by A.A. Townsend, Emmanuel College, Cambridge. Skipsmodelltankens meddelelse nr. 32, Mars 1954.
SUNMARY The paper is concerned with recent attempts to construct a coherent theory of turbulent shear flow from experimental studies of flow in wakes and channels, with particular attention to the nature of the turbulent boundary layer flow on a smooth flat
plate with no external pressure gradient. The first part describes briefly the properties and characteristics of the two main classes of turbulent shear flow, free turbulence and channel turbulence, considering the physical meaning of the general principles of flow similarity and self-preservation. The flow in, a turbulent boundary layer is interpreted in terms of these concepts, and it is shown that development of the layer is controlled by the outer part of the flow whose motion strongly resembles that in a wake. Since this part of the flow can only adjust itself very slowly to a changing wall stress, Its motion should be regarded as determined by the whole upstream distribution of wall stress and not by the local stress alone. By allowing for this effect, most of the inconsistencies
that arise from the hypothesis of fully self-preserving flow can be removed, and a relation between wall friction and Reynolds number may be derived that is expected to be accurate at high Reynolds numbers. Finally, it is shown that the observed inequality of the turbulent normal stresses over most of the layer must induce a cross-flow in a boundary layer of finite width, and the magnitudes of the consequent changes in momentum thickness and skin friction are calculated. The predictions are in reasonable agreement with experimental measurements of momentum thickness and total drag co-efficients on plates of finite aspect ratio.
1. INTRODUCTION.
Until fairly recently, the search for an adequate theory of turbulent shear flow had to be conducted without detailed information about the structure and mechanism of
turbulent flow, and, although the physical insight of L. Prandtl and G.I. Taylor led to remarkable progress in the description of the mean properties of the flow, this lack of knowledge of the turbulent motion itself led to incorrect assumptions about its nature that produced inconsistencies in the theory. The assessment of the validity of a
particular type of theory was also complicated by the discovery that some of the best confirmed predictions were common to all the formulations of the theory, and indeed are necessarily common to any theory conforming to certain fundamental principles that had been accepted, either explicitly or implicitly,in all forms of the theory. The development of a theory of turbulent shear flow based on an exact knowledge of the turbulent motion may be traced to the introduction of the hot-wire anemometer, which made possible accurate measurements of the turbulent velocity fluctuations, and to the formulation of the
statistical theory of isotropic turbulence by 0.1. Taylor, who showed for the first time that an exact treatment of a turbulent flow based on the Navjer-Stokes equations of motion might be possible. For some time after this beginning, theoretical and experimental studies were confined to isotropic turbulence, but, as the understanding of this simple flow grew, several workers began to make detailed measurements in turbulent shear flows and to seek regularities in their behaviour. As a result of this work, the structure of turbulent motion has become fairly clear, and a start has been made in the identification of the processes which cause turbulent flows to maintain their characteristic levels of shear stress and turbulent intensity. For free turbulent flows (wakes, jets and free mixing zones), it is now possible to make approximate but absolute estimates of the rate of spread which are in good agreement with observation.
-2-characteristics of free turbulent flows and of channel flows, and to interpret the flow in a boundary layer in terms of these two types of shear flow. The description so
obtained is used in a discussion of the changes in the flow with distance from the leading edge, and to discuss the probable variation of wall stress with Reynolds number.
2. NOTATION.
Rectangular axes are used, so chosen that Ox is the direction of the mean flow close to the p1ane, o , which is a plane of symmetry in a wake and a solid boundary in channel or boundary layer flow. Then
ti, V
W are the components of the mean velocity,'P v- are the components of the turbulent velocity,
P
is the mean pressure,is the kinematic viscosity of the fluid,
is the constant mean velocity in the free stream,
Txq is the stress in the O? direction across a plane with normal parallel to Oy is the shear stress at the wall,
y
o
is the total thickness of a boundary layer as defined by equation (6.4), is the Reynolds number describing the flow
at distance X from the leading edge,
2'-TO/KUI
is the local resistance parameter,K,A
are constants in the universal velocity distribution in theconstant stress layer (equation 4.5),1(7)
is a function describing the velocity distribution in a boundary layer (equation 6.1),are constants defined in equation (6.6), is a constant defined in equation (7.3),
3
is a constant defined in equation (9.15).It should be noted that the pressures and stresses used in this paper are "kinematic", that is, they are the usual mechanical ones divided by the fluid density. This procedure simplifies considerably the writing of equations if the fluid may be considered as
incompressible.
