Numerical investigation of internal high-speed viscous flows using aparabolic technique

In the numerical simulation of high Reynolds-number flow, one can frequently supply only enough grid points to resolve the viscous terms in a thin layer. As a consequence, a body-or stream-aligned coordinate system is frequently used and viscous terms in this direction are discarded. It is argued that these terms cannot be resolved and computational efficiency is gained by their neglect. Dropping the streamwise viscous terms in this manner has been termed the thin-layer approximation. The thin-layer concept is an old one, and similar viscous terms are dropped, for example, in parabolized Navier-Stokes schemes. However, such schemes also make additional assumptions so that the equations can be marched in space, and such a restriction is not usually imposed on a thin-layer model. The thin-layer approximation can be justified in much the same way as the boundary-layer approximation; it requires, therefore, a body-or stream-aligned coordinate and a high Reynolds number. Unlike the boundary-layer approximation, the same equations are used throughout, so there is no matching problem. Furthermore, the normal momentum equation is not simplified and the convection terms are not one-sided differenced for marching. Consequently, the thin-layer equations are numerically well behaved at separation and require no special treatment there. Nevertheless, the thin-layer approximation receives criticism. It has been suggested that the approximation is invalid at separation and, more recently, that it is inadequate for unsteady transonic flow. Although previous comparisons between the thin-layer and Navier-Stokes equations have been made, these comparisons have not been adequately documented.

The diffraction of an initially one-dimensional plane expansion wave over a 90° corner was explored using experiment and numerical simulation. Unlike studies of shock diffraction, expansion wave diffraction has hardly been documented previously. The planar expansion wave was produced in a shock tube by bursting a diaphragm. Two independent parameters were identified for study: (i) the initial diaphragm shock tube pressure ratio, which determines the strength (pressure ratio) of the expansion, and (ii) the position of the diaphragm from the apex of the 90° corner, which determines the width of the wave. The experimentation only considered variation in the shock tube pressure ratio whereas the simulation varied both parameters. A Navier–Stokes solver with Menter’s shear stress transport$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}k\mbox{--}\omega $turbulence model was found to adequately model the overall flow field. A number of major flow features were identified occurring in the vicinity of the corner. These were: a shear layer that originated by flow separation at the apex of the corner; a vortex within a separation bubble that remained attached to the wall, in sharp contrast to what happens in the shock wave diffraction case, where the vortex convects downstream; and a reflected compression wave arising from perturbation signals generated by the diffraction. For a narrow-width expansion wave existing prior to diffraction, the reflected compression wave steepens into an outwardly propagating, weak cylindrical shock wave. Regions of supersonic flow are identified surrounding the bubble and can extend downstream depending on the pressure ratio. Another major flow feature identified in some cases was an oblique shock located near the separation bubble. A large wake region is evident immediately downstream of the bubble and appears to consist of two distinct layers. The experimental results showed large-scale turbulent structures within the separation bubble, and shear layer instability and vortex shedding from the separation bubble were also evident.

Visualization and investigation of supersonic air micro jets (SAMJ) as weak phase objects present serious problems when done through classical visualizing methods. Usage of advanced holographic interference techniques would make the study and evaluation of characteristic densities, density gradients and jet’s structure (weak shocks followed by expansion waves) more successful. It should be remarked, that the applications of Mach-Zehnder (reference) and shear dual-hologram techniques, which were performed in earlier works, have been done separately and have a demonstrative character. In the present paper, the authors have tried to apply both the dual-hologram techniques, simultaneously. This approach makes it possible to measure the same phase object under study by reference beam and shear interference techniques using only Mach-Zehnder holograms. Finally, experimental data obtained from both the techniques were numerically compared and found to be more applicable. It was first experimentally shown that sensitivity of shear interference approach during evaluations of radial component of density gradients in a micro jet can be enhanced by using the technique of re-recording holograms. The effect of fast transition to a transonic regime accompanied by expansion and later on with disintegration of a jet was observed and jet’s densities and radii in transition region were evaluated. As a whole, method of co-joint application of reference beam and shear dual-hologram interferometry has shown its self-descriptiveness, flexibility and utility for optical diagnostics of a weak gas dynamic micro object, the supersonic air micro jet.

