Bénard Convection with Rotationand a Periodic Temperature Distribution
By: Serge D’Alessio & Kelly Ogden
Department of Applied MathematicsUniversity of Waterloo, Waterloo, Canada
AFM 2012, June 26-28
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 1 / 23
Introduction
Problem description
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 2 / 23
Introduction
Previous work
Pascal and D’Alessio [1] studied the stability of a flow with rotation and aquadratic density variation.
Schmitz and Zimmerman [2] studied the effects of a spatially varyingtemperature boundary condition as well as wavy boundaries, withoutrotation and assuming a very large Prandtl number.
Malashetty and Swamy [3] considered the effects of a temperatureboundary condition varying in time.
Basak et al. [4], studied the flow resulting from a spatially varyingtemperature boundary condition in a square cavity without rotation.
The current work takes advantage of the thinness of the fluid layer, andinvestigates flow in a rectangular domain with rotation and a varying bottomtemperature.
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 3 / 23
Governing equations
Governing equations and boundary conditions
Governing equations:~∇ · ~v = 0
∂~v∂t
+(~v · ~∇
)~v + f k̂ × ~v =
−1ρ0
~∇p − ρ
ρ0gk̂ + ν∇2~v
∂T∂t
+(~v · ~∇
)T = κ∇2T
Boundary conditions:
u = v = w = 0 at z = 0,H and y = 0, λ
T = T0 −∆T at z = H and y = 0, λ
T = T0 −∆T cos(
2πyλ
)at z = 0
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 4 / 23
Governing equations
Other conditions
Initial conditions:
u = v = w = 0 , T = T0 −∆TH
z −∆T cos(
2πyλ
)(1− z
H
)at t = 0
Density is assumed to vary according to:
ρ = (1− α [T − T0])
The flow is assumed to be uniform in the x-direction. This allows a stream function andvorticity to be defined as:
v =∂ψ
∂z, w = −∂ψ
∂y, ζ = −
(∂2ψ
∂y2 +∂2ψ
∂z2
)
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 5 / 23
Governing equations
Other conditions
Integral constraints are imposed on the vorticity and can derived using Green’sSecond Identity: ∫
V
(φ∇2χ− χ∇2φ
)dV =
∫S
(φ∂χ
∂n− χ∂φ
∂n
)dS
Here φ and χ denote arbitrary differentiable functions, ∂∂n is the normal derivative, and
S is the surface enclosing the volume V . Choosing φ to satisfy ∇2φ = 0 and lettingχ ≡ ψ, then ∇2χ = ∇2ψ = −ζ. Applying the boundary conditions ∂ψ
∂n = ψ = 0 on S,the above leads to ∫ H
0
∫ λ
0φnζdydz = 0
where
φn(y , z) = e±2nπz
{sin(2nπy)
cos(2nπy)
}for n = 1, 2, 3, · · ·
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 6 / 23
Governing equations
Scaling and dimensionless parameters
t → H2
κt , y → λy , z → Hz , ψ → κψ , ζ → κ
H2 ζ
T → (T0 −∆T ) + ∆TT , u → κ
Hu
Ra =αgH3∆T
νκRayleigh number
Ro =κ
HfλRossby number
Pr =ν
κPrandtl number
δ =Hλ
Aspect ratio
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 7 / 23
Governing equations
Dimensionless equations
∂ζ
∂t− δ ∂
∂z
(∂ψ
∂z∂2ψ
∂y∂z− ∂ψ
∂y∂2ψ
∂z2
)+ δ3 ∂
∂y
(−∂ψ∂z
∂2ψ
∂y2 +∂ψ
∂y∂2ψ
∂y∂z
)− δ
Ro∂u∂z
= δPrRa∂T∂y
+ Pr(δ2 ∂
2ζ
∂y2 +∂2ζ
∂z2
)
ζ = −(δ2 ∂
2ψ
∂y2 +∂2ψ
∂z2
)
∂u∂t
+ δ
(∂ψ
∂z∂u∂y− ∂ψ
∂y∂u∂z
)− δ
Ro∂ψ
∂z= Pr
(δ2 ∂
2u∂y2 +
∂2u∂z2
)
∂T∂t
+ δ
(∂ψ
∂z∂T∂y− ∂ψ
∂y∂T∂z
)= δ2 ∂
2T∂y2 +
∂2T∂z2
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 8 / 23
Approximate analytical solution
Approximate analytical solution
For small δ, an approximate analytical solution can be constructed:
ψ = ψ0 + δψ1 + · · · , ζ = ζ0 + δζ1 + · · ·
u = u0 + δu1 + · · · , T = T0 + δT1 + · · ·
The leading-order problem becomes:
∂T0
∂t=∂2T0
∂z2 ,∂ζ0
∂t= Pr
∂2ζ0
∂z2 + PrδRa∂T0
∂y
∂2ψ0
∂z2 = −ζ0 ,∂u0
∂t= Pr
∂2u0
∂z2
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 9 / 23
Approximate analytical solution
Approximate analytical solution
Applying the boundary, initial and integral conditions yields:
ζ0(y , z, t) = −πδRa(
z2 − z3
3− 7z
10+
110
)sin (2πy)
+∞∑
n=1
ane−n2π2Prt cos(nπz)
ψ0(y , z, t) = πδRa(
z4
12− z5
60− 7z3
60+
z2
20
)sin (2πy)
−∞∑
n=1
an
n2π2 e−n2π2Prt (1− cos(nπz))
where
an = 2πδRa sin (2πy)
∫ 1
0
(z2 − z3
3− 7z
10+
110
)cos(nπz)dz , n = 1,2, · · ·
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 10 / 23
Approximate analytical solution
Steady-state solutions
As t →∞, the following steady-state solutions emerge:
Ts = (1− z) (1− cos (2πy))
ζs = −πδRa(
z2 − z3
3− 7z
10+
110
)sin(2πy)
ψs = πδRa(
z4
12− z5
60− 7z3
60+
z2
20
)sin(2πy)
us = 0
Plotted on the next slide are the leading-order temperature and velocities(Ro = 0.0548, Pr = 0.7046 and Ra = 388.7).
