analysis of power law and log law velocity profiles in the overlap region of a turbulent wall jet

Analysis of Power Law and Log Law Velocity Profiles in the Overlap Region of a TurbulentWall JetAuthor(s): Noor AfzalSource: Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 461, No. 2058(Jun., 2005), pp. 1889-1910Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/30046356 .

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PROCEEDINGS - OF -F

THE ROYAL SOCIETY

Proc. R. Soc. A (2005) 461, 1889-1910 doi: 10. 1098/rspa. 2004.1400

Published online 26 May 2005

Analysis of power law and log law velocity profiles in the overlap region of a turbulent

wall jet BY NoOR AFZAL

Department of Mechanical Engineering, Aligarh University, Aligarh 202002, India ([email protected])

The two-dimensional turbulent wall jet on a flat surface without free stream is analysed at a large Reynolds number, using the method of matched asymptotic expansions. The open mean equations of the turbulent boundary layer are analysed in the wall and wake layers by the method of matched asymptotic expansions and the results are matched by the Izakson-Millikan-Kolmogorov hypothesis. In the overlap region, the outer wake layer is governed by the velocity defect law (based on Um, the maximum velocity) and the inner layer by the law of the wall. It is shown that the overlap region possesses a non- unique solution, where the power law region simultaneously exists along with the log law region. Analysis of the power law and log law solutions in the overlap region leads to self- consistent relations, where the power law index, a, is of the order of the non-dimensional friction velocity and the power law multiplication constant, C, is of the order of the inverse of the non-dimensional friction velocity. The lowest order wake layer equation has been closed with generalized gradient transport model and a closed form solution is obtained. A comparison of the theory with experimental data is presented.

Keywords: power law vs log law scaling; turbulent wall jets; Izakson-Millikan overlap region; turblent wall bounded flows; similarity analysis

1. Introduction

The turbulent wall jet has been extensively studied, and the work has been reviewed by Launder & Rodi (1981, 1983). The flow in a wall jet (or a moving continuous surface), has no free stream velocity ( U0 = 0) and a classical velocity defect law does not exist. Most of the authors repeatedly questioned the log law of the wall. A wide variety of values for the Karman constant, k, and the intercept, B, in the log law have been quoted by some investigators (Bradshaw & Gee 1960; Tailland & Mathieu 1967; Hammond 1982), where B varies from 5.5 to 9.5. Wygnanski et al. (1992) recently proposed k=0.41. Further, Karlsson et al. (1993), Abrahamsson et al. (1994), Eriksson et al. (1998) and Tangemann & Gretler (1998) reported interesting data. Earlier, primarily owing to inconsistencies with the trend of the experimental data, several researchers investigated alternatives to the classical theory. First, Narasimha et al. (1973) proposed a novel scaling for a streamwise flow based on the initial momentum flux at the nozzle of the wall jet and molecular

Received 8 July 2002 Accepted 14 September 2004

a 2005 The Royal Society 1889



N. Afzal

kinematic viscosity, v. Second, the existence of a power law has been investigated by Glauert (1956), Bradshaw & Gee (1960) and George et al. (2000).

Recently, George et al. (2000) proposed a theory of power law velocity profile in a turbulent wall jet based directly on the work of George & Castillo (1997) from the power law in boundary layers, in which U,, the free stream velocity, and 6, the boundary-layer thickness, have been replaced for a wall jet, by Urnm, the maximum velocity, and 61/2, the outer half-jet thickness, respectively. The shortcomings of the power law boundary-layer theory of George & Castillo (1997), its application to the wall jet by George et al. (2000) and the necessary corrections needed are described below.

Power laws in wall and wake layers in boundary layer and wall jet (George & Castillo 1997, George et al. 2000): the asymptotic expansions in inner wall layer (y+ = yu /v) and outer wake layer variables (Y= y/A), in general following Afzal (1996a,b, 1997) for a turbulent boundary layer, may be taken as:

inner layer: - = +(x, y+); (1.1)

outer layer: - = F(x, Y) - F Y). (1.2) U U

In a turbulent boundary layer and a turbulent wall jet, respectively, George & Castillo (1997) and George et al. (2000) employed asymptotic invariance principle (AIP) in the overlap region for matching the first term FQ(x, Y) in the outer expansion, equation (1.2), and inner expansion, equation (1.1), (invariant under transformation y- y+h, with h+ = hu/v and h= h/A as constants) to propose

u±(y+) = Ci(y, + h+), (1.3)

Fo(x, Y) = Co(Y + h)a, (1.4)

CRAO~ R = (1.5) U ' v

where U= U, and A =6 for the turbulent boundary layer, and U= Um and S= 61/2 for the turbulent wall jet. The power law constants, a, Ci and Co,

proposed by George & Castillo (1997) and George et al. (2000) depend on the Reynolds number as given below:

blA Co Co, (A(1 + b1) \ a - Dg)l+b C -- exp (Dgln R)b , (1.6a,b)

(In Dg R)1+b , C (DnR) /

where a, =0.0362, b1=0.46, A=2.90, Ci= 55, Co, =0.897, Dg=1, h+=-16 and

Co = Co[1 + 0.283 exp(-0.00598DgR)]. (1.6c)

The outer matching solution, equation (1.4), in the outer layer expansion, equation (1.2), for h=0 gives u(x, Y=0)=0. Further, for hi0, it gives u(x, Y) =0

Proc. R. Soc. A (2005)

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Velocity profiles in turbulent wall jets

at Y= 16/Rj, and at Y=0 it gives

u(,0) = -o1o -* 0 for R -oo. (1.6d)

