A Finite Volume Model For Simulating the Unsteady Navier-Stokes Equations Under Space Time Conservations (original) (raw)
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53rd AIAA Aerospace Sciences Meeting, 2015
One of the most important goals of this research effort is to improve the efficiencies of computational fluid dynamic (CFD) tools by focusing on the development of a robust and accurate numerical framework capable of solving the Navier-Stokes Equations under a wide variety of initial and boundary conditions. The new scheme, which was initially described in Ref. 1 and referred to as the Integro-Differential Scheme (IDS), has a number of favorable qualities. For instance, the scheme is developed on the basis of a unique combination of the differential and integral forms of the Navier-Stokes Equations (NSE). In this paper, the differential form of the NSE is used for explicit time marching and the integral form is used for spatial flux evaluations. As such, the scheme has the potential to accurately capture the complex physics of fluid flows. In addition, the Method of Consistent Averages (MCA) numerical procedure directly provides continuity of the numerical flux quantities rather than manipulating the primitive flowfield variables to ensure continuity. Coupled temporal and spatial analyses of the mass, momentum, and energy fluxes are considered at two major locations; namely, at the center of the numerical control volume, and at each of the surface making up an elementary control volume. It is also of interest to note that the IDS procedure developed herein is based on two fundamental types of control volumes. This paper elaborates on the development of the IDS procedure and presents the results of its implementation on three established fundamental high Reynolds number fluid dynamic problems. The problems of interest to this study are the supersonic rearward facing step and the supersonic cavity flow problems. A careful analysis of the results generated from the use of the IDS procedure confirms its predictive capability and supports its potential to solve a variety of fluid dynamics problems.
An integro-differential scheme for the Navier-Stokes equations
33rd Aerospace Sciences Meeting and Exhibit, 1995
Re = Revnolds number An integro-differential scheme for solving the Navier Stokes Equations is developed. The computational domain is discretized and represented by numerical control volumes. At the center of each control volume the flow field variables and their respective partial derivatives are defined. Within each control volume the flow properties are represented by a Taylor series expansion. Fluxes at the control volume interfaces are balanced through the use of exact functional expressions. obtained to the order of accuracy of the local expansions, and which identically satisfies both the integral and differential forms of the conservation laws. This process transforms the Navier Stokes equations to a system of nonlinear algebraic equations relative to the flow field variables and their respective partial derivatives.
49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 2011
The goal of this research effort is to improve the efficiencies of Computational Fluid Dynamic tools by focusing on the development of a robust, efficient, and accurate numerical framework. What is most desirable is a framework that is capable of solving a variety of complex fluid dynamics problems over a wide range of Reynolds and Mach numbers, and one with the potential to overcome several limitations of well-established schemes. Preferable, is a computational framework that might minimize the dispersive and dissipative tendencies of traditional numerical schemes. To this end, a new scheme, termed the 'Integro-Differential Scheme' (IDS) and sometimes called the 'Method of Consistent Averages' (MCA), Ref [1-2], is proposed. The MCA numerical procedure developed herein is based on two fundamental types of control volumes; namely, spatial cells and temporal cells. A typical spatial cell is developed from eight neighboring nodes, whereas, a typical temporal cell is developed from the centroid of the eight neighboring spatial cells. Further, each cell in this analysis, acts as a control volume on which the Navier-Stokes equations are integrated. The final solution at a single node is then developed from a system of carefully crafted control volumes. This paper describes the results obtained from applying the IDS method to a set of selected 2D fundamental fluid dynamic problems; namely, (1) the Hypersonic Leading Edge problem, (2) the incompressible Lid Driven Cavity problem, (3) the Shock Boundary Layer Interaction Problem and (3) the Isolator Pseudo-Shock Train problem.
2018 AIAA Aerospace Sciences Meeting (SciTech Forum), 2018
One of the most important goals of this research effort is to improve the efficiencies of computational fluid dynamic (CFD) tools by focusing on the development of a robust and accurate numerical framework capable of solving the Navier-Stokes Equations under a wide variety of initial and boundary conditions. This new scheme, called the Integro-Differential Scheme (IDS), has several favorable qualities. For instance, the scheme is developed based on a unique combination of the differential and integral forms of the Navier-Stokes Equations (NSE). In this paper, the differential form of the NSE is used for explicit time marching and the integral form is used for spatial flux evaluations. As such, the scheme has the potential to accurately capture the complex physics of fluid flows. In addition, the Method of Consistent Averages (MCA) numerical procedure directly provides continuity of the numerical flux quantities rather than manipulating the primitive flowfield variables to ensure continuity. Coupled temporal and spatial analyses of the mass, momentum, and energy fluxes are considered at two major locations; namely, at the center of the numerical control volume, and at each of the surfaces making up an elementary control volume. It is also of interest to note that the IDS procedure developed herein is based on two fundamental types of control volumes. This paper elaborates on the development of the IDS procedure and presents the results of its implementation on three different frameworks, such as 1D, quasi 1D and 2D flow problems. The problems of interest to this study are the supersonic cavity flow and the shock wave turbulent boundary layer interaction. A careful analysis of the results generated from the use of the IDS procedure confirms its predictive capabilities and its potential to solve a variety of fluid dynamics problems.
