Unsteady Compressible Navier-Stokes Equations Research Articles

Novel adjustable backflow structure in electric coolant pump for balancing hydraulic performance and reducing temperature rise

An electric coolant pump serves as the source for coolant circulation in the thermal management of electric vehicles. To address the thermal issues for the printed circuit board integrated within the electric coolant pump, a novel backflow structure with an adjustable orifice diameter is designed from the impeller outlet to the printed circuit board to promote convective heat dissipation. Eventually, this flow returns to the impeller inlet, thereby altering the hydraulic performance of pump. The balance between hydraulic performance and temperature rise is achieved by adjusting the orifice diameter. Considering the variations in coolant properties with respect to temperature, numerical simulations of the electric coolant pump are performed using the unsteady compressible Reynolds-Averaged Navier-Stokes Equations. There is good agreement between experimental and numerical simulations results in both cases with and without an orifice. On an average, by increasing the orifice diameter by 1.5 mm which results in a 1.11 % decrease in efficiency, a 0.008 decrease in pressure coefficient, and a reduction of temperature by 2.98 K on printed circuit board. As the orifice diameter increases, leakage losses in the rear side chamber increases with the improvement of heat dissipation. The clash between the backflow jet and the inflow from the suction pipe results in significant entropy generation, forming jet vortices and increasing in pressure fluctuation. However, when the orifice diameter exceeds 3 mm, the cooling advantages no longer justify the trade-off in hydraulic performance. Thus, 3 mm orifice diameter is an optimal compromise between hydraulic performance and temperature control.

Direct numerical simulations of long-range infrasound propagation: Implications for source spectra estimation.

The evolution of observed dominant frequencies from a high-intensity infrasonic pulse with receiver range and stratospheric temperature is investigated using direct numerical simulations of the two-dimensional unsteady compressible Navier-Stokes equations. There is a high level of uncertainty in estimating source dominant frequencies based on received signals at sparse points on the ground. Nonlinear propagation effects in the ground-level thermospheric arrivals are found to significantly alter dominant frequency measurements compared to stratospheric arrivals with smaller amplitude sources. With a larger amplitude source, variations in observations are minimized as a result of nonlinear effects being ubiquitous across all atmospheric components of received signals but have a greater offset to the source dominant frequency. An approach to determine the source dominant frequency and minimize atmospheric variability is presented by calculating a source-to-receiver spectral transfer function averaged across the atmospheric states. This method reduces atmospheric variability in source frequency estimates within the pseudo-linear propagation regime and the average error to the known source frequency with a large amplitude source. The reduction of errors in source frequency estimates demonstrates the feasibility of using remote infrasound measurements as an indicator of source frequency and, in turn, the explosive yield of clandestine nuclear weapon test explosions.

Numerical investigation of the self-induced unsteadiness phenomenon in the impeller of a high-loading centrifugal compressor

Unsteady flow phenomena are common in turbomachinery and have serious impacts on its stable operation. This paper reports a self-induced unsteady phenomenon inside a centrifugal impeller due to strong secondary flow that has not been previously reported. A high-loading centrifugal impeller is selected as the research object, an accurate three-dimensional model is established, and exhaustive numerical simulations are performed based on unsteady compressible Navier–Stokes equations. The unsteady performance of the compressor is measured in detail during the experiment, and the experimental results are compared with the numerical simulation results to verify the reliability of the numerical simulation method. The results of the unsteady pressure signal show the existence of unsteady phenomenon when the operating point is far away from the choke boundary. The power spectrum results of the unsteady pressure distribution at the impeller blade tip are analyzed, and two different modes of this unsteady phenomenon are confirmed. Higher-order frequency characteristics are not observed near the stall boundary. The detailed analysis of the flow field inside the impeller clarifies that the instability of the interface formed between the secondary flow and the main flow induces this unsteady phenomenon. This unsteady pressure disturbance propagates inside the blade passage through pressure waves, which contributes to the instability of the interface and the existence of this self-induced unsteady phenomenon.

