941 lines
30 KiB
Plaintext
941 lines
30 KiB
Plaintext
J . Fluid Mech. (1965), vol. 23, part 1, p p . 129-144
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129
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Printed i n Great Britain
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On the stability of steady finite amplitude convection
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BY A. SCHLUTERD, . LORTZ AND F. BUSSE
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Institute of Theoretical Physics, Munich University and Institute for Plasma Physics, Miinchen-Garching
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(Received 29 March 1965)
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The static state of a horizontal layer of fluid heated from below may become unstable. If the layer is infinitely large in horizontal extent, the Boussinesq equations admit many different steady solutions. A systematic method is presented here which yields the finite-amplitude steady solutions by means of successive approximations. It turns out that not every solution of the linear problem is an approximation to the non-linear problem, yet there are still an infinite number of finite amplitude solutions. A similar procedure has been applied to the stability problem for these steady finite amplitude solutions with the result that three-dimensional solutions are unstable but there is a class of two-dimensional flows which are stable. The problem has been treated for both rigid and free boundaries.
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1. Introduction
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When a horizontal layer of fluid is heated from below, thermal expansion causes a density gradient opposite to the direction of gravity. I n cases where the temperature gradient exceeds a certain critical value the static state of the fluid becomes unstable because the buoyancy force is sufficient to overcome the dissipative effects. It is well known that the resulting cellular convective flow is not uniquely determined by the equations of motion and the boundary conditions if the layer is infinite in horizontal extent. An infinite degeneracy was first found in the earlylinear theorieswhich apply only for infinitesimalflowamplitude. However,Malkus & Veronis(1958)have shownfor specialsolutionsthat the degeneracy persists for finite amplitude solutions. They showed that flows with rectangular or hexagonal cell pattern are finite amplitude solutions and that their number is infinite because the ratio of the side lengths of a rectangle is a free parameter.
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As important as the non-linear effects is the influence of the boundedness of the layer in horizontal extent. However, if the horizontal length of the layer is large compared to its thickness, the influence of the vertical side walls ought to be negligible at points far away from the walls.
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It appears that a stability theory is needed to explain why one or the other flow is preferred. If one recalls the mathematical difficulties that arise in the stability theory of simple flow in a channel it seems at first hopeless to apply the usual stability theory here. The steady-state solutions of finite amplitude are not even known exactly; however, the flows considered have relatively small amplitude. This enables us to treat the stability equations with the aid of successive
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9
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Fluid Mech. 23
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130
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A. Xchliiter, D. Lortx and F . B m s e
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approximations similar to those Malkus & Veronis (1968) used. At the same time we generalize the method of Malkus & Veronis for the steady state by considering the whole manifold of solutions. Furthermore, we treat the problem for both rigid and free boundaries.
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2. The fundamental equations
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Conservation of mass, momentum, and energy are described by the equation of continuity, the Navier-Stokes equations in the Boussinesq-approximation, and the heat equation, respectively, or
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aiui = 0,
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atui+uiajui= -pilaip- (p/po)ghi+vAui,
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a,T+ujaiT = KAT,
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where we have used the summation convention and the notation
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aj = alaxj, a, = apt ( j= i,2,3);
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hiis the unit vector with direction opposite t o the gravity acceleration vector,
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which is normal to the layer. All other symbols have their usual meaning.
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Suppose the bottom of the layer is held at the temperature Toand the top at the
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temperature Tl. Letting d be the depth of the layer we write the temperature
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in the form
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T-To = -/3xihj+8,
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where the first term with 4, = (To-T J / d describes the temperature distribution
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in the static state and 8 is the deviation from the linear distribution. The funda-
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mental equations must be supplemented by an equation of state which we
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approximate by
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p =po[l-a(T-TO)].
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Thus we arrive at the well-known system of equations
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aiui = 0,
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+ + a,,ui Ui aitb, = -ai G + agehi v Aui,
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a,e+ujaje= /3ujhj+KAe,
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G = p/pof gx Aj - @ q X k A, xjhi; a,g, v,K are assumedto be constants. To get a dimensionIessform of the equations
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w we set Ui = KUi/d, 8 = lJKe'/CIgd3, t = d2t'/K, Xi = d X i , = K2G'/d2.
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This yields, after dropping the primes,
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aiuj = 0,
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atui +ui aj ui = -ai 55+Peh, +P AUi, a,e+ujaie = hi-+he.
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P = v/K is the Prandtl number and R = ag/3d4/v~is the Rayleigh number.
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On the stability of steady jinite amplitude convection
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131
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Introducing the four-dimensional differential operator
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the four-dimensional vector and the matrix differential operator
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we can rewrite our equations in the form
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with the sum convention on Greek subscriptswhichrun from 0to 3. The stationary
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system has the form
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I v, a, 21, = D,, v, -a, 5, aAv, = 0.
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If we superpose infinitesimal disturbances a, onto the steady functions v, we
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derive from (2.1)the stability equations
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where we have introduced a growth rate u by
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ate, = ce,
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because the coefficients of the special linear system (2.3) are time-independent. v, is unstable if equation (2.3)has solutions for positive v.
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3. The boundary conditions
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We assume that the layer is infinite in horizontal extent and require that all
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+ functions are bounded as x2 y2+ 00. On the horizontal bounding surfaces the
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vertical component of the velocity must vanish, and since we require that the temperature has fixed values on the boundaries we have the further condition that the temperature deviation 8 must vanish.
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Since we are concerned with a viscous fluid a t rigid boundaries the horizontal components of the velocity must also vanish. For the so-called 'free' case absence of stress requires that the normal derivative of the horizontal velocity components vanishes. So we have two sets of boundary conditions
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u.=e=o
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at rigid boundaries,
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ui hi = 8, Al eiikhi uk= 6 = 0 at free boundaries,
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for the steady state as well as for the disturbances.
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9-2
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132
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A . Xchliiter, D.Lortz and F.Busse
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4. Perturbation theory in the vicinity of R = R,
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We try to solve the stationary equations (2.2)and the stability equations (2.3) for small values of R -R,, which means small amplitude convection.
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Regarding the quadratic terms in (2.2) and the interaction terms in (2.3) as perturbations, the unperturbed equations are
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where the superscript on DfA means that R is replaced by So)T.he equations are linear with constant coefficients and their solutions are well known (see, for instance, Pellew & Southwell 1940). Solutions of the non-linear equations (2.2) can be approximated by the formal expansions
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R = R(O)+ER“ +e2H2+) ...,
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(4.3)
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+ + + .. v, = €V(:) E 2 V y €32):
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. ,
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(4.4)
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where the amplitude E is a small parameter. If we substitute these series into the non-linear system (2.2) we get a set of inhomogeneous equations which are in general not solvable. We determine the R(”)from certain existence conditions for the solutions of the inhomogeneous equations. Since R is an externally given parameter, equation (4.3)defines E .
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We substitute the series (4.3),(4.4)into the stability equations which we regard as an eigenvalueproblem for the growth rate 0-. Since E is the perturbation parameter given by the steady non-linear solutions, we can apply the ordinary techniques of perturbation theory to the disturbance equations writing
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...) 0- = d O ) + € d 1 ) + € 2 d 2 ) +
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G, = 5($+€8;+~€26:’ + ....
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(4.5) (4.6)
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5. The unperturbed problems
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Equation (4.2) with do)= 0 is the same as (4.1), so we need only discuss the more general (4.3). Let v: and v: be any functions which satisfy 8, v: = a, vz = 0
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and the same boundary conditions as v,. We define the weighted scalar product
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where ( ,) means the average over the entire layer. Then for the free as well as for
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the rigid case the operator D;?has the following property of self-adjointness
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(v:, OF,v:) = R(0)P[(8’uhjj),+ ( 8 ” ~Aj;), + (u;Au;),] +P(@As”),
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= (v:, v;)
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from which we can immediately conclude that do)is real.
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On the stability of steadyjinite amplitude convection
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133
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Referring to Pellew & Southwell (1940)the vertical and horizontal dependences of the solutions of the linear equations (4.2)can be separated by assuming
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A252)+a25:) = 0,
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where a is the wave-number, and A2 = A -8, A, a, A, is the two-dimensional
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Laplacian operator in the horizontal plane. The neutral curve do=) 0 divides the (a2,Rco))-planiento a stable and an unstable region. On the neutral curve
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there is a minimum value R(O)= R, and a corresponding a,. The first-order disturbances have the highest growth rate do)= 0 if their wave-number is the
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same as that of the steady solutions. We shall prove that these disturbances will
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lead to the instability of three-dimensional steady solutions. With do)= 0 the
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two systems (4.1)and (4.2)become identical and we rewrite them explicitly:
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0 = R(o)u;il)A,+A&),
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0 = -a, 63')+P8(l)Ai+P AuLJ'.
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(5.2) (5.3)
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We first notice that h, the vertical component of the vorticity, vanishes, for if we take the curl of (5.3) we get
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Ah = 0, h hi eijrca, u:?),
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(5.4)
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with h = 0 or A, a, h = 0 on the boundaries. By multiplying (5.4)by h, averaging over the whole layer, and integrating by parts, we see that h = 0. (In the free
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case h could be a constant if the entire layer were rotating about a vertical axis,
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a case we do not want t o discuss here.)
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The velocity thus satisfies the relations
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hi Eiik a, = a, up = 0.
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(5.5)
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By introducing the operator SiE ai a, A, -hiA we write the general solution of
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( 5 . 5 )in the form
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uy = Si ,(I)
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(5.6)
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with dl)anarbitrary function. (Inthe free casewe could add a constant horizontal vector to our solution (5.6). But this would correspond to an uninteresting uniform horizontal translation.) By operating with Si on (5.3)we find
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or Then equation (5.2)yields
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(A3-R(0)A2)80) = 0
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(5.7)
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If a co-ordinate system with the origin in the middle of the layer and the z-axisin the direction of A is introduced, the solution of (5.7) has the following form (see Pellew & Sout,hwell 1940; Reid & Harris 1958):
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,(I) = ui(r)f(z)(j'(1) = A2t+) = w ( r )g(&
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(5.8)
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134
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A . Schliiter, D. Lortx and F . Busse
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A, w = -a2w,
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(5.9)
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3
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f (2) = A,coshq,z,
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n=l
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3
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g(x) = B,coshq,x,
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n=I
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(5.10)
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where r is the two-dimensional position vector in the horizontal plane. In the rigid case we have for the lowest value of R@):
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U, = 3.117, .Rho)= 1707.762;
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q1 = i.3.974, q, = 5.195-i.2.126, q 3 = A , = 1, A , = - 3.076.10-2+i. 5.194 10-2, A , = A;;
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+ Bl = 650.68, B, = 39.277 i. 0.433, B, = BZ.
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In the free case R(O)= (n2+a2)3/a2, a, = n/$ 2 ,
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q, = ni, A , = 1, A , = A , = 0, B, = (n2+a2)2, B, = B, = 0.
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We write the solutions of (5.9)in the form
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with
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2i N
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w = C,w,,
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w, = exp (ik,.r),
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n.=-N -n+o
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lknI2= a2.
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(5.11)
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I n order that (5.11)be real, there must be a k-, = -k, with C-, = (7: for each k,.
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With the normalization
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the final form of the first-order solutions is
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(A2
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+N
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vlf) = 6i)dl), v(*)=f(z) Z: C, w, for the steady motion,
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n=--N
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A2
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fif) = (&$) fi(1), 8)=f ( z )
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b, w, for the disturbances.
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n
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The last expression is a sum over the entire manifold of solutions w,.
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(5.12)
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6. Solutions of the second order A t second order the inhomogeneousequations
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vf) - a $9 = -R(1)A a,vy = 0
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must be satisfied, where A,, is the constant matrix
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O n the stability of stead2 Jinite amplitude convection
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135
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Forming the special scalar product, defined in (5.1), with the equation (6.1) and
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the functions vC)* of the first order, we derive the existence condition?
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-R(l){v:)*, A,, v?') +(vC,l)*,v~)a,v',") = (v$')*, DL~P,-)~(z):)*~)) t3,iscZ))
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- -
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(
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~
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(2)
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K9
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D(O)v(l)*=) KA A
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(
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~
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(2)
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K9
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D(O)v(U-a
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Kh K
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K
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@*)
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= 0.
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(6.3)
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We have used the facts that DF,is self-adjoint in the defined sense and the terms
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(vy-,aKa@)()v',,"' a,ZP*>
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vanish due to the boundary conditions and the continuity equation. For further reference we show that
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a (@' K ? 2I , \ h v'K"") = { v y , u3.a3.vK(l)")= 0 ,
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(6.3)
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where u, = 8, v has vanishing normal component on the boundaries and the first order functions W ~ YvC,l)"may have different horizontal dependences. By partial integration we obtain
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( v y , u, a, v',"') = ( v y , (8,.v)a, vy"') = (WSi v y, aivc,l)").
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This vanishes because, without summation over K
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8 .v(l)'a.&)" JK JK
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a. = (8.~$1)') &)" - qa, v',"') a, a, A, ~ y ) "
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3K
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3K
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+(a, A, v',)') A@)'' +(aj~2)'a), A, a,. v(,)rr
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(aj =(8, p' - a.v(l)n
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8.v(l)r=r 0.
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)3K
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3K
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The last two terms cancel because of the relation A2vi? = -a2v$? and because
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different first-order solutions have the same vertical dependence. The second term of (6.2) is of the form (6.3) and therefore vanishes, so the
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coefficient of in (6.2) is
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(UP)*,A,, up) = P(B(1)*uy)Aj),
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= -P(u$"Au~~)*=),P([aiuY)] 8,UY)*), ?= 0.
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R(l)must therefore vanish. The second-order equations of the stability problem become
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vpa, p a D(K0h)fjA(2)- K G(2) = dl)fjP)+
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+8A1)aI v(,), ah8:' = 0
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and their existence condition is
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d') (@)*, fi',) +(a:)*, 52)a, v y ) + (@*, vjll)a, gp) = 0,
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analogously to (6.2). The triple products have the same form as in (6.3), hence they are also zero and the existence condition is
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d1)= 0.
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Consequently,i t is found that in the second order no steady solution is preferred. Thisresult alsoholdsfor the unsymmetrical case of onefreeand onerigid bounding
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t We should recall that the solutionsw'," are real. The asteriskis introducedonly because
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the conclusions are also valid for complex functions .':w
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136
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A . Schliiter, D. Lortx and F . Busse
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surface, which contradicts Malkus & Veronis (1958) who conclude incorrectly that in the unsymmetrical case hexagons are preferred.
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In order to find higher approximations for the problem we must construct a particular solution of (6.I), which we rewrite explicitly for R(l)= 0
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If we take the vertical component of the curl of the equation (6.5) and use the
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identity
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E{ltl Ak 8i 0
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we get on the right-hand side
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eiklA,alUy)a,jU$') = €:ikl[(hka,(ajh,a,-hjA)V"')
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aj(aiA,a,-hiA) ?z()']
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a, a, a, = -Eikl Ak(al~ v ( 1 ) ) A, A, v(1).
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(6.6)
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The rest of the terms in the square brackets are symmetrical in i, k or i, 1 and
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hence cancel because of the antisymmetry of eikPUsing the properties
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vfl) = f (2)w ( r ) , A, w = -a2w
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of the first-order functions, one easily verifies that the last term of (6.6)vanishes. Thus Eikrhk a,Au$') = 0 and with aju:?)= 0 we can derive the velocities from a
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potential by u$,)= Sid2)in the same way as for the first-order functions. By operating with -Sion the equation (6.5),
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a, PA, + o ( 2 ) P A ~ ~ ~ =) A-,si U3P) z~(il),
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and finally by eliminating O(,)or ui2)Ajrespectively
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up (A3-R(0)A2) u3(?)A,
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(A3 -~ ( 0A) ) 8s
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= =
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-P-1ASi a, ~
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p-1~(s0iu) pai
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$
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-1 ) A,
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+ A
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up)aj @), ~ a~.ecuW.
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2
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33
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If we write the second-order solutions in the form
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then p ,q satisfy the equations
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ior with the boundary conditions
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either p"'
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P = P" =
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piv} = 0
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and
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where the primes indicate vertical differentiation. With the form (5.12) of the
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first-order solutions, the inhomogeneous terms in (6.8) become
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c,,c, a, uy)= n, m
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+ a . u(?) e(1)= -
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c,c, i 3 3
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+ w,wma4(1 #nm)
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X [f"f (1 - 24,m)f'Y' - 'a2(1 -$nm)f 'fI, (6.9)
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W , W , a,( #nm f ' g -fq'),
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12, m
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with the abbreviation
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4 n m E (kn-km)/'a2*
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O n the stability of steady Jinite amplitude convection
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137
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(6.8)has a solution of the form
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Since f and g in (6.9) are sums of hyperbolic functions according to (5.10), we
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finally obtain the formulas
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3 a2
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sinh (q,+4,) z sinh ( q -q ) z
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F q ( $ n m ,=~ )- C 2 q,($n,,,A,B,-A,B,) vp= 1
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+
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P
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Y
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A
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D-
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(6.11)
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|
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The second terms are solutions of the homogeneous part of (6.8)which must be added to satisfy the boundary conditions. The complex coefficients b,, d, are uniquely determined because the homogeneous boundary-value problem has no antisymmetrical solution if R(O)isnear RLo).It turns out that the coefficients b,, d, are zero in the free case. The case of rigid boundaries is discussed in the appendix. Sincethe calculation for the disturbances is quite analogous we write the secondorder solution
|
||
|
||
where G and F can be derived fromFpand Fqby differentiation (cf.equation (6.7)).
|
||
|
||
7. Eigenvalue perturbations to the order c3
|
||
|
||
In order to find differences in the behaviour of the various steady solutions we consider the third-order terms in the equations (2.2) and (2.3)
|
||
|
||
a, a a, D(ivf) -
|
||
|
||
- 9) sj(3) =
|
||
|
||
A + + &) ~ ( 1 ) ~ ( 2 ) K ~ AA h K
|
||
|
||
~:1),
|
||
|
||
a vp) DKOA ~)A( 3-) A j33) = d 2 )gip -R ( ~ ) A6 1,)~+ a, 67)+@(A)a, ,@)
|
||
|
||
+ ,pa, v p +6p)a h vy).
|
||
|
||
The existence conditions require that the right-hand sides of the equations be orthogonal to all first-order solutions which are represented by
|
||
|
||
138
|
||
|
||
A . Schliiter, D . Lortx and F . Busse
|
||
|
||
Since
|
||
|
||
<vi%*,v?)a,vf))= (&E)*n ) @A)aA dKl ) )= (d:A*, v~)a,fi$?=) 0,
|
||
|
||
due to equation (6.3),these conditions are
|
||
|
||
-R(2()v:%*,AKhv'A") +(v~?A*, vY)a, vP') = 0,
|
||
|
||
(7.1)
|
||
|
||
-R(2()vlfL*,AKAfip))+d2()@A*, EL')) +(~lfk*,a)aAv(,)+) ( v ~ ~ * , d ~ ) a h=i j0~. 2(7) .)2)
|
||
|
||
The second term in (7.1) can be transformed by partial integration
|
||
|
||
~ $ z * ) ~ aA J 3 (@* ,V,(1) v:~))= (vL1A*, a, vi2))= - (d:;L7 u p a,~',)*)
|
||
-- -p(@Ju3 (!ai).8n(1)*)m-R(O)((Si~ ( 2 )u) y)8, -- -p(o(2u)(!)8.8(1)*)-mR(0)( ~ ( 2S)iuy)a, u$%*),.
|
||
3 In
|
||
|
||
Substituting the representations (6.12) for the first- and second-order functions
|
||
|
||
we get
|
||
|
||
c (wz -tN
|
||
|
||
=
|
||
|
||
ck clcmL(#kl, #mn)
|
||
|
||
wk wl wm)m,
|
||
|
||
k,I,m=-N
|
||
|
||
L(#kl, #mn)
|
||
|
||
+ - ('(#kl) (fg'f # m n f ' g ) a2P R(o)F(#kl)
|
||
+ + - x [ -f"f- ( 1 2#mn)f"f'+ 2a2(1 #m,)f'fIa4(1 #mn))m*
|
||
|
||
(7.3)
|
||
|
||
The horizontal average is unequal to zero only in the following cases:
|
||
|
||
(i) k = n
|
||
1=-m
|
||
|
||
(ii) I = n k=-m
|
||
|
||
(iii) m = n k=-1
|
||
|
||
k+n
|
||
|
||
k=k n
|
||
|
||
kf-n.
|
||
|
||
Let us define the matrix
|
||
|
||
I 1 Tnm=
|
||
|
||
2L(-1, +l)+L(+l, -1)
|
||
for m = & n
|
||
2(L(#nm, - #nm) +L( -4 n m , #nm)
|
||
+L ( - 1, + 1)) otherwise
|
||
|
||
( n , m = - N ,...- 1 , 1 , . . . , + N )
|
||
|
||
which has the symmetries
|
||
|
||
Trim Trim = T-n,m = Tn,-m;
|
||
|
||
= Tmn,
|
||
|
||
since 4 n m = $mn, -4 n m = #-nm*
|
||
|
||
Note further that the diagonal elements of Tnmare equal to each other. After dividing by Cn (7.1) can then be written
|
||
|
||
-KR(2)+1- CN Tn,,GZCm= 0 , with h ' z Pa2((f"-a2f)2)m.
|
||
2 m=-N
|
||
|
||
(7.4)
|
||
|
||
From the symmetries of Tnmwe seethat only N equations of (7.4) are independent. Together with the normalization condition
|
||
|
||
cN cgc,, =
|
||
m=l
|
||
we have a system of ( N + 1 ) inhomogeneous equations, which determine the
|
||
+ ( N I ) values R(2)C, TG,, ..,,C$CN. This means that the manifold of first-order
|
||
|
||
On the stability of steady Jinite amplitude convection
|
||
|
||
139
|
||
|
||
solutions is restricted by the non-linearities of the equations. For instance, in the
|
||
‘regular ’ case, in which all angles between two neighbouring k-vectors are equal
|
||
the solution is c : c , = , . . . + c ~ c2~1N = - , R2(K~1N)m=EN=l-Tim*
|
||
|
||
The question arises as to whether further constraints arise due to the existence
|
||
|
||
conditions of the higher orders. This is certainly a complex problem in general.
|
||
|
||
I n the ‘regular’ case, however, there is no preferred k-vector so the N equations
|
||
of the vth-order existence condition, v = 3 , 4 , ..., reduce to only one equation,
|
||
which determines a@).Hence there is definitely an infinite number of steady
|
||
|
||
finite-amplitude solutions, the ‘regular’ ones, and we must find conditions for
|
||
|
||
the stability of the various solutions.
|
||
en We first restrict the class of disturbances to those whose coefficients are
|
||
|
||
zero except for k-vectors for which the C, are unequal to zero. Then (7.2)has the
|
||
|
||
form
|
||
|
||
ern + +N
|
||
|
||
Md2K’:,
|
||
|
||
T,, C zC, = 0,
|
||
|
||
m=-N
|
||
|
||
(7.5)
|
||
|
||
where we have used relation (7.3). Since Tami,s symmetrical, TnmCZC,is Hermitian and all eigenvalues d 2 ) are real. The system ( 7 . 5 ) has non-trivial
|
||
en solutions if and only if the characteristic equation
|
||
det I Md2)6,, +TnmC;lEC,l = 0
|
||
|
||
or
|
||
|
||
is satisfied.The left-hand side of ( 7 . 6 )is a real polynomialin d2w)ith 2 N real roots,
|
||
|
||
and the coefficient of (d2))mis positive. If we subtract the ( -m)th column from
|
||
the ( +m)th column and add the ( +n)th row to the ( -n)th, and use the sym-
|
||
|
||
metry properties of T,, we get a matrix with zero elements for negative m,
|
||
|
||
from which we conclude that N eigenvalues d2a)re zero.
|
||
|
||
The rest of the eigenvalues d2s)atisfy the equation
|
||
|
||
Mz:r det
|
||
|
||
+2Tnm!= 0 ( n , m = 1, ...,N ) ,
|
||
|
||
or
|
||
|
||
1 + det d2)&nm 2C, C$ T,,] = 0.
|
||
|
||
(7.7)
|
||
|
||
We show in the appendix that the matrix T,, has the additional property
|
||
|
||
+ T,, > Tnn> 0 (n m,;without summation convention),
|
||
|
||
(7.8)
|
||
|
||
from which it follows that all two-dimensional principal minors
|
||
|
||
4QnCX,cmcZ(TnnTrim- Tim)
|
||
|
||
of (7.7) are negative. Thus in the polynomial (7.7) the coefficient of (d2))Nis-2 negative. Since all zeros of ( 7 . 7 )are real, we can then conclude by the sign rule of Descartes that a t least one zero d2i)s positive, because the coefficients of a(2)N
|
||
|
||
140
|
||
|
||
A . Schliiter, D. Lortz and F . Busse
|
||
|
||
and d2)Nh-a2ve opposite sign. We have thus shown that all solutions with N > 1
|
||
are unstable. An exception is the case N = 1, the two-dimensional flow in the form of rolls,
|
||
for which the polynomial (7.7) is of degree one and has only a negative root
|
||
because the trace of Tnmis positive. In order to show that the two-dimensional
|
||
flow is not unstable with respect to any other disturbance we have to drop our original restrictions on the disturbances. We go back to equation (7.2) and consider the case that the f l m are unequal to zero for k,-vectors different from those of the steady motion. Then the horizontal average in (7.2) yields the diagonal elements
|
||
|
||
a”)~+L(-$,,,$,,)+L($,,, - 9 1 , ) - M + 1, - 11,
|
||
|
||
(7.9)
|
||
|
||
while the non-diagonal elements vanish so this part of the stability matrix can be discussed separately. I n order that the associated determinant vanishes the expression (7.9) has to be zero. As will be shown in the appendix
|
||
|
||
and hence all d2)-valuesare negative in that case, so for rolls the highest growth rate is d2)= 0. The disturbance with this growth rate turns out to be an infinitesimal horizontal translation of the steady motion normal to the axis of the rolls. This is clearly an exact solution of the stability problem with the growth rate zero, which we see by differentiatingthe steady non-linear equations.
|
||
To complete the proof for the stability of small amplitude rolls we have to consider disturbances with wave-numbers d different from the wave-number a of the rolls. Such disturbances satisfy the first-order equations with do)not necessarily zero. To second order dl)vanishes due to the vertical symmetry of the first-order functions, if we consider the cases of symmetric boundary condi-
|
||
- - - tions. At third order we get non-diagonal elements in the stability matrix, if the
|
||
condition k,+k,-3k1 = 0 , J k J= d
|
||
is satisfied, which is possible only in the case of d 2 a. If we approximate the values of the matrix element by taking the limit 16-a1 -+ 0, we find in the case of d > a the same matrix elements as in the case d = a. Hence in this limit
|
||
d -a -+ +0, the highest value of d 2 ) is zero. This means that rolls are unstable
|
||
for a < a,, because the positive value do)of c@sturbanceswith a wavelength slightly greater than a cannot be compensated by the contribution of d2k2to the growth rate (r. I n the case of d < a the non-diagonal elements vanish and in the
|
||
limit d -a -+ -0 the diagonal elements are of the form (7.9), which yield only
|
||
negative values d2)H. ence for a > a, the positive value doo)f disturbances with d < a can be compensated for finite amplitude E , while disturbances with d > a have negative values do).Thus we find that a finite-amplitude roll is stable if its horizontal scale corresponds to a value a with a,(€) > a > a, and unstable other-
|
||
wise. For free as well as for rigid boundaries figure 1 gives a qualitative picture of the stability range of finite amplitude rolls.
|
||
|
||
On the stability of steady finite amplitude convection
|
||
|
||
141
|
||
|
||
A
|
||
|
||
R
|
||
|
||
Unstable
|
||
~
|
||
|
||
RcI--- I
|
||
|
||
a,
|
||
|
||
U
|
||
|
||
FIGURE1. Stability range of rolls.
|
||
|
||
8. The convective heat transport
|
||
|
||
The difference between the total and the conduction heat transport through
|
||
|
||
a j$.e the layer is given by
|
||
|
||
= zcihj8- 8, = (uiAj 4m,
|
||
|
||
where the bars indicate a horizontal average. The first term unequal to zero is of the second order,
|
||
H = (UY)A,6’(1))m( R-R,)/R@+) ... = (K/PR(2()R) -B,.)+ ...,
|
||
|
||
giving the initial slope of the convective heat transport curve. For the free case Malkus & Veronis (1958) calculated for rolls, rectangles and hexagons. From
|
||
|
||
table 1 in the appendix one derives the following results for ‘regular’ solutions in the case of two rigid boundaries with KIP = 2904.4:
|
||
|
||
rolls
|
||
|
||
c;c1 = &;
|
||
|
||
+ g / ( R-R,) = (0.69942- 0*00472P-1 0.00832P-2)-1:
|
||
|
||
square cells
|
||
|
||
c*c c*c - 1 1 - 2 2 --4 ,I.
|
||
|
||
+ + g / ( R-R,) = (0.77890 0.03996P-1 0*06363P-2)-1:
|
||
|
||
hexagons
|
||
|
||
c:cl = c;c2= c,*c3= +;
|
||
|
||
B / ( R - R,) = (0*89360+0 ~ 0 4 9 5 9 P - ~0+*06787P-2)-1.
|
||
|
||
I n this connexionMalkus’s ‘hypothesis of maximum heat transport ’ ( 1 9 5 4 b~)~
|
||
should be mentioned. This states that, if there are several possible convective
|
||
|
||
142
|
||
|
||
A . Schluter, D . Lortx and F.Busse
|
||
|
||
motions, the fluid prefers the motion with the highest absolute value of heat transport. We prove this hypothesis for small amplitudes by showing that the
|
||
heat transport of rolls has an absolute maximum, i.e. corresponding H2h)as an
|
||
absolute minimum. Since the diagonal elements of T,, are equal to each other we derive from (7.4),using inequality (7.8) and the normalization condition,
|
||
|
||
where the equality sign is only valid for N = 1, the case of rolls.
|
||
9. Conclusions
|
||
We have found that not every linear steady solution is an approximation to the non-linear problem, but the degree of degeneracy of the finite-amplitude steady state is still extremely high. Exact formulae for the initial slope of the convective heat transport curve for a given cell pattern have been derived for rigid boundaries. Experiments of sufficient precision, especially with respect to the temperature boundary conditions,are not yet available to test these formulae.
|
||
Our systematic stability theory yields the result that three-dimensional con-
|
||
vection flows are unstable with respect to infinitesimal disturbances and that there is a class of two dimensional solutions in the form of rolls that are stable. Whether or not a given finite amplitude roll is stable depends on its wave length. The stability conclusions have been obtained at third order in an expansion in terms of the steady-state amplitude, so any small change in the Boussinesq equations being used could essentially alter the stability behaviour. This seems to be the reason why it is so difficult to produce the two-dimensional convection flows in a laboratory experiment. Stability analyses for the Boussinesq approximation in which density is the only temperature-dependent property have been extended by Palm (1960),Busse (1962),Palm & Qiann (1964),and Segel(1965), to take into account slight dependence of the other material properties on temperature. The main conclusions are that the corresponding vertical asymmetry in the layer leads to the stability of the hexagonal cell pattern in a range between the critical Rayleigh number and a certain supercritical value, beyond which rolls are stable.
|
||
REFERENCES
|
||
BUSSE,F. 1962 Dissertation,Munich University. MALKUSW , . V. R. 1954a Proc. Roy. SOCA. , 225, 185. MALKUSW, . V. R. 1954b Proc. Roy. SOCA. , 225, 196. MALKUSW, . V. R. & VERONISG, . 1958 J.Fluid Mech. 4, 225. PALM,E. 1960 J. Fluid Mech. 8, 183. PALME,. & QIANN,H. 1964 J.Fluid Mech. 19,353. PELLEWA,. & SOUTHWELLR,. V. 1940 Proc. Roy. SOCA. , 176, 132. REID,W. H. & HARRIS,D. L. 1958 PhyS. Fluids,1, 102. SEGELL, . 1965 J. Pluid Mech. 21, 359.
|
||
|
||
On the stability of steady Jinite amplitude convection
|
||
|
||
143
|
||
|
||
Appendix
|
||
We calculate here the elements of the matrix T,,. For free boundaries the sums
|
||
in the formulae (6.11)reduce to one term, because the coefficients A,, B, are zero except for one v. The boundary conditions are satisfiedwith vanishing coefficients b,, d,. Substitutingf , 9, P and G into the equation (7.3) we find
|
||
|
||
+ + + + + x (as+m2)2 P 2a2(1 ),$, B0)[(4n2 2a2(1 )$,, P-l+ 2(7r2 a2)]}.
|
||
This is a positive expression for all $nm with - 1 <,$, < 1, if the wave number
|
||
a does not deviate very much from its critical value a,, and it is zero for $%, = 1. This means that the relation (7.10) is verified and according to the definition of
|
||
T,, the inequality (7.8) holds.
|
||
I n the more realistic case of two rigid boundaries the calculation of the secondorder functions (6.11) is more complicated. Since the form of the first-order functions depends on a in this case, we restricted the calculations to the critical value a, of the wave-number and used the results of Reid & Harris (1958). In order that a solution of the homogeneouspart of (6.8)has the vertical dependence sinh [q;($,,) 21 the 4: must satisfy the equation
|
||
+ [@-2a2,(1 +4 , 2a~3,'O)~(1 +4), = 0,
|
||
|
||
which has the roots
|
||
|
||
+ 4;' = 2a2,(1+ $nm) -w,{2a2#:)(1
|
||
|
||
$nm)}',
|
||
|
||
where w, are the three cube roots of unity:
|
||
|
||
w 1 = 1, w2 = -$(1+,/3i), w3 = ~ 2 .
|
||
|
||
The coefficientsbV($,,), )$(a,
|
||
equations
|
||
|
||
are determined by the complex inhomogeneous
|
||
|
||
&I yq($nm, = p;($,3) = [d2/dz2- 'a:('+ $,,)I2 Fq($,2 )Iz=+ z= 0
|
||
|
||
and I$($, +) = Fi($,+)= I?:($, &) = 0, respectively. The expression
|
||
|
||
L($nrn?- $am)
|
||
|
||
depends in the followingway on the Prandtl number P:
|
||
|
||
,I'?- - + L($arn, $nm) = a4[-L($,,) P-l+kd$nrn) Po 4($nrn)
|
||
|
||
say-
|
||
|
||
We computed the coefficientsL, of P",which involve many complex numbers,
|
||
on the electronic computer G 3 of the Max-Planck-Institut fur Physik und Astrophysik, Munchen. The results are given in table 1, which shows that the relations (7.8) and (7.10) hold for rigid boundaries, too.
|
||
The unsymmetric case of one rigid, and one free boundary has been treated by Busse (1962) for the limit of large Prandtl number, with the same qualitative result for the relations (7.8) and (7.10).
|
||
|
||
144
|
||
|
||
A . Schbiiter, D.Lortz and F . Busse
|
||
|
||
8hwIa
|
||
|
||
-8
|
||
|
||
0-0
|
||
|
||
0.0
|
||
|
||
-7
|
||
|
||
2087.1
|
||
|
||
3125.6
|
||
|
||
-6
|
||
|
||
3630.4
|
||
|
||
4978.4
|
||
|
||
-5
|
||
|
||
4698.8
|
||
|
||
5885.7
|
||
|
||
-4
|
||
|
||
5364.3
|
||
|
||
6114.1
|
||
|
||
-3
|
||
|
||
5696.0
|
||
|
||
5873.2
|
||
|
||
-2
|
||
|
||
5759.8
|
||
|
||
5325.7
|
||
|
||
-1
|
||
|
||
5615.2
|
||
|
||
4597.8
|
||
|
||
0
|
||
|
||
5314.6
|
||
|
||
3782.2
|
||
|
||
1
|
||
|
||
4903.2
|
||
|
||
2947.4
|
||
|
||
2
|
||
|
||
4419.7
|
||
|
||
2142.4
|
||
|
||
3
|
||
|
||
3896.8
|
||
|
||
1400.4
|
||
|
||
4
|
||
|
||
3361.6
|
||
|
||
744.2
|
||
|
||
5
|
||
|
||
2836.6
|
||
|
||
186.5
|
||
|
||
6
|
||
|
||
2339.9
|
||
|
||
- 265.1
|
||
|
||
7
|
||
|
||
1886.4
|
||
|
||
- 608.3
|
||
|
||
8
|
||
|
||
1487.8
|
||
|
||
- 843.9
|
||
|
||
61,846 51,124 41,658 33,391 26,257 20,176 15,067 10,846
|
||
7,433 4,754 2,736 1,316
|
||
434 40 83
|
||
523 1,322
|
||
|
||
TABLEI The coefficients Lv($,,f,o,r,r)igid boundaries.
|
||
|