Downstream Mach Number Research Articles

Theoretical and numerical investigation of wedge and cone nose profiles at supersonic speed

This research endeavor seeks to analyze the nose shapes of a wedge and a cone through simulations conducted with the ANSYS-FLUENT software. When designing missiles and aerodynamic vehicles for high speed, the selection of nose shape is thought to be the most crucial. The aerodynamic bodies traveling at supersonic speeds experience a pronounced influence of fluid compressibility, wherein high-speed fluid particles impart their kinetic energy as heat upon interacting with the frontal surface. The primary difficulty is that missiles and aerodynamic vehicles with high aerodynamic loads depend heavily on the shape of their noses. Therefore, the purpose of this research work is to compare the shape of the wedge and the conical nose at Mach number 2.0. A numerical simulation is used to estimate aerodynamic properties such as downstream Mach number, shock wave angle, pressure ratio, density ratio, temperature ratio, and aerodynamic coefficient, and the results are compared with theoretical results, which are obtained to solve oblique shock relation and Taylor–Maccoll equation. The conical-shaped nose model exhibits superior aerodynamic characteristics compared to the wedged-shaped nose.

On the stability of piston-driven planar shocks

We present a theoretical and numerical stability analysis for a piston-driven planar shock against two-dimensional perturbations. The results agree with the well-established theory for isolated planar shocks: in the range of $h_c< h<1+2 {\mathcal {M}}_2$ , where $h$ is the D'yakov–Kontorovich (DK) parameter related to the slope of the Rankine–Hugoniot curve, $h_c$ is its critical value corresponding to the onset of the spontaneous acoustic emission (SAE) and ${\mathcal {M}}_2$ is the downstream Mach number, non-decaying oscillations of shock-front ripples occur. The effect of the piston is manifested in the presence of additional frequencies occurring by the reflection of the sonic waves on the piston surface that can reach the shock. An unstable behaviour of the shock perturbation is found to be possible when there is an external excitation source affecting the shock, whose frequency coincides with the self-induced oscillation frequency in the SAE regime, thereby being limited to the range $h_c< h<1+2 {\mathcal {M}}_2$ . An unstable evolution of the shock is also observed for planar shocks restricted to one-dimensional perturbations within the range $1< h<1+2 {\mathcal {M}}_2$ . Both numerical integration of the Euler equations via the method of characteristics and theoretical analysis via Laplace transform are employed to cross-validate the results.

Determination of the optimum nose cone geometrical shape for supersonic missile

The selection of the nose shape is considered to be one of the most crucial aspects in designing the missile as the nose cones are designed to travel at different speeds and altitude. The main challenge is that the nose shape being the foremost part of the missile, experiences high aerodynamic loads which must be resisted by the compressible motion of the fluid for optimum performance at supersonic speeds. The research aims to compare the Conical and Ogival nose cone shapes of the supersonic missile at Mach 2.0. The work includes flow characterization of these nose shapes in Ansys Fluent and for the flow being near the wall, K-omega (2 Eq.) Standard model is chosen. The aerodynamic properties such as the downstream Mach number, wave angle, pressure ratio, density ratio, temperature ratio and the aerodynamic coefficients were estimated computationally and were simultaneously compared theoretically with the oblique shock theory. The results obtained were validated within a variation of 10 percent. The conclusions were drawn and the ogival nose shape has the minimum aerodynamic resistance.

