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At least 1,099 records · Page 61

A gridless Euler/Navier-Stokes solution algorithm for complex-aircraft applications

The development of a gridless computational fluid dynamics (CFD) method for the solution of the two- and three-dimensional Euler and Navier-Stokes equations is described. The method uses only clouds of points and does not require that the points be connected to form a grid as is necessary in conventional CFD algorithms. The gridless CFD approach appears to resolve the inefficiencies encountered with structured or unstructured grid methods, and consequently offers great potential for accurately and efficiently solving viscous flows about complex aircraft configurations. The method is described in detail and calculations are presented for standard Euler and Navier-Stokes cases to assess the accuracy and efficiency of the capability.

Batina, John T.↗

A gridless Euler/Navier-Stokes solution algorithm for complex-aircraft applications

The development of a gridless computational fluid dynamics (CFD) method for the solution of the two- and three-dimensional Euler and Navier-Stokes equations is described. The method uses only clouds of points and does not require that the points be connected to form a grid as a necessary in conventional CFD algorithms. The gridless CFD approach appears to resolve the inefficiencies encountered with structured or unstructured grid methods, and consequently offers great potential for accurately and efficiently solving viscous flows about complex aircraft configurations. The method is described in detail and calculations are presented for standard Euler and Navier-Stokes cases to assess the accuracy and efficiency of the capability.

Batina, John T.↗

Semi-Analytic Reconstruction of Flux in Finite Volume Formulations

Semi-analytic reconstruction uses the analytic solution to a second-order, steady, ordinary differential equation (ODE) to simultaneously evaluate the convective and diffusive flux at all interfaces of a finite volume formulation. The second-order ODE is itself a linearized approximation to the governing first- and second- order partial differential equation conservation laws. Thus, semi-analytic reconstruction defines a family of formulations for finite volume interface fluxes using analytic solutions to approximating equations. Limiters are not applied in a conventional sense; rather, diffusivity is adjusted in the vicinity of changes in sign of eigenvalues in order to achieve a sufficiently small cell Reynolds number in the analytic formulation across critical points. Several approaches for application of semi-analytic reconstruction for the solution of one-dimensional scalar equations are introduced. Results are compared with exact analytic solutions to Burger s Equation as well as a conventional, upwind discretization using Roe s method. One approach, the end-point wave speed (EPWS) approximation, is further developed for more complex applications. One-dimensional vector equations are tested on a quasi one-dimensional nozzle application. The EPWS algorithm has a more compact difference stencil than Roe s algorithm but reconstruction time is approximately a factor of four larger than for Roe. Though both are second-order accurate schemes, Roe s method approaches a grid converged solution with fewer grid points. Reconstruction of flux in the context of multi-dimensional, vector conservation laws including effects of thermochemical nonequilibrium in the Navier-Stokes equations is developed.

Gnoffo, Peter A.↗

A Numerical Experiment on the Role of Surface Shear Stress in the Generation of Sound

The sound generated due to a localized flow over an infinite flat surface is considered. It is known that the unsteady surface pressure, while appearing in a formal solution to the Lighthill equation, does not constitute a source of sound but rather represents the effect of image quadrupoles. The question of whether a similar surface shear stress term constitutes a true source of dipole sound is less settled. Some have boldly assumed it is a true source while others have argued that, like the surface pressure, it depends on the sound field (via an acoustic boundary layer) and is therefore not a true source. A numerical experiment based on the viscous, compressible Navier-Stokes equations was undertaken to investigate the issue. A small region of a wall was oscillated tangentially. The directly computed sound field was found to to agree with an acoustic analogy based calculation which regards the surface shear as an acoustically compact dipole source of sound.

