A Deep Dive into TKE Quantification and Its Importance in Verification and Validation of CFD Simulations - Edition I

Turbulent Kinetic Energy (TKE) is one of the most important physical quantities computed computed in Computational Fluid Dynamics (CFD) - specifically when dealing with turbulent flows. It is a measure of the energy contained in the turbulent fluctuations of a fluid flow, and that's the main reason for its significance in characterizing turbulence behavior and its influence on the flow physics.

"It provides insights into the turbulence intensity, scale, and dissipation rate, making it indispensable for CFD specialists working with turbulent flows" (Tennekes and Lumley, 1972).

Why should I care about TKE as a CFD specialist?

When performing a CFD simulation of a turbulent flow, you need to remember that the main goal of what CFD is meant for: you're trying to get as close as possible solving a very complex and unsolved physical phenomenon. But how close is close? Well that's a very good question right here! You need to find the main key variables through which you can quantify your "closeness" or the lack of thereof from the real solution (i.e. the experiment data). The turbulent Kinetic Energy is one of the main physical variable that helps you do exactly that - through the following checks:

• Turbulence intensity: TKE provides a measure of turbulence intensity, which is a key parameter affecting various flow phenomena, such as mixing, heat transfer, and drag. Accurate prediction of TKE is essential to capture these effects correctly in CFD simulations.
• Energy cascade: TKE represents the energy contained in turbulent fluctuations, and its distribution across different scales of motion is central to the energy cascade process in turbulence. Understanding and predicting TKE helps in capturing the energy transfer between different scales and the eventual dissipation of energy at the smallest scales.
• Turbulence modeling: TKE is an essential component of many turbulence models used in CFD, such as the k-epsilon and k-omega models. These models require the calculation of TKE to predict the turbulent flow behavior and closure of the governing equations. Accurate TKE prediction is crucial for the overall accuracy and reliability of the CFD simulation.
• Validation and uncertainty quantification: TKE is often used as a validation metric for comparing CFD simulations against experimental data or higher-fidelity simulations, such as Direct Numerical Simulation (DNS). Comparing TKE predictions with reference data helps assess the performance of turbulence models and quantify uncertainties in the simulations.

Turbulent Kinetic Energy Mathematical Definition

Turbulent Kinetic Energy (TKE) is a scalar quantity that characterizes the energy contained in fluid turbulence. It is typically defined in the literature as:

TKE = 0.5 * (u'_i * u'_i)

where u'_i denotes the fluctuating components of the velocity vector in the i-th direction (Pope, 2000).

Quantification of TKE

Quantifying TKE requires resolving/modeling the turbulent fluctuations in the flow field. This can be achieved through various turbulence modeling approaches, such as Reynolds-Averaged Navier-Stokes (RANS), Large Eddy Simulation (LES), and Direct Numerical Simulation (DNS). Each of these methods comes with its own set of assumptions and limitations, making the choice of an appropriate model is very crucial for accurate TKE prediction (Wilcox, 2006).

1. RANS: RANS models are based on the Reynolds decomposition of flow variables, providing ensemble-averaged equations that include turbulent stresses as additional unknowns. These stresses are related to mean flow gradients through turbulence closure models, such as the k-epsilon or k-omega models, where TKE (k) is explicitly transported.
2. LES: LES involves filtering the flow field to separate large-scale (resolved) and small-scale (unresolved) motions. The subgrid-scale (SGS) stresses, which represent the effect of unresolved scales on the resolved scales, are modeled to estimate TKE (Sagaut, 2006).
3. DNS: DNS resolves all turbulence scales by numerically solving the Navier-Stokes equations without any modeling. This approach provides the most accurate TKE estimation but is computationally expensive and limited to low Reynolds number flows (Moin and Mahesh, 1998).

How Verification and Validation of TKE in CFD Simulations

Verification and validation (V&V) are essential to establish the credibility of CFD simulations. Verification ensures that the mathematical model is solved accurately, while validation ensures that the model represents the physical system of interest (Roache, 1998) - I will talk about this in more detail in a future newsletter.

• TKE verification involves comparing simulation results against a reference solution or a benchmark test case. For RANS and LES, analytical or DNS data can be used as a reference, while for DNS, grid convergence studies and solution verification techniques should be employed (Celik et al., 2008). It's worth noting that DNS is used in most cases - where experimental data is not available - as numerical experiment (for validation).
• Validation of TKE predictions requires comparison with experimental data, such as wind tunnel measurements, Particle Image Velocimetry (PIV), or Laser Doppler Velocimetry (LDV) results. When validating CFD results you have to be careful while using these experimental data; uncertainty quantification plays a vital role in this process, as it helps identify the sources of discrepancies between the turbulence model employed and experimental data (Oberkampf and Roy, 2010).