Parametric instability in propagating flames and the impact of hydrogen enrichment on the onset of secondary instability

A self-excited acoustic instability of laminar premixed flames propagating in an open-ended tube with a length of 700 mm and a radius of 10 mm was simulated by solving the reacting unsteady compressible Navier–Stokes equations, to understand the way of massive acoustic generation and its onset behaviors. Four fuel–air mixtures with an equivalence ratio of 1.2 were considered, namely, methane–air and methane–hydrogen–air mixtures, to identify the role of hydrogen in rich methane–air mixture. Parametric instability, which generated huge acoustic disturbance and violent flame pulsations, was observed only for a particular methane–hydrogen–air mixture with RH = 0.2, consistent with previously reported experimental observations. For the investigation of the reinforcement mechanism of acoustic instability under parametric instability, the flame surface area modulation was examined. It was found that violent subharmonic flame front pulsations could strongly modulate the flame surface area in the fundamental mode, resulting in a fluctuating heat release rate and increased thermoacoustic coupling. When hydrogen addition was small, attaining a higher level of primary instability, which is the precursor of the parametric instability, was more dominant than increasing the threshold level for the onset of the parametric instability. With larger hydrogen addition, the increase in the threshold level was more dominant than attaining a higher level of the primary instability. In particular, as the flame propagation time decreased, the level of the primary instability was saturated in larger hydrogen addition. This study elucidates the mechanism for the acoustic generation of propagating flames under the parametric instability, and the effects of hydrogen enrichment within rich methane–air mixtures.

Homogenization of the unsteady compressible Navier-Stokes equations for adiabatic exponent γ > 3

We consider the unsteady compressible Navier-Stokes equations in a perforated three-dimensional domain, and show that the limit system for the diameter of the holes going to zero is the same as in the perforated domain provided the perforations are small enough. The novelty of this result is the lower adiabatic exponent γ>3 instead of the known value γ>6. The proof is based on the use of two different restriction operators leading to two different types of pressure estimates. We also discuss the extension of this result for the unsteady Navier-Stokes-Fourier system as well as the optimality of the known results in arbitrary space dimension for both steady and unsteady problems.

角柱近傍を移動する渦対による音波発生について

The sound generated by a vortex pair moving close to a rectangular cylinder is studied by high-resolution simulations of the two-dimensional unsteady compressible Navier-Stokes equations at Mach numbers ranging from 0.05 to 0.3. In this study, the Mach number Ma is defined by Ma=Vt /c0, where Vt is the translation velocity of the co-moving vortex pair and c0 denotes the speed of sound. The directly computed far-field sound is compared to the prediction of the acoustic analogy proposed by Lighthill (1952) and Curle (1955). The prediction is in excellent agreement with the simulation, where the two-dimensional form of Curle’s modified solution is adopted for an accurate calculation of the sound waves with a phase difference between various vortices around the cylinder. It could be confirmed that the scaling law in two dimensions holds in this simulation. In this study, the sound generation due to the dipole and quadrupole fields is investigated by calculating the two terms in Curle’s analogy. Examining the surface distribution of dipoles on the cylinder indicates that the force exerted on the fluid by the solid boundaries of the side/upper wall of the rectangular cylinder generates the sound when the co-translating vortices approach /pass the rectangular cylinder, which is the sound due to dipoles corresponding to the directivity of the sound in the polar diagrams. The area distribution of quadrupoles which is the second time derivative of the area integral of Lighthill’s stress tensor indicates that the sound is emitted from a vortex generated at the corner of the rectangular cylinder when the vortex pair approaches the cylinder and where the interaction between many vortices around the cylinder and the co-translating vortices generates the sound when the co-translating vortices pass the cylinder.

Numerical Investigation on Transient Effect of Jet Interference Characteristics of Lateral-Jet-Controlled Spinning Missile

At present, there is no publicly published research on the unsteady interference effect in the start-up process of the lateral jet control of the spinning missile. The variation of aerodynamic characteristics during the jet start-up process of the spinning missile is still unclear. Therefore, the unsteady numerical method based on the three-dimensional unsteady compressible Navier–Stokes equations and the sliding mesh method is used to study the unsteady jet interference characteristics of the spinning missile during the starting process of the lateral jet. Based on the verification of the numerical simulation method in this paper, the jet interference flow field under the conditions of non-rotation and rotation is simulated, and the variation of the aerodynamic characteristics of the missile under the two conditions is given. The influence of rotation on the unsteady aerodynamic characteristics of the lateral-jet-controlled spinning missile is analyzed. The flow mechanism resulting in the change of the jet control characteristics and the lateral aerodynamic characteristics of the missile is analyzed through the interference flow field structure at different moments after the jet starts. The results indicate that in the start-up process of pulse jet control, the jet interference characteristics on the fins have a delay effect compared with the projectile body. The duration of the unsteady effect caused by the high-pressure region upstream of the nozzle is shorter than that caused by the low-pressure region downstream of the nozzle. The flow separation and reattachment near the nozzle have strong unsteady characteristics. The jet wake has the most obvious interference effect on Fin1. The pressure on the side of the rotation direction of Fin1 increases, while the opposite side is in contrast.

