Resolving Thermal Spectra in LES: Is Your Mesh Ready

The Challenge: Moving Beyond $Pr = 1$

When performing Large Eddy Simulations (LES) around a 3D circular cylinder, achieving a validated solution for $Pr = 1$ is a great milestone. However, increasing the Prandtl number to $Pr = 2$ introduces a tighter constraint on your thermal resolution. While the momentum mesh (velocity field) remains the same, the thermal scales shrink.

If your mesh is just fine enough to resolve 80% of the Turbulent Kinetic Energy (TKE), it might fail to resolve the same percentage of the Thermal Variance (Scalar Energy) because the thermal dissipation occurs at smaller scales when $Pr > 1$.

Why the TKE Ratio Isn't Enough

In Ansys Fluent, the standard method to verify LES quality is the Resolved TKE Ratio:

$$Ratio = \frac{k_{res}}{k_{res} + k_{sgs}}$$

However, for thermal studies, you need to monitor the Resolved Temperature Variance. Because you are changing $Pr$ via thermal conductivity ($k$), you are effectively decreasing the molecular thermal diffusivity ($\alpha$). This extends the thermal spectrum into higher wavenumbers.

How to Monitor Resolved Thermal Energy in Ansys Fluent

Fluent does not provide a "Resolved Thermal Energy Ratio" button, but you can calculate it using User Defined Functions (UDFs) or Custom Field Functions.

1. The SGS Thermal Diffusivity Approach

You correctly noted that SGS Viscosity ($\mu_{sgs}$) and SGS Diffusivity ($\alpha_{sgs}$) are linked by the Turbulent Prandtl Number ($Pr_t$):

$$\alpha_{sgs} = \frac{\mu_{sgs}}{\rho \cdot Pr_t}$$

To monitor resolution, you should track the ratio of Resolved Temperature Variance to the Total Variance. In LES, the SGS contribution to temperature fluctuations is often modeled via the SGS scalar flux.

2. The Power Spectral Density (PSD) Method

The most robust way to verify if your mesh is "fine enough" for $Pr = 2$ is to monitor the Temperature Spectra at specific monitor points (e.g., in the cylinder wake):

1. Place Point Monitors in the wake region.

2. Enable Data Sampling for Time Statistics.

3. Export the temperature time-history and perform a Fast Fourier Transform (FFT).

4. The Verification: Check if the spectrum follows the $-5/3$ slope (Obukhov-Corrsin theory). If the spectrum "piles up" or drops off prematurely at $Pr = 2$ compared to $Pr = 1$, your mesh is filtering out essential thermal scales.

3. Practical Verification: The "Grid Estimator" for Thermal LES

If you cannot easily extract the SGS scalar variance, monitor the Ratio of Molecular Conductivity to SGS Conductivity:

• Thermal Resolution Index: $\frac{k_{molecular}}{k_{sgs}}$

• If this ratio drops significantly when you switch to $Pr = 2$, it means the Sub-Grid Model is doing "too much work," indicating that your mesh is too coarse for the new thermal physics.

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Verify LES thermal resolution in Ansys Fluent. Learn how to monitor thermal energy spectra and mesh adequacy for Prandtl numbers $Pr > 1$.

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• Ansys Fluent LES Tutorial

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• Batchelor Scale vs Kolmogorov Scale CFD