APPENDIX B. GEOMETRY CONCEPTS.

BI. Definitions. A curvilinear coordinate system, {~i}. involves the covariant vectors gl = aRla~i :; R,i along the

~Lcoordinate and the contravariant vectors gl = grad~i, such that gl.gk = O:C. Contravariant and

covariant components for a vector V and a tensor T are classically defined by:

Vi = V.gl ; Vi = V.gl ; Tij = gl.(Tgj) ; Tij = gl.(Tgj) ; Tij = gl.(Tgj) ; Tij = gi.(Tgj)

with mixed (physical) T-components, li·j and t j , Then,

V = Vigl = Vigl ; T = Tij gigj = Tij glgj = Tijgigj = Tijglgj (BI)

The co variant and contravariant components of the identity tensor are gij = gl.gj, gij = gi.gj,

while mixed components are the Kronecker symbols, We may define the jacobian of the

transformation from the curvilinear coordinate system to the Cartesian coordinate system {xa }:

J = D(x,y,z) = gl.(gjxgk) ; ]-I = gl.(gJxgk) i, j, k in cyclic order. (B2) D(~ 1,~2,~3)

so that g :; det(gij) = 12, J is the volume of a small hexaedron in the physical space corresponding

to a unit cubic element of the space defined from the curvilinear system. Hence gi = J'!gjxgk'

indicating that the oriented area element, SI, on a surface ~i = Const., is simply Jgi and that

JVi = V.SI is the flux associated to V across the surface area corresponding to unit increments

dl;id~k. The so-called metric identity is satisfied: aSi/a~i = O.

The dimensions of V do not coincide with the dimensions of Vi and Vi since gi may have

the dimension L of a length - then, dirn (gi) = L'!, A contravariant basis vector gi can therefore be

normalized if it is divided by its modulus .yg;;: If multiplied by hi:; {gji; the basis vector is made

non dimensional but its modulus is not one. This observation leads to the following definitions:

g(l) = ~ ; g(l) = g!hi ; g[l] = ~ ; g[1l = giV gii (B3) 1 -.fiili

where only g(l) and gll] are unit vectors. The main common aspect of such vectors is that,

althoughthey are not gradient vectors (Le. they do not satisfy a condition, similar to gi = grad ~i,

they define a so-called anholonomic basis vectors ) they still satisfy g(l).g(j) = g[i].g[j] = öl. Hence anholonomic (or physical) vector or tensor components are easily defined using definitions

similar to those used for standard contravariant or covariant vectors. For instance, contravariant

and covariant physical components of V are then defined by :

V(i) = V.g(1) = Vihi ; V(i) = V.g(1) = ~ ; VIi] = V.gII] = Vi .. ; VIi] = V.g[i] = Vi {gli 1 -.[gii

Of course, V = V(i)g(l) = V(i)g(1) = V[i]g[l] = Uli]gll]. The physical components of V may be

regarded as associated vector components of V in a rectilinear but not necessarily rectangular

coordinate system based on the unit of lengths, {gii}, and having axes tangent at the given point to

the coordinate {~i }-lines. For olthogonal coordinates, all physical components coincide because

gii = l/hi. In a cartesian system, the difference between contravariant, covariant and physical

Appendix B Geometry Concepts 729

components vanishes. From now we shall work only with the McDonnell form, Le. with the

physical components defined with (.) (and not with those defined with [.]). The physical metric

tensors can be defined from the physical basis veetors. For instanee, g(ij) = g(l).g(J) and

g(ij) = g(l).g(j). While the diagonal components of g(ij) are equal to 1, non diagonal components

of g(ij), i;l!j are simply equal to cos 9k where 9k is the angle between the eoordinate lines!;i and!;.i

(i, j, k eyclie). It is also possible to introduee :

T(ij) = g(l).(Tg(J» ; T(ij) = g(l).(Tg(j) ; TiW = g(l).(Tg(j) ; T~6) = g(i).(Tg(j» (B4)

Notiee that the left and right mixed physieal eomponents, TiW and T~d)' differ if T is not

symmetrie. The physical eomponents satisfy again : T = T(ij)g(l)g(J) = T(ij)g(l)g(J)'

If we define (.),k == a(.)/a!;k, the eurvature of coordinate lines is defined from the

Christoffel symbols:

gl,k = {~} gn; gl,k = {ik.n} gn where {?k} = gn.R.ik; {ik.n} = ~gin,k+gkn,i-gik,n) (B5)

whieh are symmetrie in indices ik, a eonsequenee of the faet that the Christoffel symbols

associated with the holonomic basis {gd vanish in a Cartesian coordinate system. Notice the

following identities:

{rn} = (InJ),i; gjk{}k} = - J.l[Jgikh (B6)

The covariant derivative of a vector V = VPgp = V pgP involves the variations of the basis veetors

along {Si}, aceording to:

V,j = (VjVP)gp = (VjVp)gP with VjVP = VP,j + n~j}Vk; VjVp = Vp,j - {~j}Vk (B7)

and similarly:

T,j = (Vj TI"I)gpgq = (Vj T pq)gPgq with: Vj TPq = TPq,j + {kj} Tkq + {fj} TPk ;

VjTpq = Tpq,j - {rp}Tkq - {rq}Tpk; VjTÄ = TÄ,j + {fk}~ - {~}T:k' (B8)

A straightforward consequenee is the Ricci identity, Vjgpq = Vjgpq = O. In the same way:

V Tijk = Tijk + {i }TPjk + {j } Tipk + {k } Tijp m .m pm pm pm

If the basis vectors are not gradients, the associated Christoffel symbols ~(!;) and rik.n(!;)

remain defined by formulae simiJar to (B5), and the physieal Christoffel symbols result from:

g(I);(k) = r~?~)g(n); g(l);k = r(ik).(n) g(n); r~?~) = (g~gnn )1!2{?k} - KikOr (B9) !!ll

where we have noted (');(k) == a(.)/a!;(k)= hk(.),k.

The Christoffel symbols ri~) are not symmetrie in ik. They involve the geodesie curvature tensor,

Kik = [lnh~ll k = (hihk)·lhi k. The physieal eovariant derivatives are then defined by: !. - - -,-

V;(j) = V (j)V(p)g(p) = V (j)V(p)g(p) with V (j)V(p) = (hJhj)VjVP = V(p);(j) + r~~l)V(k) ....Ik)

V (j)V (p) = (hl!hj)VjVp = V(p);(j) - 1 (pj)V(k) (B1O)

730 Subjeet Index

Beeause r is not symmetrie, the seeond eovariant derivatives are not equal and V Uk)",V(kj)'

However, the laplaeian is defined in a unique way, being eontraeted with the (symmetrie) metric

tensor, g(jk). Notiee the identity: ri~i) = J-l[1gk~\k' We now list eovariant physical eomponents

whieh are required in the Reynolds-averaged Navier-Stokes equations.

B2. Vector and tensor operators.

Gradient oi a scalar: g(k).grad<l> = V(k)<I> = <I>;(k)' (B 11)

Gradient oi a vector: g(l).(VV.g(k»;: V(k)V(i) = V:i\(k) + ri~k)V(P) (B 12)

Gradient oi a tensor: g(l)gU>.(VT.g(k»;: V(k)T(ij) = T:ij.\(k)+r:~k)T(pj)+r~kl(iPJ (BI4)

Divergence oi a vector: V.V = J-I[h\cIJV(kJh (B 15)

Divergence oi a second-order tensor: V(k)TikJ = ]-I [h\cl JT(ik)h + T(jk>r~~) (B 16)

Curl oi a vector: W(i) = g(1).curIV = J-IMVj,i-Vi,j); i, j, k in eyclie order (B 17)

Div. oi a third-order tensor: V(kl(pqk) = J-1[h\cIJT(pqk)h + TUqkJri~l) + T(Pjk>ri~) (B 18)

Scalar laplacian: V2<1> = J-I[1 gik<l>.kl.i = g(ik)<I>;(ki)+f{k)<I>;(k); f{k) ;: -g(mn)h~~m {~n }(B 19)

Vector laplacian: ~V.g(i);: (~V)(i) = V2(V(i» + 2g(kP>ri~p)v:m\(kJ + Qi~JV(p) (B20)

where ()(i) [(k')r(i)] (k')r(iJ r(q) (k')r(iJ ....Iq) C(im) (mn) r(i) ~i '«p) = g J (pk) ;(j) + g J (pk) (jq) + g J (qj)1 (pk); (p) = g [(np) + K(np)Upl

(Symmetrie) second covariant derivative oi a scalar: V(i)V (k)<I> = <I>;(ik)- h~k <I>;(q)I1 (B21)

Second covariant derivative oi a vector: t'1 t'1 V() ( ). hm VIp) r m V(m) r(p) V(m) r(p) D(P) v(m) v (i) v (k) P = V p ;(Ik) - hihk • ;(m) ik + • ;(k) (mi) + • .(1) (mk) + (mkl) (B22)

where: D(P) - r(p) r(j) r(p) r(p) r U)

(mki) - (mkF(i) + (mk) (ji) - (mj) (ki)

It has the skew-symmetrie part:

(V(i)V(k)-V(k)V(i))V(p) = ci~i)v(m); Ci~ki) = Di~ki) - Di~ik) Cis the so-ealled physical Riemann-Christoffel tensor which satisfies Ci~kk) = Ci:~i) = O. Now

V.kp - V,pk = v(n)g(l)c~j) .• ) = 0 for all V. the physical Riemann-Christoffel symbol vanishes

identieally. Henee only .... c:omponents i",k and p",m of Ci~i) are non-trivially zero. They yield

the so-ealled Lame equations.

Second covariant derivative oi a tensor (B23): t'1 . t'1 () _ ( )., l!m.. (pq) (m) (pm) (q) (pm) . ....Iq) (mn) (p) v (I) v (k)T pq - T pq .(ik) - hjhk T. ;(m)r(ik) + T. ;(k)r(mi) + T. ;(1)1 (mk) + T. ;(k)r(mi)

+ T(mn) . r(p) + T<mn)(~P) r(q) r(p) ~q) ) + T(pm)D(q) + T(mq)D(P) • ;(1) (mk) (mk) (ni) + (mi) (nk) (mki) (mki)

It has the skew-symmetrie part:

(V(i)V(k)-V(k)V(i»)T(pq) = T(pm)ci~kj) + T(mqlCi~ki)

Rate oi strain: S(ik) = ~[g(iPJy:k);(p)+ g(kP)V:i\(P)+ (g(ip)ri~p)+g(kP>ri~p»)v(m)] (B24)

Appendix B Geometry Concepts 731

B3. Reynolds-averaged Navier-Stokes equations.

B3.1. General case.

Gathering previous results yields for the mean flow:

~ + V(klV~i);(kl + ri~nlv(mlv(nl = - p-lg(imlp;(ml - J-l[hklJv'(ilv'(klh - v'(ilv'(Ie)ri~L + v[ V2(V(i» + 2g(kP)r~~p)V~m\(k) + Q~~)V(P)] (B25)

and for the Reynolds-stress equations:

a '(1) '(j) .. v i)tv + V(kl( v'(1)v'(j) ;(k) + r~~k) v'(p)v'Gl + r~k) v'(1)v'(p) ] =

_ [0liV'(m) aV(j) + v'(j)v'(m) aWl + (v'(.)v'(m) I'9) + v'G)v'(m) r(i) )V(P)] a~(m) a~(m) (pm) (pm)

_ [J*(ijk) + r(il J*(pjk) + r(i) J*(ipkl+ r(m) J*(ijkl] ;(k) (pk) (pie) (km)

+ p-l[p'(g(imlv'(i);(ml+g(im)v'(il;(m» + (g(im)r1;ml + g(im)ri~ml)P'v'(Pl]

- V[g(mn) v'(i),(m)v'(i),(n) + C~~~) v'(p) v'(i),(m) + C~) v'(p) v'(i),(m) ] (B24)

where:

J*(ijk) = v'(l)v'Glv'(k) + p-lg(ik) p'V'G) + p-lg(ik) p'V'{i)-

- V g(km) [ v·(.)v·(j) ;(m» + ri~ml v'(p)v'G) + r1;m) v'(I)v'(P)] (B25)

while dissipation and turbulent kinetic energy defined by:

K=!g(· ~v· I ' 2 Ij) V \'Iv \j) (B26a.b)

E = g(ij)g(mn)[ v'(I),(m)V'(i),(n) + r~n)v'(.>'(m)v·(q) + r~~m)v'(P)v'G)'(Il) + ri~mr~n)v'(Plv'(q)]

are related by the turbulent kinetic energy equation:

aa~ + V(k)K;(k) = -g(ij)V'(1)v'(k) [V(i);(k) + r1;k)V(P)] - J-l[hk1JJ*(k)h - E (B27)

where J *(k) = ! g(ij~*(ijkl. The foregoing equations indicate that the modeling arguments laid

down in Chapters 3 and 5 do not depend on the properties of the coordinate system. However the

definition of Reynolds stresses has not. in general, the clear significance it has for Olthogonal

coordinates since v'mv'm no longer represents the turbulent stress normal to a fluid element

surface. Also the definition of transfer terms involves. because of curvature effects. not only

correlations of pressure and velocity gradient correlations. but also pressure-velocity correlations.

