Search

The aim of this work is to develop a calculation model based on the method of characteristics making it possible to study the effect of the stagnation pressure of the combustion chamber on the 2D and axisymmetric minimum length nozzle design giving a uniform and parallel flow at the exit section. The model is based on the use of the real gas approach. The co-volume and the intermolecular interaction effect are taken into account by the use of the Berthelot state equation. The effect of molecular vibration is considered in our model to evaluate the behaviour of gas at a high temperature. In this case, the stagnation pressure and the stagnation temperature are important parameters in our model. The resolution of the algebraic equations is done by the finite difference corrector predictor algorithm. The validation of the results is controlled by the convergence of the critical section ratios calculated numerically as obtained by the theory. The mass and the thrust are evaluated to improve the efficiency of the nozzle. The comparison is made with the high temperature and perfect gas models. The application is made for air.

Stability features of two-stream coaxial plasma jet simulations are investigated using numerical solutions to the spatio-temporal one-dimensional linear stability theory problem. The base states obtained from magneto-hydrodynamic simulations consider the flow as a mixture of gases in local thermodynamic equilibrium (LTE) while stability computations are performed assuming both a calorically perfect gas (CPG) model and LTE. Comparisons with solutions considering a simple CPG model show the non-negligible impact of the LTE on the stability attributes of the plasma jet. For all cases studied, a large region of absolute instability is found for the axisymmetric mode, starting at the jet inlet. The streamwise evolution of the absolute growth rate is found to depend both on the baroclinic torque and the displacement of the maximum shear toward low velocity regions of the jet, combining effects described in the literature. The jet is controlled by means of electric power and static pressure at constant mass flow. The former affects mainly the absolute growth rate through changes of the core-to-bypass stream velocity ratio, while the latter influences mostly the absolute frequency. Finally, the full impulse response reveals a competition mechanism between the absolute mixed modes dominating at low group velocities, and convective shear layer modes at higher group velocities, restricted to the first half of the chamber.

Steady states of inviscid, compressible and axisymmetric swirling flows of a single-phase, inert, thermodynamically supercritical fluid in a rotating, finite-length, straight, long circular pipe are studied. The fluid thermodynamic behaviour is modelled by the van der Waals equation of state. A nonlinear partial differential equation for the solution of the flow streamfunction is derived from the fluid equations of motion in terms of the inlet flow specific total enthalpy, specific entropy and circulation functions. This equation reflects the complicated, nonlinear thermo-physical interactions in the flows, specifically when the inlet state temperature and density profiles vary around the critical thermodynamic point, flow compressibility is significant and the inlet swirl ratio is high. Several types of solutions of the resulting nonlinear ordinary differential equation for the axially independent case describe the flow outlet state when the pipe is sufficiently long. The approach is applied to an inlet flow described by a solid-body rotation with uniform profiles of the axial velocity and temperature. The solutions are used to form the bifurcation diagrams of steady compressible flows of real fluids as the inlet swirl level and the centreline inlet density are increased at a fixed inlet Mach number and temperature. Focus is on heavy-molecule fluids with low values of $R/C_{v}$. Computed results provide theoretical predictions of the critical swirl levels for the exchange of stability of the columnar state and for the appearance of non-columnar states and of vortex breakdown states as a function of inlet centreline density. The difference in the dynamical behaviour between that of a calorically perfect gas and of a real gas is explored. The analysis sheds new fundamental light on the complex dynamics of high-Reynolds-number, compressible, subsonic swirling flows of real gases.

Some celestial bodies such as planets, moons and comets (here referred to as moons for simplicity) emit jets of material at speeds that in some instances are large enough to escape gravity. Previous investigations have shown this problem to be highly complex, e.g. involving multi-phase flows, phase changes, radiation and gas rarefaction effects. In order to learn from exploring a manageable parameter space, and to provide a limiting case, the present study considers a much simpler model situation in which the material of the jet is an inviscid, non-heat-conducting, perfect gas that issues radially at the surface of the moon with sonic velocity. Theoretical considerations show that the escape velocity of a gas is much smaller than that of a solid body. An analytical solution is obtained for the maximum height reached by a jet in steady flow. A computational parameter study of unsteady, inviscid, axisymmetric flow, including the effect of an atmosphere, provides a rich picture of the features and behaviour of the model jet. The deficit of the computed maximum steady-state penetration height below the isentropic theoretical value may be explained by the effect of the atmosphere and of dissipation in shock waves that occur in the computed flows. Many of the features of the gas jet are qualitatively mirrored in an experiment using a water flow analogy in which the gravitational field is simulated by a surface of suitable shape.

This article presents a unified approach to solve steady-state conjugate heat-transfer problem including simultaneously gas, liquid and solid regions in just one 3D domain, distinguished by their particular properties. This approach reduces approximation errors and the time to solve the problem, which characterise iterative methods based on separated domains. The formulation employs RANS equations, realisablek-ε turbulence model and near-wall treatment model. A commercial CFD code solves the pressure-based segregated algorithm combined with spatial discretisation of second order upwind. The problem consists of a convergent-divergent metallic nozzle that contains cooling channels divided in two segments along the wall. The nozzle wall insulates the high-speed hot air flow, dealt as perfect gas, from the two low-speed cold water flows, dealt as compressed liquid, both influenced by transport properties dependent of the local temperature. The verification process uses three meshes with increasing resolutions to demonstrate the independence of the results. The validation process compares the simulation results with experimental data obtained in high-enthalpy wind tunnel, demonstrating good compliance between them. Results for the bulk temperature rise of the water in the second cooling segment of the nozzle showed good agreement with available experimental data. Numerical simulations also provided wall temperature and heat flux for the gas and liquid sides. Besides, distribution of temperature, pressure, density and Mach number were plotted along the nozzle centerline showing a little disturbance downstream the throat. This phenomenon has been better visualised by means of 2D maps of those variables. The analysis of results indicates that the unified approach herein presented can make easier the task of simulating the conjugate convection-conduction heat-transfer in a class of problems related to regeneratively cooled thrust chambers.

