Jump to ContentJump to Main Navigation
Show Summary Details
In This Section

International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board Member: Armbruster, Dieter / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

8 Issues per year

IMPACT FACTOR 2016: 0.890

CiteScore 2016: 0.84

SCImago Journal Rank (SJR) 2015: 0.298
Source Normalized Impact per Paper (SNIP) 2015: 0.476

Mathematical Citation Quotient (MCQ) 2015: 0.04

See all formats and pricing
In This Section
Volume 18, Issue 1 (Feb 2017)


Study of the Shock Wave–Turbulent Boundary Layer Interaction Using a 3D von Kármán Length Scale

Jing-Lei Xu
  • National Key Laboratory of Science and Technology on Aero-Engine Aero-thermodynamics, School of Energy and Power Engineering, Beihang University, Beijing, China
/ You-Fu Song
  • National Key Laboratory of Science and Technology on Aero-Engine Aero-thermodynamics, School of Energy and Power Engineering, Beihang University, Beijing, China
/ Yang Zhang
  • Corresponding author
  • School of Aeronautics, Northwestern Polytechnical University, Xi’an, Shanxi, China
  • Email:
/ Jun-Qiang Bai
  • School of Aeronautics, Northwestern Polytechnical University, Xi’an, Shanxi, China
Published Online: 2016-12-17 | DOI: https://doi.org/10.1515/ijnsns-2016-0018


Traditional turbulence models are initially formulated and calibrated under incompressible conditions. Thus, these models are always of low fidelity when extended to high speed, complex and compressible flows. In this work, a compressible von Kármán length scale is proposed for compressible flows considering the variable densities. The length scale is the ratio between the new vorticity and its gradient. The new length scale is actually based on phenomenological theory, which is then integrated into the KDO (turbulence Kinetic energy Dependent Only) turbulence model, arriving at a compressible model called CKDO (Compressible KDO). In the CKDO turbulence model, all the extra terms produced by compressibility are modeled as dissipation. Compression corners of 8, 16, 20 and 24 angles are studied within SST, SA, KDO and CKDO. These test cases are known as the typical shock wave–boundary layer interactions. The results show that the new length scale in CKDO is able to well capture the surface pressure and skin friction distributions. Besides, compared with the standard von Kármán length scale, the new length scale in CKDO can better capture the size and position of the separation bubble. With the increase of the corner angle, CKDO shows more prominent potential for describing compressible flows.

Keywords: turbulence model; shock wave–boundary layer interaction; compressibility correction; von Kármán length scale

PACS: 76F10; 76F25; 76G25


  • [1]

    J. Ackeret, F. Feldmann, and N. Rott, Investigation of compression shocks and boundary layers in gases moving at high speed, NACA TM-1113, Jan. 1947.

  • [2]

    G. S. Settles, T. J. Fitzpatrick, and S. M. Bogdonoff, Detailed study of attached and separated compression corner flowfields in high Reynolds number supersonic flow, AIAA J. 17 (6) (1979), 579–585.

  • [3]

    G. S. Settles, S. M. Bogdonoff, and I. E. Vas, Incipient separation of a supersonic turbulent boundary layer at high Reynolds numbers, AIAA J. 14 (1) (1976), 50–56.

  • [4]

    G. S. Settles, I. E. Vas, and S. M. Bogdonoff, Details of a shock-separated turbulent boundary layer at a compression corner, AIAA J. 14 (12) (1976), 1709–1715.

  • [5]

    G. S. Settles and L. J. Dodson, Hypersonic shock/boundary-Layer interaction database, NASA CR-177577, 1991.

  • [6]

    D. S. Dolling and M. T. Murphy, Unsteadiness of the separation shock wave structure in a supersonic compression ramp flowfield, AIAA J. 21 (12) (1983), 1628–1634.

  • [7]

    D. S. Dolling and C. T. Or, Unsteadiness of the shock wave structure in attached and separated compression corner flow Fields, AIAA Paper 83–1715, July 1983.

  • [8]

    M. Wu and M. P. Martin, Direct numerical simulation of supersonic turbulent boundary layer over a compression ramp, AIAA J. 45 (4) (2007), 879–889.

  • [9]

    O. Zeman, Dilatation dissipation: The concept and application in modelling compressible mixing layers, Phys. Fluids A Fluid Dyn. 2 (1990), 178–188.

  • [10]

    S. Sarkar, G. Erlebacher, M. Y. Hussaini, H. O. Kreiss, The analysis and modelling of dilatational terms in compressible turbulence, J. Fluid Mech. 227 (1991), 473–493.

  • [11]

    D. C. Wilcox, Dilatation-dissipation corrections for advanced turbulence models, AIAA J. 30 (11) (1992), 2639–2646.

  • [12]

    S. Sarkar, The stabilizing effect of compressibility in turbulent shear flow, J. Fluid Mech. 282 (1995), 163–186.

  • [13]

    A. W. Vreman, N. D. Sandham, and K. H. Luo, Compressible mixing layer growth rate and turbulence characteristics, J. Fluid Mech. 320 (1996), 235–258.