Hence the splitting of such correlations between a diffusive contribution and apressure strain term

is rather arbitrary and both effects should rather be modeled simultaneously. Finally. it is seen that

the defmition of the dissipation term E is also rather arbitrary and that its relation to 'lJ = gik 'tkiVj Vi

is rather complex. again because of curvature effects. Also convection and production involve

extra-curvature terms. respectively:

C(ij) = V(kl(r(i) v'(p)v'(j) +~) V·(I)V·(pj)· d ij) = -(v'(I)v'(m)r(i) + v'(i)v'(m) ~i) )V(P) (pk) (pk) • + (pm) (pm)

732 Subjeet Index

While g(ij)C(ij) = 0, t g(ij)G~j) == G+ = - g(ij)V'(llv'(klr~k)Y(p)

The explicit global influenee of eurvature is usually written on the rhs of eq.(B24) as:

dij) - C(ij) == - [v'(llv'(m) (r2) + r G) ) + v'Ulv'(m) (Mi) + r(i) )]y(p) + (pm) (mp) (pm) (mp)

B3.2. Ortlwgonal curvilinear system.

In this ease g(ij) = g{ij) = Sij, so that r8J~ = KjiSjk - t<Kik + Kjk)Sij- Henee all physieal Christoffel

symbols vanish exeept :

ri~i) = - r~? = Kip. i and p different from eaeh other.

We find the following simplifieations:

f{i) = Kji + Kki - Kjj; i, j, k different from eaeh other

C <Ri) - K ' . C(il!> - K" C(ii) - K' (p,J. I') . CGi) - K" (j!) - p., (j!) - - pt, (p) - 'p ... , (i) - !!

C~~) [~%k) + ~mk)S~] = KipKii - Kl!l!Kl!i - KllPKni

so that, with k, i, p in eyelie order: (i) (j!) 2 2 2 2

<4p) = Km;(i) - Km;(j!) + Kjp Kki - Km [Km + Kkil ; Q(I!> = - ~k - Kkp - ~i - Kip

f{i) = h1h ~~h'~k) with i, j, k different from eaeh other J k o~(') •

The Laplaeian operator beeomes:

"2 1 ~o h;hk 0) 02 [K K KlO . 'k I' v =- --'--=-- = + "+ '- ,,-- sumoverI e eIe J o~(i) hi o~(i) o~(i)o~(i) J' k. 11 o~(i) , J, y

The Lame equations are given by:

(B28)

Kpn;(n) + Knp;(p) + ~n + ~p + KnqKnq - KpnKnn - KnpKpp = 0 with p "# n (3 equations)

Kpn;(q) + Kpq(Kpn-Kqn) = 0 with p, n, q different (six equations).

The Reynolds-averaged Navier-Stokes equations (Nash & PateI, 1972) are produeed from

these results.The equations used in Chapter 6 are obtain by Ietting ~l == x, ~2 == y, ~3 == z while the

physieal velocity eomponents are Y(l) == U, y(2) == Y, y(3) == W.

Mean-flow equations.

oU U oU Y oU W oU 1 oP 1 ou'2 -+--+--+--+ (K12U-K21Y)Y + (K13U-K31W)W=------ot hl OX h20Y h3 OZ phi OX hl OX

- (2K21+K31)U'2 - ~2 °ä~v' - (2KI 2+K32)U'V'

1 ou'w' (2K K )-,-, K ~ K ~ - h3 ~ - 13+ 23 u w + 21v + 31 w

+ v[ V2U _ 2(.!.ll oY _ K12 oY + K31 oW _ K13 ow) + Q1U + Q1y + Q1W]. (B29) h2 oy hl OX h3 OZ hl OX 1 2 3

and similar expressions obtained by eyclie permutation (from now, the parentheses within ~~) are

omitted).

Appendix B Geometry Concepts 733

Normal Reynolds-stress equations:

Conv: du'2 U du'2 V du'2 W du'2 - -T+hl Tx+h2dY+h3 T+ 2(K12U-K21V)U'V' + 2(K13U-K31W)U'W' =

Prod: - 2U'2(~1 ~ +K12V+K13W) - 2U'v'(~2 ~ - K21V) - 2u'w'(~3 ~- K31W)

Flux [ 1 fx<- _D'U') 1 a(-) - 1 d(-) -- -h u'3 + C"---- + -h 'L: u'2v' + 2KI2U'2v' +-h "C u'2w' + 2K13U'2w' 1 x P 2 0 Y 3 0Z

- 2K21U'v'2 - 2K31U'w'2]

Transfer + 2~(l ib!..:) u'2(l ib!..: + l dV' + l dW) P h1 dX - h1 dx h2 dy h3 Tx

Viscous 2 ['----V2' 2K21 .dV· 2K31 .dW· 2K12 .av· 2K13 .dW· + v u u - h1 udy - h3 uTz+ htuTx+ h1 uTx

+Q:u'2 +Q~u'v'+Q;u'w'] (B30)

and similar expressions obtained by cyclic permutation. In eq.(B30), we have specified the

convective terms (Conv.), the production terms (Prod.), the turbulent transport terms (Flux), the

transfer terms and the viscous terms.

Reynolds shear-stress equations.

dU'v' U dU'v' vau'v' wau'v' Conv. dt+hl dX+h2 ""dY+h3 """"dZ"+

+ (K21V-KI2U)U'2 + (KI2U-K21V)V'2 + (K23V-K32W)U'W' = Prod. - •. ( 1 dU K V K W 1 dV K U K W) -. .( 1 dV K V) - u V hl dx + 12 + 13 + h2 dY + 21 + 23 - U W h3 Tz - 32

-. .( 1 au K W) --;Z( 1 dV K U) --;Z( 1 dU K V) - v W h3 Tz - 31 - U hl dX - 12 - v h2 ay - 21

Flux 1d-V -ld-~ -ld--

- [- (u'2v'~) + K u'2v'+- (u'v'2~) + K u'v'2+--(u'v'w') h1 dx 21 h2 ihy 12 h3 dZ

Transfer

Viscous

P P

+ (K13+K23)U'V'W' - K12U'3 - K21V'3 - K31V'w'2 - K32U'w'2] +

n'(ldV' 1 dU') •• (1 dU' -1 dV' -raw') +L. + +uv- +- +-P i1idX h2dY h1 dx h2 dy h3 Tx

K (dU' :::Iv') K dV' Tu' +V[u·V2v·+ v·V2u·+r.ll u' _v,u- +2-1l(v'- -u' ) h2 dy oy h1 ox dx

2&l .ow· 2!n pw' 2~ .aw· 2K23 .aw· + h1 vTx+ h3 vTz+ h3 uTz + h2 udY + (Q:+Q~)u'v' + Q~v'w' + Q;u'w' + Q:u'2 + Q~v'2] (B31)

and similar expressions obtained by cyclic permutation.

734 Subject Index

B3.3. Cylindrical coordinates.

In the case of a cylindrical coordinate system eqtns.(6.2) result from h1 = h2 = 1, h3 = r (J = r) if

~l ;: x, ~2 ;: r, ~3 ;: e. All Kij vanish except K32 = l/r. Also, f(l) = f(3) = 0, f(2) = l/r, and all Q~~?

vanish, except Qg~ = Qm = -l/r2.

B.4. Elements of surface theory.

We suppose that Lyapounov conditions are satisfied: (i) there is a unit vector N orthogonal to each

surface point, r; (ii) The exists two values A and (1, such that for two arbitrary points, rand r ' on

the surface with corresponding normals N, N', Icos(N,N')1 < Alr-r'la ; (iii) 3 ö> 0 such that

straight lines parallel to the normal at r cut the surface inside sphere of radius ö and center r at most

one time. The most convenient way to study S is to define a parametric form, r = r(ua), a. = 1,2.

A surface will belong to the "wall family", ~2 = Const., if its parametric form is r = r(~a) with

a. = 1, 3. The vector tangent to coordinate lines at r is aa = r,a, and the first quadratic form of the

surface is aaß = aa.aß' such that the length element in the surface is ds2 = aaßd~<ld~ß.

Given a curve C on S, its curvature is defined by the so-called first Serret-Frenet formula,

dT/ds = ~M, in terms of the unit tangent T to the curve, and the unit main normal, M, to the

curve. At any point of the curve C, we have two available frames, the Serret-Frenet frame,

(T,M,B} and the frame (T,Z,N} involving the normal to the surface. The normal curvature ~N

and the surface torsion are defined from the third Serret formula, dN/ds = - ~NT - 'TsZ. Principal

directions are at point r those with zero torsion. Hence they satisfy the so-called Rodrigue theorem:

dN/ds = - 1(N T. It remains to show that 1(N is an intrinsic property of the surface. Along C we

maywrite: dT d2r d~a d~ß d2~a ----r .,--+r - (B32) ds - ds2 - ,a" ds ds ,a ds2

With the angle q, such that cosq, = M.N, we may project (B32) the foregoing equation along N:

1(cosq,ds2 = N.dTds = N.r,aßd~ad~ß = -N,ß.r,ad~ad~p = - dN.dr = N.d2r

so that we may define the second fundamental form of the surface by baß = -N,ß.r,a = N.r,aß'

Then the normal curvature to the surface at r in the T -direction is given by:

1(N = - dN.dr/ds2 = baßd~ad~ß/aaßd~ad~ß (B33)

and radii of curvature are connected by 1(= 1(~cosq,. The foregoing results have a simple

interpretation in terms of the Meunier's theorem (figure BI). The osculating circles of all surface

curves through r having the same tangent T there form a sphere. This sphere and the surface have

a common tangent plane at r, the radius of the sphere being 1(~. Hence it is sufficient to study

curves at r with M = N. Such curves, called normal or principal directions, can be thought of as

the intersection of the surface with a plane through rand containing N. Setting A. = d~2/d~1 = tana., '!(N(r;T) is a function of A. unless baß and aaß are proportional. Points r with

the property that '!(N does not depend on A. are called umbilical points. In the general case where

'!(N changes as A. changes, '!(N = '!(N(A.) is a rational quadratic function given from (B33) by.

1\NO .. ) = bll+2b13A+b33A2 all +2a13A+a33A2

B

Appendix B Geometry Concepts

Figure B 1. Meusnier's sphere viewed in the direction of T.

735

(B34)

The extreme values Kt and ~ of 1\N(A) are the roots of eq.(B35) : det IIba/l-1\NaaP 11 = O. They

occur when Al and ~ make this determinant vanish. Al and 1..2 are real and define the two

principal directions in the (~I,~3)-plane. The net of lines that have these directions at all of their

points is the net 0/ lines 0/ curvature which can be constructed from an integration of the

determinant condition and used as a parametrization of the surface (this is particularly interesting

for axisymmetric surfaces for which the net of lines of curvature is defined by the meridians and

the paralleis). At an umbilical point, the principal directions are undefined; this is the case over a

sphere : all its points are umbilical.

