One-dimensional turbulence: Application to incompressible spatially developing turbulent boundary layers

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Abstract

A map-based stochastic approach, One-Dimensional Turbulence (ODT), is applied to analyze the incompressible spatially developing turbulent boundary layer (SBL). In the present study, the SBL is formed by a plane moving wall and a free stream at rest. The flow variables are resolved on all scales along a 1-D domain. A deterministic process represents the molecular diffusion and a stochastic process is modeling the effect of turbulent advection and pressure fluctuations. Due to the reduced dimensions in the model, it achieves major cost reductions as compared to the full 3-D simulations and is, thus, able to explore large parameter regimes. The simulations are presented for momentum thickness Reynolds numbers, Reθ, varied in the range Reθ1968-8000. We have investigated various features related to the SBL, such as mean, root mean square, Reynolds shear stress, skewness, flatness and pre-multiplied velocity profiles and skin friction coefficient and shape factor, all for two bulk velocities using ODT and compared our results to the available reference Direct Numerical Simulations (DNS) and large-eddy simulations (LES) results. We have further compared the results for spatial and temporal ODT formulations used to investigate the turbulent boundary layer with the reference data at matched Reθ. The comparison presented suggests that ODT is a reasonably accurate approach for the simulations of the spatially developing turbulent boundary layers which can be further improved to yield accurate statistics at high Reynolds number by implementing Fourier transformation of the kernels which in the present formulation of the model is not implemented.

Introduction

The study and understanding of the turbulent boundary-layer-type flows are a major topic in research due to numerous applications in atmospheric sciences, Engineering and industry. Spatial and temporal approaches have been used to investigate the incompressible as well as compressible boundary layer with a particular interest shown to the canonical zero-pressure-gradient (ZPG) incompressible turbulent boundary layer over a flat plate. The spatial approach shown in Fig. 1, has been considered in Schlatter et al., 2009, Schlatter and Örlü, 2010, Schlatter and Örlü, 2012 to evaluate ZPG turbulent boundary layers from moderate Reynolds number up to computationally high Reynolds number. The temporal approach has been used to analyze the incompressible turbulent boundary layer in Kozul et al. (2016) and compressible in Martin (2007). Crucial aspect concerns the generation of initial conditions (e.g. Lund et al., 1996, Ferrante and Elghobashi, 2004, Schlatter and Örlü, 2012).

The spatially developing turbulent boundary layer (SBL) is inhomogeneous in the streamwise and wall-normal direction (Fig. 1), which results in large computational requirements discussed in Spalart, 1988, Wu and Moin, 2009, Schlatter et al., 2009, Schlatter and Örlü, 2010, Jiménez et al., 2010, Sayadi et al., 2013 particularly in the case of 3-D Direct Numerical Simulations (DNS). The spatial approach is however, physically important and it requires very high spatial resolution to resolve the full range of scales. Thus, the approach is limited to small and moderate momentum Reynolds number discussed in Schlatter and Örlü (2012).

In one dimensional turbulence (ODT) formulation, the flow variables are resolved by a deterministic process representing the molecular diffusion term and a stochastic process models the effect of nonlinear advection and fluctuating pressure gradient terms (Kerstein, 1999; Kerstein et al., 2001). The latter represents a stochastic sequence of eddy events with the aid of discrete mappings enclosing a range of scales and aims to capture the statistical properties of the turbulent flow.

In the present study, we utilize ODT (Kerstein, 1999, Kerstein et al., 2001) as a stand-alone tool for simulation of SBL. Due to reduction of dimensionality, the model enables the simulation of high Reynolds number turbulence over the full range of dynamically relevant length scales. It is difficult to capture all aspects of a full 3-D DNS with a reduced order model, but it is interesting to consider the predictability of the model for the simple possible set-up. To investigate complex flows, ODT lines can be embedded in a coarse 3-D LES mesh referred to as ODTLES (Schmidt et al., 2003, Schmidt et al., 2010), which retains the large structure information and accounts for 3-D effects. However, ODTLES formulations use temporal ODT. The primary aim for the present study is to understand the model behaviour and the flow physics captured for the simple SBL configuration. The stand-alone model is validated for the simulation set-up by comparing ODT results to available reference data at low Reynolds number and investigation is extended for high Reynolds number that are not easily computed otherwise.

The original formulation of ODT was given in Kerstein (1999) and was later extended to include pressure scrambling effects (Kerstein et al., 2001). The model was gradually extended for the simulation of variety of flows, for example channel, pipe, multi-physics and reactive flows among others. Here we will highlight some important references which are, in general, important for boundary layers. A limited validation of a case involving forcing of a boundary layer flow is presented in Lignell et al. (2013) and stably-stratified boundary layers in Kerstein and Wunsch (2006). Heat transfer from an isothermal wall using ODT have been studied in Shihn and DesJardin (2007). In Schmidt et al., 2003, Schmidt et al., 2010, Monsoon et al., 2014, Lignell et al., 2018 more complex cases were discussed. Fragner and Schmidt (2017) presented an asymptotic suction boundary layer exhibiting a temporal evolution running into a statistical steady state. Further, the temporally developing turbulent boundary layers (TBL) has been investigated in Rakhi et al. (2019), which further motivated our present study for the SBL. Most of the standalone ODT studies discussed above utilises the temporal formulation. A spatial formulation is more natural for the flow configurations studied, however, is applied less often than the temporal ODT. The spatial formulation have been utilized in Ashurst and Kerstein, 2005, Ricks et al., 2010, Shihn and DesJardin, 2007 but in a different context. In this paper, we have utilised the spatial ODT to investigation for incompressible spatially developing turbulent boundary layer.