International Journal for Numerical Methods in Fluids, 2008
We describe here a collocated finite volume scheme which was recently developed for the numerical simulation of the incompressible Navier-Stokes equations on unstructured meshes, in 2 or 3 space dimensions. We recall its convergence in the case of the linear Stokes equations, and we prove a convergence theorem for the case of the Navier-Stokes equations under the Boussinesq hypothesis. We then present several numerical studies. A comparison between a cluster-type stabilization technique and the more classical Brezzi-Pitkäranta method is performed, the numerical convergence properties are presented on both analytical solutions and benchmark problems and the scheme is finally applied to the study of the natural convection between two eccentric cylinders. 1 domain which can be gridded and more recently, finite volume schemes for the Navier-Stokes equations on triangular grids have been presented, either staggered [20], or collocated [5] where primal variables are used with a Chorin type projection method to ensure the divergence condition. Since staggered schemes have the reputation of being the most stable schemes for incompressible flows, our idea was to generalize the MAC scheme to triangular meshes. Hence we considered a scheme where the velocity unknowns were associated to the control volumes of the mesh, and the "classical" four points cell-centered scheme was applied to discretize the Laplacian of the velocities, while a Galerkin expansion was introduced for the pressure, with the pressure unknowns associated to the vertices of the mesh. Some interesting stability and convergence properties were obtained for this scheme , and [23] for a review. However, we were not able to generalize the scheme to the three-dimensional case. We then developed and studied a collocated scheme , where velocities and pressure are all collocated within the control volume.
CFR: A Finite Volume Approach for Computing Incompressible Viscous Flow
Journal of Applied Fluid Mechanics
An incompressible unsteady viscous two-dimensional Navier-Stokes solver is developed by using "Consistent Flux Reconstruction" method. In this solver, the full Navier-Stokes equations have been solved numerically using a collocated finite volume scheme. In the present investigation, numerical simulations have been carried out for unconfined flow past a single circular cylinder with both structured and unstructured grids. In structured grid, quadrilateral cells are used whereas triangular elements are used in unstructured grid. Simulations are performed at Reynolds number (Re) = 100 and 200. Flow simulation over a NACA0002 airfoil at Re = 1000 using unstructured grid based solver is also reported. The vortex shedding phenomena is mainly investigated in the flow. It is observed that the nature of flow depends strongly on the value of the Reynolds number. The present results are found to be in satisfactory agreement with several numerical results and a few experimental results available from literature.
14th Applied Aerodynamics Conference, 1996
Numerical simulations of viscous flow problems with complex moving boundaries commonly require the solution of the fluid equations on unstructured and deformable dynamic meshes. In this paper, we present a time-accurate methodology for computing the unsteady diffusive fluxes arising from such problems, and highlight its impact on the accuracy of the overall flow simulation. We illustrate this methodology with a viscous flow problem related to a supersonic aeroelastic application. The numerical schemes presented in this paper can be directly extended to the computation of turbulent flows on moving grids.
AIAA SciTech Forum, 2021
Numerical solutions of compressible turbulent flow remains a challenge. Because of the wide range of temporal and spatial length scales, different mathematical techniques have been implemented to resolve these complex flows. Large Eddy Simulation (LES) is one of the most promising techniques; however, some understanding of the expected turbulence must be known a priory and computed using sub-grid scale models. Unfortunately, there is no guarantee that the selected model will effectively capture the fine-scale turbulence. Mathematical handling of discontinuities such as shock waves typically increase the complexity and difficulty of compress-ible flow computations. To alleviate this problem, Essentially Non-Oscillatory (ENO) schemes equipped with Riemann solvers are widely used. The numerical algorithm presented in this work represents a new approach based on a consistent averaging procedure that solves an integral form of the Navier Stokes Equations. This method leads to a set of differential-algebraic equations that are solved numerically using spatial averaging. We present several computations demonstrating the flow physics capturing capabilities of the new scheme. We investigate 2D solutions of the stratified Kelvin-Helmholtz instability shear layer, the Taylor-Green Vortex, and the Riemann problem. This is our first attempt to provide a thorough study of inviscid and viscous flows. Our primary goal is to quantify the dissipative behavior, resolution characteristics , and shock-capturing capabilities of the proposed scheme. To this end, we qualitatively compared the numerical solution to reference data. Quantitatively, we use statistical techniques such as measurement of the kinetic energy spectrum and the conservation of total energy and enstrophy in the flow.
2005
An innovative and robust algorithm capable of solving a variety of complex fluid dynamic problems is developed. This so-called, Integro-Differential Scheme, (IDS) is designed to overcome known limitations of established schemes. The IDS implements a smart approach in transforming 3-D computational flowfields of fluid dynamic problems into their 2-D counterparts, while preserving their physical attributes. The strength of IDS rests on the implementation of the mean value theorem to the integral form of the conservation laws. This process transforms the integral equations into a finite difference scheme that lends itself to efficient numerical implementation. Preliminary solutions generated by IDS demonstrated its accuracy in terms of its ability to capture flowfield physics. In this paper, the results of applying the IDS to two problems; namely, the flow over a flat plate, and the shock/boundary layer interaction problem, are documented and discussed. In both cases, the results showe...