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Second-order effects in unsteady laminar compressible three-dimensional stagnation-point boundary layers

Effekte zweiter Ordnung in unstetigen laminaren kompressiblen dreidimensionalen Staupunkt-Grenzschichten

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Abstract

The flow, heat and mass transfer at the stagnation point of a three-dimensional body in unsteady laminar compressible fluid with variable properties have been studied using a second-order boundary-layer theory when the basic potential flow admits selfsimilarity. Both nodal and saddle point regions have been considered. The equations governing the flow have been solved numerically using an implicit finite-difference scheme. It is observed that the enhancement or reduction in the skin friction and heat transfer due to the second-order boundary layers depends upon the values of the parameter characterizing the unsteadiness in the free-stream velocity, the nature of the stagnation point, the variation of the density-viscosity product across the boundary layer, mass transfer and the wall temperature. The suction increases the skin friction and heat transfer whereas injection does the opposite.

Zusammenfassung

Die instationäre laminare Strömung und die Wärme- und Stoffübertragung am Staupunkt eines dreidimensionalen Körpers in kompressiblen Fluiden mit variablen Stoffwerten wurden unter Benutzung einer Grenzschicht-Theorie zweiter Ordnung untersucht für den Fall, daß die Grundpotentialströmung Modellunabhängigkeit zuläßt. Sowohl Knoten- als auch Sattelpunktbereiche wurden betrachtet. Die Gleichungen, welche die Strömung beschreiben, wurden unter Benutzung eines impliziten Finite-Differenzen-Schemas numerisch gelöst. Es wird beobachtet, daß die Verstärkung oder Reduzierung der Oberflächenreibung und der Wärmeübertragung infolge der Grenzschicht zweiter Ordnung von der Größe der Parameter abhängt, welche die Unstetigkeit in der Freistromgeschwindigkeit, die Natur des Staupunktes, die Variation des Produktes aus Dichte und Viskosität über der Grenzschicht, die Stoffübertragung und Wandtemperatur charakterisieren. Der Sog läßt die Oberflächenreibung und die Wärmeübertragung ansteigen, wogegen Einspritzung das Gegenteil bewirkt.

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Abbreviations

a 1 :

tangential momentum coefficient

c :

ratio of the velocity gradients

c 1 :

energy accommodation coefficient

c p :

specific heat at a constant pressure

c v :

specific heat at a constant volume

C fx ,C fz :

skin-friction coefficients in thex and z directions, respectively

D 1,D2,D3 :

linear differential operators defined in Eqs.(21a-c)

/,F :

first- and second-order dimensionless stream functions in thex direction, respectively

f w :

mass transfer parameter

/',F' :

first- and second-order dimensionless velocities in the x-direction, respectively

″(0), F″(0):

fi/t+rst- and second-order skin-friction parameters in the z-direction, respectively

g, G :

first- and second-order dimensionless stream function in the z-direction, respectively

g′, G′ :

first- and second-order dimensionless velocities in the z-direction, respectively

g″(0), G″(0):

first- and second-order skin-friction parameters in the z-direction, respectively

H :

second-order dimensionless temperature

H'(0) :

second-order heat-transfer parameter

k :

parameter defined in Eq. (19)

k xo ,k zo :

principal curvatures of the body

Ma∞ :

Mach number

N :

ratio of the density viscosity product in the boundary layer and the free stream

Pr :

Prandtl number

q :

rate of heat transfer

R 10 :

first-order potential flow density

Re :

Reynolds number

St :

Stanton number

t :

time

t* :

dimensionless time

T :

temperature

T 10 :

first-order potential flow temperature

u, v, w :

velocity components in thex, y andz directions, respectively

U ∞ :

free-stream velocity

U 11,U 21 :

first- and second-order potential flow velocity gradients in thex direction, respectively

W 11,W 21 :

first- and second-order potential flow velocity gradients in the z direction, respectively

x, y, z :

principal, normal and transverse direction, respectively

a, β :

integrals defined in Eq. (22)

ψ :

adiabatic constant

ɛ:

perturbation parameter defined in Eq. (19)

η :

similarity variable

θ :

first-order dimensionless temperature

θ w :

wall temperature

6"(O) :

first-order heat-transfer parameter

λ :

parameter indicating the unsteadiness in the potential flow velocity

υ :

viscosity of fluid

ϱ :

density of fluid

τ wx :

shear stresses at the wall in thex and z directions, respectively

χ :

integral defined in Eq. (22)

ω :

parameter defined in Eq. (19)

ω z1,gw x1 :

vorticity interaction parameters in thex and z direction, respectively

′ :

denotes derivatives with respect tor

0:

denotes conditions at the stagnation point

1:

denotes due to first-order effect

2:

denotes due to second-order effect

d, D :

displacement effects proportional toU21 andW21, respectively

e :

denotes conditions at the edge of the boundary layer

j :

temperature-jump effect

L:

longitudinal curvature effect

s:

velocity-slip effect

t :

transverse curvature effect

v, V :

vorticity interaction effect proportional to Ωz1 andgw x1, respectively

w:

denotes conditions on the surface

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  1. Department of Applied Mathematics, Indian Institute of Science, 560012, Bangalore, India

    M. Kumari

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Kumari, M. Second-order effects in unsteady laminar compressible three-dimensional stagnation-point boundary layers. Wärme- und Stoffübertragung 23, 219–227 (1988). https://doi.org/10.1007/BF01807324

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