A model for turbulent exchange in boundary layers
Citation for published version (APA):
Beljaars, A. C. M. (1979). A model for turbulent exchange in boundary layers. Technische Hogeschool
Eindhoven. https://doi.org/10.6100/IR99745
DOI:
10.6100/IR99745
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Published: 01/01/1979
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A MODEL FOR TURBULENT EXCHANGE IN
BOUNDAR Y LA YERS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. DR. P. VAN DER LEEDEN, VOOR EEN COMMISSIE AANGEWEZEN. DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP
DINSDAG 1.3 MAART 1979 TE 16.00 UUR.
DOOR
ANTONIUS·CORNELIUS MARIA BELJAARS
GEBOREN TE LAGEZWALUWEDit proefschrift is goedgekeurd door de promotoren:
prof.dr. D.A.de Vries prof.dr. K. Krishna Prasad
CONTENTS ABSTRACT I. 1.1 1. 2 1.3 1.4 2 2.1 2.2 3 3. I 3.2 3.3 3.4 3.5 4 4.1 4.2 4.3 4.3.1 4.3.2 5 INTRODUCTION
The prediction model approach to turbulence Recent developments in turbulence research
Linear theories related to the structure of turbulence The present investigation
A PHYSICAL MODEL FOR THE TURBULENT BOUNDARY LAYER Some experimental observations from literature Description of the model
ANALYSIS OF THE MODEL The basic equations
Order of magnitude estimations The wall layer
The outer layer
Inner-outer layer interaction
RESULTS AND DISCUSSION
The two-dimensional wall layer
A three-dimensional model for the wall layer The outer layer
Choice of parameters
Results obtained with the discrete vortex approximation
CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH
APPENDIX A THE APPLICATION OF LANDAllL's THEORY A. I Derivation of the equations
A.2 The solution method A.3 Results and conclusions
3 3 7 9 16 17 17 25 31 31 36 44 46 50 55 55 63 73 74 77 89 93 94 97 107
APPENDIX B EXPERIMENTAL RESULTS IN THE WALL LAYER 110
APPENDIX C NUMERICAL METHOD FOR THE WALL LAYER 113
LIST OF SYMBOLS 119
REPERENCES 124
NAWOORD 132
LEVENSLOOP 132
ABSTRACT
This thesis deals with turbulent structures in boundary layers along smooth walls under zero pressure gradient. The purpose of the present investigation is to develop a theory that incorporates characteristic features of the turbulent structure as revealed by recent observations. The principal characteristic features are the occurrence of
periodicity and intermittency, not only near the edge of the boundary layer, but also close to the wall.
In descrihing the turbulence, distinction is made between large-scale fluctuations of long duration and small-scale events of short
duration. The large-scale part of the turbulent fluctuations is described bymeans ofa modelbasedon approximations to the equations of motion and on empirical information •. In these approximations, two regions are distinguished : (i) a thin layer near the wall where viscous forces are important, and (ii) the remaining part of the boundary layer where inviscid equations are valid. In the equations for the large-scale motion, terros appear that represent the turbulent stresses, due to the smali-scale motion. On the basis of experimental data, it is assumed that the small-scale shear stress is limited to narrow regions where a very effective exchange of matter and momenturn takes place. These are the so called butst regions, which arise from local instahilities of the wall layer.
By transforming to a coordinate system that moves downstream with these burst· events, we obtain a stationary picture, in which laminar layers develop between two successive bursts or instability zones. In these zones, the retarded wall layer fluid is ejected into the outer region and replaced by high-momenturn fluid originating from the border of the wall layer, In this way, relatively long viscous periods
alternate with short periods in·which substantial turbulent exchange takes place. This picture shows some resemblance with the "surface renewal" concept of Einsteiri and Li. On the basis of assumptions about the velocity profile in the wall layer just after the burst event, the development of the laminar layer between successive bursts is
wel! with empirica! values. In order to obtain an unstable profile at the end of fue wall layer cycle, which is in general characterized by an inflection point, it turns out to be necessary to in.troduce counter-rotating vortices with axes in tbe streamwise direction. They also result in a periodicity in the spanwise direction which is in agreement with observations.
The outer region of the turbulent boundary layer is described on the basis of two-dimensional inviscid equations. The evolution in time of a structure, that is specified by an initial distribution of vorticity, is calculated with the discrete vortex approximation. It
turns out that the large-scale structures in the outer regi~n only transport momentum when they are fed by ejected fluid from the wall layer. This occurs in the small regions where the bursts play a role. To investigate the influence of these burst regions, a seperate model bas been developed, describing their interaction with the surrounding flow. Another mechanism that can play an important role in the outer region is the amalgamation of structures (vortex pairing). Calculations using the discrete vortex approximation clearly show that s~eh a
mechanism can exist.
It can be concluded that the momentum transport takes place ~n three stages with different transport mechanisms. First of all the fluid in the wall layer is retarded and collected in long narrow regions. The viscous forces play an important role bere. After this, a rapid exchange with the outer layer takes place. Low-momentum fluid is ejected from local instability zones and replaced by high-momentum fluid. Finally, the large-scale structures in the outer region take over the transport. It is shown that vortex pairing can plày a role in this region, both in contributing to the transport of momentum and in providing a mechanism for boundary layer growth by entrainment.
CHAPTER
INTRODUCTION
1.1. The prediation modeZ. approach
to
tuJ:>bul.enaeTurbulence is a phenomenon that arises in many applications. Flow fields occurring in practice are mostly turbulent. An important feature of turbulence is that it greatly promotea the transport of momentum, mass and heat. This is of interest in various types of turbulent flow, e.g. atmospheric flow, flow of water in oceans, rivers and lakes, flow around bodies like aerofoils and ship bodies and transport of heat in heat exchangers, ovens, etc.