  • [14]

    F. Hamba, Effects of pressure fluctuations on turbulence growth in compressible homogeneous shear flow, Phys. Fluids 11 (6) (1994-present) (1999), 1623–1635.

  • [15]

    C. Pantano and S. Sarkar, A study of compressibility effects in the high-speed turbulent shear layer using direct simulation, J. Fluid Mech. 451 (2002), 329–371.

  • [16]

    A. Yoshizawa, H. Fujiwara, F. Hamba, S. Nisizima, Y. Kumagai, Nonequilibrium turbulent-viscosity model for supersonic free-shear layer/wall bounded flows, AIAA J. 41 (6) (2003), 1029–1036.

  • [17]

    A. Yoshizawa, S. Nisizima, Y. Shimomura, Y. Shimomura, H. Kobayashi, Y. Matsuo, H. Abe, H. Fujiwara, A new methodology for Reynolds-averaged modeling based on the amalgamation of heuristic-modeling and turbulence-theory methods, Phys. Fluids 18 (3) (2006), 5109.

  • [18]

    J. Kim and S. O. Park, New compressible turbulence model for free and wall-bounded shear layers, J. Turbul. 11 (2010), N10.

  • [19]

    G. H. Kim and S. O. Park, Explicit algebraic Reynolds stress model for compressible flow turbulence, J. Turbul. 14 (5) (2013), 35–59.

  • [20]

    H. Klifi and T. Lili, A compressibility correction of the pressure strain correlation model in turbulent flow, Comptes Rendus Mécanique 341 (7) (2013), 567–580.

  • [21]

    C. A. Gomez and S. S. Girimaji, Explicit algebraic Reynolds stress model (EARSM) for compressible shear flows, Theor. Comput. Fluid Dyn. 28 (2) (2014), 171–196.

  • [22]

    S. Huang and S. Fu, Modelling of pressure–strain correlation in compressible turbulent flow, Acta Mech. Sin. 24 (1) (2008), 37–43.

  • [23]

    T. B. Gatski and T. Jongen, Nonlinear eddy viscosity and algebraic stress models for solving complex turbulent flows, Prog. Aerosp. Sci. 36 (8) (2000), 655–682.

  • [24]

    A. Favre, Statistical equations of turbulent gases, in: Problems of Hydrodynamics and Continuum Mechanics, Philadelphia: Society for Industrial and Applied Mathematics, pp. 231–266, 1969.

  • [25]

    J. L. Xu, Y. Zhang, and J. Q. Bai, One-equation turbulence model based on extended Bradshaw assumption, AIAA J. 53 (2015), 1433–1441.

  • [26]

    P. Bradshaw, D. Ferriss, and N. Atwell, Calculation of boundary layer development using the turbulent energy equation, J. Fluid Mech. 28 (3) (1967), 593–616.

  • [27]

    P. T. Harsha and S. C. Lee, Correlation between turbulent shear stress and turbulent kinetic energy, AIAA J. 8 (8) (1970), 1508–1510.

  • [28]

    P. Schlatter and R. Orlu, Assessment of direct numerical simulation data of turbulent boundary layers, J. Fluid Mech. 659 (2010), 116–126.

  • [29]

    F. Menter, M. Kuntz, and R. Langtry. Ten years of industrial experience with the SST turbulence model, Turbul. Heat Mass Trans. 4 (1) (2003), 625–632.

  • [30]

    F. Menter and Y. Egorov, A scale-adaptive simulation model using two-equation models, AIAA Paper 1095 (2005), 2005.

  • [31]

    F. Menter, The scale-adaptive simulation method for unsteady turbulent flow predictions, Part 1: theory and model description, Flow Turbul. Combust. 85 (5) (2010), 113–138.

  • [32]

    J. L. Xu, N. Hu, and G. Gao, A high-fidelity turbulence length scale for flow simulation, in: Progress in Hybrid RANS-LES Modelling, Berlin: Springer, pp. 141–145, 2012.

  • [33]

    J. L. Xu and C. Yan, A one-equation scale-adaptive simulation model, Phys. Gases 5 (1) (2010), 79–82 (in Chinese).

  • [34]

    F. R. Menter, Improved two-equation k-w turbulence models for aerodynamic flows, NASA TM-103975, 1992.

  • [35]

    P. Spalart and S. Allmaras, A one-equation turbulence model for aerodynamic flows, AIAA Paper, 92–0439, 1992.

About the article

Received: 2016-01-30

Accepted: 2016-10-07

Published Online: 2016-12-17

Published in Print: 2017-02-01

This work was supported by the National Key Laboratory of Aircraft Engine Foundation of China (9140C410505150C41002).

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2016-0018. Export Citation

Comments (0)

Please log in or register to comment.
Log in