Full Compressible Navier-Stokes Equations Research Articles

Comparison of low Mach number models for natural convection problems

We investigate in this paper two numerical methods for solving low Mach number compressible flows and their application to single-phase natural convection flow problems. The first method is based on an asymptotic model of the Navier–Stokes equations valid for small Mach numbers, whereas the second is based on the full compressible Navier–Stokes equations with particular care given to the discretization at low Mach numbers. These models are more general than the Boussinesq incompressible flow model, in the sense that they are valid even for cases in which the fluid is subjected to large temperature differences, that is when the compressibility of the fluid manifests itself through low Mach number effects. Numerical solutions are computed for a series of test problems with fixed Rayleigh number and increasing temperature differences, as well as for varying Rayleigh number for a given temperature difference. Numerical difficulties associated with low Mach number effects are discussed, as well as the accuracy of the approximations.

Efficient Parallelization of a Three-Dimensional Navier-Stokes Solver on MIMD Multiprocessors

We present an efficient parallelization strategy for speeding up the computation of a high-accuracy 3-dimensional serial Navier-Stokes solver that treats turbulent transonic high-Reynolds flows. The code solves the full compressible Navier-Stokes equations and is applicable to realistic large size aerodynamic configurations and as such requires huge computational resources in terms of computer memory and execution time. The solver can resolve the flow properly on relatively coarse grids. Since the serial code contains a complex infrastructure typical for industrial code (which ensures its flexibility and applicability to complex configurations), then the parallelization task is not straightforward. We get scalable implementation on massively parallel machines by maintaining efficiency at a fixed value by simultaneously increasing the number of processors and the size of the problem. The 3-D Navier-Stokes solver was implemented on three MIMD message-passing multiprocessors (a 64-processors IBM SP2, a 20-processors MOSIX, and a 64-processors Origin 2000). The same code written with PVM and MPI software packages was executed on all the above distinct computational platforms. The examples in the paper demonstrate that we can achieve efficiency of about 60% for as many as 64 processors on Origin 2000 on a full-size 3-D aerodynamic problem which is solved on realistic computational grids.

Combustion effects on turbulence in a partially premixed supersonic diffusion flame

Effects of chemical heat release on turbulence in a partially premixed diffusion flame are investigated using direct numerical simulation (DNS). The full three-dimensional time-dependent compressible Navier-Stokes equations are employed to simulate the coupling between supersonic turbulence and heat release from a one-step chemical reaction governed by the Arrhenius kinetics. Four reacting cases with increasing heat release and one nonreacting case have been studied. Combustion is found to produce strong coupling among fluctuations in velocity, pressure, density, and other thermochemical quantities. In the Reynolds stress budget, the pressure–strain term becomes dominant as the heat release increases. Despite its wave behavior and its alternate roles as a source and a sink, the pressure–strain serves to promote energy transfer from the streamwise to the transverse and spanwise directions, and from the diagonal to the off-diagonal components of the Reynolds stresses, thus reducing anisotropy. Moreover, the pressure–strain, which in the case of the turbulent kinetic energy budget can be split up into a mean pressure work and a pressure–dilatation, helps to convert chemical energy from combustion into turbulence energy, leading to the phenomenon of “combustion-generated turbulence.” Both the solenoidal dissipation and the dilatational dissipation in particular increase as the heat release increases, but their effects are secondary to those of the pressure–strain within the main combustion period. The mixing layer growth rate can be predicted directly through the integrated Reynolds stress generation, with the accuracy of prediction depending on the heat release rate and the Reynolds number. A simple mechanism for the interactions of combustion and turbulence is proposed, while the modeling difficulties involved are also outlined. Finally, the differences and similarities of heat release effects and compressibility effects are clarified.