The results, equations (1.3)-(1.6), by George & Castillo (1997) in a boundary- layer flow and George et al. (2000) in a turbulent wall jet, may be compared with the recent work. Firstly, equation (1.6a) shows, that for large Reynolds numbers, a approaches a limiting finite value (a = 0.0362), whereas work of Barenblatt et al. (1997) and Afzal (1997) show that, for R,-* oo, the power law index a-*0 and consequently aoo 0. Secondly, C/Ci from equation (1.6b) depends on the Reynolds number, whereas in Afzal (1997, 2001) the ratio Co/ C to the lowest order is a constant, independent of the Reynolds number. Third, the result, equation (1.6d), is not in accordance with outer layer observations of Clauser (1956) and Coles (1968). The outer matching solution, equation (1.4), predicts u(x, Y=0) =0 for h= 0 or for R, -- oo if h/Z 0, and the no slip condition at the wall (Y= 0) has been automatically satisfied and therefore the inner layer is not needed. Consequently, in the work of George & Castillo (1997) and George et al. (2000), the outer expansions should include the next order terms of order uT as in the equation (1.2) for appropriate matching with the inner layer to obtain the log law (Afzal 1996a, b) and the power law (Afzal 1997, 2001) in the turbulent boundary layer.

In the present work, the mean flow in a turbulent wall jet has been analysed in terms of two-layer theory. The outer flow is a wake layer, governed by equations of nonlinear wake. The asymptotic expansions in the outer wake layer and inner wall layer are matched to propose log laws (Izakson 1937; Millikan 1938). It is shown that the matching of the inner and outer layers in the overlap region, by the Izakson-Millikan-Kolmogorov hypothesis (Afzal 1996a,b) also possesses a power law region simultaneously with the log law region. Analysis of the power law and log law solutions in the overlap region lead to self-consistent relations for power law constants, a and C, that depend on frictional Reynolds number, R,, and constants, k and B, in the log law of the wall. It is further shown that the power law index, a, is of the order of the non-dimensional friction velocity, and power law multiplication constant, C, is of the order of -1, the inverse of the non-dimensional friction velocity. The lowest order outer wake layer equation has been closed by generalized gradient transport model involving counter- gradient flux (Lykossov 1992) and, for constant eddy viscosity, a closed form solution that corresponds to a half-jet has been obtained.

2. Momentum equation

The equations for a two-dimensional turbulent boundary layer with constant pressure on a flat plate are

Ou Ov x - 0, (2.1)

Ou Ou 02u Or u +v- v + . (2.2) ox dy dy2 pdy


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1892 N. Afzal

outer variables 0.01 Um

Y*= y/6e Y =y/5 z =y/§m

inner variable

Y+=yu,/v U/2

§ Um

Figure 1. Transverse length-scales associated with a wall jet without a free stream.

The boundary conditions for the wall jet are

y=0, = v= = 0, (2.3a)

6 , --0, T7 -+0, (2.3b)

and the initial momentum flux

J = Ufjb. (2.4)

Here, u is the velocity in the streamwise x-direction, v is the velocity in the normal y-direction, T is the appropriate Reynolds shear stress, Uj is the efflux velocity, b is the width of slot, Rj= Ujb/v is the nozzle Reynolds number and V/u, is the scale of the inner the wall layer. Various transverse outer length- scales associated with wall jet flow are described in figure 1. Here, 6(=A1/2) is the boundary layer half-defect thickness (the normal distance measured from wall to the location where the mean velocity decreases to Um/2 in the outer flow). At any streamwise location, let Um be the local maximum velocity. Further, the other outer length-scale 6m is the location of maximum velocity, u= Um in the jet and 6e is the distance of the boundary-layer edge where u/ Um is approximately 1%. The small parameter, e, and Reynolds number, R,, are defined as

e- , R, - (2.5a, b) Um V

The wall shear stress, rw, and coefficient of skin friction, Cf, are

U- C f =-, . (2.6a-c)

The Karman momentum integral of boundary-layer equations is dM 7w, M = u2 dy. (2.7a,b)

dx p Jo




Velocity profiles in turbulent wall jets 1893

The integral equation (2.7a), for the momentum flux M, gives

M = J - - dx. (2.8) oP

3. Analysis

(a) Outer wake layer

The outer wake layer variables may be defined as

Y = X = L- dx, 6 2 (3.la-c) 6' j L

and wake limit as X, Y fixed for R,- oo. The alternative forms of the outer variables, used later, are

Z- , Y=Y. (3.2a, b) 5m 5e

Here, L is the order of streamwise variations in x-direction. Based on Karman momentum integral (2.7), the scale L is of order M/ui, and the boundary layer scales become

M M 65- , L = , (3.3a,b)

The outer expansions in the wake layer variables are

U = Um[ Uo(X, Y) - e U(X, Y) +-], (3.4)

S- pU [To(X, Y)- e T(X, Y) +---]. (3.5)

The continuity and momentum along with the outer boundary conditions in terms of outer layer expansions give the outer equations:

T + (a - 3)F oF' + cF2 + F'Fox - FoFo= 0, (3.6)

F'(X, oo) = 1, To(X, o) = 0; (3.7)

T[ + (a - Pf)FoF' + 2,3cFoF1 + (a - c)Fi'F1 + F'Fox (3.8)

-F{ Fix+FoFIx - FoFx = 0,

F{(X, co) = 0, T1(X, c) = 0. (3.9)

Here, a prime and the suffix X on dependent variables denote the partial




1894 N. Afzal

differential equations with respect to coordinates X and Y, and

U0(X, Y) = Fo(X, Y), U1(X, Y) = F (X, Y), (3.10a)

c = Umx _6x . (3.