Resolved Vs Modeled TKE

A common mistake with some CFD users is looking at the wrong thing when comparing TKE values. Please remember: Modeled Turbulent Kinetic Energy is not the same as TKE! In CFD simulations, especially those employing Reynolds-Averaged Navier-Stokes (RANS) models, the turbulent fluctuations are not resolved directly but are modeled using turbulence closure models. In such cases, the TKE is not calculated explicitly from the flow field but is predicted using a modeled equation. The TKE predicted by these models is sometimes referred to as "modeled TKE" (a more illustrative figure below).

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To calculate this modeled TKE in RANS simulations, a two-equation turbulence model, such as the k-epsilon or k-omega model, is commonly used. These models involve solving two transport equations: one for the TKE (k) and another for the dissipation rate (epsilon) or the specific dissipation rate (omega). The k-equation represents the balance between the production, transport, and dissipation of TKE.

In Large Eddy Simulation (LES), the resolved TKE is calculated directly from the resolved flow field, while the unresolved (subgrid-scale) TKE is modeled using subgrid-scale (SGS) models, such as the Smagorinsky model or the dynamic Smagorinsky model. In this case, the total TKE is the sum of resolved TKE and unresolved TKE.

how do I make sure I'm capturing "enough" TKE in my CFD simulation?  

I'm happy because you just asked a REALLY good question! 🤓 Well, to answer that you have to be careful with which approach you are using to solve/ model the flow. Unfortunately there's no quick answer for that question but here I try to give a glimpse on what you should be looking at. When you locally judge the "total" TKE, there are two main things you must investigate:

1- The peak value of the TKE (magnitude)

2- The location of that peak from the wall.

The answer of the CFD simulation to these two parameters (in an ideal world) should NOT depend on the chosen turbulence modeling approach (RANS, LES, and DNS) but only on the specific problem you are addressing. But let's take a non-exhaustive look at how TKE is computed in each of the three approaches:

In RANS simulations, the quality of TKE prediction largely depends on the chosen turbulence model. To ensure that you capture enough TKE, you need to:

• Choose an appropriate turbulence model that best represents the flow physics of your problem. k-epsilon and k-omega models are popular choices, but more advanced models like Reynolds Stress Models (RSM) or Detached Eddy Simulation (DES) can be more suitable for certain complex flows.
• Use a sufficiently fine mesh to properly represent the flow, particularly in the near-wall region. This may require using wall functions or low-Reynolds number models to accurately capture the behavior of TKE near the wall.
• Validate your RANS model against experimental data or higher-fidelity simulations (e.g. DNS) for a similar flow case.

In LES, the resolved TKE is represented by the large-scale, energy-containing eddies, while the unresolved TKE is represented by the subgrid-scale (SGS) stresses. To ensure that you capture enough TKE, you need to:

• Choose an appropriate SGS model, such as the Smagorinsky, dynamic Smagorinsky, or scale-similarity models, which can accurately represent the effect of unresolved TKE on the resolved scales.
• Use a sufficiently fine mesh that resolves the large-scale, energy-containing eddies. Typically, the mesh should resolve about 80% of the total TKE in the flow. The resolution requirements can be estimated using the Kolmogorov and Taylor microscales or the grid-filter width (Pope, 2000).
• Validate your LES results against experimental data or DNS for a similar flow case, ensuring that both the resolved and unresolved TKE components are well-represented.

In DNS, all the scales of turbulence, including TKE, are resolved directly by numerically solving the Navier-Stokes equations. To ensure that you capture enough TKE, you need to:

• Use a sufficiently fine mesh to resolve the smallest scales of turbulence, typically down to the Kolmogorov scale. This often leads to high computational costs, limiting the applicability of DNS to low Reynolds number flows.
• Perform a grid convergence study to ensure that the solution is independent of the mesh resolution. Monitor the energy spectrum to verify that it follows the expected -5/3 law in the inertial subrange and ensure that the energy at the smallest resolved scales is negligible compared to the larger scales.
• Compare your DNS results against experimental data or other well-validated DNS results for a similar flow case.

Working together with the machine - ChatGPT segment

As you can tell already, the AI race is unstoppable at this point. The smartest thing is to do in my opinion is to work side by side with AI and not against it. For example, in the context of this topic, I could imagine how AI can help CFD specialists to better predict the TKE when performing a CFD simulation (apart from the potential for some emerging tools to replace CFD entirely!). Some of the dream features that we can probably think of are:

• AI tools that improves the mesh quality to guarantee a minimum threshold of resolved TKE in LES and prevents the overprediction of the modeled TKE in RANS and avoids Modeled-Stress Depletion (MSD) in DES.
• Deriving new turbulence hybrid RANS/LES models or improving existing ones to have a better treatment for the RANS/LES interface (where TKE usually have its peak value).
• Finding new correlations to model the Sub-Grid Scales (SGS) (unresolved TKE) especially when dealing with relatively coarse or non-uniform grids.

