Shock-vortex Interaction Research Articles

Hybrid compressible lattice Boltzmann method for supersonic flows with strong discontinuities

Within the framework of the hybrid recursive regularized lattice Boltzmann (HRR-LB) model, we propose a novel hybrid compressible LB method to ensure the conservation of total energy in simulating compressible flows with strong discontinuities. This method integrates a LB solver to handle the mass and momentum conservation equations via collision-streaming steps on standard lattices, while a finite volume method (FVM) is employed for the conservation of the total energy equation. The flux reconstruction in the FVM is achieved through a momentum coupled method (MCM). The interface momentum, crucial for reconstructing the convective fluxes and determining the upwind extrapolation of passive scalar quantities in MCM, is derived from the LB method. The validity and accuracy of the proposed method are evaluated through six test cases: (I) isentropic vortex convection in subsonic and supersonic regimes; (II) non-isothermal acoustic pulse; (III) one-dimensional Riemann problems; (IV) two-dimensional Riemann problem; (V) double Mach reflection of a Mach 10 shock wave; and (VI) shock–vortex interaction. Numerical results demonstrate that this method surpasses the previous HRR-LB model by Guo et al. [“Improved standard thermal lattice Boltzmann model with hybrid recursive regularization for compressible laminar and turbulent flows,” Phys. Fluids 32, 126108 (2020)] in terms of accuracy and robustness when dealing with strong shock waves.

Physical aspects of vortex-shock dynamics in delta wing configurations

Delta wing configurations with double- and triple-leading edges introduced within the North Atlantic Treaty Organization Applied Vehicle Technology -316 task group are examined to investigate the dynamics of vortices and shocks, with potential implications for the preliminary aircraft design. The numerical simulations are conducted for the configurations at Ma∞=0.85 and Re∞=12.53×106 using the Reynolds-averaged Navier–Stokes k−ω shear stress transport (SST) model across a range of incidence angles. The detailed analysis focuses on the case with α=20° using the scale-adaptive simulation based on the k−ω SST model. This study considers shock-vortex interaction and breakdown with buffeting to study the transient flow physics over the wing. Additionally, insights into vorticity strength and destruction are gained through the enstrophy transport equation. The findings reveal that the inboard vortex (IBV) development is impeded by counter-rotating secondary vortices from IBV and the midboard vortex. A key distinction is observed for the first time between the double-delta and triple-delta wings, in that the double-delta wing experiences shock-induced vortex breakdown, with the transient nature of this breakdown leading to an adjustment in the shock position, causing a shock buffet. In contrast, the breakdown in the triple-delta wing is linked to a stationary shock induced by the kink in the planform. This study highlights the crucial role of the orientation of the shock relative to the vortex axis in characterizing the aerodynamic performance of the planforms.

A general multi-objective Bayesian optimization framework for the design of hybrid schemes towards adaptive complex flow simulations

Achieving accuracy with underresolved simulation of complex compressible flows with multiscale flow structures is a challenge. Either the numerical dissipation or the resolution and thereby the numerical cost is impractically high. Also, in the design of numerical solvers, the application of a solver for specific flow classes is balanced by robustness allowing the study of a broad range of flows. In this study, we propose a hybrid fifth-order targeted essentially non-oscillatory (TENO5)-based scheme tailored to optimally simulate compressible flows with underresolved dilatational and vortical multiscale structures. For optimal design, three data-driven objectives are defined. A novel objective that derives from the numerical dissipation rate analysis is a key element to deal with underresolved complex flows in practical applications. The optimization process employs a multi-objective Bayesian optimization framework with an expected hypervolume improvement and three flow configurations representative for a broad range of two- and three-dimensional flows with genuine and non-genuine subgrid scales. The optimized hybrid scheme is validated by comparing with shock-capturing schemes of the weighted essentially non-oscillatory (WENO)- and TENO- families with flows of complex shock interactions, Kelvin-Helmholtz instabilities, shock-vortex interactions, vortical and turbulent flows

