Simulating Periodic Unsteady Flows Using a Cubic-Spline-Based Time Collocation Method

2013 ◽  
Vol 136 (4) ◽  
Author(s):  
Pengcheng Du ◽  
Fangfei Ning
Keyword(s):  
Collocation Method ◽  
Stokes Equations ◽  
Differential Quadrature Method ◽  
Unsteady Flows ◽  
Cubic Spline ◽  
Navier Stokes ◽  
Transonic Compressor ◽  
Compressor Rotor ◽  
Time Marching ◽  
Flow Variables

Time-periodical unsteady flows are typical in turbomachinery. Simulating such flows using a conventional time marching approach is the most accurate but is extremely time consuming. In order to achieve a better balance between accuracy and computational expenses, a cubic-spline-based time collocation method is proposed. In this method, the time derivatives in the Navier–Stokes equations are obtained by using the differential quadrature method, in which the periodical flow variables are approximated by cubic splines. Thus, the computation of a time-periodical flow is substituted by several coupled quasi-steady flow computations at sampled instants. The proposed method is then validated against several typical turbomachinery periodical unsteady flows, i.e., transonic compressor rotor flows under circumferential inlet distortions, single stage rotor–stator interactions, and IGV–rotor interactions. The results show that the proposed cubic-spline-based time collocation method with appropriate time sampling can well resolve the dominant unsteady effects, while the computational expenses are kept much less than the traditional time-marching simulation. More importantly, this paper provides a framework on the basis of a time collocation method in which one may choose more compatible test functions for the concerned specific unsteady flows so that better modeling of the flows can be expected.

Simulating Periodic Unsteady Flows Using Cubic-Spline Based Time Collocation Method

2013 ◽  
Author(s):  
Pengcheng Du ◽  
Fangfei Ning
Keyword(s):  
Collocation Method ◽  
Stokes Equations ◽  
Differential Quadrature Method ◽  
Unsteady Flows ◽  
Cubic Spline ◽  
Navier Stokes ◽  
Transonic Compressor ◽  
Compressor Rotor ◽  
Time Marching ◽  
Flow Variables

Time periodical unsteady flows are typical in turbomachinery. Simulating such flows using conventional time marching approach is most accurate but extremely time consuming. In order to achieve a better balance between accuracy and computational expenses, a cubic-spline based time collocation method is proposed. In this method, the time derivatives in the Navier-Stokes equations are obtained by using the differential quadrature method, in which the periodical flow variables are approximated by cubic-splines. Thus, the computation of a time-periodical flow is substituted by several coupled quasi-steady flow computations at sampled instants. The proposed method is then validated against several typical turbomachinery periodical unsteady flows, i.e., transonic compressor rotor flows under circumferential inlet distortions, single stage rotor-stator interactions and IGV-rotor interactions. The results show that the proposed cubic-spline based time collocation method with appropriate time sampling can well resolve the dominant unsteady effects, whilst the computational expenses are kept much less than the traditional time-marching simulation. More importantly, this paper provides a framework on the basis of time collocation method in which one may choose more compatible test functions for the concerned specific unsteady flows so that the better modeling of the flows can be expected.

A cubic-spline-based time collocation method applied to simulate periodic transonic flows

2016 ◽  
Vol 231 (1) ◽  
pp. 17-29
Author(s):  
Pengcheng Du ◽  
Fangfei Ning
Keyword(s):  
Collocation Method ◽  
Turbine Blades ◽  
Time Derivative ◽  
Cubic Spline ◽  
Collocation Methods ◽  
Phase Lag ◽  
Transonic Flows ◽  
Computational Mesh ◽  
Time Marching ◽  
Flow Variables

A cubic-spline-based time collocation method is used to solve periodic transonic flows for aeroelastic analysis. The time periodical flow variables are first approximated by cubic splines, and then the time-derivative terms are represented by a source term, which couples the flow solutions at all the sampled time instants. Simulation of two transonic external unsteady flows, including the oscillating NACA 0012 aerofoil and pitching rectangular supercritical wing, were carried out. The cubic-spline-based time collocation method with relatively small number of time instants sampled demonstrate high accuracy, whilst it is several times faster than the conventional time-marching method. The effect of different basis function used in the time collocation methods are also studied by comparing the results to that of the state-of-the-art time spectral method. This method was also validated against aeroelastic experimental data of the vibrating STCF 11 turbine blades. Only single passage computational mesh is involved by adopting the phase-lag periodic conditions. Good accuracy of the proposed time collocation method is also achieved and about one order of speed-up is obtained as compared with time-marching simulation.

scholarly journals Navier-Stokes Solutions of Unsteady Flow in a Compressor Rotor

1986 ◽  
Author(s):  
James N. Scott ◽  
Wilbur L. Hankey
Keyword(s):  
Unsteady Flow ◽  
Turbulence Model ◽  
Stokes Equations ◽  
Numerical Procedure ◽  
Navier Stokes ◽  
Suction Surface ◽  
Blade Passage ◽  
Radial Location ◽  
Transonic Compressor ◽  
Compressor Rotor