Shariff, Karim↗

Comparison between the PISO algorithm and preconditioning methods for compressible flow

Two widely used family of algorithms, pressure-based and density-based methods, have been developed for computational fluid dynamics (CFD) problems over the years. Pressure-based methods (such as SIMPLE and PISO) use a Poisson-like equation for updating pressure instead of the continuity equation, while density-based methods use the continuity equation to update density (an equation of state is used to provide density in pressure based schemes and pressure in density based schemes). Pressure-based methods were developed originally for incompressible flows at low Reynolds numbers and were then extended to high Reynolds numbers and compressible applications. On the other hand, density based methods were originally developed for transonic flows and have been extended down to low Mach numbers through the use of preconditioning techniques. We compare these two very different approaches to solving the Navier-Stokes equations in order to gain an understanding of their similarities and differences. Specifically, we consider the PISO scheme as a representative pressure-based method and contrast it with a recently developed preconditioning scheme. We also compare the relative performance of the PISO algorithm with a Euler implicit algorithm that is employed to solve the preconditioned equations by means of a vector stability analysis.

Merkle, Charles L.↗

A gridless Euler/Navier-Stokes solution algorithm for complex two-dimensional applications

The development of a gridless computational fluid dynamics (CFD) method for the solution of the two-dimensional Euler and Navier-Stokes equations is described. The method uses only clouds of points and does not require that the points be connected to form a grid as is necessary in conventional CFD algorithms. The gridless CFD approach appears to resolve the problems and inefficiencies encountered with structured or unstructured grid methods. As a result, the method offers the greatest potential for accurately and efficiently solving viscous flows about complex aircraft configurations. The method is described in detail, and calculations are presented for standard Euler and Navier-Stokes cases to assess the accuracy and efficiency of the capability.

Batina, John T.↗

Development of Computational Aeroacoustics Code for Jet Noise and Flow Prediction

Accurate prediction of jet fan and exhaust plume flow and noise generation and propagation is very important in developing advanced aircraft engines that will pass current and future noise regulations. In jet fan flows as well as exhaust plumes, two major sources of noise are present: large-scale, coherent instabilities and small-scale turbulent eddies. In previous work for the NASA Glenn Research Center, three strategies have been explored in an effort to computationally predict the noise radiation from supersonic jet exhaust plumes. In order from the least expensive computationally to the most expensive computationally, these are: 1) Linearized Euler equations (LEE). 2) Very Large Eddy Simulations (VLES). 3) Large Eddy Simulations (LES). The first method solves the linearized Euler equations (LEE). These equations are obtained by linearizing about a given mean flow and the neglecting viscous effects. In this way, the noise from large-scale instabilities can be found for a given mean flow. The linearized Euler equations are computationally inexpensive, and have produced good noise results for supersonic jets where the large-scale instability noise dominates, as well as for the tone noise from a jet engine blade row. However, these linear equations do not predict the absolute magnitude of the noise; instead, only the relative magnitude is predicted. Also, the predicted disturbances do not modify the mean flow, removing a physical mechanism by which the amplitude of the disturbance may be controlled. Recent research for isolated airfoils' indicates that this may not affect the solution greatly at low frequencies. The second method addresses some of the concerns raised by the LEE method. In this approach, called Very Large Eddy Simulation (VLES), the unsteady Reynolds averaged Navier-Stokes equations are solved directly using a high-accuracy computational aeroacoustics numerical scheme. With the addition of a two-equation turbulence model and the use of a relatively coarse grid, the numerical solution is effectively filtered into a directly calculated mean flow with the small-scale turbulence being modeled, and an unsteady large-scale component that is also being directly calculated. In this way, the unsteady disturbances are calculated in a nonlinear way, with a direct effect on the mean flow. This method is not as fast as the LEE approach, but does have many advantages to recommend it; however, like the LEE approach, only the effect of the largest unsteady structures will be captured. An initial calculation was performed on a supersonic jet exhaust plume, with promising results, but the calculation was hampered by the explicit time marching scheme that was employed. This explicit scheme required a very small time step to resolve the nozzle boundary layer, which caused a long run time. Current work is focused on testing a lower-order implicit time marching method to combat this problem.

