Rayleigh-Bénard convection - DiVA portal571135/...Rayleigh-Bénard convection Maja Sandberg Niclas Berg Gustav Johnsson Götgatan 78 Sarvstigen 3 Kungshamra 82 11830 Stockholm 18130

Rayleigh-Bénard convection

Maja Sandberg Niclas Berg Gustav Johnsson

Götgatan 78 Sarvstigen 3 Kungshamra 82

11830 Stockholm 18130 Lidingö 17070 Solna

+46707395394 +46735431430 +46735128711

[email protected] [email protected] [email protected]

16th May 2011

Bachelor degree project, SA104X, KTH MechanicsSupervisor: Philipp Schlatter, [email protected]

Abstract

This report considers Rayleigh-Bénard convection, i.e. the ow betweentwo large parallel plates where the lower one is heated. The change indensity due to temperature variations gives rise to a ow generated bybuoyancy. This motion is opposed by the viscous forces in the uid.The balance between these forces determines whether the ow is stableor not and the goal of this report is to nd a condition giving this limitas well as analyzing other aspects of the ow.

The starting point of the analysis is the incompressible Navier-Stokes equations and the thermal energy equation upon which theBoussinesq approximation is applied. Using linear stability analysisa condition for the stability is obtained depending solely on a non-dimensional parameter, called the Rayleigh number, for a given wavenum-ber k. This result is conrmed to be accurate after comparison withnumerical simulations using a spectral technique.

Further non-linear two- and three-dimensional simulations are alsoperformed to analyze dierent aspects of the ow for various values ofthe Rayleigh number.

Sammanfattning

I denna rapport betraktas Rayleigh-Bénardkonvektion, det vill sägaödet mellan två stora parallella plattor där den undre plattan värms.Eftersom densiteten i vätskan mellan plattorna beror av temperatu-ren kommer ett öde att uppstå på grund av ytkraften, som i sin turmotverkas av de viskösa krafterna i vätskan. Förhållandet mellan dessakrafter kommer att bestämma huruvida ödet är stabilt eller inte. Må-let med denna rapport är att hitta ett villkor som ger gränsen mellandessa fall samt att undersöka andra egenskaper hos ödet.

Analysen utgår från Navier-Stokes ekvationer för inkompressibeltöde samt ekvationen för den termiska energin, på vilka Boussinesqsapproximation tillämpas. Med hjälp av linjär stabilitetsanalys fås ettstabilitetsvillkor som endast beror av ett dimensionslöst tal, kallatRayleigh-talet, för ett givet vågtal, k. Detta resultat bekräftas ävenmed numeriska simuleringar som utfördes med en spektral metod.

Vidare utfördes även icke-linjära simulationer i två och tre dimen-sioner för att undersöka olika aspekter hos ödet för olika värden påRayleigh-talet.

Contents

1 Introduction 1

1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Theory 1

2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . 22.2 Boussinesq's approximation . . . . . . . . . . . . . . . . . . . 3

2.2.1 Navier-Stokes equations . . . . . . . . . . . . . . . . . 32.2.2 Thermal energy equation . . . . . . . . . . . . . . . . 3

3 Problem formulation 4

4 Non-dimensionalization 6

5 Stability analysis in two dimensions 7

5.1 Linear stability analysis and the method of normal modes . . 75.2 Eliminating the pressure term from the governing equations . 85.3 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.4 Streamfunction form . . . . . . . . . . . . . . . . . . . . . . . 95.5 Forming the eigenvalue problem . . . . . . . . . . . . . . . . . 105.6 Solving the eigenvalue problem . . . . . . . . . . . . . . . . . 11

6 Simulations 13

6.1 Formulation of the numerical problem . . . . . . . . . . . . . 136.2 Introduction to the numerical method of Simson . . . . . . . 146.3 Simulations in two dimensions . . . . . . . . . . . . . . . . . . 14

6.3.1 Numerical stability analysis . . . . . . . . . . . . . . . 146.3.2 Discussion of results for simulation with dierent Rayleigh

numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 166.4 Simulations in three dimensions . . . . . . . . . . . . . . . . . 16

6.4.1 Flow patterns . . . . . . . . . . . . . . . . . . . . . . . 21

7 Conclusion 22

A Matlab code for the stability curve 23

1 Introduction

Rayleigh-Bénard convection is a type of ow that is only driven by dierencesin density due to a temperature gradient. The Rayleigh-Bénard convectionoccurs in a volume of uid that is heated from below. In the case studiedin this report the uid is kept between two enclosing parallel plates andthe lower plate is kept at a higher temperature. The uid near the lowerplate will get a higher temperature and therefore a lower density than therest of the uid. Gravity will make the colder and heavier uid at the topsink but this is opposed by the viscous force in the uid. It is the balancebetween these two forces that determines if convection will occur or not. Ifthe temperature gradient, and thus the density gradient, is large enough thegravitational forces will dominate and instability will occur. It is the limit ofinstability and the occuring ow patterns that are investigated in this reportthrough both numerical and analytical methods.