The extreme values Kt and ~ are called the principal curvatures of the surface at r and a

comparison ofthe determinant condition with E - (Kt +~)K.. + Kt ~ = 0 yields:

2H=t:+t:.=.!.aaPb = allb22+a22bll-2a12b12 ·K=t:.t:-= detllbapll (B35) "l. "2 2 a/l det lIaapll ' "l. "2 det llaa/lll

K is called the Gaussian curvature, while H is the mean curvature. While Kt, ~ and H change their

sign when N is reversed, K does not. A point r of the surface may be classified according to the

sign of its gaussian curvature. Namely, the point runder consideration is called elliptic, hyperbolic

or parabolic if K > 0, K < 0 and K = 0 (H "# 0), respectively. This classification means that the

surface, in the vicinity of r, looks like a spheroid, an hyperboloid or a cylinder (if Kt and ~=O, but

H "# 0). For instance, points of the surface of a torus which face each other are hyperbolic ; points

which "look away from the torus" are elliptic ; the plane circles -orthogonal to the circular

crosssections of the torus- that separate the corresponding regions are parabolic. In the special case

where K = H = 0, the point r is aflat point and the surface looks like a plane close to r. A surface

which is everywhere such that K = 0 is called a developable surface ; it can be deformed to planar

shape without changing length measurements in it.

736 Subject Index

BS. Gauss-Weingarten formulae. The variation of the trihedra along the surface is given by the so-called Oauss-Weingarten fonnulae

which state that r,aIJ' N,n and higher derivatives can be deterrnined from r,a and N :

r,aIJ = O:IJr,o + baIJN; N,n = aolJblJar,o (B36a, b)

where O:IJ = aO'"f r,'Y.r,aIJ are the second-kind surface Christoffel symbols. Eqtns.(B36) result

from (B32) and involve surface Christoffel symbols that are defined in the same way as {.),

namely:

O:IJ = a'YOOaIJ.'Y; 0a~.'Y = r,aIJ·r,y = ~aya,~ + a~'Y,a - aa~,'Y) and share with {.) the same properties. For instance: O:IJ = a-{;;~a~IJ with a = det llaalJll.

Conditions, aa,(Jy= aa,y(J, N,aIJ = N,IJa, are integrability conditions through which the equation

of the surface can be recovered when the two surface quadratic forms are known. The integrability

conditions give three vector identities. If we develop the vectors a .. az and N, this leads to nine

scalar equations among wh ich only three are significant. The first one is the Oauss formula

(B36a). Because r,a~ can be expressed only with N and the first order derivatives of r with

respect to ~a, det IIba IJII, and therefore the gaussian curvature K, may be expressed also only by

means of aa~ and its derivatives. If the first quadratic form is given, the second one must satisfy.

det IIba~1I = a13,13 - k all,33 - k a33.ll + 0~30~3aa~ - 0~10~3aa~ (B37)

while the total curvature satisfies K = det IIba IJII / det llaa~11. This form is known sometimes as the

theorema egregium which states that K depends only on the metrics of the surface. If the surface

is twisted, its form is modified as wen as Kt and ~, but the value of K remains constant. In other

words, the gaussian curvature does not change in any length-preserving deformation of the

surface; for instance, if a plane is twisted, it gives a developable surface such that K = O. The two

other conditions are the so-called Petersson-Codazzi /ormulae 1 (1 3) 3 bll.3 - b13.2 - 013bll + 033-013 b13 + 011 b33 = 0 1 (1 3) 3 b13.3 - b33.1 - 033bll + 013-033 b13 + 013b33 = 0

(B38a)

(B38b)

Now, if we assume (i) the integrability conditions of Oauss (B36a) and of Petersson-Codazzi

(B38), (ii) the existence of aaIJ and ba~ such that a = det lIaaIJII > 0 and all> 0, then there exists a

surface r = r(~l,~3) which admits aaIJ and baIJ as its two quadratic fundamental forms. If aaIJ is

given as twice continuously differentiable functions and ba~ once continuously differentiable, then

r = r(~1,~3) is a three times continuously differentiable function and is determined in a unique

way.

B6. Boundary layer rectilinearity condition. We are only interested by the vicinity of the regular surface Sw such that ~2 =~. With a linear

~2-coordinate line, a point R near the surface is related to the vector radius, r, 0/ the interseetion 0/ the perpendicular to (Sw) passing through R by R = r+~2N. Any point, R, of (S), determined by

Appendix B Geometry Concepts 737

~2 = Const., may be identified by the two coordinates 1;1 and 1;3 of r, and the coordinate 1;2. The

covariant basis vectors for this coordinate system are: R.a. = r.a. + 1;2N.a. ; a, 13 = 1,3 ; R,2 == N,

where the r.a. are the surface covariant basis vectors. The R,\'s are used to eva1uate the covariant

metric tensor at all points within the boundary 1ayer in terms of curvature characteristics of the

wall, and because of the Weingarten formu1a,

R.a. = r.a. - 1;3 a'YKb")Ur." = (a: _1;2 a'YKbya.) r." ; a, 13 = 1, 2 ; R.2 == N (B39)

Hence

ga.~ = R.a..R.~ = aa.~ -21;2ba.~ + (1;2)2a'YKb")Ub,,~ (a, 13 = 1,3) ; ga.2 = 0 ; g22 = 1 (B40)

and with the gaussian curvature, Kw, and the mean curvature, Hw, at any point r of the wall, 2Kw = ba.~b)!yEa.)!E~V ; 2Hw = ba.~aJ.lyEIlj!E~V (E~V is the a1temator equa1 to {;.)

(Sw is convex if Kw ~ 0, and concave if Kw < 0), so that :

alC'fb")Ub,,~ = 2Hwba.~ - Kwga.~ (B42)

Substituting B42 into B40 yie1ds ga.~ for (S) in terms of fundamental forms of (Sw)

ga.~ = [1 - (~2)2Kw] aa.~ - 2~2[1 - ~2Hw]ba.~' (B43)

We may also obtain similarly the second fundamental fOlm of (S), Ba.~. using (B39, 43):

Ba.~ = N.R,a.~ = [1 - 2~2Hw] ba.~ + ~2[1 - ~2Kw] ~~. (B44)

so that gaussian and mean curvature of (S) at point Rare:

K = Kw ; H = Hw-~2Kw . 1 +(~2)2Kw-2~2Hw 1 +(~2)2Kw-2~2Hw

Similar1y, Christoffe1 symbols at R can be expressed in telms of Christoffe1 symbols at r.

{aß.K} = Ga.A" - 1;2[ba." A+2 GV AbV"] + (1;2)2[b v ba.v+b"G1 A] with b" =aKAb,,:v .... .... a.... "1 "... l'

{al3.2} = - { a2·13} = ba.~ -1;2b :b,,~ ; Ga.2.2 = 0 ; G22.k = 0 ; Ga.~.2 = ba.~

The three classes of Christoffel symbols are defined as follows. The first c1ass gathers {f1}'

{f3 }, {~3} which cannot be neglected in the boundary 1ayer approximation. They depend on

geodesic curvatures (6.60), KI and K3, of the ~2 = const. surface, respective1y in the ~3 and ~ 1

directions. a 2 2

I 1 hl 3 h l KI I h3K3 3 1 ah3 {ll} =11-1 - hlKltanÄ; {ll} = -. -; {33} =--. -; {33} =11-3 - h3K3tanÄ

I a~ h3smÄ hlsmA. 3 a~

{I} __ 1_ [ahl _ COSAah3] . {3 } __ 1_ [dh3 COSAahl] B45 13 - hlsin2Ä a~1 d~1 ' 13 - h3sin2A. O~I - O~3 ( )

The second c1ass (B46) gathers symbols involving the dependence of ga.~ with respect to ~2, and

thus intrinsic curvature properties of the surface ~2 = const.

{2 } _ ! 22aga.ß . al3 - - 2 g a~2 '

{a } =! Ia.[ogll + ag22] +! ga.30g13 . {a } =! [ la.agl3 + ga.30g33] (B46) 12 2 g a~2 a~1 2 a~2' 32 2 g a~2 a~2

738 Subject Index

The third class (B47), where index 2 occurs twice, gathers symbols expressing the dependence 01'

h2 with respect to coordinates.

{ ~ } =! 22dg22 . {IX} = _! [ ladg22 + a3dg22] 12 2 g d~i ' 22 2 g d~l g d~3 (B47)

Abnormality,513.

Accelerated flows, 328, 624.

Active motion, 353.

SUBJECT INDEX

Anisotropy, anisotropie turbulence, 72, 155, 156, 183,204,215,216, 218, 222, 236, 239, 258,

260, 279, 577, 580, 591, 697. Polarization ., 155, 156, 236. Tensorial ., 155, 156, 236.

Vorticity ., 85, 208.

Anomaly, (plane jet/round jet) 116,286, 294, 366, 389.

ASM (algebraic-stress model) 274, 279, 281, 285, 368, 384, 385, 411, 559, 560, 580, 591, 641,

645,687,696,700,701,702,703.

Aspect ratio, 330.

Axisymmetric,77, 119 .• turbulence, 342.

Azimuthai velocity component, 281, 307, 308, 346, 364, 385, 386, 387, 397,403,405,406,

440,442,448,450,451.

Backscatter, 177.

Baldwin-Lomax model, 108, 110.

Beltrami motion, 10, 12.

Bend,546,547, 693,695,696,697,698,700.

Bemoulli equation, 10.

Blockage, 436, 441,444, 449, 450, 453, 454.

Body force, 125.

Boundary layer, 119,262. 305, 306, 309, 316, 317, 318, 319, 320, 321, 322, 323, 325, 326,

327,328,333,336,337, 338, 342, 343, 344, 348, 349, 350,351,353, 355, 357, 358, 359,

360,261,362,363,366,375,378,403,410,417,421,422, 425, 426, 427, 428, 429, 431,

441,471,472,473,474,476,477,478,479,480,481,482,484, 488, 489, 490,491, 492,

493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 508, 509, 510, 511,

514,515,516,517,519,520,585,610. 3D ., 522, 523, 524, 525, 526, 527, 528, 529, 530,

531,532,537,566,567,568,581,591,613,616,632,639, 652, 659, 668, 674, 677, 678.

680,694,695,696. Flat·plate·, 113. 122,336,337,338,349,363,488,489,490,510,532

Budge~ Kbudget, 318, 342,375, 376, 377,389,475,476,489,490,505,510, 543, 548, 588,

599,615,667. Reynolds-stress " 339, 377, 489, 490,548,552,553,576,588,599,609,692.

Dissipation " 339.

Bufferlaye~ 318,319, 338, 344,351, 352, 507.

Burst, 36, 430, 442, 447, 475, 491, 507, 545, 610, 613, 671, 692.

Canopy,652,661,666,670,672.

Cascade process, 24, 181.

Cayley-Hamilton theorem. 4.

Cebeci-Smith model, 108, 109, 110.

740 Subject Index

Centreline velocity, 330, 364, 368, 377,432,449.

Channel flow, 34, 105, 113, 115, 117, 118, 184,256,262,265,269,272,330,336,337,338,

343,344,347,348,351, 352, 362, 363, 472, 489, 542, 594, 599, 605, 606, 608, 610, 612,

613,615,617,620,621,632,633,639,688,689,694,695,700.

Characteristic function, 38.

Circulicity (tensor), 54, 55.

Clauser method, • plot, (U+ vs. Iny), 326, 501, 559, 585, 603 .• constant, 106, 110, 587, 606,

636, 661. • parameter (G), 322,482,509, 552, 677.

Clearance, 555, 558.

Coherent structures, 29. Coherence spectrum, 147, 148.

Coles form, • law, 322, 325 .• function w, 324, 479, 659, 660.

Collaterality, 531, 532, 533.

Componentality, 6, 249.

Conditional averaging, 34,35,287,290,435,589.

Constraint ratio, 409.

Convection, 308, • velocity, 125,371,605.

Core, 332.

