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ABSTRACT
High swirl number flows with vortex breakdown exhibit a number of unsteady flow features, including shear-induced coherent structures and precessing recirculation zones. This article analyzes how precession influences the relationship between the reacting flows’ time-averaged and instantaneous features. Its objective is to provide interpretive insights into high fidelity computations or experimental results. It shows how precession influences three significant topological features in the time-averaged flow: (1) centerline axial jets, (2) centerline stagnation points, and (3) symmetry of the flow about the centerline. It also discusses the extent to which these first two features provide insight into the actual instantaneous flow topology. A particularly significant result of this work is in regards to aerodynamically stabilized flames. Stabilization of such flames requires a low velocity interior stagnation point(s), presumably in the vortex breakdown region. We show how precession causes systematic differences between the location of the time-averaged position of the instantaneous stagnation point, and the stagnation point of the time-averaged velocity. An important implication of this point is that a perfect prediction of the time-averaged flow field could still lead to a completely erroneous time-averaged flame position prediction. Finally, we discuss the influence of precession and coherent motion on convergence of estimated averaged quantities.
Funding
This work was partially supported by United Technologies Corporation.
Nomenclature
| Apert,z | = | axial perturbation amplitude in simple precession model |
| D | = | characteristic diameter of nozzle or centerbody (m) |
| f | = | frequency (Hz) |
| fpert,z | = | axial perturbation function |
| L | = | characteristic length (m) |
| n | = | sample size |
| n1/2,e | = | half number of elements in ensemble for autocorrelation calculation |
| N | = | population size |
| R0 | = | cross-section parabolic profile characteristic length (m) |
| Rcenter | = | cross-section center radial coordinate (m) |
| Rcs | = | cross-section radius (m) |
| Re | = | Reynolds number |
| Rp | = | radius of precession (m) |
| r(τ) | = | autocorrelation function |
| SG | = | geometric swirl number |
| Sm | = | momentum based swirl number |
| St | = | Strouhal number, fL/u |
| t | = | time axis (s) |
| T1/2,e | = | half time duration of ensemble for ensemble-averaged autocorrelation calculation |
| u | = | characteristic flow velocity (m/s) |
| Ub | = | backflow velocity (m/s) |
| Uf | = | forward velocity (m/s) |
| Uz | = | axial flow velocity component (m/s) |
| Uz,ref | = | reference axial velocity (s) used to binarize velocity field for RFP maps |
| Uz,o | = | average axial nozzle exit velocity (m/s) |
| = | in-plane average of instantaneous axial velocity leading stagnation point | |
| Zb,ave | = | in-plane average axial velocity leading stagnation point |
| = | global average of instantaneous axial velocity leading stagnation point | |
| Zb,g,ave | = | global average axial velocity leading stagnation point |
| Zb,g(t) | = | global instantaneous axial velocity leading stagnation point |
| Zb(t) | = | in-plane instantaneous axial velocity leading stagnation point |
| Greek symbols | = | |
| β | = | Uf/Ub |
| θ | = | precession angle (rad) |
| θt | = | axis of rotation tilt angle (deg) |
| ρ | = | Rp/R0 |
| σ | = | population standard deviation |
| σ2 | = | population variance |
| τ | = | time delay (s) used to define autocorrelation function |
| τc | = | integral time scale of turbulence, obtained by integrating the autocorrelation function r(τ) over all time delays τ (s) |
| ϕ0,p | = | axial disturbance initial phase (rad) |
| ϕ0.z | = | precession angular frequency (rad/s) |
| ωp | = | precession initial phase (rad) |
| ωz | = | axial perturbation angular frequency (rad/s) |