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

The evaluation of the far­field integral in 

the Green's function representation for 

steady Oseen flow

Fishwick, NJ and Chadwick, EA

http://dx.doi.org/10.1063/1.2388248

Title

The evaluation of the far­field integral in the Green's function 

representation for steady Oseen flow

Authors

Fishwick, NJ and Chadwick, EA

Type

Article

URL

This version is available at: http://usir.salford.ac.uk/389/

Published Date

2006

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 non­commercial private study or research purposes. Please check the 

manuscript for any further copyright restrictions.

PROOF COPY 016611PHF

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

共Received 24 May 2006; accepted 22 September 2006兲

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. © 2006 American Institute of Physics.

关DOI:10.1063/1.2388248兴

I. INTRODUCTION

Oseen1 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 ap-plications, the Oseen equations are used within singular per-turbation theory as a far-field matching to Stokes flow2,3. 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 equa-tions for high Reynolds number共in the sense that the Rey-nolds number is much greater than one兲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 dis-tances between aircraft flying through this wake during land-ing 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 vortic-ity. Batchelor4considers 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.5Chadwick6shows that the horse-shoe vortex in Oseen flow, whose arms are trailing line vor-tices, is equivalent to a spanwise distribution of lift Oseenlets

共a lift Oseenlet is the singular point lift solution in Oseen flow兲. 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

共see Ref.7, chapter 13兲. The requirement for a distribution of singular solutions means that an integral distribution of sin-gular 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, Chadwick8 pre-sents a slender body theory in Oseen flow where it is

as-sumed that for a streamlined body satisfying a Kutta condi-tion at the trailing edge or end seccondi-tion, that Oseen flow共the perturbation to a uniform stream兲is valid as an outer expan-sion. In the application to lift on a slender wing, Chadwick9 shows that the retention of the viscous terms in the formula-tion are important for the lift calculaformula-tion and to ensure the wake is regularized共and so is not singular as in the inviscid flow representation兲. 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 ve-locity 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 field surface such that the far-field 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 共see Ref.1, pp. 30-38兲for the Oseen velocity u, a perturbation to the uniform stream velocityUin thex1direction such that the Cartesian

coordi-nates are given by共x1,x2,x3兲, and Oseen pressurep are

␳U⳵u

⳵x1

= −ⵜp+共␮ⵜ2兲u, ⵱·u= 0, 共1兲

ⵜ2_{p= 0, 共2兲}

where␳and␮are the fluid density and dynamical coefficient of viscosity, respectively, and both are assumed to be

con-1

2

3 4

5

6 7 8

9

10

11 12 13 14 15 16

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

54

55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84

85

86 87 88 89

90

91

92 93

PROOF COPY 016611PHF

stant.ⵜdenotes the gradient operator andⵜ2_{is the Laplacian}

operator. Asr→⬁, thenu,p→0. The Oseen velocity is then represented by an integral distribution of Green’s functions called Oseenlets or Oseen fundamental solutions.1 Consider four solutions to the Oseen equations共u,p兲and共u共m兲_,p共m兲_兲,

1ⱕmⱕ3, for the Oseen velocity and pressure, respectively. From共1兲we find that

⳵

⳵y₁兵␳Uui共y兲ui 共m兲_共

x−y兲其

= − ⳵

⳵yi

兵p共y兲ui共 m兲_共

x−y兲+ui共y兲p共m兲共x−y兲其+␮

⳵ ⳵yj

⫻

再

⳵ui

⳵yj

共y兲ui共 m兲_共

x−y兲−ui共y兲

⳵ ⳵yj

ui共 m兲_共

x−y兲

冎

共3兲 for a pointy=xin the fluid. Applying Gauss’s theorem to the volume integral of the above expression gives

冕冕

S_y

冋

p共y兲u共j

m_兲共

x−y兲+uj共y兲p共m兲共x−y兲

+␮

再

ui共y兲

⳵ ⳵yj

ui共 m兲_共

x−y兲− ⳵

⳵yj ui共y兲ui共

m兲_共 x−y兲

冎

+␳Uui共y兲ui共 m_兲共

x−y兲␦j1

册

njds= 0共m兲, 共4兲

whereSy is a surface enclosing a volume of fluid, and the

integration is over they variable. The Green’s functions can be represented by the potentials␾and␹ such that

ui共 m兲_共

z兲= ⳵␾

共m_兲

⳵zi

+⳵␹

共m_兲

⳵zi

− 2k␹*␦mi,

共5兲

p共m兲共z兲= −␳U⳵␾ 共m兲

⳵z1

,

wherek=␳U/ 2␮ andz=x−y, and

␾共m_兲共

z兲= 1 4␲␳U

⳵ ⳵zm

ln共R−z1兲,

␹共m_兲共

z兲= − 1 4␲␳Ue

−k_共R−z_1兲 ⳵

⳵zm

ln共R−z1兲, 共6兲

␹*_共z兲= 1

4␲␳U e−k共R−z1兲

R ,

where兩z兩=R. So from共6兲,

⳵␹*

⳵zm

=⳵␹

共m兲

⳵z1

.

