From this, we get 5/60.08 using 0.100, 11. The same value of was obtained by 11. Integrating the momentum Eq. 17 using Eqs. and 0 yields U 5 6 exp 1 exp gb H exph K where K is a onstant of integration. Using the ondition that is bounded at 0, we obtain K/gb /H, whih leads to U 5 6 exp 1 exp 1 gb H U 1expH. (1) The ondition of self-similarity fores gb /U to be a onstant, 6, but its value is not diretly reported in the literature. However, it an be determined by fitting experimental data. For example, using Beuther et al. s 4 data we estimate gb /H U Dai et al. s 1 data similarly indiates a value of Using the latter value, 0 for 1.5, whereas the former value gives 0 for all. A negative Reynolds stress for plumes is unphysial this is also supported by the data of 1. Therefore, we use the former value of 0.04 hereafter. Thus, gb H U , implying that K0, and gb /U Setting K0 redues Eq. 1 to U 6 5 exp exp exph. Journal of Fluids Engineering SEPTEMBER 00, ol. 15 Õ 817

6 Fig. 8 Reynolds stress profiles for plumes axisymmetri plume, -- planar plume Fig. 10 Dominant kineti energy prodution term profiles for plumes axisymmetri plume, -- planar plume The behavior of shown in Fig. 8 looks similar to of axisymmetri jets. In fat the maxima lie at almost the same 0.6. As an be intuitively expeted, Reynolds stress for plumes is larger than jets. Interestingly, this similarity in for axisymmetri jets and plumes an be inferred by noting that the above expression redues to Eq. 7 for H1 although H1. for axisymmetri plumes. From the temperature Eq. 18 and using Eqs., 19, and 0 one an obtain v 5 6 1exp exph. () From the plot in Fig. 9 the ross-stream veloity-temperature orrelation remains positive for all with a maximum value of 0.06 at The prodution term is obtained as 5 exp exp exph 1. This has a maximum of 0.00 around 0.6 Fig. 10. This plot mathes that of axisymmetri jets quite losely with maxima at almost the same. As expeted the prodution term for the plumes is larger than jets for all. The dilution rate an be alulated from 1 d 5 dz b. () Comparing Eqs. 14 and, we find that entrainment, and hene dilution rate for axisymmetri plumes is about 1.6 times greater than for axisymmetri jets. The normalized eddy visosity an be obtained using Eq. 16 as T b 1 5 exp exph 1. Here again the variation with is large. In fat, the ratio of T (0) to T () Fig. 11. In addition, T z /. For plumes, a further assumption of onstant in ross-stream diretion eddy thermal diffusivity, T has been made by the researhers in the past to obtain the funtional forms of the similarity funtions where Fig. 9 eloity-temperature orrelation for plumes axisymmetri plume, -- planar plume Fig. 11 Cross-stream variation of eddy visosity for plumes axisymmetri plume, -- planar plume 818 Õ ol. 15, SEPTEMBER 00 Transations of the ASME

7 Fig. 1 Cross-stream variation of eddy thermal diffusivity for plumes axisymmetri plume, -- planar plume v T r. (4) Our approah provides the radial variation in T using Eqs. 19 and as T b 5 1H 1exp. Figure 1 reveals that T /( b) has a value of 0.09 at the enterline while it drops to 0.01 for &, i.e., the radial variation in T is substantial. Moreover, T is a funtion of downstream distane, i.e., T z /. Knowing the distribution of both T and T we an obtain the turbulent Prandtl number, Pr T as a funtion of as Pr T T 5 1exp T H 5 exp exph 1. It is interesting to note that unlike T and T,Pr T is independent of the spread rate and z. Moreover, Pr T is not suh a strong funtion of radial position as T and T.Pr T varies from 0.9 at the enterline to 0.8 at & Fig. 1. Physial arguments an be used to show that Pr T 1, 5, and our result is in good agreement. Fig. 1 Cross-stream variation of turbulent Prandtl number for plumes axisymmetri plume, -- planar plume Planar Plumes. The ontinuity equation for planar plumes is the same as for planar jets Eq. 8, while the momentum equation an be derived by adding the buoyany term to the planar jet momentum equation, 6, x z g. (5) x The temperature equation in the self-similar regime is given by 6 x z v 0. (6) x Here the enterline temperature varies as z 1, while it is interesting to note that the enterline veloity does not hange downstream, 6, i.e., presene of buoyany prevents the deay of the enterline veloity. This has an interesting onsequene for the ross-stream veloity, expressed mathematially as exp erf. Unlike axisymmetri plumes, we