INFLUENCE OF CORIOLIS FORCES ON THE DYNAMIQUE OF A
TURBULENT JET
M. Hasnaoui *
* Department of structures and materials, Air Royal School, Marrakech (Morocco)
hasnaouimohammed@hotmail.com
Key words: Turbulent free jet, Mean velocity, Linear spectral analysis, Asymptotic analysis,
Kelvin-Helmoltz instabilities, Coherent vortices.
Abstract. The influence of Coriolis forces is studied under realistic conditions on a jet in
1 INTRODUCTION
The two-point closure theories appear as less empirical than most usual models. Such theories are mainly applied to isotropic homogeneous turbulence. Their extension to homogeneous anisotropic turbulent fields allowed to enrich rapid deformation theories. These theories, also called linear models, have been previously developed following Craya's works1. However, one can think that the study of homogeneous fields remains somewhat academic when this study doesn't find applications or extensions to the inhomogeneous turbulence. In this view, a simple spectral model has been successfully applied by Bertoglio and Jeandel2 to the inhomogeneous turbulence of a boundary layer along a flat plate, showing the great practical interest of a two-point closure.
The interaction of Coriolis effects with the turbulence originated an active research several years ago. This question arises in various problems encountered in engineering science3 and geophysical applications4. At modelling point of view, a linear approach has been proposed by Bertoglio5 in order to study a homogeneous turbulent flow in rotating frame. More recently, another work dealing with a two-dimensional homogeneous turbulent flow6, 7 places in evidence the role of Coriolis forces on spectral densities, combines the interaction of a uniform shear and weak rotation of the system.
We are interested, in this paper, in the diffusion of a three-dimensional jet in a buoyant fluid turbulent in the presence of Coriolis forces. Because of the existence of a privileged direction, this deals with inhomogeneous turbulence. The problem is described by a linear spectral approach. Such an approach allows to separately taking into account the several scales of turbulence. From another point of view, since the unicity of the solutions of Navier-Stokes equations holds only for two-dimensional problems, it is of importance, in the three-dimensional case, to use asymptotic modelling to determine approximate solutions. In this view, we assume that there exists a particular direction along which the jet develops. This circumstance occurs, for instance, if the jet is emitted from a source (for instance the orifice of a chimney). In this model, there exists a small parameter ε characterizing the relative thickness of this layer. It is only recently that asymptotic tools have been combined with turbulent modelling8, 9. In the present problem the parameter ε allows to construct a boundary layer approximation as well as provide additional equations necessary for the analysis. The use of both combined spectral analysis and asymptotic tools then allows to consider practical problems : for instance, a good understanding of the influence of Coriolis forces on a turbulent structure may improve our ability to predict the mailing of rotating flows such as cited above : mixed convection generated by the rotation of the walls in a vertical annulus 3 or the rotation of the medium in atmospheric flows and oceanic flows 4 .
order to define the spectral tensor of our model. This procedure, which combines both spectral analysis and asymptotic analysis, is analogous to those followed, first, by Bertoglio, and, secondly, by Anderson and Mazumdar. The main differences between the mathematical procedures used by these authors and the present analysis are the asymptotic expansion involved here. In Sect. 4, solutions are proposed to these equations. In sect 5, some examples are considered, for which we estimate the application ranges of this model.
2 ONE-POINT CORRELATION TRANSPORT EQUATIONS
We consider the instationary jet flow of a buoyant viscous incompressible fluid moving in terrestrial frame. The ambient medium occupies a half space Ox1x2x3, (x3 ≥ 0, unit vectors i
, gravity g= −g k), itowards east direction. The rotation Ω of the Earth is located in the vertical south-north plane. The physical variables are denoted using primes. With classical notations the equations of motion are the Navier-Stokes equations with Boussinesq approximation, namely:
0 = x′ ∂ u′ ∂ δ ′ θ g β + u′ ∆′ ν = x′ ∂ p ∂ ρ 1 + u′ Ω ε + x′ ∂ u′ ∂ u′ + t′ ∂ u′ ∂ j j k 3 k k * n l ln k j k j k
(1)
in which θ′ denotes the fluctuation of temperature from its equilibrium value (Boussinesq approximation,
β
is the coefficient relating the temperature fluctuation and the density fluctuation). Theε
klnand
δ
ijare
the antisymmetric and symmetric Kronecker symbolsrespectively.
