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Some equilibrium turbulent boundary layers
Fluid Dynamics North-Holland
Research
1 (1986) 49-58
49
Some equilibrium turbulent boundary layers Itiro TAN1 Nurional Aerospace Laboratory, Received
31 January
Jindaij,
Chofu, Tokyo, 182 Japan
1986
Using the Coles additive law of the wall and law of the wake for the mean velocity profile of a Abstract. two-dimensional turbulent boundary layer, a differential equation for the friction and wake parameters is derived from the momentum integral equation with a view to finding out the conditions under which the boundary layer can exhibit equilibrium. It is predicted that equilibrium is possible for boundary layers in favorable pressure gradient over smooth as well as k-type rough walls. When the roughness height is allowed to increase linearly with the streamwise distance, equilibrium exists also in zero pressure gradient. For a d-type rough wall, equilibrium is possible for a certain range of pressure gradients, from favorable to adverse. Most of the predictions are verified by evaluating the friction and wake parameters from the available experimental data on mean velocity measurements.
1. Introduction Considerable attention has been devoted to the equilibrium turbulent boundary layer, which is characterized by similarity of the velocity-defect profiles in the course of its downstream development. The early experimental work of Clauser (1954) and the analysis of Rotta (1955, 1962) served to demonstrate that equilibrium boundary layers in both zero and adverse pressure gradients could exist at least approximately for a certain distance along a smooth wall. This has been added to by the experiments of Herring and Norbury (1967) in favorable pressure gradients and by those of Bradshaw (1967) in adverse pressure gradients. On the other hand, an exhaustive examination of experimental velocity-defect profiles has led Coles (1956) to the formulation of the additive law of the wall and law of the wake,
u _=_ UT
1
lny+C+:,
K
(
i,
1
for the region outside the viscous sublayer (y > 5Ov/u,, say), where lJ is the mean velocity in the x-direction, x and y are coordinates along and normal to the solid wall, respectively, U, is the friction velocity (square root of wall shear stress divided by fluid density), Y is the kinematic viscosity, K is the Karman constant (= 0.41), C is the smooth-wall constant (= 5.0), II is the wake parameter, and w is the universal function of y/6, 6 being the thickness of the boundary layer. There is now abundant evidence supporting the belief that the conditions under which the formulation (1) is valid are nearly equivalent to those under which the boundary layer approximation itself might be expected to apply. The wake parameter II is generally regarded as dependent on x. In the event that II is independent of x, the velocity defect relative to the free-stream velocity U,, made non-dimensional by the friction velocity u7, turns out to be a function of y/6 only, which is the property assigned to equilibrium flow. Viewed from a dimensional standpoint, however, the equilibrium turbulent boundary layer could only be reconciled with the Reynolds-averaged momentum 0169-5983/86/$3.50
0 1986, Japan
Society of Fluid Mechanics
50
I. Tani / Some equilibrium
turbulent boundary layers
equation if u/U0 and d6/dx were independent of x, namely, for example, if the flat plate were roughened in such a way that the roughness height increased linearly with the distance along the plate (Rotta 1962). These conditions are not fulfilled with boundary layers on smooth or uniformly rough flat plates. The approximate equilibrium as observed experimentally thus raises a question as to why u7/U0 and da/dx exert such a weak influence. In an extensive survey of experimental data on various boundary layer flows, Coles (1962, 1969) evaluated the wake parameter II by fitting the measured velocity profile to the analytical expression (1) and classified the boundary layer behavior by the variation of II in the x-direction. The results obtained are exceedingly illuminating in many respects, although the question of equilibrium has remained unsettled. It is connected with this question that an attempt was recently made by the present author, in collaboration with Motohashi, to devise a simple method of analysis for evaluating characteristic parameters, z = KI?&/u~ and III, from the integral thicknesses of measured velocity profile, thus dispensing with the skilled procedure of fitting, yet yielding equally reliable evaluation (Tani and Motohashi 1985a). Results of analyzing the available data on adequately tripped boundary layers would seem to negate the existence of equilibrium state so far as the pressure gradient is zero and the wall is smooth (Tani and Motohashi 1985b). The present paper aims at extending the analysis to the boundary layers in nonzero pressure gradients and also to those on rough walls, with a view to searching for the conditions under which the equilibrium state could exist.
2. Method for evaluating characteristic parameters
We confine ourselves to a two-dimensional start with the Coles formulation (1) generalized
u -=u,
1 lnl(‘Y+C-!Z+!& u U,
K
K
incompressible to the form
turbulent
boundary
layer
and
(2)
’
with AU/u, representing the shift due to roughness of the velocity profile. As regards the wake function w, the universal form suggested by Coles (1969) w(y/6) = 1 - COS(~T~/~), being incapable of satisfying the condition aU/ay = 0 at the edge of the boundary layer y = 6, a quartic polynomial due to Lewkowicz (1982)
is used for making up the deficiency. Common to most methods of modification is the to correct the wall appearance of the term proportional to 17-i, which makes it inevitable component from K-I ln(u,y/v) + C to KC* ln(u,y/v) + C - ~C’(y/tI)~(l -y/6)(1 - 2y/6) in the expression (2). The choice of a quartic in preference to a cubic is based on the smallness in magnitude of this correction. With eqs. (2) and (3) the velocity defect is expressed in the form
The displacement
and momentum
thicknesses
of the boundary
layer are then given by
(5)
I. Tani / Some equilibrium
respectively,
turbulent boundap
layers
51
where 2 -3
u, Z=K-=K u,
cf
J
a=E+fl60
8437 #8=~+ggI+g-n,
>
cf being the conventional gives
(6)
friction
2
coefficient.