Influence of gas expansion on the velocity and stability limits of stationary curved flames in channels

ABSTRACT The influence of gas expansion on the velocity of freely propagating curved stationary premixed flames in channels with free-slip adiabatic walls is studied utilizing 2D computational simulations, with the focus on the strongly nonlinear effects of the Darrieus-Landau instability on the stationary flame shape and velocity. The complete system of unsteady compressible Navier–Stokes equations is solved, including the transport properties (heat conduction, diffusion, and viscosity), and the chemical kinetics in the form of a one-step irreversible Arrhenius reaction. The parametric study focuses on the gas expansion, with a wide yet realistic range of expansion coefficients employed, and on the channel width. It is shown that the stationary curved flame velocity increases with gas expansion coefficient, in line with previous studies. The second critical channel width, at which the stationary curved flame velocity exceeds the one predicted by weakly nonlinear analyses and starts to grow rapidly with channel width, is identified with a sufficient accuracy for a wide range of gas expansion coefficients. The critical channel width for the Darrieus-Landau instability development depends on the gas expansion coefficient non-monotonically. The second critical channel width is found to decrease with the gas expansion coefficient, in a qualitative agreement with previous theoretical predictions.

Adaptive local discontinuous Galerkin methods with semi-implicit time discretizations for the Navier-Stokes equations

In this paper, we present a mesh adaptation algorithm for the unsteady compressible Navier-Stokes equations under the framework of local discontinuous Galerkin methods coupled with implicit-explicit Runge-Kutta or spectral deferred correction time discretization methods. In both of the two high order semi-implicit time integration methods, the convective flux is treated explicitly and the viscous and heat fluxes are treated implicitly. The remarkable benefits of such semi-implicit temporal discretizations are that they can not only overcome the stringent time step restriction compared with time explicit methods, but also avoid the construction of the large Jacobian matrix as is done for fully implicit methods, thus are relatively easy to implement. To save computing time as well as capture the flow structures of interest accurately, a local mesh refinement (h-adaptive) technique, in which we present detailed criteria for selecting candidate elements and complete strategies to refine and coarsen them, is also applied for the Navier-Stokes equations. Numerical experiments are provided to illustrate the high order accuracy, efficiency and capabilities of the semi-implicit schemes in combination with adaptive local discontinuous Galerkin methods for the Navier-Stokes equations.

A high-order partitioned fluid-structure interaction framework for vortex-induced vibration simulation

Vortex-induced vibration (VIV) is a highly nonlinear fluid-structure interaction (FSI) phenomenon, and its accurate prediction remains challenging. In this study, we developed a high-order partitioned FSI framework towards accurate simulation of the VIV phenomenon. The focus is on simulating the two-way coupled interaction between a flexibly mounted rigid body and its surrounding fluids. In the FSI framework, a high-order computational fluid dynamics method, i.e., flux reconstruction/correction procedure via reconstruction, is utilized to simulate the unsteady compressible Navier–Stokes equations in the arbitrary Lagrangian Eulerian format on unstructured moving grids. A linearized fluid-solid Riemann solver is employed to weakly couple fluid dynamics and rigid-body dynamics on the fluid-solid interface, which can effectively mitigate the so-called added-mass instability. Time marching is conducted through either the linearly implicit Rosenbrock–Wanner method or the explicit strong stability preserving Runge–Kutta method. The high-order FSI framework is verified with a challenging VIV phenomenon, i.e., VIV of a zero-mass cylinder in viscous flows, and has been used to study two VIV-related problems, namely, the nonlinear energy sink and oscillating foil based energy harvesting mechanism.

Wall-Resolved Large-Eddy Simulation of Smooth-Body Separated Flow

Wall-resolved large-eddy simulations were carried out for flow over a rearward-facing ramp configuration, which is representative of surfaces generating smooth-body separation. The geometry and flow conditions were motivated by an experimental investigation, which was conducted in order to provide data for validating numerical modelling. Because the experimental Reynolds number is challenging, even for approximate techniques, the present computations were initiated in order to provide benchmark results for a comparison of numerical methods. Solutions to the unsteady three-dimensional compressible Navier-Stokes equations were obtained utilising a high-fidelity computational scheme and an implicit time-marching approach. Grid resolution studies were performed to assure numerical accuracy. A number of metrics are indicated that attest to quality of the solutions. Features of the flowfields are elucidated, and comparisons are made between results at two different Reynolds numbers. A description of boundary-layer parameters and development of turbulence along the ramp surface are also provided.

