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Numerical investigation of one-parameter families of stationary

regimes in a planar filtrational convection problem.

Rostov State University, Russia

Govorukhin V.N.

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Cosymmetry theory

, 0H

Q L

Definition (Yudovich, 1991)

Let Q,L : HH be operators that act in a real Hilbert space H. We say that L is a cosymmetry of Q (and also of the differential equation) (or that Q,L is a cosymmetric pair) if for H 

Q

Definition (Yudovich, 1991)

A solution of equation is said to be cosymmetric if L=0.

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Proposition (Yudovich, 1991)Let be a solution of equation

Q = 0 and L be a cosymmetry of Q. If L0, then is a degenerate solution: L belongs to the kernel of the operator Q'*, and the operator Q'() is noninvertible. 

Theorem (V.Yudovich, 1995) Let the stability spectrum of equilibrium is changing along the cycle of equilibria. Then there doesn't exist the symmetry group for which this cycle is an orbit of its action.

It turn out that in the absence of additional degeneration the noncosymmetric solutions are no isolated – they form a one-parameter families.

This is just the case of cosymmetry.

The theory of cosymmetry was developed by V. Yudovich and L. Kurakin [1,2,3,4,…].

Was proved that families of noncosymmetric solutions may branch off from a cosymmetric solution [1]. The bifurcation of families in cosymmetric case

was studied [4], etc.

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A simple two dimensional ODE (illustrative)

2 2, , 1x xS y ayS S x y Equations:

Cosymmetry: ,ay xFamily: 2 21 0x y

Spectrum: 0, 2 1 sin , 0,2a

Examples of cosymmetric system

Cosymmetry:

Convection in porous media

Equations:

from first equation we can obtain where G is the Green operator.

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Numerical problems

Equilibria finding

The problem that equilibria finding and stability investigation are strongly degenerate, and it is impossible to handle classic methods in the

ordinary form.

Family calculation It is necessary to develop new methods of continuation of solutions on

the hidden parameter of system based on the cosymmetry theory.

Bifurcation analysis To use theoretical results for development of algorithms for bifurcation

analysis of cosymmetric systems.

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Calculation of one-parameter family of equilibria

The method bases on the cosymmetric version of implicit function theorem which was formulated by Yudovich (1991). Consider the ODE system in Rn:

Let this system possess a cosymmetry L. The equation for equilibria has the form:

( ) 0f y =

Then, the following theorem is valid:

Theorem (Yudovich, 1991)Let f, L be a cosymmetric pair and y0 be a solution of the equation f(y0 )=0. Let 0 be a simple eigenvalue of the operator fy(y0 ), and 0 be a base vector in the kernelof the operator fy(y0 ). Then equation has one-parameter family of solutions:

for any small R. depends analytically and is

defined by (,Ly0)=0.

y’=f(y)

yα=y0+α 0 + α2

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Algorithm

1. Find any point on the curve using modified Newton method.2. Calculate the kernel of Jacobi matrix at the first point, check of its non

degeneration and choice the direction of continuation.3. Do one step of Runge-Kutta method for Cauchy problem, where RHS is an

eigenvector corresponding to zero eigenvalue.4. Stability analysis of the calculated point of the family, accuracy checking of

equilibria, verification of exit conditions.5. If the accuracy of calculations is not satisfactory, correct the point,

decreasing of integration step, go to the item 3. If stability of equilibria is changed along the family, finding of a neutral point with the use of approximation of algorithm of density output of ODE solutions. Go to the item 3.

Expression in theorem defines the equilibrium branch of the system with the parameter and the vector 0 tangent to the family. Thus, the vector 0 determines an

optimal direction for continuation of the family of solutions. This equilibrium curve may be continued with respect to from the point y0, solving the following Cauchy

problem:

See Govorukhin, 1998, 1999

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Convection in porous mediaWe consider a rectangular container D = [0; a]x[0; b] filled with a porous medium

and saturated by an incompressible fluid. The dimensionless equations of gravity

convection of the fluid in the container uniformly heated from below have the form:

Here ψ(x, y, t) is the stream function, θ(x, y, t) is the temperature deviation

from the equilibrium (linear in the vertical) profile. The parameter λ is the filtrational Rayleigh number. On the boundary of the container the Dirichlet boundary conditions is specified:

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Properties of problemCosymmetry:

Discrete symmetries:

The system of equations is globally solvable, dissipative and its zero equilibrium exists for all λ¸ and it is globally stable for small λ. The critical values of the spectral problem are

To each transition of λ through the critical value corresponds the bifurcation of origin of a one-parameter family of equilibria!To the first value corresponds branches of steady-state cycle of

regimes.

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Approximation of the problemThe Galerkin method was used. The numerical solution is found in the form:

Substitution of these expressions to equations and corresponding projection operations lead to a system of ordinary differential equations of order N = nx×ny.