Although turbulent flow is very common, knowledge about the detailed mechanism of turbulent exchange is incomplete. It is generally
accepted that the time dependent Navier-Stokes equations describe turbulence, but up to now it has not been possible to find "turbulent solutions". The problems arising in solving the time dependent equations are related to the essential three-dimensional character of turbulence~ the nonlinearity of the equations and the fact that a hierarchy of scales plays an important role
Many theories have been developed to predict mean quantities, like mean velocity profiles and wall shear stresses in turbulent flow. In most of these, turbulence is treated as a purely stochästic phenomenon that has to be described in a statistical manner in terms of mean quantities and moments of the probability distribution functions. The traditional theoretical description of turbulence starts from the time dependent Navier-Stokes equations and the continuity equation for incompressible flow with constant fluid properties (Hinze, 1975)
()y
I +·VV2u
- + y.Vu -Vp p l.I. I
a:t
In these equations the velocity components and the pressure are decomposed in a mean part and a fluctuating part, e.g.
u '" u + u' l.I. 3
Inserting (1.1.3) in the equations (1.1.1) and (1.1.2), we optain, after averaging. a set of equations for the mean quantities. These are the well-known Reynolds equations.
For stationary flows, an appropriate and convenient type of averaging is time-averaging. The mea~ quantities do not change in time and their time derivatives therefore cancel in the equations. Time-averages can also be used in slowly evolving flows if the time scale of the mean flow is much larger than the largest turbulence time scale. This condition is difficult to satisfy in the atmospheric boundary layer where very low frequencies .occur in the spectrum of fluctuations.
The Reynolds equations for the mean velocity components, contain correlatio~s between velocity fluctuations that have the character of additional stresses. In literature the corresponding terms are called the Reynolds stresses. They are additional unknowns, resulting in fewer equations than unknowns. To obtain a closed set of equations, assumptions have to be made about the turbulent stress terms in the Reynolds equations. This so-called "closure prob lem" is one of the main topics in turbulence theory.
The simplest closure assumption expresses the turbulent stress as the product of the mean rate of strain and a turbulent exchange coefficient in analogy with the constitutive re lation for molecular exchange
processes. The turbulent or eddy viscosity, however. is not a property of the fluid, but of the flow (cf. Tennekes and Lumley, 1974).
More complex closure assumptions have also been developed, especially after high speed computer facilities became available, which allowed the evaluation of the consequences of these complex models. These are the so called "second order closure models", based on di:fterential
dependent Navier~Stokes equations. New unknown quantities, such as pressure strain correlations and triple velocity fluctuations, appear in these equations and for these closure assumptions have to be made as well. A number of second order models were presented in a lecture series on "prediction methods for turbulent flows" held at the Von Karmin Institute for Fluid Dynamics (Rhode-Saint-Genèse, Belgium,
1975).
The usual philosophy behind modelling of unknown terms is as follows. First, one examines the physical meaning of each term in the
equations. After that the unknown terms are replaced by expresslons involving known quantities in such a way that these expresslons have more or less the same physical effect as the original terms. This is the most critical part of the procedure. Ideas about the structure of the turbulent moëion and dimensional arguments are commonly used. Finally, empirical dimensionless factors are introduced, which have to be determined on the basis of experimental data. The procedure can be explained more easily with the aid of an example. The derivation of the mixing length model will be used by way of illustration.
Many practical floWs develop rather slowly in the streamwise
direction (the x-coordinate). This means that the mean rate of strain can be approximated by the derivative of the main flow component u with respect to the transverse coordinate y only. In the equation for
the streamwise veloei ty component Ü, the cross correlation
"ü"V'
appears as an unknown. The physical meaning of this term is that it represents the momentum exchange by turbulent motion between points with different mean velocity. In analogy with the molecular exchange
of momentum in gases, the turbulent stress is assumed to be proportional to the gradient of mean velocity. The exchange
coefficient involved has the dimensions of m2/s, which may beseen as the product of a characteristic length and a velocity scale. The scales have to be characteristic for the turbulent motion that is responsible for the momentum exchange. For boundary layers the
- 1
-friction velocity u* ~ (T /p)2,(where T is the mean wall shear stress
w w
wall are the appropriate velocity and length scales. This results in the following closure assumption
u'v' "' Kyu
*
aü
()y 1. 1. 4
The factor K is the empirica! dimensionless constant, known as the
Von Kárm!n constant in literature. This closure assumption works reasonably well for a certain range of distances from the wall
(cf. Cebeci and Smi th. 1 974).
As this example shows, modelling is far from straightforward; a certain amount of intuition is involved. The ultimate justification for the closure hypothesis, can only be given by comparing the numerical results with experimental data. So this type of modeHing is intermediate between repreaenting the physics of the problem and curve fitting.
Simple closure assumptions • for e.g. the mixing length hypothesis, have the disadvantage that they. are only vàlid for rather simple flow configurations and that the dimensionless factors have to be adapted when we pass from one flow type to the other. In the search for a more universa! model, that can handle more complex flow prob lems • second order mode ls have been deve loped. Complex mode ls contain more adjustable parameters than simple.· models, but fewer adjustments are needed when they are applied ·to different flow types. Up to now it has not been possible to construct a universa! model that is valid for all flow types. In one way or the other one has not been able to represent the physical meaning of the different terms with a sufficient degree of fidelity. This is not surprising given the complexity of turbulent flow and in view of the fact that many of the modelled terms cannot be evaluated experimentally.
The limited success of predietien models clearly shows the need for a better understanding of turbulent exchange processes. The remarkable success of the simple mixing length hypothesis also asks for an
explanation. To arrive at a better understanding of the mechanism of turbulence, several authors have considered the details of time
dependent motion instead of the mean quantities only. In this type of research extensive use bas been made of visualization techniques such as smoke in air flow and dye and hydrogen bubbles in water flow.
Same remarkable features have been observed with these techniques and same new ideas were put forward. A brief discussion of this development will be given in the next section. The corresponding
theories, dealing with the time dependent motion as well as with the mean motion are discussed insection 1.3. Insection 1.4. the
approach followed in the present work will be explained.
1,2. Reaent deveZopments in turbuZenae researah
Recent observations in free shear layers and boundary layers have resulted in some new viewpoints. These observations were not done by means of complex measuring techniques' but simply by taking moving pictures of the flow patterns, made visible by smoke, dye or hydrogen bubbles. The most important discovery was that simple turbulent flows like mixing layers, jets ànd boundary layers, are much less chaotic than was commonly believed, On occasion quasi-periodicity is observed, which means that recognizable flow patterns repeat themselves at fluctuating time intervals. Flow patterns that can be recognized and followed during their evolution in time are often called "structures" in literature. The length scale and decay time of ,such structures are of course stochastic variables. Although the same recognizable ,structures appear periodically, almost no information about them can
be derived from correlation measurements. Averaging over long time combines stiuctures with different scales which blurs the information on structures.
