2.1.1 Diagnostic methods

Experimental studies include probe measurement techniques as well as laser and optical diagnostic methods.

Steady wall pressure data acquisition is performed using National Instruments PXIe-1075 Chassis, USB 6211 and OMEGA PX-309-030AV pressure transducers (accuracy: ±0:25% BSL, max) that are flush mounted along the axis of the test section floor. The precision uncertainty is calculated using the repeatability of the results for three trial runs. Approximately 0.007 bar standard deviation $\;\sigma $ , is observed for the measurements. This is converted to a 95% confidence level uncertainty as ±2 $\sigma $ . Using the root of the sum of the squares (RSS), a total uncertainty of about ±0.017 is obtained.

Instantaneous Schlieren images are taken at 8000 fps (250µs exposure) using Phantom UHS (1280×800 pixels) with a Nikon 105mm f/2.8 macro lens and a LED light source, to visualise various quasi-steady shocks and flow features which helps in better understanding of the flow dynamics.

Particle image velocimetry (PIV) data acquisition and post-processing are done using LaVision system. A double-pulsed 532nm Nd: YAG laser from Quantel laser (Evergreen series) with a maximum pulse energy of 256 ${\rm{mJ}}$ at 10 Hz is used to create the laser sheet for the visualisation. Sheet optics and laser guiding arm from LaVision is used to obtain required laser sheet thickness of 1mm and orientation respectively. For the present test conditions, the double pulses used are of energy 35 ${\rm{mJ}}$ and are separated with a dt of 0.12µs. This can provide the minimum acceptable particle shift of 5 pixels within the interrogation window for the velocity vector calculation. Images are captured using Imager Pro X camera of 14 bit with a Nikon 105mm f/2.8 macro lens mounted on it. To achieve a narrow depth of field, the aperture of the lens is kept at maximum (f/2.8). Both laser and camera are synchronised using a programmable timing unit (PTU) in double frame mode. Data acquisition and processing are controlled using DaVis software.

There are various sources of uncertainties that can be expected during PIV data acquisition and processing from the equipment used, particle dynamics, sampling, and image analysis^{(Reference Lazar, Deblauw, Glumac, Dutton and Elliott18)}. Among them, the ability of the seeder particles to track the main flow is the most crucial one. As a result, seeding particle selection plays a vital role in PIV measurement. Previous studies on PIV in supersonic flows showed that the particles with diameters less than 1 $\;{\rm{\mu m}}$ can give agreeable results^{(Reference Melling19)}. In the present study, the validation of the PIV data is performed using the CFD simulations, which is already validated through experimental wall pressure data. The particle diameter distribution for the olive oil seeder used for the present study is obtained using Phase Doppler Particle Analyser (PDPA). It is found that about 85% of the particles are around 0.7 ${\rm{\mu m}}$ in diameter.

The present study investigates the instantaneous velocity distribution over the vertical mid-plane along the axis within the pylon-cavity wake region where the flowfield becomes subsonic. Figure 4 shows the region of interest (ROI) and the unusable regions for the vector calculation. The unusable region 1 consists of highly turbulent flow due to the pylon shock interaction. The frequency of fluctuations due to the shock presence cannot be captured by the current particle diameter. The region 2 is unusable due to lack of sufficient particles visible for the vector calculation. Surface wall reflections and shear layer oscillations make the particles in the region 2 difficult to detect. Current ROI for the vector calculation consists of 17673 vectors among which 265 (1.5%) vectors are rejected during processing. PIV data extracted along the two vertical dashed lines 1 and 2 shown in the figure are used for the validation of numerical data.

Figure 4.

Average X-velocity distribution within the ROI overlapped with the experimental image frame. The regions marked red are unusable for vector estimation with the current experimental settings. The axial velocity profiles along the dashed vertical lines 1 and 2 are used for the CFD validation purposes.

The statistical average of instantaneous X-velocity for each pair of images shows that a minimum of 200 image pairs is required to capture complete fluctuations in the flowfield (see Fig. 5).

Figure 5.

Statistical average of Vx for a different number of image pairs.

