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Hydrodynamic instability in fluid layers with uniform volumetric energy sources

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Abstract

Linear and energy theory stability criteria are presented for fluid layers of infinite horizontal extent heated internally by a uniform volumetric energy source. The thermal coupling between the layer and its environment is modeled by a general mixed boundary condition in both the conduction state and the disturbance temperature. Rigid-rigid, free-free, free-rigid, and rigid-free boundaries are considered in the computations. For a fixed ratio of upper surface Biot number to that at the lower surface, decreasing the Biot number is strictly destabilizing for both linear and energy theory criteria. A region of possible subcritical instability is found; its size is strongly dependent on Biot number and becomes small for small values of lower surface Biot number and large Biot number ratio. For two rigid surfaces and an upper and lower surface Biot number of 47.5, mean energy transport measurementswithin the convecting layer indicate a critical Reyleigh number close to that predicted by linear theory. Subcritical instability is not observed when finite amplitude disturbances are introduced at a Rayleigh number between the critical values predicted by the linear theory and the energy theory.

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Abbreviations

a :

wave number

Bi :

Biot number,hL/k f

Bi + :

effective Biot number for a boundary of finite thickness, (k w/L w)/(k f/L f)

c P :

specific heat at constant pressure of the fluid

d :

strain-rate tensor of disturbance velocity

D:

differential operator, d/dZ

D:

strain-rate tensor of the basic motion

D λ :

total dissipation integral 2 ∫ V e:e dV + λ(∫ V ∇θ·∇θ dV + ∫ Ω ΓBi 0θ2 dΩ)

e :

dimensionless strain-rate tensor for disturbance motion\(d \cdot \left[ {\frac{{L^2 v_r ({{SL} \mathord{\left/ {\vphantom {{SL} {2\alpha _r }}} \right. \kern-\nulldelimiterspace} {2\alpha _r }})}}{{g\beta \alpha _r }}} \right]^{\tfrac{1}{2}} \)

f :

the vector (0, 0, −1)

E λ :

total energy of disturbance defined in (12)

F(x, y):

plan form for disturbance velocity and temperature

g :

gravitational acceleration

h :

heat transfer coefficient on exterior of layer boundary

h int :

heat transfer coefficient on interior of layer boundary for steady convection atRa≥Ra c

H :

volumetric energy source, energy/time-volume

H(z):

z-component of disturbance temperatureθ(x, y, z, t)

I λ :

total production integral, ∫ V (μ* u·ε·u +f·uθ) dV + λ∫ V ∇ψ·uθ dV

k :

thermal conductivity

L :

height of fluid layer

m :

Rayleigh number exponent in correlation of heat transfer within the layer forRa>Ra c

M :

some characteristic value ofD in the time interval (0,t)

Nu :

Nusselt number for heat transfer within the layer at the layer surface,h int L/k f

Ra :

Rayleigh number\(Ra = \frac{{g\beta L^3 }}{{32\alpha v}}\left( {\frac{{SL^2 }}{{2\alpha }}} \right) = \frac{{g\beta }}{{\alpha v}}\left. {\frac{{L^3 }}{8}(T({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}) - T(1))} \right|_{\Gamma = 1} \)

Ra⋆:

modified Rayleigh number\(Ra^ * = (2 - \Phi )^5 Ra = \frac{{g\beta }}{{\alpha v}}L^3 (1 - {\Phi \mathord{\left/ {\vphantom {\Phi 2}} \right. \kern-\nulldelimiterspace} 2})^3 (T({\Phi \mathord{\left/ {\vphantom {\Phi 2}} \right. \kern-\nulldelimiterspace} 2}) - T(1))\)

Ra c :

critical Rayleigh number of linear theory

Ra ⋆ c :

modified critical Rayleigh number of linear theory

Re :

Reynolds number of basic state flow,L 2 M/ν

R λ :

Lagrange multiplier in energy method

S :

reduced volumetric energy source strength,H/ρc p

t :

time

T :

conduction state temperature

E ∞ :

environment temperature

u′ :

disturbance velocity (u′, v′, w′)

u :

dimensionless disturbance velocity of energy theory, (u, v, w)\(u = u' \cdot \left[ {\frac{{v_r ({{SL} \mathord{\left/ {\vphantom {{SL} {2\alpha _r }}} \right. \kern-\nulldelimiterspace} {2\alpha _r }})}}{{g\beta \alpha _r }}} \right]^{\tfrac{1}{2}} \)

V :

fluid volume

W(Z):

z-component of disturbance velocityw′ (non-dimensionalized in energy method)

x, y :

horizontal coordinates in fluid layer

z :

vertical coordinate in fluid layer, O≤z≤L

Z :

non-dimensional vertical coordinate in fluid layer,z/L

α :

thermal diffusivity of fluid

β :

isobaric coefficient of thermal expansion of fluid

Γ :

Biot number ratio,Bi 1/Bi 0

ε :

non-dimensional strain-rate tensor,D/LM

θ :

disturbance temperature of both linear and energy theory

λ :

coupling parameter in total energy of disturbance (see (12))

μ * :

\(Re/\sqrt {32Ra} \)

v :

kinematic viscosity of fluid

ρ :

density of fluid

ρ⋆−1 :

a function defined by (14)

σ :

decay constant for disturbances in linear theory

τ :

non-dimensional time,ν r t/L 2

Φ :

function defined by (4)

ψ :

non-dimensional temperature, (T−T ∞)/(SL 2/2α)

Ω :

surface of fluid volume

c:

critical value

r:

reference value

f:

fluid value

w:

wall value

0:

value atZ=0

1:

value atZ=1

∼:

energy theory value

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Authors and Affiliations

  1. Mech. Eng. Dept., The Ohio State Univ., 43210, Columbus, Ohio, U.S.A.

    F. A. Kulacki & R. J. Goldstein

  2. Mech. Eng. Dept., Univ. of Minnesota, 55455, Minneapolis, Minnesota, U.S.A.

    F. A. Kulacki & R. J. Goldstein

Authors
  1. F. A. Kulacki
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  2. R. J. Goldstein
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Kulacki, F.A., Goldstein, R.J. Hydrodynamic instability in fluid layers with uniform volumetric energy sources. Appl. Sci. Res. 31, 81–109 (1975). https://doi.org/10.1007/BF01795829

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