Nozzle Problem Research Articles

Registration-based model reduction of parameterized PDEs with spatio-parameter adaptivity

We propose an automated nonlinear model reduction and mesh adaptation framework for rapid and reliable solution of parameterized advection-dominated problems, with emphasis on compressible flows. The key features of our approach are threefold: (i) a metric-based mesh adaptation technique to generate an accurate mesh for a range of parameters, (ii) a general (i.e., independent of the underlying equations) registration procedure for the computation of a mapping Φ that tracks moving features of the solution field, and (iii) an hyper-reduced least-square Petrov-Galerkin reduced-order model for the rapid and reliable estimation of the mapped solution. We discuss a general paradigm — which mimics the refinement loop considered in mesh adaptation — to simultaneously construct the high-fidelity and the reduced-order approximations, and we discuss actionable strategies to accelerate the offline phase. We present extensive numerical investigations for a quasi-1D nozzle problem and for a two-dimensional inviscid flow past a Gaussian bump to display the many features of the methodology and to assess the performance for problems with discontinuous solutions.

Cartesian grid-based hybrid NS-DSMC methodology for continuum-rarefied gas flows around complex geometries

This work presents a coupled numerical methodology, based on a hybrid Navier-Stokes (NS) and Direct Simulation Monte Carlo (DSMC) method, for high-speed compressible gas flows—involving a combination of continuum and rarefied flow regions. Implementation details of coupling cycle are presented along with the validation and performance studies for three cases (Mach number M a 1 = 3.38 , 6.5 and 9) of normal shock waves in argon and three cases (Inlet pressure pin = 1.0 atm, 0.5 atm and 0.1 atm) of flow through a micro-scale convergent-divergent (CD) nozzle. In comparison with our DSMC method-based solutions in the complete domain, the hybrid NS-DSMC method solutions achieve almost similar accuracy in a substantially reduced computational time—44% to 50% for the normal shock problem and 67% to 85% for the micro-scale CD nozzle problem. For the first time in literature, the hybrid NS-DSMC method is presented as a Cartesian-grid method, which is becoming popular nowadays.

Enhanced Pressure Based Coupled Algorithm to Combine with Pressure–Velocity-Enthalpy for all Mach Number Flow

In this paper, a pressure-based coupled computational fluid dynamics algorithm for numerical analysis of all Mach number region flow is developed. For this purpose, an enhanced pressure based coupled algorithm was developed through the pressure–velocity coupled algorithm and the pressure-enthalpy coupled algorithm were combined. In additional, the Kurganov–Tadmor flux splitting scheme, which is mainly used in density-based solvers, was applied to a developed pressure-based coupled solver. To confirm the analytical ability of developed solver, the variety of Mach number flow problems were performed using the developed solver. It was confirmed that the developed solver had the similar analytical ability with that of the other numerical codes through the analysis of the shock tube problems. In order to verify the analytical ability for the variety Mach number flow region of the developed solver, 2D bump and nozzle problems and 3D missile and wing problems were analyzed and compared with results of experiments and other numerical analysis codes. It is confirmed that the analytical ability of developed solver in the all speed flow region is somewhat improved than the commercial analysis package and is similar to the density based in-house CFD code.

Modern discontinuous Galerkin methods for the simulation of transitional and turbulent flows in biomedical engineering: A comprehensive LES study of the FDA benchmark nozzle model.

This work uses high-order discontinuous Galerkin discretization techniques to simulate transitional and turbulent flows through medical devices. Flows through medical devices are characterized by moderate Reynolds numbers and typically involve different flow regimes such as laminar, transitional, and turbulent flows. Previous studies for the FDA benchmark nozzle model revealed limitations of Reynolds-averaged Navier-Stokes turbulence models when applied to more complex flow scenarios. Recent works based on large-eddy simulation approaches indicate that these limitations can be overcome but also highlight potential limitations due to a high sensitivity with respect to numerical parameters. The methodology presented in this work introduces two novel ingredients compared with previous studies. Firstly, we use high-order discontinuous Galerkin methods for discretization in space. The inherent dissipation and dispersion properties of high-order discontinuous Galerkin discretizations are expected to render this approach well suited for transitional and turbulent flow simulations. Secondly, to mimic blinded CFD studies, we propose to use a precursor simulation approach in order to predict the inflow boundary condition for laminar, transitional, and turbulent flow regimes instead of prescribing analytical velocity profiles at the inflow. We investigate the whole range of Reynolds numbers as suggested by the FDA benchmark nozzle problem and compare the numerical results to experimental data obtained by particle image velocimetry in order to critically assess the predictive capabilities of the solver on the one hand and the suitability of the FDA nozzle problem as a benchmark in computational fluid dynamics on the other hand.

