Three-dimensional Supersonic Flow Research Articles

Assessment of Multi-Physical Fields Reconstruction Approaches in Three-Dimensional Supersonic Flow with Shocks

The paper evaluates the application of several multi-physical fields reconstruction approaches based on the Tomographic Particle Image Velocimetry (PIV) in three-dimensional supersonic flows with shock waves, including the pressure, temperature, and density. The MacCormack method and the Flux Vector Splitting method with outstanding performance in two-dimensional supersonic flow are extended into three-dimensional forms. The conventional Poisson method is also considered because of its widespread application and high accuracy in subsonic and transonic flow. All of these approaches are established by the conservative Navier-Stokes equations and solved by the time-marching iteration process, combined with the perfect gas law and adiabatic flow assumption. The performances are evaluated by numerical velocimetry data. To simulate the typical three-dimensional features, the cases of uniform freestream at Mach 0.78 and 3 past a cone with a 20° half-angle are selected to obtain a sufficient spanwise velocity component. The results confirm the feasibility of the PIV-based reconstruction methods in the conical flow field. The conventional Poisson method performs well only in the subsonic case while the Flux Vector Splitting method has a better performance in supersonic flow, including higher accuracy, stability, and efficiency, with a low-level root-mean-square error of 0.449% and local maximum relative error of 1%.

Assessment of volume penalization and immersed boundary methods for compressible flows with various thermal boundary conditions

A continuous forcing immersed boundary method (IBM) and a volume penalization method are investigated to simulate compressible viscous flows past an isothermal or adiabatic solid obstacle using a body non-conformal Cartesian grid. The present methods are validated on a large range of flow configurations such as incompressible and compressible flows (steady or unsteady) past isothermal or adiabatic solid obstacles. An adiabatic condition with implicit treatment of the source term is derived for the penalization method, taking advantage of the framework of the ghost-cell methods. Accuracy and computational cost of the methods are discussed. Three-dimensional supersonic flow configurations are carried out with the penalization method. The coupling between the porosity parameter and the different boundary conditions of the present penalization method is investigated in order to properly reproduce the wave reflection on a solid wall. Finally, a new penalization model is designed to improve the shock wave reflection on a solid surface while significantly reducing the computational cost compared to the previous porosity model.

Space-streamline-based method of characteristics for inverse design of three-dimensional super/hypersonic flows

The inverse design of super/hypersonic flows is widely utilized in aerospace, especially in waveriders, inlets, and nozzles. However, most of the existing methods are intended for the two-dimensional (2D) problem. The inverse method for generalized three-dimensional (3D) supersonic flows is still immature and is the main purpose of the space-streamline-based method of characteristics (SMOC) presented in this paper. The key of SMOC is to integrate an additional Euler predictor–corrector algorithm for pressure gradients in the unit calculation process. In this way, the temporary orientation of the osculating plane (OP) of the space streamline is determined, and the conventional 2D axisymmetric method of characteristics can be adopted in the OP. Three common unit processes of SMOC and the posedness are introduced, and the astringency is demonstrated by corresponding algebraic calculations. With this method, inviscid super/hypersonic flows can be solved on the basis of specified flow features, such as a 3D shock surface or a 3D wall pressure distribution. The accuracy and efficiency of SMOC are verified by using an inverse design example, that is, the flow produced by an elliptic conical surface at a freestream Mach number of 6. The numerical simulation of the inverse design result indicates that the 3D shock wave geometry and the 3D wall pressure distribution match the targets completely. The relative root-mean-squared error of the surface geometry is 10−3 magnitude, and the computation time cost of the inverse design is less than that of the general direct Euler solver.