When the stagnation pressure of a perfect gas increases, the specific heat and their ratio do not remain constant anymore and start to vary with this pressure. The gas doesn't stay perfect. Its state equation changes and it becomes a real gas. In this case, the effects of molecular size and intermolecular attraction forces intervene to correct the state equation. The aim of this work is to determine the effect of stagnation pressure on the thermodynamic, physical and geometrical supersonic flow parameters in order to find a general form for real gas. With the assumptions that Berthelot's state equation accounts for molecular size and intermolecular force effects, expressions are developed for analysing the supersonic flow for thermally and calorically imperfect gas lower than the dissociation molecules threshold. The design parameters of the supersonic nozzle-like thrust coefficient depend directly on the stagnation parameters of the combustion chamber. The application made for air. A computation of error was made in this case to give a limit of the perfect gas model compared to the real gas model.

We report our systematic development of a general and exact theory for diagnosis of total force and moment exerted on a generic body moving and deforming in a calorically perfect gas. The total force and moment consist of a longitudinal part (L-force) due to compressibility and irreversible thermodynamics, and a transverse part (T-force) due to shearing. The latter exists in incompressible flow but is now modulated by the former. The theory represents a full extension of a unified incompressible diagnosis theory of the same type developed by J. Z. Wu and coworkers to compressible flow, with Mach number ranging from low-subsonic to moderate-supersonic flows. Combined with computational fluid dynamics (CFD) simulation, the theory permits quantitative identification of various complex flow structures and processes responsible for the forces, and thereby enables rational optimal configuration design and flow control. The theory is confirmed by a numerical simulation of circular-cylinder flow in the range of free-stream Mach number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}M_{\infty }$ between 0.2 and 2.0. The L-drag and T-drag of the cylinder vary with $M_{\infty }$ in different ways, the underlying physical mechanisms of which are analysed. Moreover, each L-force and T-force integrand contains a universal factor of local Mach number $M$. Our preliminary tests suggest that the possibility of finding new similarity rules for each force constituent could be quite promising.

Taylor–Couette flow in a compressible perfect gas is studied. The onset of instability is examined as a function of the Reynolds numbers of the inner and outer cylinder, the Mach number of the flow and the Prandtl number of the gas. We focus on the case of a wide gap, with radius ratio 0.5. We find new modes of instability at high Prandtl number, which can allow oscillatory axisymmetric modes to onset first. We also find that onset can occur even when the angular momentum increases outwards, so that the classical Rayleigh criterion can be violated in the compressible case. We have also considered the case of counter-rotating cylinders, where the $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}m=0$ and $m=1$ modes can onset simultaneously to give a codimension 2 bifurcation, leading to the formation of complex flow patterns. In compressible flow we also find codimension 3 points. The Mach number and the critical inner and outer Reynolds numbers can be adjusted so that the two neutral curves for the $m=0$ and $m=1$ modes touch rather than cross. Complex codimension 3 points occur more readily in the compressible case than in the Boussinesq case, and they are expected to lead to a rich nonlinear behaviour.

International audience; A detailed numerical study of the influence of dense gas effects on the large-scale dynamics of decaying homogeneous isotropic turbulence is carried out by using the van der Waals gas model. More specifically, we focus on dense gases of the Bethe–Zel’dovich–Thompson type, which may exhibit non-classical nonlinearities in the transonic and supersonic flow regimes, under suitable thermodynamic conditions. The simulations are based on the inviscid conservation equations, solved by means of a ninth-order numerical scheme. The simulations rely on the numerical viscosity of the scheme to dissipate energy at the finest scales, while leaving the larger scales mostly unaffected. The results are systematically compared with those obtained for a perfect gas. Dense gas effects are found to have a significant influence on the time evolution of the average and root mean square (r.m.s.) of the thermodynamic properties for flows characterized by sufficiently high initial turbulent Mach numbers (above 0.5), whereas the influence on kinematic properties, such as the kinetic energy and the vorticity, are smaller. However, the flow dilatational behaviour is very different, due to the non-classical variation of the speed of sound in flow regions where the dense gas is characterized by a value of the fundamental derivative of the gas dynamics (a measure of the variation of the speed of sound in isentropic compressions) smaller than one or even negative. The most significant differences between the perfect and the dense gas case are found for the repartition of dilatation levels in the flow field. For the perfect gas, strong compressions occupy a much larger volume fraction than expansion regions, leading to probability distributions of the velocity divergence highly skewed toward negative values. For the dense gas, the volume fractions occupied by strong expansion and compression regions are much more balanced; moreover, strong expansion regions are characterized by sheet-like structures, unlike the perfect gas which exhibits tubular structures. In strong compression regions, where compression shocklets may occur, both the dense and the perfect gas exhibit sheet-like structures. This suggests the possibility that expansion eddy shocklets may appear in the dense gas. This hypothesis is also supported by the fact that, in dense gas, vorticity is created with equal probability in strong compression and expansion regions, whereas for a perfect gas, vorticity is more likely to be created in the strong compression ones.

