Large eddy simulation and Reynolds-averaged Navier-Stokes calculations of supersonic impinging jets at varying nozzle-to-wall distances and impinging angles

Highlights

• Supersonic impinging jets are simulated using RANS and LES.

• Impinging angle and stand-off distances are varied and validated with experiments.

• Both methodologies are able to capture the shock structures of the jet.

• Pressure distribution on impinging plate for RANS and LES is similar for Z_n/D = 2.5.

• Transient chaotic nature and instabilities of the jet in the LES are highlighted.

Abstract

This paper utilises two different computational methods to investigate the characteristics of a supersonic impinging jet at non-dimensionalised nozzle-to-wall distances ($Z_n/D$) of 1.5 and 2.5 with the impinging angles from 0° to 45°. The static Smagorinsky subgrid-scale model was chosen for the LES and the two equation $k''–''∊$ turbulence model for the RANS. Computational parameters applied in the simulations emulated the experimental setup conducted by Risborg (2008). From the results obtained, both methodologies were able to predict the location of the first shock cell fairly accurately when compared to the steady-state shadowgraph images of Risborg (2008). However, the intensities of the shocks were significantly different between the two numerical methods, with the RANS underestimating the value of the density gradients at the shocks. The pressure distribution near the impinging plate have been investigated and found to differ between the RANS and the LES for small impinging angles (0° and 10°) when $Z_n/D=1.5$. In addition, the RANS data was not able to capture the recirculation zone for $Z_n/D=1.5$ and $0°$. The instantaneous velocity fluctuations and temperature contours of the LES were also plotted to visualise the shear layer instability and also the chaotic nature of the supersonic jet. For $Z_n/D=2.5$ and $0°$, the jet experiences high velocity fluctuations as the configuration causes the axially flapping instability. Overall, there are discrepancies between the RANS and LES but both are able to capture the key averaged flow features of the supersonic impinging jets.

Introduction

Supersonic impinging jets have various applications, ranging from short take-off and vertical landing vehicles to the manufacturing processes such as the cold gas dynamic spray process. Throughout the years, numerous experiments have been carried out to investigate the flow of a supersonic jets (Donaldson and Snedeker, 1971, Donaldson et al., 1971, Carling and Hunt, 1974, Lamont and Hunt, 1980, Alvi and Iyer, 1999, Krothapalli et al., 1999, Henderson, 2002, Henderson et al., 2005, Mitchell et al., 2012). Experimental techniques such as hot-wire anemometry (HWA), Rayleigh scattering, Schlieren photography and shadowgraphy and particle image velocimetry (PIV) has been used to analyse the flow structures of the supersonic impinging jet. The equipment used is very specialised and the experimental setup requires a vast amount of resources, both in terms of time and funding.

An alternative to analyse the structures of the supersonic impinging jet is to simulate the flow by means of computational fluid dynamics (CFD). CFD can be more cost effective than conducting experiments and with the advancement of computational power, it has become a more popular option. In addition, CFD provides information about the entire flow field which is extremely difficult to obtain experimentally, especially for supersonic flows, as summarised by Chin et al. (2013). The flow field of the fluid can be simulated from the Lagrangian frame of reference or the Eularian reference frame. In the Lagrangian reference frame, the particles of the fluid are simulated (Lattice Boltzman Method, Direct Simulation Monte Carlo) while in the Eularian reference frame, the Navier–Stokes equations, which governs the motion of fluid, is solved. There are various different methodologies developed to solve the Navier–Stokes equations (DNS, LES, MILES, DES and RANS to name a few) with each method having a different level of accuracy and computational requirement.

The Direct Numerical Simulation (DNS) method directly solves the governing equations for fluid flow and requires the highest amount of computational resources compared to other methodologies in the Eularian reference frame. In DNS, all the turbulent length scales are resolved and therefore requires a very fine computational grid. On the other hand, the large eddy simulation (LES) method only resolves the largest eddies in the flow and is therefore computationally cheaper than DNS due to the less stringent grid requirements. Results from the LES method which are three dimensional and transient have been used to better understand the aero-acoustic phenomena of supersonic impinging jets.

Nelson et al. (2012) used a high fidelity LES-based method to replicate the experimental setup of Krothapalli et al. (1999) which used a converging–diverging nozzle to accelerate the flow to supersonic velocity. Results from their simulation was able to capture the helical structure in the jet which was also observed by Krothapalli et al. (1999). Their simulation was also able to accurately predict the frequency and magnitude of the peak sound pressure level (SPL) of the experiments. However, there were discrepancies in the secondary peaks which could be due to unconverged statistics or limitations of the numerical schemes used.

