Parabolized Navier-Stokes Research Articles

Influence of nozzle geometry on torch parameters during gas combustion

In this paper, when performing and analyzing the series of calculated results of the analysis of the sides of a rectangular nozzle for the parameters of a turbulent free air jet and a flame of a propane-butane mixture based on three-dimensional parabolic Navier-Stokes bonds for multicomponent reacting jets. To estimate the turbulent viscosity, differential equations of the kinetic and dissipation of the kinetic energy of turbulence are observed with refined empirical constants that are acceptable in turbulent combustion, as in the calculations, the initial values of the kinetic energy of turbulence did not exceed 5% of the dimensionless velocity of the main jet. It was found that when an air jet flows out of a nozzle with an aspect ratio of (1:2) and (1:4) in the initial sections of the jet, its shape behaves like an ellipse and, with the distance, it turns into a round one, with dimensionless, x greater or equal to 20 for the nozzle (1:2) and x greater or equal to 30 for (1:4), and for this variant there is again a transition to an elliptical shape. Studies of the diffusion combustion of a propane-butane mixture flowing out of a rectangular nozzle with an aspect ratio of (1:2) and (1:4) showed that with the ratio of the speed of the oxidizer to fuel mu, the flame length increases when mu between 0 and 0.164, and then, with an increase of mu more than 0.41, the torch length is shortened. The axial distributions of the momentum flux density with the aspect ratio (1:4), the jet core is shortened by 2÷2.5 times, compared with the aspect ratio (1:2) with the other parameters unchanged. Setting the tangential velocity at the mouth of the nozzle (ω=6.1 m/s) leads to a decrease in the torch length by 11% compared to its value equal to zero.

Solution of Parapolic Navier-Stokes Equations for Laminar Forced Convective Flow in Entrance Region of a Flat Passage.(Dept.M)

Due to dependence of forced convective heat transfer on the hydrodynamic flow field; any improvement on the analysis of this flow field is of great importance to understand the heat transfer process. In this work the flow field is described by the parabolic Navier-Stokes equations; which are solved by local non-similarity solution-method. According to this method; with suitable definition of the problem variables, a set of ordinary differential equations are produced. This set is solved, numerically, by Runge-Kutta method accompanied with shooting method of boundary value problems. The values of local Nusselt number and local coefficient of friction are calculated for fluids of prandtl number of one. Some passages of Reynolds number (Re1) of 100, 200 & 300 are studied here. Two formulate of Nusselt number for different passage heights are proposed.

Solution of Parabolic Navier-Stokes Equations for The Entrance Region-Flow between Two Parallel Plates.(Dept.M)

The development of the flow field in the entrance region is theoretically examined, for the case of flow between two parallel flat plates. The flow field is described by the parabolic Navier-Stokes equations. Because of the nature of these equations; it is convenient to solve them by the application of the local similarity solution method. According to this method, dimensionless form of momentum equations are transferred to ordinary differential equations. These two modified equations are solved numerically by Runge-Kutta method of ordinary differential equations accompanied with shooting method of boundary value problems. The values of coefficient of friction are calculated at different positions along the passage. Also the velocity profile at any position, according to this approach, can be obtained. Three passages, in this work, are studied; with Reynolds number ( based on the half of the height of the passage) of 100, 200 and 300.

Reduced order modeling based on POD of a parabolized Navier–Stokes equations model II: Trust region POD 4D VAR data assimilation

A reduced order model based on Proper Orthogonal Decomposition (POD) 4D VAR (Four-dimensional Variational) data assimilation for the parabolized Navier–Stokes (PNS) equations is derived. Various approaches of POD implementation of the reduced order inverse problem are studied and compared including an ad-hoc POD adaptivity along with a trust region POD adaptivity. The numerical results obtained show that the trust region POD 4D VAR provides the best results amongst all the POD adaptive methods tested in all error metrics for the reduced order inverse problem of the PNS equations.

Reduced‐order modeling based on POD of a parabolized Navier–Stokes equation model I: forward model

SUMMARYA proper orthogonal decomposition (POD)‐based reduced‐order model of the parabolized Navier–Stokes (PNS) equations is derived in this article. A space‐marching finite difference method with time relaxation is used to obtain the solution of this problem, from which snapshots are obtained to generate the POD basis functions used to construct the reduced‐order model. In order to improve the accuracy and the stability of the reduced‐order model in the presence of a high Reynolds number, we applied a Sobolev H1 norm calibration to the POD construction process. Finally, some numerical tests with a high‐fidelity model as well as the POD reduced‐order model were carried out to demonstrate the efficiency and the accuracy of the reduced‐order model for solving the PNS equations compared with the full PNS model. Different inflow conditions and different selections of snapshots were experimented to test the POD reduction technique. The efficiency of the H1 norm POD calibration is illustrated for the PNS model with increasingly higher Reynolds numbers, along with the optimal dissipation coefficient derivation, yielding the best root mean square error and correlation coefficient between the full and reduced‐order PNS models. Copyright © 2011 John Wiley & Sons, Ltd.