Along the lines of the restricted Euler equation (REE) for incompressible flows, we develop homogenized Euler equation (HEE) for describing turbulent velocity gradient dynamics of an isentropic compressible calorically perfect gas. Starting from energy and state equations, an evolution equation for pressure Hessian is derived invoking uniform (homogeneous) velocity gradient assumption. Behaviour of principal strain rates, vorticity vector alignment and invariants of the normalized velocity gradient tensor is investigated conditioned on dilatation level. The HEE results agree very well with the known behaviour in the incompressible limit. Indeed, at zero dilatation HEE reproduces the incompressible anisotropic pressure Hessian behaviour very closely. When compared against compressible direct numerical simulation results, the HEE accurately captures the strain rate behaviour at different dilatation levels. The model also recovers the fixed point behaviour of pressure-released (high-Mach-number limit) Burgers turbulence.

Component maps are produced under certain environmental conditions using air as the working fluid during static ground operation. Any changes of the component characteristics when operating under different temperature conditions and/or with different working fluid are partially taken into account, because of the existence of the gas constant and the ratio of the specific heats in the non-dimensional mass flow and rotational speed. This provides a second order correction for the component characteristics, which may be adequate for the initial modeling of engines. However, for rigorous performance calculations correction factors are applied to the non-dimensional mass flow, rotational speed and pressure ratio distributions of a map, when deviations from the reference conditions under which it was extracted, are experienced. In the current study, a different approach is considered in order to eliminate the inaccuracies caused by the varying temperature and chemical composition. It makes direct use of inlet and circumferential Mach numbers based upon stagnation temperature in conjunction with dimensionless enthalpy variation. A sensitivity analysis against gas property variations is conducted to quantify the benefits gained in precision. Generally, the well-known relationships correlating the Mach number with total and static properties are based on the assumption of perfect gas and constant gas properties. Introducing dependency on temperature and/or chemical composition for the caloric properties of the semi-perfect gas, proper mean values are defined and some theoretical corrections are provided for the well-known equations. The mass flow compatibility equation is then based on the ‘corrected’ expression correlating dimensionless mass flow and Mach number and takes full account of gas property variations.

When the stagnation temperature of a perfect gas increases, the specific heats and their ratio do not remain constant and start to vary with the temperature. The gas remains perfect; its state equations remain valid, so it can be named as calorifically imperfect gas. The aim of this research is to develop the necessary thermodynamic and geometrical equations and to study the supersonic flow at high temperature, lower than the dissociation threshold. The results are found by the resolution of nonlinear algebraic equations and integration of complex analytical functions where the exact calculation is impossible. The dichotomy method is used to solve the nonlinear equations and Simpson’s algorithm for the numerical integration applied. A condensation of the nodes is used. The functions to be integrated have a high gradient at the extremity of the interval of integration. The comparison is made with the calorifically perfect gas to determine the error. The application is made for air in a supersonic nozzle.

A theoretical analysis was carried out to study the formation and propagation of a weak shock wave in a gas with longitudinal temperature gradients. An equation describing the formation and propagation of a weak shock wave through a non-uniform medium in the absence of energy dissipation was derived. An approximate analytical solution to the one-dimensional wave propagation equation is established. With this, the thermal gradient effects on the shock-wave Mach number and speed were investigated and the results were compared to earlier experiments. Numerical solutions for the same problem using Euler’s equations have also been obtained and compared to the analytical results. The analysis shows that the time of shock-wave formation from the initial disturbance, for mild temperature gradients, is independent of the gradient. The shock wave forms at a longer axial distance from the initial disturbance when the temperature gradient is positive whereas the opposite is true for a negative temperature gradient.

In Robinet et al. (2000), an instability for shock waves in gas dynamics is exhibited. The fluid was assumed to obey the polytropic perfect gas pressure law. We show in this note that such an instability does not occur, as proved in the extensive work of Majda (1983). Two arguments are developed: we give first a mathematical result that is violated by the conclusions of Robinet et al. (2000). Then we detail the results of Robinet et al. (2000) and show why they do not yield any conclusions. Finally, we develop a general calculation and show that an instability cannot occur.

We examine the effect of large bulk viscosity on the classical problem of two-dimensional shock–boundary-layer interaction. The flow is taken to be steady and supersonic over a flat adiabatic plate. The boundary layer is taken to be laminar and the fluid is modelled as a perfect gas with a bulk viscosity that is large compared with its shear viscosity. The flow details are computed using a fifth-order weighted essentially non-oscillatory finite difference scheme and a third-order Runge–Kutta scheme for the spatial and temporal discretizations. The primary result of interest is the suppression of separation when the ratio of bulk to shear viscosity is sufficiently large.

We present a theoretical and numerical investigation of longshore currents driven by breaking waves on beaches, especially barred beaches. The novel feature considered here is that the wave envelope is allowed to vary in the alongshore direction, which leads to the generation of strong dipolar vortex structures where the waves are breaking. The nonlinear evolution of these vortex structures is studied in detail using a simple analytical theory to model the effect of a sloping beach. One of our findings is that the vortex evolution provides a robust mechanism through which the preferred location of the longshore current can move shorewards from the location of wave breaking. Such current dislocation is an often-observed (but ill-understood) phenomenon on real barred beaches.To underpin our results, we present a comprehensive theoretical description of the relevant wave–mean interaction theory in the context of a shallow-water model for the beach. Therein we link the radiation-stress theory of Longuet-Higgins & Stewart to recently established results concerning the mean vorticity generation due to breaking waves. This leads to detailed results for the entire life-cycle of the mean-flow vortex evolution, from its initial generation by wave breaking until its eventual dissipative decay due to bottom friction.In order to test and illustrate our theory we also present idealized nonlinear numerical simulations of both waves and vortices using the full shallow-water equations with bottom topography. In these simulations wave breaking occurs through shock formation of the shallow-water waves. We note that because the shallow-water equations also describe the two-dimensional flow of a homentropic perfect gas, our theoretical and numerical results can also be applied to nonlinear acoustics and sound–vortex interactions.

The complex flow and wave pattern following an initially planar shock wave transmitted through a double-bend duct is studied experimentally and theoretically/numerically. Several different double-bend duct geometries are investigated in order to assess their effects on the accompanying flow and shock wave attenuation while passing through these ducts. The effect of the duct wall roughness on the shock wave attenuation is also studied. The main flow diagnostic used in the experimental part is either an interferometric study or alternating shadow–schlieren diagnostics. The photos obtained provide a detailed description of the flow evolution inside the ducts investigated. Pressure measurements were also taken in some of the experiments. In the theoretical/numerical part the conservation equations for an inviscid, perfect gas were solved numerically. It is shown that the proposed physical model (Euler equations), which is solved by using the second-order-accurate, high-resolution GRP (generalized Riemann problem) scheme, can simulate such a complex, time-dependent process very accurately. Specifically, all wave patterns are numerically simulated throughout the entire interaction process. Excellent agreement is found between the numerical simulation and the experimental results. The efficiency of a double-bend duct in providing a shock wave attenuation is clearly demonstrated.

We report on calculations and experiments with strong shocks diffracting over rigid ramps in argon. The numerical results were obtained by integrating the conservation equations that included the Navier–Stokes equations. The results predict that if the ramp angle θ is less than the angle θe that corresponds to the detachment of a shock, θ < θe, then the onset of Mach reflection (MR) will be delayed by the initial appearance of a precursor regular reflection (PRR). The PRR is subsequently swept away by an overtaking corner signal (cs) that forces the eruption of the MR which then rapidly evolves into a self-similar state. An objective was to make an experimental test of the predictions. These were confirmed by twice photographing the diffracting shock as it travelled along the ramp. We could get a PRR with the first exposure and an MR with the second. According to the von Neumann perfect gas theory, a PRR does not exist when θ < θe. A viscous length scale xint is a measure of the position on the ramp where the dynamic transition PRR → MR takes place. It is significantly larger in the experiments than in the calculations. This is attributed to the fact that fluctuations from turbulence and surface roughness were not modelled in the calculations. It was found that xint → ∞ as θ → θe. Experiments were done to find out how xint depended on the initial shock tube pressure p0. The dependence was strong but could be greatly reduced by forming a Reynolds number based on xint. Finally by definition, regular reflection (RR) never interacts with a boundary layer, while PRR always interacts; so they are different phenomena.

The theoretical linear stability of a shock wave moving in an unlimited homogeneous environment has been widely studied during the last fifty years. Important results have been obtained by Dýakov (1954), Landau & Lifchitz (1959) and then by Swan & Fowles (1975) where the fluctuating quantities are written as normal modes. More recently, numerical studies on upwind finite difference schemes have shown some instabilities in the case of the motion of an inviscid perfect gas in a rectangular channel. The purpose of this paper is first to specify a mathematical formulation for the eigenmodes and to exhibit a new mode which was not found by the previous stability analysis of shock waves. Then, this mode is confirmed by numerical simulations which may lead to a new understanding of the so-called carbuncle phenomenon.

A detailed investigation of the hydrodynamic stability to transverse linear disturbances of a steady, one-dimensional detonation in an ideal gas undergoing an irreversible, unimolecular reaction with an Arrhenius rate constant is conducted via a normal-mode analysis. The method of solution is an iterative shooting technique which integrates between the detonation shock and the reaction equilibrium point. Variations in the disturbance growth rates and frequencies with transverse wavenumber, together with two-dimensional neutral stability curves and boundaries for all unstable low- and higher frequency modes, are obtained for varying detonation bifurcation parameters. These include the detonation overdrive, chemical heat release and reaction activation energy. Spatial perturbation eigenfunction behaviour and phase and group velocities are also obtained for selected sets of unstable modes. Results are presented for both Chapman–Jouguet and overdriven detonation velocities. Comparisons between the earlier pointwise determination of stability and interpolated neutral stability boundaries obtained by Erpenbeck are made. Possible physical mechanisms which govern the wavenumber selection underlying the initial onset of either regular or irregular cell patterns are also discussed.

Non-stationary oblique shock wave reflections for fluids in the dense gas regime are examined for selected cases. A time-accurate predictor-corrector TVD scheme with reflective boundary conditions for solving the Euler equations simulates the evolution of a wave field for an inviscid van der Waals gas near the thermodynamic critical point. The simulated cases involve shock tube flows with compressive wedges and circular arcs. Non-classical phenomena, such as disintegrating shocks, expansion shocks, composite waves, etc., demonstrate significant differences from perfect gas flow fields over similar geometries. Detailed displays of wave field structures and thermodynamic states for the dense gas flow fields are presented and analysed.