High speed flows with particles on demand: Boundary conditions

The particles on demand (PonD) method is a new kinetic theory model that allows for simulation of high speed compressible flows. While the standard lattice-Boltzmann method is limited by a fixed reference frame, significantly reducing the range of Mach numbers, PonD takes advantage of adaptive reference frames to get rid of the restrictions of standard LB and is able to simulate flows at high speeds and with large temperature variation. Previously, PonD has been shown to be a viable alternative for simulation of flows with strong discontinuities and for detonation modelling. However, treatment of flows with complex boundaries has been lacking. Here, we present PonD augmented with a non-equilibrium extrapolation based boundary condition. We present several compressible test cases such as shock-vortex interaction in the Schardin’s problem and supersonic flow over a two-dimensional cylinder at Mach numbers up to 5. We observe that the results agree well with literature, paving the way for a kinetic theory based approach for simulating compressible flows in realistic scenarios.

Streamwise vortex breakdown due to the interaction with crossed shock waves

This paper presents a vortex breakdown study due to the interaction between a Batchelor vortex and the crossing of oblique shock waves with Mach numbers of 3.5 and 5.0, favourable for supersonic mixing and combustion, respectively. Numerical simulations were conducted to investigate the effects of the circulation intensity and shock angle on vortex breakdown. The results indicate that a breakdown occurs at the shock angle $\beta \geqslant 45^\circ$ or the vortex circulation $q = 0.32$ , and the configuration is a bubble structure with a recirculation region; most of the breakdowns possess a stagnation point. Furthermore, the structure differs from that of a normal shock wave and vortex interaction because the bubble region is subsonic and does not comprise a normal shock wave on the inside. Additionally, this vortex breakdown shows that the momentum flux on the centreline decreases once at the tip of the bubble owing to a sudden drop in velocity in the subsonic region. In addition, the enstrophy production resulting from vortex stretching and tilting is found to have a significant advantage in the interaction region. Based on these results, the threshold required for a bubble vortex breakdown was theoretically derived as an inequality. The numerical simulation results support the theoretical criterion obtained from the proposed inequality. Therefore, a streamwise vortex breakdown resulting from the interaction between the vortex and intersecting oblique-shocks should be reasonably predicted.

Numerical studies of shock–vortex interaction over a wedge during shock-wave diffraction—A new approach

In this study, shock wave diffraction has been investigated through a numerical simulation of a moving normal shock incident on a sharp-edged wedge. Schardin's problem is revisited using some existing and new mathematical tools. Two-dimensional compressible Navier–Stokes equation is solved using a higher-order version of the rhoCentralFoam solver in the OpenFOAM platform. Overall flow structures are captured with high efficacy. The divergence of the Lamb vector is used to probe the interaction between vorticity bearing and fluid straining motion, which increases dramatically inside the primary vortex after collision with the reflected Mach stem and increases the turbulent kinetic energy (TKE). In the separated shear layer that emerges from the wedge tip, there is a reduction of TKE after the collision between the lambda shock and accelerated shock. The vorticity pumping into the mean flow by the baroclinic torque is dominant in the separated shear layer before the above collision, whereas after collision it is only dominant inside the primary vortex. A new vector M is introduced here, which is the cross product of the vorticity vector and the pressure gradient vector. The divergence of M shows the interaction between the pressure gradient and the vorticity gradient. This interaction affects the separation bubble formed over the base wall of the wedge due to the shock-wave/boundary layer interaction induced by the lambda shock. Dynamic mode decomposition shows a dominant mode at a frequency of 125.7 Hz, which is due to low-frequency shock oscillation.