Elimination
of 6 between
the two equations
(7) of (5)
8437 ~-~~G+($&~G)fl+~-n’=o,
(8)
where (+;
1-h (
(9)
i
is the Clauser parameter. It is to be noticed that eqs. (4) to (9) are all independent of viscosity and roughness, although the flow may be affected by these variables through the boundary conditions. For further analysis it is convenient to introduce the notation S=Z-2fl--KKC. Smooth
wafl:
S= Substitution
(10)
Evaluating
6
the velocity
profile
(2) with AU/u,
= 0, at y = 8, we have
;e’.
(11)
of (11) into (5) yields
f?= a-i
/3 es v z
i
--. K
(12)
uo
Equation (8), along with the first equation of (12), affords the means for evaluating the friction parameter z and the wake parameter II from the measurable quantities S* and 8. The calculation of S* and 0 resulting in (5) is based on the assumption that the Reynolds number is high enough for the contributions from the viscous sublayer to be negligible. If this is not the case, a correction is needed. Using the sublayer velocity profile due to Spalding (1961) u7.Y 5 -=-++ V where 5 =
KU/U,,
S*=
-ye<
-
1 -
{ -
;p
-
g3
-
A[“),
(13)
K
we
obtain
olGt50.63 I
$ o,
8=((~+~-(50.63-~j)~,
instead of (12). For relatively low Reynolds numbers, the sublayer effect by using the corrected thicknesses 8; =S*-50.635,
19,=6-t 0
in place of the measured
thicknesses
50.63-p
(14) therefore,
136.41 KZ
it is necessary
v -, 1
UO
to account
for
(15)
6* and 0 in eqs. (9) and (12).
Rough wall: Two types of rough wall are considered, k-type and d-type according to the terminology of Perry, Schofield and Joubert (1969). The k-type roughness. typified by a smooth wall roughened with closely packed sand grains, or with sparsely spaced spanwise rods, has the roughness shift of the form AU -=_ u,
1 K
ln$
+ K,
(16)
I. Tuni / Some equdibrium
52
turbulent boundary layers
where h is the height scale of roughness and K is a function of u,h/v, but tends to a constant value characteristic of roughness geometry for sufficiently large values of u,h/v in the so-called fully rough regime. Evaluating the velocity profile (2) with AU/u, given by (16) at y = 6, we have 8 = heS+KK. Substitution
of (17) into (5) yields
The d-type roughness is typified by a smooth wall containing sparsely spaced narrow spanwise grooves. The roughness shift is expressed by the form independent of the roughness scale, AU _=_ u,
1
In*
K
V
(19)
+ D,
where D is a constant, characteristic of the roughness (2), with AU/u, given by (19), at y = 6, we have s=
geometry.
Evaluating
-KD.
the velocity
profile
(20)
When the constant K or D is known from other sources, eq. (8) along with the first equation of (18) or eq. (20), would afford the means for evaluating z and LI from the measured values of S* and 8. When K or D is also to be known, an auxiliary equation must be sought for determination, as will be seen later.
3. Momentum
integral equation
In the following we assume that the flow is in fully rough regime and that the k-type rough-wall constant K as well as the d-type rough-wall constant D are independent of x. Substituting the momentum thickness 9 and other quantities formulated in the preceding section into the momentum integral equation
we obtain (yz _ p + p - 136.47ee” Z
{(2a-ir)z-(2p-B)}Zg
zgi
(22)
+{2az-(~-136.47eC’)}~~=~~~e~’ 0
for the smooth-wall
boundary
layer,
+(az-p)~+(3ar-2~)~~=K2e-(SixK'
(23)
0 for the k-type
1
rough-wall
boundary
layer, and
~z_~2a+~~+~)~~+(~z-~)~+(3~~-2B)~~=K’
(24) 0
I. Tani / Some equilibrium
turbulent boundary layers
53
for the d-type rough-wall boundary layer, where the dot denotes differentiation with respect to II. Each of these equations provides a differential equation for z and II (or 13) as functions of x. In order to calculate both z and II, an auxiliary equation is required in addition to that differential equation. What is intended here, however, is simply to make use of the differential equation for finding the conditions under which the equilibrium flow could exist.
4. Possible equilibrium flows Smooth walk It has been shown in the preceding paper (Tani and Motohashi 1985b) that there exists no equilibrium state for the boundary layer on a smooth wall in zero pressure gradient. This’finding may be verified by reference to (22). With dU,/dx = 0 and dlT/dx = 0, (22) gives
(25)
/3z + (p - 136.47eC”)},
which, when integrated, yields result at variance with the experimentally established law of friction. The exercise appears to suggest that the viscosity-dependent friction might preclude the possibility of equilibrium. With dz/dx = 0 and dIT/dx = 0, then, there occurs a possibility for (22) to be satisfied, namely, (2az - (p - 136.47eCS)}$
2
= :IC~ e-“.
