Shock-free Flow Research Articles

Parametric Design Tool for Development of a Radial Guide Vane Cascade for a Variable Speed Francis Turbine

Hydropower as a part of the family of renewable energy sources represents an engineering and scientific field which inspires researchers to work on development of the systems and sub-systems in a way of optimizing the whole energy transformation process to obtain more efficient, flexible and reliable hydropower operation with the best possible water to energy ratio. This research is part of a Horizon 2020 HydroFlex project by the Norwegian University of Science and Technology (NTNU), where the main goal is development of a flexible hydropower generation. The guide vane cascade is one of the most crucial stationary sub-systems of the hydraulic turbine and is a subject of this study. Its re-design for obtaining a quality “flow-feeding” of a variable speed high head Francis turbine is developed. Having this goal in mind, a MATLAB code was generated, based on several key parameters, such as initial energy conditions as net head and turbine discharge at best efficiency point (BEP). Turbine runner geometrical constraints are taken into account during this process, while using recommendations for some initial guide vane calculations such as their number, inlet and outlet diameter, guide vane axis diameter, delivery angles etc. Using an inverse Euler turbine equation, the operating range of the turbine was calculated for a variable speed and discharge conditions, keeping the shock-free flow for all states at the runner’s inlet, as it is the most favourable inflow condition. For those operating points, the flow streamlines angles were obtained at the guide vanes leading and trailing edges. With an interpolating mathematical functions between the angles of the leading and trailing edges, the camber lines of the hydrofoils were obtained for further guide vane cascade geometry development. This algorithm can be implemented on any given runner geometry. The guide vane design is then exported into ANSYS Workbench for further numerical tests, such as CFD simulations for verifying the hydrodynamic characteristics and FEM analysis for verifying the structural integrity of this sub-system for variable speed operating conditions.

2D numerical modeling for plasma-assisted CO2 pooling in supersonic nozzles: importance of a proper nozzle contour design

Plasma assisted CO2 dissociation has recently been the topic of many studies. The production of chemical fuels from environmentally unfriendly CO2 through supersonic nozzles is one of the most prominent approaches under investigation. However, the experiments show that the theoretical conversion rates are far away from being achieved. In this study, two-dimensional fully coupled cases are investigated. This brings insights about how the CO2 dissociation can be improved; mainly by a correct design of nozzles. The proper shape of a nozzle is a fundamental aspect to be taken into account numerically and in experiments to avoid undesirable phenomena such as the occurrence of shocks. The proper design of the nozzle shape leads to a shock-free flow being uniform at the nozzle outlet. A high degree of cooling can be achieved in a shock-free nozzle. Moreover it is shown that there is no optimal value for Mach number provided that the nozzles are sized properly. If the sizing is done correctly, it is found that higher Mach numbers lead to higher degree of non-equilibrium and thereby to higher dissociation rate. The sizing of the nozzle to maximize the departure from equilibrium in the nozzle is the final key of an efficient CO2 dissociation. Finally, the results are compared with those of a semi-analytical method to conclude that if the nozzle is conceived in a proper way, simpler approaches can also give satisfactory results.

Adaptive Control of Shock Waves with a Passively Morphing Layer for Rotating Blades

Helicopter blades are known to experience complicated unsteady flow conditions during high-speed forward flight. Transonic shock takes place along the advancing side, which decreases the performance of the blades and results in impulsive noise. Adaptive airfoils seem promising for increasing performance at considerably variable flow conditions. In this paper, a blade with a passively morphing layer is presented for transonic shock/boundary layer control at forward flight conditions. The morphing layer embedded in the upper surface of the blade can form an adaptive bump, which in turn can control transonic shock/boundary layer interaction. A fluid-structure-interaction (FSI) enabled simulator is employed to study the performance of the morphing layer at two different forward conditions. The results demonstrate that the adaptive bump formed by the morphing layer could efficiently weaken the strong normal shock over the layer by producing a λ-structural shock. Moreover, the morphing layer retreats to its undeformed state to prevent detrimental effects at weaker shock or shock-free flow. To further increase the performance of the morphing layer, a morphing layer that is inclined with respect to the spanwise direction is used to replace the layer that lies parallel to the spanwise direction. The results indicate that the inclined morphing layer performs better while the shock position takes on a less uniform distribution.

