Lorenz Navier

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    International Journal of Non-Linear Mechanics 38 (2003) 629–644

    A Lorenz-like model for the horizontal convection ow

    E. Bucchignania;   ∗, A. Georgescu b, D. Mansuttic

    aC.I.R.A., Via Maiorise, 81043 Capua (CE), Italy bDepartment of Applied Mathematics, University of Pitesti, Romania

    cI.A.C.= C.N.R., Viale del Policlinico, 137 00161 Rome, Italy

    Received 18 January 2001; received in revised form 30 August 2001

    Abstract

    In this work we study a nite dynamical system for the description of the bifurcation pattern of the convection ow of a uid between two parallel horizontal planes which, under the hypothesis of the Oberbeck–Boussinesq approximation,sustains a   horizontal   gradient of temperature (horizontal convection  ow). Although in the two-dimensional casedeveloped here, literature reports a long list of analytical and numerical solutions to this problem, the peculiar aimof this work makes it worthwhile. Actually, we develop the route that Saltzman (J. Atmos. Sci. 19 (1962) 329) andLorenz (J. Atmos. Sci. 20 (1963) 130) proposed for the vertical convection ow that started successfully the approachto nite dynamical systems. We obtain steady-to-steady and steady-to-periodic bifurcations in qualitative agreement

    with already published results. At rst we adopt the non-dimensional scheme used by Saltzman and Lorenz; as weobtained huge values of the bifurcation parameters, we introduce a dierent set of reference quantities for overcomingthis drawback.  ?  2002 Elsevier Science Ltd. All rights reserved.

    1. Introduction

    The historical roots and the physical backgroundof the research topic here considered are describedin some detail by Georgescu and Mansutti in [1,2].

    We introduced briey here the main ideas. Inthermodynamics of uids [3] by convective mo-tion or, shortly, convection it is understood that auid ow which, apart from mechanical quantities(e.g., velocity), is characterized also by thermal(e.g., pressure, temperature) and= or other elds

    ∗ Corresponding author.E-mail addresses:   [email protected] (E. Bucchignani),

    [email protected] (D. Mansutti).

    (e.g., concentration, magnetic and electric eld).Conduction is a particular phenomenon character-ized by zero velocity eld. Vertical and horizon-tal convections are characterized, respectively, byvertical and horizontal temperatures and= or other 

    quantity gradients.For external vertical gradients, a conduction state

    is always possible. It can loose its stability at somecritical value of the control parameters so that con-vection sets in. For further modications of thecontrol parameters repeated bifurcations occur thatmay be referred to as the Landau–Hopf (L–H) sce-nario, [4,5], leading from regular (deterministic) toirregular (turbulent) uid motions. The L–H sce-nario cannot be proved, in general, for the modelsgoverning uid ows. However, the rst steps of 

    0020-7462/03/$ - see front matter  ?  2002 Elsevier Science Ltd. All rights reserved.P II: S 0 0 2 0 -7 4 6 2 (0 1 ) 0 0 1 2 0 -2

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    the L–H scenario were observed in specic verticalconvections [6] and in other phenomena.

    There exists a huge variety of vertical convec-

    tions according to the geometrical conguration andthe present eects acting on the uid system [6–8].In them some other scenario was also (partially)observed. We quote here the Ruelle–Takens (R–T)scenario, according to which, after a nite number of bifurcations (two or three at the most) turbulencesets in.

    For all vertical convections, the form of the ge-ometrical domain and the external inuences areincluded in the control parameters: aspect ratios,non-dimensional numbers (Prandtl, Rayleigh, Hart-mann, etc.). At present, a lot of studies concern-ing many specic types of vertical convections areavailable. In most of them, the basic conductionstate is assumed and the underlying physical mech-anism is well understood.

    The papers devoted to horizontal convection arefar less numerous than in the vertical case mainlydue to the fact that an exact basic state is not pos-sible. Actually, for the horizontal convection thereexists a basic ow and, apart from very simplesituations, its closed-form is not known. Moreover,the form of the basic ow depends on several pa-

    rameters, hence, from the very beginning we mustconne ourselves to some region in the parame-ter space. This represents an additional dicultycompared with the vertical case and claims for approximation methods.

    Mostly asymptotic approaches are used. How-ever, as each asymptotic treatment is based onspecic assumptions on the geometrical and phys-ical eects (involving the order relating the quan-tities [9]), this treatment holds in appropriateassumptions.

