# Stochastic Modelling of Turbulent Flows for Numerical Simulations

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## Abstract

**:**

## 1. Introduction

## 2. Pseudo-Stochastic Model

#### 2.1. Stochastic Formalism

#### 2.2. Pseudo-Stochastic Equations of Motion

#### 2.3. Isotropic Constant Model

#### 2.4. Near-Wall Modelling for Isotropic Model

#### 2.5. Local Variance Model

#### 2.6. Resolved Kinetic Energy Budget

- the first four terms express spatial transport;
- the fifth term is a turbulent compression/expansion term due to SUS;
- the sixth and seventh terms account for dissipation by molecular viscosity, resolved turbulence, and SUS motions;
- the eight term represents the shear production, including the contribution by the fluctuations of turbulent advection velocity;
- the last term indicates a loss due to SUS; this term is also present in the resolved kinetic energy budget.

## 3. Physical Interpretation and Comparison with Les Models

#### 3.1. Physical Interpretation

**Effective advection:**the advection velocity is corrected by an inhomogeneous turbulence contribution. As pointed out by Resseguier et al. [16], it corresponds to a velocity induced by the unresolved eddies, that is linked to the turbophoresis phenomenon detectable in geophysical flows; i.e., the tendency of the fluid-particle to migrate in the direction of less energetic turbulence.**Diffusion due to SUS:**the last two terms on the right-hand side of Equation (19) account for the turbulent diffusion; the variance tensor plays the role of a diffusion tensor similar to a generalised eddy-viscosity coefficient. Both deformation rate and rotation-rate contribute to diffusion, unlike in the classical eddy-viscosity model.**Turbulent compressibility:**the continuity Equation (20) suggests that the flow is turbulent-compressible; i.e., the unresolved turbulence induces a local fluid compression or expansion.

#### 3.2. Comparison with Les Eddy-Viscosity Models

#### 3.3. Remarks on Eddy-Viscosity Model

- The effects of unresolved scales of motion are given by ${a}_{ij}$, without imposing any constraint on the directions along which the SUS acts. Hence, the hypothesis (a) is not required.
- The tensor form of ${a}_{ij}$ allows reproducing the anisotropy of unresolved turbulence, i.e., different turbulent contributions along different directions. Thus, hypothesis (b) is not required.
- The extra terms in the governing equations account for turbulent effects, usually not considered in the classical models, namely turbulent advection and turbulent compressibility.

#### 3.4. Remarks on Smagorinsky Model

- the rotation-rate does not contribute to turbulence effects on the mean flow;
- the norm of the strain-rate tensor is almost harmonic (Laplacian is close to zero).

## 4. Simulation Methodologies

#### 4.1. Methodology and Implementation

`pisoFoam`included in the standard software distribution. The implementation details can be found in the OpenFOAM documentation and in Jasak et al. [41]. The filtered classical Navier–Stokes equations were closed by the Smagorinsky model (25), with ${c}_{s}=0.65$. The van Driest function (26) for near-wall damping is used unless otherwise specified. The optimal parameters were set as $n=m=1$, ${A}^{+}=0.26$, ${C}_{\delta}=0.158$.

`pseudoStochasticPisoFoam`, developed by the authors within the Fluminance research group at INRIA Rennes (France). The pseudo-stochastic Equation (4) were solved employing the pressure-implicit with splitting of operators (PISO) algorithm proposed by Issa et al. [42], Oliveira and Issa [43]. The variance tensor was expressed by the local variance model and the isotropic constant model (7), corrected by the near-wall damping function (11) unless otherwise specified. The isotropic constant was set to be ${c}_{m}=2{c}_{s}^{2}$ in analogy with the Smagorinsky model. The damping parameter was set to be ${y}_{B}^{+}=2/\tilde{\kappa}=12.7$ after a theoretical estimation [24], confirmed by several test simulations. In order to regularise the damped profile of ${a}_{yy}$, a smoothing filter was applied to the variance tensor.

