Large-eddy simulation of a turbulent jet

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Large-eddy simulation of a turbulent jet

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universteit Delft

op gezag van de Rector Magnificus Prof. ir. K.F. Wakker in het openbaar te verdedigen ten overstaan van een commissie,

door het College van Dekanen aangewezen. op 6 december 1994 om 16.00 uur

door

Mathieu Jean Baptiste Marie Pourquie

wiskundig ingenieur, geboren te Rotterdam.

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STELLINGEN BEHORENDE BIJ HET PROEFSCHRIFT VAN M.J.B.M. POURQUIE

The fact that the energy of the velocity fluctuations in a large-eddy simulation varies in a seemingly

random way around a seemingly well defined mean is not by itself a sufficient indicator of a stationary, turbulent velocity field.

(this thesis)

If the subgrid contribution is low in a large-eddy simulation, then this not by itself sufficient to conclude that that we have enough resolution.

(this thesis)

If one performs calculations of free turbulent flows with an inflection point in the mean velocity, one should

be careful with using feed-back in the boundary conditions in the main stream direction, certainly if at the same time a fixed calculation domain is used.

(this thesis)

In order to reproduce qualitative features of a jet flow the study of the mean velocity profile is worthwhile but is not sufficient.

(this thesis)

In turbulent flows, disturbances can have large effects at large distances from the disturbance, also on the mean values.

(this thesis)

The use of coordinate systems which contain singularities and are strongly curved at this singularity, can still give practical difficulties, even if there are no problems theoretically.

(this thesis)

The presence of a large amount of experimental data and numerical results for a particular flow geometry

is not a guarantee that this flow geometry is well understood, or even that data exist from which this

understanding can be obtained.

At this very time, large-eddy simulation of turbulent Rows is in the danger of being struck by the same fate as the time- or ensemble averaged approach, namely that it is not just used as a physics simulator

but also as a numbergenerator.

The problem of numerical advection and diff-ussion of a passive, positive scalar is an example of the fact, that linear problems are not necessarily trivial.

The fact that a flow geometry is simple, does not necessarily mean that the resulting flow field is simple. Onderzoekswerk wordt teveel belast met overhead die tot het terrein van de typograaf behoort.

19. Bij de correcties van proefschriften in de Engelse taxi bij een Nederlandse universiteit blijft het een

probleem, dat de schrijver en degene die corrigeert beide slechts een subset van het Engels beheersen, en de subsets van beide personen niet geheel overlappen.

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71

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Dit proefschrift is goedgekeurd door de. promotorenr prof. dr. jr. F.T.M. Nieuwstadt en prof. dr. ft. P. Wesseling.

Published and distributed by: Delft University Press Stevinweg 1

2628 CN Delft The Netherlands

Telephone +31 15 783254 Fax +31 15 781661

CIP-DATA KONINKLIJKE BIBLIOTHEEK, DEN HAAG Pourquie, M.J.B.M.

Large-eddy simulation of a turbulent jet / M.J.B.M. Pourquie. Delft: Delft University Press. - Ill.

Thesis Delft University ofTechnology. Withref. - With summary in Dutch,. ISBN 90-407-1061-9

NUCT 841

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission from the publisher: Delft University Press, Stevinweg I, 2628 CN Delft, The Netherlands.

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1 General description of the flow field of the free, turbulent jet

10

1.1 The near field of the jet 13

1.2 The developing region of the free, round, turbulent jet 15

1.3 Description of the far field of the jet 17

1.3.1 Similarity in the far field of the jet 17 1.4 Similarity considerations and analytical treatment for a jet 18

1.4.1 Introduction 18

1.4.2 The boundary-layer assumptions 18

1.4.3 The boundary layer equations 18

1.4.4 An integral relation for the boundary-layer equations 19

1.4.5 The boundary layer equations for a large and small ambient velocity. 20

1.4.6 Similarity 21

1.4.7 Similarity relations for a standard jet 21

1.4.8 Some classical approaches 22

1.4.9 The new approach 24

1.4.10 The derivation of Schlichting's classical result 26 1.4.11 The derivation of George for a co-flowing jet 27 1.4.12 A difference between standard jets and wakes 31

1.4.13 Discussion of George's analysis 31

1.4.14 Structures in the far field of the jet 32

1.5 Standard jet vs co-flowing jet (or wake) 33

1.6 Experimental research on jets 35

1.7 Causes for the spread in the experimental data for jets 35

1.7.1 The non-uniqueness of the experimental setup 35

1.8 'Hard' data for turbulent free jets 36

1.9 *Hard' data for turbulent co-flowing jets and wakes 39

1.10 Conclusion 40

1.11 Appendix: Stability analysis for the jet 41

1.11.2 The method used in the stability analysis 44

1.12 Appendix: The different definitions of the Reynolds number in use for

turbulent round jets 50

2 The numerical modelling of turbulence

53

2.1 Introduction 53

2.2 The time- or ensemble-averaged approach to turbulence 53

2.2.1 Some general remarks on turbulence models 54

2.2.2 The most simple closure models 55

...

....

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2.2.3 The k E model 57

2.2.4 The Reynolds stress model 57

2.2.5 Some results for the k e model and Reynolds stress model . 58

2.3 Direct Numerical Simulation (DNS) 60

2.3.1 Literature on the DNS approach to flow geometries, related to the

free, round, turbulent jet 61

2.4 Large eddy simulation (LES) as an alternative for the time/ensemble av-eraged approach and the direct numerical simulation (DNS) approach 64 2.4.1 The filter procedure for the LES and the subgrid model 65

2.4.3 The filtering operation 65

2.4.4 The subgrid model 67

2.4.5 Discussion of the Smagorinsky model 68

2.4.6 Literature on the LES approach to flow geometries, related to the

free, round, turbulent jet 70

2.5 Feasibility of an LES of a turbulent jet: resolution requirements 74

2.5.1 Rough estimate of the domain size in the r-direction 76 2.6 Appendix: derivation of the Smagorinsky model constant 77

3 The equations used in the LES

79

3.1 Introduction 79

3.2 The equations in cylindrical coordinates 79

3.2.1 Point 1: the non-constant coefficients 83

3.2.2 Point 2: the extra terms in cylindrical coordinates 83

3.2.3 Point 3: the apparent singularity for r -4 0 84

3.2.4 Cylindrical velocity components for constant Cartesian velocity

com-ponents 85

3.2.5 Stress terms in cylindrical coordinates versus stress terms in

Carte-sian coordinates 85

3.2.6 Investigation of the cylindrical stress terms near r = 0: continuous

case 89

3.2.7 Investigation of the cylindrical stress terms near r = 0: discrete case 91 3.2.8 Initial values and boundary conditions for the basic equations 93

3.2.9 The scalar transport equation 93

3.3 The choice of the spatial discretisation method 94

3.3.1 Spectral methods 94

3.3.2 Finite element methods 94

3.3.3 Finite difference methods 95

3.3.4 Finite volume methods 95

3.3.5 The choice of a discretisation method 95

3.3.6 Description of the grid 95

3.3.7 Accuracy of the finite volume integration, depending on the position

of the grid points 97

3.4 Application of the finite volume method 99

3.4.1 The discretisation of the continuity equation 99 3.5 Solution procedure for the discrete system of equations 100

3.6 The time integration 102

3.7 Appendix: The finite volume method, applied to the momentum equation

in the co-direction 104

. .

...

... .

