Abstract
A study is made of the flow of a viscous, compressible, heat conducting, and perfect gas in slender axisymmetric channels under the adiabatic wall condition. Solutions to the equations of motion for such a gas are obtained using the method of similar solutions. This approach reduces the equations of motion to a pair of coupled nonlinear ordinary differential equations which have relatively simple closed form solutions. It is found that solutions of this type are only possible in channels with divergent walls and a favorable pressure gradient.
In the present investigation it is found that the velocity profiles are insensitive to moderate variations in the Prandtl number and γ, the ratio of specific heats. The temperature profiles, total temperature profiles, and Mach number profiles are however very sensitive to changes in Prandtl number and ratio of specific heats. For a Prandtl number less than unity there is an accumulation of total energy near the channel centerline and a deficit of total energy near the channel walls. The recovery factor is found to be equal to the Prandtl number.
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Abbreviations
- A :
-
area
- c p :
-
specific heat at constant pressure
- d :
-
diameter
- f :
-
dimensionless stream function
- g :
-
dimensionless temperature
- h o :
-
stagnation enthalpy
- k :
-
thermal conductivity
- \(\dot m\) :
-
mass flow rate
- M :
-
Mach number
- n :
-
normal to the wall
- N :
-
constant
- p :
-
pressure
- Pr :
-
Prandtl number
- r :
-
wall radius
- R:
-
recovery factor
- R :
-
gas constant
- Re :
-
Reynolds number
- s :
-
dimensionless temperature function
- T :
-
temperature
- u :
-
axial component of velocity
- v :
-
transverse component of velocity
- V :
-
temperature function
- x :
-
axial coordinate
- y :
-
transverse coordinate
- α :
-
constant
- β :
-
function of ξ
- γ :
-
ratio of specific heats
- η :
-
dimensionless y coordinate
- θ :
-
wall angle
- λ :
-
constant
- μ :
-
viscosity
- ξ :
-
x coordinate
- ρ :
-
density
- χ :
-
constant
- Ψ :
-
stream function
- ω :
-
constant
- c:
-
centerline
- i:
-
initial conditions
- o:
-
stagnation conditions
- res:
-
reservoir conditions
- AW:
-
adiabatic wall conditions
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Adams, J.C., Williams, J.C. Viscous compressible laminar flow in slender axisymmetric channels with adiabatic walls. Appl. sci. Res. 21, 113–137 (1969). https://doi.org/10.1007/BF00411601
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DOI: https://doi.org/10.1007/BF00411601