Um 6

The momentum flux, M, may also be expanded as

M = Mo (X) - eMi(X) + -", (3.11)

Mo = UJ U, dY, M = 2 Ub Uo U d Y. (3.12a, b)

(b) Inner wall layer

The inner limit is defined as y+ fixed R,- oo. In the inner expansions

U = uTl (X, y+) + o(u7), (3.13)

T = pui71(X, y+) + o(U,), (3.14)

y+ = yur/v, (3.15)

the equations of motion (2.1) and (2.2) reduce to

U1 + T1 = 0. (3.16)

An integral of equation (3.16) shows that the total stress is constant in the inner wall layer,

U +T 7± = 1. (3.17)

(c) Matching

The matching of the tangential velocity component for open equations of motion leads to the functional equation

u1(X, y -* o) = e-Uo(X, Y- 0) - U1(X, Y -0). (3.18)

The solution can be obtained by the Izakson-Millikan-Kolmogorov hypothesis (Afzal 1996a,b) by differentiating with respect to Y to obtain

l u 1 Uo U1 - E- Y - Y (3.19)

dy, Y 8 Y '

as y+ - oc and Y-*0 for e-O0. The matching relation, equation (3.19), to lowest order, admits a simple solution,

Y -*0, Uo - 1 as Y - 0. (3.20) Y





The matching condition, equation (3.18), and the first-order matching relation, equation (3.19), become

u (X, y+) = E-1 - U1 (X, Y), (3.21)

y, -y- Y (3.22)

as y+ -> and Y--+0 for large Reynolds numbers. An alternative functional equation may be obtained from the ratio of the functional equations (3.21) and (3.22) as

y+ 9u Y 8 U1 ~ - Y.U- 1 (3.23)

U1 y e-1 - U1 O Y

(i) Log law velocity profile

The solution of functional equation (3.22) may be obtained by equating each side to 1/k, where k is the Karman's constant. The integration of these two equations gives

u1 = k-In y+B, y+--oo, (3.24) U = -ck- In Y + D, Y- 0. (3.25)

The matching of the velocity profile, equation (3.21), gives the skin friction law

Um/ln = k-1 In R, + B + D. (3.26)

(ii) Power law velocity profile

There exists another velocity profile: the power law solution described below. The solution of the functional equation, equation (3.23), may be obtained by equating each side to a. The integration of relations (3.23) gives

U1 = Cy, y+-- , (3.27)

U1 = C1(1 - Ya) + E, Y--0, (3.28)

where e~1= C+ E and the connection between constants a, C, C1 and E has been determined by matching. The matching relations, equations (3.27) and (3.28), require

LHS = y = Cay+, RHS = Y - C a Y, (3.29)

and the matching relation, equation (3.22), demands

C1 = CRa. (3.30)

The matching of velocity profile from equation (3.21) requires

--m CR" + E. (3.31) u,


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N. Afzal

If the ratio C1/C is a constant independent of R,, then equation (3.31) demands that Ra = const. = exp(y). Consequently, the power law index a is given by the relation

a - T C, - Cexp(y), (3.32a, b) In R,'

where the universal constant y may be determined from data. In the power law velocity profile, the multiplication constants C in inner layer and C1 in outer wake layer do depend on the Reynolds number, but the present expression, equation (3.32), regards their behaviour as analogous such that C1/ C is a constant to the lowest order.

4. Results and discussion

The lowest-order outer equation, equation (3.6), subjected to the boundary conditions, equation (3.7), momentum flux, equation (3.12a), and the matching conditions, equation (3.20), of inner wall layer, subjected to a closure model of constant eddy viscosity, is described in appendix A. The solution, equation (A12), corresponds to half of the free jet ( Y> 0), which predicts the finite slip velocity, Urn, on the wall forming the wall jet flow. The connection between the power law and log law in the overlap region is described in appendix B. The envelope of the skin friction power law giving the log law may also be explored for the possible connection between the constants (3.32a, b) in the power law and log law solutions (Afzal 2004). The uniformly valid solution for overlap log law or power law region is described in appendix C. The main results determined from matching of the inner wall layer with outer wake layers in the overlap region are described below.

(a) Log law region

Wall layer: u = k1 n y, + B, y+ - oo. (4.1) U -

Wake layer: Um - = k1 n Y + D, Y -+ 0. (4.2) UT

Skin friction: U = kl In R, + B + D. (4.3) uT

(b) Power law region

Wall layer: - Cy, + - oo. (4.4)

Wake layer: Um 1(1 - Y) + E, Y -0. (4.5) U7

Skin friction: Um = CR° + E. (4.6) U7


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(c) Power law and log law analysis

The power law and log law are analysed in the overlap region as described in Appendix B, and the predictions of the power law constants, a and C, given by equation (B la-c), are summarized below:

lR C=Cexp(),T KlkB ) C=- +B exp(-y), a - C=Cexp(y), n= 1+ \ka 7 In R, In R

(4.7)

and E=D. The above relations of power law constants, a and C, may be simplified with

the assistance of the skin friction log law, equation (4.3). The term In R, may be eliminated between equations (4.3) and (4.7) as appropriate to obtain the following relations:

1 ye 1 1 - (B + D)e u, aC- C exp(y) =--D, 7 - . k 1 - (B + D)e e 1 - D Um

(4.8)

It may be noted that the power index, a, is of the order of the non-dimensional friction velocity, e (classical scale), and the power law constant, C, is of the order of e- . Using Karman universal constant k=0.4 and for large Reynolds numbers (7= 1) the above relations become

2.5e a D' C exp(1) = e-1 D. (4.9)

This may be compared with the empirical prediction of a by Nunner (1956). The turbulent boundary-layer velocity profile data representation on log-log (power law) and lin-log (log law) plots shows the equivalence of power law and log law velocity profiles in the overlap region, rather than superiority of the log law over power law or vice versa (Osterlund et al. 2000). These results show that y7--1 and a-*0 for R,= 1000, y=0.81 and R= 1000, y=0.77.

(d) Outer wake layer closure hypothesis

The solution to the lowest-order nonlinear wake layer equations is an eigensolution. Based on the gradient transport model (Lykossov 1992), the solution described in appendix A is

Uo(X, Y) = sech2Q( Y), 2 = 0.881, U2 = , (4.10) 8ac( ~ -0(4.10) 8ac(x - e)

where ac= 0.018 is Clauser's universal constant.