find that planar plumes do not experiene an outflow near the enterline; / is small for 0.54 less than 10% of its asymptoti value of and remains negative throughout, i.e., the flow is always towards the axis Fig. 7. Qualitatively the plot is similar to planar jets. Buoyany auses planar plumes to entrain more than planar jets like their axisymmetri ounterparts. Similar to planar jets, we an obtain the oeffiient of entrainment as. Using 0.11, 16, we obtain The Reynolds stress is given by erfexp erf gb erfh. H (7) Note that the onstant of integration is zero in Eq. 7, and for similarity to exist gb /U should be a universal onstant, 6. The experimental data of Ramaprian and Chandrasekhara 16 indiates H1. idential to axisymmetri plumes. We estimate the value of gb /HU denoting this by C as whih gives max /U 0.06 at 0.6. The maximum value is very lose to the value of 0.05 predited by Malin and Spalding 6. For Ramaprian and Chandrasekhara s 16 data we obtain C 0.06, but we find that beomes negative for 1.6. As in the ase of axisymmetri plumes, should remain positive for all this is also supported by the data of 16. Hene, we will use C0.069 to obtain gb H, and gb /U 0.09 whih is very lose to the value for axisymmetri plumes. Equation 7 an then be simplified to see Fig. 8 erfexp erf erfh. While it is known that H1., an interesting result is obtained by substituting H in the above expression. Then, it is seen that for planar plumes is twie that of planar jets Eq. 1. The veloity-temperature orrelation for planar plumes is given by Journal of Fluids Engineering SEPTEMBER 00, ol. 15 Õ 819

8 v erfexph. As for axisymmetri plumes, the veloity-temperature orrelation is also positive with a maximum of 0.05 at 0.5 Fig. 9. Note that for both axisymmetri and planar plumes, the maximum value of v is to the left of the maximum for. We an obtain the prodution term as erfexp erfexp erfhexp. As for jets and axisymmetri plumes, this term has a maximum value around 0.6 Fig. 10. In fat, the maximum lies at almost the same loation as the maximum of Reynolds stresses. It should be pointed out that unlike for jets, the prodution term for planar plumes is larger than their axisymmetri ounterparts. The dilution rate for planar plumes an be obtained as 1 d dz b. (8) As expeted, the dilution rate of planar plumes is higher than planar jets ompare Eqs. 15 with 8. Interestingly, it is equal to the dilution rate of axisymmetri jets Eqs. 14 and 8. One again, we an derive expressions for T, T, and Pr T and question the validity of assigning onstant ross-stream values to them. T b erfexp erf 4 exp T b 4H erf, erfh, Pr T 1 erfexp H erfexp erferfh. Turbulent eddy visosity and thermal diffusivity are plotted in Figs. 11 and 1, respetively. Both T and T z. Pr T has a maximum value of 0.8 at the enterline while it drops to 0.74 for & Fig. 1. It is again reassuring to note that Pr T 1 as predited by physial arguments, 5. Turbulent Wakes Axisymmetri Wakes. Unlike jets and plumes, wakes have a nonlinear spread rate. The ontinuity equation for axisymmetri wakes remains the same as Eq. 1, while the momentum equation an be derived as for axisymmetri jets. However, here u/ is of order unity, i.e., the veloity flutuations are of the order of the veloity defet, 6. This simplifies the momentum equation further: U z 1 r r 0. (9) r It is interesting to note that does not appear in the streamwise momentum equation. This is beause in the momentum equation /ro(u /L), while U/zO(U 0 /L); sine /U 0 O(b/L), the former an be disarded. Using a Gaussian profile for the veloity defet, the streamwise veloity is given by Ur,z 0 zexpr /b 0 zexp. (0) Fig. 14 Cross-stream veloity and Reynolds stress profiles for wakes axisymmetri wake, -- planar wake In the self-similar region dereases as z /, while b inreases as z 1/, 6. Keeping these in mind, we an obtain from the ontinuity Eq. 1 as b z exp. (1) As for planar plumes, inflow ours for all with a maximum at Reynolds stress an be obtained using Eqs. 9 and 0 as U b 1z 1exp exp 4U 0 exp. () It should be mentioned that in this ase the integration onstant is nonzero atually Kb/1z). It is not possible to plot Eq. without knowing U 0 /. However, sine U/U 0 O(1), one an simplify the momentum equation by replaing U/z by U 0 /z, 6, to obtain: U 0 z 1 r r 0. () r On omparing Eqs. 1 and, we an see that they are idential (/U 0 replaes ). Hene it is not surprising to see b U 0 z exp, (4) whih is exatly the same expression as for /. It should, however, be noted that has been normalized with U 0 and not. /U 0 and / are plotted in Fig. 14. We will use the expression for given by Eq. to obtain the prodution term and the radial variation in turbulent eddy visosity as b 6z 1exp exp 4U 0 exp exp, 80 Õ ol. 15, SEPTEMBER 00 Transations of the ASME