Flow Field in the Vicinity of a High-Pressure-Drop Ball Valve

The testing of gas turbine combustors requires large flow rates at high pressures and elevated temperatures. To control the flow and pressure inside the combustor, some type of control valve is required in the exhaust section of the testing system. This back-pressure valve is exposed to severe operating conditions. To understand the complex flow features in the exhaust section and provide relevant information for selecting suitable low-cost valves for new large test cells, a numerical study was carried out on a back-pressure valve that has been used in a number of testing programs. The flow fields in the vicinity of the valve and the piping sections ending at the exhaust stack were resolved for three practical operating conditions. The results indicate that because of the presence of the valve with a V-shaped opening, the flow field behind experiences a series of three-dimensional expansion and shock waves. Strong interactions exist between the flow behind the blockage and the flow passing through the opening area. More importantly, it is found that most of the pressure drop occurs immediately downstream of the valve, and its values are much larger than those provided by suppliers of valves based on shock-free flow calculation. This may explain why the valve lost its ability to function during testing and its stainless-steel seat had to be removed to maintain its ability to rotate. Based on this study, it is recommended that a second pressure-drop element be installed in the exhaust section to keep the expected lifetime of the valves and reduce the noise levels. This suggestion has been implemented in the new large test cell at the Gas Turbine Laboratory.

Problem of designing an aerodynamic profile with assigned characteristics of electromagnetic scattering

At the profile trailing edge, the Joukowski–Chaplygin condition is posed that expresses the finiteness of the velocity at the flow trailing point. The inverse problem for the reconstruction of the unknown profile-surface segment S0 ⊆ S is considered in the following formulation. It is necessary to find a set of values ri, i = 1, 2, ..., L, which parametrically determine the desired surface segment S0 and provide: (i) approximation (at a reasonable level of accuracy) to the prescribed scattering diagram; (ii) shock-free incoming flow around the profile (streamline flow around the profile leading edge). We now assume that the scattering diagram is given by its complex values at a finite number of far-field points: ea(φ), φ ∈ {φj, j = 1, 2, ..., m}; φ is the polar angle; 2m = L – 1 [for H-polarization, the function ha(φ) is given]. The criterion of the approximation to the given scattering diagram (in the mean-square sense) is

A new type of shock-free axisymmetric supersonic flow

A new type of shock-free axisymmetric supersonic flow

Control of Shock Loading from a Jet in a Flexible Structure's Presence

The control of shock noise or screech from a jet near a flexible structure is discussed. The pressure from the supersonic jet consists of a shock with spiral and flapping nonaxisymmetric modes superimposed on broadband response. This shock induces a nonlinear-nonstationary loading problem associated with acoustic wave generation and propagation coupled with structural vibration. Control of the shock is achieved by placing a ring at the nozzle lip oscillating at the shock fundamental frequency. The ring prevents the shock characteristics originating in the column of the shear layer from sustaining connection with the out-of-phase surface vibration. Shock-free flow is maintained over a large pressure ratio. The peak power spectral density of the pressure due to shock fundamental and harmonics is converted to broadband level. The experiment is motivated by consideration of aircraft structural fatigue and interior noise.

Long Series in Mechanics: Janzen–Rayleigh Expansion for a Circle

We reconsider the transonic controversy – whether a given airfoil can exhibit a continuous range of shock-free supersonic flow. We employ the circle as our ‘airfoil’, and examine the Janzen–Rayleigh expansion in even powers of the free-stream Mach number, which we previously carried to 29 terms. In addition to expressing doubt about our previous conclusions, we introduce a promising modification of the graphical ratio test of Domb and Sykes.

Formation and stability of the galactic shock waves

Abstract The stability of the shocked and the shock-free gas flows in the gaseous galactic disc is discussed. It is shown that the periodic shock-free supersonic flow between the spiral gravitational potential wells is unstable due to the parametric instability and thus should be re-organized into the shocked flow. The steady-state shock at the rear side of the well (relative to the flow) is subject to the local instability. This instability is suppressed if the shock is at the front side of the well.

Steady compressible vortex flows: the hollow-core vortex array

We examine the effects of compressiblity on the structure of a single row of hollowcore, constant-pressure vortices. The problem is formulated and solved in the hodograph plane. The transformation from the physical plane to the hodograph plane results in a linear problem that is solved numerically. The numerical solution is checked via a Rayleigh-Janzen expansion. It is observed that for an appropriate choice of the parameters M∞ = q∞/c∞, and the speed ratio, a = q∞/qv, where qv is the speed on the vortex boundary, transonic shock-free flow exists. Also, for a given fixed speed ratio, a, the vortices shrink in size and get closer as the Mach number at infinity, M∞, is increased. In the limit of an evacuated vortex core, we find that all such solutions exhibit cuspidal behaviour corresponding to the onset of limit lines.

Transonic nozzle flow of dense gases

The paper deals with the flow properties of dense gases in the throat area of slender nozzles. Starting from the Navier–Stokes equations supplemented with realistic equations of state for gases which have relatively large specific heats a novel form of the viscous transonic small-perturbation equation is derived. Evaluation of the inviscid limit of this equation shows that three sonic points rather than a single sonic point may occur during isentropic expansion of such media, in contrast to the case of perfect gases. As a consequence, a shock-free transition from subsonic to supersonic speeds cannot, in general, be achieved by means of a conventional converging–diverging nozzle. Nozzles leading to shock-free flow fields must have an unusual shape consisting of two throats and an intervening antithroat. Additional new results include the computation of the internal thermoviscous structure of weak shock waves and a phenomenon referred to as impending shock splitting. Finally, the relevance of these results to the description of external transonic flows is discussed briefly.

Approximate solution of inverse shock-free transonic airfoil problem at zero incidence

The present work extends the approximate analytical solution of Niyogi [9] for shock-free transonic flow past thin symmetric airfoils at zero incidence, to the inverse problem for which the velocity distribution is prescribed and the airfoil shape is to be determined. The first step in the solution of the inverse problem leads to a singular integral equation of Betz's type solution of which has been improved iteratively using the transonic integral equation of Oswatitsch and satisfying the tangency boundary condition on the profile contour. Computational results have been presented for Nieuwland airfoils and a NACA 0012 airfoil, all of which indicate very good agreement with exact solutions.

Numerical solution of singular integral equations in a class of singular functions and the problem of blowing of the flow in aerodynamics

A numerical method for solving one of the forms of singular integral equations in a class of singular functions is presented and proved. This method enables one to determine the aerodynamic characteristics of a wing with blowing of the flow of an ideal fluid under conditions of circulating, non-circulating, and shock-free flow around the wing.

A Mixed-Flow Cascade Passage Design Procedure Based on a Power Series Expansion

Flow passage design and analysis procedures employing a TAYLOR series expansion across the flow field were first suggested as far back as 1908. With very few exceptions the early work addressed only the problem of symmetrical convergent-divergent passages. A generalization and an extension of the approach permits it to be used for the design of arbitrary cascade passages in subsonic, supersonic, or mixed-flow regimes. The method will give fast, realistic solutions in situations where shock-free flow is meaningful, and where the cascade gap-chord ratio is not excessive. It assumes inviscid, irrotational flow and employs a fourth-order TAYLOR series expansion across the cascade passage. Numerous examples are given, with comparisons with experimental data and other analytic solutions.

Spherical nonlinear wave propagation in a vibrationally relaxing gas

The shock-free flow field emanating from a pulsating sphere in a vibrationally relaxing gas without viscosity and heat conduction is obtained by numerical integration of the exact equations. The method is applicable to air when models with either one or two vibrational modes are used. The results highlight the various effects of nonlinearity, attenuation, dispersion, spherical spreading, and source compactness. The special case of isentropic flow is first investigated in some detail. This gives information about the effects of nonlinear harmonic growth and of sphere compactness. Approximate solutions are found to be useful in explaining some of the effects observed. These solutions are reviewed, improved where possible, and compared with the numerical solutions. These appear to be particularly useful because they give details of the development of the wave profile whereas the approximate theories are essentially concerned with the changes in amplitude with distance.

Example of solving transonic equations for a shock-free flow past a symmetric profile

A parametric method /1/ of solving the transonic Kármán—Fal'kovich equations is developed. The nozzle solution is generalized to the case of the flows not symmetric about the longitudinal axis of the nozzle. A procedure of passing from this solution to the case of a flow past a profile discussed in /2/ is then shown. This in fact means that the real and imaginary part of the complex function describing this flow have been obtained. The resulting solution depends on three constants determining the dimensions of the profile (length of chord and the maximum thickness) and also the flow rate at infinity. Numerical analysis is used to obtain the condition for the flow to be shock-free, and a continuous velocity field is constructed under the conditions close to the limiting state. Setting up a flow chart for the cases when the condition of no shock is violated shows that a three-sheeted fold appears, the top of which lies within the supersonic region. This confirms the conclusion made in /3–5/ that in a typical case of a flow past a profile, the shock wave originates not at the sonic stream line, but within the zone. The example constructed can be used as the basis for the theory of flow past a profile of a sufficiently general form, of a gas stream subsonic at infinity. Below the transonic Tricomi model is used to show the corresponding generalization.