Effect of shock tunnel geometry on shockwave and vortex ring formation, propagation, and head on collision

Vortex ring research primarily focuses on the formation from circular openings. Consequently, the role of tunnel geometry is less understood, despite there being numerous research studies using noncircular shock tunnels. This experimental study investigated shockwaves and vortex rings from different geometry shock tunnels from formation at the tunnel opening to head on collision with another similarly formed vortex ring using schlieren imaging and statistical analysis. The velocity of the incident shockwave was found to be consistent across all four shock tunnel geometries, which include circle, hexagon, square, and triangle of the same cross-sectional area. The velocity was 1.2 ± 0.007 Mach and was independent of the tunnel geometry. However, the velocities of the resulting vortex rings differed between the shapes, with statistical analysis indicating significant differences between the triangle and hexagon vortex velocities compared to the circle. Vortex rings from the square and circle shock tunnels were found to have statistically similar velocities. All vortex rings slowed as they traveled due to corner inversion and air drag. All shock tunnels with corners produce a wobble in the vortex rings. Vortex rings interact with opposing incident shockwaves prior to colliding with each other. Vortex velocity before and after shock–vortex interaction was measured and evaluated, showing statistically similar results. Shock–vortex interaction slows the shockwave upon interaction, while the shock–shock interaction resulted in no change in shock velocity. Although the vortex rings travel at different velocities, all head-on vortex ring collisions produce a perpendicular shockwave that travels at 1.04 ± 0.005 Mach.

An accurate and practical numerical solver for simulations of shock, vortices and turbulence interaction problems

Numerical simulations of high-speed compressible flows remain challenging in engineering as the appearance of shock waves poses difficulties for high-resolution schemes as well as explicit large eddy simulation methods. Therefore, in this work, we propose a practical numerical solver for simulations of shock-vortex and shock-turbulence problems with the implicit large eddy simulation approach and newly developed low-dissipative schemes. In order to handle the multi-scale raised in the shock-vortex and shock-turbulence problems, it is required that the flow solver can simultaneously solve the large-scale flow structures such as shock waves with numerical oscillation-free, the resolvable flow structures above the grid scale with low-dissipation, and the under-resolve isotropic sub-grid scale with numerical stability. To this end, a low-dissipative, structure-preserving scheme with rigorously adjusted numerical dissipation is employed in the proposed solver. The structure-preserving property of this scheme can ensure that the large-scale structure is solved without numerical oscillations and the low-dissipative property of this scheme can produce high-resolution results for the resolvable flow scales. Moreover, the sub-grid scale is solved and stabilized by the inherent numerical dissipation in the shock-capturing scheme. The proposed numerical solver is then applied to simulate a wide range of shock-vortex and shock-turbulence interaction problems including supersonic planar jets, transonic flows past a deep cavity and impingement of a supersonic jet on a cone mounted on a flat plate. A comparison is also made with the solver using conventional total variation diminishing schemes. The numerical results of the supersonic planar jet have demonstrated that the proposed solver can resolve the small-scale structure such as Kelvin–Helmholtz instability involved in shock and shear layer with high-resolution. Through the results of transonic flow past a deep cavity and comparisons with the experimental data, it is verified that the proposed solver can reproduce the turbulence statistical data. Furthermore, the proposed solver resolves the complex turbulent fluid mechanical phenomena in the impingement of a supersonic jet on a cone, which demonstrates the proposed solver can robustly handle irregular geometry. The proposed solver also enjoys simplicity without using the explicit sub-grid scale model and involving the complexity of very high-order schemes. Thus, this work provides an accurate and practical numerical solver for shock-vortex and shock-turbulence problems in high-speed flows.

Analysis of the vortex-dominated flow field over a delta wing at transonic speed

AbstractThe present work provides an advancement in the prediction of delta wing flow and an improved understanding of various flow physical phenomena which occur over the wing in transonic conditions. Scale-resolving simulations of the vortex-dominated flow around a sharp leading-edge VFE-2 wing have been performed using the SA-based IDDES model. The complex leading-edge vortex pattern with embedded shocks and subsequent shock-vortex interaction is investigated. A promising accuracy has been achieved using the high-fidelity flow field data provided by the scale-resolving simulation results. Besides the assessment of sensitivity to spatial and temporal resolution, physical aspects are presented, which are not accessible in experimental data in such detail and require scale-resolving simulation approaches. This includes the observation of the vortex system and the shocks in the fully three-dimensional flow field data. Finally, turbulence-related quantities such as eddy viscosity and resolved Reynolds-stresses and their behaviour during the vortex formation and sustaining process are analysed.