9 T b b 1exp exp 4z exp 4U 0 exp. Planar Wakes. The ontinuity equation for planar wakes is given by Eq. 8 while the momentum equation is, 6, U z 0. (5) x varies as z 1/, while b inreases as z 1/ for planar wakes, 6,1. Integrating the ontinuity Eq. 8 gives b z exp. (6) Comparing Eqs. 1 and 6, we see that the inward veloity is very similar. However, the deay with z is slightly different for the two ases. Reynolds stress an be obtained from the momentum Eq. 5 using Eq. 0: U b 16z 4 exp erf 8U 0 exp. The prodution term is b 8z 4 exp erf T an be obtained as 8U 0 exp exp. T b b 4 exp erf z exp 8U 0 exp. Simplifying the momentum equation as was done with axisymmetri wakes, and using it to obtain the simplified Reynolds stress b U 0 z exp, (7) whih, as expeted, is the same expression as for / for planar wakes Fig. 14. It is not possible to ompare the prodution terms for the axisymmetri and planar wakes beause U 0 is unknown. However, if we use the simplified expressions for the Reynolds stress Eqs. 4 and 7, we an write the produtions terms as U 0 b exp z axisymmetri wakes U 0 b exp planar wakes. z These two terms differ just by a onstant. The simplified expressions for T using Eqs. 16, 0, 4, and 7 are Taxisym. U 0b, 6z Tplanar U 0b. 4z Sine U 0 is invariant with z, we obtain Taxisym. wake z 1/ while Tplanar wake z 0, in agreement with Lessen s preditions, 1. Remarkably, the dependene of T on disappears for wakes. Conlusions A omprehensive analysis has been onduted for six standard ases axisymmetri and planar jets, plumes, and wakes. Expressions for ross-stream veloity, Reynolds stress, and turbulent kineti energy prodution terms are derived for these ases assuming a Gaussian streamwise veloity distribution. The plots are ompared amongst themselves and provide several insights. 1 Expressions for ross-stream veloity indiate outflow for jets and axisymmetri plumes near the axis, while inflow ours in the far field. Planar plumes do not experiene any suh outflow. Outflow of fluid near the axis is a natural onsequene of the deay of the enterline veloity with downstream distane and not beause of the assumed Gaussian veloity profile. The deay rates of axisymmetri jets, planar jets, and axisymmetri plumes are, respetively, z 1, z 1/, z 1/, while the radial extents of outflow are r/b1.1, 0.99, Moreover, the deay rate for planar plumes varies as z 0 and these do not experiene any outflow. Thus, a higher deay rate orrelates with a larger radial extent of outflow. See Agrawal et al. 0 for more disussion on the oupling between the deay of the enterline veloity and the radial extent of the outflow. Expressions for the entrainment oeffiients of planar jets and plumes are developed along the lines of their axisymmetri ounterparts. It is found that for planar jets and plumes e ; this is, however, not true for axisymmetri ases. Hene, ontrary to onventional belief, the entrainment veloity should not be equated to for axisymmetri jets and plumes. The value of the universal onstant gb /U is estimated as and 0.09 for axisymmetri and planar plumes, respetively. 4 Reynolds stress and the dominant turbulent kineti energy prodution term for jets and plumes are qualitatively the same with a maximum around 0.6. Magnitudes for jets and axisymmetri plumes are nearly the same. These are muh smaller than for planar plumes. 5 Unlike plumes and jets, the normalized ross-stream veloity and Reynolds stresses for wakes are funtions of downstream distane. In addition, the turbulent kineti energy prodution term for axisymmetri and planar wakes has the same ross-stream distribution. 6 Cross-stream variations for T, T, and Pr T are shown to be signifiant, therefore, the use of onstant values of T and T in the ross-stream diretion is unjustified. Pr T is found to lie between 0.9 and 0.7 for axisymmetri and planar plumes. 7 Our analysis reveals that the eddy visosity is independent of the ross-stream oordinate for axisymmetri and planar wakes. Aknowledgments This work was supported by National Siene Foundation, under grant NSF-ATM We thank Prof. Pablo Huq of the College of Marine Studies, University of Delaware, for enouraging us to undertake this study and helpful disussions. Nomenlature b width defined as U(b)/ e 1 for jets and plumes, and (U 0 (b))/ e 1 for wakes spread ratedb/dz T spread rate for temperature d diameter of the nozzle g aeleration due to gravity Journal of Fluids Engineering SEPTEMBER 00, ol. 15 Õ 81

