Computation Of Compressible Flows Research Articles

An efficient Adaptive Mesh Refinement (AMR) algorithm for the Discontinuous Galerkin method: Applications for the computation of compressible two-phase flows

We present an Adaptive Mesh Refinement (AMR) method suitable for hybrid unstructured meshes that allows for local refinement and de-refinement of the computational grid during the evolution of the flow. The adaptive implementation of the Discontinuous Galerkin (DG) method introduced in this work (ForestDG) is based on a topological representation of the computational mesh by a hierarchical structure consisting of oct- quad- and binary trees. Adaptive mesh refinement (h-refinement) enables us to increase the spatial resolution of the computational mesh in the vicinity of the points of interest such as interfaces, geometrical features, or flow discontinuities. The local increase in the expansion order (p-refinement) at areas of high strain rates or vorticity magnitude results in an increase of the order of accuracy in the region of shear layers and vortices.A graph of unitarian-trees, representing hexahedral, prismatic and tetrahedral elements is used for the representation of the initial domain. The ancestral elements of the mesh can be split into self-similar elements allowing each tree to grow branches to an arbitrary level of refinement. The connectivity of the elements, their genealogy and their partitioning are described by linked lists of pointers. An explicit calculation of these relations, presented in this paper, facilitates the on-the-fly splitting, merging and repartitioning of the computational mesh by rearranging the links of each node of the tree with a minimal computational overhead. The modal basis used in the DG implementation facilitates the mapping of the fluxes across the non conformal faces.The AMR methodology is presented and assessed using a series of inviscid and viscous test cases. Also, the AMR methodology is used for the modelling of the interaction between droplets and the carrier phase in a two-phase flow. This approach is applied to the analysis of a spray injected into a chamber of quiescent air, using the Eulerian–Lagrangian approach. This enables us to refine the computational mesh in the vicinity of the droplet parcels and accurately resolve the coupling between the two phases.

An accurate and efficient gridless method based on implicit, fast, and constrained weights optimization schemes for compressible flows

An accurate and efficient gridless method is presented for calculation of inviscid compressible flows in subsonic to supersonic flows. The Taylor series least squares is used for discretization of spatial derivatives at each point. Fast gridless method using the second- and fourth-order artificial dissipation terms is applied for solving the Euler equations. Two methods are proposed for increasing the accuracy; implementing the constrained weights optimization and using high-order Taylor series expansion on the fast gridless method. The explicit and implicit dual-time schemes are applied for temporal discretization. To speed up convergence, local time stepping and residual smoothing techniques are used. The capability and accuracy of the methods are compared with other gridless methods, finite volume method and AGARD data for some test cases in subsonic, transonic and supersonic flows. Results show that the use of the constraint weights optimization increases the accuracy especially with fewer point distributions in comparison with conventional gridless methods and it is more efficient than other gridless methods.

Methods for the Calculation of Thermoacoustic Stability Boundaries and Monte Carlo-Free Uncertainty Quantification

Thermoacoustic instabilities are a major threat for modern gas turbines. Frequency-domain-based stability methods, such as network models and Helmholtz solvers, are common design tools because they are fast compared to compressible flow computations. They result in an eigenvalue problem, which is nonlinear with respect to the eigenvalue. Thus, the influence of the relevant parameters on mode stability is only given implicitly. Small changes in some model parameters, may have a great impact on stability. The assessment of how parameter uncertainties propagate to system stability is therefore crucial for safe gas turbine operation. This question is addressed by uncertainty quantification. A common strategy for uncertainty quantification in thermoacoustics is risk factor analysis. One general challenge regarding uncertainty quantification is the sheer number of uncertain parameter combinations to be quantified. For instance, uncertain parameters in an annular combustor might be the equivalence ratio, convection times, geometrical parameters, boundary impedances, flame response model parameters, etc. A new and fast way to obtain algebraic parameter models in order to tackle the implicit nature of the problem is using adjoint perturbation theory. This paper aims to further utilize adjoint methods for the quantification of uncertainties. This analytical method avoids the usual random Monte Carlo (MC) simulations, making it particularly attractive for industrial purposes. Using network models and the open-source Helmholtz solver PyHoltz, it is also discussed how to apply the method with standard modeling techniques. The theory is exemplified based on a simple ducted flame and a combustor of EM2C laboratory for which experimental data are available.

A sharp‐interface immersed boundary framework for simulations of high‐speed inviscid compressible flows

SummaryA new finite‐volume flow solver based on the hybrid Cartesian immersed boundary (IB) framework is developed for the solution of high‐speed inviscid compressible flows. The IB method adopts a sharp‐interface approach, wherein the boundary conditions are enforced on the body geometry itself. A key component of the present solver is a novel reconstruction approach, in conjunction with inverse distance weighting, to compute the solutions in the vicinity of the solid‐fluid interface. We show that proposed reconstruction leads to second‐order spatial accuracy while also ensuring that the discrete conservation errors diminish linearly with grid refinement. Investigations of supersonic and hypersonic inviscid flows over different geometries are carried out for an extensive validation of the proposed flow solver. Studies on cylinder lift‐off and shape optimisation in supersonic flows further demonstrate the efficacy of the flow solver for computations with moving and shape‐changing geometries. These studies conclusively highlight the capability of the proposed IB methodology as a promising alternative for robust and accurate computations of compressible fluid flows on nonconformal Cartesian meshes.

Accuracy consideration by DRP schemes for DNS and LES of compressible flow computations

Several dispersion relation-preserving (DRP) spatially central discretizations are considered as the base scheme in the framework of the Yee & Sjögreen low dissipative nonlinear filter approach. In addition, the nonlinear filter of Yee & Sjögreen with shock-capturing and long time integration capabilities is used to replace the standard DRP linear filter for both smooth flows and flows containing discontinuities. DRP schemes for computational aeroacoustics (CAA) focus on dispersion error consideration for long time linear wave propagation rather than the formal order of accuracy of the scheme. The resulting DRP schemes usually have wider grid stencils and increased CPU operations count compared with standard central schemes of the same formal order of accuracy. For discontinuous initial data and long time wave propagation of smooth acoustic waves, various space and time DRP linear filter are needed. For acoustic waves interacting with shocks and turbulence induced noise, DRP schemes with linear filters alone usually are not capable of simulating such flows. The investigation presented in this paper is focused on the possible gain in efficiency and accuracy by spatial DRP schemes over standard central schemes having the same grid stencil width for general direct numerical simulations (DNS) and large eddy simulations (LES) of compressible flows. Representative test cases for both smooth flows and problems containing discontinuities for 3D DNS of compressible gas dynamics are included.

Implicit second-order CUSP gridless method for unsteady moving boundary simulations

Efficient first-order and second-order gridless method is presented for calculation of unsteady compressible flows. The convective upwind split pressure (CUSP) gridless scheme is applied for solving unsteady fluid flow equations in the arbitrary Lagrangian–Eulerian (ALE) formulation. In order to manage dynamic cloud and nodes movement, the nodes are moved based on the boundary movements using the segment spring analogy. Discretization of spatial derivatives are done by using the Taylor series least squares at each node. For increasing the accuracy, second-order Taylor series (SOTS) expansion is also used on the CUSP gridless method. For time advancement, implicit dual-time and explicit schemes are used. The accuracy of the methods is compared with the AGARD experimental data and finite volume results for some test cases in steady and unsteady flows. Results show that the CUSP gridless scheme is accurate in steady and unsteady flows and using the second-order discretization increases the accuracy a little bit and also the gridless schemes reduce the computational time as compared with finite volume method. Numerical results are in good agreement with the experimental data.

A compressive interface-capturing scheme for computation of compressible multi-fluid flows

Abstract In this paper, a compressive high-resolution interface-capturing scheme is presented for the computation of compressible multi-fluid flows with high-density ratios and strong shocks. The proposed scheme is coupled with a preconditioned dual-time compressible mixture solver for robust and accurate computations over a wide range of Mach numbers. The scheme is simple and relatively easy to implement. It does not require any calculations for the interface curvature and the normal vector. The numerical approximations were implemented on general, structured grids using an implicit MUSCL upwind approach. Validation tests were conducted for a single reversible vortex, advection of an air-water interface, dam-break flow, and air shock-helium bubble interaction. Finally, a three-dimensional gas-lift flow is presented to demonstrate the capability of the present scheme for handling an interface with large jumps in pressure, temperature, and density.

A novel hybrid approach with multidimensional-like effects for compressible flow computations

A multidimensional scheme achieves good resolution of strong and weak shocks irrespective of whether the discontinuities are aligned with or inclined to the grid. However, these schemes are computationally expensive. This paper achieves similar effects by hybridizing two schemes, namely, AUSM and DRLLF and coupling them through a novel shock switch that operates – unlike existing switches – on the gradient of the Mach number across the cell-interface. The schemes that are hybridized have contrasting properties. The AUSM scheme captures grid-aligned (and strong) shocks crisply but it is not so good for non-grid-aligned weaker shocks, whereas the DRLLF scheme achieves sharp resolution of non-grid-aligned weaker shocks, but is not as good for grid-aligned strong shocks. It is our experience that if conventional shock switches based on variables like density, pressure or Mach number are used to combine the schemes, the desired effect of crisp resolution of grid-aligned and non-grid-aligned discontinuities are not obtained. To circumvent this problem we design a shock switch based – for the first time – on the gradient of the cell-interface Mach number with very impressive results. Thus the strategy of hybridizing two carefully selected schemes together with the innovative design of the shock switch that couples them, affords a method that produces the effects of a multidimensional scheme with a lower computational cost. It is further seen that hybridization of the AUSM scheme with the recently developed DRLLFV scheme using the present shock switch gives another scheme that provides crisp resolution for both shocks and boundary layers. Merits of the scheme are established through a carefully selected set of numerical experiments.

Algebraic Calculation Method of One-Dimensional Steady Compressible Gas Flow

This paper describes a new method of calculation of one-dimensional steady compressible gas flows in channels with possible heat and mass exchange through perforated sidewalls. The channel is divided into small elements of a finite size for which mass, energy and momentum conservation laws are written in the integral form, assuming linear distribution of the parameters along the length. As a result, the calculation is reduced to finding the roots of a quadratic algebraic equation, thus providing an alternative to numerical methods based on differential equations. The advantage of this method is its high tolerance to coarse discretization of the calculation area as well as its good applicability for transonic flow calculations.

Kinetic energy preserving DG schemes based on summation‐by‐parts operators on interior node distributions

AbstractIn this work, a kinetic energy preserving DG scheme in one space dimension using Gauss‐Legendre nodes is presented. Stability problems will be demonstrated when using interface terms that are derived from the Lobatto nodes within the Gauss‐Legendre skew‐symmetric DG formulation. However, combined with correct interface terms, the skew‐symmetric DG scheme constructed on Gauss‐Legendre nodes is expected to yield higher accuracy compared to its Gauss‐Lobatto counterpart. This advantage is demonstrated in the case of viscous compressible flow computation. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

HLLC-type Riemann solver with approximated two-phase contact for the computation of the Baer–Nunziato two-fluid model

The computation of compressible two-phase flows with the Baer–Nunziato model is addressed. Only the convective part of the model that exhibits non-conservative products is considered and the source terms of the model that represent the exchange between phases are neglected. Based on the solver proposed by Tokareva & Toro [1], a new HLLC-type Riemann solver is built. The key idea of this new solver lies in an approximation of the two-phase contact discontinuity of the model. Thus the Riemann invariants of the wave are approximated in the “subsonic” case. A major consequence of this approximation is that the resulting solver can deal with any Equation Of State. It also allows to bypass the resolution of a non-linear equation based on those Riemann invariants. We assess the solver and compare it with others on 1D Riemann problems including grid convergence and efficiency studies. The ability of the proposed solver to deal with complex Equations Of State is also investigated. Finally, the different solvers have been compared on challenging 2D test-cases due to the presence of both material interfaces and shock waves: a shock–bubble interaction and underwater explosions. When compared with others, the present solver appears to be accurate, efficient and robust.

A third-order compact gas-kinetic scheme on unstructured meshes for compressible Navier–Stokes solutions

In this paper, for the first time a third-order compact gas-kinetic scheme is proposed on unstructured meshes for the compressible viscous flow computations. The possibility to design such a third-order compact scheme is due to the high-order gas evolution model, where a time-dependent gas distribution function at cell interface not only provides the fluxes across a cell interface, but also presents a time accurate solution for flow variables at cell interface. As a result, both cell averaged and cell interface flow variables can be used for the initial data reconstruction at the beginning of next time step. A weighted least-square procedure has been used for the initial reconstruction. Therefore, a compact third-order gas-kinetic scheme with the involvement of neighboring cells only can be developed on unstructured meshes. In comparison with other conventional high-order schemes, the current method avoids the Gaussian point integration for numerical fluxes along a cell interface and the multi-stage Runge–Kutta method for temporal accuracy. The third-order compact scheme is numerically stable under CFL condition CFL≈0.5. Due to its multidimensional gas-kinetic formulation and the coupling of inviscid and viscous terms, even with unstructured meshes, the boundary layer solution and vortex structure can be accurately captured by the current scheme. At the same time, the compact scheme can capture strong shocks as well.