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3.7.2 Discretisation of the advective term in the 0-direction 105

3.7.3 Discretisation of the diffusive term in the 0-direction 107

4 The time-jet approach to LES of a free, turbulent jet in the similarity

region 111

4.1 Motivation for an LES of a free jet, entirely in the developed region 111

4.1.1 Implementation of the time-jet approach 112

4.2 Linear advection-diffusion and the time-jet approach 113

4.3 Laminar round jets and the time-jet approach 114

4.4 The time-developing approach for a round, turbulent jet 116

4.5 The time-developing approach for LES of a round, turbulent jet 116

4.6 Boundary conditions and initial conditions for the time-developing jet . 117

4.6.1 Boundary conditions 117

4.6.2 Initial conditions 119

4.7 A time-jet simulation with forced profile 119

4.7.1 A priori discussion of the forced time-jet 120

4.7.2 The profile used 121

4.7.3 The forcing procedure for the velocity profile 122

4.7.4 First method of forcing: the instantaneous forcing method 123

4.7.5 Second method of forcing: adaption of the slab mean average with

relaxation factor 123

4.7.6 Third method of forcing: constant forcing 124

4.8 Results for the forced, time-developing jet 124

4.8.2 Numerical test calculations for the time-developing jet 124

4.9 The instantaneous forcing method, C, = 0.05 and C = 0.1 127

4.10 Domain size check: z-direction 138

4.11 Domain size check: r-direction 139

4.12 The profile of Schlichting instead of the Gaussian profile 140

4.13 Second and third method of forcing 142

4.14 A higher resolution test 144

4.15 Discussion of the forced time-jet results 145

4.16 Laminar test cases for the time-jet 150

4.17 Final conclusion for the forced profile time-jet calculations 158

A Appendix: additional forced profile tests

160

A.1 Forcing method 2: relaxation with relaxation constant 0.01 160

A.2 Forcing method 3: constant forcing 160

A.3 Some unforced tests for the wake 160

B Appendix: The TKE budget

165

3 Space- and time-developing jet

166

3.1 Introduction 166

3.2 A description of Spa1art's procedures 166

3.2.1 Spa1art's re-scaling (1988) procedure 166

3.2.2 Spalart's fringe (1993) procedure 168

. ..

..

.. .... .... , . .

. . .... ..

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3.2.3 Discussion of Spalart's procedures 168

3.3 A crude variant of the first method of Spalart 169

3.4 Some results for the crude variant of Spalart's first procedure 171

4 Conclusion

172

A Appendix: some numerical tests for advection schemes for the transport

of a passive, positive scalar

175

A.1 Introduction 175

Appendix: some numerical tests for advection schemes for the transport

of a passive, positive scalar

176

B.1 Introduction 176

B.1.1 Test problems 176

B.1.2 Introduction 182

8.1.3 The method of Smolarkiewicz 183

8.1.4 Van Leer's Second Scheme 184

B.1.5 The multi-dimensional flux corrected transport (MFCT) method . 186

B.1.6 Numerical results 188

B.1.7 Discussion and conclusion 192

C Appendix: Effect of the Asselin filter on the formal order of the

Leap-Frog discretization

205

C.1 Summary 205

C.2 Order analysis for Leap-Frog combined with the Asselin filter 205

Appendix: Neumann stability analysis for the advection-diffusion

equa-tion in rectangular coordinates

208

D.1 Summary 208

D.2 Introduction 208

D.3 The continuous equations used 208

D.4 The discretization 209

D.5 Neumann stability analysis 209

D.5.1 Substitution of the trial solution 209

D.6 Derivation of the stability conditions with the aid of Miller's theorem 210

List of symbols 213

F References

215

. . . .

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0.1 Introduction

It is difficult to imagine our society without fluid mechanics. Fluid mechanics plays a large role in problems like weather-forecasting, dispersion of pollution and efficiency of combustion, all of which play a vital role in present days society. Besides these apparently important (and complicated) examples, fluid mechanics is also occupied with flows which are simpler, like pipe or channel flows. Since these flow geometries are much simpler, they can be studied in much more depth. The results found for these simpler flows, can then in turn be used to try to understand the more difficult flows. Moreover, the approximations and conjectures made in difficult flow geometries can be tested in the simpler ones. Thus, also the study of simple(r) flow geometries is fully justified.

This thesis is about a flow geometry which is much simpler than the three examples mentioned above, namely about the numerical simulation of a free, round jet. This flow geometry is obtained, if some fluid is ejected through a round orifice in a large vessel or open space, so that there is no influence of surrounding walls. Some understanding of this flow geometry is basic to an understanding of more complicated jet flows, such as flames.

The flow field of a jet can be in different states. One such state is laminar, the

other turbulent. In the laminar state we have a smooth velocity field if there are no

perturbations because of an imperfect experimental apparatus. If a tracer is added to the jet fluid (paint or small particles), it will spread slowly and smoothly on its way from the orifice. The flow field displays cylindrical symmetry. On the other hand, by increasing the outflow velocity at the orifice sufficiently, the jet becomes turbulent. The flow field shows random and fluctuating vortices which are created and destroyed continuously. The flow pattern is chaotic and seems in-predictable. Mixing takes place much faster than in the laminar case. The a)d-symmetry of the instantaneous flow field is lost. Thus, despite the simplicity of the flow geometry (simple inlet flow, no complicating influence of

walls) the flow field is complicated. In fact, the mechanics behind the turbulence are still not understood, because it is an essentially in-stationary, three-dimensional, non-linear

problem.

In practice, conditions are such (high exit velocity, low viscosity), that nearly all jets of importance are turbulent. This thesis will therefore concentrate on the turbulent variant of the free, round jet.

The study of the round, free, turbulent jet has been pursued for quite some time.

There is an abundance of experimental data on these jets, the earliest systematic study being perhaps the study by Truepel [1915]. Not all quantities desired could be furnished, however. Some physical quantities in a turbulent flow are difficult to measure. One of these quantities is the vorticity. This vorticity is an important quantity in a turbulent flow, since it is a measure for the intensity of the eddying motions, and the dynamics of turbulence are believed to be closely connected to vortex dynamics. Therefore, alternative ways of investigation were desirable.

Experiment showed, that though the instantaneous flow field does not display cylindri-cal symmetry, the time-averaged field does. Therefore, an investigation was carried out of the time-averaged equations. Unfortunately, these equations prove to contain extra

terms which are in principle additional unknowns. Some assumption, called a turbulence model, has to be introduced in order to get a closed system of equations. Since this leads to uncertainty in the solution of these equations, there is still need for another means of 'With recent advances in flow visualisation techniques, the measurement of this quantity is coming into reach of the experimentalists

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investigation.

In view of the large computer power which is nowadays available, numerical simulation

seemed a promising means to do this. The capability of computers is large enough to

perform 3-D, dependent simulations of flows, which is desirable for a 3-D, time-dependent phenomenon like turbulence. The analysis of the resulting data of this kind of simulations has proved useful for several types of turbulent flows, such as pipe flows. Besides the fact that we have 3-D data fields at our disposal, instead of (several) one-point measurements, the simulations do not suffer from experimental difficulties such as the drift of the experimental equipment.

3-D simulations have problems of their own, however. A difficulty with 3-D simulations are still the resolution requirements. If the resolution is increased with a factor of 2, the memory requirement goes up with a factor 8, and the calculation time with a factor 16 because of a smaller time step. Thus, despite the capabilities of present computers, we are not capable of resolving all details of most flows which occur in practice, and some economical way of computing must be found which resolves only the most important features. For a turbulent flow the important features are the eddies appearing, and we have to find out which eddies are most important to know in detail.

