Supersonic Flow Research Articles

Parallel flows as a key component to interpret Super-X divertor experiments

The Super-X Divertor (SXD) is an alternative divertor configuration leveraging total flux expansion at the Outer Strike Point (OSP). While the extended 2-Point Model (2PM) predicts facilitated detachment access and control in the SXD configuration, these attractive features are not always retrieved experimentally. These discrepancies are at least partially explained by the effect of parallel flows which, when self-consistently included in the 2PM, reveal the role of total flux expansion on the pressure balance and weaken the total flux expansion effect on detachment access and control, compared to the original predictions. This new model can partially explain the discrepancies between the 2PM and experiments performed on tokamak à configuration variable (TCV), in ohmic L-mode scenarios, which are particularly apparent when scanning the OSP major radius Rt. In core density ramps in lower Single-Null (SN) configuration, the impact of Rt on the CIII emission front movement in the divertor outer leg—used as a proxy for the plasma temperature in the divertor—is substantially weaker than 2PM predictions. Furthermore, in OSP radial sweeps in lower and upper SN configurations, in ohmic L-mode scenarios with a constant core density, the peak parallel particle flux density at the OSP is almost independent of Rt, while the 2PM predicts a linear dependence. Finally, analytical and numerical modeling of parallel flows in the divertor is presented. It is shown that an increase in total flux expansion can favour supersonic flows at the OSP. Parallel flows are also shown to be relevant by analysing SOLPS-ITER simulations of TCV.

On the investigation of shock wave/boundary layer interaction with a high-order scheme based on lattice Boltzmann flux solver

Shock wave/boundary layer interaction (SWBLI) continues to pose a significant challenge in the field of aerospace engineering. This paper aims to address this issue by proposing a novel approach for predicting aerodynamic coefficients and heat transfer in viscous supersonic and hypersonic flows using a high-order flux reconstruction technique. Currently, finite volume methods are extensively employed for the computation of skin aerodynamic coefficients and heat transfer. Nevertheless, these numerical methods exhibit considerable susceptibility to a range of factors, including the inviscid flux function and the computational mesh. The application of high-order flux reconstruction techniques offers promising potential in alleviating these challenges. In contrast to other high-order methods, the flux reconstruction is combined with the lattice Boltzmann flux solver in this study. The current method evaluates the common inviscid flux at the cell interface by locally reconstructing the lattice Boltzmann equation solution from macroscopic flow variables at solution points. Consequently, this framework performs a positivity-preserving, entropy-based adaptive filtering method for shock capturing. The present approach is validated by simulating the double Mach reflection, and then simulating some typical viscous problems. The results demonstrate that the current method accurately predicts aerodynamic coefficients and heat transfer, providing valuable insights into the application of high-order methods for shock wave/boundary layer interaction.

Bi-global stability of supersonic backward-facing step flow

Supersonic backward-facing step (BFS) flow is numerically studied using direct numerical simulation (DNS) and global stability analysis (GSA) with a free stream Mach number of 2.16 and a Reynolds number of 7.938 × 105 based on the flat-plate length L and free stream conditions. Two-dimensional BFS flow becomes unstable to three-dimensional perturbations as the step height h exceeds a certain value, while no two-dimensionally unstable mode is found. Global instability occurs with the fragmentation of the primary separation vortex downstream of the step. Two stationary modes and one oscillatory unstable mode are obtained at a supercritical ratio of L/h = 32.14, among which the two stationary modes originate from the coalescence of a pair of conjugate modes. The most unstable mode manifests itself as streamwise streaks in the reattached boundary layer, which is similar to that in shock-induced separated flow, although the flow separation mechanisms are different. Without introducing any external disturbances, the DNS captures the preferred perturbations and produces a growth rate in agreement with the GSA prediction in the linear growth stage. In the quasi-steady stage, the secondary separation vortex breaks up into several small bubbles, and the number of streamwise streaks is doubled. A low-frequency unsteadiness that may be associated with the oscillatory mode is also present.

