turbulence
Julien Bodart1, Laurent Joly1, and Jean-Bernard Cazalbou1
Universit´e de Toulouse, Institut Sup´erieur de l’A´eronautique et de l’Espace (ISAE) 10 av. Edouard Belin - BP 54032 - 31055 TOULOUSE Cedex 4
Julien.Bodart@isae.fr
1 Introduction
With the objective of studying the interaction between turbulence and a solid wall, a new way to generate a statistically steady turbulent state in a DNS setup is presented.
Computation of ideal situations like turbulence diffusing from a plane/point source requires to implement mechanisms of turbulent production that, (i) do not rely on the presence of mean-velocity gradients and, (ii) can be localized in space. A first way to devise a shear-free turbulence-production mechanism has been proposed by Alvelius [1], it was based on the use of a random force field, defined in the spectral space and, consequently, not localized in the physical space. This method has been modified by Campagne et al. [2] in or-der to confine the force field in a finite-width, plane layer of fluid and study the interaction between unsheared turbulence and a free-slip surface.(Fig 1) Since the Navier-Stokes solver was based on a pseudo-spectral method, the method of Alvelius was both a convenient and natural choice.
In order to extend the work of Campagne et al. to the case of the in-teraction with a solid wall, we developed a mixed spectral/finite-difference Navier-Stokes solver. In this context, it is appealing to implement the forcing mechanism in the physical space. Such forcing methods have been recently proposed by Rosales & Meneveau [6] and Nagata et al. [5]). In the first study the random force field is linear (proportional to the instantaneous velocity field) while the second study makes use of randomly-distributed elementary force fields (blobs). In either case, the forcing is statistically homogeneous in
Fig. 1. Numerical analogy of the oscillating grid experiment, with turbulence self-diffusion from a plane source.
all the computational domain.
In this paper, we present results obtained with a forcing method which is rather similar to that of Nagata et al. However, we use a different kind of elementary force patterns, and confine the forcing inside a plane layer of fluid. The latter characteristic relies on the use of compact supports for the force patterns.
2 Random force construction
The random force fi appears as a source term in the incompressible Navier Stokes equation: ∂ui ∂t + ui· ∂uj ∂xj = − 1 ρ ∂P ∂xi + ν∂2ui ∂xj2 + fi(x, t) (1)
The forcing is localized in the physical space i.e. the vector field is com-pactly supported, and excites the large scales of the flow field. In the context of projection methods, a divergence-free vector field is preferable as it does not influence directly the pressure computation. We propose to synthetize such a vector field using any function φ of class C1and ψ of class C0and the relations:
− →f f1= φ(x1) · φ0(x2) · ψ(x3) f2= −φ0(x1) · φ(x2) · ψ(x3) f3= 0
To ensure both derivability and compact support, φ is chosen to be a second-order spline function, and ψ = φ for simplicity. Furthermore a low
order spline function contains mainly low frequencies (Fig 2). The above rela-tions generate a two-component vector field defined in a 3D cubic box of size L. We generate three vector fields by means of permutation among the three coordinate indices. Each vector field is characterized by its zero-component direction x1, x2 or x3 (Fig. 3). Using three vector fields allows to excite the flow field in every directions.
Fig. 2. Second order spline function
Fig. 3. Isomodule of the force, the 3 basis vec-tor fields
To build a plane source of turbulence (see Fig. 1), a 4x4 array of boxes divides the forced layer in the homogeneous directions. This corresponds to a turbulent input of energy at the wavenumber kL = 4 which is a good compromise between forcing at large length scales and keeping the forced layer relatively thin. At every time step, each box receives one of the basis vector fields randomly amplified, while keeping the time averaged input power constant.
To avoid middle-plane symmetry of the domain, each box is slightly and randomly translated in the x3 direction. In the homogeneous directions ran-dom translations are applied as well to the resulting vector field (Fig 4).
Fig. 4. Description of random shift in x3, and x1, x2 applied at each time step on
In the implementation point of view, the basis vector fields are computed at the preprocessing stage which keeps the simulation cost-effective. The choice and the amplitude of the basis vector field attributed to a given box at each time step as well as the amplitude of the different translations are all governed by uniform distributions.
3 Results
A MPI -based parallel solver has been developed, with an hybrid pseudo-spectral/finite difference approach. Fourier modes are used along directions x1 and x2, while 6th order compact scheme have been implemented in x3 direction [4]. Time adavancement uses a third order Runge Kutta scheme for the advective terms and a second order Cranck Nicholson for the viscous terms. The simulations are carried out using 216 fully dealiased modes in the spectral directions and 256 grid points in the finite difference direction. In the presented case, free-slip boundary conditions are applied at the lower and upper side of the domain.