3. FREE TURBULENT SHEAR FLOW.
Free turbulence is a term used to describe turbulent flows which are not
restricted in any direction by rigid boundaries, the principal shear flows in this group being wakes, jets and free mixing zones.
All
these flows are very similarin
structure and in dynamics, but the one most resembling the outer part of a boundary layer is the two-dimensional wake behind a long cylinder, and this flow will be considered below.The first and most fundamental characteristic of all turbulent flows is that they are statistically similar at all high Reynolds numbers, that is, with the exception of that part of the motion which depends on viscous forces, the mean values associated with the flow are determined by the boundary conditions and are independent of the value of the fluid viscosity. Considering the flow in the wake of an infinite circular cylinder of diameter,
d
, placed in a uniform stream of velocity £4 , the boundary conditions arenearly determined by ¿.4 and
d
, and the mean velocity distribution should be of the form,(3.1) ¿J
= ¿J
,'(x/aÇ y/d)
and the turbulent intensity distributions of the form,
and so on, where Ç , g1 are universal functions. A more exact statement of this principle is that, over that part of the flow for which the turbulent Reynolds stresses greatly exceed the mean viscous stresses, mean values describing all those aspects of the flow not directly connected with the viscous dissipation of energy are independent of the fluid viscosity and are determined by the boundary conditions alone. If the boundaries of the flow are geometrically similar, this usually means that the flow is determined by
a velocity and a length.
The existence of this similarity, which has been confirmed experimentally for a variety of flows, is a consequence of the physical nature of the process of energy
dissipation in turbulent motion, which has been likened to a cascade. This means that the energy which is extracted from the mean flow by the large energy-containing eddies that determine the Reynolds stresses is transferred to eddies one order of magnitude smaller.
These in turn transfer it to still smaller eddies, and the process continues in similar stages until the eddies receiving energy are so small that they dissipate it by working against viscous stresses, which converts it to heat. The mechanics of the process is such that the rate of transfer of energy down the "cascade" is limited by the rate of transfer in the first one or two stages whose motion is unaffected by viscous forces. The effect of an increase in viscosity is simply to remove a few stages from the "cascade', and so to increase the size of the eddies responsible for the final conversion to heat. In agreement with this view, it has often been noticed that the eddies responsible for the viscous energy dissipation are distinct in size from those containing the bulk of the turbulent energy (Taylor l93).
A second general principle concerns the development of the flow with distance from the cylinder. It is observed in this and other developing flows that lateral variations of mean velocity and turbulent intensity are much more rapid than longitudinal variations, and this implies that interaction between the turbulent fluid in different parts of the flow is almost exclusively lateral. The flow at any section has a structure that is the result of these lateral interactions on fluid arriving from upstream, and we may expect that the flow will settle down into a moving equilibrium that can be described by universal functions involving only scales of length and velocity that vary with position downstream. If this is true,
(3.3) *
Ì41/)
¿
where tie, i depend only on X , and
/
,
q
are universal functions.
This argument that a developing flow tends to a self-preserving state may be put in many different ways and there is not much doubt that such a flow has a tendency toward self-preserving development, but strict self-preservation may conflict with the equations
of notion. In the turbulent wake, self-preserving flow is only consistant with the equations of motion if the mean velocity variation across the wake is negligible compared
with
the free ztream velocity, and the approximate nature of the self-preservation leadato an appreciable modification of the spread of the wake. Also, the time taken for a flow to attain the state of moving equilibrium that Is the basis of self-preservation may be so long that self-preserving flow only occurs outside the range of normal observation.
The nature of the turbulent motion in wakes has been studied using the techniques developed for isotropic turbulence, and the most striking result Is that flow In the outer parts of the wake Is intermittently turbulent. This is a consequence of the sharp
differentiation between the region of fully turbulent, rotational flow and that of non-turbulent flow, and of the continuous distortion of the surface of separation by a group of large eddies that are nearly homogeneous in size (figure 1). Most of theturbulent energy is contained in smaller eddies of scale moderately small compared with the width of the flow, and the large eddies, whose dimensions are comparable with the flow width, contain comparatively little energy. Since the surface of separation is contorted, the flow at a point fixed with respect to the cylinder is alternately turbulent and non-turbulent as the mean stream convects the flow past the point of observation, the
relative proportion of turbulent to not:-turbulot flow decreasing with distince from the central plane of the wake (Townsend 1949).
