Convection with a First-Order Chemically Reactive Passive Scalar

  • J. R. Herring
  • J. C. Wyngaard

Abstract

Convection between horizontal, stress-free, perfectly conducting plates is examined in the turbulent regime for air. Results are presented for an additional scalar undergoing first order decay. We discuss qualitative aspects of the flow in terms of spectral and three-dimensional isosurface maps of the velocity and scalar fields. The horizontal mean profiles of scalar gradients and fluxes agree rather well with simple mixing-length concepts. Further, the mean profiles for a range of the destruction-rate parameter are shown to be nearly completely characterized by the boundary fluxes. Finally, we use the present numerical data as a basis for exploring a generalization of eddy-diffusion concepts that incorporates necessary non-local effects.

Keywords

Convection Covariance Vorticity Boulder Mirror Symmetry 

Nomenclature

C(x,y,z,t))

Concentration of decaying scalar

〈C〉)

Horizontal (or ensemble) average of C = 〈C〉 (z, t)

[C])

Volume average (over periodic domain) of C

D)

Depth of convective layer

Eυ)

Kinetic energy spectrum

g)

Acceleration of gravity

ĝ)

Unit vector along g

k)

Wave number vector

k0)

Lowest k in numerical simulation

Nu)

Heat flux through lower plate in units kΔT/D

Nc(0))

Scalar flux through lower boundary; units are same as for Nu

Nc(1))

Scalar flux through top plate; units are same as for Nu

p)

Pressure field

r)

Position vector

Ra)

Rayleigh number gα(T(0) — T(1))/(κνD 3 )

Rc)

Critical Ra for slip boundaries 27 π4/4

Rλ)

Taylor microscale Reynolds number = 〈w 2〉/〈(∂w/∂z)2½ν

T(x, y, z, t))

Temperature

〈T〉)

Horizontal (or ensemble average) of T= 〈T〉 (z, t)

T(0))

Lower plate temperature

T(1))

Top plate temperature

u (x, y, z, t))

Velocity field (x, y, z)

u(x, y, z, t))

x-component of velocity field

v(x, y, z, t))

y-component of velocity field

w(x,y,z,t))

z-component of velocity field

α)

Thermal expansivity

β)

Horizontal average of — ∂T/z

β)

C Horizontal average of — ∂C/z

γ)

Ratio of diffusivities for C and T, respectively

ΔT)

Temperature excess of lower plate over upper plate

Z)

Reacting scalar variance spectrum

ζ)

Fluctuation of C from its horizontal average

ε)

Fractional destruction rate of C in units of the thermal diffusion time

κ)

Thermometric diffusivity

κc)

Diffusivity for scalar C

ν)

Kinematic viscosity

φ1)

Toroidal velocity variance; see (6 a)

Sw)

Velocity derivative skewness [(∂w/∂z)3]/[(∂w/∂z)2]3/2

ST,C)

Scalar mixed skewness [(∂A/z)2 (∂w/z)]/[(∂A/z)2] [(∂w/∂z)2]½, A = (C, T)

(x, y, z))

Cartesian components of r

t)

Time

φ2)

Poloidal velocity variance; see (6 b)

σ)

Prandtl number (ν/ κ)

θ)

Fluctuation of T from its horizontal average

Θ)

Temperature variance spectrum

ω)

Vorticity ∇ × u

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References

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Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • J. R. Herring
    • 1
  • J. C. Wyngaard
    • 1
  1. 1.National Center for Atmospheric ResearchBoulderUSA

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