Coriolis forces, 20, 46, 52, 54, 58, 59, 88, 89, 91, 190, 200, 219, 248, 556, 558, 612, 613,

614,615,618.

Correlation coefficients, • functions, 148, 168, 169, 189, 335, 353, 354, 371, 372, 373, 374,

506,507,612,630,681,688.

Cospectrum, 147, 191.

Cubic box, 65,144, 145, 159, 172.

Cumulant, 38, 40, 42, 179.

Curvature, 281, 308, 508, 512, 546, 552, 563, 564, 568, 576, 588. Longitudinal., 518, 563,

564,565,569,570, 571, 572, 574, 575, 579, 581, 585, 586, 587, 614, 620, 632, 633, 634,

695,698,700. Transverse ., 346, 347, 433, 389,499,516,563,564,578,579,584,599,600,

601, 603, 605, 606, 608, 609, 610, 611, 629. Local ., streamline ., streamwise ., 125, 135,

414,497,499,571,577,592,608,642,684,685,693, 694. Lateral·, 516, 518.

Cyclostrophic balance, 309, 397, 513.

D'Alembert paradox, 10.

Damping function, 114.

Decay, 124, 172,449,452,675 .• rate, 365, 381, 392, 403, 450. Final period of ·,67,69,71,

172.

Defect·displacement thickness, 322.

Detachment, incipient ., 499, 500, 502 (see also separation).

Detailed balance, conservation, 161, 165, 174.

Dimensionality, directionality (tensor) 6, 54, 156,250.

Subject Index 741

Direct Numerical simulation, DNS, 26, 34, 77, 80,91,92,112,115,116,117,122,173,185,

188, 193,205,214,219, 232, 233, 247, 255, 258, 266, 269, 272, 329, 330, 338, 339, 343,

349,350,475,484,532,542,583,610,618,621,652,690.

Displacement thickness, 310.

Dissipation, 49, 57, 70, 95, 96, 97, 99, 112, 114, 115, 119, 121, 123, 129, 143, 148, 149, 174,

185, 188, 191, 213, 229, 260, 286, 293, 321, 345, 351, 355, 363, 375, 377, 438, 439, 452,

473,486,489,492,504, 508, 509, 510, 545, 552, 574, 588, 589, 592, 614, 624, 636, 642,

647,650,675,682,686.

Drag, 311, 674.

Duct, 99, 684, 685, 694, 703, 705.

Eddy, 24,55,180,191,677,698,699.

Eddy viscosity, effective viscosity, turbulent viscosity, 47, 102, 103, 105, 106, lll, 113, 115,

124, 126, 143, 186,231,274,292,294,362,367,368,393,394,407,409, 411, 417, 422,

432,437,441,445,447,449,485,487,492,496, 507, 537, 538, 552, 576, 577, 587, 596,

606,624,636,641,642,650,664,697.

Ejection, 37, 492, 692.

Ekman layer, 558, 561, 562, 612, 613.

Energy spectrum, 147,680.

Ensemble averaging, 28.

Entrainment, 290, 291, 382,476,630,653.

Enstrophy density, 6, 13,50, 149, 171, 178, 193,360.

Equilibrium boundary layers, 328, 478, 479, 482, 486, 487,509,589.

Equipartition,69.

Events, 37.

Ergodie, 13, 143. Ergodicity, 28.

Excess momentum, 310.

Explicit model, 283.

Fanning friction factor, 327.

Far wake, 279, 287, 191,311,428,430,431,432,433,435,437,448,453.

Flapping, 365, 375, 444.

Flatness, 39, 194,234,336,337,338,360,489,507,508,610,672.

Focus,8.

Frame-indifference, 18,57, 107,229,247,248,281,340.

Free shear flows, 116,281,305, 372.

Freestream turbulence, 321,436,510,673,674,675,676,677,679,680,681. - velocity, 481,

602.

Friction velocity, 92,316,317,478,501,531,532,533,534,536,605,609, 617, 688.

Fully-developed flow, 330, 346.

Galilean invariance, 28, 29, 52, 148. - transformation, 19,274.

742 Subject Index

Gap (ratio), 555, 562.

Gaussian, 41, 42. - wall curvature, 522, 523. Gaussianity, 358.

Geodesic curvature, 512, 516, 524, 564.

Gradient-diffusion, 263, 264.

Hama's form, 323, 658, 661.

Head, 10, 513.

Helicity, 13, 151, 153,513.

Helical decomposition, 156.

Helmholtz equation, 11.

Hodograph, 525,526,527, 533, 535,536, 546, 551,560,561, 650.

Homogeneous flows, - turbulence, 64, 65, 66, 88, 143.

Hot wire, 33,67, 184, 192,330,334,376, 382,415,475,481,491,498,502,532, 551, 582,

649, 652, 665, 695.

Impinging jet, 104,408,412,413,415.

Inactive motion, 354.

Inertial subrange, 182, 186, 187,188.

Indicator, 35, 43, 287.

Ingress, 555, 556.

Inhomogeneity, 54, 92, 99, 232, 272, 330.

Inflow, inlet (conditions), 365, 369, 556, 642, 645, 646, 696, 705.

Inner variables, 333, 552. - layer, - region, 343,417,422,429,443,489,496,510,583,584.

Interaction region (of ajet), 364, 377, 391, 412, 443.

Interactions; wallward, outward -, 37, 57, 474, 490. Local, distant -, 160, 162, 163, 164, 176,

183.

Intercept, 98, 317, 655, 662.

Interface, 320, 370, 476.

Intermittency (effects), 35, 39, 43, 109, 113,286,287,291,292,294,337,359,366,384,434,

437,442,471,490,507. - function, - factor, 109, 110,320,370,433,500,506,610,611.

External -, 189, 190,319. Internal -, 190, 192.

Invariant, 4, 79, 116,249. Invariant direction, 7. Invariant maps, 73, 342.

Inverse power-Iaw spectrum, 356, 357, 510, 665.

Irrotational motion, flows, 6, 7, 63, 107, 125,201,202,674.

Isotropie flows, 64.

Isotropization ofproduction, 241, 279, 281,417,440,541,542,592,646,648.

Jaumann derivative, 21.

Jet, 117, 191, 232, 279, 293, 311, 413, 414, 415. Plane -, 31, 112, 113, 126, 291, 313, 364,

365,366,373,376,377,389,390, 392,393, 394, 395,443,444,445,450, 646. Radial-, 112,

408. Round -, 112, 118,280,281,286,291,313,367,371,373,378,386,395,643,646.

Swirling -, 281, 644, 646. Jet anomaly, 116,367.

Joint-probability, 38.

Kannan-Howarth equation, 70, 170, 172, 173.

Karman-Schoenherr formula, 316.

Kelvin-Helmholtz instability, 315.

Subject Index 743

K-epsilon model, 113, 116, 119, 124, 127,344,345,346,362,366,367,368,369,383,384,

386,411,416,417,423,424,433,434,444,449,452,453,477, 491, 492, 493, 496, 541,

551, 559, 560, 561, 592, 641, 643, 645, 647, 648, 649, 663, 664, 687, 696, 699, 701, 702,

699, 702, 703.

K-omega model, 128, 129,344,345,362,367, 368, 383, 384, 386,423,424,434,489, 659,

664. - baseline, 130. - SST, 130, 494.

Kelvin's theorem, 11.

Killing theorem, 6.

Kolmogorov assumptions, 60, 181, 182, 186, 187, 193. - constant, 184, 186, 188, 356.­

scales, 25, 33, 36, 60, 92, 181, 191,320. - scaling, 334, 355, 356. - spectrum, 180, 189, 194,

620,665.

Lamb surface, 13. - vector, 10, 11.

Large-eddy simulation (LES) 27, 172, 188,216,232,336,350,618.

Laser-doppler anemometry (LDA), velocimetry, LDV, 291, 330, 333, 334, 337, 338, 386, 387,

498,502,532,543,644,696,697.

Lateral divergence, - straining, 411, 516, 518, 520, 578, 581, 622, 631, 632, 634. -

convergence, 516, 518, 520, 578, 622, 625, 627,628.

Local (Craya) frame, 153,210,219,252.

Local axisymmetry, 184,340,341, 375, 388, 439.

Local isotropy, 122, 182, 183, 184, 191, 193,340,341,375,376,388,418,439.

Logarithmic law, law of the wall, 31, 32, 98, 113,316,332, 344, 347, 362, 478, 531, 532, 558,

583,584,585,597,600,601,602,605,607,608,663,679, 687, 706. - layer, - region, 117,

118,191,256,484,492,501,504,507,531,552,650,653,654.

LRR, Launder-Reece & Rodi model, 240, 249, 254, 255, 259, 368, 390, 541, 647, 686.

Ludwieg & Tillmann correlation, 326, 501.

Matching condition, 332.

Mean wall curvature, 517, 522.

Mixing, 24. - layer, 112, 126,311,390,412,417,441,443. Plane - layer, 313, 315, 378. -

length, 106, 107, 109, 112,482,484,507,510,533,559,587,601,635,636,639,696.

Momentum thickness, 310, 311, 316, 552, 677.

Momentumless wake, 443, 444, 445, 447, 449, 452.

Moment, 38, 179.

Nash's correlation, 479, 483, 509, 625.

Near-walllayer, 316.

Negative lobe, 373, 374, 375, 395.

744 Subject Index

Noda1 point, 8, 70.

Normal-stress difference, 365, 444, 505 .• anisotropy, 124,275,571,572, 579.

Normalization condition, 272.

No-slip condition, 9.

Nozz1e, 365,377, 378, 382,400,408,413,415, 418,419, 420, 555, 705.

Objectivity, 18,21.

01droyd derivative, 21.

Organized fluctuation, 30 .• motion, 50.

Outer 1ayer, • region, • variable, 316, 319, 338, 343,417,420,422,429,489,496,500,583,

603, 606, 679.

Phase averaging, 29, 30.

Pipe flow, 113, 122,265,336,337,338,346,347,348,352, 4l3, 637, 638, 653, 668, 674,

694,705.

Poincare inequality, 12.

Po1arization decomposition, 153.

Prandtl-Batche1or theorem, 12.

Pressure-gradient effects, 119, 129, 265, 347, 425, 429, 471, 472, 474, 476, 480, 534, 624,

695,697,698,705,706. Adverse·, 111, 127,433,473,477,484,485,487,490,492,493,

494,495,501,504,507,508,509,519,533,543,548,582, 585, 625, 662, 683, 698,701. •

parameter (ß), 322, 478, 491.

Pressure hessian, 16.

Pressure diffusion, • transport, 57, 95, 125,236,267,268,438,439,455,490,588.

pressure-strain correlation, pressure redistribution effects, 165, 234, 263, 268, 270, 279, 284,

376, 390, 414, 417, 438, 440, 490, 571, 575, 576, 592, 614, 633, 646, 647, 686, 691, 692,

701.

Principa1 axes (of Reynolds stresses), 76,87,88, 103, 106,243,250,437,642.

Probability density, • distribution, 37, 38, 144, 182, 194.

Projector, 5, 146, 159.

Propeller-driven, 448, 450, 451, 452.

Pure decay, 65, 66.

Pu1sed wire, 330.

Quadrant average, 36 .• techniques, 43, 343.

Quadratic flows, 217, 218, 219, 260.

Quasi-gaussian approximation, 263.

Quiescent surrounding, 3l3, 369.

Rapid distorsion, • theory (RDT), 53, 76, 86, 160, 196, 197, 204, 205, 209, 213, 215, 216,

219,220,222,230,235,239,240,242,243,245,247, 260, 276,576.

Rapid pressure, 339 .• pressure-strain, 75, 104, 124,245,247.

Subject Index 745

Realisability, 72, 74, 75, 104, 122, 124, 185,229,235,237,243,244,245,246,247,266,

271,273,277,281,284,291. Over·, 72. Strong., 74, 246. Weak ., 75.

Regular point, 6.

Relarninarization, 471, 472, 473, 476, 602, 603, 6l3.

Return to isotropy, 75, 76, 77.

Reynolds number, bulk ., 330, 331. • effect, 332, 334, 338, 350, 353, 677.

