Consistency issues on SPH energy conservation when simulating viscous flows around solid bodies

J. L. Cercos-Pita
M. Antuono
A. Colagrossi
A. Souto-Iglesias

<jl.cercos@upm.es>

S.P. Singh, S. Mittal

Vortex-induced oscillations at low Reynolds numbers: Hysteresis and vortex-shedding modes

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$$\boldsymbol{u}_j \stackrel{\mathrm{¯\backslash\_(ツ)\_/¯}}{\longleftrightarrow} \boldsymbol{u}_{Bj}$$

$\dot{\mathcal{E}}^{SPH}_M + \dot{\mathcal{E}}^{SPH}_C - \mathcal{P}^{SPH}_V = \mathcal{P}_s^V + \mathcal{P}_s^p + \mathcal{P}_s^C$

M. Antuono, S. Marrone, A. Colagrossi, and B. Bouscasse, "Energy balance in the $\delta$-SPH scheme", Computer Methods in Applied Mechanics and Engineering, vol. 289, pp. 209–226, 2015

$$\mathcal{P}^{SPH}_V = - \frac{\mu}{2} \sum_i^{*} \sum_j^{*} V_i \, V_j \, \pi_{ij} \left(\boldsymbol{u}_j - \boldsymbol{u}_i\right) \, \nabla_i W_{ij} \le 0$$

$$\dot{\mathcal{E}}^{SPH}_M + \dot{\mathcal{E}}^{SPH}_C - \mathcal{P}^{SPH}_V = \mathcal{P}_s^V + \mathcal{P}_s^p + \mathcal{P}_s^C$$

$$\mathcal{P}_s^V = + \sum_i^{*} \overline{\sum_j} V_i \, V_j \, \mu \, \color{red}{\pi_{ij}} \, \nabla_i W_{ij} \cdot \boldsymbol{u}_i$$ $$\mathcal{P}_s^p = - \sum_i^{*} \overline{\sum_j} V_i \, V_j \, \left(p_i + \color{red}{p_j}\right) \, \nabla_i W_{ij} \cdot \boldsymbol{u}_i$$ $$\mathcal{P}_s^C = - \sum_i^{*} \overline{\sum_j} V_i \, V_j \, p_i \, \left(\color{red}{\boldsymbol{u}_j} - \boldsymbol{u}_i\right) \cdot \nabla_i W_{ij}$$

$$\mathcal{P}_s^V + \mathcal{P}_s^p + \mathcal{P}_s^C \stackrel{\color{red}{\boldsymbol{?}}}{=} \mathcal{P}^{SPH}_{body/fluid}$$

$$\mathcal{P}^{SPH}_{body/fluid} = \mathcal{P}^{p \, SPH}_{body/fluid} + \mathcal{P}^{V \, SPH}_{body/fluid}$$

$$\mathcal{P}^{p \, SPH}_{body/fluid} = - \sum_i^{*} \overline{\sum_j} V_i \, V_j \, \left(p_i + \color{red}{p_j}\right) \, \nabla_i W_{ij} \cdot \color{green}{\boldsymbol{u}_{Bj}}$$ $$\mathcal{P}^{V \, SPH}_{body/fluid} = \mu \, \sum_i^{*} \overline{\sum_j} V_i \, V_j \, \color{red}{\pi_{ij}} \, \nabla_i W_{ij} \cdot \color{green}{\boldsymbol{u}_{Bj}}$$

$$\begin{array}{lcl} \dot{\mathcal{E}}^{SPH}_M + \dot{\mathcal{E}}^{SPH}_C - \mathcal{P}^{SPH}_V & = & \mathcal{P}^{p \, SPH}_{body/fluid} + \mathcal{P}^{V \, SPH}_{body/fluid} \\ & + & \Delta \mathcal{P}^V + \Delta \mathcal{P}^p + \Delta \mathcal{P}^C \end{array}$$

$$\Delta \mathcal{P}_s^V = \, \sum_i^{*} \overline{\sum_j} V_i \, V_j \, \mu \, \color{red}{\pi_{ij}} \, \left(\boldsymbol{u}_i - \color{green}{\boldsymbol{u}_{Bj}}\right) \cdot \nabla_i W_{ij}$$ $$\Delta \mathcal{P}_s^p = - \sum_i^{*} \overline{\sum_j} V_i \, V_j \, \left(p_i + \color{red}{p_j}\right) \, \left(\boldsymbol{u}_i - \color{green}{\boldsymbol{u}_{Bj}}\right) \cdot \nabla_i W_{ij}$$ $$\Delta \mathcal{P}_s^C = \, \sum_i^{*} \overline{\sum_j} V_i \, V_j \, p_i \, \left(\boldsymbol{u}_i - \color{red}{\boldsymbol{u}_j}\right) \cdot \nabla_i W_{ij}$$

Are these terms, $\Delta \mathcal{P}^V, \Delta \mathcal{P}^p, \Delta \mathcal{P}^C$, pretending to be extra energy dissipation terms?

Moving cylinder inside a viscous fluid

The extra terms can pump energy into the system depending on the mirroring model