Substitute the Green’s functions into共4兲such thatSyconsists

of three surfaces:S0, which encloses a body surfaceSB,S␦,

which is a sphere radius␦→0 about the pointz=x, and SR,

which is a sphere radiusR→⬁共Fig.1兲.

Following1the contribution from the surfaceS_␦isum共x兲,

and if the contribution from the surfaceSR is assumed to be

zero, then we get the Green’s function integral representation in Oseen flow:

um共x兲= −

冕冕

S₀

冋

p共y兲uj共 m_兲共

x−y兲+uj共y兲p共m兲共x−y兲

+␮

再

ui共y兲

⳵ ⳵yj

ui共 m兲_共

x−y兲− ⳵

⳵yj ui共y兲ui共

m兲_共 x−y兲

冎

+␳Uui共y兲ui共 m兲_共

x−y兲␦j1

册

njds. 共7兲

III. EVALUATION OF THE FAR-FIELD INTEGRAL

The integration over the surfaceSR is given by

冕冕

S_R

冋

p共y兲u共j

m_兲共

z兲+uj共y兲p共m兲共z兲

+␮

再

ui共y兲

⳵ ⳵yj

ui共 m兲_共

z兲− ⳵

⳵yj ui共y兲ui共

m兲_共 z兲

冎

+␳Uui共y兲ui共 m兲_共

z兲␦j1

册

njds. 共8兲

The surfaceSRis such that兩z兩=R, and we want to show that

the integration over this surface tends to zero.

Taking the modulus of 共8兲 and bringing this modulus into the integrand, then we can show that共8兲tends to zero if

lim

R→⬁

再

兩

uj共y兲兩max

冕冕

S_R

兩ui共 m兲_共

y兲兩ds

冎

= 0ij共

m兲 _共

9兲 since

冏

⳵ui共 m_兲共

z兲

⳵yj

冏

ⱕAj

兩

ui共 m兲_共

z兲兩

for some constant Aj, and since 兩p共m兲共z兲兩ⱕ1 / 4␲R2, and ui共y兲→0 as R→⬁. 共In 共9兲, we define 0ij

共m_兲

= 0 for all 1

ⱕi,j,mⱕ3.兲

To evaluate 共9兲, the integration surface is divided into three共see Fig.2兲:

1. The surface Swake such that 兩z兩=R and r=

冑

z2 2

+z₃2 ⱕa0

冑

z1/k, 0⬍a01;

2. the surface Scone−wake such that 兩z兩=R and a0/

冑

kz1ⱕ␣ ⱕ␣0, 0⬍␣01;

FIG. 1. The division of the surface of the sphereSR.

1-2 N. Fishwick and E. Chadwick Phys. Fluids18, 1共2006兲 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154

PROOF COPY 016611PHF

3. and the surfaceSR−conesuch that兩z兩=R and␣⬎␣0,

wherea0and␣0are constants, and the Cartesian and

spheri-cal coordinate are such that z1=Rcos␣, z2=Rsin␣cos␪,

z3=Rsin␣sin␪. The approximations applied to the

funda-mental solutions within these three regions is given next. The surface SR is divided into the three areas SR−cone,

Scone−wake, andSwake, such that the following approximations

are made in each area.

AreaSR−cone:Within this area␣⬎␣0and so the

approxi-mation 1

R−z1 ⬍ b0

R, b0=

1 1 − cos␣0

共10兲 holds.