1.1 History

In 1900 Henri Bénard made experiments on the instability of a thin layer(≤ 1mm) of uid heated from below. The convection of Bénards experimentinvolved surface tension and thermo-capillary convection, which is calledBénard-Marangoni convection, whilst Rayleigh-Bénard convection dependssolely on a temperature gradient. But the experiment made by Bénardinspired Lord Rayleigh (J.W. Strutt) who in 1916 derived the theoreticaldemands for the temperature gradient convection, hence the combined name.Rayleigh showed that instability occurs when the temperature gradient ∆Tis large enough to make the non-dimensional Rayleigh number, Ra, exceeda certain critical value. The Rayleigh number is dened as

Ra =α∆Tgh3

νκ,

where ∆T is the temperature gradient, α is the thermal expansion coecient,g is the acceleration due to gravity, h is the distance between the plates, νis the kinematic viscosity and κ is the thermal diusivity.

2 Theory

In this section the governing conservation laws for a uid will be formulatedsuch as the ones for mass, momentum and energy. Various approximationswill then be applied to achieve a set of equations known as the Boussinesqequations.

1

2.1 Governing equations

The ow is governed by the conservation of three basic quantities, i.e. mass,momentum and energy, each yielding a conservation equation. The conser-vation of mass states that [2]

1

ρ

Dρ

Dt+∇ · u = 0, (1)

where ρ is the density, u is the velocity and D/Dt denotes the materialderivative, which can be expressed symbolically as

D

Dt=

∂

∂t+ u · ∇.

If the ow is incompressible, i.e. the density of any given uid particledoes not vary throughout the ow Dρ/Dt = 0 and (1) yields

∇ · u = 0. (2)

The law of momentum conservation yields the equation

ρDuiDt

= ρgi +∂τij∂xj

,

where gi is the gravitational acceleration and τij is the stress tensor com-posed of viscous stresses and normal stresses caused by the pressure. Forincompressible ow one can show that it takes the form

ρDu

Dt= −∇p+ ρg + µ∇2u, (3)

which is known as the incompressible Navier-Stokes equations. Here, p de-notes the pressure in the uid and µ is the viscosity.

Lastly, the energy conservation states that

ρDe

Dt= −∇ · q − p∇ · u + φ, (4)

where e is the thermal energy density, q is the heat ux density and φ is therate of dissipation through viscous eects per unit volume. For incompress-ible ow it can be calculated through φ = 2µeijeij where eij is the strainrate tensor. The strain rate tensor contains the deformations of the uidparticles in the ow and is dened as

eij =1

2

[∂ui∂xj

+∂uj∂xi

].

The set of equations (2), (3) and (4) describe the behavior and propertiesof an incompressible ow and are the starting point for the analysis in thisreport. In the next section various approximations will be applied in orderto simplify these equations.

2

2.2 Boussinesq's approximation

The so-called Boussinesq approximation was rst proposed by Joseph ValentinBoussinesq (1842-1929) during the second part of the 19th century. His es-sential observation was that in a bouyancy driven ow the density variationscan be neglected everywhere in the momentum equation except in the gravi-tation term. This will be applied to simplify the Navier-Stokes equations andfurther assumptions will be made to rewrite the thermal energy equation.

2.2.1 Navier-Stokes equations

Assume that the density variations in the uid are small in comparison withthe velocity gradients, then we can deduce from the mass conservation (1)that

∇ · u ≈ 0. (5)

Assume further that the pressure and density can both be decomposed into abackground eld in hydrostatic equilibrium and a perturbation, thus takingthe form

p = p0(y) + p′(x, t), ρ = ρ0 + ρ′(x, t),

where p0(y) and ρ0 represents the background eld. For the hydrostatic case,u = 0 and Navier-Stokes equation (3) yields

∇p0 = ρ0g. (6)

Subtracting this from the Navier-Stokes equations (3) and dividing by ρ0

gives (1 +

ρ′

ρ0

)Du

Dt= − 1

ρ0∇p′ + ρ′

ρ0g + ν∇2u, (7)

where ν = µ/ρ0 is the kinematic viscosity.For small values of ρ′/ρ0, the left hand side of the equation reduces to

Du/Dt. The bouyancy term ρ′g/ρ0 in the right hand side though cannot beneglected since the gravity is strong enough to make the density variationrelevant [3].

With these assumptions, the following equation is obtained

Du

Dt= − 1

ρ0∇p′ + ρ′

ρ0g + ν∇2u. (8)

2.2.2 Thermal energy equation

The thermal energy equation (4),

ρDe

Dt= −∇ · q − p∇ · u + φ, (9)

3

will now be treated. It can be shown that for the cases where the Boussinesqapproximation is valid, the viscous dissipation rate φ only gives a negligablecontribution to the thermal energy [3] and will thus not be included.

