A computational study of gas flow in a De‐Laval micronozzle at different throat diameters

A numerical study has been carried out to investigate the gas flows in a micronozzle using a continuum model under both slip and no‐slip boundary conditions. The governing equations were solved with a finite volume method. The numerical model was validated with available experimental data. Numerical results of exit thrust showed good agreement with experimental data except at very low Reynolds numbers. For parametric studies on the effect of geometric scaling, the nozzle throat diameter was varied from 10 to 0.1 mm, whereas throat Reynolds number was varied from 5 to 2000. A correlation has also been developed to calculate the specific impulse at specified throat diameter and Reynolds number. The effect of different gases on the specific impulse of the nozzle, such as helium, nitrogen, argon and carbon dioxide, was also examined. Copyright © 2008 John Wiley & Sons, Ltd.     

Turbulent flow in converging nozzles,part one: boundary layer solution

The boundary layer integral method is used to investigate the development of the turbulent swirling flow at the entrance region of a conical nozzle. The governing equations in the spherical coordinate system are simplified with the boundary layer assumptions and integrated through the boundary layer. The resulting sets of differential equations are then solved by the fourth-order Adams predictor-corrector method. The free vortex and uniform velocity profiles are applied for the tangential and axial velocities at the inlet region, respectively. Due to the lack of experimental data for swirling flows in converging nozzles, the developed model is validated against the numerical simulations. The results of numerical simulations demonstrate the capability of the analytical model in predicting boundary layer parameters such as the boundary layer growth, the shear rate, the boundary layer thickness, and the swirl intensity decay rate for different cone angles. The proposed method introduces a simple and robust procedure to investigate the boundary layer parameters inside the converging geometries.     

Variational boundary integral approach for asymmetric impinging jets of arbitrary two‐dimensional nozzle

We consider asymmetric impinging jets issuing from an arbitrary nozzle. The flow is assumed to be two‐dimensional, inviscid, incompressible, and irrotational. The impinging jet from an arbitrary nozzle has a couple of separated infinite free boundaries, which makes the problem hard to solve. We formulate this problem using the stream function represented with a specific single layer potential. This potential can be extended to the surrounding region of the jet flow, and this extension can be proved to be a bounded function. Using this fact, the formulation yields the boundary integral equations on the entire nozzle and free boundary. In addition, a boundary perturbation produces an extraordinary boundary integral equation for the boundary variation. Based on these variational boundary integral equations, we can provide an efficient algorithm that can treat with the asymmetric impinging jets having arbitrarily shaped nozzles. Particularly, the proposed algorithm uses the infinite computational domain instead of a truncated one. To show the convergence and accuracy of the numerical solution, we compare our solutions with the exact solutions of free jets. Numerical results on diverse impinging jets with nozzles of various shapes are also presented to demonstrate the applicability and reliability of the algorithm.     

Supersonic Flow Past a Thermal Source

The results of an experimental investigation and numerical simulation of a gasdynamic structure formed as a result of supersonic flow past a pulsating thermal source are presented. Heat was supplied to the flow by producing a limited plasma volume as a result of the breakdown of the focused radiation of a CO₂ laser operating in the pulse periodic regime. On the basis of the experimental data obtained, a thermal source model was developed and accepted for further numerical calculations. The calculations were carried out within the framework of the inviscid gas model using the TVD scheme and nonreflecting boundary conditions. The effect of the relevant gasdynamic and energetic parameters on the flow pattern associated with the studied phenomenon is established. Data on the flow parameter distributions in the wake of the thermal source are obtained as a function of the freestream Mach number.     

Solving the fully nonlinear weakly dispersive Serre equations for flows over dry beds

We describe a numerical method for solving the Serre equations that can simulate flows over dry bathymetry. The method solves the Serre equations in conservation law form with a finite volume method. A finite element method is used to solve the auxiliary elliptic equation for the depth‐averaged horizontal velocity. The numerical method is validated against the lake at rest analytic solution, demonstrating that it is well‐balanced. Since there are currently no known nonstationary analytical solutions to the Serre equation that involve bathymetry, a nonstationary forced solution, involving bathymetry was developed. The method was further validated and its convergence rate established using the developed nonstationary forced solution containing the wetting and drying of bathymetry. Finally, the method is also validated against experimental results for the run‐up of a solitary wave on a sloped beach. The finite‐volume finite‐element approach to solving the Serre equation was found to be accurate and robust.     

