Observations and some selected results from our paper, Dynamic subgrid-scale scalar-flux model based on the exact rate of production of turbulent fluxes, Shujaut H. Bader and Paul A. Durbin, Phys. Rev. Fluids 5, 114609.
I. Preliminaries
Transport of energy is one of the most fundamental processes which keeps our world going. From stellar convection and weather systems to power plants, cooling towers and automobile engines, (generation of and) transport of heat is ubiquitous. The mechanisms that govern the transport of heat depend upon the scale, environment, and the physical state of the system. For example radiation and convection both dominate in stellar cores of normal stars while conduction is inefficient as the density is too low to allow the electrons to conduct heat radially outwards. Similarly, the accumulation and redistribution of solar energy in the planetary atmospheres like ours is purely a convective-radiative affair. In systems like engines and power plants, convection, either forced or natural, plays a dominant role.
In addition to momentum, a fluid in motion can convect heat (energy) too. Convection is primarily of two different types -- natural and forced. Natural convection occurs when the temperature difference is large enough to cause a strong density differential to induce convective currents inside the fluid. These currents enable the parcels of hot fluid to rise, and the cold fluid parcels to sink. An example would be a pot of boiling water on a kitchen stove. Forced convection occurs over a hot surface when a fast moving fluid carries the heat away from it. Imagine blowing over a plate of hot food when you are hungry! It has to be noted that in forced convection, the thermal gradient must remain small enough to prevent the formation of any substantial density differential. As you can tell by now, I have a strong leaning towards discussing convective systems.
From a dynamical perspective, when the temperature variations do not affect the density of the system, the dynamics involved are labelled as passive scalar dynamics. Passive meaning the transported quantity remains passive and does not affect the flow in any way, and scalar meaning the transported quantity is defined uniquely by its concentration (magnitude) at each point in the domain. On the other hand natural convective systems involve active scalar dynamics where gravity plays an important role. Again, think about blowing over a plate of hot food versus a pot of boiling water. It is very obvious that gravity has nothing to do with the former. In this paper, we have delved into the physics and modeling of the passive scalar dynamics in the framework of wall-bounded flows.
II. Subgrid modeling
The modeling approach proposed in this work belongs to the class of subgrid turbulence models. Subgrid, as the same suggests, relates to the effects smaller than those of the resolution of the grid/filter used. In subgrid modeling the governing equations are filtered implicitly or explicitly. The filtering operation gives rise to some extra terms that need to be parameterized, analogous to but not the same as Reynolds terms in Reynolds Averaged Simulation approach. These added terms represent represent the influence of small scales on the (large) resolved scales. Let it be reiterated here that the purpose of subgrid closures is only to represent their effect on the resolved scales, and not to represent the small scales per se. Essentially these unresolved quantities are representative of turbulent dispersion at scales smaller than the filter width. The predictive power of a good Large Eddy Simulation (LES), thus, depends to a significant extent upon how effective the subgrid model is in parameterizing the physics of small scales.
III. The model
The standard closures for subgrid scalar/heat flux invoke a scalar coefficient, the turbulent eddy diffusivity. The models thus derived assume an alignment of the resolved scalar gradient and the subgrid scalar flux.
\[ h^{EDM}_{i\theta}=-\alpha_{T}\ \partial \overline{\theta}/\partial x_i \ \ \ \ \ \ Eddy \ Diffusivity \ Model \ (EDM) \]
An analogy to the Fourier's Law of heat conduction with constant thermal conductivity comes to mind. Since the coefficient is isotropic, these models fail in the significantly anisotropic flows. That error may have a small influence on the predictions of the average scalar concentration profile, because only the wall normal flux contributes to the mean flux divergence, but the structure of the fluctuating fluxes will be inaccurate. Thus there is cause to consider a more physically justified subgrid closure.