A Flux Reconstruction Stochastic Galerkin Scheme for Hyperbolic Conservation Laws

The study of uncertainty propagation poses a great challenge to design high fidelity numerical methods. Based on the stochastic Galerkin formulation, this paper addresses the idea and implementation of the first flux reconstruction scheme for hyperbolic conservation laws with random inputs. High-order numerical approximation is adopted simultaneously in physical and random space, i.e., the modal representation of solutions is based on an orthogonal polynomial basis and the nodal representation is based on solution collocation points. Therefore, the numerical behaviors of the scheme in the (physical-random) phase space can be designed and understood uniformly. A family of filters is developed in multi-dimensional cases to mitigate the Gibbs phenomenon arising from discontinuities in both physical and random space. The filter function is switched on and off by the dynamic detection of discontinuous solutions, and a slope limiter is employed to preserve the positivity of physically realizable solutions. As a result, the proposed method is able to capture the stochastic flow evolution where resolved and unresolved regions coexist. Numerical experiments including a wave propagation, a Burgers’ shock, a one-dimensional Riemann problem, and a two-dimensional shock-vortex interaction problem are presented to validate the current scheme. The order of convergence of the high-order scheme is identified. The capability of the scheme for simulating smooth and discontinuous stochastic flow dynamics is demonstrated. The open-source codes to reproduce the numerical results are available under the MIT license (Xiao et al. in FRSG: stochastic Galerkin method with flux reconstruction. https://github.com/CSMMLab/FRSG, (2021). https://doi.org/10.5281/zenodo.5588317).

Analysis of a higher-order vorticity confinement scheme in flux correction form

This article presents a detailed comparison of the classical source-term formulation of higher-order Vorticity Confinement with a flux-correction version, through an application to a series of well-documented cases. Both formulations and their numerical implementation are presented first. Then the methods are applied to the simple case of the long-time advection of an inviscid and isentropic vortex. A shock-vortex interaction case is also computed to demonstrate the compatibility of the method with the shock-capturing properties of the baseline scheme. Both methods are finally applied to a ZDES mode 3 (Wall-Modelled LES) simulation of a zero-pressure-gradient flat-plate turbulent boundary layer, showing a better preservation of smaller-scale vortical structures with respect to standard higher-order numerical schemes. Furthermore, the flux-correction form of Vorticity Confinement significantly improves the prediction of skin friction with respect to both the baseline scheme and the classical source-term formulation, thus showing that the flux-correction form of confinement is essential for a reliable prediction of this quantity.

Analysis of multipolar vortices in the interaction of a shock with a strong moving vortex

Numerical simulation results of a convecting shielded vortex interacting with a normal shock are presented. The shock and vortex Mach numbers covered are on the higher side, for which relatively little information is available. It is well known that for a weak shock the vortex survives the shock, and for stronger shocks it breaks down. We discover that there is a narrow zone of shock and vortex strengths for which the vortex neither preserves its shape, nor does it break down in the true sense, but generates a tripole or quadrupole — structures that have not so far been reported in shock-vortex interaction studies. A number of cases have been studied to determine the different shock-vortex interaction patterns depending on shock and vortex Mach numbers. A new type of interaction pattern has been determined and it has been named as Type-IV interaction.

Positivity-preserving entropy-based adaptive filtering for discontinuous spectral element methods

In this work, we present a positivity-preserving entropy-based adaptive filtering method for shock capturing in discontinuous spectral element methods. By adapting the filter strength to enforce positivity and a local discrete minimum entropy principle, the resulting approach can robustly resolve strong discontinuities with sub-element resolution, does not require problem-dependent parameter tuning, and can be easily implemented on general unstructured meshes with relatively low computational cost. The efficacy of the approach is shown in numerical experiments on hyperbolic and mixed hyperbolic-parabolic conservation laws such as the Euler and Navier-Stokes equations for problems including extreme shocks, shock-vortex interactions, and complex compressible turbulent flows.

