Analysis of wall-modelled particle/mesh PDF methods for turbulent parietal flows

Guilhem Balvet; Jean-Pierre Minier; Yelva Roustan; Martin Ferrand

doi:10.1515/mcma-2023-2017

Article

Analysis of wall-modelled particle/mesh PDF methods for turbulent parietal flows

Guilhem Balvet , Jean-Pierre Minier , Yelva Roustan and Martin Ferrand

Published/Copyright: October 19, 2023

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From the journal Monte Carlo Methods and Applications Volume 29 Issue 4

Abstract

Lagrangian stochastic methods are widely used to model turbulent flows. Scarce consideration has, however, been devoted to the treatment of the near-wall region and to the formulation of a proper wall-boundary condition. With respect to this issue, the main purpose of this paper is to present an in-depth analysis of such flows when relying on particle/mesh formulations of the probability density function (PDF) model. This is translated into three objectives. The first objective is to assess the existing an-elastic wall-boundary condition and present new validation results. The second objective is to analyse the impact of the interpolation of the mean fields at particle positions on their dynamics. The third objective is to investigate the spatial error affecting covariance estimators when they are extracted on coarse volumes. All these developments allow to ascertain that the key dynamical statistics of wall-bounded flows are properly captured even for coarse spatial resolutions.

Keywords: Interpolation scheme error; Lagrangian stochastic methods; particle/mesh methods; statistic estimation; wall-boundary condition

MSC 2010: 76M35

Funding source: Association Nationale de la Recherche et de la Technologie

Award Identifier / Grant number: 2020/1387

Funding statement: G. Balvet has received a financial support by ANRT through the EDF-CIFRE contract number 2020/1387.

A Complement on the error induced by piecewise constant interpolation

The goal in the present appendix is to further discuss the influence of the interpolation methods considered on the particle dynamics. The errors occurring when using a piecewise constant interpolation near the wall are emphasized.

Injecting the interpolated mean carrier fields at the position of the particle in the modelling of the increments of velocity (2.2b), the SLM model becomes

d ⁢ U ¯ = − 1 [ ρ ̄ ] ⁢ [ ∇ ¯ ⁢ P ̄ ] ⁢ ( t ; X ¯ ⁢ ( t ) ) ⁢ d ⁢ t − U ¯ − [ U ¯ ̄ ] ⁢ ( t ; X ¯ ⁢ ( t ) ) [ T L ̄ ] ⁢ ( t ; X ¯ ⁢ ( t ) ) ⁢ d ⁢ t + C L ⁢ C 0 ⁢ [ k ̄ ] ⁢ ( t ; X ¯ ⁢ ( t ) ) [ T L ̄ ] ⁢ ( t ; X ¯ ⁢ ( t ) ) ⁢ T L ⁢ ( t ; X ¯ ⁢ ( t ) ) ⁢ d ⁢ W ¯ ,

where, for any carrier fields Ψ, Ψ ̄ denotes the averaged value extracted from the finite volume approach and [ Ψ ̄ ] ⁢ ( t ; X ¯ ⁢ ( t ) ) its interpolation at the position of the particle X ¯ ⁢ ( t ) at the instant 𝑡. For the sake of clarity, the term ( t ; X ¯ ⁢ ( t ) ) will be discarded from now on, yet it is important to keep in mind that generally these interpolated values are space and time dependent. The corresponding equation for the first-order statistics are

(A.1) ∂ ⟨ U i ⟩ ∂ t + ⟨ U k ⟩ ⁢ ∂ ⟨ U i ⟩ ∂ x k + ∂ ⟨ u k ⁢ u i ⟩ ∂ x k = − 1 ρ r ⁢ ∂ P ̄ ∂ x i − ⟨ U i ⟩ − [ U i ̄ ] [ T L ̄ ] .

An additional relaxation term between the mean velocity extracted from the set of particles and the interpolation of the mean carrier flow at this position appears. For the streamwise mean velocity, equation (A.1) becomes

∂ ⟨ u ⁢ w ⟩ ∂ z = − ⟨ U ⟩ − [ U ̄ ] [ T L ̄ ] .

Thus, if the mean velocity associated to the particles ( ⟨ U ⟩ ) differs from the local interpolation of the mean carrier velocity fields at this position ( U ̄ ) , the local shear stress associated to the particle will not remain uniform. This effect is strengthened when approaching the wall where the Lagrangian time scale becomes small. From this equation, one can obtain the equation for the particle-averaged velocity,

⟨ U ⟩ = [ U ̄ ] − ∂ ⟨ u ⁢ w ⟩ ∂ z ⁢ [ T L ̄ ] .

In a zone where the interpolated mean carrier velocity field is differentiable, we can write

∂ ⟨ U ⟩ ∂ z = ∂ [ U ̄ ] ∂ z − ∂ 2 ⟨ u ⁢ w ⟩ ∂ z 2 ⁢ [ T L ̄ ] − ∂ ⟨ u ⁢ w ⟩ ∂ z ⁢ ∂ [ T L ̄ ] ∂ z .