Three-dimensional supersonic flows of Euler-Poisson system for potential flow

<p style="text-indent:20px;">We prove the unique existence of supersonic solutions of the Euler-Poisson system for potential flow in a three-dimensional rectangular cylinder when prescribing the velocity and the strength of electric field at the entrance. Overall, the main framework is similar to [<xref ref-type="bibr" rid="b1">1</xref>], but there are several technical differences to be taken care of vary carefully. And, it is our main goal to treat all the technical differences occurring when one considers a three dimensional supersonic solution of the steady Euler-Poisson system.</p>

Aero-optical suppression for supersonic turbulent boundary layer

Aero-optical reduction in flat-plate turbulent boundary layer has been widely concerned due to critical application for high-speed target-seeking vehicles. From the perspective of disciplinary intersection between aerodynamic and optical engineering, the turbulence-aberrated optical behaviour, in high-Reynolds-number supersonic freestream, is numerically investigated using the in-house code High-Order SimulaTor for Aerodynamics (HOSTA) and ray-tracing scheme. To evaluate the efficiency of wall cooling strategy to reduce the detrimental aero-optical distortions, the authors leverage the theoretical equation deduced from Sutton statistic model for mutual verification with the simulative results. Additionally, based on the beneficial wall cooling effects on turbulent aero-optical reduction, comparison suggests that refrigeration directly towards the turbulence section is the most effective rather than cooling the laminar or transitional region. In further discussion, wall blowing, suction and combinational blowing and suction schemes are issued. Results reveal the rationality of employing proper suction amplitude to modify the flow into laminar state, and sufficiently high suction intensity is also promising in turbulent aero-optical reduction despite the fact of low efficiency currently. The current study puts an eye on the simulation concerning three-dimensional supersonic turbulent flow in high-Reynolds-number freestream and the application of statistic model, and future investigation is suggested for higher effectiveness of aero-optical suppression.

Analytical method of nonlinear coupled constitutive relations for rarefied non-equilibrium flows

It is well known that Navier-Stokes equations are not valid for those high-Knudsen and high-Mach flows, in which the local thermodynamically non-equilibrium effects are dominant. To extend the non-equilibrium describing the ability of macroscopic equations, Nonlinear Coupled Constitutive Relation (NCCR) model was developed from Eu’s generalized hydrodynamic equations to substitute linear Newton’s law of viscosity and Fourier’s law of heat conduction in conservation laws. In the NCCR model, how to solve the decomposed constitutive equations with reasonable computational cost is a key ingredient of this scheme. In this paper, an analytic method is proposed firstly. Compared to the iterative procedure in the conventional NCCR model, the analytic method not only obtains exact roots of the decomposed constitutive polynomials, but also preserves the nonlinear constitutive relations in the original framework of NCCR methods. Numerical tests to assess the efficiency and accuracy of the proposed method are conducted for argon shock structures, Couette flows, two-dimensional hypersonic flows over a cylinder and three-dimensional supersonic flows over a three-dimensional sphere. These superior advantages of the current method are expected to render itself a powerful tool for simulating the hypersonic rarefied flows and microscale flows of high Knudsen number for engineering applications.

Affordable schlieren visualization methods for understanding three-dimensional supersonic flows

This article discusses affordable schlieren methods which can visualize hidden details in three-dimensional high-speed flow. Mainly, two schlieren methods are brought to light, namely inclined schlieren and focusing schlieren. These two methods have been applied to visualize three-dimensional flow fields like normal shock boundary layer interaction in ducts and flows with shock–shock interactions. The paper explains the possibility of measuring normal shock oscillation in spanwise/transverse direction. Shock height variation across the duct span is also illustrated, using inclined schlieren. Scanning of the focusing schlieren setup can give planar views of three-dimensional shock–shock interactions. It is also shown that the two methods can be merged to give the inclined focusing schlieren method which can also be used to understand three-dimensional flow fields. Further, the possibilities and limitations of these flow visualization methods have been discussed.

