Compressible Navier-Stokes Equations Research Articles

Gauging Centrifugal Instabilities in Compressible Free-Shear Layers via Nonlinear Boundary Region Equations

Curved free shear layers emerge in many engineering problems involving complex flow geometries, such as the flow over a backward-facing step, flows with wall injection in a boundary layer, the flow inside side-dump combustors, or wakes generated by vertical axis wind turbines, among others. Previous studies involving centrifugal instabilities have mainly focused on wall-flows where Taylor instabilities between two rotating concentric cylinders or Görtler vortices in boundary layers are generated. Curved free shear layer flows, however, have not received sufficient attention, especially in the nonlinear regime. The present work investigates the development of centrifugal instabilities in a curved free shear layer flow in the nonlinear compressible regime. The compressible Navier–Stokes equations are reduced to the nonlinear boundary region equations (BREs) in a high Reynolds number asymptotic framework, wherein the streamwise wavelength of the disturbances is assumed to be much larger than the spanwise and wall-normal counterparts. We study the effect of the freestream Mach number M∞, the shear layer thickness δ, the amplitude of the incoming disturbance A, and the relative velocity difference across the shear layer ΔV on the development of these centrifugal instabilities. Our parametric study shows that, among other things, the kinetic energy of the curved shear layer flow increases with increasing ΔV and A decreases with increasing delta. It was also found that increasing the disturbance amplitude of the incoming disturbance leads to significant growth in the mushroom-like structure’s amplitude and renders the secondary instability structures more prominent, indicating increased mixing for all Mach numbers under consideration.

Compressible Navier–Stokes equations without heat conduction in $$L^p$$-framework

Compressible Navier–Stokes equations without heat conduction in $$L^p$$-framework

Instantaneous Unboundedness of the Entropy and Uniform Positivity of the Temperature for the Compressible Navier–Stokes Equations with Fast Decay Density

Instantaneous Unboundedness of the Entropy and Uniform Positivity of the Temperature for the Compressible Navier–Stokes Equations with Fast Decay Density

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.

Modification in airfoil’s tonal noise using periodic suction-blowing excitation

This paper examines the modifications in the flow and sound fields due to the application of a periodic suction-blowing excitation strip on the top and bottom surfaces of a NACA0012 airfoil. The flow and sound fields have been directly computed (DNS approach) by solving unsteady, two-dimensional compressible Navier–Stokes equations using physical dispersion relation preserving schemes. The flow approaches the airfoil at a Reynolds number of 5000 and 5o angle of attack. In the no-excitation case, the sound is mainly created by the lift and drag dipole equivalent sources associated with vortex shedding. However, the application of suction-blowing excitation results in two additional sound sources: monopole-equivalent sound sources due to fluid movement in and out of the airfoil surface and the drag dipole-equivalent sound sources due to periodic change in the airfoil’s effective cross-sectional area. The combined effects of these sound sources result in significant mitigation of sound on the suction side of the airfoil. The excitation amplitude and forcing frequency effects on sound field modifications have been studied in depth. Beats of sound have been observed when the forcing frequency is close to the excitation frequency.

Blow‐up criterion for the modified compressible Navier–Stokes equations with density‐dependent viscosity

The dynamical behavior of one‐dimensional compressible Navier–Stokes equation with density‐dependent viscosity coefficient is considered. A new sufficient condition under which the solution blows up is established.

Entropy stable far-field boundary conditions for the compressible Navier-Stokes equations

We consider external compressible-flow problems with open boundaries. We begin by presenting the basic idea, namely the introduction of an infinite buffer zone around the computational domain, for the advection-diffusion equation. We discretise the problem with a summation-by-parts finite-volume scheme and prove convergence; a key step is the renormalisation of the solution. Next, we make the analogous rescaling of the entropy function for the Navier-Stokes equations and prove that a finite-volume scheme is entropy stable.Finally, we present numerical computations with a vortex governed by the Navier-Stokes equations interacting with the buffer zone. As proven, the scheme is stable, and with a sufficient number of grid points and with some numerical diffusion in the buffer zone, the reflections are modest.