It has been noted in several studies that the turbulent diffusivity is not independent of orientation of the mean scalar gradient. Thus, in such situations, the scalar turbulent diffusivity is grossly in error, thereby questioning the physical relevance of scalar turbulent diffusivity models. Such considerations argue for the turbulent diffusivity to be represented by an asymmetric tensor, i.e.,
\[ h_{i\theta}=-\alpha_{T_{ij}}\partial \overline{\theta}/\partial x_{j} \ \ \ \ \ \ Generalized \ Gradient \ Diffusion \ Hypothesis \]
where \(\alpha_{T_{ij}} \) is asymmetric. Models with tensor diffusivity rotate the SGS flux vector with respect to the resolved scalar gradient, and yield scalar fluxes in directions other than that of the imposed mean scalar gradient—hence the name generalized gradient diffusion hypothesis (GGDH). The widely cited GGDH model is that of Daly and Harlow where the diffusion tensor is constructed by the product of the subgrid stress tensor and a characteristic timescale,
\[ h^{GGDH}_{i\theta} = -C_{G\theta} \mathscr{T} \tau_{ij} \frac{\partial \overline{\theta}}{\partial x_{j}} \]
In the paper, we have shown that the symmetric nature of diffusion tensor in the GGDH model is inadequate to represent the geometry of the fluxes at the subgrid scale. Although a non-zero streamwise flux is produced, the predicted value is only half of that observed in direct numerical simulations. However, the predictions are still better than the Eddy Diffusivity Models, as expected.
The exact rate of production of subgrid turbulent scalar-flux contains contributions from the resolved scalar gradient as well as from the resolved shear. Taking both of these terms into account, the scalar flux can be expressed as,
\[ h_{i\theta}=-C_{\theta}\mathscr{T} \left(\tau_{ij} \frac{\partial\overline{\theta}}{\partial{x_{j}}} +h_{j\theta} \frac{\partial\overline{u}_{i}}{\partial{x_{j}}}\right) \]
where, on dimensional grounds, a suitable timescale \( \mathscr{T}\) has been multiplied. In terms of the diffusivity, the above equation can be rewritten as,
\[ \alpha_{T_{ij}} = C_{\theta} \mathscr{T} \left(\tau_{ij} - \alpha_{T_{kj}} \ \frac{\partial \overline{u}_{i}}{\partial{x_{k}}} \right) \ \ \ \ \ \ (Present \ Model) \]
Using this algebraic formula for \(\alpha_{T_{ij}} \), it is clear that the resulting formulation prodices an asymmetric diffusion tensor in contrast to eddy diffusivity models and to general gradient diffusion hypothesis models, for which the diffusivity tensor is symmetric. The explicit expression for the diffusion tensor can be obtained by several methods which are outlined in the paper.
IV. Flow physics: some selected results
The model has been validated in a plane channel flow and a backward facing step with heat transfer. In this section, some important findings are discussed.
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| Figure 1: Anisotropy (a), and asymmetry (b) of the model diffusivity tensor, \( \alpha_{T_{ij}}\), for the present model. |
Averaged values of the normalized diffusivity tensor components for the present model are shown in Fig. 1. From Fig. 1(a), it is clear that the model captures anisotropy of the diffusivity tensor. The model is also able to reproduce the expected asymmetry, \( \alpha_{T_{12}} << \alpha_{T_{21}} < 0 \) in Fig. 1(b). The asymmetry in the diffusivity tensor is due to the contribution of the velocity gradient to flux production.
The alignment angle \(\Phi \) between the resolved temperature gradient and the modeled SGS heat flux is,
\[ \Phi = cos^{-1} \left ( \frac{h_{i\theta} \cdot \overline{\theta}_{,i} }{|h_{i\theta}||\overline{\theta}_{,i} |} \right ) \]
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| Figure 2: PDF of the alignment angle between the resolved temperature gradient and the SGS heat flux for Dyn-EDM (a), Dyn-GGDH (b), and the proposed model (c). The results are shown for fully developed channel flow. |
Fig. 2 shows the PDF of the instantaneous alignment angle predicted by the Dyn-EDM, Dyn-GGDH, and the present model. Negative values of the model coefficients have been allowed for all of the described models. It is evident in Fig. 2(a) that \(\Phi\) predicted by the Dyn-EDM is either \(0^{\circ}\) or \(180^{\circ}\) corresponding to \(C_{E\theta} < 0\) or \(C_{E\theta} > 0\), respectively. The predictions are similar in the regions near the wall \((y^+ = 3)\) and near the center of the channel \((y^+ = 350)\). As the physical mechanism of heat transfer at the subgrid scale is significantly different from the simple gradient diffusion constitutive relation, the Dyn-EDM fails to accurately describe the physics of heat transfer at the such scales.