In a turbulent flow we distinguish between the large fluctuations and small fluctuations. These large scale fluctuations (related to large eddies) are supposed to have a length scale of the order of a characteristic size of the flow geometry. Thus, they are influenced by the flow geometry and we call them flow geometry dependent. On the other hand we have small fluctuations or eddies. They are much smaller than the characteristic diameter of the flow geometry. They are supposed to be more universal, i.e. they behave about the same for each different flow geometry (jet, channel, etc.). These small eddies are supposed to be more simple than the large ones. In their description one uses all kinds of simplifying assumptions, such as isotropy. For the large eddies, on the other hand, these simplifying assumptions are not valid. In order to study the large eddies properly, we will have to proceed without simplifying assumptions.

On the other hand, the large eddies are responsible for most of the mixing in a jet. Therefore, a detailed knowledge of these large eddies is desirable. Thus, in our numerical calculation the large eddies must be simulated, and cannot be modeled, whereas the small ones can. This is precisely what the numerical technique called large-eddy simulation (LES) does. Large-eddy simulation performs a 3-D, time-dependent calculation and gives 3-D, time dependent information on the large eddies. Therefore it was decided, to do a large-eddy simulation of the free, round, turbulent jet.

Large-eddy simulations of a turbulent jet had never been performed before, so that no information was available from the literature. However, the flow was not thought to give much more difficulties than other flows simulated at the laboratory. Quite another point was the transport of a possible scalar quantity which might be added to the jet flow, since experience had pointed out that special numerical treatment was necessary. The usually applied, linear schemes can lead to negative values of the concentration, or additional smearing besides the smearing caused by physical diffusion. Therefore, an investigation of some possible schemes, found in the literature, was thought necessary.

Since a large-eddy model needs a Navier-Stokes solver which can handle a non-constant viscosity, which parameterises the small eddies, a decision had to be made on the type of model. Cartesian models were already available, and could be used. However, the round jet has cylindrical symmetry, in the mean, and another PhD student (J. Eggels) was to

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cylindrical coordinates.

Some decision had to be made too on the type of simulation to be performed, and what

kind of data had to be produced. For the free, round, turbulent jet we can distinguish

several regions. At large distance from the orifice, we have the so-called far field. Following the general belief, this far field was supposed to be a relatively simple region of the free, round, turbulent jet, where it has forgotten the exact way it left the orifice. This region was supposed to be simple and universal, i.e. it should be about the same for all turbulent jets, irrespective of the way it starts at the origin. Therefore, simulation results for this far field should have some universal value too, and there would be no necessity to reproduce the exact experimental conditions. Since, in the course of the research, the problem

seemed to simplify even more if an ambient stream was added to the flow geometry, this has been done too. Concluding, it is the simulation of the far field of this so-called co-flowing jet, using the large-eddy simulation technique, which is the main subject of this thesis.

In this thesis, chapter 1 covers the findings from a literature survey on free turbulent jets and wakes. This survey had been performed at the end of the PhD-work, so that not

all ideas from this survey have actually been taken into consideration in the numerical simulations. Chapter 2 contains a literature survey of numerical modelling of turbulent flows, and contains a description of the large-eddy simulation technique plus a discussion of its potential for treating jet flows. Chapter 3 treats the discretisation of the equations, and the method of their solution. The results of some actual simulations are given in chapter 4 and 5, where it is shown that the far-field calculations show resonance phenomena. The conclusions from the simulations are in chapter 6. The schemes for scalar transport are treated in chapter 6, an appendix. Two more appendices discuss stability of the numerical method and the artificial diffusion associated with the numerical method used.

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Chapter 1 General description of the flow field

of the free, turbulent jet

This chapter contains some information on jets, obtained from a study of the literature. It is not intended to be comprehensive. The subject is too complicated for this. One of the main purposes is to show the variety of phenomena which are present in jets, and to give some idea of how divided the literature on this subject still is. Moreover, this chapter

intends to give some background to ideas and findings presented later in this thesis.

Figure 1.1: The round, free, turbulent jet

A free, round jet is obtained, if we take a circular orifice through which we eject fluid in open space or in a very large container, so that there are no effects from the boundary. If the surrounding (or ambient) fluid is quiescent, we have the so-called standard jet, also termed strong jet. If the surrounding fluid has a non-zero velocity, directed along the axis

of the jet, we have a co-flowing jet, also termed a weak jet or a compound jet. If the ambient fluid flows faster than the jet fluid, the flow geometry is called a wake. In this 'The terms weak and strong stem from the fact, that for a finite ambient flow, the ratio of jet velocity to ambient velocity goes to 1 far from the orifice, due to the spreading of the jet, see section 1.3

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orifice'

co-flowing jet

Figure 1.3: A schematic overview of the co-flowing round, free, turbulent jet

thesis, the properties of injected and ambient fluid are considered to be the same. The density of the ejected fluid and the surrounding fluid is thus equal, so that there are no

buoyancy effects. As a result, the jet is completely momentum driven.

The subsequent development of the jet after it has left the orifice dependson several parameters, such as the noise level in the surroundings, the geometry and smoothness of the orifice and also the flow conditions in the exit opening. A parameter of paramount importance here is the Reynolds number, Re =UL/v, with U and L characteristic scales of the flow geometry, and v the viscosity. Near the orifice, a natural velocity scale is the difference between the mean outflow (jet) velocity Wo and the ambient velocity W,0, i.e.

Wo W. A natural length scale is the diameter D of the orifice. We expect, that for

'large' Re the jet will be or will become turbulent.

Let us consider the influence of changing the Re on the standard jet geometry for a given experimental setup. The flow patterns observed are quite variable, and differ from 'experiment to experiment. W.C. Reynolds [1962] seems to be the first to have published a,

review of these patterns. However, his observations are made under non-ideal conditions, i.e. there is ambient noise present and the orifice is not very smooth.

For very low Re, i.e. Re < 10, we have an axi-symmetric laminar jet, whichagrees well with Schlichting's solution for a free, laminar jet. At around 10 < Re < 30, the smooth, nearly rectilinear (i.e. slowly diverging) jet breaks down at a distance of some diameters

direction axial velocity

orifice

mean axial velocity profile arrow indicates half-woe,

velocity deficit (=velocity minusc011Owj.

arrow indicateshalf-Mc:1pr

'Potential cone simIlanty region forrms velOtty (401:1 start similarity regionfor mean velocity(10D)

standard jet

Figure 1.2: A schematic overview of the standard round, free, turbulent jet

direction axial velocity

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from the orifice in a somewhat mushroom-shaped chaotic structure. This observation led Viilu [1960] to the conclusion, that a round jet becomes turbulent at Re of the order of 10. However, at higher Re, i.e. 30 < Re < 150, the nearly rectilinear part becomes progres-sively longer, with the possible formation of axi-symmetric roll-ups. The measurements of Andrade et al [1937] (made under more ideal conditions) show good agreement with the theoretical results of Schlichting [1977] for this range of Re. For 150 < Re < 300 several different flow patterns may arise. The nearly rectilinear part may become longer, until it reaches the end of the experimental confinement. Alternatively, the rectilinear part may experience a large wavelength meandering, or a break up in the same fashion as a drop of ink, injected smoothly in water, forming pedal like structures. Beyond a Re of about 300, the jet becomes fully turbulent after a rectilinear part which extends several diameters from the orifice.'