The assumption made in the last section about the small density varia-tions is not applicable here [2], we must instead use the full mass conservation(1). Multiplying this relation by p and rearranging yields

−p∇ · u =p

ρ

Dρ

Dt.

Now assume that the density variations are small enough to be approximatedby a linear temperature dependency of the form

ρ = ρ0(1− α(T − T0)), (10)

where α = −1/ρ0 · (∂ρ/∂T )p.Assume further that the uid molecules do not interact with each other,

hence we can use the ideal gas approximation p = ρRT and R = Cp − Cv,yielding

−p∇ · u ≈ p

ρ

(∂ρ

∂T

)p

DT

Dt= − p

T

DT

Dt= −ρ(Cp − Cv)

DT

Dt.

Using that for a perfect gas e = CvT the thermal energy equation becomes

ρCpDT

Dt= −∇ · q.

Fourier's law states thatq = −k∇T,

where k is the thermal conductivity.The thermal energy equation (9) nally becomes

DT

Dt= κ∇2T, (11)

where we have introduced the thermal diusivity κ ≡ k/ρCp.

3 Problem formulation

Consider two innitly large plates with a distance h between them, see gure1. The lower and upper plate have a temperature of T0 and T1 respectively,such that T0 > T1. Between the plates we have a uid with density ρ, kine-matic viscosity ν, thermal diussivity κ and thermal expansion coecientα. The domain, Ω, in which this uid is situated is dened as

Ω : 0 < y < h,−∞ < x, z <∞.

4

Figure 1: Problem domain.

The uid obeys the equations that we have derived in the earlier sections.These governing equations are

Du

Dt= − 1

ρ0∇p+

ρ

ρ0g + ν∇2u, x, y, z ∈ Ω, t > 0,

DT

Dt= κ∇2T, x, y, z ∈ Ω, t > 0,

∇ · u = 0,

ρ = ρ0(1− α(T − T0)).

(12)

As boundary conditions we use that we have a constant temperature atthe lower and upper wall along with no-slip conditions,

T (x, t)|y=0 = T0, −∞ < x, z <∞, t > 0,

T (x, t)|y=h = T1, −∞ < x, z <∞, t > 0,

u(x, t)|y=0 = u(x, t)|y=h = 0, −∞ < x, z <∞, t > 0,

(13)

and as initial conditions we have that the uid is at rest and that the tem-perature in linearly distributed,

T (y, t)|t=0 = T0 + y

h(T1 − T0), x, y, z ∈ Ω,

u(x, t)|t=0 = 0, x, y, z ∈ Ω.(14)

5

4 Non-dimensionalization

To be able to further analyze this problem and nd the characteristic quan-tities, we introduce dimensionless parameters by making the substitutions

p = U2ρ0ν

κp, u = U u, T = ∆T T + T0, x = hx, t =

h

Ut, (15)

where all the quantities with a bar are dimensionless. Here we have in-troduced a velocity scale, U , a length scale, h, a dynamic pressure scale,U2ρ0ν/κ, a time scale, h/U , and a temperature scale, ∆T = T0 − T1.

From the coordinate transformation we get that

∇ =1

h∇, ρ = ρ0(1− α∆T T ),

∂

∂t=U

h

∂

∂t,

Du

Dt=U2

h

Du

Dt, (16)

which inserted into (12) yieldsU2

h

Du

Dt= −U

2ν

hκ∇p+ α∆T Tg +

Uν

h2∇2u,

∆TU

h

DT

Dt=

∆Tκ

h2∇2T .

By rearranging and identifying non-dimensional coecients we nd thatDu

Dt= −Pr∇p+ Riα∆T Tey +

1

Re∇2u, 0 < y < 1,−∞ < x, z <∞, t > 0,

DT

Dt=

1

Pe∇2T , 0 < y < 1,−∞ < x, z <∞, t > 0,

(17)

where

Pr =ν

κ, Ri =

gh

U2, Re =

Uh

ν, Pe = Re · Pr =

Uh

κ,

are the Prandtl, Richardson, Reynolds and Péclet numbers. This form willbe used for calculation purposes. To analyse stability, however, we rewritethis into another form by introducing a dierent velocity scale

Uκ =κ

h,

which transforms the equations to1

Pr

Du

Dt= −∇p+ RaTey + ∇2u, 0 < y < 1,−∞ < x, z <∞, t > 0,

DT

Dt= ∇2T , 0 < y < 1,−∞ < x, z <∞, t > 0,

(18)

6

where Ra is the Rayleigh number dened by

Ra =α∆Tgh3

νκ,

and its physical interpretation is the ratio between the bouyancy and viscousforces since

Bouyancy force

Viscous force∼ gρ

νU/h2∼ gα∆T

νκ/h3=α∆Tgh3

νκ= Ra.