The corrugations in the distinct boundary between turbulent and non turbulent fluid at the edge of the turbulent region in free or wall bounded shear layers, are a manifestation of large-scale structures.
An important question related to this phenomenon can be formulated as f.ollows : Which process is, responsible for thè mass transport through the interface or in other words, which process governs the entrainment of turbulence-free fluid? Phillips (1972) has shown that the entrainment rate in boundary layers is one order larger than can
be axpected on the basis of viscous diffusion. The deformation of the interface by the turbulent motion enlarges the effective surface and so intensifies the entrainment process. According to Townsend (1976) the large eddies determine the entrainment rate. Visualization of jets and mixing layers bas given some more insight into the detailed entrainment mechanism (cf. for example Brown and Roshko, 1971).
A two-dimensional mixing layer is formed when fluid layers moving parallelly with different uniform velocity join.
The mixing layer starts with Helmholtz waves in the vertex sheet as can be expected from linear stability analysis. Very soon nonlinear effects become important, due to which the vertex sheet rolls up and the vorticity, that was uniformly distributed along the vertex sheet in the initial stage of development, is now concentrated in vortex-a tructures. Further downstrvortex-avortex-am the scvortex-ale increvortex-ases due to "vertex· pairing". The original vortex-structures start tuming aroun:d each ether and merge into ene new vortex-structure. The successive pairinga govem the inc.rease of the shear layer thickness. This process, which is easiest to cbserve at lew Reynolds numbers, has also been observed at high Reynolds numbers (cf. Brown and Roshko, 1971; Dimetakis and Brown, 1976). The entrainment of turbulence-free fluid takes place during the pairing process. The fluid is rolled in between two vertices and becomes turbulent by small-scale mixing and vis:cous diffusion. Browand and Weidman (1976) also conclude this on the basis of anemometer measurements, analyzed by application of a conditional sampling technique (in this technique, the averaging is done over a number of events, which are decided to have occurred when a certain condition is satisfied). The important conclusion, obtained by these simple observations, is that the entrainment in jets and mixing layers is govemed by the "vertex pairing process" even at high Reynolds numbers.
In boundary layers the situation is more complex; the mean velocity profile does not show an inflection point and is not unstable in the inviscid mode. Therefore the structures with vorticity in the spanwise direction are expected to be less dominant. Still, the same type of "vertex structures" are observed in the l:oundary la~er. Such vertices,
vorticity in the spanwise direction, have length scales of the order of the boundary layer thickness and are separated from the turbulence-free fluid by a wavy interface. The boundary layer differs from the mixing layer in that the "vortex-structures11 in the former are more
elongated in the streamwise direction and are less pronounced. Vortex pairing has never been observed in boundary layers and it is not clear whether the entrainment rate is determined by the dynamics of these "vortex-structures" or by less organized large-scale eddies. The complexity of the boundary layer is related to the interaction between the thin wall region with small-scale exchange and the outer
layer where large-scale structures are important.
Visualization of the wall region of a turbulent boundary layer has also revealed the existence of periodic phenomena. Long quiescent periods alternate with short periods, in which the fluctuations are very violent indeed. The turbulence therefore has an intermittent character. The short violent periods, wich are often called ''bursts", turn out to give the major contribution to the Reynolds shear stress. Since almost all the momentum exchange takes place during these bursts, it is important to consider them in a more detailed manner. The model presented in this thesis is based on the observed
periodicity and intermittency. A detailed description of these features will be given in chapter 2.
1.
3.Linear theories related to the structu:t'(!l of turbulence
The existence of quasi-periodic structures suggests an alternative theoretica! approach to the classical one introduced by Reynolds. Time-averaged equations do not contain any information about the details of the structure of turbulence. The effects of the structures are represented only in an overall manner in the Reynolds shear stress. A number of attempts have been made to develop models for the time dependent structures, making use of the time dependent Navier-Stokes equations. A short survey will be given here.
For free shear layers the so called discrete vortex approximation has been very successful in calculating the time dependent velocity field with the aid of digital computers. Clements {1977) gives a
survey of results obtained with this method. We shall limit the discussion here to two-dimensional mixing layers, treated by Acton
(1976). The thin vortex sheet with wavy disturbances is represented by a large number of elementary line vortices. The time evolution of the vortex positions is calculated step by step by evaluating the induced velocity at each vortex center by all others. When the velocity at a vortex center is known, its new position one time step later can be calculated. The rolling-up of the vortex sheet and the vortex pairing are clearly demonstrated by these.calculations. From this it was concluded that the two-dimensional vortex structures are dominant and that mixing layer growth is governed by the pairing mechanism. This strongly supports the conclusions of the visual observations (cf. Brown and Roshko, 1971).
For boundary layers several theories have been developed to describe the behaviour of the velocity fluctuations. Most of the theories deal with the wall layer, which is defined as the layer where viseaus stresses play a role. This includes the viseaus subl~yer and the buffer-region according to the usual nomenclature. Near the wall an important simplification can be made : the dependenee of the mean velocity on the streamwise coordinate can be neglected, which means that the mean streamlines are parallel to the wall.
One of the oldest theories for the wall region is the so called "surface renewal" model, independently developed by Einsteitt and Li (1956) and Hanratty (1956). This model is based on the idea that the wall layer is replaced periodically by fluid originating from the border of the buffer region ( y+ ~ 60, where y+ is the dimensionless distance from the wall yu*/v). By viseaus effects the wall layer will slow down until the next renewal event occurs. The fast exchange phenomenon accupies only a small portion of the complete period. The model has in fact the observed features of intermittency and
periodicity. The viscous evolution of the wall layer is calculated with a drastically simplified x-momentum equation.