When allowance is nade for the decreased relative duration of turbulent flow far from the wake centre, the main turbulent motion, i.e. the smaller scale motion which contains most of the turbulent energy, is found to have mean properties that vary surprisingly little across the flow (figure 2). This is a result of the comparatively low rate of dissipation of turbulent energy which causes the present intensity of turbulence in a small sample of turbulent fluid to derend on its energy gain from the mean flow over a previous time interval so long that the sample has had time to move
over most of the wake width and so to experience all possible rates of energy gain. Under these conditions. the present rate of energy supply to the sample has but a small
influence on the intensity. The movement of these samples of turbulent fluid across the flow is of course due to convection by the large eddies which are not confined to the turbulent fluid.
Although the turbulent motion is not in a condition cf energy equilibrium, it seems to be in a condition of structural similarity for which the ratio of the turbulent Reynolds stress to the turbulent intensity has a constant value (figure 3). This is
obviously true not too close to the wake centre where the fluid has all been subjected to a large uni-djrectjonal shear, but there is some evidence that the apparently lower values of the ratio near the wake centre are due to a mixture of both signs of Reynolds stress. The importance of the existence of this kind of structural equilibrium is that it enables the effect of the fully turbulent fluid on the motion of the large eddies to be calculated, which is necessary for a consideration of their energy balance.
The spread of turbulent flow Into the ambient undisturbed fluid depends in the first place on the diffusive movements at the bounding surface of the turbulent fluid, and proceeds at a rate per unit area of surface that is set by the intensity of the turbulence. The motion of the large eddies acts continuously to distort the bounding surface and so to increase its area and the total rate of entrainment of ambient fluid. In this way, the large eddies control the rate of entrainment of non-turbulent fluid by their distortion of the surface of separation, a higher intensity of the large eddies producing increased entrainment and an increased rate of spread of the wake. If the intensity of the large eddies is exceotionally high, the wake will spread more rapidly and convert more mean flow energy to turbulent energy. This increased production of turbulent energy allows increased Reynolds stresses that will retard the motion of the
large eddies by transferring their energy to the small scale turbulence. In this way, the large eddies control the spread of the wake ano are themselves controlled in intensity. By supPosing that the eddies are long roller-like structures with circulation in planes so directed that the eddy may extract energy from the mean flow (figure 4), the condition
for energy equilibrium can be obtained and the rate of spread of a wake can be computed without introducing any arbitrary constants. The satisfactory agreement between the observed and computed rates of spread, and the general agreement between this description and the observed turbulent structure make it almost certain that the flow really is controlled in this way (Townsend 1950).
The properties of free turbulent shear flows may be summarized as follows: A two-component structure of the whole turbulent motion, with a distinct group of large eddies controlling the spread of the flow, although the smaller eddies contain most of the turbulent energy.
Comparative homogeneity of the motion of the smaller eddies within the sharp boundary between turbulent and non-turbulent fluid.
Relative slowness of turbulent decay and permanence of eddy structures. Structural similarity of the main turbulent motion.
4. TURBULENT FLOW IN PIPES AND CHANNELS.
One essential difference between turbulent flow in a pipe or channel and free turbulent shear flows is that the oipe or channel turbulence is everywhere in a condition
(4.2)
of energy equilibrium, i.e. turbulent intensities and other mean quantities are
independent both of time and of displacement In the direction of mean flow, and questione of development of the flow do not arise. For this reason, the equations of motion take particularly simple forms and the distribution of shear stress in, say, a two-dimensional channel can be found without making any assumptions at all about the nature of the
turbulent motion. Indeed, without assuming more than similarity of the flow
at all high Reynolds numbers and similarIy of the flow very close to the walls of the channel, the greater part of the mean flow problem may be solved.
