Modeling the pressure-Hessian tensor using deep neural networks

The understanding of the dynamics of the velocity gradients in turbulent flows is critical to understanding various nonlinear turbulent processes. Several simplified dynamical equations have been proposed earlier that model the Lagrangian velocity gradient evolution equation. A robust model for the velocity gradient evolution equation can ultimately lead to the closure of the system of equations in the Lagrangian probability distribution function method. The pressure Hessian and the viscous Laplacian are the two important processes that govern the Lagrangian evolution of the velocity gradients. These processes are nonlocal in nature and unclosed from a mathematical point of view. The recent fluid deformation closure model (RFDM) has been shown to retrieve excellent statistics of the viscous process. However, the pressure Hessian modeled by the RFDM has various physical limitations. In this work, we first demonstrate such limitations of the RFDM. Subsequently, we employ a tensor basis neural network (TBNN) to model the pressure Hessian using the information about the velocity gradient tensor itself. Our neural network is trained on high-resolution data obtained from direct numerical simulation (DNS) of isotropic turbulence at the Reynolds number of 433. The predictions made by the TBNN are evaluated against several other DNS datasets. Evaluation is made in terms of the alignment statistics of the pressure-Hessian eigenvectors with the strain-rate eigenvectors. Our analysis of the predicted solution leads to the finding of ten unique coefficients of the tensor basis of the strain-rate and the rotation-rate tensors, the linear combination of which is used to accurately capture key alignment statistics of the pressure-Hessian tensor.

Figure 1

Alignment of ${\mathcal{P}}^{\mathcal{RFD}}$ eigenvectors (${\mathit{p}}_{\mathit{i}}$) with $\mathit{S}$ eigenvectors (${\mathit{s}}_{\mathit{i}}$). Here, $i$ (=$\alpha$, $\beta$, or $\gamma$) denotes the three eigenvectors corresponding to the three eigenvalues $\alpha \ge \beta \ge \gamma$.

Figure 2

Schematic of the feedforward densely connected neural network. $W^{\left(i\right)}$ and $b^{\left(i\right)}$ are the learnable parameters called weight matrix and the bias vector of the $i\mathrm{th}$ layer, respectively. $\varphi$ is a nonlinear activation function.

Figure 3

Schematic of the TBNN network. $W^{\left(i\right)}$ and $b^{\left(i\right)}$ are the weight matrix and the bias vector of the $i\mathrm{th}$ layer. Rectified linear unit (RELU) nonlinear activation function is used. Both $W^{\left(i\right)}$ and $b^{\left(i\right)}$ are the learnable parameters of the neural network, which are optimized using the RMSprop optimizer [54].

Figure 4

Decay of cost function during training for TBNN. Minibatch size = 256; 1 epoch = 924 iterations of the optimizer.

Figure 5

Alignment of ${\mathcal{P}}^{\mathcal{TBNN}}$ eigenvectors (${\mathit{p}}_{\mathit{i}}$) with $\mathit{S}$ eigenvectors (${\mathit{s}}_{\mathit{i}}$). Here, $i$ (=$\alpha$, $\beta$, or $\gamma$) denotes the three eigenvectors corresponding to the three eigenvalues $\alpha >\beta >\gamma$ (dataset A, Table 1).

Figure 6

Decay of cost function during training for the modified TBNN. Minibatch size = 256; 1 epoch = 924 iterations of the optimizer.

Figure 7

Alignment of ${\mathcal{P}}^{\mathcal{TBNN}}$ eigenvectors (${\mathit{p}}_{\mathit{i}}$) obtained from modified TBNN with $\mathit{S}$ eigenvectors (${\mathit{s}}_{\mathit{i}}$). Here, $i$ (=$\alpha$, $\beta$, or $\gamma$) denotes the three eigenvectors corresponding to the three eigenvalues $\alpha \ge \beta \ge \gamma$ (dataset A, Table 1).

Figure 8

Alignment of ${\mathcal{P}}^{\mathcal{TBNN}}$ eigenvectors (${\mathit{p}}_{\mathit{i}}$) obtained from modified TBNN with $\mathit{S}$ eigenvectors (${\mathit{s}}_{\mathit{i}}$). Here, $i$ (=$\alpha$, $\beta$, or $\gamma$) denotes the three eigenvectors corresponding to the three eigenvalues $\alpha >\beta >\gamma$. (Decaying isotropic turbulence testing dataset B, Table 1, initial Reynolds number 250, and turbulent Mach number of 0.4.) Note that the field is extracted at $t=2.7t_0$, where $t_0$ is the eddy turnover time. At this instant, the Reynolds number of the flow has decayed to the value of 67.4 and the turbulent Mach number is 0.3.

Figure 9

Alignment of ${\mathcal{P}}^{\mathcal{TBNN}}$ eigenvectors (${\mathit{p}}_{\mathit{i}}$) obtained from modified TBNN with $\mathit{S}$ eigenvectors (${\mathit{s}}_{\mathit{i}}$). Here, $i$ (=$\alpha$, $\beta$, or $\gamma$) denotes the three eigenvectors corresponding to the three eigenvalues $\alpha \ge \beta \ge \gamma$ (dataset C, Table 1).

Figure 10

Scatter plot of the ten coefficients predicted by the modified TBNN.

Figure 11

Alignment of ${\mathcal{P}}^{\mathcal{TBNN}}$ eigenvectors (${\mathit{p}}_{\mathit{i}}$) obtained using mean value of coefficients (using the methodology shown in the Appendix) with $\mathit{S}$ eigenvectors (${\mathit{s}}_{\mathit{i}}$) for testing dataset B (Table 1). Here, $i$ (=$\alpha$, $\beta$, or $\gamma$) denotes the three eigenvectors corresponding to the three eigenvalues $\alpha \ge \beta \ge \gamma$. More details in the caption of Fig. 8.

Figure 12

PDF of eigenvalues of pressure-Hessian tensor (a) $\alpha \text{eigenvalue}$, (b) $\beta \text{eigenvalue}$, and (c) $\gamma \text{eigenvalue}$. The eigenvalues are normalized by the respective standard deviation ($\sigma$).