Compressible Flow Equations Research Articles

Vanishing Viscosity Approach to the Compressible Euler Equations for Transonic Nozzle and Spherically Symmetric Flows

We are concerned with globally defined entropy solutions to the Euler equations for compressible fluid flows in transonic nozzles with general cross-sectional areas. Such nozzles include the de Laval nozzles and other more general nozzles whose cross-sectional area functions are allowed at the nozzle ends to be either zero (closed ends) or infinity (unbounded ends). To achieve this, in this paper, we develop a vanishing viscosity method to construct globally defined approximate solutions and then establish essential uniform estimates in weighted L p norms for the whole range of physical adiabatic exponents $${\gamma\in (1, \infty)}$$ , so that the viscosity approximate solutions satisfy the general L p compensated compactness framework. The viscosity method is designed to incorporate artificial viscosity terms with the natural Dirichlet boundary conditions to ensure the uniform estimates. Then such estimates lead to both the convergence of the approximate solutions and the existence theory of globally defined finite-energy entropy solutions to the Euler equations for transonic flows that may have different end-states in the class of nozzles with general cross-sectional areas for all $${\gamma\in (1, \infty)}$$ . The approach and techniques developed here apply to other problems with similar difficulties. In particular, we successfully apply them to construct globally defined spherically symmetric entropy solutions to the Euler equations for all $${\gamma\in (1, \infty)}$$ .

Converging shock flows for a Mie-Grüneisen equation of state

Previous work has shown that the one-dimensional (1D) inviscid compressible flow (Euler) equations admit a wide variety of scale-invariant solutions (including the famous Noh, Sedov, and Guderley shock solutions) when the included equation of state (EOS) closure model assumes a certain scale-invariant form. However, this scale-invariant EOS class does not include even simple models used for shock compression of crystalline solids, including many broadly applicable representations of Mie-Grüneisen EOS. Intuitively, this incompatibility naturally arises from the presence of multiple dimensional scales in the Mie-Grüneisen EOS, which are otherwise absent from scale-invariant models that feature only dimensionless parameters (such as the adiabatic index in the ideal gas EOS). The current work extends previous efforts intended to rectify this inconsistency, by using a scale-invariant EOS model to approximate a Mie-Grüneisen EOS form. To this end, the adiabatic bulk modulus for the Mie-Grüneisen EOS is constructed, and its key features are used to motivate the selection of a scale-invariant approximation form. The remaining surrogate model parameters are selected through enforcement of the Rankine-Hugoniot jump conditions for an infinitely strong shock in a Mie-Grüneisen material. Finally, the approximate EOS is used in conjunction with the 1D inviscid Euler equations to calculate a semi-analytical Guderley-like imploding shock solution in a metal sphere and to determine if and when the solution may be valid for the underlying Mie-Grüneisen EOS.

Lagrangian particle method for compressible fluid dynamics

A new Lagrangian particle method for solving Euler equations for compressible inviscid fluid or gas flows is proposed. Similar to smoothed particle hydrodynamics (SPH), the method represents fluid cells with Lagrangian particles and is suitable for the simulation of complex free surface/multiphase flows. The main contributions of our method, which is different from SPH in all other aspects, are (a) significant improvement of approximation of differential operators based on a polynomial fit via weighted least squares approximation and the convergence of prescribed order, (b) a second-order particle-based algorithm that reduces to the first-order upwind method at local extremal points, providing accuracy and long term stability, and (c) more accurate resolution of entropy discontinuities and states at free interfaces. While the method is consistent and convergent to a prescribed order, the conservation of momentum and energy is not exact and depends on the convergence order. The method is generalizable to coupled hyperbolic–elliptic systems. Numerical verification tests demonstrating the convergence order are presented as well as examples of complex multiphase flows.