Rudy and Bushnell's (1972) mixing length theory for the prediction of mean velocity fields in turbulent shear layers is extended to include supersonic free turbulent mixing. The normal momentum equation is coupled with the conventional equations of motion and is solved iteratively. Results show that transverse static pressure variation has very little effect on the mean flow variables in compressible free turbulent shear layers.

Wave processes in scramjet thrust generation

A method has been developed which can reliably compute nonequilibrium viscous shock layer over blunt-nosed bodies. This method considerably improves computational efficiency, especially for long slender bodies. The shock shape is generated as a part of the solution in this method, and in the nose region, the shock shape is calculated from an algebraic relation and corrected by global passes through that region. The current algorithm yields a considerable decrease of more than 60% in CPU time relative to other VSL methods. In the present study, both seven and eleven species air models were tested. The seven species analysis includes N, O, N2, O2, NO, NO+ and e−. The eleven species analysis includes the same species plus N+, O+, $$\text{N}_{2}^{ + }$$ , $$\text{O}_{2}^{ + }$$ . The chemical reaction models for seven and eleven species are taken from Blottner and Gupta, respectively. The governing equations are solved with a spatial-marching, implicit, finite-difference method. The solution is obtained by solving a couple of normal momentum and continuity equations. The results of the present technique show good agreement compared to the STS_2 flight data and other numerical solutions.

PurposeThe purpose of this paper is to present a non‐equilibrium viscous shock layer (VSL) solution procedure that considerably improves computational efficiency, especially for long slender bodies.Design/methodology/approachThe VSL equations are solved in a shock oriented coordinate system. The method of solution is spatial marching, implicit, finite‐difference technique, which includes coupling of the normal momentum and continuity equations. In the nose region, the shock shape is specified from an algebraic expression and corrected through global passes through that region. The shock shape is computed as part of the solution beyond the nose region and requires only a single global pass. For this study, a seven‐species (O2, N2, O, N, NO, NO+, e−) air model is used.FindingsThe present approach eliminates the need for initial shock shape, which was required by previous method of solution. This method generates its own shock shape as a part of solution and the input shock shape obtained from a different solution is not required. Therefore, in comparison with the other VSL methods, the present approach dramatically reduces the CPU time of calculations. Moreover, by using the shock oriented coordinate systems the junction point problem in sphere‐cone configurations is solved.Practical implicationsThis method is an excellent tool for parametric study and preliminary design of hypersonic vehicles.Originality/valueThe present method provides a computational capability which reduces the CPU time, and expands the range of application for the prediction of hypersonic heating rates.

We investigate experimentally the causes of jet plume instability and enhanced mixing observed in the exhaust of shock-containing convergent-divergent nozzles. Key features of the internal flow are the separation shock, separation shear layers, and pattern of alternating expansion and compression waves downstream of the shock. We focus on two possible reasons for this instability—the motion of the separation shock and the wave pattern downstream of the shock. The nozzle flow was generated in a planar facility with variable area ratio and pressure ratio, and the motion of the shock was tracked using time-resolved wall pressure measurements. The isolated effect of the wave pattern was investigated in a separate facility wherein a sonic shear layer, simulating the nozzle separation shear layer, was disturbed with compression and expansion waves emanating from a wavy wall. In both instances, the instability of the shear layer was characterized by time-resolved measurements of the total pressure. In the nozzle flow, the amplitude of shock motion increases with shock strength. Correlation of shock motion with shear layer total pressure is virtually absent for weak shocks but becomes significant for strong shocks. However, impingement of stationary waves on the shear layer had no impact on its growth rate. We conclude that the enhanced shear layer instability is strongly coupled to shock motion, and that the wave pattern by itself is not a cause of enhanced mixing. The occurrence of asymmetric separation at large shock strengths is a further contributor to the enhancement of instability.

In this paper the kinematics of a weak shock front governed by a hyperbolic system of conservation laws is studied. This is used to develop a method for solving problems, involving the propagation of nonlinear unimodal waves. It consists of first solving the nonlinear wave problem by moving along the bicharacteristics of the system and then fitting the shock into this solution field, so that it satisfies the necessary jump conditions. The kinematics of the shock leads in a natural way to the definition of ldquoshock-raysrdquo, which play the same role as the ldquoraysrdquo in a continuous flow. A special case of a circular cylinder introduced suddenly in a constant streaming flow is studied in detail. The shock fitted in the upstream region propagates with a velocity which is the mean of the velocities of the linear and the nonlinear wave fronts. In the downstream the solution is given by an expansion wave.