This being expected to apply only the case of positive boundary layers in favorable (Herring and Norbury
dU,/dx, analysis 1967), but also
is made not only for for those in adverse
r A ”
-0.4 hpredicted by(26) _____i__l___ 0
50
100
150
200
250
x cm
Fig. 1. Analysis of experimental results of Herring Norbury. Smooth wall; favorable pressure gradient.
500
750
loo00
x cm
and
Fig. 2. Analysis of experimental results Smooth wall; adverse pressure gradient.
of Clauser.
54
I. Tani / Some equihbrium
turbulent boundary layers
Fig. 3. Analysis of experimental results of Bradshaw. Smooth wall; adverse pressure gradient.
x cm
pressure gradients (Clauser 1954; Bradshaw 1967). Values of z and 17 calculated from experimental data are shown in Figs. 1-3 as functions of the streamwise distance x, along with the experimental values of S*, 8 and the pressure gradient parameter A = (s*/p~z)(dp/dx) = -(u,/u,)*(s*/U,)(dci,/dx), where p is the pressure and p is the density. When viewed in perspective, the values of z and 17 are kept nearly constant for all cases, at least in sufficiently downstream stations. In the case of adverse pressure gradient, however, the pressure gradient parameter A is positive, which cannot be reconciled with the requirement of (26). Even in the case of favorable pressure gradient, the experimental values of A are somewhat different from those predicted by (26) although the discrepancy might be accounted for by the inevitable three-dimensionality or by dz/dx = dlI/dx = 0 having been assumed in deriving (26). Be that as it may, the distinguishing feature as observed in Fig. 1 (favorable gradient) is the tendency for z, II, S* and 0 to approach equilibrium values as the flow develops downstream. It appears that the boundary layer exhibits equilibrium behavior in an unmistakable manner. Such a tendency cannot be noted from Figs. 2 and 3 (adverse gradient). It has been expected on dimensional grounds (Rotta 1962) that the conditions for equilibrium are fulfilled not with adverse pressure gradient but with favorable pressure gradient. k-type rough wall: It would
k-type
rough wall provided
be possible that
for the boundary
layer to exhibit
equilibrium
on the
(27) which results from (23) by assuming dz/dx = dIl/dx equilibrium boundary layer in zero pressure gradient roughness height h increases linearly with x so that dh dx
-_=-e
K2
= 0. This means by implication that the (dU,/dx = 0) is possible only when the
-(.s+rK)
(Yz- p
The possibility has been predicted on dimensional grounds by Rotta (1962), and realized by the experiment of Liu and Huang (1973). The roughness elements used are circular cylinders of 2
I. Tani / Some equilibrium turbulent boundary layers
55
Fig. 4. Experimental results of Liu and Huang. k-type rough wall; zero pressure gradient.
mm diameter and of variable height given by h = 0.02x (x is measured from the leading edge), attached normal to the flat plate with a spacing of 10 diameters in both streamwise and spanwise directions. Thus, the roughness geometry is not two-dimensional. In addition, the flow near the leading edge might not have been fully rough. For these reasons no analysis is made, and only the experimental values of 6, S* and z are shown in Fig. 4. Some doubt is felt as to the accuracy of determining z, but there are fair indications that the flow is likely to be approaching equilibrium. For uniform roughness (dh/dx = 0), equilibrium is possible provided (29) which is expected to apply only the boundary layer in favorable pressure gradients. This is the type of flow investigated by Coleman, Moffat and Kays (1976) with a porous wall composed of most densely packed uniform spheres (1.27 mm diameter) with and without the effect of blowing. Values of z and II calculated from the experimental data for the unblown case are shown in Fig. 5, along with the experimental values of 6*, 8 and A. The features are quite similar to those observed in Fig. 1 for the smooth-wall boundary layer in favorable pressure gradient. For analyzing the experimental data of Coleman et al., the k-type rough-wall constant K is assumed to be equal to -5.0, which has been determined from the analysis of the boundary layer measurements on the same rough wall but in zero pressure gradient (Pimenta, Moffat and Kays 1975). With dU,/dx = 0, the momentum integral equation (21) is integrated to give
where 19, is the momentum thickness at the datum station x = x0. This is used as an auxiliary equation for determining z, II and K, along with eq. (8) and the first equation of (18). For further details about the rough-wall constant and its evaluation, reference may be made to the author’s forthcoming paper, “Turbulent boundary layer development over rough surfaces”. d-type rough waN: Putting
dz/dx
= 0 in (24), we have
56
I. Tani / Some equilibrium turbuleni boundup
i0
layers
Fig. 5. Analysis of experimental results of Coleman, Moffat and Kays. k-type rough wall; favorable pressure gradient.
for the condition under which the boundary layer could exhibit equilibrium on the d-type rough wall. By virtue of (20), dz/dx = 0 is necessarily followed by dII/dx = 0. It is expected from (31) that equilibrium would exist for a certain range of pressure gradients, from favorable to adverse, although experiments have been confined to zero pressure gradient (Perry, Schofield and Joubert 1969, Wood and Antonia 1975, Osaka, Nakamura and Kageyama 1984). Evaluation of z and II is made from the measurements of Perry et al. on a flat plate containing
Fig. 6. Analysis of experimental results of Perry, Schofield and Joubert. d-type rough wall; zero pressure gradient.