Sound radiation characteristics of a beam under supersonic airflow and non-uniform temperature field

This paper investigates vibro-acoustic behaviour of an isotropic beam subjected to in-plane varying temperature field and supersonic flow. For a given elevated temperature and non-uniform temperature (NUFT) field, the aerodynamic pressure (ADP) is varied in fraction of its critical value, and the consequent effect on sound radiation characteristics is investigated. Shear and normal deformable beam theory (SNDBT) based kinematics is used, and equations of motion are obtained using Hamilton's principle. Following that, the Ritz method is used to solve the equations. To start with, aeroelastic flutter and vibro-acoustic response characteristics (as a function of critical aerodynamic pressure (λcr)) are analyzed without considering any thermal stress effect. Following that, for a given temperature field, the critical buckling temperature (ΔTcr) of the beam is obtained. Then, the impact of thermal stress caused by the NUFT field on the flutter and sound radiation characteristics is studied. The study observed that the combined effects of the NUFT field and variable ADP significantly impacted the beam's flutter and vibro-acoustic responses.

Optimal Fourth-Order Methods for Multiple Zeros: Design, Convergence Analysis and Applications

Nonlinear equations are frequently encountered in many areas of applied science and engineering, and they require efficient numerical methods to solve. To ensure quick and precise root approximation, this study presents derivative-free iterative methods for finding multiple zeros with an ideal fourth-order convergence rate. Furthermore, the study explores applications of the methods in both real-life and academic contexts. In particular, we examine the convergence of the methods by applying them to the problems, namely Van der Waals equation of state, Planck’s law of radiation, the Manning equation for isentropic supersonic flow and some academic problems. Numerical results reveal that the proposed derivative-free methods are more efficient and consistent than existing methods.

Quasi-Global Sensitivity Analysis of Supersonic Computational Fluid Dynamics Simulations

Determining the sensitivity of model outputs to input parameters is an important precursor to developing informative parameter studies, building surrogate models, and performing rigorous uncertainty quantification. Determining parameter sensitivities over a range of parameter values, termed global sensitivity analysis, requires many model evaluations sampled over the parameter space, which is intractable for many large-scale computational fluid dynamics (CFD) applications. For moderate parameter dimensions, we propose the use of Morris screening, a one-at-a-time quasi-global method for estimating parameter sensitivities over a range of parameter values for CFD simulations. The Morris method is implemented within a CFD framework that utilizes adaptive grid refinement, thereby enabling its application to state-of-the-art production-level problems. The method is shown to be viable for a model problem of supersonic flow over an axisymmetric capsule geometry with both physical and nonphysical (i.e., simulation-informing) input parameters. It is shown that Morris screening identifies relative parameter sensitivities consistent with those of more costly experimental and computational studies that were previously performed on this geometry.

Mechanism of single-mode panel flutter in low supersonic flow

Supersonic panel flutter can be of two possible types: single-mode flutter and coupled-mode flutter. Compared to coupled-mode flutter, the physical mechanism of single-mode flutter is less studied at present. In order to reveal the inducing mechanism of single-mode flutter, the present paper constructed a reduced-order fluid model by using the system identification and the Auto Regressive with eXogenous input (ARX) model. Coupling the reduced-order model (ROM) for unsteady aerodynamics in low supersonic flow with the structural equation, a highly efficient ROM-based aeroelastic model in state space is formulated, then the flutter boundaries are obtained and the stability of aeroelastic modes are investigated by the complex eigenvalue analysis. According to the physical meaning of relevant parameters in unsteady aerodynamic reduced order model ARX, it is found that the unsteady characteristic of the flow, specifically the history effects of the unsteady aerodynamic forces, plays a dominant role in causing the single-mode panel flutter at low supersonic speeds. It is also shown that the higher modes are indeed weakly unstable in the absence of any structural damping or a viscous boundary layer.

On the nonlinear dynamics and flutter response of hybrid shape memory alloy composite beams

This study presents a theoretical and numerical investigation of the dynamic response of a hybrid SMA composite beam. Particular emphasis is given to the evaluation of the equivalent damping, the aeroelastic stability, and the possibility to tune the composite’s dynamics by means of pre-strain applied locally to the SMA layers. The model of the hybrid composite beam (HCB) accounts for both the material nonlinearity associated with the phase transformation of the SMA and the geometric nonlinearity (in the von Kármán sense) due to potentially large displacements. The nonlinear governing equations of the HCB are solved by the finite element method and the dynamic behavior of the HCB is assessed for different design parameters, such as thickness, position, and pre-strain level of the SMA layers. Results help understand the role played by the different design parameters in improving the dynamic characteristics of the HCB, with particular attention to effective damping. Furthermore, it is of particular interest to understand the aeroelastic stability of hybrid layered composites due to their widespread applications in aerospace engineering. A simplified model of the nonlinear flutter response of the HCB under supersonic flow conditions is presented and analyzed with particular emphasis on bifurcations and limit cycle oscillations.

