Subsonic Viscous Flow Research Articles

Galilean-Invariant Characteristic-Based Volume Penalization Method for Supersonic Flows with Moving Boundaries

This work extends the characteristic-based volume penalization method, originally developed and demonstrated for compressible subsonic viscous flows in (J. Comput. Phys. 262, 2014), to a hyperbolic system of partial differential equations involving complex domains with moving boundaries. The proposed methodology is shown to be Galilean-invariant and can be used to impose either homogeneous or inhomogeneous Dirichlet, Neumann, and Robin type boundary conditions on immersed boundaries. Both integrated and non-integrated variables can be treated in a systematic manner that parallels the prescription of exact boundary conditions with the approximation error rigorously controlled through an a priori penalization parameter. The proposed approach is well suited for use with adaptive mesh refinement, which allows adequate resolution of the geometry without over-resolving flow structures and minimizing the number of grid points inside the solid obstacle. The extended Galilean-invariant characteristic-based volume penalization method, while being generally applicable to both compressible Navier–Stokes and Euler equations across all speed regimes, is demonstrated for a number of supersonic benchmark flows around both stationary and moving obstacles of arbitrary shape.

A high-order accurate unstructured spectral difference lattice Boltzmann method for computing inviscid and viscous compressible flows

In the present work, the spectral difference lattice Boltzmann method (SDLBM) is implemented on unstructured meshes for the solution methodology to be capable of accurately simulating the compressible flows over complex geometries. Both the inviscid and viscous compressible flows are computed by applying the unstructured SDLBM. The compressible form of the discrete Boltzmann–BGK equation with the Watari model is considered and the solution of the resulting system of equations is obtained by applying the spectral difference method on arbitrary quadrilateral meshes. The accuracy and robustness of the unstructured SDLBM for simulating the compressible flows are demonstrated by simulating four problems that are steady inviscid supersonic flow past a bump, steady inviscid subsonic flow over the two-element NACA 4412-4415 airfoil with and without the ground effect, steady viscous transonic flow around the NACA 0012 airfoil and unsteady viscous subsonic flow past two side-by-side cylinders. The results obtained by applying the unstructured SDLBM are in good agreement with those of the available high-order accurate Euler/Navier-Stokes solvers and also the experimental data. The present study introduces the unstructured SDLBM as an appropriate inviscid and viscous compressible LBM flow solver for accurately simulating fluid flows over practical problems.

Change in Separation Flow Regimes over Obstacles in Subsonic Gas Flow as a Manifestation of Viscous Forces: Numerical Results

The unsteady subsonic viscous gas flows over forward- and backward-facing steps on a flat surface are numerically simulated at a free-stream Mach number of 0.1 by applying compact difference schemes of the 14th order of accuracy. The characteristics of the boundary layer separation zones are investigated at Reynolds numbers ranging from 103 to 107. The computations are based on the nonstationary Navier–Stokes equations supplemented, if necessary, with the formulas of the one-parameter differential SA model of turbulent viscosity. The conditions for pulsing separation zones arising in the boundary layer are determined, and the subsequent transition to a turbulent flow regime is physically justified.

Meshfree Simulation of Flow around Airfoil using Different Turbulent Models

The moving least square meshfree method is used to solve the Reynolds-averaged Navier-Stokes equations with two different turbulence models: the renormalisation group (RNG) k-e model and the Wilcox k-ω model. The moving least square approach with clouds-of-points reconstruction, which can provide results from highly anisotropic clouds of points, is adopted to approximate the spatial derivatives. Both the subsonic and transonic viscous flows around RAE2822 airfoil are simulated. The pressure coefficients of the airfoil surface are calculated, the Mach number contours of the flow fields are presented, and the mean velocity profiles of the boundary layer at different locations are investigated. Some unsteady viscous flow simulation results are also presented. The simulation results show a good agreement with the experiment results and other numerical simulation results. Overall, the simulation results demonstrate that the proposed meshfree numerical algorithm is accurate and capable of modelling turbulence flows around airfoils.