The boundary and initial conditions are u (0, t) u(y,O) 0 = u = constant 0 1.3.2
Equation (1.3.1) is valid if the extent of the wall region in the x- and z-direction is sufficiently large, such that the diffusion and conveetien in these directions can be neglected. The pressure gradient imposed by the outer region on the wall layer must also be negligible. The solution of (1.3.1) is
u(y,t) =u erf [y/2(Vt)i]
0 I. 3.3
By averaging over one period the mean velocity profile and the mean wall shear stress can be calculated, For a certain distinct period
the results· agree very well wi th experimental data. This period is very close to the burst interval as publisbed in several later papers
(cf. Laufer and Badri Narayanan, 1971).
An attractive feature of the surface renewal model is that it is closely related with the observed periodicity and intermittency. Although it is still one of the best models for the wall regi~n, some weaknesses should be mentioned :
a) It is not realistic to assume, as the model does, that the entire wall layer is removed instantaneously, because there must be some distribution in a horizontal plane of "viscous" and "renewal" areas.
b) The diffusion equation can only give a crude approximation when the x- and z-scales are of limited extent.
c) The assumed initial profile leads to a singularity in the wall shear stress.
d) The turbulence intens i ty, which can be -calculated by comparing the time dependent solution (-1.3.3) with the mean velocity, rises too fast with increasing y for very small y-values and then drops too early.
Point d) should not be taken too rigidly, since the turbulence intensity has been calculated without accounting for the turbulence intensity during the burst events.
To arrive at a traetabie mathematicàl problem for the time dependent motion, a number of authors have used linear equations for the turbulence fluctuations. This is based on the assumption that the turbulence intensity is small and that the quadrätic terms can be neglected. Although this assumption is questionable in real turbulence
it is still interesting to determine the effects that can be explained by such a linear theory.
Different approximations of the linear equations .have been suggested. Sternberg (1962) used the following set of equations
--·-
1.3.4
a
tV.u'
=
0 I. 3.5The viscous diffusion in the x- and z-direction has been neglected as well as the convective terms. By this the theory is only expected to give reasonable results in a very thin layer close to the wall since here.the mean velocity is small. Shubert and Corcos (1967) used a boundary layer approximation of the linearized equations.
au.' + - au
'
,aü _ _!_!i_+ a2. 'u - - + v - V _ _ u_ at a x ay p ax a Y2.
I. 3.6
0
- l2.L
P ay I. 3. 7
aw•
- aw•
1 ap' . a2w'- - + u - - - - - + v - - 1.3.8
at a x p az a Y2.
au•
- - + ' + - -aw• = 0 I. 3.9 a x Cly az
Both sets of equations are subjected to the zero slip condition at the wall and a boundary condition outside the wall layer, which is the driving quantity for the fluctuations in the wall layer. Sternberg imposes u- and w- fluctuations at the top of the wall layer while
in which the solution is evaluated does not play an active role in the turbulence production; only the reaction of the wall layer is calculated on a turbulencefield above the wall layer, which is assumed to be known. The results of these theories can be summariZed as follows :
a) The calculated spectrum for u' (with a given spectrum for the turbulence field outside the wall layer) shows qualitative agreement with experimental data.
b) The point y+ ~ 20, where the turbulence intensity attains its maximum, is well predicted by the theory.
c) The eddies are strongly elongated in the x-direction.
d) The u'- and v'-fluctuations have the right phase relations to· cause a positive Reynolds shear stress.
The weaknesses of these theories are :
a) The wall layer plays a passive role in the theory, while it is known from experiments that bursts, essentially originating in the wall layer, are very important in the turbulence production process.
b) The magnitude of v' as predicted by the theory is one or two orders too low.
c) The increase of u' as a function of y is too slow.
d) The turbulence intensity is not sufficiently small to justify a linear analysis.
The points b) and c) are consistent with point a): an important souree of turbulent fluctuations; namely the burst events0 bas been
ignored in the theories of Sternberg and Shubert & Corcos.
The facts that bursts play an active role in the wall layer and that bursts are sma11-.~>cale effects0 led Landahl ( 1965, 1967) to suggest
a two-scale formulation for the turbulent fluctuations. Instead of distinguishing mean quantities and fluctuations, Landahl uses a decomposition in three parts : the mean quantity ~ , the large-scale fluctuation ~· and the small-scale fluctuation u". It is assumed that all large-scale quantities have small amplitudes. So nonlinear terms
in~· can be neglected. The effect of burst events, which are accompanied by large amplitudes. is reflected in the small-scale quantities.
Nonlinear terms in small-scale quantities are not neglected. The resulting equations for the large-scale fluctuations are
a~· Cl~' . (l;i
--- --- + u - +v'
-e: llp ' + \) 112~ '-
17.C!:!
"!:!") 1.3. 10 élt Cl x ély -x p 17 ~. •!:! 0 I. 3. 1l where e: -x (I ,0 ,0)So the small-scale stress terms are the driving forces for the large-scale fluctuations. Landahl solved the equations with homogeneous boundary condtions for y
=
0 and infinity, by applyingFourier-transforms in the x- and z-directions. The equations lead to t~e
well-known Orr-Sommerfeld equation, in this case with an inhomogeneous term coming from the small-scale stress term (burst events). The propagation properties of the large-scale fluctuations (or waves in the wave vector domain) are determined by the eigenvalues of the Orr-Sommerfeld equation. It is assumed that the first eigenvalue, corresponding to the weakest damped mode will be dominant. The waves with this eigenvalue will keep their identity over a long time, while
the others will decay much faster. With this approach Landahl (1965, 1967) and Bark (1974, 1975) derived the following results a) The calculated decay in the streamwise direction of pressure
fluctuations agrees very well with experiments.
b) The conveetien velocity (the real part of the first eigenvalue of the Orr-Sommerfeld equation) predicted by the theory is 30% too low.
c) The wave number at which a maximum is found in the u'-spectrum is very close to the experimental value.
d) The structures are strongly elongated in the x-direction. e) The ratiosof the turbulence intensities in the three directions
are of the right order.