Numerical Simulation of Unsteady Cavitation in a High-speed Water Jet

Abstract Concerning the numerical simulation of high-speed water jet with intensive cavitation this paper presents a practical compressible mixture flow method by coupling a simplified estimation of bubble cavitation and a compressible mixture flow computation. The mean flow of two-phase mixture is calculated by RANS for compressible fluidU he intensit. T y of cavitation in a local field is evaluated by the volume fraction of gas phase varying with the mean flow, and the effect of cavitation on the flow turbulence is considered by applying a ion to density correctthe evaluation of eddy viscosity. High-speed submerged water jets issuing from a sheathed sharp-edge orifice nozzle are treated when the cavitation number, σ = 0.1, and the computation result is compared with experimental data The result reveals that cavitation occurs initially at the entrance of orifice and bubble cloud develops gradually while flowing downstream along the shear layer. Developed bubble cloudbreaks up and then sheds downstream periodically near the sheath exit. The pattern of cavitation cloud shedding evaluated by simulation agrees experimental one, and the possibility to capture the unsteadily shedding of cavitation clouds is demonstrated. The decay of core velocity in cavitating jetis delayed greatly compared to that in no-activation jet, and the effect of the nozzle sheath is demonstrated.

Application of New Absorbing Boundary Condition with Finite Volume Schemes

In our recent paper (“Capturing Matched Layer at Absorbing Boundary with Finite Volume Schemes,” Computers and Fluids, Vol. null, Jan. 2015, pp. 79–92), a new absorbing boundary condition for compressible flow computation was proposed. It was shown that, with an appropriate finite volume scheme, a matched layer (or absorbing layer) can be automatically captured as a discontinuity across the absorbing boundary, rendering an extremely simple yet robust absorbing boundary condition. This new absorbing boundary condition is attributed to the shock-capturing capability of the finite volume scheme, with no direct relation to any existing nonreflecting boundary condition. As a sequel, the present paper focuses on application of the new absorbing boundary condition. Several more non-trivial benchmark examples are demonstrated to further show its robustness and versatility for various types of continuous and discontinuous waves in domain of different shapes. Then it is examined in practical problems with experimental data, rendering a viable tool for compressible flow and aeroacoustics computation.

О расчете двумерных сжимаемых фильтрационных течений в соответствии с методологией «сжимаемых линий тока»

О расчете двумерных сжимаемых фильтрационных течений в соответствии с методологией «сжимаемых линий тока»

Unsteady Compressible Flow Calculations with Least-Square Mesh-less Method

A dual-time implicit mesh-less method is presented for unsteady compressible flow calculations. Polynomial least-square (PLS) and Taylor series least-square (TLS) procedures are used to estimate the spatial derivatives at each node and their computational efficiencies are compared. Also, the effect of the neighbor stencil selection on the accuracy of the method is investigated. As a new approach, different neighboring stencils are used for the highly stretched point distribution inside the boundary layer region and the inviscid isotropic point distribution outside this area. The unsteady flows over stationary and moving objects at subsonic and transonic flow conditions are solved. Results indicate the computational efficiency of the method in comparison with the alternative approaches. The convergence histories of the flow solution show that the PLS method is computationally faster than TLS method. In addition, the eight point neighboring stencil inside the viscous region is more efficient than other choices.

