Transonic Hodograph Design Problems

The hodograph method is used to formulate several design problems in transonic flow using small-disturbance theory. Analytical and numerical methods give solutions to several optimum critical airfoil designs with different constraints on the tail angle. Special airfoil shapes flying at free-stream Mach number one are designed. The problem of constructing a shock-free body of revolution at subsonic speed but having a supersonic zone is formulated in the hodograph and solved numerically. Received 10 January 1997 and accepted 14 April 1997     

Numerical discussions on complete empirical correlation in Langtry's transition model

Recently, introducing a transition predicting model into the Reynolds averaged Navier-Stokes (RANS) environment has been paid more and more attention. Langtry proposed a correlation-based transition model in 2006, which was built strictly on local variables. However, two core correlations in the model had not been published until 2009. In this paper, after considerable analyses and discussions of the mechanism of this transition model and a series of numerical validations in the skin friction coefficient of flat plate boundary layers, a new correlation based on free-stream turbulence intensity is developed, and the empirical correlation of the transition onset momentum thickness Reynold number aiming at the hypersonic transition is improved. Low-speed/transonic airfoils and a hypersonic double wedge flat are tested to prove the reliability and practicability of this correlation.     

Numerical discussions on complete empirical correlation in Langtry＇s transition model

Recently, introducing a transition predicting model into the Reynolds averaged Navier-Stokes （RANS） environment has been paid more and more attention. Langtry proposed a correlation-based transition model in 2006, which was built strictly on local variables. However, two core correlations in the model had not been published until 2009. In this paper, after considerable analyses and discussions of the mechanism of this transition model and a series of numerical validations in the skin friction coefficient of flat plate boundary layers, a new correlation based on free-stream turbulence intensity is developed, and the empirical correlation of the transition onset momentum thickness Reynold number aiming at the hypersonic transition is improved. Low-speed/transonic airfoils and a hypersonic double wedge fiat are tested to prove the reliability and practicability of this correlation.     

The Effect of the Mach Number on a Turbulent Backward-Facing Step Flow

The flow around a backward-facing step in the sub-, trans- and supersonic regimes was investigated at the Trisonic Wind Tunnel Munich with particle image velocimetry and dynamic pressure measurements. These two techniques were combined to simultaneously measure and correlate the velocity fluctuations in a streamwise vertical plane with the pressure fluctuations on the reattachment surface. The results show that the dynamic loads on the reattachment surface increase from subsonic up to the transonic regime while the mean reattachment location moves downstream. As soon as the flow becomes locally supersonic aft of the backward-facing step, the mean reattachment location suddenly moves upstream while the normalized dynamic loads drastically decrease. By correlating the velocity and the dynamic pressure data, it was shown that a clear separation between outer flow and the flow close to the surface aft of the step is responsible for the drastic load reduction. Due to the large difference in pressure/density, the disturbances from the locally supersonic flow do not have an effect on the flow close to the surface. This is also reflected in the power spectral densities of the pressure fluctuations on the surface, showing that at supersonic free-stream Mach numbers a low-frequency pumping motion of the locally subsonic flow is the dominant mode, while in sub-/transonic flow Kelvin-Helmholtz instabilities and a cross-pumping motion of the shear layer dominate the dynamic loads.     

对二维分离流涡黏性系数非线性分布的新认识

以弱非线性涡黏性模型为出发点,对Delery分离流动实验结果进行分析并获得了非平衡态分离区涡黏性系数与形状因子J之间的非线性关系. 该非线性关系显示在分离起始阶段,涡黏性系数较平衡态先减小,后增大;再附阶段,涡黏性系数较平衡态数值逐渐增大,并在再附点位置接近最大,而后又逐渐减小,恢复到平衡态水平. 总结涡黏性系数的这种非线性发展数学关系式,并将它应用于BL模型,在不添加微分方程的情况下发展出一种适用于分离流动的改进代数湍流模型. 对低速平板流动,跨声速,超声速以及高超声速分离流动的计算结果表明,该改进湍流模型可以较准确地模拟各类复杂分离流动,计算精度明显优于传统代数模型以及一些两方程模型,而计算工作量仍与BL模型相当. 这表明所提出的涡黏性系数非线性发展规律是正确的,且应用在二维分离流动中具有一定的普适性.      

