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Nonasymptotic Theory of Unseparated Turbulent-Boundary-Layer — Shock-Wave Interactions with Application to Transonic Flows

  • G. R. Inger

Abstract

Shock-turbulent-boundary-layer interactions are important in the aerodynamic design of high-speed aircraft wings, and of turbine and cascade blades in turbomachinery and air-breathing-engine inlets and diffusors. Of particular importance are the features of upstream influence, boundary-layer displacement, skin friction, and incipient separation dominated by the thin interactive shear-stress disturbance layer very close to the surface. Lighthill’s pioneering study [2.1] of this region, however, takes into account only the laminar portion of the incoming turbulent-boundary-layer profile, which is inaccurate for the higher Reynolds numbers pertaining to full-scale aircraft. On the other hand, more recent work on an improved theory either has been confined to the treatment of the transonic regime by asymptotic methods [2.68, 2.69] that entail a severe limiting model of the interactive physics as Re l → ∞, or has involved approximate double-layered models for supersonic flow [2.70–2.72] with insufficient consideration of the basic flow structure in the shear-disturbance sublayer [2.73]. Consequently, there is a need for a more general theory at ordinary practical Reynolds numbers, applicable to both transonic and supersonic flow.

Keywords

Skin Friction Incident Shock Transonic Flow Total Shear Stress Incoming Boundary Layer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

Cf

Skin-friction coefficient, 2τ w /ρ e0 U e0 2

Cp

Pressure coefficient, 2p’/ρ e0 U e0 2

Hi

Incompressible shape factor, δ* i /θ* i

M

Mach number

p

Static pressure

p

Interactive pressure perturbation, p - p 1

Δp

Pressure jump across incident shock

Rel

Reynolds number based on length l

T

Absolute temperature

γ

Basic interactive wall-turbulence parameter

u’, v

Streamwise and normal interactive-isturbance-velocity components, respectively

U0

Undisturbed incoming boundary-layer velocity in x-direction

x, y

Streamwise and normal distance coordinates (origin at the inviscid shock intersection with the wall)

yw eff

Effective wall shift seen by interactive inviscid flow

β

\( \sqrt {M_1^2 - 1} \)

γ

Specific-heat ratio

δ

Boundary-layer thickness

δ*

Boundary-layer displacement thickness

εT

Kinematic turbulent eddy viscosity

μ

Ordinary coefficient of viscosity

v

μ/ρ

ω

Viscosity temperature-dependence exponent, ρT ω

ρ

Density

θ*

Boundary-layer momentum thickness

τ

Total shear stress

Subscripts

1

Undisturbed inviscid values ahead of incident shock

e

Conditions at the boundary-layer edge

inv

Inviscid-disturbance solution value

0

Undisturbed incoming-boundary-layer properties

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Copyright information

© Springer Science+Business Media New York 1982

Authors and Affiliations

  • G. R. Inger
    • 1
  1. 1.Department of Aerospace Engineering SciencesUniversity of ColoradoBoulderUSA

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