(e) Comparison with data

The data of Eriksson et al. (1998), in terms of inner wall variables u+ = u/u. versus y+, has been compared in figure 2 with the expression, equation (4.1), for the law of the wall in the overlap region. The data show that the law of the wall,


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1898 N. Afzal

equation (4.1), in the overlap region may be represented by the relation

- = 2.44 In y, + 5, (4.11) UT

where k=0.41 and B=5 correspond to universal values (Coles 1968). The first-order term of the outer wake layer, in the overlap region, gives the

solution, equation (4.2), which is a local velocity defect law (Um- u)/u, versus Y, where Um is the wall jet maximum velocity and 6 is the half-defect thickness in outer wall jet (y= 6, u= Um/2), as shown in figure 1. The data of Wygnanski et al. (1992) have been compared in figure 3 with local velocity defect law, equation (4.2). The data in the overlap region predict

Um- _ _ k-1 In Y - 5.6, k = 0.41, D = -5.6, (4.12)

which describes the data very well for Y<0.08, just below the point of velocity maxima, 6m, but above the viscous sublayer. The velocity defect law, equation (4.2), may also be expressed in terms of the location, 6m, of the point of maximum velocity, instead of half the defect thickness, 6, as stated below:

Su = k-1 In Z + Dm, Z- 0, (4.13)

UT

Z Y 6 Dm = D + klIn ; (4.14)

the skin friction relation, equation (4.3), becomes

Um = kln( + B + Dm. (4.15)

In the present work, both 6m and 6e belong to the outer layer and 6m/6e is of order unity, which is also supported by data (Wygnanski et al. 1992). A velocity defect function, q(Z), may be defined as

(Z) u-Um k In Z, (4.16)

in which the overlap region for Z- 0 from equation (4.16) demands q(Z) -> - Din. The data for /(Z) are shown in figure 4, against Y. The figure shows that its behaviour for Y-*0 limits to Dm- -1. Further, with D= -5.6, equation (4.13) predicts 6m/6= 0.158, whereas Wygnanski et al. (1992) proposed 6m/6= 0.15 from data. A careful consideration of this figure shows that the values of Dm are scattered from -0.8 to -1.2, and equation (4.16) predicts 0.14<6m/6<0.165, which may be compared with the results reported in literature (see table 1 of Launder & Rodi 1981; George et al. 2000). The skin friction, equation (4.3), may be expressed as

Cf = 8 G(A, E1) , Rm -- , (4.17) Ap V





Figure 2. Comparison of the law of the wall, equation (4.1), with Eriksson et al. (1998) data. Present work: u/7= 2.44 In y+ +5.

0.32

0.28

0.24

0.20

0.16

0.12

0.08

0.04

0

u-Um uz

Figure 3. Comparison of the outer wake layer velocity defect law, equation (4.12), with the data of Wygnanski et al. (1992) for various values of R,, the jet Reynolds number. Symbols: X, Rj= 19 000; 0, R= 15 000; , R= 10 000; 0 R= 7500; I, 0, R=5000; <1, A, R3=3700. Line is present proposal: (u- Um)/Ur=k-1 In Y-D, k=0.41, D= -5.6.

A = 2 n Rm, E = 2[ln(2k) + B + D].

The function G(A,E1) is given by

+ 2 In -E =A,

1 ( 2) (nA) G(A, E1) = 1 - (21nA-E) 1- +o A AI-1+~ A.

(4.18)

(4.19)

(4.20)

present proposal a (u-Um)u = 2.44 In Y + 5.6

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N. Afzal

0.20

0.15

Dm =-1 Y 0.10

0.05 present proposal (Z) =-Dm, -1.2 < Dm <-0.8

0 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0

¢(Z)

Figure 4. Comparison of Wygnanski et al. (1992) data with equations (4.16) for the velocity defect function q(Z)=(u- Um)/u,-2.44 in Z, with Z=u/am, against outer variable Y= y/6. p(Z)= -Dm, in the overlap region where -1.2>2Dm> -0.8. Legend as in figure 3.

For A- 00oo, equation (4.19) shows that G-*1. Further, the function G is tabulated in Gersten & Herwig (1992, p. 782). In recent works, the skin friction data have been determined from direct measurements of the mean velocity profile measured by a hot wire anemometer very near to the wall to estimate the velocity gradient. The data of Wygnanski et al. (1992) and Tailland & Mathieu (1967) do show considerably lower skin friction than the data of Eriksson et al. (1998, fig. 11, p. 54). The departure has been attributed to considerably higher values of y+, where the velocity gradient is not strictly linear (Eriksson et al. 1998).

The lowest-order, outer wake layer solution, equation (4.10), is based on constant eddy viscosity, which represents the half-jet (Y> 0) solution where slip velocity, Um, is given in terms of momentum flux J. The mean velocity data of Abrahamsson et al. (1994) for x/b= 70-150 for three different Reynolds numbers, Rj= 104, 1.5 X 104 and 2 X 104, are shown in figure 5. The comparison of the outer wake layer lowest- order solution, equation (4.10), with the data in outer layer variables shows very good agreement except for the extreme outer edge of flow (beyond Y> 1.5), where external induced flow, counter flows and measurement errors dominate.

The novel scaling was proposed by Narasimha et al. (1973): instreamwise variations of flow depend on initial momentum flux J and z, as described in appendix D. Further, the local momentum flux, M, is related to Xb=xb, by equation (D 10), which for Rj= 5000 and X= x/b=30 becomes

M/M30 = 0.61 + 0.39(30/Xb)c, c = 0.063. (4.21)

This relation, in terms of variables M/M30 against Xb, is compared in figure 6 with the data of Wygnanski et al. (1992), showing that the decay of momentum in the downstream direction prediction compares well with the data.