    In general, the most important simplicationin convection problems is the space periodicity.This enables one to consider only a periodicitycell, hence a bounded domain, therefore to useembedding inequalities. But, unlike the abstractviewpoint, in applications of a primary importanceis the form of the cell [10]. For rectangular geome-tries implying space-periodic quantities, the aspectratios are very important parameters. Various as-sumptions on their order relation will imply quitedierent treatments [11]. In one of these hypothe-

    ses in [1] we performed an investigation based onan approximate quasi-stationary basic ow (i.e.,in the rst approximation it reduces to a basic

    state).If several eects are competing (e.g., thermal,

    concentration, magnetic) there exists the possibil-ity to nd some particular relationships between theexternal thermal and other gradients such that a ba-sic state be possible [12].

    Having in view that a treatment of the horizon-tal convection on the basis of the partial dierentialequations is asymptotic in nature, we may avoidthis by using models based on ordinary dieren-tial equations. The most famous model of such akind is the Lorenz model [13,18] which describesthe vertical convection ow. It was derived fromthe Navier–Stokes–Fourier partial dierential equa-tions simplied within the Oberbeck–Boussinesqhypothesis by expanding the unknown functionsinto Fourier series with respect to space variablesand truncating the series to rst terms. In this pa- per, we shall transfer the same approach to thetreatment of horizontal convection and reduce our innite-dimensional problem to a nite dimensionaldynamical system. In the following paragraph, wedescribe the geometrical setting of the horizontal

    convection problem and introduce the mathematicalformulation. Then a section on the non-dimensionalform follows, where two schemes are proposed. TheLorenz-like dynamical system is built up in the fthsection together with the solution procedure and adiscussion of the coherence of the model. The nu-merical results and the nal remarks conclude this paper.

    2. Mathematical formulation

    We consider a viscous uid layer enclosed be-tween two horizontal walls whose mutual distanceis  H   (Fig. 1).

    A steady thermal eld with a horizontal gradi-ent T 0  is applied and induces a steady longitudi-nal ow that loses its stability when T 0  becomeslarger than a critical value. Then, as in the case of vertical convection, for T 0  increasing, several bi-furcations occur and a transition to chaos is fore-seen for T 0  approaching innity. We assume that

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    x

    y

    g

     

    H

    z

    ∆T

    Fig. 1. The ow conguration.

    the uid slips on the two horizontal walls that areimpermeable and at xed temperature.

    In this condition, the uid motion can be rep-resented by the Navier–Strokes equations for aviscous incompressible non-isothermal uid. For the sake of simplicity we adopt the followingBoussinesq–Oberbeck approximation:

    @u

    @t   + (u · ∇)u +  1

    ∇p− ∇2u − g̃  k  = 0;   (1)

    ∇ · u = 0;   (2)@

    @t   + (u · ∇)− ∇2= 0;   (3)

    where u(u;v;w);  and p are, respectively, the ve-locity, temperature and pressure elds and  is thedensity,  is the kinematical viscosity, g is the grav-itational acceleration,  is the coecient of thermalexpansion and  is the conductivity.

    Since a coherent reconstruction of this approxi-mation has been nally provided by Rajagopal et

    al. in [14], we must stress the limitations to its ap- plicability. It is known that the above equationsdescribe a mechanically incompressible and ther-mally compressible uid. This formulation can bemathematically built through a perturbation proce-dure applied to the equations for a linearly viscousnon-isothermal uid in the non-dimensional formsuggested by Chandrasekar [15]. Then by choosingthe perturbation parameter   equal to the ratio be-tween two properly chosen characteristic velocities,it has been shown that the Oberbeck–Boussinesq

    equations follow from keeping in the ow expan-sion terms up to order  4 and selecting properlyterms in the full equations to be set to zero. On

    the contrary, in a consistent perturbative formula-tion the equations would be obtained from the fullequations by equating to zero the sums of the termsof the same order up to 4.

    In the present problem, we will suppose that thevelocity eld has only two non-zero components(= 0); in this case due to the continuity equationwe can introduce the stream function     =   ( x; z )and rewrite the above equations in the followingway:

    @  

    @t    +

    @  

    @x

    @  

    @z   −@  

    @x

    @  

    @z  − ∇4

       − g@

    @x  = 0;(4)

    @

    @t  − @

    @x

    @  

    @z   +

    @

    @z 

    @  

    @x − ∇2= 0:   (5)

     Now, similarly to what is done by Gershuni etal. in [16] we consider a generic parallel base owcompatible with the prescribed physical set-up,U 0 =U 0( z ); T 0 = T 0 x + ( z ). Then the ow andtemperature elds obtained by perturbing the uid

    obey the following equations:

    @  

    @t   + U 0

    @  

    @x  −  d

    2U 0d z 2

    @  

    @x  +

    @  

    @x

    @  

    @z 

    − @  @x

    @  

    @z  − ∇4   − g@

    @x = 0;   (6)

    @

    @t   + U 0

    @

    @x −  1

    2

    @T 0@x

    @  

    @z   +

    @  

    @x

    d

    d z  − @

    @x

    @  

    @z 

    +@

    @z 

    @  

    @x − ∇2= 0;   (7)

    where now     and  relate to the perturbation eld.The boundary conditions associated with this sys-tem result

       =    = = 0 at  z = 0; H:

    Afterwards, by restricting the set of the solutions,we shall build an ordinary dierential model alongSaltzman [17] and Lorenz [18] routes for the de-scription of the vertical convection.

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    3. Non-dimensional schemes

    We study the equations in non-dimensional form.

    Here, rst we adopt scheme 1 used by Saltzmanand Lorenz for the vertical convection and then, by few modications, we obtain a new scheme 2which allows to improve sensibly our numericaldescription.

    3.1. Scheme 1

    This scheme is built up on the hypothesis thatheat transfer due to conduction is comparable withthe convection eect and was similarly adopted by

    Saltzman and Lorenz for the vertical convection problem [17,18]. Here, the reference quantities are

     L∗ = H; t ∗ = H 2

    ; ∗ =

    gH 3;

    as a consequence, we have the following combinedreference values for velocity, stream function and  :

    u∗ =

     H ;   ∗ =;     ∗ =

     H 2:

    The non-dimensional form of Eqs. (6) and (7)following from the above quantities appears:

    @  

    @t   + U 0

    @  

    @x  −  d

    2U 0d z 2

    @  

    @x  +

    @  

    @x

    @  

    @z 

    − @  @x

    @  

    @z  − Pr ∇4   −  Pr @

    @x = 0;   (8)

    @

    @t   + U 0

    @

    @x −  1

    2

    @T 0@x

    @  

    @z   + Ra

    d

    d z 

    @  

    @x − @

    @x

    @  

    @z 

    +@

    @z 

    @  

    @x − ∇2= 0 (9)

    in which  Pr   and  Ra  are, respectively, the Prandtland Rayleigh numbers, dened as

     Pr =

    ; Ra=

    gT 0 H 3

    :

    3.2. Scheme 2

    It will be clear from the numerical resultsobtained by the above scheme that, for the

    horizontal convection, a more suitable choice of thereference quantities have to be accomplished. Whenheat transfer due to convection is larger than the

    one due to conduction, this should be the case, anappropriate scheme may be built with the followingreference quantities:

     L∗ = H; u∗ =Gr 1= 2

     H ; ∗ = T ;

    as a consequence, the combined reference valuesfor time, stream function and     result:

    t ∗ = H 2

    Gr 1= 2;   ∗ = Gr 1= 2;     ∗ =

    Gr 1= 2

     H 2;

    where Gr  is the Grashof number, given by

    Gr = Ra Pr 

    :

    The non-dimensional form of Eqs. (6) and (7)following from the above quantities appears

    @  

    @t   + U 0

    @  

    @x  −  d

    2U 0d z 2

    @  

    @x  +

    @(;  )

    @( x; z )

    −   1Gr 1= 2

    ∇4   − @@x

     = 0;   (10)

    @

    @t 

      + U 0@

    @x − 1

    2

    @T 0

    @x

    @  

    @z  − d

    d z 

    @  

    @x

    − @(;   )@( x; z )

     −   1 PrGr 1= 2

    ∇2= 0:   (11)

    4. Basic ow and temperature elds

    In the literature several authors, for exampleGershuni et al. in [16], developed their analysisof horizontal convection on the basis of a basicow and temperature elds algebraically built up.

    In our case, a core region structure is required;then we modied the basic conguration suggested by Gershuni et al. in order to match with thefree-slip conditions on top and bottom walls. Weobtained the following expressions that are usedthroughout:

    U 0 = (2 z − 1)3 − 3(2 z − 1)

    6;

    T 0 = 2 x + Ra(2 z − 1)5 − 10(2 z − 1)3 + 9(2 z − 1)

    120:

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     Now by comparing with the expression of theRayleigh solution, it appears that the only terms of the Fourier series of interest are those with   1(3; 1; t )

    and 2(3; 1; t ). Moreover, a careful extension of the Saltzman and Lorenz’s results to the horizontalconvection leads to the inclusion of the term with2(0;2; t ). In this way, the discrete perturbed elds    and , and so the overall system, result to be char-acterized by three degrees of freedom which is alsothe minimal requirement to allow the description of chaotic behaviours [19].