In order to achieve more accurate predictions of unsteady flow in a transonic compressor rotor an existing numerical approach has been modified by incorporating a turbulence model. The computations are performed by solving the complete time-dependent compressible Navier-Stokes equations using MacCormack’s explicit finite difference algorithm. These equations are solved for the flow through two adjacent rotor blades at a stream surface near the blade tip subjected to the wakes emitted from upstream stators. At this radial location the flow enters the blade passage at an absolute Mach number of 0.66. The high blade curvature at this radial location produces a large region of separated flow on the suction surface with laminar flow. To more accurately resolve the features of this flow separation the Baldwin-Lomax algebraic eddy-viscosity turbulence model is incorporated into the numerical procedure in regions, near the blade surface. The unsteady flow features are represented at the inflow boundary through the use of characteristic variables involving the upstream and downstream running Riemann invariants and the entropy variation expressed in terms of the total pressure profile. At the outflow boundary the concept of a “second throat” or choke point is implemented in conjunction with supersonic outflow conditions. The results are compared with numerical results obtained without the use of a turbulence model (laminar) for a single blade passage. Improved agreement with limited experimental data is also noted.

Computation of Off-Design Flows in a Transonic Compressor Rotor

1986 ◽  
Vol 108 (1) ◽  
pp. 144-150 ◽  
Author(s):  
W. N. Dawes
Keyword(s):  
Stokes Equations ◽  
Current Paper ◽  
Navier Stokes ◽  
Experimental Measurements ◽  
Navier Stokes Equations ◽  
Interactive Method ◽  
Transonic Compressor ◽  
Compressor Rotor ◽  
Supersonic Inlet ◽  
Maximum Benefit

Recent years have seen increasing efforts to develop efficient solvers for the compressible Navier–Stokes equations. For maximum benefit to be derived from this effort, these Navier–Stokes solvers must be capable of dealing with off-design flows as readily and accurately as the on-design cases. The current paper outlines an efficient implicit algorithm developed recently by the author for solving the compressible Navier–Stokes equations in turbomachinery blade-blade flows. The Navier–Stokes solver is applied to the study of a transonic compressor rotor with supersonic inlet velocities for three cases, one on-design and two off-design. The results are compared with experimental measurements and with the predictions of a viscous-inviscid interactive method.

Navier–Stokes Solutions of Unsteady Flow in a Compressor Rotor

1986 ◽  
Vol 108 (2) ◽  
pp. 206-215 ◽  
Author(s):  
J. N. Scott ◽  
W. L. Hankey
Keyword(s):  
Unsteady Flow ◽  
Turbulence Model ◽  
Stokes Equations ◽  
Numerical Procedure ◽  
Navier Stokes ◽  
Suction Surface ◽  
Blade Passage ◽  
Radial Location ◽  
Transonic Compressor ◽  
Compressor Rotor

In order to achieve more accurate predictions of unsteady flow in a transonic compressor rotor an existing numerical approach has been modified by incorporating a turbulence model. The computations are performed by solving the complete time-dependent compressible Navier–Stokes equations using MacCormack’s explicit finite difference algorithm. These equations are solved for the flow through two adjacent rotor blades at a streamsurface near the blade tip subjected to the wakes emitted from upstream stators. At this radial location the flow enters the blade passage at an absolute Mach number of 0.66. The high blade curvature at this radial location produces a large region of separated flow on the suction surface with laminar flow. To more accurately resolve the features of this flow separation the Baldwin–Lomax algebraic eddy-viscosity turbulence model is incorporated into the numerical procedure in regions near the blade surface. The unsteady flow features are represented at the inflow boundary through the use of characteristic variables involving the upstream and downstream running Riemann invariants and the entropy variation expressed in terms of the total pressure profile. At the outflow boundary the concept of a “second throat” or choke point is implemented in conjunction with supersonic outflow conditions. The results are compared with numerical results obtained without the use of a turbulence model (laminar) for a single blade passage. Improved agreement with limited experimental data is also noted.

A Numerical Analysis of the Three-Dimensional Viscous Flow in a Transonic Compressor Rotor and Comparison With Experiment

1987 ◽  
Vol 109 (1) ◽  
pp. 83-90 ◽  
Author(s):  
W. N. Dawes
Keyword(s):  
Boundary Layer ◽  
Numerical Analysis ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Axial Flow ◽  
Tip Clearance ◽  
Navier Stokes ◽  
Viscous Effects ◽  
Transonic Compressor ◽  
Compressor Rotor

The numerical analysis of highly loaded transonic compressors continues to be of considerable interest. Although much progress has been made with inviscid analyses, viscous effects can be very significant, especially those associated with shock–boundary layer interactions. While inviscid analyses have been enhanced by the interactive inclusion of blade surf ace boundary layer calculations, it may be better in the long term to develop efficient algorithms to solve the full three-dimensional Navier–Stokes equations. Indeed, it seems that many phenomena of key interest, like tip clearance flows, may only be accessible to a Navier–Stokes solver. The present paper describes a computer program developed for solving the three-dimensional viscous compressible flow equations in turbomachine geometries. The code is applied to the study of the flowfield in an axial-flow transonic compressor rotor with an attempt to resolve the tip clearance flow. The predicted flow is compared with laser anemometry measurements and good agreement is found.