The principle of similarity of flow at all high Reynolds numbers can be applied to all that part of the channel flow where the direct viscous stresses are small compared with the Reynolds stresses, a condition that excludes only a very thin layer adjacent to the walls. Outside these viscous layers, the flow must be determined,
except for an arbitrary velocity of translation that does not affect the motion, by the Reynolds stress at the walls and the channel width. Then the mean velocity distribution in
a two-dimensional channel is of the form
(4.1)
U=U#
7 f4where
7:
is the wall stresscD
is the channel widthy
is distance from one wall.The velocity of translation, L , is usually chosen to be the velocity at the channel
centre, and it is usually a function of the Reynolds number of the flow, = The function, , will be different for different shapes of channel
section but does not depend on the flow Reynolds number. The second similarity assumption concerns the region of flow so close to the wall that the total shear Stress, the sum of the Reynolds and viscous stresses, may be considered to be constant. Within this region, It is assumed
that the motion is In a state of universal similarity and is statistically determined by universal functions involving only the wall stress, distance from the wall and the fluid viscosity. A simple argument in favour of this assumption runs as follows. Consider the
layer next the wall of thickness y small compared with JJ
. Then the boundary
conditions around this layer are similar whatever the thickness and are defined by the stress on each boundary and the layer thickness. If this is true, ordinary dimensional reasoning shows that the mean velocity distribution is
where le a universal function. This line of argument, by reducing the boundary conditions to entirely neglects boundary conditions not determined
by 2: ,
y
and Of other possible boundary conditions, lt Is easily shown that the mean pressure gradient parallel to the mean flow has a negligible effect on this thin layer, but turbulent energy transfer in appreciable amount across the outer surface might easily invalidate the analysis. The abundant evidence that equation (4.2) is valid in all sorts of wall flows (e.g. Ludwieg & Tilimann 1949) as well as direct measurements of energy flow (Laufer 1953) show that this neglect is justified at high Reynolds numbers of the flow.At moderately high Reynolds numbers, the regions of validity of the two types of similarity overlap in that part of the constant stress layer where viscous stresses are negligible, and equations (4.1) and (4.2) both apply. Then
(4.3)
rt
for all y in the range of overlap, and all Reynolds numbers. This is only possible if the functions are
and
(L.5)
f; (y 7-%)
-
L (Lo9y* M)
in this range, where K , , are absolute constants, and ?' is a constant
characteristic of two-dimensional channel flow. Equation (4.5) is equivalent to the well-known logarithmic velocity distribution which is valid in any turbulent flow close to a boundary. The conditions for its validity are now seen to be simply that the two kinds of
flow 5imilarity should exist.
The conclusions that can be drawn by considering the principles of similarity are sufficient to solve a large part of the mean flow problem, and a great part of the detailed information about the turbulent motion that has been gained from experimental studies confirms In detail the existence of the two types of similarity. This solution is not quite complete unless a method is available for calculating the various absolute constants that appear in the equations (4.4) and (4.5), and this requires a knowledge of the turbulent structure and its mechanics. Considering first the motion of the constant stress layer, it has been shown by analysing the measured correlation co-efficients between turbulent velocity fluctuations at separated points that approximately half the turbulent energy is contained in long, roller-like, attached eddies, so called because they have a fixed relation to the wall (figures 5 and 6). These attached eddies are nearly geometrically similar in structure and appear with a wide range of sizes, probably not simultaneously but in successior.. The remaining half of the turbulent energy is contained in smaller detached eddies that have no definite connection with the wall, and which provide the energy transfer mechanism necessary for the energy dissipation of the whole flow. It is interesting that a simple model of the attached eddies can be used both to predict correctly the form of the transverse correlations (figure 6), and, by considering the energy balance of the attached eddies, to obtain the estimate for the absolute
conotant, 1< , of
(4.6)
c235<K<O.5o
The observed value of the constant is 0,41
The universal equilibrium of the constant stress layer includes the existence of structural similarity of the kind defined for free turbulence, but the equilibrium
structure is decidedly different. Within this layer, it is found that
(4.7) ..1 = =
=
-0.8 .3,0 (Laufer 1953), while in free turbulence,
(4.) -.
=
-very nearly. The considerable departures from equality of intensities of the velocity components are characteristic of flow near a wall, and they extend outside the constant stress layer and over most of the channel.
Outside the constant stress layer, the influence of the wall becomes less, and in many respects the motion near the channel centre is not dissimilar to free turbulence. The change in character of the flow arises from the necessary concentration of turbulent energy production and dissipation within the constant stress layer. Within this layer, the rate of production is so large that the energy equilibrium is truly local and
determined by local conditions. Near the channel centre, rates of energy production and dissipation are low, and the apparent energy equilibrium is not due to rapid adjustment of the turbulent motion to local conditions but to homogeneity in the direction of mean flow.
5. BOUNDARY LAYER: GENERAL DESCRIPTION.
The ideal boundary layer that will be considered here is the flow around a thin, semi-infinite plane placed edge-on to the flow, and it is realised experimentally by the flow over thin bodis of rectangular plan form. Apart from the effects of finite width
-6-of the plate (which can be reduced by increasing the width-length ratio), the only serious
difficulty iii comparing experimental measurements with theoretical calculations based on the ideal flow arises from the existence of a region of laminar flow near the leading edge of the plate. If the surfaces are smooth, the extent of this region depends on the
magnitude of the external disturbances that can set off transition to turbulent flow, and these may vary considerably between one experimental arrangement and another. This problem has received a considerable amount of attention recently, and methods have been devised to induce transition from laminar to turbulent flow as soon as the laminar flow becomes unstable (Hughes 1952). In this way the effects of external disturbances are minimized, and a flow may be obtained that is determined very nearly by the free stream velocity and the fluid viscosity.