For the boundary conditions, (13), we apply a similar tranformation andget

T (x, t)|y=0 = 0, −∞ < x, z <∞, t > 0,

T (x, t)|y=1 = 1, −∞ < x, z <∞, t > 0,

u(x, t)|y=0 = u(x, t)|y=1 = 0, −∞ < x, z <∞, t > 0.

(19)

5 Stability analysis in two dimensions

An important aspect of a ow is the stability, i.e. the eect that a disturbancehas on a given state. If a perturbation is introduced and it is allowed to growup to a nite amplitude the ow might enter a new state. Further distur-bances may be able to be introduced in this state and as the disturbancesadds up the ow can reach a chaotic state, normally known as turbulence.

In this section the stability of the governing equations derived in theprevious chapters will be examined using linear stability analysis. An intro-duction to this method, called normal mode analysis, will be presented andthe equations will be rewritten in a number of steps. Finally a conditionwill be derived for the limit where the ow transitions from being stable tounstable. The analysis will be limited to two dimensions, streamwise andwall-normal.

5.1 Linear stability analysis and the method of normal modes

The general idea of linear stabilitiy analysis is to introduce an innitesimalperturbation to a background state and examine whether its amplitude growsor decays with time. Since the perturbations are small the equations can belinearized which eases the analysis. The drawback of this method is that itonly considers the initial development of the disturbance. Nevertheless, wewill see that the results found in this section agrees very well with the resultsof the numerical experiments conducted in section 6.

In the method of normal modes the perturbations are assumed to besinusoidal. As we will see later, the allowed perturbations will take the form

f = f(y)eikx+σt

7

where f is an arbitrary perturbation with complex amplitude f , wave numberk in the x-direction and growth rate σ. The development in time of thesolution will thus only depend on the sign of the real part of σ, here denotedσr.

• If σr > 0 the perturbation will grow with time, unstable solution

• If σr < 0 the perturbation will decay with time, stable solution

• σr = 0 is the so-called marginally stable case, i.e. the boundary be-tween stability and instability.

The linearity of the equations yields that any possible disturbance canbe expressed as a superpositions of these perturbations. If we can show thatfor any given k, the solution is stable, then the solution will be stable forany innitesimal disturbance.

5.2 Eliminating the pressure term from the governing equa-

tions

The governing equations from last section will now be treated. Please notethat since we will only consider non-dimensional quantities from now on wewill drop the overbar.

We have no information about the pressure distribution in the uid andtherefore we want to eliminate the pressure gradient from equation (18).Using the vector identity [2]

u · ∇u = ∇(u · u

2

)+ ω × u,

and inserting it into equation (18) yields

1

Pr

[∂u

∂t+∇

(u · u2

)+ ω × u

]= −∇p+ RaTey +∇2u. (20)

Taking the curl of this equation and using that the curl of a gradient is equalto zero, we get

1

Pr

[∂

∂t(∇× u) +∇× ω × u

]= ∇× RaTey +∇2(∇× u). (21)

Identifying the vorticity ω = ∇×u and applying ∇×ω×u = u·∇ω−ω ·∇u[2] yields

1

Pr

[∂ω

∂t+ u · ∇ω − ω · ∇u

]= ∇× RaTey +∇2ω. (22)

8

We will only consider the two-dimensional case, T = T (x, y, t) and u =ux(x, y, t)ex + uy(x, y, t)ey, hence

ω =

(∂uy∂x− ∂ux

∂y

)ez = ωez

ω · ∇u = 0

∇× Tey =∂T

∂xez.

From this we get 1

Pr

[∂ω

∂t+ u · ∇ω

]= Ra

∂T

∂x+∇2ω

∂T

∂t+ u · ∇T = ∇2T.

(23)

These equations do not contain the pressure term any longer.

5.3 Linearization

Consider small disturbances, u′, T ′ and ω′ from a background state , u = 0,T = T (y) and ω = 0. The background temperature eld, T (y), is foundfrom (23) by setting u = 0 and ω = 0, thus yielding

∇2T (y) =∂2T

∂y2= 0, (24)

applying the boundary conditions T (0) = 0, T (1) = 1 gives the solution

T (y) = y. (25)

By inserting T = T (y) + T ′, u = u′ and ω = ω′ into (23) and linearizingby neglecting higher order terms, the equations for the disturbances are foundto be

1

Pr

∂ω′

∂t= Ra

∂T ′

∂x+∇2ω′

∂T ′

∂t+ uy = ∇2T ′.

(26)

5.4 Streamfunction form

We currently have three unknowns (ω′, u′y and T′) and the three equations

((26) and the continuity equation ∇·u′ = 0). In order to reduce the numberof unknown variables we introduce the stream function, ψ′, dened by

u′x =∂ψ′

∂y, u′y = −∂ψ

′

∂x, (27)

9

which replaces the continuity equation.The vorticity then takes the form ω = −∇2ψ and upon substitution into

(26) the resulting governing equations become1

Pr

∂

∂t∇2ψ′ = −Ra∂T

′

∂x+∇2(∇2ψ′)

∂T ′

∂t− ∂ψ′

∂x= ∇2T ′.

