High-speed Compressible Flows Research Articles

Simple and robust contact-discontinuity capturing central algorithms for high speed compressible flows

The nonlinear convection terms in the governing equations of compressible fluid flows are hyperbolic in nature and are nontrivial for modelling and numerical simulation. Many numerical methods have been developed in the last few decades for this purpose and are typically based on Riemann solvers, which are strongly dependent on the underlying eigen-structure of the governing equations. Objective of the present work is to develop simple algorithms which are not dependent on the eigen-structure and yet can tackle easily the hyperbolic parts. Central schemes with smart diffusion mechanisms are apt for this purpose. For fixing the numerical diffusion, the basic ideas of satisfying the Rankine-Hugoniot (RH) conditions along with generalized Riemann invariants are proposed. Two such interesting algorithms are presented, which capture grid-aligned steady contact discontinuities exactly and yet have sufficient numerical diffusion to avoid numerical shock instabilities. Both the algorithms presented are robust in avoiding shock instabilities, apart from being accurate in capturing contact discontinuities, do not need wave speed corrections and are independent of eigen-struture of the underlying hyperbolic parts of the systems.

Multi-beam interferometric Rayleigh scattering technique for simultaneous 2-D quantitative measurement of multi-parameters in high speed flow field

In order to diagnose the high-speed compressible turbulent flow field, a two-dimensional (2-D) multi-parameters measurement method based on multi-beam interferometric Rayleigh scattering (IRS) technique was proposed. Under the beam splitting of etalon, the Rayleigh scattering signals generated by the interaction between multi-beam laser probes and the flow field gas molecules formed 2-D interference measurement lattice. Each measurement point contained the thermodynamic and velocity information of the flow field corresponding to a specific spatial location. By calibrating the effective finesse and transmittance of the etalon at different incident angles, the measurement accuracy of temperature and density of high-order interferometric points was improved. A 2-D IRS measurement system was established by combining multi-beam laser probes, amplification imaging system and solid quartz etalon. A high speed jet was used to verify the technology. The distributions of temperature, density and velocity at different positions along the radial direction and axial direction of the nozzle were obtained. The results shown that the technique could be used to simultaneously measure the 2-D multi-parameter information of the flow field.

Stabilized methods for high-speed compressible flows: toward hypersonic simulations

A stabilized finite element framework for high-speed compressible flows is presented. The Streamline-Upwind/Petrov–Galerkin formulation augmented with discontinuity-capturing (DC) are the main constituents of the framework that enable accurate, efficient, and stable simulations in this flow regime. Full- and reduced-energy formulations are employed for this class of flow problems and their relative accuracy is assessed. In addition, a recently developed DC formulation is presented and is shown to be particularly well suited for hypersonic flows. Several verification and validation cases, ranging from 1D to 3D flows and supersonic to the hypersonic regimes, show the excellent performance of the proposed framework and set the stage for its deployment on more advanced applications.

Compressible lattice Boltzmann methods with adaptive velocity stencils: An interpolation-free formulation

Adaptive lattice Boltzmann methods (LBMs) are based on velocity discretizations that self-adjust to local macroscopic conditions such as velocity and temperature. While this feature improves the accuracy and the stability of LBMs for large velocity and temperature variations, it also strongly impacts the efficiency of the algorithm due to space interpolations that are required to get populations at grid nodes. To avoid this defect, the present work proposes new formulations of adaptive LBMs that do not rely anymore on space interpolations, hence drastically improving their parallel efficiency for the simulation of high-speed compressible flows. To reach this goal, the adaptive phase discretization is restricted to particular states that are compliant with the efficient “collide-and-stream” algorithm, and as a consequence, it does not require additional interpolation steps. The development of proper state-adaptive solvers with on-grid propagation imposes new restrictions and challenges on the discrete stencils, namely, the need for an extended operability range allowing for the transition between two phase discretizations. Achieving the minimum operability range for discrete polynomial equilibria requires rather large stencils (e.g., D2Q81, D2Q121) and is therefore not competitive for compressible flow simulations. However, as shown in this article, the use of numerical equilibria can provide for overlaps in the operability ranges of neighboring discrete shifts at acceptable cost using the D2Q21 lattice. Through several numerical validations, the present approach is shown to allow for an efficient realization of discrete state-adaptive LBMs for high Mach number flows even in the low-viscosity regime.