Stability of expanding accretion shocks for an arbitrary equation of state

We present a theoretical stability analysis for an expanding accretion shock that does not involve a rarefaction wave behind it. The dispersion equation that determines the eigenvalues of the problem and the explicit formulae for the corresponding eigenfunction profiles are presented for an arbitrary equation of state and finite-strength shocks. For spherically and cylindrically expanding steady shock waves, we demonstrate the possibility of instability in a literal sense, a power-law growth of shock-front perturbations with time, in the range of$h_c< h<1+2 {\mathcal {M}}_2$, where$h$is the D'yakov-Kontorovich parameter,$h_c$is its critical value corresponding to the onset of the instability and${\mathcal {M}}_2$is the downstream Mach number. Shock divergence is a stabilizing factor and, therefore, instability is found for high angular mode numbers. As the parameter$h$increases from$h_c$to$1+2 {\mathcal {M}}_2$, the instability power index grows from zero to infinity. This result contrasts with the classic theory applicable to planar isolated shocks, which predicts spontaneous acoustic emission associated with constant-amplitude oscillations of the perturbed shock in the range$h_c< h<1+2 {\mathcal {M}}_2$. Examples are given for three different equations of state: ideal gas, van der Waals gas and three-terms constitutive equation for simple metals.

Numerical Investigation of Influence of Entropy Wave on the Acoustic and Wall Heat Transfer Characteristics of a High-Pressure Turbine Guide Vane

As an indirect noise source generated in the combustion chamber, entropy waves are widely prevalent in modern gas turbines and aero-engines. In the present work, the influence of entropy waves on the downstream flow field of a turbine guide vane is investigated. The work is mainly based on a well-known experimental configuration called LS89. Two different turbulence models are used in the simulations which are the standard k-ω model and the scale-adaptive simulation (SAS) model. In order to handle the potential transition issue, Menter’s ð-Reθ transition model is coupled with both models. The baseline cases are first simulated with the two different turbulence models without any incoming perturbation. Then one forced case with an entropy wave train set at the turbine inlet at a given frequency and amplitude is simulated. Results show that the downstream maximum Mach number is rising from 0.98 to 1.16, because the entropy waves increase the local temperature of the flow field; also, the torque of the vane varies as the entropy waves go through, the magnitude of the oscillation is 7% of the unforced case. For the wall (both suction and pressure side of the vane) heat transfer, the entropy waves make the maximum heat transfer coefficient nearly twice as the large at the leading edge, while the minimum heat transfer coefficient stays at a low level. As for the averaged normalized heat transfer coefficient, a maximum difference of 30% appears between the baseline case and the forced case. Besides, during the transmission process of entropy waves, the local pressure fluctuates with the wake vortex shedding. The oscillation magnitude of the pressure wave at the throat is found to be enhanced due to the inlet entropy wave by applying the dynamic mode decomposition (DMD) method. Moreover, the transmission coefficient of the entropy waves, and the reflection and transmission coefficients of acoustic waves are calculated.

Optimization of the LS89 Axial Turbine Profile Using a CAD and Adjoint Based Approach †

The LS89 high pressure axial turbine vane was originally designed and optimized for a downstream isentropic Mach number of 0.9. This profile has been widely used for computational fluid dynamics (CFD) validation in the open literature but very few attempts have been made to improve the already optimized design. This paper presents a sound methodology to design and optimize the LS89 using computer-aided design (CAD) at design conditions. The novelty of the study resides in the parametrization of design space, which is done at the CAD level, and the detailed analysis of the aerodynamic performance of the optimized design. Higher level constraints are imposed on the shape, such as the trailing edge thickness, the axial chord length, and G2 geometric continuity between the suction side and pressure side at the leading edge. The gradients used for the optimization are obtained by applying algorithmic differentiation to the CAD kernel and grid generator and the discrete adjoint method to the CFD solver. A reduction of almost 12% entropy generation is achieved, which is equivalent to a 16% total pressure loss reduction. The entropy generation is reduced while keeping the exit flow angle as a flow constraint, which is enforced via the penalty formulation. The resulting unconstrained optimization problem is solved by the L-BFGS-B algorithm. The flow is governed by the Reynolds-averaged Navier-Stokes equations and the one-equation transport Spalart-Allmaras turbulence model. The optimal profile is compared and benchmarked against the baseline case.