scholarly journals Viscous Flow Computations in Turbomachine Cascades

1990 ◽  
Author(s):  
P.-A. Chevrin ◽  
C. Vuillez
Keyword(s):  
Stokes Equations ◽  
Turbulence Models ◽  
Navier Stokes ◽  
Turbine Cascade ◽  
Compressor Cascade ◽  
Navier Stokes Equations ◽  
Transonic Compressor ◽  
Different Types ◽  
Time Marching ◽  
Transonic Turbine

Accurate prediction of the flow in turbomachinery requires numerical solution of the Navier-Stokes equations. A two-dimensional Navier-Stokes solver developed at ONERA for the calculation of the flow in turbine and compressor cascades was adapted at SNECMA to run on different types of grid. The solver uses an explicit, time-marching, finite-volume technique, with a multigrid acceleration scheme. A multi-domain approach is used to handle difficulties due to the geometry of the flow. An H-C grid was used in the calculations. Two turbulence models, based on the mixing length approach, were used. The flow in a transonic compressor cascade, a subsonic and a transonic turbine cascade were computed. Comparison with experiments is presented.

Effects of Inlet Distortion on the Flow Field in a Transonic Compressor Rotor

1996 ◽  
Author(s):  
Chunill Hah ◽  
Douglas C. Rabe ◽  
Thomas J. Sullivan ◽  
Aspi R. Wadia
Keyword(s):  
Boundary Layer ◽  
Flow Field ◽  
Viscous Flow ◽  
Total Pressure ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Aerodynamic Loss ◽  
Transonic Compressor ◽  
Inlet Distortion ◽  
Compressor Rotor

The effects of circumferential distortions in inlet total pressure on the flow field in a low-aspect-ratio, high-speed, high-pressure-ratio, transonic compressor rotor are investigated in this paper. The flow field was studied experimentally and numerically with and without inlet total pressure distortion. Total pressure distortion was created by screens mounted upstream from the rotor inlet. Circumferential distortions of 8 periods per revolution were investigated at two different rotor speeds. The unsteady blade surface pressures were measured with miniature pressure transducers mounted in the blade. The flow fields with and without inlet total pressure distortion were analyzed numerically by solving steady and unsteady forms of the Reynolds-averaged Navier-Stokes equations. Steady three-dimensional viscous flow calculations were performed for the flow without inlet distortion while unsteady three-dimensional viscous flow calculations were used for the flow with inlet distortion. For the time-accurate calculation, circumferential and radial variations of the inlet total pressure were used as a time-dependent inflow boundary condition. A second-order implicit scheme was used for the time integration. The experimental measurements and the numerical analysis are highly complementary for this study because of the extreme complexity of the flow field. The current investigation shows that inlet flow distortions travel through the rotor blade passage and are convected into the following stator. At a high rotor speed where the flow is transonic, the passage shock was found to oscillate by as much as 20% of the blade chord, and very strong interactions between the unsteady passage shock and the blade boundary layer were observed. This interaction increases the effective blockage of the passage, resulting in an increased aerodynamic loss and a reduced stall margin. The strong interaction between the passage shock and the blade boundary layer increases the peak aerodynamic loss by about one percent.

Flow Field Unsteadiness in the Tip Region of a Transonic Compressor Rotor

1997 ◽  
Vol 119 (1) ◽  
pp. 122-128 ◽  
Author(s):  
S. L. Puterbaugh ◽  
W. W. Copenhaver
Keyword(s):  
Flow Field ◽  
High Performance ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Vortex Interaction ◽  
Tip Leakage Vortex ◽  
Tip Leakage ◽  
Transonic Compressor ◽  
Compressor Rotor ◽  
Operating Points

An experimental investigation concerning tip flow field unsteadiness was performed for a high-performance, state-of-the-art transonic compressor rotor. Casing-mounted high frequency response pressure transducers were used to indicate both the ensemble averaged and time varying flow structure present in the tip region of the rotor at four different operating points at design speed. The ensemble averaged information revealed the shock structure as it evolved from a dual shock system at open throttle to an attached shock at peak efficiency to a detached orientation at near stall. Steady three-dimensional Navier Stokes analysis reveals the dominant flow structures in the tip region in support of the ensemble averaged measurements. A tip leakage vortex is evident at all operating points as regions of low static pressure and appears in the same location as the vortex found in the numerical solution. An unsteadiness parameter was calculated to quantify the unsteadiness in the tip cascade plane. In general, regions of peak unsteadiness appear near shocks and in the area interpreted as the shock-tip leakage vortex interaction. Local peaks of unsteadiness appear in mid-passage downstream of the shock-vortex interaction. Flow field features not evident in the ensemble averaged data are examined via a Navier-Stokes solution obtained at the near stall operating point.

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