Supersonic Axial Compressor Stage Simplified Analysis

Application of supersonic axial compressor stages is an effective way to decrease mass and volume of gas turbines. It is reported that stages with pressure ratio up to 2,8 and blade velocity 450 m/s can operate quite satisfactory. There is demand for stages with pressure ratio that exceeds 3.0. The possible efficiency of stages is vital for successful application. As a velocity coefficient at an impeller inlet of stages can be about 1.5 and more, shock wave losses can limit overall stage efficiency. The simplified model of a stage was used for calculations. Supersonic flow in elementary blade cascade with sharp leading edges of blades produces oblique shock wave with sub – or supersonic flow after it. This depends on an inlet velocity coefficient and an angle between shock wave front and flow direction. The normal shock wave follows if a flow is still supersonic after an oblique shock. The known equations are used to calculate head losses in shock waves. Losses in a subsonic part of a stage are estimated by an arbitrary appointed loss coefficient. The calculations were made for a gas with = 1,4 in a range of velocity coefficient =1,1 – 1,8. Shock wave angle was varied in a range = 900 – ?0, where ?0 is an angle of a sonic wave. There were calculated static pressure ratios, static polytrophic efficiency, loss coefficient, velocity coefficients after shock waves. There are two zones of an operation: – subsonic flow after an oblique shock wave with angles bigger than 720-620 (the bigger value corresponds to a smaller velocity coefficient); – supersonic flow after an oblique shock wave with angles smaller than 720-620 and a normal shock wave after. An efficiency of a calculated stage model with subsonic flow after an oblique shock wave is not less 0,88 when velocity coefficient is no more than 1,6. The corresponding pressure ratio is about 4,7. The problem is how far is the simplified model from a real 3-D stage with all her complications. The system of an oblique and following normal shock waves transforms kinetic energy in a pressure rise formally most effective. An optimal angle of an oblique shock wave lies in a range = 550 - 450 (the bigger value corresponds to a smaller velocity coefficient); The efficiency is about 88% for the velocity coefficient 1,8 and = 450. There is a fantastic pressure ratio 11 ion this case. In can be concluded that for stages with pressure ratio about 3,0 shock wave losses do not limit efficiency. The problem returns to trivial problems of preventing excessive separation after shock waves and to effective 3-D design

New Version of the Universal Modeling for Centrifugal Compressor Gas Dynamic Design

Decades ago at pre – computer era design process consisted of empirically based set of rules application to choose main flow path dimensions. Serious model tests were obligatory before compressor manufacturing to check delivery pressure and efficiency. Better flow physical models and computer progress made possible to develop quickly operating programs to predict gas dynamic performance curves of an arbitrary flow path. TU SPb set of computer programs was named “The Universal modeling method” and its application still in mid 1990th had lead to elimination of model tests in a design process of industrial centrifugal compressors. The Universal modeling state of the art including 4th generation loss and work input models were presented in conferences in Germany, Japan, Great Britain and Poland. Set of algebraic equations describe surface friction losses, flow separation and following mixing losses. Flow deceleration along surfaces and velocity gradient along a normal to surfaces are taken into account. By 4-th generation programs several dozens of compressor with delivery pressure up to 12,5 MPa, number of stages up to 8, power up to 25 mWt were designed for some Russian and foreign manufacturers. Amount of compressor installed exceeds 400 with total power close to 5 000 000 kWt. In all cases the design parameters were achieved without preceding model tests. The 4-th generation model was perfect enough to predict design point efficiency with accuracy about 2,5% if a single set of coefficients was applied. To raise accuracy of calculations to 1% or less different sets of empirical coefficients were necessary for stages with different flow rate and work coefficients. The proposed text is focused on scientific background and realization of model improvements. The main of them are more precise impeller size presentation, impeller velocity diagram schematization very close to non viscid diagram, surface roughness introduction in the loss model, shroud leakage influence on flow at an impeller inlet, etc. As a result 5-th generation model predicts efficiency curves of stages with different flow rate and work coefficients with mean deviation less than 0,5% at design point and 1,5% along all performance curves – with a single set of empirical coefficients. Then the compressors’ test performance curves were carefully correlated with the calculated ones by proper selection of empirical coefficients in models of pressure loss ant work coefficient. The stages of 16 tested compressors can be considered as 99 model stages with the range of gas dynamic and constructive parameters: flow rate coefficients 0,025 – 0,064, Euler work coefficients 0,40 – 0,85, relative hub diameter 0,258 – 0,483, outer relative diameter of a diffuser 1,316 -1,720. Stages polytrophic efficiency is 0,765 – 0,885 and surge limit ratio is 0,30 – 0,93 depends on a stage specific speed. There are samples of useful application of the stages. The new loss model is applied to 6-th and 7-th generation computer programs. Newly is described 3D impeller flow path shape. Q3D approach to impellers adds information for more profound optimization. New solution were checked and approved by CFD calculations

Information field dynamics for simulation scheme construction

Information field dynamics (IFD) is introduced here as a framework to derive numerical schemes for the simulation of physical and other fields without assuming a particular sub-grid structure as many schemes do. IFD constructs an ensemble of non-parametric sub-grid field configurations from the combination of the data in computer memory, representing constraints on possible field configurations, and prior assumptions on the sub-grid field statistics. Each of these field configurations can formally be evolved to a later moment since any differential operator of the dynamics can act on fields living in continuous space. However, these virtually evolved fields need again a representation by data in computer memory. The maximum entropy principle of information theory guides the construction of updated datasets via entropic matching, optimally representing these field configurations at the later time. The field dynamics thereby become represented by a finite set of evolution equations for the data that can be solved numerically. The sub-grid dynamics is treated within an auxiliary analytic consideration and the resulting scheme acts solely on the data space. It should provide a more accurate description of the physical field dynamics than simulation schemes constructed ad-hoc, due to the more rigorous accounting of sub-grid physics and the space discretization process. Assimilation of measurement data into an IFD simulation is conceptually straightforward since measurement and simulation data can just be merged. The IFD approach is illustrated using the example of a coarsely discretized representation of a thermally excited classical Klein-Gordon field. This should pave the way towards the construction of schemes for more complex systems like turbulent hydrodynamics.Comment: 19 pages, 3 color figures, accepted by Phys. Rev.

Simulating Turbulence Using the Astrophysical Discontinuous Galerkin Code TENET

In astrophysics, the two main methods traditionally in use for solving the Euler equations of ideal fluid dynamics are smoothed particle hydrodynamics and finite volume discretization on a stationary mesh. However, the goal to efficiently make use of future exascale machines with their ever higher degree of parallel concurrency motivates the search for more efficient and more accurate techniques for computing hydrodynamics. Discontinuous Galerkin (DG) methods represent a promising class of methods in this regard, as they can be straightforwardly extended to arbitrarily high order while requiring only small stencils. Especially for applications involving comparatively smooth problems, higher-order approaches promise significant gains in computational speed for reaching a desired target accuracy. Here, we introduce our new astrophysical DG code TENET designed for applications in cosmology, and discuss our first results for 3D simulations of subsonic turbulence. We show that our new DG implementation provides accurate results for subsonic turbulence, at considerably reduced computational cost compared with traditional finite volume methods. In particular, we find that DG needs about 1.8 times fewer degrees of freedom to achieve the same accuracy and at the same time is more than 1.5 times faster, confirming its substantial promise for astrophysical applications.Comment: 21 pages, 7 figures, to appear in Proceedings of the SPPEXA symposium, Lecture Notes in Computational Science and Engineering (LNCSE), Springe