The physical model parameters, discussed below are important to calibrate as these parameters depend on the physics included and the forcing mechanism used. Therefore, these parameters cannot be taken over directly. There are several studies, which includes wall-bounded flows (Kerstein, 1999, Schmidt et al., 2003, Kerstein and Wunsch, 2006), thermal convection (Wunsch and Kerstein, 2005), mixing layers (Ashurst and Kerstein, 2005) and non-reacting and reacting jets (Echekki et al., 2001, Hewson and Kerstein, 2001, Lignell et al., 2018) among others, which suggests that the ODT model parameters are not universal.

The organization of the paper is as follows. We give a brief overview of ODT with the required formulation for SBL in Section 2. Section 3 presents a comparison between ODT and other simulation approaches and finally summarises the advantages and disadvantages of the ODT, particularly focused on boundary layer investigations. In Section 4, we provide the ODT simulation set-up for the present flow configuration. In Section 5, we first find the optimal model parameter set to capture the flow dynamics in comparison with the reference data and then we analyze the influence of the domain size on the velocity statistics for different momentum thickness Reynolds numbers (Reθ) discussed in Section 6. We address the predictive capabilities of ODT for the velocity statistics and structural properties in Section 7. We compare these results with the available reference DNS and LES data from Schlatter and Örlü, 2010, Eitel-Amor et al., 2014 at Reθ2000,4000, and 8000 with Reθ=Ubθ/ν, where ν is the kinematic viscosity, Ub is the uniform velocity provided at the bottom wall and θ is the momentum layer thickness. In Section 8, we present the comparison between spatial and temporal ODT formulations used to investigate the turbulent boundary layer. Section 9 includes a short summary of the merits and limitations of the model. The conclusions of our findings are given in Section 10.

Section snippets

ODT: One-dimensional turbulence

The main features of the ODT are summarised in this section which includes the governing equations, formulation of the eddy events, their selection, and a description of the model parameters.

Predictability of the ODT model

In this section, the predictive capability of the ODT model is compared to the other simulation approaches. DNS is assumed to be the numerical reference, as in case of DNS, the Navier–Stokes equations are solved without any assumptions. These numerical experiments are completely true for statistically steady flows. However, in case of transient problems, it is expected that initial conditions might affect the results. A DNS needs to resolve every scale, i.e., the whole range of spatial and

Simulation set-up

We have utilized the simulation set-up which was used for TBL (Rakhi et al., 2019) configuration by slightly modifying the boundary and initial conditions (discussed below) for the present case, i.e., SBL (Fig. 1). We have also adjusted some of the physical as well as numerical parameters according to the problem given in Table 1. The spatial formulation of ODT explained in Lignell et al. (2013) is followed and it allows simulations of flows that are statistically 2-D and the time dimension is

Parametric analysis

In this section, we have discussed the sensitivity of the results to the physical ODT model parameters (C and Z). This is done by comparing ODT results to the reference DNS (Schlatter and Örlü, 2010) at Reθ4000. The bulk velocity used to select a suitable set of parameters for the SBL configuration is Ub=12m/s. This bulk velocity is used because we have DNS reference data available at the given bulk velocity.

Our starting point to fix the model physical parameters in the present study, i.e.,

Influence of the domain

In this section we analyze the influence of the domain size on velocity statistics for different Reθ as a prerequisite check to make sure that the simulations performed on various domain sizes within the Reynolds number variation campaign would remain unaffected by any domain size considered. The domain size is increased to capture boundary effects onto the flow and to verify if the increasing boundary layer sizes are accommodated in case of ODT similar to the reference DNS. The asymptotic

Results

In this section, we have presented various velocity statistics (up to 4thorder) at Reθ4000 and 8000 in comparison to the available reference DNS data form (Schlatter and Örlü, 2010) at Reθ4000 and LES data from Eitel-Amor et al. (2014) at Reθ8000. Note that the preliminary results for velocity statistics up to 4th order for Reθ2000 are presented in Rakhi and Schmidt (2019) and in the present work we extend our work for higher Reθ. Latter, we show some of the structural properties, like Reτ,H

Comparison between SBL and TBL

In this section we have compared TBL and SBL characteristics for various velocity statistics up to 4th order in comparison with the spatial and temporal DNS features.

Fig. 16 shows the mean streamwise velocity profile as a function of wall-normal coordinate in viscous units at Reθ1968 for the present case, i.e., the SBL flow configuration along with the TBL configuration reported in Rakhi et al. (2019). We note that the profiles for SBL and TBL overlap on to each other up to y+200 and deviate

Discussion

The ODT model allows physically sound representation of interactions between turbulent advection and microphysical processes and enables affordable simulation of high Reynolds number turbulence over the full range of dynamically relevant length scales. The model also demonstrates the degree of commonality among turbulent flow phenomena (which might not otherwise be readily apparent) by capturing diverse flow behaviour within a concise modelling framework based on broadly applicable empirical

Conclusion

In the present paper, we have applied the ODT model for the first time to investigate the incompressible spatially developing turbulent boundary layer. The model resolves the flow variables along a 1-D computational domain in which the viscosity effects are represented by the deterministic diffusion equation and the turbulent advection by stochastic mapping events. From earlier studies, we know that the ODT results are sensitive to the choice of model parameters. So, first we calibrate the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

The authors acknowledge financial support from the Graduate Research School (GRS) of the BTU Cottbus-Senftenberg. This work is part of the Cluster StochMethod SP7, ‘Stochastic Modeling of Turbulent Flow’.

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