Reynolds properties, 26, 28.

Reynolds stress, 47, 48, 56, 64, 97, 105, 124, 229, 230, 231, 233, 274, 288, 290, 380, 384,

407,408,410,422,436,439,440,446,449,451,473,475, 485, 486, 493, 505, 506, 518,

531,542,544,571,581, 590,591,592,593, 596, 608, 610, 626, 638, 689, 691, 695, 696,

697, 707 .• model (RSM) 77, 95, 115, 125,231,232, 259,274,275,276,279,281,283, 363,

383,385,386,417,477,542,560,561,591,593,621,645,646, 647, 650, 683, 687, 693,

700, 704, 706.

Richardson number, 570, 571, 572, 573, 574, 575,576,577,578,580,614,637,638.

Rodi's hypothesis, 278, 279, 280.

Rossby number, 88, 89, 208, 614, 618.

Rotation, 4, 217, 219, 220, 221, 222, 236, 248, 253, 254, 399, 612, 613, 614, 616, 618, 637,

638, 648, 692.

Rotating-disk flow, 550, 551, 552, 553, 554, 555, 557.

Rotta's term, model, 280.

Roughness, 129,652,654,655,656,657,661,663,668,669,671. • function, 653, 655, 661.

Saddle point, 8,78,435,516.

Saffman integral, 67.

sand, sandgrain, 655, 656, 661, 671.

Schwarz inequality, 104.

Secondary flow, • motion, 330, 418,512,517,534,589,595, 6l3, 684, 686, 687, 693, 694,

695,696,697,698,699,700.

Self-preservation or self-similarity, strong ., 306, 312, 315, 316, 322, 325, 327, 328, 365, 366,

370, 371, 375, 376, 378, 380, 383, 386, 389, 393, 394, 395, 397, 401, 403, 404, 414, 418,

420, 422, 425, 431, 435, 436, 437, 441, 482. Weak ., 3l3, 314 .• of a boundary layer, 327,

475, 552.

Self-propelled, 279, 448.

Separation, 111,442,477,478,479,480,481,485,492,493,494,496, 497, 498, 499,501,

502,503,507,509,516, 526, 539, 540, 541, 542, 549, 550, 554, 563, 622, 632, 644, 657,

660, 673, 674, 680, 683, 697, 700, 702. Intermittent ., 485, 499, 500, 502, 503, 508.

Shape factor, 31, 473, 482, 500, 502, 503, 505,580,585,590,597,677,

Shear-free layers, 315.

Shroud, 555.

Signal-to-noise ratio, 338.

746 Subject Index

Singular point, 6, 8.

Sink flow, 474, 475, 476.

Skew-induced generation of vorticity, 519, 706, 707.

Skewness, 39, 91, 178, 183, 185,330,336,337,338,358,359,370,371,489,507,508,608,

609, 638, 670.

Skin-friction coefficient, - factor, - data, - law, Shear-stress coefficient, 323, 325, 326, 332,

333,344,362,419,472,473,477,481,482,483,484,492, 495, 497, 501, 516, 525, 531,

533,536,541,546,558,583,585,593,597,624,635, 638,649, 652, 658,677.

Slip condition, 9.

Slow pressure, 339. Slow Pressure-strain term, 53.

Solid-body rotation, 280, 398, 399, 401, 579, 637, 640, 641, 644, 645.

Space-filling, 13.

Spectrum, 146, 150, 168, 178, 185, 200,202,207,211.

Spreading rate, 365, 366, 367, 368, 391, 395, 401, 419, 446.

Square-duct flow, 276, 684, 687, 695.

Squire-Winter-Hawthrone, 514,529,535,536,551.

SSG (Speziale-Sarkar-Gatski) model, 344, 345.

Stagnation point, - line, 100,413,414,416.

Stokes potential, 9. - pressure, 272, 339.

Strain gradient, 54. - tensor, 198, 200, 217.

Streak, (streaky) 124,488,491,545,552,613. Streaklines, 6.

Streamline, 6, 515.

Stress-driven secondary motion, 520

Stretched-induced generation of vOlticity, 519. Stretching induction, 15,205.

Stropholysis, 149, 151, 156,211,250.

Summation convention, 1.

Sublayer, 474.

Superlayer, 321, 606.

Sweeping time scale, 181, 187.

Sweeps, 37,320,338,430,474,490,545, 602,603,605,621,624,670.

Swirl, 6, 398, 405, 408, 440, 444, 448, 449, 450, 453, 644. - number, - parameter, 311, 399,

400, 556, 579, 580, 637, 642. Swirling flow, 276, 640, 641. Swirling jet, 397, 398, 399, 400,

401, 407, 45l.

Synthesis, 115, 123, 232, 234.

Taylor microscale, 61, 169,336,375.

Taylor-Görtler instability, vortices, 594, 595, 596, 599,620,621,701.

Taylor hypothesis (TH) 32, 33, 34, 62, 148,418,439,455,489,620.

Taylor-Proudman theorem, 22,49,88, 107,209,248.

Tbroughflow, 552, 561.

Top-hatshape, 379, 380, 382,404,409,415,419,442.

Trailing edge, 425, 426, 430, 43l.

Subject Index 747

Transfer term, 57, 58, 59, 235, 238, 242, 249. Spectral., 170, 174, 183,220.

Transport terms, turbulent·, 47, 63,117,266,273,281,477,488,489,490,496,513,519,

538, 542, 545, 547, 548, 549, 552, 571, 572, 587, 588, 590, 592, 594, 599, 609, 611, 615,

626,627,630,631,636,645,647,650.

Triadic interaction, 160, 161, 162, 163, 164, 166.

Tripie correlations, tripie products, third-order moments, 176, 264, 265, 266, 267, 281, 339,

377,389,423,441,592,596,610,631,635,668,671,686.

(spectral) turbulent viscosity, 177.

Variance, 39.

Variable-interval, time-averaging (VITA) 28.

Velocity defect, 311; 494, 495, 607. -law, 323.

Velocity-vorticity correlation, 48, 62, 115.

Virtual origin, 365.

Viscous sublayer, 317.

Von Karman constant, 98, 317.

Vortex 7, 197,391, 435, 446, 672, 698 .• breakdown, 399, 405, 406,644, 70l. • core, 16. Free

,,397,398,579,642,644,648. Combined " Rankine·, 397, 398, 643. Forced ., 397, 579,

642,644,647. Horseshoe ., 435, 549 .• shedding, • street, 375, 390, 404, 442, 444, 447, 673,

677· stretching, 48, 50, 67, 86, 115, 178, 203, 285, 519 .• tubes, 6, 16, 86, 193, 212, 213,

214,215,216.

Vortical region, 7.

Vorticity, 3, 24, 25, 28, 30,45,48,50,52, 53, 54, 55, 58, 61, 62, 64, 76, 83, 84, 85, 86,91,

107, 108, 110, 111, 115, 150, 193, 194, 200, 221, 321, 340, 341, 342, 343, 349, 353, 358,

359,360,377,390,397,399,428,434,435,439,442,475, 476, 478,512, 513,514,517,

518, 519, 520, 521, 525, 529, 530, 536, 538, 542, 551, 572, 595, 602, 603, 604, 609, 614,

615,616,617,690,695,707. Tilting·, 46, 54, 212, 213, 215, 218, 615.

Wake, 117,232,305, 306, 310, 313, 314, 323, 324, 345, 349, 364, 366, 368, 371, 392, 393,

397,399,417,424,425,426,427,428,429,430,431,433, 434, 435,436,437,438,439,

440,441,442,443,445,446,447,448,449,450,451,453,455, 523 .• effect, 323. Plane "

31,32, 280, 314, 371, 390, 395,425, 43l. Round " 314,440,441,443,608 .• factor, wake­

strength factor, Coles • parameter, 345, 582, 583, 590, 624, 661, 677, 707 .• strength, 346,

350. Law of the·, 535, 537.

Wall crossflow angle, 526, 527, 532, 623 .• function, 116,328,417,423,534,687,693,701,

706,707.· jet, 413, 417 .• 1ayer, 317, 332.

Wing root, • tip, 521, 528, 529.

ZPG, zero-pressure gradient (see flat-plate boundary layer)

LIST OF SYMBOLS AND PAGE OF FIRST OCCURRENCE

Latin symbols

a a

: radius of a pipe or of a cylinder, inner radius of a disk cavity : exponent for Ue = C[x-xoja

a\ : shear-stress structure parameter (= -u'v'/K = -2b\2) ai : logarithmic strain in the ith direction (= Int;) ai : expansion coefficients in the bij(VV) expansion

!lall : first fundamental form for the wall surface (= ar/a~a.ar/a~ll) A : characteristic nozzle area for a radial jet A : slope parameter for the centreline log law in a wake Ab : abnormality or torsion of neighbouring pathlines (= T.curl T) Ai : constants of asymptotic correlation regime (i = 1,2,3) Au.Ay : parameters for decay and spreading rates for a wall jet Ac : parameter for the evolution of skin friction for a wall jet A + : sublayer parameter Aa(k) : velocity components of the polarization decomposition

All : constant of the k-\ regime (= k\El\(k\)/U~) b : slot width of a wall jet b : outer radius of a disk cavity ball : second fundamental form for the wall surface bi : eigenvalues of bij bij : anisotropy tensor of velocity correlations b(e) b(Z) I" f' ij' ij : sp Ittmg 0 amsotropy B : intercept parameter for the log-Iaw centreline in the wake B : integrallayer thickness B : vector potential for the instantaneous velocity field B : binormal associated to a pathline in the Frenet formula B(i) : inner intercept of the logarithmic law

B~i) : inner intercept for the log law past a circular cylinder

B(o) : outer intercept of the logarithmic law

B(o*) : outer intercept based on 11* = y/o*

B* : vector potential for the instantaneous acceleration (= -vw) B}.B2 : constant for the skin-friction law B}.B2 : Simpson's ratios measuring turbulent transport B' \ : constant for the Fanning skin-friction law Bi : constants of asymptotic correlation regime (i = 1,2,3) B\[k+j, Br[k+]: intercepts for the Re-number- and relative-roughness forms Bijpqmn: Sixth-order tensor defined by eq.4.48b c : contraction parameter in axisymmetric strain (= e\) CD : constant for dissipation length Co : drag coefficient

Cr : skin friction coefficient (= 2't-/pU;) CK : Kolmogorov constant

CK,CE : modeling constants for transport of K and E.