Separated Flow and Buffeting Control

In transonic flow conditions, the shock wave/turbulent boundary layer interaction and the flow separations on the upper wing surfaces of civil aircraft induce flow instabilities, ‘buffet’ and then structural vibrations, ‘buffeting’. Buffeting can greatly affect aerodynamic behavior. The buffeting phenomenon appears when the aircraft's Machnumber or angle of attack increases. This phenomenon limits the aircraft's flight envelope. The objectives of this study are to cancel out or decrease the aerodynamic instabilities (unsteady separation, movement of the shock position) due to this type of flow by using control systems. The following actuators can be used: ‘Vortex Generators’ situated upstream of the shock position, a ‘Bump’ located at the shock position, and a new moving part designed by ONERA, situated on the trailing edge of the wing, the ‘Trailing Edge Deflector’ or TED. It looks like an adjustable ‘Divergent Trailing Edge’. It is an active actuator and can take different deflections or be driven by dynamic movements up to 250 Hz. Tests were performed in transonic 2D flow with models well equipped with unsteady pressure transducers. For high lift coefficients, a selected static position of the ‘Trailing Edge Deflector’ increases the wing's aerodynamic performances and delays the onset of buffet. Furthermore, in 2D flow buffet conditions, the ‘Trailing Edge Deflector’, driven by a closed-loop active control using the measurements of the unsteady wall static pressures, can greatly reduce buffet. The aerodynamic performances are not improved to the same extent by the bump actuator. From our experience, there is no effect on buffet or separated flow because of the incorrect positioning of the bump. All that can be observed is a local improvement on the intensity of the shock wave when the bump is very precisely situated at the shock position. Vortex generators have a great impact on the separated flow. The separated flow instabilities are greatly reduced and buffet is totally controlled even for strong instabilities. The aerodynamic performances of the airfoil are also greatly improved.     

A hybrid pressure–density‐based algorithm for the Euler equations at all Mach number regimes

In the present work, we propose a reformulation of the fluxes and interpolation calculations in the PISO method, a well‐known pressure‐correction solver. This new reformulation introduces the AUSM⁺ ? up flux definition as a replacement for the standard Rhie and Chow method of obtaining fluxes and central interpolation of pressure at the control volume faces. This algorithm tries to compatibilize the good efficiency of a pressure based method for low Mach number applications with the advantages of AUSM⁺ ? up at high Mach number flows. The algorithm is carefully validated using exact solutions. Results for subsonic, transonic and supersonic axisymmetric flows in a nozzle are presented and compared with exact analytical solutions. Further, we also present and discuss subsonic, transonic and supersonic results for the well known bump test‐case. The code is also benchmarked against a very tough test‐case for the supersonic and hypersonic flow over a cylinder. Copyright © 2011 John Wiley & Sons, Ltd.     

3D pressure imaging of an aircraft propeller blade-tip flow by phase-locked stereoscopic PIV

The flow field at the tip region of a scaled DHC Beaver aircraft propeller, running at transonic speed, has been investigated by means of a multi-plane stereoscopic particle image velocimetry setup. Velocity fields, phase-locked with the blade rotational motion, are acquired across several planes perpendicular to the blade axis and merged to form a 3D measurement volume. Transonic conditions have been reached at the tip region, with a revolution frequency of 19,800 rpm and a relative free-stream Mach number of 0.73 at the tip. The pressure field and the surface pressure distribution are inferred from the 3D velocity data through integration of the momentum Navier-Stokes equation in differential form, allowing for the simultaneous flow visualization and the aerodynamic loads computation, with respect to a reference frame moving with the blade. The momentum and pressure data are further integrated by means of a contour-approach to yield the aerodynamic sectional force components as well as the blade torsional moment. A steady Reynolds averaged Navier-Stokes numerical simulation of the entire propeller model has been used for comparison to the measurement data.     