Skin Friction Measurements Using Oil Film Interferometry in a 3-D Supersonic Flowfield

The application of oil film interferometry (OFI) for the measurement of global skin friction on the surface beneath a complex three-dimensional supersonic flow is explored in this study. The flowfield is produced by a fin-generated oblique shock interacting with a compressible turbulent boundary layer. Measurements were obtained by modifying a comprehensive acquisition and processing software, previously developed for two-dimensional, low-speed flows, to measure skin friction distributions underneath such flows. This approach required the local flow direction on the surface, and therefore traditional oil flow visualization was coupled with OFI measurements. A cross-correlation-based algorithm was incorporated to extract flow direction from sequences of oil flow visualization images. Photogrammetry was used with both surface oil flow and OFI to ensure proper spatial overlap for the two measurements. Results from this study demonstrate the capability of the approach in a complex, supersonic flow field that contains regions of both separation and reattachment concomitant with a wide range of skin friction values. The skin friction results show excellent homogeneity, standard deviation of 5.1% of the mean, in the undisturbed boundary layer upstream of the interaction and capture well-resolved high shear gradients in the interaction region. A comparison to legacy data from a similar interaction shows matching trends within the developed flow.

A general use adjoint formulation for compressible and incompressible inviscid fluid dynamic optimization

A discrete adjoint formulation with an ad hoc flow solver has been recently developed and tested on transonic inviscid flow optimization problems. In the present paper the formulation is extended to various compressible flow solvers as well as to a solver for incompressible flows. An approximate, dissipative flow solver is used to develop the discrete quasi-time-dependent adjoint equations. The design problem employs a progressive optimization, i.e., a sequence of operations, containing a partially converged flow solution, followed by an adjoint solution followed by an optimization step. The procedure is performed using a sequence of progressively finer grids. The sensitivity derivative variations are limited to preserve the smoothness of the progressive procedure. First, the adjoint equations are coupled with an accurate in-house flow solver to test the approach on some inverse design problems involving two- and three-dimensional supersonic, transonic, subsonic and incompressible flows. Then, the previous design test cases are re-computed coupling the extended adjoint formulation with commercial and open source flow solvers, without noticing any relevant difference in the optimization convergence histories. Finally, incompressible design test cases are successfully computed coupling the extended adjoint formulation with a commercial solver for incompressible flows. The extended compressible adjoint formulation appears to have a wide application, insofar as it allows to perform accurate and efficient design optimization using different flow conditions, different flow solvers and even a solver for incompressible flows.

On the Hysteresis of Aerodynamic Characteristics of a Cylinder with a “Fluid Flare” Immersed in a Supersonic Three-Dimensional Flow

By using the model of an ideal inviscid perfect gas, the aerodynamic characteristics of a cylinder immersed in a stationary three-dimensional supersonic flow in the case of strong axisymmetrical blowing of air from the lateral surface into the shock layer (“fluid flare”) are investigated. The aim of the present work is to compare the dependences of the aerodynamic characteristics of the cylinder with a “fluid flare” on the angle of attack on increase of the latter from zero to 10o(straight motion) and on its further decrease from 10° to zero (backward motion).

Экспериментальное исследование сверхзвукового трехмерного обтекания осесимметричного тела с кольцевым выступом на поверхности

Экспериментальное исследование сверхзвукового трехмерного обтекания осесимметричного тела с кольцевым выступом на поверхности

Influence of vorticity distribution on singularities in linearized supersonic flow

The linearized steady three-dimensional supersonic flow can be analyzed using a vector potential approach which transforms the governing equation to a standard form of two-dimensional wave equation. Of particular interest are the canonical horseshoe line-vortex distribution and the resulting induced velocity field in supersonic flow. In this case, the singularities are present at the vortex line itself and also at the surface of the cone of influence originating from the vertices of the horseshoe structure. This is a characteristic of the hyperbolic nature of the flow which renders the study of supersonic vortex dynamics a challenging task. It is conjectured in this work that the presence of the singularity at the cone of influence is associated with the step-function nature of the vorticity distribution specified in the canonical case. At the phenomenological level, if one considers the three-dimensional steady supersonic flow, then a sudden appearance of a line-vortex will generate a ripple of singularities in the induced velocity field which convect downstream and laterally spread, at the most, to the surface of the cone of influence. Based on these findings, this work includes an exploration of potential candidates for vorticity distributions that eliminate the singularities at the cone of influence. The analysis of the resulting induced velocity field is then compared with the canonical case, and it is observed that the singularities were successfully eliminated. The manuscript includes an application of the proposed method to study the induced velocity field in a confined supersonic flow.