The high-enthalpy, hypersonic flow over a compression corner has been examined experimentally and theoretically. Surface static pressure and heat transfer distributions, along with some flow visualization data, were obtained in a free-piston shock tunnel operating at enthalpies ranging from 3 MJ kg−1 to 19 MJ kg−1, with the Mach number varying from 7.5 to 9.0 and the Reynolds number based on upstream fetch from 2.7×104 to 2.7×105. The flow was laminar throughout. The experimental data compared well with theories valid for perfect gas flow and with other relevant low-to-moderate enthalpy data, suggesting that for the current experimental conditions, the real gas effects on shock wave/boundary layer interaction are negligible. The flat-plate similarity theory has been extended to include equilibrium real gas effects. While this theory is not applicable to the current experimental conditions, it has been employed here to determine the potential maximum effect of real gas behaviour. For the flat plate, only small differences between perfect gas and equilibrium gas flows are predicted, consistent with experimental observations. For the compression corner, a more rapid rise to the maximum pressure and heat transfer on the ramp face is predicted in the real gas flows, with the pressure lying slightly below, and the heat transfer slightly above, the perfect gas prediction. The increase in peak heat transfer is attributed to the reduction in boundary layer displacement thickness due to real gas effects.

Formation and evolution of N (-like) waves is studied without the restriction of low amplitude, namely weak nonlinearity. To this end, the classical piston problem of gasdynamics is investigated, in which the wave is radiated by a piston executing a single cycle of harmonic oscillation into an inviscid perfect gas. The method of analysis is based on the simple-wave theory up to the shock formation time, and beyond that time on the numerical calculation by a high-resolution TVD upwind scheme. The initial sinusoid-like wave profile is rapidly distorted as the wave propagates, and this leads to the formation of head and tail shocks. The main effects of strong nonlinearity may be listed as follows: (i) entropy production at shock fronts, (ii) the existence of waves reflected from shocks, (iii) an asymmetric wave profile stemming from the boundary condition at the source of the strongly nonlinear problem. As the result, the strongly nonlinear wave possesses the following remarkable distinctive features, in contrast to its counterpart in the weakly nonlinear regime. The tail shock is not formed at the tail of the wave, and the expansion wave behind the head shock has non-uniform intensity. The N (-like) wave propagates with some excess mass. Thereby a region with low density, associated with the entropy production, appears in the vicinity of the source.

The mechanisms of heat and mass transport in a side-heated square cavity filled with a near-critical fluid are explored, with special emphasis on the interplay between buoyancy-driven convection and the Piston Effect. The Navier–Stokes equations for a near-critical van der Waals gas are solved numerically by means of an acoustically filtered, finite-volume method. The results have revealed some striking behaviour compared with that obtained for normally compressible gases: (i) heat equilibration is still achieved rapidly, as under zero-g conditions, by the Piston Effect before convection has time to enhance heat transport; (ii) mass equilibration is achieved on a much longer time scale by quasi-isothermal buoyant convection; (iii) due to the very high compressibility, a stagnation-point effect similar to that encountered in high-speed flows provokes an overheating of the upper wall; and (iv) a significant difference to the convective single-roll pattern generated under the same conditions in normal CO2 is found, in the form of a double-roll convective structure.

New subgrid-scale models for the large-eddy simulation of compressible turbulent flows are developed and tested based on the Favre-filtered equations of motion for an ideal gas. A compressible generalization of the linear combination of the Smagorinsky model and scale-similarity model, in terms of Favre-filtered fields, is obtained for the subgrid-scale stress tensor. An analogous thermal linear combination model is also developed for the subgrid-scale heat flux vector. The two dimensionless constants associated with these subgrid-scale models are obtained by correlating with the results of direct numerical simulations of compressible isotropic turbulence performed on a963grid using Fourier collocation methods. Extensive comparisons between the direct and modelled subgrid-scale fields are provided in order to validate the models. A large-eddy simulation of the decay of compressible isotropic turbulence – conducted on a coarse323grid – is shown to yield results that are in excellent agreement with the fine-grid direct simulation. Future applications of these compressible subgrid-scale models to the large-eddy simulation of more complex supersonic flows are discussed briefly.

Spin-up flows of a compressible gas in a finite, closed cylinder from an initial state of rest are studied, The flow is characterized by small reference Ekman numbers, and the peripheral Mach number is O(1). Comprehensive numerical solutions have been obtained for the full, time-dependent compressible Navier-Stokes equations. The details of the flow, temperature, and density evolution are described. In the early phase of spin-up, owing to the thermoacoustic disturbances caused by the compressible Rayleigh effect, the flows are oscillatory, and this oscillatory behaviour is pronounced at higher Mach numbers. The principal dynamical role of the Ekman layer is dominant over moderate times of orders of the homogeneous spin-up timescales. Owing to the density stratification in the radial direction, the Ekman layer is thicker in the central region of the interior. The interior azimuthal flows are mainly uniform in the axial direction. As the Mach number increases, the rate of spin-up in the interior becomes slower, and the propagating shear front is more diffusive. Explicit comparisons with the results for an infinite cylinder are made to ascertain the contributions of the endwall disks. In contrast to the usual incompressible spin-up from rest, the viscous effects are relatively more important for the case of a compressible fluid.

