The evaluation of the far field integral in the Green's function representation for steady Oseen flow

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The evaluation of the far field integral in the Green's function representation for steady Oseen flow Fishwick, NJ and Chadwick, EA http://dx.doi.org/0.063/.

1 The evaluation of the far field integral in the Green's function representation for steady Oseen flow Fishwick, NJ and Chadwick, EA Title Authors Type URL The evaluation of the far field integral in the Green's function representation for steady Oseen flow Fishwick, NJ and Chadwick, EA Article Published Date 006 This version is available at: USIR is a digital collection of the research output of the University of Salford. Where copyright permits, full text material held in the repository is made freely available online and can be read, downloaded and copied for non commercial private study or research purposes. Please check the manuscript for any further copyright restrictions. For more information, including our policy and submission procedure, please contact the Repository Team at: usir@salford.ac.uk.

2 PHYSICS OF FLUIDS 8, 006 The evaluation of the far-field integral in the Green s function representation for steady Oseen flow Nina Fishwick and Edmund Chadwick School of Computing, Science and Engineering, University of Salford, Salford M5 4WT, United Kingdom Received 4 May 006; accepted September 006 Consider the Green s function representation of an exterior problem in steady Oseen flow. The far-field integral in the formulation is shown to be zero. 006 American Institute of Physics. DOI: 0.063/ I. INTRODUCTION Oseen gives the Green s function representation of the exterior problem in steady Oseen flow, but assumes that the far-field integral in the formulation is zero without proof. It is essential to show that this integral is zero for the Oseen representation to be valid. In low Reynolds number flow applications, the Oseen equations are used within singular perturbation theory as a far-field matching to Stokes flow,3.in this case, only the singular point solution is required and so the integral is satisfied trivially. However, there are at least two increasingly important applications of the Oseen equations for high Reynolds number in the sense that the Reynolds number is much greater than one flows. The first application is the decay of the trailing vortex wake behind an aircraft. This has attracted significant recent interest with the advent of superheavy class aircraft, such as the Airbus A380, and the stipulation of safe separation distances between aircraft flying through this wake during landing and takeoff. However, the line vortex in inviscid flow has a constant strength and profile. So in order to model vortex decay, viscosity must be modelled which diffuses the vorticity. Batchelor 4 considers far-field Oseen flow to represent the trailing vorticity as the Oseen formulation is a linearization to a uniform stream of the Navier-Stokes equations, and so retains the viscous term. The Batchelor vortex has been the focus of work on stability analysis for the trailing vorticity, a review given by Delbende. 5 Chadwick 6 shows that the horseshoe vortex in Oseen flow, whose arms are trailing line vortices, is equivalent to a spanwise distribution of lift Oseenlets a lift Oseenlet is the singular point lift solution in Oseen flow. Furthermore, the trailing vortex behind an aircraft has been developed from rollup of the vortex sheet, and even at large distances behind an aircraft the representation by a line vortex is insufficient and instead a distribution is required see Ref. 7, chapter 3. The requirement for a distribution of singular solutions means that an integral distribution of singular solutions over a surface, as formulated by Oseen, is necessary. In this case, it is then necessary to show that the far-field integral arising from Oseen s representation is zero. The second application is in the field of slender body theory and related theories. The usual approach is for inner and outer expansions around the boundary layer, with the inner region being purely viscous. However, Chadwick 8 presents a slender body theory in Oseen flow where it is assumed that for a streamlined body satisfying a Kutta condition at the trailing edge or end section, that Oseen flow the perturbation to a uniform stream is valid as an outer expansion. In the application to lift on a slender wing, Chadwick 9 shows that the retention of the viscous terms in the formulation are important for the lift calculation and to ensure the wake is regularized and so is not singular as in the inviscid flow representation. Again, the Oseen representation is given by a distribution of solutions over a Green s integral surface rather than reducing to point solutions, as in the case of low Reynolds number singular perturbation theory. It is therefore essential to show that the far-field Green s integral arising from Oseen s representation of the Oseen velocity is zero, for the Oseen representation to be valid for both these important problems. One would assume that a likely way to proceed would be to represent the far-field integral surface as the surface of a sphere, and divide this surface into an interior wake surface and exterior surface where appropriate approximations can be made. However, when this is done then it can only be shown that the far-field integral is bounded by a constant. So, the idea is to find an appropriate division of the far-field surface such that the farfield integral tends to zero as the radius of the sphere tends to infinity. In the present paper, this is achieved by dividing the surface of the sphere into three surfaces by: the intersection of a cone subtended by a small angle and enclosing the wake; and also by the intersection of the wake boundary. Making appropriate approximations within the three regions then enables us to show that the far-field integral in the Green s function formulation of steady Oseen flow is indeed zero as expected. II. THE OSEEN FORMULATION The steady Oseen equations see Ref., pp for the Oseen velocity u, a perturbation to the uniform stream velocity U in the x direction such that the Cartesian coordinates are given by x,x,x 3, and Oseen pressure p are U u x p + u, u 0, p 0, 9 where and are the fluid density and dynamical coefficient of viscosity, respectively, and both are assumed to be con /006/8//0/$3.00 8, American Institute of Physics