One-dimensional model analysis and performance assessment of Tesla turbine

Tesla turbine is characterized by the bladeless design that makes it easy to be manufactured and operated. It offers an attractive option for power output in small and micro scale systems if an efficient design can be achieved. One-dimensional model is useful as it can adequately represent the flow characteristics in the Tesla turbine and allow parametric exploration for early design analysis. This paper improves the one-dimensional model for Tesla turbine. The limit expansion ratio of the nozzle is introduced, which is related to the geometry angle and the working fluid properties. The flow loss in the nozzle is evaluated instead of assuming an empirical velocity coefficient. The flow is regarded as turbulence rather than laminar, and friction factor is determined by the Reynolds number based on Moody Figure. In the rotor, the governing equations for compressible flow between adjacent discs are used. In addition, the radial pressure gradient effect in the rotor gap spacing is considered. The improved model show better agreement with the experimental data than the original model. The flow characteristics in the Tesla turbine is analyzed and the streamline of the bulk flow in the rotor is derived based on the one-dimensional model. Tesla turbine can yield considerable efficiency and it can be regarded as a potential choice to be applied in small and micro scale systems.

Adjoint analysis of Buckley-Leverett and two-phase flow equations

This paper analyzes the adjoint equations and boundary conditions for porous media flow models, specifically the Buckley-Leverett equation, and the compressible two-phase flow equations in mass conservation form. An adjoint analysis of a general scalar hyperbolic conservation law whose primal solutions include a shock jump is initially presented, and the results are later specialized to the Buckley-Leverett equation. The non-convexity of the Buckley-Leverett flux function results in adjoint characteristics that are parallel to the shock front upstream of the shock and emerge from the shock front downstream of the shock. Thus, in contrast to the behavior of Burgers’ equation where the adjoint is continuous at a shock, the Buckley-Leverett adjoint, in general, contains a discontinuous jump across the shock. Discrete adjoint solutions from space-time discontinuous Galerkin finite element approximations of the Buckley-Leverett equation are shown to be consistent with the derived closed-form analytical solutions. Furthermore, a general result relating the adjoint equations for different (though equivalent) primal equations is used to relate the two-phase flow adjoints to the Buckley-Leverett adjoint. Adjoint solutions from space-time discontinuous Galerkin finite element approximations of the two-phase flow equations are observed to obey this relationship.

Hydraulic analogy and visualisation of two-dimensional compressible fluid flows: part 1: theoretical aspects

The principles of the well-known, hydraulic analogy are explained (see Courant and Friedricks, 1948; Loh, 1969). The governing equations of two-dimensional compressible fluid two-dimensional flows in non-dimensional form, based on conservation laws are first discussed. Then, the isentropic flow condition is introduced to produce the isentropic Euler equations. On the other hand, the equations governing surface waves on thin water layers over a flat surface are derived in non-dimensional form, using the assumption of hydrostatic pressure across the water layer, hence the analogy between the two problems is established. The normalised density of the compressible flow corresponds to the normalised height of the thin water layer and the speed of sound corresponds to the speed of surface waves in water, hence, the Mach number corresponds to Froude number. Finally, it is shown that the analogy can be used to visualise supersonic and transonic two-dimensional flow patterns, including shock waves and expansion fans around airfoils and in convergent/divergent nozzles. Also, nonlinear water waves of finite amplitude, in dispersive media, are discussed. In part 2 of this study, water table experiments are presented together with qualitative and quantitative measurement techniques.

Towards unification of the Vorticity Confinement and Shock Capturing (TVD and ENO/WENO) methods

New multidimensional extensions of the TVD and finite difference ENO/WENO methods for the compressible flow equations are proposed. The novelty of the approach is in the discretization schemes that acquire by means of a single mechanism both shock-capturing and vorticity confinement capabilities. Thus, the new method can be interpreted as a unification of the two methodologies, intended initially for different purposes.

Numerical Modeling of Gas-Liquid Compressible Pipe Flow Based on the Theory of Thermodynamically Compatible Systems

Based on the theory of thermodynamically compatible systems, we formulate the governing equations of a gas-liquid compressible pipe flow and develop the Runge–Kutta–TVD and Runge–Kutta–WENO high accuracy methods. The computational model is used to solve a series of test cases demonstrating its efficiency and capability to be applied to slug flows modeling.