In a recent paper Landahl (1975) uses the theory developed earlier, to explain the periodic occurrence of bursts. It is assumed that a burst causes a disturbance in the large-scale velocity field and
conditions for a local instability. It is known (cf. Blackwelder and Kaplan, 1972) that bursts are preceded by a locally inflectional profile, which leads often to instability as is known from elementary stability theory. The question is whether the time dependent
development of the large-scale field will be such that an inflectional profile is formed somewhere downstream. Landahl gives qualitative arguments to show that equations (1.3.10) and (1.3.11), solved .as an
--initial value problem with !:!":!:!" = 0 and the velocity field just after the occurrence of a burst as initial condition, do indeed predict unstable conditions downstream. The numerical results of Landahl and Bark are all given in the wave-vector domain and obtained by
representing the effect of bursts in a statistica! manner. The averaging involved here, implies that no information can be derived any more about the individual events. Transformation to the space domain of these numerical results only leads to correlation functions which are difficult to interpret.
A certain amount of computational work has been carried out by the present author (Appendix A) to verify the qualitative arguments of Landahl. The numerical results are not conclusive and lead to doubts about the way the problem has been posed. This has to do with the assumption that the small-scale souree terms can be neglected after the occurrence of a burst. This may be locally true, but is certainly not right for the whole x, z-plane. The burst regions also move down-stream with a certain velocity. With the actual approach (cf. Appendix ·A) the velocity defect in the initial condition increases first and
disappears afterwards, which is inconsistent with the observations.
Although Landahl's theory explains certain features reasonably well, a number of questions remain unanswered. Some of them are :
a) Which mechanism is responsible for forming locally unstable condi ti ons?
b) What are the sealing parameters for the burst interval? c) How does interaction between inner and outer layer take place?
1.4. The present inveatigation
As already mentioned, it is believed that the observed structures are very important for the transport processes. The existing theories and models that deal with 11structures" in boundary layers have a very
limited validity and most of them do not deal with the transport process.
In this thesis a three-dimensional, nonlinear model will be formulated that is based on the observed "structures". In chapter 2, the model will be described in conneetion with the relevant experimental data. The concepts of periodicity and intermittency are particularly important in the model. The purpose is to find out whether the transport of momenturn can be explained by this simple deterministic model. Chapter 3 contains the analysis of the equations in a two-dimensional approximation. Approximations and salution methods are given for different regions in the boundary layer. In chapter 4 the numerical results are presented and discussed. It will be shown that the model leads to a quantitative explanation of the wall shear stress However it is shown that for regeneratien of burst events, consideration of three-dimensional aspects is necessary. Although the model can be easily extended to heat and mass transport, the discussion in this thesis will be limited to the transport of momentum. It is believed that the present model leads to an improved insight into the
CHAPTER 2
A PHYSICAL MODEL FOR THE TURBULENT BOUNDARY LAYER
In the first section of this chapter, we shall present some
experimental results that are relevant for the understanding of the behaviour of the "turbulence structures". Based on these results, a simplified picture of the turbulent boundary layer will be presented in section 2.2. It will be shown in chapter 3 that this picture leads to a traetabie mathematica! problem.
2.1. Sane e::cperimental observations from litemture
Kline et al. (1967) and Corino and Brodkey (1969) carried out visualization studies on the wall region of a boundary layer and pipe flow respectively. Both studies emphasize the intermittent and quasi-periodic nature of the turbulence. Short periods with intense turbulence alternate with long relatively quiescent periods. The flow in the wall region is called quasi-periodic because recognizable flow patterns present themselves repeatedly with time intervals that vary in a stochastic menner. The short periods with intense turbulence (often called burst events) are only present during a small fraction of the total time. However, they provide the major contribution to the Reynolds shear stress and therefore are very important for the transport process.
y
~·
zr/7777777777777777777777 7777,
According to Kline et al. (1967) the following stages can be distinguished in one cycle :
a) An elongated narrow layer slows down. This is the so called "lew-speed streak", which has dimensions of about the boundary layer thickness ê in the x-direction and about 20 v/u* in the z-direction. (cf. Fig. 2.1 for the coordinate system).
b) The thickness of this retarded layer increases. The deceleration of a layer with increasing thickness, combined with the
acceleration of the fluid above this layer leads to a horizontal shear layer with a steep velocity gradient. The local profile shows an inflection point which is known from elementary stability analysis to be very sensitive for instabilities.
c) The wall region becomes unstable : small-scale fluctuations build up quickly to large amplitudes, which results in the ejectiÓn of low-momentum fluid out of the retarded layer into the outer region. d) The ejection is followed by a replacement of ejected fluid by
high-momentum fluid. This is often called the "sweep event".
The intermittent character of the flow is caused by the stages c and d, which occupy only a small fraction of the total time taken by one cycle. During the ejection c and the sweep d, the Reynolds stress can attain 10 - 20 times its mean value. During the ejection u' is
negative and v' positive; during the sweep u' is positive and v' is negative. In Fig. 2.2 an illustration is given of the local velocity profiles befere and after the burst, which occurs at about t+
=
0. The burst event includes the ejection as well as the sweep: it standsfor the short violent activity. The figure clearly shows the inflectional profile befere the occurrence of a burst at about
t + = - 3.
The elongated lew-speed regions of stage (a) show up periodically in the z-direction with a wavelengthof about 100 v/u*. This number turns out to be independent of the Reynolds number of the flow (based on u00 and the momenturn loss thickness). The argument often 'used to explain the z-periodicity is that longitudinal
counter-rotating vertices exist. These vertices, which have never been observed directly (only their effect on the streamwise component has
been observed), have a diameter of about 50v/u* and so are really part of the wall layer.
100
y+y+
so
0
Fig. 2.2 Instantaneous velocity profile (---), obtained by means of conditional averaging, compared with the mean velocity profile (---), (Blackwelder and Kaplan, 1972).
The fact that bursts are preeerled by an inflectional profile, suggests that a burst has to be seen as a local instability. Therefore,one would expect the mean burst interval to scale on the wall parameters u* and v. Most experiments, however, point in the direction of a
universa! sealing on u~ and ó. In a review of experimental data, obtained by different methods, Laufer and Badri Narayanan (1971) state that the dimensionless mean period Tbu
00
/Ó
equals 5 in the walllayer and 2,5 in the outer region.
