Simulation Of Compressible Viscous Flows Research Articles

Double distribution function-based lattice Boltzmann flux solver for simulation of compressible viscous flows

In this work, a double distribution function-based lattice Boltzmann flux solver (LBFS) is proposed for simulating compressible viscous flows. This approach utilizes the double distribution function compressible lattice Boltzmann model and employs Chapman–Enskog expansion analysis to connect the lattice Boltzmann equation (LBE) with the Navier–Stokes (N–S) equations. Unlike conventional computational fluid dynamics methods that compute inviscid and viscous fluxes separately, the present method simultaneously evaluates both types of fluxes at the cell interface by locally reconstructing the solution of the LBE. Recognizing the significance of considering the non-equilibrium part of distribution functions for viscous flows, a straightforward method is introduced to calculate this component. This facilitates the derivation of computational expressions for macroscopic conservative variables and fluxes in the N–S equations. To validate the accuracy and stability of the present numerical scheme, various benchmark problems, including shock tube problem, Couette flow, lid-driven cavity flow, and flow around the NACA0012 airfoil, are tested. The obtained numerical results are compared with analytical solutions or existing reference data, confirming the capability of the proposed LBFS to deliver accurate and stable numerical results for compressible flows. Moreover, this method demonstrates effectiveness in handling viscous flow problems on non-uniform grids and with curved boundaries.

Development of a Moving-Least-Square-based Immersed Boundary Method for the simulation of viscous Compressible Flows

In this paper, a Moving-Least-Squares (MLS)-based immersed boundary method (IBM) for simulating viscous compressible flows is proposed. For this method, the compressible lattice Boltzmann flux solver (LBFS) is used to solve the flow field on the Eulerian mesh and the MLS-based IBM introduced in present paper is used to implement boundary condition. The discrete Lagrangian points are used to represent the Solid boundaries. A second-order MLS interpolation is used for numerical exchange between two grids of IBM. The forced implicit boundary condition is adopted to ensure that the physical boundary condition can be accurately satisfied. The present method is able to achieve a higher numerical accuracy than traditional IBM, which use the Delta function for interpolation. Some numerical validation tests are used for the validation of proposed method. The order and accuracy of the MLS approximation are tested by using a classical example of flow past a cylinder. Test of flow past the NACA0012 airfoil at different Mach numbers, angles of attack and Reynolds numbers is applied to verified the accuracy.

Constrained least-squares based adaptive-order finite-volume WENO scheme for the simulation of viscous compressible flows on unstructured grids

We develop a novel constrained least-squares technique for fourth-order weighted essentially non-oscillatory (WENO) polynomial reconstruction on general unstructured grids composed of curved-edged quadrilateral elements. Our technique relies on a segregation of the conditions that match the spatial averages of the candidate WENO polynomial and the reconstructed state variable over stencil-forming cells into two sets. The first set consists of exactly enforced linearly independent constraints that are applied only over the central cell and its edge-sharing neighbors. The second set consists of the remaining conditions and are enforced only approximately in a least-squares sense. We develop a specialized technique that reduces the computational burden associated with the numerical solution of the constrained least-squares system for the WENO interpolation coefficients. For a comparable computational expense, our constrained reconstruction approach yields pointwise estimates that are consistently more accurate than the conventional least-squares based estimates over a wide spectrum of spatial wavenumbers. This enhanced spectral resolution yields a significant, over two-fold increase in the accuracy with which unsteady smooth flow features, such as an advecting isentropic vortex, are captured over highly distorted curvilinear unstructured grids. Our constraint based reconstruction framework is generic, facilitates implementation of high-order boundary closures, and is amenable to large scale parallelization over several thousand CPU cores. Numerical tests over a range of inviscid and viscous flow configurations are presented to demonstrate the accuracy and robustness of our technique. Our constrained least-squares based adaptive WENO discretization faithfully captures intricate features that arise in strongly shocked, and highly unsteady and separated complex geometry flows. Our results suggest exact preservation of the match between the cell averages of the interpolant and the reconstructed state variable over the nearest neighbors to be quite an effective strategy that dramatically enhances the accuracy and spectral resolution of the high-order finite-volume reconstruction operation.