Supposing piecewise uniform interpolation denoted ( [ . ] 0 ) , away from the faces of the cells, the gradients of the interpolated mean carrier fields are well defined and are null. We get

(A.2) ∂ ⟨ U ⟩ ∂ z = − ∂ 2 ⟨ u ⁢ w ⟩ ∂ z 2 ⁢ [ T L ̄ ] 0 .

The derivation of the Reynolds tensor remains formally unchanged; however, the shear stress being now non-uniform, we will also consider the turbulent diffusion terms in the equation

∂ ⟨ u i ⁢ u j ⁢ w ⟩ ∂ z = − δ x ⁢ j ⁢ ⟨ u i ⁢ w ⟩ ⁢ ∂ ⟨ U ⟩ ∂ z − δ i ⁢ x ⁢ ⟨ u j ⁢ w ⟩ ⁢ ∂ ⟨ U ⟩ ∂ z − 2 ⁢ ⟨ u i ⁢ u j ⟩ T L + C 0 ⁢ C L ⁢ k T L ⁢ δ i ⁢ j .

Injecting the mean velocity gradient (A.2) in the shear stress equation, we have

∂ ⟨ u ⁢ w ⁢ w ⟩ ∂ z = ⟨ w ⁢ w ⟩ ⁢ ∂ 2 ⟨ u ⁢ w ⟩ ∂ z 2 ⁢ [ T L ̄ ] 0 − 2 ⁢ ⟨ u ⁢ w ⟩ [ T L ̄ ] 0 ,

(A.3) ⟨ u ⁢ w ⟩ = − [ T L ̄ ] 0 2 ⁢ ∂ ⟨ u ⁢ w ⁢ w ⟩ ∂ z + ⟨ w ⁢ w ⟩ ⁢ ∂ 2 ⟨ u ⁢ w ⟩ ∂ z 2 ⁢ ( [ T L ̄ ] 0 ) 2 2 .

Injecting the mean velocity gradient (A.2) and the shear stress (A.3) in the streamwise kinetic energy equation, we get

∂ ⟨ u ⁢ u ⁢ w ⟩ ∂ z = ( − [ T L ̄ ] 0 ⁢ ∂ ⟨ u ⁢ w ⁢ w ⟩ ∂ z + ⟨ w ⁢ w ⟩ ⁢ ∂ 2 ⟨ u ⁢ w ⟩ ∂ z 2 ⁢ ( [ T L ̄ ] 0 ) 2 ) ⁢ ( ∂ 2 ⟨ u ⁢ w ⟩ ∂ z 2 ⁢ [ T L ̄ ] 0 ) − 2 ⁢ ⟨ u ⁢ u ⟩ [ T L ̄ ] 0 + C L ⁢ C 0 ⁢ [ k ̄ ] 0 [ T L ̄ ] 0 ,

(A.4) ⟨ u ⁢ u ⟩ = C L ⁢ C 0 ⁢ [ k ̄ ] 0 2 ⏟ ⟨ w ⁢ w ⟩ = ⟨ v ⁢ v ⟩ = 2 3 ⁢ k iso − [ T L ̄ ] 0 2 ⁢ ∂ ⟨ u ⁢ u ⁢ w ⟩ ∂ z + ( [ T L ̄ ] 0 ) 2 2 ⁢ ∂ 2 ⟨ u ⁢ w ⟩ ∂ z 2 ⁢ ( − ∂ ⟨ u ⁢ w ⁢ w ⟩ ∂ z + ⟨ w ⁢ w ⟩ ⁢ ∂ 2 ⟨ u ⁢ w ⟩ ∂ z 2 ⁢ [ T L ̄ ] 0 ) .

Assuming that we are going close to the wall where the Lagrangian time scale tends towards zero, equations (A.2), (A.3), (A.4) imply that the particle-averaged velocity gradient and the shear stress tend towards zero whereas the streamwise kinetic energy tends towards the normal kinetic energy, i.e. the one obtained for maintained isotropic turbulence. This spurious behaviour is schematized in Figure 7 and demonstrated in Figure 14.

Figure 14

Vertical profiles of the dimensionless mean streamwise velocity (a), the four non-null components of the dimensionless Reynolds tensor (b) in the few cells near the wall using a piecewise uniform interpolation scheme 1 ( ◆ ) (note that, in the spanwise and normal direction, the Reynolds tensor components are equal; only the latter one is plotted). These statistics are compared with analytical solution (black dashed line). In each cell of the FV simulation (whose faces are schematized by the grey dotted lines), the statistics are first estimated into 100 finer bins. The results plotted are an agglomeration of these statistics based on a spatial average over ten bins.

(a)

Dimensionless streamwise velocity

(b)

Dimensionless Reynolds tensor

Acknowledgements

The authors acknowledge the infrastructures at EDF R&D and the CEREA laboratory for providing access to computational resources.

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Received: 2023-06-29

Revised: 2023-09-18

Accepted: 2023-09-19

Published Online: 2023-10-19

Published in Print: 2023-12-01

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Articles in the same Issue

https://doi.org/10.1515/mcma-2023-2017

Keywords for this article

Interpolation scheme error; Lagrangian stochastic methods; particle/mesh methods; statistic estimation; wall-boundary condition