Streamline Curvature Method Research Articles

Improved Approach to the Streamline Curvature Method in Turbomachinery

Quasi three dimensional blade-to-blade solutions for stators and rotors of turbomachines are obtained using the Streamline Curvature Method (SLCM). The first-order velocity gradient equation of the SLCM, traditionally solved for the velocity field, is reformulated as a second-order elliptic differential equation and employed in tracing the streamtubes throughout the flow field. The equation of continuity is then used to calculate the velocity. The present method has the following advantages. First, it preserves the ellipticity of the flow field in the solution of the second-order velocity gradient equation. Second, it eliminates the need for curve fitting and strong smoothing under-relaxation in the classical SLCM. Third, the prediction of the stagnation streamlines is a straightforward matter which does not complicate the present procedure. Finally, body-fitted curvilinear coordinates (streamlines and orthogonals or quasi-orthogonals) are naturally generated in the method. Numerical solutions are obtained for inviscid incompressible flow in rotating and non-rotating passages and the results are compared with experimental data.

Ill‐posed design methods for the computation of inviscid flows

AbstractIt is shown here that the solutions of the governing partial differential equations fail to behave well for three well‐known methods of analysis and design for the computation of inviscid flows in turbomachines and for time‐dependent transonic airflows. The first two examples are related to flows in turbomachines: the so‐called ‘streamline curvature methods’ and the proposition of C. H. Wu to reduce the solution of an essentially three‐dimensional internal flow problem to successive two‐dimensional ones. Finally, time‐dependent compressible and irrotational flows which employ the density and the velocity potential as dependent variables are considered.

A Quasi-Three-Dimensional Turbomachinery Blade Design System: Part I—Throughflow Analysis

The purpose of this work has been to develop a quasi-three-dimensional blade design and analysis system incorporating fully linked throughflow, blade-to-blade and blade section stacking programs. In Part I of the paper, the throughflow analysis is developed. This is based on a rigorous passage averaging technique to derive throughflow equations valid inside a blade row. The advantages of this approach are that the meridional streamsurface does not have to be of a prescribed shape, and by introducing density weighted averages the continuity equation is of an exact form. Included in the equations are the effects of blade blockage, blade forces, blade-to-blade variations and loss. The solution of the equations is developed for the well-known streamline curvature method, and the contributions from these extra effects on the radial equilibrium equation are discussed. Part II of the paper incorporates the analysis into a quasi-three-dimensional computing system and demonstrates its operational feasibility.

Comparison of design techniques for pumps with skewed blades

Two different approaches, developed over a period of approximately 12 years, of including blade skew effects in the design of a pump rotor are presented. Performance comparisons are included between the radial and skewed bladed rotor for each of the pump designs. The streamline curvature method which incorporated skew effects and secondary flow losses produced a skewed rotor that satisfactorily met the design performance goals, whereas a design procedure using a bound vorticity approach did not.

Laboratory Slurry Erosion Tests and Pump Wear Rate Calculations

From order-of-magnitude calculations, the erosion mechanism in slurry pumps for mining applications was found to be a scouring type. Impact erosion was found to be unlikely to occur. The erosion model assumes a “sliding bed” type of flow pattern and neglects the effect of turbulent diffusion of slurry particles. This choice was based on flow regime data from slurry pipeline flow and was not verified experimentally in pumps. A new centrifugal-erosion laboratory testing fixture was designed and fabricated at Allis-Chalmers which reproduces this type of erosion. Since flow conditions were closely controlled, an absolute rate of erosion could be determined. For mild steel (Rockwell C = 10) and aluminum oxide particles, a specific erosion energy of 1.5 × 108 in-lb/in3 (1000 joules/mm3) was measured. Flow pattern and slurry concentrations were calculated in a pump impeller using a quasi-three-dimensional streamline curvature method. These were combined with the above erosion model and the local erosion rate was estimated. Actual operating data had to be estimated for a quarry application of the pump because field test data was not available. The uncertainties of the field test data did not allow an exact numerical verification of the wear rate calculation, but order of magnitude agreement was obtained. The wear rate distribution agreed very well with the wear pattern observed on the impeller used in the quarry and the locations of excessive wear could be identified.

Three-Dimensional Inviscid Computation of an Impeller Flow

The flow in a centrifugal impeller is analyzed by a quasi-three-dimensional streamline curvature method, by a three-dimensional Euler code, and by a three-dimensional finite element potential flow method. Comparison with Eckardt’s published data for a backswept impeller shows that full three-dimensional methods better predict the loading at the hub and shroud.

A Streamline Curvature Method for Calculating S1 Stream Surface Flow

A streamline curvature method for calculating S1 surface flow in turbines is presented. The authors propose a simple method in which a domain of calculation can be changed into an orderly rectangle without making coordinate transformations. Calculation results obtained on subsonic and transonic turbine cascades have been compared with those of experiment and another theory. Good agreement has been found. When calculating blade-to-blade flow velocity at subsonic speed, a function approximation technique can be used in lieu of iteration method in order to reduce calculation time. If the calculated flow section is of a mixed (subsonic-supersonic) flow type, a Boolean expression obtained from the truth table of flow states is proposed to judge the integrated character of the mixed flow section. Similarly, another Boolean expression is used to determine whether there exists a “choking” of the relevant section. Periodical conditions are satisfied by iterating the first-order derivative of stagnation streamline, which is formed simultaneously. It can be proved that the stagnation streamline formed in this way is unique.