Continuous and discontinuous compressible flows in a converging–diverging channel solved by physics-informed neural networks without exogenous data

Physics-informed neural networks (PINNs) are employed to solve the classical compressible flow problem in a converging–diverging nozzle. This problem represents a typical example described by the Euler equations, a thorough understanding of which serves as a guide for solving more general compressible flows. Given a geometry of the channel, analytical solutions for the steady states do indeed exist, and they depend on the ratio between the back pressure of the outlet and the stagnation pressure of the inlet. Moreover, in the diverging region, the solution may branch into subsonic flow, supersonic flow, or a mixture of both with a discontinuous transition where a normal shock occurs. Classical numerical schemes with shock fitting and capturing methods have been developed to solve this type of problem effectively, whereas the original PINNs are unable to predict the flows correctly. We make a first attempt to exploit the power of PINNs to solve this problem directly by adjusting the weights of different components of the loss function to acquire physical solutions and in the meantime, avoid trivial solutions. With a universal setting yet no exogenous data, we are able to solve this problem accurately; that is, for different given pressure ratios, PINNs provide different branches of solutions at both steady and unsteady states, some of which are discontinuous in nature. For an inverse problem such as unknown specific-heat ratio, it works effectively as well.

Existence of transonic shocks in a cylindrical nozzle with given exit normal momentums

In this paper, we study the existence of transonic shocks in a cylindrical nozzle. More precisely, given the compatible outer normal momentum at the exit of a cylindrical nozzle, there exists a transonic shock solution such that the incoming supersonic flow is compressed to some subsonic flow. The shock position is uniquely determined by the exit normal momentum when we fix one point of the shock. This problem can be reduced to solve a nonlinear free boundary value problem for a hyperbolic–elliptic coupled system and the key ingredient is to solve a second‐order quasilinear elliptic equation with a nonlocal term by the method of separation of variables.

A Vlasov-Fokker-Planck-Landau code for the simulation of colliding supersonic dense plasma flows

A Vlasov-Fokker-Planck-Landau (VFPL) code is developed for the study of colliding supersonic dense plasma flows, in which the VFPL equations for both electrons and ions are solved by the time splitting strategy in the two-dimensional Cartesian coordinate space and the two-dimensional velocity space (2D2V). To accurately handle two colliding supersonic plasma flows with their velocities even much higher than the thermal velocities of ions, the full Fokker-Planck-Landau operator is employed for the collision terms. Using advanced numerical methods such as the fast spectral method and the asymptotic-preserving scheme, the ion-ion and electron-electron collision terms can be properly solved with a relatively large time step so that both high accuracy and efficiency of the developed code can be achieved. Further, the quantum degeneracy effect due to the high density and relatively low initial temperature of the electrons is included in the model. In addition, both the energy and momentum conservation are well satisfied. The developed code provides a unique numerical tool to study the interactions between high-velocity high-density plasma flows, which may be encountered in some advanced schemes of inertial confinement fusion and laser-driven laboratory astrophysics experiments.

Development and Validation of a Compressible Reacting Gas-Dynamic Flow Solver for Supersonic Combustion

An approach based on the OpenFOAM library has been developed to solve a high-speed, multicomponent mixture of a reacting, compressible flow. This work presents comprehensive validation of the newly developed solver, called compressibleCentralReactingFoam, with different supersonic flows, including shocks, expansion waves, and turbulence–combustion interaction. The comparisons of the simulation results with experimental and computational data confirm the fidelity of this solver for problems involving multicomponent high-speed reactive flows. The gas dynamics of turbulence–chemistry interaction are modeled using a partially stirred reactor formulation and provide promising results to better understand the complex physics involved in supersonic combustors. A time-scale analysis based on local Damköhler numbers reveals different regimes of turbulent combustion. In the core of the jet flow, the Damköhler number is relatively high, indicating that the reaction time scale is smaller than the turbulent mixing time scale. This means that the combustion is controlled by turbulent mixing. In the shear layer, where the heat release rate and the scalar dissipation rate have the highest value, the flame is stabilized due to finite rate chemistry with small Damköhler numbers and a limited fraction of fine structure. This solver allows three-dimensional gas dynamic simulation of high-speed multicomponent reactive flows relevant to practical combustion applications.