Similar experiments have been performed by McNaughton et al [1966], who worked under more ideal conditions than Reynolds, and covered a Re range of 100 28000. An irregular breakdown, corresponding to the pedal-like instability of Reynolds is found for 100 < Re <300, but a rectilinear, laminar jet is obtained at 300 < Re < 1000. This must probably be attributed to the more ideal conditions compared to Reynolds' experiments. Moreover, the large-wavelength meandering seen by Reynolds was not observed. Above Re = 1000 the jet becomes turbulent. For 1000 < Re < 3000 we still have a rectilinear part extending from the orifice. The length of this part (which may take up 0(10D))

decreases steadily for increasing Re. For Re > 3000 the rectilinear part is negligible. For higher Re, the gross appearance of the jet does not change any more.

Crow and Champagne [1971] describe the jet flow near the orifice for Re = 100 to Re = 1000. The first instability at Re = 100 has the form of a flat (2D) meandering. For higher Re this becomes a helical instability, and at still higher Re the helix falls apart in

a series of (approximately) aid-symmetric puffs. No pedal-like instability is observed. The observations of Reynolds, McNaughton and Crow et al discussed above do not agree entirely on the kind of phenomena appearing, which indicates that the observations depend on the experimental set-up. Nevertheless, we have obtained an indication at what Re a free, round jet becomes turbulent. It may become turbulent at Re = 300 under

non-ideal conditions, but at Re = 3000 a fully turbulent jet is seen to occur at under any circumstances. 3

For an axi-symmetric co-flowing jet or wake, a similar detailed report on the develop-ment with Re was not available. For a large ratio of jet orifice velocity to ambient fluid velocity, we expect some similarity to the standard jet's development.

Our main interest is the case of the (fully) turbulent, round, free jet. Let us therefore describe this case in more detail. First of all consider the standard jet. See figures (1.1) and (1.2) for a photograph of a turbulent jet and a schematic picture of the (Re = 0(1000)) jet. The photograph shows a smoke-laden standard jet, so that we can easily distinguish

jet fluid and surrounding fluid. The free jet starts with a top-hat profile, followed by

a rectilinear column. On this initial profile (approximately) axi-symmetric disturbances develop. The disturbances form roll-ups, leading to the formation of vortex rings. After this initial stage, the jet becomes fully turbulent and shows a linear spread (in the mean). So we may distinguish three typical regions in a jet. Near the orifice we have the near field, with the rectilinear column and its (approximately) axi-symmetric disturbances. For a 2Compare this Re with the Re at which a pipe flow becomes turbulent, i.e. 2000, with a Re based on

mean velocity, pipe diameter and viscosity.

'For pipe flows laminar flows have been obtained for Re as high as 100.000, see Pfenninger et al [1961]. This is connected with the fact, that pipe flows are in-viscid stable.

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high Re jet length of this region is about 5D. Far from the orifice we have the so-called far field or developed field, where the turbulence is fully developed and the jet spreads Linearly. We take this region at SOD and beyond. In between we have the developing region. In the developing region and the far field, scaling is usually done with the local, mean axial centerline velocity deficit and the local width of the jet. The centerline velocity deficit is defined as Wei W, with Wci the mean axial centerline velocity and Woo the ambient velocity, i.e. the velocity at a large distance from the jet axis. As a measure of the local width one usually takes the so-called half-width, 6. This is the radial distance from the axis, at which the mean axial velocity reaches half its value on the centerline.

From the photograph we can also see, that there appears to be a sharp boundary

between jet fluid and surrounding fluid. This boundary is rather irregular, and because of this we have so-called intermittency. Intermittency means, that if we stay at one and the same position, then part of the time the fluid is turbulent (i.e. it contains vorticity), and

part of the time it is non-turbulent (i.e. _{ii-rotational flow). The fluctuation level in the}

jet. i.e. the local PAIS value of the velocity fluctuations, divided by the local mean axial velocity (w'/W) can be very large (>> 1) near the edge of the jet. This high fluctuation level is one of the causes of the experimental and numerical difficulties encountered in the study of free jets.

For a co-flowing jet, the picture is somewhat different. We have of course the devel-opment of turbulence, like for the standard jet. If the ratio Wo Woo to Woo is not too small, the flow field will be like that of a standard jet if we are not too far from the orifice. However, the ratio Wo Wa, to W becomes ever smaller if we travel away from the

orifice. If it becomes 'small', say 0.1, the co-flowing jet will have behave differently from a standard jet. For instance, it does not spread linearly in the fully developed field, but instead the width increases more slowly with the distance from the orifice. The fluctua-tion level in the far field, relative to the ambient velocity, will also be much less than for the standard jet.

The differences between a co-flowing and a standard jet are further elucidated in the next sections, in which we describe in somewhat more detail the three regions which we

mentioned above. The differences between a co-flowing jet and a standard jet are found to be most pronounced in the far field. For this reason, we point out the differences to be discussed mainly in section (1.3). We will describe the different regions in a (co-flowing) jet, travelling in the main flow direction, starting at the orifice.

1.1 The near field of the jet

It is recognised nowadays, that turbulent flows are not completely random. Instead, orderly patterns usually characterized by concentrated vorticity can be discerned. These are called coherent structures (see Fiedler [1987] for a more precise description of the notion of coherent structure). The form of these structures depends on the type of flow

(wall boundary layer, plane jet, round jet). They can play an important role in the

dynamics of the turbulence. For this reason, some information is given on these coherent structures.

The structure of the near field has been studied extensively, qualitatively as well as quantitatively. Pictures of the structures in the near field can already be found in

publications dating from the 1930's (Brown [1935]). The first quantitative study at high Re is that of Crow et al [1971].

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hat profile for the exit velocity, with value Wo, say. For most experiments the turbulence level is of the order 0.1%, which is more than an order of magnitude less than for a fully turbulent pipe flow. Directly following the orifice we distinguish the potential cone, also called potential core. In this core, the fluid still has the velocity Wo and is essentially non-viscous (non-rotational). The potential cone is surrounded by an axi-symmetric, growing mixing layer. After some 5 D from the orifice the mixing layer has reached the centerline, the potential cone has vanished and the fluid is rotational over the entire diameter. The axi-symmetric mixing layer does not have a smooth boundary. It contains surface ripples, which scale with the thickness of the initial (laminar) boundary layer in the pipe before the orifice.

The ripples mentioned are have a typical frequency, given in dimension-less form by the Strouhal number f 001W0 = 0.01 0.018, with f the frequency and Bo the momentum thickness of the shear layer at the orifice. This thickness varies as 1/VVITo for a laminar

boundary layer, or rather 1/ViTe. The large spread in the Strouhal number is caused

by the sensitivity of the vortex-ripple formation for even very small disturbances, given

that they are are coherent. (Gutmark et al [19831 report that coherent disturbances of

0(5.10-5W0 (!!)) in the flow before it leaves the orifice already give appreciable effects!) The ripples initiate roll-ups, leading to vortex rings. The vortex rings are easy to visualise

for lower and moderate Re (say, between 150 and 3000). At higher Re visualisation

becomes much less easy, but the vortex rings still exist (Petersen [1978], Crow et al

[1971]).

As we travel away from the orifice the vortex rings get involved in a pairing process. Thus, at the end of the potential cone the Strouhal number for the vortex ring frequency has become smaller (fD/Wo = 0.24-0.64.) Furthermore, the vortex rings are unstable to waves in the d.-direction (Wydnall et al [1972). These instabilitiesdeform the vortex rings. The rings are also tilted and deformed by the developing surrounding turbulence. Thus,

it becomes more difficult to identify the ring structure. According to Petersen [1978],

for the First 4-5 diameters the dominant coherent structures are ring-shaped. Thereafter, there is no longer strong experimental evidence for ring-shaped structures. Instead, single and double helical structures are observed. The same was found by Mattingly et al [1973]

by experimental means as well as by a stability analysis.