A procedure for the calculation of flow through axisymmetric ducts

A procedure for the calculation of flow through axisymmetric ducts as are typically found in turbomachines, is presented. The procedure is based on a streamline curvature method with the governing equations formulated along quasi-orthogonals in the flow field. This formulation allows the procedure to be used for segments of a duct wherein the flow direction is predominantly radial. It is assumed that the flow on specific stream surfaces is isentropic, but normal entropy gradients may exist because of processes which took place upstream of the duct.

A transonic quasi‐3D analysis for gas turbine engines including split‐flow capability for turbofans

AbstractA numerical approximation is taken to the solution of the complex flows existing in gas turbine engines with transonic blading. The quasi‐3D approach decouples the problem into through‐flow and blade‐to‐blade solutions. An industrially practical finite element through‐flow solution is developed and for blade‐to‐blade solutions a transonic finite areas method is utilized. The finite element code developed is capable of operating in an analysis or a design mode. In both modes a dynamic relaxation factor is employed and considerable reduction in solution time can be achieved. Comparisons to streamline curvature methods are carried out for simple analytical and complex industrial problems.

Analysis of Transonic Cascade Flow Using Method of Streamlines

A numerical method is proposed for analysing the transonic flows through an arbitrary shaped cascade on an arbitrary surface of revolution. The method which is an improvement of the streamline curvature method developed by Katsanis and Wilkinson needs very short computer time and has practically sufficient accuracy. The method is applicable to the flow conditions which considerably deviate from the shockless entry, to the transonic flows with locally supersonic regions and without shock wave, and to the blades which do not have large radius of curvature at the leading edge. As numerical examples, the pressure distributions on the blade surface, the streamlines, and the contours of Mach number for compressor and turbine cascades are calculated. The pressure distributions agree well with the previous experimental and numerical results.

Performance of Rotating Cascades under the Inlet-Distortions : 1st Report, Performance under the Radial Distortions

Some experimental and theoretical studies of performances of rotating cascades operating under the radial-distortions are shown. The transmission of the distorted velocity profiles through the cascades, changes in characteristic curves and aerodynamic loading on the blade elements are obtained experimentally, and some discussions of the criterion of the onset of stall in the rotating cascades are made. The flow fields near the rotating cascades are analyzed by stream-line curvature method and the results are compared with the experimental ones. The rotating cascades have effects to attenuate the distorted velocity profile both at the design point and off- design points. The rotating cascades operate stable even when the aerodynamic loading near the root exceeds the stall limit of the two-dimensional cascades. The stall limit of the rotating cascade is determined by the equivalent diffusion ratio at a specific radius near the tip. The analytical results of transmission of the distorted velocity profiles through the rotating cascades, changes in the aerodynamic loading and stall limits agree with the experimental ones with sufficient accuracy.

Fluid Dynamic Excitation of Centrifugal Compressor Rotor Vibrations

A mechanism by which compressor rotor lateral vibration perturbs the mass flow-rate, the velocity and the pressure distributions within impeller passages is postulated. Such a perturbation will develop an unbalanced force on the rotor which, if it enhances the rotor vibration, is termed self-exciting. The concepts of rotor orbital velocity, the virtual center of shaft rotation, the reduction of unsteady flow to quasi-steady flow, the fluid dynamic force coefficient, mechanical orbital stability and the stability increment are introduced. The ideas are imposed on the streamline curvature method of quasi-three dimensional analysis of passage flow and a computer program has been assembled to carry out computation. No generalized guidelines have been found as yet but rather individual passage calculations are needed to determine the potentially exciting or damping character of the induced fluid dynamic forces. The average stability increment per stage for nine industrial multistage centrifugal compressors has been determined and compared with known operating experience. Important engineering characteristics of two of the compressors are shown in an example of the analysis. A provisional limit of the stability increment per stage ⩽ 1.85 lbf-s/in. (323.9N-s/m) is suggested, below which unstable nonsynchronous vibration of the compressor rotor can be expected.

A Comparison of the Matrix and Streamline Curvature Methods of Axial Flow Turbomachinery Analysis, From a User’s Point of View

In recent years two general methods for flow analysis in turbomachinery have been developed, one generally called the Streamline Curvature Method, the other the Matrix Through-Flow Method. Both methods solve the same flow equations but the differences in technique introduce different operational constraints and difficulties. A comparative assessment of the relative merits of the two methods has been difficult because the various authors did not use similar cascade models to represent cascade loss and deviation, a necessary adjunct to each technique. This paper outlines the two methods, and a common cascade model for both, and compares two programs written to implement the two techniques for ease of use, computer time and storage requirements, flexibility and inherent limitations. The programs are used to compute the flow field in three axial flow compressor applications: an interconnecting duct, a transonic fan, and three stage axial compressor. The predicted performance for the above machines was fairly good, although no attempt was made to “tune” the cascade model for the specific type of machine, as the relative merits of each method were of interest. It is concluded that there is a small operational advantage to the matrix method.