An implicit algorithm based on iterative modified approximate factorization method coupling with characteristic boundary conditions for solving subsonic viscous flows

An algorithm composed of an iterative modified approximate factorization (MAF(k)) method with Navier-Stokes characteristic boundary conditions (NSCBC) is proposed for solving subsonic viscous flows. A transformation on the matrix equation in MAF(k) is made in order to impose the implicit boundary conditions properly. To be in consistent with the implicit solver for the interior domain, an implicit scheme for NSCBC is formulated. The performance of the developed algorithm is investigated using spatially evolving zero pressure gradient boundary layer over a flat plate and a wall jet mixing with a cross flow over a flat plate with a square hole as the test cases. The numerical results are compared to the existing experimental datasets and a number of general correlations, together with other available numerical solutions, which demonstrate that the developed algorithm possesses promising capacity for simulating the subsonic viscous flows with large CFL number.

An implicit discontinuous Galerkin method for the unsteady compressible Navier–Stokes equations

A high-order implicit discontinuous Galerkin method is developed for the time-accurate solutions to the compressible Navier–Stokes equations. The spatial discretization is carried out using a high order discontinuous Galerkin method, where polynomial solutions are represented using a Taylor basis. A second order implicit method is applied for temporal discretization to the resulting ordinary differential equations. The resulting non-linear system of equations is solved at each time step using a pseudo-time marching approach. A newly developed fast, p-multigrid is then used to obtain the steady state solution to the pseudo-time system. The developed method is applied to compute a variety of unsteady subsonic viscous flow problems. The numerical results obtained indicate that the use of this implicit method leads to significant improvements in performance over its explicit counterpart, while without significant increase in memory requirements.

Nonreflecting boundary conditions and numerical simulation of external flows

The specification of conditions on artificial boundaries of the computational domain in the simulation of subsonic viscous gas flows is considered. The steps in the construction and implementation of nonreflecting boundary conditions on the path from one-dimensional linearized Euler equations to real-life problems are described. The technique is intended for flow simulation at low Mach numbers. Numerical results for the essentially subsonic flow over a flat plate are presented

Effect of Small-Scale Output Unsteadiness on Adjoint-Based Sensitivity

This paper investigates the impact of small-scale unsteadiness on adjoint-based output sensitivity analysis. In particular, when iterative methods for nonlinear flows fail to converge to a steady state, it is demonstrated that the resulting sensitivity analysis can be highly inaccurate, even when the unsteadiness in the outputs is small. The specific example considered is the viscous subsonic flow around an airfoil over a range of angles of attack. If a strengthened solver is used to solve the adjoint equation (even though the flow equations did not fully converge), it is demonstrated that the sensitivity of the lift with respect to angle of attack can vary significantly, due to linearizing about different solution iterates of the steady flow solver. Further, the unsteady iterates from the time-inaccurate steady-state solver to the time-accurate solution are compared. The unsteadiness of the time-accurate solution is markedly different from the iterate solutions of the steady-state solver. If a strengthened solver is applied to the nonlinear flow equations, steady solutions can be achieved whose lift is significantly different from the time-averaged lift of the time-accurate simulations. Time-accurate unsteady adjoint analysis is then shown to provide accurate sensitivities for the time-averaged lift.

Numerical Validation on The Performance of A Two Dimensional Curved Diffuser

This paper deals with the investigation of the characteristic of subsonic viscous flow through a curved diffuser numerically with commercial code for computational fluid dynamics (CFD) Fluent Inc. version 6.3. The diffuser flow is a two-dimensional, turbulent, incompressible and fully developed. The investigations are based on the Spalart-Allmaras turbulent model. A 2-D quadrilateral grid is generated by the grid generator GAMBIT. Obtained results are compared with the available experimental data and found to give good agreement. The effects of curvature angle, area ratio and adding tail channel with constant area on the diffuser performance and flow pattern are studied and revealed by the pressure contour, velocity vector, and variation of the pressure recovery factor for all above mentioned parameters.

A generalized framework for high order anisotropic mesh adaptation

We present a method for anisotropically adapting meshes for high order ( p > 2 ) finite volume methods. We accomplish this by assuming a polynomial error measure based on the local reconstruction. We then use Fourier series to choose a metric function that approximates our error measure. This approach is theoretically valid for any solution order, with any number of variables and any number of dimensions. Using both second and third order solvers, we present examples of two-dimensional subsonic viscous and inviscid flow around the NACA-0012 airfoil. These test cases demonstrate that mesh refinement based on our metric converges to an accurate solution much faster than with uniform refinement. For subsonic viscous flows, we also show that anisotropic refinement is more efficient than isotropic refinement. By limiting the magnitude of our error measure for a transonic inviscid flow, we demonstrate that our metric can be effective even in the presence of discontinuities. We also examined using second derivatives to estimate the error for third order solutions. While this is less theoretically sound, we found little difference in the resulting accuracy.