(28)

The boundary conditions transform to

ux(0) = uy(0) =∂uy∂y

(0) = 0 =⇒ ∂ψ

∂y(0) = 0

ux(1) = uy(1) =∂uy∂y

(1) = 0 =⇒ ∂ψ

∂y(1) = 0

T (0) = 0, T (1) = 1, T ′ = T − T =⇒ T ′(0) = T ′(1) = 0.

5.5 Forming the eigenvalue problem

The set of dierential equations (28) is linear and the coecients do neitherdepend on x nor t. This allow us to make the following normal mode ansatzfor ψ′ and T ′

ψ′ = ψ(y)eikx+σt

T ′ = T (y)eikx+σt,

where ψ and T are the complex amplitudes of ψ′ and T ′, k is the wavenumberand σ is the growth rate. Insertion into the governing equations yields

1

Prσ

[d2

dy2− k2

]ψ = −RaikT +

[d2

dy2− k2

]2

ψ,

σT − ikψ =

[d2

dy2− k2

]T .

(29)

σ and k can be shown to be real [2]. The sign of σ will, as previously stated,determine whether the solution is stable or not.

Consider the limit between these cases, σ = 0, the so-called marginallystable case. This case can only be consistent with the governing equations if

0 = −RaikT +

[d2

dy2− k2

]2

ψ,

−ikψ =

[d2

dy2− k2

]T .

(30)

10

Solving for T in the rst and sustituting into the second yields

−ikψ =

[d2

dy2− k2

]1

Raik

[d2

dy2− k2

]2

ψ, (31)

which can be simplied to

k2ψ =1

Ra

[d2

dy2− k2

]3

ψ. (32)

This forms an eigenvalue problem with eigenvalue Ra and eigenfunctionψ. The eigenvalues for a given k will give thus give the Rayleigh numbersyielding a neutrally stable solution of the equations of motion.

In order to be able to nd the possible solutions, the boundary conditionsmust be transformed again

T ′(0) = T ′(1) = 0 =⇒ T (0) = T (1) = 0, (33)

∂ψ

∂y(0) =

∂ψ

∂y(1) = 0 =⇒ dψ

dy(0) =

dψ

dy(1) = 0. (34)

The two last boundary conditions are found by considering (30) and using(33), resulting in[

d2

dy2− k2

]2

ψ(0) =

[d2

dy2− k2

]2

ψ(1) = 0. (35)

5.6 Solving the eigenvalue problem

The equation (32) is a linear homogenous ODE with constant coecientsand we can make the ansatz ψ(y) = eqy, where q is a constant. By insertingthis into the equation (32) we get

k2 +1

Ra

[q2 − k2

]3= 0, (36)

which has the six rootsiq0 = ±ik

(−1 +

(Rak4

)1/3)1/2

q1 = ±k(

1 +(Rak4

)1/3 (12 + i

√3

2

))1/2

q2 = ±k(

1 +(Rak4

)1/3 (12 − i

√3

2

))1/2,

(37)

and the solution takes the form

ψ(y) = Aeq0y +Be−q0y + Ceq1y +De−q1y + Eeq2y + Fe−q2y (38)

11

where A, B, C, D, E and F are constants. The boundary conditions (33)-(35)demand that

ψ(0) = A+B + C +D + E + F = 0

ψ(1) = Aeq0 +Be−q0 + Ceq1 +De−q1 + Eeq2 + Fe−q2 = 0

ψ′(0) = Aq0 −Bq0 + Cq1 −Dq1 + Eq2 − Fq2 = 0

ψ′(1) = Aq0eq0 −Bq0e

−q0 + Cq1eq1 −Dq1e

−q1 + Eq2eq2 − Fq2e

−q2 = 0[d2

dy2− k2

]2ψ(0) = (q2

0 − k2)2A− (q20 − k2)2B − (q2

1 − k2)2C

−(q21 − k2)2D − (q2

2 − k2)2E − (q22 − k2)2E = 0[

d2

dy2− k2

]2ψ(1) = (q2

0 − k2)2Aeq0 − (q20 − k2)2Beq0 − (q2

1 − k2)2Ceq1

−(q21 − k2)2Deq1 − (q2

2 − k2)2Eeq2 − (q22 − k2)2Eeq2 = 0.