Measurements of weak and moderate oblique shock-vortex interactions in supersonic flow

Stereoscopic particle image velocimetry measurements have been performed to quantify the centerline flow field associated with oblique shock-vortex interactions. This novel data set provides a unique view into the time-averaged velocity and turbulence kinetic energy distribution of weak and moderate interactions. Traditionally visualized using qualitative techniques, these data demonstrate that even weak interactions will result in measurable alterations to the flow field.

An Efficient Mapped WENO Scheme Using Approximate Constant Mapping

We present a novel mapping approach for WENO schemes through the use of an approximate constant mapping function which is constructed by employing an approximation of the classic signum function. The new approximate constant mapping function is designed to meet the overall criteria for a proper mapping function required in the design of the WENO-PM6 scheme. The WENO-PM6 scheme was proposed to overcome the potential loss of accuracy of the WENO-M scheme which was developed to recover the optimal convergence order of the WENO-JS scheme at critical points. Our new mapped WENO scheme, denoted as WENO-ACM, maintains almost all advantages of the WENO-PM6 scheme, including low dissipation and high resolution, while decreases the number of mathematical operations remarkably in every mapping process leading to a significant improvement of efficiency. The convergence rates of the WENO-ACM scheme have been shown through one-dimensional linear advection equation with various initial conditions. Numerical results of one-dimensional Euler equations for the Riemann problems, the Mach 3 shock-density wave interaction and the Woodward-Colella interacting blastwaves are improved in comparison with the results obtained by the WENO-JS, WENO-M and WENO-PM6 schemes. Numerical experiments with two-dimensional problems as the 2D Riemann problem, the shock-vortex interaction, the 2D explosion problem, the double Mach reflection and the forward-facing step problem modeled via the two dimensional Euler equations have been conducted to demonstrate the high resolution and the effectiveness of the WENO-ACM scheme. The WENO-ACM scheme provides significantly better resolution than the WENO-M scheme and slightly better resolution than the WENO-PM6 scheme, and compared to the WENO-M and WENO-PM6 schemes, the extra computational cost is reduced by more than 83% and 93%, respectively.

Restoring the conservativity of characteristic-based segregated models: Application to the hybrid lattice Boltzmann method

A general methodology is introduced to build conservative numerical models for fluid simulations based on segregated schemes, where mass, momentum, and energy equations are solved by different methods. It is especially designed here for developing new numerical discretizations of the total energy equation and adapted to a thermal coupling with the lattice Boltzmann method (LBM). The proposed methodology is based on a linear equivalence with standard discretizations of the entropy equation, which, as a characteristic variable of the Euler system, allows efficiently decoupling the energy equation with the LBM. To this extent, any LBM scheme is equivalently written under a finite-volume formulation involving fluxes, which are further included in the total energy equation as numerical corrections. The viscous heat production is implicitly considered thanks to the knowledge of the LBM momentum flux. Three models are subsequently derived: a first-order upwind, a Lax–Wendroff, and a third-order Godunov-type schemes. They are assessed on standard academic test cases: a Couette flow, entropy spot and vortex convections, a Sod shock tube, several two-dimensional Riemann problems, and a shock–vortex interaction. Three key features are then exhibited: (1) the models are conservative by construction, recovering correct jump relations across shock waves; (2) the stability and accuracy of entropy modes can be explicitly controlled; and (3) the low dissipation of the LBM for isentropic phenomena is preserved.

A Legendre spectral viscosity (LSV) method applied to shock capturing for high-order flux reconstruction schemes

