Dynamics of PIV seeding particles in turbulent premixed flames

Abstract

Particle image velocimetry (PIV) estimates the fluid velocity field measuring the displacement of small dispersed particles between two successive instants separated by a small time interval. The accuracy of the measurements depends on the ability of the particles to accommodate their velocity to the fluid fluctuations. When the fluid is subjected to extreme accelerations, the small but finite inertia prevents the particles from following the fluid, originating a substantial relative velocity. This effect is shown to be crucial for applications of PIV to turbulent premixed combustion, particularly in the product region at locations just behind the instantaneous flame front. The issuing inaccuracy may easily spoil the estimate of certain statistical observables which are of crucial importance in the theory of turbulent premixed combustion. By exploiting the direct numerical simulation of a model air/methane flame, a suitable criterion for proper particle seeding is validated and compared with the corresponding experiments with a combined PIV/OH-LIF (laser-induced fluorescence) system. The proposed parameter, the flamelet Stokes number, depends on particle properties and thermochemical conditions of the flame and substantially restricts the particle dimensions required for a reliable estimate of the relevant flow statistics.

Appendix A: Eulerian formulation of particle velocity

In this appendix, we discuss a procedure to derive a continuum particle velocity field in terms of particle velocities in the limit of small τ _p . Starting from Eq. (2), we obtain the fluid velocity along the particle trajectory as the solution of dx _p /dt = v _p ,

$$ u[x_p(t),t]=v_p+\tau_p \dot{v}_p. $$

(14)

This allows to express the time derivative of the fluid velocity experienced by the particle as

$$ \left. {\frac{du} {dt}}\right|_p= {\frac{\partial u} {\partial t}}+ \dot{x}_p \cdot \nabla{u}= \dot{v}_p+\tau_p\ddot{v}_p. $$

(15)

The above expression yields

$$ {\frac{Du} {Dt}}= (u-v_p) \cdot \nabla{u}+ \dot{v}_p+\tau_p\ddot{v}_p, $$

(16)

which, rearranged as an expression for $\dot{v}_p$ to be inserted in (14), provides the fluid velocity as:

$$ \begin{aligned} u&=v_p+\tau_p\left[ {\frac{Du} {Dt}}+ (v_p-u) \cdot\nabla{u}-\tau_p\ddot{v}_p\right]\\ &=v_p+ \tau_p {\frac{Du}{Dt}}+ \tau_p^2\left( \dot{v}_p \cdot \nabla{u}-\ddot{v}_p \right).\end{aligned} $$

(17)

It follows

$$ v_p=u-\tau_p {\frac{Du}{Dt}}+{ O}(\tau_p^2). $$

(18)

Since both fluid velocity and acceleration are Eulerian fields, Eq. (18) defines an equivalent Eulerian particle velocity field by neglecting corrections of order $\tau_p^2,$

$$ v(x,t)=u(x,t)-\tau_p {\frac{Du} {Dt}}(x,t). $$

(19)

In this approximation, the particle velocities are treated as a single valued field. In its use one should be aware of the possible existence of caustics corresponding to a multi-valued particle velocity at the same position (Bec et al. 2005).