The first R cr as a possible measure of the entrainment length in a 2D steady wake
aa r X i v : . [ phy s i c s . f l u - dyn ] J u l The first R cr as a possible measure of the entrainmentlength in a 2D steady wake D. Tordella ♯ and S. Scarsoglio ♯ ♯ Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di Torino, 10129Torino, Italy
Abstract
At a fixed distance from the body which creates the wake, entrainment is onlyseen to increase with the Reynolds number ( R ) up to a distance of almost 20 bodyscales. This increase levels up to a Reynolds number close to the critical value forthe onset of the first instability. The entrainment is observed to be almost extin-guished at a distance which is nearly the same for all the steady wakes within the R range here considered, i.e. [20-100], which indicates that supercritical steadywakes have the same entrainment length as the subcritical ones. It is observedthat this distance is equal to a number of body lengths that is equal to the valueof the critical Reynolds number ( ∼ A fortiori of these findings, we propose to interpret the un-steady bifurcation as a process that allows a smooth increase-redistribution of theentrainment along the wake according to the weight of the convection over thediffusion. The entrainment variation along the steady wake has been determinedusing a matched asymptotic expansion of the Navier-Stokes velocity field [Tordellaand Belan,
Physics of Fluids , 15(2003)] built on criteria that include the matchingof the transversal velocity produced by the entrainment process.
Keywords : 2D steady wake, entrainment, critical Reynolds number, first instabil-ity Introduction
The dynamics of entrainment and mixing is of considerable interest in engineering appli-cations concerning pollutant dispersal or combustion, but it is also of great relevance ingeophysical and atmospherical situations. In all these instances, flows tend to be complex.In most cases, entrainment is a time dependent multistage process in both the laminarand turbulent regime of motion.The entrainment of external fluid in a shear flow such as that of a wake or a jet isa convective-diffusive process which is ubiquitous when the Reynolds number is greaterthan about a decade. It is a key phenomenon associated to the lateral momentum trans-port in flows which evolve about a main spatial direction. However, quantitative dataconcerning the spatial evolution of entrainment are not frequent in literature and aredifficult to determine experimentally. Quantitative experimental observations are verycumbersome to obtain either in the laboratory or in the numerical simulation context. Insome cases, such as, for instance, fluid entrainment by isolated vortex rings, theoreticalstudies (Maxworthy 1972[1]) predate experimental observations (Baird, Wairegi and Loo1977[2]; M¨uller and Didden 1980[3]; Dabiri and Gharib 2004[4]).It is interesting to note that more attention has been paid to complex unsteady andhighly turbulent configurations in literature than to their fundamentally simpler steadycounterparts.In unsteady situations, entrainment is believed to consist of repeated cycles of viscousdiffusion and circulatory transport. In turbulent flows, a sequence of processes is observed,where the exterior fluid is first ingested by the highly stretched and twisted interiorturbulent motion (large-scale stirring) and is then mixed to the molecular level by theaction of the small-scale velocity fluctuations, see for instance the recent experimentalworks carried out on free jets by Grinstein 2001[5] or on a plane turbulent wake by Kopp,Giralt and Keffer 2002[6].In steady laminar shear flows, stretching dynamics is generally absent (as in 2D flows)or is close to its onset. In this case, entrainment is determined by the balance between thelongitudinal and lateral nonlinear convective transport and the mainly lateral moleculardiffusion.Air entrainment in free-surface flows is another important instance of the entrainment2 x x=7 x=17 u(x,y) u(x,y) U D