3 - N. Fishwick and E. Chadwick Phys. Fluids 8, stant. denotes the gradient operator and is the Laplacian operator. As r, then u, p 0. The Oseen velocity is then represented by an integral distribution of Green s functions called Oseenlets or Oseen fundamental solutions. Consider four solutions to the Oseen equations u, p and u m, p m, m3, for the Oseen velocity and pressure, respectively. From we find that y Uu i yu i m x y pyu m y i x y + u i yp m x y + 0 i y j u i yu m y i x y u i y u m j y i x j y for a point yx in the fluid. Applying Gauss s theorem to the 05 volume integral of the above expression gives S ypyu j m x y + u j yp m x y + u i y y j u i m x y y j u i yu i m x y + Uu i yu i m x y jn j ds 0 m, 4 FIG.. The division of the surface of the sphere S R. u m x S 0pyu j m x y + u j yp m x y + u i y y j u i m x y y j u i yu i m x y + Uu i yu i m x y jn j ds where S y is a surface enclosing a volume of fluid, and the integration is over the y variable. The Green s functions can be represented by the potentials and such that u i m z m z i + m z i k * mi, 3 p m z U m, 4 z 5 where ku/ and zx y, and 6 7 m z lnr z, 4U z m m z 4U e kr z lnr z, z m * z, 8 4U R 9 where zr. So from 6, e kr z * m. 0 z m z Substitute the Green s functions into 4 such that S y consists of three surfaces: S 0, which encloses a body surface S B, S, 3 which is a sphere radius 0 about the point zx, and S R, 4 which is a sphere radius R Fig.. 5 Following the contribution from the surface S is u m x, 6 and if the contribution from the surface S R is assumed to be 7 zero, then we get the Green s function integral representation 8 in Oseen flow: 5 6 III. EVALUATION OF THE FAR-FIELD INTEGRAL The integration over the surface S R is given by S Rpyu j m z + u j yp m z + u i y u m y i z u i yu m j y i j z + Uu i yu i m z jn j ds. 8 The surface S R is such that zr, and we want to show that the integration over this surface tends to zero. Taking the modulus of 8 and bringing this modulus into the integrand, then we can show that 8 tends to zero if R u jy u m i maxs R yds 0 m ij 9 since u i m z y j A j u m i z for some constant A j, and since p m z/4r, and 44 u i y 0 as R. In 9, we define 0 m ij 0 for all 45 i, j,m3. 46 To evaluate 9, the integration surface is divided into 47 three see Fig. : 48. The surface S wake such that zr and r z +z 3 a 0 z /k, 0a 0 ;. the surface S cone wake such that zr and a 0 / kz 0,0 0 ;

4 -3 The evaluation of the far-field integral Phys. Fluids 8, 006 FIG.. The surface S y. O r z O r z in the far-field region S wake as z. The integral calculation of 9 is now evaluated over the three regions of the integral surface for the varying index values i,m3. However, since u i m u m i, u has similar form to u 3, and u has similar form to u 3 3, then it is sufficient to consider the four permutations i,m,3,,,,, and,. Permutation i,m,3: Over the area S R cone, applying the approximation 0 to the Oseenlet given by 6 in the region S R cone gives and the surface S R cone such that zr and 0, where a 0 and 0 are constants, and the Cartesian and spherical coordinate are such that z R cos, z R sin cos, z 3 R sin sin. The approximations applied to the fundamental solutions within these three regions is given next. The surface S R is divided into the three areas S R cone, S cone wake, and S wake, such that the following approximations are made in each area. Area S R cone : Within this area 0 and so the approximation R z b 0 R, b 0 0 cos 0 holds. Area S cone wake : In this region r/z 0 and so we can apply the approximation and so z 3 Similarly and so and so z 3 b 0+b 0 4UR 5 b 0+kR + b 0 4UR e kr/b 0 7 R S R cone z 3 ds R S R cone ds b 0+b 0. 6 z 3 U R z z + r / z r r4 3 z z 8z + Or6 /z 5, where O means of order of. So, 7 e kr z e kr /z +Or /z u j y u 3 zds 0 maxs R cone R 9 since u j y max 0asR. Over the area S cone wake, applying the approximation to the Oseenlet given by 6 in the region S cone wake gives AQ: # e kr /z +Or /z kr /z e kr /z +or /z where o means of order less than, since +a b+ and so +a b +a for a0, b0. Finally, an element of area s over the surface is approximated by 77 s R sin rr+or /z Area S wake : In this region kr /z a 0 /, and so 79 from Ref. 0, p. 69, Sec. 4.., since e kr z kr z + k R z + OR z 3! kr z + k r 4 8z + Or6 /z 3, 4 z 3 Ur 0 06 and so using the approximation for elements of the surface 07 3 gives 08 R S cone wake 0 z a ds z 3 0 a 0 z /k Ur drd U ln 0z lna 0 z / k for some constant a. We note that if this integration was continued into the wake then the right-hand side of would approach infinity and no bound would be obtained, which demonstrates the necessity for dividing the surface of the sphere up such that there is a wake region. Similarly