Area Scone−wake:In this region r/z1ⱕ␣0 and so we can

apply the approximation

R−z1=z1

再

1 +

r2 z₁2

冎

1/2

−z1=

r2

2z1

− r

4

8z₁3+O共r

6_/z 1 5_兲

, 共11兲 whereOmeans “of order of.” So,

e−k共R−z1兲_=e−共kr2/2z1兲共1+O共r2/z1 2_兲兲

=e−kr2/2z1共_{1 +O}共_r2_/z

1 2_兲兲−kr2/2z₁

=e−kr2/2z1共_{1 +o}共_r2_/z

1

2_{兲兲 共12兲}

whereo means “of order less than,” since共1 +a兲b+1_{⬎1 and}

so 共1 +a兲−b_{⬍1 +a for a⬎0, b⬎0. Finally, an element of}

area⌬sover the surface is approximated by

⌬s=R2sin␣⌬␣⌬␪=r⌬r⌬␪共1 +O共r2/z₁2兲兲. 共13兲

Area Swake: In this region kr2/ 2z1ⱕa02/ 21, and so

from Ref.10, p. 69, Sec. 4.2.1,

e−k共R−z1兲_{= 1 −k}共_R−z

1兲+

k2共R−z1兲2

2! +O共关R−z1兴

3_兲

= 1 −kr

2

2z1

+k

2_r4

8z₁2 +O共r

6_/z 1

3_{兲, 共14兲}

since

O

冉

r

2

z₁2

冊

O

冉

r2 z1

冊

in the far-field regionSwakeasz1→⬁.

The integral calculation of共9兲is now evaluated over the three regions of the integral surface for the varying index values 1ⱕi,mⱕ3. However, sinceu_i共m兲=u_m共i兲,u₂共1兲has similar form tou₃共1兲, and u共₂2兲 has similar form tou₃共3兲, then it is suf-ficient to consider the four permutations共i,m兲=共2 , 3兲,共2 , 2兲,

共1 , 2兲, and共1 , 1兲.

Permutation共i,m兲=共2 , 3兲:Over the areaSR−cone,

apply-ing the approximation共10兲 to the Oseenlet given by 共6兲 in the regionSR−conegives

冏

⳵␾共2兲

⳵z3

冏

⬍b0共1 +b0兲

4␲␳UR2 共15兲

and so lim

R→⬁

冕冕

_S R−cone

冏

⳵␾共2兲

⳵z3

冏

ds⬍b0共1 +b0兲

␳U . 共16兲

Similarly

冏

⳵␹共2兲

⳵z3

冏

⬍b0共1 +kR+b0兲

4␲␳UR2 e

−kR/b_{0 共}

17兲 and so

lim

R→⬁

冕冕

_S R−cone

冏

⳵␹共2兲

⳵z3

冏

ds= 0 共18兲 and so

lim

R→⬁兩

uj共y兲兩max

冕冕

S_R−cone

兩u₃共2兲共z兲兩ds= 0 共19兲 since兩uj共y兲兩max→0 asR→⬁.

Over the areaScone−wake, applying the approximation共11兲

to the Oseenlet given by共6兲in the regionScone−wakegives

冏

⳵␾共2兲

⳵z3

冏

⬍ 1

␲␳Ur2 共20兲

and so using the approximation for elements of the surface

共13兲gives lim

R→⬁

冕冕

S_cone−wake

冏

⳵␾共2兲

⳵z3

冏

ds⬍

冕

0 2␲

冕

a₀冑z₁/k

␣0z1 _a

2

␲␳Urdrd␪

= 2

␳U兵ln共␣0z1兲

− ln共a0

冑

z1/

冑

k兲其 共21兲

for some constant a2. We note that if this integration was

continued into the wake then the right-hand side of 共21兲 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

FIG. 2. The surfaceSy.

155

156 157 158 159 160 161 162 163 164

165

166 167 168

169

170

171

172

173

174 175 176

177

178 179

180

181

182 AQ: #1

183

184 185 186

187 188 189 190 191 192 193

194

195

196

197

198

199

200

201

202

203 204 205

206

207 208

209

210

211

PROOF COPY 016611PHF

冏

⳵␹共2兲

⳵z3

冏

⬍ a3

z1

e−kr2/2z1 共₂₂兲

for some a3 independent of the coordinate variables, since

1 /r2_ⱕa 0 2_/z

1. So using the approximation for elements of the

surface共13兲gives lim

R→⬁

冕冕

S_cone−wake

冏

⳵␹共2兲

⳵z3

冏

ds⬍

冕

0 2␲

冕

a₀冑z₁/k

␣0z1 _a

3

z1

e−kr2/2z1_rdrd␪

=2␲a3

k e

−ka₀2/2_{, 共23兲}

which is bounded. In the far field, we expect the fluid veloc-ityuj共y兲to behave as a combination of the fundamental

so-lutions u_j共m兲共y兲 to leading order. So we expect that

兩uj共y兲兩max→0 faster than 1 / lnR as R→⬁. This means that

combining the results共21兲and共23兲we expect lim

R_→⬁兩

uj共y兲兩max

冕冕

S_cone−wake

兩u₃共2兲共z兲兩ds= 0. 共24兲 Over the area Swake, making use of the approximation

共11兲, gives an approximation for␾共2兲in this region

␾共2兲₌ 1

4␲␳UR z2

R−z1

= z2 2␲␳Ur2

冉

1 +

r2

2z₁2

冊

−1

⫻

冉

1 − r

2

4z12

冊

−1

共1 +O共r4/z₁4兲兲

= z2 2␲␳Ur2

冉

1 −

r2

4z₁2

冊

共1 +O共r

4_/z 1 4_兲兲.