The effect of core size on the speed of compressible hollow vortex streets

The effect of weak compressibility on the speed of steadily translating staggered vortex streets of hollow vortices in isentropic subsonic flow is studied. A small-Mach-number perturbation expansion about the incompressible solutions for staggered streets of hollow vortices found recently by Crowdy & Green (Phys. Fluids, 2011, vol. 23, 126602) is carried out; the latter solutions provide a desingularization of the classical point vortex streets of von Kármán. The first-order compressible flow correction is calculated. We employ a novel scheme based on a complex variable formulation of the compressible flow equations (the Imai–Lamla method) combined with conformal mapping theory to track the vortex shape in this free boundary problem. The analysis to find the perturbed streamfunction and compressible vortex shapes is greatly facilitated by exploiting a calculus based on use of the Schottky–Klein prime function of a conformally equivalent parametric annulus. It is found that, for a vortex street of specified aspect ratio comprising vortices of specified circulation, the vortex core size is a key determinant of whether compressibility increases or decreases the steady propagation speed (relative to the incompressible street with the same parameters) and that both eventualities are possible. We focus attention on streets with aspect ratios around 0.28, which is close to the neutrally stable case for incompressible flow, and find that a critical vortex core size exists at which compressibility does not affect the speed of the street at first order in the (squared) Mach number. Streets comprising vortices with core size below the critical value speed up due to compressibility; larger vortices slow down.

Three-Dimensional Numerical Simulation of Aerodynamic Drag on High-speed Train in Tunnel

The purpose of the paper is to derive variation law of aerodynamic drag introduced by high-speed train passing through a tunnel and to offer guidance for design of high-speed railway longitudinal section and high-speed train head shape. The paper presents a cell-centered finite volume scheme for numerical integration of Navier-Stokes equations for three-dimensional unsteady viscous compressible nonhomentropic flow to simulate the aerodynamic drag introduced by high-speed train passing through a tunnel. The aerodynamic drag curve is analysed and explained in detail in accord with the train positions．

Symmetries of the gas dynamics equations using the differential form method

A brief review of the theory of exterior differential systems and isovector symmetry analysis methods is presented in the context of the one-dimensional inviscid compressible flow equations. These equations are formulated as an exterior differential system with equation of state (EOS) closure provided in terms of an adiabatic bulk modulus. The scaling symmetry generators—and corresponding EOS constraints—otherwise appearing in the existing literature are recovered through the application and invariance under Lie derivative dragging operations.

Incipient Separation in Laminar Ramp-Induced Shock-Wave/Boundary-Layer Interactions

Incipient separation in laminar, ramp-induced shock-wave/boundary-layer interactions is examined using numerical solutions of the triple-deck equations and the full set of governing equations for compressible flows. Validation of the numerical methods is presented and the results are compared. A new comparison between solutions to the triple-deck equations and the compressible Navier–Stokes equations is presented. It was found that the triple-deck trends qualitatively match the trends produced by numerical solutions to the compressible Navier–Stokes equations. An expression for true incipient separation is derived and the trends are discussed.

A Lagrangian meshfree method applied to linear and nonlinear elasticity.

The repeated replacement method (RRM) is a Lagrangian meshfree method which we have previously applied to the Euler equations for compressible fluid flow. In this paper we present new enhancements to RRM, and we apply the enhanced method to both linear and nonlinear elasticity. We compare the results of ten test problems to those of analytic solvers, to demonstrate that RRM can successfully simulate these elastic systems without many of the requirements of traditional numerical methods such as numerical derivatives, equation system solvers, or Riemann solvers. We also show the relationship between error and computational effort for RRM on these systems, and compare RRM to other methods to highlight its strengths and weaknesses. And to further explain the two elastic equations used in the paper, we demonstrate the mathematical procedure used to create Riemann and Sedov-Taylor solvers for them, and detail the numerical techniques needed to embody those solvers in code.