Figure 1: - Sketch of the physical problem. Longitudinal velocity profiles (solid lines) at R = 60 and at stations x = 7, x = 17.process. The mechanism is complex and is also significant in nominally steady flows e.g.,a waterfall, or a steady jet. In such flows, entrainment is produced through the generationof cavities that can entrap air. The cavities are due to the impingement of the fallingjet, which free-surface is usually strongly disturbed, over the liquid surface of the pool.As observed in Ohl, Oguz, Prosperetti 2000[7], the generation process takes advantage ofboth the kinetic energy of the jet surface disturbances and of part of the actual energyin the jet.In this letter, we consider the steady two-dimensional (2D) wake flow past a circularcylinder. We deduce the entrainment as the longitudinal variation of the volume flowdefect using a matched Navier-Stokes asymptotic solution determined in terms of inversepowers of the space variables (Belan and Tordella 2002[8]; Tordella and Belan 2003[9]), seeSection 2. This approximated (2D) solution was obtained by recognizing the existenceof a longitudinal intermediate region, which introduces the adoption of the thin shearlayer hypothesis and supports a differentiation of the behaviour of the intermediate flowwith respect to its infinite asymptotics. The streamwise behaviour of the entrainment ispresented in Section 3. The concluding remarks are given in Section 4.3 Analytical approximation of the velocity field, ve-locity flow rate defect and entrainment
For an incompressible, viscous flow behind a bluff body, the adimensional continuity andNavier-Stokes equations are expressed as u∂ x u + v∂ y u + ∂ x p = R − ∇ u (1) u∂ x v + v∂ y v + ∂ y p = R − ∇ v (2) ∂ x u + ∂ y v = 0 (3)where ( x, y ) are the adimensional longitudinal and normal coordinates, ( u, v ) the adi-mensional components of the velocity field, p the pressure and R the Reynolds number.The physical quantities involved in the adimensionalization are the length D of the bodythat generates the wake, the density ρ and the velocity U of the free stream, see the flowschematic in fig. 1. The Reynolds number is defined as R = ρU D/µ , where µ is thedynamic viscosity of the fluid.The velocity field for the intermediate region of the 2 D steady wake behind a circularcylinder was obtained by matching an inner solution - a Navier-Stokes expansion innegative powers of the inverse of the longitudinal coordinate xf i = f i ( η ) + x − / f i ( η ) + x − f i ( η ) + · · · (4)where f is a generic dependent variable and where the quasi-similar transformation η = x − / y is introduced, and an outer solution, which is a Navier-Stokes asymptoticexpansion in powers of the inverse of the distance r from the body f o = f o ( s ) + r − / f o ( s ) + r − f o ( s ) + · · · (5)where r = p x + y and s = y/x .The wake mass-flow deficit of the inner field was considered by means of an infield boundary condition carefully accounting for it. In fact, this condition is placed at thebeginning of the intermediate flow region which inherits the full dynamics properties ofnear field. To this aim, we took advantage of experimental velocity and pressure profiles,as usually done in many physical contexts and as suggested, in the present context, byStewartson (1957)[10]. Further details about the use of this infield condition are given4elow. It should be noted that the matched expansion in ranging from minus infinityto plus infinity in the transversal flow direction and that the concept of wake flow isclearly defined downstream from the intermediate region where the thin layer hypothesisstarts to apply. The relevant boundary conditions involve, aside the infield condition,symmetry to the longitudinal coordinate and uniformity at infinity, both laterally andlongitudinally. For details on the expansion term determination, the reader can refer toTordella and Belan 2003[9].The physical quantities involved in the matching criteria are the vorticity, the lon-gitudinal pressure gradients generated by the flow and the transversal velocity pro-duced by the mass entrainment process. The composite expansion is defined as f c = f i + f o − f common , where f common is the common part of the inner and outer expansions.In Tordella and Belan 2003[9] the explicit inner and outer velocity and pressure expan-sions can be found up to order four (i.e. O ( x − ) and O ( r − ), for the inner and outer wake,respectively), the composite approximation has been shown graphically. In this work, weapproximate the wake flow with the composite solution obtained by truncating the innerand outer expansions at the third order term and then by determining their common partby taking the inner limit of the outer approximation. For the reader’s convenience, theinner and outer velocity component expansion terms are listed below (see equations 9 -23). The common part has not been included because it has an analytical representationwhich alone would take