Influence of volume and frequency parameters on opposing plasma synthetic jet for drag reduction in supersonic flow

Shock wave drag reduction is of great significance to improve the performance and range of supersonic/hypersonic vehicles. As a new active flow control method, plasma synthetic jet actuator (PSJA) shows great potential in the field of shock wave drag reduction. The influence of volume and frequency parameters on opposing plasma synthetic jet (PSJ) for shock wave drag reduction of a hemisphere in M = 3 flow is investigated using numerical method. The first part of this study focuses on the effects of PSJA cavity volume and hemisphere volume on the drag reduction characteristics of opposing PSJ. As the input energy is constant, the average drag reduction increases with the increase of PSJA cavity volume but decreases with the increase of hemisphere volume. In addition, it is found that the trend of the average drag reduction as a function of the volume ratio (the ratio of PSJA cavity volume and hemisphere volume) fits a logarithmic law. Next, the drag reduction characteristics of high frequency opposing PSJ, and the effect of discharge frequency are investigated. There is a “self-balancing process” when opposing PSJA operates at high frequency. The physical meaning of the “self-balancing process” is that the PSJA automatically prolongs the suction recovery time and shortens the jet ejection time in order to maintain continuous operation. After the “self-balancing process”, the PSJA eventually maintains a pseudo-steady operating state. The dimensionless frequency (the ratio of discharge frequency and Helmholtz resonance frequency) is an important parameter that affects opposing PSJ. The average drag reduction is basically equal when the dimensionless frequency is less than 1, about 11%, but decreases significantly when the dimensionless frequency is about 1, about 8%. The effects of volume and frequency parameters on the flow field characteristics of opposing PSJ are also analyzed.

Numerical simulation of thermal environment characteristics within the streamwise gap induced by supersonic flow

Gaps represent a common feature on the surface of supersonic vehicles, and their presence engenders localized thermal phenomena, including elevated heat flux and thermal invasion. These occurrences significantly imperil the aircraft's thermal stability and safety. This study employs numerical simulation to investigate supersonic streamwise gap flow and offers a comprehensive analysis of the impact of geometric and leakage on the distribution of wall heat flux. The computational results manifest a high alignment between regions of high heat flux and high Mach number. Consequently, these findings underscore the pivotal role of convective heat transfer governing the distribution of heat flux within the gap. Most notably, the sensitivity of wall heat flux is most pronounced with respect to gap width. A reduction in gap width yields a substantial reduction in wall heat flux, with this alteration primarily stemming from its impact on the shear interactions between the supersonic mainstream and the vortices within the gaps. Furthermore, changes in gap length influence the invasion characteristics of the supersonic mainstream, thereby instigating consequential shifts in the distribution of heat flux density along the inner surfaces of the gaps. In contrast, variations in gap depth exert a comparatively modest impact on the distribution of heat flux density. The introduction of leakage effects at the base of the gaps results in a significant intensification of supersonic mainstream penetration and concomitant convective heat transfer in the bottom exit region. This phenomenon engenders localized zones characterized by elevated heat flux density, thus severely compromising the aircraft's thermal sealing performance.

