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

Compressibility Rosen Boulder 

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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

Personalised recommendations