Determination of fluid flow adjacent to a gas/liquid interface using particle tracking velocimetry (PTV) and a high-quality tessellation approach

Abstract

PTV velocity vectors, with high spatial resolution in the flow field, can be used to calculate important flow parameters such as pressure. Determination of such a parameter, which is a function of velocity gradients, entails the velocity vectors to be interconnected by a network of nodes. A tessellated network of the flow field can use the original positions of the PTV particles. This results in a mesh with elements of large aspect ratios, which can produce large numerical errors in the calculation of velocity gradients. In flow scenarios with a moving solid boundary or two-phase interface, a tessellation method is required that can attune to the dynamic topology while capturing the details of the near interface region. Here, we develop a methodology to tessellate two-dimensional (2D) unsteady PTV fields using high quality triangular dynamic meshes, with fine control of the mesh density close to the moving boundaries. To examine the applicability of the method, an experimental setup based on particle shadow velocimetry was conducted, with air and a water/glycerol mixture as the working fluids. Two flow channels of a straight, with a square cross-section of 3 × 3 mm² and wavy, with a throat width of 0.5 mm were utilized to capture the dynamics of relatively large bubbles with quasi-steady and highly deformable moving interfaces. The versatility of the method was successfully demonstrated by the generation of a high-quality mesh, with controlled sizes and determination of the radial and tangential velocity components at the near interface region for different flow conditions.

Graphic abstract

Appendix: Calculation of depth of field and depth of correlation

Depth of field (Butterfield 1978) and depth of correlation (Olsen and Adrian 2000) were determined using:

$$\delta_{{{\text{of}}}} = \frac{n\lambda }{{\left( {{\text{NA}}} \right)^{2} }} + \frac{ne}{{M_{0} \left( {{\text{NA}}} \right)}},$$

(15)

$$\delta_{{{\text{oc}}}} = 2\left[ {\frac{1 - \sqrt \varepsilon }{{\sqrt \varepsilon }}\left( {f_{\# }^{2} D_{{\text{p}}}^{2} + \frac{{5.95\left( {M_{0}^{2} + 1} \right)^{2} \lambda^{2} f_{\# }^{4} }}{{M_{0}^{2} }}} \right)} \right]^{1/2} .$$

(16)

Here, n is the refractive index of the immersion medium between the object and objective lens (here air), λ is the wavelength of light (here green light), NA is the numerical aperture of the objective lens, e is the pixel size of the camera sensor and M₀ is the total magnification factor. $\varepsilon$ is a constant intensity threshold and was taken as ~ 0.01 (Kloosterman et al. 2011). f_# is the f-number of the objective lens, defined by Meinhart and Wereley (2003) as

$$f_{\# } = \frac{1}{2}\left[ {\left( \frac{n}{NA} \right)^{2} - 1} \right]^{1/2} .$$

(17)

Here, n = 1, λ = 532 nm, e = 10 μm and NA = 0.176. For the square capillary experiments, M₀ = 0.32, and for the wavy channel tests, M₀ = 0.70. Introducing these values into Eqs. (15)–(17), the depth of field and depth of correlation were calculated as δ_of ≈ 195 μm and δ_oc ≈ 250 μm for the square channel, and δ_of ≈ 98 μm and δ_oc ≈ 283 μm for the wavy channel.