Research Papers

Space–Time Gradient Method for Unsteady Bladerow Interaction—Part I: Basic Methodology and Verification

J. Yi and L. He
[+] Author and Article Information
J. Yi

Department of Engineering Science,
University of Oxford,
Oxford, OX2 0ES, UK
e-mail: Junsok.Yi@eng.ox.ac.uk

L. He

Department of Engineering Science,
University of Oxford,
Oxford, OX2 0ES, UK
e-mail: Li.He@eng.ox.ac.uk

1Corresponding author.

Manuscript received July 14, 2015; final manuscript received July 24, 2015; published online September 16, 2015. Editor: Kenneth C. Hall.

J. Turbomach 137(11), 111008 (Sep 16, 2015) (13 pages) Paper No: TURBO-15-1144; doi: 10.1115/1.4031342 History: Received July 14, 2015; Revised July 24, 2015

For advanced turbomachinery development, there is increasing interest to carry out unsteady analyses for flows through multiple bladerows during a design stage. Even with the huge increase in computer processing power currently available, direct unsteady calculations in a whole annulus domain are still very time consuming. Efficient alternative methods with truncations in time and/or in space have been developed for unsteady turbomachinery flows in the past 20 years or so, but they all are associated with related limitations. The present development is motivated to maintain as many modeling fidelities of direct unsteady solution methods as possible while still have a significant speed-up. To this end, a new steady-solution-like unsteady time-domain methodology has been developed for bladerow interactions. No circumferential domain truncation is required so that a direct periodic (repeating) condition can be applied. For each mesh cell, the temporal gradient term as required to balance the discretized unsteady flow equation is obtained by specially sequenced spatial variations of corresponding cells in multiple-passages. Consequently, the rotor–stator interface treatment becomes completely compatible to that of a direct unsteady solution. Thus a fully conservative interface is easily achieved, in contrast to existing truncated models where interface treatments tend to be complicated and nonconservative. The simultaneous solution procedure with the space–time gradient (STG) link enables an unsteady flow solution to converge at a rate compatible to a steady solution. The background, motivation/justification, basic methodology, and some preliminary verifications are described in this paper as Part I. Further validations and applications with extension to more complex configurations and flow conditions will be presented in Part II.

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Fig. 1

Instantaneous entropy contours of turbine stage: (a) mixing plane solution, (b) frozen rotor solution, and (c) direct unsteady solution

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Fig. 2

Dependence of flow solution on relative clocking

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Fig. 3

Aliased mode 1 and mode 14

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Fig. 4

Passage sample points and sequencing: (a) passages indexed before sequencing and (b) passages ordered after sequencing for R1

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Fig. 5

Convergence histories

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Fig. 6

Domain partitions for parallelization (partitioned domains are indexed by the numbers): (a) conventional and (b)modified

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Fig. 7

Multidomains in STG (blade counts 1:1) (a) direct unsteady solution (100 × 10 × 20 iterations) and (b) present (STG) solution (100 × 500 iterations)

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Fig. 8

Comparison of unsteady entropy: (a) direct unsteady solution and (b) present (STG) solution

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Fig. 9

Difference in parallelization between direct unsteady and present (STG) solutions: (a) direct unsteady solution (inherently sequential in time, parallelizable in space only) and (b) present (STG) solution (only need to be parallelizable in space)

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Fig. 10

Instantaneous entropy contours: (a) direct unsteady solution and (b) present (STG) solution (direct unsteady versus STG solution)

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Fig. 11

Time-averaged entropy contours: (a) direct unsteady solution and (b) present (STG) solution (direct unsteady versus STG solution)

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Fig. 12

Total temperature profiles at exit

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Fig. 13

Static temperatures on blade surfaces: (a) NGV and (b) rotor

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Fig. 14

Time-/space-averaged entropy contours at exit

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Fig. 15

Instantaneous entropy contours at midspan: (a) mixing plane solution, (b) frozen rotor solution, (c) direct unsteady solution, and (d) present (STG) solution

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Fig. 16

Instantaneous entropy contours at 90% span (a) mixing plane solution, (b) frozen rotor solution, (c) direct unsteady solution, and (d) present (STG) solution

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Fig. 17

Stream-traces in near-tip region: (a) steady solution and (b) unsteady solution (1—tip leakage vortex; 2—passage vortex)



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