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Number of modes controlling fluid flows from experimentally measurable quantities
14 September 1981
PHYSICS LE’MERS
Volume 85A, number 2
NUMBER OF MODES CONTROLLING FLUID FLOWS FROM EtiERIMENTALLY
MEASURABLE QUANTITIES
YM. TREVE La Jolla Insthe,
La Jolla, CA 92038, USA
Received 21 May 1981
It is showp by means of an example (the B&ard problem) that the number of modes N which control the qualitative behavior of stationary fluid flows governed by the Navier-Stokes equations can be determined exactly from experimentally measurable qnantities. For timedependent flows the method provides a lower bound for N.
As time tends to infinity, the behavior of the solutions of the Navier-Stokes equations is controlled by a finite number of Fourier modes, say N. First established in ref. [l] ~for twodimensional solutions and recently extended to the three-dimensional case [2], this property has already been discussed and exploited in refs. [3,4]. In particular it has been shown in ref. [4] that the theory presented in ref. [l] can be used to estimate the minimum number of modes required for the description of the roll-like solutions of the Boussinesq model of BBnard convection. In this letter we want to point out that the same theory allows one to determine the minimum number of modes N from experimentally measurable quantities in the case of stationary Bbnard convection and probably in that of other steady fluid motions. For time-dependent flows the method yields a lower bound for N. In the interest of brevity the reader is referred to refs. [3] or [4] for the mathematical details and symbol definitions. Let Pnp,udenote the projection of the velocity field on the linear space spanned by the first n eigenfunctions wl, WI, .... w,, of the associated Stokes pro-, blem, i.e. Pn u = Z&ci wi, and let in general
where vi is the ith component of the vector u and
dVd is the d-dimensional volume element. Then according to ref. [ 11, with hk the kth eigenvalue of the associated Stokes problem, if N is the smallest n for which the inequality xn+l > (2/rJ> lim supIv”l; t+-
(1)
holds and if in the limit t + 00the projection yn;pnu, n >N, is stationary, periodic, quasi-periodic, or almost periodic (in the usual mathematical sense), so is u respectively. In the case of 3-D solutions the corresponding inequality is (2) where co is a number smaller than 27 Now since
we see that if the solution is stationary the rhs must vanish and hence
Thus the minimum number of modes N can be determined whenever the integral appearing in the rhs can
81
Volume 85A, number 2
PHYSICS LETTERS
be evaluated either analytically or from experimentally measurable quantities. We shall show that such an evaluation is indeed possible in the case of steady Bénard convection in a finite container. We assume that the container is made of a rigid material and has two plane, parallel, and horizontal sides a distance H apart. The lower and upper sides are maintained at constant and uniform temperatures
T0 and T1 = T0
~ T < T0, respectively. The vertical sides whose shape will be specified later are assumed to be thermally insulating. Under the assumption that (1) the Boussinesq approximation provides a valid description of the flow and temperature distribution and (2) the kinematic andconstant, thermometric conductivity vviscosity and ~ are the equations for thecoefficients velocity u (t, x y, z) and temperature T(t, x y z) are [5] —
,
,
,
14 September 1981
and purely conductive solution, ~( is the area of the top side of the container, and dS is the surface element. The rhs of eq. (5) is proportional to the per. turbed heat flux leaving the container through the top which is equal to the difference between the total heat flux, a measurable quantity (see for instance ref. [6]), and that due to conduction alone [ g (L~T/H)szq]. Hence the values of IV u needed in inequalitites (1) and (2) for the determination of the minimum number of modes can be obtained from physical experiments. Inequalities (1) and (2) assume interesting and convenient forms in terms of the usual parameters of the problem, the Rayleigh number ge 3/, namely the Prandtl number Pr P/K, Ra and=the x ~TH number Nu = ratio of the amount of heat Nusselt transferred by conduction and convection to that transferred by conduction alone. ~‘,
p 0 [au/at +2 (u• u, V)u] = —Vp +p0 [1 + e(T0 T)]g (3a) + p0vV ‘ô + T— 2T ‘3b’ ai’, t u V KV
We situations. Firstconsider assume two that different the container is a rectangular
—
—
“
‘
box of horizontal dimensionsL in which steady rolls parallel to 1the andL2 shorter (L2side
where p0 is the (constant) average density, e the thermal expansion coefficient, and g = (0, 0, —g) is the vector acceleration due to gravity. Any sufficiently smooth solution of eqs. (3) satisfies the energy balance equation
observed. The motion is then two dimensional and the Nusselt number is given by
ff(pov2/2
so that, after using eq. (5), inequality (1) becomes 2 (6) A~1>2a~ Ra(Nu 1)/Pr where ~2)the~ two-dimensional H2X~)is the nthgeometry dimensionless value for undereigenconsid-
—
p0egzT)dV~
—
Nu
1
f
H d L1 L~T~ az z—~H -
,
—
=
p0 eg~JzV2TdV d
—
p
2d
(4)
,
0vIVuI
eration, and a = H/L where z is the vertical coordinate and d = 2 or 3 according to whether the observed motion is two- or three-dimensional, respectively. Under the assumption that the motion is stationary the rhs of eq. (4) must vanish. Hence 2T dVd
Ik7uI~= _eg’~~f ZV
=
—eg-~-Hf
1 is the aspect ratio. Second suppose that the vertical wall bounding the fluid has an arbitrary shape (e.g. is cylindrical as in the experiments reportedin ref. [6]) and that a steady motion can be inferred from the observed constancy of the heat flux. Then inequality (2) applies and can be written as 2)2Ra2(Nu 1)2/Pr4 (7) A~1~ c0(~(/H where A~3~ (—H2A~)is the nth dimensionless eigenvalue for the three-dimensional container used. The actual determination of the minimum number —
8 ~z~HdS,
where
,
o defT~ [T 0 (~T/H)z] —
is the deviation of the temperature from the static 82
ofmodesNwhich control the behavior of the flows is thus reduced to the calculation of the corresponding eigenvalues of the associated Stokes problem. Note
Volume 85A, number 2
PHYSICS LETTERS
that only few eigenvalues of low order would probably require numerical determination since their behavior for large n is known to be of the form A~’~Cn21’1 where C is a constant (see ref. [7], formula (1.8)). As pointed out to us by 02. Manley the foregoing arguments can be extended to provide a lower bound for N in the case where the asymptotic regime (t is time dependent. To see this it suffices to take the time average of eq. (4) and note that the resulting ~ vanishes. Hence eqs. (5) still hold on average and since the supremum of’IVul~is larger than or equal 00)
to its time average it is easy to see that inequalities
(6) and (7) give a lower bound for N when the measured time average of the Nusselt number is substituted for Nu. To conclude we remark that our method of estimating N can probably be applied to other problems,
14 September 1981
This work was supported by the US Department of Energy under contract DE-ASO3-78ET5 3072. References [1] C. Foias and G. Prodi, Rend. Semin. Mat. Univ. Padova 39 (1967) 1. C. Foias and Y.M. Treve, Phys. Lett. 85A (1981)35. Y.M. Treve, J. Comput. Phys. 41(1981)127. OP. Manley and Y.M. Treve,Phys. Lett. 82A (1981) 88. S. Chandrasekhar, Hydrodynamic and hydromagnetic stability (Clarendon Press, Oxford, 1961). [61 E.L. Koschmieder and S.G. Pailas, Intern. J. Heat Transfer 17 (1974) 991. [7] C. Foias and R. Temam, J. Math. Pures Appl. 58 (1979) 339. [2] [3] [4] [5]
e.g. to MHD flows, and that the knowledge of N should prove very useful for the numerical simulation of the related experiments.
83
PHYSICS LE’MERS
Volume 85A, number 2
NUMBER OF MODES CONTROLLING FLUID FLOWS FROM EtiERIMENTALLY
MEASURABLE QUANTITIES
YM. TREVE La Jolla Insthe,
La Jolla, CA 92038, USA
Received 21 May 1981
It is showp by means of an example (the B&ard problem) that the number of modes N which control the qualitative behavior of stationary fluid flows governed by the Navier-Stokes equations can be determined exactly from experimentally measurable qnantities. For timedependent flows the method provides a lower bound for N.
As time tends to infinity, the behavior of the solutions of the Navier-Stokes equations is controlled by a finite number of Fourier modes, say N. First established in ref. [l] ~for twodimensional solutions and recently extended to the three-dimensional case [2], this property has already been discussed and exploited in refs. [3,4]. In particular it has been shown in ref. [4] that the theory presented in ref. [l] can be used to estimate the minimum number of modes required for the description of the roll-like solutions of the Boussinesq model of BBnard convection. In this letter we want to point out that the same theory allows one to determine the minimum number of modes N from experimentally measurable quantities in the case of stationary Bbnard convection and probably in that of other steady fluid motions. For time-dependent flows the method yields a lower bound for N. In the interest of brevity the reader is referred to refs. [3] or [4] for the mathematical details and symbol definitions. Let Pnp,udenote the projection of the velocity field on the linear space spanned by the first n eigenfunctions wl, WI, .... w,, of the associated Stokes pro-, blem, i.e. Pn u = Z&ci wi, and let in general
where vi is the ith component of the vector u and
dVd is the d-dimensional volume element. Then according to ref. [ 11, with hk the kth eigenvalue of the associated Stokes problem, if N is the smallest n for which the inequality xn+l > (2/rJ> lim supIv”l; t+-
(1)
holds and if in the limit t + 00the projection yn;pnu, n >N, is stationary, periodic, quasi-periodic, or almost periodic (in the usual mathematical sense), so is u respectively. In the case of 3-D solutions the corresponding inequality is (2) where co is a number smaller than 27 Now since
we see that if the solution is stationary the rhs must vanish and hence
Thus the minimum number of modes N can be determined whenever the integral appearing in the rhs can
81
Volume 85A, number 2
PHYSICS LETTERS
be evaluated either analytically or from experimentally measurable quantities. We shall show that such an evaluation is indeed possible in the case of steady Bénard convection in a finite container. We assume that the container is made of a rigid material and has two plane, parallel, and horizontal sides a distance H apart. The lower and upper sides are maintained at constant and uniform temperatures
T0 and T1 = T0
~ T < T0, respectively. The vertical sides whose shape will be specified later are assumed to be thermally insulating. Under the assumption that (1) the Boussinesq approximation provides a valid description of the flow and temperature distribution and (2) the kinematic andconstant, thermometric conductivity vviscosity and ~ are the equations for thecoefficients velocity u (t, x y, z) and temperature T(t, x y z) are [5] —
,
,
,
14 September 1981
and purely conductive solution, ~( is the area of the top side of the container, and dS is the surface element. The rhs of eq. (5) is proportional to the per. turbed heat flux leaving the container through the top which is equal to the difference between the total heat flux, a measurable quantity (see for instance ref. [6]), and that due to conduction alone [ g (L~T/H)szq]. Hence the values of IV u needed in inequalitites (1) and (2) for the determination of the minimum number of modes can be obtained from physical experiments. Inequalities (1) and (2) assume interesting and convenient forms in terms of the usual parameters of the problem, the Rayleigh number ge 3/, namely the Prandtl number Pr P/K, Ra and=the x ~TH number Nu = ratio of the amount of heat Nusselt transferred by conduction and convection to that transferred by conduction alone. ~‘,
p 0 [au/at +2 (u• u, V)u] = —Vp +p0 [1 + e(T0 T)]g (3a) + p0vV ‘ô + T— 2T ‘3b’ ai’, t u V KV
We situations. Firstconsider assume two that different the container is a rectangular
—
—
“
‘
box of horizontal dimensionsL in which steady rolls parallel to 1the andL2 shorter (L2side
where p0 is the (constant) average density, e the thermal expansion coefficient, and g = (0, 0, —g) is the vector acceleration due to gravity. Any sufficiently smooth solution of eqs. (3) satisfies the energy balance equation
observed. The motion is then two dimensional and the Nusselt number is given by
ff(pov2/2
so that, after using eq. (5), inequality (1) becomes 2 (6) A~1>2a~ Ra(Nu 1)/Pr where ~2)the~ two-dimensional H2X~)is the nthgeometry dimensionless value for undereigenconsid-
—
p0egzT)dV~
—
Nu
1
f
H d L1 L~T~ az z—~H -
,
—
=
p0 eg~JzV2TdV d
—
p
2d
(4)
,
0vIVuI
eration, and a = H/L where z is the vertical coordinate and d = 2 or 3 according to whether the observed motion is two- or three-dimensional, respectively. Under the assumption that the motion is stationary the rhs of eq. (4) must vanish. Hence 2T dVd
Ik7uI~= _eg’~~f ZV
=
—eg-~-Hf
1 is the aspect ratio. Second suppose that the vertical wall bounding the fluid has an arbitrary shape (e.g. is cylindrical as in the experiments reportedin ref. [6]) and that a steady motion can be inferred from the observed constancy of the heat flux. Then inequality (2) applies and can be written as 2)2Ra2(Nu 1)2/Pr4 (7) A~1~ c0(~(/H where A~3~ (—H2A~)is the nth dimensionless eigenvalue for the three-dimensional container used. The actual determination of the minimum number —
8 ~z~HdS,
where
,
o defT~ [T 0 (~T/H)z] —
is the deviation of the temperature from the static 82
ofmodesNwhich control the behavior of the flows is thus reduced to the calculation of the corresponding eigenvalues of the associated Stokes problem. Note
Volume 85A, number 2
PHYSICS LETTERS
that only few eigenvalues of low order would probably require numerical determination since their behavior for large n is known to be of the form A~’~Cn21’1 where C is a constant (see ref. [7], formula (1.8)). As pointed out to us by 02. Manley the foregoing arguments can be extended to provide a lower bound for N in the case where the asymptotic regime (t is time dependent. To see this it suffices to take the time average of eq. (4) and note that the resulting ~ vanishes. Hence eqs. (5) still hold on average and since the supremum of’IVul~is larger than or equal 00)
to its time average it is easy to see that inequalities
(6) and (7) give a lower bound for N when the measured time average of the Nusselt number is substituted for Nu. To conclude we remark that our method of estimating N can probably be applied to other problems,
14 September 1981
This work was supported by the US Department of Energy under contract DE-ASO3-78ET5 3072. References [1] C. Foias and G. Prodi, Rend. Semin. Mat. Univ. Padova 39 (1967) 1. C. Foias and Y.M. Treve, Phys. Lett. 85A (1981)35. Y.M. Treve, J. Comput. Phys. 41(1981)127. OP. Manley and Y.M. Treve,Phys. Lett. 82A (1981) 88. S. Chandrasekhar, Hydrodynamic and hydromagnetic stability (Clarendon Press, Oxford, 1961). [61 E.L. Koschmieder and S.G. Pailas, Intern. J. Heat Transfer 17 (1974) 991. [7] C. Foias and R. Temam, J. Math. Pures Appl. 58 (1979) 339. [2] [3] [4] [5]
e.g. to MHD flows, and that the knowledge of N should prove very useful for the numerical simulation of the related experiments.
83