A sharp-interface immersed boundary method for simulating high-speed compressible inviscid flows

This paper presents a robust sharp-interface immersed boundary method for simulating inviscid compressible flows over stationary and moving bodies. The flow field is governed by Euler equations, which are solved by using the open source library OpenFOAM. Discontinuities such as those introduced by shock waves are captured by using Kurganov and Tadmor divergence scheme. Wall-slip boundary conditions are enforced at the boundary of body through reconstructing flow variables at some ghost points. Their values are obtained indirectly by interpolating from their mirror points. A bilinear interpolation is employed to determine the variables at the mirror points from boundary conditions and flow conditions around the boundary. To validate the efficiency and accuracy of this method for simulation of high-speed inviscid compressible flows, four cases have been simulated as follows: supersonic flow over a 15 ° angle wedge, transonic flow past a stationary airfoil, a piston moving with supersonic velocity in a shock tube and a rigid circular cylinder lift-off from a flat surface triggered by a shock wave. Compared to the exact analytical solutions or the results in literature, good agreement can be achieved.

Investigation and back-propagation modeling of base pressure at sonic and supersonic Mach numbers

The experimental analysis of base pressure in a high-speed compressible flow is carried out. The flow is made to expand abruptly from the nozzle into an enlarged duct at fifteen sonic and supersonic Mach numbers. The analysis is made for variation in the nozzle pressure ratio (NPR), length to diameter ratio, and area ratio. The effect of active micro-jets on the base and wall pressure is assessed. The data visualization of the huge experimental data generated is performed using heat maps. For the first time, six back-propagation neural network models (BPMs) are developed based on input and output possibilities to predict the pressure in high-speed flows. The experimental analysis revealed that depending upon the type of expansion, the base pressure changes. A jet of air blown at the base using micro-jets is found to be effective in increasing the base pressure during the under-expansion regime, while the wall pressure remains unaffected. The data visualization provided an insight into the highest impact on the base pressure by the NPR. The six BPMs with two hidden layers having four neurons per layer are found to be most suitable for the regression analysis. BPM 5 and BPM 6 accurately predict the highly non-linear data of the base and wall pressure.

250 kHz three-component Doppler velocimetry at 32 simultaneous points: a new capability for high speed flows

Time-resolved Doppler global velocimetry (TRDGV) is used to demonstrate three-component velocity measurements of 32 planar points, acquired simultaneously at 250 kHz. Photomultiplier tube arrays are used to detect the flow signal, with each array having eight rows of four pixels. The system utilizes frequency scanning of a continuous wave laser, allowing for simple calibration of the signals used to determine the incident and Doppler-shifted frequency. The TRDGV system is used to measure an unheated, Mach 0.91 free jet downstream of the potential core collapse. Axial mean velocities were measured within a root-mean-square error of 7 m s−1 compared to Pitot probe validation measurements. Additionally, axial turbulence intensities showed agreement to hot wire validation measurements and exhibited typical magnitudes reported in the literature. Velocity spectra were obtained for the three components of velocity, revealing a − 5/3 decay in the axial velocity spectra in the inertial subrange frequencies, and generally less decay present in the transverse component spectra. Hot wire measurements and spectra from literature were used for validation of the TRDGV spectral results, showing broadband spectral agreement. The instrument described in this work shows promising capability for multi-point, time-resolved velocity vector measurement in high speed compressible flows with acceptable levels of uncertainty.

Grid refinement in the three-dimensional hybrid recursive regularized lattice Boltzmann method for compressible aerodynamics.

Grid refinement techniques are of paramount importance for computational fluid dynamics approaches relying on the use of Cartesian grids. This is especially true of solvers dedicated to aerodynamics, in which the capture of thin shear layers require the use of small cells. In this paper, a three-dimensional grid refinement technique is developed within the framework of hybrid recursive regularized lattice Boltzmann method (HRR-LBM) for compressible high-speed flows, which is an efficient collide-stream-type method on a compact D3Q19 stencil. The proposed method is successfully assessed considering several test cases, namely, an isentropic vortex propagating through transition interface, shock-vortex interaction with intersection between grid refinement interface and shock corrugation, and transonic flows over three-dimensional DLR-M6 wing with seven levels of grid refinement.

An efficient lattice Boltzmann method for compressible aerodynamics on D3Q19 lattice

An efficient lattice Boltzmann (LB) model relying on a hybrid recursive regularization (HRR) collision operator on D3Q19 stencil is proposed for the simulation of three-dimensional high-speed compressible flows in both subsonic and supersonic regimes. An improved thermal equilibrium distribution function on D3Q19 lattice is derived to reduce the complexity of correcting terms. A simple shock capturing scheme and an upwind biased discretization of correction terms are implemented for supersonic flows with shocks. Mass and momentum equations are recovered by an efficient streaming, collision and forcing process on D3Q19 lattice. Then a non-conservative formulation of the entropy evolution equation is used, that is solved using a finite volume method. The proposed method is assessed considering the simulation of i) 2D isentropic vortex convection, ii) 3D non-isothermal acoustic pulse, iii) 2D supersonic flow over a bump, iv) 3D shock explosion in a box, v) 2D vortex interaction with shock wave, vi) 2D laminar flows over a flat plate at Ma of 0.5, 1.0 and 1.5.