A parallel pressure implicit splitting of operators algorithm applied to flows at all speeds

A parallel implementation of the pressure‐based implicit splitting of operators (PISO) method is described and applied to both compressible and incompressible flows. The treatment of variables at the interfaces between adjacent blocks is highlighted, and, for compressible flow, a straightforward method for the implicit handling of density is described. Steady state and oscillatory flow through a sudden expansion are considered at low speeds for both two‐ and three‐dimensional geometries. Extension of the incompressible method to compressible flow is assessed for subsonic, transonic and supersonic flow through a two‐dimensional bump. Although good accuracy is achieved in these high‐speed flows, including the automatic capturing of shock waves, the method is deemed unsuitable for simulating steady state high‐speed flows on fine grids due to the requirement of very small time steps. Copyright © 2001 John Wiley & Sons, Ltd.     

The Multi-point Optimization of Shock Control Bump with Constant-Lift Constraint Enhanced with Suction and Blowing for a Supercritical Airfoil

Both shock control bump (SCB) and suction and blowing are flow control methods used to control the shock wave/boundary layer interaction (SWBLI) in order to reduce the resulting wave drag in transonic flows. A SCB uses a small local surface deformation to reduce the shock-wave strength, while suction decreases the boundary-layer thickness and blowing delays the flow separation. Here a multi-point optimization method under a constant-lift-coefficient constraint is used to find the optimum design of SCB and suction and blowing. These flow control methods are used separately or together on a RAE-2822 supercritical airfoil for a wide range of off-design transonic Mach numbers. The RANS flow equations are solved using Roe’s averages scheme and a gradient-based adjoint algorithm is used to find the optimum location and shape of all devices. It is shown that the simultaneous application of blowing and SCB (hybrid blowing/SCB) improves the average aerodynamic efficiency at off-design conditions by 18.2 % in comparison with the clean airfoil, while this increase is only 16.9 % for the hybrid suction/SCB. We have also studied the SWBLI and how the optimization algorithm makes the flow wave structure and interactions of the shock wave with the boundary layer favorable.     

The application of thin-film technology to measure turbine-vane heat transfer and effectiveness in a film-cooled, engine-simulated environment

Thin-film technology has been used to measure the heat transfer coefficient and cooling effectiveness over heavily film cooled nozzle guide vanes (NGVs). The measurements were performed in a transonic annular cascade which has a wide operating range and simulates the flow in the gas turbine jet engine. Engine-representative Mach and Reynolds numbers were employed and the upstream free-stream turbulence intensity was 13%. The aerodynamic and thermodynamic characteristics of the coolant flow (momentum flux and density ratio between the coolant and mainstream) have been modelled to represent engine conditions by using a foreign gas mixture of SF₆ and Argon. Engine-level values of heat transfer coefficient and cooling effectiveness have been obtained by correcting for the different molecular (thermal) properties of the gases used in the engine-simulated experiments to those which exist in the true engine environment. This paper presents the best combined heat transfer coefficient and effectiveness data currently available for a fully cooled, three-dimensional NGVs at engine conditions.     

The application of the gradient-based adjoint multi-point optimization of single and double shock control bumps for transonic airfoils

A shock control bump (SCB) is a flow control method that uses local small deformations in a flexible wing surface to considerably reduce the strength of shock waves and the resulting wave drag in transonic flows. Most of the reported research is devoted to optimization in a single flow condition. Here, we have used a multi-point adjoint optimization scheme to optimize shape and location of the SCB. Practically, this introduces transonic airfoils equipped with the SCB that are simultaneously optimized for different off-design transonic flight conditions. Here, we use this optimization algorithm to enhance and optimize the performance of SCBs in two benchmark airfoils, i.e., RAE-2822 and NACA-64-A010, over a wide range of off-design Mach numbers. All results are compared with the usual single-point optimization. We use numerical simulation of the turbulent viscous flow and a gradient-based adjoint algorithm to find the optimum location and shape of the SCB. We show that the application of SCBs may increase the aerodynamic performance of an RAE-2822 airfoil by 21.9 and by 22.8 % for a NACA-64-A010 airfoil compared to the no-bump design in a particular flight condition. We have also investigated the simultaneous usage of two bumps for the upper and the lower surfaces of the airfoil. This has resulted in a 26.1 % improvement for the RAE-2822 compared to the clean airfoil in one flight condition.     