Inclusion modification in Al-killed valve spring steel by Na2CO3 addition

ABSTRACTHard inclusions with high melting temperatures such as Al2O3 (2054°C) and MgO·Al2O3 (2135°C) generate nozzle blockage problems during continuous casting of Al-killed valve spring steel and are very detrimental with respect to fatigue properties. In the present paper, inclusion modification in Al-killed valve spring steel by Na2CO3 addition was investigated in the laboratory using a graphite tube resistance furnace. The results show that inclusions with high melting temperature can be successfully modified into Na2O-containing inclusions with lower melting temperatures by the addition of Na2CO3. The effectiveness of inclusion modification can be enhanced by increasing the Na2CO3 addition and/or decreasing the amount of Al. This suggests that Na2CO3 addition could possibly be a substitute for Ca treatment as a method for preventing nozzle blockage during continuous casting of Al-killed steel.

Comparison among unstructured TVD, ENO and UNO schemes in two- and three-dimensions

This study focuses on unstructured TVD, ENO and UNO schemes applied to solve the Euler equations in two- and three-dimensions. They are implemented on a finite volume context and cell centered data base. The algorithms of Yee, Warming and Harten 1982; Harten; Yee and Kutler; Yee Warming and Harten 1985; Yee; Yee and Harten; Harten and Osher; Yang 1990, Hughson and Beran; Yang 1991; and Yang and Hsu are implemented to solve such system of equations in two- and three-dimensions. All schemes are flux difference splitting and good resolution is expected. This study deals with calorically perfect gas model and in so on the cold gas formulation has been employed. Two problems are studied, namely: the transonic convergent-divergent symmetrical nozzle, and the supersonic ramp. A spatially variable time step is implemented to accelerate the convergence process. The results highlights the excellent performance of the Yang 1990 TVD scheme, yielding an excellent pressure distribution at the two-dimensional nozzle wall, whereas the Harten and Osher scheme yields accurate values to the angle of the oblique shock wave and the best wall pressure distributions in the two-dimensional ramp problem. On the other hand, the excellent performance of the Harten scheme in the three-dimensional nozzle problem, yielding an excellent pressure distribution at the nozzle wall, and the Yee and Harten scheme yielding an accurate value to the angle of the oblique shock wave and the best wall pressure distribution in the three-dimensional ramp problem are of good quality.

Numerical Simulations of Canted Nozzle and Scarfed Nozzle Flow Fields

Computational fluid dynamics (CFD) techniques are used for the analysis of issues concerning non-conventional (canted and scarfed) nozzle flow fields. Numerical simulations are carried out for the quality of flow in terms of axisymmetric nature at the inlet of canted nozzles of a rocket motor. Two different nozzle geometries are examined. The analysis of these simulation results shows that the flow field at the entry of the nozzles is non axisymmetric at the start of the motor. With time this asymmetry diminishes, also the flow becomes symmetric before the nozzle throat, indicating no misalignment of thrust vector with the nozzle axis. The qualitative flow fields at the inlet of the nozzles are used in selecting the geometry with lesser flow asymmetry. Further CFD methodology is used to analyse flow field of a scarfed nozzle for the evaluation of thrust developed and its direction. This work demonstrates the capability of the CFD based methods for the nozzle analysis problems which were earlier solved only approximately by making simplifying assumptions and semi empirical methods.

A Very Efficient Class of HOC Schemes for the One-Dimensional Euler Equations of Gas Dynamics

This manuscript introduces a class of higher order compact schemes for the solution of one dimensional (1-D) Euler equations of gas dynamics. These schemes are fourth order accurate in space and second or lower order accurate in time, depending on a weighted average parameter μ. The robustness and efficiency of our proposed schemes have been validated by applying them to three different shock-tube problems of gas dynamics, including the famous SOD shock-tube problem. Later on, the 1-D convergent-divergent nozzle problem (De laval nozzle problem) is also considered and numerical simulations are performed. In all the cases, our computed numerical solutions are found to be in excellent match with the exact solutions or available results in the existing literature. Overall the schemes are found to be efficient and accurate.