The sealing property of the mean time interval between two successive bursts, suggests that there is a synchronization mechanism between outer region and wall region. To obtain more insight into this mechanism some experiments will be discussed that deal with the behaviour of the flow near the burst events. Corino and Brodkey (1969)
discovered that the instability in the wall region is preceded by acceleration of the fluid-mass upstream of the lew-speed streak. This is the so called "sweep event". Later observations by Nychas, Hersbey and Brodkey (1973) have shown that the scale of the sweeps is much larger than that of the low-speed streaks. This suggests that the sweep event is part of the large-scale motion in the outer layer. The high-momenturn fluid does not interact directly with the low-speed streaks but moves partly over them. Corino and Brodkey sometimes observed a "two layer effect", which means that t.he high-momenturn fluid overtakes the streak at a spanwise position. After this the low-momentum fluid of the wall region is ejected into the outer region in the form of small-scale jets that interact violently with the surrounding fluid. The fact that high-momenturn fluid is observed above and next to the lew-speed streak, before Elieetion occurs, ~means
that the streak acts as a wall-attached obstruction for the high-momenturn fluid. This effect is also seen in experimental results of Blackwelder and Kaplan (1972, 1976) and those obtained at the Eindhoven Univarsity of Technology (see Appendix B). In both experiments the velocity in the streamwise direction bas been measured near the burst event by means of a conditionat averaging technique. Blackwelder and Kaplan (1972) found that the acceleration is first observed at large y+- values (cf. Fig. 2.2}. In Appendix B it is shown that the acceleration occurs earlier at z+ • 60 than at z+ = 0 where the burst event is located.
Also in the outer region certain "structures" can be recognized. The wavy interface between turbulent and non-turbulent fluid at the edge of the boundary layer are a manifestation of these structures. According to Blackwelder and Kovasznay ( 1972) large-scale "vortex structures" are responsible for these waves. These vortices have their axes in the spanwise direction. Correlation measurements have shown that the vortices keep their identity over large distances downstraam (about 206). Laufer (1972) suggests that vortices as shown in Fig. 2.3 roll over the wall and induce instabilities in the wall region. At the interface the entrainment takes place by small-scale mixing. However, Phillips (1972) and Townsend (1970) attribute entrainment to the engulfment of turbulence-free fluid by the large-scale turbulent
Fig. 2.3 Conceptual picture of the outer region of a turbulent boundary layer after Laufer (1972)
Kovasznay.Kibens and Blackwelder (1970) studied the outer region by means of a conditional averaging technique. The passage of fronts and backs of the bulges in the turbulent, non-turbulent interface were detected, and these detections were used as reference for taking samples of the velocity components at different positions relative to the bulges. The velocity field inside the bulge measured in this way shows that the v-component inside a bulge can be split in a symmetrie and an antisymmetrie part (the axis of symmetry is a line in the y-direction through the top of the bulge). The symmetrie part ·has a short life-time and is responsible for the outward movement of the bulge, while the antisymmetrie part has a long life-time and is responsible for the slow rotation inside the bulge. Kovasznayet al. interpret this experimental result as follows. A new bulge is formed by a violent outward moving fluid parcel that derives its momentum and vorticity from its point of origin. The fluid parcel grows, loses momentum and distributes the vorticity over the layer it travels
through. S'o in a new bulge the outward motion disappears quickly while the rotation stays. The bulges are very passive, and the slow rotation does not contribute to the Reynolds shear stress, while the violent outward motion is very important as a mechanism of momentum transport.
is related to the wall ejections, but a comparison with the observations of Kline et al. ( 1967), showing that wall ejections travel through the entire outer region, seems to suggest such a conclusion.
Falco (1974, 1977) visualized boundary layers by filling them.with smoke and observed what .he defined as "typical eddies". These eddies move like discrete vortex rings through the outer tayer and cause corrugations at the backs of the large-scale bulges (cf. Fig. 2.4). The dimensions of the typical eddies scale on wall parameters and are about 100 v/u* to 200 v/u*' which is an order of magnitude smaller
than the boundary layer thickness at moderate Reynolds numbers. According to Falco, the typical eddies are formed in the boundary
u.
-Fig. 2.4. Faleo's observations in the turbulent boundary layer.
layer at different distances from the wall as the result of a local instability. The present author believes, however, that Faleo's experiments are open to another interpretation. The sealing behaviour of the typical eddies can only be explained if these eddies really originate in the wall layer. Unfortunately, this was not visible in Faleo's experiments as the technique employed made it possible to visualize only the regions with significant smoke gradients (which are really large only near the edge of the boundary layer). The present interpretation, taken together with the observations of Kline et al. (1967) mentioned earlier, leading to the conclusion that the ejected fluid travela through the entire boundary layer, suggests that the typical eddies are "grown out" ejections from the
Brown and Thomas (1977) did an experiment to obtain more insight into the interaction between wall region and outer region. They measured the time dependent wall shear stress in correlation with the velocity at different positions in the outer region. The overall picture that emerges from their experiments is shown in Fig. 2.5, where the form of the large-scale structures has been derived from Faleo's (1974) smoke pictures. These large-scale structures rotate slowly and bend
Q2U.,
-
-wal! shear stress
Fig. 2.5 Flow pattem proposed by Brown and Thomas (1977), as seen by an ob server moving at a speed of 0, 8 u· and the correspondirtg wall shear stress. m
over in the streamwise direction, making an angle of about 18° with the x-axis. Their dimension in the x-di reetion is about 2ó. The high frequency part as well as the low frequency part of the wall shear stress fluctuations are correlated with the large-scale motion in the outer layer, The violent fluctuations in the wall shear stress (high frequency), which are interpreted as burst events, are followed by a maximum in the wall shear stress (sweep). These experiments clearly show the phase relation between inner and outer region. Another important conclusion is that the picture is almost stationary when we· move downstream with a constant "convection veloei ty" of about 0,8 u00 (cf. Fig. 2.5)
Offen and Kline (1974, 1975) studied the wall layer of a t.urbulent boundary layer by combining a visualization technique and anemometer measurements. They presented the following ideas about the initiation of a burst event. When a low-speed streak has been formed, it is
passed by a transverse vortex (the diameter is about 50 v/u*) with the vorticityin the samedirection as that of the nlean flow. According to the authors this vortex induces an adverse pressure gradient beneath it and so.provokes separation of the low-speed streak in a convected frameof reference. This separation is associated with a new vortex beneath the lifted low-speed streak. The old and the new vortex merge to.form a slightly larger one that initiatea the next burst event further downstream. According to observations the low-speed streaks in the next cycle are shifted over
tÀ
in the z-directionz
with respect to the previous ones.