10 H / T K onstant of integration L downstream distane sale p time-averaged pressure Pr T turbulent Prandtl number r ross-stream oordinate used for axisymmetri ase T temperature T flutuating temperature T 0 ambient fluid temperature assumed onstant T time-averaged enterline temperature u, v flutuating omponents of veloity u flutuating veloity sale U time-averaged streamwise veloity U 0 free stream veloity time-averaged enterline veloity for wakes veloity defet at the enterline time-averaged ross-stream veloity e time-averaged entrainment veloity x ross-stream oordinate used for planar ase z oordinate along the axis oeffiient of entrainment oeffiient of thermal expansion thermal diffusivity T turbulent thermal diffusivity TT 0 TT 0 T T 0 volume flux T eddy visosity nondimensional ross-stream oordinate (r/b for axisymmetri ase, x/b for planar ase. See Fig. density Referenes 1 Dai, Z., Tseng, L. K., and Faeth, G. M., 1995, eloity Statistis of Round, Fully Developed, Buoyant Turbulent Plumes, ASME J. Heat Transfer, 117, pp George, W. K., Alpert, R. L., and Tamanini, F., 1977, Turbulene Measurements in an Axisymmetri Buoyant Plume, Int. J. Heat Mass Transfer, 0, pp Bhat, G. S., and Narasimha, R., 1996, A olumetrially Heated Jet: Large Eddy Struture and Entrainment Charateristis, J. Fluid Meh., 5, pp Hussein, H. J., Capp, S. P., and George, W. K., 1994, eloity Measurements in a High Reynolds Number, Momentum-Conserving Axisymmetri Turbulent Jet, J. Fluid Meh., 58, pp Wygnanski, I., and Fiedler, H., 1969, Some Measurements in a Self- Preserving Jet, J. Fluid Meh., 8, pp Tennekes, H., and Lumley, J. L., 197, A First Course in Turbulene, MIT Press, Cambridge, MA, pp White, F. M., 1974, isous Fluid Flow, MGraw-Hill, New York, pp Townsend, A. A., 1976, The Struture of Turbulent Shear Flow, Cambridge Univ. Press, Cambridge, UK, pp Pope, S. B., 000, Turbulent Flows, Cambridge Univ. Press, Cambridge, UK, pp Lesieur, M., 1990, Turbulene in Fluids, Kluwer, Dordreht, The Netherlands, pp Turner, J. S., 1986, Turbulent Entrainment: The Development of the Entrainment Assumption, and Its Appliation to Geophysial Flows, J. Fluid Meh., 17, pp List, E. J., 198, Turbulent Jets and Plumes, Annu. Rev. Fluid Meh., 14, pp Chen, C. J., and Rodi, W., 1980, ertial Turbulent Buoyant Jets A Review of Experimental Data, Pergamon Press, Oxford, UK, pp Gebhart, B., Jaluria, Y., Mahajan, R. L., and Sammakia, B., 1988, Buoyany- Indued Flows and Transport, Hemisphere, Washington, DC, pp Ramaprian, B. R., and Chandrasekhara, M. S., 1985, LDA Measurements in Plane Turbulent Jets, ASME J. Fluids Eng., 107, pp Ramaprian, B. R., and Chandrasekhara, M. S., 1989, Measurements in ertial Plane Turbulent Plumes, ASME J. Fluids Eng., 111, pp Agrawal, A., and Prasad, A. K., 00, Properties of orties in the Self- Similar Turbulent Jet, Exp. Fluids,, pp Gutmark, E., and Wygnanski, I., 1976, The Planar Turbulent Jet, J. Fluid Meh., 7, pp Narasimha, R., 001, private ommuniation. 0 Agrawal, A., Sreenivas, K. R., and Prasad, A. K., 00, eloity and Temperature Measurements in an Axisymmetri Jet With Cloud-Like Off-Soure Heating, Int. J. Heat Mass Transfer, to appear. 1 Lessen, M., 1978, On the Power Laws for Turbulent Jets, Wakes and Shearing Layers and Their Relationship to the Prinipal of Marginal Instability, J. Fluid Meh., 88, pp Sreenivas, K. R., and Prasad, A. K., 000, ortex-dynamis Model for Entrainment in Jets and Plumes, Phys. Fluids, 1, pp Lumley, J. L., 1971, Explanation of Thermal Plume Growth Rates, Phys. Fluids, 14, pp Beuther, P. D., Capp, S. P., and George, W. K., Jr., 1979, Momentum and Temperature Balane Measurements in an Axisymmetri Turbulent Plumes, ASME Paper No. 79-HT-4. 5 Kays, W. M., and Crawford, M. E., 1980, Convetive Heat and Mass Transfer, MGraw-Hill, New York, pp Malin, M. R., and Spalding, D. B., 1984, The Predition of Turbulent Jets and Plumes by Use of the k- Model of Turbulene, PhysioChemial Hydrodynamis, 5, pp Õ ol. 15, SEPTEMBER 00 Transations of the ASME