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Numerical Simulations of Turbulent Plane Shear Layers

  • Conference paper
Turbulent Shear Flows 6

Abstract

Two-dimensional turbulent plane shear layers are simulated, using a finite-difference ψ — ω unstationary numerical simulation. Vorticity and temperature (within the passive scalar approximation) fields are visualized. Direct numerical simulations (Reynolds number of 1000) and large-eddy simulations are performed. The following situations are envisaged:

  1. a)

    A temporally growing mixing layer (periodic in the basic flow direction) developing from a hyperbolic tangent velocity profile to which is superposed an infinitesimal white-noise perturbation. One shows that broad-band kinetic energy and temperature spectra develop after the first pairing. Evidence is given that the growth of the coherent structures is not simply monitored by spatial period doubling. The influence of the spatial resolution is also studied.

  2. b)

    A Bickley jet developing from a basic sech2y velocity profile. We study the temporal case. In the free case, this basic flow is perturbed by a small white noise. The forced case will correspond to the latter situation, to which a supplementary small deterministic perturbation at the most unstable sinuous mode is added. In the forced case, it is shown that the jet evolves towards a very stable Karman-like street if the deterministic perturbation is strong enough. By contrast, the street of vortices exhibits, in the free case, a turbulent behaviour with possibilities of pairing for eddies of same sign. The eventual state in this latter case is very similar to nearly isotropic two-dimensional turbulence with concentrated coherent vortices.

  3. c)

    A spatially growing mixing-layer, where the upstream basic flow is perturbed by a time-dependant white noise of small amplitude. The calculation shows a spreading rate of the layer in very good agreement with the experimental unperturbed mixing-layer data.

  4. d)

    A temporally growing mixing layer submitted to a differential rotation (β-effect). It is shown that the first pairing is delayed as β increases, and is inhibited for values of β larger than 0.05.

The Grenoble Institute of Mechanics is sponsored by the CNRS.

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Abbreviations

ccomplex phase speed of the perturbation E(k):

two-dimensional kinetic-energy spectrum

E(k x ,t):

one-dimensional kinetic-energy spectrum

E θ (k x ,t):

one-dimensional temperature spectrum

f(y):

Gaussian filter and also Coriolis parameter in section on differential rotation

J :

Jacobian operator

k a :

most amplified wavenumber

l R :

Rhines scale

L :

size of the computational domain if square

L x , L y :

respectively: length and width of the domain if rectangular

m i :

number of fundamental modes in the computational domain n i = ε 2 /ε 1 deterministic to stochastic perturbation ratio

Re:

Reynolds number

S(t):

global enstrophy

u(x, y, t) = u(x, y, t)x + v(x, y, t)y :

velocity field

ū(y):

basic longitudinal velocity profile

ū(y, t):

x-averaged longitudinal velocity

U :

unit of velocity

U 0 :

advection velocity in spatially growing mixing layers

ṽ(y):

amplitude of the perturbation in Orr-Sommerfeld equation

β = ∂f/∂y :

defined in section on differential rotation

δ(t):

current vorticity thickness in temporal calculations

δ i :

initial vorticity thickness

δ 0 :

unit of length

δ viz :

visual thermal thickness of spatially growing mixing layers

δ w :

vorticity thickness of spatially growing mixing layers

Δt :

timestep

Δx :

spacestep

Δx pair :

advection distance during a pairing

\(\epsilon_1 \hat \psi\) :

white-noise perturbation

\(\epsilon_2 \tilde \psi (x)\) initial deterministic perturbation η = U 0 /U :

velocity ratio

θ(x, y, t):

passive temperature

\(\overline {\theta}(y)\) :

basic temperature profile

\(\overline {\theta}(y, t)\) :

x-averaged temperature

\(\hat \theta (k_z,y,t)\) :

one-dimensional Fourier transform of the temperature

κ :

molecular conductivity

λ a :

most amplified wavelength

v :

molecular viscosity

ξ :

transverse coordinate

ϱ = ϱ 0 + ϱ´:

density

σ :

non-dimensioned empiric constant

ψ(x, y, t):

stream-function

\(\bar \psi (y)\) :

stream-function of the basic flow

\(\bar \psi (y, t)\) :

x-averaged stream-function

ω(x, y, t) = ω(x, y, t):

z vorticity field

ω f(x, y, t) = ω(x, y, t):

z potential vorticity field

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Author information

Authors and Affiliations

  1. Institut de Mécanique de Grenoble, BP 68, 38402, Saint-Martin d’Hères, France

    P. Comte, M. Lesieur, H. Laroche & X. Normand

Authors
  1. P. Comte
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  2. M. Lesieur
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  3. H. Laroche
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  4. X. Normand
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Editor information

Editors and Affiliations

  1. Centre National de Recherches Météorologiques, 42, Avenue Coriolis, 31057, Toulouse Cedex, France

    Jean-Claude André

  2. O.N.E.R.A./C.E.R.T., 2, Avenue Edouard Belin, 31055, Toulouse Cedex, France

    Jean Cousteix

  3. Lehrstuhl für Strömungsmechanik, Universität Erlangen-Nürnberg, Egerlandstraße 13, 8520, Erlangen, Fed. Rep. of Germany

    Franz Durst

  4. Department of Mechanical Engineering, University of Manchester, Institute of Science and Technology, PO Box 88, Manchester, M60 1QD, England

    Brian E. Launder

  5. Mechanical Engineering Department, The Pennsylvania State University, University Park, PA, 16802, USA

    Frank W. Schmidt

  6. Department of Mechanical Engineering, Imperial College of Science and Technology, Exhibition Road, London, SW7 2BX, England

    James H. Whitelaw

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© 1989 Springer-Verlag Berlin Heidelberg

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Comte, P., Lesieur, M., Laroche, H., Normand, X. (1989). Numerical Simulations of Turbulent Plane Shear Layers. In: André, JC., Cousteix, J., Durst, F., Launder, B.E., Schmidt, F.W., Whitelaw, J.H. (eds) Turbulent Shear Flows 6. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-73948-4_29

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  • DOI: https://doi.org/10.1007/978-3-642-73948-4_29

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-73950-7

  • Online ISBN: 978-3-642-73948-4

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