    5.2. The Lorenz-like model 

    As in the Fourier series expansion of     and onlythe terms in   1(3; 1; t ); 2(3; 1; t ) and2(0;2; t ) have been retained, the governing equations result highlysimplied. Actually, for the rst non-dimensionalscheme, they assume the following expressions:

    c1 ̇  3;1

    1   +

    u0( z )c2 +

     d2u0d z 2

    c3 +  Pr c5

      3;11

    + Pr c43;12   = 0;   (12)

    d1 ̇3;1

    2   + d2 ̇0;2

    2   +1

    2

    @T 0@x   ( z )d4 − Ra

    d

    d z ( z )d5  

    3;11

    + [d6 − u0( z )d3]3;12   + d70;22   − d8  3;11 3;12−d9  3;11 0;22   = 0;   (13)

    where we have dened   3;11   =   1(3;1; t ); 3;12   =

    2(3;1; t ) and 0;22   =2(0;2; t ) and the coecients

    ci and di are known functions of the space variables x  and  z :

    c1 =

    1 +6 H 

     L2

    sin

    6 H 

     L  x;

    c2 =2 6 H 

     L

    1 +

    6 H 

     L

    2cos

    6 H 

     L x

    ;

    c3 = 6 H 

     L  cos

    6 H 

     L x

    ;

    c4 = 6 H 

     L  sin

    6 H 

     L x

    ;

    c5 =

    6 H 

     L

    43+2

    6 H 

     L

    23+3

    sin

    6 H 

     L x

    ;

    d1 = cos6 H  L

     x

    sin(z );

    d2 = 0:5 sin(2z );

    d3 = 6 H 

     L sin

    6 H 

     L x

    sin(z );

    d4 = sin

    6 H 

     L x

    cos(z );

    d5 = 6 H 

     L cos6 H 

     L x sin(z );

    d6 =

    1 +

    6 H 

     L

    22 cos

    6 H 

     L x

    sin(z );

    d7 = 22 sin(2z );

    d8 = 42

    6 H 

     L

    sin(z ) cos(z );

    d9 = 42 6 H 

     L  cos

    6 H 

     L x sin(z )cos(2z ):

    According to the truncation operated on theFourier series of      and , we are requested to col-locate the stream function transport equation on asingle point and the energy equation on two points.By choosing in a suitable way the collocation points, it is possible to annihilate a large number of coecients ci   and di   and to obtain a system veryclose to the Lorenz one [18], a rst-order ordinarydierential system of three equations in normalform. By setting:

     x =  L12 H 

    ;    z = 0:5;

    Eq. (12) simplies into:

    c1 ̇  3;1

    1   + Pr  c5  3;1

    1   + Pr   c43;12   = 0 (14)

    and Eq. (13) simplies into:

    d2 ̇0;2

    2   + 1

    2

    @T 0@x

      ( z ) d4  3;1

    1   − u0( z ) d33;12

    + d70;22   −   d8  3;11 3;12   = 0;   (15)

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    where we indicated by ci = ci(  x;   z ); i= 1; : : : ;5, andd j =d j(  x;   z ); j = 1; : : : ;9. As a second collocation

     point for Eq. (13) we choose:

     x = L

    6 H ;    z = 0:5;

    where Eq. (13) becomes

    d1 ̇3;1

    2   − Rad

    d z ( z ) d5  

    3;11   +

      d63;12   −   d9  3;11 0;22   = 0;

    (16)

    where   ci = ci(  x;   z ); i = 1; : : : ;5, and  d j =d j(  x;   z );

     j = 1; : : : ;9. Using the Lorenz notation [18]:

     X  =   3;11 ;

    Y  = 3;12 ;

    Z  = 0;22

    and taking into account that   12@T 0=@x =1, the full

    system becomes

    c1   ˙ X   + Pr   c5 X  + Pr   c4Y  = 0;

    d1  Ẏ  − Ra  dd z 

    ( z ) d5 X  +   d6Y  −   d9 XZ  = 0;

    d2 Ż  +

      d4 X  − u0( z )

     d3Y  +

      d7Z  −

      d8 XY  = 0 (17)

    resembling clearly the characteristics of the famousLorenz model.

