Wärme - und Stoffübertragung

, Volume 12, Issue 3–4, pp 211–220 | Cite as

Turbulent boundary layers with large free-stream to wall temperature ratio

  • T. K. Bose
Article

Abstract

Turbulent boundary layers with free-stream temperatures between 300°K and 3000°K, and wall temperature 300°K is studied numerically for air at 1 bar. Solution is started at one plane for a laminar, local similar boundary layer by solving differential equations by Runge-Kutta method. Velocity and enthalpy profiles are obtained at downstream planes by an implicit finite-difference iterative procedure. Effects of free-stream Mach number, sudden acceleration or deceleration, surface roughness, and uniform blowing or suction through the wall are studied and numerical results are compared with those available in open literatures.

Keywords

Boundary Layer Enthalpy Surface Roughness Mach Number Wall Temperature 

Nomenclature

A, B

damping constants, see Appendix

a

coefficient matrix, Eq. (12)

ā

coefficient element or sub-matrix, Eq.(11)

b

right hand vector, Eq.(12)

¯b

right hand value or sub-vector, Eq. (11)

C

density-viscosity ratio, Eq. (8c)

cf

friction coefficient, Eq.(13a)

Ec

Eckert number, Eq.(8i)

F, G, H

functions defined in Eqs. (8e-g)

f

dimensionless stream function

h

specific enthalpy, J/kg

h0

specific total enthalpy, J/kg

h0*

dimensionless total enthalpy

K1 - K4

constants, see Appendix

k

surface roughness, m

1

Prandtl mixing length, m

M

Mach number

N

a pressure gradient parameter, see Appendix

Pr

Prandtl number

p

pressure, bar or N/m2

qw

heat flux on wall, W/m2

Re

Reynolds number=[ρeUμe]x/μ e 2

Rθ

Reynolds number based on θ=ρe Uθ/μe

r

local radius for an axi-symmetric body

r0

wall radius for an axi-symmetric body

St

Stanton number, Eq.(13b)

s

transformed coordinate, Eq. (6a)

U

free-stream gas velocity, m/sec

u, v

gas velocity components along and normal to wall, m/sec

uτ

friction speed, m/sec

V

dimensionless normal velocity=√2sψs

x, y

physical coordinates along and normal to wall, m

z

gas flow properties vector, Eq. (12)

z

a gas flow property value or sub-vector, Eq. (11)

α

slope on the body surface, Eq. (2)

α

outer eddy viscosity coefficient constant

β

pressure gradient parameter, Eq. (8b)

λ

intermittency factor

°

boundary layer thickness

δ*k

kinematic displacement thickness=\(\mathop \smallint \limits_0^\infty \) [1− (u/U)]dy, m

ɛ

eddy viscosity, m2/sec

ɛh

eddy conductivity, m2/sec

η

transformed coordinate, Eq. (6b)

θ

momentum thickness=\(\mathop \smallint \limits_0^\infty \) [ρu/(ρeU)] × 1−(u/U)dy, m

χ, χh

mixing length constants

μ

dynamic viscosity, kg/m-sec

ν

kinematic viscosity, m2/sec

ψ

stream function, kg/m-sec

ρ

mass density, kg/m3

τW

wall shear stress, N/m2

Subscripts and Superscripts

\(\mathop {()}\limits^\_ \)

time-averaged quantity

e

inviscid flow property

i, j

indices

1

an index for a two-dimensional or an axi-symmetric body

+

dimensionless quantity

()′

time-dependent quantity

()″

derivative with respect to η

Turbulente Grenzschichten mit einem hohen Verhältnis von Freistromzu Wandtemperatur

Zusammenfassung

Turbulente Grenzschichten werden für den Fall der Freistromtemperatur von 300°K bis 3000°K, und der Wandtemperatur von 300°K für Luft bei 1 bar einer numerischen Analyse unterzogen. Das Verfahren geht von der Lösung der Differentialgleichungen durch das Runge-Kutta Verfahren in einer Ebene einer örtlich ähnlichen laminaren Grenzschicht aus. Geschwindigkeits- und Enthalpie-Profile für strom-abgelegene Ebenen werden mit Hilfe einer impliziten finiten Differenzeniteration ermittelt. Es wird der Einfluß der Freistrom-Machzahl, einer plötzlichen Beschleunigung oder Verzögerung, der Oberflächenrauhigkeit, des Ausblasens in die Grenzschicht sowie des Absaugens aus der Grenzschicht untersucht und die numerischen Ergebnisse mit denjenigen in der Literatur verfügbaren verglichen.

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References

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Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • T. K. Bose
    • 1
  1. 1.Indian Institute of TechnologyMadrasIndia

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