Skip to main content Accessibility help

Fractal scaling of the turbulence interface in gravity currents

  • Dominik Krug (a1), Markus Holzner (a2), Ivan Marusic (a1) and Maarten van Reeuwijk (a3)


It was previously observed by Krug et al. (J. Fluid Mech., vol. 765, 2015, pp. 303–324) that the surface area $A_{\unicode[STIX]{x1D702}}$ of the turbulent/non-turbulent interface (TNTI) in gravity currents decreases with increasing stratification, significantly reducing the entrainment rate. Here, we consider the multiscale properties of this effect using direct numerical simulations of temporal gravity currents with different gradient Richardson numbers $Ri_{g}$ . Our results indicate that the reduction of $A_{\unicode[STIX]{x1D702}}$ is caused by a decrease of the fractal scaling exponent $\unicode[STIX]{x1D6FD}$ , while the scaling range remains largely unaffected. We further find that convolutions of the TNTI are characterized by different length scales in the streamwise and wall-normal directions, namely the integral scale $h$ and the shear scale $l_{Sk}=k^{1/2}/S$ (formed using the mean shear $S$ and the turbulent kinetic energy  $k$ ) respectively. By recognizing that the anisotropy implied by the different scaling relations increases with increasing $Ri_{g}$ , we are able to model the $Ri_{g}$ dependence of $\unicode[STIX]{x1D6FD}$ in good agreement with the data.


Corresponding author

Email address for correspondence:


Hide All
Corrsin, S. & Kistler, A.1954 The free-stream boundaries of turbulent flows. NACA TN-3133, TR-1244, 1033–1064.
Craske, J. & van Reeuwijk, M. 2015 Energy dispersion in turbulent jets. Part 1. Direct simulation of steady and unsteady jets. J. Fluid Mech. 763, 500537.
Da Silva, C. B., Hunt, J. C. R., Eames, I. & Westerweel, J. 2014 Interfacial layers between regions of different turbulence intensity. Annu. Rev. Fluid Mech. 46, 567590.
Dimotakis, P. E. & Catrakis, H. J. 1999 Turbulence, fractals, and mixing. In Mixing: Chaos and Turbulence (ed. Chaté, H., Villermaux, E. & Chomaz, J.-M.), pp. 59143. Springer.
Ellison, T. H. & Turner, J. S. 1959 Turbulent entrainment in stratified flows. J. Fluid Mech. 6 (03), 423448.
Holzner, M. & Lüthi, B. 2011 Laminar superlayer at the turbulence boundary. Phys. Rev. Lett. 106 (13), 134503.
Kneller, B., Nasr-Azadani, M. M., Radhakrishnan, S. & Meiburg, E. 2016 Long-range sediment transport in the world’s oceans by stably stratified turbidity currents. J. Geophys. Res. Oceans 121, 86088620.
Krug, D., Holzner, M., Lüthi, B., Wolf, M., Kinzelbach, W. & Tsinober, A. 2015 The turbulent/non-turbulent interface in an inclined dense gravity current. J. Fluid Mech. 765, 303324.
Mandelbrot, B. B. 1982 The Fractal Geometry of Nature. W.H. Freeman.
Mater, P. D. & Venayagamoorthy, S. K. 2014 A unifying framework for parameterizing stably stratified shear-flow turbulence. Phys. Fluids 26 (3), 036601.
Mistry, D., Philip, J., Dawson, J. R. & Marusic, I. 2016 Entrainment at multi-scales across the turbulent/non-turbulent interface in an axisymmetric jet. J. Fluid Mech. 802, 690725.
Paizis, S. T. & Schwarz, W. H. 1974 An investigation of the topography and motion of the turbulent interface. J. Fluid Mech. 63 (02), 315343.
Pollard, R. T., Rhines, P. B. & Thompson, R. O. R. Y. 1972 The deepening of the wind-mixed layer. Geophys. Astrophys. Fluid Dyn. 4 (1), 381404.
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.
van Reeuwijk, M. & Holzner, M. 2014 The turbulence boundary of a temporal jet. J. Fluid Mech. 739, 254275.
van Reeuwijk, M., Krug, D. & Holzner, M. 2017 Small-scale entrainment in inclined gravity currents. Environ. Fluid Mech. doi:10.1007/s10652-017-9514-3.
Schmid, P. J. & Henningson, D. S. 2000 Stability and Transition in Shear Flows. Springer.
Sequeiros, O. E. 2012 Estimating turbidity current conditions from channel morphology: a Froude number approach. J. Geophys. Res. 117, C04003.
Sequeiros, O. E., Spinewine, B., Beaubouef, R. T., Sun, T., García, M. H. & Parker, G. 2010 Characteristics of velocity and excess density profiles of saline underflows and turbidity currents flowing over a mobile bed. J. Hydraul. Engng ASCE 136 (7), 412433.
de Silva, C. M., Philip, J., Chauhan, K., Meneveau, C. & Marusic, I. 2013 Multiscale geometry and scaling of the turbulent–nonturbulent interface in high Reynolds number boundary layers. Phys. Rev. Lett. 111, 044501.
Sreenivasan, K. R. 1991 Fractals and multifractals in fluid turbulence. Annu. Rev. Fluid Mech. 23 (1), 539604.
Sreenivasan, K. R., Ramshankar, R. & Meneveau, C. 1989 Mixing, entrainment and fractal dimensions of surfaces in turbulent flows. Proc. R. Soc. Lond. A 421 (1860), 79108.
Thiesset, F., Maurice, G., Halter, F., Mazellier, N., Chauveau, C. & Gökalp, I. 2016 Geometrical properties of turbulent premixed flames and other corrugated interfaces. Phys. Rev. E 93 (1), 013116.
Turner, J. S. 1986 Turbulent entrainment – the development of the entrainment assumption, and its application to geophysical flows. J. Fluid Mech. 173, 431471.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

Fractal scaling of the turbulence interface in gravity currents

  • Dominik Krug (a1), Markus Holzner (a2), Ivan Marusic (a1) and Maarten van Reeuwijk (a3)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.