New time-marching methods for compressible Navier-Stokes equations with applications to aeroacoustics problems

This paper derives a new class of time-marching methods of Runge-Kutta type (CERK) for the simulations of the two and three-dimensional compressible Navier-Stokes equations. Despite being implicit, the developed CERK methods do not require any numerical or analytical inversion of the coefficient matrix computationally explicitly. The efficiency and robustness of the developed methods are validated by solving the convection-diffusion equation and the unsteady compressible Navier-Stokes (NS) equations, which display stiff dynamical behavior at low Mach numbers. The performance of the developed methods is also compared with the representative explicit and implicit time-marching methods discussed in the literature. Several benchmark problems in computational aeroacoustics are analyzed by solving the NS equations using the developed time-marching methods. The computed results display an excellent match with the numerical and experimental results reported in the literature. For the computational aeroacoustics (CAA) problems, the computational costs required for the present methods are also compared with the methods noted in the literature.

On the development, verification, and validation of a discontinuous Galerkin solver for the Navier-Stokes equations

We present maDG, a newly developed code designed for the efficient parallel implicit solution of the 3D unstructured nodal discontinuous Galerkin discretization of the unsteady compressible Navier-Stokes equations. The code is being developed to provide an efficient, modular, and maintainable software platform for studying algorithmic and modeling issues arising in high-resolution CFD. In this paper we address some issues pertinent to the software architecture and testing implementation used towards the goal of implicit time integration of discontinuous Galerkin (DG) approximations.This effort is largely inspired by the life work of Antony Jameson and by his legendary FLO codes, which for more than four decades have provided a unique platform for the development of CFD methods, and enabled path-breaking advances in aerodynamics. We dedicate this paper to Antony on the occasion of his 85th birthday, with the utmost gratitude for his teaching and his friendship.

On the development, verification, and validation of a discontinuous Galerkin solver for the Navier–Stokes equations

We present maDG, a newly developed code designed for the efficient parallel implicit solution of the 3D unstructured nodal discontinuous Galerkin discretization of the unsteady compressible Navier–Stokes equations. The code is being developed to provide an efficient, modular, and maintainable software platform for studying algorithmic and modeling issues arising in high-resolution CFD. In this paper we address some issues pertinent to the software architecture and testing implementation used towards the goal of implicit time integration of discontinuous Galerkin (DG) approximations.This effort is largely inspired by the life work of Antony Jameson and by his legendary FLO codes, which for more than four decades have provided a unique platform for the development of CFD methods, and enabled path-breaking advances in aerodynamics. We dedicate this paper to Antony on the occasion of his 85th birthday, with the utmost gratitude for his teaching and his friendship.

Effect of cross passage on aerodynamic characteristics of super-high-speed evacuated tube transportation

Based on 3- D, unsteady compressible Navier-Stokes equations and SST k-ω two-equation IDDES turbulence model, the effect of cross passage on aerodynamic characteristics of evacuated tube transportation is simulated by overset mesh technology. The flow structure, aerodynamic force and aerothermal environment in three train/tube configurations are studied. The results show that there is a linear relationship between the length of the disturbed region of normal shock wave and running time. Cross passage can reduce both aerodynamic drag and aerothermal environment in the case of a single train passing. The transport channel and diffusion channel are formed to transport the air from tube A (B) to tube B (A). However, the lateral force fluctuates greatly. After the train leaves the cross passage, the lateral force coefficient periodically decays and oscillates. While, in the case of two train crossing, the existence of the cross passage could aggravate pressure field and aerothermal environment in the tube. Besides, the aerodynamic drag coefficient fluctuates during the intersection of two trains and the lateral force coefficient of the train fluctuates greatly and the maximum lateral force coefficient of two trains crossing is 2.86 times that of a single train passing.

A numerical method for multi-body separation with collisions

Multi-body separation with collisions is ubiquitous in aerospace engineering. In this paper, a numerical method for this problem is developed. Coupling with the equations for six-degree-of-freedom rigid-body motion, the unsteady compressible Reynolds-Averaged Navier-Stokes equations are solved. The dynamic overset unstructured grid based on the criteria of wall distance is developed to handle the relative motion of multiple bodies. The collision among rigid bodies is modeled by using an impulse-based method. Efficient algorithms for collision detection and contact determination utilizing hole cutting information of overset unstructured grid are proposed. In the simulation of six-degree-of-freedom rigid-body motion, a collision event can be robustly detected and the collision time can be accurately determined, without needing additional data structures. Test cases of store separation and ball collision are used to validate the accuracy of the dynamic solvers and the collision algorithms. An application to the release of a large number of projectiles from a carrier assesses the capability of the present method in simulating the problems of complex multi-body separation with collisions.