Dual-code solution procedure for efficient computing equilibrium hypersonic axisymmetric transitional/turbulent flows

An appropriate combination of the thin-layer Navier–Stokes (TLNS) and parabolized Navier–Stokes (PNS) solvers is used to accurately and efficiently compute hypersonic transitional/turbulent flowfields of perfect gas and equilibrium air around blunt-body configurations. The TLNS equations are solved in the nose region to provide the initial data plane needed for the solution of the PNS equations. Then the PNS equations are employed to efficiently compute the flowfield for the afterbody region by using a space marching technique. Both the TLNS and the PNS equations are numerically solved by using the implicit non-iterative finite-difference algorithm of Beam and Warming. A shock fitting procedure is used in both the TLNS and PNS codes to obtain accurate solution in the vicinity of the shock. For turbulent flow simulations, both the Cebeci–Smith (CS) and the Baldwin–Lomax (BL) turbulence models are analyzed with the present technique for the case of long slender blunt bodies. The Baldwin–Lomax turbulence model, which does not need the determination of the edge of the boundary layer, is modified in the present work to include the effects of variable properties and damping factor to accurately calculate flowfield characteristics in turbulent regions. Detailed comparisons are made with other numerical and experimental results to assess the accuracy and efficiency of the present solution procedure for computing hypersonic transitional/turbulent flows over long slender blunt bodies. The results of these computations are found to be in good agreement with available data. The effects of real gas on the flowfield characteristics are also studied.

Implementation of high‐order compact finite‐difference method to parabolized Navier–Stokes schemes

AbstractThe numerical solution to the parabolized Navier–Stokes (PNS) and globally iterated PNS (IPNS) equations for accurate computation of hypersonic axisymmetric flowfields is obtained by using the fourth‐order compact finite‐difference method. The PNS and IPNS equations in the general curvilinear coordinates are solved by using the implicit finite‐difference algorithm of Beam and Warming type with a high‐order compact accuracy. A shock‐fitting procedure is utilized in both compact PNS and IPNS schemes to obtain accurate solutions in the vicinity of the shock. The main advantage of the present formulation is that the basic flow variables and their first and second derivatives are simultaneously computed with the fourth‐order accuracy. The computations are carried out for a benchmark case: hypersonic axisymmetric flow over a blunt cone at Mach 8. A sensitivity study is performed for the basic flowfield, including profiles and their derivatives obtained from the fourth‐order compact PNS and IPNS solutions, and the effects of grid size and numerical dissipation term used are discussed. The present results for the flowfield variables and also their derivatives are compared with those of other basic flow models to demonstrate the accuracy and efficiency of the proposed method. The present work represents the first known application of a high‐order compact finite‐difference method to the PNS schemes, which are computationally more efficient than Navier–Stokes solutions. Copyright © 2008 John Wiley & Sons, Ltd.

Dual-code solution procedure for efficient computing equilibrium hypersonic axisymmetric laminar flows

An appropriate combination of the thin-layer Navier–Stokes (TLNS) and parabolized Navier–Stokes (PNS) solvers is used to accurately and efficiently compute hypersonic flowfields of equilibrium air around blunt-body configurations. The TLNS equations are solved in the nose region to provide the initial data plane needed for the solution of the PNS equations. Then the PNS equations are employed to efficiently compute the flowfield for the afterbody region by using a space marching procedure. Both the TLNS and PNS equations are numerically solved by using the efficient implicit non-iterative finite-difference algorithm of Beam and Warming. A shock fitting technique is used in both the TLNS and PNS codes to obtain accurate solution in the vicinity of the shock. In the regions of the flowfield with upstream influences, the globally iterated PNS (IPNS) equations, which are computationally more efficient than the TLNS equations, can be used to improve the accuracy of the results. To validate the results of the developed TLNS code and study real gas effects on the flowfield, hypersonic laminar flow over a sphere at Mach number of 11.26 is computed. The computations are performed for hypersonic laminar flow over 5° long slender blunt cone at Mach number of 19.25 to demonstrate the accuracy and efficiency of using the present TLNS-PNS methodology. To show the capability and robustness of the present solution procedure, the comparisons are also made with the wall heat-transfer data on a 12.84 ° / 7 ° bicone at Mach number of 6.89 and freestream temperature of 1604 K. The results of these computations are found to be in good agreement with available numerical and experimental data. The results of the perfect gas for these cases are also presented and the effects of real gas on the flowfield characteristics are studied in both the TLNS and PNS solutions.