I. Tani / Some equilibrium
turbulent boundary layers
57
spanwise grooves of 3.1 mm depth and 2.8 mm width, spaced along a 28 mm wavelength. The results are shown in Fig. 6, along with the experimental values of S* and 8. The value determined for the d-type rough-wall constant D is - 13.7. A little disturbing is the scatter of z and IT from station to station, but there are fair indications that the flow is approaching equilibrium, in which state z, II, dS*/dx and d9/dx are all constant.
5. Conclusion
Recapitulating, we assume that the mean velocity profile outside the viscous sublayer of a turbulent boundary layer is expressed with sufficient accuracy by the Coles additive law of the wall and law of the wake, with modification for roughness as given in eq. (2). We also define the boundary layer to be in equilibrium when the wake parameter II is constant in the streamwise direction, dII/dx = 0. With this formulation, a differential equation for z and 71 is derived from the momentum integral of the boundary layer equation. The differential equation, though incapable of solution because of closure difficulty, serves to identify the conditions under which the boundary layer could exhibit equilibrium. We predict that equilibrium is possible for boundary layers in favorable pressure gradient over smooth as well as k-type rough surfaces. If the roughness were allowed to increase in height linearly with the streamwise distance, equilibrium would be possible in zero pressure gradient. If the roughness is of d-type, equilibrium would exist for a certain range of pressure gradients, from favorable to adverse. For all these possible equilibrium flows, dII/dx = 0 is accompanied by dz/dx = 0. The prediction is verified by evaluating z and II from available experimental results of mean velocity measurements, except for the case of d-type roughness subject to nonzero pressure gradient. Some of the evaluation indicates values of z and IT more or less distant from the equilibrium value, presumably affected by the initial conditions, but approaching equilibrium as the flow develops downstream. This observation is worthy of mention in that the approach to equilibrium is implicit in the definition of equilibrium and that the like behavior cannot be noted in non-equilibrium boundary layer flows. It is to be noted, however, that all the equilibrium flows listed above, except for the d-type rough-wall flows, are those anticipated by Rotta (1962) on dimensional grounds. The contribution of the present study to the subject is simply to demonstrate the existence of equilibrium by evaluating z and II from the experimental data, and to discriminate between the equilibrium and non-equilibrium behavior of boundary layer flows. The conclusions reached are in direct variance with what is commonly believed, but it must be remembered that most of the previous investigations has been kinematic in nature, not dynamic. This means by implication that the examination of equilibrium has been confined to the streamwise variation of z and IT, which are evaluated from experimental data by means of equations of a kinematic nature. No attempt has been made, to the author’s knowledge, to apply the momentum integral to discussing the equilibrium problem, except in connection with an approximate analysis based on a postulate of doubtful validity.
Acknowledgment The author would like to thank Professor Hugh W. Coleman, Mississippi State University, for directing his notice to the Stanford experiments on the rough-wall boundary layer in favorable pressure gradients (Coleman, Moffat and Kays 1976).
I. Tani / Some equilibrium
turbulent boundary layers
References Bradshaw, P. (1967) J. Fluid Mech. 29, 625-645. Clauser, F.H. (1954) J. Aeron. Sci. 21, 91-108. Coleman, H.W., Moffat, ‘R.J., Kays, W.M. (1976) Thermosc. Div., Mech. HMT-24; J. Fluid Mech. 82, 507-528 (1977). Coles, D. (1956) J. Fluid Mech. I, 191-226. Coles, D. (1962) RAND Corp. Rep. R-403-PR. Coles, D. (1969) Proc. 1968 AFOSR-IFP-Stanford Conference on Computation Stanford Univ. Herring, H.J., Norbury, J.F. (1967) J. Nuid Mech. 27, 541-549. Lewkowicz, A.K. (1982) Z. Flugtviss. Weltraumforsch. 6, 261-266. Liu, C.Y., Huang, M.Y. (1973) Aeron. J. 77, 192-194. Osaka, H., Nakamura, I., Kageyama, Y. (1984) Trans. Jpn. Sot. Mech. Engrs. Perry, A.E., Schofield, W.H., Joubert, P.N. (1969) J. Fluid Mech. 37, 383-413. Pimenta, M.M., Moffat, R.J., Kays, W.M. (1975) Thermosc. Div., Mech. HMT-21. Rotta, J.C. (1955) J. Aeron. Sci. 22, 215-216. Rotta, J.C. (1962) Progr. Aeron. Sci. 2, 1-219. Spalding, D.B. (1961) ASME J. Appl. Mech. 28, 455-458. Tani, I., Motohashi, T. (1985a) Proc. Japan Acad. 61 B, 333-336. Tani, I., Motohashi, T. (1985b) Proc. Japan Acad. 61 B, 337-340. Wood, D.H., Antonia, R.A. (1975) ASME J. Appl. Mech. 42, 591-597.
Eng.
Dept.,
of Turbulent
Stanford
Boundary
Univ.,
Layers
Rep.
No.
2, l-45,
50B, 2299-2306. Eng.