Acoustic streaming in turbulent compressible channel flow for heat transfer enhancement

Acoustic streaming in high-speed compressible channel flow and its impact on heat and momentum transfer is analysed numerically at two different Mach numbers, and 1.5, and moderate Reynolds numbers, and 6000. An external time-periodic forcing function is implemented to model the effect of acoustic drivers placed on the sidewalls. The excitation frequency is chosen according to the linear stability analysis of the background (unexcited) flow. High-fidelity numerical simulations performed at the optimal resonant condition reveal an initially exponential growth of perturbations followed by a nonlinear regime leading to the limit-cycle oscillations. In the last stage, we observe an acoustic (steady) streaming appearing as a result of nonlinear interactions between the periodic external wave and the background flow. This causes a steady enhancement in heat transfer at a rate higher than the skin-friction augmentation. We also show that perturbations of similar amplitude, but at suboptimal frequencies, may not lead to such limit-cycle oscillations and cannot make any noticeable modifications to the time-averaged flow quantities. The present research is the first study to demonstrate the acoustic streaming in compressible turbulent flows, and it introduces a novel technique towards enhancing the heat transfer with minimal skin-friction production.

Development of a Parallel Explicit Finite-Volume Euler Equation Solver using the Immersed Boundary Method with Hybrid MPI-CUDA Paradigm

ABSTRACTThis study proposed the application of a novel immersed boundary method (IBM) for the treatment of irregular geometries using Cartesian computational grids for high speed compressible gas flows modelled using the unsteady Euler equations. Furthermore, the method is accelerated through the use of multiple Graphics Processing Units – specifically using Nvidia’s CUDA together with MPI - due to the computationally intensive nature associated with the numerical solution to multi-dimensional continuity equations. Due to the high degree of locality required for efficient multiple GPU computation, the Split Harten-Lax-van-Leer (SHLL) scheme is employed for vector splitting of fluxes across cell interfaces. NVIDIA visual profiler shows that our proposed method having a computational speed of 98.6 GFLOPS and 61% efficiency based on the Roofline analysis that provides the theoretical computing speed of reaching 160 GLOPS with an average 2.225 operations/byte. To demonstrate the validity of the method, results from several benchmark problems covering both subsonic and supersonic flow regimes are presented. Performance testing using 96 GPU devices demonstrates a speed up of 89 times that of a single GPU (i.e. 92% efficiency) for a benchmark problem employing 48 million cells. Discussions regarding communication overhead and parallel efficiency for varying problem sizes are also presented.

Implicit high-order flux reconstruction solver for high-speed compressible flows

The present paper addresses the development and implementation of the first high-order Flux Reconstruction (FR) solver for high-speed flows within the open-source COOLFluiD (Computational Object-Oriented Libraries for Fluid Dynamics) platform. The resulting solver is fully implicit and able to simulate compressible flow problems governed by either the Euler or the Navier–Stokes equations in two and three dimensions. Furthermore, it can run in parallel on multiple CPU-cores and is designed to handle unstructured grids consisting of both straight and curved edged quadrilateral or hexahedral elements. While most of the implementation relies on state-of-the-art FR algorithms, an improved and more case-independent shock capturing scheme has been developed in order to tackle the first viscous hypersonic simulations using the FR method. Extensive verification of the FR solver has been performed through the use of reproducible benchmark test cases with flow speeds ranging from subsonic to hypersonic, up to Mach 17.6. The obtained results have been favorably compared to those available in literature. Furthermore, so-called “super-accuracy” is retrieved for certain cases when solving the Euler equations. The strengths of the FR solver in terms of computational accuracy per degree of freedom are also illustrated and its computation cost per degree of freedom is assessed for different orders. Finally, the influence of the characterizing parameters of the FR method as well as the influence of the novel shock capturing scheme on the accuracy of the developed solver is discussed.