On the Use of High-Order Accurate Solutions of PNS Schemes as Basic Flows for Stability Analysis of Hypersonic Axisymmetric Flows

Abstract High-order accurate solutions of parabolized Navier–Stokes (PNS) schemes are used as basic flow models for stability analysis of hypersonic axisymmetric flows over blunt and sharp cones at Mach 8. Both the PNS and the globally iterated PNS (IPNS) schemes are utilized. The IPNS scheme can provide the basic flow field and stability results comparable with those of the thin-layer Navier–Stokes (TLNS) scheme. As a result, using the fourth-order compact IPNS scheme, a high-order accurate basic flow model suitable for stability analysis and transition prediction can be efficiently provided. The numerical solution of the PNS equations is based on an implicit algorithm with a shock fitting procedure in which the basic flow variables and their first and second derivatives required for the stability calculations are automatically obtained with the fourth-order accuracy. In addition, consistent with the solution of the basic flow, a fourth-order compact finite-difference scheme, which does not need higher derivatives of the basic flow, is efficiently implemented to solve the parallel-flow linear stability equations in intrinsic orthogonal coordinates. A sensitivity analysis is also conducted to evaluate the effects of numerical dissipation and grid size and also accuracy of computing the basic flow derivatives on the stability results. The present results demonstrate the efficiency and accuracy of using high-order compact solutions of the PNS schemes as basic flow models for stability and transition prediction in hypersonic flows. Moreover, indications are that high-order compact methods used for basic-flow computations are sensitive to the grid size and especially the numerical dissipation terms, and therefore, more careful attention must be kept to obtain an accurate solution of the stability and transition results.

Elimination AD applied to Jacobian assembly for an implicit compressible CFD solver

AbstractIn CFD, Newton solvers have the attractive property of quadratic convergence but they require derivative information. An efficient way of computing derivatives is by algorithmic differentiation (AD) also known as automatic differentiation or computational differentiation. AD allows us to evaluate derivatives, usually at a cheap cost, without the truncation errors associated with finite‐differencing. Recently, efficient and reliable AD tools for evaluating derivatives have been published. In this paper, we use some of the best AD tools currently available to build up the system Jacobian involved in the solution of a finite‐volume parabolized Navier–Stokes (PNS) solver. Our aim is to direct scientists and engineers confronted with the calculation of derivatives to the use of AD and to highlight those AD tools that they should try. Moreover, we introduce an AD tool that produces Jacobian code that runs usually twice as fast as that from conventional AD tools. We further show that the use of AD increases the performance of a Newton‐like solver for the PNS equations. Copyright © 2005 John Wiley & Sons, Ltd.

Accuracy of Parabolized Navier-Stokes Schemes for Stability Analysis of Hypersonic Axisymmetric Flows

The accuracy of the parabolized Navier ‐Stokes (PNS) and globally iterated PNS (IPNS) schemes for use as basic e ows is studied for stability analysis of hypersonic axisymmetric e ow over blunt and sharp slender cones at Mach 8. The numerical solution of the PNS and IPNS models is obtained by using the implicit e nite difference algorithm of Beam and Warming with a shock-e tting procedure. Both the PNS and IPNS solutions are thoroughly verie ed and compared with the thin-layer Navier ‐Stokes (TLNS) solution and experimental data. The stability analysis and transition prediction are performed using theparallel-e owlinearstability theory together with the eN method.Asystematicsensitivitystudyisperformedforbasicstatesolutions,includingproe lesandtheirderivatives, and spatial growth rates obtained from the PNS and IPNS models. The basic e ow and stability results and the predicted transition location based on the PNS model are qualitatively in good agreement with those of other theoretical investigations using various basic e ow models. Improved agreement with the TLNS model for both basic state solution and stability results is achieved when the IPNS model is applied. Attention is concentrated on those features of the numerical methods of the basic e ow computation that need particular care if adequate stability results are to be obtained. The study shows that the number of grid points, the grid distribution, and the amount of numerical dissipation have signie cant effects on the stability results. Moreover, indications are that carefulnumericalsolutionsofthePNSandIPNSschemesaslaminarbasice owcalculationsleadto amorepractical transition prediction tool in hypersonic e ows.