Rarefied gas electro jet (RGEJ) micro-thruster for space propulsion

This article numerically investigates a micro-thruster for small satellites which utilizes plasma actuators to heat and accelerate the flow in a micro-channel with rarefied gas in the slip flow regime. The inlet plenum condition is considered at 1 Torr with flow discharging to near vacuum conditions (<0.05 Torr). The Knudsen numbers at the inlet and exit planes are ~0.01 and ~0.1, respectively. Although several studies have been performed in micro-hallow cathode discharges at constant pressure, to our knowledge, an integrated study of the glow discharge physics and resulting fluid flow of a plasma thruster under these low pressure and low Knudsen number conditions is yet to be reported. Numerical simulations of the charge distribution due to gas ionization processes and the resulting rarefied gas flow are performed using an in-house code. The mass flow rate, thrust, specific impulse, power consumption and the thrust effectiveness of the thruster are predicted based on these results. The ionized gas is modelled using local mean energy approximation. An electrically induced body force and a thermal heating source are calculated based on the space separated charge distribution and the ion Joule heating, respectively. The rarefied gas flow with these electric force and heating source is modelled using density-based compressible flow equations with slip flow boundary conditions. The results show that a significant improvement of specific impulse can be achieved over highly optimized cold gas thrusters using the same propellant.

Experimental Studies on the Effect of Suction Chamber Angle on the Entrainment of Passive Fluid in a Steam Ejector

In this experimental study, the impact of suction chamber angle (SCA) on the entrainment ratio of a steam ejector refrigeration system (ERS) of 700 W was investigated. The basic dimensions of the ejector were derived from the compressible fluid flow equations using matlab. The system was tested with six different SCAs with various operating conditions, and its performance was analyzed. It is inferred that the entrainment of passive fluid from the evaporator is the strong function of the SCA. For all the active steam pressures, the entrainment of the passive fluid increases up to 12 deg of SCA, and above that the performance decreases significantly. Optimum angle of suction chamber increases the entrainment ratio for at least 49.96%. It is also found that the SCA has a minor influence on the back pressure.

Verification Assessment of Piston Boundary Conditions for Lagrangian Simulation of the Guderley Problem

This work is concerned with the use of Guderley's converging shock wave solution of the inviscid compressible flow equations as a verification test problem for compressible flow simulation software. In practice, this effort is complicated by both the semi-analytical nature and infinite spatial/temporal extent of this solution. Methods can be devised with the intention of ameliorating this inconsistency with the finite nature of computational simulation; the exact strategy will depend on the code and problem archetypes under investigation. For example, scale-invariant shock wave propagation can be represented in Lagrangian compressible flow simulations as rigid boundary-driven flow, even if no such “piston” is present in the counterpart mathematical similarity solution. The purpose of this work is to investigate in detail the methodology of representing scale-invariant shock wave propagation as a piston-driven flow in the context of the Guderley problem, which features a semi-analytical solution of infinite spatial/temporal extent. The semi-analytical solution allows for the derivation of a similarly semi-analytical piston boundary condition (BC) for use in Lagrangian compressible flow solvers. The consequences of utilizing this BC (as opposed to directly initializing the Guderley solution in a computational spatial grid at a fixed time) are investigated in terms of common code verification analysis metrics (e.g., shock strength/position errors, global convergence rates). For the examples considered in this work, the piston-driven initialization approach is demonstrated to be a viable alternative to the more traditional, direct initialization approach.

Space–time adaptive ADER-DG schemes for dissipative flows: Compressible Navier–Stokes and resistive MHD equations

This paper presents an arbitrary high-order accurate ADER Discontinuous Galerkin (DG) method on space–time adaptive meshes (AMR) for the solution of two important families of non-linear time dependent partial differential equations for compressible dissipative flows : the compressible Navier–Stokes equations and the equations of viscous and resistive magnetohydrodynamics in two and three space-dimensions.The work continues a recent series of papers concerning the development and application of a proper a posteriori subcell finite volume limiting procedure suitable for discontinuous Galerkin methods (Dumbser et al., 2014, Zanotti et al., 2015 [40,41]). It is a well known fact that a major weakness of high order DG methods lies in the difficulty of limiting discontinuous solutions, which generate spurious oscillations, namely the so-called ‘Gibbs phenomenon’. In the present work, a nonlinear stabilization of the scheme is sequentially and locally introduced only for troubled cells on the basis of a novel a posteriori detection criterion, i.e. the MOOD approach. The main benefits of the MOOD paradigm, i.e. the computational robustness even in the presence of strong shocks, are preserved and the numerical diffusion is considerably reduced also for the limited cells by resorting to a proper sub-grid. In practice the method first produces a so-called candidate solution by using a high order accurate unlimited DG scheme. Then, a set of numerical and physical detection criteria is applied to the candidate solution, namely: positivity of pressure and density, absence of floating point errors and satisfaction of a discrete maximum principle in the sense of polynomials. Furthermore, in those cells where at least one of these criteria is violated the computed candidate solution is detected as troubled and is locally rejected. Subsequently, a more reliable numerical solution is recomputed a posteriori by employing a more robust but still very accurate ADER-WENO finite volume scheme on the subgrid averages within that troubled cell. Finally, a high order DG polynomial is reconstructed back from the evolved subcell averages.We apply the whole approach for the first time to the equations of compressible gas dynamics and magnetohydrodynamics in the presence of viscosity, thermal conductivity and magnetic resistivity, therefore extending our family of adaptive ADER-DG schemes to cases for which the numerical fluxes also depend on the gradient of the state vector. The distinguished high-resolution properties of the presented numerical scheme standout against a wide number of non-trivial test cases both for the compressible Navier–Stokes and the viscous and resistive magnetohydrodynamics equations. The present results show clearly that the shock-capturing capability of the news schemes is significantly enhanced within a cell-by-cell Adaptive Mesh Refinement (AMR) implementation together with time accurate local time stepping (LTS).

Gas permeability measurements from pressure pulse decay laboratory data using pseudo-pressure and pseudo-time transformations

This paper presents a pseudo-pressure and pseudo-time straight-line approach to interpret laboratory pulse decay data in order to estimate rock core permeability using gas as the pore fluid. The implementation of the straight-line approach provides a practical method to estimate gas permeability from experimental data, as long as changes in gas viscosity and compressibility are negligible. On the other hand, pseudo-pressure and pseudo-time allow the transformation of the compressible flow equation from its highly nonlinear form to a quasi-linear partial differential equation, where changes in gas viscosity, gas compressibility and compressibility factor are accounted for. The purpose of this work is to combine both, pseudo-functions and the straight-line approach to estimate gas permeability from pressure pulse laboratory data with a more rigorous treatment of gas properties. Five pulse decay experiments were performed in Marcellus shale cores at pore pressures ranging from 130 to 700 psi in a triaxial cell to estimate permeability and porosity of ultra-low permeability cores. The experiments were made in an increasing order of equilibration pressure starting from 130 until 700 psi, and vertical and radial stresses kept constant at 1500 psi. Permeability estimates were compared against the P2-approach to show the validity of the proposed method at low gas pressures.

Interaction of a centered simple wave and a planar rarefaction wave of the two-dimensional Euler equations for pseudo-steady compressible flow

In this paper, we study the expansion problem which arises as two-dimensional (2D) pseudo-steady supersonic flow turns a sharp corner and expands into vacuum. The problem catches interaction of a centered simple wave and a backward planar rarefaction wave, which is deduced a Goursat problem for 2D self-similar Euler equations for compressible flow. By the methods of characteristic decomposition and invariant regions, we get the hyperbolicity in the wave interaction domain and prior C1 estimates of solutions to the Goursat problem. The global solution up to the interface of gas with vacuum to the expansion problem is obtained constructively.

Numerical analysis of gas leakage in the piston-cylinder clearance of reciprocating compressors considering compressibility effects

Leakage is a major source of inefficiency in low-capacity reciprocating compressors. Not only does it lower the mass flow rate provided by the compressor, reducing its volumetric efficiency, but also gives rise to outflux of energy that decreases the isentropic efficiency. Leakage in the piston-cylinder clearance of reciprocating compressors is driven by the piston motion and pressure difference between the compression chamber and the shell internal environment. In compressors adopted for domestic refrigeration, such a clearance is usually filled by a mixture of refrigerant and lubricating oil. Besides its lubricating function, the oil also acts as sealing element for the piston-cylinder clearance, and hence leakage is expected to be more detrimental to oil-free compressors. This paper presents a model based on the Reynolds equation for compressible fluid flow to predict leakage in oil-free reciprocating compressors. The model is solved throughout the compression cycle so as to assess the effect of the clearance geometry and piston velocity on leakage and compressor efficiency. The results show that compressible fluid flow formulation must be considered for predictions of gas leakage in the cylinder-piston clearance.