Onset conditions for vortex breakdown in supersonic flows

This study proposes an onset condition of shock-free supersonic vortex breakdown from the axial momentum variation, which applies in the presence or absence of a stagnation point. The condition is derived from a comprehensive approach to vortex breakdown. Supersonic breakdown appeared when the swirl parameter and Mach number were small. Moreover, bubble-type breakdowns with a stagnation point, which occur in subsonic conditions, could not occur under the supersonic condition in the present analysis. The predicted breakdowns under this condition were consistent with the results of the three-dimensional numerical simulations for Mach numbers ranging from 1.5 to 5.0. Supersonic vortex breakdowns were clearly captured by the helicity contours in the numerical results. The threshold of the downstream Mach number required for spiral breakdown with no stagnation point was also theoretically derived and verified in numerical results. These findings provide new insights into vortex breakdown in supersonic flows.

On the RH relations across weak and strong shocks in van der Waals gases

The Rankine–Hugoniot (RH) jump relations for normal shock waves in van der Waals fluids have been studied in order to improve a theoretical understanding of those shock related phenomena as observed in a real atmosphere which cannot be accounted for by the ideal gas model. The RH jump relations for the pressure, density, particle velocity, temperature, speed of sound, adiabatic compressibility and change-in-entropy across the shock front have been analysed in terms of the non-idealness parameter of the gas, downstream Mach number and adiabatic index of the gas. Further, as the strength of shock waves may range from weak to strong, the convenient forms of RH jump relations for weak and strong shock waves have been discussed, simultaneously. Finally, the effects on the flow-field behind the shock front have been explored due to the non-idealness parameter of the gas, downstream Mach number and adiabatic index of the gas.

Prediction of the unsteady turbine trailing edge wake flow characteristics and comparison with experimental data

The trailing edge base pressure is known to play a significant role in the profile losses of turbine blades in particular at transonic outlet Mach numbers. Recent tests have shown that the assumption of an isobaric near wake region is wrong because of the highly unsteady character of the near wake flow which affects directly the trailing edge base pressure. New experimental data show the evolution of the trailing edge base pressure in function of the downstream Mach number from a uniform base pressure at moderate subsonic Mach numbers to an increasingly strong non-uniform distribution up into the transonic range followed by a sudden return to an isobaric base trailing edge pressure. To get more insight into the unsteady flow features, numerical simulations are carried out. A LES approach is validated and employed in this work to analyze three different operating conditions for the profile, representing three configurations of the flow and the trailing edge shock system.

Effects of Circumferential Casing Grooves on the Performance of a Transonic Axial Compressor

AbstractThe casing groove is a highly effective method for improving the stall margin with the least detrimental effect on the peak efficiency in an axial compressor. In this work, a single casing groove with different heights and positions was numerically tested to evaluate effects of each geometric parameter on the stall margin in a transonic compressor. Validation between the simulation and experiment was conducted with a smooth casing as a reference case, and the computed and experimental results were compared in terms of performance curves, downstream flow properties and Mach number contours. Subsequently, the performance curves for each test case with a single casing groove were obtained from the numerical results and compared with each other to determine the appropriate position and height of the groove for increasing the stall margin. A casing groove installed near the leading edge was found to be effective for expanding the stable operating range in a transonic compressor, giving about 3% point improvement in the stall margin at the cost of a small drop in efficiency at the design point. An attempt was also made to understand the underlying mechanism of the casing groove based on the analysis of the numerical flow data.

One-dimensional analysis of thermal choking in case of heat addition in ducts

The thermal choking phenomenon is of great importance in an inlet isolator in dual-mode ram jet/scramjet combustor. In some cases the choked flow creates a pseudo-shock wave including a shock train in it at the engine inlet and causes large amounts of drag and radically reduces the performance of the engine at high flight Mach numbers. The present paper describes a one-dimensional flow model taking account of the upstream boundary-layer as well as heat addition by using a mass-weighted averaging technique. The simple relationships for the flow field in a constant area duct in which the effect of the upstream boundary-layer is considered but the effect of the wall friction in the duct can be neglected are presented. The results of the calculation such as the maximum heat addition when the thermal choking occurs, the downstream Mach number and the static pressure ratio are presented and examined in detail.