Dauptain et al. (2010) used unstructured tetrahedra grids and the static Smagorinsky sub-grid scale model when simulating the supersonic impinging jet. Unlike Nelson et al. (2012) which only analysed the pressure distribution on the impinging plate, Dauptain et al. (2010) probed the pressure at various locations along the supersonic impinging jet. He found that the largest pressure fluctuations occur at the surface of the impinging plate while halfway between the lip of the exit nozzle and the impinging plate the SPL remained constant for a range of frequencies from 0 to 25,000 Hz. At the lip of the exit nozzle, the SPL decreases as the frequency increases. In addition, the velocity vector field on the impinging plate was plotted and showed similar resemblance to the grease-streak photograph of Donaldson and Snedeker (1971).

Erwin et al. (2012) also analysed the SPL of cold and hot supersonic impinging jet using LES and FW-H (Ffowes Williams and Hawkings) transformation method to predict the far-field noise. They also conducted experiments to validate their simulations and found that the broadband levels of noise prediction are in good agreement with the experiments.

In addition, complex supersonic impinging jet flows with dual stream jet has also been simulated by Brès et al. (2011). Simulations were carried out at different Nozzle Pressure Ratio (NPR) and the SPL showed a convincing collapse with the experimental findings of Gustavsson et al. (2010) with the frequency and magnitude of maximum SPL almost matching.

In all of the LES studies mentioned above, the flow of the jet is perpendicular to impinging plate. Nonomura et al. (2011) on the other hand, conducted LES at an impinging angle of 45° at different stand-off distances for a cold and hot supersonic jet. They observed that there are three distinct acoustic waves which were generated by the shear layer of the supersonic jet, the impinging tones and the shear layer of the supersonic flow downstream of the jet impingement.

The Reynolds-averaged Navier–Stokes (RANS) method has also been used to simulate supersonic impinging jet flows. This method requires the least amount of computational resources as it only resolves the largest turbulent eddies in the flow. Due to its low computational requirement, it has been the preferred choice for engineering applications. The RANS method has been widely used to simulate supersonic impinging jets with the focus on the cold gas dynamic spray process. The cold gas dynamic process is a material deposition process where powder particles of sizes ranging from 1 to 50 $\mathrm{\mu }\text{m}$ in diameter are accelerated to high speeds (300–1200 ${\text{ms}}^{-1}$) by a supersonic jet (Papyrin et al., 2006). High impact velocity of particles is favourable as it results in better quality coating. There are various operating parameters in the cold spray process (particle size and density, carrier gas, nozzle length and shape, nozzle exit to substrate distance, NPR and substrate angle) which affect the impact velocity of the particles and it has been the focus of researchers to optimise these operating parameters.

Jen et al. (2005) simulated an uncoupled two-phase flow using the RNG $k''–''∊$ turbulence model. In this steady-state RANS simulation, particles of sizes ranging from 100 nm to 50 $\mathrm{\mu }\text{m}$ were seeded into the flow and two different carrier gases, Nitrogen and Helium were used. They found that the extended straight section of the supersonic nozzle might not be needed for sub-microsize particles which contradicts the findings of Dykhuizen and Smith (1998) which used the one-dimensional analytical model for compressible flow and particle dynamics.

Samareh et al. (2009) used a two way coupling method to analyse the interaction between the carrier gas and the particles. A two-dimensional axisymmetric computational domain was used and the flow field was simulated using the non-linear Reynold’s stress model (RSM) turbulence model. They found that increasing mass flow rate of the particles decreases the velocity of the carrier gas and the axial velocity of the particles, even at modest powder feed rates of less than 3 g/s.

Karimi et al. (2006) conducted two phase flow using steady-state RANS with $k''–''∊$ turbulence model with an oval-shaped nozzle. Results from the simulation showed that the particle velocity at the nozzle exit plane is lower than the values obtained experimentally but showed similar trends. Yin et al. (2010) also used the same turbulence model to simulate the flow of supersonic impinging jets and found that in general, the impact velocity of particles, which is perpendicular to the impinging plate, reduces as impinging angle increases. However, for particle size of 1 $\mathrm{\mu }\text{m}$, the impact velocity increases at an impinging angle of 15° before decreasing again at larger impinging angles. This is because the strength of the bow shock decreases and sub-micron particles are greatly affected by the shock structures of the supersonic jet.

Next, Chun et al. (2012) investigated the impact velocity of nano-sized particles at different stand-off distances for $\mathit{NPR}=10$. Using steady-state RANS with $k''–''\omega$ SST turbulence model, they found that increasing the stand-off distance of the impinging plate up to $Z_n/D=2.33$ increases the impact velocity of the particles as the intensity of the bow shock reduces.

While there has been extensive research on supersonic jets which has been conducted experimentally and numerically, there is limited work done in investigating both the stand-off distance and impinging angle of supersonic impinging jets. The results in this paper is a continuation to prior work conducted by Chan et al., 2012, Chin et al., 2013 and aims to investigate the validity of both the LES and RANS methodology in simulating supersonic impinging jet flows at $Z_n/D$ of 1.5 and 2.5 for impinging angles $0°\text{,}10°\text{,}30°$ and $45°$. Results from the simulation will primarily focus on analysing the characteristics and mean properties of the supersonic impinging jet.