Nonlinear turbulence closures were developed that improve the prediction accuracy of wake mixing in low-pressure turbine (LPT) flows. First, Reynolds-averaged Navier–Stokes (RANS) calculations using five linear turbulence closures were performed for the T106A LPT profile at isentropic exit Reynolds numbers 60,000 and 100,000. None of these RANS models were able to accurately reproduce wake loss profiles, a crucial parameter in LPT design, from direct numerical simulation (DNS) reference data. However, the recently proposed k v 2 ¯ ω transition model was found to produce the best agreement with DNS data in terms of blade loading and boundary layer behavior and thus was selected as baseline model for turbulence closure development. Analysis of the DNS data revealed that the linear stress–strain coupling constitutes one of the main model form errors. Hence, a gene-expression programming (GEP) based machine-learning technique was applied to the high-fidelity DNS data to train nonlinear explicit algebraic Reynolds stress models (EARSM), using different training regions. The trained models were first assessed in an a priori sense (without running any RANS calculations) and showed much improved alignment of the trained models in the region of training. Additional RANS calculations were then performed using the trained models. Importantly, to assess their robustness, the trained models were tested both on the cases they were trained for and on testing, i.e., previously not seen, cases with different flow features. The developed models improved prediction of the Reynolds stress, turbulent kinetic energy (TKE) production, wake-loss profiles, and wake maturity, across all cases.

Machine learning was applied to large-eddy simulation (LES) data to develop nonlinear turbulence stress and heat flux closures with increased prediction accuracy for trailing-edge cooling slot cases. The LES data were generated for a thick and a thin trailing-edge slot and shown to agree well with experimental data, thus providing suitable training data for model development. A gene expression programming (GEP) based algorithm was used to symbolically regress novel nonlinear explicit algebraic stress models and heat-flux closures based on either the gradient diffusion or the generalized gradient diffusion approaches. Steady Reynolds-averaged Navier–Stokes (RANS) calculations were then conducted with the new explicit algebraic stress models. The best overall agreement with LES data was found when selecting the near wall region, where high levels of anisotropy exist, as training region, and using the mean squared error of the anisotropy tensor as cost function. For the thin lip geometry, the adiabatic wall effectiveness was predicted in good agreement with the LES and experimental data when combining the GEP-trained model with the standard eddy-diffusivity model. Crucially, the same model combination also produced significant improvement in the predictive accuracy of adiabatic wall effectiveness for different blowing ratios (BRs), despite not having seen those in the training process. For the thick lip case, the match with reference values deteriorated due to the presence of large-scale, relative to slot height, vortex shedding. A GEP-trained scalar flux model, in conjunction with a trained RANS model, was found to significantly improve the prediction of the adiabatic wall effectiveness.

The complex structures in the flow field of gas turbine film cooling increase the anisotropy of turbulence making it difficult to accurately compute turbulent eddy viscosity and scalar diffusivity. An algebraic anisotropic turbulence model is developed while aiming at a more accurate modeling of the Reynolds stress and turbulent scalar-flux. In this study, the algebraic anisotropic model is validated by two in-house experiments. One is a leading edge with showerhead film cooling and the other is a vane with full coverage film cooling. Adiabatic film cooling effectiveness under different blowing ratios, density ratios, and film cooling arrangements were measured using pressure sensitive paint (PSP) technique. Four different turbulence models are tested and detailed analyses of computational simulations are performed. Among all the turbulence models investigated, the algebraic anisotropic model shows better agreement with the experimental data qualitatively and quantitatively. The algebraic anisotropic model gives a good prediction of the vortex strength and turbulence mixing of the jet, therefore improves the prediction of the scalar field.