An immersed boundary method with implicit body force for compressible viscous flow

We present a new immersed boundary method for the simulation of compressible viscous flow. The method is based on the singular source approach where the immersed boundary adds surface singularities to the governing equations. The strengths of the surface singularities are treated as algebraic variables under the framework of differential-algebraic equations and are implicitly calculated to enforce the corresponding boundary conditions. Discrete delta functions are used to approximate the surface singularities in the Eulerian grid and interpolate variables back to the surface mesh. A half-explicit Runge-Kutta method is used to achieve desired temporal accuracy. The method has been generalized from the well-known no-slip and isothermal condition to stress and heat flux conditions. The latter necessitates the introduction of further singularities in the stresses and heat flux, which is a new feature of the formulation. Furthermore, the approach has been extended to porous surfaces. The method is also applicable to flow-structure interaction problems. Two- and three-dimensional numerical tests are carried out to demonstrate accuracy. The results are compared with previous numerical and experimental studies or analytical solutions. Finally, a fully three-dimensional fluid-structure interaction simulation of a subsonic parachute is showcased to demonstrate applicability to a real problem.

A discontinuous Petrov-Galerkin method for compressible Navier-Stokes equations in three dimensions

Application of a Discontinuous Petrov-Galerkin (DPG) method for simulation of compressible viscous flows in three dimensions is presented. The approach enables construction of stable schemes for problems with a small perturbation parameter. The main idea of the method is a weak formulation with a relaxed interelement continuity of the solution. The formulation satisfies the inf-sup condition with the stability constant independent of the small perturbation parameter, which here is the viscosity constant for the compressible Navier-Stokes equations. The DPG discrete formulation uses the specially designed so-called optimal test functions. They do not compromise the inf-sup stability of the continuous formulation. DPG does not use any artificial dissipation for the compressible Navier-Stokes equations. Being a residual minimization method it has got a built-in a posteriori error estimation which allows for mesh adaptivity leading to resolving reliable viscous fluxes, the major difficult task in simulations of viscous flows. We illustrate the method with a few steady state laminar solutions.

Immersed boundary conditions for hypersonic flows using ENO-like least-square reconstruction

In this study, we present an immersed boundary reconstruction method for the simulation of viscous compressible flows presenting strong discontinuities in the vicinity of the immersed body. The technique we propose is based on a weighted least-square reconstruction. Its major feature is that the weights are automatically adapted to the presence of a discontinuity within the least-square stencil by using a shock-sensor approach. In so doing, the reconstruction becomes ENO-like and mathematically consistent even in the presence of strong discontinuities. The objective of this paper is to demonstrate that it then becomes possible to use an immersed boundary method for the simulation of hypersonic flows and that it is robust, even for flows around complex geometries. To do so, two-dimensional compressible Navier-Stokes equations are solved to study canonical hypersonic flows and it is showed that the results are in good agreement with either experimental data and/or body-fitted approaches and that the impact of the immersed boundary technique on the overall spatial order of convergence is reasonable.

A generalized sphere function-based gas kinetic scheme for simulation of compressible viscous flows

In this paper, a generalized sphere function-based gas kinetic scheme (GKS) is developed for simulation of two-dimensional compressible viscous flows. This work aims to improve the existing simplified GKS, in which a simple two-dimensional/three-dimensional distribution function is used as the equilibrium state. The present scheme applies the finite volume method to discretize Navier–Stokes equations and the numerical flux at cell interface is evaluated by the local reconstruction of solution for the continuous Boltzmann equation with [Formula: see text]-dimensional sphere function. This local solution contains both the equilibrium part and nonequilibrium part, and the contribution of the nonequilibrium part is controlled by a switch function. It is found that the present scheme has the same expression as the circular function-based GKS for the two-dimensional case and the sphere function-based GKS for the three-dimensional case. Therefore, these two GKSs can be unified, and the developed method provides another way for the simple distribution function.

A diffuse‐interface immersed boundary method for simulation of compressible viscous flows with stationary and moving boundaries

SummaryIn this paper, a diffuse‐interface immersed boundary method (IBM) is proposed for simulation of compressible viscous flows with stationary and moving boundaries. In the method, the solution of flow field and the implementation of boundary conditions are decoupled into two steps by applying the fractional step technique, ie, the predictor step and the corrector step. Firstly, in the predictor step, the intermediate flow field is resolved by a recently developed gas kinetic flux solver (GKFS) without consideration of the solid boundary. The GKFS is a finite volume approach that solves the Navier‐Stokes equations for the flow variables at cell centers. In GKFS, the inviscid and viscous fluxes are evaluated as a single entity by reconstructing the local solution of continuous Boltzmann equation. Secondly, in the corrector step, the intermediate flow field is corrected by the present diffuse‐interface IBM. During this process, the velocity field is firstly corrected by the implicit boundary condition–enforced IBM so that the no‐slip boundary condition can be accurately satisfied. After that, the density correction is made by an iterative approach with the help of the continuity equation. Finally, the correction of the temperature field is made in the same way as that of the velocity field. Good agreements between the present simulations and the reference data in literature demonstrate the reliability of the proposed method.