2.2.1 Numerical scheme

Steady state computational data is obtained using compressed real gas Reynolds-averaged Navier-Stokes (RANS) equations, solved by coupled, implicit, second-order upwind solver. The ability of the RANS model to resolve the flow field for similar cases has been proven already in previous studies^{(Reference Huang, Li and Yan20,Reference Huang21)} . CFD++ provides the Harten-Laxvan Leer-Contact (HLLC) scheme to solve the cell fluxes. For simulating the multi-species problem, the following transport equations are solved in three-dimensional Cartesian coordinates^{(Reference Kuo22)}:

1. a) Mass conservation

   (1) \begin{equation}\frac{{\partial \rho }}{{\partial t}} + \frac{{\partial\!\left( {\rho {u_i}} \right)}}{{\partial {x_i}}} = 0\end{equation}

2. b) Momentum conservation

   (2) \begin{equation}\frac{{\partial\!\left( {\rho {u_i}} \right)}}{{\partial t}} + \frac{{\partial\!\left( {\rho {u_i}{u_j}} \right)}}{{\partial {x_i}}} = \frac{{\partial p}}{{\partial {x_i}}} + \frac{{\partial {\tau _{ij}}}}{{\partial {x_i}}}\end{equation}
   where,
   $${\tau _{ij}} = \mu \left( {\frac{{\partial {u_i}}}{{\partial {x_j}}} + \frac{{\partial {u_j}}}{{\partial {x_i}}} - \frac{2}{3}{\delta _{ij}}\frac{{\partial {u_k}}}{{\partial {x_k}}}} \right)$$

3. c) Energy conservation

   (3) \begin{equation}\frac{{\partial\!\left( {\rho e} \right)}}{{\partial t}} + \frac{{\partial\!\left( {\rho {u_i}e} \right)}}{{\partial {x_i}}} = \frac{{\partial p}}{{\partial t}} + \frac{\partial }{{\partial {x_i}}}\left[ {\mu \left( {\frac{1}{{{S_{{c_k}}}}} - \frac{1}{{{P_r}}}} \right)\mathop \sum \nolimits_{k = 1}^N {e_k}\frac{{\partial {Y_k}}}{{\partial {x_i}}} + \frac{\mu }{{{P_r}}}\frac{{\partial e}}{{\partial {x_i}}}} \right]\end{equation}
   where ${e_k}$ is the specific enthalpy of the species, ${Y_k}$ is the mass fraction of the species, Schmidt number ${S_{{c_k}}}$ = $\mu $ / $\rho {D_k}$ = 0.7, Prandtl number ${P_r}$ = $\mu {C_p}$ / $\lambda $ = 0.707^{(Reference Hyslop23)}.

4. d) Species transport

   (4) \begin{equation}\frac{{\partial\!\left( {\rho {Y_k}} \right)}}{{\partial t}} + \frac{{\partial\!\left( {\rho {u_i}{Y_k}} \right)}}{{\partial {x_i}}} = \frac{\partial }{{\partial {x_i}}}\left( {\rho D\frac{{\partial {Y_k}}}{{\partial {x_i}}}} \right)\end{equation}

   Using Dalton’s law, the working fluids for the current cold flow study are treated as real gases with no reactions modelled. This is represented as,

   (5) \begin{equation}P = \rho RT\mathop \sum \nolimits_{k = 1}^N \left( {\frac{{{Y_k}}}{{M{W_k}}}} \right)\end{equation}

The two-equation Menter’s Shear Stress Transport (SST) $\kappa\omega$ model has been selected for turbulence modelling over $\kappa\omega$ or $\kappa\varepsilon$ models due to the improved performance in adverse pressure gradient flow^{(Reference Freeborn14,Reference Menter, Kuntz and Langtry24–Reference Huang, Liu, Li, Xia, Liu and Wang27)} . It solves the transport equations for the turbulent kinetic energy $\kappa$ and for the turbulence inverse time-scale, $\omega$ .

Turbulence kinetic energy transport equation is,

(6) \begin{align}\frac{\partial\rho \kappa}{\partial t} + \frac{\partial}{\partial x_{i}} (u_{i}\rho \kappa) = \tilde{P}_{\kappa} - \beta^{*} \rho \kappa\omega + \frac{\partial}{\partial x_{i}} \left[(\mu + \sigma_{\kappa}\mu_{t})\frac{\partial\kappa}{\partial x_{i}}\right] \end{align}

and turbulence inverse time-scale transport equation is,

(7) \begin{align} \frac{{\partial \rho \omega }}{{\partial t}} + \frac{\partial }{{\partial {x_i}}}\left( {{u_i}\rho \omega } \right) & = \gamma \rho {S^2} - \beta \rho {\omega ^2} + \frac{\partial }{{\partial {x_i}}}\left[ {\left( {{\rm{\mu }} + {\sigma _\omega }{{\rm{\mu }}_t}} \right)\frac{{\partial \omega }}{{\partial {x_i}}}} \right]\nonumber\\[8pt] & + {\rm{\;}}2\left( {1 - {F_1}} \right)\frac{{\rho {\sigma _{\omega 2}}}}{\omega }\frac{{\partial \kappa }}{{\partial {x_i}}}\frac{{\partial \omega }}{{\partial {x_i}}}\end{align}

where, closure coefficients are ${\sigma _{\kappa 1}}$ = 0.85, ${\sigma _{\omega 1}}$ = 0.5, β = 0.075 for kω and ${\sigma _{\kappa 2}}$ = 1.0, ${\sigma _{\omega 2}}$ = 0.856, β = 0.0828 for k $\varepsilon$ . ${\mu _t}\;$ is the turbulent eddy viscosity, F1 is the blending function, S is the invariant measure of the strain rate and is the production limiter used in the SST model.