A novel approach to shock capturing for high-order flux reconstruction schemes is derived based on the mathematical formalism of the filtered governing equations. While the latter perspective is only typically used for turbulence modeling in the context of Large-Eddy Simulations (LES), the novel Legendre Spectral Viscosity (LSV) subfilter scale (SFS) closure model is capable of performing simulations in the presence of shock-discontinuities. The LSV method exploits the set of hierarchical basis functions formed by the Legendre polynomials to extract the information on the energy content near the resolution limit and estimate the overall magnitude of the required SFS dissipative terms, resulting in a scheme that dynamically activates only in cells where nonlinear behavior is important. Additionally, the modulation of such terms in the Legendre spectral space allows for the concentration of the dissipative action at small scales. The proposed method is tested in canonical shock-dominated flow setups in both one and two dimensions. These include the 1D Burgers' problem, a 1D shock tube, a 1D shock-entropy wave interaction, a 2D inviscid shock-vortex interaction and a 2D double Mach reflection. Results showcase a high-degree of resolution power, achieving accurate results with a small number of degrees of freedom, and robustness, being able to capture shocks associated with the Burgers' equation and the 1D shock tube within a single cell with orders 120 and higher.

Investigation of transonic aerodynamics on a triple-delta wing in side slip conditions

The vortex-dominated flow around the triple-delta wing ADS-NA2-W1 aircraft is investigated in order to achieve a better understanding of the flow physics phenomena that occur over the aircraft particularly at the transonic speed condition. Both URANS and scale-resolving DDES have been employed in order to explore the range of suitability of current CFD methods. The Spalart–Allmaras One-Equation Model with corrections for negative turbulent viscosity and Rotation/Curvature (SA-negRC) is employed to close the RANS equations, whereas the SAneg-based DDES model is applied in the scale-resolving computations. The DLR TAU-Code is used to perform the numerical simulations. The deficiencies of the URANS results are illustrated and promising improvements are reached employing the SAneg-DDES numerical method. The hybrid method results show great advancement in the prediction of the multiple-delta wing flow by revealing physical aspects which have not been seen from URANS with sufficient accuracy like vortex–vortex interaction and shock-vortex interaction. These phenomena furthermore explain in a clear way the improved prediction of the surface pressure coefficient over the aircraft and consequently of the aerodynamic force and moment coefficients.

Low dissipative finite difference hybrid scheme by discontinuity sensor of detecting shock and material interface in multi-component compressible flows

A discontinuity sensor for shock waves and material interfaces is proposed to construct a low dissipative scheme that consists of a weighted compact finite nonlinear scheme (WCNS) and central difference to address compressible multi-component flows. The Harten-Lax-van Leer and contact (HLLC) scheme is applied to capture shock waves through material interfaces. At the material interface, the overestimated quasi-conservative WCNS maintains the equilibriums of velocity, pressure, and temperature. The discontinuity sensor is composed of a Larrson shock sensor and a material interface sensor to distinguish between smooth and discontinuous regions, respectively. In the single-component shock-vortex interaction problem, the present scheme successfully captures shock waves with low numerical dissipation. In the multi-component shock-bubble interaction problem, Richtmyer-Meshkov instability problem, and triple-point problem, the simultaneous capturing of shock waves and material interfaces by the present scheme is demonstrated. The proposed scheme suppresses the numerical oscillation by shock waves and material interfaces; additionally, it successfully reduces the numerical dissipation in smooth regions.

Experiments in Shock-Vortex Interactions

Studies of shock-vortex interactions in the past have predominantly been numerical, with a number of idealizations such as assuming an isolated vortex and a plane shock wave. In the present case the vortex is generated from flow separation at a corner. A shear layer results which wraps up into a spiral vortex. The flow is impulsively initiated by the diffraction of a shock wave over the edge. The strength of the shock determines the nature of the flow at the corner and that induced behind the diffracted wave. A wide variety of cases are considered using different experimental arrangements such as having two independent shock waves arriving at the corner at different times, to reflecting the diffracting wave off different surfaces back into the vortex, and to examining the flow around bends where the reflection off the far wall reflects back onto the vortex. The majority of studies have shown that the vortex normally retains its integrity after shock transit. Some studies with curved shock waves and numerous traverses have shown evidence of vortex breakup and the development of turbulent patches in the flow, as well as significant vortex stretching. Depending on the direction of approach of the shock wave it refracts through the shear layer thereby changing the strength and direction of both. Of particular note is that the two diffracted waves which emerge from the vortex as the incident wave passes through interact with each other resulting in a pressure spike of considerable magnitude. An additional spike is also identified.