An implicit dual-time stepping spectral difference lattice Boltzmann method for simulation of viscous compressible flows on structured meshes

In this work, the spectral difference lattice Boltzmann method (SDLBM) is extended and applied for accurately computing two-dimensional viscous compressible flows on structured meshes. Here, the compressible form of the discrete Boltzmann-BGK equation with the Watari model is considered and the numerical solution of the resulting LB equation is obtained by using the spectral difference method. The main benefit of the use of the LB method in simulating compressible flows is that a same formulation can be applied to compute the inviscid and viscous portions of the flowfield. Note that the LB formulation for simulating the viscous flows is the same as that used for the inviscid ones, however, the wall boundary conditions are changed in a manner to consider the no-slip conditions. Here, the SDLBM is also extended to use curved-edge cells for properly representing curved wall boundaries. In addition, to enhance the solution of the SDLBM the time integration is efficiently performed by implementing an implicit dual-time stepping method which does not require a matrix inversion. Both steady and unsteady flows are simulated. Different test cases including the Couette flow, the viscous shock–vortex interaction, the flow over a circular cylinder and the flow over a NACA-0012 airfoil are simulated to assess the accuracy and robustness of the solution procedure proposed based on the SDLBM in computing steady and unsteady viscous compressible flows. The present results obtained by applying the SDLBM exhibit good agreement compared to the analytical and available high-order accurate solutions of the LB and Navier–Stokes equations.

Development of a new OpenFOAM solver using regularized gas dynamic equations

The paper introduces the development of a new OpenFOAM solver QGDFoam for the numerical simulation of viscous compressible flows within a wide range of Mach numbers in the framework of the OpenFOAM formalism. The new solver is based on the implementation of regularized, or quasi-gas dynamic (QGD) equations. The mixed finite-volume and finite-difference approximation is constructed on unstructured space grids with co-located variables storage and explicit time scheme for convection approximation. The solver has been tested for a range of 1D Riemann problems and 2D cases, comparing results with analytic solutions and OpenFOAM’s implementation of the Kurganov–Tadmor scheme known as rhoCentralFoam.

Simulation of moving boundaries interacting with compressible reacting flows using a second-order adaptive Cartesian cut-cell method

A high-order adaptive Cartesian cut-cell method, developed in the past by the authors [1] for simulation of compressible viscous flow over static embedded boundaries, is now extended for reacting flow simulations over moving interfaces. The main difficulty related to simulation of moving boundary problems using immersed boundary techniques is the loss of conservation of mass, momentum and energy during the transition of numerical grid cells from solid to fluid and vice versa. Gas phase reactions near solid boundaries can produce huge source terms to the governing equations, which if not properly treated for moving boundaries, can result in inaccuracies in numerical predictions. The small cell clustering algorithm proposed in our previous work is now extended to handle moving boundaries enforcing strict conservation. In addition, the cell clustering algorithm also preserves the smoothness of solution near moving surfaces. A second order Runge–Kutta scheme where the boundaries are allowed to change during the sub-time steps is employed. This scheme improves the time accuracy of the calculations when the body motion is driven by hydrodynamic forces. Simple one dimensional reacting and non-reacting studies of moving piston are first performed in order to demonstrate the accuracy of the proposed method. Results are then reported for flow past moving cylinders at subsonic and supersonic velocities in a viscous compressible flow and are compared with theoretical and previously available experimental data. The ability of the scheme to handle deforming boundaries and interaction of hydrodynamic forces with rigid body motion is demonstrated using different test cases. Finally, the method is applied to investigate the detonation initiation and stabilization mechanisms on a cylinder and a sphere, when they are launched into a detonable mixture. The effect of the filling pressure on the detonation stabilization mechanisms over a hyper-velocity sphere launched into a hydrogen–oxygen–argon mixture is studied and a qualitative comparison of the results with the experimental data are made. Results indicate that the current method is able to correctly reproduce the different regimes of combustion observed in the experiments. Through the various examples it is demonstrated that our method is robust and accurate for simulation of compressible viscous reacting flow problems with moving/deforming boundaries.