Discussion: “A Comparison of the Matrix and Streamline Curvature Methods of Axial Flow Turbomachinery Analysis, From a User’s Point of View” (Davis, W. R., and Millar, D. A. J., 1975, ASME J. Eng. Power, 97, pp. 549–558)

Discussion: “A Comparison of the Matrix and Streamline Curvature Methods of Axial Flow Turbomachinery Analysis, From a User’s Point of View” (Davis, W. R., and Millar, D. A. J., 1975, ASME J. Eng. Power, 97, pp. 549–558)

Closure to “Discussion of ‘A Comparison of the Matrix and Streamline Curvature Methods of Axial Flow Turbomachinery Analysis, From a User’s Point of View’” (1975, ASME J. Eng. Power, 97, pp. 558–559)

Closure to “Discussion of ‘A Comparison of the Matrix and Streamline Curvature Methods of Axial Flow Turbomachinery Analysis, From a User’s Point of View’” (1975, ASME J. Eng. Power, 97, pp. 558–559)

Experimental Evaluation of a Streamline-Curvature Method for Flared Subsonic Cascades and for Transonic Cascades

The accuracy of the streamline-curvature method has hitherto not been evaluated for curved-axis convergent-divergent flow as found in transonic turbine cascades and for flow in cascades with flow divergence or flare. To provide the data for evaluation, two simple single-channel rectangular-section wind tunnels were built to simulate the flow between turbine blades. The supersonic channel was successfully and economically powered by a large steam ejector. In transonic and supersonic flow, an examination of the sonic and M = 1·1 lines was used as a basis for evaluating the predictions. The correlation was found to be good near the convex surface, but showed a deterioration towards the concave surface. Both test and prediction revealed the unexpected fact that the sonic line contour is not smooth, but wavy. The correlation at the M = 1·1 line was positionally the same as the sonic line, but the experimental contour was considerably more wavy. The accuracy deteriorated further into the supersonic region. The discrepancies in the transonic region are thought to be caused by a very slow rate of change of area, which produces a complex boundary layer interaction in the throat region, affecting both the location of the throat and the area ratios in that zone. The prediction method was shown to be extremely accurate for flared-cascade flow even up to the very exacting situation of a 40 per cent blade height increase, the only error being that due to boundary layers in the apparatus that effect the velocity distribution over the latter half of the suction surface in a highly staggered cascade.

Stability and convergence of streamline curvature flow analysis procedures

AbstractA convergence technique for use with the streamline curvature method of cascade and channel flow is presented which produces stable and rapid convergence for subsonic, transonic and supersonic flow and also in the downstream region where the Kutta–Joukowski condition is applied to a cascade flow. The convergence method is fully two‐dimensional in nature, the iterative changes being determined via a model which takes into account the actual streamwise error distribution and the real normal changes in the grid.

Streamline Curvature Analysis of Compressible and High Mach Number Cascade Flows

A blade to blade design tool for analysing the subsonic, transonic and supersonic flows in axial turbine blades is presented using the streamline curvature method. The work includes a numerical convergence analysis which produced convergence of the digital computer programme for all cases attempted. Computed results are compared with experiment and other theory. Comparisons for turbine type cascades up to Mach 10 show a satisfactory agreement and techniques introduced give improved leading edge results and allow the Kutta–Joukowski condition to be applied at the trailing edge for the accurate determination of the velocity distribution and the cascade deviation. Predictions for convergent-divergent supersonic wind tunnel type nozzles agree closely with experiment and with other theories. Numerical difficulties are found to be more acute for subsonic/supersonic turbine cascades and experimental agreement, while acceptable, is inferior to that obtained for subsonic cascades.

Paper 35: Stability, Convergence, and Accuracy of Two-Dimensional Streamline Curvature Methods Using Quasi-Orthogonals

The streamline curvature technique for the numerical solution of the equations of motion for two-dimensional flow has been analysed. By considering a simple flow case an optimum damping factor for the quasi-orthogonal method was derived as a function of grid aspect ratio, Mach number, and differentiation formula, and the number of iterations required for a given accuracy was found. The stability and accuracy properties of numerical methods for finding the second derivative of a curve were also investigated, and it was found that, in this context, finite difference methods using polynomials were the best and spline methods the worst.

Paper 8: Calculations of the Flow Distribution within a Radial Turbine Rotor

The distribution of pressure and velocity within the rotor of a radial flow gas turbine is computed using a two-dimensional streamline curvature method. Solutions are obtained on the meridional plane and also on three blade-to-blade surfaces at the hub, mid-blade position, and at the shroud. The torque, as calculated from the blade-to-blade flows, agrees with the measured torque over a speed range of approximately 2:1 with sufficient accuracy to support the usefulness of the flow structure as indicated by two-dimensional calculation methods applied to radial turbomachines.