Parabolized Navier-Stokes Algorithm for Solving Supersonic Flows with Upstream Influences

A new parabolized Navier ‐Stokes (PNS) algorithm has been developed to efe ciently compute supersonic e ows with embedded regions that produce upstream effects. Innovative techniques are used to automatically detect and measure the extent of the embedded regions. Within the embedded regions, the PNS equations are globally iterated to duplicate the results that would be obtained with the complete Navier ‐Stokes (NS) equations. Once an embedded region is computed, the algorithm returns to the standard space-marching PNS mode until the next embedded region is encountered. This procedure has been successfully incorporated into NASA’ s upwind PNS code and it has been validated by its application to several two-dimensional test cases. All of these test cases include embedded regions that produce signie cant upstream effects. The present numerical results arein excellent agreement with previous NS computations and experimental data. In addition, these calculations demonstrate the signie cant reduction in computer time and storage that can be achieved by the use of this approach.

Parabolized Navier--Stokes code for hypersonic flows in thermo-chemical equilibrium or nonequilibrium

A parabolized Navier–Stokes (PNS) code has been developed to efficiently compute three-dimensional flows in thermo-chemical equilibrium or nonequilibrium. The new code is an extension of NASA’s three-dimensional upwind PNS (UPS) code. The chemical, vibrational, and electronic nonequilibrium effects have been incorporated via a loosely-coupled approach. The new code has been used to compute air flows in thermo-chemical equilibrium and nonequilibrium through expanding hypersonic nozzles. The results are in good agreement with existing numerical and experimental results.

Calculation of exhaust plume structure and emissions of the ER 2 aircraft in the stratosphere

Calculations are presented for the NASA ER 2 aircraft at high altitude to simulate measurements taken of its own emissions during a wake‐crossing event. Results are presented for a Mach = 0.71 case in the lower stratosphere with an engine NOx emission index of 4.6 corresponding to the measured value. A series of codes was used in the analysis to calculate the flow field and chemical kinetics, from the engine combustor out to a distance of about 20.2 km (97 s). Initial plume properties were calculated with a two‐dimensional computational fluid dynamics (CFD) code with finite rate chemistry. The results of the plume code initialized a three‐dimensional parabolized Navier‐Stokes (PNS) reacting flow solution, where the plume dynamics interacting with the aircraft wake were calculated out to the region of plume breakup. Results show that the early shape and mixing rate of the engine exhaust plume are dominated by the presence of the aircraft vortex wake. Model results for NOY emissions compare well to in situ measurements taken in the field. Calculated exhaust species evolutions predict several species ratios in good agreement with field data. The mixing rate of the engine plume was also predicted to be consistent with dilution measured in the field.

Data-parallel total variation diminishing method for sonic boom calculations

Sonic boom predictions are shown for the near and midfield and comparisons are made with experimental data. The computations are performed on the Thinking Machines' CM-5 massively parallel supercomputer to utilize its large available memory and high floating point performance. A second-order-accurate total variation diminishing scheme is used to solve the Euler equations in the computations. Additionally, a recently developed implicit method, based on the LU-SGS algorithm, is used to speed the convergence and accuracy of the steady- state computations. The method is shown to work well on near- and midfield sonic boom predictions for several test cases. HE projected use of the high speed civil transport (HSCT) has drawn attention to the problems associated with the noise due to sonic booms.1 An accurate and efficient sonic boom prediction methodology is needed for the assessment of various proposed HSCT configurations. One such method that utilizes the power of massively parallel supercomputers is presented here. A review of current sonic boom prediction methods is given by Plotkin.2 Many of these methods are based on the modified linear analysis of Whitham 3 and Walkden.4 Other methods have been developed based on a modified method of char- acteristics that approximately account for the effects of three- dimensional flows. Experimental and analytical studies have shown that these methods lose effectiveness as freestream Mach number approaches 3. In high Mach number regions there are strong shock waves with significant entropy gen- eration. The linear-based methods neglect these production terms.5 A modified method of characteristi cs has been de- veloped that can account for these terms,6 but it requires nonlinear near-field initial data that is difficult to obtain com- putationally or experimentall y. Several authors have developed prediction methods based on near-field solutions of the Euler or Navier-Stokes equa- tions.7-8 Most of these methods involve marching in one spatial dimension with an implicit Euler scheme or a parabolized Navier-Stokes (PNS) algorithm. These solutions are used in conjuction with Whitham's F-function to evaluate the far-field pressure signature. The existing prediction methods are un- attractive for several reasons. First, additional accuracy in the marching direction greatly increases the computational time. Also, they do not employ methods designed to resolve shock waves effectively. Finally, they are not easily implemented on massively parallel supercomputers. Accurate shock resolution is needed in the computational fluid dynamic (CFD) methods to determine the far-field pres- sure signature effectively. Since the prediction methods are dependent on shock resolution it is desirable to use a com- putational method that is designed to capture shock waves. The total variation diminishing (TVD) schemes developed by Harten,9 Yee,10 and others are well suited for this purpose. In this study a second-order-accurate TVD scheme is used to determine the pressure field about several models. These so- lutions are then compared with experiment.

