A comparison of laminar-turbulent boundary-layer transitions induced by deterministic and random oblique waves at Mach 3

Significance Statement

High-speed laminar-turbulent boundary layer transitions are commonly observed in external flows around rockets transporting space shuttle or satellite into the space or internal ones ejected to initiate combustion process. In such engineering applications laminar-turbulent transitions significantly influence rocket flight paths before and after releasing transported vehicles or the mixing of fuel and oxidizer in combustion initiation process. Hence, it is important to understand the relevant transition process to achieve satisfactory engineering goals, controlling the rocket flight paths or enhancing the efficiency of combustors. In most of theoretical or numerical studies of high-speed laminar-turbulent transitions it is assumed that the source of disturbance has deterministic frequency component(s). However, it is not guaranteed in all real engineering applications that disturbances are generated deterministically. Hence, the objective of this paper is to numerically investigate a randomly induced transition scenario and further suggest a way, RMS-averaging the amplitude of Fourier transformed velocity components, to quantitatively characterize the disturbance growth in linear and nonlinear instability stages. Based on the RMS-averaging, the randomly induced transition process is compared with the classical one initiated by forcing oblique waves with a fixed frequency and spanwise wave number. Two similarities between the two transition scenarios are presented as

  • the fundamental disturbance grows exponentially in the deterministic forcing (DF) case in the linear

instability stage, whereas the RMS-averaged mode in the random forcing (RF) case also shows the

exponential growth.

  • The expected evolution of longitudinal vortices and their quadratic nonlinear interactions with

the fundamental oblique waves are observed in the nonlinear insatiability regime in case DF, whereas

the streaky structures and its quadratic nonlinear interaction are also seen considering the RMS averaged modes in case RF.

Moreover, the two differences are also shown as

  • the stationary longitudinal vortex structure is observed in the nonlinear instability regime in case DF, whereas localized patches containing vortical structures are evolved into turbulent spots and then convected downstream in case RF.
  • The transition location is fixed in case DF, whereas in the stochastic sense in case RF.

Our findings above can shed lights on future studies of laminar-turbulent transitions in realistic sense.

comparison of laminar-turbulent boundary-layer transitions induced by deterministic and random oblique waves at Mach 3- advances in engineering

Journal Reference

International Journal of Heat and Fluid Flow, Volume 56, December 2015, Pages 218–232.

Sungmin Ryu1, Olaf Marxen2,  Gianluca Iaccarino1

[expand title=”Show Affiliations”]
  1. Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA
  2. Department of Mechanical Engineering Sciences, University of Surrey, Guildford, Surrey GU2 7XH, UK
[/expand]

Abstract

A numerical investigation of transition processes initiated by deterministic and random disturbances is presented for a Mach 3 flat-plate boundary layer. In both cases, disturbance forcing is localized slightly downstream of the leading edge of the flat plate. The targeted kind of disturbance for laminar-turbulent transitions is an oblique wave but it is introduced with two different ways: deterministic suction and blowing at the wall, or random volume forcing at the edge of the boundary layer. Moreover, the forced perturbations are sinusoidal in spanwise direction with a single fixed wavenumber in the deterministic case and multiple harmonics in the random case. In the latter case, the random disturbance evolution is characterized by the RMS values of Fourier transformed velocities in a band of frequency to cover the amplifications of multiple frequency components. The observed path to turbulence with respect to the two cases are compared in three stages: linear stage, non-linear regime, and breakdown to turbulence. In the initial stage, the amplitude of unsteady disturbances grows exponentially due to a linear instability of the boundary layer, as it could be observed in the deterministic forcing case. This exponential growth is also observed after considering a broad band of frequencies in the random forcing case. In the second stage, non-linearity leads to the formation of streamwise streaks via the lift-up effect. In the deterministic case, these streaks are steady, while they take the form of low-frequency traveling waves in the case of random forcing. However, in the random forcing case a streak instability could not clearly be identified. In the final stage, sudden breakdown to turbulence occurs at a fixed streamwise location in the deterministic case, marked by a sharp rise in skin friction. Non-periodicities appear only downstream of the breakdown location. In the random forcing case, breakdown takes place within a transition zone in which one can observe the formation of distinct turbulent spots.

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