Implicit preconditioned WENO scheme for steady viscous flow computation

A class of lower–upper symmetric Gauss–Seidel implicit weighted essentially nonoscillatory (WENO) schemes is developed for solving the preconditioned Navier–Stokes equations of primitive variables with Spalart–Allmaras one-equation turbulence model. The numerical flux of the present preconditioned WENO schemes consists of a first-order part and high-order part. For first-order part, we adopt the preconditioned Roe scheme and for the high-order part, we employ preconditioned WENO methods. For comparison purpose, a preconditioned TVD scheme is also given and tested. A time-derivative preconditioning algorithm is devised and a discriminant is devised for adjusting the preconditioning parameters at low Mach numbers and turning off the preconditioning at intermediate or high Mach numbers. The computations are performed for the two-dimensional lid driven cavity flow, low subsonic viscous flow over S809 airfoil, three-dimensional low speed viscous flow over 6:1 prolate spheroid, transonic flow over ONERA-M6 wing and hypersonic flow over HB-2 model. The solutions of the present algorithms are in good agreement with the experimental data. The application of the preconditioned WENO schemes to viscous flows at all speeds not only enhances the accuracy and robustness of resolving shock and discontinuities for supersonic flows, but also improves the accuracy of low Mach number flow with complicated smooth solution structures.

Control of the aerodynamic characteristics of a profile oscillating with respect to the incidence angle in subsonic viscous flow

The results of a numerical simulation of the unsteady subsonic viscous gas flow around a two-dimensional profile oscillating with respect to the incidence angle are presented and the possibility of controlling the nonstationary aerodynamic characteristics is considered. The hysteresis phenomena typical of oscillatory profile motions are investigated, the dependence of the lift force and drag is found for various laws of periodic variation of the incidence angle with time, and the effect of the frequency and amplitude of the angular profile oscillations on the shape of the hysteresis curves is studied. The calculations were based on the numerical solution of the nonstationary Navier-Stokes equations averaged in the Reynolds sense (Reynolds equations) which were closed using the k-ω turbulence model with modeling of the laminar/turbulent transition.

An inverse blade design method for subsonic and transonic viscous flow in compressors and turbines

An inverse blade design method applicable to two- and three-dimensional inviscid and viscous flow in turbomachinery cascades is presented and is applied to design cascades in two-dimensional viscous flow. The pressure distribution along the blade surfaces is prescribed and is reached by modifying the initial guess of the blade geometry. The geometry modification is computed from a virtual velocity distribution derived from the difference between the current and the target pressure along the blade surfaces. The inverse method is implemented into and is consistent with the unsteady Reynolds-averaged Navier-Stokes (RANS) equations where an arbitrary Lagrangian–Eulerian (ALE) formulation on a moving and deforming grid is used. The grid velocities are determined from the space conservation law (SCL), which ensures a fully conservative computational procedure. The unsteady RANS equations are discretized using a cell-vertex finite volume method and the time accuracy is achieved using a dual time stepping scheme. An algebraic Baldwin–Lomax model is used for turbulence closure. The design method is first validated, and then its robustness, flexibility and usefulness are demonstrated on the redesign of recent compressor and turbine blade geometries used in modern gas turbine engines.

Design and optimization of a two-element wing airfoil in a subsonic viscous flow

A numerical and analytical solution of the problem of designing a two-element wing airfoil providing maximum lift-drag ratio in a subsonic viscous flow is presented. In order to bring the theoretical results closer to the facts, viscosity and compressibility are taken into account within the framework of boundary layer theory and the Chaplygin gas model, respectively.

Acoustic Properties of Continuous and Discrete Models in Fluid Dynamics

The article investigates the propagation of small perturbations in fluids whose dynamics is described by Euler and Navier–Stokes equations or by a quasi-fluid-dynamic system derived from the difference approximation of the Boltzmann equation. The problem of wave reflection from the artificial boundaries of the numerical region is solved using various boundary conditions. The analysis is repeated for difference approximations of fluid-dynamic equations. The procedure is tested for viscous subsonic flow past a plate.