The components Ωjof the rotation have the values Ω1 = 0, Ω2 = Ω/tg ϕ, Ω3 = Ω
(Ω is the vertical component of the Earth's rotation, ϕ is the latitude). The main flow is assumed to be a jet along the x1-axis.
The whole velocity field splits up into mean part and turbulent part:u′k=uk′+uˆ′k,
pˆ p p∗ = ′+ ′
and θ ′=θ ′+θˆ ′
.
In this decomposition the averaged variables u′kare governedby equations analogous to (1), but involving in their right hand side the double correlationu uˆ ˆk′ ′ . The fluctuations are solutions of the difference between these equations and j
k j k j j k i k m j m i m m 2 k j k j 1 j 1k m m j k j 1 f k j m m T ( )u uˆ ˆ ( )u uˆ ˆ u u + [( ˆ ˆ ) ( ˆ ˆ ) ] u u u u u t x x x ( )u uˆ ˆ - p[ˆ uˆ uˆ ] p[ˆ uˆ uˆ ] x x x x x x ˆ ˆ u u R +2 ( ) ( ) x x ∂ ∂ ∂ ∂ + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ − + − δ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ − ∂ ∂ σ
(
)
3 3 3 3 k 3 3 2 k 1 3 j 2 j 2 lj 2 k l 3lj 3 2 l 2 k 1 1 o u u uˆ ˆ x 1 u u = - ( ) [ u uˆ ˆ 2 ] + [ ( ) 2 u uˆ ˆ ( ) u uˆ ˆ ] x x R ∂ δ ∂ ∂ ∂ ε δ + δ ε ω + ε ω δ ∂ ∂ (2)Where we have set:
ω
1= 0; ω
2= 1/tgϕ ; ω
3= 1 and k=1, 2, 3; m=2,3
(3)In which uk denotes the components of mean velocity field; k1, k2 et k3 are the components of
the wave vector along x1, x2 and x3 respectively. k ( k2⊥ = 22+k )23 is the transverse wavenumber,
and M(=kjuj) measures the interaction between the wave-vector and the mean velocity field.
o o
R =u ΩLis the Rossby number (L is the horizontal length scale, uo is a characteristic speed
of the turbulent fluctuations); ωjare the components of the Coriolis forces.
( )
xu N G R 3 3 2 2 R f ∂ ∂= is the flux Richardson number associated with the shear of the mean
flow (where GR is the Grashof number, and N is the Brunt-Vàisâlà frequency of the ambient
medium). σT is the turbulent Prandtl number (usually is approximately chosen11 equal to 0.9).
The small parameter ε =d / L(d is the thickness of the jet) characterising the relative thickness of this layer, and this parameter allows to construct a boundary layer approximation as well as to provide additional equations necessary for the analysis. Moreover the system (2) is valid under the asymptotic restriction:
1 e
R−
ε = (4) relating the Reynolds number of the flow and the stratification of the medium. The relation (4) seems to be restrictive, but, from a mathematical point of view, it characterizes a phenomenon where the stratification and the dissipation originate competing influences.
3 LINEAR SPECTRAL ANALYSIS
We now introduce the two-point correlations of the velocitiesu u . These correlations are ˆ ˆk j defined using two points nearby x, say x' and x", in the following manner:
) x ′′ + x′ ( 2 1 = x ; x′ -x ′′ = r ; x′ -x ′′ = r ) t , x ′′ ( uˆ = u ′ˆ′ ; ) t , x′ ( uˆ = u′ˆ ; u ′ˆ′ u′ˆ = R k k k k k k k j k kj (5)
We obtain, for the two-point correlations, and after some calculation, the equation satisfied by these correlations, namely:
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ δ ∂ ∂ ∂ ∂ + - [ u [ k kj i kj i km j m jm k m k m m kj m m j j k k j j R t u R x R u x R u x j R x x R r r p x p u x p u x p u x + + + − ′′ ′ ′ + ′ ′′ ′′ − ′′ ′ ′ + ′ ′′ ′′ [ ] [ ] ] ] 1 2 2 2 2 1 1 1 - = - + + , f T j k k j j o lj kl lj l k kj R R u x R u x u x R R R W σ ∂ ∂ δ ε ∂ ∂ δ ∂ ∂ δ ε ω ε ω δ 3 3 3 3 1 3 1 3 2 1 2 2 2 3 3 2 2 2 1 2 [ ] [ ] + + (6)
where Wkj is the inhomogeneous spectral transfer. In this study, this term will be neglected.