Rankine–Hugoniot analysis of two-phase flow with inter-phase slip

This paper is one in a series of papers considering different characteristics of two-phase flow. The previous analyses were conducted to determine the momentum flux and the nozzle design for a two-phase supersonic cleanser, where the focus of this paper is on the general gas dynamic relationships of the two-phase mixture across a normal shock wave. Historically, normal shock analyses have provided closed form solutions for the downstream state in terms of the upstream state for perfect gases, i.e. the Rankine–Hugoniot analysis. This analysis examines the effect of the mass injection ratio and the inter-phase slip for a homogeneous, two-phase mixture by applying a control volume approach from the state immediately preceding the shock wave to the state immediately after the shock wave where the liquid phase has not had time to react, and from the state immediately after the shock wave to a state where the gas and liquid phases have had sufficient time to become re-equilibrated. The results show that the downstream Mach number decreases while the ratios of pressure, density, and temperature increase for increases in the mass injection ratio. The same trend is also shown for increases in the slip parameter. Whereas the previous analyses applied mainly to the characteristics of the industrial cleanser mentioned before, this analysis has far reaching implications ranging from two-phase particulate flow in solid rocket motors to sand blasting applications.

Acoustically Induced Structural Fatigue of Piping Systems

Piping systems handling high-pressure and high-velocity steam and various process and hydrocarbon gases through a pressure-reducing device can produce severe acoustic vibration and metal fatigue in the system. It has been previously shown that the acoustic fatigue of the piping system is governed by the relationship between fluid pressure drop and downstream Mach number, and the dimensionless pipe diameter/wall thickness geometry parameter. In this paper, the devised relationship is extended to cover acoustic fatigue considerations of medium and smaller-diameter piping systems.

The Influence of Endwall Contouring on the Performance of a Turbine Nozzle Guide Vane

The paper presents the results of a detailed investigation of the flow field in a gas turbine linear cascade. A comparison between a contoured and a planar configuration of the same cascade has been performed, and differences in the three-dimensional flow field are here analyzed and discussed. The flow evolution downstream of the trailing edge was surveyed by means of probe traversing while a three-dimensional Navier–Stokes solver was employed to obtain information on flow structures inside the vaned passages. The experimental measurements and the numerical simulation of the three-dimensional flow field have been performed for two cascades; one with planar endwalls, and the other with one planar and one profiled endwall, so as to present a reduction of the nozzle height. The investigation was carried out at an isentropic downstream Mach number of 0.6. Airfoils of both cascades were scaled from the same high-pressure gas turbine inlet guide vane. Measurements of the three-dimensional flow field have been performed on five planes downstream of the cascades by means of a miniaturized five-hole pressure probe. The presence of endwall contouring strongly influences the secondary effects; the vortex generation and their development are inhibited by the stronger acceleration taking place throughout the cascade. The results show that the secondary effects on the contoured side of the passage are confined in the endwall region, while on the flat side the secondary vortices display characteristics similar to the ones occurring downstream of the planar cascade. The spanwise outlet angle distribution presents a linear variation for most of the nozzle height, with quite low values approaching the contoured endwall. The analysis of mass-averaged losses shows a significant performance improvement in the contoured cascade. This can be ascribed not only to lower secondary losses but also to a reduction of the profile losses.

Piping system providing minimal acoustically-induced structural vibration and fatigue

Piping systems adapted for handling fluids such as steam and various process and hydrocarbon gases through a pressure-reducing device at high pressure and velocity conditions can produce severe acoustic vibration and metal fatigue in the system. It has been determined that such vibrations and fatigue are minimized by relating the acoustic power level (PWL) generated in the piping system to being a function of the ratio of downstream pipe inside diameter D 2 to its wall thickness t 2 . Additionally, such vibration and metal fatigue can be further minimized by relating the fluid pressure drop Δp and downstream Mach number M 2 to being a function of the ratio of downstream piping inside diameter D 2 to the pipe wall thickness t 2 , as expressed by M 2 Δp=f (D 2 /t 2 ). Pressure-reducing piping systems designed according to these criteria exhibit minimal vibrations and metal fatigue failures and have long operating life.