Keith, Theo G., Jr.↗

The Burnett shock structures in low density hypersonic flows

One-dimensional shock structures in low density hypersonic flows have been studied numerically by solving the Burnett and Navier-Stokes equations. A new lower-upper scheme has been developed to obtain the steady and stationary normal shock solutions by the time-marching technique. The new scheme directly employs the implicit finite difference method without computing the eigenvalues, and it is first-order accurate in time and second-order accurate in space. The computed shock structures have been compared with available experimental data, and they are in good agreements. Numerical results show that the Burnett calculations has less numerical diffusion than the Navier-Stokes calculations does. As to the shock thickness, the Burnett solutions are apparently superior than the Navier-Stokes solutions.

Liaw, Goang-Shin↗

Entropy Splitting for High Order Numerical Simulation of Compressible Turbulence

A stable high-order numerical scheme for direct numerical simulation (DNS) of shock-free compressible turbulence is presented. The method is applicable to general geometries. It contains no upwinding, artificial dissipation, or filtering. Instead the method relies on the stabilizing mechanisms of an appropriate conditioning of the governing equations and the use of compatible spatial difference operators for the interior points (interior scheme) as well as the boundary points (boundary scheme). An entropy-splitting approach splits the inviscid flux derivatives into conservative and nonconservative portions. The spatial difference operators satisfy a summation-by-parts condition, leading to a stable scheme (combined interior and boundary schemes) for the initial boundary value problem using a generalized energy estimate. A Laplacian formulation of the viscous and heat conduction terms on the right hand side of the Navier–Stokes equations is used to ensure that any tendency to odd–even decoupling associated with central schemes can be countered by the fluid viscosity. The resulting methods are able to minimize the spurious high-frequency oscillations associated with pure central schemes, especially for long time integration applications such as DNS. For validation purposes, the methods are tested in a DNS of compressible turbulent plane channel flow at low values of friction Mach number, where reference turbulence data bases exist. It is demonstrated that the methods are robust in terms of grid resolution, and in good agreement with published channel data. Accurate turbulence statistics can be obtained with moderate grid sizes. Stability limits on the range of the splitting parameter are determined from numerical tests.

N.D. Sandham↗

Grid-Independent Large-Eddy Simulation in Turbulent Channel Flow using Three-Dimensional Explicit Filtering

In this paper, turbulence-closure models are evaluated using the 'true' LES approach in turbulent channel flow. The study is an extension of the work presented by Gullbrand (2001), where fourth-order commutative filter functions are applied in three dimensions in a fourth-order finite-difference code. The true LES solution is the grid-independent solution to the filtered governing equations. The solution is obtained by keeping the filter width constant while the computational grid is refined. As the grid is refined, the solution converges towards the true LES solution. The true LES solution will depend on the filter width used, but will be independent of the grid resolution. In traditional LES, because the filter is implicit and directly connected to the grid spacing, the solution converges towards a direct numerical simulation (DNS) as the grid is refined, and not towards the solution of the filtered Navier-Stokes equations. The effect of turbulence-closure models is therefore difficult to determine in traditional LES because, as the grid is refined, more turbulence length scales are resolved and less influence from the models is expected. In contrast, in the true LES formulation, the explicit filter eliminates all scales that are smaller than the filter cutoff, regardless of the grid resolution. This ensures that the resolved length-scales do not vary as the grid resolution is changed. In true LES, the cell size must be smaller than or equal to the cutoff length scale of the filter function. The turbulence-closure models investigated are the dynamic Smagorinsky model (DSM), the dynamic mixed model (DMM), and the dynamic reconstruction model (DRM). These turbulence models were previously studied using two-dimensional explicit filtering in turbulent channel flow by Gullbrand & Chow (2002). The DSM by Germano et al. (1991) is used as the USFS model in all the simulations. This enables evaluation of different reconstruction models for the RSFS stresses. The DMM consists of the scale-similarity model (SSM) by Bardina et al. (1983), which is an RSFS model, in linear combination with the DSM. In the DRM, the RSFS stresses are modeled by using an estimate of the unfiltered velocity in the unclosed term, while the USFS stresses are modeled by the DSM. The DSM and the DMM are two commonly used turbulence-closure models, while the DRM is a more recent model.

Gullbrand, Jessica↗

Computing viscous flows

Although the goals and techniques of computational aerodynamics and computational fluid dynamics differ, advancement in the physical and mathematical aspects of the latter are required for progress in aerodynamic computation. The most attractive approach is the use of hybrid methods where both the equations treated and the solution algorithms reflect the local character of the flow. A working general turbulence model that is only peripherally related to the availability of large fast computers would provide a significant breakthrough in computational aerodynamics. There is no unanimity of opinion as to what may be the optimum algorithm or family of algorithms in the next decade. While it is premature to develop an optimum processor, such a machine dedicated to study the structure of solutions to the three-dimensional time-dependent Navier-Stokes equations and to the computability of turbulence would be very valuable.

Murphy, J. D.↗

On the use of wall functions as boundary conditions for two-dimensional separated compressible flows

A new and improved wall function method for compressible turbulent flows has been developed and tested. This method is applicable to attached and separated flows, to both high- and low-Reynolds number flows, and to flows with adiabatic and nonadiabatic surfaces. This wall function method has been applied to the Launder-Spalding k-epsilon two-equation model of turbulence. The tests consist of comparisons of calculated and experimental results for: (1) an axisymmetrical transonic shock-wave/boundary-wave interaction flow at low Reynolds number in an adiabatic tube, (2) an axisymmetrical high-Reynolds number transonic flow over a nonadiabatic bump, and (3) a two-dimensional supersonic high-Reynolds number flow on a nonadiabatic deflected flap. Each of these experiments had significant regions of flow separation. The calculations are performed with an implicit algorithm that solves the Reynolds-averaged Navier-Stokes equations. It is shown that the results obtained agree very well with the data for the complex compressible flows tested.

Viegas, J. R.↗

A least-squares finite element method for incompressible Navier-Stokes problem

Most finite element schemes for solving the Navier-Stokes equations can be categorized into the Galerkin mixed method and the penalty method. The mixed method leads to a saddle-point problem. In order to guarantee the existence of a solution, the combination of velocity and pressure interpolations requires satisfaction of the Ladyzhenskaya Babuska Brezzi (LBB) consistency condition which precludes the use of equal order interpolations and many seemingly natural pairs of velocity and pressure elements. In a previous paper a least-squares finite element method based on the first order velocity-pressure-vorticity formulation for the Stokes problem was proposed. This method leads to a minimization problem. The choice of combination of elements is thus not subject to the LBB condition. The numerical experiments exhibit the optimal rate of convergence for all variables with equal order interpolations. A theoretical error analysis supports the numerical results. In this paper the least-squares finite element method is extended to solving the incompressible Navier-Stokes problem.

Jiang, Bonan↗

Discussion summary: Fictitious domain methods

Fictitious Domain methods are constructed in the following manner: Suppose a partial differential equation is to be solved on an open bounded set, Omega, in 2-D or 3-D. Let R be a rectangle domain containing the closure of Omega. The partial differential equation is first solved on R. Using the solution on R, the solution of the equation on Omega is then recovered by some procedure. The advantage of the fictitious domain method is that in many cases the solution of a partial differential equation on a rectangular region is easier to compute than on a nonrectangular region. Fictitious domain methods for solving elliptic PDEs on general regions are also very efficient when used on a parallel computer. The reason is that one can use the many domain decomposition methods that are available for solving the PDE on the fictitious rectangular region. The discussion on fictitious domain methods began with a talk by R. Glowinski in which he gave some examples of a variational approach to ficititious domain methods for solving the Helmholtz and Navier-Stokes equations.