The structure of the ring-shaped vortices has been the subject of several experimental and numerical studies. According to Agin et al [1989] they are part of mushroom-shaped structures which consist of the vortex rings which are connected by braids of longitudinal vorticity, which are accompanied by high shear. The vortex rings have been studied

longitudinal braids roll-up

mushroom shaped structure

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theoretically too, with stability theory, see section (1.11).

Let us now turn to axi-symmetric co-flowing jets and wakes. For these flow geometries, the instabilities which are visible at Re < 3000 have not been documented as much as for the standard jet. The structure of co-flowing jets and wakes at Re below 1000 is studied at some length in Perry et al [1978, 1980]), who describe the development of vortex

roll-ups and study their topology. In their equipment, the ambient velocity was varied, so

that wakes and co-flowing jets could be simulated. Re varied between 300 and 1000. It is interesting that Perry et al find differences in the development of jets and wakes. Some differences are not surprising, for instance in the way the initial vortex rings roll up. This is shown in figure (1.5), from which we see that the vortex roll-ups are in the ambient flow direction for a wake and in the opposite direction for a co-flowing jet. When a co-flowing jet and wake are studied, the jet may show alternative flow patterns, which mostly break up in mushroom-shaped vortex rings, interconnected by longitudinal braids. A wake does not seem to show this alternative pattern (or its break-up form). It is not clear whether this difference between jets and wakes is in general present, or whether it is typical for this experiment.

Flow

Figure 1.5: The initial vortex roll-ups for a wake (upper picture) and a co-flowing jet

(lower picture)

1.2 The developing region of the free, round,

turbu-lent jet

In the first 10 diameters, the mean velocity profile of a standard (Woo = 0) jet approaches so-called similarity. Similarity for a jet or wake means, that variables such as the velocity scaled with the centerline velocity deficit, are a unique function of the radial distance, scaled with an appropriate length scale. For this length scale we can e.g. take the half-width 6. Similarity for the mean axial velocity TT) implies IT = (14/d Woo)f (r/5),

and for the mean ELMS fluctuation of the radial velocity 1/, vi = (Wel Woo) g(r /6).)

Very often, results for standard jets are expressed as a function of r/z, with z the

distance from a virtual origin of the jet. The virtual origin is found by considering the half-width as a function of z. If we are at a distance from the orifice where the jet spreads

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spreads linearly at the same rate everywhere along the z-axis. The point where the width of the jet would be 0 is called the virtual origin of the jet. Scaling with the distance from

the virtual origin makes the implicit assumption, that all standard jets have the same

half width, a fact which has been accepted until quite recently. Now this assumption is subjected to some doubt (see section (1.3)). The virtual origin does not have to coincide with the orifice, and may be different for different quantities such as axial velocity and

concentration. The virtual origin is typically situated at a few diameters before the

orifice. Its exact location depends on Re (see Flora et al [1969] for a discussion of the Re dependence of the virtual origin for a plane jet).

The form of the mean, axial velocity profile turns smoothly from the top-hat like

profile to a profile which can be approximated by a Gaussian curve, having the functional form (Hinze [1975]):

w = Wdexp(-108

(:)2)

which is equivalent to

w = Weiexp(-1n2r)2 (1.2)

if we accept 5 = 0.08z. (Roth [1975] recommends 96 instead of 108 in expression (1.1), but we have used 108 in the simulations of chapter 4 and 5).

Whereas we have argued above that for the mean velocities similarity is valid after about 10 diameters, it is found that for the turbulent axial velocity fluctuations similarity is only attained after some 40 diameters (see Wygnanski and Fiedler [1969]). After some 70 diameters the fluctuations of the other two velocity components become similar too. Apparently, the fluctuations of the normal and angular velocity components are produced by a redistribution of the fluctuations of the axial component, which has to attain simi-larity before the other two components can be similar too. For a passive scalar added to the jet about 80 diameters are needed to reach similarity (see Dowling [1990]). The first 80 diameters may thus be called the developing region. The structure of the developing region has been studied, among others, by Komori and Ueda [1985], who were especially interested in the intermittency region. They describe a structure consisting of outward burst of rotational fluid, accompanied by turbulent reverse motion and inward flow from

outside the rotational region. They found, that the instantaneous motion at two

posi-tions differing 7r in angular position were almost independent, suggesting the absence of axi-symmetry or helical symmetry. This is somewhat surprising, since normally some evidence for aid-symmetric or helical structures is found (see section (1.3) on the far field of the jet)

Let us now turn to the developing region of the wd-symmetric co-flowing jet and wake. We have said before, that here we have two kinds of development, namely of the turbulence and of the ratio (Wd Woo) /Woo. Because of this fact alone, the length of the developing region can differ strongly between various flows. Carmody [1964] mentions 15D for the wake behind a circular disk, and Chevray [1968] finds 3D for the wake behind a smooth, 6:1 ellipsoid. For his co-flowing jets, Antonia [1973] mentions, that a co-flowing jet with

(W0 Woo) /Woo = 2 has a self-similar axial RMS at zID = 100, but that this ELMS is

still rising at zID = 250 for a co-flowing jet with (W0 W) /W 3.5. The results

from Chevray (who did measurements up to zID = 18 and Carmody (who worked up to SOD) did not show self-similarity for the RiMS and shear stress. Thus, we may conclude that none of the authors mentioned have shown that their results apply to the developed,

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axi-symmetric co-flowing jet or wake, but probably apply only to the developing region. (According to Hinze [1975], Hwang et al [1966] did observations in the developed field.)

1.3 Description of the far field of the jet

In the far field, the standard jet is fully turbulent for r/z < 0.22 (r < 2.756).

The

region r/z < 0.22 is therefore called the turbulent core. The flow is intermittent for 0.22 < r/z < 0.25 and non-turbulent (but still fluctuating!) outside r/z = 0.25. The largest eddies are assumed to be of the order of magnitude of the turbulent core (i.e.

0.44z).

The far, or developed field, is first of all characterized by the fact, that the turbulence is by definition fully developed. For standard jets, it is furthermore assumed that it is self-similar. Regarding the co-flowing jet or wake, there is some discussion about the existence of similarity, but here we assume similarity too. Similarity is also used in a numerical model for the space-developing jet, so we discuss it here in some detail. Another aspect of the far field is the possible presence of coherent structures. We encountered them too in the simulations, so we will discuss the experimental evidence for these structures in section (1.4.14).

1.3.1 Similarity in the far field of the jet

As to the far field, there is general acceptance that a standard jet becomes self-similar far from the orifice. This means, that all turbulent and mean quantities are a function of r/6 only, when scaled with a proper scaling quantity which is a function of z only (6 is the half width). Though similarity is widely used for the standard jet, there is still some discussion about it. For aid-symmetric wakes, the existence of similarity is generally recognised, but the existence of universally valid (i.e. the same for all wakes) profiles is not. For an aid-symmetric co-flowing jet, there is even some doubt in the literature whether similarity exists at all (Roth [1975], Antonia [1973]).

The discussion around similarity for a standard jet is related to the universality of

some scaling assumptions which have been made in the classical treatments of jet flows. Let us make this more precise.

According to the classical thoughts, the profile for the mean velocity has the (universal) form f(r/z) when scaled with the mean centerline velocity. This is confirmed rather well by all experiments. The mean centerline velocity should be of the form Wa A/z, with A universal, i.e. the same far all standard jets. However, there is some disagreement regarding the value of A (see Panchapakesan et al [19931). Moreover, profiles for other quantities, such as RAMS values for the velocity components, do not always have the same

form (see section 1.8). There may be various causes for this disagreement. They may

be a consequence of imperfect experimental conditions or of using different experimental techniques, as described in section (1.8). Even the assumption of similarity itself may be doubted. Alternatively, George [1991] uses dimensional arguments to show, that the

constant A may be dependent on the initial conditions (for instance, the form of the

velocity profile at the orifice), so that a different A can be found for different experiments.