We study the linear temporal hydrodynamic stability in the Rayleigh-Benard problem for a compressible fluid (a perfect gas) under marginally super-adiabatic conditions, i.e. when the ambient temperature gradient only slightly exceeds the adiabatic gradient and then only within the fluid adjacent to the upper (cold) wall. The onset of convection in this limit demonstrates some unique features which differ qualitatively from those of the familiar Boussinesq approximation. Thus, the ensuing convection is effectively confined to a narrow domain of the fluid close to the upper wall and is characterized by large wavenumbers. Furthermore, these distinct attributes persist with diminishing temperature difference, implying that the prevailing generalized Boussinesq approximation (based on the use of the potential temperature gradient) is non-uniform in the present limit. This non-uniformity is resolved in terms of the small yet significant variations of fluid properties (which are commonly neglected). We comment on the analogy between the present problem and the Taylor-Couette problem for a viscous incompressible fluid within a narrow gap between counter-rotating cylinders. We briefly discuss the potential relevance of the present limit to some recent observations of the onset of convection within near-critical fluids.

The unsteady expansion of an ideal gas into a vacuum is studied in one-dimensional planar and spherical geometries. The free-molecular expansion of a Maxwell-distributed gas is compared to the continuum expansion of a perfect gas with for planar flow and tends to zero for spherical flow. Continuum thermodynamic temperature pulses, on the other hand, rise and fall in simple relation to continuum density. The far-field peak wall pressure in both Knudsen-number limits is found to decrease in inverse (or inverse cubic) proportion to the distance from the initial planar (or spherical) region. This result for the spherical case is at odds with the experiments of Ahrens, Allen & Kovach (1971) which indicate a more rapid (ξ−3.5) fall-off of peak overpressure with distance from a point source in a vacuum.

We present results from our experiments with the irregular reflection of shock waves in argon. We compare the data with the results we obtained numerically; the assumptions for the computational code were that we had unsteady, two-dimensional, compressible, inviscid, flow of a perfect gas. When precautions were taken to reduce the effects of the gas viscosity on the experimental data, we obtained very good agreement between the numerical and the experimental results for the ramp Mach number and the trajectory path triple-point angle, but there were discrepancies with the wave-angle data. The discrepancies were ascribed to the sensitivity of the data to both viscosity and to a singularity. We show that there are actually two weak irregular wave reflections, namely a classic Mach reflection (MR) and a new type, that we call a von Neumann reflection (NR). The structure of the NR is discussed in some detail, and so are the transition criteria for the various wave systems.

We consider the effects of critical-layer nonlinearity on spatially growing oblique instability waves on compressible shear layers between two parallel streams. The analysis shows that mean temperature non-uniformities cause nonlinearity to occur at much smaller amplitudes than it does when the flow is isothermal. The nonlinear instability wave growth rate effects are described by an integro-differential equation which bears some resemblance, to the Landau equation in that it involves a cubic-type nonlinearity. The numerical solutions to this equation are worked out and discussed in some detail. We show that inviscid solutions always end in a singularity at a finite downstream distance but that viscosity can eliminate this singularity for certain parameter ranges.

The shock wave equations for a perfect gas often provide more than one solution to a problem. In an attempt to find out which solution appears in a given physical situation, we present a linearized analysis of the equations of motion of a flow field with a shock boundary. It is found that a solution will be stable when there is supersonic flow downstream of the shock, and asymptotically unstable when there is subsonic flow downstream of it. It is interesting that both flows are found to be stable against disturbances of the d'Alembert type which grow from point sources; it is only when larger-scale line sources are considered that one can discriminate between the stabilities of the two types of flow. The results are applicable to supersonic flow over flat plates at incidence, to wedges, and to some cases of regular reflexion, diffraction and refraction of shocks.

The interaction of a rarefaction wave with a gradual monotonic area reduction of finite length in a duct, which produces transmitted and reflected rarefaction waves and other possible rarefaction and shock waves, was studied both analytically and numerically. A quasi-steady flow analysis which is analytical for an inviscid flow of a perfect gas was used first to determine the domains of and boundaries between four different wave patterns that occur at late times, after all local transient disturbances from the interaction process have subsided. These boundaries and the final constant strengths of the transmitted, reflected and other waves are shown as a function of both the incident rarefaction-wave strength and area-reduction ratio, for the case of diatomic gases and air with a specific-heat ratio of 7/5. The random-choice method was then used to solve numerically the conservation equations governing the one-dimensional non-stationary gas flow for many different combinations of rarefaction-wave strengths and area-reduction ratios. These numerical results show clearly how the transmitted, reflected and other waves develop and evolve with time, until they eventually attain constant strengths, in agreement with quasi-steady flow predictions for the asymptotic wave patterns. Note that in all of this work the gas in the area reduction is initially at rest.

Special relativistic magnetohydrodynamic shock waves in a perfect gas of infinite conductivity and constant adiabatic index are analysed. It is shown that the Rankine-Hugoniot equations for such shocks may be reduced to a seventh degree polynomial for the downstream dynamical volume ω, with the polynomial coefficients depending on the upstream state (ω equals the specific volume times the ratio of the energy density of the fluid (omitting electromagnetic terms) to the fluid rest mass energy density). In the non-relativistic limit, the polynomial equation reduces to a relation between the upstream and downstream Alfvénic Mach numbers, previously obtained by Cabannes. The equation for ω classifies in a natural way both shocks in which the electric field may be eliminated by transforming to the de Hoffman–Teller frame, and shocks for which this is not possible. The equation is used to determine the downstream state of relativistic shocks for a given upstream state as specified by the plasma beta, magnetic field obliquity, and flow speed.

SynopsisA priori estimates on the solution to the complete system of equations governing a heat-conductive, viscous reactive perfect gas confined between two infinite parallel plates are derived. From these estimates, global existence of both weak and classical solutions is obtained.