Effect of pressure-dilatation on energy spectrum evolution in compressible turbulence

The effect of internal-kinetic energy exchange on transient spectral energy transfer in compressible turbulence is investigated. We derive the spectral evolution equations for kinetic energy and pressure fields to highlight the key mechanisms that affect the turbulence spectral evolution. Direct numerical simulations of decaying isotropic turbulence are performed from solenoidal, dilatational, and mixed velocity initial conditions. It is shown that internal-kinetic energy exchange arising due to pressure-dilatation renders the dilatational kinetic energy amplitudes at large scales of motion to be oscillatory. The oscillatory behavior of amplitude diminishes with increasing wavenumber. The dilatational energy spectrum also exhibits a wider range of scales due to its inherent tendency to form shocks. The findings are expected to lead to an improved understanding of energy dynamics in high-speed compressible flows.

Numerical simulation of shock-disturbances interaction in high-speed compressible inviscid flow over a blunt nose using weighted essentially non-oscillatory scheme

In the present study, shock-disturbances interaction in high-speed compressible inviscid flow is simulated utilizing the weighted essentially non-oscillatory (WENO) scheme by employing the shock-capturing technique. For this aim, the two-dimensional Euler equations in strong conservative form are discretized by using the explicit third-order TVD Runge–Kutta scheme in time and the fifth-order WENO finite difference scheme in space. The main advantage of using the WENO scheme is its capability for properly solving the discontinuities in the domain without needing any artificial viscosity, limiter function or filter. Hence, this scheme is stable, and thus, it is suitable for simulating very high-speed flows with strong shocks. Three numerical tests are presented to demonstrate the remarkable capability of the WENO scheme and the corresponding numerical results are compared with the available analytical and numerical studies in the literature. The final and main test case is the steady and unsteady numerical simulation of the high-speed compressible inviscid flow past a two-dimensional circular cylinder. For this test case, firstly, the steady numerical solution at Mach=8.03 is provided. Then, the disturbances field is added to the upstream flow field and the unsteady numerical solution is performed for the fast and slow acoustic and also entropy waves. A sensitivity study is also performed to check the effects of the grid size and the type of flux splitting method on the solution. Both the steady and unsteady numerical solutions are compared with the existing analytical and numerical solutions and some conclusions are made.

An accurate and robust AUSM-family scheme on two-dimensional triangular grids

Most numerical schemes for solving high-speed compressible flow problems exhibit an instability that usually occurs inside the numerical shock structure in low-dissipative shock-capturing finite volume methods. In examining several test cases, the flux-difference splitting and the AUSM family of schemes cannot satisfy the robustness requirement, which manifests as the carbuncle phenomenon on two-dimensional triangular grids. This paper presents an accurate and robust AUSM-family scheme ( $$\hbox {AUSMDV}^+$$ scheme) that is verified against shock-induced anomalies on two-dimensional triangular grids. The linearized discrete analysis of an odd–even decoupling problem is applied to investigate the perturbation damping mechanism of these schemes. The corresponding recursive equations show that the $$\hbox {AUSMDV}^+$$ scheme is less sensitive to these anomalies than are other schemes in the AUSM family. Finally, the presented scheme is extended to achieve second-order solution accuracy. Its robustness and efficiency are then evaluated on both structured and unstructured triangular grids. The $$\hbox {AUSMDV}^+$$ scheme yields a physically meaningful solution for all test cases without introducing an additional shock fix technique.

History and Progress of Boundary-Layer Transition on a Mach-6 Flared Cone

Results from experimental, theoretical, and computational efforts are combined in a comprehensive review to summarize the improved understanding of the underlying physical mechanisms causing second-mode dominated laminar–turbulent transition on a 3 m circular-arc flared cone for the Boeing/U.S. Air Force Office of Scientific Research Mach 6 Quiet Tunnel (BAM6QT) conditions. Qualitatively similar patterns of streamwise streaks have been observed in experiments, using temperature sensitive paint, and in high-resolved “controlled” breakdown direct numerical simulations (DNS). The experimental and computational results using nonlinear parabolized stability equations (NPSEs) and DNS revealed that the maximum second-mode pressure fluctuation reaches extremely large values (30–60% of the surface pressure) before the breakdown to turbulence. These values are significantly higher as compared to what has been observed in incompressible flow, therefore providing an explanation for an elongated transition/breakdown regime in high-speed compressible flow. A sudden spectral broadening in the disturbance content has been observed in the three different approaches once the primary wave saturates nonlinearly, indicating a sudden nonlinear amplification of a broad spectrum of frequencies and wave numbers when transition begins. Good agreement was found between NPSE calculations and experiments when comparing the spectral broadening onset location, the total disturbance amplitude, and the amplitude ratio between the primary disturbances and the first harmonics, which shows that NPSEs can accurately reproduce/predict the experimental results. The DNS results revealed that the streaks on the surface of the cone are caused by steady streamwise vortices that are generated by nonlinear interaction of the primary and secondary disturbance wave. The similarities in the streak patterns suggest that the mechanisms observed in the controlled breakdown DNS are likely the same mechanisms leading to laminar–turbulent transition in the natural transition scenario observed in the BAM6QT. The more in-depth understanding of the transition process, and especially the understanding of the mechanisms leading to the streak patterns, can be used to explore potential flow control strategies to mitigate the detrimental effects of the massive heat overshoots over the turbulent values arising as a consequence of these streaks.