On the immersed boundary‐lattice Boltzmann simulations of incompressible flows with freely moving objects

For simulating freely moving problems, conventional immersed boundary‐lattice Boltzmann methods encounter two major difficulties of an extremely large flow domain and the incompressible limit. To remove these two difficulties, this work proposes an immersed boundary‐lattice Boltzmann flux solver (IB‐LBFS) in the arbitrary Lagragian–Eulerian (ALE) coordinates and establishes a dynamic similarity theory. In the ALE‐based IB‐LBFS, the flow filed is obtained by using the LBFS on a moving Cartesian mesh, and the no‐slip boundary condition is implemented by using the boundary condition‐enforced immersed boundary method. The velocity of the Cartesian mesh is set the same as the translational velocity of the freely moving object so that there is no relative motion between the plate center and the mesh. This enables the ALE‐based IB‐LBFS to study flows with a freely moving object in a large open flow domain. By normalizing the governing equations for the flow domain and the motion of rigid body, six non‐dimensional parameters are derived and maintained to be the same in both physical systems and the lattice Boltzmann framework. This similarity algorithm enables the lattice Boltzmann equation‐based solver to study a general freely moving problem within the incompressible limit. The proposed solver and dynamic similarity theory have been successfully validated by simulating the flow around an in‐line oscillating cylinder, single particle sedimentation, and flows with a freely falling plate. The obtained results agree well with both numerical and experimental data. Copyright © 2016 John Wiley & Sons, Ltd.     

An experimental investigation of hypervelocity flow in a conical nozzle

The flow in a conical nozzle is examined experimentally for a range of hypervelocity conditions in a free-piston shock tunnel. The pitot pressure levels compare reasonably well with an inviscid numerical prediction which includes a correction for the growth of the nozzle wall boundary layer. The size of the nozzle wall boundary layer seems to be well predicted by semi-empirical expressions developed for perfect gas flows, as do data from other free-piston shock tunnels.     

Approximate nonreflecting inflow-outflow boundary conditions for the calculation of navier-stokes equations in internal flows

In this paper, the nonreflecting boundary conditions based upon fundamental ideas of the linear analysis are developed for gas dynamic equations, and the modified boundary conditions for Navier-Stokes equations are proposed as a substitute of the nonreflecting boundary conditions inside boundary layers near rigid walls. These derived boundary conditions are then applied to calculations both for the Euler equations and the Navier-Stokes equations to determine if they can produce acceptable results for the subsonic flows in channels. The numerical results obtained by an implicit second-order upwind difference scheme show the effectiveness and generality of the boundary conditions. Furthermore, the formulae and the analysis performed here may be extended to three dimensional problems. recommended by Prof. Cui Erjie     

Investigation of a Gas Flow with Solid Particles in a Supersonic Nozzle

A two-phase flow with high Reynolds numbers in the subsonic, transonic, and supersonic parts of the nozzle is considered within the framework of the Prandtl model, i.e., the flow is divided into an inviscid core and a thin boundary layer. Mutual influence of the gas and solid particles is taken into account. The Euler equations are solved for the gas in the flow core, and the boundary-layer equations are used in the near-wall region. The particle motion in the inviscid region is described by the Lagrangian approach, and trajectories and temperatures of particle packets are tracked. The behavior of particles in the boundary layer is described by the Euler equations for volume-averaged parameters of particles. The computed particle-velocity distributions are compared with experiments in a plane nozzle. It is noted that particles inserted in the subsonic part of the nozzle are focused at the nozzle centerline, which leads to substantial flow deceleration in the supersonic part of the nozzle. The effect of various boundary conditions for the flow of particles in the inviscid region is considered. For an axisymmetric nozzle, the influence of the contour of the subsonic part of the nozzle, the loading ratio, and the particle diameter on the particle-flow parameters in the inviscid region and in the boundary layer is studied. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 65–77, November–December, 2005.     

Computation of a drastic flow pattern change in an annular swirling jet caused by a small decrease in inlet swirl

This paper investigates the flow pattern change in an annular jet caused by a sudden change in the level of inlet swirl. The jet geometry consists of an annular channel followed by a specially designed stepped‐conical nozzle, which allows the existence of four different flow patterns as a function of the inlet swirl number. This paper reports on the transition between two of them, called the ‘open jet flow high swirl’ and the ‘Coanda jet flow.’ It is shown that a small sudden decrease of 4% in inlet swirl results in a drastic and irreversible change in flow pattern. The objective of this paper is to reveal the underlying physical mechanisms in this transition by means of numerical simulations. The flow is simulated using the unsteady Reynolds‐averaged Navier–Stokes (URANS) approach for incompressible flow with a Reynolds stress turbulence model. The analysis of the numerical results is based on a study of different forces on a control volume, which consists of the jet boundaries. The analysis of these forces shows that the flow pattern change consists of three different regimes: an immediate response regime, a quasi‐static regime and a Coanda regime. The simulation reveals that the pressure–tangential velocity coupling during the quasi‐static regime and the Coanda effect at the nozzle outlet during the Coanda regime are the driving forces behind the flow pattern change. These physical mechanisms are validated with time‐resolved stereo‐PIV measurements, which confirm the numerical simulations. Copyright © 2008 John Wiley & Sons, Ltd.     