In the classification of turbulent shear flows, the boundary layer occupies a position intermediate between channel flow and free turbulence. Arguments similar to those used for channel flow show that most of the turbulent energy production and dissipation take place in a thin layer close to the wall over which the shear stress may
be considered constant. This suggests and many experimental measurements confirm that the turbulent flow there is in a condition of universal similarity determined bythe wall stress and the fluid viscosity, just as is the constant stress layer ina channel. On the other hand, the outer part of the flow is very similar to wake flow, showing the same characteristics of intermittently turbulent flow, relatively slow turbulent decay rate and near homogeneity of the turbulent fluid.
The differing properties of the outer and inner parts of a boundary layer may be described in terms of the conversion, transfer and dissipation of flow energy.
Fundamentally, the turbulent motion obtains its energy by removing kinetic energy from the mean stream, and this conversion of mean stream kinetic energy is not only more intense in the outer part of the layer where the nican stream kinetic energy is a maximum, but it i dependent on continual entrainment of non-turbulent fluid by the outer layer. Although the extraction of kinetic energy from the mean flow is concentrated in the outer layer, the energy so obtained is not kept in this layer but is transferred for the most part to
the inner layer through the action of the Reynolds stresses. There it is converted to turbulent energy and dissipated. This description indicates that the outer part of the layer controls the energy supply to the boundary layer as a whole, and although turbulent energy does diffuse from the inner layer to the outer, the inner layer has little effect on the rest of the flow and merely adjusts Its structure to absorb an energy flow deter-mined by the entrainment processes of the outer layer. The independence and
characteristic stability of the outer flow is well shown if the boundary layer is
subjected to an adverse pressure gradient, when the stress distribution in the outer half of the layer is found to be nearly unchanged although the wall stress may be reduced almost to zero (Schubauer & Klebanoff 1950).
If this view of the relations between the various parts of the flow is accepted, it is clear that a physical theory of the boundary layer flow must consider the outer, wake-like flow as establishing its own equilibrium and not as accepting one dictated by the constant Stress layer. As the motion in the constant stress layer is certainly determined by the local wall stress, this indicates that the scales of velocity and length in the two parts of the layer may not be simply related.
6. SELF-PRESERVATION IN A BOUNDARY LAYER.
The boundary layer is a developing flow, and it is natural to suppose that it may be self-preserving in structure over the whole of the flow excluding the viscous layer
next the wall. If this complete self-preservation exists, it must be determined bya velocity scale which is the square root of the wall Stress and a length scale proportional to the thickness of the layer. In view of the slowness with which the outer flow adjusts itself to changing conditions, it would be rather surprising if the local flowstructure were really determined by the local wall stress as complete self-preservation demands.
This ouestion can be discussed most conveniently by assuming complete self-preservation and deducing the corresponding variation of wall stress. Then substitution of this variation in the equations of motion will make clear the nature of the departures from
fully self-preserving flow.
If the mean velocity distribution is self-preserving in form,
(6.1)
V
= 2'
where
2' =
2:
is the local wall Stress'( is the universal constant of equation (4.5) is the thickness of the layer.
Since equation (6.1) describes also the outer part of the constant stress layer, it must be identical with
(6.2) ¿1
= 3-
(/ofor small values of
y/6
, and so(6.3) Ì(y/ó) /oyì/6 4'
in this region. By a suitable choice of ¿ , the constant A' may be made zero, and
(6)
S
The mean velocity variation and the wall stress are related by the momentum equation, which, omitting some small terms involving the turbulent intensities, is
(6.5) v(u,-IL)dy
ï;
Substituting in this e'uat1on the self-rreserving velocity distribution
(6.1)
and the value of S given by equation (6.4), it is found that(6.6)
where
¿= _Jf('x)a'x
=
and = Lx/ is the Reynolds number describing the flow at distance X from the leading edge. The solution of this equation is
(6.7)
where is a constant of' integration, and the total drag co-efficient, defined by
(6.)
is related to by (6.9)
If the function and the values of 2' given by equation (6.7) are inserted in the equation of mean motion
r-(6.12)
/
7;('d7'
7Z121
r
r
6.10) iJ it becomes (6.11)S
/
everywhere except within the constant stress la-er where
= y/1 « /
This condition can only be satisfied when the stress rarameter is less than 0.05, corresponding to extremely high Reynolds numbers, probebly in excess of 1O. It seems likely that fully self-preserving flow is not established within the ordinary range of experimental measurements.
