Stream Tube Thickness Research Articles

A continuous adjoint method with objective function derivatives based on boundary integrals, for inviscid and viscous flows

A continuous adjoint formulation for inverse design problems in external aerodynamics and turbomachinery is presented. The advantage of the proposed formulation is that the objective function gradient does not depend upon the variation of field geometrical quantities, such as metrics variations in the case of structured grids. The final expression for the objective function gradient includes only boundary integrals which can readily be calculated in both structured and unstructured grids; this is feasible in design problems where the objective function is either a boundary integral (pressure deviation along the solid walls) or a field integral (the entropy generation over the flow domain). The formulation governs inviscid and viscous flows; it takes into account the streamtube thickness variation terms in quasi-3D cascade designs or rotational terms in rotating blade design problems. The application of the method is illustrated through a number of design problems concerning isolated airfoils, a 3D duct, 2D, quasi-3D and 3D, stationary and rotating turbomachinery blades.

Numerical and experimental investigation of unsteady flow interaction in a low-pressure multistage turbine

The paper presents results of unsteady viscous flow calculations and corresponding cold flow experiments on a three-stage low-pressure turbine. The investigation emphasizes the study of unsteady flow interaction. A time-accurate, Reynolds-averaged Navier-Stokes solver is applied for the computations. Turbulence is modelled using the Spalart-Allmaras one-equation turbulence model. The influence of modern transition models on the unsteady flow predictions is investigated. Integration of the governing equations in time is performed with a four-stage Runge-Kutta scheme, which is accelerated by a two-grid method in the viscous boundary layer around the blades. At the inlet and outlet, non-reflecting boundary conditions are used. The quasi-three-dimensional calculations are conducted on a stream surface around mid-span, allowing a varying stream tube thickness. A three-stage, low-pressure turbine rig of a modern commercial jet engine is used for a study of the unsteady flow interaction. The numerical method is able to capture important unsteady effects found in the experiments, i.e. unsteady transition as well as the bladerow interaction. In particular, the flowfield with respect to time-averaged and unsteady quantities such as surface pressure, vorticity and turbulence intensity is compared with the experiments conducted in the cold airflow test rig.

One- and two-equation turbulence models for the prediction of complex cascade flows using unstructured grids

One- and two-equation, low-Reynolds eddy-viscosity turbulence models are employed in the context of a primitive variable, finite volume, Navier–Stokes solver for unstructured grids. Through the study of the complex flow in a controlled-diffusion compressor cascade at off-design conditions, the ability of the models under consideration to predict the laminar separation bubble close to the leading edge and the boundary layer development is investigated. In order to control the unphysical growth of turbulent kinetic energy near the leading edge stagnation point, appropriate modifications to the conventional models are employed and tested. All of them improve the leading edge flow patterns and significantly affect the size of the predicted laminar separation bubble. The use of an adequately refined mesh around the airfoil, that is formed by triangles placed in a quasi-structured way, allows for the generation of grid elements of moderate aspect ratios. This helps to readily overcome any relevant problems of accuracy; a second-order upwind scheme without flux limiters or least squares approximations is successfully employed for the gradients. The test case includes quasi-3D effects by considering the streamtube thickness variation in the governing equations.

Computation of Mistuning Effects on Cascade Flutter

A computational method is described for predicting eutter of turbomachinery cascades with mistuned blades. Themethodsolves theunsteadyEuler/Navier‐Stokesequationsformultiple-bladepassageson aparallel computer usingthemessagepassinginterface. Asecond-orderimplicit schemewithdualtime-steppingand multigrid isused. Each individual blade is capable of moving with its own independent frequency and phase angle, thus modeling a cascade with mistuned blades. Flutter predictions are performed through the energy method. Both phase-angle and frequency mistuning are studied. It is found that phase-angle mistuning has little effect on stability, whereas frequencymistuningsignie cantlychanges theaerodynamicdamping.Theimportant effectoffrequencymistuning is to average out the aerodynamic damping of the tuned blade row over the whole range of interbladephase angles (IBPA). If a tuned blade row is stable over most of the IBPA range, the blades can be stabilized for the complete IBPA range through appropriate frequency mistuning. Nomenclature Ch = coefecient of aerodynamic force in h direction CW = aerodynamic work coefe cient c = blade chord length E = total energy per unit mass h = translational blade displacement p = static pressure T = e utter period t = time u, v = e ow velocity components in x and y directions ub, vb = grid velocity components in x and y directions ¯ u, ¯ v = relative velocity components (u i ub, v i vb) a = rotational blade displacement h = stream tube thickness ratio in x direction N = e utter damping coefe cient q = density r = interblade phase angle x = e utter frequency