CM : torque on a disk of radius R (= 2M1pQ2R5) Cs : c10sure coefficient for third-order diffusive terms Cv : specific heat at constant volume

327,558 477

105, eq.2.225 204

275, eq.4.86

522 409 426, eq.5.163 513 357, eq.5.105 419 419 97, eq.2.196 153, eq.3.64

356, eq.5.102

418 556 522 72, eq.2.145 47, eq.2.59

155, eq.3.73 426, eq.5.163 494 9, eq.1.31 512 98, eq.2.201

600,eq.6.141c

323, eq.5.53 328, eq.5.67c

10 326, eq.5.59 548, eq.6.96 327, eq.5.64 357, eq.5.105 653, eq.6.176 249, eq.4.48a 203 107, eq.2.229 311, eq.5.16b

316, eq.5.34 184, eq.K41a

117, eq.2.276

550 261 49

List of Symbols 749

C2 : Isotropization of production coefficient C3,C4 : coefficients of the model of Launder et AI.(l975) ChC2,C3:coefficients of non-linear eddy-viscosity models Co, Ce, CI. C3, C2: coefficients for self-preservation C, Ci : constants of asymptotic correlation regime (i = 1, 2, 3) Cjk : cumulant of the characteristic function

Cij : inhomogeneity tensor (- '!"i,n,!,'n)

C~j) : curvature-induced convection terms in Reynolds-stress equations

Cwmin : non dimensional mass flow rate supplied to a disk cavity (= rhJl1b)

C.IoC.2: closure coefficients for «llij C'~ : Eddy-viscosity constant C~ : Eddy-viscosity constant Cel,Ce2: modeling constants for the K-E model Cv1oCv2: modeling constants for the eddy-viscosity model

CrohCro2: modeling constants for the K-ro model Cik : cospectrum d : height or diameter of the exit-nozzle of a jet D : exit diameter of a round jet D : side of a square duct o : drag of an obstacle 'D : viscous rate of dissipation (= p-1't:S) 'D : true turbulent rate of dissipation (= ~/2)the most often Dh : hydraulic diameter of a duct (= 4xcrossection/perimeter) ~. : true dissipation of velocity correlations o/äB : along-the-binormal derivative (= B.grad) oloM : along-the-normal derivative (= M.grad) olos : along-the-streamline derivative (= T.grad) IX : Dean number (= Reb[D/2Rcp/2) detA : determinant of A d/dt : material derivative (= (JI(Jt + v.V)

Mdt duldt

duldt rJi'D. d~(i)

e E E(k)

E(Tl)

: time derivative following the mean flow : Oldroyd derivative

: mean 01droyd derivative : Jaumann derivative

: physical variations of the curvilinear coordinates (= hid~i) : instantaneous intemal energy : entrainment rate for a round jet (= -27tyV[y~oo]) : turbulent kinetic energy spectrum : self-similar dissipation (= eO/ u 3)

Ei-(k) : spectrum of V'iV'· E~ Ei : unit (axial) vectof for an axisymmetric turbulence

e : (= «llkk(k)/2)

e(k)NL non linear contribution of the non-averaged modal energy ei strain in the ith direction ( = exp[tlJ) ei ith vector of the cartesian frame of reference el(k) : local orthonormal frame (Craya frame) eK : Kth unit vector in a fixed frame

241, eq.4.25 241, eq.4.29 276, eq.4.88 312 357, eq.5.105 38, eq.2.22b

54, eq.2.84b

569

556 77, eq.2.160a 111, eq.2.253 113, eq.2.261 116, eq.2.273 127, eq.2.293

128 147, eq.3.30 365, eq.5.1l3 379 695 311, eq.5.16a

7 49, eq.2.68, 684 57, eq.2.90d 512 512 512 695 3, eq.l.lOb 5, eq.l.22a

47, eq.2.58a 21, eq.1.75

58, eq.2.93 21, eq.1.76

512 49, eq.2.65 382, eq.5.122 68, 167 312, eq.5.18

189, eq.3.201 153 154, eq.3.68

160, eq.3.91 201 1 152, eq.3.63 18

750 List of Symbols

ei : ith unit veetor in a moving frame

f : Fanning frietion faetor (= 8U~/U~) f : low frequeney eharaeteristie f : body force F : mean strain gradient (FiK = i)x/aXil f(r) : longitudinal two-point velocity eorrelation funetion

f(Tl> : selfsimilar velocity defeet for the axisymmetrie ease (= F'(l1)/ll

F(l1) : self-similar velocity defeet (= [Ue-U]/Uo)

t(k\) : one-dimensional spectrum defined from f(r) fv : Van Driest damping funetion fl! : eddy-viscosity damping funetion

fij(k\o): eddy-speetrum sealing (= Eij(k\)/U~O) F : dissipative-range speetra1 funetion F : ratio of normal-stress production 10 shear-streaa production Ff : Flatness faetor of f Fi : momentum exchange vector Fijpq : velocity-gradient eorrelation tensor F"fi : flux of intermitteney g(r) : lateral eorrelation funetion in homogeneous isotropie turbulenee

g(k\) : one-dimensional speetrum defined from g(r) g : gravity vector

gl : eovariant basis veetor (i = 1,2,3, gl = aRlaSi)

gl : contravariant basis veetor (i = 1,2,3, gl = gradsi)

g(i) : physieal eovariant basis veetor (i = 1,2,3, g(i) = hi\aRlasi)

g(i) : physical contravariant basis veetor (i = 1,2,3, g(i) = higradsi) gij : eovariant metrie tensor of a eurvilinear system (= gl.gj) giJ : eovariant metrie tensor of a eurvilinear system (= gl.gJ) g(ij) : physieal eovariant metrie tensor of a eurvilinear system (= g(i).g(j) . g(ij) : physica1 contravariant metrie tensor of a eurvilinear system (= g(i).gU»

gij{l1) : self-similar Reynolds stress (= v'iv'jlu 2)

gij(k\y): eddy-speetrum scaling (= Eij(kl)/U~Y) G G G G G

G G Ge GE GiE

Gij dij)

+ Gr

h h

: weighting funetion for the defmition of mean value : production of turbulent kinetie energy (= Gi/2) : Green's funetion for the half-spaee Poisson problem : angular momentum integral : Clauser parameter : axial pressure gradient in a duet (= -p.laP/ax) : gap ratio of a disk eavity (= s/b) : shroud-clearanee ratio of a disk eavity : generation of dissipation : i = 1,2, 3,4 eontributions to generation of dissipation

: production of velocity eorrelations

: eurvature-indueed produetion terms in the Reynolds-stress equations : Görtler number

: total head (= p*/p + v212) : half-width of aplane ehannel

18

327

353 5, eq.1.22b 54, eq.2.80 167, eq.3.122

380, eq.5.118

312, eq.5.18

168, eq.3.125 97, eq.2.196 113, eq.2.261

355, eq.5.98

184, eq.K41a 505, eq.6.28 39, eq.2.29 288,eq.4.120 62, eq.2.1l7 288, eq.4.1l7 167, eq.3.122

168, eq.3.125 5 512

512

512

512 512 512 512 512

312, eq.5.18

355, eq.5.99

26, eq.2.1, 2 59, eq.2.103a 270, eq.4.76 311, eq.5.12 322, eq.5.49

684, eq.6.214 556 556 63, eq.2.123 63, eq.2.123

56, eq.2.90a

569 596

10, eq.1.38 327

List of Symbols 751

hi : i = 1,2,3; metric e1ements(= &~12, no summation)

hij(k:(T)): eddy-spectrum scaling (= ~i<kl)/[v/E]lI2rl) H : ho1esize

H : shape factor (= olle) H : distance between the nozzle of an impinging jet and the wall H : integral angular momentum invariant for momentumless wake

H : mean total head (= p-IP*+VzI2) H(k) : helicity spectrum H : helicity of a fluid volume HA : A-helicity of a fluid volume

H i : dissipation flux (= H~T + rrE - VE i) 1 1 '

H J : indicator function for a quadrant conditional average Hw : mean wall curvature H ij : tensor connected to the pressure hessian Hij(k) : vorticity spectrum tensor

H ijpq i 11, III IA IL In IIA lIlA IIL IIIL

: (= 'I''i,p'l''j,q) : (",eI) unit vector along XI : second and third invariants of bij (I = 0) : first invariant associated to the second-order tensor A (= trace A) : Loitsanskii integral

: integral of PO(11) : second invariant associated to the second-order tensor A : third invariant associated to the second-order tensor A (= det A) : second velocity-gradient invariant (= - Vm,kVk,n/2)

: third velocity-gradient invariant (= detVv)

IIR,IIIR: second and third invariant associated to v'v' j : index for plane flow Ci = 0) or axisymmetric flow Ci = 1)

J : jacobian of the transformation curvilinear system ~ cartesian system h : turbulent kinetic energy flux (= Jiil/2) J*k : turbulent kinetic energy flux (= J*iil/2) J ~k : transport of velocity correlations J ijk : transport of velocity correlations + Jijk

j k

k.! kes kq

: transport of velocity correlations : (",e2) unit vector along Xz : mean roughness height

: Kolmogorov wave number (= 11-1 = [Elv3]l/4) : equivalent sand roughness : amplification parameter (= 2Ucdq/[ldu/dylqdx])

k;,k; : values of k+ limiting the tmsitionally rough regime

k* : (= [S3/E]lI2) k : ("'e3) unit vector along XI k, ki : wave number of modulus k kL,kM,ks: long-, medium, short wave numbers in a triadic interaction K : turbulent kinetic energy K : initial wave number = k(t=O). t : instability parameter (= 1.57Wmax/Uo)

Kmax,Kmin: principal curvatures of a surface 'l( : curvature of a pathline or a streamline 'X: : energy partition parameter

512

355, eq.5.100 36

310 412 448,eq.5.181b

513 151, eq.3.50 12, eq.1.42 13, eq.1.44

63, eq.2.124

36, eq.2.17 517 15, eq.1.51 149, eq.3.40

150, eq.3.43b 1 72, eq2.144 4 171, eq.3.163

323, eq.5.22c 4 4 7

7

73, eq.2.152 307

524 59, eq.2.103c 60, eq.2.107 56, eq.2.90b 57, eq.2.91b

57, eq.2.92b 1 657, eq.6.173

184 656 573

655

191 1 68, 144 160 47, eq.2.49 198 400 564 512 83, eq.2.176

752 List of Symbols

K : kinetie energy of a fluid volume Kacc : acceleration pressure-gradient parameter Ki : isotropy measures (i = 1,2,3,4) Kn : mesure of anisotropy of normal stresses Kv : aeceleration parameter Kw : gaussian eurvature of the wall surfaee KloK3 : geodesie eurvature of eurves z = const. and x = const. on the wall K13, K31: wall eurvature parameters K : Stratford parameter for the one-half power law K 1 : associated intereept for the one-half power law Ki : parameters for deeay and spreading in a weak swirling jet (i = I, 2, 3) Kp : pressure gradient parameter (= -dP+/dx+)

K't : relaminarization eriterion (= [Ch+/oY+]w) Kij : geodesie eurvature tensor Kp.i : two-point eorrelation between p'(X) and v'i(X')

Ki.p : two-point eorrelation between V'i(X) and p'(X') L : length seale of the eddy-viseosity model L : streamwise extent of the shear layer L lo Lz : longitudinal, transverse integrallength seale L lo Lz : longitudinal and laterallength seales

L~ : outer dissipation length seale L3 : length seale associated to Rl1(O,O,r) I : size of large, energetie eddies { : distanee from the eentreline where U = Umin (momentumless wake) {1I2 : half-width of streamwise turbulent intensity {1I2 : half-width of turbulent kinetie energy

Im: mixing length defined by [VTS-l] 112

I ~ : integral seale of turbulenee defined by VT/{K I E : integral dissipation seale of turbulenee In : eigenvalues associated with S2+W2, 11 ;:: 12 ;:: h. L : box size for the Fourier transform of random amplitudes Lp : streamwise pressure-gradient length se ale (= [dlnU.,Idx]-I) Ln : length of the entry tangent of abend

4 : dissipation length seale (= [-u'V'PI2/E)

LGk) : integrallength seale of turbulenee m : unit veetor along a line (= dp/dp) M : mesh of a grid generating turbulenee M : flux of momentum M : normal assoeiated to a pathline in the Frenet formula

Mjk : moment iiJvIC ~~,M;: linear spectral operators for velocity and vOiticity amplitude equations M ijpq : fourth order tensor involved in rapid-pressure strain

mjk n N N N N(T])

: eentral moment u'Jv'k : exponent for the decay of turbulent kinetie energy : integral momentum invariant for a laminar momentumless wake : distance from the wall of the maximum baekflow velocity : rotation rate of a pipe wall (= W wlUb)

: self-similar eddy viseosity

12, eq.1.42 119.473 343, eq.5.87 608 624 522, eq.6.58 524, eq.6.60 524, eq.6.61 493, eq.6.14 493, eq.6.14a 401, eq.5.149 474, eq.6.1b

474 512, eq.6.37b

157

157 127, eq.2.293 306 65,90 169, eq.3.131

675, eq.6.209 373 26 443 443 444

107, eq.2.228

106, eq.2.227 107, eq.2.229 16 144 471 696

510

148, eq.3.32 17 65 310, eq.5.lOa 512

38, eq.2.22a

199 150, eq.3.43a

39 66, eq.2.127 445,eq.5.178b 503 637

383, eq.5.124

List of Symbols 753

Ne Ni(k)

