1. Hydraulic Analogy for Compressible flow Simulation and comparison with experimental data

2. Hydraulic Analogy Compressible FlowFree Surface Flow h P2P2 h Sound speedSurface wave speed MachFroude Shock WaveHydraulic Jump Subsonic FlowSubcritical Flow Sonic FlowCritical Flow Supersonic FlowSupercritical Flow

3. Solved equations and variables The general transport equation: The general transport equation: Is solved in 2-D for the variables: P1 P1 U1 U1 V1 V1 Standard k- turbulence model is activated.

4. Implementation in PHOENICS The following settings must be made in the Q1 file in order to activate the hydraulic analogy ecuations. The following settings must be made in the Q1 file in order to activate the hydraulic analogy ecuations.

5. Subcritical flow over a bump. Geometry. Geometry. A 7m long, 2.1m width channel with a 0.1m high and 1m long bump was considered.

6. Subcritical flow over a bump. Inlet conditions. Inlet conditions. Initial depth h=1m. Initial depth h=1m. Initial velocity v=1.5m/s Initial velocity v=1.5m/s Initial Froude Fr=0.479 Initial Froude Fr=0.479 Turbulence intensity 5% Turbulence intensity 5% The bump is simulated with a porous object, set with a sine function with a minimum porosity of 0.9 The bump is simulated with a porous object, set with a sine function with a minimum porosity of 0.9

7. Simulation Results Velocity and depth in the middle of the channel. Velocity and depth in the middle of the channel.

8. Simulation Results 3-D representation of the free surface. 3-D representation of the free surface. Comparision with analytical results Comparision with analytical results Analitical results Simulation Depth [m]Velocity [m/s]Depth [m]Velocity [m/s] Initial conditions11.50.991.51 Bump0.8591.7450.8741.731

9. Simulation of supercritical flow near an abrupt wall deflection. Geometry. Geometry. A 2.5m long and 0.5m wide with a variable 1m long deflection was simulated. Experimental reference: Hager W., Jimenez O., et al. Supercritical flow near an abrupt wall deflection Journal of Hydraulic Research. V32-1. 1994.

10. Simulation of supercritical flow near an abrupt wall deflection. Inlet with Fr=4.0 Inlet with Fr=4.0 Initial depth h=50mm. Initial depth h=50mm. Turbulence intensity 5%. Turbulence intensity 5%. Simulations were performed with the same inlet conditions. Four different deflection widths were considered. 50, 100, 150 and 200mm. Simulations were performed with the same inlet conditions. Four different deflection widths were considered. 50, 100, 150 and 200mm. Simulations results are compared with experimental data. Simulations results are compared with experimental data.

11. Comparison with experimental data in the deflection area. Comparison for dimensionless depth for 50mm deflection.

12. Transverse comparison Dimensionless depth profile at 40cm from the origin of the deflection wall.

13. Transverse comparison Dimensionless depth profile at 80cm from the origin of the deflection wall.

14. Comparison with experimental data in the deflection area. Comparison for dimensionless depth for 100mm deflection.

15. Transverse comparison Dimensionless depth profile at 40cm from the origin of the deflection wall.

16. Transverse comparison Dimensionless depth profile at 80cm from the origin of the deflection wall.

17. Comparison with experimental data in the deflection area. Comparison for dimensionless depth for 150mm deflection.

18. Transverse comparison Dimensionless depth profile at 40cm from the origin of the deflection wall.

19. Transverse comparison Dimensionless depth profile at 80cm from the origin of the deflection wall.

20. Comparison with experimental data in the deflection area. Comparison for dimensionless depth for 200mm deflection.

21. Transverse comparison Dimensionless depth profile at 40cm from the origin of the deflection wall.

22. Transverse comparison Dimensionless depth profile at 80cm from the origin of the deflection wall.

23. 3-D representation of the free surface

24. Simulation of supercritical flow at channel expansions. Geometry Geometry A 14m long, 2.1m witdth channel was considered. Expansion length is 3.0m. Expansion ratio is 1.1667.

25. Simulation of supercritical flow at channel expansions. Standard k- turbulence model is activated. Standard k- turbulence model is activated. Inlet conditions. Inlet conditions. Initial depth h=0.3m Initial depth h=0.3m Initial velocity u=8.577m/s Initial velocity u=8.577m/s Initial Froude Fr=5.0 Initial Froude Fr=5.0 Turbulence intensity 5% Turbulence intensity 5%

26. Depth and Froude results

27. 3-D representation of the free surface.