Dept.,
Stanford
Univ.,
Rep.
No.
Research
1 (1986) 49-58
49
Some equilibrium turbulent boundary layers Itiro TAN1 Nurional Aerospace Laboratory, Received
31 January
Jindaij,
Chofu, Tokyo, 182 Japan
1986
Using the Coles additive law of the wall and law of the wake for the mean velocity profile of a Abstract. two-dimensional turbulent boundary layer, a differential equation for the friction and wake parameters is derived from the momentum integral equation with a view to finding out the conditions under which the boundary layer can exhibit equilibrium. It is predicted that equilibrium is possible for boundary layers in favorable pressure gradient over smooth as well as k-type rough walls. When the roughness height is allowed to increase linearly with the streamwise distance, equilibrium exists also in zero pressure gradient. For a d-type rough wall, equilibrium is possible for a certain range of pressure gradients, from favorable to adverse. Most of the predictions are verified by evaluating the friction and wake parameters from the available experimental data on mean velocity measurements.
1. Introduction Considerable attention has been devoted to the equilibrium turbulent boundary layer, which is characterized by similarity of the velocity-defect profiles in the course of its downstream development. The early experimental work of Clauser (1954) and the analysis of Rotta (1955, 1962) served to demonstrate that equilibrium boundary layers in both zero and adverse pressure gradients could exist at least approximately for a certain distance along a smooth wall. This has been added to by the experiments of Herring and Norbury (1967) in favorable pressure gradients and by those of Bradshaw (1967) in adverse pressure gradients. On the other hand, an exhaustive examination of experimental velocity-defect profiles has led Coles (1956) to the formulation of the additive law of the wall and law of the wake,
u _=_ UT
1
lny+C+:,
K
(
i,
1
for the region outside the viscous sublayer (y > 5Ov/u,, say), where lJ is the mean velocity in the x-direction, x and y are coordinates along and normal to the solid wall, respectively, U, is the friction velocity (square root of wall shear stress divided by fluid density), Y is the kinematic viscosity, K is the Karman constant (= 0.41), C is the smooth-wall constant (= 5.0), II is the wake parameter, and w is the universal function of y/6, 6 being the thickness of the boundary layer. There is now abundant evidence supporting the belief that the conditions under which the formulation (1) is valid are nearly equivalent to those under which the boundary layer approximation itself might be expected to apply. The wake parameter II is generally regarded as dependent on x. In the event that II is independent of x, the velocity defect relative to the free-stream velocity U,, made non-dimensional by the friction velocity u7, turns out to be a function of y/6 only, which is the property assigned to equilibrium flow. Viewed from a dimensional standpoint, however, the equilibrium turbulent boundary layer could only be reconciled with the Reynolds-averaged momentum 0169-5983/86/$3.50
0 1986, Japan
Society of Fluid Mechanics
50
I. Tani / Some equilibrium
turbulent boundary layers
equation if u/U0 and d6/dx were independent of x, namely, for example, if the flat plate were roughened in such a way that the roughness height increased linearly with the distance along the plate (Rotta 1962). These conditions are not fulfilled with boundary layers on smooth or uniformly rough flat plates. The approximate equilibrium as observed experimentally thus raises a question as to why u7/U0 and da/dx exert such a weak influence. In an extensive survey of experimental data on various boundary layer flows, Coles (1962, 1969) evaluated the wake parameter II by fitting the measured velocity profile to the analytical expression (1) and classified the boundary layer behavior by the variation of II in the x-direction. The results obtained are exceedingly illuminating in many respects, although the question of equilibrium has remained unsettled. It is connected with this question that an attempt was recently made by the present author, in collaboration with Motohashi, to devise a simple method of analysis for evaluating characteristic parameters, z = KI?&/u~ and III, from the integral thicknesses of measured velocity profile, thus dispensing with the skilled procedure of fitting, yet yielding equally reliable evaluation (Tani and Motohashi 1985a). Results of analyzing the available data on adequately tripped boundary layers would seem to negate the existence of equilibrium state so far as the pressure gradient is zero and the wall is smooth (Tani and Motohashi 1985b). The present paper aims at extending the analysis to the boundary layers in nonzero pressure gradients and also to those on rough walls, with a view to searching for the conditions under which the equilibrium state could exist.
2. Method for evaluating characteristic parameters
We confine ourselves to a two-dimensional start with the Coles formulation (1) generalized
u -=u,
1 lnl(‘Y+C-!Z+!& u U,
K
K
incompressible to the form
turbulent
boundary
layer
and
(2)
’
with AU/u, representing the shift due to roughness of the velocity profile. As regards the wake function w, the universal form suggested by Coles (1969) w(y/6) = 1 - COS(~T~/~), being incapable of satisfying the condition aU/ay = 0 at the edge of the boundary layer y = 6, a quartic polynomial due to Lewkowicz (1982)
is used for making up the deficiency. Common to most methods of modification is the to correct the wall appearance of the term proportional to 17-i, which makes it inevitable component from K-I ln(u,y/v) + C to KC* ln(u,y/v) + C - ~C’(y/tI)~(l -y/6)(1 - 2y/6) in the expression (2). The choice of a quartic in preference to a cubic is based on the smallness in magnitude of this correction. With eqs. (2) and (3) the velocity defect is expressed in the form
The displacement
and momentum
thicknesses
of the boundary
layer are then given by
(5)
I. Tani / Some equilibrium
respectively,
turbulent boundap
layers
51
where 2 -3
u, Z=K-=K u,
cf
J
a=E+fl60
8437 #8=~+ggI+g-n,
>
cf being the conventional gives
(6)
friction
2
coefficient.