The shock conditions for a perfect gas in aligned-fields magnetogasdynamics are solved to give all variables downstream as rational functions of the density ratio. The thermodynamically admissible, evolutionary shock solutions are exhaustively classified into seven shock polar types, including a new type of supersonic, sub -Alfvénic polar. Example polars of each type are presented in the magnetograph (dimensionless magnetic induction vector plane) and in the Taniuti–Resler diagram (M, Aplane) in order to display, respectively, the shock geometry and the state of the gas. A short discussion of the relation of the shock properties to flow over bodies is presented.

It is shown that for the steady isoenergetic rotational flow of an ideal gas, both the specific enthalpy and the speed of sound can be expressed as functions of the velocity. As a result, it is possible to formulate the equations of motion so that the velocity is the only dependent variable. For a gas whose enthalpy and sound speed are functionally related, the results are a generalization of those for a perfect gas. If the enthalpy and sound speed are independent variables, the new formulation leads to a single vector equation whose solution completely determines the flow.

The plane, steady, laminar vortex flow of a viscous, heat-conducting perfect gas is treated. Simple relations are obtained for the flow quantities in the irrotational vortex, for arbitrary Prandtl numbers. When the Prandtl number is ½, the irrotational vortex is also isentropic. When the temperature dependence of the viscosity coefficient is taken into account, the vortex flow is rotational. An exact solution for the rotational vortex is obtained which is suitable for numerical evaluation by successive approximations. Distributions of velocity, temperature, pressure, density, and stagnation temperature through the rotational vortex are given for a typical case.

This is a sequel to a recent paper [1] on the construction by the hodograph method of trans-sonic nozzle-flows of a perfect gas. At the end of that paper it was shown how we can obtain regular flows that are ultimately uniform (as required in the test section of a supersonic wind tunnel), and the object now is to give some quantitative examples of such flows. The gas is supposed to have the polytropic equation of state Pρ−γ = constant, and the calculations have been made for the case γ = 1.4, with the Mach number M = 2.25 at the test section. The results, which are exhibited graphically, are indicative of what may be expected for other supersonic values of M, and it is hoped that they may be significant for the design of wind tunnels.

The error minimization technique is introduced as a method of obtaining the simultaneous solution to a set of partial differential equations. Because this numerical technique is at worst neutrally stable, it is believed to have fundamental advantages over existing techniques. The method is formulated in general for the steady fluid dynamic conservation equations and then applied to a specific case of irrotational isentropic flow of a perfect gas through a nozzle. Flow-field solutions were found in nozzle contours having a throat radius of curvature as low as 0·5. Comparison between experimental Mach number contours and the theoretical solution for a finite inlet nozzle with a conical exit is excellent.

The compression of a perfect gas between a uniformaly moving piston and a rigid wall is discussed in the one-dimensional case. If the piston moves with a finite speed, it will initiate a shock in the gas which will reflect successively from rigid wall and piston and cause the compression process to deviate from a reversible adiabatic process. Expressions are derived for the relative changes in pressure and density at each shock reflection. Then values of density and pressure after any number of shock reflections are computed relative to their initial values, and, in terms of these, the corresponding values of temperature and entropy, as well as shock speeds, are determined. The limiting value of the entropy change, as the number of reflections goes to infinity, is obtained as a function of the ratio of specific heats of the gas and the strength of the initial shock. Hence it is possible to estimate an upper limit to the deviation of the shock compression process from a reversible adiabatic process. Some illustrative numerical examples are given.

SummaryThe paper deals with the behaviour of three shock waves meeting at a point in a perfect gas. It is shown that the equations of motion can be reduced to a single polynomial equation of degree 10. The real roots of this equation are studied to determine their physical significance. In addition, the appearance of degenerate shock systems is shown to be associated with the formation of certain multiple roots of the polynomial equation.

In a recent letter to Nature F. London(3) has called attention to a discontinuity in the derivative dCV/dT of the specific heat with respect to temperature which arises in a perfect Einstein-Bose gas at low temperatures. When discussing the properties of a perfect gas satisfying the Bose statistics, Einstein(1) remarked that at low temperatures something resembling a condensed phase should appear. The temperature at which this condensation should begin is given by the equationwhere n is the number of particles of mass m per unit volume and the other symbols have their usual significance.

The viscous hypersonic flow past an axisymmetric blunt body is analysed based upon the Navier-Stokes equations. It is assumed that the fluid is a perfect gas having constant specific heats, a constant Prandtl number, P, whose numerical value is of order one, and a viscosity coefficient varying as a power, ω, of the absolute temperature. Limiting forms of solutions are studied as the free-stream Mach number, M, and the free-stream Reynolds number based on the body nose radius, R, go to infinity, and ε = (γ − 1)/(γ + 1), where γ is the ratio of the specific heats, and δ = 1/(γ − 1) M2 go to zero.

The appearance of a component of vorticity in the direction of flow materially alters the pattern of flow of a fluid in three dimensions. Expressions are obtained for this secondary vorticity in an inviscid compressible fluid flowing under the action of body forces. They are applied to examples such as a liquid under gravity and gas flow behind a curved shock. In compressible gas flow with varying temperature but constant stagnation pressure no secondary circulation appears. In a perfect gas atmosphere it is shown that secondary circulation may appear because of nou-adiabatic lapse rates as well as wind-speed gradients. It is also shown that in a liquid with a density gradient, gravitational effects can give rise to secondary vorticity components.