On the stability of stationary compressible Navier–Stokes flows in 3D

AbstractThis paper studies the stability of the stationary solution of the compressible Navier–Stokes equation in the 3D whole space with an external force which decays at spatial infinity. We obtain the global existence result of the non-stationary problem under the smallness assumptions on the initial perturbation around the small stationary solution. We also derive the time decay rates of the perturbations under the smallness assumption on the initial perturbations, and show the optimality of the time decay rates. The proofs are based on the combination of the spectral analysis and energy method in the framework of Besov spaces. The time-space integral estimate for the linearized semigroup around the constant state in some Besov spaces plays a crucial role in the proofs.

Existence and stability of steady noncharacteristic solutions on a finite interval of full compressible Navier–Stokes equations

Existence and stability of steady noncharacteristic solutions on a finite interval of full compressible Navier–Stokes equations

Flow–acoustic resonance mechanism in tandem deep cavities coupled with acoustic eigenmodes in turbulent shear layers

This study presents the interplay of flow and acoustics within tandem deep cavities, focusing on the resonance mechanism occurring between turbulent shear layers and acoustic eigenmodes. The arrangement inside the tandem deep cavities includes both close and remote configurations. A combined fully coupled and decoupled aeroacoustic simulation strategy was devised. Employing an advanced high-order spectral/hp element method in conjunction with implicit large eddy simulation, the nonlinear compressible Navier–Stokes equations were solved to acquire internal flow–acoustic resonant field. In parallel, the linearized Navier–Stokes equations were tackled to determine coherent shear layer perturbations with external acoustic forcing. Based on acoustic measurements, the mainstream Reynolds number approaches approximately $R{e_{in}} = {O}({10^5})$ , where we identified the presence of frequency lock-in and a resonance range. Aeroacoustic noise sources were examined by implementing spectral proper orthogonal decomposition to decompose the pressure fields into hydrodynamic and acoustic components. As feedback intensified, the flow characteristics by the acoustic forcing effect and the flow-interactive effect were categorized according to the development of concurrent turbulent shear layers. Subsequently, the alternating and synchronous behaviours of concurrent shear layers resonated with the out-of-phase and in-phase acoustic eigenmodes were identified, and the corresponding large-scale counter-rotating vortex pairs and co-rotating vortex structures at the cavity entrances were extracted. The acoustic power generated by the Coriolis force was calculated using Howe's vortex-sound analogy, and the aeroacoustic energy transfer mechanism between large-scale shear layer vortices with acoustic eigenmodes was further explored. Finally, a linear response of coherent perturbations of the concurrent shear layers by external acoustic forcing was established. The amplification of flow in the streamwise direction toward the main duct led to the formation of coherent vortex structures, accompanied by separation bubbles into the main duct.

A general positivity-preserving algorithm for implicit high-order finite volume schemes solving the Euler and Navier-Stokes equations

This paper presents a general positivity-preserving algorithm for implicit high-order finite volume schemes that solve compressible Euler and Navier-Stokes equations to ensure the positivity of density and internal energy (or pressure). Previous positivity-preserving algorithms are mainly based on the slope limiting or flux limiting technique, which rely on the existence of low-order positivity-preserving schemes. This dependency poses serious restrictions on extending these algorithms to temporally implicit schemes since it is difficult to know if a low-order implicit scheme is positivity-preserving. In the present paper, a new positivity-preserving algorithm is proposed in terms of the flux correction technique. And the factors of the flux correction are determined by a residual correction procedure. For a finite volume scheme that is capable of achieving a converged solution, we show that the correction factors are in the order of unity with additional high-order terms corresponding to the spatial and temporal rates of convergence. Therefore, the proposed positivity-preserving algorithm is accuracy-reserving and asymptotically consistent. The notable advantage of this method is that it does not rely on the existence of low-order positivity-preserving baseline schemes. Therefore, it can be applied to the implicit schemes solving Euler and especially Navier-Stokes equations. In the present paper, the proposed technique is applied to an implicit dual time-stepping finite volume scheme with temporal second-order and spatial high-order accuracy. The present positivity-preserving algorithm is implemented in an iterative manner to ensure that the dual time-stepping iteration will converge to the positivity-preserving solution. Another similar correction technique is also proposed to ensure that the solution remains positivity-preserving at each sub-iteration. Numerical results demonstrate that the proposed algorithm preserves positive density and pressure in all test cases and significantly improves the robustness of the numerical schemes.