In order to convert the correlation equations into spectral forms, we now introduce three- dimensional Fourier transforms with respect to r in the following forms:
Where k=(k , k , k )1 2 3
is the wavevector and
I= (-1)
1/2 is the complex number. The tensor Φ (of elementsΦ ) is the spectral tensor. After some calculation the Fourier transforms take ijthe simplified forms:
kj kj k j 2 f 3 mj km kj 3 j i 3k i m m T 3 2 1 3 k mj j km 1k 1 2 j j 12 1 3 j 2 m 2 3 4 k1 u u R u [ ] 2 u k t x x x x 2 M M M + (k k ) [ (k k ) k ] k x x x - ⊥ ⊥ ∂Φ + ∂Φ = − ∂ + ∂ − + ∂ δ Φ Φ Φ Φ ∂ ∂ ∂ ∂ σ ∂ ∂ ∂ ∂ + −δ + + Φ Φ Φ Φ Φ ∂ ∂ ∂ = ε 3 2 kl 2 l 3 j 2 j 2lj 2 3lj 3 2 k 1 1 o 2 3 k 1 j j k1 1k 1 1 j 2 o 1 1 u u [ 2 ] [ 2 ] x x R 1 k k tg M [( 2 ) (k k k )] k R x ∂ ∂ + + + δ δ ε ω ε ω δ Φ Φ Φ ∂ ∂ − ϕ ∂ + − Φ + Φ − δ Φ ∂ (8)
On the other hand the incompressibility equations read: 1 2 x I 1 2 x I m 1 ∂ ∂ ε ∂ ∂ km m km k k k k Φ Φ Φ Φ + = − [ 1+ 1 1] (9)
The different terms of the right hand side of Eq.(8) express the contributions of various processes to the rate of change of the spectral tensor : the first term is the rate of production by work done on the turbulence by mean velocity gradient. The second term represents the rate of viscous dissipation. The third term is the rate of production due to the vertical heat flux. The fourth term is the rate of linear transfer between pressure and turbulent velocity gradient. The remaining terms (proportional toε in Eqs.(8)) express, on one hand, the rotational production associated with the action of Coriolis forces, and, on the other hand, the rates of production associated with a transfer and production due to geometrical effects.
4 A CASE OF ANALYTICAL SOLUTION
In this view, we assume that theΦ 's admits the following asymptotic expansions: KJ
Φkj =Φokj+ε Φ1kj+ε2 Φkj2 + ⋅⋅ ⋅ (10)
the general form of this equation may be written as:
Lkj( )Φ = − ε Akj (11)
Lkj( )Φ = Lo(Φkj)+ L1km(Φmj)+ L1jm(Φkm)+Fkj( )Φ (12)
The procedure for solving (11) is, in general, purely numerical. However, there exist a case where L acts on kj Φ only. This case occurs when the two equalities: KJ
2 32 2 3 2 k k M x u x ⊥ = ∂ ∂ ∂ ∂ ; 2 2 2 3 2 3 k k M x u x ⊥ = ∂ ∂ ∂ ∂ (13) are satisfied simultaneously. Under this assumption, a solution of Eqs.(11) may now be looked for using a generalized asymptotic expansion (10).
By prescribing that 1 kj
εΦ is uniformly small before 0 kj
εΦ in Eqs. (11), we obtain, for the terms of order 1, the non-secularity condition:
Akj= (14) 0
The relations (13) and (14) permit to define the velocity field associated to the spectral tensorΦ . At order 0, these equations now read:
kj kj o
L (Φ )= 0 (15) All equations (15) now appear as first-order linear homogeneous equations with no constant coefficients. The spectral tensor of our model can be deduced.