Da p p p(u) p(U,v)

p*

: eddy-viscosity ratio (= vTivTx) : unit complex vector : a. = 1, 2, 3 eigenvectors associated to Aa in the reference frame : pressure : pitch between roughness elements : pdf attached to a random variable UI taking possible values u. : joint pdf : (= p + pgz)

p* : (= p* + x. 'ti + IQxr1212) p'(r) : rapid pressure fluctuation p'(s) : slow pressure fluctuation

P : anisotropy of the Craya frame (= [Ij>'12+Ij>'2dl2) P : production integral for the jet problem eq.5.l34b * * Px' P z

PI P(N)

Pij(k) q

q qeff

q2

q(k)

Q Q Q

Q(T]) Q* Ql,Q3 Qs

Oe

: non-dimensional pressure-gradient parameters

: normalized pressure gradient in a backflow region (= N2dP/[pvIUNI)dx) : orthogonal projector (= 1- NN) : orthogonal projector (= Öij - kik/k2) : measure of the divergence of inviscid-flow streamlines : (=divr) : effective heat flux

: (= V'iV'i) = 2K) twice the turbulent kinetic energy : Third-order spectral function of homogeneous isotropic turbulence

: anisotropy of the Craya frame (= [lJ>'12-Ij>'21l12 = H/2k) : second invariant for instability : mean velocity modulus : self-similar turbulent kinetic energy (= 2K1u 2) : velocity scale obeying the law of the wall : functions in the pure-shear problem of rapid-distorsion theory : modulus of the secondary velocity (= [V2+ W2pl2)

: wall-friction velocity modulus (= [1I'tw IVp)1I2)

QiK(t) : matrix of cosinus of angle between ei (t) and eK Qf. g : two-point correlation between fand g

Qi.k : two-point velocity correlation between V'i(X) and V'k(X')

Qi.j.k : three-point velocity correlation between V'i(X), v'iX'), V'k(X")

Qij.k : three-point velocity correlation between V'i(X), v'iX), V'k(X')

Qi.jk : three-point velocity correlation between V'i(X), V'j(X'), V'k(X') Q.k : quadrature spectrum r : distance between measming points X and X' (= X'-X)

r*. : wall variation of v'iv'Ji2K 1J

r : radius in the cylindrical coordinate system ro : local radius of an axisymmetric body rj,ro : inner and outer radius of abend

R(N,8):tensor of the rotation around axis N and of angle 8

R : measure of rotation (= [2-Rii(N,8))/4)

537, eq.6.91 155 4 5, eq.1.23a 656 37 38

5

20 53, eq.2.75a 53, eq.2.76a

154, eq.3.68 393,

531

504, eq.6.25 5 154, eq.3.69 525 10, eq.1.35a 49, eq.2.67

47, eq.2.59 170, eq.3.137

154, eq.3.68 198 531

312, eq.5.l8 531 206 517

536

19 143, eq.3.l

146, eq.3.17

151, eq.3.54

152, eq.3.55a

152, eq.3.55b 147, eq.3.30 144

268

307 516 695

4, eq.1.l8.

4, eq.1.20

754 List of Symbols

Re Rf Rg

Rl

RL Rp

RT

mean radius of abend (= [ro+rill2) Riehardson number gradient-Riehardson number

: energetie Reynolds number (= ullv = unnsL,Iv)

: box-size Reynolds number (= ELi/u;ms) : propeller radius in a self-propelled wake : turbulent Reynolds number (= K2/VE)

Rs : momentum Reynolds number (= Ue91v)

Re : eentreline velocity Reynolds number for a ehannel flow (= 2hUmax/v)

Re(x) : Reynolds number (= Uo(x)L(x)lv)

Re.

ReA

Reh ReJ

Rek

ReL ReM R~

Re)..

Ren Re,;

Re, Reloc

Re+ Ri

ROT Ro*

ROm

: Reynolds number ofthe flow along a eircular eylinder (= Uealv)

: Reynolds number of a radial jet (= UdAIv)

: bulk Reynolds number (= 2hU~, 2aU~)

: Reynolds number of a wall jet (= U Jb/v)

: Reynolds number (= vk2t(k» for an eddy of size O(k-')

: Reynolds number (= U'tUv) : Mesh Reynolds number behind a grid of mesh M

: Shear-layer Reynolds number (= UoÖlv, = UOYI/2/VT)

: turbulent mieroseale Reynolds number (= unnsAlv)

: loeal Reynolds number for a rotating disk (= Qr2/v)

: skin-frietion Reynolds number for a ehannel flow (= 2hU't/v)

: Displacement Reynolds number (= Ueö,/v) : loeal Reynolds number on a rotating disk (= [rlbFRen)

: Reynolds number of a duet (= <U't>Di/v) : eurvature Richardson number : mieroscopie Rossby number (= u Im)

: maeroseopic Rossby number (= unn.t2QL)

: vortieity Rossby number (= wnn.t2Q)

rij : normalized Reynolds stress (= v';v'jlq2) r ik : velocity eorrelation for organized motion Rij(r,x): two-point eorrelation funetion

Rft)('t), R l1('t): eorrelation eoeffieients s : arc length along a pathline or a streamline s : distance between the two eoaxial rotating disks of a eavity flow s : distanee along the wall around the perimeter of a transition duet Sc : clearance Sref : one quarter of the eross-sectional perimeter of a transition duet Sx : aspect ratio of the roughness (= lxlk), Ix: length of bars along x

So:

S S S[k+] Si S, S2, Su Sf

: a. = 1,2,3, eigenvalues of the rate of strain tensor, s, ~ S2 ~ S3 : shear intensity (in pure shear) : swirl parameter (= G/MR)

: roughness funetion of the K-ro model : momentum souree term : spreading rate of a jet : decay eonstant of a jet : skewness faetor assoeiated with f.

695 573, eq_6.113 574 61, eq.2.110

196 451 65

316

330

312

602

409

327

418

181

317 65 309, 383

65

550

330

326, eq.5.59b 555

684 572,637 88 89

92

55 50, eq.2.72 335

374 512 556 706 556 706 656 15 80, eq.2.169 311

664 288, eq.4.115 365, eq.5.114 365, eq.5.114 39, eq.2.28

SK SL Sw Sy

SE Sm

Si.i

s S* S*(t) S

S

S t(N) t(k) ts(k)

1m tFG

t~~) 1)

T T TD T· T(k) ~~)

1)

~r) 1)

(s) Tijpq

T rr TUoo U u' U

Ud Ue Um UN Up Us Us

Us

Ux

U't Uo U* U* V V'

V

: decay rate for K : Saffman integral ; swirl number : entrainment eontribution to intelmitteney equation

: deeay rate for e ; swirl intensity

List of Symbols 755

368 172, eq.3.165 637, eq.6.166 290, eq.4.127

368 637, eq.6.166

:third order unknown eorrelation in homogeneous isotropie turbulenee 170, eq.3.146

: shear intensity (= [2S ijS ijPI2)

: mean shear parameter (= 2SKle) : mean strain-rate parameter

: rate of strain tensor (symmetrie part of Vv tensor)

: mean rate of strain tensor (symmetrie part of VV tensor)

: dimensionless ratio (= -2nt[dU/dy]) : stress veetor on da as a funetion of the normal N to da : lifetime of an eddy of size O(k-l) : sweeping time scale (= [urmsk]-l)

: eharaeteristie time for molecular diffusion (= t.;/v)

: pressure-gradient time scale (= [po(dP/dx)-1]1I2

: non linear advection term : averaging time : mean temperature : mean period of distorsion : transfer term : spectral transfer term

: rapid transfer term

: slow transfer term

: slow term defined by eq.(4.48b) : unit veetor tangent to the pathline in the Frenet formula : reciproeal of the torsion of a pathline or of a streamline : freestream turbulence parameter (= urmsoJUoo) : mean axial- (or streamwise-) velocity eomponent : fluetuating axial- (or streamwise-) velocity eomponent : order of magnitude of the random velocity scale : (= U(y)-Uc), for a momentumless wake : freestream mean velocity : eentreline velocity of a jet : maximum baekflow velocity : velocity seale in the pipe-flow problem : eonvection velocity : velocity seale in Sehofield's theory

: mean velocity where "fuw = 0.5 : x-derivative of mean streamwise velocity (= aUlax)

: skin frietion veloeity (= ['tw/p] 112)

: order of magnitude of the veloeity exeess or defeet

: veloeity seale (= UiUmax, = [vAlU~2] at y+ = 3) : veloeity seale : mean radial- (or nOlmal) velocity eomponent : fluetuating radial- (or normal) velocity eomponent : instantaneous velocity field (field in the moving frame)

105, eq.2.224

191 196, eq.3.203

3 50, eq.2.69

614 10 181 181

47

479, eq.6.8

165, eq.3.110 28, eq.2.6b 49 196 56, eq.2.90e 170, eq.3.151

53

53

249, eq.4.48a 512 512 675 307, eq.5.1a. 307, eq.5.1a. 48 445 106 364 503 347, eq.5.91 371 494 502 528

92 306

332, 491,resp. 611, eq.6.149 307, eq.5.1 b 307, eq.5.1b 5, eq.1.22

756 List of Symbols

Va : instantaneous absolute velocity field VH : (= P(Wil)v) V : mean-velocity modulus (= IIVII) V : mean velocity V q' V uv,V vw: turbulent transpOlt velocities

V ij

V m•n

V'iV'j

f-b w w' W W' Wo w' w

wer]) W(k) W Wa

W

: vorticity correlation tensor (= w'iwj)

: vector potential

: one-point velocity correlation

: Kolmogorov velocity seale (= [ve]l/4) : streamwise wavelength of roughness : gap spacing : mean azimuthal- (or spanwise-) velocity component : (=i)W/h3dZ) : Narasimha & Prabhu's prameter for the self-preserved wake : fluctuating azimuthal- (or spanwise-) velocity component : instantaneous vorticity field (= curl v)

: Coles defect function : energy transfer through the wave number k : mean vorticity field : mean absolute vorticity field (= W + 2il)

: mean tilting vorticity (= W + 4il)

W : instantaneous rate of spin tensor (skewsymmetric Palt of Vv tensor) .lYa : mean absolute rate of spin tensor W'x,W'y,W'z: x, y, z-components of the vorticity fluctuation 'Ub : dynamic vOiticity number

'nie X X X

: kinematic vOiticity number : a reference pruticle : matrix of eigenvectors of S : place of the particle X in its reference configuration

Xij : circulicity tensor (= 'I"i,n'l"j,n) xR : abcissa for the eye of recirculation zone in a swirling jet Xij : normalized circulicity (= Xij/Xpp)

xsO,xuo : virtual origins x : actual place of the particle X x,y,z : cartesian coordinates in the ex,ey,ez frame Ym : distance from the wall where the streamwise velocity is maximum Yml2 : distance from the wall where the velocity is half its maximum value yo : roughness length seale

Y : normalized outer-flow distance to the wall (= y/ö) Y : distance aabove the crests of roughness elements YE : destruction of dissipation

Yij : dimensionality tensor (- 'I"n,i'l"n,j) Yij : normalized dimensionality (= Yij/Y pp) Z : decay parameter Z : anisotropy measure

Z(k) : structural anisotropy parameter (= ~-iP)

~~,Z~ : transfer functions from initial to actual velocity and vorticity ampitudes

{Jk} : Christoffel symbols

19 20 512 46, eq.2.56 545, eq.6.95

62, eq.2.118

157, eq.3.82

47, eq.2.58a

60, eq.2.108 656 656 307, eq.5.1c 623, eq.6.157 431,eq.5.169a 307, eq.5.lc 3, eq.1.l2a

324, eq.5.55a 175, eq.3.177 48, eq.2.61 49

54

3 58 359, eq.5.106 9 7, eq.1.28. 5 15 5

54, eq.2.84c 400 55 365, eq.5.114 5 1 418 418 654, eq.6.179

319 652, eq.6.172 63, eq.2.124c

54, eq.2.84a 55 70, eq.2.137 542, eq.6.93

156, eq.3.76

199, eq.3.215

514

a a a* aE

ai ai

ß ß ß ßj ß* ßi 'Y 'Yg 'Yp

'Ytr 'Y.