Variable density flow in porous media. A study by means of pore level numerical simulations

This paper is devoted to the modelling of isothermal low Reynolds and Mach numbers transient compressible flow through porous media. Traditionally, this type of flow at the macroscopic level is described by the classical Darcy's law combined with a mass balance that includes the transient term. This model is called the 'classic model'. The aim of this paper is to explore the validity of this classic model. To this end, the flow of an ideal gas is considered within two-dimensional model porous media. The flow is due to the imposed pressure variations at the outlet of the fluid domain. At the microscopic level, the flow is computed by solving the full compressible Navier-Stokes equations in two dimensions. Special attention is given to the outlet boundary conditions. The analysis is based on the comparison between the macroscopic data, obtained on the one hand by spatially averaging the microscopic results, and on the other hand by solving the problem directly at the macroscopic level. The various behaviours are discussed in terms of the various time scales controlling the flow and are explained by analysing the flow structure at pore level

Direct Numerical Simulation of Flow in a Ribbed Channel

The issue of turbine lifetime is an important one, particularly for modern turbines operating at high temperature regimes. A cooling design such as ribs may achieve an improved lifetime and complex mechanisms of heat transfer need to be well studied. In this paper, a Direct Numerical Simulation (DNS) is presented for a 3-D channel flow with two square ribs on the lower wall. The full unsteady compressible Navier-Stokes equations are solved with an original hybrid finite difference/finite element scheme. The Reynolds number of the simulation is 7 000 based on the bulk velocity at the inlet and the channel height. The present study is mainly devoted to understand the mechanism of heat transfer at the wall through the topological analysis of the flow and the temperature flux. Results show that the large-scale structures generated by obstacles splash onto the lower surface and induce longitudinal vortices which enhance heat transfer at the wall. A comprehensive data base including 56 correlations was set up for testing and improving turbulence models for this complex, separated flow.

Combustion-induced pressure effects in supersonic diffusion flames

A turbulent diffusion flame at a convective Mach number 1.2 was investigated using direct numerical simulation (DNS). The DNS employed the full time-dependent compressible Navier-Stokes equations coupled with a one-step chemical reaction governed by the Arrhenius kinetics. Detailed study of combustion-induced pressure effects on turbulence generation, conserved scalar, and stagnation enthalpy transport was conducted. Local countergradient diffusion (CGD) of a conserved scalar flux was observed for the first time in a diffusion flame where heat release was strong enough while gradient diffusion prevailed when heat release was zero or weak. The CGD occurred in spite of the absence of an externally imposed mean pressure gradient and was attributed to combustion-induced pressure fluctuations. The balance of the turbulent kinetic energy budget was strongly influenced by the pressure dilatation and the (combustion-induced) mean pressure work when heat release was strong. Both terms can be a source or a sink of turbulence, depending on the intricate interactions between turbulence and combustion. However, the temporal change in pressure ϱp/ϱt had an insignificant influence on the stagnation enthalpy transport. A linear relation between the stagnation enthalpy and the mixture fraction was confirmed, which could lead to considerable simplification in modeling high-speed turbulent combustion.

Shock-wave/boundary-layer interactions with bleed through rows of holes

Computations were performed to investigate three-dimensional, shock-wave/bo undary-layer interactions on a flat plate with bleed through four staggered rows of circular holes that discharge into the same plenum. The focus of the computations was to examine how bleed through rows of holes affect bleed rate and the pressure and Mach number distributions. The effects of the following parameters on the flow were investigated: 1) with and without shock-wave impingement on the boundary layer and 2) spacings between bleed holes in the streamwise and spanwise directions. Results show that just two rows of bleed holes arranged in a staggered fashion placed upstream of the incident shock are adequate in blocking the shock-induced adverse pressure gradient from propagating further upstream. Results also show that the spacings between the centers of holes can exceed the hole diameter not just in the streamwise direction, but also in the spanwise direction, and still be able to control shock-induced flow separation. This study is based on the ensemble-averaged, full compressible Navier-Stokes (N-S) equations closed by the Baldwin-Lomax algebraic turbulence model. Solutions to the ensemble-averaged N-S equations were obtained by an implicit, partially split, two-factored algorithm with flux-vector splitting in the streamwise direction on a chimera overlapping grid.

Structure of a turbulent double-fin interaction at Mach 4

This paper examines the three-dimensional shock-wave/turbulent boundary-layer interaction in an inletlike geometry consisting of symmetrically placed 15-deg fins on a plate in an oncoming Mach 4 flow. Two different implicit numerical methods are employed to solve the full three-dimensional mean compressible Navier-Stokes equations: 1) a high resolution upwind scheme with the Baldwin-Lomax algebraic turbulence model and 2) the traditional Beam-Warming approach with a k-e model. The comparison with experiment is very good for surface pressures, pitot pressure surveys, and surface shear patterns, although discrepancies with skin friction data are evident in the trailing portion of the interaction. Both computational schemes, despite major differences in modeling, predict similar flow structures in agreement with previous results at Mach 8. The flow is described in terms of a separated, nonreattaching boundary layer, a vortex interaction, longitudinal centerline vortices, and entrainment. The computed and experimental shock patterns are related to the streamline structure to provide a unified understanding of the flowfield.

Control of shock-wave/boundary-layer interactions by bleed

This numerical study investigates the effectiveness of bleed in controlling shock-wave/boundary-layer interactions on a flat plate with a focus on understanding how bleed-hole angle, presence of upstream and downstream bleed holes, and pressure ratio across bleed holes affect structure of barrier shock, surface pressure distribution, and bleed rate (in terms of flow coefficient). The bleed-hole angles investigated are 30 deg slanted and 90 deg normal, which give rise to two different types of barrier shocks. The influence of upstream and downstream bleed holes were investigated by studying the bleed process through an isolated hole and through three holes arranged in tandem along the streamwise direction. The plenum/freestream pressure ratios investigated range from 0.3 to 1.7, which produced choked and unchoked flows in the bleed holes. This study is based on the ensemble-averaged, full compressible Navier-Stokes equations closed by the Baldwin-Lomax model with solutions obtained by an implicit finite volume method on an overlapping Chimera grid.

Structure of a double-fin turbulent interaction at high speed

This study examines the interaction of a Mach 8.3 turbulent boundary layer with intersecting oblique shock waves generated by 15-deg sharp fins mounted symmetrically on a flat plate. The full three-dimensional mass-averaged compressible Navier-Stokes equations are solved with a high-resolution implicit finite-volume scheme. Turbulence closure is achieved with variations of the Baldwin-Lomax algebraic model. Excellent agreement with experimental data is observed for plate surface pressure. However, accurate heat transfer rates are obtained only near the plane of symmetry. Some quantitative details are dependent on the manner in which the turbulence model is implemented. Within this limitation, the overall computed mean flow structure remains similar and mesh independent and compares well with available field surveys. The incoming flat plate boundary layer separates along the entire spanwise width and does not reattach in the domain of computation. Beneath the separated boundary layer are a vortex interaction with an off-surface stagnation point, a centerline longitudinal vortex, and an entrainment flow originating from the essentially inviscid stream near the fin leading edge.

Flux-limited schemes for the compressible Navier-Stokes equations

Several high-resolution schemes are formulated with the goal of improving the accuracy of solutions to the full compressible Navier-Stokes equations. Calculations of laminar boundary layers at subsonic, transonic, and supersonic speeds are carried out to validate the proposed schemes. It is concluded that these schemes, which were originally tailored for nonoscillatory shock capturing, yield accurate solutions for viscous flows. The results of this study suggest that the formulation of the limiting process is more important than the choice of a particular flux splitting technique in determining the accuracy of computed viscous flows. Symmetric limited positive and upstream limited positive schemes hold the promise of improving the accuracy of the results, especially on coarser grids. HE calculation of compressible flows at transonic, supersonic, and hypersonic Mach numbers requires the implementation of nonoscillatory discrete schemes which combine high accuracy with high resolution of shock waves and contact discontinuities. These schemes must also be formulated in such a way that they facilitate the treatment of complex geometric shapes. In the past decade numerous schemes have been developed to meet these requirements in conjunction with the solution of the Euler equations.l More recently, the application of such schemes to the Navier-Stokes equations has produced algorithms which have progressively gained acceptance as analysis tools in the aerospace industry. There remains, however, a need to understand and improve Navier-Stokes schemes beyond the current state of the art. The most compelling reason for this rests on the fact that shock capturing requires the construction of schemes which are numerically dissipative, a requirement which could affect the global accuracy of the solution of the physical viscous problem. In a recent paper2 Jameson has shown that a theory of nonoscillatory schemes can be developed for scalar conservation laws based upon the local extremum diminishing (LED) principle that maxima should not increase and minima should not decrease. Moreover, although it is equivalent to the total variation diminishing principle (TVD) for one-dimensional problems, the LED principle can be applied naturally to multidimensional problems on both structured and unstructured meshes. This recent development has shed new light on the principles underlying the construction of both high-resolution switched and flux-limited dissipation schemes. In particular, it allowed the new formulation of two families of flux-limited schemes denominated, symmetric limited positive (SLIP) and upstream limited positive (USLIP), respectively. The present work merges several dissipation schemes based on both the SLIP and USLIP construction with a well-developed cell-centered, finite-volume formulation for solving the two-dimensional Navier-Stokes equations.3 The aim is to analyze and validate these new discretizations for the solution of viscous flow problems. In Sec. II the design principles of nonoscillator y discrete approximations to a scalar convection equation are reviewed together

A numerical/experimental study of the dynamic structure of a buoyant jet diffusion flame

An overview of a joint numerical/experimental investigation of the dynamic structure of a low-speed buoyant jet diffusion flame is presented. The dynamic interactions between the flame surface and the surrounding fluid mechanical structures are studied by means of a direct numerical simulation closely coordinated with experiments. The numerical simulation employs the full compressible axisymmetric Navier-Stokes equations coupled with a flame sheet model. Counterrotating vortex structures both internal and external to the flame surface are seen to move upward along with flame sheet bulges. These buoyancy-driven dynamic features compare well with those observed experimentally by means of phase-locked flow visualizations over entire flame-flickering cycles. The flicker frequencies measured both computationally and experimentally also compare well. Other aspects of this investigation which are discussed include sudden jumps in flicker frequency with increasing coflow velocity and the utilization of background pressure changes to simulate gravitational force variations experimentally.

Zonal-local solution method for the turbulent Navier-Stokes equations

AZONAL-LOCAL solution method is developed for the solution of the compressible Navier-Stokes equations. The main feature of the method is coupling of the NavierStokes equations with the Euler equations and the local mesh solution procedure. Mesh sequencing (multilevel) technique is also used for the improvement of the efficiency of the algorithm. The methodology is applied to turbulent flowfields past an airfoil. A flux vector splitting method with an upwind scheme up to fourth-order accuracy is used for the discretization of the inviscid fluxes. The system of the equations is solved by an unfactored implicit method using Gauss-Seidel relaxation. Contents Techological improvements in supercomputer speed and memory size provided the means to solve the full compressible Navier-Stokes equations for turbulent flowfields and complex geometries. However, large amounts of computer time are required for the solution of the equations especially for problems in design practice. To reduce the computational time, a zonal-local solution method is presented for the solution of the two-dimensional Navier-Stokes equations in high Reynolds number flows. The equations are solved in time-dependent form and for a curvilinear body-fitted coordinate system JUt ^ + (Gvis)r] (1) where /is the Jacobian of the transformation of the Cartesian coordinates (x, z) to curvilinear coordinates (£, f), the subscripts inv, vis indicate the inviscid and viscous fluxes, respectively, and U is the conservative variables matrix. The equations are solved by a finite volume Navier-Stokes code1 which uses a modified 1 Steger-Warming flux vector splitting (FVS) method for the discretization of the inviscid fluxes. The FVS method split the fluxes into two parts, positive and negative, in accordance with the sign of the eigenvalues (2)

Three-dimensional shock-wave/boundary-layer interactions with bleed

Computations were performed to investigate the physics of three-dimensi onal, shock-wave/boundary-layer interactions on a flat plate in which fluid in the boundary layer was bled through a circular hole into a plenum to control shock-wave induced flow separation. This study revealed two underlying mechanisms by which bleed holes can control shock-wave/boundary-layer interactions. It also showed how bleed-hole placement relative to where the incident shock wave impinges affects upstream, spanwise, and downstream influence lengths. This study is based on the ensemble-averaged, full compressible Navier-Stokes equations closed by the BaldwinLomax turbulence model. Solutions to these equations were obtained by an implicit, partially split, two-factored method with flux-vector splitting on a chimera overlapping grid.

Interaction between crossing oblique shocks and a turbulent boundarylayer

A numerical study of the three-dimensional interaction between crossing shocks generated by symmetric sharp fins and a flat-plate turbulent boundary layer is presented. The full mean compressible Navier-Stokes equations, incorporating a turbulent eddy-viscosity model, are solved. Computed results for the flow past (11 deg, 11 deg) symmetric fins at a freestream Mach number Mw = 2.95 and Reynolds number Re^ = 2.5 X 10s (based on the undisturbed boundary-layer thickness 600) show general agreement with experimental measurements for flatplate surface pressure and surface flow visualization. Analysis of the computed flowfield reveals a complex interaction involving the collision of two slowly counter-rotating vortical structures generated by the initial shock/boundary-layer interaction due to each fin. Associated with the streamline structure of the interaction is the formation and growth of a region of low energy near the centerline, downstream of the crossed shocks. A first look at the shock structure of the interaction is provided.