After a transient, the power induced by the forcing field is balanced by the dissipation rate in the whole domain and a statistically steady state is reached. Turbulence self-diffuses out of the forced layer, as presented in Fig 5.
0 0.25 0.5 0.75 1 E / E 0 5 10 15 20 25 Computational time Whole domain Forced layer
Fig. 5. Turbulent kinetic energy evo-lution in time. 0 2.5 5 7.5 10 z /L 0 0.5 1 1.5 τ = (ν/)12 Forced layer
Fig. 6. Kolmogorov time scale evolu-tion in the wall normal direcevolu-tion
Statistically steady state is necessary in every horizontal plane, and start-ing gatherstart-ing statistics requires a particular care. Indeed, (i) the turbulence self-diffusion has to reach the wall (ii) the time scale is growing in the diffusive layer (Fig 6), and so does the transient.
This artificially generated turbulence remains physically consistent, as proved by the skewness and flatness factors of the velocity derivatives (Fig. 8) in the forced layer. We find Su' -0.5 and Fu' 5 in the middle plane, which is consistant with well known values of isotropic homogeneous turbulence.
0 2.5 5 7.5 10 z /L 0 0.25 0.5 0.75 1 1.25 I =w0 u0 Forced layer Pure diffusion 0 2.5 5 7.5 10 z /L 0 2 4 6 8 l/L = (k32/)/L l/L ∝ −0.09 · z/L Forced layer Pure diffusion
Fig. 7. Isotropy factor evolution an turbulent length scale evolution, averaged in x1x2 planes. 0 2.5 5 7.5 10 z /L −2 −1.5 −1 −0.5 0 0.5 ˙ Sui= ∂ui ∂xi 3 /∂ui ∂xi 2 3 2 St Sn Forced layer 0 2.5 5 7.5 10 z /L 0 10 20 30 40 ˙ Fui= ∂ui ∂xi 4 /∂ui ∂xi 22 Ft Fn Forced layer
Fig. 8. Skewness and flatness factor of transverse and normal velocity derivatives evolution, averaged in x1x2 planes.
The resulting turbulent Reynolds number at the edge of the forced layer is Ret' 150. Since the forced layer can be seen as the numerical analogue of the turbulence grid in oscillating-grid experiments, we shall examine our results with reference to the qualitative behaviour observed in these experiments. Characteristics of the pure-diffusion region are retrieved: a linear increase of the turbulent length scale and a nearly constant anisotropy ratio. The values found are presented in the table 1 and compare very well with the experiments of De Silva & Fernando[3].
4 Conclusion
We demonstrated that this confined forcing is a well adapted tool to carry out Direct Numerical Simulation of self diffusion of unsheared turbulence. In our
Experiment[3] DNS I =„ u3 2 u12 «12 1.1-1.2 1.12-1.15 C = ∂k 3 2/ε ∂x3 0.10 0.09
Table 1. Comparisons of the self-similarity characteristics in the pure diffusion layer.
case, a plane source is built which fulfill all the requirements of our numerical experiment, i.e. to study kinematic-blocking effect in the viscinity of a solid wall. The energy input can be adjusted, as well as the forcing field anisotropy by changing the relative amplification level of each basis vector field.
A pure diffusive state is recovered between the forced layer and the block-ing layer and compares very well with oscillatblock-ing grid experiments. Assumblock-ing the box size and the stroke length S of the grid in the experiments are compa-rable, this state is recovered much sooner. In our case less than 0.5L is needed while more than 4S in the experiment[3] which allows direct numerical simu-lations to be performed at a reasonable cost.
References
1. K. Alvelius. Random forcing of three-dimensional homogeneous turbulence. Physics of Fluids, 11:1880–1889, July 1999.
2. G. Campagne, J. B. Cazalbou, L. Joly, and P. Chassaing. Direct numerical sim-ulation of the interaction between unsheared turbulence and a free-slip surface. In ECCOMAS CFD, 2006.
3. I. P. D. De Silva and H. J. S. Fernando. Oscillating grids as a source of nearly isotropic turbulence. Physics of Fluids, 6(7):2455–2464, 1994.
4. Sanjiva K. Lele. Compact finite difference schemes with spectral-like resolution. Journal of Computational Physics, 103(1):16–42, November 1992.
5. K. Nagata, P. A. Davidson, J. C. R. Hunt, Y. Sakai, and S. Komori. Direct numerical simulation of surface blocking effects on isotropic and axisymmetric turbulence. In 5th International Symposium on Turbulence and Shear Flow Phe-nomena (TSFP-5 Conference), August 2007.
6. Carlos Rosales and Charles Meneveau. Linear forcing in numerical simulations of isotropic turbulence: Physical space implementations and convergence properties. Physics of Fluids, 17(9), 2005.
