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Maximum density effects on convective instability of horizontal plane poiseuille flows in the thermal entrance region

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Abstract

A linear stability analysis is used to study the conditions marking the onset of secondary flow in the form of longitudinal vortices for plane Poiseuille flow of water in the thermal entrance region of a horizontal parallel-plate channel by a numerical method. The water temperature range under consideration is 0∼30°C and the maximum density effect at 4°C is of primary interest. The basic flow solution for temperature includes axial heat conduction effect and the entrance temperature is taken to be uniform at far upstream location jackie=−∞ to allow for the upstream heat penetration through thermal entrance jackie=0. Numerical results for critical Rayleigh number are obtained for Peclet numbers 1, 10, 50 and thermal condition parameters (λ 1, λ 2) in the range of −2.0≤λ 1≤−0.5 and −1.0≤λ 2≤1.4. The analysis is motivated by a desire to determine the free convection effect on freezing or thawing in channel flow of water.

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Abbreviations

A :

temperature difference ratio, (T 1−T max)/ΔT

A n, Bn, Cn, Dn :

coefficients of infinite series defined by eqs. (5) and (6)

a :

dimensionless wave number

D :

operator, d/dz

f :

(1−λ 1 φ+λ 2 φ 2)

Gr :

Grashof number defined below eq. (14)

g :

gravitational acceleration

L, l :

height of channel and L/2

P, P b :

liquid pressure (P b+P′) and basic flow pressure

Pe :

Peclet number, 4U m l/α=Re Pr

Pr :

Prandtl number, ν/α

p :

dimensionless perturbation pressure, P′/(ρν 2/L 2)

Ra :

Rayleigh number, Pr Gr

Re :

Reynolds number, 4U m l/ν

T, T b, T m, T 0 :

water temperature (T b+θ′), basic flow temperature, (T 1+T 2)/2 and uniform upstream temperature

T 1, T 2, T max :

constant lower and upper plate temperatures, and maximum density temperature (4°C)

U b, U m, u b :

axial and mean velocities and (U b/U m) of basic flow

u, v, w :

dimensionless perturbation velocity components, (U′, V′, W′)/(ν/L)

X, Y, Z :

Cartesian coordinates with origin at lower plate

x, y, z :

(X, Y, Z)/L

x′, z′ :

(X′/(3/8)lPe, Z′/l)

jackie :

transformed coordinates, (x/(3Pe/16), z)

Y n, Rn, Fn, Zn :

eigenfunctions

Z′ :

transverse coordinate with origin at center of channel

α :

thermal diffusivity

α n, βn, εn, γn :

eigenvalues

γ 1, γ 2 :

temperature coefficients for density-temperature relationship

θ :

dimensionless temperature disturbance, θ′/ΔT

θ b, θ 0 :

dimensionless temperature and uniform entrance temperature, (T b−T m)/(T 2−T m) and (T 0−T m)/(T 2−T m)

λ 1, λ 2 :

thermal condition parameters defined below eq. (13)

ν :

kinematic viscosity

ρ, ρ max :

density and maximum density at 4°C

φ, φ u, φ θ :

(φ θ −1), dimensionless basic velocity and temperature profiles, (1/2)u b=3(z−z 2), (1−θ b)/2

ΔT :

(T 1−T 2)

o:

perturbation quantity

o:

amplitude of disturbance quantity

*:

transformed perturbation variable or critical value

b:

basic flow quantity in unperturbed state

1, 2:

upstream and downstream regions

f:

fully developed value

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Authors and Affiliations

  1. Dept. of Mech. Eng., Univ. of Alberta, Edmonton, Alberta, Canada

    K. C. Cheng & Ray-Shing Wu

Authors
  1. K. C. Cheng
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  2. Ray-Shing Wu
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Cheng, K.C., Wu, RS. Maximum density effects on convective instability of horizontal plane poiseuille flows in the thermal entrance region. Appl. Sci. Res. 33, 405–425 (1977). https://doi.org/10.1007/BF00411822

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  • DOI: https://doi.org/10.1007/BF00411822

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