If the forms of the shear stress distributions for various values of
7
are compared (figure 7), it is observed that the stress distributions over the outer three-quarters of the layer are similar in form and could be brought into near coincidence by a suitable adjustment of the vertical scale. This suggests that a good approximation to the actual behaviour of the flow could be obtained by postulating inner and outer self-preservation, the velocity scale for the outer flow being different from the scale for the inner flow.7. SELF-PRESERVATION 0F THE OUTER FLOW.
It has been shown that the boundary layer flow can only be self-preserving at extremely high Reynolds numbers outside the ordinary range of extieriiental measurements, and that the deviations from comolete self-preservation at ordinary Reynolds numbers
suggest that the outer three-quarters of the flow is nearly self-rreserving but has a velocity scale greater than that of the constant stress layer. That this should be so is readily understood by considering the different natures of self-preservation in these two parts of the flow. Within the constant stress layer, the "self-preservation" is due to an absolute energy equilibrium which adjusts itself, virtually instantaneously, to the local wall stress. Self-preservation in the outer flow is the result of a moving
equilibrium in fluid carried along by the mean flow, and the local scales of velocity and length which specify the flow are dependent on conditions in the upstream flow, in
particular on the energy supply to the turbulence. Since this energy suoply to the turbulence is approximately proportional to the wall stress, the velocity scale in the outer flow may be regarded as determined by the upstream distribution of wall stress, or, to a first approximation, by the wall stress at a distance
L
upstream, such thatappreciable turbulent dissipation takes place while fluid is transported by the mean flow over this distance. Then approximately,
(7.1)
L=
¿where o( is an absolute constant of order unity,
é
is the layer thickness.and is the velocity scale of the outer flow. If the velocity scale depends only on this wall Stress,
(7.2)
or, to the same aooroximation
( z./_/ small),
(7.3)with a suitable definition of
z., . To obtain a relation between the layer thickness and
the velocity scale, it is assumed that the layer thickness corresponds to that of a completely self-preserving layer of wall stress , an assumption that is consistent
with the probable existence of complete self-preservation at very high Reynolds numbers where y/( . Evidently, the distrbton of shear stress is only seif-oreserving,
i.e. is of the forni (6.13)
if
lo
-and with the dependence of the outer flow on the wall stress at a distance
L
upstream. If this is assumed,(7.z)
é=
- ewhere
»
We now have that the mean velocity distribution for the outer part of the flow is given by
(7.5) * /#c ((y/é)
where , and
S
are defined by equations (7.3) and (7.lj. If the contribution of the inner flow to the momentum integral may be neglected, which is possible at fairly highReynolds numbers, the equation for the momentum integral is
(7.6)
In terms of the total drag co-efficient, the solution of this equation is
(7.7)
c7r(f_
0/R)k
%(Ct/) # ,q-t,'
which differs from the relation obtained on the assumption of complete self-preservation only by the presence of the additional constant
If the distribution of shear stress , is computed for any reasonable form
of the velocity distribution function with the variation of
(-'i given by equation (7.6), it is found to be nearly self-preserving, i.e. of the functional form
(7.) =
over the outer three-quarters of the flow, with a suitable choice of o The proper
value of o depends on the form of the velocity distribution function, but it is usually of order unity in agreement with its physical meaning.
. FRICTIONAL RESISTANCE AT HIGH REYNOLDS NUMBERS.
The account of the development of a turbulent boundary layer given in the
previous section is consistent both with the equations of mean motion and with our present knowledge of the physical characteristics of turbulent flow, and it is hard to believe that the inferred relation between skin friction arid Reynolds number is not accurate when
the various assumptions are valid. These assumptions fall in two classes, the first class including the assumptions of self-preservation of the outer flow and similarity of the flow near the wall, and the second class various mathematical approximations whose errors can be estimated. The validity of the first class of assumptions depends on general theoretical arguments based on experimental observations at comparatively low
Reynolds numbers, but they are unlikely to be in error ori this account. For Reynolds numbers greater than iO7, the various approximations introduce negligible errors, and the relation between drag co-efficient and Reynolds number should be
(s.l)
OE1(f-/)
=
It is useful to consider the probable values of the various constants.