Nonlinear Instability in the Region of Laminar-Turbulent Transition in Supersonic Three-Dimensional Flow over a Flat Plate

Direct numerical simulation is applied to obtain laminar-turbulent transition in supersonic flow over a flat plate. It is shown that, due to the nonlinear instability, Tollmien–Schlichting waves generated in the boundary layer lead to the formation of oblique disturbances in the flow. These represent a combination of compression and expansion waves, whose intensities can be two orders higher than that of external harmonic disturbances. The patterns of the three-dimensional flow over the plate are presented and the structures of the turbulent flat-plate boundary layers are described for the freestream Mach numbers M = 2 and 4.

A combined ghost-point-forcing / direct-forcing immersed boundary method (IBM) for compressible flow simulations

The present manuscript describes the implementation of an immersed boundary method (IBM) into a compressible reactive multicomponent Navier-Stokes solver. The latter makes use of high-order accuracy discretization schemes applied on structured Cartesian grids. For the immersed boundary method, both direct-forcing and ghost-point-forcing schemes are considered. The direct-forcing source term, which is added to the three components of the Navier–Stokes (NS) equations, i.e. the momentum equations, is used to impose the no-slip boundary conditions at the fluid-body interface. It is combined with a ghost-point-forcing scheme to enforce both mass conservation and boundary conditions in the scalar equations (density, energy and species mass fractions). The discretization of the three-dimensional surface associated to the solid geometry is obtained from a triangulation, which is handled within the stereo-lithography (STL) data format. The immersed boundary is thus represented with a finite number of Lagrangian points, which are associated to the centers of gravity of the elementary triangles. The major components of the IBM procedure include the data structure itself, the determination of the status of the computational nodes, i.e. fluid, ghost or solid, the interpolation rules and the forcing model. The performance of the resulting immersed boundary scheme is investigated by considering a large set of two-dimensional benchmarks including a cylinder placed in either subsonic or supersonic flows, a subsonic airflow on a flat plate, moving-shock interactions with various obstacles. Numerical results are satisfactorily compared with available theoretical, experimental, and numerical data. The capabilities of the method are then illustrated on more representative flowfields such as two-dimensional non-reactive and reactive high-speed flows developing over rectangular cavities and three-dimensional supersonic flows over a rigid body.

Shock-conforming mesh generation for aerodynamic analyses at supersonic regimes

A shock estimation approach is proposed in the context of the generation of shock-conforming meshes for the numerical analysis of inviscid, steady, supersonic and hypersonic flows. For given flow conditions and vehicle’s geometry, the method provides a fast estimation of the shock waves pattern such that grid points can be clustered along the shock waves in a judicious manner. In this way, the uncertainty on mesh generation for shock-dominated flows is reduced and the use of adaptive mesh refinement could be made more efficient or, in some cases, even considered not necessary. The approach is verified against two- and three-dimensional supersonic flows for conceptual exemplary geometries like wedges and revolution bodies and more real-world vehicles configurations like rockets and hypersonic aircraft. Qualitative and quantitative assessment of the solution-mesh pair quality is proposed to evaluate the quality of the resulting shock-conforming meshes.