Bifurcations of transonic flow past simple airfoils with elliptic and wedge-shaped noses

A turbulent transonic flow past two symmetric airfoils with flat midparts is studied numerically. Using the Reynolds-averaged Navier-Stokes equations, we analyze the flow past a 9% thick airfoil with an elliptic nose. A range of the free-stream Mach number M_∞, in which flow bifurcations occur, is determined. Values of M_∞ that give rise to significant changes in the lift coefficient with variations of the angle of attack are specified. Flow bifurcations are also revealed for a thin double wedge, i.e., a sort of a hexagon.     

On the Design of Lifting Airfoils with High Critical Mach Number Using Full Potential Theory

 We wish to construct airfoils that have the highest free-stream Mach number for a given set of geometric constraints for which the flow is nowhere supersonic. Nonlifting airfoils that maximize the critical Mach number for a given cross-sectional area are known to possess long sonic segments at their critical speed. To construct lifting airfoils, we proceed under the conjecture that an airfoil with a high value of has the longest possible arc length of sonic velocity over its upper and lower surface. In Kropinski et al. (1995) the lifting problem was tackled in transonic small-disturbance theory. In this paper we numerically construct lifting airfoils with high using the full potential theory and we show that these airfoils have significantly higher than some standard airfoils. We also construct airfoils with higher values of the lift coefficient, by relaxing the speed constraint on the lower surface of the airfoil to have a value less than sonic. Received 13 May 1996 accepted 12 September 1996)     

Film cooling on a convex surface: influence of external pressure gradient and Mach number on film cooling performance

 The film cooling performance on a convex surface subjected to zero and favourable pressure gradient free-stream flow was investigated. Adiabatic film cooling effectiveness values were obtained for five different injection geometries, three with cylindrical holes and two with shaped holes. Heat transfer coefficients were derived for selected injection configurations. CO₂ was used as coolant to simulate density ratios between coolant and free-stream close to gas turbine engine conditions. The film cooling effectiveness results indicate a strong dependency on the free-stream Mach number level. Results obtained at the higher free-stream Mach number show for cylindrical holes generally and for shaped holes at moderate blowing rates significant higher film cooling effectiveness values compared to the lower free-stream Mach number data. Free-stream acceleration generally reduced adiabatic film cooling effectiveness relative to constant free-stream flow conditions. The different free-stream conditions investigated indicate no significant effects on the corresponding heat transfer increase due to film injection. The determined heat flux ratios or film cooling performance indicated that coolant injection with shaped film cooling holes is much more efficient than with cylindrical holes especially at higher blowing rates. Heat flux penalties can occur at high blowing rates when using cylindrical holes. Received on 29 May 2000     

A second-order upwind finite-volume method for the Euler solution on unstructured triangular meshes

A scheme for the numerical solution of the two-dimensional (2D) Euler equations on unstructured triangular meshes has been developed. The basic first-order scheme is a cell-centred upwind finite-volume scheme utilizing Roe's approximate Riemann solver. To obtain second-order accuracy, a new gradient based on the weighted average of Barth and Jespersen's three-point support gradient model is used to reconstruct the cell interface values. Characteristic variables in the direction of local pressure gradient are used in the limiter to minimize the numerical oscillation around solution discontinuities. An Approximate LU (ALU) factorization scheme originally developed for structured grid methods is adopted for implicit time integration and shows good convergence characterisitics in the test. To eliminate the data dependency which prohibits vectorization in the inversion process, a black-gray-white colouring and numbering technique on unstructured triangular meshes is developed for the ALU factorization scheme. This results in a high degree of vectorization of the final code. Numerical experiments on transonic Ringleb flow, transonic channel flow with circular bump, supersonic shock reflection flow and subsonic flow over multielement aerofoils are calculated to validate the methodology.     