An analysis of the exact equations of the inviscid flow of a perfect gas over cusped concave bodies is described. The field is examined in the limit of infinite free-stream Mach numberM∞. The slope of the shock wave in a small region adjacent to the leading edge is strongly dependent onM∞, while much further downstream the shock-wave slope is controlled primarily by the body slope. Consequently the region near the leading edge introduces into the field downstream a thin layer of gas, adjacent to the body, where the entropy is much lower than that of the gas above it. This layer is so dense that the gas velocity along it is not appreciably slowed by the pressure gradient along the body. However, it is so thin that there is little pressure change across it.The well-known self-similar solutions to the hypersonic small-disturbance equations have previously only been used to study the flow on blunted slender convex surfaces. They are known to behave singularly at the body. It is shown that there is a region on concave power-law shapes where the self-similar solutions are the correct first approximation to the exact inviscid equations in the limitM∞→ ∞; and that, further, they predict the correct first-order surface pressure.Numerical results for surface pressure from the similar solutions are presented, and comparisons are made with certain approximate theories available for more general shapes. Pressure measurements taken on a cubic surface in the Imperial College gun tunnel are presented and compared with the theoretical distributions.

In a previous paper charts were given which related the flow Mach numbers on either side of a wave to a dimensionless wave speed. Charts were given for shock waves, isentropic expansion waves and isentropic non-steep pressure waves in a perfect gas. In this note it is shown that the lines of constant state ratio plotted on the charts constitute families of straight lines; this fact is particularly important in the case of the shock wave chart. The wave domains are also established and compared diagrammatically.

SummaryThe paper deals with a class of shock-wave systems that are of importance in the design and operation of supersonic intakes. It is shown that, when n− 1 wedge shocks, all of the same family, meet at a point, together with a tail shock and a bow shock, then the equations of motion which describe the flow near the shock confluence or junction are reducible to a polynomial equation of degree 10. Numerical results are obtained for the special case of the four-shock confluence and it is shown that, the number of solutions of physical interest, m4, may be 0, 2 or 4. It is found that the m4=0 solution can be interpreted physically in three different ways. These are a Guderley type flow, a confluence of three shocks with a Prandtl-Meyer expansion fan or the appearance of a shock system containing more than one confluence point. The methods developed also permit the discussion of multi-confluence wave systems and a number of examples are given. Finally, the Prandtl-Meyer compression fan is brought within the scope of the method by approximating it with a system of shock waves of weak intensity and minimum entropy.

An analytical description for the initial phases of collapse of a spherical or cylindrical shock wave in a perfect gas is given in the present paper. The shock wave is initiated by the instantaneous and uniform deposition of a finite quantity of energy per unit surface area at a finite radiusR0. For the initial shock motion wherexs= (Rs-R0)/R0is small, analytical solutions are obtained by a power-series expansion of the dependent variables inxs. The classical self-similar solution for a strong planar blast wave is recovered as the present zero-order solution. Non-similar effects arising from both finite shock strengths and the presence of a characteristic lengthR0are accounted for simultaneously in the present perturbation scheme. The analysis is carried out up to third order inxs. For very large values of the initiation energy where the shock wave remains strong throughout its collapse, it is found that the present perturbation solution can adequately describe a significant portion of the collapse processes. The solution indicates that the shock decays rather rapidly initially and later begins to accelerate as a result of the additional adiabatic compression of the shocked states due to flow-area convergence. However, for weak initiation where the energy released is small, the present perturbation solution is an asymptotic series and diverges very rapidly as the shock propagates away from the wall. The range of validity then is limited to very small values ofxs.

SummaryThe paper contains an analysis of the flow of a perfect gas with constant specific heats through a rotating channel of constant cross-sectional area, as used in certain helicopter propulsion systems and wind-driven gas turbines. The analysis is restricted to the adiabatic one-dimensional treatment, the Coriolis accelerations acting across a section being disregarded.The equations of motion and energy are deduced and, together with the equation of continuity, reduced to an ordinary non-linear differential equation of the first order, involving a dimensionless form of velocity and distance.The patterns of the integral curves of the differential equation are discussed and sketched by examining their asymptotic behaviour as well as that in the neighbourhood of singular points. It is shown that there exists a critical value of the angular velocity below which the flow remains subsonic in the pipe, if the entrance velocity is subsonic; it may, however, become sonic at the exit of the pipe. For supercritical angular velocities the flow may become sonic or supersonic in the pipe if the entrance velocity attains a given “ correct” but subsonic value. A method of examining for the possibility of shock formation is indicated.The initial conditions for the differential equation are deduced for the design and for the performance problem, two new flow functions being introduced and tabulated to facilitate practical calculations. Formulae are also deduced for the calculation of the pressure and Mach number variation from the previously calculated velocity variation along the pipe.Finally an approximate solution in closed terms is given for the case of small entrance velocities.

For investigating the steady irrotational isentropic flow of a perfect gas in two dimensions, the hodograph method is to determine in the first instance the position coordinates x, y and the stream function ψ as functions of velocity compoments, conveniently taken as q (the speed) and θ (direction angle). Inversion then gives ψ, q, θ as functions of x, y. The method has the great advantage that its field equations are linear, so that it is practicable to obtain exact solutions, and from any two solutions an infinity of others are obtainable by superposition. For problems of flow past fixed boundaries the linearity of the field equations is usually offset by non-linearity in the boundary conditions, but this objection does not arise in problems of transsonic nozzle design, where the rigid boundary is the end-point of the investigation.