A numerical study has been conducted to investigate the effects of surface curvature on cooling effectiveness using three-dimensional film cooling geometries that included the mainflow, injection hole, and supply plenum regions. Three surfaces were considered in this study, namely, convex, concave, and flat surfaces. The fully elliptic, three-dimensional Navier–Stokes equations were solved over a body-fitted grid. The effects of streamline curvature were taken into account by using algebraic relations for the turbulent viscosity and the turbulent Prandtl number in a modified k–ε turbulence model. Computations were performed for blowing ratios of 0.5, 1.0, and 1.5 at a density ratio of 2.0. The computed and experimental cooling effectiveness results were compared. For the most part, the cooling effectiveness was predicted quite well. A comparison of the cooling performances over the three surfaces reveals that the effect of streamline curvature on cooling effectiveness is very significant. For the low blowing ratios considered, the convex surface resulted in a higher cooling effectiveness than both the flat and concave surfaces. The flow structures over the three surfaces also exhibited important differences. On the concave surface, the flow involved a stronger vorticity and greater mixing of the coolant jet with the mainstream gases. On the convex surface, the counterrotating vortices were suppressed and the coolant jet pressed to the surface by a strong cross-stream pressure gradient.

Computations have been conducted on curved, three-dimensional discrete-hole film cooling geometries that included the mainflow, injection hole, and supply plenum regions. Both convex and concave film cooling geometries were studied. The effects of several film cooling parameters have been investigated, including the effects of blowing ratio, injection angle, hole length, hole spacing, and hole staggering. The blowing ratio was varied from 0.5 to 1.5, the injection angle from 35 to 65 deg, the hole length from 1.75D to 6.0D, and the hole spacing from 2D to 3D. The staggered-hole arrangement considered included two rows. The computations were performed by solving the fully elliptic, three-dimensional Navier–Stokes equations over a body-fitted grid. Turbulence closure was achieved using a modified k–ε model in which algebraic relations were used for the turbulent viscosity and the turbulent Prandtl number. The results presented and discussed include plots of adiabatic effectiveness as well as plots of velocity contours and velocity vectors in cross-stream planes. The present study reveals that the blowing ratio, hole spacing, and hole staggering are among the most significant film cooling parameters. Furthermore: (1) The optimum blowing ratios for curved surfaces are higher than those for flat surfaces, (2) a reduction of hole spacing from 3D to 2D resulted in a very significant increase in adiabatic effectiveness, especially on the concave surface, (3) the increase in cooling effectiveness with decreasing hole spacing was found to be due to not only the increased coolant mass per unit area, but also the smaller jet penetration and the weaker counterrotating vortices, (4) for all practical purposes, the hole length was found to be a much less significant film cooling parameter.

External heat transfer predictions are performed for two-dimensional turbine blade cascades. The Reynolds-averaged Navier–Stokes equations with algebraic (Arnone and Pacciani, 1998), one-equation (Spalart and Allmaras, 1994), and two-equation (low-Re k–ε, Biswas and Fukuyama, 1994) turbulence closures are solved with a fully implicit time-marching finite volume method. Comparisons with measurements (Arts et al., 1990; Arts, 1994) for a highly loaded transonic turbine nozzle guide vane cascade show good agreement in some cases, but also reveal problems with transition prediction and turbulence modeling. Special attention has been focused on the low-Re k–ε model concerning the influence of the inlet boundary condition for the ε-equation and problems in the stagnation point region.

A three-dimensional Navier–Stokes procedure has been used to compute the three-dimensional viscous flow through the turbine nozzle passage of a single-stage turbine. A low-Reynolds-number k–ε model and a zonal k-ε/ARSM (algebraic Reynolds stress model) are utilized for turbulence closure. The algebraic Reynolds stress model is used only in the endwall region to represent the anisotropy of turbulence. A four-stage Runge–Kutta scheme is used for time integration of both the mean-flow and the turbulence transport equations. For the turbine nozzle flow, comprehensive comparisons between the predictions and the experimental data obtained at Penn State show that most features of the vortex-dominated endwall flow, as well as nozzle wake structure, have been captured well by the numerical procedure. An assessment of the performance of the turbulence models has been carried out. The two models are found to provide similar predictions for the mean flow parameters, although slight improvement in the prediction of some secondary flow quantities has been obtained by the ARSM model.