AN IMPLICIT ALGORITHM OF THIN LAYER EQUATIONS IN VISCOUS，TRANSONIC，TWO－PHASE NOZZLE FLOW

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An immersed boundary solver for inviscid compressible flows

In this paper, a simple and efficient immersed boundary (IB) method is developed for the numerical simulation of inviscid compressible Euler equations. We propose a method based on coordinate transformation to calculate the unknowns of ghost points. In the present study, the body‐grid intercept points are used to build a complete bilinear (2‐D)/trilinear (3‐D) interpolation. A third‐order weighted essentially nonoscillation scheme with a new reference smoothness indicator is proposed to improve the accuracy at the extrema and discontinuity region. The dynamic blocked structured adaptive mesh is used to enhance the computational efficiency. The parallel computation with loading balance is applied to save the computational cost for 3‐D problems. Numerical tests show that the present method has second‐order overall spatial accuracy. The double Mach reflection test indicates that the present IB method gives almost identical solution as that of the boundary‐fitted method. The accuracy of the solver is further validated by subsonic and transonic flow past NACA2012 airfoil. Finally, the present IB method with adaptive mesh is validated by simulation of transonic flow past 3‐D ONERA M6 Wing. Global agreement with experimental and other numerical results are obtained.     

Experimental study of heat transfer characteristics due to confined impinging two-dimensional jets

Experimental results are presented for characteristics of impingement heat transfer caused by three slot jets. Experimental values were obtained for the dimensionless distance H = 0.5−3, dimensionless pitch P = 6−16, and Reynolds number Re = 500−8000. For laminar impinging flow, they were compared with numerical results. For turbulent impinging flow, two peaks of the local Nusselt number were obtained behind the second nozzle. The position of the second peak approached the nozzle as the space between nozzle and impinged surface decreased. The average Nusselt number between the central and second nozzles was determined from the ratio P/H and the Reynolds number based on the pitch of the nozzles.     

Time‐domain BEM solution of convection–diffusion‐type MHD equations

The two‐dimensional convection–diffusion‐type equations are solved by using the boundary element method (BEM) based on the time‐dependent fundamental solution. The emphasis is given on the solution of magnetohydrodynamic (MHD) duct flow problems with arbitrary wall conductivity. The boundary and time integrals in the BEM formulation are computed numerically assuming constant variations of the unknowns on both the boundary elements and the time intervals. Then, the solution is advanced to the steady‐state iteratively. Thus, it is possible to use quite large time increments and stability problems are not encountered. The time‐domain BEM solution procedure is tested on some convection–diffusion problems and the MHD duct flow problem with insulated walls to establish the validity of the approach. The numerical results for these sample problems compare very well to analytical results. Then, the BEM formulation of the MHD duct flow problem with arbitrary wall conductivity is obtained for the first time in such a way that the equations are solved together with the coupled boundary conditions. The use of time‐dependent fundamental solution enables us to obtain numerical solutions for this problem for the Hartmann number values up to 300 and for several values of conductivity parameter. Copyright © 2007 John Wiley & Sons, Ltd.     

Dissipative particle dynamics for complex geometries using non‐orthogonal transformation

Dissipative particle dynamics (DPD) was applied to fluid flow in irregular geometries using non‐orthogonal transformation, where an irregular domain is transformed into a simple rectangular domain. Transformation for position and velocity was used to relate the physical and computational domains. This approach was described by simulating fluid flow inside a two‐dimensional convergent–divergent nozzle. The nozzle geometry is controlled by the contraction ratio (CR) in the middle of the channel. The range of Reynolds number and CR, in this paper, was Re = 10hbox??200 and CR = 0.8 and 0.6, respectively. The DPD results were validated against in‐house computational fluid dynamic (CFD) finite volume code based on the stream function vorticity approach. The results revealed an excellent agreement between DPD and CFD. The maximum deviation between the DPD and CFD results was within 2%. Local and average coefficients of friction was calculated and it compared well with the CFD results. Copyright © 2011 John Wiley & Sons, Ltd.     