The Implementation of Cell-Centred Finite Volume Method over Five Nozzle Models

The continued research and development of high-order methods in Computational Fluid Dynamics (CFD) is primarily motivated by their potential to significantly reduce the computational cost and memory usage required to obtain a solution to a desired level of accuracy. The present work presents the developed computer code based on Finite Volume Methods (FVM) Cell-centred Finite Volume Method applied for the case of Quasi One dimensional Inviscid Compressible flow, namely the flow pass through a convergent divergent nozzle. In absence of the viscosity, the governing equation of fluid motion is well known as Euler equation. This equation can behave as Elliptic or as Hyperbolic partial differential equation depended on the local value of its flow Mach number. As result, along the flow domain, governed by two types of partial differential equation, in the region in which the local mach number is less than one, the governing equation is elliptic while the other part is hyperbolic due to the local Mach number is a higher than one. Such a mixed type of equation is difficult to be solved since the boundary between those two flow domains is not clear. However by treating as time dependent flow problems, in respect to time, the Euler equation becomes a hyperbolic partial differential equation over the whole flow domain. There are various Finite Volume Methods can be used for solving hyperbolic type of equation, such as Cell-centered scheme, Cusp Scheme Roe Upwind Scheme and TVD Scheme. The present work will concentrate on the case of one dimensional flow problem through five nozzle models. The results of implementation of Cell Centred Finite Volume method to these five flow nozzle problems are very encouraging. This approach able to capture the presence of shock wave with very good results.

Subsonic to Supersonic Nozzle Flows

Steady, irrotational flow of a compressible fluid through a two-dimensional planar and axisymmetric nozzle is formulated and solved numerically in the hodograph or velocity plane. The Legendre potential is used to express the governing equation which for planar flow is linear and for axisymmetric flow nonlinear. The hodograph transformation method is extended to solve the nozzle problem for supersonic flow. A rectangular numerical domain for the supersonic case results from the way the information travels in the region that is the image of the flow that occurs around the lip of the nozzle surface. Knowledge of the characteristic's exact location in the supercritical region of the domain is not required, but only the general direction in which information propagates.

Padé–Legendre approximants for uncertainty analysis with discontinuous response surfaces

A novel uncertainty propagation method for problems characterized by highly non-linear or discontinuous system responses is presented. The approach is based on a Padé–Legendre (PL) formalism which does not require modifications to existing computational tools (non-intrusive approach) and it is a global method. The paper presents a novel PL method for problems in multiple dimensions, which is non-trivial in the Padé literature. In addition, a filtering procedure is developed in order to minimize the errors introduced in the approximation close to the discontinuities. The numerical examples include fluid dynamic problems characterized by shock waves: a simple dual throat nozzle problem with uncertain initial state, and the turbulent transonic flow over a transonic airfoil where the flight conditions are assumed to be uncertain. Results are presented in terms of statistics of both shock position and strength and are compared to Monte Carlo simulations.

Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections

We establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross-sections, including a slowly varying de Laval nozzle. The transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance, uniform subsonic downstream flow at the exit at infinity, and the slip boundary condition on the nozzle boundary. Our results indicate that, if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow, there exists a solution that consists of a C1,α subsonic flow in the unbounded downstream region, converging to a uniform velocity state at infinity, and a C1,α multidimensional transonic shock separating the subsonic flow from the supersonic upstream flow; the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow; and the shock is orthogonal to the nozzle boundary at every point of their intersection. In order to construct such a transonic flow, we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase, in which the equation is elliptic and the free boundary is a transonic shock. The free boundary conditions are determined by the Rankine–Hugoniot conditions along the shock. We further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain. We also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance.

Water Nozzles

The following type of nozzle problem is found in some introductory-level physics textbooks.1–3 The (average) flow speed of water through and out a hose with a cross-sectional area AH is vH. If a nozzle with an exit area AN < AH is attached to the hose, what is the speed vN of the water out of it? The books simply apply the continuity equation AHvH = ANvN to the nozzle to obtain vN = (AH/AN)vH. This solution is not correct because it does not take account of the fact that attaching the nozzle to the hose reduces the flow speed in it. So the books' values of vN must always be too high, sometimes by large amounts. It should not seem surprising that it takes more time to fill a watering can with a garden hose when there is a nozzle at the end of it than when there isn't. This paper will explain how a water nozzle actually works, and for a situation with a simple water source, the correct flow speed from a nozzle will be derived.