up a few pages. However, the c (cid:13) Mathematica file that describesits analytical structure and which allows its computation is given in the EPAPS onlinerepository[11]. The common expansion was obtained by writing the inner and outer ex-pansions in the primitive independent variables and by taking the inner limit of the outerexpansion, that is, by taking the limit for s → r → ∞ . To this end, the Laurentseries of the outer expansion about x → ∞ was considered up to the first order. Thecomposite expansion - which is, by construction, a continuous curve, since it is obtainedby the additive composition of three continuous curves, the sum of the inner and outerexpansions minus the part they have in common - is accurate if the common expansion isaccurate. This is always obtained if, at each order, the distance δ n = | f i,n − f o,n | betweenthe inner and the outer expansions is bounded and is at most of the same order as therange of f i and f o . In the present matching, we have verified that in the matching region- that is, in the region where the composite connects the inner and the outer expansions5 the distance δ n is not only bounded, but is small with respect to the ranges of f i and f o . The velocity approximation is shown in figures 2 and 3, where the longitudinal andtransversal components of the composite solution for the velocity field are plotted fordifferent longitudinal stations and Reynolds numbers.It should be noted, that in this analytical flow representation, a few key propertiesof the wake flow have been taken into account. These properties can help an accuratedescription of the entrainment process to be obtained. These properties are:i) The existence of intermediate asymptotics for the wake flow, in the general sense asgiven by Barenblatt and Zeldovich[12]. This is an important point, because the existenceof the intermediate region supports the adoption of the thin shear layer hypothesis andrelevant near-similar variable transformations for the inner flow, while, at the same time,it also supports a differentiation of the behavior of the intermediate flow with respect toits infinite asymptotics (Oseen’s flow).ii) The use of an in-field boundary condition which consists of the distribution ofthe momentum and pressure at a given section along the mainstream of the flow inopposition to the use of integral field quantities. This kind of boundary condition is notnew in literature[10], and presents the evident advantage of having a higher degree offield information with respect to the use of integral quantities such as the drag or the liftcoefficients (a given integral value can be obtained from many different distributions).iii) The acknowledgment of the fact that in free flows, such as low Reynolds numbers- 2D or axis-symmetric - wakes or jets developing in an otherwise homogeneous andinfinite expanse of a fluid, the main role in shaping the flow is played by the inner flow.This directly inherits the main portion of the convective and diffusive transport of thevorticity, which is created, at the solid boundaries, by the motion of the fluid relativeto the body. For these flows, it is physically opportune to denote the ”inner” flow asthe straightforward or basic approximation. This means that, up to the first order,the inner solution is independent of the outer solution. According to this, the Navier-Stokes model, coupled with the thin layer hypothesis, very naturally yields the orderof the field pressure variations O ( x − ). Pressure variations were often overestimatedat O ( x − )[13],[9]. This was due to the use, in the inner expansion, of the assumptionthat the field can accommodate an inner pressure which is independent of the lateral6oordinate, which however varies at the leading orders along the x coordinate. However,at intermediate values of y and for fixed x , this assumption is responsible for an anomalousrise in the composite expansion, due to the central plateau that appears in the outerexpansion. The outer solution is in fact biased at finite values of x to values greaterthan 1 and forces the composite expansion to assume inaccurate values – with respect toexperimental results – mostly in the region around y/D ≈ y/D = 20the longitudinal velocity is still appreciably different from U ). For details, the readermay refer to Section IV and fig.6 in Tordella and Belan 2003[9].iv) Last, we would like to point out that we have used the Navier-Stokes equations inthe whole field, without the addition