Numerical and Experimental Investigation of Unsteady Flow Interaction in a Low-Pressure Multistage Turbine

This paper presents results of unsteady viscous flow calculations and corresponding cold flow experiments of a three-stage low-pressure turbine. The investigation emphasizes the study of unsteady flow interaction. A time-accurate Reynolds-averaged Navier–Stokes solver is applied for the computations. Turbulence is modeled using the Spalart–Allmaras one-equation turbulence model and the influence of modern transition models on the unsteady flow predictions is investigated. The integration of the governing equations in time is performed with a four-stage Runge–Kutta scheme, which is accelerated by a two-grid method in the viscous boundary layer around the blades. At the inlet and outlet, nonreflecting boundary conditions are used. The quasi-three-dimensional calculations are conducted on a stream surface around midspan, allowing a varying stream tube thickness. In order to study the unsteady flow interaction, a three-stage low-pressure turbine rig of a modern commercial jet engine is built up. In addition to the design point, the Reynolds number, the wheel speed, and the pressure ratio are also varied in the tests. The numerical method is able to capture important unsteady effects found in the experiments, i.e., unsteady transition as well as the blade row interaction. In particular, the flow field with respect to time-averaged and unsteady quantities such as surface pressure, entropy, and skin friction is compared with the experiments conducted in the cold air flow test rig. [S0889-504X(00)02004-3]

Compressor Cascade Flow with Strong Shock-Wave/Boundary-Layer Interaction

A two-dimensional multiblock Navier ‐Stokes solver has been successfully applied to a supersonic compressor cascade e ow with strong shock-wave/boundary-layer interaction. The cascade model investigated is typical of a fan blade section with precompression design, low e ow turning, and a static pressure ratio around 2.0. Numerical calculations have been performed on a very e ne grid to guarantee sufe cient resolution of the supersonic/transonic e owe eldwithacomplexwavepatternandshock-inducedboundary-layerseparation.Highaccuracyandrobustness of the numerical scheme is achieved by the second-order Roe upwind ‐total variation diminishing scheme and a one-equation turbulence model. Investigations were performed for inlet Mach numbers from 1.28 to 1.53 and a Reynolds number of 2 :6 ££ 106. The results are compared with those of previous experiments in a supersonic cascade wind tunnel that provide proe le Mach number distributions, wake traverse data, and measured suction surface boundary-layer proe les throughout the strong interaction region. Furthermore, calculated discontinuities of Mach number and e ow direction across the shock system at blade passage entrance are compared with laser anemometer data. Finally the ine uence of inlet Mach number and axial stream tube thickness variation on the shock-wave/boundary-layer interaction mechanism and blade performance is presented.

Validation of a Numerical Method for Unsteady Flow Calculations

This paper describes and validates a numerical method for the calculation of unsteady inviscid and viscous flows. A companion paper compares experimental measurements of unsteady heat transfer on a transonic rotor with the corresponding computational results. The mathematical model is the Reynolds-averaged unsteady Navier–Stokes equations for a compressible ideal gas. Quasi-three-dimensionality is included through the use of a variable streamtube thickness. The numerical algorithm is unusual in two respects: (a) For reasons of efficiency and flexibility, it uses a hybrid Navier–Stokes/Euler method, and (b) to allow for the computation of stator/rotor combinations with arbitrary pitch ratio, a novel space–time coordinate transformation is used. Several test cases are presented to validate the performance of the computer program, UNSFLO. These include: (a) unsteady, inviscid flat plate cascade flows (b) steady and unsteady, viscous flat plate cascade flows, (c) steady turbine heat transfer and loss prediction. In the first two sets of cases comparisons are made with theory, and in the third the comparison is with experimental data.