Turbulence Models and Heat Transfer in Nozzle Flows

Introduction S IGNIFICANT resources have been spent on creating accurate and robust turbulence models for flows with adverse pressure gradients. Capturing correct boundary-layer profiles under adverse pressure gradients is of particular importance to the accurate modeling of separated flows. In favorable pressure-gradient flows, smallscale features of boundary layers generally do not couple to largerscale flow features such as flow separations. As a result, problems that can occur in regions of favorable pressure gradients are more difficult to spot in standard benchmarks. However, accurate modeling of such flows will be required for reliable predictions of surface heat fluxes or shear stresses. Here we investigate the performance of Menter’s two equation shear-stress-transport (SST) turbulence model1 for heat-transfer computation under strongly favorable pressure gradients found in choked nozzle flows (such as required in rocket nozzle analysis and design). We have observed anomalous behavior of the SST turbulence model for strongly favorable pressure-gradient flows. We shall demonstrate the anomaly using an experimental choked nozzle problem published in the open literature.2 The anomaly in the computed heat transfer appears to be caused by the shear-stress-transport correction term of Menter’s model. By design, this term should be active in near-wall regions under adverse pressure gradients. However, it also appears to incorrectly become active under strongly favorable pressure-gradient conditions.

A Novel Amylose Corn-Starch Dispersion as an Aqueous Film Coating for Tablets

A novel aqueous coating dispersion of amylose-rich corn starch (Hylon VII) was evaluated in an aqueous film-coating process of tablets using an instrumented laboratory-scale pan-coating apparatus. The influence of two independent process variables, the coating temperature and the atomizing air pressure, on the properties of the coated tablets were investigated. The pre-use stability of aqueous coating dispersion (i.e., amylose corn-starch precipitate) was studied using a powder X-ray diffraction (XRD) technique. The crystallinity of amylose starch in the coating dispersion was found to increase slightly during 9 months of storage (in a refrigerator 6±2°C). The film coatings of an aqueous amylose-rich starch dispersion were successfully applied onto tablets without any significant drawbacks, such as nozzle blockage or related problems. It was found that the temperature in the coating pan had a significant influence on the film surface roughness, mechanical strength, and drug release in vitro. When the lowest coating temperature (30°C) was used, rougher film coatings were obtained due to overwetting. At higher temperatures (up to 50–60°C), lower surface roughness and higher mechanical strength values for the coated tablets were obtained. With the present amylose starch dispersion, the atomizing air pressure had a minor influence on the quality of the coating. Under appropriate coating conditions, a smooth tablet film coating was produced with this new, natural, and inexpensive amylose starch dispersion.

Parallel evolutionary algorithms for optimization problems in aerospace engineering

This paper presents the recent developments in hierarchical genetic algorithms (HGAs) to speed up the optimization of aerodynamic shapes. It first introduces HGAs, a particular instance of parallel GAs based on the notion of interconnected sub-populations evolving independently. Previous studies have shown the advantages of introducing a multi-layered hierarchical topology in parallel GAs. Such a topology allows the use of multiple models for optimization problems, and shows that it is possible to mix fast low-fidelity models for exploration and expensive high-fidelity models for exploitation. Finally, a new class of multi-objective optimizers mixing HGAs and Nash Game Theory is defined. These methods are tested for solving design optimization problems in aerodynamics. A parallel version of this approach running a cluster of PCs demonstrate the convergence speed up on an inverse nozzle problem and a high-lift problem for a multiple element airfoil.

Solution of aerospace problems using structured and unstructured strategies

Products developed at industries, institutes and research centers are expected to have high level of quality and performance, having a minimum waste, which require efficient and robust tools to numerically simulate stringent project conditions with great reliability. In this context, Computational Fluid Dynamics (CFD) plays an important role and the present work shows two numerical algorithms that are used in the CFD community to solve the Euler and Navier-Stokes equations applied to typical aerospace and aeronautical problems. Particularly, unstructured discretization of the spatial domain has gained special attention by the international community due to its ease in discretizing complex spatial domains. This work has the main objective of illustrating some advantages and disadvantages of numerical algorithms using structured and unstructured spatial discretization of the flow governing equations. Numerical methods include a finite volume formulation and the Euler and Navier-Stokes equations are applied to solve a transonic nozzle problem, a low supersonic airfoil problem and a hypersonic inlet problem. In a structured context, these problems are solved using MacCormack’s implicit algorithm with Steger and Warming’s flux vector splitting technique, while, in an unstructured context, Jameson and Mavriplis’ explicit algorithm is used. Convergence acceleration is obtained using a spatially variable time stepping procedure.