of any further restrictive axiomatic position, such asthe principle of exponential decay. This did not prevent our approximated solution fromspontaneously showing the properties of rapid decay and irrotationality at the first andsecond orders for the inner and the outer flows, respectively. At the higher orders, whichmainly influence the intermediate region, the decay becomes a fast algebraic decay.For an unitary spanwise length, the defect of the volumetric flow rate D is defined as D ( x ) = Z + ∞−∞ (1 − u ( x, y )) dy (6)and is approximated through u c = u c ( x, y ), the composite solution for the velocity field,as D ( x ) ≈ Z + ∞−∞ (1 − u c ( x, y )) dy. (7)Entrainment is the quantity that takes into account the variation of the volumetricflow rate in the streamwise direction, and is defined as E ( x ) = | dD ( x ) dx | . (8)The sequence of the first four terms of the inner and outer approximation for thestreamwise velocity and the transversal velocity is given in the following. Zero order, n=0, u i ( x, y ) = c (9) v i ( x, y ) = 0 (10) u o ( x, y ) = k (11) v o ( x, y ) = 0 (12)7ith c = 1 , k = 1. First order, n=1 u i ( x, y ) = − Ac e − Ry / (4 x ) x − / (13) v i ( x, y ) = 0 (14) u o ( x, y ) = 0 (15) v o ( x, y ) = 0 (16)with c = 1, while the constant A is related to the drag coefficient ( A = ( R/π ) / c D ( R )). Second order, n=2 u i ( x, y ) = − A e − Ry / (4 x ) [e − Ry / (4 x ) + 12 y √ x √ πR erf( 12 r Rx y )] x − (17) v i ( x, y ) = − A y √ x e − Ry / (4 x ) x − (18) u o ( x, y ) = 0 (19) v o ( x, y ) = 0 (20) Third order, n=3 u i ( x, y ) = A e − Ry / (4 x ) (2 − R y x )[ 12 c − RF ( x, y )] x − / v i ( x, y ) = − A {− y √ x e − Ry / (2 x ) − r π R erf( r R x y ) + ( 12 r πR + − √ πR y x )e − Ry / (4 x ) erf( 12 r Rx y ) } x − / (21) u o ( x, y ) = Re ( i k e (3 i/ arctan ( s ) + k s / s / + 12 k s − / s / × { p (1 + is ) s ( − ii + s )2( i + s ) + ( − / √ log [ ( i − √ + √ s )( i − √ − (1 − i ) √ is + √ s )( − i √ + √ s )( − i √ + (1 − i ) √ is + √ s ) ) v o ( x, y ) = Re ( e (3 i/ arctan ( s ) [ k + k s + s + i s − i ) ])where c = − . . R − . R + 0 . R , F is the third order of thefunction F n ( x, y ) = 1 √ x Z y e Rζ / (4 x ) Hr n − ( x, ζ ) G n ( x, ζ ) dζ (22) G n ( x, y ) = A − n √ x Z y M n ( x, ζ )Hr n − ( x, ζ ) dζ (23)where M n ( x, y ) is the sum of the non homogeneous terms of the general ordinary differen-tial equation for the inner solution coefficients ( φ n ), n ≥
1, obtained from the x component8f the Navier-Stokes equation[8] , [9], and Hr n − ( x, y ) = H n − ( 12 r Rx y ), where H n are Her-mite polynomials. In the outer terms, the variables r, s, s ± are defined as r = p x + y , s = y/x , s ± = (1 + s ) ± / and the relevant constants are k = ± A / p π/ (2 R ), k = 3 ik , k = 0. Before describing the entrainment features we have observed, let us first discuss theasymptotic behaviour of the inner expansion in the lateral far field, since this aspect isimportant to determine the entrainment decay. At finite values of x , the inner streamwisevelocity decays to zero as a Gaussian law for n = 1 and as a power law of exponent − n = 2 and of exponent − n ≥
3. The cross-stream inner velocity goes to zero for n = 0 , n ≥
2. This allows v to vanish as x − / for x → ∞ .When x → ∞ this approximation coincides with the Gaussian representation given bythe Oseen approximation. It can be concluded that, at Reynolds numbers as low as thefirst critical value and where the non-parallelism of the streamlines is not yet negligible,the division of the field into two basic parts - an inner vortical boundary layer flow and anouter potential flow - is spontaneously shown up to the second order of accuracy ( n = 1).At higher orders in the expansion, the vorticity is first convected and then diffused in theouter field. This is the dynamical context in which the entrainment process takes place.In figures 2 (a) and 3 (a), the longitudinal velocity profiles are contrasted with theexperimental data available for steady flows by Berrone[14], Paranthoen et al.(1999)[15],Nishioka & Sato (1974)[16] and Kovasznay (1948)[17]. The accuracy on the velocitydistributions, between the analytical data and the laboratory ones is lower than 5%.This estimate was obtained by contrasting the longitudinal velocity distribution u withthe laboratory and numerical distributions, considered as the reference distribution. Tothis end, we computed the deviation ∆ ref = k u − u ref k ,x / k − u ref k ,x . At R = 34,a deviation ≃ .
5% was obtained for the laboratory results by Kowasznay, where x isthe station at 20 diameters from the center of the cylinder, see fig.2. As for the databy Berrone, we find a ∆ ref of about 1 . x ∼
10 we have a comparison withParanthoen et al. and Nishioka and Sato, that yields a deviation ∆ ref of about 2 .
5% and1 . D = D ( x ; R ) and the entrainment E = E ( x ; R ) obtained from the composite expansion. It can be observed that the volumetricdefect flow rate slowly decreases with the distance from the body (fig.4 a). This decreaseis faster at the beginning of the intermediate wake and at the higher Reynolds values.Considering a fixed position x (fig.4 c), the flow defect decreases with the Reynoldsnumber. Fig.4 (a) includes data from the laboratory experiments by Paranthoen et al.(1999, R = 53 .
3) and Kovasznay (1948, R = 56), both carried out at a slight supercritical R (unsteady regime). The difference between their results in not small, but it should berecognized that the difficulties in measuring at small values of the Reynolds number areexceptionally high. By considering the arithmetic mean between these two sets of data,an increase of more than 50% with regards to the values of the steady configuration, for x <
20, is observed.Parts (b, d) of fig.4 concern the entrainment, that is, the spatial rate of change ofthe wake velocity defect. The important points are: - the initial high variation at thebeginning of the intermediate part of the wake, which increases with R , - the higherexperimental mean value near x = 10 (2 .
45 10 − . − ), - for all the R , theexhaust of the entrainment at a distance of about 50 body lengths, - the collection ofexperimentally determined values of the critical R number that has a median value of46 .
6: a fact that relates the entrainment exhaust length -
EEL - to R cr with a simplescaling, such as EEL ∼ R ncr with n = 1. In fig.4 (d), one can also observe that at aconstant distance x from the body, the entrainment stops growing beyond around R cr .10hough the connection between the entrainment length and the instability cannotbe direct: - the first can be deduced as an integral property of the steady fully nonlinear version of the motion equations, - the second from the linear theory of stability,which is conceived to highlight the role of the perturbation characteristics and not of theintegral properties of the basic flow, these results could be a fortiori used to interpret thebifurcation to the unsteady flow condition at R cr as a process that allows the wake totune the entrainment, and, possibly, to redistribute it on a larger wake portion, accordingto the actual R value.It can be noticed that the decay distance is of the same order of magnitude as R cr and this shows that the scaling used in recent stability analyses[19],[18] to represent theslow time and space wake evolution - τ = εt and ξ = εx , where ε = 1 R ∼ R cr - is linkedto the exhaust of the entrainment process. In fact, one can say that the unit value of theslow time and spatial scales is reached where the entrainment nearly ends. The entrainment is observed to be intense in the intermediate wake downstream fromthe separation region where the two-symmetric standing eddies are situated. Here, thedependence on the Reynolds number is clear. The entrainment grows six-fold when R isincreased from 20 to 100. The subsequent downstream evolution presents a continuousdecrement of the entrainment. For all the R here considered, it has been observed thatthis decrease is almost accomplished at a distance from the body of about 50 diameters,which is a value that is close to the critical value R cr for the onset of the first instabilityand the subsequent set up of the unsteady regime (the median value in literature being R cr = 46 . eferences [1] Maxworthy, T. The structure and stability of vortex rings. J. Fluid Mech. , 51:15–32, 1972.[2] Baird, M. H. I., T. Wairegi and H. J. Loo. Velocity and momentum of vortex ringsin relation to formation parameters.
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Comput. Meth. Appl. Mech. Engrg. , 195: 6046-6058, 2006.14 y x =20 u (a) R Berrone (2001)Kovasznay (1948) −5 0 50.40.60.81 y u R (b) x =80 −5 0 5−0.01−0.00500.0050.01 y x =20 v (c) R −5 0 5−0.01−0.00500.0050.01 v y (d) R x =80 Figure 2: - Velocity profiles at the downstream stations x = 20, x = 80 and for R =20 , , ,
80 and 100. (a)-(b) Longitudinal velocity u , x = 20 and x = 80, (c)-(d)transversal velocity v , x = 20 and x = 80. The comparison with the numerical results byBerrone (2001) (triangles, R = 34, x = 20) and the laboratory data by Kovasznay (1948)(circles, R = 34, x = 20) is shown in part (a).15 y xu R=30(a)
Nishioka & Sato (1974)Paranthoen et al. (1999)Takami & Keller (1969) −5 0 50.20.40.60.81 xu R=60(b) y −5 0 5−0.02−0.0100.010.02 x R=30 v (c) y −5 0 5−0.02−0.0100.010.02 x R=60 v (d) y Figure 3: - Velocity profiles for R = 30 ,
60 plotted at stations x = 10 , , , ,
80 and100. (a)-(b) Longitudinal velocity u , R = 30 and R = 60, (c)-(d) transversal velocity v , R = 30 and R = 60. The comparison with the experimental data by Nishioka & Sato(1974) (squares, R = 40, x = 7) and Paranthoen et al. (1999) (triangles, R = 34, x = 10)is shown in part (a). 16
20 40 60 80 1000 0.20.40.60.81 1.2D x (a) R
Kovasznay (1948) R=56 (unsteady flow)Paranthoen et al. (1999) R=53.3 (unsteady flow) Takami & Keller(1969) R=40 x, R (b) R
Paranthoen et al. (1999) R=53.3 (unsteady flow) Kovasznay (1948) R=56 (unsteady flow) R cr x e Takami & Keller(1969) R=40 cr Figure 4: - (a)-(b): Downstream distribution of the volumetric flow rate defect D and entrain-ment E for R = 20 , , ,
80 and 100. (c)-(d): Volumetric flow rate defect D and entrainment E as a function of the R for different stations ( x = 8 , , , , D for the oscillating (supercritical) wake, as inferred from experimental data byKovasznay (1948, R = 56) and Paranthoen et al. (1999, R = 53 . x e , equal to R cr , see part (b). Position x e isobserved to be the wake length where the entrainment is almost extinguished ∀ R ∈ [20 , N ), Zebib 1987 ( ♦ ), Pier 2001 ( × ), Williamson1989 ( △ ), Leweke & Provansal 1995 (+), Strykowski & Sreenivasan 1990 ( ∗ ), Coutanceau &Bouard 1977 ( ◦ ), Elsenlhor & Eckelmann 1989 ( • ), Hammache & Gharib 1989 ( (cid:4) ), Jackson1987 ( (cid:3) ), Ding & Kawahara 1999 ( (cid:7) ), Morzynski et al. 1999 ( ▽ ), Kumar & Mittal 2006 ( H ).The solid line in parts (b) and (d) indicates the median value ( R cr ≈ .
6) of these data.6) of these data.