The constants, and q , are known from measurements of the turbulent flow
in pipes, and are
/(=441,
4=23
The constant,. , is the integral over the whole flow of the non-dimensional function
specifying the shape of the velocity distribution in the outer layer, and its value, estimated from observed velocity profiles, is near 0.5 . The remaining constant, o(
describes the lag of the outer layer, and its value is estimated as about 0.8. Substituting these values in equation (8.1) gives
transition, available experiments indicate that it is of order -2.1O . If it is zero,
the relation resembles in form the Schoenherr equation, but the predicted values of total drag are considerably less for the same Reynolds numbers (figure 8). This seems to be in agreement with the measurements of Hughes (1952). It should be remarked that the greater slope of the friction line in the figure, compared with the accepted Schoenherr line
(8.3) ¿2 e = ç.
is due to accepting the value of 0.41 for the constant ,4 . The Scheenherr line
corresponds to tK = O39
The theoretical prediction of equation (8.2) depends only on the validity of a few well-established generalizations about the nature of turbulent flow, and for this reason the author feels considerable confidence that it describes the variation of drag co-efficient for flat plates of effectively infinite aspect ration at Reynolds numbers in excess of lO . Its relation to experimental measurements of the drag of real plates of
small aspect ratio is difficult to discuss without a proper knowledge of the effects of departures from the conditions assumed by the theory.
9. EFFECTS OF FINITE WIDTH ON THE DRAG OF A FLAT PLATE.
In order that measurements of the frictional drag due to a turbulent boundary layer may be made at the largest attainable Reynolds numbers, it has been necessary to use rectangular plates for which the ratio of length to breadth is large and not small as assumed in the theory. These effects have been studied by Hughes, and the change in drag seems to be larger than cari be explained as an edge effect analogous to that calculated for laminar flow. The suggestion that the finite aspect ratio has an effect not only on the flow near the edges of the plate but also on the whole flow is confirmed by
measurements of the meen velocity profiles made by Allan & Cutland (1953). These measurements show that the mean flow distribution hardly varies over most of the plate breadth, although its form depends on the aspect ratio. A theory of this part of the edge effect can be developed by considering the equations of mean motion for the flow.
Consider the flow over a thin plate of breadth ¿D . For this flow, Oz is
the leading edge of the plate, and Ox runs along the centre-line. Then, neglecting terms that are known to be small, the three equations of mean motion are
(9.1)
(9.2)
(9.3)
The second equation can be integrated to give
(9.4)
p -
=where is the pressure outside the boundary layer and is constant over the whole flow. The third equation, which represents the momentum balance in the Ox direction, may be integrated along a line parallel t Ox to give
(9.5) / -- a'
-
-òy
Mutual consistency of "equations (9.4) and (9.5) then requires that
(9.6)
/-dz
-where the values of V and are taken at z = O . It is known that within the
-
12
-arid this difference in intensity persists, in diminished form, over most of the boundary layer. For this reason, the integral of over the breadth of the plate is of order , and , which is the Reynolds stress in the
O
direction across aplane parallel to the plate, is not everywhere zero as it would be for a plate of infinite aspect ratio. This amounts to saying that the stress acting across planes parallel to the plate is riot everywhere parallel to the free stream, but is directed at an angle to the flow. If the non-zero values of are concentrated near the edges of the plate, the values of are necessarily high and must induce rapid accelerations of the flow in this region. No sign has ever been observed of the large disturbances of the flow that would be expected if these accelerations existed, and it is more reasonable to suppose that has non-zero values distributed over most of the plate width, approximately as
/?&'
(9.7) 09' J=.D .D
This stress gradient must induce a cross-flow in the boundary layer from the centre-line outwards, as shown in figure 9, and this requires an inflow over the whole surface of the layer. This inflow increases the rate of entrairunent of fluid from the free stream, and so increases the shear stress.
The effect of this secondary flow may be estimated in the following way. If the amount of cross-flow is small, the equation of mean motion for the downstream component may be written
(9.8)
and the integral with respect to j is
(9.9)
-
7:
where the second term on the left represents the loss of momemtum due to the cross-flow. It is a reasonable approximation to suppose that the component of the mean velocity parallel to the plate is everywhere parallel to the direction of shear stress in the same plane, that is,
(9.10)
W
With the further assumption that the distribution of the down stream component of the mean velocity does not vary much across the width of the plate, it can be shown that
(9.11) =
-7:
o "a
where the integrals are to be understood as mean values over the width of the plate. If the values of are distributed as in equation (9.7), comparison with equation (9.6) shows that
(9.12)
(-4L=
and the distribution of the Reynolds stress. - , is of the form shown in figure 10. The second term in equation (9.11) may be regarded as the ordinary momentum integral modified by the inclusion of a weighting factor which is of order at the wall and becomes small in the outer part of the layer (
4
is the extent of the mean velocityvariation in the ¿7,i direction, which is about one-third of
L
as defined above).The momentum equation may now be written
(9.13)
-/(4-U)ai
-.34
of' self-rreserving velocity rrofiles similar to those of' ecuation (6.1). nd it is found that
(9.14)
/ =
-//n&
9
where
R=
the various symbolo having the same meanings as in section 6, and
whe,"e
.B=3'
The changes in momentum thickness and drag co-efficient due to finite asrect ratio are given very nearly by
(9.15)
9
= 9 1nA
oLand
(9.16) C,E
= C[f*-C
'9(
ztth
where the suffix
in':icates the value for infinite as'ect ratio at the same Reynolds number. In terms of CL.
