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Incipient separation in shock wave/boundary layer interactions as induced by sharp fin

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Abstract

The incipient separation induced by the shock wave/turbulent boundary layer interaction at the sharp fin is the subject of the present study. Existing theories for the prediction of incipient separation, such as those put forward by McCabe (1966) and Dou and Deng (1992), can thus far only predic the direction of surface streamline and tend to overpredict the incipient separation condition based on the Stanbrook’s criterion. In this paper, the incipient separation is first predicted with Dou and Deng (1992)’s theory and then compared with Lu and Settles’ (1990) experimental data. The physical mechanism of the incipient separation as induced by the shock wave/turbulent boundary layer interactions at sharp fin is explained via surface flow pattern analysis. Furthermore, the reason for the observed discrepancy between the predicted and experimental incipient separation conditions is clarified. It is found that when the wall-limiting streamlines behind the shock wave becomes aligned with one ray from the virtual origin as the strength of the shock wave increases, the incipient separation line is formed at which the wall-limiting streamline becomes perpendicular to the local pressure gradient. The formation of this incipient separation line is the beginning of the separation process. The effects of Reynolds number and Mach number on incipient separation are also discussed. Finally, a correlation for the correction of the incipient separation angle as predicted by the theory is also given.

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Abbreviations

A :

parameter in the “triangular model”

C fx :

component of local skin friction coefficient in the mainflow (streamwise) direction (dimensionless)

k :

specific heat ratio (dimensionless)

M :

Mach number (dimensionless)

M ∞(or M 1):

incoming Mach number (dimensionless)

p :

static pressure (N/m2)

R :

temperature recovery factor at wall (dimensionless)

Re θ :

Reynolds number based on momentum thickness (dimensionless)

T :

absolute temperature (K)

u :

velocity component in the mainflow direction in the boundary layer (m/s)

w :

velocity component in the crossflow direction in the boundary layer (m/s)

x :

coordinate in the streamwise direction (m)

y :

coordinate in the direction normal to the flat plate (m)

z :

coordinate in the direction normal to the streamwise direction (m)

α:

turning angle of the main flow, measured relative to the direction of velocity vectorat the beginning of interaction (radian or degree)

β:

angle measured from the incoming freestream direction, centred at the virtual origin (near the fin apex) (degree)

β 0 :

angle of shock wave (degree)

γ w :

wall shear angle, i.e. angle between the direction of the wall-limiting streamline and the external streamline direction of boundary layer (degree)

θ:

momentum thickness of boundary layer (m)

ν:

kinematic viscosity (m2/s)

ν(M):

Prandtl–Meyer function (radian), see (7)

ρ:

density (kg/m3)

σ:

α  + γ w, turning angle of surface streamline, i.e. the limiting streamline direction on the wall measured from the incoming flow direction (degree)

τ:

shear stress (N/m2)

e:

external of the boundary layer

i:

incipient separation

p:

apex of “triangular model,” Fig.5

w:

at walls

x :

component in the direction of the main flow

z :

component in the direction of the cross-flow

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Authors and Affiliations

  1. Temasek Laboratories, National University of Singapore, Singapore, 117508, Singapore

    Hua-Shu Dou

  2. Department of Mechanical Engineering, National University of Singapore, Singapore, 119260, Singapore

    Boo Cheong Khoo & Khoon Seng Yeo

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  1. Hua-Shu Dou
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  2. Boo Cheong Khoo
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  3. Khoon Seng Yeo
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Correspondence to Hua-Shu Dou.

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Communicated by K. Takayama.

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Dou, HS., Khoo, B.C. & Yeo, K.S. Incipient separation in shock wave/boundary layer interactions as induced by sharp fin. Shock Waves 15, 425–436 (2006). https://doi.org/10.1007/s00193-006-0044-z

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