4.1 Determination of the mean velocity field
In what follows, an approximated solution is proposed, related to a nonuniform mean speed profile. The study is made in the spectral space.
At large Reynolds number, it is shown using the results of Hasnaoui12 that the mean velocity field is of the form:
1 3 1 1 1 1 1 3 2 2 o 2 1 1 3 3 J 3 o ( , , t) U f ( ) (t) u x x x U U * x ( , , t) (t) u x x U R k x ( , , t) W (t) u x x U tg( ) R ∞ ∞ ∞ ∞ ∞ = + η + σ = − − ξ + = − − ξ + + ϕ (16)
In (16), h (=h*/r) denotes the relative height of the chimney (h* is the dimensional height of the chimney and r is its radius), *
s
U ( U h u )∞ = is the adimensional velocity of wind
( *
U denotes the dimensional speed of wind, u is the speed in the chimney (of order 4 at 5 s
m/s)), *
J J s
W ( W / u )= is the non-dimensional speed at the exit of the chimney ( * J
W is the dimensional speed at the exit). 1(t), 2(t), 3(t) are the arbitrary functions, namely :
3 2 1 3 1 2 2 2 3 2 2 2 3 2 2 3 2 2 2 2 2 o 3 2 o 3 2 k a b (t) (t) ; (t) (t) 2 2k (k k ) 3 (k k k tg( )) 3 (k k k tg( )) a ; b ( ) ( ) k⊥ R k k R k k + = = − − − ϕ − ϕ = = − − (18)
Observation of the smoke emitted from the chimney shows that the flow possess a certain organization in sufficiently periodic manner. The smoke occupies the areas of similar form. Also, the visualization of turbulent boundary layer shows that the flow is formed by structures which are of repetitive allure. In all cases, there exists an interaction between the turbulence and its boundary. It is only near the frontier that plays the training mechanism. So, we admit, in this region, that the function f ( )η represents the law of deficiencies velocities, namely:
f ( )η →0 when x ,1 ξ → (19) 0 The velocity field satisfies boundary conditions and matching conditions with the outer medium (out of the boundary layer), namely:
1 2 3 J 1 * 1 1 u u 0 ; u W for x t 0 u U x∞ U (t) if ∞ ∞ and 1 = = = = ξ = = = + ξ → ξ η → (20)
Moreover, we note the existence of a temporal function1(t): this part is responsible for
the intermittency process. In a free flow, the boundaries vary in the time in significant manner, thus originating boundary intermittency13, 14.
In order to illustrate the fundamental characteristics of the boundary intermittency, the functions 1(t) and f ( )η are assumed, taking into account of Broze's model
4.2 Determination of the spectral density 0 kj
Ψ
It is now recognized that the wavelengths (phase, amplitudes) correlated vortical structures within a single realization determine many properties of the turbulence. In order to understand the role of these coherent structures in turbulence, one requires experimental and mathematical tools capable of detecting and analyzing such structures.
Thus, in this study, the Fourier transform allows one to investigate simultaneously the structure of a signal in both physical and wavenumber space. It is ideally suited to the investigation of coherent structures in turbulence because it allows the definition of the local 'Fourier energy spectrum'.
The Fourier energy spectrum integrated over the whole flow (spectral density) may to determine how that structure contributes to the energy spectrum of the flow as a whole. The Fourier transform can also be used to detect coherent structures in turbulent flows. Strong, isolated cones in the Fourier transform are, therefore, the signature of eddies or other approximately singular structures in the flow. Coherent structures in a turbulent flow can thus be both detected (by looking for cones) and characterized (by finding the order of singularity and local energy spectrum) using the Fourier transform16.