Evaluation of breakup models for liquid side jets in supersonic cross flows

In this study, numerical simulations of the interaction between supersonic crossflow and a side liquid jet were performed. The processes of the liquid jet breakup, flow separation and shock waves resulting from the interaction of the gas flows and liquid droplets were predicted. In the numerical simulation, a two-way coupling Eulerian-Lagrangian method was used. The two-dimensional unsteady compressible Navier-Stokes Equations were solved based on the MUSCL type (Monotone Upstream-Centred Scheme for Conservation Laws) spatial differencing coupling with Roe-type flux-difference splitting and a third-order accurate Runge-Kutta method for time-splitting. The atomization of liquid droplets was simulated using the combined KH-RT breakup model. The modified breakup model considered both the interface instability and the sudden deformation of a liquid spray induced by the compressible flows. Shock waves, recirculation zones and breakup processes of droplet particles were captured. The predicted droplet distributions at various operating conditions were discussed. Satisfied agreement between simulations and measurements is shown in the current work.

Numerical investigation of compressibility effect on dynamic stall

A rotorcraft main-rotor blade experiences a broad range of the Mach number in high-speed forward flights or rapid maneuvers. Near the boundary of the flight envelope, the rotor blade often encounters severe dynamic stall which limits the overall aerodynamic performance. The effects of the flow compressibility on dynamic stall due to the variation of the freestream Mach number are studied here. A rotor airfoil, VR-12, is computationally investigated for both static and dynamic stall conditions in the relevant range of the Mach number, 0.2 to 0.4. For the small enough Mach number such as 0.2, both static and dynamic stalls are significantly influenced by flow separation due to adverse pressure gradient with little compressibility effect. For the highest Mach number 0.4, compressibility near the leading edge is no longer negligible, forming a local supersonic pocket there. The shock-induced separation occurs near the leading edge, which dominates both the static and dynamic stall at the high Mach number. At the intermediate Mach number 0.3, it is observed that both adverse pressure gradient and weak shock affect the stall. Unsteady compressible Reynolds-averaged Navier-Stokes equations are numerically solved with the Spalart-Allmaras turbulence model in the current study. Numerical results are compared to relevant wind-tunnel test data.

A hybrid discontinuous spectral element method and filtered mass density function solver for turbulent reacting flows

We present a novel hybrid scheme for the large eddy simulation (LES) of turbulent reacting flows. The scheme couples the discontinuous spectral element method (DSEM) solver for the unsteady compressible Navier-Stokes equations with a Monte Carlo particle filtered mass density function (FMDF) solver for the transport of reacting species. The method is capable of high-order simulations on unstructured grids. Mean particle estimate construction mimics the DSEM numerical procedure and utilizes variable basis functions. The scheme is tested on non-reacting and reacting Taylor-Green vortex flows. Studies of varying polynomial order, different basis functions for constructing particle estimates, and varying particle quantities are conducted. We demonstrate that a tent kernel, in conjunction with high polynomial order, produces the most accurate results. The chemically reacting simulations validate the hybrid scheme and demonstrate its applicability across a range of reaction regimes. The hybrid scheme's computational cost is 2.1 times the DSEM-LES solver.

Sound generation by a pair of co-rotating vortices using spectral acoustic analogy

The spectral acoustic analogy (AA) theory is used to study the acoustic generation by a pair of co-rotating vortices from the decomposition of wave packets. The use of a two-dimensional spectral AA formulation is based on the data obtained by conducting a direct simulation of unsteady compressible Navier-Stokes equations. By way of the spectral AA theory and convolution filtering operations, the role played by the wave packets of the vortices in the generation of the sound is thoroughly discussed. It is found that the far-field sound is generated by the linear superimposition of the interactions between the base flow and the remaining wave packets. Moreover, a notable wave packet of the co-rotating vortices, called the acoustic-related wave packet herein, is identified in the spectral domain, which not only contains all radiating components but also contributes to a discernible acoustic source. It potentially bridges the gap between the near-field wave packets and the far-field sound. The present methodology based on the spectral AA theory shows promise in providing insights into the mechanisms of sound generation in turbulent or unsteady vortical flows.