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2.0

proposed solution 1.5 u/Um = sech2(2Y), 2= 0.881

Y 1.0

0.5

0 0.2 0.4 0.6 0.8 1.0 1.2 u/Um

Figure 5. Comparison of lowest order outer wake layer solution, equation (4.10), with the data of Abrahamson et al. (1994). Symbols: x/b=150: A, Rj=104, X, Rj=1.5X104, [, RN=2X104; x/b= 125: 0, R= 104, V, R= 1.5X104, +, R-= 2X 104; x/b=70: 0, R-= 104, 0, 1.5X104. Proposed

solution, equation (4.10), from gradient transport model: u/ Um= sech2(Q Y), =0.881.

1.2

1.1 -

1.0

0.9

0.8

0.7

m/m30

0 20 40 60 80 100 Xb

Figure 6. Comparison of Wygnanski et al. (1992) data for the decay of wall jet integral momentum, equation (4.21), estimated by three methods: (open circle), estimation from wall shear; (filled left triangle), integral momentum equation; (open diamond), Preston tube measurements. Present work: M/M30o=0.61+0.39(30/Xb)c, c=0.063.

5. Conclusions

(i) The functional equation has been matched in the Izaksom-Millikan overlap region. This gives two functional solutions: the power law and log law profiles of the velocity and friction factor. This is not surprising as we are dealing with open functional equations of turbulent motion without any closure model (such as eddy viscosity, mixing length, k-e).

(ii) The power law constants, a and C, may be estimated from equation (4.8), provided the local skin friction coefficient, Cf, is known. The alternative equation (4.7) depends of friction Reynolds numbers, R,.

(iii) The classical privileged log law solution, where log law constants, k and B, are universal numbers (independent of Reynolds number), is equivalent to


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N. Afzal

the power law solution, in the limiting situation of very large Reynolds numbers, R,. Further, for lower Reynolds numbers, R,, the log law solution is not equivalent to the power law solution.

(iv) The lowest-order outer wake layer equation has been closed with the generalized gradient transport model with constant eddy viscosity. The closed-form solution gives the half-jet-like solution, where slip velocity corresponds to maximum velocity, Um, on the wall constituting the wall jet flow. The comparison with experimental data support the predictions.

Note added in proof. Eaton & Nagib (2004) proposed that Afzal (2004) highlighted a number of limitations in the asymptotic analysis leading to the power law for the boundary layer theory by George & Castillo (1997) and George et al. (2000). In particular, the last of terms shown here is ignored by them and leads to inappropriate limits:

= 1 + U( Y)- UU( Y ). U U

Appendix A. Lowest order outer wake layer

The lowest-order outer layer equation, equation (3.3), boundary conditions, equation (3.7), and matching conditions along with constant momentum flux, equation (3.12a), are summarized as

To + (a - Oc)FoF' + OcFl2 + FoFox - FFx = 0, (A 1)

Y-oo, F'- 0, To -0, (A2a)

Y-+0, Fo - , F -1, YFg-0, T0- 1, (A 2b)

Mo = U6 F2 d Y =J. (A 3)

The constancy of momentum flux, J, requires Umo6-1/2, =a/2. Further, there is no free stream velocity, the equations (A 1) and (A 2) admit eigen- solutions, and the multiplicative constant, Um, is determined from the specified momentum flux relation, equation (A 3).

For the Reynolds stress term, a closure model is needed. In a wall jet flow, all turbulent models based on the eddy viscosity concept fail because the location of the zero shear stress and zero velocity gradients do not coincide. The modified eddy viscosity closure is needed to account for non-coincidence of the positions of maximum velocity and zero Reynolds stress. Various models have been described (Launder & Rodi 1981, 1983; Lykossov 1992; Tangemann & Gretler 1998). In the present work, the eddy viscosity closure model is

r(, y) = pv + S , (A 4)


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Velocity profiles in turbulent wall jets 1903

with an additional function, S(x, y), that has been introduced to account for non- coincidence of the positions of maximum velocity and zero Reynolds stress. The generalization of the gradient transport model, equation (A 4), was considered by Lykossov (1992), where S was defined as counter-gradient flux that depends on flow shear, Brunt-Vaisala frequency, relaxation time and third-order covari- ances. The dissipation effect of velocity profile curvature and jet axis can be compensated by non-local (counter-gradient) turbulent transport of momentum. Further, for the Reynolds heat flux, a model, analogous to equation (A 4) was also proposed by Townsend (1956), Sreenivasan et al. (1982) and Lykossov (1992). In the present work, as a first approximation, the counter-gradient flux, S, is regarded as a function of x. The outer layer relation, equation (A 4), to lowest order becomes

S6S T(X, Y) (F + So), So(X) = . (A 5)

Uma Um

The constant eddy viscosity closure (analogous to Clauser 1956) may be taken as

v=acUm , - dY = Q , 6 =4ac-(x - xo), (A 6a-c) SJo Um

where ac is the Clauser constant, and equation (A3) yields

2_ 3J aUm = (X- (A7) 8ac(z - zo)

Based on above relations, the lowest-order wake layer equations, equations (A1)-(A3), under equilibrium Fo(X, Y) =g( Y), are simplified as

g"' + 2Q2(gg + 2) = 0, (A 8)

2 g(0)=0, g"(0)+So - , '() = =0. (A9a-c)

ac

The integral condition, equation (A6b), becomes

" 2 jg/2 d Y - (A 10)

If one adopts

QE2 g"(0)=0, So(X) - , (A 11)

equations (A8) and (A9) become an eigenvalue problem, which differs from that of a laminar wall jet (Glauert 1956; Afzal 1980). The eigensolution to equation (A8), subjected to the boundary conditions, equations (A9a,c) and (All), is given by

g( Y) = Q-1 tanh(Q Y), (A 12)




N. Afzal

which predicts finite slip velocity g'(0)=1 from the outer solution at the wall. The tangential velocity component is given by

Uo(X, Y) = sech2(Q Y). (A 13)

The half-defect thickness, 6, by definition implies that at Y= 1, Uo= 1/2 and we get

Q = tanh- (1//V) = 0.881. (A 14)