A numerical solution of 3D inviscid rotational flow in turbines and ducts

The numerical solutions of inviscid rotational (Euler) flows were obtained using an explicit hexahedral unstructured cell vertex finite volume method. A second-order-accurate, one-step Lax–Wendroff scheme was used to solve the unsteady governing equations discretized in conservative form. The transonic circular bump, in which the location and the strength of the captured shock are well predicted, was used as the first test case. The nozzle guide vanes of the VKI low-speed turbine facility were used to validate the Euler code in highly 3D environment. Despite the high turning and the secondary flows which develop, close agreements have been obtained with experimental and numerical results associated with these test cases. © 1998 John Wiley & Sons, Ltd.     

Kriging‐based optimization applied to flow control

The automatic optimization of flow control devices is a delicate issue because of the drastic computational time related to unsteady high‐fidelity flow analyses and the possible multimodality of the objective function. Thus, we experiment in this article the use of kriging‐based algorithms to optimize flow control parameters because these methods have shown their efficiency for global optimization at moderate cost. Navier–Stokes simulations, carried out for different control parameters, are used to build iteratively a kriging model. At each step, a statistical analysis is performed to enrich the model with new simulation results by exploring the most promising areas, until optimal flow control parameters are found. This approach is validated and demonstrated on two problems, including comparisons with similar studies: the control of the flow around an oscillatory rotating cylinder and the reduction of the intensity of a shock wave for a transonic airfoil by adding a bump to the airfoil profile. Copyright © 2011 John Wiley & Sons, Ltd.     

PIV study on a shock-induced separation in a transonic flow

A transonic interaction between a steady shock wave and a turbulent boundary layer in a Mach 1.4 channel flow is experimentally investigated by means of particle image velocimetry (PIV). In the test section, the lower wall is equipped with a contour profile shaped as a bump allowing flow separation. The transonic interaction, characterized by the existence in the outer flow of a lambda shock pattern, causes the separation of the boundary layer, and a low-speed recirculating bubble is observed downstream of the shock foot. Two-component PIV velocity measurements have been performed using an iterative gradient-based cross-correlation algorithm, providing high-speed and flexible calculations, instead of the classic multi-pass processing with FFT-based cross-correlation. The experiments are performed discussing all the hypotheses linked to the experimental set-up and the technique of investigation such as the two-dimensionality assumption of the flow, the particle response assessment, the seeding system, and the PIV correlation uncertainty. Mean velocity fields are presented for the whole interaction with particular attention for the recirculating bubble downstream of the detachment, especially in the mixing layer zone where the effects of the shear stress are most relevant. Turbulence is discussed in details, the results are compared to previous study, and new results are given for the turbulent production term and the return to isotropy mechanism. Finally, using different camera lens, a zoom in the vicinity of the wall presents mean and turbulent velocity fields for the incoming boundary layer.     

Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections

We establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross-sections, including a slowly varying de Laval nozzle. The transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance, uniform subsonic downstream flow at the exit at infinity, and the slip boundary condition on the nozzle boundary. Our results indicate that, if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow, there exists a solution that consists of a C ^1,α subsonic flow in the unbounded downstream region, converging to a uniform velocity state at infinity, and a C ^1,α multidimensional transonic shock separating the subsonic flow from the supersonic upstream flow; the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow; and the shock is orthogonal to the nozzle boundary at every point of their intersection. In order to construct such a transonic flow, we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase, in which the equation is elliptic and the free boundary is a transonic shock. The free boundary conditions are determined by the Rankine–Hugoniot conditions along the shock. We further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain. We also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance.