1. Lecture 19-20: Natural convection in a plane layer. Principles of linear theory of hydrodynamic stability 1 z x Governing equations: T=0T=0 T=AT=A h =1 We will distinguish two cases: A =-1 – layer heated from above A =1 – layer heated from below

2. Quiescent state 2 1. Quiescent state:

3. Linear stability of a quiescent state 3 Let us analyse the time evolution of a small perturbation of the quiescence The linearised equations for a perturbation read v 0 =0, p 0 and T 0 is the basic state; v, θ and p’ is a small perturbation

4. 4 For 2D flow, we may introduce the stream-function defined as The continuity equation is satisfied automatically Taking x - and z -projections of the Navier-Stokes equation gives

5. 5 The Navier-Stokes and heat transfer equation can be re-written as z -derivative of the first equation minus x -derivative of the second equations gives Boundary conditions: For temperature, at the upper and lower plate: For velocity, at the upper and lower plate (rigid walls): or, in terms of stream-function

6. 6 Time dependence of a perturbation: In general, If λ r >0 then the perturbation will exponentially grow (with the rate λ r ). If λ r <0 then the perturbation will exponentially decay (with the rate λ r ). If λ i ≠ 0 then the growth (or decay) is oscillatory. If λ i = 0 then the growth (or decay) is monotonic. Basic idea of the stability analysis: (i)Seek a solution in the form of the normal modes. (ii)Find λ that satisfy the equations (we may find several discrete values of λ or even continuous spectrum of λ ). (iii)If at least one λ has positive real part then the considered basic state is unstable. We will analyse stability of the quiescent state only in respect to normal modes:

7. 7 For this problem it can be shown that perturbations develop monotonically, i.e. λ i = 0. Substitution of the normal modes gives Boundary conditions: Next, system (*) together with the above boundary conditions can be solved numerically. For the case, when the upper and lower plates are both free surfaces (which is not a good assumption as both plates cannot be free surfaces), the solution can be obtained analytically. (*) At a free boundary: In terms of the stream-function, for normal perturbations:

8. 8 Functions and satisfy the boundary conditions. Let us substitute these functions into system (*): or We obtained the homogeneous system of linear equations, This system has a non-trivial (non-zero) solution if or (**)

9. 9 The last condition written for equations (**) is or This quadratic equation can be written in the following form:

10. 10 A basic state is unstable if λ r >0. For this, we need This equation will have two solutions defined by the formula: or Finally, For the layer heated from above, A =1. The instability may occur if But Ra >0, this condition is never satisfied. Hence, the layer heated from above is hydrodynamically stable. Fluid will remain at the quiescent state. -- Rayleigh number

11. 11 For the layer heated from below, A =-1. The instability may occur if Ra unstable stable neutral curve, λ r =0 (stability curve) k kckc Ra c Let us determine Ra c. Condition of minimum: Quiescence becomes unstable for the layer heated from below if the temperature difference between the plates is high enough for Ra > Ra c. Convective rolls with dimensions of will be observed.

12. 12 For the case of rigid-rigid boundaries, the stability diagram is very similar but For the free-rigid boundaries,

13. Cloud streets 13 Horizontal convective rolls producing cloud streets (lower left portion of the image) over the Bering Sea Simple schematic of the production of cloud streets by horizontal convective rolls Good pictures: http://www.meteorologyn ews.com/2009/10/29/clo ud-streets- photographed-over-gulf- of-mexico/

14. 14 Remarks: (i) The method used to define the instability threshold is universal. This method can be used for finding the thresholds of instability of any solution for any partial differential equations. (ii) Next, we can take the convective rolls as a basic state; represent all physical quantities as sums of a basic state with small disturbances; linearise the equations; and determine the conditions when the found rolls become unstable. (iii) At the first instability threshold, the state of quiescence is replaced by convective rolls with a typical horizontal size k c. Passing the next instability threshold, the convective motion will represent the combination of the rolls of two different sizes. And so on. At large Gr~10 5, convective motion becomes turbulent: superposition of the rolls with the sizes determined by continuous spectrum of k.

15. 15 John William Strutt, 3rd Baron Rayleigh (12 November 1842 – 30 June 1919) was an English physicist who, with William Ramsay, discovered the element argon, an achievement for which he earned the Nobel Prize in Physics in 1904. He also discovered the phenomenon now called Rayleigh scattering, explaining why the sky is blue, and others.