Simplified Navier-Stokes Equations for Internal Mixed Flows and a Numerical Method of Solution

Simplified Navier-Stokes equations, of the elliptic and hyperbolic type in the subsonic and supersonic flow regions, respectively, are derived for viscous flows in channels and nozzles with curved walls whose local radii of longitudinal curvature are comparable with the transverse channel dimensions. A new numerical method is developed for the system of equations obtained. This method is of the evolution type along the longitudinal coordinate and includes global iterations of the streamline direction field and the longitudinal pressure gradient field. The effectiveness of the method is illustrated with reference to the solution of the direct Laval nozzle problem for an air flow at Reynolds numbers Re≈104 and 106 in conical nozzles with throat curvatures Kw=1.0 and 1.6 (Kw is the curvature divided by the inverse radius of the nozzle throat). Two iterations are sufficient to calculate the nozzle flow rate and power correct to 0.01%.

Analysis of high speed non-equilibrium chemically reacting gas flows. Part I. Physical modeling and sensitivity studies

In this part, a theoretical model for high speed flow of chemically reacting gases out of thermal and chemical equilibrium is presented. The main features of the physical model are discussed together with details for a new form of the kinetic rate coefficients for non-equilibrium flows and presentation of a two-layer radiation model used for a plasma torch problem. This model is implemented in a new hybrid finite volume/finite element scheme, which is developed in Part II. Results from this physical model are compared with experiments and other results in the literature for an arcjet and non-equilibrium nozzle test case. Sensitivity studies are included for the nozzle problem to simulate the influence of the rate coefficients. Copyright © 2000 John Wiley & Sons, Ltd.

Solving the nozzle direct problem by iterations along streamlines

1. Methods for solving the problem. Numerically solving the Laval-nozzle direct problem for viscous gas in the framework of the steady-state Navier‐Stokes equations is hampered by a necessity to determine, in the course of solving, the critical gas-flow rate. Another difficulty consists in the fact that, in the case of high Reynolds numbers, these equations are of a mixed elliptic-hyperbolic type [1, 2]. Therefore, the basic numerical method for solving the direct problem in the framework of the Navier‐Stokes equations is the relaxation method [3‐5]. Besides an increase in the dimension of the problem under consideration, the disadvantage of this method is the slow (requiring several hundreds time steps) flow relaxation caused by a weak attenuation of wave processes in the subsonic part of nozzles. Methods based on different simplified forms of the Navier‐Stokes equations [2, 6], which are valid at certain restrictions imposed on the nozzle contour and the character of the flow, are sufficiently more efficient These simplified equations are of the evolution type with respect to the longitudinal coordinate (along the main flow direction) and, therefore, can be integrated by the use of rapid marching algorithms. For example, the marching calculation method based on the slenderduct approximation allows us to calculate the entire flow field in a nozzle with small inclinations of the contour to the main flow direction and a small dimensionless duct curvature K w [2, 7]. The smooth-duct approximation [8‐12] based on using an orthogonal coordinate system adapted to the duct geometry provides a better accuracy. This approximation accounts for the transverse pressure gradient associated with the centrifugal force. The accuracy of the method is determined by deviations of directions and curvatures of streamlines and longitudinal lines of the coordinate network used. The smooth-duct approximation makes it possible to calculate with an acceptθ tan able accuracy viscous flow in nozzles with moderate values of ≤ 1 and K w ≤ 0.5 [9, 11]. Below, we propose an improved modification of the smooth-duct model. It well describes the entire field of viscous flow in nozzles with a considerable duct curvature ( K w ≤ 2 ) and thus, covers all requirements of practice. In contrast to both the parabolic slender-duct model and smooth-duct model, the set of equations describing the new model is elliptic in the subsonic zones and hyperbolic in supersonic ones. For numerically solving the direct problem in the framework of these equations, an efficient marching method was developed. This method is based on the global iteration along streamlines and the longitudinal component of the pressure gradient. The method makes it possible to calculate, by a unified manner, viscous flow in the subsonic and supersonic zones. It is by orders of magnitude more efficient with respect to time consumed and computer memory required compared to the relaxation methods. Two global iterations are sufficient to determine within the accuracy of 0.01% such integral characteristics as the critical flow rate and the nozzle thrust. 2. The flow model. We consider the steady laminar flow of viscous gas in a plane or axial-symmetric Laval nozzle. The system of the simplified Navier‐Stokes equations in terms of the adapted ( ξ , η ) coordinate system [8, 9, 11] and natural variables has the following form. The ξ -projection of the momentum equation is