A Hybrid Lattice Boltzmann Flux Solver for Simulation of Viscous Compressible Flows

AbstractIn this paper, a hybrid lattice Boltzmann flux solver (LBFS) is proposed for simulation of viscous compressible flows. In the solver, the finite volume method is applied to solve the Navier-Stokes equations. Different from conventional Navier-Stokes solvers, in this work, the inviscid flux across the cell interface is evaluated by local reconstruction of solution using one-dimensional lattice Boltzmann model, while the viscous flux is still approximated by conventional smooth function approximation. The present work overcomes the two major drawbacks of existing LBFS [28–31], which is used for simulation of inviscid flows. The first one is its ability to simulate viscous flows by including evaluation of viscous flux. The second one is its ability to effectively capture both strong shock waves and thin boundary layers through introduction of a switch function for evaluation of inviscid flux, which takes a value close to zero in the boundary layer and one around the strong shock wave. Numerical experiments demonstrate that the present solver can accurately and effectively simulate hypersonic viscous flows.

A high-order adaptive Cartesian cut-cell method for simulation of compressible viscous flow over immersed bodies

A new adaptive finite volume conservative cut-cell method that is third-order accurate for simulation of compressible viscous flows is presented. A high-order reconstruction approach using cell centered piecewise polynomial approximation of flow quantities, developed in the past for body-fitted grids, is now extended to the Cartesian based cut-cell method. It is shown that the presence of cut-cells of very low volume results in numerical oscillations in the flow solution near the embedded boundaries when standard small cell treatment techniques are employed. A novel cell clustering approach for polynomial reconstruction in the vicinity of the small cells is proposed and is shown to achieve smooth representation of flow field quantities and their derivatives on immersed interfaces. It is further shown through numerical examples that the proposed clustering method achieves the design order of accuracy and is fairly insensitive to the cluster size. Results are presented for canonical flow past a single cylinder and a sphere at different flow Reynolds numbers to verify the accuracy of the scheme. Investigations are then performed for flow over two staggered cylinders and the results are compared with prior data for the same configuration. All the simulations are carried out with both quadratic and cubic reconstruction, and the results indicate a clear improvement with the cubic reconstruction. The new cut-cell approach with cell clustering is able to predict accurate results even at relatively low resolutions. The ability of the high-order cut-cell method in handling sharp geometrical corners and narrow gaps is also demonstrated using various examples. Finally, three-dimensional flow interactions between a pair of spheres in cross flow is investigated using the proposed cut-cell scheme. The results are shown to be in excellent agreement with past studies, which employed body-fitted grids for studying this complex case.

Improved Navier–Stokes Boundary Conditions Based on Generalized Characteristics

In this paper, the three-dimensional Navier–Stokes characteristic boundary conditions for large-eddy and aeroacoustic simulations are extended to curvilinear coordinates formulations. A robust way of treating the transverse and gradient terms on boundary planes is presented which is different from previous generalized characteristic boundary conditions. The performance of the new formulation is examined via four test problems: an inviscid convective vortex, a two-dimensional mixing layer, a Mach 0.75 round jet, and a Mach 0.51 nozzle/jet. For each test problem, the numerical schemes used to implement the boundary conditions, the numerical parameters employed, and the predicted three-dimensional flow fields are presented. Based on the numerical experiments conducted, the new boundary conditions show promise for high-fidelity simulations of compressible viscous flows.

The discontinuous Galerkin method for the numerical simulation of compressible viscous flow

In this paper we deal with numerical simulation of the compressible viscous flow. The mathematical model of flow is represented by the system of non-stationary compressible Navier-Stokes equations. This system of equations is discretized by the discontinuous Galerkin finite element method in space and in time using piecewise polynomial discontinuous approximations. We present some numerical experiments to demonstrate the applicability of the method using own-developed code.

Simulation of compressible viscous flow in time-dependent domains

The paper is concerned with the simulation of viscous compressible flow in time dependent domains. The dependence on time of the domain occupied by the fluid is taken into account with the aid of the Arbitrary Lagrangian–Eulerian (ALE) formulation of the compressible Navier–Stokes equations. They are discretized by the discontinuous Galerkin finite element method using piecewise polynomial discontinuous approximations. The time discretization is based on a semi-implicit linearized scheme, which leads to the solution of a linear algebraic system on each time level. A suitable treatment of boundary conditions and shock capturing are used, allowing the solution of flow with a wide range of Mach numbers. The applicability of the developed method is demonstrated by computational results obtained for compressible viscous flow in a channel with moving walls and flow induced airfoil vibrations.