RESEARCH OF SUPERSONIC FLOW IN SHORTENED NOZZLES OF ROCKET ENGINES WITH A BELL-SHAPED TIP

The flow in a shortened nozzle with a bell-shaped tip is considered. A comparison of the wave structures of the supersonic gas flow in shortened nozzles with short and long tips formed by compression and stretching of the original bell-shaped nozzle for connection, respectively, with the long and short conical part of the base nozzle at the same nozzle length was carried out. Under operation conditions at sea level and low pressure at the nozzle inlet (P0<50·105 Ра), a large-scale vortex structure, starting from the corner point of the nozzle inlet, is observed in both nozzles. In addition, in the long tip, a small-scale vortex is observed on the wall near its cut. A barrel-shaped wave structure of hanging jumps with a closing Mach disc is formed in the flow in both nozzles, inside which a "saddle-shaped" wave structure of low intensity is noticed. In the separation flow in the tip (when Р0<50·105 Ра and Рн = 1·105 Ра), the pressure on the wall in the separation zone is slightly lower (by ≈ 5-10%) than the external pressure Рн. When the engine is operating in the upper layers of the atmosphere, the static pressure on the section of both tips is proportional to the pressure at the entrance of the nozzle. In the cross-section, starting from the axis of the nozzle to ~0.89 R/Ra (the ratio of the current value of the radius R to the radius of the nozzle wall at the outlet Ra), the pressure decreases to a value proportional to the pressure at the nozzle inlet. Then, it increases linearly to the value of the pressure on the tip wall, which is proportional to the pressure at the nozzle inlet. This is due to the wave structure of the flow inside the nozzle. It was established that with a decrease in the length of the nozzle conical part, the impulse coefficient of the nozzle decreases significantly for operating at sea level and slightly decreases for operating in the upper layers of the atmosphere. The results of calculations correlate satisfactorily with the experimental study results of the flows in shortened nozzles with a bell-shaped tip

Multiline molecular tagging velocimetry of nitric oxide at 100 kHz using an injection-seeded burst-mode OPO.

An injection-seeded, burst-mode optical parametric oscillator (OPO) operating at a repetition rate of 100 kHz is used to demonstrate the multiline molecular tagging velocimetry of an underexpanded jet using nitric oxide fluorescence. The very narrow linewidth of the OPO system, along with the relatively high pulse energies of the burst-mode system, enables efficient single-photon excitation of nitric oxide along multiple laser beam lines at a high repetition rate. Simultaneous one-dimensional velocity profile measurements were obtained of an underexpanded jet system at six different locations using a reference initial image and single-shot delayed images. A methodology for calculating the uncertainty of single-shot velocity is also described. Mean and root-mean-square velocity profiles are obtained at multiple locations simultaneously over a sampling time of 1 ms. The high-repetition-rate velocity measurements also appear to capture the onset of velocity oscillations and has the potential to reveal velocity frequency content occurring in the tens of kHz. The demonstrated velocimetry technique could be paired with other emerging burst-mode laser capabilities for a quantitative multiparameter gas property or multicomponent gas velocity measurements for supersonic and hypersonic flows, especially within ground test facilities that are limited to very short run durations.

Effects of inflow Mach numbers on shock train dynamics and turbulence features in a backpressured supersonic channel flow

Investigations on shock train dynamics and relevant turbulence features in a backpressured supersonic channel flow are carried out using direct numerical simulation for three inflow Mach numbers of Ma0= 1.61, 2.0, and 2.45. As Ma0 increases, the shock train undergoes a structural change characterized by the leading shock which changes from the symmetric “λ” (Ma0=1.61) to the symmetric “X” (Ma0=2.00) and then to the asymmetric “X” pattern (Ma0=2.45). The symmetry breaking of shock structures induces asymmetric separation, which significantly alters the distribution characteristics of wall variables such as wall pressure and friction. To examine the unsteady behaviors of the shock train, a mode decomposition technique, namely, reduced-order variational mode decomposition [Liao et al., J. Fluid Mech. 966, A7 (2023)], is adopted taking its merit of adaptively extracting time-frequency features of dynamic systems. The modal analysis reveals that the shock train system exhibits significant centralization of low-frequency energy. Specifically, two basic types of low-frequency oscillation modes dominate the unsteady motion of the shock train: one depicts overall translating oscillation while another represents accordion-like oscillation. The analysis of turbulent kinetic energy shows that turbulence amplification is mainly dominated by the interaction of the decelerating mean flow with streamwise velocity fluctuations in the vicinity of the leading shock for all three cases, which is unaffected by the symmetry breaking of shock structures.