(39)In order to nd the non-trivial solutions to this system of equations we writeit on matrix form, Mb = 0, where

MT =

1 eq0 q0 q0eq0 (q2

0 − k2)2 (q20 − k2)2eq0

1 e−q0 −q0 −q0e−q0 (q2

0 − k2)2 (q20 − k2)2e−q0

1 eq1 q1 q1eq1 (q2

1 − k2)2 (q21 − k2)2eq1

1 e−q1 −q1 −q1e−q1 (q2

1 − k2)2 (q21 − k2)2e−q1

1 eq2 q2 q2eq2 (q2

2 − k2)2 (q22 − k2)2eq2

1 e−q2 −q2 −q2e−q2 (q2

2 − k2)2 (q22 − k2)2e−q2

, b =

ABCDEF

,

and solve for a zero-determinant of M.The equation det(M) = 0 was solved numerically for a range of x values

of k. When k is x the value of M depends solely on the value of theRayleigh number yielding a one-dimensional equation det(M(Ra))|k = 0.This equation was solved in Matlab using Newton's method which, givena start guess Ra0, applies the following iteration scheme

Ran+1 = Ran −d(Ran)

d′(Ran),

where we have let d = det(M) for brewity.The derivative, d′(Ra), was evaluated using a dierence approximation

on the form

d′(Ra) ≈ d(Ra + ∆)− d(Ra−∆)

2∆

where ∆ was choosen to be small.The complete code can be found in appendix A and the result for a range

of k-values are presented in gure 2. We can see the highest Rayleigh numberyielding a stable solution for all values of k is Ra = 1708.

12

1000 2000 3000 4000 5000 6000 7000 80001

2

3

4

5

6

7

8

Ra

k

Unstable, σ > 0

Stable, σ < 0

Figure 2: Stability region

6 Simulations

In this section, the previously derived equations will be solved using Sim-son [1], a code developed at KTH Mechanics during the last 20 years. Thenumerical method will be introduced briey and the derived analytic condi-tions for stability will be compared to numerical results in two-dimensionalsimulations. Dierent solutions in two and three dimensions will also bepresented and discussed.

6.1 Formulation of the numerical problem

The starting point for the numerical simulations are the set of equations (17)from section 4 and the continuity equation. However, the innite region isreplaced by a nite domain

Ω = (x, y, z) ; 0 < x < Lx,−1 < y < 1, 0 < z < Lz.

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where Lx and Lz are the dimensions of the domain in the x- and z-direction.We have that

Du

Dt= −Pr∇p+ Riα∆TTey +

1

Re∇2u, x ∈ Ω, t > 0,

Dt

Dt=

1

Pe∇2t, x ∈ Ω, t > 0,

∇ · u = 0, x ∈ Ω, t > 0,

The boundary conditions in the x- and z-directions are replaced by pe-riodicity, yielding

T (x, t)|y=0 = T0, x ∈ [0, Lx], z ∈ [0, Lz], t > 0,

T (x, t)|y=h = T1, x ∈ [0, Lx], z ∈ [0, Lz], t > 0,

u(x, t)|y=0 = u(x, t)|y=h = 0, x ∈ [0, Lx], z ∈ [0, Lz], t > 0,

u(x, t) = u(x + Lxex, t), y ∈ [−1, 1], z ∈ [0, Lz], t > 0,

u(x, t) = u(x + Lzez, t), x ∈ [0, Lx], y ∈ [0, Lx], t > 0.

The intial conditions used are a linear temperature distribution and avelocity eld with random noise, u0(x), satisfying the continuity equation,hence

T (y, 0)|t=0 = T0 + yh(T1 − T0), x ∈ Ω,

u(x, 0)|t=0 = u0(x), x ∈ Ω.

6.2 Introduction to the numerical method of Simson

Simson is a pseudo-spectral solver for incompressible ows. The solver ap-plies a Fourier expansion of the solution in the span- and streamwise direc-tions and an expansion in Chebychev polynomials in the normal direction.Time is discretized using a combination of the Crank-Nicholson and Runge-Kutta iteration schemes. To avoid having to evaluate the convolution ap-pearing from the non-linear terms in the dierential equations at each timestep this value is calculated in physical space and then transformed back intoFourier-Chebychev space. For more information about the solution scheme,see Chevalier, Schlatter, Lundbladh and Henningson [1].

6.3 Simulations in two dimensions

6.3.1 Numerical stability analysis

The stability criteria found in section 5 will now be veried by numericalexperiments conducted with Simson. The resolution used in the experimentsare 32 and 33 spectral modes in the x- and y-direction respectively.

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Since we have periodic boundary conditions, the wavenumbers for thenormal modes in two dimensions will be a multiple of 2π/Lx where Lx is thelength of the domain in the x-direction. This allows us to set a value of kfor which the limiting value of the Rayleigh number can be found.

The procedure used was rst choosing a value of Ra and then makingsimulations for dierent k:s until the limit of stability was found. This wasdone for nine dierent values of Ra. The values found are plotted togetherwith the curve found in section 5 in gure 3. We were however unable toobtain results for k < 2 since the numerical method failed to converge in thiscase.

1000 2000 3000 4000 5000 6000 7000 80001

2

3

4

5

6

7

8

9

Ra

k

Unstable, σ > 0

Stable, σ < 0

Figure 3: Stability region with numerically found values.