Appendix B. The power law and log law analysis in overlap region

A power law region may simultaneously exist along with a classical log law region (Osterlund et al. 2000; Buschmann & Gad-el-Hak 2003) in the overlap region of the classical inner and outer layers. The common overlap domain may be explored from the results obtained here by the Izakson-Millikan-Kolmogorov hypothesis. The distribution of U+ and a U+ /R from the skin friction power law, equation (4.6), and classical skin friction log law, equation (4.3), are equated to give the following relations:

a=- i ' C - + B exp(-), (Bla)

y=1-k(B+D-E)a, or y= (i+ B + -E) (B1b,c) In R, "

It is interesting to note that the relation (Bla) is exactly same as relation (3.32a), obtained earlier purely by asymptotic consideration. The above results show that for large Reynolds numbers, y-1l and a-*0. Further, with B=5.5, D=E and R,=10 000, we get y=0.81, and, with R,=1000, we get y=0.77.

In the Izakson-Millikan-Kolmogorov overlap region (1lyK + <R), the velocity distribution u+ and its normal derivative au+ /y+ from power law (4.4) are equated with those obtained by using the log law (4.1). This predicts the power law constants a and C in the overlap domain as

I - , C - +B exp(-8), (B2a, b) In y' \ka

3= 1- kBa, or = ( +lIn (B2c)

The observations of Osterlund et al. (2000) about the insensitivity of the turbulent boundary-layer velocity profile data representation on log-log (power law) and lin-log (log law) plots are shown to be due to the equivalence of the power law and log law velocity profiles in the common domain of the overlap region, rather than the superiority of log law over power law or vice versa. The equation (B2a) is estimated at the pipe axis or the edge of the boundary layer (y= 6, u+ = U+ ), and relation (Bla) predicts

3= 7. (B3)


1904




Further, the equations (Bic) and (B2c) may imply E=D. The relations, equations (B2a-c), determined from analysis of the matching relations, are valid in the domain of the overlap region 1 < y+ <K R, but, in view of the particular form of equations (Bla-c), the equations (B2a-c) also hold at Y= 1 or y+ = R,. Therefore, in the outer wake region, Y, < Y< 1, the equations (B2a-c) are not valid where Y* is the upper limiting value of Yin the overlap domain. Further, in the situation which Barenblatt calls the 'main body of flow' approximation, the pipe flow exhibits a weak velocity defect layer, where the wake layer ( Y < Y< 1) may be neglected, and equations (B2a-c) hold in the entire region (R71 < Y < 1).

The equation (B2a) a versus y+, y+ may be compared with the relation ab versus y+ estimated from the Barenblatt (1993) power law envelop relation, which shows that aB decreases with increasing wall distance y+. This behaviour of data, initially observed by Osterlund et al. (2000b), was contradicted by Barenblatt et al. (2000). Later, however, by examining extensive velocity profile data Osterlund et al. (2000a) and Zanoun et al. (2002) clearly observed that a monotonically decreases with increasing wall distance, y+. It was proposed that the functional behaviour of the velocity profile data is different from that of a power law, where a must be a constant. In these works, it was concluded that the power law is far from being able to describe the mean velocity profile in the overlap region, as no universal value can be assigned to the power law, constants a and C.

In the present work, the matching of inner and outer layers leads to power law relations in a limited overlap domain, where the power law constants a and C may be assigned locally similar values, in contrast to log law relations with universal constants having a substantially larger overlap region. It may be pointed out that equations (B2a-c), determined from the velocity profile matching relations, could be deleted without affecting the argument. Further, the power law constants a and C depend on the R, that are adopted from relations (Bla-c) by comparison of the experimental data from various sources.

On the basis of power law velocity, given by equation (3.25) and skin friction, given by (3.29), the left-hand side of the matching relation, equation (3.21), requires

y - Caya, R CaR. (B 4a, b) + y+ ' R,

Based on the a and C proposal of Barenblatt (1993), as shown in fig. 21 of Zagarola & Smits (1998), the relation (B5a) becomes unbounded for large y+. The present a and C relations, equation (B3), on the other hand, express the matching relations, equations (B4a, b), as

y + Ba)exp(-P)y - + (B5a)

aU 7 1 B R, + Ba e xp^(-7) R- I7 n- , (B 5b)

which tend to a finite limit at large y+ and R,. Further, in view of equation (B3), the constant f and y may be eliminated from (B5a, b) to yield


1905



N. Afzal

Ou+ 1 9U+ 1 Y+ k' O R (B 6a,b)

and their integration leads to the classical log laws. The present results show that the non-unique solutions of the power law and log law matching functions do approach the same limit 1/k at large y+ in the overlap region.

Certain expressions for power law constants have been proposed by George & Castillo (1997), in the zero pressure gradient boundary layer; by George et al. (2000) in the wall jet; and by Barenblatt (1993), Kailasnath (1993) and Zagarola & Smits (1998) in fully developed pipe flow. These are based on certain analogies, and, of course, are supported by experimental data, but are arbitrary in nature (Afzal 2004). This has certain implications on their predictions. The envelope of the skin friction power law giving the log law may also be explored for the possible connection between the constants in the power law and log law solutions (Afzal 2004).

Appendix C. Uniformly valid solution

The uniformly valid solution for the velocity distribution, u, may be obtained when the common part, uc, is subtracted from the union of the inner solution, uj, and outer solution, no, giving u= U + Uo- Uc.