The suggestion by Offen and Kline that burst events can be seen as local separated regions, is different from the idea that bursts are local instabilities. It should be noted that backflow bas never been observed. Obviously, the word separation has not been used in the normal sense by Offen and Kline. The only conclusion that can be drawn, is that bursts are initiated by effects coming from outside the wall region.
The most obvious quantity that may be responsible for synchronizing the wall cycle with the periodicity in the outer region is the pressure, Willmarth (1975) reports an experiment in which the wall pressure has been measured as a function of time, before and after the occurrence of a burst. It turns out that the wall pressure decreases before the burst occurs, has a minimum at the burst event and increases afterwards. This means that a favourablepressure gradient exists before the burst occurs, which is opposite to the interpretat1on given by Offen and Kline.
Although many questions remain unanswered, some conclusions can be drawn from this literature survey :
(i) The burst events in the wall region occupy a small fraction of the time and provide the major contribution to the Reynolds shear stress.
(ii) The wall region shows periodicity in the x- and in the z-direction.
2. 2 Description of the modsZ
An idealized pbenomenological picture of tbe turbulent boundary layer is derived bere from tbe experimental features discussed in tbe previous section. Tbe main purpose is to make the processes amenable to a relatively simple mathematica! treatment wbile retaining as many as possible of the detailed aspects of the real turbulent motion.
The model represents the burst events by moving instability zones with spacing À x and À z in the x- and z-directions respectively (cf. Fig. 2.6).
Fig. 2.6 The moving instability zones in the wall layer
·The velocity of these instability zones is assentially the convection. velocity Uc ( ~ 0,8 ua) of tbe large-scale structures in tbe outer region. This model is based on the idea that a mecbanism exists that synchronizes the burst events with the periodic structures in the outer region. Since the z-scale of the structures in the outer region is much larger than À , one "outer region structure" wi 11 be related
z
to a number of spanwise periods in the wall region. Thus one structure in the outer region will move in pbase with a line of instability zones.
· The moving instability zones have small dimensions compared with the wavelength Àx' The model represents the intermittency as weli as the pe·riodi<:ity of these zones for an observer at rest in the laboratory
frame. The spatial periodicity in the moving plane of Fig. 2.6 is observed as a periodicity in time and the instability zones with
dimensions that are small as compared to Àx show up as an intermittency in the records obtained in a fixed observation point.
The instability zones represent the burst events of the visualization studies and have the effect of removing low-momentum fluid from the wall region and replacing it by high-momenturn fluid from the border of the wall region (y+ ~ 60). Substantial momenturn exchange takes place in the instability zone during a very short time.
Between two successive lines of instability zones the initially high momenturn fluid is retarded by viscous forces until a new instability zone removes the retarded fluid again. This means that viscous diffusion in the developing wall layer over one wavelength Àx alternates withstrong turbulent exchange in.the instability zones. It takes almost the complete period for viscous forces to build up a retarded layer while it takes only a very short time for the instability zones to remove this layer.
The periodicity in the z-direction is assumed to arise out of counter-rotating longitudinal vortices in the wall region between two lines of instability zones. They cause an accumulation of low-momentum fluid in streaks and give the instability zone a local character in the z-direction as well.
Since the instability zdnes move downstream with a constant convection velocity Uc1 the picture is stationary in a coordinate system that
moves downstream with this velocity. What exactly happens in the wall layer is easier to understand in the convected frame (Fig. 2.7).
We assume that atposition A along the x-axis, just after the passage of an instability zone, the velocity profile is known. Since a strong
smeU-scale exchange mechanism (the burst) supplied the wall layer with fluid from the edge of the buffer region, the velocity in the wall layer will be higher than the loc al mean velocity. A visèous boundary layer develops nowbetween
A
and B,' in which the momenturnFig. 2.7 Developing wall layer between two instability zones as seen by an observer in the moving coordinate system.
disturb the viscous boundary layer. At the z-locations where the secondary motion is outwards a streak with wall velocity will develop (corresponding toa low-speed streak in fixed coordinates); at z-locations where the longitudinal vorticity produces a wallward motion, the boundary layer development is suppressed (corresponding
to high-speed streaks in the laboratory frame). Almost all the low-momentum fluid retarded by viscous forces is swept together in a low-speed streak, on top of which an inflectional profile arises.
A very important feature emerging from experimental data is the univeraal sealing of the burst period on outer layer parameters. In the present model it is assumed that the vortex structures
(cf. Fig. 2.8) in the outer region initiate the burst events in the wall layer. This is supported by the experiments of Brown and Thomas (1977) whohave shown that the burst events run in phase with the structures in the outer region. In accordance with Nychas et al. (1973) it is believed that sweep events are part of these structures and that they initiate the burst events.
7
I/ I I I I I I 11 I I I I I I I I I I I I IJ I I I I 11 I I I I I I I I I I I I I I I I I 11Fig. 2.8 Structures in the outer region seen by an observer in the moving coordinate system.
The difference by a factor of 2 between the period in the
wall region and that in the outer region, can be explained on the basis of experimental results of Offen and Kline (1974, 1975).
They observed that the low-speed streaks in two successive cycles are shifted over
!À
with respect to each other in the z-direction.z
Fig. 2.9 The relative position of low-speed s~reaks (the edge is marked by a dashed line) in two successive cycles. The arrows 1 and 3 indicate the di reetion of rotation of the longitudinal vortices.