In gure 3 we can observe the lowest value of the Rayleigh number onthe stability limit is Ra = 1708 for k = 3.12. This means that we willhave a stable ow for all k-values where Ra < 1708. An illustration ofthe temperature and velocity in the x-direction at this point is presented ingure 4. The temperature is here linearly distributed vertically, the velocityeld though has small values. To know if the ow is stable for this Rayleighnumber we will have to investigate if the velocity eld increases or decreaseswith time i.e. if the growth rate σr is positive or negative. To see thislog urms and log vrms was plotted against time for some Rayleigh numbersaround 1708. The slope of these curves gives σr, as discussed in the stabilitysection. In gure 5 we can see that for Ra = 1707 σr < 0, the velocity goesto zero and we have a stable solution. For Ra = 1708 σr > 0 and we havean unstable solution. The conclusion is that the limit σr = 0 lies betweenthese two Rayleigh numbers and we have a stable ow for Ra < 1708.

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6.3.2 Discussion of results for simulation with dierent Rayleigh

numbers

It is also interesting to investigate the behavior of the uid for dierentRayleigh numbers. Studies where made for a x k-value, k ≈ 2.24 and Ra =1000, 4000 and 10000. For the higher Rayleigh numbers the non-linearity isexpected to play an important role, leading to a so-called saturated state.All of the plots of these simulations are for 70 time units.

For Ra = 1000 we would expect, as concluded earlier, a stable ow. Thetemperature and velocity distibution can be seen in gure 6. Here we can seethat just as for Ra = 1708 the temperature is linearly distributed verticallywhile we have a small irregular velocity eld.

For Ra = 4000 we obtained a more unstable ow, as can bee seen ingure 7. The higher ratio between bouyancy and viscous forces results inthis wave-shaped temperature distribution. We can also see on the velocitylevel curves that we now have a ow with clear two-dimensional rolls in thevelocity eld.

The last study is for Rayleigh number, Ra = 10000. In gure 12 we cansee the characteristic mushroomed-shaped temperature distributions and ahigh velocity in the ow.

To better understand how the velocity is distributed we also plot thevelocity eld in the x-y-plane. In gure 9 we can see the vector elds forRa = 1000, 4000 and 10000. For Ra = 1000 we only have a small randommotion due to the decaying eect of the initial random disturbances. ForRa = 4000 and 10000 the uid is circulating and as the length of the arrowsindicates we have a much higher velocity for the higher Rayleigh number.

6.4 Simulations in three dimensions

Simson was also used for simulating the ow in three dimensions for dif-ferent values of the Rayleigh number. The resolution was now incresed to64 spectral nodes in the x- and z-directions and 33 in the y-direction. ForRayleigh numbers in the region 2000 ≤ Ra ≤ 5000 we observed that theow converged to convection rolls in the velocity eld and the characteristicmushrooms in the temperature eld. Examples of this can be seen in gure10 where temperature distribution and streamlines of the ow are shown.This agrees well with theory that states that stationary patterns should oc-cur when the kinetic energy generated due to the cooling and heating of theplates is balanced by viscous dissipation [2].

For Ra = 4500 and 4750 the ow never converged for a Lx = Lz = 10simaluation area, but when it was increased to Lx = Lz = 20 the owconverged to rolls. This could be because the domain was too small withLx = Lz = 10 which inhibited the growth of the rolls with the properwavelength.

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(a) T (b) u

Figure 4: Level curves for T and u for Ra = 1708, k ≈ 3.12.

(a) Ra = 1707 (b) Ra = 1708

Figure 5: Plots of log urms and log vrms.

(a) T

(b) u

Figure 6: Level curves for T and u for Ra = 1000.

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(a) T

(b) u

Figure 7: Level curves for T and u for Ra = 4000.

(a) T

(b) u

Figure 8: Level curves for T and u for Ra = 10000.

18

(a) Ra = 1000

(b) Ra = 4000

(c) Ra = 10000

Figure 9: Velocity elds for three dierent values of Ra.

19

(a) Temperature distribution (b) Streamlines

Figure 10: Ra = 3000.

(a) Temperature distribution (b) Streamlines

Figure 11: Ra = 4500.

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For higher Rayleigh numbers, in the region 5000 < Ra ≤ 10000, theow never converged into a stationary pattern. Instead, the pattern took anoscillating sinusodial form, which can be seen in gure 12.

(a) Temperaturedistribution overthe center plane.

(b) Isocontours for the tem-perature eld.

(c) Isocontours for the veloc-ity in the normal direction.

Figure 12: Ra = 10000 simulated over a Lx = Lz = 20 area.

When the Rayleigh number reached 20000 the motion was initially totallyrandom, but after some time it merged into travelling wave-like patterns.Illustrations of the temperature eld at Ra = 20000 is presented in gure 13.This simulation was done with 128 spectral nodes in the x- and z-directionsand 65 in the y-direction.

(a) 30 time units (b) 60 time units (c) 90 time units

Figure 13: Ra = 20000 for three time units.

For higher Rayleigh numbers a less structured ow will appear and tur-bulence will occur at Ra > 50000 [2].