The spectral analysis allows taking into account the several scales of turbulence. Indeed, integration is made over a sphere, whose radius is equal to the modules of the wavenumber, accordingly turbulent structures are only characterised by their sizes. Consequently, the spectral components computed with a spherical average operation, namely:
Ψkjo( , )x ki =k2∫ ∫Φokj( , ) sin( )x ki 0 2 0 θ φ θ π π d d (22) where: 3 1
k (k sin cos , k sin sin , k cos ); is the angle between k and the x axis; is the angle between k and the x axis
= θ φ θ φ θ θ − φ − (23)
3 / 2 o 2 2 33 i 33 a J 2 n n 1/ 2 3 1 2 n 1 o n 1 / 2 n 0 J 2 3 1 o 2h ( , k)x 2 k B (k) exp[ (R k )] W 2 x [2k[ h ]] J x x 2hk tg( ) R U tg( ) W x 2 k[x h x] tg( ) R U tg( ) + = ∞ + = ∞ = π Λ − Ψ − − − ϕ ϕ ∑ − − − ϕ ϕ (27)
Where Γ(n)is the Euler function and J(n) are the Bessel functions. The B (k) are kj
homogeneous functions18. R is the Rayleigh number and a Λ ( 2
0
N d g
= Θ ∆θ , N is the
Brünt-Väisälä frequency of the ambient medium, Θ is the potential temperature and ∆θ is the 0 difference of temperature between the jet and the ambient medium) is a characteristic freauency .
We note, that the spectral tensor Ψ and the mean velocity field associated u, are valid under the regularity condition:
U∞ 1 (28)
5 APPLICATIONS
In order to validate the preceding results, some examples (with the software Maple) for which we estimate the applications range of these models. In these examples, the adimensional velocity U∞is defined as: 0.1≤U∞≤0.2 and the adimensional speed at the exit of the chimney W is defined as: J 0.8≤WJ≤ . The latitude is fixed: 1 ϕ = π 6. A vortex size λ
measured by a correlation is therefore associated to a wavenumber k such: k λ = Ο(1). So, the large structure corresponds to large λ (small k), while the small structure corresponds to small λ (large k).
Finally, the Coriolis force is characterized by the Rossby numberR . So, the weak rotation 0
(a) (b)
(c) (d)
Figure.1. Modulus of velocity. Ro=10,Re=105 and z∈[-1, 1];. U∞ =0.1, WJ =1, h=100, φ = π/6. k2=2/30,
k3=1/30, t=1. a) x∈[0,1] ; b) x∈[0,3]; c) x∈[0, 8]; d) x∈[0, 13].
(a) (b)
(c) (d)
Figure.2. Modulus of vorticity. Re=105. U∞ =0.1, WJ =1, h=100, φ = π/6. k2=2/30, k3=1/30, t=1. x∈[0,1] and
z∈[-1, 1]. a) Ro = 1000; b) Ro = 100; c) Ro = 50; d) Ro = 10.
Figure.3. Density spectralΨ12. k=1/50. U∞ =0.1, WJ =1, h=100, φ = π/6. x∈[0,1] and z∈[-3, 3]. a) Ro=20,
b) Ro=8, c) Ro=2.
The figure (3) shows the influence of the Coriolis forces on the coherent structures. Approximate singularities in the flow appear in the plane of the density spectral Ψ as long, 12
slender cones pointing towards the location of the big structures. The intensity of these structures decreases when the rotation increases.
6 CONCLUSION
Using a model of inhomogeneous turbulence, we exhibit some consequences of the vortex dynamics applied to a free jet submitted to Coriolis forces. The most important effect is the appearance of Kelvin-Helmholtz instability and the appearance of coherent three-dimensional
vortices induced by the jet itself. The effect of the Coriolis force on the spatial evolution of these structures along the jet is observed.
We have presented some simple vortex dynamics arguments allowing us to understand qualitatively the formation of coherent vortices in the fully developed turbulence. Of particular interest is the vortex interaction such as: roll up of a vortex sheet.
The results obtained in the present study globally show the importance of the Coriolis forces in the modelling of the inhomogeneous turbulence, and the power of the spectral approach and the asymptotic analysis for this modelling. The several characteristics usually encountered in the inhomogeneous turbulence are correctly predicted by our model. Matching of predictions, numerical simulations and experimental results19, 20, 21, 22 remains quantitatively acceptable.
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