Tensor diffusivity models, however, are able to predict a SGS heat flux vector which is rotated with respect to the temperature gradient, giving rise to a wide range of alignment angles. In the case of Dyn-GGDH, as shown in Fig. 2(b), the PDF of the instantaneous alignment angle is quite scattered with a substantial contribution from (\Phi=180^{\circ}\). The near wall predictions are better than those away from the wall.
In the proposed SGS heat flux model, Fig. 2(c), a single angle of \(\Phi \approx 125^{\circ}\) is preferred in the near wall region. This property is in accordance with the physical intuition that in the near wall region, the direction of the flux is governed by both the wall-normal temperature gradient and the advection of temperature parallel to the wall, giving rise to the SGS heat flux vector aligned at the angles weighted toward \(\Phi =90^{\circ}\). In the core region of the channel, the proposed model predicts \(\Phi\) closer to \(150^{\circ}\) which is still better than both the Dyn-EDM and Dyn-GGDH model predictions.
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| Figure 3: Effective SGS diffusivity for Dyn-EDM, Dyn-GGDH and the proposed model. |
In order to compare the SGS effects of tensor diffusivity models with the Dyn-EDM, it is reasonable to define the effective SGS diffusivity as,
\[ \alpha^{eff}_{sgs} = -{h_{2\theta}}/({{\partial \overline{\theta}}/{\partial y}}) \]
The ratio of the averaged effective SGS diffusivity to the molecular diffusivity is quantified for all the three models in Fig. 3. This ratio signifies how important the SGS diffusion is, compared to molecular diffusion. It is observed that the degree of variability in the effective SGS diffusivity is greatest in the case of the proposed model. The value of \(\alpha^{eff}_{sgs}/\alpha\) predicted by Dyn-EDM and Dyn-GGDH models ranges from 0 at the wall to a maximum of \(\approx 0.4\) and \(\approx 0.9\) in the core region respectively. However, the effective diffusivity of the present model is close to \(1.5\) over most of the channel, and has a maximum of about \(6.5\). It is to be noted that in case of the Dyn-EDM, diffusivity is expressed simply by a scalar \(\alpha_T\) and there is no question of defining an effective diffusivity.
The effectiveness of any SGS heat flux model is best gauged by how well it can represent the geometrical structure of the fluxes at subgrid level. Considering the case of resolved fluxes, it is evident that the magnitude of streamwise fluxes is always greater than the magnitude of wall-normal fluxes, i.e. \(|\overline{u^{'} \theta^{'}}| > |\overline{v^{'} \theta^{'}}|\); see Fig. 10(a) in the paper. A good SGS heat flux model should be able to preserve this structure for the SGS heat fluxes, i.e. \(|h_{1\theta}| > |h_{2\theta}|\).
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| Figure 4: Comparison of time and span averaged SGS heat fluxes in streamwise and wall-normal direction between Dyn-EDM (a), Dyn-GGDH (b), and the proposed model. The values are normalized by friction velocity and friction temperature. |
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| Figure 5: Resolved temperature predictions in fully developed channel flow at \(Re_{\tau} = 395\). |
To highlight the effect of modeling the SGS scalar fluxes, in Fig. 5, the resolved mean temperature prediction of the proposed model is compared with those of Dyn-EDM and Dyn-GGDH. It is observed that the prediction of the wall-normal distribution of the mean temperature by the proposed model is more accurate than the others, especially in the logarithmic layer. It is important to note here that the present model reverts back to Dyn-GGDH if the velocity gradient term is neglected. Hence it is clear that the contribution of this term is very important in improving the resolved mean temperature predictions. Furthermore, this confirms that there is no physical reason for the diffusivity tensor to be symmetric.
Summing up, this paper confirmed the experimental observations of the asymmetric nature of the diffusion tensor and its importance in anisotropic flows. The observations further motivate a tensor diffusivity model whose predictions are seen to be superior than Standard Gradient Diffusion and Generalized Gradient Diffusion Models.
Cite as
1. Blog: Bader, Shujaut H., “Scalar turbulence: physics and modeling." Backscatter, November 25, 2020, https://backscatterblog.blogspot.com/2020/11/scalar-turbulence-physics-and-modeling.html
2. Paper: Bader, Shujaut H., and Paul A., Durbin. "Dynamic subgrid-scale scalar-flux model based on the exact rate of production of turbulent fluxes". Phys. Rev. Fluids 5 (2020): 114609. doi: 10.1103/PhysRevFluids.5.114609.
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