#### 4.2. Case Geometry and Settings

## 5. Results and Discussion

`LES vanDriest`and

`LES smagConst`; two PSS using the isotropic constant model with and without the LU near-wall model, labelled respectively

`PSS isoConst`and

`PSS isoLUWall`; one PSS using the local variance model labelled

`PSS variance`. They were compared with the DNS by Moser et al. [40].

`LES smagConst`and the

`PSS isoConst`exhibit a very similar profile. This is because the isotropic model is derived by analogy with the classical Smagorisnky model. The main difference between the two is the presence in the PSS equations of turbulent advection (21) and turbulent compressibility (22); however, in this case, they are not strong enough to produce remarkable results on the mean flow (see discussion of Figure 4-left-panel). Both models underestimate the velocity: this is a well-known shortcoming of Smagorinsky model when ${c}_{s}^{2}$ is constant, and it is inherited by the constant isotropic model. Second, we compare the

`LES vanDriest`and the

`PSS isoLUWall`, which implement a near-wall damping function for ${\nu}_{SGS}$ and ${a}_{ij}$ (respectively). The damping reduces the turbulent dissipation in the buffer layer, preventing the underestimation of velocity. When the near-wall models are activated, velocity is well captured in the viscous and buffer layers but is slightly overestimate in the logarithmic and far-wall regions (${y}^{+}>50$). Remarkably, the LU near-wall model here proposed is as accurate as the well-established van Driest wall function in reproducing the velocity profile. Further analysis of such a damping model is reported in the discussion of Figure 3. Third, the

`PSS variance`is the most accurate model and exhibits a very good agreement with the reference profile. Notice that the local variance model is not corrected with a damping function.

`LES smagConst`and the

`PSS isoConst`and between the

`LES vanDriest`and the

`PSS isoLUWall`; hence, just the LES profiles are displayed. The PSS with local variance model reproduces the results of the

`LES vanDriest`, which is more accurate than the

`LES smagConst`, as expected.

`PSS isoLUWall`. A large SGS viscosity in the buffer layer entails an underestimation of the speed of the fluid (see Figure 1). The same shortcoming affects the PSS with isotropic constant model (not reported), where ${a}_{ij}\sim {\nu}_{SGS}{\delta}_{ij}$ by definition. In the

`LES vanDriest`, the van Driest function damps the SGS viscosity in the buffer layer (${y}^{+}\sim 10$), increasing the accuracy of the simulation. In the

`PSS isoLUWall`, the LU near-wall model scales the component ${a}_{yy}$ similarly to the ${\nu}_{SGS}^{+}$. As a result, the accuracy of this model is equivalent to the

`LES vanDriest`. Surprisingly, the values of stream-wise and spanwise components ${a}_{xx}$ and ${a}_{yy}$ seem not to affect the overall fluid dynamics, which is mainly influenced by the wall-normal component ${a}_{yy}$. In the

`PSS variance`model, the off-diagonal components (${a}_{xy},{a}_{yz},{a}_{xz}$) assume low values and they are not reported. The diagonal components of variance tensor have a different behaviour. The component ${a}_{yy}$ goes to zero near the wall. The profile starts to decrease at the beginning of the buffer layer (${y}^{+}\sim 50$) and scales like $O\left({y}^{3}\right)$. Instead, the components ${a}_{xx}$ and ${a}_{zz}$ have larger values and do not approach zero near the wall. However, as previously noticed, they do not influence the final velocity statistics.

`PSS isoConst`model shows an increase in computational time of about $16\%$ compared to

`LES smagConst`, while the

`PSS variance`has an increment of $18\%$. This increment is mainly ascribed to the additional computations due to the extra terms in the governing equations, along with the numerical manipulation of the variance tensor. Notice that such an estimate depends on the particular software used and on the numerical implementation; the optimisation of the numerical solver can lead to better performances.