In fact, the spread in the constant A found by different experimentalists is one of the

4According to Rodi [1975), similarity for the RMS of the velocity fluctuations had only been reached at zIR = 500 for the plane wake behind a cylinder (radius R), placed at right angles to the main flow

direction

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facts he uses to support his arguments. He also mentions experiments such as those

by Gutmark and Ho [1983], in which different initial conditions in otherwise the same experimental setups give different spreading rates (and consequently different A). This shows the importance of initial conditions. In view of the impact of George's point of

view, part of his derivation of similarity relations are repeated here in section (1.4).

1.4 Similarity considerations and analytical treatment

for a jet

1.4.1 Introduction

Jets are often studied using similarity concepts, so that these deserve some discussion. Similarity is for instance used in analytic treatments of jets. The analytic findings such as Schlichting's [1979] are often compared to experimental and numerical results for free jets. Moreover, we shall sometimes make use of the analytical findings in our numerical treatment of jets. Because the analytical treatment of jets relies heavily on similarity considerations, this section was considered appropriate to treat it.

In analytical treatments of the free jet a number of assumptions are made. One is

usually that the boundary-layer form of the N-S equations applies, and another one is, that similarity applies. Let us make a few remarks about both of these two assumptions.

1.4.2 The boundary-layer assumptions

The boundary-layer (B-L) concept is discussed in Hinze [1975] and Schlichting [1979]. According to Hinze, it applies to a flow having the following characteristics:

there is a clear main flow direction.

the profile is slender, that is, the transverse dimensions are much smaller than

the dimensions in the direction of the main flow. If we denote the dimension in transverse direction by (L2) and the dimension in main stream direction (L1) then we should have: L2/L1 <<1

The transverse flow (in the direction normal to the main flow) is supposed to be small in comparison to the main flow.

the change of quantities in the main flow direction is much smaller than the change of quantities in the transverse direction

the variation with transverse distance of the pressure is connected with the velocity variations, whereas the pressure change in the main flow direction is connected with the pressure change in the medium surrounding the boundary layer flow.

1.4.3 The boundary layer equations

The EL equations for an axi-symmetric flow in cylindrical coordinates can be found in Hinze [1975]. First of all we have the continuity equation:

1 arfi 0

(21)

For the transverse momentum we have ap

a

-71

u - v

-p-o-r + wrie + - 01

This can be integrated over r. The result is further rewritten by the introduction of po, the so-called free-stream pressure. This is the pressure in the absence of a jet or wake. Po is taken to be a function of z only, po = po(z). We get

1

171-dr -

Po

-p + 7.71+ I

-P

The BL equation for the axial momentum is given by:'

OuT Ow 1 ap Oj7 1 Ore&

Or

17 az +17 car .7 p az or

r

(1.5) With the aid of the transverse momentum equation this can be put in the form:

Ow Ow 1 dpo

aii7'-7

1 aruqui

iv- (1.6)

az

ar

p dz Or

With po is the free-stream pressure. The term

a

ur2 -17

dr

02J

r

has been neglected, so that the RMS of the velocity fluctuations in 0- and r-directiori

are supposed equal. Equation (1.6) is valid for jets and wakes, with or without

co-flow. Additionally we have the following boundary conditions (with We, the free stream velocity):

Ou

it =0 and

=u at r

Or

w = Wee for r oo

1.4.4 An integral relation for the boundary-layer equations

From equation (1:6) we can derive an integral relation. First (1.6) is multiplied by r and then integrated from 0 to oo. This gives for the first term on the left:

Ow

dr =

f

ce -1ri 2dr

(1.75

f

r---0.

r

OZ r=0 2 az

The second term on the left is rewritten with the aid of partial integration and the'

continuity equation:

30 Ow

(partial integration)

fr 07'

cc aril

.= rUthi f rirdr = (continuity equation)

r.o 07'

co ow

coo r, vv emar + f Wdr r=0 CZ r=0 az

-

,

- 160

-

r

VV °oar

f

r=0 OZ r=0 2 Oz

The pressure' term on the right can be written as:

(1.4)

0.8j

= -

_r = 0

dr =

- (1.9) r = (1.10) r - (1.11)

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1 dPo

_{_w}

c°

p dz dz

The integration of the turbulent stress term gives:

(1.12)

1 aru'w'rdr

₌

_(1.13)

r

ar

mite 1,to= 0 (1.14)

so that the turbulent stress terms integrate out to 0. We thus end up with:

0.0 au2 _oo _87w _oo oo aw ut.2

dr = 0 (1.15)

r, dr -

f

Wodr - f rW--- dr +

r

Lo

az r=o az r=0 dz r=o az

This can be further rewritten as: tiff 7 W

Jr

ti7oo

°° wa 12 - te2

)dr + facr (IT wccdr +

f

r dr = 0 (1.16)

=o az r=0 dz r=0 az

or:

fcc riff (if

-

Wco)dr+dW'r f

(IT Ww) dr+

r

-

u'2) dr = 0(1.17)

dz r=0

0 r=0 az --o

From this we find, that:

frrI T <Tv - Woo) dr +

f

r (W(2 222) dr (1.18)

=0

is constant for all free jets and wakes with an ambient velocity which does not vary with

z. If the term

r

(wa

-

7) dr

(1.19)

fr-io

is neglected, we get the well-known relation:

Jr=0 - Woo) dr = constant (1.20)

1.4.5 The boundary layer equations for a large and small

ambi-ent velocity.

The equations simplify somewhat for the standard jet, which has ambient velocity 0, and for the far field of the co-flowing jet or wake, where the ambient velocity is much greater than the difference between jet (wake) velocity and ambient velocity. The equations for the free, standard jet can be written as:

arv 1 aru'w'

Tv +

=

Oz

ar

r

ar

atT; 1 art-1

=0

az

_{r ar}

Additionally, for a free, round jet we have the following boundary conditions:

= 0 and = 0 at r = 0

ar

= 0 for r 4 oo (1.21) (1.22) Wab)

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-Equation (1.20) reduces to

2irfss

v?rdr=K

(4.23)

Equations (1.21) and (1.23) will be used in the derivation of George (section 1.4.9). Next we give the equations for the far field of the free, co-flowing jet or wake with

constant W. We use the fact, that (IT Wee) /Wei, << 1 and ft/Wec, << 1. We get:

air)

1 artilw' W., Oz

r

Sr cati7 _{_}1 arri = !() dz

r ar

Additionally, we have the following boundary conditions:

= 0 and 0 at r =.0 (1.26)

Or

Er = W. for r co (1.27)

Equation (1:20),' reduces to

2n: 0

W.,)rdr

(1.28)

Equations,(1.25), and (1.28) will 'be used in section (1.4.14,

1,4.6

Similarity

Let us now discuss the concept of similarity. For our applications, similarity means the following. Take any quantity, related to the time-averaged form of the equations of motion. This can be the axial velocity to, the radial velocity u or the uw-stress, etc. We say that such is quantity is similar, if we can write it as a scale function, which varies only with z, multiplied by a function of r/5, with 5 the local half width, and in principle 6 = 5(z). For instance, = uw,g(rI6), with mu. = uw(z).

