Self-similar states in turbulent mixing layers

Large-eddy simulations of temporally evolving turbulent mixing layers have been carried out. The effect of the initial conditions and the size of the computational box on the turbulent statistics and structures is examined in detail. A series of calculations was initialized using two different realizations of a spatially developing turbulent boundary-layer with their free streams moving in opposite directions. Computations initialized with mean flow plus random perturbations with prescribed moments were also conducted. In all cases, the initial transitional stage, from boundary-layer turbulence or random noise to mixing-layer turbulence, was followed by a self-similar period. The self-similar periods, however, differed considerably: the growth rates and turbulence intensities showed differences, and were affected both by the initial condition and by the computational domain size. In all simulations the presence of quasi-two-dimensional spanwise rollers was clear, together with ‘braid’ regions with quasi-streamwise vortices. The development of these structures, however, was different: if strong rollers were formed early (as in the cases initialized by random noise), a well-organized pattern persisted throughout the self-similar period. The presence of boundary layer turbulence, on the other hand, inhibited the growth of the inviscid instability, and delayed the formation of the roller–braid patterns. Increasing the domain size tended to make the flow more three-dimensional.

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The interaction between the mean flow and coherent structures in turbulent mixing layers

Journal of Fluid Mechanics ◽

10.1017/s0022112094003447 ◽

1994 ◽

Vol 260 ◽

pp. 81-94 ◽

Cited By ~ 8

Author(s):

J. Cohen ◽

B. Marasli ◽

V. Levinski

Keyword(s):

Velocity Profile ◽

Coherent Structures ◽

Turbulent Mixing ◽

Mixing Layer ◽

Mixing Layers ◽

Mean Flow ◽

Mean Velocity ◽

Turbulent Mixing Layer ◽

Self Similar ◽

The Mean

The nonlinear interaction between the mean flow and a coherent disturbance in a two-dimensional turbulent mixing layer is addressed. Based on considerations from stability theory, previous experimental results, in particular the modification of the mean velocity profile, the peculiar growth of the forced shear-layer thickness and the spatial growth of the disturbance amplitude, are explained. A model that assumes a quasi-parallel mean flow having a self-similar mean velocity profile is developed. The model is capable of predicting the downstream evolution of turbulent mixing layers subjected to external excitations.

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Frontogenesis and frontal arrest of a dense filament in the oceanic surface boundary layer

Journal of Fluid Mechanics ◽

10.1017/jfm.2017.833 ◽

2017 ◽

Vol 837 ◽

pp. 341-380 ◽

Cited By ~ 32

Author(s):

Peter P. Sullivan ◽

James C. McWilliams

Keyword(s):

Boundary Layer ◽

Turbulent Mixing ◽

Shear Instability ◽

Upper Ocean ◽

Surface Boundary ◽

Horizontal Shear ◽

Boundary Layer Turbulence ◽

Filament Axis ◽

Small Width ◽

The Cross

The evolution of upper ocean currents involves a set of complex, poorly understood interactions between submesoscale turbulence (e.g. density fronts and filaments and coherent vortices) and smaller-scale boundary-layer turbulence. Here we simulate the lifecycle of a cold (dense) filament undergoing frontogenesis in the presence of turbulence generated by surface stress and/or buoyancy loss. This phenomenon is examined in large-eddy simulations with resolved turbulent motions in large horizontal domains using${\sim}10^{10}$grid points. Steady winds are oriented in directions perpendicular or parallel to the filament axis. Due to turbulent vertical momentum mixing, cold filaments generate a potent two-celled secondary circulation in the cross-filament plane that is frontogenetic, sharpens the cross-filament buoyancy and horizontal velocity gradients and blocks Ekman buoyancy flux across the cold filament core towards the warm filament edge. Within less than a day, the frontogenesis is arrested at a small width,${\approx}100~\text{m}$, primarily by an enhancement of the turbulence through a small submesoscale, horizontal shear instability of the sharpened filament, followed by a subsequent slow decay of the filament by further turbulent mixing. The boundary-layer turbulence is inhomogeneous and non-stationary in relation to the evolving submesoscale currents and density stratification. The occurrence of frontogenesis and arrest are qualitatively similar with varying stress direction or with convective cooling, but the detailed evolution and flow structure differ among the cases. Thus submesoscale filament frontogenesis caused by boundary-layer turbulence, frontal arrest by frontal instability and frontal decay by forward energy cascade, and turbulent mixing are generic processes in the upper ocean.