Three-dimensional boundary conditions for direct and large-eddy simulation of compressible viscous flows

Navier–Stokes characteristic boundary conditions (NSCBC) usually assume the flow to be normal to the boundary plane. In this paper, NSCBC is extended to account for convection and pressure gradients in boundary planes, resulting in a 3D-NSCBC approach. The introduction of these additional transverse terms requires a specific treatment for the computational domain’s edges and corners, as well as a suited set of compatibility conditions for boundaries joining regions associated to different flow properties, as inlet, outlet or wall. A systematic strategy for dealing with edges and corners is derived and compatibility conditions for inlet/outlet and wall/outlet boundaries are proposed. Direct numerical simulation (DNS) tests are carried out on simplified flow configurations at first. 3D-NSCBC brings a drastic reduction of flow distortion and numerical reflection, even in regions of strong transverse convection; the accuracy and convergence rate toward target values of flow quantities is also improved. Then, 3D-NSCBC is used for large-eddy simulation (LES) of a free jet and an impinging round-jet. Edge and corner boundary treatment, combining multidirectional characteristics and compatibility conditions, yields stable and accurate solutions even with mixed boundaries characterized by bad posedness issues (e.g. inlet/outlet). LES confirms the effectiveness of the proposed boundary treatment in reproducing mean flow velocity and turbulent fluctuations up to the computational domain limits.

Partially reflecting and non-reflecting boundary conditions for simulation of compressible viscous flow

For numerical simulation of compressible viscous flow, the characteristics-based NSCBC boundary conditions proposed by Poinsot and Lele [T. Poinsot, S.K. Lele, Boundary conditions for direct simulation of compressible viscous flows, J. Comput. Phys. 101 (1992) 104–129] are frequently employed. This formulation is analyzed analytically and it is found that the linear relaxation term proposed by Rudy and Strikwerda [D.H. Rudy, J.C. Strikwerda, A nonreflecting outflow boundary condition for subsonic Navier–Stokes calculations, J. Comput. Phys. 36 (1) (1980) 55–70] to suppress slow “drift” of flow variables results in a non-zero reflection coefficient for acoustic waves. Indeed, although the NSCBC formulation of boundary conditions is often called “non-reflecting”, the magnitude of the reflection coefficient approaches unity for low frequencies. A modification of the NSCBC boundary conditions and in particular the linear relaxation term is proposed, which should appear fully non-reflecting to plane acoustic waves with normal incidence on the boundaries for all frequencies. The new formulation is implemented and successfully validated in large eddy simulation of turbulent channel flow.

Actual Impedance of Nonreflecting Boundary Conditions: Implications for Computation of Resonators

Nonreflecting boundary conditions are essential elements in the computation of many compressible flows. Such simulations are very sensitive to the treatment of acoustic waves at boundaries. Nonreflecting conditions allow acoustic waves to propagate through boundaries with zero or small levels of reflection into the domain. However, perfectly nonreflecting conditions must be avoided because they can lead to ill-posed problems for the mean flow. Various methods have been proposed to construct boundary conditions that can be sufficiently nonreflecting for the acoustic field while still making the mean flow problem well-posed. widely used technique for nonreflecting outlets is analyzed (Poinsot, T., and Lele, S., Conditions for Direct Simulations of Compressible Viscous Flows, Journal of Computational Physics, Vol. 101, No.1,1992, pp. 104-129; Rudy, D. H., and Strikwerda, J. C., A Non-Reflecting Outflow Boundary Condition for Subsonic Navier-Stokes Calculations, Journal of Computational Physics, Vol. 36, 1980, pp. 55-70). It shows that the correction introduced by these authors can lead to large reflection levels and resonant behavior that cannot be observed in the experiment. simple scaling is proposed to evaluate the relaxation coefficient used in these methods for a nonreflecting outlet. The proposed scaling is tested for simple cases (ducts) both theoretically and numerically.

D05 ベーシック型パルス管冷凍機における熱流動数値シミュレーション : パルス管管壁と作動ガスの熱移動に関する解析

Numerical simulation of viscous compressible flow in basic pulse tube refrigerator is conducted to make clear the working principle of refrigeration. Transient axisymmetric two-dimensional equations of continuity, momentum and energy were solved by means of TVD method. The physical model of the pulse tube with a regenerator is used for numerical simulation. Heat exchange between the pulse tube wall and the working gas in the pulse tube is assumed as a convective heat transfer. In this paper the transient behaviors of pressure and gas temperature, moving distances and temperature changes of gas elements in the pulse tube, and the effect on Surface Heat Pumping are analyzed.