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Flow reversal in a combined convection flow of Newtonian fluid in a vertical duct

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Summary

Numerical solutions are presented for the problem of steady laminar combined convection flows in vertical parallel-plate ducts, with linearly varying wall temperatures. Neglecting streamwise diffusion in the analysis leads to a parabolic set of governing equations. These are solved using a marching technique for an implicit finite-difference scheme with vorticity, streamfunction and temperature as variables. A constant wall temperature is applied after a certain height to mimic downhole conditions in an oil well. Various values of the governing parameter,Gr, are considered, including the forced convection solution,Gr=0, whilstPr is set at a value of unity in order to present the numerical method. As |Gr| increases reverse flow regions appear and these are dealt with using a modification of the standard marching technique. Results are obtained in terms of velocity profiles, local Nusselt numbers, flow average temperatures and friction factors, with particular attention applied to the case where the wall temperature decreases with height.

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Abbreviations

d :

half width of the duct

f :

friction factor

g :

gravitational acceleration

Gr :

Grashof number,gβλd 3/v3

h :

local heat transfer coefficient of the fluid

H :

finite difference step size across the duct

k :

thermal conductivity of the fluid

K :

finite difference step size along the duct

N :

number of finite difference steps across the duct

NN :

number of finite difference steps along the duct

Nu :

Nusselt number,hd/k

P :

dimensionless pressure,p/(ϱ0υ 2m )−gy/υ 2m

Pr :

Prandtl number,v/α

Ra :

Rayleigh number,Gr Pr

Re :

Reynolds number,dv m/v

T :

temperature

u :

transverse velocity

U :

dimensionless transverse velocity,u/v m

ν:

streamwise velocity

V :

dimensionless streamwise velocity,v/v m

x :

transverse coordinate

X :

dimensionless streamwise velocity,x/d

y :

streamwise coordinate

Y :

dimensionless streamwise coordinate,y/(d Re)

α:

molecular thermal diffusivity of the fluid

β:

coefficient of thermal expansivity of the fluid, (−1/ϱ0)(∂ϱ/∂T)

ν:

non-dimensional vertical distance at which constant temperature is applied

λ:

applied temperature gradient

v :

kinematic viscosity of the fluid

ϱ:

density of the fluid

θ:

dimensionless temperature, (T−T 0/(Re λ)+y/(d Re)

ψ:

dimensionless streamfunction

Ω:

dimensionless vorticity

b :

value at the beginning of the reverse flow region

e :

entry value

f :

value at the end of the reverse flow region

i :

transverse finite difference suffix

j :

streamwise finite difference suffix

m :

flow average value

w :

value at the wall

O :

value at the wall aty=0

∞:

value asy → ∞

d :

value calculated during the downstream iteration sweep

s :

number of the iteration in the reverse flow region

u :

value calculated during the upstream iteration sweep

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Author notes

  1. A. T. Jones & D. B. Ingham

    Present address: Department of Applied Mathematical Studies, Leeds University, L52 9JT, Leeds, UK

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    1. A. T. Jones
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    2. D. B. Ingham
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    Jones, A.T., Ingham, D.B. Flow reversal in a combined convection flow of Newtonian fluid in a vertical duct. Acta Mechanica 99, 135–153 (1993). https://doi.org/10.1007/BF01177241

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    • DOI: https://doi.org/10.1007/BF01177241

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