Glowinski, Rowland↗

Comparative Study of Advanced Turbulence Models for Turbomachinery

A computational study has been undertaken to study the performance of advanced phenomenological turbulence models coded in a modular form to describe incompressible turbulent flow behavior in two dimensional/axisymmetric and three dimensional complex geometry. The models include a variety of two equation models (single and multi-scale k-epsilon models with different near wall treatments) and second moment algebraic and full Reynolds stress closure models. These models were systematically assessed to evaluate their performance in complex flows with rotation, curvature and separation. The models are coded as self contained modules that can be interfaced with a number of flow solvers. These modules are stand alone satellite programs that come with their own formulation, finite-volume discretization scheme, solver and boundary condition implementation. They will take as input (from any generic Navier-Stokes solver) the velocity field, grid (structured H-type grid) and computational domain specification (boundary conditions), and will deliver, depending on the model used, turbulent viscosity, or the components of the Reynolds stress tensor. There are separate 2D/axisymmetric and/or 3D decks for each module considered. The modules are tested using Rocketdyn's proprietary code REACT. The code utilizes an efficient solution procedure to solve Navier-Stokes equations in a non-orthogonal body-fitted coordinate system. The differential equations are discretized over a finite-volume grid using a non-staggered variable arrangement and an efficient solution procedure based on the SIMPLE algorithm for the velocity-pressure coupling is used. The modules developed have been interfaced and tested using finite-volume, pressure-correction CFD solvers which are widely used in the CFD community. Other solvers can also be used to test these modules since they are independently structured with their own discretization scheme and solver methodology. Many of these modules have been independently tested by Professor C.P. Chen and his group at the University of Alabama at Huntsville (UAH) by interfacing them with own flow solver (MAST).

Hadid, Ali H.↗

A shock-layer theory based on thirteen-moment equations and DSMC calculations of rarefied hypersonic flows

Grad's thirteen-moment equations are applied to the flow behind a bow shock under the formalism of a thin shock layer. Comparison of this version of the theory with Direct Simulation Monte Carlo calculations of flows about a flat plate at finite attack angle has lent support to the approach as a useful extension of the continuum model for studying translational nonequilibrium in the shock layer. This paper reassesses the physical basis and limitations of the development with additional calculations and comparisons. The streamline correlation principle, which allows transformation of the 13-moment based system to one based on the Navier-Stokes equations, is extended to a three-dimensional formulation. The development yields a strip theory for planar lifting surfaces at finite incidences. Examples reveal that the lift-to-drag ratio is little influenced by planform geometry and varies with altitudes according to a 'bridging function' determined by correlated two-dimensional calculations.

Cheng, H. K.↗

On the inversion of block tridiagonals without storage constraints

A strategy was developed to permit trade-offs between the number of floating point operations required and the storage requirements for the solution of certain difference problems, such as block tridiagonal systems of equations. This is done by recomputing some intermediate results instead of storing them. Reducing the storage to the square root of the current requirement roughly doubles the number of computations. Reducing the storage more than this tends to make the number of computations prohibitively large. In theory, if m is the order of each sub-matrix in the block tridiagonal matrix, one can solve any linear system with only 5m(2) + 1 temporary storage cells. In many cases m is a constant and quite small. For example, in solving a factored form of the three-dimensional Navier-Stokes equations, the size m of the block tridiagonals is 5. This method lends itself to efficient use on computers with parallel processing or vector processing architectures. On these computers the larger number of floating point operations is more than offset by the decrease in I/O and the increased percentage of vector operations made possible by this algorithm.

Merriam, M. L.↗

Navier Stokes solution of the flowfield over ice accretion shapes

The numerical simulation of flow about ice accretion shapes has been accomplished by solving the Navier-Stokes equations using MacCormack's explicit finite difference scheme. The computations were performed on a CRAY-XMP computer. The influence of turbulence is taken into account by means of an algebraic eddy-viscosity model. In order to optimize the grid spacing and to achieve near orthogonality at the surface of the complex ice shapes, a hyperbolic grid generation scheme is utilized. Particular attention is given to the heat transfer process for which good agreement between the numerical and experimental results is achieved. In addition, liquid water droplet trajectories are coupled within the flowfield along with the resulting collection efficiencies using a parabolized Navier-Stokes formulation.

Scott, J. N.↗