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Natural convection flow far from a horizontal plate

Journal of Fluid Mechanics ◽

10.1017/s0022112099004462 ◽

1999 ◽

Vol 387 ◽

pp. 227-254 ◽

Cited By ~ 6

Author(s):

VALOD NOSHADI ◽

WILHELM SCHNEIDER

Keyword(s):

Boundary Layer ◽

Prandtl Number ◽

Stokes Equations ◽

Axisymmetric Flow ◽

The Self ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Boundary Layer Equations ◽

Self Similar ◽

Statistical characteristics of turbulent mixing in spherical and cylindrical converging Richtmyer–Meshkov instabilities

Journal of Fluid Mechanics ◽

10.1017/jfm.2021.818 ◽

2021 ◽

Vol 928 ◽

Author(s):

Xinliang Li ◽

Yaowei Fu ◽

Changping Yu ◽

Li Li

Keyword(s):

Turbulent Mixing ◽

Initial Perturbation ◽

Great Influence ◽

Mixing Layers ◽

Statistical Characteristics ◽

Computational Domain ◽

Heavy Fluid ◽

Node Number ◽

Geometric Effect ◽

Atwood Number

In this paper, the Richtmyer–Meshkov instabilities in spherical and cylindrical converging geometries with a Mach number of approximately 1.5 are investigated by using the high resolution implicit large eddy simulation method, and the influence of the geometric effect on the turbulent mixing is investigated. The heavy fluid is sulphur hexafluoride (SF6), and the light fluid is nitrogen (N2). The shock wave converges from the heavy fluid into the light fluid. The Atwood number is 0.678. The total structured and uniform Cartesian grid node number in the main computational domain is 20483. In addition, to avoid the influence of boundary reflection, a sufficiently long sponge layer with 50 non-uniform coarse grids is added for each non-periodic boundary. Present numerical simulations have high and nonlinear initial perturbation levels, which rapidly lead to turbulent mixing in the mixing layers. Firstly, some physical-variable mean profiles, including mass fraction, Taylor Reynolds number, turbulent kinetic energy, enstrophy and helicity, are provided. Second, the mixing characteristics in the spherical and cylindrical turbulent mixing layers are investigated, such as molecular mixing fraction, efficiency Atwood number, turbulent mass-flux velocity and density self-correlation. Then, Reynolds stress and anisotropy are also investigated. Finally, the radial velocity, velocity divergence and enstrophy in the spherical and cylindrical turbulent mixing layers are studied using the method of conditional statistical analysis. Present numerical results show that the geometric effect has a great influence on the converging Richtmyer–Meshkov instability mixing layers.

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Self-similar compressible turbulent boundary layers with pressure gradients. Part 1. Direct numerical simulation and assessment of Morkovin’s hypothesis

Journal of Fluid Mechanics ◽

10.1017/jfm.2019.670 ◽

2019 ◽

Vol 880 ◽

pp. 239-283 ◽

Cited By ~ 3

Author(s):

Christoph Wenzel ◽

Tobias Gibis ◽

Markus Kloker ◽

Ulrich Rist

Keyword(s):

Numerical Simulation ◽

Boundary Layer ◽

Mach Number ◽

Direct Numerical Simulation ◽

Boundary Layers ◽

Turbulent Boundary Layers ◽

Shear Stresses ◽

Pressure Gradients ◽

Mean Flow ◽

Self Similar

A direct numerical simulation study of self-similar compressible flat-plate turbulent boundary layers (TBLs) with pressure gradients (PGs) has been performed for inflow Mach numbers of 0.5 and 2.0. All cases are computed with smooth PGs for both favourable and adverse PG distributions (FPG, APG) and thus are akin to experiments using a reflected-wave set-up. The equilibrium character allows for a systematic comparison between sub- and supersonic cases, enabling the isolation of pure PG effects from Mach-number effects and thus an investigation of the validity of common compressibility transformations for compressible PG TBLs. It turned out that the kinematic Rotta–Clauser parameter $\unicode[STIX]{x1D6FD}_{K}$ calculated using the incompressible form of the boundary-layer displacement thickness as length scale is the appropriate similarity parameter to compare both sub- and supersonic cases. Whereas the subsonic APG cases show trends known from incompressible flow, the interpretation of the supersonic PG cases is intricate. Both sub- and supersonic regions exist in the boundary layer, which counteract in their spatial evolution. The boundary-layer thickness $\unicode[STIX]{x1D6FF}_{99}$ and the skin-friction coefficient $c_{f}$, for instance, are therefore in a comparable range for all compressible APG cases. The evaluation of local non-dimensionalized total and turbulent shear stresses shows an almost identical behaviour for both sub- and supersonic cases characterized by similar $\unicode[STIX]{x1D6FD}_{K}$, which indicates the (approximate) validity of Morkovin’s scaling/hypothesis also for compressible PG TBLs. Likewise, the local non-dimensionalized distributions of the mean-flow pressure and the pressure fluctuations are virtually invariant to the local Mach number for same $\unicode[STIX]{x1D6FD}_{K}$-cases. In the inner layer, the van Driest transformation collapses compressible mean-flow data of the streamwise velocity component well into their nearly incompressible counterparts with the same $\unicode[STIX]{x1D6FD}_{K}$. However, noticeable differences can be observed in the wake region of the velocity profiles, depending on the strength of the PG. For both sub- and supersonic cases the recovery factor was found to be significantly decreased by APGs and increased by FPGs, but also to remain virtually constant in regions of approximated equilibrium.