Explicit discontinuous spectral element method with entropy generation based artificial viscosity for shocked viscous flows

A spatio-temporal adaptive artificial viscosity (AV) based shock-capturing scheme is proposed for the solution of both inviscid and viscous compressible flows using a high-order parallel Discontinuous Spectral Element Method (DSEM). The artificial viscosity and artificial thermal conduction coefficients are proportional to the viscous and thermal entropy generating terms, respectively, in the viscous entropy conservation law. The magnitude of AV is limited based on the explicit stable CFL criterion, so that the stable artificial viscous time step size is greater than the convective stable time step size. To further ensure the stability of this explicit approach, an adaptive variable order exponential filter is applied, if necessary, in elements where the AV has been limited. In viscous flow computations a modified Jameson's sensor (Ducros et al., 1999 [61]) limits the AV to small values in viscous shear regions, so as to maintain a high-order resolution in smooth regions and an essentially non-oscillatory behavior near sharp gradients/shocks regions. We have performed a systematic and extensive validation of the algorithm with one-dimensional problems (inviscid moving shock and viscous shock-structure interaction), two-dimensional problems (inviscid steady and unsteady shocked flows and viscous shock-boundary layer interaction), and a three-dimensional supersonic turbulent flow over a ramped cavity. These examples demonstrate that the explicit DSEM scheme with adaptive artificial viscosity terms is stable, accurate and efficient.

Supersonic steady aerodynamics of a cylinder with a fluid flare

Basing upon the ideal inviscid perfect gas model the aerodynamic characteristics and the gasdynamic function distributions over the surface of a cylinder in a steady supersonic three-dimensional flow are studied in the absence and the presence of combined intense air injection from the flat-ended nose and the lateral surface of the body into the shock layer. The purpose of the study is to investigate the influence of gas injection from different regions of the cylinder surface on supersonic flow past the cylinder at different angles of attack.

Three-dimensional supersonic flow around double compression ramp with finite span

Three-dimensional flows of Mach number 3 around a double-compression ramp with finite span have been investigated numerically. Shadowgraph visualisation images obtained in a supersonic wind tunnel are used for comparison. A three-dimensional Reynolds-averaged Navier–Stokes solver was used to obtain steady numerical solutions. Two-dimensional numerical results are also compared. Four different cases were studied: two different second ramp angles of $${30}^{\circ }$$ and $${45}^{\circ }$$ in configurations with and without sidewalls, respectively. Results showed that there is a leakage of mass and momentum fluxes heading outwards in the spanwise direction for three-dimensional cases without sidewalls. The leakage changed the flow characteristics of the shock-induced boundary layer and resulted in the discrepancy between the experimental data and two-dimensional numerical results. It is found that suppressing the flow leakage by attaching the sidewalls enhances the two-dimensionality of the experimental data for the double-compression ramp flow.

Investigation of Vortex Structures for Supersonic Jet Interaction Flowfield

The three-dimensional supersonic turbulent flow in presence of symmetric transverse injection of round jet is simulated numerically. The simulation is based on the Favre-averaged Navier-Stokes equations coupled with Wilcox’s turbulence model. The numerical solution is performed using ENO scheme and is validated with the experimental data that include the pressure distribution on the wall in front of the jet in the plane symmetry. The numerical simulation is used to investigate in detail the flow physics for a range of the pressure ratio . The well-known primary shock formations are observed (a barrel shock, a bow shock, and the system of λ-shock waves), and the vortices are identified (horseshoe vortex, an upper vortex, two trailing vortices formed in the separation region and aft of the bow shock wave, two trailing vortices that merge together into one single rotational motion). During the experiment the presence of the new vortices near the wall behind the jet for the pressure ratio is revealed.

Vortical structures behind a transverse jet in a supersonic flow at high jet to crossflow pressure ratios

A three-dimensional supersonic turbulent flow with symmetric normal injection of circular jets from the channel walls is numerically simulated. The initial Favre-averaged Navier–Stokes equations closed by the k–ω turbulence model are solved by an algorithm based on an ENO scheme. The mechanism of the formation of vortical structures due to the interaction of the jet with the free stream is studied for jet to crossflow total pressure ratios ranging from 3 to 50. It is known from experiments reported in the literature that, for n ⩾ 10, mixing of the jet with the high-velocity flow leads to the formation of a pair of vortices and of an additional separation zone near the wall behind the jet. It is demonstrated that the present numerical results are consistent with such findings and that the pressure distribution on the wall ahead of the jet in the plane of symmetry is also in reasonable agreement with available experimental data.