The similarity concept applies as well to the full Reynolds averaged equations (i.e.

not in the boundary layer approximation), as to these equations in the boundary layer

formulation (Hinze [1975]).

A special case of similarity is so-called complete similarity, which is often used in classical derivations of analytical solutions for jet flows. For complete similarity, we have

only one length scale and only one velocity scale, with which we may form the scale.

functions such as :awe..

.1.4:7

Similarity relations for a standard jet

With the concept of similarity applied to the B-L approximation of the N-S equations,

a number of analytical relations can be derived. In this thesis we distinguish between

the classical derivations, such as Schlichting's [19791, and modern derivations, such as 'discussed by George. The differences and its consequences are treated in George [19911. Because of its impact on the interpretation of experimental or numerical data, we include some results and conclusions taken from the article by George.

(1.24) (1.25)

+

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The derivation of George starts in the same way as the classical derivations, with the following similarity assumptions:

= Waf(n)

(1.29)

UW = Rag(77)

7'

with 5 = 5(z) the half-width, Wd = Wd(z) the mean axial centerline velocity and R, = R5(z) a scale function for the uw-stress. Substitution of relations (1.30) in the

BL equations (1.21 and 1.22) gives:

dz

f

6 dWd 2

S dWd d6 f2.) fridri

Wd dz dz J .10 wc3 77

R, (779)'

This must be true for all z. Equation (1.31) yields:

which can be written: Wd =B./7K

with a suitable definition of B. This can be substituted in equation (1.30), with as a

result:

[1(-P

f 177(M)

dz 77 o

This is to hold for all z, so that the terms in square brackets must be proportional, and as a result we must have:

do. Re

-d; Wel

So by now we have obtained equations (1.33), (1.34) and (1.35), with B and C con-stants which are still unspecified. Now we are at the point where the difference from the classical derivations starts.

1.4.8 Some classical approaches

In the classical derivations some additional assumption is made regarding the relevant length scales. Monin and Yaglom postulate, that we have:

Wd = Wd(z,K)

(1.36)

= Ra(z,K)

(1.37)

= 6(z,K)

(1.38) (1.39) W5227

f fzori

0 j

_

[R,

](779)' wei2 (1.30) (1.34) (1.35) i(1.33) K (1.31)

0.32)

c

5

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so that the problem is determined by one length scale (z) and a scale for the momentum entering through the orifice (K). This gives:

(1.40)

R, (1.41)

z2

z (1.42)

Since no other length scales than z enter this problem, the proportionality constant for must be universal, and the same goes for the other relationships. In other approaches the assumption is, that the solution depends on a single (far field) length scale and a single (far field) velocity scale (Townsend [1976], Tennekes and Lumley [1972]). Taking Wed as this velocity scale we have

R, (1.43)

Only the length scale z is relevant for the determination of 5, and therefore 5 is again the same for all standard jets.

Once more, these two ways of scaling the far field do not take into account any near field length scales. As a consequence, any variation in orifice velocity Wo and diameter

D do not change the far field properties, as long as K stays the same. Substitution in

equation(1.34) for both ways of scaling leads to

N/7K

Mic.2 _(1.44)

=

5,1

= z

where proportionality factors are absorbed in the universal profile functions f and g.

Thus, there is only one universal far field, which depends only on K, the momentum flux through the orifice.

In Hinze [1975] scaling is done with length scales and velocity scales of the form:

L = Lo (zLo)

(1.45)

\

U = Wo (nq

_Lo/

(1.46)

(1.47) with Lo, Wo a length- and a velocity scale, for instance the orifice diameter and the mean axial velocity at the orifice, and with p,q integers. The integers p and q may differ for each term in the governing equations, so that more than one length scale and more than one velocity scale is possible. The dependence on z of each length scale and each velocity scale is supposed to be some power of z.

Hinze arrives at the solution given above in (1.45) by first substituting his scaled

quantities in the equations for continuity and momentum (1.21) , and the integral relation (1.23). He obtains relations for the exponents p,q occurring in all terms, and needs still one additional relation to get a unique solution for p and q. For plane wakes and round jets, this can be the assumption that TM should be scaled with the same velocity scale as Tr. This is equivalent to equation (1.43). Alternatively however, Hinze also tries the

assumption of a constant eddy viscosity. As it happens, both assumptions lead to the

same solution, which is the same as for Monin et al's or Tennekes et al's given in (1.45).

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1.4.9 The new approach

In equations (1.33), (1.34) and (1.35), B and C are constants which are still free. Contrary to the classical line of thought, we do not make additional scaling assumptions at this moment. Even without making additional assumptions we can proceed further.

First we consider the profiles f and g. It is interesting to know, whether the profiles

for f, g are the same for all jets. The answer can be found from equation (1.34). This

equation is rewritten as:

(-

f2 +

f'7 AO) =[-

-1 1(u)1 77 0

(1.48)

and this equation shows, that the profile for the mean centerline velocity will be the same

for all jets if the mean axial velocity and the u-Tg-stress are scaled with Wa and WI

respectively. The u-ru-stress will differ from jet to jet by the amplitude factor R,

(1.49)

dz

which is independent of z according to equation (1.35).

We have thus found, that the mean profile and the uw-stress have always a similar shape, when they are scaled with the aid of the mean centerline velocity. It is of some interest to see, whether the same is true for other turbulence quantities, such as the RMS values for the velocity fluctuations. Moreover, it is interesting to investigate, whether something more can be said about the functional relationship 5(z). To this end the simi-larity assumptions will be used in the turbulent kinetic energy (TKE) equation, consisting of the following terms:

dissipation (1.50) [17,

+

+Wa)] az

1 a r

+

-r La

+ v' +

advection

a

az

1 a

(1.51) w' (1/12 + v + w`2) + -r arru' (u12 +1/2 + w12) diffusion air) (1.52) tilwi az az production 2 p pressure (1.53) r &yip' 1 5ril/pi-1 az

r ar

transport 2c (1.54) 2

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Denoting 0 + wi2 by F, we introduce the following scaling assumptions: W Wcif(n) &to' R3g(77) Ks(z)k(n) 2 (piui +1q2u1) T(z)t(n) E D3(z)e(77) 8(z)

Substituting in the TKE equation leads to:

do R.

dz KT,

do 8K,

dz K

In addition to these two relations, equation (1.35) gives: d8 1

R.

dzW7

Combining (1.57) and (1.58) we have:

KW

Finally, then, the TKE equation gives:

do Rs D,c5 Ts

(1.61)

dz Mr; W;

In order to proceed we need to consider more equations or use additional assumptions. For D. we can make the usual scaling assumption

W3 D, (1.62) ci which leads to do

R, ,Ts6

WC

(1.63) dz 1/1/

with C a constant, or, equivalently:

141 W3 (1.64) (1.65) (1.55) (1.56) (1.57) (1.58) (1.59) (1.60)

D

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which is just like the traditional scaling, however with no universal constants, so that for instance the spreading rate can differ from jet to jet.

By inspecting the TKE equation with the similarity relations (1.61) substituted we

see, that profiles like k for the RMS values need not be the same from jet to jet. (The

similarity functions do not appear in the same, simple form as f and g in (1.48), where g is determined, once f is given. Instead, the TKE equation allows different similarity

functions k,e,t for given f and g.)

1.4.10 The derivation of Schlichting's classical result

In this derivation of the velocity profile for a free jet, the assumption is, that there is only one relevant length scale, namely the distance z from the origin. Let us assume, that the z-dependence of the various terms in the BL equations is of the form z', with n still to be determined. To arrive at Schlichting's solution, take 8 en, and for the velocities take a stream function of the form znf (r/(5). (See Batchelor [1979] for the definition of a stream

function) This assumed form for 8 and the stream function is substituted in equations

(1.21). The ute-term is further rewritten with the aid of an eddy viscosity eo as

coft.