    When the second non-dimensional scheme isadopted, the Lorenz-like model assumes the form:

    c1   ˙ X   +  1

    Gr 1= 2 c5 X   + c4Y  = 0;

    d1  Ẏ −GrPr  dd z 

    ( z ) d5 X  +  1

     PrGr 1= 2d6Y − d9 XZ =0;

    d2 Ż + d4 X −u0( z ) d3Y  +  1

     PrGr 1= 2 d7Z −  d8 XY  = 0:(18)

    5.3. Coherence of the model 

    In this section, we show that the truncationof the series expansions that is behind the con-struction of the above nite dynamical systemhas not introduced undesirable singularities in themodel.

    One aspect that has to be veried is the negativityof the divergence of the vector  v = ( ˙ X ;  Ẏ ;  Ż ); thisfact ensures the contraction of the volumes of the

     phase space invaded by the motion of the systemso that the system corresponding to the truncatedmodel is as dissipative as the original model. Let usexemplify this aspect for the rst non-dimensionalscheme and obtain the expression of the divergencefrom the equations of the truncated model:

    ∇ · v = @ ˙ X 

    @X   +

    @ Ẏ 

    @Y   +

    @ Ż 

    @Z  = − Pr   c5

    c1−

    d6d1−

    d7d2:

    As the coecients c;   d;   d are positive due to the

    chosen collocation points, it follows that ∇ · v   isnegative at any time.Regarding the coherence of the model, it is also

    relevant to show that no solution is admitted thatdescribes in the phase space  R3 = ( X; Y; Z ) a pathtowards innity; in other words, for each solutionand at any surface surrounding the origin in the phase space, the vector  v = ( ˙ X ;  Ẏ ;  Ż ) has to be ev-erywhere directed inside the surface. This issue is presently under study.

    6. Numerical algorithm

    The system of non-linear ordinary dierentialequations to be solved has the following form:

    dV idt 

      = jk 

    C ijk V  jV k 

    with i ;j ;k     varying from 1 to 3 and being(V 1; V 2; V 3)   ≡   ( X; Y; Z ). Let us introduce adiscretization of the time t    and indicate with

    n

    i ; i = 1;2;3 the value of the variables V i; i= 1; 2; 3at the generic time step t n. For the numerical in-tegration, we adopt the Lorenz’s double approxi-mation technique that is particularly suitable whenthe deterministic nature of the solution is not in-sured [18]; in this case at each step t n+1   the un-known quantities are computed by averaging the

    sum of the values V ni   and   V n+2

    i   in the followingmanner:

    V n+1i   =  12

    [V ni   +   V n+2

    i   ];

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    -0.2

    0

    0.2

    0.4

    0.6

    0.8

    1

    0 4e-05 8e-05 0.00012

          Y

    t

    Fig. 2. Y vs. t at  Ra= 4:0× 1010.

    -1

    -0.8

    -0.6

    -0.4

    -0.2

    0

    0.2

    0.4

    0.6

    0.8

    1

    0 0.0004 0.0008 0.0012 0.0016

          Y

    t

    Fig. 3. Y vs. t at  Ra= 8:3× 1010.

    For values of   Ra   smaller than 4 ×  1010 the perturbation is damped to zero. Fig. 2 shows thetransient history of  Y   as a function of time. For larger values of  Ra, for example at  Ra= 8:3× 1010(Fig. 3), the perturbation tends to zero via anoscillatory damped periodic regime.

    -1.5

    -1

    -0.5

    0

    0.5

    1

    1.5

    0 0.0004 0.0008 0.0012

          Y

    t

    Fig. 4. Y vs. t at  Ra= 8:4128375× 1010.

    0.1

    1

    10

    100

    1000

    10000

    100000

    1e+06

    0 10000 30000 50000 70000 90000

          A

    f

    Fig. 5. FFT of Y(t) at  Ra= 8:4128375× 1010.

    At  Ra= 8:4128375 ×  1010, a Hopf bifurcationoccurs and the ow becomes oscillatory peri-odic (Fig. 4) (the frequency spectrum is shownin Fig. 5). The signal has been observed for anon-dimensional time interval equal to 2.62144,corresponding to 524288 samples. The frequency

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    -6

    -4

    -2

    0

    2

    4

    6

    8

    0 0.05 0.1 0.15 0.2 0.25 0.3

          Y

    t

    Fig. 6. Y vs. t at  Ra= 8:413× 1010.

    content of this signal has been evaluated by meansof a FFT (frequency step f= 0:381). The energyspectrum highlights a unique frequency, equal tof = 83406:83. However, it is easy to understand, by observing Fig. 4, that this is not the unique fre-quency in this complex signal, but there are other 

    lower frequencies, whose energy content is not reg-istered by the previous FFT. In order to highlightthese frequencies, the signal has been ltered inseveral ways, considering a sample every 10, every100 and so on. So the frequencies registered are

    full signal : 83406:83;

    every 10 : 3406:9824;

    every 100 : 593:261;

    every 1000 : 7:031250:

    Unfortunately, it is not possible to investigate the behaviour of the system for larger values of  Ra, asthe system does not seem to support such valuesof  Ra: in fact, as shown in Fig. 6, at  Ra= 8:413 ×1010 the ow is oscillatory periodic again, but theamplitudes of the oscillations are becoming faster and faster and lose physical meaning.