Out of curiosity I asked ChatGPT the following question:

How can AI help us predict turbulent kinetic energy in complex CFD simulations? [the answer literally shocked me!]

Before you see the answer, try to think yourself of some wishes you'd like AI to achieve for you 😊

>> Answer from ChatGPT-4

Artificial intelligence (AI) and machine learning (ML) techniques have gained significant attention for their potential to improve the prediction of turbulent kinetic energy (TKE) and other flow quantities in complex CFD simulations. AI can help in various aspects of turbulence modeling, including model development, model parameterization, and uncertainty quantification. Here are some ways in which AI can contribute to predicting TKE in complex CFD simulations:

1. Data-driven turbulence models: AI can be used to develop data-driven turbulence models based on the statistical relationships between flow quantities learned from high-fidelity simulations or experimental data. These data-driven models can provide more accurate predictions of TKE and other turbulence-related quantities than traditional models, especially in complex flow scenarios where standard models may underperform.
2. Model improvements and parameterization: AI techniques can be used to identify and correct deficiencies in existing turbulence models by learning the optimal model coefficients, functional forms, or model improvements based on high-fidelity data. This can lead to better predictions of TKE in regions where traditional models struggle, such as flow separation, transition, and complex geometries.
3. Hybrid modeling: AI can help in creating hybrid models that combine the strengths of various turbulence modeling approaches. For example, AI can be employed to blend RANS and LES predictions in a way that optimizes the balance between computational cost and accuracy, improving TKE prediction in complex flows. [Here my expectation was a little more precise 😌]
4. Subgrid-scale modeling for LES: AI can be used to develop data-driven subgrid-scale (SGS) models for LES that are capable of adapting to local flow conditions and better capturing the effects of unresolved scales on the resolved TKE. This can result in more accurate and efficient LES simulations for complex flows. [not bad at all 👏 ]
5. Uncertainty quantification: AI techniques can be used to quantify the uncertainties associated with TKE predictions in CFD simulations, enabling a better understanding of the reliability of the predictions and guiding model improvements. AI can also help in estimating uncertainties in model parameters, initial and boundary conditions, and discretization errors.
6. Surrogate modeling and reduced-order modeling: AI can be employed to develop surrogate models or reduced-order models that approximate the behavior of complex CFD simulations with significantly lower computational cost. These surrogate models can be used for real-time TKE prediction, optimization, and control of turbulent flow systems.

In summary, artificial intelligence has the potential to significantly improve TKE prediction in complex CFD simulations by aiding in model development, parameterization, and uncertainty quantification. As more high-fidelity data and advanced AI techniques become available, the role of AI in turbulence modeling and CFD will continue to expand, offering new opportunities for more accurate and efficient simulations of turbulent flows.

As you can see, there's no question we're living in a super interesting time! 😁

AI will soon revolutionise the way we do CFD forever - sooner than you think.

References

• Celik, I. B., Ghia, U., Roache, P. J., Freitas, C. J., Coleman, H., & Raad, P. E. (2008). Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications. Journal of Fluids Engineering, 130(7), 078001.
• Moin, P., & Mahesh, K. (1998). Direct numerical simulation: A tool in turbulence research. Annual Review of Fluid Mechanics, 30(1), 539-578.
• Oberkampf, W. L., & Roy, C. J. (2010). Verification and Validation in Scientific Computing. Cambridge University Press.
• Pope, S. B. (2000). Turbulent Flows. Cambridge University Press.
• Roache, P. J. (1998). Verification and Validation in Computational Science and Engineering. Hermosa Publishers.
• Sagaut, P. (2006). Large Eddy Simulation for Incompressible Flows: An Introduction (3rd ed.). Springer.
• Tennekes, H., & Lumley, J. L. (1972). A First Course in Turbulence. MIT Press.
• Wilcox, D. C. (2006). Turbulence Modeling for CFD (3rd ed.). DCW Industries, Inc.
• Sayed, M. A., Dehbi, A., Niceno, B., & Mikityuk, K. (2021-b). Particle subgrid scale modeling in hubrid rans/les of turbulent channel flow at low-to-moderate reynolds number. J. Powder Technology. https://doi.org/https://doi.org/10.1016/j.powtec.2021.11.057

Disclaimer! This newsletter is not meant to contain ALL the technical details you may need about the topic, however, it should serve CFD enthusiasts as an eye-opener to refresh and compound knowledge in the field.

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