Investigation of aerodynamic effects of coolant ejection at the trailing edge of a turbine blade model by PIV and pressure measurements

In order to simulate the thick trailing edges of turbine blades a slotted plate profile together with a newly designed nozzle was installed into the high-speed wind tunnel of the DLR Gottingen. At different supersonic Mach numbers and at four coolant flow rates in the range of 0–2.5% pressure distribution measurements and probe measurements were performed. The flow field was visualized by schlieren photos and the instantaneous velocity field was quantitatively investigated by Particle Image Velocimetry (PIV). The measurements of the velocity field gave an insight into stationary effects, for example the change of shock strength with coolant flow rate, and instationary effects such as the existence of a vortex street in the wake. The PIV technique offers special advantages for the investigation of transonic flow fields, but also yields to special experimental difficulties, which are also described in this article. Measured losses display a maximum at the downstream Mach number 1. This is strongly related to the behaviour of the base pressure. A loss minimum is achieved at moderate coolant flow rates, showing that an optimum coolant flow rate exists. The loss was analysed and separated into the loss contributions from the profile upstream of the trailing edge and the mixing loss due to the coolant flow.

Designing Piping Systems Against Acoustically Induced Structural Fatigue

Piping systems adapted for handling fluids such as steam and various process and hydrocarbon gases through a pressure-reducing device at high pressure and velocity conditions can produce severe acoustic vibration and metal fatigue in the system. It has been determined that such vibrations and fatigue are minimized by relating the acoustic power level (PWL) to being a function of the ratio of downstream pipe inside diameter D2 to its thickness t2. Additionally, such vibration and fatigue can be further minimized by relating the fluid pressure drop and downstream Mach number to a function of the ratio of downstream piping inside diameter to the pipe wall thickness, as expressed by M2 Δp = f(D2/t2). Pressure-reducing piping systems designed according to these criteria exhibit minimal vibrations and metal fatigue failures and have long operating life.

One-Dimensional Flow Model of a Pseudo-Shock Wave in a Constant-Area Duct.

One-dimensional flow model for a pseudo-shock wave in a constant-area duct is presented where the boundary layer at the initial section of the pseudo-shock region is considered. The conservation equations of mass, momentum and energy across the pseudo-shock are treated one-dimensionally by assuming an appropriate mean flow over a given cross section of the flow. For given conditions upstream of the pseudo-shock, the characteristic properties through the pseudo-shock such as the downstream Mach number and the static and stagnation pressure ratios across the pseudo-shock calculated using the present model agree well with the previous experimental results.

Oblique shock wave with sweep

An attached planar, oblique shock with sweep is investigated for the inviscid flow of a perfect gas. The ratio of specific heats, freestream Mach number, and wedge angle in the plane of the freestream velocity are prescribed, with the sweep angle as a free parameter. Explicit relations are provided for jump and detachment conditions. A number of trends, some non-intuitive, are discussed, e.g., the downstream Mach number may increase with sweep.

Plane steady shock waves in Isreal-Stewart fluids

The plane steady shock wave solutions of the Israel-Stewart theory of relativistic dissipative fluids are studied. The Israel-Stewart theory is shown to fail to adequately describe the structure of strong shock waves. Physically acceptable solutions do not exist above a maximum upstream Mach number in any thermally nonconducting, viscous fluid described by the Israel-Stewart theory because the solutions become multiple-valued when the characteristic velocity is exceeded. It is also proven that physically acceptable solutions do not exist for thermally conducting, viscous fluids above either a maximum upstream Mach number, or else below a minimum downstream Mach number (or both). These limiting Mach numbers again correspond to the characteristic velocities of the fluid. Only extremely weak plane steady continuous shock solutions exist in the Israel-Stewart theory for the ultrarelativistic Boltzmann gas, or for the degenerate free Fermi gas.