Averaging method for nonlinear laminar Ekman layers

Transcription

1 Unde consideation fo publication in J. Fluid Mech. 1 Aveaging method fo nonlinea lamina Ekman layes By A. A N D E R S E N 1, 2, B. L A U T R U P 3 A N D T. B O H R 1 1 The Technical Univesity of Denmak, Depatment of Physics, DK-28 Kgs. Lyngby, Denmak 2 Risø National Laboatoy, Optics and Fluid Dynamics Depatment, DK-4 Roskilde, Denmak 3 The Niels Boh Institute, Blegdamsvej 17, DK-21 Copenhagen Ø, Denmak (Received Octobe 1, 22) We study lamina Ekman bounday layes in otating systems using an aveaging method simila to the technique of von Kámán and Pohlhausen. The method allows us to exploe nonlinea coections to the standad Ekman theoy even at lage Rossby numbes. We conside both the standad self-simila ansatz fo the velocity pofile, which assumes that a single length scale descibes the bounday laye stuctue, and a new non self-simila ansatz in which the decay and the oscillations of the bounday laye ae descibed by two diffeent length scales. Fo both pofiles we calculate the up-flow in a votex coe in solid body otation analytically. We compae the quantitative pedictions of the model with von Kámán s exact similaity solution and find that the esults fo the non self-simila pofile ae in almost pefect quantitative ageement with the exact solution and pefoms makedly bette than the self-simila pofile. 1. Intoduction Ekman layes ae bounday layes which fom in otating systems at eithe fee o solid boundaies, typically nomal to the axis of otation. Such bounday layes play impotant oles in many geophysical and technical flows including lage atmospheic votices and souce-sink flows in tubines (Lugt 1995). In the bulk, fa fom any walls, such flows ae essentially two-dimensional with velocities mainly pependicula to the axis of otation. Due to the Ekman laye, howeve, thee is also a small but impotant velocity component along the axis of otation, which is efeed to as eithe Ekman pumping o suction. A cyclone (lage absolute otation ate in the bulk than at the bounday) ceates Ekman pumping wheeas an anti-cyclone (smalle absolute otation ate in the bulk than at the bounday) ceates Ekman suction. The popeties of Ekman layes ae well-descibed by linea diffeential equations when the flow in the otating efeence system is weak compaed with the backgound otation, i.e., at low Rossby numbe. The Rossby numbe is defined as the chaacteistic value of the atio of the nonlinea tems and the Coiolis tems in the Navie-Stokes equations in the fame of efeence otating with the backgound otation ate (Batchelo 1967). At lage values of the Rossby numbe the nonlinea tems become impotant, which makes it difficult to solve the govening equations and ceates a need fo appoximate methods. We model the nonlinea Ekman layes at solid boundaies fo flows with otational symmety using an aveaging method simila to the technique intoduced by von Kámán (1921) and Pohlhausen (1921). Owen, Pincombe & Roges (1985) applied an aveaging

2 2 A. Andesen, B. Lautup and T. Boh method to descibe lamina and tubulent bounday layes in otating fluids. They used a self-simila bounday laye pofile (descibed by a single length scale) and focused on a paticula geomety in which up-flow fom the bounday laye does not occu because of the symmety of the poblem. We extend thei equations to include the tems descibing up-flow fom the bounday laye. The self-simila pofile has the same damped oscillatoy stuctue as the solution to the linea Ekman equations but with a bounday laye thickness which depends on the local Rossby numbe. To investigate the applicability of the method we compae the esults with the exact similaity solution due to von Kámán (1921) fo a fluid above an infinite otating disk. The esults agee qualitatively, but it tuns out that the quantitative ageement is poo, leading, e.g., to a consideable oveestimate of the Ekman pumping. We tace this discepancy to the assumption that a self-simila ansatz fo the velocity pofile based on a single length scale can descibe both the oscillatoy and the exponential components of the bounday laye. This is coect fo the linea solution but fo the nonlinea case thee is no basis fo such an assumption. We thus intoduce a non self-simila pofile which involves two length scales and show that the bounday laye stuctue and the up-flow fo the non self-simila pofile agee almost pefectly with von Kámán s exact solution at abitay (positive) Rossby numbe. The layout of the pape is as follows. In 2 we biefly outline the linea Ekman theoy. In 3 we pesent the two diffeent aveaging methods and deive analytical expessions fo the cental up-flow velocity in a votex coe in solid body otation. In 4 we pesent numeical solutions of the aveaged equations fo a geneic souce-sink votex. Finally, in 5 we apply the aveaging techniques to the von Kámán flow and compae the esults with the exact solution. 2. Linea Ekman theoy We focus on the flow in a containe with otational symmety which is otating about its vetical axis of symmety, and we thus apply the Navie-Stokes equations in pola coodinates witten in the fame of efeence co-otating with the containe. We let u, v, and w denote the adial, the azimuthal, and the vetical velocity component, espectively. In the bulk of the fluid the flow is appoximately two-dimensional, independent of the vetical coodinate z and we denote the azimuthal velocity thee by v. At the bottom of the containe thee is a mismatch between the bulk velocity and the igidly otating solid bottom, and theefoe an Ekman laye is fomed (see, e.g., Batchelo 1967). We assume that the vetical velocity component is small and fom the vetical Navie-Stokes equation it thus follows that the effective pessue is independent of z and equal to its value in the geostophic bulk of the fluid, i.e., fa above the bottom Ekman laye. At small Rossby numbe the govening equations educe to the following linea equations = ν 2 u z Ω (v v ) (2.1) = ν 2 v z 2 2 Ω u, (2.2) whee Ω is the angula velocity of the containe. With the bounday conditions u(, z) = v(, z) = at z =, u(, z) as z, and v(, z) v () as z, the solution is u(, z) = v () e z/δ sin(z/δ) (2.3) v(, z) = v () [1 e z/δ cos(z/δ)], (2.4)

3 Aveaging method fo nonlinea lamina Ekman layes 3 whee δ = ν/ω is the bounday laye thickness. The continuity equation links u and w 1 (u) and since w(, z) = fo z = we have + w z =, (2.5) w(, z) = δ d(v ) ( 1 e z/δ [sin(z/δ) + cos(z/δ)] ). (2.6) 2 d It follows that the vetical velocity component in the bulk of the fluid is popotional to the z-component of the voticity thee w = δ d(v ). (2.7) 2 d Depending on the sign of w, this is efeed to as Ekman pumping and Ekman suction. If the fluid is otating as a solid body with v = C it follows that w = ( ν Ω ) 1/2 C. (2.8) A cyclonic votex of this kind thus geneates up-flow wheeas and anti-cyclonic votex gives ise to down-flow. In the following we calculate nonlinea coections to (2.8) using the aveaged equations. 3. Aveaged bounday laye equations 3.1. Govening equations The linea equations (2.1) and (2.2) ae only valid when the flow in the co-otating efeence system is weak and the nonlinea tems can be neglected compaed to the Coiolis tems, i.e., at small Rossby numbe. At lage values of the Rossby numbe we appoximate the adial and the azimuthal Navie-Stokes equations by bounday laye equations in which we neglect the deivatives with espect to in the viscous tems u u + w u z v2 v 2 = ν 2 u z 2 + 2Ω(v v ) (3.1) u v + w v z + uv = ν 2 v 2Ωu, (3.2) z2 whee v(, z) v () when z. Using the continuity equation (2.5) we ewite the Navie-Stokes equations in a fom which is easy to integate with espect to z 1 (u 2 ) + (uw) v2 v 2 z 1 ( 2 uv) 2 + (vw) z = ν 2 u z 2 + 2Ω(v v ) (3.3) = ν 2 v 2Ωu. (3.4) z2 These two equations ae used in the following as the stating point in the deivation of the aveaged equations.

4 4 A. Andesen, B. Lautup and T. Boh 3.2. Self-simila pofile In the aveaging method we make an ansatz on the z-dependence of the velocity components, i.e., on the bounday laye stuctue. Following Owen, Pincombe & Roges (1985) we make a staight fowad genealization of (2.3) (2.4) and conside the self-simila pofile whee u, v, and δ ae functions of, and whee u(, z) = u f(z/δ) (3.5) v(, z) = v [1 g(z/δ)], (3.6) f(η) exp( η) sin η (3.7) g(η) exp( η) cos η. (3.8) The tem self-simila is used hee to emphasize that the pofile does not depend on both and z but only on the combination z/δ(). The govening equations (3.3) and (3.4) ae integated with espect to z fom zeo to infinity, and the esulting aveaged equations ead 1 d(u 2 I δ) 2 + (2I 4 I 5 ) v2 d δ = νu δ f () 2I 4 Ωv δ (3.9) (I 1 I 3 ) 1 2 d( 2 u v δ) d I 1 v d(u δ) d whee the following definitions of integals ae used I 1 = I 3 = I 5 = dηf(η) = 1 2, I 2 = dηf(η)g(η) = 1 8, I 4 = = νv δ g () 2I 1 Ωu δ, (3.1) dηf 2 (η) = 1 8 dηg(η) = 1 2 (3.11) (3.12) dηg 2 (η) = 3 8. (3.13) Equations (3.9) and (3.1) ae witten fo a geneal self-simila pofile as defined by (3.5) and (3.6). The integals can be evaluated as shown in (3.11) (3.13) fo the ansatz (3.7) and (3.8), and the govening equations become 1 d(u 2 δ) + 5v2 8 d 8 δ = νu Ωv δ (3.14) δ 3 d( 2 u v δ) 8 2 v d(u δ) = νv d 2 d δ Ωu δ. (3.15) Notice that w does not appea in the aveaged equations. The flow ate, q, of the adial inflow is q 2π dz u = 2πu dzf(z/δ) = πu δ. (3.16)

5 Aveaging method fo nonlinea lamina Ekman layes 5 Owen, Pincombe & Roges (1985) assumed that q is independent of and theefoe the second tem on the left hand side of (3.15) does not appea in thei aveaged azimuthal equation. With linea velocity pofiles u = A and v = C, we find that δ becomes independent of and the aveaged equations (3.14) and (3.15) educe to the algebaic equations 3 8 A C2 = ν A δ 2 Ω C (3.17) 1 2 A C = ν C δ 2 + Ω A, (3.18) In this special case the Rossby numbe is by definition Ro C 2 Ω. (3.19) The vetical velocity component in the bulk of the fluid, w, is though the continuity equation (2.5) elated to the adial velocity component u w = 1 d(u δ), (3.2) 2 d and thus it follows that the assumption of a linea adial velocity pofile leads to w = δ A. (3.21) By solving (3.17) and (3.18) fo δ and A it follows that the nonlinea Ekman pumping velocity is educed in compaison with the linea esult as the Rossby numbe is inceased [ ] 1/4 w (ν/ω) 1/2 C = Ro (4 + Ro)(1 + Ro) 2. (3.22) At Ro = the ight hand side is equal to one in ageement with linea Ekman theoy (2.8), and in the limit of infinite Rossby numbe w is independent of Ω and we have w = 2 1/4 ν C ν C Non self-simila pofile As we will show in 5 the self-simila pofile leads to pedictions which ae not in quantitative ageement with von Kámán s exact similaity solution. We thus intoduce the following non self-simila bounday laye pofile u(, z) = v e z/δ 1 sin(z/δ 2 ) (3.23) v(, z) = v [ 1 e z/δ 1 cos(z/δ 2 ) ], (3.24) whee v, δ 1, and δ 2 ae functions of. Like the self-simila pofile this is a genealization of (2.3) (2.4) but now with two length scales descibing espectively the decay and In this expession we have not allowed fo diffeent amplitudes on the adial and azimuthal fields. Allowing this equies anothe condition to close the system of equations. We have investigated the case whee this othe condition is the standad wall cuvatue compatibility condition, ν 2 u z 2 z= = 2Ωv + v2. The esulting equations ae consideably moe complex, and lead in the end to no appeciable impovement in the poblems that we have consideed.

6 6 A. Andesen, B. Lautup and T. Boh the oscillations in the bounday laye stuctue. At low Rossby numbe we have δ 1 = δ 2 = ν/ω but in geneal the two length scales ae diffeent. The pesence of two length scales in the solution is appaent in the exact asymptotic solution at lage z as shown by Bödewadt (194) and Roges & Lance (196). Ou ansatz has the same stuctue as the special asymptotic solution which also satisfies the no-slip bounday condition at z =. The aveaged equations ae in this case whee we define 1 d(vk 2 2 ) d 1 d( 2 vk 2 3 ) 2 v K 1 d + v2 (2K 4 K 5 ) = νv δ 2 2Ωv K 4 (3.25) d(v ) d = νv δ 1 + 2Ωv K 1, (3.26) K 1 = 4K 3 = δ2 1 δ 2 δ1 2 +, K 2 = δ2 2 δ 3 1 4(δ δ2 2 ) (3.27) K 4 = δ 1 δ2 2 δ1 2 +, K 5 = K δ2 2 2 K 4. (3.28) With the linea velocity pofile v = C, the aveaged equations educe to C ( 1 2 δ ) 2 δ2 2 = ν δ2 1 + δ2 2 2 Ω δ2 2 (3.29) δ 1 δ 2 C δ 2 1 = ν δ2 1 + δ 2 2 δ 1 δ Ω δ 2 1, (3.3) unde the assumption that δ 1 and δ 2 do not depend on. Using the non self-simila pofile we thus pedict that the nonlinea Ekman pumping velocity is educed in the following way in compaison with the linea esult w (ν/ω) 1/2 C = ((2 + Ro)(2 + 3Ro))1/4 2(1 + Ro) (3.31) Simila to (3.22) the ight hand side is equal to one when Ro =, and in the limit of infinite Ro we have w = 3 1/4 ν C ν C. The non self-simila pofile thus gives a weake up-flow than the self-simila pofile in the limit of lage Rossby numbe. 4. Numeical solution fo souce-sink votex Souce-sink flows in a otating cylindical containe ae impotant examples of votex flows contolled by Ekman layes. In this paagaph we conside a geneic souce-sink votex and assume an azimuthal velocity pofile of the fom v = Γ 2π [ 1 exp ( 2 κ 2 )]. (4.1) The velocity pofile is linea when κ, and it is like a line votex with ciculation Γ when κ. This is simila to a Rankine votex, and the pofile is thus like a smooth Rankine votex with a soft tansition between the votex coe and the outside velocity

7 Aveaging method fo nonlinea lamina Ekman layes 7 field. It follows by definition that the z-component of voticity has a Gaussian pofile and in linea Ekman theoy we thus pedict that the up-flow is ) ν/ω Γ w = 2πκ 2 exp ( 2 κ 2. (4.2) The vetical velocity component in the bulk of the fluid, w, is though the continuity equation elated to q w = 1 dq 2π d, (4.3) and integation gives the following expession fo q in tems of w q() = Q 2π dxw (x)x, (4.4) whee Q is defined as the limit value of q fo lage. At κ linea Ekman theoy is valid since the local Rossby numbe thee is small, and the ciculation Γ is thus linked to the flow-ate Q. The velocity pofiles now become Q v = π ν/ω w = Q πκ 2 exp [ 1 exp ( 2 κ 2 ) ( 2 κ 2 )] (4.5). (4.6) To obtain numeical solutions fo the self-simila pofile we eliminate the bounday laye thickness δ in the aveaged equations (3.9) and (3.1) and use u, v, and q as the dependent vaiables. We wite the equations in the following fom assuming the azimuthal velocity component v to be known u = 3u v 3u v 5v2 16π2 ν 2 u 2 ( u u q 2 8Ω + v ) v u q = 3qv v (4.7) + 3q + 8π2 ν 2 u + 8Ωq, (4.8) q v and supplement the equations by bounday conditions at lage = R: Q u (R) = π ν/ω R (4.9) q(r) = Q. (4.1) To solve the equations fo the non self-simila pofile numeically we use the functions α K 1 and β δ 1 /δ 2 and ewite equations (3.25) and (3.26) in tems of them as α = 2 [ ( v + v ) v + 4Ω α 2ν ] β = 1 2αv δ 1 [( 5v + 4v + 16Ω ) (4.11) ( ) 3v α αβ + + 4Ω β 1ν ]. (4.12) δ 2 We supplement the equations by the following bounday conditions at lage = R:

8 8 A. Andesen, B. Lautup and T. Boh (a) -.5 v.75 u (b) (c).1 delta 1 w.1.5 delta.8 delta 2.6 (d) Figue 1. Numeical solution of the aveaged equations fo the souce-sink votex. (a) The assumed azimuthal velocity v, (b) the adial velocity u, (c) the vetical velocity w, and (d) the bounday laye thickness δ all computed at a cental Rossby numbe of 1. The solid cuves show the linea theoy, the dashed cuves the esult obtained with the self-simila pofile, and the dot-dashed cuves the esult obtained with the non self-simila pofile. In (b) the solid and the dot-dashed cuves ae identical by assumption and thus only the solid cuve is shown. α(r) = 1 2 ν Ω (4.13) β(r) = 1. (4.14) Figue 1 shows the solution fo u, w, and δ with ν =.1, Ω = 1, κ = 1, and Q = π/5, which coesponds to a cental Rossby numbe of 1. At lage values of the solution is well-descibed by the linea theoy, wheeas the bounday laye thickness deceases close to the votex cente compaed to the constant linea Ekman laye thickness. The cental up-flow velocities ae descibed quantitatively by (3.22) and (3.31) with Ro = The similaity solution by von Kámán In this paagaph we investigate the applicability of the self-simila and the non selfsimila bounday laye pofile by compaing them with the exact similaity solution intoduced by von Kámán. The flow above a flat otating disk of infinite extension was studied by von Kámán (1921) who intoduced the following ansatz fo the velocity field

9 Aveaging method fo nonlinea lamina Ekman layes 9 u(, z) = F (z) (5.1) v(, z) = G(z) (5.2) w(, z) = H(z). (5.3) With this ansatz the Navie-Stokes equations educe to a set of odinay diffeential equations fo F, G, and H. The Navie-Stokes equations and the continuity equation in the fame of efeence otating with angula velocity Ω become F 2 + H F G 2 + C 2 = ν F + 2 Ω (G C) (5.4) 2 F G + H G = ν G 2 Ω F (5.5) H H = P + ν H (5.6) 2 F + H =, (5.7) which ae to be supplemented by the five bounday conditions F () = F ( ) = (5.8) G() = G( ) = C (5.9) H() = (5.1) In the oiginal poblem consideed by von Kámán the azimuthal velocity goes to zeo fa above the disk in laboatoy fame coesponding to C = Ω. Similaly the poblem consideed by Bödewadt (194) of a fluid in solid body otation above a fixed disk is obtained with Ω =. Linea Ekman theoy (2.3), (2.4), and (2.6) is valid at small values of C. Figue 2 shows numeical solutions (symbols) of von Kámán s equations (5.4) (5.7) togethe with esults of the aveaging method with the self-simila pofile (dashed cuves) and the non self-simila pofile (solid cuves). The solution with the non self-simila pofile has δ 1 δ 2 and it captues the damped oscillatoy stuctue of the exact solution at Ro = 4 makedly bette than the self-simila pofile. This is paticulaly evident fo the vetical velocity component shown in figue 2(d) fo Rossby numbes between and 8. The asymptotic behavio of the exact solution in the limit of infinite Rossby numbe was descibed by Bödewadt (194) who found that w = ν C. The pediction, w ν C, obtained using the non self-simila pofile agees well with this esult wheeas the method using the self-simila pofile oveestimates the up-flow velocity, w ν C. 6. Conclusions We have analyzed two methods based on aveaging fo computing the stuctue of nonlinea Ekman layes. We find that a non self-simila velocity pofile with sepaate length scales fo the decay and the oscillations, espectively, give vey accuate pedictions. We believe that this method will be useful fo many impotant flow configuations and hope that it will be confonted with accuate measuements of the up-flow in wellcontolled laboatoy expeiments and ideally with velocity measuements esolving the Ekman laye stuctue.

10 1 A. Andesen, B. Lautup and T. Boh (b) F -2 G (a) z z.6 1 H (c) up-flow velocity (d) z Ro Figue 2. Numeical solutions of the aveaged equations fo the von Kámán flow at Ro = 4. (a) (c) the bounday laye stuctue of u, v, and w, and (d) the atio between the nonlinea up-flow velocity and the esult fom linea Ekman theoy as function of Ro. The symbols show the numeical solution of (5.4) (5.7), the dashed cuves the esult obtained with the self-simila pofile, and the solid cuves the esult obtained with the non self-simila pofile. 7. Acknowledgments We thank Jens Juul Rasmussen and Bjane Stenum fo many valuable discussions. REFERENCES Batchelo, G. K An Intoduction to Fluid Dynamics. Cambidge Univesity Pess. Bödewadt, U. T. 194 Die Dehstömung übe festem Gunde. Z. angew. Math. Mech. 2, Kámán, T. von 1921 Übe laminae und tubulente Reibung. Z. angew. Math. Mech. 1, Lugt, H. J Votex Flow in Natue and Technology. Kiege. Owen, J. M., Pincombe, J. R. & Roges, R. H Souce-sink flow inside a otating cylindical cavity. J. Fluid Mech. 155, Pohlhausen, K Zu näheungsweisen Integation de Diffeentialgleichung de laminaen Genzschicht. Z. angew. Math. Mech. 1, Roges, M. H. & Lance, G. N. 196 The otationally symmetic flow of a viscous fluid in the pesence of an infinite otating disk. J. Fluid Mech. 7,