greek symbols

: Clauser eonstant

: freestream turbulence parameter

: eddy-viscosity eonstant for the K-co model

: entrainment coeffieient (= FJ27tbW)

: eoeffieients of the funetional for Mijpq(b)

: coefficients in the Speziale model

: Clauser pressure gradient parameter (= 51dP/'twdx)

: erossflow angle (= tan-1[W1U])

: Monin-Obukhov parameter

: half-width spread angle of a round jet

: dissipation eonstant for the K-co model

: eoeffieients of the funetional for Tij(b,S,W)

: intermitteney faetor

: mean-strain-rate angle (= tan-1[dWtaU])

: pressure-strain angle (= tan-1[~r1t;i]) : transition intermitteney factor

: empirieal intermitteney faetor

'Yr : mean-turbulent-stress angle (= tan-1[v'w'/u'v'])

'Yu : mean time fraction during whieh u > 0

'Y't : mean-total-stress angle (= tan-1['tJ't,J)

r : acceleration

r : circulation of the velocity field

r : ratio of disk angular velocities in a disk eavity (= Q,.!!4)

r(t) : integral ofvk2(t) in rapid-distorsion theory

~~~ : physical Christoffel symbols

5, 50.99: boundary layer thiekness

51 : (streamwise) displaeement thiekness

5* : Rotta defect-displaeement thiekness

5p : pressure-gradient length seale (= pU~/[dP/dx])

5ij : Kronecker symbol, (5ij = 1 if i = j, ~j = 0 if i * j)

11 : anisotropy of the Craya tensor

11(.) : inerease or reduetion due to freestream turbulence

110 : Narasimha & Prabhu's prameter for the self-preserved wake

l1a : indieator for axisymmetrie turbulenee (= 4113+271112)

I1La : Cardano funetion (= 4II~ + 27111~) I1p : indieator for 2C turbulenee (= 1 +911+27111)

List of Symbols 757

106

677,eq.6.211

129,eq.2.297

382

238, eq.4.19a

273, eq.4.84

322

531

616

380

129, eq.2.297

238, eq.4.17

35, eq.2.15

536, eq.6.83

540, eq.6.92

109, eq.2.242

361, eq.5.107

536, eq.6.83

499

536, eq.6.83

9, eq.1.29

10, eq.1.40

555

201, eq.3.227

512

306

106,310

322, eq.5.47

480, eq.6.5a

2 154, eq.3.68

677

431,eq.5.169a

73, eq.2.150.

7

72, eq.2.147

758 List of Symbols

ßUc : centreline velocity exeess in a momentumless wake

ßUmax :maximum velocity differenee (= Umax-Umin), self-propelled wake

ßU/U't : roughness funetion

ßy : distance ofthe origin below the rougness erest

ßE P

E

: indieator for 2C state for Eij

: small parameter (= [U*Re/2]-l)

E : rate of dissipation of turbulent kinetie energy (= E;/2 = VV'iJV'ij)

E, E', E": polarities associated to the helical modes

~ : eorrected dissipation

443 443 653, eq.6.177

657

268

332

60, eq.2.106

162

96, eq.2.194

Eij : destruetion of velocity eorrelations 57, eq.2.91a

Eijk : alternator, Eijk= 1 if {i,j,k} = perm{I,2,3}, Eijk=-1 if {i,j,k} = perm{2,I,3},

Eijk = 0 otherwise 2

E;jpq : destruetion tenn defined by eq.(4A8b) 249, eqAA8a

6 : momentum boundary layer thiekness 310, eq.5.11b

6 : eolatitude in spherieal integration 146, eq.3.23

!p(k) : angle between the triad plane and k 163

!Pi : expansion eoefficients of a symmetrie tensor in tenns of two tensors

!Pij : velocity-vorticity one-point eorrelation (= V'iW'rV'jW'i)

$uu.i : spectral density for the indueed eomponent U'i

$ : longitude in spherieal integration

$'ap : Craya tensor

$10$2 : closure eoeffieients of the algebraix-stress model

<Pi : (= Eijk!Pk/2) <P(~;rt) : Charaeteristie funetion (Fourier transfonn of the joint pdf

<Pi : funetional fonns of the velocity gradient for pressure-gradient effects

<Pij : traeeless non linear transfer between Reynolds stress eomponents

<Pfg : eospectrum of f and g

<Pij(k) : velocity spectrum tensor

<Pij.k(k): Fourier transform of Qij.k(r)

<Pi.jk(k): Fourier transform of Qi.jk(r)

Tl : Kolmogorov length seale (= [V3/E]l/4)

Tl : similarity vruiable

K : Von Kannan eonstant ("" 0041)

A. A. A. A. A. A.

: second dynamie viseosity (= -2J.113) : orientation of the triad plane around k

: Taylor length scale

: strength ratio for a jet (= Um/Ue)

: roughness parameter (= k/w)

: Lamb vector ( = wxv)

240, eqA.23

48

674, eq.6.206

146, eq.3.23

154, eq.3.66b

279, eqA.94

48, eq.2.62

38, eq.2.20

480, eq.6.9

76, eq.2.157

145, eq.3.16

146, eq.3.18

152, eq.3.59

152, eq.3.59

60, eq.2.108

312

98, eq.2.201

5, eq.1.230nly

166, eq.3.114

61, eq.2.112

364

654

9, eq.1.29

List of Symbols 759

A.I, '-2: Longitudinal and lateral Taylor length scales

A.a : a. = 1, 2, 3 eigenvalues such that A.I ~ A.2 ~ A.3 A : unit vector orthogonal to the triad plane

A : relaminarization criterion (= -&JP/'twodx)

A J.1 V

Vr * vT

: equivalent sand roughness density parameter : dynamic viscosity : kinematic viscosity (= !lf P ) : turbulent viscosity

: non-dimensional eddy viscosity (= v-rlUeÖI)

VTx,VTz: turbulent viscosities (= -u'v'/[aU/ay], = -w'v'/[aw/ay])

Vq, V't

va n ns

: eddy diffusivities

: a. = 1,2,3 eigenvectors of the rate of strain tensor S : angular velocity of a rotating frame or of the solid-body rotation

: instantaneous rotation of the principal strains

nr,ns : rapid and slow angular velocity of two rotating disks

0> : = 41t, 21t, 0

COsv : Brunt-Vaisala frequency

&;j n n n* no nij

1ti

P Pu 't

't

'tI(k)

'tK

'tmax

'ts(k) 't

'tw

'If

'If'i

'" ~ ~i

: instantaneous rotation tensor of the moving framelfixed frame : scalar potential for the instantaneous velocity field : Coles factor : scalar potential for the acceleration (= p.lp*) : order of magnitude of the pressure term in the thin-layer approximation

: velocity-pressure gradient correlation (= p.I[V'iP'j+V'jp',d)

: eigenvalues of the pressure second gradient, 1t 1 ~ 1t2 ~ 1t3 : density of the fluid

: correlation coefficient

: total shear stress : non dimensional time (= tElK)

: eddy tum-over (inertial) time scale (= [Ek2].I/3)

: Kolmogorov time scale (= [V/E]1/2)

-" /U2 : parameter -u V m.x 0

: sweeping time scale (= [urmsk]·I) : viscous stress tensor : wall skin friction

: streamfunction of a two-dimensional velocity field

: fluctuating streamfunction vector

: non dimensional decay parameter

: dimensionless streamwise distance for the wall jet problem (= xM/v2)

: set of curvilinear coordinates

(i = 1, ~I=x, streamwise; 2, ~2=y, normal to the wall; 3, ~3=z,spanwise)

169, eq.3,133

4. 162

474

660 5, eq,1.23b 5 48

487

537, eq.6.90

635, eq.6.164

13 19, eq.1.65

17

555

9, eq.1.32, 33

614

19, eq.1.65 6,9, eq.1.31 110,324 10 310, eq.5.8

95

16

5, eq.1.21b

373

107, eq.2.231 282

186

35, eq.2.108

394

187 5, eq.1.23a 92, eq.2.184b

10

54, eq.2.83

67, eq.2.130d

421, eq.5.162

512

760 List of Symbols

Xfg

Xl e e eo

: measure of the organized motion to fg

: indicator function of the property J

: polar angle of a cylindrical coordinate system

: instantaneous temperature

: initial jet shear layer momentum thickness

ex,exll : momentum thicknesses

e ij : traceless velocity-pressure gradient correlation

V : Nabla (gradient operator)

Vv : velocity gradient tensor

Vi: covariant derivative

V (i) : physical covariant derivative

Cf : non dimensional growth factor for sheared turbulence

CfK : Prandtl-type constant for diffusion of turbulent kinetic energy

CfE : Prandtl-type constant for diffusion of dissipation

Cfv : Prandtl-type constant for diffusion of eddy viscosity

Cf Cl) : Prandtl-type constant for diffusion of (j)

1: : structural anisotropy vector

1: : swirl parameter

0' : stress tensor

2 -Cff : (= ['2 ) variance of f.

S± : helical components of the velocity amplitude

~ : (= w.k) vorticity modulus in the case of a 2D flow

~ : integral of normal-stress differences

~~r(k) : axisymmetric tensor of Chandrasekhar

yrot(k) I" . '>ij : po anzatlOn amsotropy tensor

Subscripts

a : absolute ave : surface-averaged value b : bulk quantity C, CL : centreline value D : indicate a divergence parameter diff : relative to second derivative of normal stress difference e : edge-boundary layer quantity h : homogeneous quantity

in J min max p pot

: i = 1, 2, 3; latin index for cartesian or curvilinear coordinates : inhomogeneous quantity : exit section of a jet : minimum value : maximum value : pressure side : potential quantity

31, eq.2.13

35, eq.2.14

307

49, eq.2.65

390

639, eq.6.170

57, eq.2.92a

3

3, eq.l.ll

514

514

81, eq.2.173.

112, eq.2.255

117, eq.2.277

127, eq.2.293

128

155

581, eq.6.134

5, eq.1.22b

39, eq.2.27

156, eq.3.75

12

309, eq.5.3c

154, eq.3.69

154, eq.3.69

19, eq.1.67 167, eq.3.120 47 397 625 520 106 157, eq.3.81c I, eq.l 157, eq.3.81c 365, eq.5.113 427 427 617 481

List of Symbols 761

lms : root-mean-square value T : refers to a turbulent quantity tot : refers to constant upstream dynamic conditions s : structural reference for a frequency, suction side w : wall value o : initial condition or condition prior to the imposition of extra-strain 1/2 : half-velocity distance 00 : asymptotic turbulent state, or far field reference

1: : total-stress reference

D e i L P pot (r) (s) w

:-

(.)*

* * (.)0

<.> (.) , 1\

1\

+

Superseripts

: (before a letter A) deviator of A = A - ItrA13 : index for outer boundary conditions for a boundary layer : i = I, 2, 3; latin index for coordinates in the actual frame : index for invariants associated to the instantaneous velocity gradient : P = 1,2,3; upper-case latin index for coordinates in fixed frame : index for the potential formulation. : rapid terms : slow terms : wall value.

Other symbols

: single-dot product of two tensors : double-dot product of two tensors

: upper dot: time rate

: associated to moving frame

: associated to mean value

: associated with the organized fluctuation : tilted quantities (associated with vorticity or mean gradient)

: conditional average for the turbulent zone : conditional average for the non-turbulent zone

: fluctuation associated to the turbulent zone : viscous-sublayer quantities non dimensionalized with u* : non dimensionalized quantities with curvature length scale qldUldyl"

: fluctuation associated to the non-turbulent zone

: phase average

: randorn fluctuation : Fourier transformation : variable-interval, time average. : viscous-sublayer scaling (using U't and v)

334 48 481 371,617 92, eq.2.84b 582 364 81

350, eq.5.93

77 309, eq.5.3 1 7, eq.1.27 19 481 53 53 92

2, eq.1.6. 2, eq.1.7

160

18

26. eq.2.1

30, eq.2.9 46. eq.2.55 287 287

287 489 573

287

29. eq.2.8

26, eq.2.56 27 28, eq.2.6b

93