
I consider 3D viscous compressible flow of a viscous heatconducting compressible gas through multistage turbine or compressor. Such a flow is described by the Reynoldsaveraged NavierStokes equations (RANS), which include the continuity equation, three Cartesian projections of the momentum equation and the energy equation. Governing equations are transformed to a bodyfitted curvilinear coordinate system, rotating at a constant speed.
The working medium properties is modelled via one of the following equation of state:
 the perfect gas state equation,
 the Tammann equation of state,
 the van der Waals equation of state.
For each of these equations, specific heat capacities are considered to be either constant or linearly dependent on the temperature.
Turbulence is modeled using a family of the twoequation kω models, which includes
 the standard Wilcox's kω model,
 the Menter's kω BSL model,
 the Menter's kω SST model.
The Wilcox's LowReynoldsNumber modification can be used for each of these models.
To ensure physically plausible solutions the realizability constraints are applied on turbulent Reynolds stresses.
Laminartoturbulent transition is described using the LowReynoldsnumber form of the kω SST model and the algebraic PTM (Production Term Modifier) model, suggested by Langtry. The transition model introduces the turbulence production limiter that is analogous to the turbulence intermittency factor.
A computational domain includes one bladetoblade channel of each blade row and is supplemented with inlet and exit regions of axial gaps.
The following boundary conditions are prescribed.
At the inlet:
 absolute total pressure and temperature, and the angles of flow in absolute motion;
At the exit:
 static pressure or axial velocity;
At the walls:
 impermeability condition and heat transfer condition (adiabaticity, temperature, or heat flux);
At the periodicity boundaries:
 the condition of the flow periodicity;
At the mixing boundaries between blade rows:
 flow parameters averaged in the circumferential direction;
In the holes on solid surfaces (blades and endwalls):
 inflowing or outflowing massflowrate of the working medium.
The initial conditions are specified using the approximate calculation of onedimensional flow in the flowpath.
A number of simplified flow problems such as calculations of the flow through twodimensional or threedimensional isolated turbomachinery cascades, simulations of the flow around isolated wing airfoils, etc, can be solved using suggested approach.
The governing differential equations are solved using an implicit iterative TVD and ENO finite volume schemes of Godunov's type. The numerical approximation is derived in two steps. During the first "explicit" step, increments of conservative variables are calculated using a finite volume method and an exact Riemann solver, which are applied to the convective terms only. Diffusion derivatives, however, are approximated using finite differences. During the second "implicit" step, an iterative procedure, which is similar to the local Newton's method, is used to update the values at the current time step. As a result of this twostep process, the increments are computed with high accuracy. If the implicit step is terminated after one iteration, then the proposed algorithm reduces to the wellknown BeamWarming implicit scheme.
To accelerate the convergence, a local time step technique and simplified multigrid algorithm can be used. To increase the stability of calculations, I take into account the cell elongation in determining the local time step.
Both the proposed mathematical model and its numerical approximation are implemented in software package F, which includes a shell program (a preprocessor for geometry and gasdynamic initial data input and a postprocessor for visualization of computed results), and actually a CFD solver of RANS equations.
The overwhelming majority of software modules are written in Fortran95 algorithmic language. The shell program runs in Windows or using Wine emulator in Linux. The CFD solver runs in any OS given software modules can be compiled using Intel Fortran.
The following results were obtained using the CFD solver of the software package F. I used the RANS equation and Menter's kω SST turbulence model.
To simulate transitional flows the lowReynolds kω SST model together with Langtry's algebraic model PTM were used.
I used 3D meshes of sizes from five hundred thousand to forty four million cells in a single bladetoblade channel.
Typically, a tangentional section of such mesh contains between seven thousand and one hundred twenty thousand cells.
Calculations of a 2D flow around isolated airfoils were performed using meshes of sizes from one hundred twenty five thousand to five hundred thousand cells.
In film cooling simulations, between six and sixteen cells per single surface hole are used.
In most cases, visualizations are made using the mentioned postprocessor, which is part of the software package. Some of the results were exported to Paraview, an open graphical crossplatform package for interactive visualization.
The author of the project expresses gratitude to his former colleagues, namely Ph.D. Yakovlev VA, Ph.D. Grizun MN, Derevyanko AI, Kozyrets DA and so on.,
which provided a great assistance in the researches and data processing.
The flow around the isolated NACA0012 airfoil
Subsonic flow  Supersonic flow 
The flow around the isolated cylinder
The flow in a compressor cascade from surge to choking conditions("numerical Schlieren"  density gradient contours)
The flow near the root endwall of the Hodson turbine cascade
Experiment, H.P.Hodson и R.G.Dominy, 1987  Calculated limiting streamlines (visualization using Paraview) 
The vortex street visualization behind the VKI1 turbine cascade
Entropy fluctuations contours  Mach number contours and fluctuations of velocity vectors 
The separated flow within a compressor cascade (visualization using Paraview)
Hub separation  Tip separation 
B  blade surface; T  tip surface; H  hub surface; s  saddle points; f  focuses; n  nodes (due to insufficient grid resolution, the visualization does not detect on the blade surface a focus, corresponding to focus f1) 
Flow through turbine cascade at offdesign angle (limiting streamlines, visualization using Paraview)
The tip surface and upper portion of the blade  An enlarged fragment near the leading edge 
s  saddle points; f  focuses; n  nodes; streamline ("quasivortex line") connecting focuses f1 and f2 is shown 
Unsteady statorrotorstator interaction within 1 and 1/2 turbine stage cascades (entropy fluctuations contours)
The flow through a filmcooled turbine cascade
3D geometric model of the blade with holes (visualization using Paraview) 
Without film cooling  With film cooling 
Mach number contours 
Without film cooling  With film cooling 
An enlarged fragment near the leading edge 
Mass flow rate distribution along the turbine axis 
red line  without film cooling; blue line  with film cooling 
The flow through an axiallyradial compressor
Simulation of a laminarturbulent transition in a turbine cascade
The fully turbulent flow  The transitional flow 
Turbulence kinetic energy contours 
The friction coefficient at the blade surface; red line  fully turbulent flow; blue line  transitional flow 
The turbulence kinetic energy along a grid line near the blade suction surface; red line  fully turbulent flow; blue line  transitional flow; LE  leading edge; TE  trailing edge; local extrema behind trailing edge correspond to the intersections of grid line and wake 
The adiabatic Mach number along the blade suction surface near the shock wave; red line  fully turbulent flow; blue line  transitional flow 
Law of the wall at the suction side of a subsonic turbine cascade
The laminar region of the boundary layer  The transitional region of the boundary layer  The turbulent region of the boundary layer 
The secondary flow pattern in a subsonic turbine cascade
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