Research using CFD solver F

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Problem statement

I consider 3D viscous compressible flow of a viscous heat-conducting compressible gas through multi-stage turbine or compressor. Such a flow is described by the Reynolds-averaged Navier-Stokes equations (RANS), which include the continuity equation, three Cartesian projections of the momentum equation and the energy equation. Governing equations are transformed to a body-fitted 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 two-equation k-ω models, which includes
- the standard Wilcox's k-ω model,
- the Menter's k-ω BSL model,
- the Menter's k-ω SST model.
The Wilcox's Low-Reynolds-Number modification can be used for each of these models.
To ensure physically plausible solutions the realizability constraints are applied on turbulent Reynolds stresses.

Laminar-to-turbulent transition is described using the Low-Reynolds-number form of the k-ω 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 blade-to-blade 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 mass-flow-rate of the working medium.

The initial conditions are specified using the approximate calculation of one-dimensional flow in the flowpath.

A number of simplified flow problems such as calculations of the flow through two-dimensional or three-dimensional isolated turbomachinery cascades, simulations of the flow around isolated wing airfoils, etc, can be solved using suggested approach.

Numerical approximation

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 two-step process, the increments are computed with high accuracy. If the implicit step is terminated after one iteration, then the proposed algorithm reduces to the well-known Beam-Warming 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.

CFD Solver

Both the proposed mathematical model and its numerical approximation are implemented in software package F, which includes a shell program (a pre-processor for geometry and gasdynamic initial data input and a post-processor for visualization of computed results), and actually a CFD solver of RANS equations. The overwhelming majority of software modules are written in Fortran-95 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.

Gallery of numerical results

The following results were obtained using the CFD solver of the software package F. I used the RANS equation and Menter's k-ω SST turbulence model. To simulate transitional flows the low-Reynolds k-ω 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 blade-to-blade 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 post-processor, which is part of the software package. Some of the results were exported to Paraview, an open graphical cross-platform 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

Flow around NACA0012, transonic mode Flow around NACA0012, supersonic mode
Subsonic flow Supersonic flow




The flow around the isolated cylinder

The flow around the isolated cylinder




The flow in a compressor cascade from surge to choking conditions("numerical Schlieren" - density gradient contours)

Flow through compressor cascade, mode #1 Flow through compressor cascade, mode #2 Flow through compressor cascade, mode #3 Flow through compressor cascade, mode #4
Flow through compressor cascade, mode #5 Flow through compressor cascade, mode #6 Flow through compressor cascade, mode #7 Flow through compressor cascade, mode #8
Flow through compressor cascade, mode #9 Flow through compressor cascade, mode #10 Flow through compressor cascade, mode #11 Flow through compressor cascade, mode #12
Flow through compressor cascade, mode #13 Flow through compressor cascade, mode #14 Flow through compressor cascade, mode #15 Flow through compressor cascade, mode #16




The flow near the root endwall of the Hodson turbine cascade

Flow visualisation, Hodson cascade
Experiment, H.P.Hodson и R.G.Dominy, 1987 Calculated limiting streamlines
(visualization using Paraview)




The vortex street visualization behind the VKI-1 turbine cascade

Unsteady flow in VKI-1 cascade flow, Karman street Unsteady flow in VKI-1 cascade flow, Karman street
Entropy fluctuations contours Mach number contours and fluctuations of velocity vectors




The separated flow within a compressor cascade (visualization using Paraview)

Hub separation in compressor cascade, visualisation of vortices Tip separation in compressor cascade, visualisation of vortices
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 off-design angle (limiting streamlines, visualization using Paraview)

Tip separation in turbine cascade, limiting strimlines Near leading edge of turbine cascade, visualisation of vortex
The tip surface and upper portion of the blade An enlarged fragment near the leading edge
s - saddle points; f - focuses; n - nodes; streamline ("quasi-vortex line") connecting focuses f1 and f2 is shown




Unsteady stator-rotor-stator interaction within 1 and 1/2 turbine stage cascades (entropy fluctuations contours)

Unsteady flow through a 1 and 1/2 turbine stage



The flow through a film-cooled turbine cascade

3d blade of film-cooled turbine cascade
3D geometric model of the blade with holes (visualization using Paraview)


Mach number contours without film cooling Mach number contours with film cooling
Without film cooling With film cooling
Mach number contours


Mach number contours without film cooling Mach number contours with film cooling
Without film cooling With film cooling
An enlarged fragment near the leading edge


Mass flow rate distribution along the turbine axis
Mass flow rate distribution along the turbine axis
red line - without film cooling; blue line - with film cooling




The flow through an axially-radial compressor

Axi-symetric pressure





Simulation of a laminar-turbulent transition in a turbine cascade

The fully turbulent flow. Hub separation The fully turbulent flow. Tip separation
The fully turbulent flow The transitional flow
Turbulence kinetic energy contours



skin friction
The friction coefficient at the blade surface;
red line - fully turbulent flow; blue line - transitional flow



Turbulence kinetic energy rise
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



Adiabatic Mach number
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 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

secondary flow pattern



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