High-temperature effects in hypersonic flows

The hypersonic (M=25) flow past a 10° right circular cone at 0° angle of attack has been computed using an implicit, noniterative, finite-difference scheme for the Parabolized Navier-Stokes (PNS) equations. Three different gas models have been considered in the study — (1) ideal gas (γ = constant), (2) equilibrium air, and (3) five species (N2, O2, NO, N, O) chemical nonequilibrium air. For the nonequilibrium air case, three different models have been considered — (a) one in which the constituent species have no internal structure (i.e., species are ideal gases), (b) one in which the constituent species have internal structure (i.e., species are thermally perfect gases), and (c) one in which the species have internal structure and their thermodynamic properties have been obtained from quantum mechanical and spectroscopic calculations.

An implicit-explicit upwind algorithm for the parabolized Navier-Stokes equations

In this paper, a second-order implicit-explicit upwind algorithm has been developed for three-dimensional Parabolized Navier-Stokes(PNS) equations. The agreement between the results of the new upwind algorithm and those of the im- plicit upwind algorithm and its ability in marching a long distance along the stream- wise direction have been shown for the supersonic viscous flow past a sphere-cone body. The CPU time is greatly reduced.

Three-dimensional upwind parabolized Navier-Stokes code for supersonic combustion flowfields

A new upwind, parabolized Navier-Stokes (PNS) code has been developed to compute the three-dimensional chemically reacting flow in scramjet (supersonic combustion ramjet) engines. The code is a modification of the three-dimensional upwind PNS (UPS) airflow code which has been extended in the present study to permit internal flow calculations with hydrogen-air chemistry. With these additions, the new code has the capability of computing both aerodynamic and propulsive flowfields. The algorithm solves the PNS equations using a finite-volume, upwind TVD method based on Roe's approximate Riemann solver that has been modified to account for nonequilibrium effects. The fluid medium is assumed to be a chemically reacting mixture of thermally perfect (but calorically imperfect) gases in thermal equilibrium. The new code has been applied to two test cases. These include the Burrows-Kurkov supersonic combustion experiment and a three-dimensional shockinduced combustion flowfleld. The computed results compare favorably with the available experimental data.

Computational study of advanced exhaust system transition ducts withexperimental validation

The current study is an application of CFD to a 'real' design and analysis environment. A subsonic, three-dimensional parabolized Navier-Stokes (PNS) code is used to construct stall margin design charts for optimum-length advanced exhaust systems' circular-to-rectangular transition ducts. Computer code validation has been conducted to examine the capability of wall static pressure predictions. The comparison of measured and computed wall static pressures indicates a reasonable accuracy of the PNS computer code results. Computations have also been conducted on 15 transition ducts, three area ratios, and five aspect ratios. The three area ratios investigated are constant area ratio of unity, moderate contracting area ratio of 0.8, and highly contracting area ratio of 0.5. The degree of mean flow acceleration is identified as a dominant parameter in establishing the minimum duct length requirement. The effect of increasing aspect ratio in the minimum length transition duct is to increase the length requirement, as well as to increase the mass-averaged total pressure losses. The design guidelines constructed from this investigation may aid in the design and manufacture of advanced exhaust systems for modern fighter aircraft.

Numerical modeling of wall-injected scramjet experiments

A wall-injected, hydrogen-fueled scramjet is modeled numerically using a parabolic Navier-Stokes computer code with a k-epsilon turbulence model and finite-rate chemistry. The numerical results are compared to experimental scramjet data taken in a shock tunnel and are found to be in reasonable agreement. Under the conditions studied, the numerical results show that combustion in the scramjet is mainly mixing limited.