Viscous vortex domains method

The viscous vortex domains (VVD) method is a mesh-free method of computational fluid dynamics for directly numerically solving 2D Navier-Stokes equations in Lagrange coordinates ^[1][2] It doesn't implement any turbulence model and free of arbitrary parameters. The main idea of this method is to present vorticity field with discrete regions (domains), which travel with diffusive velocity relatively to fluid and conserve their circulation. The same approach was used in Diffusion Velocity method of Ogami and Akamatsu ,^[3] but VVD uses other discrete formulas

Features

The VVD method deals with viscous incompressible fluid. The viscosity and density of fluid is considered to be constant. Method can be extended for simulation of heat conductive fluid flows (viscous vortex-heat domains method)

The main features are:

• Direct solving Navier-Stokes equations (DNS)
• Calculation of the friction force at the body surfaces
• Proper description of the boundary layers (even turbulent)
• Infinite computation region
• Convenient simulation of deforming boundaries^[4]
• Investigation of the flow-structure interaction,^[5] even in case of zero mass
• Estimated numerical diffusion and stability criteria ^[6]

Governing equations

Scheme of VVD method

The VVD method is based on a theorem,^[1] that circulation in viscous fluid is conserved on contours travelling with speed

$\mathbf{u} = \mathbf{V} + \mathbf{V}_d; ~~~\mathbf{V}_d = -\nu \dfrac{\nabla\mathbf{\Omega}}{|\mathbf{\Omega}|}; ~~~\mathbf{\Omega} = [\nabla \times \mathbf{V}]$,

where V is fluid velocity, V_d — diffusion velocity, ν — kinematic viscosity. This theorem shows resemblance with Kelvin's circulation theorem, but it works for viscid flows.

Basing on this theorem, flow region with non-zero circulation is presented with number of domains (small regions with finite volumes), which move with velocity u and thus their circulation $\gamma$ remains constant. The actual boundaries of every domain are not tracked, but coordinates of the only tracking point in every domain is saved. Array of domains' coordinates and circulations is known either from boundary conditions or from initial conditions. Such a motion results in vorticity evolution and satisfies Navier-Stokes equations.

Discrete formulas

Diffusive vortex-vortex interaction

Diffusive body-vortex interaction

Fluid velocity V in point r can be calculated with help of Biot-savart law

$\mathbf{V}(\mathbf{r}) = \dfrac{1}{2\pi} \sum_i \gamma_i \cdot \left[\mathbf{e}_z \times \dfrac{\mathbf{r}-\mathbf{r}_i}{(\mathbf{r}-\mathbf{r}_i)^2 + \delta^2}\right]$

where i indexes domains in flow, r_i — tracking point of domain and γ_i — his circulation. δ is a so-called "radius of discreteness" — small value that smooths the vortex and helps to get rid of singularity in the domain tracking point.^[6] It equals to mean distance between domains.

Calculation of diffusion velocity is more difficult^[1][4]

$\mathbf{V}_d(\mathbf{r}) = \nu\left( \dfrac{\mathbf{I}_2(\mathbf r)}{I_1(\mathbf r)} + \dfrac{\mathbf{I}_3(\mathbf r)}{2\pi\varepsilon^2 - I_0(\mathbf r)} \right)$

First fraction produces vortex-vortex interaction (i — vortex index).

$\mathbf{I}_2(\mathbf r) = \sum\limits_i \dfrac{\mathbf{r}-\mathbf{r}_i}{\varepsilon \left|\mathbf{r}-\mathbf{r}_i\right|} \cdot \gamma_i \cdot \exp(-\left|\mathbf{r}-\mathbf{r}_i\right|/\varepsilon)$

$I_1(\mathbf r) = {\sum\limits_i \gamma_i \cdot \exp(-\left|\mathbf{r}-\mathbf{r}_i\right|/\varepsilon)}$

And second fraction represents vortex-boundary repulsion. It helps to calculate ∇Ω near body surface and properly describe boundary layer.

$\mathbf{I}_3(\mathbf r) = {\sum\limits_k d\mathbf S_k\cdot \exp(-\left|\mathbf{r}-\mathbf{r}_k\right|/\varepsilon)}$

$I_0(\mathbf r) = {\varepsilon^2\sum\limits_k \dfrac{\left|\mathbf{r}-\mathbf{r}_k\right| /\varepsilon +1}{(\mathbf{r}-\mathbf{r}_k)^2} \cdot((\mathbf{r}-\mathbf{r}_k) \cdot d\mathbf S_k)\cdot \exp(-\left|\mathbf{r}-\mathbf{r}_k\right|/\varepsilon)}$

Here k indexes boundary segments, r_k — its center, dS_k — its normal.

References

1. Dynnikova, G. Ya. (1 November 2004). "The Lagrangian approach to solving the time-dependent Navier-Stokes. equations". Doklady Physics 49 (11): 648–652. Bibcode:2004DokPh..49..648D. doi:10.1134/1.1831530.
2. Dynnikova, G.Ya. (16–21 May 2010). "The Viscous Vortex Domains (VVD) method for non-stationary viscous incompressible flow simulation" (PDF). Proceedings of IV European Conference on Computational Mechanics, Paris, France.
3. Ogami, Yoshifumi; Akamatsu, Teruaki (31 December 1990). "Viscous flow simulation using the discrete vortex model—the diffusion velocity method". Computers & Fluids 19 (3-4): 433–441. doi:10.1016/0045-7930(91)90068-S.
4. Guvernyuk, S. V.; Dynnikova, G. Ya. (31 January 2007). "Modeling the flow past an oscillating airfoil by the method of viscous vortex domains". Fluid Dynamics 42 (1): 1–11. doi:10.1134/S0015462807010012.
5. Andronov, P. R.; Grigorenko, D. A.; Guvernyuk, S. V.; Dynnikova, G. Ya. (1 October 2007). "Numerical simulation of plate autorotation in a viscous fluid flow". Fluid Dynamics 42 (5): 719–731. Bibcode:2007FlDy...42..719A. doi:10.1134/S0015462807050055.
6. Dynnikov, Ya. A.; Dynnikova, G. Ya. (12 October 2011). "Numerical stability and numerical viscosity in certain meshless vortex methods as applied to the Navier-Stokes and heat equations". Computational Mathematics and Mathematical Physics 51 (10): 1792–1804. Bibcode:2011CMMPh..51.1792D. doi:10.1134/S096554251110006X.

External links

• YouTube channel with some VVD computations