Elimination
of 6 between
the two equations
(7) of (5)
8437 ~-~~G+($&~G)fl+~-n’=o,
(8)
where (+;
1-h (
(9)
i
is the Clauser parameter. It is to be noticed that eqs. (4) to (9) are all independent of viscosity and roughness, although the flow may be affected by these variables through the boundary conditions. For further analysis it is convenient to introduce the notation S=Z-2fl--KKC. Smooth
wafl:
S= Substitution
(10)
Evaluating
6
the velocity
profile
(2) with AU/u,
= 0, at y = 8, we have
;e’.
(11)
of (11) into (5) yields
f?= a-i
/3 es v z
i
--. K
(12)
uo
Equation (8), along with the first equation of (12), affords the means for evaluating the friction parameter z and the wake parameter II from the measurable quantities S* and 8. The calculation of S* and 0 resulting in (5) is based on the assumption that the Reynolds number is high enough for the contributions from the viscous sublayer to be negligible. If this is not the case, a correction is needed. Using the sublayer velocity profile due to Spalding (1961) u7.Y 5 -=-++ V where 5 =
KU/U,,
S*=
-ye<
-
1 -
{ -
;p
-
g3
-
A[“),
(13)
K
we
obtain
olGt50.63 I
$ o,
8=((~+~-(50.63-~j)~,
instead of (12). For relatively low Reynolds numbers, the sublayer effect by using the corrected thicknesses 8; =S*-50.635,
19,=6-t 0
in place of the measured
thicknesses
50.63-p
(14) therefore,
136.41 KZ
it is necessary
v -, 1
UO
to account
for
(15)
6* and 0 in eqs. (9) and (12).
Rough wall: Two types of rough wall are considered, k-type and d-type according to the terminology of Perry, Schofield and Joubert (1969). The k-type roughness. typified by a smooth wall roughened with closely packed sand grains, or with sparsely spaced spanwise rods, has the roughness shift of the form AU -=_ u,
1 K
ln$
+ K,
(16)
I. Tuni / Some equdibrium
52
turbulent boundary layers
where h is the height scale of roughness and K is a function of u,h/v, but tends to a constant value characteristic of roughness geometry for sufficiently large values of u,h/v in the so-called fully rough regime. Evaluating the velocity profile (2) with AU/u, given by (16) at y = 6, we have 8 = heS+KK. Substitution
of (17) into (5) yields
The d-type roughness is typified by a smooth wall containing sparsely spaced narrow spanwise grooves. The roughness shift is expressed by the form independent of the roughness scale, AU _=_ u,
1
In*
K
V
(19)
+ D,
where D is a constant, characteristic of the roughness (2), with AU/u, given by (19), at y = 6, we have s=
geometry.
Evaluating
-KD.
the velocity
profile
(20)
When the constant K or D is known from other sources, eq. (8) along with the first equation of (18) or eq. (20), would afford the means for evaluating z and LI from the measured values of S* and 8. When K or D is also to be known, an auxiliary equation must be sought for determination, as will be seen later.
3. Momentum
integral equation
In the following we assume that the flow is in fully rough regime and that the k-type rough-wall constant K as well as the d-type rough-wall constant D are independent of x. Substituting the momentum thickness 9 and other quantities formulated in the preceding section into the momentum integral equation
we obtain (yz _ p + p - 136.47ee” Z
{(2a-ir)z-(2p-B)}Zg
zgi
(22)
+{2az-(~-136.47eC’)}~~=~~~e~’ 0
for the smooth-wall
boundary
layer,
+(az-p)~+(3ar-2~)~~=K2e-(SixK'
(23)
0 for the k-type
1
rough-wall
boundary
layer, and
~z_~2a+~~+~)~~+(~z-~)~+(3~~-2B)~~=K’
(24) 0
I. Tani / Some equilibrium
turbulent boundary layers
53
for the d-type rough-wall boundary layer, where the dot denotes differentiation with respect to II. Each of these equations provides a differential equation for z and II (or 13) as functions of x. In order to calculate both z and II, an auxiliary equation is required in addition to that differential equation. What is intended here, however, is simply to make use of the differential equation for finding the conditions under which the equilibrium flow could exist.
4. Possible equilibrium flows Smooth walk It has been shown in the preceding paper (Tani and Motohashi 1985b) that there exists no equilibrium state for the boundary layer on a smooth wall in zero pressure gradient. This’finding may be verified by reference to (22). With dU,/dx = 0 and dlT/dx = 0, (22) gives
(25)
/3z + (p - 136.47eC”)},
which, when integrated, yields result at variance with the experimentally established law of friction. The exercise appears to suggest that the viscosity-dependent friction might preclude the possibility of equilibrium. With dz/dx = 0 and dIT/dx = 0, then, there occurs a possibility for (22) to be satisfied, namely, (2az - (p - 136.47eCS)}$
2
= :IC~ e-“.