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 11 / 23
Approximate analytical solution
Steady-state solutions
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
297.5
298
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
246810
x 10−4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
−10−505
x 10−4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
−2−101
x 10−4
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 12 / 23
Approximate analytical solution
Steady-state solutions
At leading order us = 0, the O(δ) solution for us is:
us =1
720πδRaRoPr
sin (2πy)z(z − 1)(2z4 − 10z3 + 11z2 − z − 1
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
−5
0
5x 10
−4
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 13 / 23
Approximate analytical solution
Transient development
v - velocity development with time for Ra = 1, Pr = 1 at times t = 0.01,0.1,1from top to bottom.
5 10 15 20 25 30 35 40 45 50
20
40
−10
−5
0
5
x 10−3
5 10 15 20 25 30 35 40 45 50
20
40
−0.02
0
0.02
5 10 15 20 25 30 35 40 45 50
20
40
−0.02
0
0.02
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 14 / 23
Approximate analytical solution
Transient development
w- velocity development with time for Ra = 1, Pr = 1 at times t = 0.01,0.1,1from top to bottom.
5 10 15 20 25 30 35 40 45 50
20
40
−0.01
0
0.01
5 10 15 20 25 30 35 40 45 50
20
40
−0.04−0.0200.02
5 10 15 20 25 30 35 40 45 50
20
40
−0.04−0.0200.020.04
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 15 / 23
Approximate analytical solution
Steady-state numerical solutions
The steady-state solution was also be determined numerically using thecommercial software package CFX. All results shown use air at 298K with adomain having a length of 20 cm and height of 2 cm with periodicity imposedat the ends. For cases with rotation, the angular velocity is 0.05s−1. Thenon-dimensional values are Ro = 0.0548 and Pr = 0.7046. The Rayleighnumber will depend on ∆T .
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 16 / 23
Approximate analytical solution
Steady-state numerical solutions
The temperature distribution for a case without rotation or modulated bottomheating (top) is compared to a case with these effects (∆T = 0.5K ,Ra = 388.7).
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 17 / 23
Approximate analytical solution
Steady-state numerical solutions
The velocity for a case without rotation or modulated bottom heating (top) iscompared to a case with these effects (∆T = 0.5K , Ra = 388.7).
Note that the pattern and magnitude of the temperature and velocity agreewell with the approximate analytical solutions.
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 18 / 23
Unstable solutions
Unstable numerical solutions
∆T = 0.5K , Ra = 388.7
∆T = 4K , Ra = 3109
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 19 / 23
Unstable solutions
Unstable numerical solutions
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 20 / 23
Unstable solutions
Unstable numerical solutions
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 21 / 23
Summary
Summary
Conclusions
Contrary to the Bénard problem, a non-zero background flow was found.
The approximate analytical solution agrees well with the fully numericalsolution.
While rotation is known to stabilize the flow, a variable bottomtemperature also influences the stability of the flow.
Interesting features emerging from the unstable numerical simulationswere observed.
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 22 / 23
References
References
[1] Pascal, J.P. & D’Alessio, S.J.D., The effects of density extremum and rotation on the onsetof thermal instability, International Journal of Numerical Methods for Heat & Fluid Flow 13,pp. 266 - 285, 2003.
[2] Schmitz, R & Zimmerman, W., Spatially periodic modulated Rayleigh-Bénard convection,Physical Review E 53, pp. 5993 - 6011, 1996.
[3] Malashetty, M.S., & Swamy, M., Effect of thermal modulation on the onset of convection in arotating fluid layer, International Journal of Heat and Mass Transfer 51, pp. 2814 - 2823,2008.
[4] Basak, T., Roy, S. & Balakrishnan, A.R., Effects of thermal boundary conditions on naturalconvection flows within a square cavity, International Journal of Heat and Mass Transfer 49,pp. 4525 - 4535, 2006.
By: Serge D’Alessio & Kelly Ogden (Waterloo) 2012 23 / 23