Investigation of the flow structure around a rapidly pitching airfoil

A numerical study is presented for unsteady laminar flow past a NACA 0015 airfoil that is pitched, at a nominally constant rate, from zero incidence to a very high angle of attack. The flowfield simulation is obtained by solving the full two-dimensional compressible Navier-Stokes equations on a moving grid employing an implicit approximate-factorization algorithm. An evaluation of the accuracy of the computed solutions is presented, and the numerical results are shown to be of sufficient quality to merit physical interpretation. The highly unsteady flowfield structure is described and is found to be in qualitative agreement with available experimental observations. A discussion is provided for the effects of pitch rate and pitch axis location on the induced vortical structures and on the airfoil aerodynamic forces.

The interaction of an oblique shock wave with a laminar boundary layer revisited. An experimental and numerical study

An investigation of an oblique shock wave/laminar boundary layer interaction is presented. The Mach number was 2.15, the Reynolds number was 105 and the overall pressure ratio was 1.55. The interation has been demonstrated to be laminar and nominally two-dimensional. Experimental results include pressure distributions on the plate and single component laser-Doppler velocimetry velocity measurements both in the attached and separated regions.The numerical results have been obtained by solving the full compressible Navier-Stokes equations with the implicit approximate factorization algorithm by Beam & Warming (1980). Comparison with experimental data shows good agreement in terms of pressure distributions, positions of separation and reattachment and velocity profiles.

Improved Calculation of High Speed Inlet Flows. Part II: Results

An improved numerical algorithm that solves the full mean compressible Navier-Stokes equations has been applied to the calculation of the flowfield in three separate configurations of a simulated high speed aircraft inlet. The inlet geometry consists of a converging supersonic diffuser, formed by two nonparallel plates, followed by a constant height throat. For all cases, the freestream Mach number is 3.51, and the Reynolds number is 13.6 X10 based on the inlet length. The three configurations are characterized by different values of the angle of the converging supersonic diffuser and different boundary-layer bleed schedules. The computed results are compared with detailed experimental data for the ramp and cowl surface pressure distribution, and the boundary-layer pitot profiles at seven different streamwise locations. The agreement with the experimental results is generally good, although the experimental data display evidence of three dimensionality over a portion of the inlet flowfield.

Improved Calculation of High Speed Inlet Flows: Part I. Numerical Algorithm

An improved numerical algorithm has been developed for the calculation of flowfields in two-dimensional high speed inlets. The full mean compressible Navier-Stokes equations are employed, with turbulence represented by an algebraic turbulent eddy viscosity. A body-oriented coordinate transformation is used to facilitate treatment of arbitrary inlet contours. The explicit finite-difference algorithm of MacCormack is utilized. Several well-known techniques for improving computational efficiency are incorporated, including time-splitting of the finite-difference operators and splitting of the mesh into several regions in the cross-stream direction. A number of new computational techniques are introduced; namely, a procedure for automatic determination of the optimal mesh splitting, and a separate treatment of the viscous sublayer and transition wall region of the turbulent boundary layers. The accuracy and efficiency of the approach is demonstrated for the specific examples of the development of a turbulent boundary layer on a flat plate, and the interaction of a shock wave with a flat plate turbulent boundary layer. In all cases, the results compare very favorably with previous numerical calculations and experimental results.

Numerical Solution for Perpendicular Sonic Hydrogen Injection into a Ducted Supersonic Airstream

This note discusses a computer program being developed to study the flow field near opposing perpendicular fuel injectors in scramjets. The MacCormack time-split, finite difference relaxation technique was used to solve the full two-dimensional compressible Navier-Stokes equations along with energy and species equations. By using this technique, a program was developed to consider the turbulent nonreacting flow of hydrogen and air in a rectangular duct. A damping term, proportional to the second derivative of pressure and temperature, was used to produce a stable solution behind the hydrogen jet in the neighborhood of the recompression shock. A case using actual conditions encountered in current scramjet design was analyzed, with results agreeing qualitatively with experimental observations.