Finite element simulation of fully non‐linear interaction between vertical cylinders and steep waves. Part 1: methodology and numerical procedure

A methodology for computing three‐dimensional interaction between waves and fixed bodies is developed based on a fully non‐linear potential flow theory. The associated boundary value problem is solved using a finite element method (FEM). A recovery technique has been implemented to improve the FEM solution. The velocity is calculated by a numerical differentiation technique. The corresponding algebraic equations are solved by the conjugate gradient method with a symmetric successive overrelaxation (SSOR) preconditioner. The radiation condition at a truncated boundary is imposed based on the combination of a damping zone and the Sommerfeld condition. This paper (Part 1) focuses on the technical procedure, while Part 2 [Finite element simulation of fully non‐linear interaction between vertical cylinders and steep waves. Part 2. Numerical results and validation. International Journal for Numerical Methods in Fluids 2001] gives detailed numerical results, including validation, for the cases of steep waves interacting with one or two vertical cylinders. Copyright © 2001 John Wiley & Sons, Ltd.     

An immersed boundary method for simulation of inviscid compressible flows

In this paper, an immersed boundary method for simulating inviscid compressible flows governed by Euler equations is presented. All the mesh points are classified as interior computed points, immersed boundary points (interior points closest to the solid boundary), and exterior points that are blanked out of computation. The flow variables at an immersed boundary point are determined via the approximate form of solution in the direction normal to the wall boundary. The normal velocity is evaluated by applying the no‐penetration boundary condition, and therefore, the influence of solid wall in the inviscid flow is taken into account. The pressure is computed with the local simplified momentum equation, and the density and the tangential velocity are evaluated by using the constant‐entropy relation and the constant‐total‐enthalpy relation, respectively. With a local coordinate system, the present method has been extended easily to the three‐dimensional case. The present work is the first endeavor to extend the idea of hybrid Cartesian/immersed boundary approach to compressible inviscid flows. The tedious task of handling multi‐valued points can be eliminated, and the overshoot resulting from the extrapolation for the evaluation of flow variables at exterior points can also be avoided. In order to validate the present method, inviscid compressible flows over fixed and moving bodies have been simulated. All the obtained numerical results show good agreement with available data in the literature. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of aerodynamic problems with geometrically unspecified boundaries using an enhanced Lagrangian method

This paper presents solutions for several 2-D aerodynamic problems with geometrically unspecified boundaries. These solutions are obtained with an enhanced Lagrangian method based on stream function and Lagrangian-distance coordinates, which includes special procedures to substantially improve the numerical resolution of the shock waves and for the numerical implementation of the aerodynamic conditions defining the geometrically unspecified solid or fluid boundaries. The method is first validated for flows with specified solid boundaries by comparison with exact analytical solutions and with previous computational results obtained by numerical methods using Eulerian formulations. In all cases, this enhanced Lagrangian method displayed a very good accuracy and computational efficiency and a sharp numerical resolution of shock waves. Then this method is used to obtain solutions for problems with geometrically unspecified boundaries, such as: (i) indirect problems of determining the geometrical shape of airfoils and nozzle walls for a specified pressure distribution; (ii) supersonic nozzle design problem for a specified uniform flow at the nozzle outlet based on reflection-suppression condition; (iii) analysis of flexible-membrane airfoils; and (iv) analysis of jet-flapped airfoils.     

The shallow flow equations solved on adaptive quadtree grids

This paper describes an adaptive quadtree grid‐based solver of the depth‐averaged shallow water equations. The model is designed to approximate flows in complicated large‐scale shallow domains while focusing on important smaller‐scale localized flow features. Quadtree grids are created automatically by recursive subdivision of a rectangle about discretized boundary, bathymetric or flow‐related seeding points. It can be fitted in a fractal‐like sense by local grid refinement to any boundary, however distorted, provided absolute convergence to the boundary is not required and a low level of stepped boundary can be tolerated. Grid information is stored as a tree data structure, with a novel indexing system used to link information on the quadtree to a finite volume discretization of the governing equations. As the flow field develops, the grids may be adapted using a parameter based on vorticity and grid cell size. The numerical model is validated using standard benchmark tests, including seiches, Coriolis‐induced set‐up, jet‐forced flow in a circular reservoir, and wetting and drying. Wind‐induced flow in the Nichupté Lagoon, México, provides an illustrative example of an application to flow in extremely complicated multi‐connected regions. Copyright © 2001 John Wiley & Sons, Ltd.     

Calculation of the reflected wave from a pipe with a nozzle end by the Lax-Wendroff method

The solution of gas flow problems in pipes with nozzle ends is discussed. The Lax-Wendroff method, with a hybrid boundary condition approximation, is used to compute the numerical solutions to some test problems. The accuracy of the solutions obtained by this method is assessed by a comparison with theoretical solutions.