The

ar

eddy viscosity is assumed to be no function of r/5, and to scale as co --:6 Wd. Substituting

in equation (1.21) gives in = 1 and n = 1, so that the eddy viscosity co is constant.

From the momentum balance follows a differential equation for the function f occurring in the expression for the stream function. The appropriate boundary conditions for this equation are found from (1.23). Schlichting found an analytic solution to this equation,

namely:

The velocity in the r-direction:

1 3 ,,/k." 1713

Tr Z (1 + i,r12)2

The velocity in the z-direction:

3K

1

=

8760.Z (1+ 1772)2 Here, 77 is given by:

3 r

17=4

ir co

The eddy viscosity co, which is constant in Schlichtings model solution, is given by: co = 0.01611/TC

Observe, that this solution leads to the result that all round, turbulent jets with the same total momentum efflux K have completely the same velocity profile.

Schlichting's analytical result should be re-examined to verify, that all assumptions made are valid. First of all, a dominating velocity component in axial direction is sup-posed. Schlichting's solution shows, that the ratio between the radial and the axial velocity component is 0 on the axis. This ratio is equal to:

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Table 1.1: Ratio between radial velocity rz and axial velocity iii, asia function of r/z

A quick impression of this ratio can be obtained from table 1.1. The boundary layer

assumption seems satisfied well enough for the region of interest. (Experiments usually do not extend further than r/z = 0.25, and the calculations will not go beyond r/z = 2)1 Still, there will be some distance from the axis where the axial velocity component is no longer dominating. In other words: the solution is valid only for a finite range of

1.4.11 _{The derivation of George for a co-flowing jet}

The similarity considerations for the co-flowing jet or wake differ from those of the

stan-dard jet. In the following we shall see, that our simulations resemble rather such a

co-flowing jet. Therefore, some of the analysis of George [1987] related to co-flowing jets is repeated here.

For co-flowing jets, the fact that there is not a universal far field has already been

noticed for some time. This is, because differences in observed spreading rates etc. were much larger for wakes than for standard jets. Moreover, for co-flowing jets some authors

have even expressed their doubt that there is similarity in the far field (Antonia et al

f1.973]). The reason for this was, that the predictions of the classical similarity considera-tions did not give results which correspond to experimental results. For instance, classical scaling suggested a decay as X-113, but the experiments rather suggested

(Ra-jaratnam [1978]). Contrary to the standard jet case, the deviation between theory and

experiments was not just a question of the magnitude of some constant, but also of func-tional form!: A possible explanation is, that co-flowing jet measurements simply havenot, been performed far enough from the origin (see Roth [1975] and section (1.9))

Some consequences of the assumption of similarity are derived here, following the same: kind of reasoning as for the standard jet. We take the governing equations (1.25), and (11.28), which use the fact, that (IF

W.) /W. c< 1 and Fi/W. << 1:

aw 1 anew' Woo

az

r

(167) 191U 1 arff

+-- o

(1:68) Oz

r ar

We have the following boundary conditions:'

= 0 and

0 at r =

(L69

ar

tT, = Wco for r co _.(1.70)

plus the integral relation:

2r f00 (Ye

W.) rdr = k

,(1.71)1.

r/z

ratio VT) 1.01 2.47500E-02 2.0 .0 3.0 -.12375 4.0 1 -.396 8r 0

(30)

This equation can be rewritten with the introduction of the momentum thickness 0 as:

Woc,( W,)rdr = 7rW.2 02

(1.72)

Let us substitute the similarity assumptions

(v Won) = W3i.(77)

(1.73)

mu = R,g(77)

17=

5(z)

(1.74) Substitution in equation (1.68) gives:

I 5 dW31 d51 , [.Tiz] 71' R. (rig)" (1.75) dz 77 (1.76) Since this must hold for all z, we have:

do SdW,, (1.77) dz R. dd W, dz (1.78)

The integral relation (1.72) becomes:

{W,W3,3821 2/1- f f77d77= irW02 (1.79)

from which we get:

WI'

147,,092 (1.80)

If we use this relation to eliminate derivatives of W, in equation (1.75) we get:

5](2771 +772f1)=[ dz R, (1.81) ](779)1 W,cW. which can be reduced to:

(7721 [ R. ds]

(Iry

_(1.82)

which can be integrated to:

(iif) R,

rig

_(1.83)

(1.84) Just as for the standard jet, a universal profile f is possible for the mean velocity, if it is scaled with W,. Given such a universal f, the function g may still differ from jet (wake) to jet by the amplitude factor

1.417W,

[ R, 1

(31)

The functional dependence on z of Ws and iS is still free, and we resort to the TKE,

equation to get additional relations. The following terms in the TKE will be considered::

a

az[w,

+171+ advection

+ v +

'2 zuf2) + -1

(0 + v'2 + /0)

az

r ar

diffusion 2 I[teteaz] production

P ir

Or 2 P. arITT'l pressure transport 2e

0.80

(1.87> (1.88) (4.89) dissipation

Note, that some terms have been dropped' when compared to the TKE equation for the

standard jet (equation (1.51)). Denoting 0 + 0 +2E'2 by F, we introduce the following scalings: Plisf(n) u w _R.,g(n) 1 -1 -2q

= K,(z)k(n)

= (pi

+

= T,(z)t(n)

2

C = D,(z)e(q)

Substitution in the TKE equation gives:

do dK, dz

K, dz

do RAW,

Irs 7, Tv:

do dz d6

D, 6

,crz

Equations i(1.78) and (1,94) give:

(1.97) (1.92) (1.93) (1.94) (495) (1.96) (1.90)

=

= (1.91)

(32)

and if thistle substituted in (1.94), then using equation (1.80) we find that it is equivalent to equation (1.77), so that equation1.77 can be omitted. Using relation (1.97) in (1.93) we find that this equation is equivalent to equation (1.77) again, so that equation (1.93) can be omitted too. We retain:

do R, 198 crz

WW,

do (1.99) crz dO D, (1.100) W3(52 Woo02

Further assumptions are needed to proceed further. Two kinds of assumptions are tested, i.e. two possible scalings are studied for the dissipation scale function D,:

W3 6 (1.102) D VV2 D, (1.103) 6-2

The first relation (a scaling according to Kolmogorov's hypothesis) leads to:

Woo,

M

Wa5 (Wc.0) (

z)=31-Ii ii

)

and the second one (a scaling which applies to a laminar jet) leads to:

6

(ti( z)i

14T0)

)

(

v ( 0) Woo )

z)

W,5

(W0)1

(z-ii ) 0)

Together with the half-width and centerline velocity as a function of z, we have also given the Reynolds number (equations (1.106), (1.109) as a function of z for future reference. The second alternative corresponds to the solution for a laminar wake (Schlichting [19791), which should not come as a surprise, since the dissipation has been scaled as for a laminar

flow. It should be noted, that we have arrived at the relations above without making

assumptions about the near field dependence of the amplitude constants.

Hinze [1975] finds the same alternatives as George, which are given above. After substituting his relations (1.47) in the equations of motion and the integral relation, he still needs an additional equation to find a unique value for the exponents p, q. The first alternative of George is found, by assuming R, W. 5 The second alternative is found

5The eddy-viscosity in this first case proves to vary as l/zi.

(1.101)

(1.104)

(1.105)

(1.106)

Large-eddy simulation of a turbulent jet