#### PSS Extra Terms

`PSS isoConst`. The stream-wise component of ${u}_{\mathrm{TA}}$ is practically zero, as well as the span-wise component; thus they are not displayed. The vertical component profile reveals low negative values, with a climax at ${y}^{+}\cong 10$. Quantitatively, the turbulent advection is not strong enough to produce remarkable results on the mean flow; however, it generates a weak vertical velocity ${w}_{y}$ directed from the centre of the channel to the wall (not reported). Hence, ${u}_{\mathrm{TA}}$ is qualified as a weak turbophoresis velocity: it advects the flow from the buffer region to the log–law region, i.e., in the direction of decreasing turbulence levels. The turbulent compressibility ${\mathsf{\Phi}}_{\mathrm{TC}}$ is practically zero, but it assumes slightly negative and positive values in the viscous sub-layer and the buffer layer, that can be qualitatively related to weak turbulent compression and expansion. The profiles of the other PSSs performed are not shown because the magnitude of ${\mathsf{\Phi}}_{\mathrm{TC}}$ is negligible and, overall, does not have an impact on the velocity statistics.

`PSS variance`, the turbulent advection and compressibility assume low values, possibly because of the regular profile of ${a}_{ij}$. Nevertheless, the instantaneous structures linked to these quantities can be analysed. Figure 4-central-panel shows the instantaneous distribution of $|{u}_{TA}^{+}|$ at a vertical slice. Spots of non-zero turbulent advection are localised in the region ${y}^{+}<100$; they are more intense in the buffer layer and logarithmic region ${y}^{+}<50$. As for the

`PSS isoConst`above discussed, the velocity ${u}_{TA}^{+}$ generates a weak instantaneous and localised turbophoresis velocity in the direction of lower turbulent areas.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Mathematical Derivation of Local Variance Model

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**Figure 1.**Non-dimensional stream-wise velocity profiles along wall-normal direction for turbulent channel at $R{e}_{\tau}=395$. Left panel: solid black, direct numerical simulation (DNS) by Moser et al. [40]; dash violet, analytical profile (11) derived from location uncertainty (LU) by Pinier et al. [24]. Right panel: comparison between the large-eddy simulation (LES) and pseudo-stochastic simulation (PSS) (see label definition in the text).

**Figure 2.**Left-panel: averaged non-dimensional root-mean square of velocity components along wall-normal direction for three selected simulations (same label as Figure 1). Right-panel: non-dimensional Turbulent Kinetic Energy (TKE) budget (18) along the wall-normal direction. Blue lines, LES with van Driest damping (

`LES vanDriest`); Red lines with symbols, PSS with local variance model (

`PSS variance`).

**Figure 3.**Non-dimensional mean ${\nu}_{SGS}$ and ${a}_{ij}$ along wall-normal direction. Three selected simulations are reported: LES with van Driest damping (

`LES vanDriest`), PSS isotropic constant with LU near-wall model (

`PSS isoLUWall`), and PSS with local variance model (

`PSS variance`). Black solid line: the slope ${\nu}^{+}\sim {y+}^{3}$.

**Figure 4.**Top-panel: averaged non-dimensional turbulent advection (21) and compressibility (22) appearing in PSS with constant isotropic model (

`PSS isoConst`). Central-panel: contourplot of the instantaneous turbulent advection magnitude $|{u}_{TA}^{+}|$ at a vertical slice for the simulation

`PSS variance`. Bottom-panel: contourplot of the instantaneous turbulent compressibility ${\mathsf{\Phi}}_{TC}^{+}$ at the channel bottom wall for the simulation

`PSS variance`.

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**MDPI and ACS Style**

Cintolesi, C.; Mémin, E.
Stochastic Modelling of Turbulent Flows for Numerical Simulations. *Fluids* **2020**, *5*, 108.
https://doi.org/10.3390/fluids5030108

**AMA Style**

Cintolesi C, Mémin E.
Stochastic Modelling of Turbulent Flows for Numerical Simulations. *Fluids*. 2020; 5(3):108.
https://doi.org/10.3390/fluids5030108

**Chicago/Turabian Style**

Cintolesi, Carlo, and Etienne Mémin.
2020. "Stochastic Modelling of Turbulent Flows for Numerical Simulations" *Fluids* 5, no. 3: 108.
https://doi.org/10.3390/fluids5030108