A order of the system N was considered sufficient if doubling N leads to

changing the numerical results not greater than 5%. The system of order

N=400 used for the numerical investigation of the families. Such an approximation order guarantees admissible divergence of results.

Finite-dimensional approximations conserve the discrete symmetries and cosymmetry of the input problem.

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Numerical resultsNumerical analysis of one-parameter families, convective regimes and their subsequent rebuilding were carried out at various values of the parameter 8≤b≤75, and a fixed value of the parameter a=20.

The numerical results are presented graphically in terms of the vertical heat flow through the bottom boundary and the horizontal heat flow through the middle of the rectangle:

For families calculation was applied described above algorithm.

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Evolution of familiesNarrow rectangle →

→ Wide rectangle

If λ increases then both minimal and maximal values of averaging vertical heat flow grow in case of wide rectangle but for narrow

case only maximal ones!

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Loss of stability on a families

The loss of stability on a families of equilibria has a various character and can be both monotonic and oscillatory and may arise in

two, four, six or eight points. It depends on rectangle size. Simultaneous loss of stability by an even number of regimes is

explained by the presence of two discrete symmetries.

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Critical curves corresponding to monotonic (square) and oscillatory (triangle) instabilities.

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Evolution of instabilityAfter the first loss of stability and further increasing the parameter λ the

amount of unstable regimes grow by forming unstable arcs on the families.

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Bifurcation of families

Subdivision

Complicated families

…etc.

There was found a set of bifurcations of one-parameter families of equilibria.

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Convective regimes

Stream function Temperature Heat transfer

To each point of a family of equilibria corresponds stationary driving of a liquid.

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Stream function Temperature Heat transfer

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Stream function Temperature Heat transfer

The evolution of convective regimes from the origin of the family to appearance of instability is the following: for the narrow container at increasing λ the shape of the existing rolls is complicated and new convective cells are formed inside the existing ones; for the wide vessel new convective rolls appear. Both local and average heat transfer of the regimes of the wide vessel do not differ very much neither qualitatively nor quantitatively. This difference is essential for the narrow vessel since there exist regimes with high and low heat transfer at the top and at the bottom of the vessel.

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Selection of regimesQuestion: which stable regime on family will be realized from different initial conditions?

We found that two regimes can be realized from the initial conditions near mechanical equilibrium.

The selection mechanism was pretty obvious in this case for λ close λ11. Along a stable invariant manifold, the motions are

attracted to a two-dimensional invariant unstable manifold. Any motion that has spent long enough in the neighborhood of the zero equilibrium will be attracted to one of two main unstable

directions.

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These regimes marks on the pictures marked by letters c and d.

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References[1] Yudovich V.I.,”Cosymmetry, degeneration of solutions of operator equations, and onset off a filtration convection.” // Translated in england from Matematicheskie Zametki,Vol.49,N 5,pp.142-148, May, 1991.[2] Yudovich V.I, “Secondary cycle of equilibria in a system with cosymmetry, its creation bybifurcation and impossibility of symmetric treatment of it.” // Chaos N2, (1995) 402-411. [3] Yudovich V.I., “Implicit function theorem for cosymmetric equations.” // Mat. Zametky, V. 60, N 2, 313 (1995). [4] Kurakin L.G., Yudovich V.I, “Bifurcations accompanying monotonic instability of an equilibriumof a cosymmetric dynamical system.” // Chaos V. 10, N. 2 (2000) 311--330.[5] Govorukhin V.N., “Numerical Simulation of the Loss of Stability for Secondary Steady Regimes in the Darcy Plane-Convection Problem”.// Doklady Akademii Nauk, V. 363, N. 4-6 (1998) 806-808.[6] Govorukhin V.N., Yudovich V.I.,”Bifurcations and selection of equilibria in a simple cosymmetric model of filtrational convection.” // Chaos, V.9, N 2, (1999) 403--412.[7] Govorukhin V.N., “Analysis of families of secondary steady-state regimes in the problem of plane flow through a porous medium in a rectangular vessel.” // Fluid Dynamics, V. 34, N 5, 1999, 652-659.[8] Govorukhin V.N., “Calculation of one-parameter families of stationary regimes in a cosymmetric case and analysis of plane filtrational convection problem.” In: D.Henry, A.Bergeon (Eds.), Notes on Numerical Fluid Mechanics, Vol. 74, Vieweg Publ. 2000, 133--144.[9] Govorukhin V.N., Shevchenko I.V. “Numerical solution of the plane convection Darcy problem on a computer with distributed memory.” // Vychisl. Tekhnol. 6, No.1, 3-12 (2001).[10] Govorukhin, V. N., Tsybulin V. G., Karasözen B. “Dynamics of numerical methods for cosymmetric ordinary differential equations.” // Internat. J. Bifur. Chaos Appl. Sci. Engrg. 11 (2001), no. 9, 2339--2357. [11] Govorukhin V.N., Shevchenko I.V. “Second bifurcation transition in planar filtrational convection problem.” // // Fluid Dynamics, N 5, 2003.