Predictions of turbine vane and endwall heat transfer and pressure distributions are compared with experimental measurements for two vane geometries. The differences in geometries were due to differences in the hub profile, and both geometries were derived from the design of a high rim speed turbine (HRST). The experiments were conducted in the Isentropic Light Piston Facility (ILPF) at Pyestock at a Reynolds number of 5.3 x 10 6 , a Mach number of 1.2, and a wall-to-gas temperature ratio of 0.66. Predictions are given for two different steady-state three-dimensional Navier–Stokes computational analyses. C-type meshes were used, and algebraic models were employed to calculate the turbulent eddy viscosity. The effects of different turbulence modeling assumptions on the predicted results are examined. Comparisons are also given between predicted and measured total pressure distributions behind the vane. The combination of realistic engine geometries and flow conditions proved to be quite demanding in terms of the convergence of the CFD solutions. An appropriate method of grid generation, which resulted in consistently converged CFD solutions, was identified.

The fluid mechanics and heat transfer characteristics of film cooling are three-dimensional and highly complex. To understand this problem better, an experimental study was conducted in a low-speed wind tunnel on a row of six rectangular jets injected at 90 deg to the crossflow (mainstream flow). The jet-to-crossflow velocity ratios (blowing ratios) examined were 0.5, 1.0, and 1.5, and the jet spacing-to-jet width ratio was 3.0. No significant temperature difference between jet and crossflow air was introduced. Mean velocities and six flow stresses were measured using a three-component laser-Doppler velocimeter operating in coincidence mode. Seeding of both jet and cross-stream air was achieved with a commercially available smoke generator. Flow statistics are reported in the form of vector plots, contours, and x-y graphs, showing velocity, turbulence intensity, and Reynolds stresses. To complement the detailed measurements, flow visualization was accomplished by transmitting the laser beam through a cylindrical lens, thereby generating a narrow, intense sheet of light. Jet air only was seeded with smoke, which was illuminated in the plane of the light sheet. Therefore, it was possible to record on video tape the trajectory and penetration of the jets in the crossflow. Selected still images from the recordings are presented. Numerical simulations of the observed flow field were made by using a multigrid, segmented, k–ε CFD code. Special near-wall treatment included a nonisotropic formulation for the effective viscosity, a low-Re model for k, and an algebraic model for the length scale. Comparisons between the measured and computed velocities show good agreement for the nonuniform mean flow at the jet exit plane. Velocities and stresses on the jet centerline downstream of the orifice are less well predicted, probably because of inadequate turbulence modeling, while values off the centerline match those of the experiments much more closely.

The present investigation is devoted to one of the most difficult problems in the gas turbine industry: predicting the heat transfer to turbine blades. It is known that one of the important factors that affects heat transfer coefficients is a significant level of turbulence in the flow that surrounds a turbine blade. The influence of free-stream turbulence on heat transfer coefficients for a flat plate boundary layer with zero pressure gradient or in the vicinity of the stagnation point of a circular cylinder is investigated numerically. An algebraic relaxation-length model of turbulence is applied in order to simulate real situations in flows with a high level of free-stream turbulence. The results, temperature and velocity profiles, and heat transfer and drag coefficients, are compared with available experimental data. The proposed method is recommended for practical calculations of heat transfer coefficients on turbine blades.

A Navier–Stokes solver is applied to investigate the three-dimensional viscous flow in a low-speed linear compressor cascade with tip clearance at design and off-design conditions with two different meshes. The algebraic turbulence model of Baldwin–Lomax is used for closure. Relative motion between the blades and wall is simulated for one flow coefficient. Comparisons with experimental data, including flow structure, static and total pressures, velocity profiles, secondary flows and vorticity, are presented for the stationary wall case. It is shown that the code predicts well the flow structure observed in experiments and shows the details of the tip leakage flow and the leading edge horseshoe vortex.

The effect of transition modeling on the heat transfer predictions from rotating turbine blades was investigated. Three-dimensional computations using a Reynolds-averaged Navier–Stokes code were performed. The code utilized the Baldwin–Lomax algebraic turbulence model, which was supplemented with a simple algebraic model for transition. The heat transfer results obtained on the blade surface and the hub endwall were compared with experimental data for two Reynolds numbers and their corresponding rotational speeds. The prediction of heat transfer on the blade surfaces was found to improve with the inclusion of the transition length model and wake-induced transition effects over the simple abrupt transition model.

