Implicit molecular stresses in weakly compressible particle-based discretization methods for fluid flow

Weakly compressible particle-based discretization methods, utilized for the solution of the subsonic Navier-Stokes equation, are gaining increasing popularity in the fluid dynamics community. One of the most popular among these methods is the weakly compressible smoothed particle hydrodynamics. Since the dynamics of a single numerical particle is determined by fluid dynamic transport equations, the particle per definition should represent a homogeneous fluid element. However, it can be easily argued that a single particle behaves only pseudo-Lagrangian as it is affected by volume partition errors and can hardly adapt its shape to the actual fluid flow. Therefore, we will assume that the kernel support provides a better representative of an actual fluid element. By means of nonequilibrium molecular dynamics (NEMD) analysis, we derive isothermal transport equations for a kernel-based fluid element. The main discovery of the NEMD analysis is a molecular stress tensor, which may serve to explain current problems encountered in applications of weakly compressible particle-based discretization methods.

Figure 1

Generic one-dimensional, axisymmetric nozzle flow considered to demonstrate the effect of the molecular stress tensor on a kernel-based fluid element $K$. The element at two different times $t_1$ and $t_2$ along its Lagrangian flow path is depicted as a gray patch.

Figure 2

Kernel-based fluid element $K$ exposed to a quadratically increasing velocity. The corresponding acceleration is highlighted as a blue dashed line.

Figure 3

Snapshots of the statistically steady turbulent flow field at time $t^{*}=4.1\text{s}$. (a) Snapshot of the velocity magnitude. (b) Snapshot of the vorticity field.

Figure 4

Eulerian turbulent kinetic energy spectrum at time $t^{*}=4.1\text{s}$.

Figure 5

Visualization of the molecular stress tensor ${\mathit{\tau }}_{\text{mol}}$ at time $t^{*}=4.1\text{s}$ by eigendecomposition. For (d)–(f) the corresponding vorticity field is superimposed in the background. (a) Snapshot of the first eigenvalue ${\tau }_{\text{I}}$. (b) Snapshot of the second eigenvalue ${\tau }_{\text{II}}$. (c) Snapshot of the first eigenvector ${\mathbf{t}}_{\text{I}}$ with the region considered for a detailed view. (d) Snapshot of the second eigenvector ${\mathbf{t}}_{\text{II}}$ with the region considered for a detailed view. (e) Detail of the first eigenvector ${\mathbf{t}}_{\text{I}}$. (f) Detail of the second eigenvector ${\mathbf{t}}_{\text{II}}$.

Figure 6

Juxtaposition of (a) the detail of the second eigenvector ${\mathbf{t}}_{\text{II}}$ and (b) the detail of the velocity field $\mathbf{v}$ for the statistically steady turbulent flow field at time $t^{*}=4.1\text{s}$. The corresponding vorticity field is superimposed in the background. A comparison reveals that the eigenvectors inside the vortex cores are dominantly oriented perpendicular to the flow field, whereas on high vorticity filaments, in the vortex surrounding, the eigenvectors are dominantly oriented tangentially with respect to the main flow.

Figure 7

Schematic illustration of the modes extracted from the molecular stress tensor ${\mathit{\tau }}_{\text{mol}}$. Shear mode: two parallel velocity vectors in the eigenbasis defined by Eq. (24) are depicted. The molecular energy transfer is indicated by a white arrow. Stretch mode: two orthogonal velocity vectors in the eigenbasis defined by Eq. (24) are depicted. The molecular energy transfer is indicated by a white arrow.