Three-dimensional temporally resolved measurements of turbulence–flame interactions using orthogonal-plane cinema-stereoscopic PIV

Abstract

A new orthogonal-plane cinema-stereoscopic particle image velocimetry (OPCS-PIV) diagnostic has been used to measure the dynamics of three-dimensional turbulence–flame interactions. The diagnostic employed two orthogonal PIV planes, with one aligned perpendicular and one aligned parallel to the streamwise flow direction. In the plane normal to the flow, temporally resolved slices of the nine-component velocity gradient tensor were determined using Taylor’s hypothesis. Volumetric reconstruction of the 3D turbulence was performed using these slices. The PIV plane parallel to the streamwise flow direction was then used to measure the evolution of the turbulence; the path and strength of 3D turbulent structures as they interacted with the flame were determined from their image in this second plane. Structures of both vorticity and strain-rate magnitude were extracted from the flow. The geometry of these structures agreed well with predictions from direct numerical simulations. The interaction of turbulent structures with the flame also was observed. In three dimensions, these interactions had complex geometries that could not be reflected in either planar measurements or simple flame–vortex configurations.

Appendix

A general transport equation for the strain rate components in a reacting flow can be derived by taking the spatial derivative the equations for conservation of momentum:

$$ {\frac{\partial} {\partial{x_j}}}\,{\frac{\partial{u_i}}{\partial{t}}} + {\frac{\partial}{\partial{x_j}}}\left(u_k{\frac{\partial{u_i}}{\partial{x_k}}}\right) =\frac{\partial{g}} {\partial{x_j}} -{\frac{\partial} {\partial{x_j}}}\left({\frac{1}{\rho}}\,{\frac{\partial{p}} {\partial{x_i}}}\right) +{\frac{\partial} {\partial{x_j}}}\left(\nu{\frac{\partial^2u_i}{\partial{x_k}\partial{x_k}}}\right)$$

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where g is the gravitational body force, which is assumed constant. Setting $A_{ij} =\partial{u_i}/\partial{x_j}$ , this can be expanded as:

$$ \begin{aligned} {\frac{\partial{A_{ij}}} {\partial{t}}} + u_k{\frac{\partial{A_{ij}}} {\partial{x_k}}} =& -A_{ik}A_{kj}-{\frac{1} {\rho}}\,{\frac{\partial^2{p}} {\partial{x_i}\partial{x_j}}} + {\frac{1} {\rho^2}}\,{\frac{\partial{p}} {\partial{x_i}}}{\frac{\partial{\rho}} {\partial{x_j}}} \\ & + \nu{\frac{\partial^2A_{ij}}{\partial{x_k}\partial{x_k}}} + {\frac{\partial\nu} {\partial{x_j}}}\,{\frac{\partial^2u_i} {\partial{x_k}\partial{x_k}}} \end{aligned} $$

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Recognizing that S _ij  = 1/2(A _ij  + A _ji ), the appropriate forms of Eq. 11 can be summed to yield:

$$ \begin{aligned} {\frac{\partial{S_{ij}}} {\partial{t}}} + u_k{\frac{\partial{S_{ij}}} {\partial{x_k}}} =& -\left({\frac{A_{ik}A_{kj}+A_{jk}A_{ki}} {2}}\right)-{\frac{1} {\rho}}\,{\frac{\partial^2{p}} {\partial{x_i}\partial{x_j}}} \\ &+ \nu{\frac{\partial^2S_{ij}} {\partial{x_k}\partial{x_k}}}+ {\frac{1} {2\rho^2}}\left({\frac{\partial{p}} {\partial{x_i}}}\,{\frac{\partial{\rho}} {\partial{x_j}}}+{\frac{\partial{p}} {\partial{x_j}}}\,{\frac{\partial{\rho}} {\partial{x_i}}}\right)\\ &+ {\frac{1} {2}}\left({\frac{\partial\nu} {\partial{x_j}}}\,{\frac{\partial^2u_i} {\partial{x_k}\partial{x_k}}}+{\frac{\partial\nu} {\partial{x_i}}}\,{\frac{\partial^2u_j} {\partial{x_k}\partial{x_k}}}\right) \end{aligned} $$

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The first term on the right can be written in terms of $\underline {S}$ and ω by recognizing that A _ij  = S _ij  − Ω _ij and equating the terms of $\underline {\Upomega}$ with the various vorticity components. This yields the final transport equation:

$$ \begin{aligned} {\frac{\partial{S_{ij}}} {\partial{t}}} + u_k{\frac{\partial{S_{ij}}} {\partial{x_k}}} =& -S_{ik}S_{kj}-{\frac{1} {4}}\left(\omega_i\omega_j-\delta_{ij}\omega_k\omega_k\right)\\ & -{\frac{1} {\rho}}\,{\frac{\partial^2{p}} {\partial{x_i}\partial{x_j}}}+ \nu{\frac{\partial^2S_{ij}} {\partial{x_k}\partial{x_k}}} + {\frac{1} {2\rho^2}}\left({\frac{\partial{p}} {\partial{x_i}}}\,{\frac{\partial{\rho}} {\partial{x_j}}}+{\frac{\partial{p}} {\partial{x_j}}}\,{\frac{\partial{\rho}} {\partial{x_i}}}\right)\\ &+ {\frac{1} {2}}\left({\frac{\partial\nu} {\partial{x_j}}}\,{\frac{\partial^2u_i} {\partial{x_k}\partial{x_k}}}+{\frac{\partial\nu} {\partial{x_i}}}\,{\frac{\partial^2u_j} {\partial{x_k}\partial{x_k}}}\right) \end{aligned} $$

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