Finite-volume predictions are presented for the convective heat transfer rates in a rotating cavity, formed by two corotating plane disks and a peripheral shroud, and subjected to a radial outflow of cooling air. The heating of the disks is asymmetric, the air entering the cavity through a central hole in the cooler (upstream) disk. The predicted Nusselt number distributions for each disk are compared with unpublished data from the University of Sussex for dimensionless mass-flow rates in the range 2800 ⩽ C w ⩽ 14,000 and rotational Reynolds numbers, Re θ , up to 5.2 × 10 6 . A single-grid elliptic procedure was used with turbulent transport represented via a low-Reynolds-number k–ε model and the turbulence Prandtl number concept. In comparing the predicted and measured convective heat fluxes, it is important to consider the radiative heat exchange between the disks. This is estimated using a conventional view-factor approach based on black-body emission. Under conditions of asymmetric heating, rotationally induced buoyancy forces can exert significant effect on the flow structure, the induced motion tending to oppose that imposed by the radial outflow. Indeed, flow visualization studies have revealed that, as the rotational Reynolds number is increased (for a fixed value of C w ), the flow in the source region initially becomes oscillatory in nature, leading eventually to the onset of chaotic flow in which the usual Ekman layer structure does not persist in all angular planes. The extent to which the effects of such flow behavior can be captured by the steady, axisymmetric calculation approach used here is questionable, but it is found that the turbulence model (used previously for the prediction of heat transfer in symmetrically heated cavities) still leads to good (± 10 percent) predictive accuracy for the heated (downstream) disk. However, the predicted Nusselt numbers for the cooler (upstream) disk generally show little accord with experimental data, often signifying heat flow into the disk instead of vice versa. It is concluded that the modeling of the turbulent heat transport across the core region of the flow is erroneous, especially at high rotational Reynolds numbers: This is attributed to overestimated turbulence energy production in that region due to the action of the radial-circumferential component of shear stress ( νw ). Adoption of an algebraic-stress model for this shear stress is partly successful in removing the discrepancies between prediction and experiment.

A numerical study of developing flow through a heated duct of square cross section rotating in orthogonal mode is reported. The two main aims are to explore the effects of rotational buoyancy on the flow development and to assess the ability of available turbulence models to predict such flows. Two test cases have been computed corresponding to values of the rotation number, Ro, of 0.12 and 0.24, which are typical of operating conditions in internal cooling passages of gas turbine blades. Computations from three turbulence models are presented: a k–ε eddy viscosity (EVM) model matched to a low-Reynolds-number one-equation EVM in the near-wall region; a low-Re k–ε EVM and a low-Re algebraic stress model (ASM). Additional computations in which the fluid density is assumed to remain constant allow the distinct contributions from buoyancy and Coriolis forces to be separated. It is thus shown that rotational buoyancy can have a substantial influence on the flow development and that, in the case of outward flow, it leads to a considerable increase of the side-averaged heat transfer coefficient. The Coriolis-induced secondary motion leads to an augmentation of the mean heat transfer coefficient on the pressure surface and a reduction on the suction side. The k–ε/one-equation EVM produces a mostly reasonable set of heat transfer predictions, but some deficiencies do emerge at the higher rotation number. In contrast, predictions with the low-Re k–ε EVM return a spectacularly unrealistic behavior while the low-Re ASM thermal predictions are in encouragingly close agreement with available measurements.

The viscous flow through a controlled diffusion (CD) compressor cascade was calculated and compared with the measured data for two different test conditions. A control volume method was used, which has been developed for a generalized nonorthogonal coordinate system. The discretized equations for the physical covariant velocity components were obtained by an algebraic manipulation of the discretized equations for the Cartesian velocity components. Low Reynolds number k–ε turbulence models were used to obtain the eddy viscosity. The numerical scheme using the low Reynolds number k–ε turbulence model reasonably predicted the general performance, i.e, mean outlet flow angle and loss coefficients. The development of the shear layer along the pressure and suction sides was well estimated, and the physical features found in the experiment were reasonably well confirmed in the simulation. However, the calculated profiles of mean velocity and turbulent kinetic energy in the near wake show considerable disagreement with the measured values.