Figure 7.

Boundary conditions for the computational domain^{(Reference Oamjee and Sadanandan28)}.

Figure 7 shows the boundary conditions applied for the computational domain. Tables. 2 and 3 give the details regarding the boundary conditions and grid resolution, respectively^{(Reference Oamjee and Sadanandan28)}.

Figure 8 shows the grid independence study on local and global variables over five different meshes with varying grid density and wall spacing. Figure 9 shows the grid generated for the computational domain. It is found that a total of 4.8 million cells with a wall spacing of 0.001mm with a ${Y^ + }$ value of 1, can give an acceptable result. The computational domain is initially discretised with 0.5 million cells and refined to finer grid densities until convergence with an error below 1% in local and global variables is observed. CFL number is ramped from 0.1 to 5 for the initial 100 steps. A convergence of the residual that drops by six orders of magnitude can be obtained with less than 20000 iterations (see Fig. 10).

Table 2

Boundary conditions and input variables

Table 3

Grid resolution

Figure 8.

(a) Local pressure variation along the centre line of the test section (b) mixing efficiency (c) mean X-velocity d) H₂ mass fraction variations for various grid refinements for the baseline geometry case^{(Reference Oamjee and Sadanandan28)}.

Figure 9.

Computational domain grid (top) and the enlarged view of grid in the pylon vicinity shown within the red boundary (bottom)^{(Reference Oamjee and Sadanandan28)}.

Figure 10.

Residual plot for baseline configuration.

2.2.2 Performance parameters definition

To obtain the data for the performance analysis, the computational domain is divided into 800 sections along the X-axis. Data is extracted along the length of the test section by taking averages over the respective cross sections as defined in^{(Reference Huang, Li and Yan20)}. The parameters used for the performance analysis are total pressure loss, mixing efficiency, and flammable plume extent along with fuel jet penetration h/d, where h is the maximum height at which the fuel jet core travels in each cross section and d is the fuel injector diameter.

The present study is considered as a globally fuel-lean system due to a single injection in a large duct configuration. Mixing efficiency is defined as the fraction of the least available reactant that would react if the fuel-air mixture was brought to chemical equilibrium without additional local or global mixing^{(Reference Fuller, Wu, Nejad and Schetz29)}. Fuel dispersion magnitude at each cross-section defines the mixing effectiveness ${\eta _m}$ given by Equation (8).

(8) \begin{equation}{\eta _m} = \frac{{{{\dot m}_{fuel,mixed}}}}{{{{\dot m}_{fuel,total}}}} = \frac{{\smallint {\alpha _R}\rho UdA}}{{\smallint \alpha \rho UdA}}\end{equation}

(9) \begin{equation}\alpha = \frac{f}{{f + 1}} = \frac{{\phi {f_s}}}{{\phi {f_s} + 1}}{\rm{\;}}\end{equation}

(10) \begin{equation}{\alpha _R} = \left\{ {\begin{array}{*{20}{c@{\quad}c}}\alpha , & \alpha \le {\alpha _s}\\[5pt] \frac{\alpha _s \left( {1 - \alpha } \right)}{1 - \alpha _s}, & \alpha \gt {\alpha _s}\end{array}} \right\}\end{equation}

Mixing effectiveness alone cannot give the complete picture regarding the mixing efficiency rather it provides only the information regarding how well the fuel gets dispersed. To understand mixing efficiency even further, an additional parameter known as fuel plume area fraction $F{P_f}$ is taken into account. It is defined as the fuel plume area having fuel mass fractions that fall within the flammability limit, normalised by the total fuel plume area for each cross section.

The presence of pylon-cavity and transverse injection results in total pressure loss. It is one of the key parameters which decides the efficiency of a wall-based flameholder configuration when compared to the instream injection models. Equation (11) gives the total pressure loss $\Lambda$ , which is calculated by the mass averaged integral of total pressure at each cross section. ${P_{Tunnel}}$ is the total pressure at the inlet.

(11) \begin{equation}\Lambda = 1 - \frac{\overline{P_T}}{P_{Tunnel}}\end{equation}

Where,

(12) \begin{equation}\overline{P}_{T} = \frac{\smallint{P_T}\rho UdA}{\smallint \rho UdA}\end{equation}