    Although not acceptable form the physical view- point, just to investigate the behaviour of the math-ematical system, a series of numerical simulationshas also been performed for negative values of  Ra.

    Table 2

     Non-dimensional scheme 1: main solutions observed at nega-

    tive values of  Ra

     Ra   Solution

     Ramin   Steady

    −4:4× 109 Oscillatory damped and thensteady ( S̃ 0)

    −7:1× 109 Oscillatory damped and then

    steady−7:2× 109 One oscillation and then steady

    to zero−4:5× 1010 Oscillatory damped and then

    steady to zero−5:036230917 × 1010 Oscillatory periodic¡− 5:036230917 × 1010 Periodic divergent (blow up)

    52000

    54000

    56000

    58000

    60000

    62000

    64000

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

          Y

    t

    Fig. 7. Y vs. t at  Ra=− 4:3625× 109.

    The rst simulation has been executed at Ra= Ramin, initializing the run with the non-zero

    steady solution ( S̃ 0) calculated in the previous paragraph. This simulation converges at such asolution as steady state.

    Still starting from this steady perturbation, thevalue of  Ra   has been decreased. Table 2 reportsthe main solutions observed.

    For values of   Ra   slightly smaller than  Raminthe ow remains steady. Fig. 7 shows the tran-sient history of Y  as a function of time at  Ra= −4:3625 × 109. For smaller values of  Ra   the solu-tion is oscillatory damped towards the same steady

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    50000

    100000

    150000

    200000

    250000

    300000

    350000

    400000

    450000

    500000

    550000

    0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

          Y

    t

    Fig. 8. Y vs. t at  Ra=− 4:4× 109.

    -3e+08

    -2e+08

    -1e+08

    0

    1e+08

    2e+08

    3e+08

    4e+08

    5e+08

    6e+08

    0 0.0001 0.0002 0.0003

          Y

    t

    Fig. 9. Y vs. t at  Ra=− 5× 1010.

    conguration. Fig. 8 shows the transient history of Y   at Ra=−4:4×109. At  Ra=−7:2×109 a changeoccurs when the solution tends monotonically torest. Close to Ra=−4:5×1010, the solution becomesoscillatory damped around the zero solution. This isevident in Fig. 9, where the variable Y  is shown at Ra=−5:0×1010. At  Ra=−5:0362309170×1010,the ow becomes oscillatory periodic (Fig. 10).However, it can be observed that this regime lasts

    -1.5e+08

    -1e+08

    -5e+07

    0

    5e+07

    1e+08

    1.5e+08

    0.001 0.002 0.003

          Y

    t

    Fig. 10. Y vs. t at  Ra=− 5:036230917 × 1010.

    -4e+07

    -3e+07

    -2e+07

    -1e+07

    0

    1e+07

    2e+07

    3e+07

    4e+07

    -6 -4 - 2 0 2 4 6 8

          Y

    X

    Fig. 11. Phase trajectory at  Ra=− 5:036230917× 1010.

    for a limited time interval (0.003 time units) af-ter which the ow tends to rest. In Fig. 11, the projection of the phase trajectory onto the plane X  – Y   is reported: it looks very complex and sug-gests that the number of frequencies is high. In order to evaluate the frequency content, a FFT has beenexecuted on the samples belonging to the time in-terval (0.0002–0.0027). The frequency spectrum is

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    1e+06

    1e+07

    1e+08

    1e+09

    1e+10

    0 20000 40000 60000 80000 1 00000

          A

    f

    Fig. 12. FFT of Y(t) at  Ra=− 5:036230917 × 1010.

    -1.5e+08

    -1e+08

    -5e+07

    0

    5e+07

    1e+08

    1.5e+08

    0.001 0.002 0.003 0.004

          Y

    t

    Fig. 13. Y vs. t at  Ra=− 5:0362309176× 1010.

    shown in Fig. 12. The number of samples is rather small and allows us to register only the largest fre-quency (83203), while the other ones are not high-lighted. For values of  Ra   slightly smaller, for ex-ample at  Ra= − 5:0362309176 × 1010, the initialoscillations become faster and faster (Fig. 13) tillthey lose physical meaning. The present numericalmodel does not support such values of  Ra.