5 -4 N. Fishwick and E. Chadwick Phys. Fluids 8, z 3 a 3 e kr /z z u j y u 3 zds 0 R maxs R for some a 3 independent of the coordinate variables, since /r a 0 /z. So using the approximation for elements of the surface 3 gives R S cone wake 0 z a 3 ds e z 3 0 a kr /z rdrd 0 z /k z a 3 e ka 0 /, k 3 3 which is bounded. In the far field, we expect the fluid velocity u j y to behave as a combination of the fundamental so- 4 5 lutions u m j y to leading order. So we expect that 6 u j y max 0 faster than /ln R as R. This means that 7 combining the results and 3 we expect u j y u 3 zds 0. 4 R 8 maxs cone wake 9 Over the area S wake, making use of the approximation 30, gives an approximation for in this region z z r 4UR R z Ur+ z 3 r 4z +Or 4 /z z r Ur 4z +Or 4 /z Further, making use of the approximation 4 for e kr z in 35 this region then gives + z kr k r 4 Ur z 8z + Or6 /z 3, so + k z z 3 z 3 8Uz +Or /z Therefore u 3 zds R S wake k R 4Uz 0 k a 0 4 6U +Oa 0. a 0 z r 3 dr+or /z 7 8 Combining all results together over the three surfaces S R cone, S cone wake, and S wake on the surface of the sphere S R then gives as expected. Permutation i,m,: Over the area S R cone, following the same approximations as for the permutation i,m,3, then in this region we have z a 4 R, z a 5 R e kr/a 0, * a 6 R e kr/a 0 30 for some constants a 4, a 5, and a 6. So, using the same argument as for the permutation i,m,3, then in this region we have u j y u zds 0 3 R maxs R cone 55 since u j y max 0asR. 56 Over the area S cone wake, following the same approximations as for the permutation i,m,3, then in this region we have 59 z a 7 r, z a 8 e kr /z, 60 z /z 3 * a 9 e kr z 6 for some constants a 7, a 8, and a 9. So, using the same argument as for the permutation i,m,3, then in this area 63 6 we have 64 u j y u zds 0 33 R maxs R cone 65 since u j y max 0 faster than /ln R as R. 66 Over the area S wake, making use of the approximations 67 for and the approximation 4 for e kr z in this 68 region gives 69 + z z kz +Or /z 4Uz 70 Therefore k 4Uz +Or /z. u zds ka 0 R S wake 4U +Oa 0, which is bounded. Also in this region, * a 0 /z and so combining all results together over the three surfaces S R cone, S cone wake, and S wake on the surface of the sphere S R then gives u j y u zds 0 R maxs R as expected

6 -5 The evaluation of the far-field integral Phys. Fluids 8, Permutations i,m, and i,m,3: The 8 analysis for these permutations give similar bounds, with the 8 added simplification that z lnr z /R. This means 83 that the condition 9 given by R u jy u m i maxs R yds 0 m 84 ij holds for all i, j, and m. So the evaluation of the far-field integral in the Green s function representation for steady Oseen flow is zero as expected. 88 ACKNOWLEDGMENT This work was carried out during an EPSRC sponsored studentship for a doctorate in applied mathematics at the University of Salford. C. W. Oseen, Neure Methoden und Ergebnisse in der Hydrodynamik Akad. Verlagsgesellschaft, Leipzig, 97. S. Kaplun and P. A. Lagerstrom, Low Reynolds number flow past a circular cylinder, J. Math. Mech. 6, I. Proudman and J. R. A. Pearson, Expansions of small Reynolds number for the flow past a sphere and a circular cylinder, J. Fluid Mech., G. K. Batchelor, Axial flow in trailing line vortices, J. Fluid Mech. 0, I. Delbende, et al., Various aspects of fluid vortices, C. R. Mec. 33, E. Chadwick, The vortex line in steady, incompressible Oseen flow, Proc. R. Soc. London, Ser. A 46, P. G. Saffman, Vortex Dynamics Cambridge University Press, Cambridge, E. Chadwick, A slender-body theory in Oseen flow, Proc. R. Soc. London, Ser. A 458, E. Chadwick, A slender wing theory in potential flow, Proc. R. Soc. London, Ser. A 46, M. Abramowitz and I. N. Stegun, Handbook of Mathematical Functions Dover, New York,

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