Direct numerical simulation of supersonic nanoparticles flow in free-molecule regime using the angular coefficient method

We employ molecular flow methods to numerically simulate the supersonic nanoparticles flow in free-molecule regime. To streamline the computational complexity, interaction forces between the gas and solid particles are disregarded. We first develop a discrete phase model (DPM) method that integrates the non-rigid body collision model, enabling an accurate simulation of nanoparticle diffusion under the influence of the drag force and Brownian motion force. The nanoparticles considered in this study have sizes below 10 nm, and the accuracy of the DPM method is verified by comparing its results with experimental data. Subsequently, we theoretically and numerically investigate the transmission probability and number density of N2 molecules flowing through two-dimensional (2D) channels and three-dimensional (3D) tubes by using the angular coefficient (AC) method and the direct simulation Monte Carlo (DSMC) method. The findings indicate that as the diameter of the nanoparticle (dp) decreases to 1 nm, the diffusion coefficient (D) and the root mean square displacement (x) of nanoparticles approach the N2 molecules. The microscopic velocity of most N2 molecules falls within the range of 62–1400 m/s, and the macroscopic velocity of N2 flow falls within the range of Ma = 1.28–1.35. In contrast to the DSMC method, the AC method exhibits enhanced accuracy even with a reduced number of grids and obviates the process for large-scale sampling. Additionally, the solution time required by the AC method is approximately 1/10 and 1/13–1/32 of the DSMC method in 3D cylindrical tubes and 2D channels, respectively. Moreover, the AC method demonstrates superior adaptability when dealing with complex geometries.

Mixing enhancement of transverse jets in supersonic crossflow using an actively controlled novel fluidic oscillator

This study investigates the fluid dynamics and mixing characteristics of an oscillating sonic jet injected into a supersonic cross flow of Mach 2.1 using experimental and computational techniques. The oscillating jet is produced by a novel fluidic oscillator, which consists of a primary rectangular duct that expands into an outer duct with sudden expansion. Control jets are injected in the lateral direction from the side walls of the sudden expansion in an out-of-phase manner to oscillate the injected jet in the spanwise direction of the crossflow. Experimental and numerical investigations based on wall static pressure and mass fraction fluctuations, respectively, revealed that the injected jet oscillation frequency matches the control jet frequency. The iso-surface of lambda-2 criterion showed the presence of various dominant vortex structures, such as counter-rotating vortex pairs, horseshoe vortex, sidewall vortices, and trailing vortices. Helicity contour plots showed that the streamwise vortices oscillate in the spanwise direction with the control strategy and promote the spread of the injected jet in the spanwise direction. The spatiotemporal reconstruction (z–t plot) of the density gradients at a particular streamwise location revealed that the bow shock produced by the interaction of the injected jet and the crossflow oscillates with the actuation of the control strategy. The power spectral density of the z–t plot revealed that the shock wave oscillation frequency matches the control jet frequency. The oscillating jet produced by the control strategy showed significant mixing enhancement in supersonic crossflow compared to a simple rectangular injection.

Multi-objective optimization of a hypersonic airbreathing vehicle

Multi-objective optimization of a hypersonic airbreathing engine (scramjet technology) was carried out with the aim of maximizing thrust and minimizing drag while satisfying a series of design constraints, such as avoiding unstart (blockage of supersonic flow within the combustion chamber) by ensuring that the pressure ratio across the shock waves remains below the adverse pressure gradient given by the Korkegi limit, geometry correction to achieve shock on-lip condition, and temperature and pressure requirements at the inlet exit. Using the relations presented in the literature, pressure and viscous drag are estimated analytically. The analytical approach is verified against computational fluid dynamics data from Ansys Fluent to solve two-dimensional compressible Reynolds-averaged Navier–Stokes flow equations, with transition shear stress transport as the turbulence closure model. Comparing the total drag and the flow properties at the combustion chamber entrance shows the model's feasibility for the optimization approach. Three different approaches were conducted to formulate the multi-objective function to determine the one that can find the highest number of geometries satisfying the Korkegi limit with the highest net thrust. The best approach was the multi-objective function formulated with the uninstalled thrust, total pressure recovery, and pressure drag, concentrating the search in the region with greater uninstalled thrust and lower drag and nearly doubling the value of net thrust compared to the first formulation, which uses the uninstalled thrust, pressure drag, and viscous drag.