(a) Log law

The uniformly valid solution for velocity distribution in wall jet is

S= Um Uo(X, Y) + [u (Z, y) + W(X, Y) , (C 1)

where the wake function W(X, Y) based on the log law, w, in the overlap regions is described below. The common part, Uc, from the relations given in equations (3.24) and (3.25) is

Uc = u(k-1 In y+ + B) = Um + u(k-1 In Y - D). (C 2)

The wake function W(X, Y) is given as

W(X, Y) = (-U1 - k-1 In Y + D)/D,, D. = D + k-1 ln(6/6) = H/k, (C 3)

as Y-+0, W- 0 and Y- 6/6e, W- 1. Here, 6e is the boundary-layer thickness of the wall jet, where the velocity, u/Urn, is 1%. The relation (C3) may also be expressed as

W(X, Y,) = (-U1 - k-1 In Y, + D,)/D,, Y, = y/6. (C 4)

The uniformly valid velocity distribution, equation (Cl), above the sublayer of the wall jet becomes

U = UUo(X, Y) + ln y + B + W(X Y). (C5)


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(b) Power law

The uniformly valid solution for velocity distribution in wall jet is

S= Um Uo(X, Y)+ u4 u(X, y) + Jo(X, Y) , (C 6)

where the wake function w(X, Y) based on power law in the overlap region is described below. The common part, Uc, from equations (3.27) and (3.28) is

U = u,Cy = Um+ u[C( Y - 1) - E]. (C7)

The power function, o(x, Y), based on the power law and the boundary conditions is

0(X, Y) = [- UI(, Y) - C,( Y" - 1) + E]I/E*, (C 8)

E* = C [1 - (6e/6)"] + E, E* = tr/k, (C 9)

Y -0, -,0; Y- 1, w-- 1. (C 10)

The uniformly valid velocity distribution, equation (C1), above the sublayer of the wall jet becomes

u U=mUo(X, Y) +u[Cy + (X, Y)]. (Ci1)

In fact, Buschmann & Gad-el-Hak (2003) analysed the extensive experimental data for velocity distribution, and proposed that the existing data, covering a wide range of Reynolds numbers, supports the log law and power law with equal measure throughout most of the overlap region. Further, fig. 7 on p. 571 of Buschmann & Gad-el-Hak (2003) displays common overlap region marked COR. Above COR, there is region marked PPR, where the power law shows better agreement with experimental data. However, below COR, there is another region marked PLR, where the log law is in better agreement with experimental data.

The classical log law solution for large Reynolds numbers Ro has the advantage that its slope, k, the Karman constant and additive term B are universal numbers (independent of the Reynolds number), which is equivalent to the power law solution in the overlap region for large Reynolds numbers (say R, or Re).

Appendix D. Novel scaling

Narasimha et al. (1973) suggested that the traditional scaling of relevant distance in a wall jet by the characteristic dimension of the nozzle might be erroneous. They proposed scaling of the streamwise evaluation of the flow by the initial momentum flux, J, and the molecular kinematic viscosity, v. Based on the proposal of Narasimha et al. (1973), we obtain

UmV 6J -- - Fm () = Aun, - Fn () = Aym, (D 1) J v


1907



1908 N. Afzal

7p \2 J - - F() = AP, =x:2. (D2)

The power indices n, m and p determined from data analysis (Narasimha et al. 1973; Wygnanski et al. 1992) are

n = -0.472, A = 1.473, m = 0.881, (D 3) (D 3)

Ay = 1.445, p= -1.07, A, = 0.146

The parameters in the skin friction relation, equation (4.3), may be estimated from equations (Dl) and (D2), as given below:

u_ A_ n- T/2, Rr =Ay A m+P/2. (D4)

The Karman momentum integral, equation (2.5), may be expressed as

d I ( dY= w- . (D5) d J UmJo P

The right-hand side in equation (D5), based on expressions (D1) and (D2), becomes

d M - -A, (D 6)

d( J

where local momentum flux, M, is given by

M = 6UmI1, I, = ( dY. (D7)

The integration of the momentum integral, equation (D7), gives

M A M-_ 1 l, l+p<0. (D8)

J l+p

If, at a certain location, z= l, and the momentum M= MaI, then the relation in equation (D\hskip 2pt8) may be expressed as

M-J (X)C (D9) M - J \x '

where c=-(1+p)= 0.063. The relation (D9) shows that for large x, M approaches a constant value J. In Wygnanski et al. (1992), for data at z/b= 30, the nozzle Reynolds number Rej= Ub/= 5000 and = (x/b)R=- 7.5 X 10, we obtain J/M1i= 0.609 and therefore equation (D9) may be expressed as

M lc - 0.61 + 0.39 , (D 10)

which has been compared in figure 6 with the data.





References

Abrahamsson, H., Johansson, B. & Lofdahl, L. 1994 A turbulent plane two dimensional wall-jet in a quiescent surrounding. Eur. J. Mech. B/Fluids 13, 533-556.

Afzal, N. 1980 Second order effects in the wall jet. J. Mecanique 19, 387-402. Afzal, N. 1996a Wake layer in a turbulent boundary layer: a new approach. In Proceedings of

IUTAM Symposium on Asymptotic Methods for Turbulent Shear Flows at High Reynolds Numbers (ed. K. Gersten), Invited Lecture, pp. 96-115. Dordrecht: Kluwer Academic Publisher.

Afzal, N. 1996b Turbulent boundary layer on a moving continuous plate. Fluid Dyn. Res. 17, 181-194.

Afzal, N. 1997 Power law in wall and wake layers of a turbulent boundary layer. In Proceedings of Seventh Asian Congress of Fluid Mechanics, pp. 805-808. New Delhi: Allied Publishers.

Afzal, N. 2001 Power law and log law velocity profiles in turbulent boundary layer flow: equivalent relations at large Reynolds numbers. Acta Mechanica 151, 171-183.

Afzal, N. 2004 Outer nonlinear wake layer in turbulent boundary layers & Izakson-Millikan overlap domain: power law and log law solutions. In Second International Workshop on Wall- Bounded Turbulent Flows, 2-5 November 2004, Abdus Salam Internatioal Center for Theoretical Physics, Trieste (org. H. M. Nagib & A. J. Smits), Invited Lecture.

Afzal, N. 2005 Scaling of power law velocity profile in wall-bounded turbulent shear flows. In Proc. 43rd AIAA Aerospace Sciences Meeting and Exhibit, 10-13 January 2005, Reno, Nevada, Paper no. AIAA-2005-0109. Reston, VA: American Institute of Aeronautics and Astronautics.