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Heat and mass transfer characteristics of the self-similar boundary-layer flows induced by continuous surfaces stretched with rapidly decreasing velocities

Heat and Mass Transfer ◽

10.1007/s002310000126 ◽

2001 ◽

Vol 38 (1-2) ◽

pp. 65-74 ◽

Cited By ~ 51

Author(s):

E. Magyari ◽

M. E. Ali ◽

B. Keller

Keyword(s):

Boundary Layer ◽

Mass Transfer ◽

Heat And Mass Transfer ◽

The Self ◽

Boundary Layer Flows ◽

Transfer Characteristics ◽

Self Similar

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Skin-friction generation by attached eddies in turbulent channel flow

Journal of Fluid Mechanics ◽

10.1017/jfm.2016.665 ◽

2016 ◽

Vol 808 ◽

pp. 511-538 ◽

Cited By ~ 36

Author(s):

Matteo de Giovanetti ◽

Yongyun Hwang ◽

Haecheon Choi

Keyword(s):

Reynolds Number ◽

Skin Friction ◽

Large Scale ◽

Reynolds Numbers ◽

The Self ◽

Friction Reduction ◽

Computational Domain ◽

Cutoff Wavelength ◽

Self Similar ◽

Attached Eddies

Despite a growing body of recent evidence on the hierarchical organization of the self-similar energy-containing motions in the form of Townsend’s attached eddies in wall-bounded turbulent flows, their role in turbulent skin-friction generation is currently not well understood. In this paper, the contribution of each of these self-similar energy-containing motions to turbulent skin friction is explored up to $Re_{\unicode[STIX]{x1D70F}}\simeq 4000$. Three different approaches are employed to quantify the skin-friction generation by the motions, the spanwise length scale of which is smaller than a given cutoff wavelength: (i) FIK (Fukagata, Iwamoto, Kasagi) identity in combination with the spanwise wavenumber spectra of the Reynolds shear stress; (ii) confinement of the spanwise computational domain; (iii) artificial damping of the motions to be examined. The near-wall motions are found to continuously reduce their role in skin-friction generation on increasing the Reynolds number, consistent with the previous finding at low Reynolds numbers. The largest structures given in the form of very-large-scale and large-scale motions are also found to be of limited importance: due to a non-trivial scale interaction process, their complete removal yields only a 5–8 % skin-friction reduction at all of the Reynolds numbers considered, although they are found to be responsible for 20–30 % of total skin friction at $Re_{\unicode[STIX]{x1D70F}}\simeq 2000$. Application of all the three approaches consistently reveals that the largest amount of skin friction is generated by the self-similar motions populating the logarithmic region. It is further shown that the contribution of these motions to turbulent skin friction gradually increases with the Reynolds number, and that these coherent structures are eventually responsible for most of turbulent skin-friction generation at sufficiently high Reynolds numbers.

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A model for inhomogeneous turbulent flow

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1970.0125 ◽

1970 ◽

Vol 317 (1530) ◽

pp. 417-433 ◽

Cited By ~ 150

Keyword(s):

Boundary Layer ◽

Eddy Viscosity ◽

The Self ◽

Diffusion Equations ◽

Mean Flow ◽

Mean Velocity ◽

Closed Set ◽

Law Of The Wall ◽

The Mean ◽

Model Equations

A set of model equations is given to describe the gross features of a statistically steady or 'slowly varying’ inhomogeneous field of turbulence and the mean velocity distribution. The equations are based on the idea that turbulence can be characterized by ‘densities’ which obey nonlinear diffusion equations. The diffusion equations contain terms to describe the convection by the mean flow, the amplification due to interaction with a mean velocity gradient, the dissipation due to the interaction of the turbulence with itself, and the diffusion also due to the self interaction. The equations are similar to a set proposed by Kolmogorov (1942). It is assumed that both an ‘energy density’ and a ‘vorticity density’ satisfy diffusion equations, and that the self diffusion is described by an eddy viscosity which is a function of the energy and vorticity densities; the eddy viscosity is also assumed to describe the diffusion of mean momentum by the turbulent fluctuations. It is shown that with simple and plausible assumptions about the nature of the interaction terms, the equations form a closed set. The appropriate boundary conditions at a solid wall and a turbulent interface, with and without entrainment, are discussed. It is shown that the dimensionless constants which appear in the equations can all be estimated by general arguments. The equations are then found to predict the von Kármán constant in the law of the wall with reasonable accuracy. An analytical solution is given for Couette flow, and the result of a numerical study of plane Poiseuille flow is described. The equations are also applied to free turbulent flows. It is shown that the model equations completely determine the structure of the similarity solutions, with the rate of spread, for instance, determined by the solution of a nonlinear eigenvalue problem. Numerical solutions have been obtained for the two-dimensional wake and jet. The agreement with experiment is good. The solutions have a sharp interface between turbulent and non-turbulent regions and the mean velocity in the turbulent part varies linearly with distance from the interface. The equations are applied qualitatively to the accelerating boundary layer in flow towards a line sink, and the decelerating boundary layer with zero skin friction. In the latter case, the equations predict that the mean velocity should vary near the wall like the 5/3 power of the distance. It is shown that viscosity can be incorporated formally into the model equations and that a structure can be given to the interface between turbulent and non-turbulent parts of the flow.

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