A theoretical and experimental investigation on the aerodynamic forces generated by a single gland labyrinth seal executing a spinning/whirling motion has been conducted. A lumped parameter model, which includes the kinetic energy carryover effect, is presented along with a linear perturbation solution technique. The resulting system is nondimensionalized and the physical significance of the reduced parameters is discussed. Closed-form algebraic formulas are given for some simple limiting cases. It is shown that the total cross force predicted by this model can be represented as the sum of an ideal component due to an inviscid flow with entry swirl and a viscous part due to the change in swirl created by friction inside the gland. The frequency-dependent ideal part is solely responsible for the rotordynamic direct damping. The facility designed and built to measure these frequency dependent forces is described. Experimental data confirm the validity and usefulness of this ideal/viscous decomposition. A method for calculating the damping coefficients based on the force decomposition using the static measurements only is presented.

An interactive boundary-layer method is described for computing unsteady incompressible flows over airfoils, including the initiation of dynamic stall. The inviscid unsteady panel method developed by Platzer and Teng is extended to include viscous effects. The solutions of the boundary-layer equations are obtained with an inverse finite-difference method employing an interaction law based on the Hilbert integral, and the algebraic eddy-viscosity formulation of Cebeci and Smith. The method is applied to airfoils subject to periodic and ramp-type motions and its abilities are examined for a range of angles of attack, reduced frequency, and pitch rate.

An explicit Navier–Stokes solver has been written with the option of using one of two types of turbulence model. One is the Baldwin–Lomax algebraic model and the other is an implicit k –ε model which has been coupled with the explicit Navier–Stokes solver in a novel way. This type of coupling, which uses two different solution methods, is unique and combines the overall robustness of the implicit k –ε solver with the simplicity of the explicit solver. The resulting code has been applied to the solution of the flow in a transonic fan rotor, which has been experimentally investigated by Wennerstrom. Five separate solutions, each identical except for the turbulence modeling details, have been obtained and compared with the experimental results. The five different turbulence models run were: the standard Baldwin–Lomax model both with and without wall functions, the Baldwin–Lomax model with modified constants and wall functions, a standard k –ε model, and an extended k –ε model, which accounts for multiple time scales by adding an extra term to the dissipation equation. In general, as the model includes more of the physics, the computed shock position becomes closer to the experimental results.

Time-resolved turbine rotor blade heat transfer data are compared with ab initio numerical calculations. The data were taken on a transonic, 4-to-1 pressure ratio, uncooled, single-stage turbine in a short-duration turbine test facility. The data consist of the time history of the heat transfer distribution about the rotor chord at midspan. The numerical calculation is a time accurate, two-dimensional, thin shear layer, multiblade row code known as UNSFLO. UNSFLO uses Ni’s Lax-Wendroff algorithm, conservative boundary conditions, and a time tilting algorithm to facilitate the calculation of the flow in multiple blade rows of arbitrary pitch ratio with relatively little computer time. The version used for this work had a simple algebraic Baldwin-Lomax turbulence model. The code is shown to do a good job of predicting the quantitative time history of the heat flux distribution. The wake/boundary layer and transonic interaction regions for suction and pressure surfaces are identified and the shortcomings of the current algebraic turbulence modeling in the code are discussed. The influence of hardware manufacturing tolerance on rotor heat transfer variation is discussed. A physical reasoning explaining the discrepancies between the unsteady measurement and the calculations for both the suction and pressure surfaces are given, which may be of use in improving future calculations and design procedures.

An implicit upwind scheme has been developed for Navier–Stokes simulations of unsteady flows in transonic cascades. The two-dimensional, Reynolds-averaged Navier–Stokes equations are discretized in space using a cell-centered finite volume formulation and in time using the Euler implicit method. The inviscid fluxes are evaluated using a highly accurate upwind scheme based on a TVD formulation with the Roe’s approximate Riemann solver, and the viscous fluxes are determined in a central differencing manner. The algebraic turbulence model of Baldwin and Lomax is employed. To simplify grid generations, a zonal approach with a composite zonal grid system is implemented, in which periodic boundaries are treated as zonal boundaries. A new time linearization of the inviscid fluxes evaluated by Roe’s approximate Riemann solver is presented in detail. No approximate factorization is introduced, and unfactored equations are solved by a pointwise relaxation method. To obtain time-accurate solutions, 30 linear iterations are performed at each time step. Numerical examples are presented for unsteady flows in a transonic turbine cascade where periodic unsteadiness is caused by the trailing edge vortex shedding.