In the present work the shock wave propagation in a kinked duct was investigated to understand effects of compression and expansion waves on shock reflection. A finite volume method for two‐dimensional flow with the upwind TVD scheme was applied to solve the time‐dependent Euler equations. Two types of the flow in the duct were analyzed namely, the flow observed in a shock tube and that produced by a wire explosion. The results were compared with the available experimental data on weak shock waves. The flow field produced by wire explosion in a kinked duct was simpler than that in a shock tube, so that the shock dynamics theory may be available for engineering practice as an economical method.

This work presents a numerical simulation of the flow field in a liquid propellant rocket engine chamber and exit nozzle using techniques to allow the results to be taken as starting points for designing those propulsive systems. This was done using a Finite Volume method simulating the different flow regimes which usually take place in those systems. As the flow field has regions ranging from the low subsonic to the supersonic regimes, the numerical code used, initially developed for compressible flows only, was modified to work proficiently in the whole velocity range. It is well known that codes have been developed in CFD, for either compressible or incompressible flows, the joint treatment of both together being complex even today, given the small number of references available in this area. Here an existing code for compressible flow was used and primitive variables, the pressure, the Cartesian components of the velocity and the temperature instead of the conserved variables were introduced in the Euler and Navier-Stokes equations. This was done to permit the treatment at any Mach number. Unstructured meshes with adaptive refinements were employed here. The convective terms were treated with upwind first and second order methods. The numerical stability was kept with artificial dissipation and in the spatial coverage one used a five stage Runge-Kutta scheme for the Fluid Mechanics and the VODE (Value of Ordinary Differential Equations) scheme along with the Chemkin II in the chemical reacting solution. During the development of this code simulating the flow in a rocket engine, comparison tests were made with several different types of internal and external flows, at different velocities, seeking to establish the confidence level of the techniques being used. These comparisons were done with existing theoretical results and with other codes already validated and well accepted by the CFD community.

The success of the implementation of the normal equations approach of interior point methods (IPM) for linear programming depends on the quality of its analysis phase, i.e. reordering for sparsity. The goal of this analysis is to find a permutation matrix P such that the Cholesky factor of PAD 2 A T P T is the sparsest possible. In practice, heuristics are used to solve this problem because finding an optimal permutation is an NP-complete problem. Two such heuristics, namely the minimum degree and the minimum local fill—in orderings are particularly useful in the context of IPM implementations. In this paper a parametric set of symbolic orderings is presented, which connects these two major approaches. It will be shown that in the “neighborhood” of the minimum degree ordering a practically efficient method exist. Implementation details will be discussed as well, and on a demonstrative set of linear programming test problems the performance of the new method will be compared with Sparspak’s GENQMD subroutine which was for a long time public accesible from NETLIB, and with the minimum local fill—in ordering implementation of CPLEX version 4.0.KeywordsMinimum DegreeInterior Point MethodCholesky FactorSparsity PatternTrue DegreeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Semi-implicit discontinuous Galerkin methods for the incompressible Navier–Stokes equations on adaptive staggered Cartesian grids

An Arbitrary-Lagrangian-Eulerian hybrid finite volume/finite element method on moving unstructured meshes for the Navier-Stokes equations

For the time being, the required computational cost to solve the 3D time dependent flow prevents the use of such methods for internal flows at high Reynolds number in complex geometries. In this work we present a method based on a numerical length scale analysis to get a rational reduction of the full 3D governing equations for turbulent pipe flows. The length scale analysis quantifies the terms of the governing equations after changing the coordinate system into a curvilinear coordinate system with one coordinate aligned with the flow path. By retaining the most important terms or neglecting the (significantly) smallest terms, different reductions may be attained. The results for a double bent pipe, used to illustrate the approach, show that the most significant component of the viscous terms is the normal component. The convective terms are all important. The normal component is significant in the bends of the pipe due to centrifugal forces, while the spanwise component is most significant after the second bend due to a swirling motion.KeywordsLarge Eddy SimulationModel ReductionConvective TermStreamwise DirectionSpanwise DirectionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.