6.4.1 Flow patterns

We have seen that the unstable ow can converge into stationary ow pat-terns. The most commonly occuring ow pattern in our simulations is con-vections rolls but other patterns such as hexagonal convections cells may alsooccur. The hexagonal cells often occur in the beginning of a simulation andthen merge into convetion rolls. Since the problem is horizontally isotropicthe convection patterns will form in a horizontally random direction i.e. ei-ther in the x-, z- or a diagonal direction. This can be seen if you comparegure 10 and gure 14 where the rolls have lined up in dierent directions.An example of a hexagonal cell can be seen in gure 11. In this example apoint source located in the middle of the hexagonal cell was used to visualisethe streamlines.

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(a) Temperature distribution (b) Streamlines

Figure 14: Ra = 5000.

7 Conclusion

The general behavior of the uid, in the aspect of Rayleigh-Bénard, is onlygoverned by the dimensionless Rayleigh number. The stability analysisshowed that the ow will be stable for any wavenumber if the Rayleighnumber is smaller than 1708, this was later veried by the numerical simula-tions. The analytically derived stability curve agrees well with the stabilityfound numerically. This implies that the linear stability analysis is accurateand a good model to predict the behavior of the ow.

The Rayleigh number also aected other aspects of the ow. We sawthat when the Rayleigh number was 2000 ≤ Ra ≤ 5000 the ow converged toconvection rolls which are steady and strictly two-dimensional, when 5000 <Ra ≤ 10000 these rolls oscillated in a sinusodial way instead of convergingto a steady pattern. For Ra = 20000 the behavior of the ow becametotally random at rst but eventually took a more structured form. Anincreasing Rayleigh number resulted in higher velocities and less structuredow patterns.

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A Matlab code for the stability curve

The value of det(M) is calculated with the following function

1 function ret = detA(Ra)2 global k3 q0= 1 i ∗k∗(−1+(Ra/k^4) ^(1/3) ) ^(1/2) ;4 q1= k∗(1+(Ra/k^4) ^(1/3) ∗(1/2+1 i ∗sqrt (3 ) /2) ) ^(1/2) ;5 q2= k∗(1+(Ra/k^4) ^(1/3) ∗(1/2−1 i ∗sqrt (3 ) /2) ) ^(1/2) ;67 A=[1 1 1 1 1 18 exp(q0) exp(−q0) exp(q1) exp(−q1) exp(q2)

exp(−q2)9 q0 −q0 q1 −q1 q2 −q210 q0∗exp(q0) −q0∗exp(−q0) q1∗exp(q1) −q1∗exp(−q1)

q2∗exp(q2) −q2∗exp(−q2)11 (q0^2−k^2)^2 (q0^2−k^2)^2 (q1^2−k^2)^2

(q1^2−k^2)^2 (q2^2−k^2)^2 (q2^2−k^2)^212 (q0^2−k^2)^2∗exp(q0) (q0^2−k^2)^2∗exp(−q0)

(q1^2−k^2)^2∗exp(q1) (q1^2−k^2)^2∗exp(−q1)(q2^2−k^2)^2∗exp(q2) (q2^2−k^2)^2∗exp(−q2) ] ;

13 ret=det (A) ;14 end

and is thereafter used in the following program to nd the solution of det(M) =0 for a range of values of k in the interval [1, 8].

1 clear , cl f , clc2 global k34 ks=8:−0.01:1 ; %range o f ks to s o l v e f o r5 Ra=7000; %s t a r t guess f o r k = 86 delta_Ra = 0 . 0 0 1 ; %step s i z e f o r d e r i v a t i v e

approximation7 tolerance = 0 . 0 0 1 ; %to l e r an c e f o r the abso lu t e e r r o r

o f Ra8 Result=[ ] ; %vecto r f o r r e s u l t s910 for k=ks11 dR = 1 ; %dummy value12 %Solve with Newton ' s method un t i l the abso lu t e

va lue e r r o r13 %approximation dR reaches a value below the

t o l e r an c e14 while abs (dR) > tolerance15 f = detA(Ra) ;16 df =

(detA(Ra+delta_Ra)−detA(Ra−delta_Ra) ) /(2∗delta_Ra) ;17 dR = − f /df ;18 Ra = Ra + dR;19 end

23

20 Result=[Result Ra ] ;21 end22 plot (Result , ks )23 xlabel ('Ra' ) , ylabel ('k' )

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References

[1] M. Chevalier, P. Schlatter, A. Lundbladh, and D. Henningson. Simson - apseudo-spectral solver for incompressible boundary layer ows. Technicalreport, KTH Mechanics, 2007.

[2] P. K. Kundu and I. M. Cohen. Fluid Mechanics. Academic press, 4thedition, 2008.

[3] E. A. Spiegel and G. Veronis. On the boussinesq appoximation for acompressible uid. Astrophysical Journal, 131:442447, 1960.

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