Vanishing viscosity limit to the planar rarefaction wave with vacuum for 3-D full compressible Navier–Stokes equations with temperature-dependent transport coefficients

Vanishing viscosity limit to the planar rarefaction wave with vacuum for 3-D full compressible Navier–Stokes equations with temperature-dependent transport coefficients

A centered limited finite volume approximation of the momentum convection operator for low‐order nonconforming face‐centered discretizations

AbstractWe propose in this article a discretization of the momentum convection operator for fluid flow simulations on quadrangular or generalized hexahedral meshes. The space discretization is performed by the low‐order nonconforming Rannacher–Turek finite element: the scalar unknowns are associated with the cells of the mesh while the velocities unknowns are associated with the edges or faces. The momentum convection operator is of finite volume type, and its expression is derived, as in MUSCL schemes, by a two‐step technique: computation of a tentative flux, here, with a centered approximation of the velocity, and limitation of this flux using monotonicity arguments. The limitation procedure is of algebraic type, in the sense that its does not invoke any slope reconstruction, and is independent from the geometry of the cells. The derived discrete convection operator applies both to constant or variable density flows and may thus be implemented in a scheme for incompressible or compressible flows. To achieve this goal, we derive a discrete analogue of the computation (with the velocity, one of its component, the density, and assuming that the mass balance holds) and discuss two applications of this result: first, we obtain stability results for a semi‐implicit in time scheme for incompressible and barotropic compressible flows; second, we build a consistent, semi‐implicit in time scheme that is based on the discretization of the internal energy balance rather than the total energy. The performance of the proposed discrete convection operator is assessed by numerical tests on the incompressible Navier–Stokes equations, the barotropic and the full compressible Navier–Stokes equations and the compressible Euler equations.

The existence of a weak solution for a compressible multicomponent fluid structure interaction problem

We analyze a system of PDEs governing the interaction between two compressible mutually noninteracting fluids and a shell of Koiter type encompassing a time dependent 3D domain filled by the fluids. The dynamics of the fluids is modeled by a system resembling compressible Navier-Stokes equations with a physically realistic pressure depending on densities of both the fluids. The shell possesses a non-linear, non-convex Koiter energy. Considering that the densities are comparable initially we prove the existence of a weak solution until the degeneracy of the energy or the self-intersection of the structure occurs for two cases. In the first case the adiabatic exponents are assumed to satisfy max⁡{γ,β}>2, min⁡{γ,β}>0, and the structure involved is assumed to be non-dissipative. For the second case we assume the critical case max⁡{γ,β}≥2 and min⁡{γ,β}>0 and the dissipativity of the structure. The result is achieved in several steps involving, extension of the physical domain, penalization of the interface condition, artificial regularization of the shell energy and the pressure, the almost compactness argument, added structural dissipation and suitable limit passages depending on uniform estimates.

Analytical solutions to the 1D compressible isothermal Navier–Stokes equations with Maxwell’s law

In this paper, we present some analytical solutions to the one-dimensional compressible isothermal Navier–Stokes equations with Maxwell’s law in the real line. First, we construct two analytical solutions by using a self-similar ansatz, one blows up in finite time and the other exists globally-in-time. Second, we construct two global analytical solutions with different large initial data by using a non-self-similar ansatz.