    7.2. Non-dimensional scheme 2

    7.2.1. Steady solutions

     Neglecting the three time derivatives of system(18), we obtain the following algebraic non-linear system for the steady ows:

    1

    Gr 1= 2 c5 X  + c4Y  = 0;

    Gr Pr  d

    d z ( z ) d5 X  −   1

     PrGr 1= 2d6Y  +   d9 XZ  = 0;

    − d4 X  + u0( z ) d3Y  −   1 PrGr 1= 2

    d7Z  +  d8 XY  = 0:

    (20)

    This system admits, of course, the zero solution X  =Y  =Z  = 0. In order to compute the non-trivialsolutions, we combine the three equations and ob-tain the following non-linear equation for Y :

    Gr 2 Pr 2 d9 c24

    d8d7Y 2 −   d9 c4

    ×Gr 2 Pr 2

    d4d7

    c4 + Gr 3= 2 Pr 2

    d3d7

    ×Y  − Gr 2 Pr 2 d5 c4 −   d6 = 0:It is simple to verify that with the values of the

     parameters adopted, this equation has real roots onlyif Gr¿Gr min  with

    Gr min = 15:

    For Gr =Gr min, the system has a unique non-zerosolution given by

     X  = − 1:43 × 10−2;

    Y  = 15159:55;

    Z  = 347383:12:

    For any value of Gr  larger than Gr min, the systemadmits two distinct non-zero solutions.

    7.2.2. Unsteady solutionsThen we solved the unsteady case by the pre-

    viously described algorithm. The time step chosenwas t = 10−5.

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    Table 3

     Non-dimensional scheme 2: main solutions

    Gr Solution

    Gr min   Steady ow (S 1)

    102 After one oscillation perturbation to rest (S 0)

    103 –107 Oscillatory damped ow towards (S 0)7:835× 107 Periodic ow ( P 2)

    8× 107 Periodic divergent (blow-up)

    -50000

    0

    50000

    100000

    150000

    200000

    250000

    300000

    0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

          Y

    t

    Fig. 14. Y vs. t at Gr = 102.

    The rst simulation at Gr =Gr min   has been de-veloped by initializing the run with the non-zerosteady ow just obtained by exact calculation. Thenumerical simulation leads to such a ow.

    Starting from this solution, the value of Gr   has been increased in order to observe the bifurcationsequence. Table 3 reports the main solutions ob-

    tained.At Gr = 102, the perturbation tends to rest

    after one oscillation: Fig. 14 shows the transienthistory of  Y   as a function of time. At Gr = 103

    (Fig. 15), the solution tends to zero through anoscillatory damped transient regime. This is con-rmed by the plot in Fig. 16, in which the pro- jection of the phase trajectory onto the plane X  – Y  is reported: the picture shows that the trajec-tory moves along a spiral towards the xed point(0; 0) (stable spiral node). At higher values of Gr 

    -300000

    -200000

    -100000

    0

    100000

    200000

    300000

    400000

    0 0.005 0. 01 0.015 0.02 0.025 0.03

          Y

    t

    Fig. 15. Y vs. t at Gr = 103.

    -300000

    -200000

    -100000

    0

    100000

    200000

    300000

    400000

    -1.5 -1 -0.5 0 0.5 1

          Y

    X

    Fig. 16. Phase trajectory at Gr = 103

    .

    (e.g., Gr = 105) the spiralling approach of thetrajectories to the xed point (0; 0) is even more pronounced (Fig. 17) and a much longer time isneeded to end the transient regime.

    At Gr = 7:835×107 the projection onto the phase plane ( X; Y ) of the solution trajectory exhibits alimit cycle in Fig. 18, i.e. the perturbed ow uc-tuates periodically around the basic ow. This is

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    -400000

    -300000

    -200000

    -100000

    0

    100000

    200000

    300000

    400000

    -2.5 - 2 -1.5 - 1 -0.5 0 0.5 1 1.5 2

          Y

    X

    Fig. 17. Phase trajectory at Gr = 105.

    -30000

    -20000

    -10000

    0

    10000

    20000

    30000

    -0.02 -0.01 0 0.01 0.02

          Y

    X

    Fig. 18. Phase trajectory at Gr = 7:835× 107.

    conrmed by the transient history of Y  (Fig. 19). AFFT executed on this signal (Fig. 20) shows the ex-istence of a main frequency f  = 1696:1. This FFThas been executed using 262,144 samples with fre-quency step f = 0:3814.