Shock-Free Wing Design

A simple numerical method for generating wing shapes that will be shock free at a specified Mach number is described. The method involves using a fictitious gas law for the supersonic domain to make the governing equations elliptic. Requirements on this gas law are detailed and a method for computing the real flow in the supersonic domain, given initial data on the embedded sonic surface, is described. The failure of the method to yield a shock-free flow when a limit surface occurs in the supersonic flow, and the difficulties that arise because the initial-value problem for the supersonic domain is ill-posed, are delineated. Finally, a small perturbation algorithm is used to illustrate the procedure and results are given for a simple baseline wing. NCREASED fuel efficiency, and in the case of commercial aircraft, productivity, can be achieved by operating aircraft at Mach numbers, provided that shock waves can be avoided or made acceptably weak. Two-dimensional procedures for prescribing airfoil sections that are shock free have already provided improvements in aircraft efficiency by employing these airfoils on swept wings. Three-dimensional effects have compromised such designs to some extent, and extensive wind tunnel development tests have been required to recapture the benefits of these supercritical airfoils. Sobieczky et al. l demonstrated a method of modifying baseline configurations so that they would be shock free at a prescribed Mach number and lift coefficient. This procedure provides a special opportunity for improving aircraft per- formance through a careful selection of the baseline con- figuration in order to provide wings and wing-body com- binations that are shock free at Mach numbers, and that have acceptable off-design performance. Yu2 and Yu and Rubbert3 have also documented that this procedure is possible and demonstrated its application. As was first described by Sobieczky,4 a numerical algorithm is used to solve a fictitious set of equations for the flow past the baseline configuration. These equations are identical to the correct equations for subsonic portions of the flow, but they are modified when the flow becomes super- sonic, so that even though the flow speed is larger than the local speed of sound the equations themselves remain elliptic. This procedure generates a numerical solution that satisfies the appropriate equations where the flow is subsonic, and the appropriate boundary conditions on the configuration outside of the supersonic zone. The results of this calculation provide the flowfield at the sonic surface. This surface and flowfield define an ill-posed initial value problem for the supersonic domain that is to be solved using the correct equations. Because this problem is ill-posed in three dimensions any numerical method must, in principle, be unstable. This in- stability, however, is of no consequence for moderate to high aspect ratios. However, if the detailed definition of the spanwise modifications required to make the wing shock free

Recent developments in integral equation method in transonic flow

The present work gives a brief review of the integral equation method in transonic aerodynamics, with particular emphasis on the works of Nϕrstrud, Nixon and Niyogi. The use of integral equation method can result in significant reduction in computer time, roughly by a factor of 60, compared to the finite difference relaxation procedure. In shock-free symmetrical super-critical flows, the simple solution of Niyogi which delivers results with less than 5 % overall error, may be conveniently used for analysis as well as for design purposes. The direct iteration scheme, applicable to shock-free flow, as well as to flows with shocks, emerges as particularly attractive due to its favourable convergence behaviour and computational simplicity. It is expected to be even more effective for three-dimensional problems. Further, the simple model suggested by Oswatitsch for thin symmetrical profile flow is sufficiently accurate for most practical purposes. For results of high accuracy, a hybrid direct iteration/finite difference procedure is suggested, which would use the converged direct iteration solution of the simple Oswatitsch model as the starting solution for the finite difference procedure.

Efficient Transonic Shock-Free Wing Redesign Procedure Using a Fictitious Gas Method

A new method for transonic shock-free three-dimensional wing design based on the full potential tqutuion is presented. The method locally modifies (i.e., redesigns) the wing surface geometry beneath the supersonic region to produce a shock-free flow. Results for a redesigned nonlifting rectangular wing and for a redesigned lifting wing of ONERA M6 planform are illustrated. Significant drag reduction and improved lift-drag characteristics are achieved from the present redesign procedures. Methods of compensating for wing thickness reduction experienced in the redesign process and of correcting for viscous effects are discussed.

Iterative computation of transonic shock-free profile flow at zero incidence

An approximate form of the transonic integral equation of Oswatitsch for steady inviscid flow past a thin symmetric profile at zero incidence has been studied iteratively. A direct iteration scheme has been presented, which converges to the same shock-free supercritical flow for four different starting solutions. Another square-root iteration scheme, which converges in subcritical flow, has been discussed. Also, the computational results delivered by the Schubert-Schleiff series [2] are presented briefly. Computational results show that, for a parabolic arc profile, shock-free supercritical flow exists for values of the reduced thickness ratio τ in the range 0.596<τ<0.64.