共25兲 Further, making use of the approximation共14兲fore−k共R−z1兲_in

this region then gives

␾共2兲_+␹共2兲₌ z2

2␲␳Ur2

再

kr2

2z₁− k2r4

8z₁2 +O共r

6_/z 1 3_兲

冎

, 共26兲 so

⳵ ⳵z3

共␾共2兲_+␹共2兲_{兲= −} k 2_z

2z3

8␲␳Uz₁2共1 +O共r

2_/z

1兲兲. 共27兲

Therefore lim

R_→⬁

冕冕

_S

wake

兩u₃共2兲共z兲兩ds

= lim

R→⬁ k2

4␳Uz₁2

冕

0

a₀冑z₁

r3dr共1 +O共r2/z1兲兲

= k

2_a 0 4

16␳U共1 +O共a0

2_{兲兲. 共28兲}

Combining all results together over the three surfacesSR−cone,

Scone−wake, and Swake on the surface of the sphere SR then

gives

lim

R→⬁兩

uj共y兲兩max

冕冕

S_R

兩u₃共2兲共z兲兩ds= 0 共29兲 as expected.

Permutation 共i,m兲=共2 , 2兲: Over the area SR−cone,

fol-lowing the same approximations as for the permutation

共i,m兲=共2 , 3兲, then in this region we have

冏

⳵␾共2兲

⳵z2

冏

ⱕ a4

R2,

冏

⳵␹共2兲

⳵z2

冏

ⱕa5

Re

−kR/a₀

, 兩␹*兩ⱕa6

Re

−kR/a₀

共30兲 for some constantsa4,a5, anda6. So, using the same

argu-ment as for the permutation共i,m兲=共2 , 3兲, then in this region we have

lim

R→⬁兩

uj共y兲兩max

冕冕

S_R−cone

兩u₂共2兲共z兲兩ds= 0 共31兲 since兩uj共y兲兩max→0 asR→⬁.

Over the areaScone−wake, following the same

approxima-tions as for the permutation共i,m兲=共2 , 3兲, then in this region we have

冏

⳵␾共2兲

⳵z2

冏

ⱕa7

r2,

冏

⳵␹共2兲

⳵z2

冏

ⱕa8

z1

e−kr2/2z1_,

兩␹*_兩ⱕ a9

z1

e−kr2/2z1 共₃₂兲

for some constantsa7,a8, anda9. So, using the same

argu-ment as for the permutation 共i,m兲=共2 , 3兲, then in this area we have

lim

R→⬁兩

uj共y兲兩max

冕冕

S_R−cone

兩u₂共2兲共z兲兩ds= 0 共33兲 since兩uj共y兲兩max→0 faster than 1 / lnR asR→⬁.

Over the area Swake, making use of the approximations

共11兲for ␾共2兲 and the approximation共14兲for e−k共R−z1兲 _{in this}

region gives

⳵ ⳵z2

共␾共2兲_+␹共2兲_兲= ⳵

⳵z2

再

kz2

4␲␳Uz1

共1 +O共r2/z1兲兲

冎

= k 4␲␳Uz1

共1 +O共r2/z1兲兲. 共34兲

Therefore lim

R→⬁

冕冕

_S

wake

兩u₂共2兲共z兲兩ds= ka0

2

4␳U共1 +O共a0

2_兲兲

, 共35兲 which is bounded. Also in this region, 兩␹*兩ⱕa10/z1 and so

combining all results together over the three surfacesSR−cone,

Scone−wake, and Swake on the surface of the sphere SR then

gives lim

R→⬁兩

uj共y兲兩max

冕冕

S_R

兩u₂共2兲共z兲兩ds= 0 共36兲 as expected.

1-4 N. Fishwick and E. Chadwick Phys. Fluids18, 1共2006兲

217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279

PROOF COPY 016611PHF

Permutations 共i,m兲=共1 , 1兲 and 共i,m兲=共1 , 3兲: The analysis for these permutations give similar bounds, with the added simplification that _⳵⳵_z

1ln共R−z1兲= −1 /R. This means

that the condition共9兲given by lim

R_→⬁

再

兩

uj共y兲兩max

冕冕

S_R

兩ui共 m兲_共

y兲兩ds

冎

= 0ij共

m兲 _共

37兲 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.

ACKNOWLEDGMENT

This work was carried out during an EPSRC sponsored studentship for a doctorate in applied mathematics at the University of Salford.

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