Nonlinear Effects During Perturbation Propagation in Strong Supersonic Interaction
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NONLINEAR EFFECTS DURING PERTURBATION PROPAGATION IN STRONG SUPERSONIC INTERACTION I. I. Lipatova,∗ and V. K. Famb
UDC 532.526
Abstract: Nonlinear interaction and energy loss by harmonics of perturbations propagating upstream are investigated. It is shown how nonlinear interaction of harmonics can form new harmonics. Keywords: boundary layer, perturbation propagation, strong viscous–inviscid interaction theory, nonlinear interaction. DOI: 10.1134/S0021894420030153
INTRODUCTION Perturbations develop because of hydrodynamic stability. It is necessary to analyze perturbation propagation in a boundary layer in order to correctly formulate a boundary-value problem for a system of equations of an unsteady boundary layer and construct computational models. The analysis is carried out as part of the study of resistance to long-wave perturbations. In this study, the nonlinear interaction of perturbation harmonics during their propagation in an unsteady two-dimensional boundary layer is considered.
1. FORMULATION OF THE PROBLEM This study touches upon an unsteady hypersonic flow of viscous gas around a flat surface (including a plate and a wedge) located at a zero angle of attack to an incoming flow. In this case, the Mach number of the incoming flow is assumed to be large, and there is strong viscous–inviscid interaction: M∞ 1,
M∞ τ 1
(M∞ is the Mach number of the incoming flow, and τ is the dimensionless thickness of the laminar boundary layer). The OX axis of the Cartesian coordinate system, bound with the plate, is directed along the plate surface, and the OY axis is directed along the normal to it. The following denotations are introduced: lx and ly are the coordinates counted along the plate surface and along the normal to it, respectively; lt/u∞ is time, u∞ u, u∞ v, and u∞ w are velocity vector components, ρ0 ρ is the density, ρ∞ u2∞ p is pressure, H∞ g is the total enthalpy, μ0 μ is viscosity, l is the characteristic length of the body in the flow, τ = O(ρ0 u∞ l/μ0 )−1/2 , and μ0 is the dynamic viscosity at a stagnation temperature; subscript ∞ matches values in the incoming flow.
a Central Aerohydrodynamic Institute, Zhukovskiy, 140180 Russia; ∗ igor [email protected]. b Moscow Institute of Physics and Technology, Dolgoprudny, 141700 Russia; [email protected]. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 61, No. 3, pp. 140–143, May–June, 2020. Original article submitted December 16, 2019; revision submitted December 16, 2019; accepted for publication January 27, 2020. ∗ Corresponding author.
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Pressure harmonics and their frequencies during perturbation propagation upstream
Harmonic
f
Harmonic corresponding to a linear combination of frequencies of the main harmonics
f1 f2 f3 f0 f23 f113 f12 f11 f13
0.6 1.6 2.0 0 0.4 0.8 1.0 1.2 1.4
f1 f2 f3 fi − fi f3 − f2 f3 − 2f1 f2 − f1 f1 + f1 f3 − f1
A 1.000
Harmonic
f
Harmonic corresponding to a linear combination of frequencies of the main harmonics
f111 f21 f31 f112 f123
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