Barenblatt, G. I. 1993 Scaling laws for fully developed turbulent shear flows. Part I. Basic hypothesis and analysis. J. Fluid Mech. 248, 513-520.

Barenblatt, G. I., Chorin, A. J. & Prostokishin, V. M. 1997 Scaling laws for fully developed turbulent flow in pipes. Appl. Mech. Rev. 90, 413-429.

Barenblatt, G. I., Chorin, A. J. & Prostokishin, V. M. 2000 A note on the overlap region in turbulent boundary layers. Phys. Fluids 12, 2159-2161.

Bradshaw, P. & Gee, M. Y. 1960 The wall jet with or without free stream. Aero. Res. Counc. Rep. Mem., 3252.

Buschmann, M. H. & Gad-el-Hak, K. 2003 The debate concerning the mean velocity profile of a turbulent boundary layer. AIAA J. 41, 565-572.

Clauser, F. H. 1956 The turbulent boundary layers. Advances in Applied Mechanics, vol. 4, pp. 1- 51. New York: Academic Press.

Coles, D. 1968 The young person's guide to the data. In Proceedings of Computations of Turbulent Boundary Layer-1968 AFOSR-IPF-Stanford Conference, vol. II. Stanford, CA: Stanford University.

Eaton, J. K. & Nagib, H. M. 2004 Summary report. In Second International Workshop on Wall- Bounded Turbulent Flows, 2-5 November 2004, Abdus Salam International Center for Theoretical Physics, Trieste (org. H. M. Nagib & A. J. Smits).

Eriksson, J., Karlsson, R. & Pearson, J. 1998 An experimental study of two dimensional plane turbulent wall jet. Exp. Fluids 25, 50-60.

George, W. & Castillo, L. 1997 Zero pressure gradient turbulent boundary layer. Appl. Mech. Rev. 50, 689-729 (see also Technical Report No. TRL-153 p. 20, Nov. 1996).

George, W., Abrahamsson, H., Eriksson, J., Karlsson, R. L., Lofdahl, L. & Wosnik, M. 2000 A similarity theory for the turbulent plane wall jet without external stream. J. Fluid Mech. 425, 367-411.

Gersten, K. & Herwig, H. 1992 Stromungs mechanik. Wiesbaden: Verlag Vieweg. Glauert, M. B. 1956 The wall jet. J. Fluid Mech. 1, 625. Hammond, G. P. 1982 Complete velocity profile and optimum skin friction formula for wall jet.

J. Fluid Eng. 104, 59. Izakson, A. A. 1937 On formula for the velocity distribution near walls (Engl. Trans.). Tech. Phys.

USSR 4, 155-1590.


1909



N. Afzal

Kailasnath, P. 1993 Reynolds number effect and the momentum flux in turbulent boundary layer. Ph.D. thesis, Mason Laboratory, Yale University.

Karlsson, R., Eriksson, J. & Pearson, J. 1993 An experimental study of a two dimensional plane turbulent wall jet Tech Rep VU-S93-836. Vattenfall Utvekling AB. Sweden: Alvkarleby Laboratory.

Launder, B. E. & Rodi, W. 1981 The turbulent wall jets. Prog. Aerospace Sci. 9, 81. Launder, B. E. & Rodi, W. 1983 The turbulent wall jet: measurements and modeling. Annu. Rev.

Fluid Mech. 15, 429. Lykossov, V. N. 1992 The momentum turbulent counter-gradient and transfer. Adv. Atmos. Sci. 9,

191-200. Millikan, C. B. 1938 A Critical discussion of turbulent flow in channels and circular tubes. Proc.

5th Int. Cong. Appl. Mech., 386-392. Narasimha, R., Narayan, K. Y. & Parthasarathy, S. P. 1973 Parametric analysis of turbulent wall

jet in still air. Aeronaut. J. 77, 335. Nunner, W. 1956 Warmeubergang und Druckabfall in rauhen Rohren. Forschungsheft Nr 455.

Diisseldorf: Verein Deutscher Ingenieure e.V. Osterlund, J. M., Johansson, A. A. & Hites, M. H. 2000a Comment on A note on the overlap region

in turbulent boundary layer [Phys. Fluids 12, 2159 (2000)]. Phys. Fluids 12, 2360-2363. Osterlund, J. M., Johansson, A. A., Nagib, H. M. & Hites, M. H. 2000b Comment on A note on the

overlap region in turbulent boundary layer. Phys. Fluids 12, 2159-2162. Sreenivasan, K. R., Tavoularis, S. & Corrsin, S. 1982 A test of gradient transport and it

generalization. Turbulent shear flow 3. Berlin: Springer. Tailland, A. & Mathieu, J. 1967 Jet parietal. J. Mecanique 6, 103-131. Tangemann, R. & Gretler, W. 1998 The assessment of different turbulence models for a wall jet.

ZAMM, 78. Townsend, A. A. 1956 The structure of turbulent shear flow. Cambridge: Cambridge University

Press. Wygnanski, I., Katz, Y. & Horev, E. 1992 On the applicability of various scaling laws to the

turbulent wall jet. J. Fluid Mech. 234, 669-690. Zagarola, M. V. & Smits, A. J. 1998 Mean flow scaling in turbulent pipe flow. J. Fluid Mech. 373,

33-79. Zanoun, E. S., Nagib, H., Drust, F. & Monkewitz, P. 2002 Higher Reynolds number channel data

and their comparison to recent asymptotic theory. In Proc. 40th AIAA Aerospace Sciences Meeting and Exhibit, 14-17 January 2002, Reno, Nevada, Paper no. AIAA-2002-1102. Reston, VA: American Institute of Aeronautics and Astronautics.


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analysis of power law and log law velocity profiles in the overlap region of a turbulent wall jet

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