Fluid–acoustic interactions around an expanding pipe with orifice plates

Expanding pipes with orifice plates are often utilized as silencers for fluid machinery. However, intense tonal sounds can be generated from a flow through such expanding pipes. To clarify the mechanism of tonal sound from a flow through a circular expanding pipe with two orifice plates and the conditions for intense acoustic radiation, the flow and acoustic fields are directly solved based on the compressible Navier–Stokes equations. Phase-averaged flow fields indicate the occurrence of periodic vortex shedding in the free shear layers of the expanding pipe, resulting in acoustic radiation. The effects of the orifice radius and freestream Mach number on the acoustic radiation are focused on. The computational results demonstrate that vortex rings or spiral vortices are generated in the cavities formed by the orifice plates, where the primary vortical shape changes, depending on the freestream Mach number and orifice radius. The collision of the vortex ring and spiral vortex with the orifice plate or downstream edge of the expanding pipe leads to the occurrence of circumferentially in-phase and one-wavelength-mode pressure fluctuations, respectively. The orifice radius also affects the convective velocity of vortices and the position of the acoustic source, varying the frequency of the acoustic radiation. The findings of this research provide the first clarifications of fluid–acoustic interactions in an expanding pipe with orifice plates.

Numerical simulation of sound attenuation in an acoustically lined duct in high-temperature air flows

Noise reduction structures are important for the vibration and noise reduction design of aerospace engines. The design of noise-reducing structures often needs to be quickly evaluated via numerical simulations. Hence, the simulation results of the corresponding system are very important for guiding the design of noise-reducing structures. High temperature is one of the key environmental factors that need to be considered when evaluating the sound attenuation process via numerical methods. In this study, numerical simulations of acoustic wave propagation on an acoustic liner structure considering air temperature variations are carried out by using compressible Navier–Stokes equations and the ideal gas equation of state. The results showed that the effect of temperature on sound attenuation under grazing flow conditions is complex. Moreover, an increase in temperature will reduce the transmission loss of the acoustic liner in the grazing flow at high air speed.

Self-similar solutions and large time behavior to the 2D isentropic compressible Navier–Stokes equations with vacuum free boundary and rotation

This paper is concerned with a class of self-similar analytical solutions to the 2D isentropic compressible Navier–Stokes equations with vacuum free boundary for polytropic gases. It is shown that the free boundary will grow linearly in time (see Eq. (1.11)), moreover, both angular velocity uϕ and its derivatives, and the derivative of radial velocity ur will tend to zero as t→+∞, while radial velocity itself is bounded. In particular, it is interesting to see that when the adiabatic exponent γ>2 and for a suitably large time, the rotation plays a dominant role in accelerating the growth of the free boundary compared with the effect of pressure.

A new re-redistribution scheme for weighted state redistribution with adaptive mesh refinement

State redistribution (SRD) is a recently developed technique for stabilizing cut cells that result from finite-volume embedded boundary methods. SRD has been successfully applied to a variety of compressible and incompressible flow problems. When used in conjunction with adaptive mesh refinement (AMR), additional steps are needed to preserve the accuracy and conservation properties of the solution if the embedded boundary is not restricted to a single level of the mesh hierarchy. In this work, we extend the weighted state redistribution algorithm to cases where cut cells live at or near a coarse-fine interface within the domain. We present numerical results that demonstrate that the algorithm is conservative when the coarse-fine interface intersects the embedded boundary. Additionally we compare the numerical solution of the Sod shock tube problem in an inclined cylinder with the analytic solution, and we compare the simulation of a shock hitting a cylindrical obstacle with experimental data. Finally we demonstrate the methodology for simulation of the multicomponent compressible Navier-Stokes equations in a piston-bowl geometry, and discuss the computational efficiency gained by not requiring the entire embedded boundary to be defined at the finest level.

Global Solutions of 3D Isentropic Compressible Navier–Stokes Equations with Two Slow Variables

Global Solutions of 3D Isentropic Compressible Navier–Stokes Equations with Two Slow Variables