Acoustics in viscous subsonic flow models with nonreflecting boundary conditions

The article investigates acoustic waves in a subsonic viscous thermally conducting gas. Two descriptions of the medium are compared: a description in terms of Navier-Stokes equations and a description as a quasi-gas-dynamic system based on a difference approximation of the Boltzmann equation. Reflection of acoustic waves from artificial boundaries in the numerical region is considered, and special boundary conditions are constructed to ensure damping of nonphysical modes. A difference approximation of such boundary conditions is investigated. The procedure is tested in application to viscous flow around plates.

PARALLEL IMPLEMENTATION FOR SOLVING THE TWO-DIMENSIONAL NAVIER-STOKES EQUATIONS ON A CELL-CENTERED, UNSTRUCTURED GRID

A compressible, two-dimensional, unstructured-grid Navier-Stokes solver has been developed for subsonic viscous flows. Pseudo-time preconditioning is applied to improve low Much number convergence for primitive as well as conserved variables. The scheme was implemented on a workstation and on the Ncube2s, which utilizes the multiple-instruction, multiple-data (MIMD) approach to parallel computing. A detailed description of the implementation procedure for an unstructured-grid flow solver on a MIMD computer is given. Test cases include steady as well as unsteady flows.

An investigation of artificial dissipation for the cell‐vertex finite volume method

AbstractThis paper is concerned with an investigation of artificial dissipation models that are used with the cell‐vertex finite volume approximation of the compressible Euler and Navier‐Stokes equations. Based on the observation that first and second‐order upwind schemes can be written as a central discretization plus an appropriately scaled dissipative flux, a matrix scaling of second and fourth‐differences is implemented in an artificial dissipation model that also uses a procedure to limit the applicability of shock‐capturing dissipation. Analysis of the model and the associated boundary conditions is given to determine the effect on accuracy. Numerical results are given for transonic Euler flow past a NACA0012 aerofoil profile which demonstrate the improved shock‐capturing capability of the model. Results for laminar subsonic viscous flow over a flat plate show that the matrix‐dissipation model reduces the amount of spurious artificial dissipation within boundary layers.

PARALLEL COMPUTATION OF THE 2-D NAVIER-STOKES FLOWFIELD OF A PITCHING AIRFOIL

SUMMARY An unsteady, compressible, two dimensional Navier-Stokes solver has been developed for the Connection Machine (CM-2 and CM-5). The solver has been applied to the computations of subsonic viscous flow past a pitching airfoil at Rec = 1044, (based on airfoil chord length) M∞ = 0.2, non-dimensional pitching rate Ωυ + = 0.2 and 0.4. The algorithm for solving the Navier-Stokes equations is based on an unstructured grid of triangles. Roe's flux difference split method is employed for the inviseid fluxes. A discrete representation of Gauss' theorem is used for the viscous fluxes and heat transfer. A four stage modified Runge-Kutta algorithm is employed for the temporal integration. The algorithm is second order accurate in space and time. The implementation of the algorilhm on the Connection Machine and the investigation of the incipient boundary layer separation for a pitching airfoil are presented. The investigation of the effect of pitching rate on incipient boundary layer separation for a pitching ai...

A fully implicit Navier-Stokes algorithm using an unstructured grid and flux difference splitting

An implicit algorithm is developed for the two-dimensional, compressible, laminar Navier-Stokes equations using an unstructured grid of triangles. A sell-centered data structure is employed with the flow variables stored at the centroids of the triangles. The algorithm is based on Roe's flux difference split method for the inviscid fluxes, and a discrete representation of the viscous fluxes and heat transfer using Gauss' Theorem. Linear reconstruction of the flow variables to the cell faces, employed for the inviscid terms, provides second-order spatial accuracy. Interpolation of the flow variables to the nodes is achieved using a second-order accurate method. Temporal discretization employs Euler, trapezoidal or 3-point backward differencing. The complete, exact Jacobian of the inviscid and viscous terms is derived. The algorithm is applied to the Riemann Shock tube problem, a supersonic laminar boundary layer on a flat plate, and subsonic viscous flow past an NACA0012 airfoil. Results are in excellent agreement with theory and previous computations.