Three-dimensional building-cube method for inviscid compressible flow computations

In this study, the Building-Cube method (BCM) devised for calculating large-scale flows and getting rid of the grid dependencies of computational results is extended to perform 3D inviscid compressible fluid flow simulations. The proposed scheme divides a computational domain into a large number of cubes with different sizes, and each individual cube is a sub-flow field simulated with a Cartesian grid of uniform spacing and an equal number of cells and nodes. Through the proposed scheme, we determine the geometric size of individual cube by Adapting the flow characteristics and geometrical shapes using an adaptively refining Cartesian grid approaches. The uniform spacing and equal number of cells and nodes in the Cartesian grid of individual cubes ensure a good performance for parallel computations; large result data can be handled efficiently. Further, an algorithm to solve the inviscid flow equations on the Building-Cube mesh for three-dimensional (3D) geometries is presented. The validation and performance of the proposed 3D BCM are demonstrated through comparisons of the computed results with the experimental data for the ONERA M6 wing and ONERA M5 wing-fuselage configurations.

On the computation of the Baer–Nunziato model using ALE formulation with HLL- and HLLC-type solvers towards fluid–structure interactions

Computation of compressible two-phase flows with the unsteady compressible Baer–Nunziato model in conjunction with the moving grid approach is discussed in this paper. Both HLL- and HLLC-type Finite-Volume methods are presented and implemented in the context of Arbitrary Lagrangian–Eulerian formulation in a multidimensional framework. The construction of suitable numerical methods is linked to proper approximations of the non-conservative terms on moving grids. The HLL discretization follows global conservation properties such as free-stream preservation and uniform pressure and velocity profiles preservation on moving grids. The HLLC solver initially proposed by Tokareva and Toro [1] for the Baer–Nunziato model is based on an approximate solution of local Riemann problems containing all the characteristic fields present in the exact solution. Both “subsonic” and “supersonic” configurations are considered in the construction of the present HLLC solver. In addition, an adaptive 6-wave HLLC scheme is also proposed for computational efficiency. The methods are first assessed on a variety of 1-D Riemann problems including both fixed and moving grids applications. The methods are finally tested on 2-D and 3-D applications: 2-D Riemann problems, a 2-D shock–bubble interaction and finally a 3-D fluid–structure interaction problem with a good agreement with the experiments.

Unsteady compressible flow computations using an adaptive multiresolution technique coupled with a high-order one-step shock-capturing scheme

This article deals with the development of adaptive multiresolution coupled with a one-step shock-capturing scheme for the numerical simulation of unsteady compressible flows in the transonic and supersonic regimes with high frequency oscillations. The discretization of the convective terms is based on a coupled time and space approach by using a one-step (OS) scheme, developed following the Lax–Wendroff approach by correcting the successive modified equations. A monotonicity preserving (MP) criterion is added in order to locally relax the TVD constraints for such schemes. The adaptive strategy relies on the Harten cell-average multiresolution analysis, with a dynamical data structure organized as a graded tree that dynamically evolves in time. We apply the method to several prototype test-cases of shock-wave propagation interaction. We validate this approach on 2D inviscid advection of a vortex. We then present 2D viscous test-cases of shock-shear layer interactions and a 3D spherical Riemann problem to demonstrate the capability of the present method. Results demonstrate that 7th order OSMP schemes coupled with adaptive grid refinement gives very accurate results in comparison with more classical schemes applied on a single grid. We then propose an appropriate MR threshold parameter value that ensures accurate results while achieving drastic gains on the CPU time and memory usage.

Simulation of the Taylor–Green Vortex Using High-Order Flux Reconstruction Schemes

In this paper, the ability of high-order flux reconstruction numerical schemes to perform accurate and stable computations of compressible turbulent flows on coarse meshes is investigated. Two new flux reconstruction schemes, which are optimized for wave dissipation and dispersion properties, are compared to the nodal discontinuous Galerkin and spectral difference methods recovered via the energy-stable flux reconstruction method. The Taylor–Green vortex benchmark problem at is used as a simple a priori test of the numerics. Dissipation rates computed from kinetic energy, vorticity, and pressure dilatation are plotted against reference solutions. Results show that, at low mesh resolution, the flux reconstruction schemes are highly accurate across a range of orders of accuracy, although oscillations can appear in the solution at orders of six and above. Although the flux reconstruction method has a built-in stabilization mechanism, an additional means of damping these instabilities is required. The schemes vary in the amount of numerical dissipation and resolution of the turbulent spectrum. One of the optimized flux reconstruction schemes (the optimized flux reconstruction scheme) is shown to have greater spectral accuracy than any of the others tested, motivating its future usage for high-order high-fidelity computational fluid dynamics.