This being expected to apply only the case of positive boundary layers in favorable (Herring and Norbury
dU,/dx, analysis 1967), but also
is made not only for for those in adverse
r A ”
-0.4 hpredicted by(26) _____i__l___ 0
50
100
150
200
250
x cm
Fig. 1. Analysis of experimental results of Herring Norbury. Smooth wall; favorable pressure gradient.
500
750
loo00
x cm
and
Fig. 2. Analysis of experimental results Smooth wall; adverse pressure gradient.
of Clauser.
54
I. Tani / Some equihbrium
turbulent boundary layers
Fig. 3. Analysis of experimental results of Bradshaw. Smooth wall; adverse pressure gradient.
x cm
pressure gradients (Clauser 1954; Bradshaw 1967). Values of z and 17 calculated from experimental data are shown in Figs. 1-3 as functions of the streamwise distance x, along with the experimental values of S*, 8 and the pressure gradient parameter A = (s*/p~z)(dp/dx) = -(u,/u,)*(s*/U,)(dci,/dx), where p is the pressure and p is the density. When viewed in perspective, the values of z and 17 are kept nearly constant for all cases, at least in sufficiently downstream stations. In the case of adverse pressure gradient, however, the pressure gradient parameter A is positive, which cannot be reconciled with the requirement of (26). Even in the case of favorable pressure gradient, the experimental values of A are somewhat different from those predicted by (26) although the discrepancy might be accounted for by the inevitable three-dimensionality or by dz/dx = dlI/dx = 0 having been assumed in deriving (26). Be that as it may, the distinguishing feature as observed in Fig. 1 (favorable gradient) is the tendency for z, II, S* and 0 to approach equilibrium values as the flow develops downstream. It appears that the boundary layer exhibits equilibrium behavior in an unmistakable manner. Such a tendency cannot be noted from Figs. 2 and 3 (adverse gradient). It has been expected on dimensional grounds (Rotta 1962) that the conditions for equilibrium are fulfilled not with adverse pressure gradient but with favorable pressure gradient. k-type rough wall: It would
k-type
rough wall provided
be possible that
for the boundary
layer to exhibit
equilibrium
on the
(27) which results from (23) by assuming dz/dx = dIl/dx equilibrium boundary layer in zero pressure gradient roughness height h increases linearly with x so that dh dx
-_=-e
K2
= 0. This means by implication that the (dU,/dx = 0) is possible only when the
-(.s+rK)
(Yz- p
The possibility has been predicted on dimensional grounds by Rotta (1962), and realized by the experiment of Liu and Huang (1973). The roughness elements used are circular cylinders of 2
I. Tani / Some equilibrium turbulent boundary layers
55
Fig. 4. Experimental results of Liu and Huang. k-type rough wall; zero pressure gradient.
mm diameter and of variable height given by h = 0.02x (x is measured from the leading edge), attached normal to the flat plate with a spacing of 10 diameters in both streamwise and spanwise directions. Thus, the roughness geometry is not two-dimensional. In addition, the flow near the leading edge might not have been fully rough. For these reasons no analysis is made, and only the experimental values of 6, S* and z are shown in Fig. 4. Some doubt is felt as to the accuracy of determining z, but there are fair indications that the flow is likely to be approaching equilibrium. For uniform roughness (dh/dx = 0), equilibrium is possible provided (29) which is expected to apply only the boundary layer in favorable pressure gradients. This is the type of flow investigated by Coleman, Moffat and Kays (1976) with a porous wall composed of most densely packed uniform spheres (1.27 mm diameter) with and without the effect of blowing. Values of z and II calculated from the experimental data for the unblown case are shown in Fig. 5, along with the experimental values of 6*, 8 and A. The features are quite similar to those observed in Fig. 1 for the smooth-wall boundary layer in favorable pressure gradient. For analyzing the experimental data of Coleman et al., the k-type rough-wall constant K is assumed to be equal to -5.0, which has been determined from the analysis of the boundary layer measurements on the same rough wall but in zero pressure gradient (Pimenta, Moffat and Kays 1975). With dU,/dx = 0, the momentum integral equation (21) is integrated to give
where 19, is the momentum thickness at the datum station x = x0. This is used as an auxiliary equation for determining z, II and K, along with eq. (8) and the first equation of (18). For further details about the rough-wall constant and its evaluation, reference may be made to the author’s forthcoming paper, “Turbulent boundary layer development over rough surfaces”. d-type rough waN: Putting
dz/dx
= 0 in (24), we have
56
I. Tani / Some equilibrium turbuleni boundup
i0
layers
Fig. 5. Analysis of experimental results of Coleman, Moffat and Kays. k-type rough wall; favorable pressure gradient.
for the condition under which the boundary layer could exhibit equilibrium on the d-type rough wall. By virtue of (20), dz/dx = 0 is necessarily followed by dII/dx = 0. It is expected from (31) that equilibrium would exist for a certain range of pressure gradients, from favorable to adverse, although experiments have been confined to zero pressure gradient (Perry, Schofield and Joubert 1969, Wood and Antonia 1975, Osaka, Nakamura and Kageyama 1984). Evaluation of z and II is made from the measurements of Perry et al. on a flat plate containing
Fig. 6. Analysis of experimental results of Perry, Schofield and Joubert. d-type rough wall; zero pressure gradient.