(9.17)
c)
In fipure 11, the variation of with aspect ratio is shown for c1O,Coa5 and cQrnpared with some measurements of' the momentum thicknets at different aspect ratios made by Allan and Outland (1953). The agreement is moderately good if
V
is put equal to 0.040 , corresronding to a value of .3/_
=Q08
. This same value of
.B/
is
consistent with measurements of total drag made by Hughes (1952) at Reynolds numbers near lO7 (figure 12). Since
.3
is less than 3' in the ratio4/f
, and ..B' was expected to be rather leos than one, this value of ..3=QO8..-OOis not inconsistent with the physical ideas underlying the theory.REFERENC ES
Allan, J.F. & Jutland, N.S. (1953) Meeting of N.E. Joast Inst. Engineers and Shipbuilders, p. 245.
i3atchelor, O.K. (1950) J. Aero. Sci. 17, 441. Hughes, G. (19c2) Trans. I.N.A., . 287.
Lauf'er, J. (1953 Nat. Adv. Otee. Aero. Tech. Note 2954. I:;dwieg, H. & Tilimann, W. (1949) Ingenieur-Archiv 17, 288.
Schubauer, G.R. & Klehanoff, P.S.
(1950)
Nat. Adv. Otee. Aero. Tech. Report 1030. Taylor. 0.1. (193e) Proc. Roy. Soc. A, i6t. 476.Townsend, A.A. (1Q49) Froc. Roy. Soc. A, 197, 124. Townsend. A.A. (1750) Phil. Mag. 41, RQO.
Y O u, D o O C 0 0 o o o
00 O
0 O 0 0 0 00 00 0 00 0
0 0 0 0 0 x 0 00 0000 0
00 0000
OoO00O
4FIGURE 1 SECTIONS OF WAKE FLOW. (CIRCLES
INDI-CATE TURBULENT FLUID, ARROWS MOTION OF BOUNDARY)
FIGURE 2. TURBULENT INTENSITIES AND SCALES
WflHIN THE TURBULENT FLUID
B
-14-u.-u
FIGURE 3. SHEAR CO-EFFCIENT IN A WAKE [---soo 135O]
BOUNDARY OF TURBULENT FLU ID y N N Id ¶0 cs O 15
FIGURE 4. DIAGRAMMATIC PRESENTATION OF LARGE EDDIES
u&)+)
0,5
o
u,
FIGURE 5. ATTACHED EDDY
\
N.
N+
CORRELATION FOR ATTHED EDDIES (COMPUTE D)'
- CORRELATDIN rcATTACHED EDDIES (COMPUTED).
-O.2o i
FIGURE 6. CORRELATIONS IN CONSTANT STRESS LAYER LINE OF CONTACT Cfld SHEAR STRESS O
FIGURE 7. STRESS DISTRIBUTIONS FOR SELF-PRESERVING VELOCITY
PROFILES R- O /SCHOENHERR LINE 2-10' 5-10' iû 2-1O R
°
108 2-10FIGURE 6. PROBABLE VARIATION OF DRAG Cs-EFFICIENT WITH REYNOLDS
(EQUATION 8-i) 5-1 NUMBER s +5 DI
FIGURE 9. SECONDARY FLOW IN BOUNDARY LAYER -16--to 0.9 0,8 0,7 3.4 3,3 3.2 3,1 Cf'1 ao 2,9 2,8 0.5 1,0 1,5
--- (-o,00)
FIGURE 11. VARIATON 0F MOMENTUM THICKNESS (R-4.1
RUNS 13-17 + RUNS 8-12 EQUATION (9.16) I t
+
O 1 2(a-
0,042)FKE Q. VARATN OF DRAG CO-EFFICIENT (WGHES 1952)
O-O 05
FkURE 10. DISTRIBUTION 0F LDI'4GTUDINALE AND TRANSVERSE STRESSES