    Our model does not support larger values of Gr as it is not able to simulate the ow: actually, asshown in Fig. 21, at  Ra= 8 × 107 the ow appearsoscillatory periodic again but the amplitudes of the

    -30000

    -20000

    -10000

    0

    10000

    20000

    30000

    0 0.005 0.01 0.015 0.02

          Y

    t

    Fig. 19. Y vs. t at Gr = 7:835× 107.

    10000

    100000

    1e+06

    1e+07

    1e+08

    1e+09

    1e+10

    0 1000 2000 3000 4000 5000

          A

    f

    Fig. 20. FFT of Y(t) at Gr = 7:835× 107.

    oscillations become higher and higher till they lose physical meaning and blow up.

    8. Conclusions

    We have solved a nite dynamical system de-scribing the ow and thermal eld of a viscousincompressible uid between two horizontal par-allel free planes in the presence of a horizontal

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    -80000

    -60000

    -40000

    -20000

    0

    20000

    40000

    60000

    80000

    0 1 2 3 4 5 6 7 8 9 10

          Y

    t

    Fig. 21. Y vs. t at Gr = 8× 107.

    temperature gradient and terrestrial gravity eld.The Navier–Stokes–Fourier equations in terms of stream function within the Boussinesq–Oberbeck approximation represent the rst-step model. It isknown that this physical and geometrical congu-ration does not admit a basic state. Then, we built

    up the polynomial expression of a basic core owand developed the equations of the perturbed owthat are the object of our study. Following Saltz-man and Lorenz analysis of the Benard convectionow between two horizontal planes, we formulatethe nal model by adopting a collocation methodwith the Fourier expansion of the unknowns. Asthe nature of the present ow is very close to theone considered in [17,18], in such a expansion,we retained the sample triple of frequencies andnally obtained a three-dimensional nite dynam-

    ical system whose structure resembles closely theone studied by Lorenz. We solved numerically by a double step nite dierence scheme. Twonon-dimensional schemes are considered. The rstone is that adopted by Lorenz, that, in our case, pro-vides very large critical values of the Rayleigh num- ber ( Ra=O(1010)). Then, although the solutionsobtained do not appear spoiled by numerical errors,we properly selected a second non-dimensionalform that allows safer computations and yieldedto critical values of the Grashof number of lower 

    order of magnitude. The bifurcation pattern ob-tained by numerical simulation highlights a changeof stability from steady regime to periodic regime

    that occurs at  Ra= 8:41283751010

    in the rstnon-dimensional scheme and at Gr = 7:835107 inthe second non-dimensional scheme. We can con-clude that the core ow that we have built as a basic ow becomes extremely stable with respectto the three selected Fourier modes. The analysisof the evolution of a perturbation including themode m= 4, which will be the subject of a future paper, is under development. We believe that theapproach presented here is basically valuable asit extends the pioneering work by Saltzman andLorenz on Benard convection to the horizontal con-vection. Nevertheless, we are convinced that withthe great impulse given to computing resources bythen, more accurate numerical techniques may beemployed at an aordable cost.

    Acknowledgements

    This work has been developed within the projectI.A.C.= C.N.R. “Dierential and numerical modelsfor uid dynamics and material science”, 1999.

    Moreover, the authors acknowledge the nancialsupport by A.S.I. (Agenzia Spaziale Italiana), project “Modelli matematici, analisi sperimentalee numerica di alcuni aspetti della cristallizzazioneda fuso in microgravita”, 1998.

    References

    [1] A. Georgescu, D. Mansutti, Coincidence of the linear 

    and nonlinear stability bounds in horizontal thermalconvection problems, Int. J. Non-Linear Mech. 34 (4)

    (1999) 603–613.[2] A. Georgescu, D. Mansutti, The Rayleigh-like solution for 

    the horizontal convection, Int. J. Non-Linear Mechanics,

    to be submitted.

    [3] I. Muller, Thermodynamics, Pitman, London, 1975.

    [4] L. Landau, On the problem of turbulence, C. R. Acad.

    Sci. U.S.S.R 44 (1944) 311–314.

    [5] E. Hopf, A mathematical example displaying features of 

    turbulence, Comm. Appl. Math. 1 (4) (1948) 303–322.

    [6] M. Dubois, P. Berge, Velocity eld in the Rayleigh– 

    Benard instability: transitions to turbulence, in: A.Packott, C. Vidal (Eds.), Synergetics—Far from

    Equilibrium, Springer, Berlin, 1979, pp. 85–93.

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