I. Tani / Some equilibrium
turbulent boundary layers
57
spanwise grooves of 3.1 mm depth and 2.8 mm width, spaced along a 28 mm wavelength. The results are shown in Fig. 6, along with the experimental values of S* and 8. The value determined for the d-type rough-wall constant D is - 13.7. A little disturbing is the scatter of z and IT from station to station, but there are fair indications that the flow is approaching equilibrium, in which state z, II, dS*/dx and d9/dx are all constant.
5. Conclusion
Recapitulating, we assume that the mean velocity profile outside the viscous sublayer of a turbulent boundary layer is expressed with sufficient accuracy by the Coles additive law of the wall and law of the wake, with modification for roughness as given in eq. (2). We also define the boundary layer to be in equilibrium when the wake parameter II is constant in the streamwise direction, dII/dx = 0. With this formulation, a differential equation for z and 71 is derived from the momentum integral of the boundary layer equation. The differential equation, though incapable of solution because of closure difficulty, serves to identify the conditions under which the boundary layer could exhibit equilibrium. We predict that equilibrium is possible for boundary layers in favorable pressure gradient over smooth as well as k-type rough surfaces. If the roughness were allowed to increase in height linearly with the streamwise distance, equilibrium would be possible in zero pressure gradient. If the roughness is of d-type, equilibrium would exist for a certain range of pressure gradients, from favorable to adverse. For all these possible equilibrium flows, dII/dx = 0 is accompanied by dz/dx = 0. The prediction is verified by evaluating z and II from available experimental results of mean velocity measurements, except for the case of d-type roughness subject to nonzero pressure gradient. Some of the evaluation indicates values of z and IT more or less distant from the equilibrium value, presumably affected by the initial conditions, but approaching equilibrium as the flow develops downstream. This observation is worthy of mention in that the approach to equilibrium is implicit in the definition of equilibrium and that the like behavior cannot be noted in non-equilibrium boundary layer flows. It is to be noted, however, that all the equilibrium flows listed above, except for the d-type rough-wall flows, are those anticipated by Rotta (1962) on dimensional grounds. The contribution of the present study to the subject is simply to demonstrate the existence of equilibrium by evaluating z and II from the experimental data, and to discriminate between the equilibrium and non-equilibrium behavior of boundary layer flows. The conclusions reached are in direct variance with what is commonly believed, but it must be remembered that most of the previous investigations has been kinematic in nature, not dynamic. This means by implication that the examination of equilibrium has been confined to the streamwise variation of z and IT, which are evaluated from experimental data by means of equations of a kinematic nature. No attempt has been made, to the author’s knowledge, to apply the momentum integral to discussing the equilibrium problem, except in connection with an approximate analysis based on a postulate of doubtful validity.
Acknowledgment The author would like to thank Professor Hugh W. Coleman, Mississippi State University, for directing his notice to the Stanford experiments on the rough-wall boundary layer in favorable pressure gradients (Coleman, Moffat and Kays 1976).
I. Tani / Some equilibrium
turbulent boundary layers
References Bradshaw, P. (1967) J. Fluid Mech. 29, 625-645. Clauser, F.H. (1954) J. Aeron. Sci. 21, 91-108. Coleman, H.W., Moffat, ‘R.J., Kays, W.M. (1976) Thermosc. Div., Mech. HMT-24; J. Fluid Mech. 82, 507-528 (1977). Coles, D. (1956) J. Fluid Mech. I, 191-226. Coles, D. (1962) RAND Corp. Rep. R-403-PR. Coles, D. (1969) Proc. 1968 AFOSR-IFP-Stanford Conference on Computation Stanford Univ. Herring, H.J., Norbury, J.F. (1967) J. Nuid Mech. 27, 541-549. Lewkowicz, A.K. (1982) Z. Flugtviss. Weltraumforsch. 6, 261-266. Liu, C.Y., Huang, M.Y. (1973) Aeron. J. 77, 192-194. Osaka, H., Nakamura, I., Kageyama, Y. (1984) Trans. Jpn. Sot. Mech. Engrs. Perry, A.E., Schofield, W.H., Joubert, P.N. (1969) J. Fluid Mech. 37, 383-413. Pimenta, M.M., Moffat, R.J., Kays, W.M. (1975) Thermosc. Div., Mech. HMT-21. Rotta, J.C. (1955) J. Aeron. Sci. 22, 215-216. Rotta, J.C. (1962) Progr. Aeron. Sci. 2, 1-219. Spalding, D.B. (1961) ASME J. Appl. Mech. 28, 455-458. Tani, I., Motohashi, T. (1985a) Proc. Japan Acad. 61 B, 333-336. Tani, I., Motohashi, T. (1985b) Proc. Japan Acad. 61 B, 337-340. Wood, D.H., Antonia, R.A. (1975) ASME J. Appl. Mech. 42, 591-597.
Eng.
Dept.,
of Turbulent
Stanford
Boundary
Univ.,
Layers
Rep.
No.
2, l-45,
50B, 2299-2306. Eng.
Dept.,
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Rep.
No.