LOCALIZED RADIAL BASIS FUNCTIONS AND DIFFERENTIAL QUADRATURE-MESHLESS METHOD FOR SIMULATING COMPRESSIBLE FLOWS

A numerical approach based on the meshless method is used to simulate compressible flow. The meshless, or mesh-free, method circumvents the need to generate a mesh. Since there is no connectivity among the nodes, the method can be easily implemented for any geometry. However, one of the most fundamental issues in numerically simulating compressible flow is the lack of conservation, which can be a source of unpredictable errors in the solution process. This problem is particularly evident in the presence of steep gradient regions and shocks that frequently occur in highspeed compressible flow problems. To resolve this issue, a conservative localized meshless method based on radial basis functions and differential quadrature (RBF-DQ) has been developed. An upwinding scheme, based on the Roe method, is added to capture steep gradients and shocks. In addition, a blended RBF is used to decrease the dissipation ensuing from the use of low shape parameters. A set of test problems are used to confirm the accuracy and reliability of the algorithm, and the method applied to the solution of Euler's equation.

Numerical investigation of a chemically reacting and rarefied hypersonic flow field

Numerical simulations are carried out in the non-continuum flow regime to analyze flow features in the shock layer of a reentry vehicle. A new solver, rarefiedHypersonicFoam, has been developed based on the OpenFOAM platform, which can simulate the intermediate hypersonic reacting flow regime, where chemical non-equilibrium effects are imperative. The solver accommodates features to model air chemistry, multispecies transport, thermodynamic properties of high-temperature air, and non-equilibrium boundary conditions. The solver is validated with ballistic range experimental data for shock standoff distance and heat flux values over a conical reentry vehicle. Results have exhibited good agreement with the experimental data and show significant improvement when compared with the conventional high-speed compressible flow solver. The modified solver is used to analyze hypersonic flow over a bi-conic reentry capsule at different altitudes and velocities in the rarefied hypersonic flow regime. The results show that at lower altitude, chemical reactions absorb a considerable amount of heat compared to higher altitude. The rate of reaction reduces with the decrease in the flow velocity, which results in reduced heat flux values. It is observed that, if only rarefaction effects are considered in the solver, it overpredicts the heat flux values. Therefore, incorporation of chemical reactions while analyzing rarefied hypersonic flow fields is imperative.

Conduction, convection, and radiation thermal exchange with discretized fixed, contacting particles in a high speed compressible flow

A novel technique is proposed to simulate the thermal interactions occurring within a compressible fluid flow with particles. Thermal exchanges between particles, fluid, and flow paths including particle-particle and particle-solid body contact are included. Contact conduction with a contact resistance, convection, and radiation modes of heat transfer are accounted for without the need for heat transfer coefficient correlations or other simplifying assumptions since the particles are discretized within the fluid domain. The discretization of the particles in the flow allows for the local temperature and flow field data to be utilized in calculating the thermal interactions and facilitates the exploration of temperature dependent phenomena based on local instead of bulk interface conditions. So that these developed techniques can be applied to computational fluid dynamics simulations with moving objects, a fluid gap space is retained around the particles while the thermal conductive contact effects are simulated. The method is applied to an eleven fixed particle configuration with the flow driver strengths and limited radiative emissivities varied. The results demonstrate that for the conditions considered, the conductive contact dominates the temperature field development, with the radiative and convective cooling taking a lesser role. The new method clearly improves capabilities for the study of the thermal interaction in particle laden high speed gas flow and thus of temperature dependent phenomena for a more comprehensive simulation.

CFD Analysis of the Transonic Flow over a NACA 0012 Airfoil

Computational Fluid Dynamics (CFD) incorporates mathematical relations and algorithms to analyze and solve the problems regarding fluid flow. CFD analysis of an airfoil produces results such as lift and drag forces which determines the ability of an airfoil. In this paper a transonic flow will be modelled over a NACA 0012 airfoil for which experimental data has been published, so that a comparison can be made. The flow to be considered is compressible and turbulent and the solver used is the density based implicit solver, which gives good results for high speed compressible flows. The results show that the predicted lift, drag and pressure coefficients are in good agreement with experimental data.