This is illustrated in Fig. 2.9. Because of this shift, an observer in the laboratory frame will alternately see a high-speed region and a low-speed region. After the build-up of a low-speed region an instability or burst will occur, which is only once in every two periods of the outer layer. This is believed to be the reason that experimental burst dateetion techniques indicate 2 Àx for the
wavelength in the wall region and À in the outer region. The observed
x
phase shift in the z-direction between successive cycles implies that the direction of rotation of the longitudinal vortices reverses near a line of instahilities (cf .. Fig. 2.9 and Fig. 2.6). The reason for
this is not yet clear; the interaction between low-speed streaks and the large-scale sweeps probably plays a crucial role bere. Although the processas are not completely understood, the conclusion for the present model is that the outer layer waveleng tb À x bas to be used instead of the experimentally found wall layer wavelength 2 Àx·
The structures in the outer region are assumed to be two-dimensional. This is based on the experimental observations that the spanwise "vortex structures" play an important role in the outer layer
(cf. Brown and Thomas, 1977; Falco, 1977 and Kovasznay et al., 1970). The burst events have a very local character and the resulting fluid ejection from the wall region is an important coupling mechanism between wall region and outer region. Since the burst events run in phase with the large-scale structures in the outer region, the
trajectory of the ejected fluid will have an almost fixed position · with respect to the large-scale structures, in which they will create·
new vorticity. Without synchronization between wall region and outer region, the vorticity created would be randomly distributed along the x-axis. Now the vorticity injected can accuroulate at certain
x-positions and so make the periodicity in the outer layer more pronounced.
In Fig. 2.10 a schematic representation of the complete boundary layer is shown. The figure clearly indicates the phase relation between inner and outer layer.
y
Fig. 2.10 A composite picture of the proposed structures in a turbulent boundary layer. The dasbed linea are the trajectories of the ejected fluid parcels. The shaded areas repreaent the low-speed streaks.
CHAPTER 3
ANALYSIS OF THE MODEL
In this chapter, the equations of motion will be used to derive equations corresponding to the physical model described in Chapter 2.
First of all insection 3.1 equations will be derived for the large-scale motion •. In section 3.2 these equations will be simplified for different regions in the flow field; making use of order of magnitude considerations. This leads to an approximation for a thin layer close
to the wall (the wall layer) and an approximation for the remaining part of the boundary layer (the outer region). The metbod of salution for the two regions will be discussed in sections 3.3 and 3.4.
Finally in sec ti on 3.5 a model will be proposed for burst events, which play an important role in the interaction between wall layer and outer region.
3. 1. The bas ia equationa
The time dependent Navier-Stokes equations are assumed to describe turbulent motion. For constant p and
v
they take the following farma
u I+ '!:!·17'!:! - 17p +V 172!:! 3, t, I
a
t p17.J:! - 0 3. 1.2
where '!:! = (u, v, w)
The coordinate system has been presentedalready in Fig. 2.1. Fora boundary layer under zero pressure gradient, the boundary conditions are
u U00 = constant for y -+ oo
u 0 for y 0 3. l. 3
The next step in most theories is to split the velocity components into mean flow and fluctuations. In combination with long time averaging
this procedure excludes any possibility of obtaining information about the detai led time dependent turbulent motion. So an alternative decomposition procedure is used consisting of a decomposition of the velocity components in a large-scale part and a small-scale part. The
large-scale quantities have as a typical length scale the mean distance between two bursts Àx. The small-scale fluctuations have a length scale À ' and are assumed to be present only during the burst events, This concept is based on the observation that long quiescent periods alternate with short periods with violent activity, The different scales are indicated in Fig. 3.1. The burst events have an extent of about lb in the x-direction,
- - - À x
Fig. 3.1. The different scales in a typical cycle. À is the mean distance between two bursts, lb is the width of a burst region and À1 is the length scale of the fluctuations in
a burst.
The large-scale quantities can be obtained by averaging over a short time or by averaging over a certain length in the xdirection. The -latter type of averaging is preferable because this procedure retains its meaning in the moving frame of reference. The large-scale quantities are defined as
p
I= ï.
1 x+!l. 1f
x+U.
1 !:! dxJ
p dx x-jl. 1 3 .1.4The small-scale fluc tuations are defined as
u" u - îi
p" = p - p 3. 1. 5
The integration length li defines the resolution that can be obtained in the large-scale quantities. When li is taken large, more detail of the turbulent fluctuations is blurred. The decomposition procedure used is equivalent to high pass and· low pass filtering of turbulence signals. a technique that is commonly used in experiments to isolate large-scale and small-scale fluctuations (cf. Blackwelder and Kaplan, 1972 and Rao et al,, 1971).
Averaging over length li• the equations (3.1.1) and (3.1.2) transform to
au.
ot
V.~
=
0
Here the following approximations have been made
x+
U.
1 IT.
f
u u dx .. u u 1 IT.
x-U.
1 x+U.
1f
!:!"~ dx = 1x-u.
1 uî.
1x-U.
1 u"dx = 0This is only valid when 1. is much larger than the scale
1
on which u" varies (1. >>À 1
) and much smaller than
- 1
the scale on whièh
u
varies (1. ·.«À ) • The two- 1 x
3. 1.6
3. I. 7
3. I .8
conditions can only be satisfied if À »À'. According to Blackwelder x
and Kaplan (1972) the ratio Àx/~ is about 20. Since there are only a few small-scale wavelengtbs À1 in one burst event, the ratio
À /À 1 is
x
Although the observed intermittency in boundary layers is a clear indication of the existence of distinct scales, the spectra of turbulent fluctuations are rather smooth, This must be attributed to the strongly fluctuating distances between successive burst events.
The equations (3.1.6) and (3.1.7) can be interpreted as time ~ependent Reynolds equations with
~
as the turbulent shear stress caused by small-scale fluctuations. As in all stochastic turbulence theories, we encounter the closure problem. The advantage of equatións (3,1.6) and(3.1.7) over the usual Reynolds equations is that they still contain information about the time behaviour of the large-scale motion. Suitable assumptions have to be made to arrive at a closed set of equations. Here we shall employ the picture that emerges from the experimental studies described in chapter 2. The turbulent stress is zero most of the time and shows peaks during bursts. The burst regions travel downstream with a constant convection velocity Uc according to the model described in chapter 2. So the following form is adopted.
:!:":!:"= fuu (y,z,t) Ö [x- J!b(y,z,t) + ri Àx] , n=l,2,3 ••• 3. I. 9
In this expression fuu represents the strengthof the burst and J!b the trajectory of the ejected fluid (The dashed line in Fig. 2.10). The ö-function indicates the local character (local in space and
instantaneous. in time) of the .!l"!:!" correlation. Because of the constant convection velocity, an almost stationary picture will be obtained in coordinates that move downstream with this velocity. The following transformations are applied
u
cu
c-
u, vc v,w
= w, Pc =p,
cx c
